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# A Trust-Centric Privacy-Preserving Blockchain for Dynamic Spectrum
Management in IoT Networks
Jingwei Ye, Xin Kang, , Ying-Chang Liang, , and Sumei Sun J. Ye, X. Kang and
Y.-C. Liang are with the Center for Intelligent Networking and Communications
(CINC), University of Electronic Science and Technology of China (UESTC),
Chengdu 611731, China (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>liangyc@ieee.org).S. Sun is with the Institute for Infocomm Research, Agency
for Science, Technology and Research, Singapore (e-mail:
sunsm@i2r.a-star.edu.sg).
###### Abstract
In this paper, we propose a trust-centric privacy-preserving blockchain for
dynamic spectrum access in IoT networks. To be specific, we propose a trust
evaluation mechanism to evaluate the trustworthiness of sensing nodes and
design a Proof-of-Trust (PoT) consensus mechanism to build a scalable
blockchain with high transaction-per-second (TPS). Moreover, a privacy
protection scheme is proposed to protect sensors’ real-time geolocatioin
information when they upload sensing data to the blockchain. Two smart
contracts are designed to make the whole procedure (spectrum sensing, spectrum
auction, and spectrum allocation) run automatically. Simulation results
demonstrate the expected computation cost of the PoT consensus algorithm for
reliable sensing nodes is low, and the cooperative sensing performance is
improved with the help of trust value evaluation mechanism. In addition,
incentivization and security are also analyzed, which show that our design not
only can encourage nodes’ participation, but also resist to many kinds of
attacks which are frequently encountered in trust-based blockchain systems.
###### Index Terms:
Dynamic spectrum access, blockchain, trust model, consensus algorithm,
cooperative spectrum sensing.
## I Introduction
Recent years have witnessed the exponential growth of mobile data traffic. The
rapid development of new wireless applications such as autonomous vehicles,
remote healthcare, augmented and virtual reality will continue driving the
demand for radio spectrum[1]. However, the current static spectrum management
method has resulted in under-utilization of spectrum resources[2]. To this
end, dynamic spectrum access (DSA) has been proposed to allow secondary users
(SUs) sense and then use the idle spectrum bands. Since single SU’s sensing
capability is usually limited, cooperative sensing[3] is proposed to fuse the
sensing results of multiple SUs to improve the sensing accuracy. However, the
traditional usage of a centre to collect and fuse sensing results faces the
risk of single point of failure and the increasing management and regulatory
costs. Moreover, SUs usually need to upload their sensing data to the fusion
center at the risk of leakage of their privacy, such as identity and location.
As an emerging technology, blockchain has attracted attention from both
academia and industry. By using blockchain, a decentralized resource
management system can be established without the requirement of trustworthy
central management agencies[4, 5, 6, 7, 8]. Moreover, smart contracts which
are implemented on a blockchain can be used to replace traditional central
management agencies and facilitate the cooperation of users [9]. Recently,
researchers have investigated how to apply the blockchain technology to DSA.
[10] summarizes the types of blockchains applicable in different spectrum
sharing scenarios, and the possible advantages and disadvantages of applying
blockchain technology to DSA. With the help of cryptocurrency issued by the
blockchain and smart contracts, flexible and automatic spectrum trading
markets are made possible for spectrum sellers and buyers. [11, 12, 13, 14,
15].
However, the integration of blockchain and DSA still faces many challenges.
Firstly, although the data recorded on a blockchain is prevented from being
tampered, the quality or value of it cannot be guaranteed. Especially, in a
public blockchain, a malicious user can easily join the blockchain and record
their data in the blockchain. The data from such a malicious user or an
unreliable user is valueless or even harmful to cooperation tasks based on the
data, e.g., cooperative sensing. Therefore, there is compelling need to
evaluate the quality of the data from each user. Secondly, though the
communication among Ethereum external accounts can be protected by encryption
and decryption with their public and private key pairs, a contract account is
not equipped with a key pair hence such protection method is not applicable to
smart contracts. As a result, sensing results uploaded to the smart contract
cannot be encrypted, and the sensitive information in the sensing results
(such as real-time geolocatioin), can thus be accessed by malicious users.
Thirdly, traditional consensus algorithms like Proof of Work (PoW) used in
public blockchain introduce much computation overhead and thus make the
transaction processing speed very low. As a result, it is inefficient to
directly use existing consensus algorithms (such as PoW) for spectrum
management, especially considering the limited computation capabilities of IoT
devices.
In this paper, we consider the DSA for an IoT network where IoT devices
opportunistically access the licensed spectrum bands through cooperative
sensing. Blockchain is used as a platform for dynamic spectrum management.
Specifically, key component designs for such a blockchain-enabled DSA system,
including the trust evaluation mechanism, privacy protection, consensus
algorithm and smart contracts, are studied. The main contributions of this
paper are summarized as follows.
* •
We propose a trust evaluation mechanism to evaluate the trustworthiness of
sensing nodes which participate in the collaborative sensing. We show that the
proposed trust evaluation mechanism is effective in incentivizing sensing
nodes to be honest, and thus improving the spectrum sensing result.
* •
We propose a strong privacy protection mechanism by combining ring signature
with the commit-and-reveal scheme for providing sensing nodes’ privacy such as
their real-time location. The ring signature is used to protect sensing nodes’
identities when they upload sensing data. The commit-and-reveal scheme is
proposed to figure out the connection between sensing data packets and its
originated sensing node after the fusion of sensing results, so that the trust
value of each sensing node is updated and the incentive tokens are
distributed.
* •
We propose a new consensus algorithm named as Proof-of-Trust (PoT) by
connecting the mining difficulty with a miner’s trust value. We show that the
proposed PoT can greatly reduce the computation cost of honest nodes and
enhance the scalability of current blockchian-based DSA system.
* •
We design the system protocol and two smart contracts to make the whole DSA
procedure, including spectrum sensing, spectrum auction, and spectrum
allocation, run automatically. In addition, we implement and verify a
prototype of our proposed protocol and smart contracts using Solidity[16] on
the Ethereum test net.
The rest of the paper is organized as follows. In Section II, we introduce the
basic concept of blockchain and smart contract. In Section III, we describe
the works related to the application of blockchain to DSA. In Section IV, we
introduce our system model. In Section V, we describe the design of our
proposed protocol including the block structure, the trust evaluation
mechanism, the consensus algorithm and the privacy protection scheme. In
section VI, the design of smart contract and workflow are introduced. In
Section VII, we discuss the performance of the trust evaluation mechanism, and
analyze the security of the system. Finally, Section VIII concludes this
paper.
## II Preliminaries
### II-A Blockchain
Blockchain is a growing chain of data blocks with a specific data structure
constructed using cryptographic algorithms. It uses hash pointers to connect
the data blocks to form an entangled chain. In this way, the integrity of the
data can be protected. Blockchain uses a measurable and verifiable mechanism
to reach a consensus among nodes in the network on the generation of new
blocks. Such a mechanism is usually referred to as the consensus algorithm. On
the premise that the data block entanglement is guaranteed, the structure and
the consensus algorithm of the blockchain can be flexibly designed based on
the demand of application scenarios.
Blockchain can be roughly classified into three types, depending on degree of
openness, which are public blockchain, private blockchain, and consortium
blockchain. Public blockchain is designed to enable and to record the
transactions in a public network, which means that any node can freely join or
leave the blockchain network without authorization. Bitcoin, the most famous
digital cryptocurrency, is generated and managed by a public blockchain.
Consortium blockchain is a permissioned blockchain, which is open to
authorized members of external institutions in limited roles and functions.
The authorization of the node to join are all determined by an authorized
organization. The nodes in a private blockchain network trust each other.
Thus, private chains can simplify operations in data authentication and the
consensus algorithm to improve the efficiency.
### II-B Smart Contract
In the 1990s, Nick Szabo first proposed the concept of smart contracts[17]. It
was defined as computerized transaction protocols that execute terms of a
contract. In the field of cryptocurrency, a smart contract is defined as an
application or a program that runs on a blockchain. Normally, it contains a
set of digital agreements with specific rules. These rules are predefined in
the form of computer source codes, and all network nodes will copy and execute
these computer source codes independently. Smart contracts are highly
customizable and can be flexibly designed to provide different services and
solutions.
### II-C Ring Signature
Ring signatures were first introduced in 2001[18] which follow a ring-like
structure of the signature algorithm. It is a type of digital signature that
can be performed by any member of a group of users without the agreement of
others. Therefore, the only “open” information is that a message signed with a
ring signature is endorsed by someone in a particular group of people. Ring
signatures are deliberately designed so that it is computationally infeasible
to determine which of the group members’ keys was used to produce the
signature. Ring signatures are similar to group signatures but differ in two
key ways: firstly, there is no group administrator to revoke the anonymity of
an individual signature; secondly, any set of users can be used as a group by
the signer without additional setup.
## III Related work
Blockchain was first proposed as a secure decentralized ledger for recording
spectrum transaction data and analyzing overall spectrum usage in dynamic
spectrum management. In [19], the authors proposed to use the blockchain as a
distributed database to securely store the transactions in dynamic spectrum
sharing, and to use the digital currency issued by the blockchain as the
currency for spectrum auctions. In [15], a smart contract running on Ethereum
was proposed as a platform to provide spectrum sensing services.
In [13], a blockchain-based spectrum trading and sharing scheme was proposed
for Mobile Network Operators (MNOs) to lease their spectrum to secondary
Unmanned Aerial Vehicle (UAV). In [20], spectrum sharing contracts deployed on
a permissioned blockchain platform are constructed for multi-operators
spectrum trading. In [14], the authors proposeds to use the smart contract to
securely store and then autonomously implement the spectrum leasing
agreements, thus to avoid the interference from the disordered spectrum
access.
However, these works did not make any improvement to the existing blockchain.
Thus, the TPS of the blockchain is low, limiting the performance of the
blockchain-based DSA platform. Thus,
in [21], the authors proposed a consensus algorithm named as "blockchain-KM
protocol" by modifying the PoW, to speed up the transaction processing for the
license-free spectrum sharing. In [22], a novel consensus algorithm named as
proof-of-strategy was proposed, where the best strategy to allocate the
spectrum bands in the whole network is the proof of authority to publish a new
block. Such a consensus algorithm can not only regulate the transaction
generation, but also optimize the spectrum allocation.
In [23], by modifying the PoW, the authors proposed a consensus algorithm
named as Proof-of-Device, which selects a device, represented by its unique
identifier, to publish a new block in a way like lottery. It thus eliminates
the heavy computation cost in PoW.
On the other hand, blockchain system is an open system, hence the reliability
of data source needs to be guaranteed. In [24], the data uploaded by the
sensors are compared with the data collected by the trustworthy validator to
determine the credibility degree of the sensor, in this way, the system can
evaluate the credibility of nodes for better performance. However, this method
is a semi-centralized method which needs to deploy trustworthy validators.
Different from the above works, we not only delve into the design of block
structure and consensus algorithm, intending to overcome the scalability
problem of the blockchain technology, but also design a decentralized trust
evaluation mechanism and a privacy protection mechanism for the openness and
transparency of blockchain system while protecting the data’s confidentiality
and the sensors’ privacy. To the best of our knowledge, this is the first work
of joint trust evaluation and privacy protection mechanism in cooperative
sensing in a blockchain-enabled DSA system.
## IV System Model, challenges and design guidelines
In this section, we introduce the dynamic spectrum sharing scenario in the
future IoT network, and list the difficulties and challenges in applying the
blockchain to this scenario. Finally, we give the guidelines on how to design
a blockchain-based DSA system.
### IV-A Blockchain-enabled DSA System Model
In this paper, we consider the DSA in a two-tier cognitive radio network
consisting of licensed/primary users (PUs) and unlicensed/secondary users
(SUs). In this network, PUs are inactive users who may not use their spectrum
all the time so SUs can detect these spectrum holes and use them. As
illustrated in Fig.2, SUs can be a variety of heterogeneous IoT devices, which
form a mesh network. SUs access the spectrum bands of a PU in an opportunistic
manner based on cooperative sensing, and a blockchain is used as the platform
of the cooperative spectrum sensing decision and the spectrum allocation.
Figure 1: System model
Basically, as shown in Fig. 3, there are four phases in our blockchain-enabled
DSA system.
1. 1.
Individual Sensing: The sensors perform spectrum sensing to detect the state
of the PU’s spectrum band.
2. 2.
Sensing Fusion: Each of the sensor broadcasts its sensing data which mainly
consists of the sensing result and the geolocation where the sensing is
performed. These data can be raw data or quantized bits, which will be fed to
a smart contract implemented upon a blockchain. Then, the smart contract
provides the final sensing result according to predefined fusion rules.
3. 3.
Spectrum Allocation: When the final sensing result indicates that PU is
inactive, it should be decided which spectrum requester can access the
spectrum. Spectrum auction, as a common and fair way for spectrum allocation,
is chosen to achieve that. In the spectrum auction, the digital
cryptocurrency, which is issued by the blockchain through incentive mechanisms
for spectrum sensing and mining, is commonly considered.
4. 4.
Spectrum Access: The SU who obtains the access right through spectrum auction
accesses the idle spectrum for data transmission.
Compared with traditional cooperative spectrum sensing and spectrum allocation
methods, blockchain-enabled DSA has the following benefits:
* •
Decentralized: Blockchain-based dynamic spectrum access does not require the
deployment of trusted central nodes which avoids the single point of failure.
* •
Transparency: The blockchain ledger can record the whole process of DSA.
* •
Automatic: Using smart contract instead of traditional contracts, we can
achieve automatic spectrum management and on-chain payment settlement.
* •
Flexibility: For spectrum trading on the blockchain platform, diversified
spectrum trading rules can be dynamically enforced by adjusting parameters of
smart contracts. Compared with the current fixed paper-based contracts signed
between mobile users and operators, this method has higher flexibility.
Figure 2: Blockchain based DSA
### IV-B Difficulties and Challenges
Traditional cooperative spectrum sensing is divided into two modes:
centralized mode and distributed mode. Centralized collaborative spectrum
sensing requires the deployment of a central fusion node to process the data
collected from sensors which usually contain sensors’ geolocation information.
Therefore, this mode not only requires redundant regulation fees (usually
charged to maintain the operation of the central node), but also faces the
risk of data leakage when the central node is attacked. While the distributed
mode often suffers from the fake reporting issue caused by malicious nodes and
the iterative-based malicious node detection algorithms [25] introduce extra
computation cost.
Therefore, how to implement the cooperative sensing in a secure and effective
way is a challenge. The decentralized nature of the blockchain system is
promising to realize distributed cooperative sensing and encourage all sensors
to share spectrum sensing results. However, there are still some challenges
when applying blockchain into spectrum management. Firstly, IoT devices
usually lack sufficient computing and storage resources to maintain and store
the whole blockchain ledger. Secondly, a malicious node can easily join a
public blockchain network and threaten the reliability of the cooperative
spectrum sensing. Moreover, the sensing-related data recorded on blockchain or
collected by smart contracts is usually in an unencrypted format which may
leak the private information of IoT devices contained in their sensing data
packets.
### IV-C Design Guidelines
Considering above challenges, our discussion on system design mainly includes
the following four parts.
* •
_Trust Evaluation_ : Specially, in a public blockchain network, anyone can
join or exit the system at will, and one user can create multiple accounts in
the system. Therefore, if there is no suitable node evaluation mechanism,
malicious users can launch sybil attacks easily. This is because, malicious
nodes can keep on applying for accounts and continue launching attacks.
Therefore, the system should be able to evaluate the trust values for nodes,
and identity the malicious nodes based on the their trust values.
* •
_Lightweight Consensus Algorithm_ : Due to the limitation of the computing
power and storage resources of IoT devices, maintaining a normal public
blockchain system is a heavy burden. To reduce the computing cost, it is
necessary to invent a new consensus algorithm with low computational and
storage complexity.
On the other hand, as time goes by, the length of the ledger keeps on
increasing, resulting in increased node storage. To reduce the storage cost,
methods such as edge storage can alleviate storage pressure on the blockchain
nodes but may introduce additional budget. Thus, a method that can directly
reduce the storage cost of the blockchain is preferred.
* •
_Privacy Protection Mechanism_ : Smart contracts does not have public-and-
private key pair and thus can not communicate with external accounts in an
encrypted manner. Therefore, when a smart contract is used as the data fusion
center of cooperative spectrum sensing, a privacy protection mechanism is
needed to ensure that the private information of the sensors will not be
leaked during the interaction between sensors and the smart contract.
## V Blockchain Design, Trust Evaluation and Privacy Protection
In this section, to improve the accuracy of cooperative sensing, we first
introduce the block structure of our blockchain. We then propose the trust
evaluation mechanism to evaluate the trustworthiness of each SU on spectrum
sensing. After that, a high-efficiency consensus algorithm is proposed for our
blockchain. Finally, a privacy protection mechanism is proposed for protecting
participants’ privacy.
TABLE I: List of notations Notations | Descriptions
---|---
$N_{i,w}(n)$ | Accumulated number of wrong sensing results of $sensor_{i}$ at round $n$
$N_{i,r}(n)$ | Accumulated number of right sensing results of $sensor_{i}$ at round $n$
$P$ | Length of window for calculating $N_{i,w}(n)$
$\mathcal{D}^{i}_{n}$ | Mining difficulty of node $i$ at block $n$
$\mathcal{H(\cdot)}$ | Hash operation
$R_{sleep}$ | Number of inactive rounds since last active round
$RND_{i}$ | Random number picked by $sensor_{i}$
$SR$ | Sensing result of unknown but legal sensors
$SR_{i}$ | Sensing result of $sensor_{i}$
$N_{1}$ | Number of required sensors in Cooperative Sensing Contract ($CSC$)
$N_{2}$ | Number of required bidders in Spectrum Auction Contract ($SAC$)
$TV_{thr}$ | Trust value threshold set in $CSC$
$T_{ddl}$ | Sensing deadline in $SAC$
$d_{s}$ | Deposit for sensing
$d_{a}$ | Deposit for auction
$T_{self-d}$ | Time for $SAC$ to self-destruct
$TS$ | Timestamp
$loc$ | Location of unknown but legal sensors
### V-A Block Structure
The block structure should be tailor-designed for the blockchain-based DSA
system. Some of the unnecessary components in the block header can be removed
to reduce the overheads on the block transmission and storage and other
necessary information, such as the trust value of the block generator, needs
to be recorded in the block header to prevent possible attacks in the DSA
system.
* •
Block header: As shown in Fig. 3, The block header mainly contains (i) the
merkle root of transactions; (ii) the merkle root of account states; (iii) the
hash value of the header of the previous block, as the hash pointer of the
chain structure; (iv) the trust value of the miner who mines this block; (v)
the signature of the miner who mines this block; (vi) a Nonce, which is the
solution of Hash puzzle of the consensus algorithm; (vii) the timestamp when
the block is mined.
* •
Block body: The block body mainly stores transaction generated during spectrum
allocation using merkle tree. Unlike the unspent transaction output (UTXO)
[26] model used in Bitcoin, the system based on the account model usually
contains an account tree to record the account states such as the balance and
the trust value of each account. Merkle Patricia Trie (MPT) [27] which is a
commonly adopted data structure, is used here to store the account states.
Figure 3: Block structure
### V-B Trust Evaluation Mechanism
Although the blockchain can guarantee that the data recorded in a block will
not be tampered, it cannot guarantee that the data source is trustworthy.
Especially, in a public blockchain, a malicious node can upload misleading
data which may interfere the cooperation sensing based on such data. In such
cases, the performance of the cooperative sensing will be degraded, and the
honest and reliable sensors will be discouraged from participating in
cooperative sensing. To this end, we design a trust evaluation mechanism to
evaluate the credibility of every sensing node in the blockchain. Instead of
using a central authority to record and maintain every account’s trust value,
we propose to include the trust value as an attribute of the blockchain
account, which is recorded in the block and maintained by all the miners. In
the following, we will discuss the design of trust value updating method.
Intuitively, the trust value of a sensing node should be adjusted based on its
performance on the current and historical cooperative sensing. In detail, if
it plays a positive role in the cooperative sensing, its trust value should be
increased, and vice versa. Here, we define that the effect of a sensor is
positive when its sensing result is consistent with the cooperative sensing
result, and is negative when its sensing result is inconsistent with the
cooperative sensing result, or when the sensing node does not participate in
the cooperative sensing. Denote the number of times when the sensing result of
the i-th node is consistent with the cooperative sensing result as $N_{i,r}$;
the number of times when the sensing result of a node is inconsistent with the
cooperative sensing result as $N_{i,w}$. Inspired by [28], we propose a new
model to calculate trust value of $sensor_{i}$ denoted by $TV_{i}^{(1)}(n)$ in
sensing round $n$, where $n$ is also the number of sensing tasks published
since this system was created.
$\displaystyle TV_{i}^{(1)}(n)=e^{-\rho N_{i,w}(n)}\left(1-e^{-\eta
N_{i,r}(n)}\right),$ (1)
where $\rho>0$ and $\eta>0$ are coefficient that determine how fast the trust
value changes with respect to $N_{i,w}$ and $N_{i,r}$ , respectively.
Moreover, we let $N_{i,w}(n)$ decay by $\frac{1}{P}$ every time when the node
participates in sensing, and we only consider the latest $P$ sensing rounds of
each node. In this way, the effect of a wrong sensing result on the trust
value will gradually be degraded over time. As a result, a node that
unintentionally submitting an inconsistent result will be gradually forgotten.
Mathematically, $N_{i,w}(n)$ can be defined as
$N_{i,w}(n)=\sum\limits_{m=n-P}^{n}r_{i}(m)\left(1-\frac{n-m}{P}\right),$ (2)
where $r_{i}(m)=1$ if $sensor_{i}$ broadcasts the wrong sensing result in
round $m$, and $r_{i}(m)=0$ otherwise.
Besides, we denote the number of rounds when a sensing node is inactive as
$R_{sleep}$, and we add a function $f(R_{sleep})$ which models the negative
impact of being inactive on the trust value. It is designed to satisfy the
following criteria.
* •
${\forall}R_{sleep}\in[1,+\infty)$, $f(R_{sleep})\in(0,1]$,
* •
$\frac{\partial f_{dcv}}{\partial R_{sleep}}<0$,
The first criterion is for normalization; the second criterion is to ensure
that the negative effect increases as $R_{sleep}$ increases. Moreover, as
$R_{sleep}$ increases, the downward trend of the function is first gentle, and
then becomes severe. This is designed to punish the peers that does not
participate in the sensing process. The punishment is light when the peer does
not participate only a few times, but becomes severe when the node always does
not participate in the sensing. $f(R_{sleep})$ will reach a certain low value
when $R_{sleep}$ is big enough. For example, the trust value of a node will
reduce to $r_{1}$ ($r_{1}<1$) of its original value for $k_{1}$ consecutive
non-participation, and to $r_{2}$ ($r_{2}<r_{1}<1$) for $k_{2}$ consecutive
non-participation. Here, a piecewise function consisting of multiple linear
functions is utilized as our $f(R_{sleep})$.
$f(R_{sleep})=\left\\{\begin{array}[]{lcl}\frac{k_{1}-R_{sleep}}{k_{1}}\cdot(1-r_{1})+r_{1}&&{0\leq
R_{sleep}\leq k_{1}}\\\ \\\
\frac{R_{sleep}-k_{2}}{k_{1}-k_{2}}\cdot(r_{1}-r_{2})+r_{2}&&{k_{1}\leq
R_{sleep}\leq k_{2}}\\\ \\\ r_{2}&&{k_{2}<R_{sleep}}\\\ \end{array}\right.$
(3)
These parameters can be set flexibly. If the actual deployment requires
stricter penalties, then all these parameters should be set smaller.
By adding the attenuation function, the model in (1) can be modified as:
$TV^{(2)}_{i}(n)=\begin{cases}TV_{i}^{(2)}(n-1)+\Delta TV_{i}\cdot
f(R_{sleep}),&\mbox{if $\Delta TV_{i}>0$}\\\ \\\ TV_{i}^{(1)}(n),&\mbox{if
$\Delta TV_{i}<0$}\\\ \\\ TV_{i}^{(2)}(n-1)\cdot f(R_{sleep}),&\mbox{if
inactive}\end{cases}$ (4)
where $\Delta TV_{i}$ = $TV_{i}^{(1)}(n)-TV_{i}^{(2)}(n-1)$ is defined as the
original increment of trust value, which can be either positive or negative.
The design criterion of (4) is explained as follows. We consider three cases.
First of all, when the sensor senses correctly, the original increment of
trust value is positive, i.e., $\Delta TV_{i}>0$. Such increment is first
degraded by the negative effect of sleeps in the previous sensing activities,
i.e., $f(R_{sleep})$, and then added to the trust value. Secondly, when the
sensor gives the wrong sensing result, the original increment of trust value
is negative, i.e., $\Delta TV_{i}<0$. In this case, we choose not to consider
the negative effect of the sleep in previous sensing rounds, in order to
encourage the sensors which unintentionally derive the wrong sensing result in
this time. Finally, for sensors who do not participate in sensing, their trust
value will not remain unchanged but gradually decrease as their sleep time
increases. This design is to ensure that sensors can only maintain its high
trust value by actively participating in spectrum sensing.
### V-C Consensus Algorithm, Forking and Block Compression
Based on the last block of the block chain, nodes in the blockchain network
will compete for publishing the next block, and only one block will be
accepted as the next valid block.
The traditional consensus algorithm for public blockchain like PoW is
computation-intensive, which may be inapplicable in IoT networks, where IoT
nodes are usually with limited computational capabilities. One reason why the
PoW is designed to be computation-insensitive is that it assumes there is
nearly no trust among nodes in a blockchain network. Nevertheless, with the
trust value mechanism proposed in this paper, we can evaluate the credibility
of a node on spectrum sensing in the blockchain network. Therefore, based on
the trust value, we can optimize our design of the blockchain consensus
algorithm, forking solution and blockchain compression.
Consensus Algorithm: Intuitively, the nodes with higher trust value on sensing
are more likely to be honest in publishing a new block. Therefore, we can
reduce the difficulty of such nodes to mine a new block. In this way, the
computation consumption for reliable nodes will be decreased, and a block will
thus be more quickly mined and published.
Based on PoW and trust value, we propose a Proof of Trust (PoT) consensus
algorithm. Similar to the Proof-of-Stake (PoS), PoT assigns different nodes
different mining difficulties. To be specific, for successful mining, the
miner with the higher trust value is required to find a hash value with fewer
leading zeros and vice versa.
Mathematically, assuming that the difficulty of node $i$ at block $n$ is
denoted as $\mathcal{D}^{i}_{n}$, which can be denoted as
$\mathcal{D}^{i}_{n}=\beta_{n}\cdot\left(1-sin\left(\frac{\pi}{2}\cdot
TV_{i}\right)\right),$ (5)
where $\beta_{n}$ denotes the base difficulty of block $n$. $TV_{i}$ is the
trust value of node $i$. The initial difficulty is denoted as $\beta_{0}$.
$\beta_{0}$ can be determined by evaluating the computing power of actual IoT
devices. We will discuss this problem in the simulation section. A node with
$TV_{i}=0.8$ has about $0.049\cdot\beta_{n}$ difficulty, which is 17 times
less than a node with $TV_{i}=0.1$ whose difficulty is about
$0.844\cdot\beta_{n}$.
We denote the timestamp of block $n$ as $T_{n}$, the ideal time interval of
two consecutive blocks is $T_{0}$. Then,we have
$q=\left\lfloor\frac{T_{n-1}-T_{n-2}}{T_{0}}\right\rfloor,$ (6)
where $q$ is defined as adaptive adjustment factor of base difficulty, and $g$
is defined as the difficulty adjustment granularity, which can be written as
$g=\left\lfloor\frac{\beta_{n-1}}{128}\right\rfloor,$ (7)
The updating of $\beta_{n}$ can be denoted as
$\beta_{n}=\beta_{n-1}-q\cdot g,$ (8)
Forking Solution: As the mining speed increases, there may be multiple blocks
mined at nearly the same time. Because of the communication latency, the block
that is first mined might not be the first to be received by all the nodes in
the blockchain network. Instead, the block that is first received and
recognized by different nodes might be different. In this case, the blockchain
forking occurs. It is harmful and thus needs to be solved. In Algorithm 1, we
propose a trust value based forking solution. It first compares the trust
values in the head of blocks, and it then validates the block with the highest
trust value. If the trust values of multiple blocks are the same, the block
whose timestamp is earlier is selected as the valid block. If the timestamp is
the same, it compares the hash value of the blocks, and then selects the block
with the smallest hash value as the valid block. Since the probability of hash
collision is negligible, Algorithm 1 can select one valid block eventually.
Blockchain Compression: Since IoT devices are usually with limited storage
space, each time when the blockchain grows by $L$ blocks, compression will be
performed. For the compression of blockchain, in the existing literature, the
RSA accumulator as a data structure, which functions similarly to that of a
Merkle tree, can be used to compress the blockchain [29]. Another approach to
compress blockchain is to use the chameleon hash function to replace the
traditional hash function of the blockchain [30]. Here, using the trust value,
we propose a compression method called as Trust-based Compression (TBC). The
node with the highest current trust value will be authorized to compress the
blockchain. To be specific, such a node will first extract the account tree in
the last block as the body of the new block, calculate a hash value for the
body, and finally combine the obtained nonce, miner’s signature, and timestamp
to form the block header. The obtained block is the new genesis block. Since
each node stores the original blockchain, it is easy to check whether the
account status has been tampered by the selected node during the compression
process. After a node verifies the new genius block, it will clear the
original blockchain.
Algorithm 1 Block Selection
Input: the block to be compared: $Block_{i}$, $Block_{j}$;
Output: The winner block;
if $TV_{i}<TV_{j}$ then
$Block_{i}$ wins.
else if $TV_{i}>TV_{j}$ then
$Block_{j}$ wins.
else
if $Timestamp_{i}<Timestamp_{j}$ then
$Block_{i}$ wins.
else if $Timestamp_{i}>Timestamp_{j}$ then
$Block_{j}$ wins.
else
if $\mathcal{H}(Block_{i})<\mathcal{H}(Block_{j})$ then
$Block_{i}$ wins.
else
$Block_{j}$ wins.
end if
end if
end if
### V-D Privacy Protection Mechanism
The location where a sensing node senses the spectrum bands of interest is
useful for the fusion centre to cluster the sensing nodes and to improve the
cooperative sensing accuracy [31]. Therefore, alongside the sensing result,
the location is needed to be uploaded by a sensing node. However, it is
difficult to protect the location information of a sensing node from being
leaked when a smart contract is used as the fusion centre. This is because a
smart contract account is not equipped with a key pair which can be used by
sensors to encrypt their upload data packet.
The privacy protection issue in this case is to hide the source of a sensing
packet, i.e., from which sensing node the sensing packet comes. However, the
sensing node cannot be allowed to be totally anonymous when submitting the
sensing packet because this will make the cooperative sensing system
vulnerable to malicious attacks. To this end, we propose the use of the ring
signature [18] by each sensing node to hide the source of a sensing packet in
a group of valid sensing nodes. Also, the smart contract as the fusion centre
can identify the validity of each received sensing packet. Moreover, since
sensing packets are unencrypted, fusing these sensing results can be carried
out directly and automatically in the smart contract.
In the following, we illustrate the procedures of one sensor, denoted by
$Sensor_{s}$, in generating the ring signature.
* 1.
$Sensor_{s}$ selects $n-1$ legal sensors to form a group and collect their
public keys. The format of a sensing data packet, denoted as msg, is given as
follows
$msg=\\{\mathcal{H}(msgID),SR,time,location\\},$ (9)
where $\mathcal{H}(msgID)$ is used in the update of the trust value and $SR$
is the sensing result denoted by one bit.
* 2.
$Sensor_{s}$ uses one-way hash function to compute $k=\mathcal{H}(msg)$, which
is a symmetric key of the symmetric encryption function $E_{k}$.
* 3.
$Sensor_{s}$ generates a random value for each of other members in the group
where it belongs to. Specifically, the random number $x_{i}$ is first
generated for the $i$-th node in the group. It then calculates corresponding
$y_{i}$=$g_{i}(x_{i})$ using corresponding public key, where the function
$g_{i}(\cdot)$ is the encryption function encrypted with the public key
$pk_{i}$.
* 4.
$Sensor_{s}$ finds the solution to the ring equation (10) and gets the
undetermined parameter $y_{s}$, where $v$ is a random value chosen by
$Sensor_{s}$. Then, $Sensor_{s}$ calculates $x_{s}$ using its own privacy key:
$x_{s}$=$g_{s}^{-1}(y_{s})$. To find solution, a private key of a sensor in
this group is needed , anyone who is not in the group cannot generate a legal
ring signature.
$\begin{split}C_{k,v}(y_{1},y_{2},...,y_{n})&=E_{k}(y_{n}\oplus
E_{k}(y_{n-1}\oplus E_{k}(...\\\ &\oplus E_{k}(y_{1}\oplus
v)...)))=v,\end{split}$ (10)
Finally, a valid ring signature, denoted as $Ring_{sig}$, can be generated by
$Sensor_{s}$.
$Ring_{sig}=(pk_{1},pk_{2},...,pk_{n},v,x_{1},x_{2},...,x_{n}),$ (11)
The signature verification by the smart contract as the fusion centre involves
the following three steps.
* 1.
Calculate all $y_{i}$ using corresponding public key $pk_{i}$ and $x_{i}$;
* 2.
Calculate the symmetric key $k=\mathcal{H}(msg)$;
* 3.
Verify that if $C_{k,v}(y_{1},y_{2},...,y_{n})=v$ holds.
If the verification succeeds, the smart contract recognizes that $msg$ is sent
by a valid sensing node from the group
$\\{Sensor_{1},Sensor_{2},...,Sensor_{n}\\}$. In this way, we can cut the
connection between the data packet and its corresponding owner.
However, for the trust value update and payments assignment process, it is
necessary to know the mapping of the sensing result and its corresponding
sensor. To this end, we propose a two stage commitment scheme as follows.
This scheme consists of two stages including Commit Stage and Reveal Stage. As
shown in Fig. 4, at the commit stage, each sensor participating in the
cooperative sensing needs to submit the $msg$ whose privacy is protected by
the ring signature, and the field “hash of $msgID$”, i.e.
$\mathcal{H}(msgID)$, is used as a commitment. $msgID$ is an identifier which
is the input of hash function, and can be any message such as "I am User 3" or
"I like apples". At the reveal stage, each sensor directly uploads the
unhashed $msgID$ for miners to verify the consistency. This checking mechanism
is effective since hash function is preimage resistance and second-preimage
resistance [32], i.e., given a hash output $\mathcal{R}$=$\mathcal{H}(msgID)$,
it is difficulty to find the input $msgID$ or another input $msgID^{\prime}$
such that $\mathcal{H}(msgID)$ = $\mathcal{R}$ or
$\mathcal{H}(msgID^{\prime})=\mathcal{R}$.
Figure 4: Two-Stage commitment scheme.
## VI Smart Contracts and DSA Protocol Design
Smart contracts are computer codes that run on the blockchain platform, which
are automatically executed when the predefined conditions are satisfied.
Moreover, smart contracts are not controlled by any third party. Therefore, we
propose to use smart contracts to realize the automated operation of
cooperative spectrum sensing and spectrum auction on blockchain. In this
section, we will give the designs of the corresponding smart contracts, and
then propose the DSA protocol based on these smart contracts.
### VI-A Smart Contract Design
We first discuss the design of our smart contracts for cooperative sensing and
spectrum auction. In the following, we describe the parameters and functions
in the two smart contracts, respectively.
_Cooperative Sensing Contract (CSC)_ : The parameters in CSC that need to be
predefined are $T_{ddl}$, $d_{s}$, $N_{1}$ and $TV_{thr}$, which are
introduced as follows.
* •
$T_{ddl}$: The deadline for a sensor to send its sensing packet to CSC;
* •
$d_{s}$: The deposit that a sensor needs to pay before participating in
sensing, which is used to be a guarantee for acting honestly. The deposit can
be withdrawn only if the sensor uploads the sensing result which is consistent
to the cooperative sensing result..
* •
$N_{1}$: The maximum number of sensors needed in cooperative sensing. If there
are more than $N_{1}$ sensors who apply to participate in cooperative sensing,
those with $N_{1}$ highest trust value will be selected.
* •
$TV_{thr}$: the minimum trust value of a sensor is needed to participate in
cooperative sensing.
The functions in CSC are introduced as follows.
* •
$SensorRegister(\cdot)$: The inputs to this function are address $Addr$,
deposit $dpt$ and trust value $TV$ of the node who invokes this function. This
function will decide whether this node can register successfully by checking
its trust value.
* •
$CheckRegisterQuality(\cdot)$: This function is used to check node’s quality
of participating in sensing by checking the parameters passed into it, i.e.,
node’s trust value $TV_{i}$ and deposit $deposit_{i}$. For new users,
participating in sensing is the only way to improve their trust value. This
function offers users a way to convert their tokens to additional trust value,
which enables the users whose trust value is below threshold to participate in
sensing by paying more deposit.
* •
$Fusion(\cdot)$: The input to this function is $msgList$, which is a list
consisting of all the sensing results from the registered sensors. This
function outputs the final sensing result according to the pre-defined fusion
rule.
* •
$UploadSensingData(\cdot)$: The inputs to this function are $msg$ and $RSIG$ ,
which are the sensing packets in (9) and the ring signature of the node,
respectively. This function is invoked by legal sensors to upload their
sensing result.
The pseudocode of the CSC is summarized in Algorithm 2.
_Spectrum Auction Contract (SAC)_ : The parameters in SAC are as follows.
* •
$CSC_{id}$: The identity of corresponding CSC, which is used to identify the
connection between SAC and CSC since every SAC is associated with a particular
CSC.
* •
$N_{2}$: The maximum number of the bidders, which is used to prevent too many
nodes from sending bidding message.
* •
$T_{self-d}$: The time when this SAC will conduct the self-destruct operation
to release memory.
* •
$d_{a}$: The amount of tokens that the bidders need to deposit before the
final sensing result is released.
The functions in SAC are introduced as follows.
* •
$BidderRegister(\cdot)$: Bidders invoke this function to register in SAC.
* •
$Commit(\cdot)$: The input to this function is $bldBid$ which denotes the
commitment of $bidder_{i}$. Note that since the account balance is
transparent, nodes can identify others’ bid by checking their balance[33].
Therefore, everyone makes several commits in order to prevent others from
inferring your bid based on changes of their account balance.
* •
$Reveal(\cdot)$: The inputs to this function include $Dps$, $Bools$ and
$RNDs$. $Dps$ is the list of deposit ($Dp$) the bidders make; $Bools$ is the
list of boolean values which indicate whether these bids are valid or not;
$RNDs$ are random values to make the hash of $\\{Dp$, $Bool$, $RND\\}$ hard to
guess. A $Commit$ is the hash of these three parameters, i.e., $Commit$=
$\mathcal{H}$$\\{$$Dp$, $Bool$, $RND\\}$. $bidsList[msg.sender]$ is the list
of the invoker’s commits. By comparing the reveal messages and the commits,
this function can identify which bids are revealed correctly. Valid bids are
added together as the bidder’s total bid, invalid but correctly revealed bids
are allowed to be withdrawn, bids that are not correctly revealed will not be
returned.
* •
$Win(\cdot)$: The input to this function is $bidderList$, which is the list of
all valid bids. Here we adopt the second-price sealed-bid auction for better
economic profit [34]. Accordingly, this function selects the bidder with the
highest bid as the winner, who only needs to pay the second highest bid.
* •
$EndOfAuction(\cdot)$: SAC will execute self-destruct operation at the preset
time $T_{self-d}$.
The pseudocode of SAC is shown in Algorithm 3.
Algorithm 2 Cooperative Sensing Contract
1:function init($TV_{thr}$, $d_{s}$, $N_{1}$)
2: require(msg.sender==ContractOwner);
3: initialize $TV_{thr}$, $d_{s}$, $N_{1}$;
4:end function
5:
6:function SensorRegister($Addr,dpt,TV$)
7: $sensor_{num}\leftarrow 0$;
8: if checkRegisterQuality($TV_{i}$, $addr_{i}$) then
9: $sensorMap\leftarrow sensor_{i}$;
10: $sensor_{num}\leftarrow sensor_{num}+1$;
11: EMIT event("Registration success!");
12: else
13: EMIT event("Registration failed, please checkout the trust value and
deposit.");
14: end if
15: if $sensor_{num}>N_{1}$ then
16: Select top $N_{1}$ sensors according to trust value;
17: end if
18:end function
19:
20:function Fusion($msgList$)
21: Decision fusion in majority rule;
22: return Cooperative sensing result;
23:end function
24:
25:function checkRegisterQuality($TV_{i}$,$deposit_{i}$)
26: if $deposit_{i}>d_{s}$ then
27: $TV^{{}^{\prime}}_{i}$ $\leftarrow$ $convert(deposit_{i}-d_{s})$
28: if ($TV_{i}+TV^{{}^{\prime}}_{i})>TV_{thr}$ then
29: if $totalNum<N_{1}$ or $TV_{i}>lowestTV$ then
30: Register Successful;
31: end if
32: else
33: Register Failed;
34: end if
35: end if
36:end function
37:
38:function UploadSensingData($msg$, $RSIG$)
39: if $RSIG$ is legal then
40: $msgList\leftarrow msg$;
41: EMIT event("Upload successfully!");
42: else
43: EMIT event("Upload failed, illegal sensor!");
44: end if
45:end function
Algorithm 3 Sealed Spectrum Auction Contract
1:$Bid\leftarrow{bldBid_{i},msg.value}$;
2:$bidsList\leftarrow mapping(address=>Bid[])$;
3:$refund\leftarrow 0$
4:
5:function Commit($bldBid$)
6: $bidsList[msg.sender].push(bldBid);$
7:end function
8:
9:function Reveal($Dps,Bools,RNDs$)
10: $len\leftarrow bidsList[msg.sender].length$;
11: while $len>0$ do
12: $Commit\leftarrow\mathcal{H}\\{Dps[len],Bools[len],RNDs[len]\\}$;
13: if $Commit!=bidsList[msg.sender][len]$ then
14: EMIT event("Illegal Reveal!");
15: Continue;
16: else if $Commit==bidsList[msg.sender][len]$ and $Bools[len]==true$ then
17: $refund\leftarrow refund+Dps[len]$;
18: else if $Commit==bidsList[msg.sender][len]$ and $Bools[len]==false$ then
19: withdraw invalid but reveal correctly bid;
20: end if
21: $len\leftarrow len-1$;
22: end while
23:end function
24:
25:function Win($bidderList$)
26: return Bidder with the highest bid;
27:end function
28:
29:function EndofAuction($T_{self-d}$)
30: Execute self-destruct operation;
31:end function
### VI-B The DSA Protocol
In this part, we illustrate our proposed blockchain-based DSA protocol. To
make it clearer, we elaborate this protocol in Fig. 5.
* •
Phase 1 (_Spectrum Sensing Request_): To ask for the channel state, SUs need
to send a request message to Task Issuer (TI) through the control channel. TI
is played by the node whose trust value is currently the highest. The role of
TI is set to prevent multiple contracts from appearing in the network, which
results in users participate in different contracts. TI will then creates and
deploys the CSC and the corresponding SAC in the blockchain.
* •
Phase 2 (_Smart Contracts and Nodes Register_): The CSC and SAC are
instantiated when TI issues the transaction
$CSC=\\{CSC_{id}\mid T_{ddl}\mid N_{1}\mid TV_{thr}\mid Fusion(\cdot)\mid
d_{s}\\}$ (12)
with the signature $Sig_{TI}(STC)$, and the transaction
$SAC=\\{CSC_{id}\mid SAC_{id}\mid N_{2}\mid T_{self-d}\mid Win(\cdot)\mid
d_{a}\\}$ (13)
with the signature $Sig_{TI}$=$(AC)$. To register to CSC, a sensor issues the
transaction
$Deposit_{CSC_{id}}=\\{pk_{i}\mid TV_{i}\mid CSC_{id}\mid d_{s}\\}$ (14)
with the signature $Sig_{sk_{i}}(Deposit_{CSC_{id}})$ to make deposits.
Similarly, a SU who intends to participate in SAC makes its deposit by issuing
$Bty_{SAC_{id}}=\\{pk_{i}\mid TV_{i}\mid SAC_{id}\mid
d_{a}\mid\mathcal{H}(bid_{i},RND_{i})\\}$ (15)
with the signature $Sig_{sk_{i}}(Bty_{SAC_{id}})$. The
$\mathcal{H}(bid_{i},RND_{i})$ denotes the bidding commitment this SU makes
for the sealed auction.
* •
Phase 3 (_Players selection_): Players (including sensors and bidder) are
selected based on $SensorRegister(\cdot)$ and $BidderRegister(\cdot)$,
respectively. Finally, an event will be triggered to provide an appropriate
notification to every eventually selected nodes.
* •
Phase 4 (_Sensing phase_): Sensors upload the sensing data by issuing the
transaction
$msg_{tx}=\\{CSC_{id}\mid\mathcal{H}(msgID)\mid TS\mid SR\mid loc\\}$ (16)
with the ring signature generated by (11) before $T_{ddl}$, and $TS$ denotes
timestamp, $SR$ denotes the sensing result. In this process, we use the ring
signature to protect the SU’s identity. Meanwhile, these sensors need to make
commitments mentioned in Section V-D by sending the transaction
$Commit_{i}=\\{CSC_{id}\mid\mathcal{H}(SR,RND_{i},msgID)\mid pk_{i})\ $ (17)
with $Sig_{sk_{i}}(Commit_{i})$. After the sensing deadline, data fusion is
done, and the final sensing result will be published in the blockchain
network.
* •
Phase 5 (_Auction if necessary_): If the cooperative sensing result indicates
that the corresponding spectrum bands are idle, the SAC will be invoked. Users
who have been selected will reveal their bids for the spectrum by sending bids
and random numbers. Then the winner will be published on the blockchain.
* •
Phase 6 (_Trust value updating_): In this phase, the trust value of each node
is updated using equation (4) by miners in the network.
Figure 5: The workflow of our proposed blockchain-based DSA.
## VII Implementation and Performance Analysis
### VII-A Implementation of Smart Contract
We implement the proposed smart contracts using Solidity[16] in the Ethereum
Virtual Machine (EVM). The smart contracts are compiled and deployed in the
Remix IDE[35] which is used for testing. After that, the nodes in the
blockchain network can send transactions to deploy the smart contracts and
invoke functions in these contracts. In the cooperative sensing phase, five
sensors are considered and the information of their accounts in the blockchain
network are listed in table II. In the following, we describe the
implementation procedures in detail.
1. 1.
_Smart Contract Preparation_ : The smart contract is created and deployed by
the node with the highest trust value, i.e., the fifth account in table II.
The relevant parameters and functions which need to be predefined are set as
follows: $N_{1}$ is set as 3, the trust value threshold $TV_{thr}$ is set as
900 where we magnify the trust values by 100 times and make they as integers,
since fixed point numbers are not fully supported by Solidity yet. $d_{s}$ is
set as 100 wei,
2. 2.
_Smart Contract Deployment_ : Fig.6 shows the details about the deployed smart
contract. The field “from" is the account address of TI. The field “decoded
input" shows all of our preset parameters.
3. 3.
_Sensor Registration_ : All the five sensors send their registration request
to the smart contract. Then, three sensors are selected by the function
$SensorRegister(\cdot)$.
4. 4.
_Sensing Results Fusion_ : The function $Fusion(\cdot)$, where the majority
rule is selected, is used to fuse the sensing results from three legal
sensors.
If the spectrum is detected as idle, the auction phase initiates. In this
phase, we consider two bidders: each bidder has one true bid and one false
bid. The related information is listed in Table III. The first bidder has two
bids: one is a true bid with 100 wei, the other one is a false bid with 200
wei. The second bidder also has two bids: one is true with 150 wei, the other
is false with 300 wei.
TABLE II: Related information of sensor accounts
Account addresses | Trust value | Sensing Result
---|---|---
0x5B38Da6a701c568545dCfcB03FcB875f56beddC4 | 0.91 | 0
0xAb8483F64d9C6d1EcF9b849Ae677dD3315835cb2 | 0.92 | 1
0x4B20993Bc481177ec7E8f571ceCaE8A9e22C02db | 0.87 | 1
0x78731D3Ca6b7E34aC0F824c42a7cC18A495cabaB | 0.93 | 0
0x5B38Da6a701c568545dCfcB03FcB875f56beddC4 | 0.94 | 1
Figure 6: Initialization of Smart Contract
1. 1.
_Commit_ : Bidders invoke $Commit(\cdot)$ function to make commits. Fig. 7
shows the log of commit made by bidder 2. There are address information and
bid information about this commit, thus others can know that bidder 2 make a
bid with 150 wei, but they can not identify whether this is a real commit or
not.
2. 2.
_Reveal_ : After the commit phase, bidders invoke the function $Reveal(\cdot)$
to reveal its commits. The field “decoded input” in Fig. 8 shows the whole
information about this commit. The smart contract will automatically verify
the information uploaded in the two phase and the final bidding result can be
accessed by all nodes in the system.
TABLE III: Related Information of Bidders
Bidder address | Bid | isReal | Commit
---|---|---|---
0xAb84…5835cb2 | 100wei | yes | 0x591a291ad67…1b62c96a2092
200wei | no | 0x4871a30b671…4d08cfaa8bc1
0x4B20…2C02db | 150wei | yes | 0x4e6a24e35c7be…23643d68efa
300wei | no | 0x0c98edecac…74d24d971ec88
Figure 7: The commit made by bidder 2
Figure 8: The reveal made by bidder 2.
### VII-B Performance Analysis of The Proposed PoT Consensus Algorithm
In this part, we evaluate the performance of the proposed PoT consensus
algorithm. We first discuss how to set the initial mining difficulty at the
beginning when every node’s trust value is 0. We simulate the mining process
in our personal computer as a reference to explain how to select a suitable
mining difficulty. We use a given string to represent the transactions in a
block in the blockchain network. The result is shown in Fig. 9. The running
time increases exponentially when the number of leading zeros is larger than
20. Supposing that the hash rate of the IoT devices is similar to that of our
personal computer, the interval between two blocks $T_{0}$ is about 1 second,
then the mining difficulty should be less than 18 leading zeros, the
$\beta_{0}$ is about $2^{18}=262144$.
It’s worth mentioning that even under the same mining difficulty and
experimental environment, the mining time may be different. For example, given
the mining difficulty where 32 leading zeros are needed in the target hash, it
sometimes take tens of seconds, and sometimes can take more than an hour. In
order to eliminate the impact of luck, we convert the mining difficulty into
the expected mining cost and evaluate the expected mining cost instead of
running time. Theoretically, assuming that the output of each hash calculation
is unpredictable, each time when we increase the leading zero by one, the
success probability of each hash trial will be halved. The trust value is
associated with the mining difficulty. Therefore, we first convert the trust
value into the number of leading zeros, and then the number of leading zeros
is converted into the expected mining cost proportionally. Simulation is
conducted for 1000 time slots, and the average expected mining cost of every
type of nodes is calculated in Fig.11. To be specific, we simulate a network
with 20 nodes and 4 types of nodes are considered. Rnode means a node with
high performance and always act honestly; UAnode means a node with high
performance and act honestly, however it participates in sensing infrequently.
OOnode is a node who conducts malicious behavior periodically. We assume that
OOnode will act maliciously after every two normal sensing rounds. The Lnode
is a node who randomly provides binary sensing result without sensing. There
are 12 Rnode with probability of detection $p_{d}$ = 0.90 and probability of
false alarm $p_{f}$ = 0.15 [36]; 3 OOnode whose $p_{d}$ = 0.90 and $p_{f}$ =
0.15; 3 Lnode with $p_{d}$ = 0.5 and $p_{f}$ = 0.5; and 2 UAnode with $p_{d}$
= 0.90 and $p_{f}$ = 0.15. Besides, it is assumed that they only have a 50%
chance to sign up in CSS each sensing round.
In Fig.11, it can be observed that the reliable node performs best with
respect to the average expected mining cost per timeslot, which is about one
third of the other three types of nodes. The expected mining cost of a on-off
node is a little smaller than Lnode and UAnode but still cost three times than
Rnode. This proves the effectiveness of our proposed consensus algorithm.
Figure 9: Running time (second) with increasing number of leading zeros Figure
10: Curve of trust value Figure 11: Expected mining consumption of different
nodes
### VII-C Performance of Cooperative Sensing
In this part, we evaluate the performance of cooperative spectrum sensing with
the help of our proposed trust evaluation mechanism. In traditional blockchain
network, there is no trust among nodes hence anyone can record the sensing
information into blockchain, including nodes with poor performance or
malicious behavior. However, due to the introduction of the trust value
mechanism, the cooperative sensing contract can exclude bad nodes according to
nodes’ trust value, thus the system’s sensing performance can be improved. For
comparison, we consider 3 kinds of selection schemes to select the candidate
nodes for the cooperative sensing. (i) random selection scheme; (ii) select
according to register time; (iii) select according to sensor’s trust value. We
simulate a 20 nodes network with the same setup in VII-B.
It can be seen from Fig.12 that when the number of needed sensors $N_{1}$ is
small, $p_{d}$ and $p_{f}$ of cooperative sensing in the first two schemes
which do not take advantage of trust value perform worse than that of the last
selection scheme. This indicates that our proposed trust value mechanism can
effectively improve the cooperative sensing performance of the system.
Moreover, in the last selection scheme, $p_{d}$ is close to $1$ when $N_{1}$
is about one fourth of all network nodes. The fewer nodes involved, the
smaller the total monetary reward that the network needs to pay to sensors,
and the smaller the economic burden for spectrum buyers. As $N_{1}$ grows, the
performance of different schemes will gradually close, since most of the
network nodes (including bad nodes) will participate in cooperative sensing.
(a) $P_{d}$ of cooperative sensing (b) $P_{f}$ of cooperative sensing
Figure 12: Performance of cooperative sensing under three selection schemes
### VII-D Incentive Mechanism Analysis
Since both sensing and mining will consume SU’s computing power, an incentive
mechanism is needed to reward nodes for the work they have done. The users who
upload a sensing result that is consistent to the final cooperative sensing
result will be rewarded with $R_{s}$ tokens; The users who successfully mine
will be rewarded with $R_{m}$ tokens. The tokens can be used in auction to bid
for spectrum resources. In implementation, the tokens rewarded for accurate
spectrum sensing and successful mining, i.e., $R_{s}$ and $R_{m}$, need to be
designed based on the evaluation of the computation consumption of spectrum
sensing and mining.
This incentive mechanism not only encourages the nodes to participate in
spectrum sensing, but also encourage them to behave honestly and accurately in
spectrum sensing. This is because, on the one hand, an honest and accurate
sensing node is more likely to derive a sensing result that is consistent to
the final cooperative sensing result and obtain tokens rewarded for spectrum
sensing. On the other hand, with the proposed consensus algorithm, the node
with a higher trust value is easier to mine successfully so that they have a
higher probability of obtaining tokens for mining. On the contrary, the
dishonest nodes will be discouraged since they are less likely to obtain
rewards since they need to spend more computing resources on mining.
### VII-E Security Analysis
* •
_Distributed Denial of Service (DDoS) Attack_ : The DDoS attack here means
that malicious users try to make the sensing service unavailable to other
users. Our system is resist to this attack since we adopt the deposit
mechanism. Thus, under our scheme, the cost of launching large-scale DDoS
attack is very high since the attackers need to obtain lots of tokens in order
to launch the attack.
* •
_Spoofing Attack_ : Spoofing attack means someone tries to masquerade others
to create forged transactions. Secure Elliptic Curve Digital Signature
Algorithm (ECDSA)[37] used in our blockchain can prevent this attack on the
premise that attack does not have the user’s private key.
* •
_Free-riding Attack (lazy node)_ : The Free-riding attack here means lazy
users may directly copy others’ sensing results at the phase of uploading
sensing data. Firstly, there is no motivation for sensors to submit the
sensing result before sensing deadline. Thus, when the lazy nodes get the
sensing result, it is difficult for them to repack the sensing result and
submit before the deadline. Secondly, even if a few sensors submit sensing
results in advance, the connection between a user’s identity and sensing data
is cut by the ring signature, thus the lazy users cannot determine the owner
of the sensing data, and thus the credibility of the data cannot be
guaranteed. Thus, it is no better than submit a sensing result randomly.
* •
_On-off Attack_ : On-off attack means that a node performs malicious behaviors
periodically. If a trust management mechanism satisfy the condition that the
dropping rate of trust value is larger than its increasing rate, it is
considered to be resist to the on-off attack [28]. However, the model in [28]
may cause the trust value drop a lot even the misbehavior is unintentional,
since the increasing rate is much smaller than dropping rate. In our proposed
model, as show in Fig.13, the unintentional mistakes made by sensors can be
compensated by making right sensing decision. When the parameters $\tau$ and
$\eta$ satisfy $\rho\textgreater\frac{\eta}{1-e^{-\eta}}-\eta$, our proposed
trust value management mechanism is considered to be resist to the on-off
attack, the proof is given in Appendix A.
(a) Existing model (b) Proposed model
Figure 13: Trust value of different types of nodes under two model
## VIII Conclusion
In this paper, we have proposed a blockchain-based dynamic spectrum sharing
protocol. This protocol mainly consists of three parts: the first part is the
trust value management mechanism, which is designed to evaluate the
credibility of nodes; the second part is the PoT consensus algorithm, which
make the mining difficulty for malicious nodes greatly increased. The
combination of node’s trust value and their mining difficulty can motivate
nodes to be more willing to behave honestly; the third part is the privacy
protection mechanism, in which we combine the ring signature and the commit-
reveal scheme to solve the problem of privacy issue in the process of
cooperative spectrum sensing. Finally, we implemented the prototype of our
proposed smart contracts and analyzed the performance of PoT consensus
algorithm and the improvement in cooperative sensing. Security analysis of the
system show that our framework can resist many kinds of attacks which are
frequently encountered in trust-based blockchain systems.
## Appendix A Proof of the resistance of on-off attack
###### Proposition 1.
The proposed trust model given in (1) is resistant to on–off attack when
$\rho\textgreater\frac{\eta}{1-e^{-\eta}}-\eta$.
###### Proof.
Let $f(N_{r},N_{w})=e^{-\rho\cdot N_{w}}\cdot\left(1-e^{-\eta\cdot
N_{r}}\right)$ denotes the trust value update function in (1). Because when
$N_{r}=0$, $f(N_{r},N_{w})$ can only increase, this situation is not discussed
here. At any other points $(N_{r},N_{w})$ in this function, the decreasing
rate should be larger than the increasing rate, that is:
$\left\lVert\frac{\partial f}{\partial
N_{w}}\right\rVert>\left\lVert\frac{\partial f}{\partial N_{r}}\right\rVert,$
(18)
where
$\left\lVert\frac{\partial f}{\partial N_{w}}\right\|=\rho\cdot e^{-\rho\cdot
N_{w}}\cdot\left(1-e^{-\eta\cdot N_{r}}\right)$ (19)
and
$\left\|\frac{\partial f}{\partial N_{r}}\right\|=\eta\cdot e^{-\rho\cdot
N_{w}}\cdot\left(e^{-\eta\cdot N_{r}}\right),$ (20)
then we have
$\left\|\frac{\partial f}{\partial N_{w}}\right\|>\left\|\frac{\partial
f}{\partial N_{r}}\right\|\Leftrightarrow\frac{\rho}{\eta}>\frac{e^{-\eta\cdot
N_{r}}}{1-e^{-\eta\cdot N_{r}}}$ (21)
which can also be denoted as:
$\left\|\frac{\partial f}{\partial N_{w}}\right\|>\left\|\frac{\partial
f}{\partial
N_{r}}\right\|\Leftrightarrow\frac{\rho}{\eta}>\frac{1}{1-e^{-\eta\cdot
N_{r}}}-1$ (22)
When $N_{r}\geq 1$,
$\frac{1}{1-e^{-\eta\cdot N_{r}}}-1\leq\frac{1}{1-e^{-\eta}}-1,$ (23)
Therefore, as long as it is satisfied
$\frac{\rho}{\eta}>\frac{1}{1-e^{-\eta}}-1$, the above inequality holds. ∎
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.gifpng.pngconvert gif:#1 png: .gif
# The DAQ system of the 12,000 Channel CMS High Granularity Calorimeter
Prototype
B. Acar G. Adamov C. Adloff S. Afanasiev N. Akchurin B. Akgün M.
Alhusseini J. Alison G. Altopp M. Alyari S. An S. Anagul I. Andreev M.
Andrews P. Aspell I. A. Atakisi O. Bach A. Baden G. Bakas A. Bakshi P.
Bargassa D. Barney E. Becheva P. Behera A. Belloni T. Bergauer M.
Besancon S. Bhattacharya S. Bhattacharya D. Bhowmik P. Bloch A. Bodek M.
Bonanomi A. Bonnemaison S. Bonomally J. Borg F. Bouyjou D. Braga J.
Brashear E. Brondolin P. Bryant J. Bueghly B. Bilki B. Burkle A. Butler-
Nalin S. Callier D. Calvet X. Cao B. Caraway S. Caregari L. Ceard Y. C.
Cekmecelioglu G. Cerminara N. Charitonidis R. Chatterjee Y. M. Chen Z.
Chen K. y. Cheng S. Chernichenko H. Cheung C. H. Chien S. Choudhury D.
Čoko G. Collura F. Couderc I. Dumanoglu D. Dannheim P. Dauncey A. David
G. Davies E. Day P. DeBarbaro F. De Guio C. de La Taille M. De Silva P.
Debbins E. Delagnes J. M. Deltoro G. Derylo P.G. Dias de Almeida D. Diaz
P. Dinaucourt J. Dittmann M. Dragicevic S. Dugad V. Dutta S. Dutta J.
Eckdahl T. K. Edberg M. El Berni S. C. Eno Yu. Ershov P. Everaerts S.
Extier F. Fahim C. Fallon B. A. Fontana Santos Alves E. Frahm G. Franzoni
J. Freeman T. French E. Gurpinar Guler Y. Guler M. Gagnan P. Gandhi S.
Ganjour A. Garcia-Bellido Z. Gecse Y. Geerebaert H. Gerwig O. Gevin W.
Gilbert A. Gilbert K. Gill C. Gingu S. Gninenko A. Golunov I. Golutvin
T. Gonzalez N. Gorbounov L. Gouskos Y. Gu F. Guilloux E. Gülmez M.
Hammer A. Harilal K. Hatakeyama A. Heering V. Hegde U. Heintz V. Hinger
N. Hinton J. Hirschauer J. Hoff W. S. Hou C. Isik J. Incandela S. Jain
H. R. Jheng U. Joshi O. Kara V. Kachanov A. Kalinin R. Kameshwar A.
Kaminskiy A. Karneyeu O. Kaya M. Kaya A. Khukhunaishvili S. Kim K. Koetz
T. Kolberg A. Kristić M. Krohn K. Krüger N. Kulagin S. Kulis S. Kunori
C. M. Kuo V. Kuryatkov S. Kyre O. K. Köseyan Y. Lai K. Lamichhane G.
Landsberg J. Langford M. Y. Lee A. Levin A. Li B. Li J.-H. Li H. Liao
D. Lincoln L. Linssen R. Lipton Y. Liu A. Lobanov R. S. Lu I. Lysova A.
M. Magnan F. Magniette A. A. Maier A. Malakhov I. Mandjavize M. Mannelli
J. Mans A. Marchioro A. Martelli P. Masterson B. Meng T. Mengke A.
Mestvirishvili I. Mirza S. Moccia I. Morrissey T. Mudholkar J. Musić I.
Musienko S. Nabili A. Nagar A. Nikitenko D. Noonan M. Noy K. Nurdan C.
Ochando B. Odegard N. Odell Y. Onel W. Ortez J. Ozegović L. Pacheco
Rodriguez E. Paganis D. Pagenkopf V. Palladino S. Pandey F. Pantaleo C.
Papageorgakis I. Papakrivopoulos J. Parshook N. Pastika M. Paulini P.
Paulitsch T. Peltola R. Pereira Gomes H. Perkins P. Petiot F. Pitters F.
Pitters H. Prosper M. Prvan I. Puljak T. Quast R. Quinn M. Quinnan K.
Rapacz L. Raux G. Reichenbach M. Reinecke M. Revering A. Rodriguez T.
Romanteau A. Rose M. Rovere A. Roy P. Rubinov R. Rusack A. E. Simsek U.
Sozbilir O. M. Sahin A. Sanchez R. Saradhy T. Sarkar M. A. Sarkisla J.
B. Sauvan I. Schmidt M. Schmitt E. Scott C. Seez F. Sefkow S. Sharma I.
Shein A. Shenai R. Shukla E. Sicking P. Sieberer Y. Sirois V. Smirnov
E. Spencer A. Steen J. Strait T. Strebler N. Strobbe J. W. Su E. Sukhov
L. Sun M. Sun C. Syal B. Tali U. G. Tok A. Kayis Topaksu C. L. Tan I.
Tastan T. Tatli R. Thaus S. Tekten D. Thienpont T. Pierre-Emile E. Tiras
M. Titov D. Tlisov J. Troska Z. Tsamalaidze G. Tsipolitis A. Tsirou N.
Tyurin S. Undleeb D. Urbanski V. Ustinov A. Uzunian M. van de Klundert
J. Varela M. Velasco M. Vicente Barreto Pinto P. M. da Silva T. Virdee R.
Vizinho de Oliveira J. Voelker E. Voirin Z. Wang X. Wang F. Wang M.
Wayne S. N. Webb M. Weinberg A. Whitbeck D. White R. Wickwire J. S.
Wilson H. Y. Wu L. Wu C. H Yeh R. Yohay G. B. Yu S. S. Yu D. Yu F.
Yumiceva A. Zacharopoulou N. Zamiatin A. Zarubin S. Zenz H. Zhang J.
Zhang
###### Abstract
The CMS experiment at the CERN LHC will be upgraded to accommodate the 5-fold
increase in the instantaneous luminosity expected at the High-Luminosity LHC
(HL-LHC) [1]. Concomitant with this increase will be an increase in the number
of interactions in each bunch crossing and a significant increase in the total
ionising dose and fluence. One part of this upgrade is the replacement of the
current endcap calorimeters with a high granularity sampling calorimeter
equipped with silicon sensors, designed to manage the high collision rates
[2]. As part of the development of this calorimeter, a series of beam tests
have been conducted with different sampling configurations using prototype
segmented silicon detectors. In the most recent of these tests, conducted in
late 2018 at the CERN SPS, the performance of a prototype calorimeter equipped
with ${\approx}12,000\rm{~{}channels}$ of silicon sensors was studied with
beams of high-energy electrons, pions and muons. This paper describes the
custom-built scalable data acquisition system that was built with readily
available FPGA mezzanines and low-cost Raspberry PI computers.
## 1 Introduction
The HL-LHC at CERN is planned to operate with an instantaneous luminosity of
$5\times 10^{34}$ cm-2s-1 or higher, delivering up to ten times more
integrated luminosity than is expected in the current LHC programme. This
increase poses significant challenges in the design and operation of the
detectors at the HL-LHC. In particular, in the forward direction the absorbed
dose will be up as much as 2 MGy and the fluence in the innermost region is
expected to reach $10^{16}$ neq/cm2, which is an unprecedented level in high
energy collider experiments. Additionally there will be ${\approx}140$ proton-
proton interactions occurring (pile up) in every bunch crossing, which happens
at a rate of 40 MHz. This considerably complicates the reconstruction of
events. To contend with these conditions, the CMS Collaboration is planning a
series of upgrades to some of the existing detector components, and replacing
others with new detectors designed specifically to mitigate the effect of the
high pile up [1]. As part of this upgrade programme the current
electromagnetic and hadronic calorimeters in the endcaps will be replaced with
a new calorimeter, known as the ‘High-Granularity Calorimeter’ (HGCAL) [2].
This new sampling calorimeter (CE) will be sub-divided into two sections, the
electromagnetic (CE-E) and the hadronic (CE-H), in an arrangement shown in
Figure 1. The CE-E will be equipped with silicon sensors, while the CE-H will
be equipped with both silicon sensors and scintillator tiles read out directly
with SiPMs. In the CE-H the silicon sensors will be at small radii, close to
the beam where the radiation levels are highest, and scintillator tiles at the
larger radii. The absorber of CE-E will be a mixture of lead, copper and
sintered copper-tungsten, while in CE-H the absorber plates will be stainless
steel. The hexagonal silicon sensors will be subdivided into hexagonal cells
with areas of ${\approx}1.1\,\rm{cm}^{2}$ or 0.5 cm2, with the sensors with
smaller cells placed at small radii. The full calorimeter will be operated at
-30∘C to reduce the dark current in the silicon sensors and in the SiPMs.
There will be 28 sampling layers in CE-E and 22 in CE-H. This high-transverse
granularity, combined with the high longitudinal segmentation of the
calorimeter has been selected, within the constraints of cost and available
space, to optimise the identification and measurement of hadronic and
electromagnetic showers in the presence of the high pile up.
Figure 1: Schematic view of the CMS high granularity Endcap Calorimeter.
The basic detector unit is a silicon module. A module consists of a silicon
sensor, glued to a baseplate on one side and to a printed circuit board (PCB)
for the readout on the opposite side. The individual cells of the sensor are
connected electrically to a readout ASIC on the PCB with wire-bonds that pass
through holes in the PCB. The ASIC amplifies and digitises the analogue
signals and transmits them to the off-detector electronics on receipt of an
external trigger signal. By 2018 more than 100 prototype silicon modules have
been produced using 6-inch hexagonal silicon sensors subdivided into cells
with areas of ${\approx}$1.1 cm2. A module is shown in Fig. 2 (left). Further
details of the construction and assembly of the silicon modules used in these
tests may be found in [3]. The modules were assembled into a prototype of
HGCAL that was tested with beams of electrons, pions and muons at the CERN
SPS.
The ASIC used to read out of the signals from the silicon cells was the
Skiroc2-CMS [4], a custom ASIC designed by the OMEGA group at Ecole
Polytechnique. The function of this 64-channel ASIC was to measure both the
amplitude of the signal and its time of arrival, so not only the energy
response can be measured, but the timing performance can be characterised.
This Skiroc2-CMS ASIC has many of the features of the HGCAL front-end readout
ASIC in development.
In the latest beam test in late 2018 the HGCAL prototype was equipped with
94-hexagonal silicon modules arranged into a 26 radiation length
electromagnetic section and 5 nuclear interaction length hadronic section.
Behind the prototype calorimeter we placed the Analog Hadronic Calorimeter
(AHCAL) prototype, developed by the CALICE Collaboration [5]. This calorimeter
is a scintillator-based sampling calorimeter, similar in design to the
proposed design of the HGCAL [2], but with much finer longitudinal
segmentation. In the final test at the CERN SPS, data were taken with beams of
muons, charged hadrons and electrons with energies ranging from 20 to 300 GeV
at the H2 beam line of the CERN-SPS over a period of two weeks in October
2018.
The data acquisition (DAQ) system for the beam tests needed to be flexible and
scalable to control and read out the increasing number of prototype silicon
modules as they became available. It was designed with readily available FPGA
mezzanines and low-cost Raspberry PIs, and scaled up to work with
${\approx}$12,000 channels for the final test.
This paper describes the DAQ system and is structured as follows: the overall
architecture of the system is described in section 2; the data format and the
back-end DAQ components are described in section 3; the DAQ software is
explained in section 4; in section 5 the detector systems used for system
synchronisation is discussed and the operational experience is discussed in
section 6.
## 2 DAQ System Architecture
Each hexagonal cell of the silicon sensor was connected to the 64-channel
Skiroc2-CMS ASIC. Each channel of this ASIC had a low noise pre-amplifier
followed by high- and low-gain shapers, with a shaping time of 40 ns, and
time-over-threshold (ToT) and time-of-arrival (ToA) circuits. Both of the
shapers had analogue-to-digital converters (ADCs) that sampled the signal
every 25 ns. It also had a circuit to measure the ToA of large amplitude
signals (${}>3\rm{fC}$) with a precision of 50 ps. Further details of the
design can be found in [4].
To simplify routing of the signals in a very dense board, four Skiroc2-CMS
ASICs were used to readout the 128 channels of each silicon sensor, leaving
half of the channels unused. The PCB also had a MAX®10 field programmable gate
array (FPGA) to control the readout of the module. It received the clock,
trigger and busy signals from the off-detector electronics, aggregated the
data from the Skiroc2-CMS ASICs and transmitted it to the off-detector
electronics. Figure 2 (left) is a photograph of a prototype module, with the
Skiroc2-CMS ASICs marked with rectangles, and the MAX®10 FPGA, on the top
left, is indicated with a white circle.
Figure 2: (left) A prototype module used in beam tests. The Skiroc2-CMS ASICs
are marked with white rectangles. The MAX®10 FPGA is marked with a white
circle. The red grounding wire (on the left side) and, red and violet bias
wires (on the right side) are soldered on the PCB. The micro HDMI (uHDMI)
cable is connected on the top. (right) An interposer board used in beam tests.
The HDMI cable and bias voltage wires, connecting the interposer board to the
prototype module, are connected on the top. The HDMI and RG174 cables,
connecting the interposer board to the DAQ board, are connected on the bottom.
Each of the prototype silicon modules was connected to the off-detector DAQ
boards through an interposer board. These boards regulated the 5 V output from
the DAQ boards to the 3.3 V needed by the prototype modules via HDMI cables.
They also filtered and transmitted the bias voltage coming from the DAQ boards
through RG174 cables to wires soldered to the prototype modules. A photograph
of an interposer board is shown in Figure 2 (right).
The off-detector electronics consisted of a set of custom 9U readout boards,
each of which could receive data from up to eight silicon modules. All the the
readout boards were controlled by a single custom 9U synchronisation board,
the ‘sync board’. The sync board distributed the clock, trigger and busy
signals to all the readout boards. The readout boards communicated with the
data acquisition computer through Ethernet, with a 100 Mbit/s output of the
readout boards connected to a Gigabit Ethernet switch, from which data were
sent to the DAQ computer for processing. In the tests with 94 silicon modules,
one sync board and fourteen readout boards were mounted in two custom air-
cooled crates. The crate that was equipped with one sync board and seven
readout boards is shown in Figure 3.
Figure 3: A crate, used in HGCAL beam tests, populated with one sync board
and seven readout boards. The sync board (in the leftmost slot) distributed
the clock, trigger and busy signals to the readout boards via HDMI cables, not
shown. The readout boards were used to send control data to the silicon
modules, to receive data from them, and to transmit the data to the
acquisition computer through an Ethernet switch (on the right side of the
readout boards). The readout and sync boards were powered by the power modules
located under the DAQ crate.
Figure 4 shows a schematic view of the inter-connectivity of the DAQ system.
The readout boards were connected to the prototype silicon modules by HDMI
cables both between the readout boards and the interposer boards and between
the interposer boards and the modules. The trigger signal was formed from a
coincidence of signals from two scintillation counters located upstream of the
calorimeter. The 40 MHz system clock, generated on the sync board and the
trigger signals were transmitted to the readout boards with HDMI cables. Since
the beam was not synchronised with the clock, the trigger and the clock were
asynchronous. The bias voltage was distributed to the silicon sensors through
the readout boards with separate RG174 cables.
Figure 4: Schematic view of the DAQ system used in HGCAL beam tests.
## 3 Data format and back-end DAQ components
### 3.1 Skiroc2-CMS ASIC and prototype module data format
In operation, the analog signals were stored every 25 ns in a switched-
capacitor array (SCA) with a depth of 13 cells. When a trigger was received,
the updating of the SCA was halted and the two values for the ToA were stored,
one referenced to the next falling edge of the 40 MHz clock, and the other to
the next rising clock edge. Two values of the ToT were also kept, one with a
fast ramp time-to-digital converter and another with a slow ramp.
[bitwidth=1.6em]16 []5 & 0,1,2,3,4,15
1924 $\times$ 16-bit
integers []5$\times$64 channels 111 110 110 11$H_{A}$ 1212Low gain ADC (SCA0)
[]5$\times$64 channels 111 110 110 11$H_{A}$ 1212High gain ADC (SCA0)
[]5 1616…$\times$13 SCA cells
[]5$\times$64 channels 111 110 110 11$H_{A}$ 1212Low gain ADC (SCA12)
[]5$\times$64 channels 111 110 110 11$H_{A}$ 1212High gain ADC (SCA12)
[]5$\times$64 channels 111 110 110 11$H_{A}$ 1212ToA (stop falling clk)
[]5$\times$64 channels 111 110 110 11$H_{A}$ 1212ToA (stop rising clk)
[]5$\times$64 channels 111 110 110 11$H_{T}$ 1212ToT (fast ramp)
[]5$\times$64 channels 111 110 110 11$H_{T}$ 1212ToT (slow ramp)
[]5 110 110 110 1313Roll position (13-bit)
[]5 110 110 1414Global timestamp MSB (14-bit)
[]5 110 110 110 1212Global timestamp LSB (12-bit) 110
[]5 111 111 110 110 110 110 110 110 88Chip ID (8-bit)
Figure 5: Data format of the Skiroc2-CMS ASIC. $H_{A}$ is the hit bit for ToA
and is set to ’1’ when ToA is fired. Similarly, $H_{T}$ is the hit bit for ToT
and is set to ’1’ when ToT is fired. The 13-bit roll position is used to
reorder the SCA cells in time.
Figure 5 shows the data format of the Skiroc2-CMS ASIC.
When a trigger was received, by the MAX®10 FPGA of the hexaboard the four
Skiroc2-CMS ASICs converted the data in analogue memory to the digital data
format. These data were then read out by the MAX®10 FPGA and packaged as shown
in Figure 6. The bits $b_{Ai}$ belong to the ASIC "i" which followed the
Skiroc2-CMS data format shown in Fig. 5. For every event 30784 bytes of data
were transmitted from each hexaboard. These data were then gathered, via the
HDMI-uHDMI cables, by the back-end readout boards.
[bitwidth=3.0em]8 0,3,4,7
44HEADER & 441 bit per ASIC
1924 $\times$ 16 bytes 11 0 0 0 $b_{A0}$ $b_{A1}$ $b_{A2}$ $b_{A3}$
88…
11 0 0 0 $b_{A0}$ $b_{A1}$ $b_{A2}$ $b_{A3}$
Figure 6: Hexaboard data format. The bits $b_{Ai}$ belong to the ASIC "i"
which followed the Skiroc2-CMS data format described by Figure 5.
### 3.2 Back-end DAQ electronics
The DAQ system was designed to be easily scalable to provide a readout for
different numbers of silicon modules. To minimise costs, readily available
commercial components were used. Additionally, optical receiver modules (oRMs)
[6], recovered from the CMS level-1 trigger system, when it was upgraded with
faster electronics, were used. Each oRM was equipped with a Kintex-7 FPGA, 4.8
Mbits of block RAM, two 6.6 Gbit/s bi-directional serial ports, and a 128 Mbit
FLASH memory for configuration. The connection from the Kintex-7 FPGA to the
gigabit Ethernet switch was made with SFP to RJ-45 adapters.
#### 3.2.1 Readout board
In the final beam test in October 2018, there were 94 silicon modules that
were read out with 14 readout boards mounted in two racks, controlled by a
sync board. The readout boards performed the following tasks:
* •
Loading firmware on the Max®10 FPGAs and module initialisation and reset.
* •
Generating and distributing control signals for the prototype silicon modules.
* •
Accumulating the data received from the prototype silicon modules.
* •
Distributing the clock, busy and trigger signals.
* •
Distributing the low voltage power and the bias voltage for the prototype
silicon modules.
The readout board, shown in Figure 7, was a custom PCB equipped with five oRMs
and a Raspberry Pi. Each board had eight HDMI ports on the front panel
connected to the silicon modules through the interposers and one HDMI port on
the back panel for connection to the sync board. On the front of each board
there were eight RG174 connectors that were used to distribute, after
filtering, the bias voltage through the interposers to each detector module.
Figure 7: The readout board used to send control data to the silicon modules,
to receive data from them and to transmit it to the data acquisition computer.
It readout data from up to eight silicon modules via HDMI connectors. It also
supplied the bias voltage for up to eight prototype silicon modules via
standard RG174 connectors. It was equipped with one control and four data oRMs
and one Raspberry Pi.
The overall readout cycle was controlled via helper processes running on the
Raspberry Pi, which communicated with each oRM through the SPI bus. Each Pi
was connected to the central DAQ server through its Ethernet port, from which
it also received ‘Start’, ‘Stop’, and other commands.
A single readout board was equipped with five oRMs: one control (CTL) oRM, and
four DATA oRMs. The DATA oRMs were responsible for reading the data from up to
two silicon modules, while the CTL oRM received data from the DATA oRMs and
transferred it to the central server. It also managed the communication with
the sync board. The firmware installed on the CTL oRM’s FPGA included the
IPBus firmware [7, 8] for this purpose. The IPBus IP and MAC addresses of the
oRM were set by the Raspberry Pi, as well as other parameters used by the CTL
oRM as it combined the four streams of data.
Before a run started, the helper processes on the Raspberry Pis first
configured the ASICs on the prototype silicon modules, and data collection was
initiated. The Trigger signal was broadcast from the sync board to the readout
boards, from where it was forwarded on to the modules. On the readout boards,
the helper processes running on the Raspberry Pis after receiving the trigger
signal, prompted the ASICs to initiate data transmission. The data from the
ASICs were then sent unprocessed, via the MAX®10 FPGA, to the DATA oRMs, where
it was then merged into a single data stream by the CTL oRM. The 4-bit headers
of Figure 6 were then dropped and 32-bit integers were built with the data
from up to eight modules, corresponding to 32 ASICs. These 32-bit integers
were written to a FIFO to be readout by the central server using the IPBus
protocol over a gigabit Ethernet link. When ready, a flag was set inside the
CTL’s RAM to indicate that the data were ready for transfer. In Fig. 8 the
output data format of the CTL oRM FIFO is shown.
Once the data had been fully read out by the server, the helper processes on
the Pis reset the ASICs, and sent a start acquisition signal. The CTL oRM then
sent a ‘ReadoutDone’ signal to the sync board, indicating the boards had
finished their cycles and were ready to receive the next trigger. The firmware
block diagram of the readout board is shown in Figure 9.
[bitwidth=2.0em,bitformatting=16 0-15
30784 $\times$ 32-bit
integers 1 $b_{0,0}$ $b_{0,1}$ $b_{0,2}$ $b_{0,3}$ 88… 1$b_{7,0}$ $b_{7,1}$
$b_{7,2}$ $b_{7,3}$
1616…
1 $b_{0,0}$ $b_{0,1}$ $b_{0,2}$ $b_{0,3}$ 88… 1$b_{7,0}$ $b_{7,1}$ $b_{7,2}$
$b_{7,3}$
Figure 8: Data format of the CTL oRM FIFO readout by the central server using
the IPBus protocol. The bits $b_{i,j}$ correspond to the data of ASIC "j" of
module "i". Figure 9: The firmware block diagram of the readout board.
#### 3.2.2 Sync board
The function of the sync board, shown in Figure 10, was the distribution of
the common signals to the readout boards and to synchronise the flow of data
from the readout boards. The sync board generated the 40 MHz system clock on a
small mezzanine card mounted at the rear of the board. On the same mezzanine
there were four 50 $\Omega$ RG174 connectors. Two were for the Trigger and a
Veto signal inputs and two for the Clock and a copy of the Trigger signal
outputs. The Veto signal was not used in these tests. Processing on the sync
boards was handled by a Raspberry Pi as well as a central (SYNC) oRM. One sync
board could control up to 15 readout boards through 15 HDMI ports mounted on
the front. An extra HDMI port was mounted on the front to allow for the
possibility of daisy-chaining two or more sync boards together when more than
15 readout boards are to be readout.
Figure 10: The sync board was used to control up to 15 readout boards. It
received the Veto and the Trigger signals and distributed control signals to
up to 15 readout boards. On top of the 15 HDMI ports for readout board control
there was one extra port for connection to another sync board for daisy-
chaining. It was equipped with a Raspberry Pi and an Kintex-7 FPGA for control
and communication. It was also equipped with a mezzanine card for clock
generation and receiving external signals.
At the start of a readout cycle, the sync board waited for an asynchronous
External Trigger signal. Then this signal was synchronized with the on-board
40 MHz clock and sent to the readout boards to be distributed to the silicon
modules. The sync board then waited for a ‘ReadoutDone’ signal from each
readout board. Once this signal was received from all the readout boards, the
sync board made itself ready to process the next available trigger. The
firmware block diagram of the sync board is shown in Figure 11.
Figure 11: The firmware block diagram of the sync board.
## 4 Data acquisition software
The DAQ software selected for these tests was based on the EUDAQ [9]
framework. This framework, written in C++, was developed specifically for
small-to-medium scale systems with significantly less overhead than frameworks
used in large scale experiments, like XDAQ [10]. Additionally, in separate
earlier tests of the AHCAL prototype, the EUDAQ system had already been used
successfully.
The EUDAQ framework was designed to be modular and portable. It was structured
so that software for the readout of specific detector components was kept
separate and distinct from the core processes. For this each detector
component that produced data had a ‘Producer’ process running. The functions
performed by the Producer was to initialise, configure, issue stops and starts
to the component, and to collect the data and forward it to the core process.
### 4.1 CMS-CE EUDAQ Producer
A Readout Producer was developed to read out the data from one or more readout
boards in parallel. During the combined beam test of October 2018, the DAQ had
seven of these Readout Producers, each connected to two readout boards. The
$\mu$TCA Hardware Access Library ($\mu$HAL) was used to read the IPBus UDP
transactions from the readout boards. The sequence of operations of the
Readout Producer were as follows:
1. 1.
Wait until each $\mu$HAL interface is notified that a trigger occurred (by
checking an IPBus register of the CTL oRM boards).
2. 2.
Read out the FIFO of the CTL oRM board from each readout board and fill raw
data containers. The data format of this FIFO is described in Section 3.2.1. A
time-stamp – the number of 40 MHz clock cycles in a 64-bit integers since the
last configuration – is read out with the data.
3. 3.
‘ReadoutDone’ signal is sent from CTL oRM to Sync oRM.
4. 4.
Create an event block containing the raw data from each readout board, the
time-stamp of the readout board and the event ID.
5. 5.
Forward the event block to the EUDAQ data collector.
6. 6.
Increment the event ID.
7. 7.
Return to step 1 and wait for the next trigger.
Once the readout was complete, data from each of the event blocks from each of
the Readout Producers were combined with data from the Producers connected to
other detector components to form a complete event data block.
#### 4.1.1 EUDAQ online data monitoring
Part of the EUDAQ framework were tools to monitor the data collection. Online
analyses were developed to monitor in real time the stability of the pedestal
values, noise and occupancies of each of the silicon sensor channels.
#### 4.1.2 Data unpacking and first analysis steps
The first step of the data analysis consisted of unpacking the CMS CE EUDAQ
events. For this purpose a C++ library embedded in the EUDAQ framework has
been developed.
An initial data quality check was performed before unpacking the raw data by
comparing the difference in time-stamp with the time-stamp of the previous
event as a check of the synchronisation for all the readout boards. During the
October data taking only a few runs had events with a synchronization failure.
After this test the data were unpacked and the data from each ASIC were sorted
into tables of 16-bit integers with the structure shown in Figure 5 and stored
in a ROOT [11] file. As zero-suppression was not used for simplicity, these
tables contained the data from every channel of the ASICs, including those not
connected to a detector channel. The data for each cell in an event contained
data and pointer information from the 13 SCA cells for both the high- and the
low-gain slow shapers, the ToA and the ToT measurements. The pointer was the
address of first SCA cell data for the event, which allowed the ordering of
the SCA data in time. This was required for the data reconstruction since the
trigger was asynchronous with the 40 MHz clock. The data analysis workflow,
which was developed in the CMSSW framework [12], and transformed the tables of
16-bit integers into a collection of calibarated hits for data analysis used
the ROOT files as input.
## 5 System synchronization
### 5.1 Beam-characterization detectors
The tests of HGCAL and AHCAL prototypes with particle beams in the H2 area at
CERN had been complemented by the readout of various beam-characterization
detectors. Four delay wire chambers (DWC) [13] measured the trajectory and
impact of the particles in the beam, two scintillator detectors served as
external trigger source and two micro-channel plates (MCP) had been used to
provide fast signals for reference timing measurement of the incident
particles [14]. For this purpose, two 16-channel CAEN v1290N TDCs and one
v1742 digitiser were integrated into the HGCAL prototype DAQ.
From each DWC four signals were separately discriminated at a threshold of -30
mV and fed as inputs to the TDC. Since the binning of the time-stamp
digitization should be less than 1 ns corresponding to an optimal resolution
of the position measurement of 200$~{}\mu$m [13], a binning of 25 ps had been
chosen. For proper event synchronization, the trigger for all CAEN modules
stemmed from the duplicated TTL trigger signal issued by the synchronization
board. After conversion to NIM, it was copied three times and fed into each
module individually. After receiving a trigger, events were built, were
labeled with trigger time-stamps and subsequently stored in a local buffer.
Two dedicated EUDAQ producers [9] had been developed. They ran on a separate
computer and communicated to these modules through optical link and VMEbus.
These producers polled the event data from the buffers at a (configurable)
frequency of 500 Hz (HGCAL DAQ rate $\approx 40~{}\text{Hz}$), converted the
raw data into the EUDAQ format and sent it to the main DAQ for storage and
online data monitoring.
### 5.2 AHCAL
The front-end electronics of the AHCAL prototype were designed for low power
operation under a specific ILC accelerator timing with a less than 1 ms long
spill followed by 199 ms idle time. The operation of the Spiroc ASIC [15] was
therefore split in 3 phases: 1) acquisition phase, where self-triggered events
were stored into up to 16 analog memory columns; 2) conversion phase, where up
to 16 events were sequentially digitized by internal ADC; 3) readout phase,
where the digitized data was read out. Detailed timing was described in [16].
For beam test purposes, the acquisition phase length was extended to 16 ms and
the readout phase varied typically between 2 to 20 ms, depending on the hit
occupancy. Any external trigger from the sync board stopped the acquisition
for immediate readout, as shown in Fig 12.
Figure 12: The time diagram of AHCAL acquisition and readout phases.
Due to the ‘self-trigger’ design of the ASIC, the AHCAL did not require an
external trigger for data taking. All the hits (including noise hits) were
read out and referenced by a number of bunch-crossing clock cycles (4 us
period, called BXID) from the start of the acquisition phase. In order to
assign an external trigger to the hits in the AHCAL, the DAQ internally
samples the external trigger number (with a time-stamp, 48-bit counter with a
25 ns resolution) and the time-stamp of the start of the relevant acquisition
phase. The trigger was assigned to one of the self-triggered events in the
acquisition cycle. Assignment was based on the startup time from the start of
the acquisition phase to the beginning of the first BXID and the additional
delay due to the length of the trigger cable.
Figure 13: The organization of self-triggered particles in AHCAL BXIDs,
showing the BXID ambiguity for an example of split events due to the time of
arrival with respect to the BXID clock.
The collation of events according to the BXID might have however led to an
existence of incomplete events for particles, that arrived close to the BXID
counter value switching in the ASICs. An example for such a split event is
shown in Figure 13 (particle no. 2). Several factors contribute to the BXID
ambiguity: time walk of the signals, clock skew due to board and ASIC
location, clock tree distribution through the FPGAs and clock jitter. The
internal ASIC TDC [15] had also a region of non-linearity around the BXID
change. Therefore, particles arriving close to the BXID change, within 10 ns,
needed to be excluded from the data analysis.
The AHCAL provided two means of synchronization with the HGCAL data: the
trigger number and the trigger time-stamp, which used the 40 MHz clock from
the HGCAL sync board. Both pieces of information were accessible in the data
file.
## 6 Collected Data
The DAQ was operated at a readout rate of 40 Hz. At this rate the amount of
data readout for a typical 5 second spill of the SPS for the HGCAL prototype
was approx 300 MB, as each of the 94 silicon modules sent 16 KB of data per
event.
In the last run, over a period of two weeks, six million events were collected
with beams of charged hadrons, electrons, and muons with momenta from 20 to
300 GeV/c, with different detector configurations. Figure 14 shows accumulated
events for different detectors over the beam-test campaign.
Figure 14: The accumulated events for different detectors during beam-test
campaign in October 2018.
## 7 Summary
In the upgrade of the CMS detector for when the HL-LHC is operational, the two
endcap calorimeters will be replaced with high granularity sampling
calorimeters equipped with silicon sensors. As part of the development of this
calorimeter, a series of beam tests have been conducted with different
sampling configurations using prototype segmented silicon detectors readout
with a low-cost custom scalable data acquisition system. The software
framework used for the run control and data collection was the portable
modular EUDAQ framework. In the most recent of the tests conducted in late
2018 at the CERN SPS in 2018, the performance of a prototype calorimeter
equipped with ${\approx}12,000\rm{~{}channels}$ of silicon sensors, in
conjunction with the CALICE prototype analogue hadron calorimeter, was studied
with beams of high-energy electrons, pions and muons, with six million events
collected over a two week period.
## Acknowledgments
We thank the technical and administrative staffs at CERN and at other CMS
institutes for their contributions to the success of the CMS effort. We
acknowledge the enduring support provided by the following funding agencies:
BMBWF and FWF (Austria); CERN; CAS, MoST, and NSFC (China); MSES and CSF
(Croatia); CEA and CNRS/IN2P3 (France); SRNSF (Georgia); BMBF, DFG, and HGF
(Germany); GSRT (Greece); DAE and DST (India); MES (Latvia); MOE and UM
(Malaysia); MOS (Montenegro); PAEC (Pakistan); FCT (Portugal); JINR (Dubna);
MON, RosAtom, RAS, RFBR, and NRC KI (Russia); MST (Taipei); ThEPCenter, IPST,
STAR, and NSTDA (Thailand); TUBITAK and TENMAK (Turkey); STFC (United
Kingdom); DOE (USA).
## References
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* [2] C. Collaboration, “The Phase-2 Upgrade of the CMS Endcap Calorimeter,” Tech. Rep. CERN-LHCC-2017-023. CMS-TDR-019, CERN, Geneva, Nov 2017. Technical Design Report of the endcap calorimeter for the Phase-2 upgrade of the CMS experiment, in view of the HL-LHC run.
* [3] A. Steen, B. Akgun, et al., “Construction, commissioning and response equalisation of cms ce prototype silicon modules,” In preparation for JINST.
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* [8] J. Mans and E. Frahm, “Status report on a MicroTCA card for HCAL trigger and readout at SLHC,” JINST, vol. 5, no. 12, pp. C12027–C12027, 2010\.
* [9] E. D. Team, “EUDAQ User Manual - Last update on October 2016 for EUDAQ version 1.7,” 2016.
* [10] V. B. et al., “Using xdaq in application scenarios of the cms experiment,” FERMILAB-CONF-03-293, CHEP-2003-MOGT008 (2003).
* [11] “https://root.cern.ch/guides/reference-guide,”
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* [13] J. Spanggaard, “Delay Wire Chambers - A Users Guide,” SL-Note-98-023, 1998\.
* [14] N. Akchurin et al., “First beam tests of prototype silicon modules for the CMS High Granularity Endcap Calorimeter,” JINST, vol. 13, p. P10023, 2018.
* [15] S. Lorenzo, S. Callier, J. Fleury, F. Dulucq, C. de La Taille, G. Martin Chassard, L. Raux, and N. Seguin-Moreau, “SPIROC: design and performances of a dedicated very front-end electronics for an ILC Analog Hadronic CALorimeter (AHCAL) prototype with SiPM read-out,” Journal of Instrumentation, vol. 8, p. C01027, 01 2013.
* [16] J. Kvasnicka, “Data acquisition system for the CALICE AHCAL calorimeter,” Journal of Instrumentation, vol. 12, no. 03, p. C03043, 2017.
|
# Causal dynamics of null horizons under linear perturbations
Peter K S Dunsby<EMAIL_ADDRESS>Cosmology and Gravity Group,
Department of Mathematics and Applied Mathematics, University of Cape Town,
Rondebosch 7701, Cape Town, South Africa South African Astronomical
Observatory, Observatory 7925, Cape Town, South Africa Centre for Space
Research, North-West University, Potchefstroom 2520, South Africa Seoktae Koh
<EMAIL_ADDRESS>Department of Science Education, Jeju National
University, Jeju, 63243, South Korea Abbas M Sherif<EMAIL_ADDRESS>Department of Science Education, Jeju National University, Jeju, 63243, South
Korea
###### Abstract
We study the causal dynamics of an embedded null horizon foliated by
marginally outer trapped surfaces (MOTS) for a locally rotationally symmetric
background spacetime subjected to linear perturbations. We introduce a simple
procedure which characterizes the transition of the causal character of the
null horizon. We apply our characterization scheme to non-dissipative
perturbations of the Schwarzschild and spatially homogeneous backgrounds. For
the latter, a linear equation of state was imposed. Assuming a harmonic
decomposition of the linearized field equations, we clarify the variables of a
formal solution to the linearized system that determine how the null horizon
evolves. For both classes of backgrounds, the shear and vorticity 2-vectors
are essential to the characterization, and their roles are made precise.
Finally, we discuss aspects of the relationship between the characterizing
conditions. Various properties related to the self-adjointness of the MOTS
stability operator are extensively discussed.
## I Introduction
Background and motivation: The discovery of supermassive black holes as well
as the detection of gravitational waves emitted by inspiralling black holes
have ignited renewed interest in black hole physics. A key concept in
understanding the dynamics of black holes is the notion of a trapped surface,
those on which both ingoing and outgoing null rays are converging, introduced
in the proof of the 1965 singularity theorem by Roger Penrose [1].
Marginally outer trapped surfaces (MOTS) are the marginal case of trapped
surfaces, closed surfaces on which the outgoing null ray is neither converging
nor diverging. Slight iterations to the definition of MOTS can be found in the
literature from Wald’s marginally trapped surface [2] (note the “outer” here
has been dropped) which rather imposes that both null rays are non-diverging,
to Hayward’s marginal surface [3] which is in fact what we call a MOTS here,
but in Hayward’s case the choice is which of the null ray that is neither
converging nor diverging is a not fixed. (See the references [4, 5] for
excellent reviews and additional details. The notion of a MOTS will also be
rigorously defined in Section III).
MOTS foliate 3-dimensional hypersurfaces known as marginally outer trapped
tubes (MOTTs) [8], and where the causal character is fixed at all points of a
MOTT it will be referred to by the common name horizon, which under certain
conditions bound trapped regions containing trapped surfaces. Dynamical and
null horizons are well known cases of horizons (see for example [9]). The
timelike MOTTs are usually referred to simply as membranes as they, by
definition, cannot enclose trapped surfaces.
MOTS have also been fundamental to understanding the formation of black holes
due to gravitational collapse, as well as their dynamics and evolution. MOTS
admit a notion of stability [6, 7] (consequently leading to a notion of
stability of black holes). This is a way to determine whether one can deform a
MOTS $\mathcal{S}$ to another $\mathcal{S^{\prime}}$ along a unit direction
normal to $\mathcal{S}$ that points along the slice in which $\mathcal{S}$ is
embedded. This stability condition is given in terms of conditions on the
principal eigenvalue of a second order linear elliptic operator. It has been
shown that under physically reasonable conditions, strict stability, where the
principal eigenvalue is positive, a MOTS will evolve to foliate a DH [6] (the
same conclusion follows in the case that the operator has no zero eigenvalue).
Given a horizon embedded in a background spacetime which is subjected to a
perturbation, one expects the perturbation to in general affect the dynamics,
and hence causal character of the horizon. Perturbation of horizons have been
examined variously in different contexts using different approaches. For
example, in [10] linear perturbations of a null horizon by a gravitational
field was studied and it was established that such perturbations do not affect
the causal character of the horizon. This feature of the invariance of the
causal character, as have already been mentioned, will not be the case in
general, and this was neatly discussed in [11], where generic first and higher
order perturbations of non-expanding horizons were considered.
A more explicit example was considered by Pilkington et al. in [12]. Embedding
a stationary metric in the Weyl solution is a way of smoothly distorting
stationary blackholes (see for example [13]). The Weyl potentials tune the
behavior of horizons and are referred to as distortion potentials. The authors
demonstrated that while for small distortions the horizon is apparently ‘well
behaved’, for large distortions, even the MOTS structure of the horizon is not
assured, let alone the causal character.
Objectives: Our aim here is to introduce an approach which characterizes the
evolution of MOTS in a linearly perturbed locally rotationally symmetric (LRS)
background spacetime [14, 15]. More precisely we consider the following
problem: How does a null horizon foliated by MOTS, embedded in a LRS
background solution, evolve when the background is subjected to linear
perturbations?
We will employ the 1+1+2 covariant formalism [16, 17], which has been used in
several works to study MOTS and their evolution in background LRS spacetimes
[18, 19, 20, 21]. After formulating the characterization scheme, we will adapt
our approach to cases of some well known background solutions. Specifically,
we consider the null event horizon in the Schwarzschild background and null
horizons in hypersurface orthogonal and spatially homogeneous background
solutions. As will be seen, at least in these examples to which our scheme
will be applied, the shear and vorticity 2-vectors appear to consistently play
the pivotal roles in determining the causal behavior of the null horizons
under perturbations. (The roles of these 2-vectors will be made more precise
in the text).
We also aim to consider aspects of the self-adjointness of the stability
operator. As it will be seen, the 1-form that characterizes the self-
adjointness of the operator is defined in terms of the shear and vorticity
2-vectors as well, and so it is quite natural to consider this in relation to
the evolution of the horizons.
Structure: Section II of the paper gives a quick overview of the covariant
formalism that is used, and the procedure for linearizing a LRS background
solution, which provides the mapping between a background solution and a
perturbed one, is outlined. The process to harmonically decompose first order
scalars, 2-vectors and 2-tensor, in this formalism is briefly reviewed
following the standard literature. In Section III, we introduce an approach to
characterize MOTS evolution in the perturbed solution.
We start by briefly introducing the notion of a MOTS, which is well
proliferated in the literature but nonetheless necessary for the flow of the
paper. We then go on to introduce the characterization scheme where it becomes
transparent how the required gauge choice forces us to consider a null horizon
in the background. That is, our scheme does not work for a non-null horizon.
Section IV applies the characterization scheme to static and spatially
homogeneous LRS backgrounds. In both cases the evolution of the null horizon
is characterized, with the case of the latter an equations of state being
imposed and a partial characterization provided. In Section V, we discuss some
simple implications of the results of Section IV for the self-adjointness of
the MOTS stability. Several additional properties of the MOTS and aspects of
the stability operator are discussed. We conclude with discussion of results
in Section VI and discuss potential avenues for future research.
## II Preliminaries
In this section we briefly introduce the 1+1+2 covariant formalism. We then
describe the general prescription for the perturbative scheme that have been
used extensively over the last few years, to carry out perturbations of
background LRS solutions.
### II.1 Covariant decomposition
There is by now a well established trove of literature on the 1+1+2 covariant
formalism, with several references providing details of the steps in the
decomposition. A reader interested in more details is referred to the
following standard texts [16, 17] (Also see the references [22, 23, 24] for
some recent corrections to the full set of equations). As such, the
introduction here would be very brief, only necessary enough for the work
carried out in this paper.
In the well known and powerful 1+3 approach, a unit tangent direction - call
this $u^{a}$ \- along which timelike observers flow, is what threads the
spacetime. The field equations are decomposed along $u^{a}$ plus some
constraint equations. If there exists some unit spacelike vector $n^{a}$
obeying $u_{a}n^{a}=0$, the 3-space from the previous split can be further
decomposed along this direction as a $1+2$ product manifold. This, in addition
to the evolution equations along $u^{a}$, introduces a set of propagation
equations along $n^{a}$, as well as some constraint equations. The splitting
naturally introduces new derivative along $n^{a}$ and on the 2-space
(generally referred to as the sheet):
* •
Along $u^{a}$:
$\dot{\psi}^{a\cdots b}_{\ \ \ \ c\cdots d}=u^{f}\nabla_{f}\psi^{a\cdots b}_{\
\ \ \ c\cdots d}$;
* •
Along $n^{a}$:
$\hat{\psi}^{a\cdots b}_{\ \ \ \ c\cdots d}=n^{f}\nabla_{f}\psi^{a\cdots b}_{\
\ \ \ c\cdots d}$;
* •
Along the sheet:
$\delta_{f}\psi^{a\cdots b}_{\ \ \ \ c\cdots d}=N^{\bar{c}}_{\ c}\cdots
N^{\bar{d}}_{\ d}N^{a}_{\ \bar{a}}\cdots N^{b}_{\ \bar{b}}N^{e}_{\
f}\nabla_{e}\psi^{\bar{a}\cdots\bar{b}}_{\ \ \ \ \bar{c}\cdots\bar{d}}$,
for any tensor $\psi^{a\cdots b}_{\ \ \ \ c\cdots d}$, where
$N^{ab}=g^{ab}+u^{a}u^{b}-n^{a}n^{b}$ projects vectors and tensors to the
sheet (it is the metric induced from the splitting, on the sheet).
A spacetime vector $\psi^{a}$ can accordingly be decomposed as $\psi^{a}=\Psi
u^{a}+\bar{\Psi}n^{a}+\Psi^{a}$ ($\Psi^{a}$ is the sheet component of
$\psi^{a}$), and a fully projected tensor ${}^{3}\psi_{ab}$ on the 3-space
decomposes as
${}^{3}\psi_{ab}=\Psi\left(n_{a}n_{b}-\frac{1}{2}N_{ab}\right)+\Psi_{(a}n_{b)}+\Psi_{ab},$
(1)
with $\Psi=-^{3}\psi_{ab}N^{ab}$, $\Psi_{a}=N_{a}^{\ b}\ {}^{3}\psi_{bc}n^{c}$
and
$\displaystyle\Psi_{ab}=\left(N_{(a}^{\ c}N_{b)}^{\
d}-\frac{1}{2}N_{ab}N^{cd}\right)\ ^{3}\psi_{cd},$ (2)
is the symmetric, fully projected and trace-free part of ${}^{3}\psi_{ab}$.
The above can be seen as the 1+1+2 analogue of the 1+3 orthogonally projected
symmetric trace-free part of a 4-tensor:
$\displaystyle\psi^{\langle ab\rangle}=\left(h^{(a}_{\ c}h^{b)}_{\
d}-\frac{1}{3}h^{ab}h_{cd}\right)\psi^{cd},$ (3)
where $h_{ab}=g_{ab}+u_{a}u_{b}$ which introduces the derivative operator
$\displaystyle D_{f}\psi^{a\cdots b}_{\ \ \ \ c\cdots d}=h^{\bar{c}}_{\
c}\cdots h^{\bar{d}}_{\ d}h^{a}_{\ \bar{a}}\cdots h^{b}_{\ \bar{b}}h^{e}_{\
f}\nabla_{e}\psi^{\bar{a}\cdots\bar{b}}_{\ \ \ \ \bar{c}\cdots\bar{d}}.$
So, for example the shear, electric and magnetic Weyl 3-tensors decompose
respectively as
$\displaystyle\sigma_{ab}$
$\displaystyle=\sigma\left(n_{a}n_{b}-\frac{1}{2}N_{ab}\right)+\Sigma_{(a}n_{b)}+\Sigma_{ab},$
(4) $\displaystyle E_{ab}$
$\displaystyle=\mathcal{E}\left(n_{a}n_{b}-\frac{1}{2}N_{ab}\right)+\mathcal{E}_{(a}n_{b)}+\mathcal{E}_{ab},$
(5) $\displaystyle H_{ab}$
$\displaystyle=\mathcal{H}\left(n_{a}n_{b}-\frac{1}{2}N_{ab}\right)+\mathcal{H}_{(a}n_{b)}+\mathcal{H}_{ab}.$
(6)
Similarly, gradients of scalars decompose along these directions:
$\displaystyle\nabla_{a}\psi=-\dot{\psi}u_{a}+\hat{\psi}n_{a}+\delta_{a}\psi.$
(7)
The energy momentum tensor takes the covariant form
$\displaystyle T_{ab}=\rho u_{a}u_{b}+ph_{ab}+2q_{(a}u_{b)}+\pi_{ab}.$
Here, $\rho=T_{ab}u^{a}u^{b}$ is the (local) energy density, $3p=T_{ab}h^{ab}$
is the (isotropic) pressure, $q_{a}=T_{bc}h^{b}_{\ a}u^{c}$ is the heat flux
vector ($q_{a}=Qn_{a}+Q_{a}$), and $\pi_{ab}$, which can be decomposed using
(1) (with $\Pi=T_{ab}n^{a}n^{b}-p$), captures the degree of anisotropy.
From the Einstein field equations with cosmological constant
$\displaystyle R_{ab}-\frac{1}{2}Rg_{ab}+\Lambda g_{ab}=T_{ab},$ (8)
where $R_{ab}$ and $R$ are respectively the spacetime Ricci tensor and scalar
curvature, one can write down the Ricci tensor of the spacetime in the
covariant form
$\displaystyle R_{ab}$
$\displaystyle=g_{1}u_{a}u_{b}+g_{2}n_{a}n_{b}+2\left(Qu_{(a}+\Pi_{(a}\right)n_{b)}$
$\displaystyle+\left(g_{2}-\frac{3}{2}\Pi\right)N_{ab}+2Q_{(a}u_{b)}+\Pi_{ab},$
where
$\displaystyle
g_{1}=\frac{1}{2}\left(\rho+3p-2\Lambda\right)\quad\mbox{and}\quad
g_{2}=\frac{1}{2}\left(\rho-p+2\Lambda+2\Pi\right).$
The covariant derivatives of the unit vector fields can be written down in
terms of the kinematic (geometrical) variables:
$\displaystyle\nabla_{a}u_{b}$
$\displaystyle=-u_{a}\left(\mathcal{A}e_{b}+\mathcal{A}_{b}\right)+\left(\frac{1}{3}\theta+\sigma\right)n_{a}n_{b}$
$\displaystyle+\frac{1}{2}\left(\frac{2}{3}\theta-\sigma\right)N_{ab}+\Omega\varepsilon_{ab}+\Sigma_{ab}$
$\displaystyle+2\left(n_{(a}\Sigma_{b)}+n_{[a}\varepsilon_{b]c}\Omega^{c}\right),$
(9a) $\displaystyle\nabla_{a}n_{b}$
$\displaystyle=-u_{a}\left(\mathcal{A}u_{b}+\alpha_{b}\right)+\left(\frac{1}{3}\theta+\sigma\right)n_{a}u_{b}$
$\displaystyle+\frac{1}{2}\phi
N_{ab}+\xi\varepsilon_{ab}+\zeta_{ab}+n_{a}a_{b}$
$\displaystyle+\left(\Sigma_{a}-\varepsilon_{ac}\Omega^{c}\right)u_{b}.$ (9b)
$\mathcal{A}=n_{a}\dot{u}^{a}$ is acceleration,
$\theta=h^{ab}\bar{D}_{a}u_{b}$ is expansion of $u^{a}$, $\phi$ is expansion
of $n^{a}$ (called sheet expansion), $\sigma=\sigma_{ab}n^{a}n^{b}$ is the
shear, $2\Omega=\varepsilon^{ab}\nabla_{a}u_{b}$ and
$2\xi=\varepsilon^{ab}\nabla_{a}n_{b}$ are the respective twists of $u^{a}$
and $n^{a}$ ($\Omega$ is usually referred to as vorticity or rotation, and
$\varepsilon_{ab}$ is the 2-dimensional alternating tensor); $\mathcal{A}^{a}$
is the sheet component of $\dot{u}_{a}$; $\alpha_{a}$ is the sheet component
of $\dot{n}_{a}$; $a_{a}=\hat{n}_{a}$ is the acceleration of $n^{a}$ and;
$\zeta_{ab}=\delta_{\\{a}n_{b\\}}$ is the shear of $n^{a}$.
The field equations then takes a covariant form as a set of evolution and
propagation equations of the covariant scalars, vectors and tensors, along
with some constraint equations. These are obtained from the field equations
themselves, and the Ricci and Bianchi identities for the unit fields $u^{a}$
and $n^{a}$. We do not need all of these equations here and will not enlist
them. The few equations that will be needed for the current work will be
introduced accordingly.
### II.2 Mapping the background and ‘true’ spacetimes
Perturbative schemes to study LRS cosmologies have been recently implemented
using the 1+1+2 formalism. They have largely been applied to the LRS II class
of spacetimes, with a recent consideration being a generalization which
introduces dissipation of the fluid via the non-vanishing of the anisotropic
stress on the background (see for example these recent works [22, 23]).
In this section, we begin by first introducing LRS spacetimes in context of
the 1+1+2 formulation, and we briefly discuss the linearization procedure
(more details can be found in the references [16, 17]), up to where it is
relevant to this paper.
Locally rotationally symmetric (LRS) spacetimes admit a continuous isotropy
group at each point, as well as a preferred spatial direction (see the
references [14, 15] for more details).
The symmetry of these spacetimes allows one to write the metric in local
coordinates $(t,\mathcal{R},y,z)$ as (see [15] for example)
$\displaystyle ds^{2}$
$\displaystyle=-A_{1}^{2}dt^{2}+A_{2}^{2}dr^{2}+A_{3}^{2}dy^{2}$
$\displaystyle+\left(\left(DA_{3}\right)^{2}+\left(A_{2}h\right)^{2}-\left(A_{1}g\right)^{2}\right)dz^{2}$
$\displaystyle+2\left(A_{1}^{2}gdt-A_{2}^{2}hdr\right)dz,$ (10)
where $A_{i}=A_{i}\left(t,r\right)$, and $g=g(y)$, $h=h(y)$. $D=D(y,k)$ where
$k$ is a constant and specified $D$: For $k=-1$, $D=\sinh y$; for $k=0$, $D=y$
and; for $k=1$, $D=\sin y$.
In the limiting case that $g=0=h$, we recover the well studied LRS II class of
spacetimes, which generalizes spherically symmetric solutions to the Einstein
field equations. This class includes FLRW models, LTB spacetime model,
Schwardschild solution, Oppenheimer–Snyder dust model, etc.
The covariant set
$\displaystyle\mathcal{D}:\equiv\\{{\mathcal{A},\theta,\sigma,\phi,\xi,\Omega,\rho,p,Q,\Pi,\mathcal{E},\mathcal{H}\\}},$
(11)
specifies a LRS spacetime.
In general, the mapping between the background and perturbed spacetime
generally depends on the problem of physical interest. And for the specific
background under consideration, the linearization procedure may or may not be
particularly cumbersome. Essentially, the problem reduces to determining which
quantities vanish in the background and which do not. Here, we outline the
procedure for the perturbation. (A detailed discussion on this is found in
[17], formulated differently but unchanging underlying concept.)
* •
We first define the background spacetime: start with a background LRS
spacetime $M$, and let the collection of scalars
$\displaystyle\mathcal{D}_{1}:\equiv\\{{\varphi^{i}\\}}_{i\in I},$ (12)
for some indexing set $I$, specify $M$, i.e.
$\mathcal{D}_{1}\subset\mathcal{D}$. (Note that all 2-vector and 2-tensor
quantities, those defined on the sheet, vanish in the background).
* •
Define the set
$\displaystyle\mathcal{D}_{2}:\equiv\mathcal{D}\setminus\mathcal{D}_{1},$ (13)
of those covariant scalars in $\mathcal{D}$ that vanish in the background.
* •
Further introduce the set of 2-vectors and 2-tensors:
$\displaystyle\mathcal{D}_{3}:\equiv\\{{\Psi_{a}^{j},\Psi_{ab}^{w}\\}}_{j\in
J,w\in W},$ (14)
with $J$ and $W$ being indexing sets, which also vanish in the background.
* •
The full set of gauge invariant quantities is then
$\displaystyle\mathcal{D}_{2}\cup\mathcal{D}_{3},$ (15)
which follows from the Stewart-Walker Lemma [25]. One can then use the Ricci
and Bianchi identities for $u^{a}$ and the direction of anisotropy $n^{a}$ to
obtain the linearized equations for the set
$\bar{D}:\equiv\mathcal{D}_{1}\cup\mathcal{D}_{2}\cup\mathcal{D}_{3}$.
The linearized system is generally not gauge invariant. The evolution and/or
propagation equations of the background quantities may contain terms that do
not vanish in the background. This is resolved by introducing $\delta$
gradients of the background quantities whose evolution and propagation
equations replace those of the background scalars.
* •
We note that the ‘type’ of perturbation being carried out will further
simplify the collection $\mathcal{D}_{2}\cup\mathcal{D}_{3}$. For example, one
may consider an irrotational or a shear-free perturbation, eliminating all
shear and vorticity quantities.
There is also a degree of freedom in choosing the frame as $n^{a}$ can be
freely chosen. This choice will depend on the particular physical
consideration at hand. One common choice when dealing with black hole
perturbation is to set the sheet component of the acceleration to zero [16]
which corresponds to a hovering observer. (At some point in this work we will
make this choice of frame to transparently demonstrate our characterization
scheme.) Other choices include the aligning of $n^{a}$ along the vorticity or
the electric Weyl scalar. That is, one makes the respective frame choices
$\Omega^{a}=0$ and $\mathcal{E}^{a}=0$.
* •
In general the linearized equations are quite messy and seldom tractible,
analytically. This generally results from the presence of the angular
‘$\delta$’ derivative. One approach is to decompose first order variables into
harmonics, which converts the linearized equations into ordinary differential
equations which in some cases can be solved exactly.
In this work we do not seek to solve the linearized system. Rather, we suppose
that the harmonically decomposed linearized system can be solved exactly for
an appropriate basis. Then, we specify the roles that certain components of
the even (electric) vector harmonics (to be defined shortly) play in the
evolution of a null horizon embedded in a LRS background which is subject to a
perturbation.
### II.3 Harmonic decomposition
We now present a quick overview of the harmonic decomposition procedure
adapted to the formalism. Here we again follow the general setup in the
references [16, 17].
One starts by introducing dimensionless harmonic functions that satisfy
$\displaystyle\delta^{2}\mathcal{Q}=-\frac{\bar{k}^{2}}{r^{2}}\mathcal{Q},\quad\hat{\mathcal{Q}}=\dot{\mathcal{Q}}=0,\quad\quad\quad\quad\mbox{($0\leq\bar{k}^{2}$)}$
(16)
on any LRS background, i.e. they are eigenfunctions of the sheet Laplacian.
The function $r$ is covariantly specified by the relations
$\displaystyle\dot{r}=\frac{r}{2}\left(\frac{2}{3}\theta-\sigma\right),\quad\hat{r}=\frac{r}{2}\phi,\quad\delta_{a}r=0.$
(17)
One may decompose any first order scalar $\Psi$ as
$\displaystyle\psi^{(1)}=\sum\psi_{S}^{(\bar{k})}\mathcal{Q}^{(\bar{k})},$
(18)
usually written simply as $\psi^{(1)}=\psi_{S}\mathcal{Q}$, with the sum
implicit over $\bar{k}$, and the $S$ subscript signalling a scalar
decomposition.
The 2-vectors are also expanded in harmonics via the following defined basis
$\displaystyle\mathcal{Q}^{\bar{k}}_{a}$
$\displaystyle=r\delta_{a}\mathcal{Q}^{\bar{k}}\implies N^{b}_{\
a}\hat{\mathcal{Q}}_{b}=0=N^{b}_{\ a}\dot{\mathcal{Q}}_{b},$
$\displaystyle\delta^{2}\mathcal{Q}_{a}$
$\displaystyle=\frac{1}{r^{2}}\left(1-\bar{k}^{2}\right)\mathcal{Q}_{a};$
(19a) $\displaystyle\bar{\mathcal{Q}}^{\bar{k}}_{a}$
$\displaystyle=r\varepsilon_{ab}\delta^{b}\bar{\mathcal{Q}}^{\bar{k}}\implies
N^{b}_{\ a}\hat{\bar{\mathcal{Q}}}_{b}=0=N^{b}_{\
a}\dot{\bar{\mathcal{Q}}}_{b},$ $\displaystyle\delta^{2}\bar{\mathcal{Q}}_{a}$
$\displaystyle=\frac{1}{r^{2}}\left(1-\bar{k}^{2}\right)\bar{\mathcal{Q}}_{a},$
(19b)
respectively named even (electric) and odd (magnetic) vector harmonics. (For
all that are to follow, there should be no confusion that is to arise if we
drop the superscript $\bar{k}$ in relevant expressions.)
Notice that the harmonic basis vectors $\mathcal{Q}_{a}$ and
$\bar{\mathcal{Q}}_{a}$ can be obtained from one another, up to sign reversal,
by applying $\varepsilon_{ab}$:
$\bar{\mathcal{Q}}_{a}=\varepsilon_{ab}\mathcal{Q}^{b}\iff\mathcal{Q}_{a}=-\varepsilon_{ab}\bar{\mathcal{Q}}^{b}$.
For this reason, $\varepsilon_{ab}$ is understood as a parity operator, in the
sense of the harmonics introduced here. A quick check also shows that
$\displaystyle\delta_{a}\bar{\mathcal{Q}}^{a}$
$\displaystyle=0=\varepsilon_{ab}\delta^{a}\mathcal{Q}^{b},$
$\displaystyle\delta_{a}\mathcal{Q}^{a}$
$\displaystyle=-\frac{\bar{k}^{2}}{r}\mathcal{Q}=-\varepsilon_{ab}\delta^{a}\bar{\mathcal{Q}}^{b}.$
Since for each $\bar{k}$, $\mathcal{Q}_{a}\bar{\mathcal{Q}}^{a}=0$, first
order vectors ($2$-vectors) can always be decomposed as
$\displaystyle\Psi_{a}=\sum\Psi_{V}^{(\bar{k})}\mathcal{Q}_{a}+\bar{\Psi}_{V}^{(\bar{k})}\bar{\mathcal{Q}}_{a}=\Psi_{V}\mathcal{Q}_{a}+\bar{\Psi}_{V}\bar{\mathcal{Q}}_{a},$
(21)
again, where the sum over $\bar{k}$ is implicit, with the $V$ signalling a
vector decomposition. We note the following important point: for any first
order 2-vector $\Psi^{a}$, a quick check gives its sheet divergence as
$\displaystyle\delta_{a}\Psi^{a}=-\frac{\bar{k}^{2}}{r}\Psi_{V}Q.$ (22)
Thus, $\Psi^{a}$ is gradient if and only if $\Psi_{V}$ vanishes.
For a $2$-tensor $\Psi_{ab}$, the even and odd parity tensor harmonics are
defined respectively as
$\displaystyle\mathcal{Q}_{ab}$
$\displaystyle=r^{2}\delta_{\\{a}\delta_{b\\}}\mathcal{Q}\implies\hat{\mathcal{Q}}_{ab}=0=\dot{\mathcal{Q}}_{ab},$
(23a) $\displaystyle\bar{\mathcal{Q}}_{ab}$
$\displaystyle=r^{2}\varepsilon_{c\\{a}\delta^{c}\delta_{b\\}}\mathcal{Q}\implies\hat{\bar{\mathcal{Q}}}_{ab}=0=\dot{\bar{\mathcal{Q}}}_{ab}.$
(23b)
Again, we see that $\varepsilon_{ab}$ is a parity operator here:
$\bar{\mathcal{Q}}_{ab}=\varepsilon_{c\\{a}\mathcal{Q}_{b\\}}^{\ \
c}\iff\mathcal{Q}_{ab}=-\varepsilon_{c\\{a}\bar{\mathcal{Q}}_{b\\}}^{\ \ c}$.
Indeed, for each $\bar{k}$ $\mathcal{Q}_{ab}\bar{\mathcal{Q}}^{ab}=0$ so that
any $\Psi_{ab}$ can be decomposed as
$\displaystyle\Psi_{ab}=\Psi_{T}\mathcal{Q}_{ab}+\bar{\Psi}_{T}\bar{\mathcal{Q}}_{ab},$
with the $T$ signalling a tensor decomposition.
## III Characterizing MOTS evolution
We introduce a characterization scheme for the evolution of MOTS orthogonal to
$u^{a}$ and $n^{a}$. We are interested in how MOTS in constant time slices in
linearized LRS background geometries evolve. We begin with a quick definition
of MOTS, after which we introduce the characterization. As will be seen, the
particular problem we are trying to address, or more precisely, how our
objective is formulated, is imposed by the required gauge invariance of the
evolution characterizing function.
### III.1 MOTS
For the surfaces under consideration, the following must hold in order for
‘2-space’ to be a genuine surface rather than simply a collection of tangent
planes (see the reference [17] for discussion):
$\displaystyle\xi=\Omega=a^{a}$ $\displaystyle=0;$
$\displaystyle\Sigma^{a}+\varepsilon^{ab}\Omega_{b}-\alpha^{a}$
$\displaystyle=0.$ (24)
(The left hand side of the second line is known as the Greenberg vector.) In
particular, these conditions ensure that the Lie bracket of $u^{a}$ and
$n^{a}$ vanishes, i.e. $u^{a}$ and $n^{a}$ are surface forming, and that the
operator $\delta_{a}$ is a ‘true’ covariant derivative for the surface [17].
Consider a 2-surface $S$ in spacelike constant $t$ slices, with unit spacelike
normal $n^{a}$. Outgoing and ingoing null normals are respectively
$\displaystyle k^{a}=u^{a}+n^{a};\quad
l^{a}=\frac{1}{2}\left(u^{a}-n^{a}\right),$
normalized so that $l_{a}k^{a}=-1$. There is a degree of freedom to rescale
the null vectors as $k^{a}\rightarrow fk^{a},l^{a}\rightarrow f^{-1}l^{a}$,
for $f>0$. But as will be noted shortly, the MOTS character relevant here is
invariant under such scaling.
The induced metric on $S$ is
$\displaystyle\mathcal{F}_{ab}=g_{ab}+2k_{(a}l_{b)},$ (25)
and the ingoing and outgoing expansions are, respectively,
$\displaystyle\chi=\mathcal{F}^{ab}\nabla_{a}k_{b};\qquad\bar{\chi}=\mathcal{F}^{ab}\nabla_{a}l_{b}.$
(For our current purposes, $\mathcal{F}_{ab}$ here is just $N_{ab}$.) By MOTS,
we mean that on $S$, one has $\chi=0$. In the case that the sign of
$\bar{\chi}$ is constrained as $\bar{\chi}<0$ on $S$, MOTS is abbreviated to
MTS. (A MTS is generally considered to be associated to horizons that enclose
black holes in dynamical spacetimes.) One sees that, under the rescaling, the
expansion becomes $\chi_{f}=f\chi$ (similarly,
$\bar{\chi}_{f}=f^{-1}\bar{\chi}$). Thus, it is clear that any sign conditions
on $\chi$ and $\bar{\chi}$ is seen to be invariant under the freedom to scale
the null vectors.
### III.2 MOTS evolution
Because of the MOTS condition $\chi=0$, we know that the gradient
$\nabla_{a}\chi$ is orthogonal to the MOTS. Hence the norm, which we notate by
$z$, specifies the causal character, pointwise:
$\displaystyle z>0\quad\mbox{(Timelike)};\quad z<0\quad\mbox{(Spacelike)};\
z=0\quad\mbox{(Null)}.$
Let us decompose the expansion $\chi$ into zeroth (background) and first order
scalars:
$\displaystyle\chi=\chi_{0}+\chi_{1},$
with the subscripts $\ast_{0}$ and $\ast_{1}$ denoting, respectively, zeroth
and first orders. (In general this will be the case as this is a linear
combination of the covariant scalars, some of which may or may not vanish in
the background.) We expand the norm $z$ as
$\displaystyle z$
$\displaystyle=\left(-\dot{\chi}_{0}^{2}+\hat{\chi}_{0}^{2}\right)+\left(-\dot{\chi}_{1}^{2}+\hat{\chi}_{1}^{2}\right)+2\left(-\dot{\chi}_{0}\dot{\chi}_{1}+\hat{\chi}_{0}\hat{\chi}_{1}\right)$
$\displaystyle+\left(\delta_{a}\chi_{0}\delta^{a}\chi_{0}+\delta_{a}\chi_{1}\delta^{a}\chi_{1}+2\delta_{a}\chi_{0}\delta^{a}\chi_{1}\right).$
(26)
As we neglect the products of first order quantities, the second and fourth
parenthesized terms of (III.2) vanish and we have
$\displaystyle
z=\left(-\dot{\chi}_{0}^{2}+\hat{\chi}_{0}^{2}\right)+2\left(-\dot{\chi}_{0}\dot{\chi}_{1}+\hat{\chi}_{0}\hat{\chi}_{1}\right).$
(27)
Indeed, it is clear that depending on the particular background solution one
begins with, the above expression will simplify as will later be observed.
Of course there will be some cases where even extending to the nonlinear
regime, i.e. in cases of nonlinear perturbations, the characterization can
proceed using the form of $z$ (27), for example when $z$ is optical, i.e. when
$\delta_{a}z\delta^{a}z$ vanishes on the MOTS. Let us consider some cases
where this is true for a particular frame choice or a class of backgrounds:
The commutation relations between the ‘$\delta_{a}$’, and the dot and hat
derivatives, acting on a scalar $\psi$, are given respectively as (we set
$\Omega=\xi=a_{a}=0$)
$\displaystyle\delta_{a}\dot{\psi}-N^{\
b}_{a}\left(\delta_{b}\psi\right)^{\cdot}$
$\displaystyle=-\mathcal{A}_{a}\dot{\psi}+\left(\Sigma_{a}-\varepsilon_{ab}\Omega^{b}+\alpha_{a}\right)\hat{\psi}$
$\displaystyle+\frac{1}{2}\left(\frac{2}{3}\theta-\sigma\right)\delta_{a}\psi,$
(28) $\displaystyle\delta_{a}\hat{\psi}-N^{\
b}_{a}\widehat{\left(\delta_{b}\psi\right)}$
$\displaystyle=-\left(\Sigma_{a}-\varepsilon_{ab}\Omega^{b}\right)\dot{\psi}+\frac{1}{2}\phi\delta_{a}\psi.$
(29)
Now notice that the function $r$, introduced in the harmonic decomposition
obeys
$\displaystyle\dot{r}+\hat{r}=\frac{r}{2}\chi.$
And upon using the equations (28) and (29) with $\psi=r$ we find that
$\displaystyle\delta_{a}\chi=\left(\mathcal{A}_{a}+\alpha_{a}\right)\phi.$
(30)
So, a frame choice $\mathcal{A}_{a}=\alpha_{a}=0$, $\delta_{a}\chi=0$. Also,
on any background with a vanishing sheet expansion, it is true that
$\delta_{a}\chi=0$ on the MOTS. Thus, analyzing MOTS in perturbed spatially
homogeneous or static LRS backgrounds, the form of the function $z$ suffices
even in nonlinear regimes, although the linearized equations is now more
formidable. However, we will keep the forthcoming applications modest and
restrict ourselves to linear perturbations.
As is easily seen from (27), the scalar $z$ will generally not be gauge
invariant: Indeed, the second parenthesized terms do vanish in the background
due to the presence of $\chi_{1}$ and derivatives thereof, but the first
parenthesized ones do not. Characterization of the evolution of the MOTS in
the perturbed spacetime obviously requires a gauge invariant $z$. In fact, $z$
is gauge invariant if and only if the combination of terms in the first
parenthesis vanishes. It indeed follows that, were one to start with a null
horizon $\bar{H}$, foliated by MOTS, in the background, $z$ is gauge invariant
with respect to $\bar{H}$ since the vanishing of the sum in the first
parenthesis characterizes the nullity of $\bar{H}$ in the first place. For
this reason, we can more precisely frame what is being done in this work:
Consider an exact LRS background spacetime $M$ and let $\bar{H}$ be a null
horizon in $M$, foliated by MOTS. Let us suppose that $M$ is subject to a
perturbation of linear order. How do we characterize the evolution of the
horizon $\bar{H}$ under the perturbation?
Now we emphasize again that we are not interested in solving linearized
system. It is assumed that an appropriate gauge (and possibly frame) choice
has been made, so that one has a mapping between the background and the
perturbed spacetimes. Independent of the gauge choice, Then, by simply
imposing the vanishing condition
$\displaystyle-\dot{\chi}_{0}^{2}+\hat{\chi}_{0}^{2}=0,$
the function $z$ will characterize the evolution of a null horizon from the
background. The gauge choice that maps the background and perturbed spacetimes
of course determines the simplification of $z$ as it is constructed from
elements of the linearized equations.
Now, to ensure a consistent sign of $z$ along a horizon foliated by the MOTS
under consideration, one requires $\delta_{a}z=0$. Explicitly, this is the
requirement that
$\displaystyle\dot{\chi}_{0}\delta_{a}\dot{\chi}-\hat{\chi}_{0}\delta_{a}\hat{\chi}=0.$
(31)
Again, once a background is specified, the expression (31) simplifies as will
be seen in Section IV.
* Remark 1.
We point out that if one relaxes the $\delta_{a}z=0$ condition, one allows for
a range of possibilities, for example, horizons with different portions
exhibiting varying causal characters. This is beyond the scope of the current
work and will be deferred for a subsequent paper.
Let us now compute the forms of the outgoing and ingoing null expansion
scalars:
$\displaystyle\chi=\frac{2}{3}\theta-\sigma+\phi,\qquad\bar{\chi}=\frac{1}{2}\left(\frac{2}{3}\theta-\sigma-\phi\right).$
(32)
Thus, a MOTS is specified by the covariant triple $(\theta,\sigma,\phi)$, and
for any particular background, this splits into the background and first order
parts.
If our interest is in MOTS that are MTS, then the requirement that
$\bar{\chi}<0$ implies that necessarily $\phi>0$. For our purposes here,
however, we will not impose such restriction and will instead consider the
more general ‘MOTS’.
We note that the form of $\chi$, first equation of (32), is form invariant
with respect to the background and so the gauge problem is automatically
fixed. This is a consequence of the particular class of MOTS we are
considering for this work. For a different class of MOTS, this correspondence
may be broken and some additional condition may have to be imposed to fix the
gauge, in general. However, up to linear order for null horizons as is
considered here (or more generally nonexpanding horizons) $\chi$ vanishes on
the perturbed horizon as was established in [11].
* Remark 2.
Before proceeding, we point out that caution is to be exercised when
considering the vanishing of the first parenthesized terms of (III.2). For
example if one is to consider a static background, then obviously,
$\dot{\chi}^{2}_{0}$ should vanish. However, $\hat{\chi}^{2}_{0}$ does not
vanish on a null horizon in the background. So, the terms should be taken
together. In fact, its easily seen that the first parenthesized terms of
(III.2) – we denote this as $F$ – factors as
$\displaystyle F=-\mathcal{L}_{k}\chi\mathcal{L}_{l}\chi.$ (33)
The vanishing condition in the background, assuming the null energy condition
is then $\mathcal{L}_{k}\chi=0$. So, one has to take care of the function $F$,
so that once a background is specified, $F$ vanishes on a null horizon in the
background.
The evolution and propagation equations for $\phi$ and $(2/3)\theta-\sigma$,
up to linear order, are [16]
$\displaystyle\dot{\phi}$
$\displaystyle=\left(\frac{2}{3}\theta-\sigma\right)\left(\mathcal{A}-\frac{1}{2}\phi\right)+Q+\delta_{a}\alpha^{a},$
(34a) $\displaystyle\hat{\phi}$
$\displaystyle=-\frac{1}{2}\phi^{2}+\left(\frac{2}{3}\theta-\sigma\right)\left(\frac{1}{3}\theta+\sigma\right)-\mathcal{E}-\frac{1}{2}\Pi$
$\displaystyle-\frac{2}{3}\left(\rho+\Lambda\right)+\delta_{a}a^{a},$ (34b)
$\displaystyle\frac{2}{3}\dot{\theta}-\dot{\sigma}$
$\displaystyle=\mathcal{A}\phi-\frac{1}{2}\left(\frac{2}{3}\theta-\sigma\right)^{2}+\mathcal{E}-\frac{1}{2}\Pi$
$\displaystyle-\frac{1}{3}\left(\rho+3p-2\Lambda\right)+\delta_{a}\mathcal{A}^{a},$
(34c) $\displaystyle\ \frac{2}{3}\hat{\theta}-\hat{\sigma}$
$\displaystyle=\frac{3}{2}\phi\sigma+Q+\delta_{a}\left(\Sigma^{a}+\varepsilon^{ab}\Omega_{b}\right).$
(34d)
Equation (34d) is obtained by projecting along $n^{a}$ the shear divergence
equation [26]
$\displaystyle
D^{b}\sigma_{ba}-\frac{2}{3}D_{a}\theta+\varepsilon_{abc}\left(D^{b}+2\dot{u}^{b}\right)\omega^{c}+q_{a}=0,$
where
$\displaystyle\omega^{a}=\Omega n^{a}+\Omega^{a},\quad q^{a}=Qn^{a}+Q^{a}.$
Equations (34a), (34b), and (34c) are obtained, respectively, by applying
$u^{a}N^{bc}$, $n^{a}N^{bc}$, and $-n^{a}u^{b}u^{c}$ to the Ricci identities
for $n^{a}$.
We can write down the evolution of the null expansion along the null rays as
$\displaystyle\mathcal{L}_{k}\chi$
$\displaystyle=\chi\left(\mathcal{A}-\frac{1}{2}\phi+\frac{3}{2}\sigma\right)-\left(\rho+p+\Pi\right)$
$\displaystyle+2Q+\delta_{a}\left(\alpha^{a}+\mathcal{A}^{a}+a^{a}+\Sigma^{a}+\varepsilon^{ab}\Omega_{b}\right),$
(35) $\displaystyle\mathcal{L}_{l}\chi$
$\displaystyle=-\frac{1}{2}\chi\left(\chi-\frac{1}{2}\phi+\frac{3}{2}\sigma\right)+\frac{1}{3}\left(\rho-3p\right)+2\mathcal{E}$
$\displaystyle+\delta_{a}\left(\alpha^{a}+\mathcal{A}a-a^{a}-\Sigma^{a}-\varepsilon^{ab}\Omega_{b}\right).$
(36)
Note that $\chi$ vanishes on the horizon and hence the above are simplified
there. The scalar $F$ now expands, up to linear order on the horizon, to the
form
$\displaystyle F$
$\displaystyle=\left(\frac{1}{3}\left(\rho-3p\right)+2\mathcal{E}\right)$
$\displaystyle\times\left(\left(\rho+p+\Pi\right)-2Q-\delta_{a}Z^{a}\right)$
(37)
where we have defined
$\displaystyle
Z^{a}=\alpha^{a}+\mathcal{A}^{a}+a^{a}+\Sigma^{a}+\varepsilon^{ab}\Omega_{b}.$
(38)
Depending on the particular class of background, and how one chooses the
$\\{u^{a}-n^{a}\\}$ frame by appropriately fixing the $n^{a}$ direction, the
scalar $F$ can be considerably simplified as will be seen in the next section.
#### The future outer trapping condition
A MOTS $\mathcal{S}$ on which the ingoing expansion $\bar{\chi}$ is strictly
negative and $\mathcal{L}_{l}\chi<0$ is known as a future outer trapped
surface (FOTS). A horizon foliated by FOTS is a future outer trapping horizon
(FOTH). These were introduced by Hayward in [3] and have since be extensively
studied [27, 28].) In the dynamical and isolated cases, respectively, assuming
the null energy condition, one has that $\mathcal{L}_{k}\chi<0$ and
$\mathcal{L}_{k}\chi=0$. In this case we have a dynamical FOTH and an isolated
FOTH. Crucially, the FOTH condition immediately implies the presence of a
black hole as it ensures the existence of trapped surfaces just to the inside
of the horizon. Thus, the conditions characterizing a FOTH for the dynamical
and isolated cases are $\bar{\chi}<0$, (31), plus $\mathcal{L}_{k}\chi<0$ and
$\mathcal{L}_{k}\chi=0$, respectively,
$\displaystyle z$ $\displaystyle<0,\dot{\chi}<0,\hat{\chi}<0,$ (39)
$\displaystyle z$ $\displaystyle=0,\dot{\chi}<0,\hat{\chi}>0.$ (40)
Henceforth, when specifying to a FOTH the condition $\bar{\chi}<0$ will be
assumed.
In the next section, we consider some backgrounds as particular examples, to
apply the horizon characterization scheme developed in this section to
horizons in the corresponding perturbed spacetime.
## IV Applications
In this section we consider some particular background LRS solutions.
Specifically, we consider static and spatially homogeneous LRS backgrounds,
with restriction to the class II. We will also briefly comment on the case of
a background with both $t$ and $r$ dependence (see Remark 3).
### IV.1 Static background
Let us take as an example a non-dissipative linear perturbation of the
Schwarzschild background with metric (we set the mass to unity here, which
does not, in any serious way, affect the calculations)
$\displaystyle
ds^{2}=-\left(1-\frac{2}{r}\right)dt^{2}+\left(1-\frac{2}{r}\right)^{-1}dr^{2}+r^{2}d\varsigma^{2},$
(41)
where $d\varsigma^{2}$ is the metric on the unit 2-sphere. We know the above
is time-symmetric, i.e. $\theta=0=\sigma$. Hence, we have the zeroth and first
orders decomposition of the null expansion $\chi$ as
$\displaystyle\chi_{0}=\phi\qquad\mbox{and}\qquad\chi_{1}=\frac{2}{3}\theta-\sigma.$
Therefore, $\dot{\chi}_{0}$ and $\dot{\chi}_{1}$ are first order quantities so
that $\dot{\chi}_{0}^{2}=\dot{\chi}_{0}\dot{\chi}_{1}=0$. The only non-
vanishing background covariant scalars are $\mathcal{A},\phi,\mathcal{E}$,
given in coordinate forms as
$\displaystyle\mathcal{A}=\frac{1}{r^{2}}\left(\sqrt{1-\frac{2}{r}}\right)^{-1};\quad\phi=\frac{2}{r}\sqrt{1-\frac{2}{r}};\quad\mathcal{E}=-\frac{2}{r^{3}}.$
In this background the MOTS is located at $\phi=0\iff r=2$. Now, the norm $z$
becomes
$\displaystyle z$ $\displaystyle=F+2\hat{\chi}_{0}\hat{\chi}_{1}$
$\displaystyle=-4\mathcal{E}\delta_{a}\left(Z^{a}+\Sigma^{a}+\varepsilon^{ab}\Omega_{b}\right).$
(42)
Additionally one also requires, from (31), that (we have now set $a^{a}=0$ on
the horizon as it should be)
$\displaystyle
0=\hat{\chi}_{0}\delta_{a}\hat{\chi}=\delta_{a}\mathcal{E}+\delta_{a}\left[\delta_{b}\left(\Sigma^{b}+\varepsilon^{bc}\Omega_{c}\right)\right].$
(43)
Since $\mathcal{E}<0$, coupled with (43), the $r=2$ horizon in the
Schwarzschild background evolves into an isolated (resp. dynamical) horizon
provided that
$\displaystyle\delta_{a}\left(Z^{a}+\Sigma^{a}+\varepsilon^{ab}\Omega_{b}\right)=\mbox{({resp.}
$>$)}0.$ (44)
Separately, we examine the two cases of (44).
#### IV.1.1 Null case
We start with the null criterion, with the vanishing of (44), in harmonics,
yielding
$\displaystyle
0=-\frac{\bar{k}^{2}}{r^{2}}\left[\mathcal{A}_{V}+2\left(\Sigma_{V}-\bar{\Omega}_{V}\right)\right],$
(45)
which should hold for all $\bar{k}$.
By defining $Y_{a}=\delta_{a}\mathcal{E}$ and decomposing the equation (43)
into harmonics, we write it down as the pair of equations
$\displaystyle
Y_{V}=-\frac{\bar{k}^{2}}{r^{2}}\left(\Sigma_{V}-\bar{\Omega}_{V}\right);\quad\bar{Y}_{V}=0.$
(46)
Upon substituting the first equation of (46) into (45) gives
$\displaystyle 0=-\frac{\bar{k}^{2}}{r^{2}}\mathcal{A}_{V}+2Y_{V}.$ (47)
Evoking the vanishing of the Greenberg vector, (46) is
$r^{2}Y_{V}+\bar{k}^{2}\alpha_{V}=0$, so that (47) reduces to the condition
$\displaystyle\mathcal{A}_{V}+2\alpha_{V}=0.$ (48)
This is the required condition that preserves the null character of the null
horizon $r=2$ under linear perturbation.
Were one to fix the frame by setting $\mathcal{A}^{a}=0$ (so that $n^{a}$
points along $\dot{u}^{a}$), we will have that the condition (48) now becomes
$\alpha_{V}=0$. Otherwise, the horizon does not retain its null character.
On the other hand, by (48), the vanishing condition
$\Sigma_{V}-\bar{\Omega}_{V}=0$ would lead to the nullity condition fixing the
frame: $\mathcal{A}_{V}=0$.
FOTH: Let us consider the FOTH condition, $\dot{\chi}<0$ and $\hat{\chi}>0$.
If one fixes the frame by setting $\mathcal{A}^{a}=0$, the FOTH condition
trivially follows: Explicitly, the FOTH condition is the pair
$\displaystyle\mathcal{E}$ $\displaystyle<-\delta_{a}\alpha^{a},$ (49)
$\displaystyle\mathcal{E}$
$\displaystyle<\delta_{a}a^{a}+\delta_{a}\alpha^{a},$ (50)
where we have used the fact that the Greenberg vector must vanish. So, again,
how the black hole evolves under linear perturbation depends crucially on how
the frame is choosen, as one would expect.
Again, $a^{a}$ must vanish for our MOTS to be a genuine surface. As the
specification of the frame choice $\mathcal{A}^{a}=0$ gives $\alpha_{V}=0$,
the FOTH condition is simply the requirement that $\mathcal{E}<0$, which
always holds.
#### IV.1.2 Dynamical case
Let us now consider the case where the $r=2m$ null horizon possibly evolves to
acquire a different character. (Again for simplicity let us make the same
frame choice here by setting $\mathcal{A}^{a}=0$.) Then, with all
considerations of the vanishing Greenberg vector, as well as the necessity of
the vanishing of $a^{a}$, the characterizing norm $z$ takes the form
$\displaystyle
z=-12\frac{\bar{k}^{2}}{r}\mathcal{E}\left(\Sigma_{V}-\bar{\Omega}_{V}\right).$
(51)
We note that from the field equations the vectors $\Sigma_{a}$ and
$\Omega_{a}$ do not evolve independently [16, 17], and so the presence of one
ensures the non-vanishing of the other. Hence, we see that as long as there is
a nonvanishing shear, $z\neq 0$ for $\bar{k}^{2}\neq 0$, and if there is no
shear the horizon stays null as was seen in the previous case.
We do expect that under the perturbation and in the presence of a nonzero
shear the Schwarzschild event horizon becomes spacelike. This then imposes
that
$\displaystyle\Sigma_{V}-\bar{\Omega}_{V}<0,$ (52)
thereby imposing a constraint on the solution to the harmonically decomposed
linearized system, when restricted to the horizon. We summarize the above
discussion as follows:
Let $\tilde{M}$ be the spacetime obtained by linearly perturbing a
Schwarzschild background. Then the $r=2$ horizon is null in $\tilde{M}$ if and
only if the shear and vorticity 2-vectors obey
$\Sigma_{V}-\bar{\Omega}_{V}=0$, but otherwise spacelike in $\tilde{M}$ with
$\Sigma_{V}-\bar{\Omega}_{V}<0$.
Note: A nondissipative perturbation is assumed here. The above conclusion
drawn therefore may or may not be true in the presence of a nonzero $Q$ and
$\Pi$, depending on their relative magnitudes.
### IV.2 Spatially homogeneous background
We consider a hypersurface orthogonal LRS II solution, which is spatially
homogeneous, with metric
$\displaystyle
ds^{2}=-dt^{2}+\bar{a}^{2}_{1}(t)dr^{2}+\bar{a}^{2}_{2}(t)d\bar{\varsigma}^{2}.$
(53)
(The 2-surface with metric $d\bar{\varsigma}^{2}$ is allowed different
geometries including a toroidal one.) In this case we have that
$\mathcal{A}=\phi=0$, with the non-vanishing background kinematic quantities
being the expansion and shear:
$\displaystyle\theta=\frac{\bar{a}_{1t}}{\bar{a}_{1}}+2\frac{\bar{a}_{2t}}{\bar{a}_{2}};\qquad\sigma=\frac{2}{3}\left(\frac{\bar{a}_{1t}}{\bar{a}_{1}}-\frac{\bar{a}_{2t}}{\bar{a}_{2}}\right),$
(54)
where the underscore ‘$t$’ indicates partial differentiation. Hence,
$\displaystyle\chi_{0}=\frac{2}{3}\theta-\sigma\qquad\mbox{and}\qquad\chi_{1}=\phi.$
Indeed, $\dot{\chi}_{1}$ and $\hat{\chi}_{0}$ are first order quantities so
that $\hat{\chi}_{0}^{2}=\hat{\chi}_{0}\hat{\chi}_{1}=0$. For simplicity, we
specify to a conformally flat perturbation, and set $\Lambda=0$. We further
restrict to the nondissipative case. This then gives the norm $z$ as (we
recall that this is always evaluated at $\chi=0$)
$\displaystyle z$ $\displaystyle=F-2\dot{\chi}_{0}\dot{\chi}_{1}$
$\displaystyle=\frac{1}{3}\left(\rho-3p\right)\left[\left(\rho+p\right)-\delta_{a}Z^{a}\right]-\frac{4}{3}\rho\delta_{a}\alpha^{a}.$
(55)
Also, the vanishing condition (31) becomes (again evaluating at $\chi=0$)
$\displaystyle 0=\dot{\chi}_{0}\delta_{a}\dot{\chi}$
$\displaystyle=-\frac{1}{3}\left(\rho+3p\right)\delta_{a}\left(\delta_{b}\mathcal{A}^{b}+\delta_{b}\alpha^{b}\right)$
$\displaystyle=\frac{\bar{k}^{2}}{3r^{2}}\left(\rho+3p\right)\left(\mathcal{A}_{V}+\alpha_{V}\right)\mathcal{Q}_{a}.$
(56)
So, either we fix the equation of state $\rho+3p=0$ or
$\mathcal{A}_{V}+\alpha_{V}=0$.
Now, for the former case this would require that $\rho$ (and consequently $p$)
vanishes in the background to ensure the gauge invariance of $z$, in which
case the horizon is always null.
Let us consider the latter case, and consider the particular frame choice
$\mathcal{A}^{a}=0$. This would impose that $\alpha_{V}=0$. (Note again that
this condition is compatible with $\Sigma_{V}-\bar{\Omega}_{V}=0$ by virtue of
the vanishing of the Greenberg vector.) Therefore, the characterizing function
$z$ reduces to
$\displaystyle z=\frac{1}{3}\left(\rho-3p\right)\left(\rho+p\right).$ (57)
Hence, assuming the weak energy condition, the character of a null horizon,
evolving under linear perturbation, behaves according to the sign of
$\rho-3p$: the horizon stays null, or becomes spacelike or timelike provided
$\rho=3p,\rho<3p$ or $\rho>3p$. So, for example, with a linear equation of
state (EoS) $p=w\rho$, the above conditions are the respective requirements
$w=0,w>0$ and $w<0$.
Of course, a choice different from $\mathcal{A}^{a}=0$ may be used to fix the
frame. In this case, we will have $z$ assuming the form
$\displaystyle
z=\frac{1}{3}\left[\left(\rho-3p\right)\left(\rho+p\right)+\frac{\bar{k}^{2}}{r}\left(5\rho-3p\right)\alpha_{V}\right].$
(58)
We see that even with a linear EoS of the form $p=w\rho$, the signature of $z$
above acquires some complexity an carries a crucial dependence on sign of the
component $\alpha_{V}$ (Note that $\alpha_{V}=\Sigma_{V}-\bar{\Omega}_{V}$).
We may however still comment on some possible characters of the perturbed
horizon (we are assuming a positive $\rho$). For example, it is not difficult
to see that for the range
$\displaystyle-1\leq w\leq\frac{1}{3},$ (59)
$\alpha_{V}>0$ precludes a spacelike or null character, allowing for only a
timelike character. This indeed differs from the timelike characterization of
the background as the extremes $w=-1$ and $w=1/3$ allow only for the null
character. (See for example the references [29, 19].) If we look outside the
bound (59) in the case $\alpha_{V}>0$, for $w<-1$, the horizon character will
depend on the magnitude of $\alpha_{V}$ at each $r$ in relation to each
$\bar{k}^{2}$. (In the background, this is quite straightforward as outside
the bound (59), the horizon is spacelike. Again, see the references [29, 19].)
This will also be the case for the bound
$\displaystyle\frac{1}{3}<w<\frac{5}{3},$ (60)
but for $w\geq 5/3$, only the timelike character is possible.
Now consider the case for $\alpha_{V}<0$. Then, a timelike or null character
is ruled out within the bound (60), allowing for only a spacelike character.
For the bounds
$\displaystyle-1<w<\frac{1}{3}\qquad\mbox{and}\qquad w>\frac{5}{3},$ (61)
the character again depends on the magnitude of $\alpha_{V}$ at each $r$ in
relation to each $\bar{k}^{2}$.
At the extreme $w=1/3$ the horizon is spacelike and timelike for
$\displaystyle\frac{1}{3}<w\leq\frac{5}{3},$ (62)
And for values $w<-1$, a timelike or null character is prohibited. (Notice
that at the critical point $w=5/3$ the horizon signature is insensitive to the
sign of $\alpha_{V}$.)
We summarize the above results as follows:
Let $\tilde{M}$ be the spacetime obtained by linearly perturbing a spatially
homogeneous background $M$ with EoS $p=w\rho$, and let $\mathcal{T}$ be a null
horizon foliated by MOTS in $M$. Then for $\Sigma_{V}-\bar{\Omega}_{V}=0$,
$\mathcal{T}$ is null, spacelike or timelike in $\tilde{M}$ provided $w=0,w>0$
or $w<0$, respectively. Otherwise, for $\Sigma_{V}-\bar{\Omega}_{V}\neq 0$,
see Table 1. (Note that $\alpha_{V}=\Sigma_{V}-\bar{\Omega}_{V}$ on the MOTS.)
Null horizon character under linear perturbation
---
$\alpha_{V}$ | $w$ | Character
| $(-1\leq w\leq\frac{1}{3})\cup(w\geq\frac{5}{3})$ | Timelike
$\alpha_{V}>0$ | |
| $(\frac{1}{3}<w<\frac{5}{3})\cup(w<-1)$ | –
| $(-1<w<\frac{1}{3})\cup(w>\frac{5}{3})$ | –
$\alpha_{V}<0$ | $\frac{1}{3}<w\leq\frac{5}{3}$ | Timelike
| $(w\leq-1)\cup(w=\frac{1}{3})$ | Spacelike
Table 1: The behavior of a null horizon evolving under linear perturbation.
Assuming a linear equation of state $p=w\rho$, the above table shows the
character of the horizon for different ranges of the equation of state
parameter $w$, and the relationship with the sheet vector $\alpha^{a}$
decomposed harmonically. The ‘$-$’ indicates a non-trivial relationship which
is dependent on the magnitudes of $\alpha_{V}$ and $\rho$ as well as the value
of the nonnegative integer $\bar{k}^{2}$.
Let it be emphasized that the linear choice of the EoS is for demonstrable
purposes for our particular application and there is no restriction here. Any
EoS is in principle allowable with (possible) caveat being the the analysis
acquiring a more complicated character.
We will now conclude this section with the following remark:
* Remark 3.
In a general inhomogeneous case, $\chi_{1}=0$ and we see that the associated
norm is simplified:
$\displaystyle z=F,$
so that the characterization proceeds from (III.2). However, the consideration
of the $\delta_{a}z=0$ criterion is now significantly more complicated.
Relaxing this condition of course then gives a simple pointwise
characterization via (III.2). If one does not impose $\delta_{a}z=0$ a priori,
this allows for the possibility of varying character across the horizon. This
is clearly a more involving case and this is worth considering for a
subsequent work. However, as was just mentioned, while generally these cases
are expected to pose problems, we see that in a general inhomogeneous
background the condition provides a certain analytic simplicity, at least
point-wise.
## V On the MOTS stabiliity operator
As have been seen from the previous section, the 2-vectors associated to the
shear and vorticity play a crucial role in how a null horizon in a LRS
background spacetime evolves under a linear perturbation. This particular
combination of the shear and vorticity evolves simultaneously according to the
field equations and have no independent evolution or propagation equations as
was mentioned in the previous section. This relationship has implications for
the self-adjointness of the MOTS stability operator as these same 2-vectors
also control the self-adjointness of the operator. We will begin by
introducing the MOTS stability operator after which we provide some commentary
on this relationship. We will then delve into several properties of the
operator in context of our current considerations as it relates to self-
adjointness. The evolution, propagation and sheet equations that will be
utilized here have been obtained in [16, 17] (with corrections in [22, 23] as
stated in Section III). So, especially the reference [16] is implicit wherever
these equations appear.
### V.1 The MOTS stability operator
Given a MOTS $\mathcal{S}$ and a normal foliation by a collection
$\mathcal{S}_{v}$, with the subscript ‘$v$’ labelling the foliation, one may
define the tangent vector to curves generating the deformation as
$\displaystyle\partial_{v}=\bar{\psi}n^{a},$
for some smooth function $\bar{\psi}$ on $\mathcal{S}$. Then, stability of
$\mathcal{S}$ is captured via the variation of the expansion $\chi$ along
$\partial_{v}$ (several discussions of this can be found in the original
papers [6, 7], as well as, for example, the works [30, 31] and references
therein):
$\displaystyle\bar{\delta}_{\bar{\psi}n}\chi=L_{\mathcal{S}}\bar{\psi},$
where the operator $L_{\mathcal{S}}$ is
$\displaystyle L_{\mathcal{S}}=-\delta^{2}+2\tilde{s}^{a}\delta_{a}+\bar{F},$
(63)
with the definition
$\displaystyle\bar{F}=\frac{1}{2}\mathcal{R}_{\mathcal{S}}-\tilde{s}_{a}\tilde{s}^{a}+\delta_{a}s^{a}-G_{ab}k^{a}l^{b},$
(64)
(It is important to note that the scalar $\bar{\psi}$ completely characterizes
the variation of $\mathcal{S}$.) The one-form $\tilde{s}_{a}=-N^{c}_{\
a}l_{b}\nabla_{c}k^{b}$ is the projection of the torsion of $k^{a}$ onto
$\mathcal{S}$, $G_{ab}$ is the Einstein tensor and $\mathcal{R}_{\mathcal{S}}$
is the scalar curvature of $\mathcal{S}$.
$\mathcal{S}$ is said to be strictly or marginally stable if there exists a
nonnegative (and not identically zero) $\bar{\psi}$ obeying respectively
$L_{\mathcal{S}}\bar{\psi}>0$ or $L_{\mathcal{S}}\bar{\psi}=0$, and unstable
otherwise.
For a 1+1+2 decomposed spacetime, $\tilde{s}_{a}$ is explicitly
$\displaystyle\tilde{s}_{a}=\Sigma_{a}-\varepsilon_{ab}\Omega^{b},$ (65)
And in the harmonic basis,
$\displaystyle\tilde{s}_{a}=\underbrace{\left(\Sigma_{V}+\bar{\Omega}_{V}\right)}_{\text{$\tilde{s}_{V}$}}\mathcal{Q}_{a}+\underbrace{\left(\bar{\Sigma}_{V}-\Omega_{V}\right)}_{\text{$\bar{\tilde{s}}_{V}$}}\bar{\mathcal{Q}}_{a}.$
(66)
In general the operator $L_{\mathcal{S}}$ is not self-adjoint, which results
from the presence of the linear term $\tilde{s}^{a}\delta_{a}$ so that
$L_{\mathcal{S}}$ is allowed to have complex eigenvalues. The imaginary part
encodes information about rotation through the Komar angular momentum,
obtained by integrating over the MOTS the inner product of $\tilde{s}_{a}$ and
an axial Killing field.
Still, the principal eigenvalue, which we denote as $\bar{\lambda}$, is always
real [6]. The stability of a MOTS $\mathcal{S}$ then reduces to the sign of
$\bar{\lambda}$: $\mathcal{S}$ is strictly or marginally stable if
$\bar{\lambda}>0$ or $\bar{\lambda}=0$, and unstable otherwise.
Here, we are interested in the implications of the characterization conditions
obtained for the backgrounds in the previous sections, for the self-
adjointness of the stability operator. Crucially, the dependence on the sign
of $\alpha_{V}$ (or equivalently $\Sigma_{V}-\bar{\Omega}_{V}$).
Now, it is known that the self-adjointness of the MOTS stability operator is
guaranteed by the one-form $\tilde{s}_{a}$ being a gradient [31]. Particular,
if the 1-form $\tilde{s}_{a}$ is gradient, then the operator $L_{\mathcal{S}}$
is similar to a self-adjoint operator. (Of course, the case of vanishing
$\tilde{s}_{a}$ is trivial, and so we consider this possibility
$\tilde{s}_{a}\neq 0$.) This self-adjointness in the case that $\tilde{s}_{a}$
is gradient can be seen here in a rather trivial way. To see this, let us
consider the action of the operator $L_{\mathcal{S}}$ on a first order scalar
$\psi^{(1)}=\psi_{S}\mathcal{Q}$:
$\displaystyle
L_{\mathcal{S}}y^{(1)}=\left[\left(-\delta^{2}+\bar{F}\right)\mathcal{Q}+\frac{1}{r}\tilde{s}_{V}\right]\psi_{S}.$
(67)
(The normalization $\mathcal{Q}_{a}\mathcal{Q}^{a}=1$ has been assumed for
simplicity.) Since if the $\tilde{s}_{a}$ is gradient it has a vanishing
divergence, it is then clear that this requires the vanishing of
$\tilde{s}_{V}$, i.e.
$\displaystyle\Sigma_{V}+\bar{\Omega}_{V}=0.$ (68)
Thus, the eigenvalue problem for the MOTS stability operator simply reduces to
the following eigenvalue problem for $\mathcal{Q}$:
$\displaystyle-\delta^{2}\mathcal{Q}+F\mathcal{Q}=\lambda\mathcal{Q}.$ (69)
That is, the problem reduces to the eigenvalue problem for the self-adjoint
operator
$L^{\prime}_{\mathcal{S}}\mathcal{Q}=\left(-\delta^{2}+F\right)\mathcal{Q}=\lambda\mathcal{Q}$
so that
$\displaystyle\lambda_{\bar{k}^{2}}=\frac{\bar{k}^{2}}{r}+\bar{F},$ (70)
i.e. the eigenvalues are parametrized by $\bar{k}^{2}$. Thus, strict stability
is always guaranteed for
$\displaystyle\bar{F}=\frac{1}{2}\left(\mathcal{R}_{\mathcal{S}}-R\right)-\bar{\tilde{s}}_{V}^{2}-\left(p+Q\right)+2\Lambda>0.$
(71)
(recall that $R$ is the four dimensional scalar curvature.)
It will later be seen that $\bar{\tilde{s}}_{V}$ is necessarily zero on the
MOTS in the case that $\tilde{s}_{V}=0$, i.e. $\tilde{s}_{a}$ is a gradient.
So, in this case for $p+Q\leq 0$ and $\Lambda\geq 0$, whenever
$\mathcal{R}_{\mathcal{S}}>R$, the operator has no negative eigenvalue. The
principal eigenvalue and the MOTS is strictly stable. This is indeed the case,
say, for vacuum solutions undergoing a shear-free and a nondissipative
perturbation where one will have $\tilde{s}_{V}$ vanishing.
On the other hand, consider an irradiating ($Q=0$) spacetime with a positive
scalar curvature, a vanishing $\Lambda$ and a nonnegative pressure. Then, for
a sufficiently large $p$ it is possible that $\bar{F}<0$, so that the
principal eigenvalue is always negative and hence the MOTS will be unstable.
* Remark 4.
Actually, in the linear regime any perturbation to the MOTS scalar curvature
$\mathcal{R}_{\mathcal{S}}$ is encoded in first order scalars. Particularly,
if the scalars $\mathcal{E},\Pi$, and $\Lambda$ (if $\Lambda$ is included as a
perturbation variable) are nonvanishing in the background, then
$\mathcal{R}_{\mathcal{S}}$ is not perturbed. However, nonlinear effects are
present under perturbation of the background as
$\displaystyle\mathcal{R}_{\mathcal{S}}=\
^{0}\mathcal{R}_{\mathcal{S}}+\mathcal{Z},$ (72)
where $\mathcal{Z}=\Sigma_{ab}\Sigma^{ab}-\zeta_{ab}\zeta^{ab}$ is just twice
the inner product of the projected shears of $k^{a}$ and $l^{a}$. So, it is
clear that once nonlinear effects are factored in any analysis, we do expect
the behavior of the eigenvalues to be modified.
Now, the commutation relations of the dot and hat derivatives for an arbitrary
2-vector $\Psi_{a}$ (imposing the vanishing of the Greenberg vector and taken
up to linear order)
$\displaystyle\hat{\dot{\Psi}}_{a}-\dot{\hat{\Psi}}_{a}=-\mathcal{A}\dot{\Psi}_{a}+\left(\frac{1}{3}\theta+\sigma\right)\hat{\Psi}_{a}+\mathcal{H}\varepsilon_{ab}\Psi^{b}.$
(73)
Harmonically decomposing the above relation and applying to the one-form
$\tilde{s}_{a}$ while setting the component $\tilde{s}_{V}=0$ says that
$\displaystyle\hat{\dot{\bar{\tilde{s}}}}_{V}-\dot{\hat{\bar{\tilde{s}}}}_{V}$
$\displaystyle=-\mathcal{A}\dot{\bar{\tilde{s}}}_{V}+\left(\frac{1}{3}\theta+\sigma\right)\hat{\bar{\tilde{s}}}_{V},$
(74) $\displaystyle\mathcal{H}\bar{\tilde{s}}_{V}$ $\displaystyle=0.$ (75)
The above is meant to suggest that for a magnetized ($\mathcal{H}\neq 0$)
linear perturbation $\tilde{M}$ of a LRS background spacetime $M$, if the
1-form $\tilde{s}_{a}$ is gradient in which case the MOTS stability operator
is self-adjoint (remember this is for the class of MOTS under consideration),
then $\tilde{s}_{a}$ necessarily vanishes on the MOTS. But as it will be seen
shortly, the quantities $\bar{\tilde{s}}_{V}$ and $\mathcal{H}$ are
proportional so that in the linear regime on any MOTS it is true that both
$\bar{\tilde{s}}_{V}$ and $\mathcal{H}$ vanish. More specifically,
$\mathcal{H}$ must be first order.
### V.2 Comments on relations to the characterization of null horizons
Let us now specify our discussions to the backgrounds of the previous section.
#### V.2.1 Schwarzschild background
To begin with, consider the case of the Schwarzschild background. We will
establish the following statement (C1) and the corresponding implication (C2):
1. (C1).
For a linear perturbation $\tilde{M}$ of a Schwarzschild background, the
1-form $\tilde{s}_{a}$ is gradient (in which case the MOTS stability operator
is self-adjoint) if and only if $\tilde{s}_{a}$ is identically zero;
$\implies$
2. (C2).
For a linear perturbation $\tilde{M}$ of a Schwarzschild background, if the
one-form $\tilde{s}_{a}\neq 0$ and the vorticity 2-vector has a nonzero
contribution from the odd sector, the $r=2$ null horizon necessarily evolves
to a spacelike character.
Note: As was mentioned in the previous section, away from the null case the
horizon can only evolve to the spacelike case with the even parity component
of $\Sigma_{a}+\varepsilon_{ab}\Omega^{a}$ being negative. (A timelike
character would imply that timelike curves do enter the trapped region, which
is of course ruled out.) Thus, the second statement above, (C2), follows from
the fact that $\tilde{s}_{V},\bar{\Omega}_{V}\neq 0$ ensures that
$\Sigma_{V}-\bar{\Omega}_{V}\neq 0$.
Let us apply the tensor $\varepsilon^{ab}n^{c}$ to the Ricci identities for
the unit field $n^{a}$ to get
$\displaystyle\delta_{a}\left(\Omega^{a}+\varepsilon^{ab}\Sigma_{b}\right)=\left(2\mathcal{A}-\phi\right)\Omega+\mathcal{H},$
(76)
and upon noting $\Omega$ should be zero on the MOTS, the above reduces to
$\displaystyle-\frac{\bar{k}^{2}}{r}\bar{\tilde{s}}_{V}\mathcal{Q}=\mathcal{H},$
(77)
so that $\mathcal{H}$ is a first order quantity that vanishes on the
background, i.e. $\mathcal{H}$ is decomposed into the harmonic $\mathcal{Q}$
basis.
Then, by (75) it follows that $\mathcal{H}=\bar{\tilde{s}}_{V}=0$. It
therefore follows that $\tilde{s}_{V}=0\iff\tilde{s}_{a}=0$, i.e.
$\tilde{s}_{a}$ is gradient if and only if $\tilde{s}_{a}$ vanishes
identically.
Finally, the linearized evolution and propagation equations of $\mathcal{H}$
are respectively given by (these are respectively the $u^{a}$ and $n^{a}$
components of the magnetic Weyl divergence equation [26])
$\displaystyle\dot{\mathcal{H}}$
$\displaystyle=-3\mathcal{E}\xi-\varepsilon_{ab}\delta^{a}\mathcal{E}^{b},$
(78) $\displaystyle\hat{\mathcal{H}}$
$\displaystyle=-3\mathcal{E}\Omega-3\phi\mathcal{H}-\delta_{a}\mathcal{H}^{a},$
(79)
which translates to, by the vanishing of $\mathcal{H}$ as well as noting that
$\Omega=\xi=0$ on the MOTS,
$\displaystyle\mathcal{H}_{V}=0;\quad\bar{\mathcal{E}}_{V}=0.$ (80)
This is to say that, necessarily, on the MOTS one requires that
$\mathcal{H}^{a}$ is solenoidal (divergence-free) and that $\mathcal{E}^{a}$
has no odd parity contribution on the MOTS.
While there appears to be freedom in the quantities $\mathcal{E}_{V}$ and
$\bar{\mathcal{H}}_{V}$, they are actually constrained by a coupling of the
sheet expansion $\phi$ and the 1-form $\tilde{s}_{a}$ (see the discussions of
the next subsection). For example, for a vanishing $\tilde{s}_{a}$ in the case
of a nondissipative perturbation, it follows that one requires (again, see the
next subsection for more details on this)
$\displaystyle\mathcal{E}_{V}+\bar{\mathcal{H}}_{V}=0.$
Of course, we do not rule out self-adjointness of the operator in the case of
a spacelike character. This is in principle possible depending on the nature
of the perturbation. This case will however impose that $\Sigma_{V}<0$.
On the other hand the null character does not necessarily imply the self-
adjointness of the operator. The self-adjointness due to vanishing of
$\tilde{s}_{a}$, in this case, would imply (and is implied by)
$\Sigma_{V}=\bar{\Omega}_{V}=0$.
#### V.2.2 Hypersurface orthogonal and spatially homogeneous background
We now consider the spatially homogeneous case of the previous section, and
stick with a nondissipative perturbation. In this case the equations
corresponding to (78), (79) and (76) are respectively
$\displaystyle\dot{\mathcal{H}}$
$\displaystyle=-\frac{3}{2}\left(\frac{2}{3}\theta-\sigma\right)\mathcal{H}-3\mathcal{E}\xi-\varepsilon_{ab}\delta^{a}\mathcal{E}^{b},$
(81) $\displaystyle\hat{\mathcal{H}}$
$\displaystyle=-3\left(\mathcal{E}+\rho+p\right)\Omega-3\phi\mathcal{H}-\delta_{a}\mathcal{H}^{a},$
(82) $\displaystyle\mathcal{H}$
$\displaystyle=\delta_{a}\left(\Omega^{a}+\varepsilon^{ab}\Sigma_{b}\right)+3\xi\sigma.$
(83)
The vanishing of $\Omega,\xi=0$ and $\mathcal{H}$ then implies the result
$\mathcal{H}_{V},\bar{\mathcal{E}}_{V},\bar{\tilde{s}}_{V}=0$ follows as in
the Schwarzschild case. For this reason, the statement (C1) of the previous
subsection also holds here, although for obvious reasons the statement (C2)
does not hold.
* Remark 5.
We do expect these three vanishing properties to be generic to the MOTS we are
considering in the sense that they are independent of the particular
background one works with. It will be quite interesting to see how much these
vanishing conditions further constrains the MOTS in full generality. This is
however beyond the scope of the current work.
In the self-adjoint case with a vanishing $\tilde{s}_{a}$, obviously either
one of the components $\Sigma_{V}$ or $\bar{\Omega}_{V}$, paired with the
parameter ranges suffice for the characterization of the horizon.
On the other hand, if one is to impose the condition that
$|\Sigma_{V}|\neq|\bar{\Omega}_{V}|$, this of course ensures that
$\tilde{s}_{a}$ is nonvanishing, in which case one may then check which
conditions this further impose on the horizon character. However, this does
not in by itself rule out self-adjointness of the operator. In principle, if
one can expand a nonvanishing $\tilde{s}_{V}$ in the harmonic $\mathcal{Q}$
basis, then the operator can still be brought to a self-adjoint form. In the
forthcoming subsection we will see a particular class of perturbations where
this may be possible.
### V.3 More on the MOTS and stability operator
We obtain some additional results required to be satisfied on the MOTS, given
properties derived from the one-form $\tilde{s}_{a}$. As we shall see, this
will provide some interesting insights into properties of the 1-form
$\tilde{s}_{a}$.
The magnetic Weyl tensor is written down as a constraint to the field equation
[26]:
$\displaystyle H_{ab}=\varepsilon_{cd\langle a}D^{c}\sigma_{b\rangle}^{\
d}-2\dot{u}_{\langle a}\omega_{b\rangle}-D_{\langle a}\omega_{b\rangle}.$ (84)
One may interpret the above as the magnetic Weyl tensor measuring the degree
of distortion of vorticity.
Taking the $n^{a}$ component of (84) produces the constraint equation, up to
linear order,
$\displaystyle-\delta_{a}\left(\frac{2}{3}\theta-\sigma\right)$
$\displaystyle=-2\delta^{b}\Sigma_{ab}-\phi\tilde{s}_{a}$
$\displaystyle-2\varepsilon_{ab}\left[\mathcal{H}^{b}+\left(\delta^{b}+2\mathcal{A}^{b}\right)\Omega\right]$
$\displaystyle+\xi\left(\Omega_{a}-3\varepsilon_{ab}\Sigma^{b}\right).$ (85)
By setting $\Omega=\xi=0$ we have
$\displaystyle-\delta_{a}\left(\frac{2}{3}\theta-\sigma\right)=-2\left(\delta^{b}\Sigma_{ab}+\varepsilon_{ab}\mathcal{H}^{b}\right)-\phi\tilde{s}_{a}.$
(86)
Also, fully projecting the Ricci identities for $n^{a}$ taken up to linear
order and imposing the MOTS condition and the vanishing of $\xi$ and $\Omega$
as required we obtain
$\displaystyle-\delta_{a}\phi=2\left(\mathcal{E}_{a}-\delta^{b}\zeta_{ab}\right)-\Pi_{a}-\phi\tilde{s}_{a},$
(87)
and upon combining with (86) and (87) we have
$\displaystyle-\cancel{\delta}_{a}\chi=2\left[\mathcal{E}_{a}-\delta^{b}\left(\Sigma_{ab}+\zeta_{ab}\right)-\varepsilon_{ab}\mathcal{H}^{b}\right]-\phi\tilde{s}_{a},$
(88)
where we have written $\cancel{\delta}_{a}=\delta_{a}-\tilde{s}_{a}$. One may
then use the Codazzi equations to establish the following
$\displaystyle-
Q_{a}+Z_{ab}k^{b}=2\left(\mathcal{E}_{a}-\varepsilon_{ab}\mathcal{H}^{b}\right)-\phi\tilde{s}_{a},$
(89)
where $Z_{ab}=N^{\ c}_{a}C_{cbde}k^{d}l^{e}$, with $C_{abcd}$ being the
spacetime Weyl tensor.
We will now utilize a curvature property of the MOTS to establish that the
second term on the left hand side of (89) is of nonlinear order on the MOTS.
With this we will have established a relationship between the Weyl and heat
flux 2-vectors and the rotation 1-form.
The curvature 2-form on the MOTS, i.e. the curvature of $\tilde{s}_{a}$ –
which we denote by $\tilde{\Omega}_{ab}$ – contracted along the null direction
$k^{a}$ can be shown to obey the relation
$\displaystyle k^{b}\tilde{\Omega}_{ba}=-N^{\ c}_{a}Z_{cb}k^{b},$
so that
$\displaystyle
k^{b}\tilde{\Omega}_{ba}=-\left[2\left(\mathcal{E}_{a}-\varepsilon_{ab}\mathcal{H}^{b}\right)-\phi\tilde{s}_{a}+Q_{a}\right].$
(90)
Explicitly, the left hand side of (90) can be reduced to
$\displaystyle k^{b}\tilde{\Omega}_{ba}$
$\displaystyle=-\frac{1}{2}\Sigma_{b}\delta_{a}k^{b}$
$\displaystyle=-\frac{1}{2}\left(\Sigma_{ab}+\zeta_{ab}\right)\Sigma^{b},$
which we ignore as it is of second order. Thus,
$\displaystyle\phi\tilde{s}_{a}=Q_{a}+2\left(\mathcal{E}_{a}-\varepsilon_{ab}\mathcal{H}^{b}\right).$
(91)
Furthermore, imposing the MOTS condition on (29) we have
$\displaystyle\cancel{\delta}_{a}\phi=0,$
so that by (87) it follows
$\displaystyle\Pi_{a}=2\left(\mathcal{E}_{a}-\delta^{b}\zeta_{ab}\right).$
(92)
Clearly, away from the case of a vanishing $\phi$, the following statement is
concluded:
The MOTS stability operator in a conformally flat and nondissipative (this
imposes that $\zeta_{ab}$ is divergence-free) linearly perturbed inhomogeneous
LRS background is self-adjoint.
So for example, the above will be true for a nondissipative perturbation of a
Lemaitre-Tolman-Bondi type solutions.
Notice that if the tensor $\delta_{a}n_{b}$ is pure trace, dissipation on the
MOTS is generated entirely by contributions from the Weyl tensor.
On the other hand, without restricting to the nondissipative case, using the
vanishing of $\bar{\mathcal{E}}_{V},\mathcal{H}_{V}$ and $\bar{\tilde{s}}_{V}$
we have the following pair of equations from (91).
$\displaystyle\bar{Q}_{V}$ $\displaystyle=0,$ (93)
$\displaystyle\phi\tilde{s}_{V}$
$\displaystyle=Q_{V}+2\left(\mathcal{E}_{V}+\bar{\mathcal{H}}_{V}\right).$
(94)
In any case, we can draw some immediate conclusions from both (91) and (92)
about the generation of rotation on a horizon:
* •
In a conformally flat linearly perturbed inhomogeneous LRS background,
rotation of a horizon foliated by MOTS can be entirely sourced by the heat
flux 2-vector along the MOTS. This then implies that self-adjointness of the
MOTS stability operator in this particular case may be obtained by switching
off any heat flux contribution along the MOTS.
* •
In a nondissipative linearly perturbed inhomogeneous LRS background, rotation
of a horizon foliated by MOTS can be sourced by the 2-vector components of the
Weyl curvature tensor. This is to say that in this case the contributing
sources are tidal forces acting along the MOTS.
* Remark 6.
We remark here that the form of (94) suggests that it is possible, in
principle, to have a nonvanishing $\tilde{s}_{V}$ – i.e. a nonvanishing
$\tilde{s}_{a}$ – even in the case of a conformally flat perturbation with a
vanishing heat flux and vanishing $\phi$. Thus, in the first bulleted point
above we were quite cautious with semantics when we used “may be” when
switching off the heat flux contribution along the MOTS to obtain self-
adjointness of the stability operator. Only when $\phi$ is nonzero in the
perturbed spacetime that this is true, at least in our current context.
Finally, let us describe a simple case where $\tilde{s}_{V}$ is non-vanishing
but the stability operator may take a self-adjoint form. More specifically, we
may be able to express $\tilde{s}_{V}$ in terms of the $\mathcal{Q}$ basis. We
shall consider a static background, i.e. the shear and expansion scalars
vanish on the background.
We note that for any 2-tensor $\Psi_{ab}$ one has as the divergence
$\displaystyle\delta^{b}\Psi_{ab}=\frac{1}{r}\left(\bar{k}^{2}-2\right)\left(-\bar{\Psi}_{T}\mathcal{Q}_{a}+\Psi_{T}\bar{\mathcal{Q}}_{a}\right).$
(95)
Thus, one can harmonically decompose (86) to obtain the pair of constraint
equations
$\displaystyle 0$
$\displaystyle=\frac{1}{r}\left(\bar{k}^{2}-2\right)\Sigma_{T}+\mathcal{H}_{V},$
(96) $\displaystyle 0$
$\displaystyle=\frac{1}{r}\left(\bar{k}^{2}-2\right)\bar{\Sigma}_{T}+\bar{\mathcal{H}}_{V}$
$\displaystyle+\frac{1}{2r}\left(\frac{2}{3}\theta_{S}-\sigma_{S}\right)-\phi\tilde{s}_{V}.$
(97)
Indeed, from the first equation (96) it follows that the tensorial part
$\Sigma_{T}$, for $\bar{k}^{2}\neq 2$, vanishes by virtue of the vanishing of
the quantity $\mathcal{H}_{V}$. Furthermore, by imposing the MOTS condition
$\chi=0$, the second equation (97) can be recast as
$\displaystyle
0=\frac{1}{r}\left(\bar{k}^{2}-2\right)\bar{\Sigma}_{T}+\bar{\mathcal{H}}_{V}+\phi\left(\frac{1}{2r\mathcal{Q}}-\tilde{s}_{V}\right).$
(98)
Therefore, as long as the MOTS is nonminimal in the perturbed spacetime
(recall that by nonminimal it is meant that $\phi$ and $(2/3)\theta-\sigma$
cannot vanish simultaneously on the MOTS), for a perturbation with no
contributions from the odd vector harmonics part of $\mathcal{H}_{a}$ and the
shear 2-tensor $\Sigma_{ab}$, the 1-form component $\tilde{s}_{V}$ can
effectively be cast as
$\displaystyle\tilde{s}_{V}=\left(\frac{1}{2r\mathcal{Q}^{2}}\right)\mathcal{Q}.$
(99)
(Of course $\tilde{s}_{V}$ is nonzero and so $\tilde{s}_{a}$ is not a
gradient.) This then allows us to write down the eigenvalues of the operator
as
$\displaystyle\lambda_{\bar{k}^{2}}=\frac{\bar{k}^{2}}{r}+\frac{1}{2r^{2}\mathcal{Q}^{2}}+\bar{F}.$
(100)
Consequently the positivity of the modification term means that $\bar{F}>0$
again ensures nonnegativity of the eigenvalues.
## VI Summary and outlook
Summary – In this work we have considered the behavior of a null horizon
foliated by MOTS in a LRS background spacetime subjected to linear
perturbations. The problem is fixed by the gauge invariant requirement of the
characterizing function. More precisely, we set out to characterize the
evolution of a MOTS in a linearized spacetime admitting a 1+1+2 decomposition,
which lies in the slice orthogonal to the $u^{a}$. The gauge invariance of the
chacterizing function introduced is possible if and only if a certain function
vanishes in the background. This vanishing condition is equivalent to the
condition for a horizon in the background to be null, thus phrasing the
problem in terms of the evolutionary behavior of a null horizon in a LRS
background subjected to linear perturbations. In other words – and emphasis is
placed on this – this characterization fails in the case of a non-null horizon
in the background. While this work has been dedicated to the case of linear
perturbations, it was demonstrated that the characterizing function is form
invariant even when extending to non-linear regimes, for a certain frame
choice or provided that the perturbed spacetime has a vanishing sheet
expansion.
We focused the application of our characterization to non-dissipative
perturbations. As was described in the introduction, perturbing a seed metric
generally affects the dynamics of an embedded horizon, and that this is the
case of the null event horizon in the Schwarzschild spacetime. The presence of
shear ensures a spacelike transition of the null horizon. We have assumed a
harmonic decomposition of first order scalars $\Psi$ and 2-vectors $\Psi_{a}$,
where the notations $\Psi_{V}$ and $\bar{\Psi}_{V}$ are used respectively for
the components of $\Psi_{a}$ along the even and odd vector harmonics. It is
established here that this spacelike transition requires that a particular
relationship between the shear and the vorticity 2-vectors holds,
specifically, $\Sigma_{V}-\bar{\Omega}_{V}<0$. In order for the causal
preservation of the horizon character, $\Sigma_{V}-\bar{\Omega}_{V}=0$. The
result was obtained for the particular consideration of a non-dissipative
perturbation where the scalars, tensors and vector quantities associated to
the heat flux and anisotropic pressure are vanish.
We also considered the case of a spatially homogeneous background solution and
wrote down the form of the horizon characterizing function. In the
Schwarzschild case, any transition from a null character is necessarily
spacelike as the horizon encloses a black hole (note that we are assuming the
MOTS condition is preserved). However, for the spatially homogeneous case null
horizons do not enclose black holes, and therefore the causal character could
transition to a timelike one under the perturbation. Imposing a linear
equation of state, for a particular frame choice, i.e., fixing the spatial
unit vector to align with the acceleration of the temporal unit vector (which
coincides with the vanishing of the difference between the 2-vector
components, i.e., $\Sigma_{V}-\bar{\Omega}_{V}=0$), the sign of the equation
of state parameter $w$ alone characterizes the evolution: null if $w=0$,
spacelike if $w>0$, and timelike if $w<0$. Without this frame choice, the
evolution of the horizon is a bit more involved. We nonetheless present a
partial characterization of the horizon dynamics in yhje presence of
perturbations. In particular, where the difference between these vector
components is strictly negative or strictly positive, we find the ranges of
the equation of state parameter where the induced metric on the horizon has
absolute sign, i.e., the causal character is timelike or spacelike. However,
there are ranges of the parameter for which the causal character is determined
by a complex relationship between the 2-vectors, the energy density and the
eigenvalues of MOTS Laplacian, which we could not analytically determine.
Finally, we considered the relationship between the shear and vorticity
2-vectors which play a crucial role in the characterization of the horizon, to
the MOTS stability operator. These 2-vectors also specify the 1-form
$\tilde{s}_{a}$ which controls the self-adjointness of the operator, and
points to an obvious connection to the characterization we have introduced. We
established that the component $\bar{\tilde{s}}_{V}$ vanishes for our
particular consideration in this work, and that
$\tilde{s}_{V}=\Sigma_{V}+\bar{\Omega}_{V}$. This implied that for our
consideration the 1-form $\tilde{s}_{a}$ is a gradient (so that the MOTS
stability operator is self-adjoint) if and only if $\tilde{s}_{a}$ vanishes
identically. As a consequence, in the case of perturbation of a Schwarzschild
background, it follows that when the stability operator is not self-adjoint
and $\bar{\Omega}_{V}\neq 0$, the event horizon necessarily transitions to a
spacelike character. In the case of a spatially homogeneous background, in the
self-adjoint case with a vanishing $\tilde{s}_{a}$, it is clear that only one
of the components $\Sigma_{V}$ or $\bar{\Omega}_{V}$ is needed for the
characterization.
Several additional results were also obtained, restricting the form of
2-tensors and 2-vectors on the MOTS. For example, it was established that (and
this appears independent of the choice of background) the components of the
even and odd vector harmonics, respectively, of the magnetic and electric Weyl
2-vectors necessarily vanish on the MOTS. In fact, it was demonstrated that
the bi-implication $\bar{\tilde{s}}_{V}=0\iff\mathcal{H}=0$ follows, which is
due to the fact that $\mathcal{H}$ is necessarily a first order quantity.
Furthermore, the component of the odd tensor harmonics of the shear 2-tensor
also vanishes on the MOTS.
As the 1-form $\tilde{s}_{a}$ is tied to rotation generation on the horizon,
we provide a clear picture, at least in the linear regime, of the precise
source of rotation on the horizon: for a non-dissipative linear perturbation
of a LRS background spacetime, rotation is entirely sourced by contributions
from the Weyl 2-vectors on the MOTS. If one includes dissipation, then
rotation gains a contribution from the heat flux 2-vector. Actually, there
could be dissipation from the anisotropic 2-vector and no contribution from
the heat flux 2-vector, and it remains true that horizon rotation will be
entirely sourced by Weyl contributions. It then follows that for a conformally
flat and non-dissipative perturbation of an inhomogeneous LRS background
spacetime, the stability operator is necessarily self-adjoint.
We also presented a class of perturbations for which the 1-form
$\tilde{s}_{a}$ is non-vanishing but one may nonetheless expand the component
$\bar{\tilde{s}}_{V}$ in the harmonic basis so that the MOTS stability
operator can be brought to a self-adjoint form.
Outlook – There are several interesting directions that could be explored. In
this work, we have restricted ourselves to first order perturbations. What
happens when non-linearities are incorporated into the analysis of the horizon
evolution? Naturally the problem becomes a bit (or a lot) more complicated as
seen from the characterizing function $z$ in (27). For example, there will be
additional contributions from the non-linearities to the scalar curvature of
the MOTS, so that large distortions are introduced on the MOTS. These
distortions are not present at first order as was discussed earlier.
Modification to the eigenvalue in the self-adjoint case will present in the
form of the inclusion of dissipation and the cosmological constant $\Lambda$
(if $\Lambda$ is considered a perturbation ‘variable’). Any non-linearity
introduced is an extra contribution to the principal eigenvalue through the
scalar curvature of the MOTS.
Another example is the following. At every stage of our analysis we have
imposed the MOTS condition $\chi=0$, meaning that the built-in assumption here
is that a background horizon, under perturbations of its ambient background
spacetime, evolves to a horizon foliated by MOTS. However, it is quite
possible that under the perturbation the horizon is no longer sliced into
MOTS. This is the case of the Weyl-distorted Schwarzschild geometry discussed
in the introduction, where by the tuning up the distortion parameter the $r=2$
horizon is no longer a MOTS [12]. This degeneracy may be a feature of non-
linearities under the perturbation and so might not happen in the linear
perturbative case considered in this work. However, as mentioned in the second
point above, extending our analysis to non-linear perturbations, it would
suffice as a more general consideration, not to impose the MOTS condition and
analyze the non-linear equations.
## VII Acknowledgement
AS and SK research is supported by the Basic Science Research Program through
the National Research Foundation of Korea (NRF) funded by the Ministry of
education (grant numbers) (NRF-2022R1I1A1A01053784) and
(NRF-2021R1A2C1005748). PKSD acknowledges support through the First Rand Bank,
South Africa. AS would also like to thank the IBS Center for Theoretical
Physics of the Universe, Daejeon, South Korea, for its hospitality, where a
part of this work was carried out.
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|
# Edit Distance between Merge Trees
Raghavendra Sridharamurthy, , Talha Bin Masood, Adhitya Kamakshidasan, and
Vijay Natarajan R. Sridharamurthy, T.B. Masood, A. Kamakshidasan, and V.
Natarajan are with the Department of Computer Science and Automation, Indian
Institute of Science, Bangalore, 560012.
E-mail<EMAIL_ADDRESS>
###### Abstract
Topological structures such as the merge tree provide an abstract and succinct
representation of scalar fields. They facilitate effective visualization and
interactive exploration of feature-rich data. A merge tree captures the
topology of sub-level and super-level sets in a scalar field. Estimating the
similarity between merge trees is an important problem with applications to
feature-directed visualization of time-varying data. We present an approach
based on tree edit distance to compare merge trees. The comparison measure
satisfies metric properties, it can be computed efficiently, and the cost
model for the edit operations is both intuitive and captures well-known
properties of merge trees. Experimental results on time-varying scalar fields,
3D cryo electron microscopy data, shape data, and various synthetic datasets
show the utility of the edit distance towards a feature-driven analysis of
scalar fields.
###### Index Terms:
Merge tree, scalar field, distance measure, persistence, edit distance.
## 1 Introduction
The study of the behavior of physical quantities over time helps in
understanding underlying scientific processes. Physical quantities are either
measured using imaging devices or computed via simulation. In either case,
they are often modeled as scalar functions (also referred to as scalar
fields). Direct analysis and visualization of such a scalar function using
isosurfaces or volume rendering provides a good overview but is limited by two
factors. First, increasing size of data makes storage and retrieval
inefficient. Second, the analysis often requires a sweep over a large subset
of the domain or range of the function even when the features of interest may
be contained within a small region. These limitations are amplified when we
consider time-varying scalar functions. Thus, these techniques are not well
suited for feature directed analysis and visualization. Topological structures
such as the _merge tree_ [1] shown in Figure 1 provide a succinct
representation of the scalar function, support feature-directed visualization
and exploration, and hence enable the user to quickly identify patterns and
gain insights. Multiple scenarios demand a method for comparing scalar
functions. For example, a distance or similarity measure between scalar
functions is essential for detecting periodicity in a time-varying dataset.
The matrix of distances between all pairs of time steps will display a
characteristic pattern if the function is periodic. A method for comparing
scalar functions is also useful for tracking features in time-varying
phenomena [2], topological shape matching [3], detecting symmetry/asymmetry in
scalar fields [4, 5, 6, 7, 8], or clustering [9], computing temporal summaries
of large data sets, to identify features that are preserved in ensemble
simulations or multi-field data, or to compare simulated data against measured
data [10, 11, 12, 13]. In the above-mentioned scenarios, the similarity or
dissimilarity between scalar functions is often captured by a distance measure
between topological structures that represent the functions. We want such a
distance measure to satisfy useful theoretical properties and be efficiently
computable in order to be applicable in practice.
(a) 2D scalar field
(b) left: join tree and right: split tree
Figure 1: Merge trees. (a) A 2D scalar field (b) A merge tree tracks the
connectivity of sub-level sets (preimage of $f^{-1}(-\infty,c]$) or the super-
level sets (preimage of $f^{-1}[c,\infty)$).
### 1.1 Related Work
Assuming identical domains, RMS distance, Chebyschev distance, and other norms
such as $L_{p},1\leq p\leq\infty$ can be used for point-to-point comparisons.
However, a direct comparison of the two scalar functions may not be
appropriate because of its sensitivity to noise and minor perturbations.
Distance measures between various topological structures have been studied in
the literature, beginning with the bottleneck distance between persistence
diagrams [14]. A topological feature is often represented by a creator-
destructor pair, a critical point pair in the case of scalar functions. For
example, a minimum creates a 0-dimensional topological feature (connected
component) in the sub-level set that is destroyed by a saddle. A persistence
diagram (see Figure 3) depicts the persistence or “lifetime” of all
topological features by plotting their orresponding time of creation (birth)
and destruction (death) as points in $\mathbb{R}^{2}$. The bottleneck distance
($D_{B}$) between two persistence diagrams is equal to the weight of the
minimum weight mapping between points of the two diagrams. The weight of a
mapping is equal to the largest $L_{\infty}$ distance between a point and its
image under the mapping. We say that the persistence diagram is _stable_ with
respect to a distance measure if it is bounded above by the $L_{\infty}$
distance between the two scalar functions. Intuitively, we require that small
perturbations to the scalar functions translate to small changes in the
distance between the respective persistence diagrams. The persistence diagram
is stable with respect to $D_{B}$. But, the persistence diagram is only a
multiset. It does not capture the spatial configuration of critical points,
which reduces its discriminative capability. Morozov et al. [15] proposed the
interleaving distance between merge trees. This distance is defined by a
continuous map that shifts points of one merge tree onto the other and vice-
versa. The distance is equal to the smallest value of the shift such that the
map satisfies certain compatibility conditions. The merge tree is stable under
this distance and the distance measure is more discriminative compared to the
bottleneck distance but computing it is not a tractable problem. Beketayev et
al. [16] define a distance measure between merge trees that can be computed by
considering all possible branch decompositions. This measure can be computed
in polynomial time but provide no guarantees on stability.
Distance measures have been also defined for other topological structures. The
_Reeb graph_ captures the topology of both sub-level sets and super-level sets
of scalar functions defined on a manifold [17]. Bauer et al. [18] imposed a
metric on Reeb graphs called the functional distortion distance. They proved
its stability and connections with other distances such as the bottleneck and
interleaving distance. The computation depends on the Gromov-Hausdorff
distance, which is proven to be NP-hard [19] to even approximate up to a
constant factor for general metric graphs. Di Fabio and Landi [20] defined an
edit distance for Reeb graphs on surfaces. They also proved its stability and
showed connections with interleaving distance and function distortion distance
but there is no polynomial time algorithm to compute the distance. Dey et al.
[21] defined the persistence distortion distance to compare metric graphs,
proved its stability, and described a polynomial time algorithm with
asymptotic running time $O(m^{12}\log n)$ (continuous version) and
$O(n^{2}m^{1.5}\log m)$ (discrete version), where $m$ is the number of edges
and $n$ is the number of vertices in the larger graph. They also reported
applications to shape matching.
Narayanan et al. [22] defined a distance measure to compare _extremum graphs_
, whose nodes corresponds to critical points of the scalar function and arcs
correspond to integral lines. The distance measure is based on the maximum
weight common subgraph and they use pruning techniques to speedup the
computation. While stability is not guaranteed, they present many experimental
results on time-varying data to demonstrate its application to time-varying
data analysis and visualization.
In contrast to the rigorous definitions of distance measures introduced in the
above-mentioned works, simpler but practical similarity measures have also
been studied. Saikia et al. [7] introduced the extended branch decomposition
graph (eBDG) that describes a hierarchical representation of all subtrees of a
join/split tree and designed an efficient algorithm to compare them. They also
present experimental results on time-varying data. Saikia et al. [23] studied
a measure that compared histograms that are constructed together with the
merge trees. As in the case of bottleneck distance, this measure ignores the
structure but it can be computed efficiently and is therefore useful in
practice. Saikia and Weinkauf [2] later extended this measure and demonstrated
applications to feature tracking in time-varying data.
Edit distances and alignment distances for trees are inspired by edit
distances defined on strings. They have found various applications, such as
comparing neuronal trees [24], comparing shapes [25], comparing music genre
taxonomy [26], analysis of glycan structures [27], comparing RNA structures
[28], and comparing plant architectures [29]. Given two strings, one is
transformed into the other via a sequence of operations where each operation
has a non-negative associated cost. The distance is defined as the minimum
cost over all such transformations. Similar distance measure may be defined
for labeled trees with edit operations like relabeling, addition, and deletion
of nodes. Zhang and Shasha [30] described an algorithm to compute the tree
edit distance for ordered labeled trees. Later, Zhang [31] proposed a new
algorithm for constrained tree edit distance for ordered labeled trees. The
computation of tree edit distance for unordered labeled trees is NP-complete
[32]. However, the constrained version of the problem can be solved in
polynomial time using a dynamic programming based algorithm [33]. A gap
corresponds to a collection of nodes that are inserted / deleted during a
sequence of edit operations. Edit distance with arbitrary gap costs were first
proposed by Touzet [34], who showed that the distance computation is NP-hard.
But, the distance between ordered labeled binary trees can be computed in
polynomial time [35].
While tree edit distance based algorithms have been employed in many
applications, they have not been well studied for comparing topological
structures like merge trees except in very recent work. Riecke et al. [36]
defined a hierarchy of persistence pairs and a tree edit distance based
dissimilarity measure to compare hierarchies. Sridharamurthy et al. [37] adapt
Xu’s algorithm [35] for computing distance between ordered labeled binary
trees to the case of the general subtree gap model that preserve the merge
tree structure. The general subtree gap model allows for interior nodes to be
inserted / deleted while retaining the child nodes. The cost model is
intuitive and they present preliminary experimental results to show its
utility. However, this method has multiple shortcomings:
* •
The gap model is too general. In the case of merge trees, we require a
constrained version that considers gaps as persistence pairs in order to
preserve the structural integrity of the tree. These pairs depend on function
values. So, the constraints are ad hoc, difficult to express directly and to
incorporate into the dynamic programming based algorithm that is used to
compute the measure.
* •
The above-mentioned pairs are not stable under perturbations to the scalar
function.
* •
Merge trees constructed on real world data are not necessarily binary trees.
* •
Absence of a natural left-to-right ordering of children of a node in the merge
tree. The algorithm requires such an ordering, random or canonical orderings
lead to instabilities.
* •
The running time of the algorithm is approximately $O(n^{5})$, where $n$ is
the number of nodes in the tree. This is very slow for practical applications.
In summary, existing work either propose a rigorous definition of distance
with theoretical guarantees but without practical value (with very few
exceptions) or describe a similarity / dissimilarity measure with practical
applications but without theoretical analysis. Further, existing methods do
not provide natural support for a fine grained analysis of similar/dissimilar
regions.
### 1.2 Contributions
In this paper, we propose a tree edit distance based approach to compare merge
trees. The distance measure is an adaptation of the constrained unordered tree
edit distance [33], but is a significant modification that caters to merge
trees and alleviates the shortcomings of the measure proposed by
Sridharamurthy et al. [37]. Individual edits correspond to topological
features. The edit operations may be subsequently studied for a fine grained
analysis. The paper makes the following key contributions:
1. 1.
An intuitive and mathematically sound cost model for the individual edit
operations.
2. 2.
A proof that the distance measure is a metric under the proposed cost model.
3. 3.
A computational solution to handle instabilities.
4. 4.
Experiments to demonstrate the practical value of the distance measure using
various applications — 2D time-varying data analysis by detecting periodicity,
summarization to support visualization of 3D time-varying data, detection of
symmetry and asymmetry in scalar fields, study of topological effects of
subsampling and smoothing, and shape matching.
In addition, we describe a comprehensive set of validation experiments that
are designed to help understand the properties of the measure.
## 2 Background
In this section, we introduce necessary definitions and background on merge
trees, list some desirable properties of distance measures, and describe three
edit operations on a merge tree that define a tree edit distance.
### 2.1 Merge tree
Figure 2: Persistence pairs in the join (left) and split (right) trees.
A merge tree [1] captures the connectivity of sub-level sets (_join tree_) or
super-level sets (_split tree_) of a scalar function
$f:\mathbb{X}\longrightarrow\mathbb{R}$ defined on a manifold domain
$\mathbb{X}$, see Figure 1. A value $c$ in the range of $f$ is called an
_isovalue_. Given an isovalue, an _isocontour_ is defined as the collection of
all points $x\in\mathbb{X}$ such that $f(x)=c$. Nodes of join trees consist of
minima $M=\\{m_{i}\\}$, saddles $S=\\{s_{j}\\}$, and the global maximum. In
theory, the structure of a join tree is simple. Excluding the global maximum,
which is the root of the tree, every node has either 0 (minimum) or 2 children
(saddle). All minima are paired with saddles based on the notion of
topological persistence [38] except for one which is paired to the lone global
maximum. Each such pair $(m,s)$ represents a topological feature and its
persistence is defined as $pers(m)=pers(s)=f(s)-f(m)$. In practice, saddles
may have more than two children. We discuss how to handle them in Section 4.4.
A split tree is defined likewise. It contains a set of maxima and saddles
together with the global minimum. Figure 2 shows the persistence pairing for
the trees from Figure 1. Several fast algorithms have been developed for
computing merge trees (join or split) for piecewise linear functions defined
on simply connected domains [1, 39, 40].
Figure 3: A 1D scalar function (left) and the persistence diagram of the
function (right). Each birth-death pair $(b_{i},d_{i})$ is a feature of the
scalar function and its persistence is defined as $d_{i}-b_{i}$. Each pair is
represented as a point in $\mathbb{R}^{2}$.
### 2.2 Distance measures
Designing distance measures is a well studied problem and has several
applications in data analysis, visualization, pattern recognition, data
mining, and machine learning. A distance measure
$D:\mathbb{X}\times\mathbb{X}\longrightarrow\mathbb{R}$ on a domain
$\mathbb{X}$ satisfies the metric properties:
1. 1.
_Non-negativity_ : $D(x,y)\geq 0$
2. 2.
_Identity of indiscernibles_ : $D(x,y)=0$ iff $x=y$
3. 3.
_Symmetry_ : $D(x,y)=D(y,x)$
4. 4.
_Triangle inequality_ : $D(x,z)\leq D(x,y)+D(y,z)$
When metric properties such as triangle inequality are relaxed, we get a
_dissimilarity measure_ rather than a distance measure. Further, while
comparing scalar fields or topological structures constructed based on scalar
fields, it is desirable that the distance measure satisfies two additional
properties – stability and discrimination. In the following discussion, we
will use $D$ to refer to the distance between topological structures that
represent the functions. Given two scalar functions $f,g$,
1. 1.
_Stability_ : $D(f,g)\leq\|f-g\|_{\infty}$
2. 2.
_Discrimination_ : $D_{B}(f,g)\leq D(f,g)$.
Intuitively, stability requires that if the functions are not too “different”
in terms of the $L_{\infty}$ norm of the difference between the functions then
the distance measure between the topological structures representing the
scalar functions should also be small. Discrimination, on the other hand,
requires that however “small” the difference between two functions, it should
be captured by the distance measure. Specifically, the distance measure equals
$0$ should imply that the functions are equal. Since the bottleneck distance
$D_{B}$ between persistence diagrams of $f$ and $g$ was among the first
measures defined between topological structures, we typically state this
property in terms of how the distance $D$ is related to bottleneck distance.
Figure 4 shows an example where $D_{B}=0$ for a pair of functions that are not
equal, which implies that $D_{B}$ is not discriminative enough. One reason for
the low discriminative power is that the persistence diagram, and hence
$D_{B}$, does not incorporate the connectivity between critical points as in
the merge tree. We wish to design a distance measure that satisfies the
following property:
$\displaystyle D_{B}(f,g)\leq D(f,g)\leq\|f-g\|_{\infty}$ (1)
From the computational perspective, $D$ should be computable either exactly or
within a constant factor of approximation in polynomial time in order for it
to be useful in a practical application.
Figure 4: The discriminative power of the bottleneck distance $D_{B}$ is low.
Two scalar functions (blue and red) and the corresponding persistence diagrams
and merge trees. Even though the scalar functions are different, $D_{B}$ is
not able to capture the difference because the persistence diagrams are equal.
A distance measure that considers the structure of the merge tree would
discriminate the two scalar functions.
### 2.3 Tree edit distance
Tree edit distances have been studied extensively in the past few decades
[41]. All these measures typically employ a set of edit operations with
associated costs and try to minimize the total cost over the set of all edit
operations. Let $T$ be a rooted tree with node set $V$ and edge set $E$. For a
node $v\in V$, $deg(v)$ is the number of children of $v$, and $parent(v)$ is
its parent in the tree. The maximum degree of a node in the tree is denoted as
$deg(T)$. We denote an empty tree by $\theta$. Since we are interested in
labeled trees, let $\Sigma$ be the set of labels, and $\lambda\notin\Sigma$
denote the null or empty character, which corresponds to a gap. In the
following discussion, we use notations and definitions from Zhang [33].
Edit operations. The edit operations differ based on the gap model. For this
discussion we consider edit operations that modify the tree, one node at a
time. Xu [35] gives a detailed discussion of general gaps where edits modify
multiple nodes. We consider a total of three edit operations as shown in
Figure 5.
(a) delete
(b) insert
(c) relabel
Figure 5: Three different tree edit operations. Each edit affects only one
node in the tree. The null character $\lambda$ corresponds to a gap.
1. 1.
relabel: A relabel $a\longrightarrow b$ corresponds to an operation where the
label $a\in\Sigma$ of a node is changed to a label $b\in\Sigma$.
2. 2.
delete: A delete operation $a\longrightarrow\lambda$ removes a node $n$ with
label $a\in\Sigma$ and all the children of $n$ are made the children of
$parent(n)$.
3. 3.
insert: An insert operation $\lambda\longrightarrow b$ inserts a node $n$ with
label $b\in\Sigma$ as a child of another node $m$ by moving all the children
of $m$ to children of $n$.
We define a cost function $\gamma$ that assigns a non-negative real number to
each edit operation of the form $a\longrightarrow b$. It is useful if the cost
function $\gamma$ satisfies metric properties i.e. $\forall
a,b,c\in\Sigma\cup\\{\lambda\\}$
1. 1.
$\gamma(a\longrightarrow b)\geq 0,\ \gamma(a\longrightarrow a)=0$
2. 2.
$\gamma(a\longrightarrow b)=\gamma(b\longrightarrow a)$
3. 3.
$\gamma(a\longrightarrow c)\leq\gamma(a\longrightarrow
b)+\gamma(b\longrightarrow c)$
In particular, Zhang [33] proved that if $\gamma$ is a metric then the edit
distance is also a metric, else it will be merely a dissimilarity measure.
Given a tree $T_{1}$, we can apply a sequence of edit operations to transform
it into another tree $T_{2}$. If $S=s_{1},s_{2},\ldots,s_{k}$ is a sequence of
edit operations, where each $s_{i}$ is an edit, we can extend the cost
function to $S$ by defining $\gamma(S)=\Sigma_{i=1}^{|S|}\gamma(s_{i})$.
Edit distance. Formally, the distance between two trees $T_{1},T_{2}$ is
defined as
$\displaystyle D_{e}(T_{1},T_{2})=\min_{S}\\{\gamma(S)\\}$ (2)
where $S$ is an edit operation sequence from $T_{1}$ to $T_{2}$.
## 3 Edit Distance Mappings
Computing the edit distance between merge trees is a minimization problem with
a huge search space. In order to understand this search space and how it
affects the computation, we first define some edit distance mappings –
unconstrained, constrained, and restricted – and their properties as described
by Zhang [33]. We refer the reader to the supplementary material for
additional description and illustrations of the mappings.
### 3.1 Unconstrained edit distance mapping
The sequence of edit operations performed to transform $T_{1}$ into $T_{2}$
determines a mapping between the two trees. For convenience, we order the
nodes of both the trees. This ordering does not affect the distance. Let
$t_{1}$ and $t_{2}$ denote the ordering of nodes in $T_{1}$ and $T_{2}$,
respectively, and $t_{1}[i]$ represents the $i$th node in the ordering. Let
$M_{e}$ denote a collection of ordered integer pairs $(i,j)$. A triple
$(M_{e},T_{1},T_{2})$ defines the _edit distance mapping_ from $T_{1}$ to
$T_{2}$, where each pair $(i_{1},j_{1}),(i_{2},j_{2})\in M_{e}$ satisfies the
following properties:
* •
$i_{1}=i_{2}$ iff $j_{1}=j_{2}$ (one-to-one)
* •
$t_{1}[i_{1}]$ is an ancestor of $t_{1}[i_{2}]$ iff $t_{2}[j_{1}]$ is an
ancestor of $t_{2}[j_{2}]$ (ancestor ordering).
The cost of transforming $T_{1}$ into $T_{2}$ can be expressed through the
mapping as
$\displaystyle\gamma(M_{e})$ $\displaystyle=\sum\limits_{(i,j)\in
M_{e}}\gamma(t_{1}[i]\longrightarrow t_{2}[j])$ (3)
$\displaystyle+\sum\limits_{\\{i|\nexists j,(i,j)\in
M_{e}\\}}\gamma(t_{1}[i]\longrightarrow\lambda)$
$\displaystyle+\sum\limits_{\\{j|\nexists i,(i,j)\in
M_{e}\\}}\gamma(\lambda\longrightarrow t_{2}[j])$
Given a sequence of edit operations $S$ that transforms $T_{1}$ into $T_{2}$,
there exists a mapping $M_{e}$ such that $\gamma(M_{e})\leq\gamma(S)$.
Conversely, given an edit distance mapping $M_{e}$ ,there exists a sequence of
edit operations $S$ such that $\gamma(S)=\gamma(M_{e})$. Using the above, it
can be shown that
$\displaystyle D_{e}(T_{1},T_{2})=\min_{M_{e}}\\{\gamma(M_{e})\\}$ (4)
where $(M_{e},T_{1},T_{2})$ defines the _edit distance mapping_ from $T_{1}$
to $T_{2}$. Zhang et al. [32] showed that computing $D_{e}(T_{1},T_{2})$ is
NP-complete even when the trees are binary and $|\Sigma|=2$.
### 3.2 Constrained and restricted mappings
Adding constraints to the edit distance mapping brings it within the
computationally tractable realm. The main constraint imposed is that disjoint
subtrees are mapped to disjoint subtrees. Let $T[i]$ denote the subtree rooted
at the node with label $i$ and $F[i]$ denote the unordered forest obtained by
deleting the node $t[i]$ from $T[i]$. A node $t_{1}[i]$ is a _proper ancestor_
of $t_{1}[j]$ if $t_{1}[i]$ lies on the path from the root to $t_{1}[j]$ and
$t_{1}[i]\neq t_{1}[j]$. The triple $(M_{c},T_{1},T_{2})$ is called a
_constrained edit distance mapping_ if,
* •
$(M_{c},T_{1},T_{2})$ is an edit distance mapping, and
* •
Given three pairs $(i_{1},j_{1}),(i_{2},j_{2}),(i_{3},j_{3})\in M_{c}$, the
_least common ancestor_ $lca(t_{1}[i_{1}],t_{1}[i_{2}])$ is a proper ancestor
of $t_{1}[i_{3}]$ iff $lca(t_{2}[j_{1}],t_{2}[j_{2}])$ is a proper ancestor of
$t_{2}[j_{3}]$.
The constrained edit distance mappings can be composed. Given two constrained
edit distance mappings $M_{c_{1}}$ from $T_{1}$ to $T_{2}$ and $M_{c_{2}}$
from $T_{2}$ to $T_{3}$, $M_{c_{2}}\circ M_{c_{1}}$ is a constrained edit
distance mapping between $T_{1}$ and $T_{3}$. Also,
$\displaystyle\gamma(M_{c_{2}}\circ
M_{c_{1}})\leq\gamma(M_{c_{1}})+\gamma(M_{c_{2}})$ (5)
which can be proven using the triangle inequality imposed on the edit
operation costs. This leads to the definition of constrained edit distance
$\displaystyle D_{c}(T_{1},T_{2})=\min_{M_{c}}\\{\gamma(M_{c})\\}$ (6)
$D_{c}$ also satisfies metric properties. Both $M_{e}$ and $M_{c}$ deal with
mapping between unordered trees. Similar mappings work for forests. We define
a _restricted mapping_ $M_{r}(i,j)$ between $F_{1}[i]$ and $F_{2}[j]$ as
follows:
* •
$M_{r}(i,j)$ corresponds to a constrained edit distance mapping between
$F_{1}[i]$ and $F_{2}[j]$.
* •
Given two pairs $(i_{1},j_{1}),(i_{2},j_{2})\in M_{c}$, $t_{1}[l_{1}]$ and
$t_{1}[l_{2}]$ belong to a common tree in $F_{1}[i]$ if and only if
$t_{2}[j_{1}]$ and $t_{2}[j_{2}]$ belong to a common tree in $F_{2}[i]$.
Essentially, nodes within different trees of $F_{1}$ are mapped to nodes lying
in different trees of $F_{2}$.
### 3.3 Constrained edit distance
We recall the properties of $D_{c}$. Let
$t_{1}[i_{1}],t_{1}[i_{2}],\ldots,t_{1}[i_{n_{i}}]$ be the children of
$t_{1}[i]$ and $t_{2}[j_{1}],t_{2}[j_{2}],\ldots,t_{2}[j_{n_{j}}]$ be the
children of $t_{2}[j]$. Further, let $\theta$ denote the empty tree. Then,
$\displaystyle D_{c}(\theta,\theta)$ $\displaystyle=0,$ (7) $\displaystyle
D_{c}(F_{1}[i],\theta)$
$\displaystyle=\sum\limits_{k=1}^{n_{i}}D_{c}(T_{1}[i_{k}],\theta),$ (8)
$\displaystyle D_{c}(T_{1}[i],\theta)$
$\displaystyle=D_{c}(F_{1}[i],\theta)+\gamma(t_{1}[i]\longrightarrow\lambda),$
(9) $\displaystyle D_{c}(\theta,F_{2}[j])$
$\displaystyle=\sum\limits_{k=1}^{n_{j}}D_{c}(\theta,T_{2}[j_{k}]),$ (10)
$\displaystyle D_{c}(\theta,T_{2}[j])$
$\displaystyle=D_{c}(\theta,F_{2}[j])+\gamma(\lambda\longrightarrow
t_{2}[j]),$ (11) $\displaystyle\scriptsize D_{c}(T_{1}[i],T_{2}[j])$
$\displaystyle=\scriptsize\min\begin{cases}D_{c}(\theta,T_{2}[j])+\min\limits_{1\leq
t\leq n_{j}}\\{D_{c}(T_{1}[i],T_{2}[j_{t}])-D_{c}(\theta,T_{2}[j_{t}])\\},\\\
D_{c}(T_{1}[i],\theta)+\min\limits_{1\leq s\leq
n_{i}}\\{D_{c}(T_{1}[i_{s}],T_{2}[j])-D_{c}(T_{1}[i_{s}],\theta)\\},\\\
D_{c}(F_{1}[i],F_{2}[j])+\gamma(t_{1}[i]\longrightarrow t_{2}[j]).\end{cases}$
(12)
If the cost is not a metric, we need to include one additional case, namely
$D_{c}(F_{1}[i],F_{2}[j])+\gamma(t_{1}[i]\longrightarrow\lambda)+\gamma(\lambda\longrightarrow
t_{2}[j])$. The distance between two forests is given by
$\displaystyle\scriptsize D_{c}(F_{1}[i],F_{2}[j])$
$\displaystyle=\scriptsize\min\begin{cases}D_{c}(\theta,F_{2}[j])+\min\limits_{1\leq
t\leq n_{j}}\\{D_{c}(F_{1}[i],F_{2}[j_{t}])-D_{c}(\theta,F_{2}[j_{t}])\\},\\\
D_{c}(F_{1}[i],\theta)+\min\limits_{1\leq s\leq
n_{i}}\\{D_{c}(F_{1}[i_{s}],F_{2}[j])-D_{c}(F_{1}[i_{s}],\theta)\\},\\\
\min\limits_{M_{r}(i,j)}\gamma(M_{r}(i,j)).\end{cases}$ (13)
The minimum restricted mapping may be computed by constructing a weighted
bipartite graph in such a way that the cost of the minimum weight maximum
matching $MM(i,j)$ is exactly the same as the cost of the minimum restricted
mapping $M_{r}(i,j)$,
$\displaystyle\min\limits_{M_{r}(i,j)}\gamma(M_{r}(i,j))=\min\limits_{MM(i,j)}\gamma(MM(i,j))$
(14)
### 3.4 Algorithm
Zhang described an algorithm for computing the tree edit distance for labeled
unordered trees [33]. It is a dynamic programming based algorithm that follows
from the properties discussed in Section 3.3. The pseudo code is presented in
the supplementary material (Section 2). The entry $D(T_{1}[m],T_{2}[n])$ in
the table with $m=|T_{1}|$ and $n=|T_{2}|$ corresponds to the final result.
The algorithm computes the distance in
$O(|T_{1}|\times|T_{2}|\times(deg(T_{1})+deg(T_{2}))\times
log_{2}(deg(T_{1})+deg(T_{2})))$ time in the worst case.
## 4 Tree Edit Distance
We now describe a new tree edit distance that is appropriate for comparing
merge trees, discuss its properties, and an algorithm for computing the
distance measure.
### 4.1 Comparing merge trees
Our proposed measure is based on a variant of tree edit distance that applies
to unordered general trees as opposed to ordered binary trees. This variant is
appropriate because
* •
Merge trees are unordered trees.
* •
Merge trees are not binary in general.
* •
Persistence pairs represent topological features. So, it is natural that the
edit operations are defined in terms of persistent pairs.
* •
The pairs do not fit into any subtree gap model that has been studied in the
literature.
Consider the properties of edit distance mapping mentioned in Section 3.1, but
now in the context of merge trees. The one-to-one property is applicable but
ancestor ordering might not hold in all cases. Small perturbations in the
function value may result in swaps similar to rotations in AVL or red-black
trees [42, Chapter 13], which violate the ancestor ordering. Such violations
also result in instabilities i.e., cause significant fluctuations in the
distance (see Section 4.4). Computing the edit distance with the ancestor
order preserving mappings is already infeasible. Removing that constraint will
make the computation more difficult. We introduce a stability parameter to
ensure that ancestor order preserving mappings are identified in practically
all cases. More details on this computational solution to handling
instabilities can be found in Section 4.4. This solution does discard some
mappings and may lead us away from the optimum solution. But, the
stabilization ensures that the mapping remains meaningful and helps reduce the
search space thereby making the problem tractable.
To summarize, $D_{c}$ between unordered trees with suitable modifications
seems to be a good candidate for comparing merge trees. In this section, we
describe one such distance measure and demonstrate its use in the following
section. The additional constraint of mapping disjoint subtrees to disjoint
subtrees may seem limiting. Also, $D_{e}(T_{1},T_{2})\leq D_{c}(T_{1},T_{2})$,
which implies that the constrained edit distance may not be optimal in many
cases. But, we observe that, in practice, it is not as limiting and gives good
results in many applications.
### 4.2 Cost model
The edit distance mapping $M_{e}$ and the constrained edit distance mapping
$M_{c}$ need to be suitably modified so that they are applicable for comparing
merge trees. We begin by considering the edit operations as applicable to
merge trees together with appropriate cost models. The literature on tree edit
distances study generic trees and hence do not describe particular cost
models. The following discussions focus on join trees but all results hold for
split trees also.
Tree edit operations on the join tree need to preserve the structural
integrity of the join tree. This reduces the number of operations, say
insertions and deletions. Consider a min-saddle pair $(m_{2},s_{1})$ in Figure
6. If $s_{1}$ is deleted, then it’s pair $m_{2}$ should also be deleted, and
vice-versa. After deletion, $m_{1}$ is adjacent to $s_{2}$. But deletion of
$s_{1}$ does not necessarily require that the entire subtree rooted at $s_{1}$
be deleted. In fact, deleting the entire subtree may not result in a valid
join tree as illustrated in Figure 6. In this particular illustration, we
consider the pairing imposed by the persistence. But, in general, we may
consider other pairings based on say volume, hyper-volume, etc.
Figure 6: Permitted and forbidden edit operations. (left) A gap is introduced
by removing a persistence pair. (right) An edit operation that is permitted
for generic trees but is invalid for a join tree. Nodes and arcs are
repositioned to improve the tree layout.
Gaps in the join tree can be represented as a collection of min-saddle pairs.
In Figure 6, we can transform the first tree into the last tree by deleting
the pairs $\\{(m_{2},s_{1}),(m_{3},s_{2})\\}$. We propose two cost models that
capture the preservation of topological features and are applicable for join
trees. Consider nodes $p\in T_{1}$ and $q\in T_{2}$. Then $p$ and $q$ are
creators or destroyers of topological features in $T_{1}$ and $T_{2}$,
respectively. Let the birth and death times of these features be
$(b_{p},d_{p})$ and $(b_{q},d_{q})$, respectively. These birth-death pairs
correspond to points in the persistence diagrams. Alternatively, they are
represented as closed intervals $[b_{p},d_{p}]$ and $[b_{q},d_{q}]$ in a
persistence barcode.
#### 4.2.1 $L_{\infty}$ cost $C_{W}$
$\displaystyle\gamma(p\longrightarrow q)$
$\displaystyle=\min\begin{cases}\max(|b_{q}-b_{p}|,|d_{q}-d_{p}|),\\\
\frac{(|d_{p}-b_{p}|+|d_{q}-b_{q}|)}{2}\end{cases}$ (15)
$\displaystyle\gamma(p\longrightarrow\lambda)$
$\displaystyle=\frac{|d_{p}-b_{p}|}{2}$ (16)
$\displaystyle\gamma(\lambda\longrightarrow q)$
$\displaystyle=\frac{|d_{q}-b_{q}|}{2}$ (17)
This cost model is based on the bottleneck and Wasserstein distances. Note
that the insert / delete cost is based on the $L_{\infty}$-distance of the
points $p$ (or $q$) from the diagonal in the persistence diagram. The relabel
cost is the minimum of the $L_{\infty}$-distance between the points $p$ and
$q$ and the sum of the $L_{\infty}$-distance from the points $p$ (or $q$) to
the diagonal. This corresponds to the scenario where transforming $p$ to $q$
($p\longrightarrow q$) by deleting $p$ and inserting $q$
($p\longrightarrow\lambda$ and $\lambda\longrightarrow q$) has a lower cost in
some cases. Figure 7 shows how these costs can be derived from the persistence
diagram when there is no overlap in the barcodes and when there is overlap
between the barcodes.
#### 4.2.2 Overhang cost $C_{O}$
$\displaystyle\gamma(p\longrightarrow q)$
$\displaystyle=\min\begin{cases}|b_{q}-b_{p}|+|d_{q}-d_{p}|,\\\
|d_{p}-b_{p}|+|d_{q}-b_{q}|\end{cases}$ (18)
$\displaystyle\gamma(s\longrightarrow\lambda)$ $\displaystyle=|d_{p}-b_{p}|$
(19) $\displaystyle\gamma(\lambda\longrightarrow t)$
$\displaystyle=|d_{q}-b_{q}|$ (20)
This cost model is based on the overlap of the barcodes or the intervals. We
consider the lengths of the overhang or the non-overlapping section to
determine the costs. Consider $p\longrightarrow\lambda$, the interval
corresponding to $p$ is given by $[b_{p},d_{p}]$ with length $|d_{p}-b_{p}|$
and the interval corresponding to $\lambda$ is $\emptyset$ with length $0$.
Since there is no overlap, the cost is $|d_{p}-b_{p}|+0=|d_{p}-b_{p}|$. The
cost of $\lambda\longrightarrow q$ can be derived similarly. Let us now
consider the cost of $p\longrightarrow q$. If there is an overlap, we discard
the overlap and obtain $|b_{q}-b_{p}|+|d_{q}-d_{p}|$. If there is no overlap
then the cost is equal to $|d_{p}-b_{p}|+|d_{q}-b_{q}|$. The minimum of the
two expressions is the relabel cost. The barcodes are shown in the fourth
quadrant of the persistence diagrams in Figure 7.
Figure 7: Illustration of the cost models when there is no overlap in the
barcodes (left) and when there is overlap (right). We distinguish between
birth and death events in the barcode by using different glyphs for the start
and the end of the intervals.
### 4.3 Metric properties
Metric property enables us to study the space of all the trees, compute the
mean, and also compose transformations between merge trees. From Sections 2.3
and 3.2, we know that if the cost model satisfies the metric property then the
distance measure is also a metric. We now prove the metric properties for our
cost model.
The overhang cost is similar to symmetric difference, which is a well-known
metric [43]. We now prove that the $L_{\infty}$ cost $C_{W}$ is a metric.
#### 4.3.1 $C_{W}$ is a metric
We show that the cost $C_{W}$ is equal to the Wasserstein distance between two
corresponding persistence diagrams. Let $N$ denote the set of all nodes in the
merge trees and $\lambda$ denote a node corresponding to the null character.
We define a mapping $\mathcal{M}:N\cup\\{\lambda\\}\longrightarrow Dgm$, where
$Dgm$ is set of all persistence diagrams, as follows:
1. 1.
$\forall p\in N,\ \mathcal{M}(p)=\\{(b_{p},d_{p})\\}\cup\\{(x,x),x\geq 0\\}$,
2. 2.
$\mathcal{M}(\lambda)=\\{(x,x),x\geq 0\\}$.
Define the distance on the set $N\cup\\{\lambda\\}$ as the Wasserstein
distance of the first order _i.e._ , given $p,q\in N\cup\\{\lambda\\}$
$d(p,q)=W_{1}(\mathcal{M}(p),\mathcal{M}(q))$
Now, the cost $C_{W}$ can be rewritten as
$\displaystyle\gamma(p\longrightarrow q)$
$\displaystyle=W_{1}(\mathcal{M}(p),\mathcal{M}(q))$ (21)
$\displaystyle\gamma(p\longrightarrow\lambda)$
$\displaystyle=W_{1}(\mathcal{M}(p),\mathcal{M}(\lambda))$ (22)
$\displaystyle\gamma(\lambda\longrightarrow q)$
$\displaystyle=W_{1}(\mathcal{M}(\lambda),\mathcal{M}(q)).$ (23)
Since the Wasserstein distance $W_{1}(\cdot,\cdot)$ between persistence
diagrams is known to be a metric [44, Chapter 6], $C_{W}$ is also a metric.
However, this proof of the metric property is for general distributions. We
have an alternative proof for merge trees from first principles with the aim
to better understand the cost.
#### 4.3.2 $C_{W}$ is a metric : proof from first principles
Non-negativity and symmetry follows by definition because $C_{W}$ is based on
sum, max, min of absolute values.
Figure 8: The cost of edit operations can be reformulated as the weight of a
minimum weight maximum matching in a bipartite graph. Bipartite graph for a
relabel operation $p\longrightarrow q$ (left) and a delete operation (right).
To prove the triangle inequality, we first reformulate the cost of the edit
operations as the weight of a minimum weight maximum matching. The matching is
defined in a bipartite graph. Nodes of the bipartite graph consists of the
merge tree nodes together with an equal number of copies of $\lambda$. We
collect the nodes of the graph to construct sets of the form
$P=\\{p,\lambda\\}$ and a special multiset $\Lambda=\\{\lambda,\lambda\\}$,
see Figure 8. All pairs of nodes from different multisets are connected by an
edge. The edge weight $c$ is given by the $L_{\infty}$ distance between the
corresponding points in the persistence diagram:
$\displaystyle c_{pq}=L_{\infty}(p,q)$
$\displaystyle=\max(|b_{q}-b_{p}|,|d_{q}-d_{p}|)$ (24) $\displaystyle
c_{p\lambda}=L_{\infty}(p,\lambda)$ $\displaystyle=\frac{|d_{p}-b_{p}|}{2}$
(25) $\displaystyle c_{\lambda q}=L_{\infty}(\lambda,q)$
$\displaystyle=\frac{|d_{q}-b_{q}|}{2}$ (26) $\displaystyle
c_{\lambda\lambda}=L_{\infty}(\lambda,\lambda)$ $\displaystyle=0$ (27)
The cost of the edit operations is equal to the cost of the minimum weight
maximum matching $MM$ in this bipartite graph. In Figure 8, one of the two
matchings will determine the cost of the edit operation.
$\displaystyle\gamma(p\longrightarrow q)=MM(P,Q)$
$\displaystyle=\min\begin{cases}c_{pq}+c_{\lambda\lambda},\\\
c_{p\lambda}+c_{\lambda q}\end{cases}$ (28)
$\displaystyle\gamma(p\longrightarrow\lambda)=MM(P,\Lambda)$
$\displaystyle=\min\begin{cases}c_{p\lambda}+c_{\lambda\lambda},\\\
c_{\lambda\lambda}+c_{p\lambda}\end{cases}$ (29)
$\displaystyle\gamma(\lambda\longrightarrow q)=MM(\Lambda,Q)$
$\displaystyle=\min\begin{cases}c_{\lambda\lambda}+c_{\lambda q},\\\
c_{\lambda q}+c_{\lambda\lambda}\end{cases}$ (30)
Figure 9: The cost function satisfies triangle inequality. (left) The blue or
the red matching may be the minimum weight maximum matching that corresponds
to the cost of the edit operation. (right) The matching between $P$ and $R$ is
a composition of matching between $P,Q$ and $Q,R$. The inequality can be
proved via case analysis by considering all possible compositions.
Consider three multisets as shown in Figure 9. Using the above construction,
we prove triangle inequality by considering the two cases, namely when the
minimum weight matching is equal to either the red or blue matching.
Case red: $MM(P,R)$ is given by the red matching. The cost of the relabel
$\gamma(p\longrightarrow r)=c_{pr}+c_{\lambda\lambda}=c_{pr}$. Two different
paths from $p$ lead to $r$, $p\longrightarrow q\longrightarrow r$ and
$p\longrightarrow\lambda\longrightarrow r$. Consider the first path,
$\displaystyle\gamma(p\longrightarrow r)=c_{pr}=L_{\infty}(p,r)$
$\displaystyle\leq L_{\infty}(p,q)+L_{\infty}(q,r)$ (31)
$\displaystyle=c_{pq}+c_{qr}$ (32) $\displaystyle=\gamma(p\longrightarrow
q)+\gamma(q\longrightarrow r)$ (33)
Now, let us consider the second path. If $c_{pr}\leq c_{p\lambda}+c_{\lambda
r}$ then $c_{pr}\leq c_{p\lambda}+c_{\lambda q}+c_{q\lambda}+c_{\lambda r}$
and we are done. Else, we have two sub-cases
$\displaystyle c_{pr}$ $\displaystyle>c_{p\lambda}+c_{\lambda
q}+c_{q\lambda}+c_{\lambda r}>c_{p\lambda}+c_{\lambda r},\mbox{ or}$ (34)
$\displaystyle c_{pr}$ $\displaystyle>c_{p\lambda}+c_{\lambda r}\mbox{ but
}c_{pr}<c_{p\lambda}+c_{\lambda q}+c_{q\lambda}+c_{\lambda r}.$ (35)
In both sub-cases, we have a matching with weight
$MM^{\prime}(P,R)=c_{p\lambda}+c_{\lambda r}\leq
MM(P,R)=c_{pr}+c_{\lambda\lambda}$, which contradicts our assumption.
The cost $\gamma(\lambda\longrightarrow\lambda)=c_{\lambda\lambda}=0$. Both
paths via $Q$, $\lambda\longrightarrow\lambda\longrightarrow\lambda$ and
$\lambda\longrightarrow q\longrightarrow\lambda$, should necessarily have a
non-zero total cost. So, the inequality holds trivially.
Case blue: $MM(P,R)$ is given by the blue matching. The cost of the relabel is
equal to the sum $c_{p\lambda}+c_{\lambda r}$. We consider the two weights
individually.
Two paths from $p\in P$ lead to $\lambda\in R$ via a node in $Q$,
$p\longrightarrow\lambda\longrightarrow\lambda$ and $p\longrightarrow
q\longrightarrow\lambda$. Similarly, two paths from $\lambda\in P$ lead to
$r\in R$, $\lambda\longrightarrow\lambda\longrightarrow r$ and
$\lambda\longrightarrow q\longrightarrow r$.
In both cases, triangle inequality holds trivially for the first path via $Q$,
$c_{p\lambda}\leq c_{p\lambda}+c_{\lambda\lambda}$ and $c_{\lambda r}\leq
c_{\lambda\lambda}+c_{\lambda r}$. We need to show that the inequality holds
for the second paths as well. From the persistence diagram, we observe that
$L_{\infty}(p,\lambda)\leq L_{\infty}(p,q)+L_{\infty}(q,\lambda)$ for all $q$,
even when $q$ lies on the perpendicular from $p$ onto the diagonal. So,
$c_{p\lambda}\leq c_{pq}+c_{q\lambda}$. A similar argument can be used to show
that $c_{\lambda r}\leq c_{\lambda q}+c_{qr}$.
The red and blue cases together imply that the cost $C_{W}$ satisfies the
triangle inequality and is therefore a metric. It follows that the tree edit
distance measure $D$ is also a metric.
### 4.4 Handling instabilities
Saikia et al. [7] discuss two kinds of instabilities, _vertical_ and
_horizontal_ , that affect branch decompositions and hence the distance
measures. Figure 10 illustrates how horizontal instability can occur. In our
case, the horizontal stability has a more drastic effect on the measure
because
Figure 10: Illustrating instabilities. Since the difference in the function
values between $s_{1}$ and $s_{2}$ is small, a slight perturbation leads to a
change in the structure of the tree, which affects the distance measure.
* •
It changes the persistence pairing, which in turn affects the cost.
* •
It also changes the subtrees thereby affecting the matching found by the
algorithm.
We employ a strategy similar to the one used for branch decompositions by
Thomas and Natarajan [4] and apply it to merge trees. We introduce a stability
parameter $\varepsilon$ and use it to determine how to merge simple saddles
into a multi-saddle where instabilities occur. We merge the saddles in a
bottom-up manner as follows. Begin from the lower saddle $s_{l}$ that is
further from the root and merge it into a higher saddle $s_{h}$ that is nearer
to the root if the function difference $|f(s_{h})-f(s_{l})|<\varepsilon$.
Repeat this process until none of the saddles satisfy the merging condition.
In the implementation, the multi-saddle is represented by the saddle with the
highest persistence. We compute the distance between the stabilized trees.
Optionally, a fixed value may be added to the final distance to incorporate
the cost incurred due to the stabilization. In Section 5.2, we experimentally
analyze how varying the stability parameter $\varepsilon$ affects the distance
measure.
### 4.5 Algorithm
We adapt Zhang’s algorithm [33] (See supplementary material, Section 2) with
the edit costs discussed in Section 4.2 to compute the tree edit distance
between merge trees. The input to this algorithm is a pair of merge trees that
are stabilized using the strategy described in Section 4.4.
### 4.6 Implementation
The computation proceeds in a bottom up manner. Distances for the subtrees are
computed and stored in a table. These are next used for computing distances
between subtrees at higher levels of the merge trees. This proof of concept
implementation does not include code and memory optimizations for efficiently
computing and storing the dynamic programming tables. We use the simple Kuhn-
Munkres algorithm [45] for computing $MM(i,j)$. We still observe reasonable
running times for most of the data sets as reported in the individual
experiments in the following section.
## 5 Experiments and case studies
We demonstrate the utility of the tree edit distance measure by applying it to
analyze time-varying data, to study symmetry in scalar fields, for summarizing
data, and for shape matching. We use the Recon library [46] to compute merge
trees, the algorithm described in Section 4.5 to compute the tree edit
distance between the merge trees, and Paraview [47] together with the Topology
ToolKit TTK [48] to generate renderings of the merge trees together with the
scalar fields. We uniformly use the $L_{\infty}$ cost $C_{W}$ (4.2.1). All
experiments were performed on a machine with an Intel Xeon CPU with $8$ cores
running at $2.0$ GHz and $16$ GB main memory.
### 5.1 Understanding the distance measure
(a) Three scalar field $f_{1},f_{2},f_{3}$
(b) Merge tree driven segmentation for each field.
(c) Mapping determined by $W_{1}$ for ($f_{1},f_{2}$) and ($f_{2},f_{3}$)
(d) Mapping determined by $D$ for ($f_{1},f_{2}$) and ($f_{2},f_{3}$)
Figure 11: Comparing mappings established by tree edit distance measure $D$
and Wasserstein distance $W_{1}$. 11(a) Three scalar functions
$f_{1},f_{2},f_{3}$ in the synthetic data set. 11(b) Regions corresponding to
the maxima and arcs incident on them in the merge trees of
$f_{1},f_{2},f_{3}$. Each region is assigned a unique color. 11(c) Mapping
determined by $W_{1}$ between ($f_{1},f_{2}$) and between ($f_{2},f_{3}$).
11(d) Mapping determined by the tree edit distance $D$ between ($f_{1},f_{2}$)
and between ($f_{2},f_{3}$). Merge tree nodes and their corresponding spatial
regions have the same color.
We construct three synthetic datasets to understand the difference between the
tree edit distance $D$ and other well known distances between topological
structures. The scalar functions $f_{1},f_{2},f_{3}$ are sums of gaussians
whose extrema are fixed in space. The scalar values change in a controlled
manner for the three functions so that the values at the extrema increase /
decrease monotonically as we step from $f_{1}$ to $f_{2}$ to $f_{3}$. We
compute the tree edit distances together with the corresponding mapping for
each pair $(f_{1},f_{2})$ and $(f_{2},f_{3})$.
Figure 11 shows the three scalar functions $f_{1},f_{2},f_{3}$. We observe in
Figure 11(d) that $D$ establishes intuitively correct mappings. The mappings
also preserve the tree hierarchy. On the other hand, the Wasserstein distance
$W_{1}$ (Figure 11(c), left) maps the brown regions to the null character. The
tree edit distance prefers the relabel over a sequence of delete-insert
operations. The reason $W_{1}$ does not find the correspondence between the
two brown nodes is because their birth-death intervals do not overlap. As a
result, these nodes are mapped to the null character _i.e._ , inserted or
deleted. The intervals corresponding to the two nodes in question are
$[1.17,1.39]$ in $f_{1}$ and $[1.06,1.08]$ in $f_{2}$.
We also see from Figure 11(c) that $W_{1}$ maps the brown region to the
magenta region, thereby mapping nodes that lie within different subtrees. This
also causes a pair of nodes being mapped to the null character. The tree edit
distance $D$ is constrained to map disjoint subtrees to disjoint subtrees and
establishes a better mapping. To summarize, $D$ in general establishes
mappings that are better than $D_{B}$ and $W_{1}$ because it is aware of the
structure of the merge tree and preserves the hierarchy captured in the tree.
### 5.2 Comparison with other distance measures
$0$$50$$100$$150$$200$$0$$0.5$$1$TED
$D$$W_{1}$$D_{B}$$0$$50$$100$$150$$200$$0$$0.5$$1$$\varepsilon$ =
0%$\varepsilon$ = 1.5%$\varepsilon$ =
5%$0$$50$$100$$150$$200$$0$$0.5$$1$$\varepsilon$ = 5%$\varepsilon$ =
10%$\varepsilon$ = 20%$0$$50$$100$$150$$200$$0$$0.5$$1$Timesteps$\varepsilon$
= 20%$\varepsilon$ = 50%$\varepsilon$ = 100%$W_{1}$ Figure 12: Comparing
distance measures on the von Kármán vortex street dataset. (top) Plot of
distance measures between the first time step and others when stability
parameter $\varepsilon$ is set to $0$ and comparison with the Wasserstein
distance $W_{1}$ and bottleneck distance $D_{B}$. (rows 2-4) Effect of
stabilization parameter $\varepsilon=0,1.5,5,10,20,50,100\%$ and comparison
with Wasserstein distance $W_{1}$. Results are shown in three plots to reduce
clutter.
We compare the proposed tree edit distance measure $D$ with existing measures
such as bottleneck distance $D_{B}$ and Wasserstein distance $W_{1}$ via
computational experiments on the 2D Bénard-von Kármán vortex street dataset
[49]. Figure 13(a) shows a few time steps of the data, which represents flow
around a cylinder. The dataset contains the velocity magnitude on a $400\times
50$ grid over $1001$ time steps. Each split tree contains approximately
$55-65$ nodes. We calculate $D$ and plot it together with the bottleneck and
Wasserstein distance, see top row of Figure 12. The tree edit distance $D$ is
always greater than $W_{1}$ and $D_{B}$. Indeed, $D$ is likely to be more
discriminative than $W_{1}$ and $D_{B}$ because it incorporates the structure
of the merge tree in addition to the persistence pairs.
We also compute and plot $D$ for increasing values of the stability parameter
$\varepsilon$. The values of $\varepsilon$ are reported as a percentage of the
maximum persistence of the particular dataset. While there are some anomalies
for small values of $\varepsilon$, in general we observe in Figure 12 that
with increase in $\varepsilon$, $D$ tends towards $W_{1}$. For a high enough
value of $\varepsilon$, $D$ becomes almost equal to $W_{1}$. The reason for
this behavior is that the bottleneck/Wasserstein distance does not consider
the structure of the trees. Increasing the stability parameter transforms the
tree to become more like a bush. Finally, all the nodes become children of the
root thereby simplifying and eliminating the tree structure. Varying
$\varepsilon$ from $0-5\%$ results in a decrease of up to $25$ nodes in the
split tree. Further increasing $\varepsilon$ led to an additional reduction by
only $1-2$ nodes. We observe this trend in the distance plots also.
### 5.3 Periodicity in time-varying data
(a) Three time steps from the flow around a cylinder simulation.
(b) Distance matrix highlights the periodicity.
Figure 13: 13(a) Time step 0 (top), 37 (middle) and 74 (bottom) of the von
Kármán vortex street dataset. The split tree and critical points are overlaid.
13(b) A truncated version of the DM showing the tree edit distance measure
between all pairs of time steps. Blue bands indicate periodicity with time
period 74-75. A half period of 37, corresponding to the alternating nature of
vortex shedding, is also visible.
Earlier studies of the Bénard-von Kármán vortex street dataset have
successfully identified periodicity in the dataset. Narayanan et al. [22]
detect both a half period of 38 and the full period of 75. We also aim to
identify periodicity. Towards this, we compare the split tree of time step $1$
with the remaining $1000$ time steps of the dataset. We plot the tree edit
distance for time steps $1-220$, see top plot of Figure 12. We rerun the
experiment and compare all $1000$ time steps with all other time steps. The
distances are stored in a distance matrix (DM). Each split tree contains
approximately $55-65$ nodes. The distances were computed in parallel using 12
threads and took approximately $25$ minutes. A truncated version is shown in
Figure 13(b) for clarity. From Figure 13(b), we can also observe a periodicity
of 37, which matches with the results reported by Narayanan et al. [22]. The
tree edit distance was computed in this experiment without stabilization.
### 5.4 Topological effects of subsampling and smoothing
(a) $f_{1}$
(b) subsampled $f_{1}$
(c) smoothened $f_{1}$
(d) $f_{2}$
(e) subsampled $f_{2}$
(f) smoothened $f_{2}$
(g) DM for $f_{1}$, original and subsampled
(h) DM for $f_{2}$, original and subsampled
(i) DM for $f_{2}$, $\varepsilon=0.5\%$
(j) DM for $f_{1}$, original and smoothened
(k) DM for $f_{2}$, original and smoothened
(l) DM for $f_{2}$, $\varepsilon=0.5\%$
Figure 14: Measuring the effect of subsampling and smoothing. 14(a),14(d) Two
synthetic functions sampled over a $300\times 300$ grid. 14(b),14(e)
Subsampled down to $30\times 30$ over $9$ iterations. 14(c),14(f) Smoothed in
$9$ iterations. 14(g)-14(i) DMs showing distance between all pairs of
subsampled datasets without and with stabilization. 14(j)-14(l) DMs showing
distances for all pairs of smoothed datasets. Row and column indices
correspond to the iteration number, $0$ corresponds to the original, $9$
corresponds to the lowest resolution/extreme smoothing. Red indicates high and
blue indicates low values. Colormaps for $f_{1}$ and $f_{2}$ are not on the
same scale.
The size of datasets are ever increasing and this mandates the use of
subsampling and/or smoothing of the data as a preprocessing step. The aim of
this preprocessing is to reduce the data size while ensuring a limited effect
on geometric accuracy. However, the effect on the topological features of the
scalar field is often not quantified. We want to observe how the tree edit
distance measure captures these topological effects.
We consider two synthetically generated datasets of size $300\times 300$
(iteration $0$), see Figures 14(a), 14(d). The data is downsampled over $9$
iterations to a $30\times 30$ grid by reducing the number of samples in each
dimension by $30$ within each iteration. We also apply $9$ iterations of
laplacian smoothing on both $300\times 300$ datasets. Next, we compare all
merge trees corresponding to the subsampled and smoothed datasets pairwise.
The distance matrix (DM) for the function $f_{1}$ indicates that the distances
are monotonic, which conforms to the expected behavior. But we see a different
pattern in the case of function $f_{2}$. A small stabilization applied on
$f_{2}$ with $\varepsilon=0.5\%$ results in distance matrices that conform to
the expected behavior. This indicates that the stabilization may indeed be
required, particularly when the scalar functions contain flat regions and
multi-saddles. In both datasets, we notice that the distances between the
lowest resolution ($30\times 30$) dataset and others is relatively high. We
identified two reasons for the high values. First, the number of critical
points reduces significantly between iterations $8$ and $9$. For example, in
the case of $f_{2}$, it goes down from $66-70$ in earlier iterations to $58$
in iteration $9$. Second, the function value at the critical points in the
lowest resolution dataset are also different. Hence, the relabel costs
increase significantly, up to a factor of 1.5 in some cases.
### 5.5 Detecting symmetry / asymmetry
Identifying symmetric or repeating patterns in scalar fields enables feature-
directed visualization. For example, it supports applications such as
symmetry-aware transfer function design for volume rendering, anomaly
detection, and query-driven exploration. A distance measure is central to any
method for identifying symmetry. Consider the synthetic dataset in Figure 15
that contains six regions corresponding to six subtrees of the merge tree.
Four regions colored green in Figure 15(b) are symmetric copies. The remaining
two regions, colored orange and magenta, are slightly perturbed to cause
asymmetry. We compute the tree edit distance measure to compare each subtree
corresponding to a region with other subtrees. The measure clearly
distinguishes between symmetric and asymmetric regions as can be seen from the
distance matrix (DM) in Figure 15(c). These results are consistent with the
premise upon which the data is generated.
We present additional case studies that demonstrate the applicability of the
tree edit distance measure to symmetry identification.
EMDB111https://www.ebi.ac.uk/pdbe/emdb/ contains 3D electron microscopy
density data of macromolecules, subcellular structures, and viruses. Some of
these structures contain symmetric subunits. We study two structures, EMDB
1654, and 1897, see Figure 16. First, we compute the split tree for each
structure. We then use a semi-automated method to extract sub-trees
corresponding to significant features from the merge tree based on user
specified persistence and minimum scalar value thresholds. Next, we compute
the tree edit distance measure between sub-trees corresponding to these
regions of interest. We observe two distinct groups from the DMs. The tree
edit distance measure clearly identifies two groups with $4$ and $8$ regions
each in EMDB 1654, see Figure 16(a). Similarly, it identifies two groups
containing $3$ and $6$ symmetric regions each in EMDB 1897, see Figure 16(b).
(a) Synthetic field
(b) Segmentation
(c) Distance matrix
Figure 15: Identifying symmetry and asymmetry. 15(a) Sum of 2D gaussians.
15(b) The DM indicates presence of a symmetric group containing 4 regions. Two
regions are correctly identified as being different from the rest. 15(c) DM
between various subtrees of the merge tree.
(a) EMDB 1654
(b) EMDB 1897
Figure 16: Detecting groups of symmetric regions in EMDB datasets. (centre) DM
showing tree edit distance between various pairs of subtrees of the merge
tree. Low values are mapped to blue and high values to red. The DM indicates
the presence of two distinct groups. All regions within a group are symmetric
copies of each other. (left, right) Volume rendering where one region from
each symmetric group is highlighted.
### 5.6 Shape matching
Shape matching involves comparing geometric shapes and finding similarity
between them. A good distance measure helps quantify this notion of similarity
more concretely. The TOSCA non-rigid world
dataset222http://tosca.cs.technion.ac.il/book/resources_data.html contains a
set of different shapes, see Figure 17. The shapes are in different poses and
the project aims to develop methods to identify similarity between shapes in a
pose invariant manner. We compute the average geodesic distance field [3] on
the surface mesh. This field is well studied in the literature and is known to
be a good shape descriptor. We apply a persistence simplification threshold of
1% on the merge trees both to remove topological noise and to reduce the
number of nodes. Next, we compute the tree edit distance measure between all
pairs of shapes. It takes around $15$ seconds to generate the distance matrix
with the same setup used for the periodicity experiment. Figure 18 shows the
distance matrix. Each collection of shape appears as a blue block irrespective
of variations in pose. We also observe higher values for a pair of shapes that
are different. Note the blue blocks away from the diagonal. They correspond to
Michael vs Victoria, David vs Victoria, David vs Michael, and David vs
Gorilla. These pairs have similar shapes, which is more apparent in a few
poses. Not all poses are shown in Figure 17.
Figure 17: Collection of shapes from the TOSCA non-rigid world dataset. The
average geodesic distance field [3] is computed on the surface. Each shape is
available in multiple poses (number of poses mentioned within parenthesis),
only one pose is shown here. Figure 18: Tree edit distance matrix for all
pairs of shapes from the TOSCA non-rigid world dataset. Blocks of low values
(blue) correspond to similar shapes but in different poses.
### 5.7 Data Summarization
Exploring large scientific data, particularly time-varying data, and
identifying patterns of interest is often time consuming even with good
visualization tools. Well designed abstract representations provide good
overviews of the data and direct the user to features of interest.
Abstractions such as the merge trees present a summary of spatial features.
Temporal summaries enable effective visualization of time-varying data.
Central to the design of a temporal summary is a good distance measure that
can distinguish between periods of significant activity and inactive time
periods.
In this experiment, we consider the 3D Bénard-von Kármán vortex street
dataset. The velocity magnitude is available as a scalar field on a $192\times
64\times 48$ grid over $102$ time steps [50]. Figure 19 shows volume
renderings and isosurfaces for a few time steps. Topological features of the
velocity magnitude scalar field are represented using the split tree. Each
split tree had approximately $180-200$ nodes. We compute the tree edit
distance between all pairs of time steps. It takes around $4$ seconds to
generate the distance matrix (DM) with the same setup used for the periodicity
experiment.
Figure 19: The 3D Bénard-von Kármán vortex street dataset. (top) Volume
rendering of the velocity magnitude field for time steps $15,35,58,91,98$
ordered left to right. (bottom) Isosurfaces at isovalue $0.7$ extracted for
the above time steps. Figure 20: Tree edit distance matrix for all time steps
of the 3D Bénard-von Kármán vortex street dataset. Columns corresponding to
time steps $15,35,58,91,98$ are highlighted. Patterns that help in generating
a temporal summary are highlighted using black and green boxes.
The DM shown in Figure 20 contains multiple patterns. A fluid dynamics expert
helped study and interpret the results. The distance between time steps $2-28$
are small because the flow does not contain any vortices and the features do
not change. The top left blue block in the matrix corresponds to this time
period. This is followed by the period when new vortex structures are formed
(small block highlighted in green that corresponds to time steps $29-39$).
Next, the vortices exhibit shedding, which is shown by the repeating patterns
present in the larger green block in the matrix (time steps $40-85$). Finally,
the vortices are significantly distorted, which is captured by the high values
of distance in the bottom right block. Thus we can use the patterns that
emerge in the distance matrix to distinguish between different types of
behavior and summarize the scientific phenomena using these patterns.
## 6 Conclusions
We described a distance measure between two scalar fields that compares their
merge trees. The distance measure is defined as the minimum cost of a set of
restricted edit operations that transforms one tree into another. The edit
operations and the associated costs are both intuitive and mathematically
sound. The measure satisfies metric properties, can be efficiently computed,
and is useful in practice. We study the properties of the measure and
demonstrate its application to data analysis and visualization using various
computational experiments. In future work, we plan to develop a theoretical
analysis of the stability properties of the measure. Developing a comparative
visualization framework based on the tree edit distance measure is also an
interesting problem with potential applications to time-varying data and
multifield data visualization.
## Acknowledgments
This work is supported by the Department of Science and Technology, India
(DST/SJF/ETA-02/2015-16), and Joint Advanced Technology Programme, Indian
Institute of Science (JATP/RG/PROJ/2015/16), and the Robert Bosch Centre for
Cyber Physical Systems, Indian Institute of Science. We thank Shrisha Rao for
discussions on the data summarization experiment.
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| Raghavendra Sridharamurthy is a PhD candidate in computer science at Indian
Institute of Science, Bangalore. He received BE degree in information
technology from National Institute of Technology Karnataka, Surathkal and M.Sc
degree in computer science from Indian Institute of Science. His research
interests include scientific visualization, computational topology and its
applications.
---|---
| Talha Bin Masood is a PhD candidate in computer science at Indian Institute
of Science, Bangalore. He received B.Tech degree from Aligarh Muslim
University and ME degree in computer science from Indian Institute of Science.
His research interests include scientific visualization, computational
geometry, computational topology and its applications to various scientific
domains.
---|---
| Adhitya Kamakshidasan is a Junior Research Fellow at Visualization and
Graphics Lab, Indian Institute of Science. He holds a B.Tech degree in
computer science from Visvesvaraya National Institute of Technology, Nagpur.
His research interests include fluid simulation, information visualization and
cartography.
---|---
| Vijay Natarajan is an associate professor in the Department of Computer
Science and Automation at the Indian Institute of Science, Bangalore. He
received the Ph.D. degree in computer science from Duke University in 2004.
His research interests include scientific visualization, computational
topology, and geometry processing.
---|---
|
Higher order hesitant fuzzy Choquet integral operator and its application to
multiple criteria decision making
B. Farhadinia111Corresponding author. U. Aickelin† H.A. Khorshidi†
Dept. Math., Quchan University of Technology, Iran.
<EMAIL_ADDRESS>
† Dept. Computing and Information Systems, University of Melbourne, Australia.
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
Abstract. Generally, the criteria involved in a decision making problem are
interactive or inter-dependent, and therefore aggregating them by the use of
traditional operators which are based on additive measures is not logical.
This verifies that we have to implement fuzzy measures for modelling the
interaction phenomena among the criteria. On the other hand, based on the
recent extension of hesitant fuzzy set, called higher order hesitant fuzzy set
(HOHFS) which allows the membership of a given element to be defined in forms
of several possible generalized types of fuzzy set, we encourage to propose
the higher order hesitant fuzzy (HOHF) Choquet integral operator. This concept
not only considers the importance of the higher order hesitant fuzzy
arguments, but also it can reflect the correlations among those arguments.
Then, a detailed discussion on the aggregation properties of the HOHF Choquet
integral operator will be presented. To enhance the application of HOHF
Choquet integral operator in decision making, we first assess the appropriate
energy policy for the socio-economic development. Then, the efficiency of the
proposed HOHF Choquet integral operator-based technique over a number of
exiting techniques is further verified by employing another decision making
problem associated with the technique of TODIM (an acronym in Portuguese of
Interactive and Multicriteria Decision Making).
Keywords: Higher order hesitant fuzzy set (HOHFS), Choquet integral, Multiple
criteria decision making.
## 1 Introduction
The theory of extending and generalizing fuzzy sets is very prospective tool
in different research domains. One of interesting extensions is the concept of
hesitant fuzzy set (HFS) which was proposed by Torra and Narukawa [28]. HFS is
quite suit for the case where the decision maker has a set of possible values,
rather than a margin of error (like that in intuitionistic fuzzy sets [2]) or
some possibility distribution on the possible values (like that is considered
in defining type-2 fuzzy sets [19]). By the way, HFS [28] and its extension
which is called the generalized HFS (G-HFS) [22], have their own drawbacks
because of the reason that the membership degrees of an element to a given set
can be expressed only by crisp numbers or intuitionistic fuzzy sets. The
evidence to this fact is that in a real and practical decision making problem
where the information provided by a group of experts may not be described only
by hesitant fuzzy sets (HFSs) or just by one kind of HFS extensions. A more
specific situation occurs when a number of experts are intended to return
their evaluations in the form of hesitant multiplicative sets, and others are
interested to describe their evaluation by the help of hesitant fuzzy
linguistic term sets. Therefore, it is difficult for the decision maker to
provide exact hesitant multiplicative set or hesitant fuzzy linguistic term
set for the membership degrees. Such a difficulty is avoided by considering
the higher order HFS (HOHFS) [10] in order to describe the membership degrees.
The concept of HOHFS not only encompasses fuzzy sets, intuitionistic fuzzy
sets, type 2 fuzzy sets and HFSs, but also it extends the concept of
generalized hesitant fuzzy set (G-HFS) [22].
It does worth to say that the topic of developing information aggregations in
fuzzy multiple criteria decision making (MCDM) has been an interesting
research topic [8, 9, 11, 24, 30] so far. As it is known, most of the existing
fuzzy aggregation operators only consider those situations in which all the
elements are independent. Moreover, in many real-world situations, we observe
that the elements are usually interactive or interdependent, that is, they are
correlative.
The Choquet integral [7] with respect to fuzzy measures [25] is an aggregation
function that has been employed successfully in MCDM problems where it
determines the relative importance of decision criteria as well as their
interactions [17]. There exist a growing number of studies on the Choquet
integral, and it has been investigated by many researchers. Yager [41]
proposed the induced Choquet ordered averaging operator, and moreover, he
introduced the concept of Choquet aggregation based on the order induced
aggregation. By extending the concept of Choquet integral to that for fuzzy
context, Meyer and Roubens [18] applied this concept to multiple criteria
decision making problems. Yu et al. [35] defined a hesitant fuzzy aggregation
operator by taking the concept of Choquet integral into account, and they
implemented it in modelling a hesitant-based multiple criteria decision making
problem. Tan and Chen [27] and Tan [26] proposed respectively the
intuitionistic and the interval-valued intuitionistic fuzzy Choquet integral
operators. Bustince et al. [3] investigated a multiple criteria decision
making technique by considering the concept of interval-valued Choquet
integral. Wang et al. [31] investigated multiple criteria group decision
making problems in which the Choquet integral aggregation operators together
with interval two-tuple linguistic information play the main role. Some more
studies have been given by Abdullah et al. [1], Candeloro et al. [4], Pasi et
al. [20], etc.
By the way, inspired by the concept of Choquet integral, we will develop here
a higher order hesitant fuzzy Choquet integral operator for aggregating higher
order hesitant fuzzy (HOHF) information in MCDM, and investigate different
properties of this operator. As will be shown in Section 3, such a proposed
concept not only is well applicable to the case where the decision makers have
a hesitation among several possible memberships with uncertainties in a
general form (i.e., HOHFSs), but also it models the interaction phenomena
among the criteria (i.e., the Choquet integral property).
The present paper is organized as follows: The concept of higher order HFS
(HOHFS) as an extension of HFSs is reviewed in Section 2, and then the higher
order hesitant fuzzy (HOHF) Choquet integral operator is presented together
with a detailed discussion on the aggregation properties of the HOHF Choquet
integral operator. In Section 3, we apply the proposed HOHF Choquet integral
operator to MCDM problems involving the higher order hesitant fuzzy
information, and then a comparative analysis is then given to demonstrate the
effectiveness of the proposed operator. Finally, conclusion is drown in
Section 4.
## 2 The HOHF Choquet integral operator
Among the several extensions of the theory of HFSs, we deal in brief with the
higher order hesitant fuzzy set (HOHFS) which was first introduced by
Farhadinia [10].
###### Definition 2.1.
[10] Let $X$ be the universe of discourse. A generalized type of fuzzy set
(G-Type FS) on $X$ is defined as
$\displaystyle\widetilde{A}=\\{\langle x,\widetilde{A}(x)\rangle:x\in X\\},$
(1)
where
$\displaystyle\widetilde{A}:X\rightarrow\psi([0,1]).$
Here, $\psi([0,1])$ denotes a family of crisp or fuzzy sets that can be
defined with in the universal set $[0,1]$.
We mention here some special cases of G-Type FS which are encountered by Klir
and Yuan in [15]:
* •
if $\psi([0,1])=[0,1]$, then the G-Type FS $\widetilde{A}$ reduces to an
ordinary fuzzy set;
* •
if $\psi([0,1])=\varepsilon([0,1])$ denoting the set of all closed intervals,
then the G-Type FS $\widetilde{A}$ reduces to an interval-valued fuzzy set;
* •
if $\psi([0,1])={\cal F}([0,1])$ denoting the set of all ordinary fuzzy sets,
then the G-Type FS $\widetilde{A}$ reduces to a type II fuzzy set;
* •
if $\psi([0,1])=L$ denoting a partially ordered Lattice, then the G-Type FS
$\widetilde{A}$ reduces to a lattice fuzzy set.
###### Definition 2.2.
[28] Let $X$ be the universe of discourse. A hesitant fuzzy set (HFS) on $X$
is symbolized by
$H=\\{\langle x,h(x)\rangle:x\in X\\},$
where $h(x)$, referred to as the hesitant fuzzy element (HFE), is a set of
some values in $[0,1]$ denoting the possible membership degree of the element
$x\in X$ to the set $H$.
As can be seen from Definition 2.2, HFS expresses the membership degrees of an
element to a given set only by several real numbers between 0 and 1, while in
many real-world situations assigning exact values to the membership degrees
does not describe properly the imprecise or uncertain decision information.
Thus, it seems to be difficult for the decision makers to rely on HFSs for
expressing uncertainty of an element.
To overcome the difficulty associated with expressing uncertainty of an
element to a given set, the concept of HOHFS allows the membership degrees of
an element to be expressed by several possible G-Type FSs.
###### Definition 2.3.
[10] Let $X$ be the universe of discourse. A higher order hesitant fuzzy set
(HOHFS) on $X$ is defined in terms of a function when it is applied to $X$, it
then returns a set of G-Type FSs. A HOHFS is denoted by
$\displaystyle\widetilde{H}=\\{\langle x,\widetilde{h}(x)\rangle:x\in X\\},$
(2)
where $\widetilde{h}(x)$, referred to as the higher order hesitant fuzzy
element (HOHFE), is a set of some G-Type FSs denoting the possible membership
degree of the element $x\in X$ to the set $\widetilde{H}$. In this regards,
the HOHFS $\widetilde{H}$ is also represented as
$\displaystyle\widetilde{H}=\\{\langle
x,\\{\widetilde{h_{\widetilde{h}}}^{(1)}(x),...,\widetilde{h_{\widetilde{h}}}^{(|\widetilde{h}(x)|)}(x)\\}\rangle:x\in
X\\},$
where all
$\widetilde{h_{\widetilde{h}}}^{(1)}(x),...,\widetilde{h_{\widetilde{h}}}^{(|\widetilde{h}(x)|)}(x)$
are G-Type FSs on $X$, and $|\widetilde{h}(x)|$ is the number of G-type FSs in
$\widetilde{h}(x)$.
The noteworthy feature about Definition 2.3 is that it encompasses all the
above-mentioned generalizations of ordinary fuzzy sets which have been
discussed in detail by Farhadinia [10].
Having introduced HOHFEs of a HOHFS $\widetilde{H}=\\{\langle
x,\widetilde{{h}}(x)\rangle:x\in X\\}$, we now turn our attention to the
interpretation of HOHFS $\widetilde{H}$ as the union of all HOHFEs i.e.,
$\displaystyle\widetilde{H}=\bigcup_{\widetilde{{h}}(x)\in\widetilde{H}}\\{\widetilde{{h}}(x)\\}$
(3)
which is fundamental in the study of HOHFS aggregation operators within the
next parts of the paper.
Hereafter, to simplify the notation, we use only $\widetilde{{h}}$ instead of
$\widetilde{{h}}(x)$ in the following descriptions, that is, we assume that
$\displaystyle\widetilde{H}=\bigcup_{\widetilde{{h}}\in\widetilde{H}}\\{\widetilde{{h}}\\}.$
###### Definition 2.4.
Let
$\widetilde{H}_{1}=\bigcup_{\widetilde{{h}}_{1}\in\widetilde{H}_{1}}\\{\widetilde{{h}_{1}}\\}$
and
$\widetilde{H}_{2}=\bigcup_{\widetilde{{h}}_{2}\in\widetilde{H}_{2}}\\{\widetilde{{h}_{2}}\\}$
be two HOHFSs. We give the definition of operation $\odot$ on HOHFSs based on
its counterpart on HOHFEs as follows:
$\displaystyle\widetilde{H}_{1}\odot\widetilde{H}_{2}=\bigcup_{\widetilde{h_{1}}\in\widetilde{H_{1}},\widetilde{h_{2}}\in\widetilde{H_{2}}}\\{\widetilde{{h_{1}}}\odot\widetilde{{h_{2}}}\\}=\bigcup_{\widetilde{h_{1}}\in\widetilde{H_{1}},\widetilde{h_{2}}\in\widetilde{H_{2}}}\\{\bigcup_{\widetilde{h_{h1}}\in\widetilde{h_{1}},\widetilde{h_{h2}}\in\widetilde{h_{2}}}\\{\widetilde{{h_{h1}}}\odot\widetilde{{h_{h2}}}\\}~{}\\},$
(4)
where the last operation $\odot$ stands for the operation on G-type FSs.
Note that, if there is no confusion, all the operations performing over
HOHFSs, HOHFEs and G-type FSs are denoted by the same symbol. Furthermore,
that property holds for G-type FSs is denoted by $\widetilde{(p_{G})}$, and
that is defined on HOHFEs is symbolized by $\widetilde{(p)}$.
###### Theorem 2.5.
Let
$\widetilde{h}_{1}=\bigcup_{\widetilde{{h}}_{h1}\in\widetilde{h}_{1}}\\{\widetilde{{h}_{h1}}\\}$
and
$\widetilde{h}_{2}=\bigcup_{\widetilde{{h}}_{h2}\in\widetilde{h}_{2}}\\{\widetilde{{h}_{h2}}\\}$
be two HOHFEs. If the operation $\odot$ on G-type FSs has the property
$\widetilde{(p_{G})}$, then the operation $\odot$ on HOHFEs has the
counterpart property, denoted by $\widetilde{(p)}$.
Proof. The proof is obvious from Definition 2.4 Suppose that, for example, the
operation $\odot$ on G-type FSs is commutative, that is, for any G-type FSs
$\widetilde{h_{h1}}\in\widetilde{h_{1}}$ and
$\widetilde{h_{h2}}\in\widetilde{h_{2}}$, we have
$\displaystyle\widetilde{{h}_{h1}}\odot\widetilde{{h}_{h2}}=\widetilde{{h}_{h2}}\odot\widetilde{{h}_{h1}}.\quad\quad(\textrm{Property}~{}\widetilde{(p_{G})})$
Then, one can see from Definition 2.4 that
$\displaystyle\widetilde{h}_{1}\odot\widetilde{h}_{2}=\bigcup_{\widetilde{h_{h1}}\in\widetilde{h_{1}},\widetilde{h_{h2}}\in\widetilde{h_{2}}}\\{\widetilde{{h_{h1}}}\odot\widetilde{{h_{h2}}}\\}=\bigcup_{\widetilde{h_{h1}}\in\widetilde{h_{1}},\widetilde{h_{h2}}\in\widetilde{h_{2}}}\\{\widetilde{{h_{h2}}}\odot\widetilde{{h_{h1}}}\\}=\widetilde{h}_{2}\odot\widetilde{h}_{1}.\quad\quad(\textrm{Property}~{}\widetilde{(p)})$
This means that the operation $\odot$ on HOHFEs is accordingly commutative.
By similar reasoning, one can easily prove that the HOHFE operator inherits
all the properties of the G-type FS operator. $\Box$
As a corollary of Theorem 2.5 , it can be observed that HOHFS operators
inherit subsequently all operational properties of G-type FS operators.
###### Corollary 2.6.
Let
$\widetilde{H}_{1}=\bigcup_{\widetilde{{h}}_{1}\in\widetilde{H}_{1}}\\{\widetilde{{h}_{1}}\\}$
and
$\widetilde{H}_{2}=\bigcup_{\widetilde{{h}}_{2}\in\widetilde{H}_{2}}\\{\widetilde{{h}_{2}}\\}$
be two HOHFSs. If the operation $\odot$ on HOHFEs has the property
$\widetilde{(p)}$, then the operation $\odot$ on HOHFSs has the mentioned
property, denoted by $\widetilde{(P)}$.
Ranking fuzzy information play a key role in fuzzy decision-making procedure.
This fact is very encouraging for introducing a score function of HOHFSs. To
compare the HOHFSs, we define the following comparison laws:
###### Definition 2.7.
Let $X=\\{x_{1},x_{2}...,x_{n}\\}$. For HOHFS $\widetilde{H}=\\{\langle
x,\widetilde{{h}}(x)\rangle:x\in
X\\}=\bigcup_{\widetilde{h}\in\widetilde{H}}\\{\bigcup_{\widetilde{h_{h}}\in\widetilde{h}}\\{\widetilde{{h_{h}}}\\}\\}$,
the score function ${\cal S}(.)$ is defined by
$\displaystyle{\cal
S}(\widetilde{H})=\frac{1}{n}\sum_{i=1}^{n}S_{\widetilde{h}}(\widetilde{h}(x_{i}))=\frac{1}{n}\sum_{i=1}^{n}(~{}\frac{1}{|\widetilde{h}(x_{i})|}\sum_{\widetilde{h_{h}}\in\widetilde{h}(x_{i})}S_{\widetilde{{h_{h}}}}(\widetilde{{h_{h}}}(x_{i}))~{}),$
(5)
where $S_{\widetilde{h}}(.)$ is a score function of HOHFEs,
$|\widetilde{h}(x_{i})|$ is the number of G-type FSs in $\widetilde{h}(x_{i})$
and $S_{\widetilde{{h_{h}}}}(.)$ is a score function of G-type FSs. For two
HOHFSs $\widetilde{H_{1}}$ and $\widetilde{H_{2}}$, if ${\cal
S}(\widetilde{H}_{1})<{\cal S}(\widetilde{H}_{2})$ then
$\widetilde{H_{1}}\prec\widetilde{H_{2}}$; if ${\cal
S}(\widetilde{H}_{1})={\cal S}(\widetilde{H}_{2})$ then
$\widetilde{H_{1}}\approx\widetilde{H_{2}}$.
Similar comparison rules can be considered for HOHFEs and G-type FSs such that
* •
for two HOHFEs $\widetilde{h_{1}}$ and $\widetilde{h_{2}}$ if
$S_{\widetilde{h}}(\widetilde{h}_{1})<S_{\widetilde{h}}(\widetilde{h}_{2})$
then $\widetilde{h_{1}}\prec\widetilde{h_{2}}$; if
$S_{\widetilde{h}}(\widetilde{h}_{1})=S_{\widetilde{h}}(\widetilde{h}_{2})$
then $\widetilde{h_{1}}\approx\widetilde{h_{2}}$;
* •
for two G-type FSs $\widetilde{h_{h}}^{(1)}$ and $\widetilde{h_{h}}^{(2)}$ if
$S_{\widetilde{h_{h}}}(\widetilde{h}_{h}^{(1)})<S_{\widetilde{h_{h}}}(\widetilde{h}_{h}^{(2)})$
then $\widetilde{h_{h}}^{(1)}\prec\widetilde{h_{h}}^{(2)}$; if
$S_{\widetilde{h_{h}}}(\widetilde{h}_{h}^{(1)})=S_{\widetilde{h_{h}}}(\widetilde{h}_{h}^{(2)})$
then $\widetilde{h_{h}}^{(1)}\approx\widetilde{h_{h}}^{(2)}$.
In most of the real world applications, we usually face to the elements in the
universe of discourse which may have a different importance. This impulses us
to consider the weight of each element $x_{i}\in X$. Assume that the weight of
$x_{i}\in X$ is $\mu(\\{x_{i}\\})$, $(i=1,...,n)$ where $\mu$ is a fuzzy
measure.
###### Definition 2.8.
[25] A fuzzy measure on $X$ is a set function $\mu$ from the power set of $X$
into $[0,1]$, satisfying the following conditions:
$\displaystyle(1)~{}\mu(\emptyset)=0,~{}\mu(X)=1;$
$\displaystyle(2)~{}\textrm{For}~{}\textrm{any}~{}A,B\subseteq
X,~{}\textrm{if}~{}A\subseteq B,~{}\textrm{then}~{}\mu(A)\leq\mu(B);$
$\displaystyle(3)~{}\textrm{For}~{}\textrm{any}~{}A,B\subseteq X,~{}\mu(A\cup
B)=\mu(A)+\mu(B)-\rho\mu(A)\mu(B),~{}\textrm{where}~{}A\cap
B=\emptyset~{}\textrm{and}~{}\rho\in(-1,\infty).$
A fuzzy measure $\mu$ on $X$ is said to be
* •
Additive if $\mu(A\cup B)=\mu(A)+\mu(B)$ for all $A,B\subseteq X$ where $A\cap
B=\emptyset$;
* •
Subadditive if $\mu(A\cup B)\leq\mu(A)+\mu(B)$ for all $A,B\subseteq X$ where
$A\cap B=\emptyset$;
* •
Superadditive if $\mu(A\cup B)\geq\mu(A)+\mu(B)$ for all $A,B\subseteq X$
where $A\cap B=\emptyset$.
Sometimes, in MCDM, we need an aggregation function to determine the relative
importance of decision criteria considering their interactions. Such a need
could be satisfied by implementing the Choquet integral [7] which is defined
on the basis of fuzzy measures.
###### Definition 2.9.
[7] Let $f$ be a positive real-valued function on $X$, and $\mu$ be a fuzzy
measure on $X$. The Choquet integral of $f$ with respect to $\mu$ is defined
by
$\displaystyle
C_{\mu}(f)=\sum_{i=1}^{n}(\mu(A_{\sigma(i)})-\mu(A_{\sigma(i-1)})f_{\sigma(i)}$
(6)
where $(\sigma(1),\sigma(2),...,\sigma(n))$ is a permutation of $(1,2,...,n)$
such that $f_{\sigma(1)}\geq f_{\sigma(2)}\geq...\geq f_{\sigma(n)}$,
$A_{\sigma(k)}=\\{x_{\sigma(1)},x_{\sigma(2)},...,x_{\sigma(k)}\\}$ for $k\geq
1$, and $A_{\sigma(0)}=\emptyset$.
The Choquet integral $C_{\mu}(.)$ has some considerable aggregation
properties, such as idempotence, compensativeness and comonotonic additivity
[17]. Moreover, it generalizes the weighted arithmetic mean (WAM) and the
ordered weighted averaging (OWA) [40]. In view of the Choquet integral
formulation (6), we here introduce the HOHF Choquet integral:
###### Definition 2.10.
Let $\mu$ be a fuzzy measure on $X$, and $\widetilde{h}(x_{i}),~{}(i=1,...,n)$
be a collection of HOHFEs on $X$. The HOHF Choquet integral of
$\widetilde{h}(x_{i}),~{}(i=1,...,n)$ with respect to $\mu$ is defined by
$\displaystyle
HOHFEC_{\mu}(\widetilde{h}(x_{1}),...,\widetilde{h}(x_{n}))=\oplus_{i=1}^{n}[(\mu(A_{\sigma(i)})-\mu(A_{\sigma(i-1)})\widetilde{h}(x_{\sigma(i)})]\hskip
142.26378pt$ $\displaystyle\hskip
28.45274pt=\bigcup_{\widetilde{h_{h}}(x_{\sigma(1)})\in\widetilde{h}(x_{\sigma(1)}),...,\widetilde{h_{h}}(x_{\sigma(n)})\in\widetilde{h}(x_{\sigma(n)})}\\{\oplus_{i=1}^{n}[(\mu(A_{\sigma(i)})-\mu(A_{\sigma(i-1)})\widetilde{h_{h}}(x_{\sigma(i)})]\\},$
(7)
where $(\sigma(1),\sigma(2),...,\sigma(n))$ is a permutation of $(1,2,...,n)$
such that
$\widetilde{h}(x_{\sigma(1)})\succ\widetilde{h}(x_{\sigma(2)})\succ...\succ\widetilde{h}(x_{\sigma(n)})$,
$A_{\sigma(k)}=\\{x_{\sigma(1)},x_{\sigma(2)},...,x_{\sigma(k)}\\}$ for $k\geq
1$, and $A_{\sigma(0)}=\emptyset$.
###### Proposition 2.11.
If for any G-type FSs $\widetilde{h_{h}}^{(1)}$ and $\widetilde{h_{h}}^{(2)}$
of HOHFE $\widetilde{h}$, the addition
$\widetilde{h_{h}}^{(1)}\oplus\widetilde{h_{h}}^{(2)}$ and the scalar
multiplication $\lambda\widetilde{h_{h}}^{(1)}$ where $\lambda>0$ are also
G-type FSs, then the HOHF Choquet integral of
$\widetilde{h}(x_{i}),~{}(i=1,...,n)$ with respect to $\mu$ introduced in
Definition 2.10 is a HOHFE.
Proof. The proof is obvious by putting together the definition of the HOHF
Choquet integral of $\widetilde{h}(x_{i}),~{}(i=1,...,n)$ with respect to
$\mu$ introduced in Definition 2.10 and Theorem 2.5. $\Box$
###### Example 2.12.
Let $X=\\{x_{1},x_{2}\\}$, and $\widetilde{h}(x_{i}),~{}(i=1,2)$ be a
collection of HOHFEs on $X$ whose G-type FSs are in the form of IFSs given by
$\displaystyle\widetilde{h}(x_{1})=\\{\widetilde{h_{h}}^{(1)}(x_{1})=\langle
0.2,0.5\rangle,\widetilde{h_{h}}^{(2)}(x_{1})=\langle
0.3,0.4\rangle\\},\quad\quad\widetilde{h}(x_{2})=\\{\widetilde{h_{h}}^{(1)}(x_{2})=\langle
0.3,0.5\rangle\\},$
and a fuzzy measure $\mu:X\rightarrow[0,1]$ is given by
$\displaystyle\mu(\emptyset)=0,\quad\mu(\\{x_{1}\\})=0.2,\quad\mu(\\{x_{2}\\})=0.3,\quad\mu(\\{x_{1},x_{2}\\})=0.6.$
If we take the G-type FSs score function as the hesitancy degree, i.e.,
$\displaystyle
S_{\widetilde{h_{h}}}(\widetilde{h_{h}})=S_{\widetilde{h_{h}}}(\langle\mu_{\widetilde{{h_{h}}}},\nu_{\widetilde{{h_{h}}}}\rangle)=1-\mu_{\widetilde{{h_{h}}}}-\nu_{\widetilde{{h_{h}}}},$
then the HOHFE scores $S_{\widetilde{h}}(\widetilde{h}(x_{1}))=0.3$ and
$S_{\widetilde{h}}(\widetilde{h}(x_{2}))=0.1$ indicate that
$\widetilde{h}(x_{1})\succ\widetilde{h}(x_{2})$ and therefore
${\sigma(1)}=1,{\sigma(2)}=2$.
Now from (7) the HOHF Choquet integral of $\widetilde{h}(x_{i}),~{}(i=1,2)$
with respect to $\mu$ is calculated as follows:
$\displaystyle
HOHFEC_{\mu}(\widetilde{h}(x_{1}),\widetilde{h}(x_{2}))=\bigcup_{\widetilde{h_{h}}(x_{1})\in\widetilde{h}(x_{1}),\widetilde{h_{h}}(x_{2})\in\widetilde{h}(x_{2})}\\{\oplus_{i=1}^{2}[(\mu(A_{\sigma(i)})-\mu(A_{\sigma(i-1)})\widetilde{h_{h}}(x_{\sigma(i)})]\\}$
$\displaystyle=\\{[(\mu(A_{1})-\mu(A_{0})]\widetilde{h_{h}}^{(1)}(x_{1})\oplus[(\mu(A_{2})-\mu(A_{1})]\widetilde{h_{h}}^{(1)}(x_{2}),$
$\displaystyle\hskip
113.81102pt[(\mu(A_{1})-\mu(A_{0})]\widetilde{h_{h}}^{(2)}(x_{1})\oplus[(\mu(A_{2})-\mu(A_{1})]\widetilde{h_{h}}^{(1)}(x_{2})\\}$
$\displaystyle=\\{[(\mu(\\{x_{1}\\})-\mu(\emptyset)]\widetilde{h_{h}}^{(1)}(x_{1})\oplus[(\mu(\\{x_{2}\\})-\mu(\\{x_{1}\\})]\widetilde{h_{h}}^{(1)}(x_{2}),$
$\displaystyle\hskip
113.81102pt[(\mu(\\{x_{1}\\})-\mu(\emptyset)]\widetilde{h_{h}}^{(2)}(x_{1})\oplus[(\mu(\\{x_{2}\\})-\mu(\\{x_{1}\\})]\widetilde{h_{h}}^{(1)}(x_{2})\\}$
$\displaystyle=\\{[0.2]\langle 0.2,0.5\rangle\oplus[0.6-0.2]\langle
0.3,0.5\rangle,$ $\displaystyle\hskip 113.81102pt[0.2]\langle
0.3,0.4\rangle\oplus[0.6-0.2]\langle 0.3,0.5\rangle\\}.$
By the use of the following relations (see e.g. [44])
$\displaystyle\widetilde{{h_{h1}}}\oplus\widetilde{{h_{h2}}}=\langle\mu_{\widetilde{{h_{h1}}}},\nu_{\widetilde{{h_{h1}}}}\rangle\oplus\langle\mu_{\widetilde{{h_{h2}}}},\nu_{\widetilde{{h_{h2}}}}\rangle=\langle\mu_{\widetilde{{h_{h1}}}}+\mu_{\widetilde{{h_{h2}}}}-\mu_{\widetilde{{h_{h1}}}}.\mu_{\widetilde{{h_{h2}}}},~{}~{}\nu_{\widetilde{{h_{h1}}}}.\nu_{\widetilde{{h_{h2}}}}\rangle;$
$\displaystyle\widetilde{{h_{h1}}}^{\lambda}=\langle(\mu_{\widetilde{{h_{h1}}}})^{\lambda},1-(1-\nu_{\widetilde{{h_{h1}}}})^{\lambda}\rangle,\quad\lambda>0,$
in which $\mu_{\widetilde{{h_{h2}}}}$ and $\nu_{\widetilde{{h_{h2}}}}$ stand
respectively for the membership and the non-membership functions, we get
$\displaystyle\widetilde{h}_{{}_{HOHFEC}}:=HOHFEC_{\mu}(\widetilde{h}(x_{1}),\widetilde{h}(x_{2}))=\\{\langle
0.1708,0.6598\rangle,\langle 0.1927,0.6310\rangle\\}.$
###### Proposition 2.13.
(Idempotency). If for any G-type FS $\widetilde{h_{h}}$ of HOHFE
$\widetilde{h}$, it holds
$\lambda_{1}\widetilde{h_{h}}\oplus\lambda_{2}\widetilde{h_{h}}=(\lambda_{1}+\lambda_{2})\widetilde{h_{h}}$
where $\lambda_{1},\lambda_{2}>0$, and if all HOHFEs
$\widetilde{h}(x_{i}),~{}(i=1,...,n)$ are equal, that is, for all $i$,
$\widetilde{h}(x_{i})=\overline{\widetilde{h}}$, then the HOHF Choquet
integral of $\widetilde{h}(x_{i}),~{}(i=1,...,n)$ with respect to $\mu$
introduced in Definition 2.10 is equal to the HOHFE
$\overline{\widetilde{h}}$.
Proof. Suppose that for all $i,~{}(i=1,...,n)$, we have
$\widetilde{h}(x_{i})=\overline{\widetilde{h}}$. From definition of the HOHF
Choquet integral of $\widetilde{h}(x_{i}),~{}(i=1,...,n)$ with respect to
$\mu$ introduced in Definition 2.10, we have
$\displaystyle HOHFEC_{\mu}(\widetilde{h}(x_{1}),...,\widetilde{h}(x_{n}))$
$\displaystyle=$
$\displaystyle\oplus_{i=1}^{n}[(\mu(A_{\sigma(i)})-\mu(A_{\sigma(i-1)})\widetilde{h}(x_{\sigma(i)})]$
$\displaystyle=[(\mu(A_{\sigma(1)})-\mu(A_{\sigma(0)})]\overline{\widetilde{h}}\oplus...\oplus[(\mu(A_{\sigma(n)})-\mu(A_{\sigma(n-1)})]\overline{\widetilde{h}}.$
If for any G-type FS $\widetilde{h_{h}}$ of HOHFE $\widetilde{h}$, it holds
$\lambda_{1}\widetilde{h_{h}}\oplus\lambda_{2}\widetilde{h_{h}}=(\lambda_{1}+\lambda_{2})\widetilde{h_{h}}$
where $\lambda_{1},\lambda_{2}>0$, then by Theorem 2.5 and the fact that
$\mu(A_{\sigma(0)}=\emptyset)=0,\mu(A_{\sigma(n)}=X)=1$, one gets
$\displaystyle HOHFEC_{\mu}(\widetilde{h}(x_{1}),...,\widetilde{h}(x_{n}))$
$\displaystyle=([(\mu(A_{\sigma(1)})-\mu(A_{\sigma(0)})]+...+[(\mu(A_{\sigma(n)})-\mu(A_{\sigma(n-1)})])\overline{\widetilde{h}}=\overline{\widetilde{h}}.~{}\Box$
###### Lemma 2.14.
Suppose that the score functions $S_{\widetilde{h}}$ on HOHFEs and
$S_{\widetilde{{h_{h}}}}$ on G-type FSs are those introduced in Definition
2.7, that is,
$\displaystyle
S_{\widetilde{h}}(\widetilde{h})=\frac{1}{|\widetilde{h}|}\sum_{\widetilde{h_{h}}\in\widetilde{h}}S_{\widetilde{{h_{h}}}}(\widetilde{{h_{h}}}).$
(8)
If for each HOHFE
$\widetilde{h}=\bigcup_{\widetilde{h_{h}}\in\widetilde{h}}\\{\widetilde{h_{h}}\\}$,
$S_{\widetilde{{h_{h}}}}$ satisfies
$S_{\widetilde{{h_{h}}}}(\lambda\widetilde{{h_{h}}})=\lambda
S_{\widetilde{{h_{h}}}}(\widetilde{{h_{h}}})$, for any $\lambda>0$. Then,
$S_{\widetilde{h}}$ also satisfies the latter property, i.e.,
$\displaystyle S_{\widetilde{h}}(\lambda\widetilde{h})=\lambda
S_{\widetilde{h}}(\widetilde{h}),\quad\forall\lambda>0.$ (9)
Proof. From definition of $S_{\widetilde{h}}$ given by (8) and the mentioned
property of $S_{\widetilde{{h_{h}}}}$, one gets for any $\lambda>0$
$\displaystyle
S_{\widetilde{h}}(\lambda\widetilde{h})=\frac{1}{|\lambda\widetilde{h}|}\sum_{\widetilde{h_{h}}\in\widetilde{h}}S_{\widetilde{{h_{h}}}}(\lambda\widetilde{{h_{h}}})=\frac{1}{|\widetilde{h}|}\sum_{\widetilde{h_{h}}\in\widetilde{h}}S_{\widetilde{{h_{h}}}}(\lambda\widetilde{{h_{h}}})=\frac{1}{|\widetilde{h}|}\sum_{\widetilde{h_{h}}\in\widetilde{h}}\lambda
S_{\widetilde{{h_{h}}}}(\widetilde{{h_{h}}})=\lambda
S_{\widetilde{h}}(\widetilde{h}).~{}\Box$
Hereafter, the HOHFE score function $S_{\widetilde{h}}$ is said to be
$\lambda$-invariance if it satisfies the property (9).
###### Lemma 2.15.
Assume that the HOHFE score function $S_{\widetilde{h}}$ is
$\lambda$-invariance. For any two HOHFEs $\widetilde{h}_{1}$ and
$\widetilde{h}_{2}$, if $\widetilde{h}_{1}\succ\widetilde{h}_{2}$, then
$\lambda\widetilde{h}_{1}\succ\lambda\widetilde{h}_{2}$, for any $\lambda>0$.
Proof. As follows from Definition 2.7, we find that
$\widetilde{h}_{1}\succ\widetilde{h}_{2}$ whenever
$S_{\widetilde{h}}(\widetilde{h}_{1})>S_{\widetilde{h}}(\widetilde{h}_{2})$.
On the other hand, with the $\lambda$-invariance property of the HOHFE score
function $S_{\widetilde{h}}$ in mind, the relation
$S_{\widetilde{h}}(\widetilde{h}_{1})>S_{\widetilde{h}}(\widetilde{h}_{2})$
gives rise to
$S_{\widetilde{h}}(\lambda\widetilde{h}_{1})>S_{\widetilde{h}}(\lambda\widetilde{h}_{2})$
for any $\lambda>0$, which implies that
$\lambda\widetilde{h}_{1}\succ\lambda\widetilde{h}_{2}$. This completes the
proof. $\Box$
###### Proposition 2.16.
(Monotonicity). Let $\mu$ be a fuzzy measure on $X$, and
$\widetilde{h}_{1}(x_{i})$ and $\widetilde{h}_{2}(x_{i}),~{}(i=1,...,n)$ be
two collection of HOHFEs on $X$. Assume that the HOHFE score function
$S_{\widetilde{h}}$ is $\lambda$-invariance, and
$(\sigma(1),\sigma(2),...,\sigma(n))$ denotes a permutation of $(1,2,...,n)$
such that
$\widetilde{h}_{1}(x_{\sigma(1)})\succ\widetilde{h}_{1}(x_{\sigma(2)})\succ...\succ\widetilde{h}_{1}(x_{\sigma(n)})$
and
$\widetilde{h}_{2}(x_{\sigma(1)})\succ\widetilde{h}_{2}(x_{\sigma(2)})\succ...\succ\widetilde{h}_{2}(x_{\sigma(n)})$.
If $\widetilde{h}_{1}(x_{\sigma(i)})\succ\widetilde{h}_{2}(x_{\sigma(i)})$ for
$(i=1,...,n)$, then
$\displaystyle
HOHFEC_{\mu}(\widetilde{h}_{1}(x_{1}),...,\widetilde{h}_{1}(x_{n}))\succ
HOHFEC_{\mu}(\widetilde{h}_{2}(x_{1}),...,\widetilde{h}_{2}(x_{n})).$
Proof. The assertion follows by applying Lemma 2.15 to Definition 2.10. $\Box$
###### Proposition 2.17.
(Boundedness). Let $\mu$ be a fuzzy measure on $X$,
$\widetilde{h}(x_{i}),~{}(i=1,...,n)$ be a collection of HOHFEs on $X$, and
$(\sigma(1),\sigma(2),...,\sigma(n))$ be a permutation of $(1,2,...,n)$ such
that
$\widetilde{h}_{1}(x_{\sigma(1)})\succ\widetilde{h}_{1}(x_{\sigma(2)})\succ...\succ\widetilde{h}_{1}(x_{\sigma(n)})$.
Then, the $\lambda$-invariance property of the HOHFE score function
$S_{\widetilde{h}}$ gives rise to
$\displaystyle\widetilde{h}_{Max}\succ
HOHFEC_{\mu}(\widetilde{h}_{1}(x_{1}),...,\widetilde{h}_{1}(x_{n}))\succ\widetilde{h}_{Min},$
where
$\widetilde{h}_{Min}=\min\\{\widetilde{h}_{1}(x_{1}),...,\widetilde{h}_{1}(x_{n})\\}=\widetilde{h}_{1}(x_{\sigma(n)})$
and
$\widetilde{h}_{Max}=\max\\{\widetilde{h}_{1}(x_{1}),...,\widetilde{h}_{1}(x_{n})\\}=\widetilde{h}_{1}(x_{\sigma(1)})$.
Proof. The application of Proposition 2.16 gives rise to the assertion. $\Box$
###### Proposition 2.18.
If for any G-type FS $\widetilde{h_{h}}$ of HOHFE $\widetilde{h}$, it holds
$(\widetilde{h_{h}}^{(1)}\oplus\widetilde{h_{h}}^{(2)})\oplus\widetilde{h_{h}}^{(3)}=\widetilde{h_{h}}^{(1)}\oplus(\widetilde{h_{h}}^{(2)}\oplus\widetilde{h_{h}}^{(3)})$
and
$\lambda_{1}\widetilde{h_{h}}\oplus\lambda_{2}\widetilde{h_{h}}=(\lambda_{1}+\lambda_{2})\widetilde{h_{h}}$
where $\lambda_{1},\lambda_{2}>0$, then the HOHF Choquet integral of
$\widetilde{h}(x_{i}),~{}(i=1,...,n)$ with respect to $\mu$ introduced in
Definition 2.10 satisfies
$\displaystyle
HOHFEC_{\mu}(\kappa\widetilde{h}_{1}(x_{1})\oplus\widetilde{\overline{h}},...,\kappa\widetilde{h}_{1}(x_{n})\oplus\widetilde{\overline{h}})=\kappa
HOHFEC_{\mu}(\widetilde{h}_{1}(x_{1}),...,\widetilde{h}_{1}(x_{n}))\oplus\widetilde{\overline{h}},\quad\kappa>0.$
Proof. The proof is much like that of Proposition 2.13 and therefore is
omitted. $\Box$
## 3 MCDM with the HOHF Choquet integral operator
In this section, we present a new method for MCDM, in which the evaluations of
the alternatives are given by HOHFEs and the interaction among the criteria
are allowed.
The MCDM procedure with HOHF Choquet integral operator can be described as
follows: suppose that $Y=\\{y_{1},y_{2},...,y_{m}\\}$ is an alternative set of
$m$ alternatives. In a MCDM problem, the decision maker is interested in
choosing the best one(s) from $Y$ according to the criteria set
$X=\\{x_{1},x_{2},...,x_{n}\\}$. In the HOHFE MCDM, the evaluation of each
alternative on each criterion is a HOHFE. By using the score function
$S_{\widetilde{h}}$ introduced in Definition 2.7, all HOHFEs associated with
each alternative are re-ordered such that
$\widetilde{h}(x_{\sigma(1)})\succ\widetilde{h}(x_{\sigma(2)})\succ...\succ\widetilde{h}(x_{\sigma(n)})$
where $(\sigma(1),\sigma(2),...,\sigma(n))$ is a permutation of $(1,2,...,n)$.
Taking into account the correlations of the HOHFEs, the evaluations of an
alternative can be aggregated to its overall evaluation by the HOHF Choquet
integral operator. According to a given total order relation on aggregated
HOHFEs, the decision maker can rank those overall evaluations and get the best
alternative(s).
### 3.1 A practical example
In what follows, we are going to demonstrate the practicality of implementing
the proposed concept of HOHFE in a higher order hesitant fuzzy multi-attribute
decision making problem.
Energy plays an important role in socio-economic development. Thus selecting
an appropriate policy for energy is critical for economic development and
environment. Assume that there exist five alternatives (energy projects)
${y}_{i},(i=1,2,3,4,5)$ which are invested in accordance with four criteria:
$x_{1}$: technological; $x_{2}$: environmental; $x_{3}$: socio-political;
$x_{4}$: economic. A number of decision makers are invited to evaluate the
performance of the above five alternatives. For an alternative under a
criterion, the decision makers give their evaluations anonymously in the form
of HOHFSs. In this regard, the results evaluated by the decision makers are
contained in a higher order hesitant fuzzy decision matrix which is shown in
Table 1.
Table 1. Higher order hesitant fuzzy decision matrix
| $x_{1}$ | $x_{2}$ | $x_{3}$ | $x_{4}$
---|---|---|---|---
${y}_{1}$ | $\\{(0.3,0.4,0.5)\\}$ | $\\{(0.4,0.5,0.6),\\{0.7,0.8,0.9\\}\\}$ | $\\{(0.5,0.7,0.7),(0.7,0.8,0.9)\\}$ | $\\{(0.2,0.3,0.4),\\{0.3,0.4,0.5\\}\\}$
${y}_{2}$ | $\\{(0.1,0.2,0.3),(0.2,0.3,0.4)\\}$ | $\\{(0.5,0.6,0.7),\\{0.3,0.4,0.5\\}\\}$ | $\\{(0.2,0.4,0.6),(0.7,0.8,0.9)\\}$ | $\\{(0.1,0.4,0.7),(0.6,0.7,0.8)\\}$
${y}_{3}$ | $\\{(0.1,0.2,0.3),\\{0.3,0.4,0.5\\}\\}$ | $\\{0.7,0.8,0.9\\}$ | $\\{(0.2,0.3,0.4),(0.5,0.6,0.7)\\}$ | $\\{(0.4,0.5,0.6)\\}$
${y}_{4}$ | $\\{0.2,0.3,0.4\\}$ | $\\{(0.3,0.4,0.5),(0.2,0.4,0.6)\\}$ | $\\{(0.5,0.6,0.7),\\{0.3,0.4,0.5\\}\\}$ | $\\{(0.1,0.2,0.3),(0.3,0.4,0.5)\\}$
${y}_{5}$ | $\\{(0.2,0.4,0.6),(0.7,08,0.9)\\}$ | $\\{0.7,0.8,0.9\\}$ | $\\{(0.3,0.4,0.5)\\}$ | $\\{(0.5,0.7,0.7),(0.7,0.8,0.9)\\}$
In Table 1, the notation $A=(a_{1},a_{2},a_{3})$ is used to describe a fuzzy
event by the help of a triangular fuzzy number in which the values $a_{1}$,
$a_{2}$ and $a_{3}$ denote respectively the smallest possible value, the most
promising value and the largest possible value, and moreover,
$\\{\widetilde{h_{\widetilde{h}}}^{(1)},\widetilde{h_{\widetilde{h}}}^{(2)},\widetilde{h_{\widetilde{h}}}^{(3)}\\}$
indicates a HFE. In this setting and as a consequence, we observe that the
second and third elements of the last row of Table 1 are conceptually
different. The second array of Table 1 indicates a HFE, while, the third array
is used to indicate a triangular fuzzy number.
Suppose that the fuzzy measures of criteria $X=\\{x_{1},x_{2},x_{3},x_{4}\\}$
are given as follows:
$\displaystyle\mu(\emptyset)=0;$
$\displaystyle\mu(\\{x_{1}\\})=0.2,~{}\mu(\\{x_{2}\\})=0.4,~{}\mu(\\{x_{3}\\})=0.3,~{}\mu(\\{x_{4}\\})=0.1;$
$\displaystyle\mu(\\{x_{1},x_{2}\\})=0.4,~{}\mu(\\{x_{1},x_{3}\\})=0.2,~{}\mu(\\{x_{1},x_{4}\\})=0.1,~{}\mu(\\{x_{2},x_{3}\\})=0.3,$
$\displaystyle\mu(\\{x_{2},x_{4}\\})=0.3,~{}\mu(\\{x_{3},x_{4}\\})=0.1;$
$\displaystyle\mu(\\{x_{1},x_{2},x_{3}\\})=0.3,~{}\mu(\\{x_{1},x_{2},x_{4}\\})=0.1,~{}\mu(\\{x_{1},x_{3},x_{4}\\})=0.2,~{}\mu(\\{x_{2},x_{3},x_{4}\\})=0.2;$
$\displaystyle\mu({X})=1.$
For re-ordering HOHFEs given in Table 1, we recall the HOHFE score function
$S_{\widetilde{h}}$ introduced in Definition 2.7, where
$\displaystyle
S_{\widetilde{h}}(\widetilde{h})=\frac{1}{|\widetilde{h}|}\sum_{\widetilde{h_{h}}\in\widetilde{h}}S_{\widetilde{{h_{h}}}}(\widetilde{{h_{h}}}).$
(10)
Here, the G-type FS score function $S_{\widetilde{{h_{h}}}}$ is taken as
* •
(For triangular fuzzy number):
$\displaystyle
S_{\widetilde{{h_{h}}}}({(a_{1},a_{2},a_{3})})=\frac{1}{3}(a_{1}+a_{2}+a_{3}),$
(11)
which was proposed by Lee and Li in [16];
* •
(For hesitant fuzzy element):
$\displaystyle
S_{\widetilde{{h_{h}}}}(\\{\widetilde{h_{\widetilde{h}}}^{(1)},...,\widetilde{h_{\widetilde{h}}}^{(|\widetilde{h}|)}\\})=\frac{1}{|\widetilde{h}|}(\\{\widetilde{h_{\widetilde{h}}}^{(1)}+...+\widetilde{h_{\widetilde{h}}}^{(|\widetilde{h}|)}\\}),$
(12)
Although, the corresponding score functions of HFEs and triangular fuzzy
numbers may differ from each other, but in this example, both of them have a
same rule. However, by using the HOHFE score function $S_{\widetilde{h}}$
given by (10), we re-arrange the HOHFEs corresponding to each criterion in
descending order as follows:
$\displaystyle
y_{1}:\quad\widetilde{h}(x_{\sigma(1)}=x_{3})\succ\widetilde{h}(x_{\sigma(2)}=x_{2})\succ\widetilde{h}(x_{\sigma(3)}=x_{1})\succ\widetilde{h}(x_{\sigma(4)}=x_{4});$
$\displaystyle
y_{2}:\quad\widetilde{h}(x_{\sigma(1)}=x_{3})\succ\widetilde{h}(x_{\sigma(2)}=x_{4})\succ\widetilde{h}(x_{\sigma(3)}=x_{2})\succ\widetilde{h}(x_{\sigma(4)}=x_{1});$
$\displaystyle
y_{3}:\quad\widetilde{h}(x_{\sigma(1)}=x_{2})\succ\widetilde{h}(x_{\sigma(2)}=x_{4})\succ\widetilde{h}(x_{\sigma(3)}=x_{3})\succ\widetilde{h}(x_{\sigma(4)}=x_{1});$
$\displaystyle
y_{4}:\quad\widetilde{h}(x_{\sigma(1)}=x_{3})\succ\widetilde{h}(x_{\sigma(2)}=x_{2})\succ\widetilde{h}(x_{\sigma(3)}=x_{1})\succ\widetilde{h}(x_{\sigma(4)}=x_{4});$
$\displaystyle
y_{5}:\quad\widetilde{h}(x_{\sigma(1)}=x_{2})\succ\widetilde{h}(x_{\sigma(2)}=x_{4})\succ\widetilde{h}(x_{\sigma(3)}=x_{1})\succ\widetilde{h}(x_{\sigma(4)}=x_{3}).$
If we employ the HOHF Choquet integral operator, we then are able to aggregate
all HOHFEs in the i-$th$ row of the HOHFE decision matrix into an overall
value. For instance,
$\displaystyle\widetilde{h}_{y_{1}}=HOHFEC_{\mu}(\widetilde{h}(x_{1}),...,\widetilde{h}(x_{4}))=$
$\displaystyle[(\mu(A_{\sigma(1)})-\mu(A_{\sigma(0)}))\widetilde{h}(x_{\sigma(1)})]\oplus[(\mu(A_{\sigma(2)})-\mu(A_{\sigma(1)}))\widetilde{h}(x_{\sigma(2)})]$
$\displaystyle\oplus[(\mu(A_{\sigma(3)})-\mu(A_{\sigma(2)}))\widetilde{h}(x_{\sigma(3)})]\oplus[(\mu(A_{\sigma(4)})-\mu(A_{\sigma(3)}))\widetilde{h}(x_{\sigma(4)})]$
$\displaystyle=[(\mu(\\{x_{\sigma(1)}\\})-\mu(\emptyset))\widetilde{h}(x_{\sigma(1)})]\oplus[(\mu(\\{x_{\sigma(1)},x_{\sigma(2)}\\})-\mu(\\{x_{\sigma(1)}\\}))\widetilde{h}(x_{\sigma(2)})]$
$\displaystyle\oplus[(\mu(\\{x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)}\\})-\mu(\\{x_{\sigma(1)},x_{\sigma(2)}\\}))\widetilde{h}(x_{\sigma(3)})]\oplus[(\mu(X)-\mu(\\{x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)}\\}))\widetilde{h}(x_{\sigma(4)})]$
$\displaystyle=[(\mu(\\{x_{3}\\})-\mu(\emptyset))\widetilde{h}(x_{3})]\oplus[(\mu(\\{x_{3},x_{2}\\})-\mu(\\{x_{3}\\}))\widetilde{h}(x_{2})]$
$\displaystyle\oplus[(\mu(\\{x_{3},x_{2},x_{1}\\})-\mu(\\{x_{3},x_{2}\\}))\widetilde{h}(x_{1})]\oplus[(\mu(X)-\mu(\\{x_{3},x_{2},x_{1}\\}))\widetilde{h}(x_{4})]$
$\displaystyle=[(0.3-0)\\{(0.5,0.7,0.7),(0.7,0.8,0.9)\\}]\oplus[(0.3-0.3)\\{(0.4,0.5,0.6),\\{0.7,0.8,0.9\\}\\}]$
$\displaystyle\oplus[(0.3-0.3)\\{(0.3,0.4,0.5)\\}]\oplus[(1-0.3)\\{(0.2,0.3,0.4),\\{0.3,0.4,0.5\\}\\}].$
Let us now apply the following arithmetic operations
* •
(For triangular fuzzy number [14]):
$\displaystyle\lambda A=\lambda(a_{1},a_{2},a_{3})=(\lambda a_{1},\lambda
a_{2},\lambda a_{3}),$ $\displaystyle A\oplus
B=(a_{1},a_{2},a_{3})\oplus(b_{1},b_{2},b_{3})=(a_{1}+b_{1}-a_{1}b_{1},a_{2}+b_{2}-a_{2}b_{2},a_{3}+b_{3}-a_{3}b_{3});$
* •
(For hesitant fuzzy element [11]):
$\displaystyle\lambda\\{\widetilde{h_{\widetilde{h}}}^{(1)},...,\widetilde{h_{\widetilde{h}}}^{(|\widetilde{h}|)}\\}=\\{\lambda\widetilde{h_{\widetilde{h}}}^{(1)},...,\lambda\widetilde{h_{\widetilde{h}}}^{(|\widetilde{h}|)}\\},\quad\lambda\geq
0,$
$\displaystyle\\{\widetilde{h_{\widetilde{h_{1}}}}^{(1)},...,\widetilde{h_{\widetilde{h_{1}}}}^{(|\widetilde{h}|)}\\}\oplus\\{\widetilde{h_{\widetilde{h_{2}}}}^{(1)},...,\widetilde{h_{\widetilde{h_{2}}}}^{(|\widetilde{h}|)}\\}$
$\displaystyle\quad\quad=\\{\widetilde{h_{\widetilde{h_{1}}}}^{(1)}+\widetilde{h_{\widetilde{h_{2}}}}^{(1)}-\widetilde{h_{\widetilde{h_{1}}}}^{(1)}\widetilde{h_{\widetilde{h_{2}}}}^{(1)},...,\widetilde{h_{\widetilde{h_{1}}}}^{(|\widetilde{h}|)}+\widetilde{h_{\widetilde{h_{2}}}}^{(|\widetilde{h}|)}-\widetilde{h_{\widetilde{h_{1}}}}^{(|\widetilde{h}|)}\widetilde{h_{\widetilde{h_{2}}}}^{(|\widetilde{h}|)}\\},$
to the latter findings. Then, one achieves that
$\displaystyle\widetilde{h}_{y_{1}}$ $\displaystyle=$
$\displaystyle[\\{(0.15,0.21,0.21),(0.21,0.24,0.27)\\}]\oplus[\\{(0.14,0.21,0.28),\\{0.21,0.28,0.35\\}\\}]$
$\displaystyle=$
$\displaystyle\\{(0.27,0.37,0.43),(0.32,0.40,0.47),\\{0.21,0.28,0.35\\}\\}.$
The HOHFE score value of $\widetilde{h}_{y_{1}}$ is then equal to
$\displaystyle
S_{\widetilde{h}}(\widetilde{h}_{y_{1}})=\frac{1}{|\widetilde{h}_{y_{1}}|}\sum_{\widetilde{h_{{h}_{y_{1}}}}\in\widetilde{h}_{y_{1}}}S_{\widetilde{{h_{h}}}}(\widetilde{{h_{{h}_{y_{1}}}}})=0.3456.$
Similarly, the other HOHFE score values are obtained as:
$\displaystyle S_{\widetilde{h}}(\widetilde{h}_{y_{2}})=0.4915,\quad
S_{\widetilde{h}}(\widetilde{h}_{y_{3}})=0.4364,\quad
S_{\widetilde{h}}(\widetilde{h}_{y_{4}})=0.2739,\quad
S_{\widetilde{h}}(\widetilde{h}_{y_{5}})=0.5601.$
Therefore,
$\displaystyle
S_{\widetilde{h}}(\widetilde{h}_{y_{5}})>S_{\widetilde{h}}(\widetilde{h}_{y_{2}})>S_{\widetilde{h}}(\widetilde{h}_{y_{3}})>S_{\widetilde{h}}(\widetilde{h}_{y_{1}})>S_{\widetilde{h}}(\widetilde{h}_{y_{4}}),$
which are led to
$\displaystyle y_{5}\succ y_{2}\succ y_{3}\succ y_{1}\succ y_{4},$
and hence the most appropriate energy project is $y_{5}$.
### 3.2 Comparisons and further discussions
Here, we adopt a multiple criteria group decision making problem from [21]
which is originally based on extending TODIM method by encountering the
Choquet integral within a multiset hesitant fuzzy environment.
Peng et al. [21] considered an investment company for investing in a project.
They supposed five alternatives ${y}_{1}$: car company; ${y}_{2}$: food
company; ${y}_{3}$: computer company; ${y}_{4}$: arms company; and ${y}_{5}$:
TV company which need to be invested. In this setting, the desired decision
has to be made according to the four criteria including $x_{1}$: the
environment impact which is referred to as the impact on the companys
environment $x_{2}$: the risk factor including product risk and development
environment risk; $x_{3}$: the growth prospects including increased
profitability and returns; and $x_{4}$: social-political impact that is
referred to as the governments and local residents support for company.
Peng et al. [21] used multiset hesitant fuzzy sets to evaluate alternatives by
two decision makers. In this contribution to have a complete picture about the
performance of the proposed approach and that of Peng et al. [21], we re-state
Peng et al’s [21] decision matrix in the following form of higher order
hesitant fuzzy decision matrix:
Table 2. Higher order hesitant fuzzy decision matrix
| $x_{1}$ | $x_{2}$ | $x_{3}$ | $x_{4}$
---|---|---|---|---
${y}_{1}$ | $\\{(0.4,0.5,0.7)\\}$ | $\\{(0.5,0.5,0.8)\\}$ | $\\{(0.6,0.6,0.9)\\}$ | $\\{0.5,0.6\\}$
${y}_{2}$ | $\\{(0.6,0.7,0.8)\\}$ | $\\{0.5,0.6\\}$ | $\\{(0.6,0.7,0.7)\\}$ | $\\{0.4,0.5\\}$
${y}_{3}$ | $\\{0.6,0.8\\}$ | $\\{(0.2,0.3,0.5)\\}$ | $\\{0.6,0.6\\}$ | $\\{0.5,0.7\\}$
${y}_{4}$ | $\\{(0.5,0.5,0.7)\\}$ | $\\{0.4,0.5\\}$ | $\\{0.8,0.9\\}$ | $\\{(0.3,0.4,0.5)\\}$
${y}_{5}$ | $\\{0.6,0.7\\}$ | $\\{0.5,0.7\\}$ | $\\{0.7,0.8\\}$ | $\\{(0.3,0.3,0.4)\\}$
some of its arrays are in form of triangular fuzzy numbers and some of them
are HFEs. Needless to say that preserving both forms of arrays appears to
maintain the implication of HOHFEs.
According to Peng et al’s [21] assumption, the fuzzy measures of criteria
$X=\\{x_{1},x_{2},x_{3},x_{4}\\}$ are as follows:
$\displaystyle\mu(\emptyset)=0;$
$\displaystyle\mu(\\{x_{1}\\})=0.40,~{}\mu(\\{x_{2}\\})=0.25,~{}\mu(\\{x_{3}\\})=0.37,~{}\mu(\\{x_{4}\\})=0.20;$
$\displaystyle\mu(\\{x_{1},x_{2}\\})=0.60,~{}\mu(\\{x_{1},x_{3}\\})=0.70,~{}\mu(\\{x_{1},x_{4}\\})=0.56,~{}\mu(\\{x_{2},x_{3}\\})=0.68,$
$\displaystyle\quad\quad\quad\mu(\\{x_{2},x_{4}\\})=0.43,~{}\mu(\\{x_{3},x_{4}\\})=0.54;$
$\displaystyle\mu(\\{x_{1},x_{2},x_{3}\\})=0.88,~{}\mu(\\{x_{1},x_{2},x_{4}\\})=0.75,~{}\mu(\\{x_{1},x_{3},x_{4}\\})=0.84,~{}\mu(\\{x_{2},x_{3},x_{4}\\})=0.73;$
$\displaystyle\mu(X)=1.$
We re-arrange the HOHFEs corresponding to each criterion in descending order
by the use of HOHFE score function $S_{\widetilde{h}}$ given by (10) as
follows:
$\displaystyle
y_{1}:\quad\widetilde{h}(x_{\sigma(1)}=x_{3})\succ\widetilde{h}(x_{\sigma(2)}=x_{2})\succ\widetilde{h}(x_{\sigma(3)}=x_{1})\succ\widetilde{h}(x_{\sigma(4)}=x_{4});$
$\displaystyle
y_{2}:\quad\widetilde{h}(x_{\sigma(1)}=x_{1})\succ\widetilde{h}(x_{\sigma(2)}=x_{3})\succ\widetilde{h}(x_{\sigma(3)}=x_{2})\succ\widetilde{h}(x_{\sigma(4)}=x_{4});$
$\displaystyle
y_{3}:\quad\widetilde{h}(x_{\sigma(1)}=x_{1})\succ\widetilde{h}(x_{\sigma(2)}=x_{3})\succ\widetilde{h}(x_{\sigma(3)}=x_{4})\succ\widetilde{h}(x_{\sigma(4)}=x_{2});$
$\displaystyle
y_{4}:\quad\widetilde{h}(x_{\sigma(1)}=x_{3})\succ\widetilde{h}(x_{\sigma(2)}=x_{1})\succ\widetilde{h}(x_{\sigma(3)}=x_{2})\succ\widetilde{h}(x_{\sigma(4)}=x_{4});$
$\displaystyle
y_{5}:\quad\widetilde{h}(x_{\sigma(1)}=x_{3})\succ\widetilde{h}(x_{\sigma(2)}=x_{1})\succ\widetilde{h}(x_{\sigma(3)}=x_{2})\succ\widetilde{h}(x_{\sigma(4)}=x_{4}).$
Now, by the help of HOHF Choquet integral operator, we aggregate all HOHFEs in
the i-$th$ row of the HOHFE decision matrix into an overall value, for
example,
$\displaystyle\widetilde{h}_{y_{1}}=HOHFEC_{\mu}(\widetilde{h}(x_{1}),...,\widetilde{h}(x_{4}))=$
$\displaystyle[(\mu(A_{\sigma(1)})-\mu(A_{\sigma(0)}))\widetilde{h}(x_{\sigma(1)})]\oplus[(\mu(A_{\sigma(2)})-\mu(A_{\sigma(1)}))\widetilde{h}(x_{\sigma(2)})]$
$\displaystyle\oplus[(\mu(A_{\sigma(3)})-\mu(A_{\sigma(2)}))\widetilde{h}(x_{\sigma(3)})]\oplus[(\mu(A_{\sigma(4)})-\mu(A_{\sigma(3)}))\widetilde{h}(x_{\sigma(4)})]$
$\displaystyle=[(\mu(\\{x_{\sigma(1)}\\})-\mu(\emptyset))\widetilde{h}(x_{\sigma(1)})]\oplus[(\mu(\\{x_{\sigma(1)},x_{\sigma(2)}\\})-\mu(\\{x_{\sigma(1)}\\}))\widetilde{h}(x_{\sigma(2)})]$
$\displaystyle\oplus[(\mu(\\{x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)}\\})-\mu(\\{x_{\sigma(1)},x_{\sigma(2)}\\}))\widetilde{h}(x_{\sigma(3)})]\oplus[(\mu(X)-\mu(\\{x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)}\\}))\widetilde{h}(x_{\sigma(4)})]$
$\displaystyle=[(\mu(\\{x_{3}\\})-\mu(\emptyset))\widetilde{h}(x_{3})]\oplus[(\mu(\\{x_{3},x_{2}\\})-\mu(\\{x_{3}\\}))\widetilde{h}(x_{2})]$
$\displaystyle\oplus[(\mu(\\{x_{3},x_{2},x_{1}\\})-\mu(\\{x_{3},x_{2}\\}))\widetilde{h}(x_{1})]\oplus[(\mu(X)-\mu(\\{x_{3},x_{2},x_{1}\\}))\widetilde{h}(x_{4})]$
$\displaystyle=\\{(0.0765,0.0795,0.1205),\\{0.0100,0.0120\\}\\}.$
By applying the same procedure, we achieve
$\displaystyle\widetilde{h}_{y_{2}}=\\{(0.0550,0.0642,0.0672),\\{0.0533,0.0645\\}\\},$
$\displaystyle\widetilde{h}_{y_{3}}=\\{(0.0110,0.0165,0.0275),\\{0.0650,0.0750\\}\\},$
$\displaystyle\widetilde{h}_{y_{4}}=\\{(0.0210,0.0230,0.0310),\\{0.0713,0.0830\\}\\},$
$\displaystyle\widetilde{h}_{y_{5}}=\\{(0.0060,0.0060,0.0080),\\{0.1330,0.1633\\}\\}.$
Moreover, the HOHFE score function (11) results in
$\displaystyle S_{\widetilde{h}}(\widetilde{h}_{y_{1}})=0.2985,\quad
S_{\widetilde{h}}(\widetilde{h}_{y_{2}})=0.3041,\quad
S_{\widetilde{h}}(\widetilde{h}_{y_{3}})=0.1950,\quad
S_{\widetilde{h}}(\widetilde{h}_{y_{4}})=0.2293,\quad
S_{\widetilde{h}}(\widetilde{h}_{y_{5}})=0.3163.$
Therefore,
$\displaystyle
S_{\widetilde{h}}(\widetilde{h}_{y_{5}})>S_{\widetilde{h}}(\widetilde{h}_{y_{2}})>S_{\widetilde{h}}(\widetilde{h}_{y_{1}})>S_{\widetilde{h}}(\widetilde{h}_{y_{4}})>S_{\widetilde{h}}(\widetilde{h}_{y_{3}}).$
These findings indicate that
$\displaystyle y_{5}\succ y_{2}\succ y_{1}\succ y_{4}\succ y_{3},$
and hence the most appropriate energy project is $y_{5}$.
In what follows, we are going to verify the validity of proposed technique
compared to the other existing techniques including Xu’s [37], Wei’s [36],
Zhang et al.’s [46], Chen et al.’s [5], Xu’s [39], Farhadinia’s [10], Zhang
and Wei’s [47], Zhang and Xu’s [48], Wang et al.’s [33] and Peng et al.’s [21]
from a perspective of performance. Indeed, from the performance stand point of
view, we intend to discuss the resulted rankings of techniques macroscopy, and
not microscopy, that is, we do not intend to investigate the pros and cons of
each technique.
Before going more in depth with future analysis, let us first re-state here
from Farhadinia [12] the algorithm being implemented to sort decision making
techniques based on their outcomes.
###### Algorithm 3.1.
[12] (Sorting of decision making techniques based on their outcomes)
Step 1.
Take into account each outcome of decision making techniques (i.e., the
resulted ranking order of alternatives) as an individual preference $R_{k}$
$(k=1,...,m)$ together with the associated dominance-vector $\pi_{R_{k}}$.
Step 2.
Add up all the individual preference matrices component-wisely to get the
collective preference matrix $\overline{R}=[\overline{r}_{ij}]_{n\times n}$.
Step 3.
Extract the collective preference from the matrix
$\overline{R}=[\overline{r}_{ij}]_{n\times n}$ with respect to the collective
majority decision rule.
Step 4.
Characterize the corresponding dominance-vector hesitant fuzzy sets to the
individual and the collective preferences. Then, determine the importance
weight of decision making techniques in accordance with the distance value of
their individual preference from the collective preference using a
corresponding distance measure.
Now, for determining the importance weights of decision making techniques
according to their distance values of their individual rankings from the
collective one, we implement here Algorithm 3.1 above. Indeed, Algorithm 3.1
is allocated to assess the best outcome(s) without regarding of how the
preference ordering is obtained or from what numerical values one concludes
the preference ordering.
Now, by considering the total ranking order $R$ on
$Y=\\{y_{1},y_{2},y_{3},y_{4},y_{5}\\}$, we can correspond to each of them the
dominance-vectors $\pi_{R_{k}}$ (for $k=X1,W1,...,Pro$) which are given in
Table 3.
Table 3. The dominance-vectors corresponding to the different techniques.
Techniques | Ranking orders | Dominance-vector
---|---|---
$(e_{X1})$: Xu’s [37] technique | $R_{X1}$: $y_{4}\succ y_{5}\succ y_{2}\succ y_{1}\succ y_{3}$ | $\pi_{R_{X1}}=(2,3,1,4,5)$
$(e_{W1})$: Wei’s [36] technique | $R_{W1}$: $y_{2}\succ y_{5}\succ y_{4}\succ y_{1}\succ y_{3}$ | $\pi_{R_{W1}}=(2,5,1,3,4)$
$(e_{Z1})$: Zhang et al.’s [46] technique | $R_{Z1}$: $y_{5}\succ y_{2}\succ y_{1}\succ y_{4}\succ y_{3}$ | $\pi_{R_{Z1}}=(3,4,1,2,5)$
$(e_{C})$: Chen et al.’s [5] technique | $R_{C}$: $y_{5}\succ y_{1}\succ y_{2}\succ y_{4}\succ y_{3}$ | $\pi_{R_{C}}=(4,3,1,2,5)$
$(e_{X2})$: Xu’s [39] technique | $R_{X2}$: $y_{5}\succ y_{2}\succ y_{4}\succ y_{1}\succ y_{3}$ | $\pi_{R_{X2}}=(2,4,1,3,5)$
$(e_{F})$: Farhadinia’s [10] technique | $R_{F}$: $y_{5}\succ y_{1}\succ y_{2}\succ y_{4}\succ y_{3}$ | $\pi_{R_{F}}=(4,3,1,2,5)$
$(e_{Z2})$: Zhang and Wei’s [47] technique | $R_{Z2}$: $y_{5}\succ y_{1}\succ y_{2}\succ y_{4}\succ y_{3}$ | $\pi_{R_{Z2}}=(4,3,1,2,5)$
$(e_{Z3})$: Zhang and Xu’s [48] technique | $R_{Z3}$: $y_{5}\succ y_{1}\succ y_{2}\succ y_{3}\succ y_{4}$ | $\pi_{R_{Z3}}=(4,3,2,1,5)$
$(e_{W2})$: Wang et al.’s [33] technique | $R_{W2}$: $y_{5}\succ y_{1}\succ y_{2}\succ y_{3}\succ y_{4}$ | $\pi_{R_{W2}}=(4,3,2,1,5)$
$(e_{P})$: Peng et al.’s [21] technique | $R_{P}$: $y_{5}\succ y_{2}\succ y_{1}\succ y_{3}\succ y_{4}$ | $\pi_{R_{P}}=(3,4,2,1,5)$
$(e_{Pro})$: The proposed technique | $R_{Pro}$: $y_{5}\succ y_{2}\succ y_{1}\succ y_{4}\succ y_{3}$ | $\pi_{R_{Pro}}=(3,4,1,2,5)$
By considering all the individual preference matrices $R_{k}$
$(k=X1,W1,...,Pro)$, for instance, $R_{X1}$: $y_{4}\succ y_{5}\succ y_{2}\succ
y_{1}\succ y_{3}$ which is given by
$\displaystyle R_{X1}=\left(\begin{array}[]{ccccc}0&-1&1&-1&-1\\\
1&0&1&-1&-1\\\ -1&-1&0&-1&-1\\\ 1&1&1&0&1\\\ 1&1&1&-1&0\\\
\end{array}\right),$
the collective preference $\overline{R}$ is then achieved through the
aggregation process in which all the individual preference matrices are added
up component-wisely. The result is as follows:
$\displaystyle\overline{R}=\left(\begin{array}[]{ccccc}0&-1&11&5&-11\\\
1&0&11&9&-9\\\ -11&-11&0&-5&-11\\\ -5&-9&5&0&-9\\\ 11&9&11&9&0\\\
\end{array}\right),$
which returns $y_{5}\succ y_{2}\succ y_{1}\succ y_{4}\succ y_{3}$ as the
collective preference.
Now, with the distance
$\displaystyle{\mathcal{D}}({\pi_{R_{k}}},{\pi_{\overline{R}}})=\max\\{\max_{\pi_{R_{k}}}\min_{\pi_{\overline{R}}}|\pi_{R_{k}},\pi_{\overline{R}}|~{},~{}\min_{\pi_{R_{k}}}\max_{\pi_{\overline{R}}}|\pi_{R_{k}},\pi_{\overline{R}}|\\},\quad(k=X1,W1,...,Pro)$
(15)
at hand, we may obtain the distance of individual preferences from the
collective preference as those given in Table 4.
Table 4. The distance of individual preferences from the collective
preference.
Techniques | Ranking orders | Distance values
---|---|---
$(e_{X1})$: Xu’s [37] technique | $R_{X1}$: $y_{4}\succ y_{5}\succ y_{2}\succ y_{1}\succ y_{3}$ | ${{\cal{D}}}(\pi_{R_{X1}},{\pi}_{\overline{R}})=4$
$(e_{W1})$: Wei’s [36] technique | $R_{W1}$: $y_{2}\succ y_{5}\succ y_{4}\succ y_{1}\succ y_{3}$ | ${{\cal{D}}}(\pi_{R_{W1}},{\pi}_{\overline{R}})=4$
$(e_{Z1})$: Zhang et al.’s [46] technique | $R_{Z1}$: $y_{5}\succ y_{2}\succ y_{1}\succ y_{4}\succ y_{3}$ | ${{\cal{D}}}(\pi_{R_{Z1}},{\pi}_{\overline{R}})=0$
$(e_{C})$: Chen et al.’s [5] technique | $R_{C}$: $y_{5}\succ y_{1}\succ y_{2}\succ y_{4}\succ y_{3}$ | ${{\cal{D}}}(\pi_{R_{C}},{\pi}_{\overline{R}})=2$
$(e_{X2})$: Xu’s [39] technique | $R_{X2}$: $y_{5}\succ y_{2}\succ y_{4}\succ y_{1}\succ y_{3}$ | ${{\cal{D}}}(\pi_{R_{X2}},{\pi}_{\overline{R}})=2$
$(e_{F})$: Farhadinia’s [10] technique | $R_{F}$: $y_{5}\succ y_{1}\succ y_{2}\succ y_{4}\succ y_{3}$ | ${{\cal{D}}}(\pi_{R_{F}},{\pi}_{\overline{R}})=2$
$(e_{Z2})$: Zhang and Wei’s [47] technique | $R_{Z2}$: $y_{5}\succ y_{1}\succ y_{2}\succ y_{4}\succ y_{3}$ | ${{\cal{D}}}(\pi_{R_{Z2}},{\pi}_{\overline{R}})=2$
$(e_{Z3})$: Zhang and Xu’s [48] technique | $R_{Z3}$: $y_{5}\succ y_{1}\succ y_{2}\succ y_{3}\succ y_{4}$ | ${{\cal{D}}}(\pi_{R_{Z3}},{\pi}_{\overline{R}})=4$
$(e_{W2})$: Wang et al.’s [33] technique | $R_{W2}$: $y_{5}\succ y_{1}\succ y_{2}\succ y_{3}\succ y_{4}$ | ${{\cal{D}}}(\pi_{R_{W2}},{\pi}_{\overline{R}})=4$
$(e_{P})$: Peng et al.’s [21] technique | $R_{P}$: $y_{5}\succ y_{2}\succ y_{1}\succ y_{3}\succ y_{4}$ | ${{\cal{D}}}(\pi_{R_{P}},{\pi}_{\overline{R}})=2$
$(e_{Pro})$: The proposed technique | $R_{Pro}$: $y_{5}\succ y_{2}\succ y_{1}\succ y_{4}\succ y_{3}$ | ${{\cal{D}}}(\pi_{R_{Pro}},{\pi}_{\overline{R}})=0$
The larger distance ${{\cal{D}}}$ indicates the more dis-similar between each
individual preference and the collective preference. This implies that the
larger value of distance shows lower performance of decision-making technique.
In view of Table 4, we are able to sort the considered techniques which are
labelled by $k=X1,W1,...,Pro$ as the following:
$\displaystyle\\{(e_{Pro}),(e_{Z1})\\}>\\{(e_{C}),(e_{X2}),(e_{F}),(e_{Z2}),(e_{P})\\}>\\{(e_{X1}),(e_{W1}),(e_{Z3}),(e_{W2})\\}.$
The above results verify that among the above decision-making techniques, the
proposed technique can be chosen as the one of the most suitable techniques.
However, these findings indicate that although the procedure of obtaining the
most appropriate alternative using different techniques may be different, but
their outcomes can be used for constructing an admissible assessment which was
called here as the collective preference.
## 4 Conclusion
The current contribution proposed the higher order hesitant fuzzy (HOHF)
Choquet integral operator that not only considers the importance of the
elements, but also it can reflect the correlations among the elements. Then,
we employed the HOHF Choquet integral operator to aggregate HOHFEs in a higher
order hesitant fuzzy MCDM problem. Eventually, a comparative analysis of
different techniques with the HOHF Choquet integral-based technique presented,
and it indicates that the latter technique is enough efficient. In future
work, we plan to study the other aggregation operators in the higher order
hesitant fuzzy setting for handling MCDM with higher order hesitant fuzzy
information. In addition, since the application potentials of HOHFS are
diverse, it can be investigated in other domains such as clustering, pattern
recognition, image processing, etc.
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|
# Consistency of sample-based stationary points for infinite-dimensional
stochastic optimization
Johannes Milz H. Milton Stewart School of Industrial and Systems Engineering,
Georgia Institute of Technology, Atlanta, GA 30332, USA
<EMAIL_ADDRESS>
(June 19, 2023)
###### Abstract
We consider stochastic optimization problems with possibly nonsmooth
integrands posed in Banach spaces and approximate these stochastic programs
via a sample-based approaches. We establish the consistency of approximate
Clarke stationary points of the sample-based approximations. Our framework is
applied to risk-averse semilinear PDE-constrained optimization using the
average value-at-risk and to risk-neutral bilinear PDE-constrained
optimization.
###### keywords:
stochastic programming, sample average approximation, optimization under
uncertainty, PDE-constrained optimization, uncertainty quantification,
bilinear optimal control
AMS subject classifications. 65C05, 90C15, 35R60, 90C48, 90C30, 60H25, 49M41,
35Q93
## 1 Introduction
Infinite-dimensional optimization problems arise in a plethora of research
fields such as dynamic programming [56], statistical estimation [26], feedback
stabilization of dynamical systems [54], and optimization problems governed by
partial differential equations (PDEs) [49]. PDE-constrained optimization is an
active research field with a focus on modeling, analyzing and solving complex
optimization problems with PDE constraints. For example, numerous applications
in the field of renewable and sustainable energy yield challenging PDE-
constrained optimization problems, such as wind plant layout optimization
[44], wind turbine blade planform design [1], and tidal stream array
optimization [28]. Parameters in PDEs may be uncertain, such as diffusion
coefficients and boundary conditions, and as such can have a significant
influence on the simulation output. Using parameter-dependent PDEs for
decision making naturally leads to decision making under uncertainty. A common
approach in optimization under uncertainty models uncertain parameters as
random variables with known probability distribution. Using this stochastic
programming approach, we can formulate an optimization problem having
solutions performing best on average; its objective function is the expected
value of a parameterized cost function, the integrand. Such optimization
problems are nowadays often referred to as risk-neutral problems to
distinguish them from risk-averse formulations. Since high-dimensional
integrals cannot be accurately evaluated and solving PDEs requires numerical
computations, we approximate the expectations using a Monte Carlo sample-based
approximation, the sample average approximation (SAA) approach [45, 72, 74],
and other schemes approximating the true probability distribution via weakly
convergent probability distributions. In this manuscript, we demonstrate the
consistency of Clarke-(C-)stationary points of the approximated problems
towards the risk-neutral problem’s set of C-stationary points.
Motivated by infinite-dimensional optimization problems governed by complex
physical systems under uncertainty, we consider the risk-neutral composite
optimization problem
$\displaystyle\min_{x\in
X}\,\int_{\Xi}\,F_{\xi}(x)\,\mathrm{d}\mathbb{P}(\xi)+\psi(x),$ (1.1)
where $X$ is a reflexive Banach space, $\psi\colon X\to(-\infty,\infty]$ is a
proper, convex, lower semicontinuous function having the so-called Kadec
property, $\mathbb{P}$ is a probability measure on the complete, separable
metric space $\Xi$, and $F_{\xi}$ is a locally Lipschitz continuous integrand
with integrable Lipschitz constants.
In this manuscript, we consider two types of approximations of the
expectations in (1.1). While our main focus is on analyzing the SAA approach
[45, 72], we also consider approximations of $\mathbb{P}$ via weakly
converging probability measures [56, 22]. To introduce, the SAA approach, let
$\xi^{1}$, $\xi^{2},\ldots$ be independent identically distributed
$\Xi$-valued random elements such that each $\xi^{i}$ has distribution
$\mathbb{P}$. The SAA problem with sample size $N\in\mathbb{N}$ corresponding
to (1.1) is given by
$\displaystyle\min_{x\in X}\,\frac{1}{N}\sum_{i=1}^{N}F_{\xi^{i}}(x)+\psi(x).$
(1.2)
The principle contributions of this manuscript are twofold.
1. (i)
We demonstrate the consistency of approximate C-stationary points of the SAA
problem (1.2) for a class of risk-neutral nonlinear optimization problems
(1.1) by adapting and extending a framework developed in [59], where the
consistency of SAA optimal values and SAA solutions has been demonstrated.
More specifically, our analysis relies on the identity
$\displaystyle\partial_{C}F_{\xi}(x)=BM_{\xi}(x)\quad\text{for
all}\quad(x,\xi)\in X\times\Xi,$ (1.3)
where $\partial_{C}F_{\xi}(x)$ is Clarke’s subdifferential of $F_{\xi}$ at
$x$, $V$ is a Banach space, $B\colon V\to X^{*}$ is linear and compact, and
$M_{\xi}\colon X\rightrightarrows V$ is a multifunction for each $\xi\in\Xi$.
Moreover, using proof techniques developed in [62], we establish consistency
statements for the significantly larger class of functions $\psi$ than those
considered in [59], so-called functions having the Kadec property [9].
2. (ii)
We apply our framework to risk-averse semilinear PDE-constrained optimization
problems using the average value-at-risk and to steady-state elliptic bilinear
optimal control problems under uncertainty. While the literature on risk-
neutral and risk-averse semilinear PDE-constrained optimization is extensive
[24, 48, 50, 64], bilinear PDE-constrained problems under uncertainty have not
been analyzed, but serve as yet another class of nonconvex problems and arise
in many applications.
The consistency of SAA solutions and SAA stationary points of nonlinear
possibly infinite-dimensional stochastic programs are typically based on some
form of the feasible set’s compactness. In particular, the consistency results
for SAA stationary points of infinite-dimensional stochastic programs
established in [5, 78] require the feasible set’s compactness. However,
compact sets in infinite dimensions may regarded as rare phenomena, given the
fact that closed unit balls in Banach spaces are compact if and only if the
spaces are finite-dimensional. This fact complicates the consistency analysis
for nonconvex stochastic programs. For strongly convex risk-neutral PDE-
constrained optimization, the SAA approach has been analyzed in [38, 58, 71,
60, 61] without requiring the compactness of the feasible set.
Contributions to analyzing the SAA approach for optimization problems governed
by nonlinear operator equations with random inputs are relatively recent. The
SAA objective function’s almost sure epiconvergence and the SAA stationary
points’ weak consistency is analyzed in [68] for optimization problems
governed by ordinary differential equations with random inputs. Utilizing
epiconvergence, performance guarantees of optimal values of PDE-constrained
optimization problems under uncertainty are provided in [16] with respect to
sample average approximations and control and state space discretizations, for
example. The asymptotic consistency of SAA optimal values and SAA solutions to
risk-averse PDE-constrained optimization problems has recently been
demonstrated in [62] using (Mosco-)epiconvergence, and sample size estimates
for risk-neutral semilinear PDE-constrained problems are derived in [64].
We use the compact operator $B$ in (1.3) to mathematically model problem
structure typically found in PDE-constrained optimization problems. Related
problem characteristics are used for establishing error estimates for finite-
dimensional approximations to PDE-constrained optimization problems [13, 20,
52] and for algorithmic design and analysis in function space [33, 36, 35,
80], for example.
The analysis of bilinear PDE-constrained optimization problems is generally
more challenging than those of their linear counterparts, as bilinear PDEs may
lack solutions for each point in the control space and bilinear control
problems are generally nonconvex. Bilinear control problems can be governed by
steady-state elliptic PDEs [52, 81], convection-diffusion equations [11], and
advection-reaction-diffusion equations [27], for instance. A substantial body
of literature is available on the analysis, algorithmic design, and
applications of deterministic bilinear control problems. We refer the reader
to [14, 27, 31] for the analysis of deterministic bilinear control problems.
Finite element error estimates are established in [14, 23, 52, 82, 84] and
algorithms are designed in [11, 27, 41, 43, 53, 81]. Bilinear control problems
arise in medical image analysis [57], damping design [75], groundwater
remediation system design [30], optical flow [40], PDE-constrained regression
problems [65], and chemorepulsion production [31], for example.
## Outline
We introduce notation and terminology in section 2. Section 3 establishes
consistency of C-stationary points of optimization problems obtained through
approximations of infinite-dimensional nonconvex optimization posed in
reflexive Banach spaces. This basic result is used to establish consistency of
SAA C-stationary points in section 4 and of C-stationary points of approximate
stochastic programs defined by weakly convergent probability measures in
section 5. In section 6, we apply the framework developed in section 4 to
risk-averse semilinear PDE-constrained optimization using the average value-
at-risk. Section 7 discusses the application of our theory to a risk-neutral
bilinear PDE-constrained optimization problem. We summarize our contributions
and discuss open research questions in section 8.
## 2 Notation and preliminaries
Metric spaces are defined over the real numbers and equipped with their Borel
sigma-algebra if not specified otherwise. For a Hilbert space $H$, we denote
by $(\cdot,\cdot)_{H}$ its inner product. Let $X_{1}$ and $X_{2}$ be Banach
spaces. The space of linear, bounded operators from $X_{1}$ to $X_{2}$ is
denoted by $\mathscr{L}(X_{1},X_{2})$. An operator
$A\in\mathscr{L}(X_{1},X_{2})$ is compact if the image $A(W)$ is precompact in
$X_{2}$ for each bounded set $W\subset X_{1}$. The operator
$A^{*}\in\mathscr{L}(X_{2}^{*},X_{1}^{*})$ is the adjoint operator of
$A\in\mathscr{L}(X_{1},X_{2})$. We denote by $\mathrm{D}f$ the Fréchet
derivative of $f$ and by $\mathrm{D}_{y}f$ or $f_{y}$ the Fréchet derivative
with respect to $y$. The dual to a Banach space $\Lambda$ is $\Lambda^{*}$ and
we use $\langle\cdot,\cdot\rangle_{{\Lambda}^{*}\\!,\Lambda}$ to denote the
dual pairing between $\Lambda^{*}$ and $\Lambda$. Let $X_{0}\subset X$ be an
open, nonempty subset of a Banach space $X$. We denote Clarke’s generalized
directional derivative of a locally Lipschitz continuous mapping $f\colon
X_{0}\to$ at $x\in X_{0}$ in the direction $h\in X$ by $f^{\circ}(x;h)$ and
Clarke’s subdifferential at $x$ by $\partial_{C}f(x)\subset X^{*}$ [18, pp. 25
and 27]. We refer the reader to [18, p. 39] for the definition of Carke
regularity of a function $f$. We use $\partial f$ to denote the usual convex
subdifferential of a function $f$. We define the distance
$\mathrm{dist}({v,\Upsilon})$ from $v\in\Lambda\subset X$ to $\Upsilon\subset
X$ by
$\displaystyle\mathrm{dist}({v,\Upsilon})=\inf_{w\in\Upsilon}\,\|v-w\|_{X}\quad\text{if}\quad\Upsilon\neq\emptyset,\quad\text{and}\quad\mathrm{dist}({v,\Upsilon})=\infty\quad\text{otherwise}.$
and the deviation $\mathbb{D}({\Lambda,\Upsilon})$ between the sets $\Lambda$
and $\Upsilon$ by
$\mathbb{D}({\Lambda,\Upsilon})=\sup_{v\in\Lambda}\,\mathrm{dist}({v,\Upsilon})$.
Here $\|\cdot\|_{X}$ is a norm of $X$. If $\phi\colon X\to(-\infty,\infty]$,
then $\mathrm{dom}(\phi)=\\{x\in X\colon\phi(x)<\infty\\}$ denotes its domain.
Following [9, p. 147] and [10, p. 306], we say that a proper function
$\phi\colon X\to(-\infty,\infty]$ has the Kadec property if for each
$\bar{x}\in\mathrm{dom}(\phi)$, we have $x_{k}\to\bar{x}$ whenever
$\phi(x_{k})\to\phi(\bar{x})$, $(x_{k})\subset\mathrm{dom}(\psi)$, and
$x_{k}\rightharpoonup\bar{x}$. Here $\to$ denotes strong convergence and
$\rightharpoonup$ denotes weak convergence. For $x\in X$ and $\varepsilon\geq
0$, we denote by $\mathbb{B}_{X}(x;\varepsilon)$ the open ball about $x$ in
$X$ with radius $\varepsilon$ and by
$\overline{\mathbb{B}}_{X}(x;\varepsilon)$ its closure in $X$.
The following notation is used in sections 7 and 6. Let $D\subset^{d}$ be a
bounded domain. The space $C^{0,1}(\bar{D})$ is the set of Lipschitz
continuous functions on $\bar{D}$, $L^{p}(D)$ ($1\leq p\leq\infty$) and
$H^{1}(D)$, $H^{2}(D)$, and $H_{0}^{1}(D)$ are the “usual” Lebesgue and
Sobolev spaces, respectively. The space $H_{0}^{1}(D)$ is equipped with the
norm $\|v\|_{H_{0}^{1}(D)}=\|\nabla v\|_{L^{2}(D)^{d}}$ and the dual to
$H_{0}^{1}(D)$ is denoted by $H^{-1}(D)$. Moreover, let $C_{D}\in(0,\infty)$
be Friedrichs’ constant of $D$. If $X$ is a reflexive Banach space, then we
identify $(X^{*})^{*}$ with $X$ and write $(X^{*})^{*}=X$. Moreover, we
identify $L^{2}(D)^{*}$ with $L^{2}(D)$ and write $L^{2}(D)^{*}=L^{2}(D)$. We
denote by $\iota_{0}\colon H_{0}^{1}(D)\to L^{2}(D)$ the embedding operator of
the compact embedding $H_{0}^{1}(D)\xhookrightarrow{}L^{2}(D)$ and by
$\iota_{1}\colon H^{1}(D)\to L^{2}(D)$ that of
$H^{1}(D)\xhookrightarrow{}L^{2}(D)$. We introduce further application-
specific notation in section 7.
## 3 Consistency of C-stationary points
We consider the optimization problem
$\displaystyle\min_{x\in\mathrm{dom}(\psi)}\,f(x)+\psi(x)$ (3.1)
and its approximations
$\displaystyle\min_{x\in\mathrm{dom}(\psi)}\,f_{k}(x)+\psi(x).$ (3.2)
We explicitly add $\mathrm{dom}(\psi)$ as a constraint sets in (3.1) and
(3.2), as $f$ and $f_{k}$ may only be finite-valued on an open neighborhood of
$\mathrm{dom}(\psi)$. We provide conditions sufficient for the asymptotic
consistency of C-stationary points of the approximated problems (3.2) to those
of “true” problem (3.1). Our analysis is inspired by those in [72, 74, 17, 62]
and based on technical assumptions used in [59]. Using these consistency
results, we establish asymptotic consistency of C-stationary points of
infinite-dimensional stochastic programs in sections 4 and 5.
###### Assumption 3.1.
1. 1.
The spaces $X$ and $V$ are Banach spaces, $X$ is reflexive, and $X_{0}\subset
X$ is nonempty and open.
2. 2.
The function $\psi\colon X\to(-\infty,\infty]$ is proper, convex, and lower
semicontinuous. Moreover, $\psi$ has the Kadec property and
$\mathrm{dom}(\psi)\subset X_{0}$.
3. 3.
The functions $f\colon X_{0}\to$ and $f_{k}\colon X_{0}\to$, $k\in\mathbb{N}$,
are locally Lipschitz continuous.
4. 4.
The operator $B\in\mathscr{L}(V,X^{*})$ is compact,
$M_{k}\colon\mathrm{dom}(\psi)\rightrightarrows V$ is a set-valued mapping,
and $\partial_{C}f_{k}(x)=BM_{k}(x)$ for all $x\in\mathrm{dom}(\psi)$ and
$k\in\mathbb{N}$.
5. 5.
For each $\bar{x}\in\mathrm{dom}(\psi)$, each sequence
$(x_{k})\subset\mathrm{dom}(\psi)$ with $x_{k}\rightharpoonup\bar{x}$, and
each sequence $(v_{k})$ with $v_{k}\in M_{k}(x_{k})$ for all $k\in\mathbb{N}$,
$\limsup_{k\to\infty}\,\|v_{k}\|_{V}<\infty$.
6. 6.
For each $h\in X$, each $\bar{x}\in\mathrm{dom}(\psi)$, and each sequence
$(x_{k})\subset\mathrm{dom}(\psi)$ with $x_{k}\to\bar{x}$, it holds that
$\limsup_{k\to\infty}f_{k}^{\circ}(x_{k};h)\leq f^{\circ}(\bar{x};h)$.
Under 3.1.1, 3.1.3, and 3.1.2, Clarke’s subdifferential $\partial_{C}f_{k}(x)$
for each $x\in X_{0}$ is nonempty, convex, bounded, and weakly∗-compact [18,
Prop. 2.1.2]. Hence the set-valued mapping $M_{k}$ in 3.1.4 is nonempty-
valued. If $M_{k}$ has bounded images, then 3.1.4 ensures that
$\partial_{C}f_{k}(x)$ is precompact-valued. For Monte Carlo sample-based
approximations of stochastic programs, we verify 3.1.6 using an epigraphical
law of large numbers [3], basic properties of Clarke’s generalized directional
derivative, and by imposing Clarke regularity of integrands studied in section
4.
3.1.4 is satisfied for large problem classes as demonstrated next.
###### Example 3.2.
Let 3.1.1, 3.1.2, and 3.1.3 hold. If $W$ is a Banach space,
$\iota\in\mathscr{L}(X,W)$ is compact, $F_{k}\colon W\to$ is locally Lipschitz
continuous and Clarke regular, and $f_{k}\colon X\to$ is defined by
$f_{k}(x)=F_{k}(\iota x)$, then
$\partial_{C}f_{k}(x)=\iota^{*}\partial_{C}F_{k}(\iota x)$ [18, Thm. 2.3.10],
and we may choose $B=\iota^{*}$ and $M_{k}(x)=\partial_{C}F_{k}(\iota x)$ to
satisfy 3.1.4. Composite objective functions with compact linear operators
arise in nonsmooth optimal control [32, 33, 35, 76], for example.
Next we discuss 3.1.5 and 3.1.4 when $X=^{n}$.
###### Remark 3.3.
Let 3.1.1, 3.1.2, and 3.1.3 hold. Let $X=^{n}$ (equipped with the Euclidean
norm $\|\cdot\|_{2}$) and let $X^{*}=V$. We identify the dual to n with n. Let
$B\colon^{n}\to^{n}$ be the identity mapping. Then 3.1.4 holds true with
$M_{k}=\partial_{C}f_{k}$. Next we discuss 3.1.5. Let
$\bar{x}\in\mathrm{dom}(\psi)$ and let $L_{k}\geq 0$ be Lipschitz constant of
$f_{k}$ on $\mathbb{B}_{X}(\bar{x};\varepsilon)$, where
$\varepsilon=\varepsilon(\bar{x})>0$. If $\limsup_{k\to\infty}\,L_{k}<\infty$,
then 3.1.5 holds true. Let us verify this assertion. Let
$(x_{k})\subset\mathrm{dom}(\psi)$ with $x_{k}\rightharpoonup\bar{x}$, and let
$(v_{k})$ with $v_{k}\in M_{k}(x_{k})$ for all $k\in\mathbb{N}$. Then
$x_{k}\to\bar{x}$ and there exists $K=K(\bar{x})$ such that
$x_{k}\in\mathbb{B}_{X}(\bar{x};\varepsilon)$ for all $k\geq K$. Since
$\partial_{C}f_{k}(x_{k})=M_{k}(x_{k})$, [18, Prop. 2.1.2] ensures
$\|v_{k}\|_{2}\leq L_{k}$ for all $k\geq K$. Hence
$\limsup_{k\to\infty}\,\|v_{k}\|_{2}<\infty$.
Theorem 3.4 establishes consistency of approximate C-stationary points. Let
$\mathcal{C}$ be the set of C-stationary points of (3.1), that is,
$\mathcal{C}$ is the set of all points $x\in\mathrm{dom}(\psi)$ with
$0\in\partial_{C}f(x)+\partial\psi(x)$. Let $\varepsilon\geq 0$ and let
$\mathcal{C}_{k}^{\varepsilon}$ be the set of all points
$x\in\mathrm{dom}(\psi)$ with
$0\in\bigcup_{y\in\overline{\mathbb{B}}_{X}(x;\varepsilon)\cap\mathrm{dom}(\psi)}\partial_{C}f_{k}(y)+\partial\psi(x)$
(cf. [74, eqns. (3.2) and (4.2)]). Each point in
$\mathcal{C}_{k}^{\varepsilon}$ is referred to as an approximate C-stationary
point of (3.2).
###### Theorem 3.4.
Let Assumption 3.1 hold and let $(\varepsilon_{k})\subset[0,\infty)$ satisfy
$\varepsilon_{k}\to 0$ as $k\to\infty$. Moreover, let
$\mathcal{C}_{k}^{\varepsilon_{k}}$ be nonempty for each $k\in\mathbb{N}$.
1. (i)
If $(x_{k})$ is a bounded sequence with
$x_{k}\in\mathcal{C}_{k}^{\varepsilon_{k}}$ for each $k\in\mathbb{N}$, then
$\mathcal{C}$ is nonempty and $\mathrm{dist}({x_{k},\mathcal{C}})\to 0$ as
$k\to\infty$.
2. (ii)
If there exists a bounded set $\mathscr{C}\subset X$ with
$\mathcal{C}_{k}^{\varepsilon_{k}}\subset\mathscr{C}$ for all sufficiently
large $k\in\mathbb{N}$, then
$\mathbb{D}({\mathcal{C}_{k}^{\varepsilon_{k}},\mathcal{C}})\to 0$ as
$k\to\infty$.
###### Proof.
3.1.3 ensures that the Clarke subdifferentials of $f$ and $f_{k}$ are well-
defined [18, Prop. 2.1.2].
(i) Let $(\mathrm{dist}({x_{k},\mathcal{C}}))_{K_{0}}$ be a subsequence of
$(\mathrm{dist}({x_{k},\mathcal{C}}))\subset[0,\infty]$. Since
$x_{k}\in\mathcal{\mathcal{C}}_{k}^{\varepsilon_{k}}$, there exists
$y_{k}\in\mathrm{dom}(\psi)$ with $\|x_{k}-y_{k}\|_{X}\leq\varepsilon_{k}$ and
$g_{k}\in\partial_{C}f_{k}(y_{k})$ with
$\displaystyle\langle
g_{k},x-x_{k}\rangle_{{X}^{*}\\!,X}+\psi(x)\geq\psi(x_{k})\quad\text{for
all}\quad x\in X.$ (3.3)
The boundedness of $(x_{k})$ implies that of $(x_{k})_{K_{0}}$. Combined with
the fact that $X$ is reflexive (see 3.1.1) and that $\mathrm{dom}(\psi)$ is
weakly sequentially closed (see 3.1.2 and [7, Thm. 2.23 (ii)]), we find that
$(x_{k})_{K_{0}}$ has a subsequence $(x_{k})_{K_{1}}$ with
$x_{k}\rightharpoonup\bar{x}\in\mathrm{dom}(\psi)$ as $K_{1}\ni k\to\infty$.
Since $\|x_{k}-y_{k}\|_{X}\leq\varepsilon_{k}$ and $\varepsilon_{k}\to 0$, we
find that $y_{k}\rightharpoonup\bar{x}$ as $K_{1}\ni k\to\infty$. Since
$g_{k}\in\partial_{C}f_{k}(x_{k})$, 3.1.4 ensures $g_{k}=Bv_{k}$ for some
$v_{k}\in M_{k}(y_{k})$. 3.1.5 implies that $(v_{K})_{K_{1}}$ is bounded.
Combined with the compactness of $B$ (see 3.1.4), we obtain that
$(g_{k})_{K_{1}}\subset X^{*}$ has a further subsequence $(g_{k})_{K_{2}}$
with $g_{k}\to\bar{g}$ as $K_{2}\ni k\to\infty$ [51, Thm. 8.1-5]. Hence
$\langle
g_{k},x-x_{k}\rangle_{{X}^{*}\\!,X}\to\langle\bar{g},x-\bar{x}\rangle_{{X}^{*}\\!,X}$
as $K_{2}\ni k\to\infty$ [7, Thm. 2.23 (iv)]. 3.1.2 implies that $\psi$ is
weakly lower semicontinuous. Hence $\liminf_{K_{2}\ni
k\to\infty}\,\psi(x_{k})\geq\psi(\bar{x})$. Putting together the pieces and
using (3.3), we have
$\displaystyle\langle\bar{g},x-\bar{x}\rangle_{{X}^{*}\\!,X}+\psi(x)\geq\psi(\bar{x})\quad\text{for
all}\quad x\in X.$ (3.4)
Using (3.3) once more, we obtain $\psi(\bar{x})=\lim_{K_{2}\ni
k\to\infty}\langle
g_{k},\bar{x}-x_{k}\rangle_{{X}^{*}\\!,X}+\psi(\bar{x})\geq\limsup_{K_{2}\ni
k\to\infty}\psi(x_{k})$. Hence $\psi(x_{k})\to\psi(\bar{x})$ as $K_{2}\ni
k\to\infty$. Since $\psi$ is Kadec according to 3.1.2, $x_{k}\to\bar{x}$ as
$K_{2}\ni k\to\infty$. Since $\|x_{k}-y_{k}\|_{X}\leq\varepsilon_{k}$, we also
have $y_{k}\to\bar{x}$ as $K_{2}\ni k\to\infty$.
We define $\widetilde{x}_{k}=y_{k}$ if $k\in K_{2}$ and
$\widetilde{x}_{k}=\bar{x}$ otherwise. We have $\widetilde{x}_{k}\to\bar{x}$
as $k\to\infty$. Fix $h\in X$. Since $g_{k}\in\partial_{C}f_{k}(y_{k})$,
$f_{k}^{\circ}(y_{k};h)\geq\langle g_{k},h\rangle_{{X}^{*}\\!,X}$ [18, Prop.
2.1.5]. Combined with 3.1.6, we have
$\displaystyle\langle\bar{g},h\rangle_{{X}^{*}\\!,X}=\lim_{K_{2}\ni
k\to\infty}\,\langle g_{k},h\rangle_{{X}^{*}\\!,X}\leq\limsup_{K_{2}\ni
k\to\infty}\,f_{k}^{\circ}(y_{k};h)\leq\limsup_{k\to\infty}\,f_{k}^{\circ}(\widetilde{x}_{k};h)\leq
f^{\circ}(\bar{x};h).$
Hence $\bar{g}\in\partial_{C}f(\bar{x})$ [18, Prop. 2.1.5]. Now (3.4) ensures
$0\in\partial_{C}f(\bar{x})+\partial\psi(\bar{x})$, that is,
$\bar{x}\in\mathcal{C}$. Hence
$\mathrm{dist}({x_{k},\mathcal{C}})\leq\|x_{k}-\bar{x}\|_{X}\to 0$ as
$K_{2}\ni k\to 0$, and $(\mathrm{dist}({x_{k},\mathcal{C}}))$ is bounded. Our
derivations also show that each subsequence of
$(\mathrm{dist}({x_{k},\mathcal{C}}))$ has a further subsequence converging to
zero. Hence $\mathrm{dist}({x_{k},\mathcal{C}})\to 0$ as $k\to\infty$.
(ii) Since $\mathcal{C}_{k}^{\varepsilon_{k}}\subset\mathscr{C}$, and
$\mathcal{C}_{k}^{\varepsilon_{k}}$ and $\mathcal{C}$ are nonempty, and
$\mathscr{C}$ is bounded, we have
$\mathbb{D}({\mathcal{C}_{k}^{\varepsilon_{k}},\mathcal{C}})\leq\mathbb{D}({\mathscr{C},\mathcal{C}})<\infty$
for all sufficiently large $k\in\mathbb{N}$. Hence for all sufficiently large
$k\in\mathbb{N}$, there exists $x_{k}\in\mathcal{C}_{k}^{\varepsilon_{k}}$
with
$\mathbb{D}({\mathcal{C}_{k}^{\varepsilon_{k}},\mathcal{C}})\leq\mathrm{dist}({x_{k},\mathcal{C}})+1/k$.
Using part (i), we have $\mathrm{dist}({x_{k},\mathcal{C}})\to 0$ as
$k\to\infty$. Hence
$\mathbb{D}({\mathcal{C}_{k}^{\varepsilon_{k}},\mathcal{C}})\to 0$ as
$k\to\infty$. ∎
## 4 Empirical approximations via Monte Carlo sampling
Let $\Xi$ be a complete, separable metric space, and let
$(\Xi,\mathcal{A},\mathbb{P})$ be a complete probability space. Let $\xi^{1}$,
$\xi^{2},\ldots$ be independent identically distributed $\Xi$-valued random
elements defined on a complete probability space $(\Omega,\mathcal{F},P)$ such
that each $\xi^{i}$ has distribution $\mathbb{P}$.
We consider the risk-neutral optimization problem
$\displaystyle\min_{x\in\mathrm{dom}(\psi)}\,\int_{\Xi}\,F_{\xi}(x)\,\mathrm{d}\mathbb{P}(\xi)+\psi(x)$
(4.1)
and its SAA problem
$\displaystyle\min_{x\in\mathrm{dom}(\psi)}\,\frac{1}{N}\sum_{i=1}^{N}F_{\xi^{i}}(x)+\psi(x).$
(4.2)
We study the almost sure convergence of SAA C-stationary points of (4.2) to
the C-stationary point’s set of (4.1). We analyze the consistency under
conditions related to those in Assumption 3.1.
###### Assumption 4.1.
1. 1.
The spaces $X$ and $V$ are separable Banach spaces, $X$ is reflexive, and
$X_{0}\subset X$ is nonempty and open.
2. 2.
The function $\psi\colon X\to(-\infty,\infty]$ is proper, convex, and lower
semicontinuous. Moreover, $\psi$ has the Kadec property and
$\mathrm{dom}(\psi)\subset X_{0}$.
3. 3.
For each $\bar{x}\in\mathrm{dom}(\psi)$, there exists an open neighborhood
$\mathcal{V}_{\bar{x}}\subset X_{0}$ of $\bar{x}$ and a
$\mathbb{P}$-integrable random variable $L_{\bar{x}}\colon\Xi\to[0,\infty)$
such that for each $\xi\in\Xi$, $F_{\xi}\colon X_{0}\to$ is Lipschitz
continuous on $\mathcal{V}_{\bar{x}}$ with Lipschitz constant
$L_{\bar{x}}(\xi)$.
4. 4.
For each $\xi\in\Xi$, $F_{\xi}\colon X_{0}\to$ is Clarke regular.
5. 5.
For each $x\in X_{0}$, $\Xi\ni\xi\mapsto F_{\xi}(x)\in$ is measurable and
$\mathbb{P}$-integrable.
6. 6.
The operator $B\in\mathscr{L}(V,X^{*})$ is compact,
$M_{\xi}\colon\mathrm{dom}(\psi)\rightrightarrows V$ is a set-valued map with
nonempty images for each $\xi\in\Xi$, and $\partial_{C}F_{\xi}(x)=BM_{\xi}(x)$
for each $(x,\xi)\in\mathrm{dom}(\psi)\times\Xi$.
7. 7.
There exists $\Omega_{0}\subset\Omega$ with $\Omega_{0}\in\mathcal{F}$ and
$P(\Omega_{0})=1$ such that for each $\bar{x}\in\mathrm{dom}(\psi)$, each
sequence $(x_{N})\subset\mathrm{dom}(\psi)$ with
$x_{N}\rightharpoonup\bar{x}$, and each sequence $(v_{N})\subset V$, we have
$\limsup_{N\to\infty}\,\|v_{N}\|_{V}<\infty$ whenever $\omega\in\Omega_{0}$
and $v_{N}\in(1/N)\sum_{i=1}^{N}M_{\xi^{i}(\omega)}(x_{N})$ for each
$N\in\mathbb{N}$.
4.1.1 and 4.1.2 correspond to 3.1.1 and 3.1.2, but we impose in addition
separability of $X$ and $V$. 4.1.1, 4.1.4, and 4.1.3 allow us to use Clarke’s
calculus for generalized gradients of integral functions [18, sect. 2.7]. To
apply the epigraphical law of large numbers derived in [3], we use 4.1.4 and
4.1.3. 4.1.6 and 4.1.7 generalize technical conditions used in [59].
###### Lemma 4.2.
Let $\zeta\colon\Xi\to[0,\infty)$ be measurable and $\mathbb{P}$-integrable,
and let 4.1.1, 4.1.6, and 4.1.2 hold. If $\|m_{\xi}(x)\|_{V}\leq\zeta(\xi)$
for all $m_{\xi}(x)\in M_{\xi}(x)$, each $x\in\mathrm{dom}(\psi)$, and every
$\xi\in\Xi$, then 4.1.7 holds true.
###### Proof.
Since $\zeta(\xi^{1}),\zeta(\xi^{2}),\ldots$ are independent, the law of large
numbers ensures that $(1/N)\sum_{i=1}^{N}\zeta(\xi^{i})$ converges almost
surely to $\int_{\Xi}\zeta(\xi)\,\mathrm{d}\mathbb{P}(\xi)$ as $N\to\infty$.
Hence there exists a set $\Omega_{0}\subset\Omega$ with
$\Omega_{0}\in\mathcal{F}$, $P(\Omega_{0})=1$, and for each
$\omega\in\Omega_{0}$, we have
$(1/N)\sum_{i=1}^{N}\zeta(\xi^{i}(\omega))\to\int_{\Xi}\zeta(\xi)\,\mathrm{d}\mathbb{P}(\xi)$
as $N\to\infty$.
Fix $\omega\in\Omega_{0}$. Let $\bar{x}\in\mathrm{dom}(\psi)$, let
$(x_{N})\subset\mathrm{dom}(\psi)$ fulfill $x_{N}\rightharpoonup\bar{x}$, and
let $(v_{N})$ satisfy $v_{N}\in(1/N)\sum_{i=1}^{N}M_{\xi^{i}(\omega)}(x_{N})$
for all $N\in\mathbb{N}$. The latter ensures the existence of $v_{N}^{i}\in
M_{\xi^{i}(\omega)}(x_{N})$ with $v_{N}=(1/N)\sum_{i=1}^{N}v_{N}^{i}$. Now the
lemma’s hypotheses ensure $\|v_{N}^{i}\|_{V}\leq\zeta(\xi^{i})$. Hence
$\|v_{N}\|_{V}\leq(1/N)\sum_{i=1}^{N}\|v_{N}^{i}\|_{V}\leq(1/N)\sum_{i=1}^{N}\zeta(\xi^{i})$.
We obtain $\limsup_{N\to\infty}\,\|v_{N}\|_{V}<\infty$. ∎
We impose a technical measurability condition on Clarke’s generalized
directional derivative of $F_{\xi}$ in Theorem 4.3. The measurability of
Clarke’s subdifferential as a function of the decision variables and
parameters has been analyzed, for example, in [66, Lem. 4] and [83, Prop.
2.1].
Let $\mathcal{C}$ be the set of all C-stationary points of (4.1), that is, the
set of all $x\in\mathrm{dom}(\psi)$ with
$0\in\partial_{C}\big{[}\int_{\Xi}\,F_{\xi}(x)\,\mathrm{d}\mathbb{P}(\xi)\big{]}+\partial\psi(x)$.
Let $\varepsilon\geq 0$ and for each $\omega\in\Omega$, let
$\hat{\mathcal{C}}_{N}^{\varepsilon}(\omega)$ be the set of all points
$x\in\mathrm{dom}(\psi)$ with
$0\in\bigcup_{v\in\overline{\mathbb{B}}_{X}(x;\varepsilon)\cap\mathrm{dom}(\psi)}\partial_{C}\Big{[}\frac{1}{N}\sum_{i=1}^{N}F_{\xi^{i}(\omega)}(v)\Big{]}+\partial\psi(x);$
cf. [74, eqns. (3.2) and (4.2)]. Each element in
$\hat{\mathcal{C}}_{N}^{\varepsilon}(\omega)$ is referred to as an approximate
C-stationary point of (4.2).
###### Theorem 4.3.
Let Assumption 4.1 hold true and let $(\varepsilon_{N})\subset[0,\infty)$ be a
sequence with $\varepsilon_{N}\to 0$ as $N\to\infty$. Suppose that
$\mathrm{dom}(\psi)\times X\times\Xi\ni(x,h,\xi)\mapsto F_{\xi}^{\circ}(x;h)$
is measurable, and let $\mathcal{C}$ and
$\hat{\mathcal{C}}_{N}^{\varepsilon_{N}}(\omega)$ be nonempty for each
$N\in\mathbb{N}$ and $\omega\in\Omega$.
1. (i)
If $\Omega_{1}\subset\Omega$ with $\Omega_{1}\in\mathcal{F}$ and
$P(\Omega_{1})=1$ and for each $\omega\in\Omega_{1}$, $(x_{N}(\omega))$ is a
bounded sequence with
$x_{N}(\omega)\in\hat{\mathcal{C}}_{N}^{\varepsilon_{N}}(\omega)$ for all
sufficiently large $N\in\mathbb{N}$, then with probability one (w.p. $1$),
$\mathrm{dist}({x_{N},\mathcal{C}})\to 0$ as $N\to\infty$.
2. (ii)
If there exists a bounded set $\mathscr{C}\subset X$ such that w.p. $1$,
$\hat{\mathcal{C}}_{N}^{\varepsilon_{N}}\subset\mathscr{C}$ for all
sufficiently large $N\in\mathbb{N}$, then w.p. $1$,
$\mathbb{D}({\hat{\mathcal{C}}_{N}^{\varepsilon_{N}},\mathcal{C}})\to 0$ as
$N\to\infty$.
###### Proof.
We establish the assertions via applications of Theorem 3.4.
First, we observe that for each $\omega\in\Omega_{0}$, 4.1.7 implies 3.1.5
with $M_{k}(\cdot)$ replaced by
$\hat{M}_{N}(\cdot,\omega)=(1/N)\sum_{i=1}^{N}M_{\xi^{i}(\omega)}(\cdot)$.
Second, 4.1.4, 4.1.3, and 4.1.6, and the sum rule [18, Cor. 3 on p. 40] ensure
for all $\omega\in\Omega$, $x\in X_{0}$, and every $N\in\mathbb{N}$,
$\displaystyle\partial_{C}\Big{[}\frac{1}{N}\sum_{i=1}^{N}F_{\xi^{i}(\omega)}(x)\Big{]}=\frac{1}{N}\sum_{i=1}^{N}\partial_{C}F_{\xi^{i}(\omega)}(x)=\frac{1}{N}\sum_{i=1}^{N}BM_{\xi^{i}(\omega)}(x)=B\hat{M}_{N}(x,\omega).$
Hence for all $\omega\in\Omega$, 3.1.4 holds true with $M_{k}(\cdot)$ replaced
by $\hat{M}_{N}(\cdot,\omega)$.
Third, we show that for almost every $\omega\in\Omega$, 3.1.6 is satisfied
with $f(\cdot)$ replaced by the expectation function
$\int_{\Xi}F_{\xi}(\cdot)\,\mathrm{d}\mathbb{P}(\xi)$ and with $f_{k}(\cdot)$
replaced by the SAA objective function
$(1/N)\sum_{i=1}^{N}F_{\xi^{i}(\omega)}(\cdot)$. For each $\xi\in\Xi$,
$X_{0}\times X\ni(x,h)\mapsto F_{\xi}^{\circ}(x;h)$ is upper semicontinuous
[18, Prop. 2.1.1]. Fix $\bar{x}\in\mathrm{dom}(\psi)$. Let
$\mathcal{V}_{\bar{x}}$ be an open neighborhood and let $L_{\bar{x}}$ be a
Lipschitz constant given by 4.1.3. Using [18, Prop. 2.1.1], we have
$|F_{\xi}^{\circ}(x;h)|\leq L_{\bar{x}}(\xi)$ for each
$(x,h,\xi)\in\mathcal{V}_{\bar{x}}\times X\times\Xi$. Moreover
$(x,h,\xi)\mapsto F_{\xi}^{\circ}(x;h)$ is measurable by assumption. Combined
with [3, Thm. 2.3], we find that w.p. $1$, $\mathrm{dom}(\psi)\times
X\ni(x,h)\mapsto(1/N)\sum_{i=1}^{N}F_{\xi^{i}}^{\circ}(x;h)$ hypoconverges to
$(x,h)\mapsto\int_{\Xi}F_{\xi}^{\circ}(x;h)\,\mathrm{d}\mathbb{P}(\xi)$.111We
refer the reader to [3, p. 3] for a definition of epiconvergence of a sequence
of functions $(g_{k})$ to some function $g$. A sequence of functions $(g_{k})$
hypoconverges to $g$ if (and only if) $(-g_{k})$ epiconverges to $-g$ [70, pp.
242–243]. Using 4.1.1, [18, Cor. 3 on p. 40 and Thm. 2.7.2] and Clarke
regularity of $F_{\xi}$ for each $\xi\in\Xi$ (see 4.1.4), we have the
identities
$\int_{\Xi}F_{\xi}^{\circ}(x;h)\,\mathrm{d}\mathbb{P}(\xi)=(\int_{\Xi}F_{\xi}(\cdot)\,\mathrm{d}\mathbb{P}(\xi))^{\circ}(x;h)$
and
$(1/N)\sum_{i=1}^{N}F_{\xi^{i}}^{\circ}(x;h)=\big{(}(1/N)\sum_{i=1}^{N}F_{\xi^{i}}(\cdot)\big{)}^{\circ}(x;h)$
for every $(x,h)\in X_{0}\times X$. Hence there exists
$\Omega_{2}\subset\Omega$ with $\Omega_{2}\in\mathcal{F}$ and
$P(\Omega_{2})=1$ such that for each $\omega\in\Omega_{2}$, all $h\in X$ and
each $(x_{N})\subset\mathrm{dom}(\psi)$ with $x_{N}\to x$, we have
$\limsup_{N\to\infty}\,\Big{(}\frac{1}{N}\sum_{i=1}^{N}F_{\xi^{i}(\omega)}(\cdot)\Big{)}^{\circ}(x_{N};h)\leq\Big{(}\int_{\Xi}F_{\xi}(\cdot)\,\mathrm{d}\mathbb{P}(\xi)\Big{)}^{\circ}(x;h).$
Now we establish parts (i) and (ii).
(i) By assumption and construction, we have
$\Omega_{0}\cap\Omega_{1}\cap\Omega_{2}\in\mathcal{F}$. Moreover this event
happens w.p. $1$. Combining our derivations, Theorem 3.4 ensures for each
$\omega\in\Omega_{0}\cap\Omega_{1}\cap\Omega_{2}$, we have
$\mathrm{dist}({x_{N}(\omega),\mathcal{C}})\to 0$ as $N\to\infty$.
(ii) Since w.p. $1$,
$\hat{\mathcal{C}}_{N}^{\varepsilon_{N}}\subset\mathscr{C}$ for all
sufficiently large $N\in\mathbb{N}$, there exists a set
$\Omega_{3}\in\mathcal{F}$ with $P(\Omega_{3})=1$ such that for each
$\omega\in\Omega_{3}$, there exists $n(\omega)\in\mathbb{N}$ with
$\hat{\mathcal{C}}_{N}^{\varepsilon_{N}}(\omega)\subset\mathscr{C}$ for all
$N\geq n(\omega)$. In particular
$\Omega_{0}\cap\Omega_{2}\cap\Omega_{3}\in\mathcal{F}$ and
$P(\Omega_{0}\cap\Omega_{2}\cap\Omega_{3})=1$. Now Theorem 3.4 ensures for
each $\omega\in\Omega_{0}\cap\Omega_{2}\cap\Omega_{3}$,
$\mathbb{D}({\hat{\mathcal{C}}_{N}^{\varepsilon_{N}}(\omega),\mathcal{C}})\to
0$ as $N\to\infty$. ∎
## 5 Approximations via weakly converging probability measures
Motivated by recent contributions on epiconvergent discretizations of
stochastic programs [19, 22, 34, 67], we establish consistency of
approximations to the stochastic program (4.1) which result from approximating
$\mathbb{P}$ by a sequence of weakly convergent probability measures
$(\mathbb{P}_{N})$. Our main interest in considering weakly convergent
probability measures stems from those considered in [67, sect. 2] and [46,
sect. 3]. Throughout the section, we assume $\Xi$ be a complete, separable
metric space and $\mathbb{P}$ be a probability measure on $\Xi$. We refer the
reader to [42, p. 65] for the definition of weak convergence of probability
measures.
For a sequence $(\mathbb{P}_{N})$ of probability measures defined on $\Xi$, we
approximate (4.1) via the optimization problems
$\displaystyle\min_{x\in\mathrm{dom}(\psi)}\,\int_{\Xi}\,F_{\xi}(x)\,\mathrm{d}\mathbb{P}_{N}(\xi)+\psi(x).$
(5.1)
Our consistency analysis is built on the conditions in assumption 5.1 and
those needed to apply [56, Cor. 3.4].
###### Assumption 5.1.
In addition to 4.1.1, 4.1.2, 4.1.6, and 4.1.5, we consider the following
conditions.
1. 1.
For each $N\in\mathbb{N}$, $\mathbb{P}_{N}$ is a probability measure on $\Xi$,
and $(\mathbb{P}_{N})$ weakly converges to $\mathbb{P}$ as $N\to\infty$.
2. 2.
For each $\xi\in\Xi$, $F_{\xi}\colon X_{0}\to$ is continuously differentiable.
For every $x\in X_{0}$, $\Xi\ni\xi\mapsto F_{\xi}(x)\in$ is
$\mathbb{P}_{N}$-integrable. For all $h\in X$,
$X_{0}\times\Xi\ni(x,\xi)\mapsto\langle\mathrm{D}F_{\xi}(x),h\rangle_{{X}^{*}\\!,X}\in$
is upper semicontinuous.
3. 3.
4.1.3 holds true with $L_{\bar{x}}$ being continuous,
$\mathbb{P}_{N}$-integrable for each $N\in\mathbb{N}$, and satisfying
$\limsup_{N\to\infty}\,\int_{\Xi}L_{\bar{x}}(\xi)\,\mathrm{d}\mathbb{P}_{N}(\xi)\leq\int_{\Xi}L_{\bar{x}}(\xi)\,\mathrm{d}\mathbb{P}(\xi)$.
4. 4.
For each $\bar{x}\in\mathrm{dom}(\psi)$, each sequence
$(x_{N})\subset\mathrm{dom}(\psi)$ with $x_{N}\rightharpoonup\bar{x}$, and
each sequence $(v_{N})$ with
$v_{N}\in\int_{\Xi}M_{\xi}(x_{N})\,\mathrm{d}\mathbb{P}_{N}(\xi)$ for all
$N\in\mathbb{N}$, $\limsup_{N\to\infty}\,\|v_{N}\|_{V}<\infty$.
While 5.1.2 imposes continuity and differentiablity, these conditions are
satisfied for certain problem classes [63]. However the continuity conditions
may be relaxed in view of [2, Thms. 3.6 and 5.2]. Under 5.1.2, the
continuously differentiable function $F_{\xi}\colon X_{0}\to$ is Clarke
regular for each $\xi\in\Xi$ [18, Prop. 2.3.6] and
$\partial_{C}F_{\xi}(x)=\\{\,\mathrm{D}F_{\xi}(x)\,\\}$ [18, Cor. on p. 32 and
Prop. 2.2.4]. Using Assumption 5.1 and [24, Lem. C.3], we find that $X_{0}\ni
x\mapsto\int_{\Xi}\,F_{\xi}(x)\,\mathrm{d}\mathbb{P}(\xi)$ and $X_{0}\ni
x\mapsto\int_{\Xi}\,F_{\xi}(x)\,\mathrm{d}\mathbb{P}_{N}(\xi)$ are Fréchet
differentiable and we can interchange derivatives and integrals.
Let $\mathfrak{C}$ be the set of all $x\in\mathrm{dom}(\psi)$ with
$0\in\mathrm{D}\big{[}\int_{\Xi}\,F_{\xi}(x)\,\mathrm{d}\mathbb{P}(\xi)\big{]}+\partial\psi(x)$.
Let $\varepsilon\geq 0$ and for each $\omega\in\Omega$, let
$\mathfrak{C}_{N}^{\varepsilon}$ be the set of all points
$x\in\mathrm{dom}(\psi)$ with
$0\in\bigcup_{v\in\overline{\mathbb{B}}_{X}(x;\varepsilon)\cap\mathrm{dom}(\psi)}\mathrm{D}\big{[}\int_{\Xi}\,F_{\xi}(v)\,\mathrm{d}\mathbb{P}_{N}(\xi)\big{]}+\partial\psi(x)$.
###### Theorem 5.2.
Let Assumption 5.1 hold true and let $(\varepsilon_{N})\subset[0,\infty)$ be a
sequence with $\varepsilon_{N}\to 0$ as $N\to\infty$. Let $\mathfrak{C}$ and
$\mathfrak{C}_{N}^{\varepsilon_{N}}$ be nonempty for each $N\in\mathbb{N}$.
1. (i)
If $(x_{N})$ is bounded with $x_{N}\in\mathfrak{C}_{N}^{\varepsilon_{N}}$ for
each $N\in\mathbb{N}$, then $\mathrm{dist}({x_{N},\mathcal{C}})\to 0$ as
$N\to\infty$.
2. (ii)
If there exists a bounded set $\mathscr{C}\subset X$ such that
$\mathfrak{C}_{N}^{\varepsilon_{N}}\subset\mathscr{C}$ for all sufficiently
large $N\in\mathbb{N}$, then
$\mathbb{D}({\mathfrak{C}_{N}^{\varepsilon_{N}},\mathfrak{C}})\to 0$ as
$N\to\infty$.
###### Proof.
Using [56, Cor. 3.4], we show that 3.1.6 holds true with $f(\cdot)$ replaced
by $\int_{\Xi}\,F_{\xi}(\cdot)\,\mathrm{d}\mathbb{P}(\xi)$, and $f_{k}(\cdot)$
replaced by $\int_{\Xi}\,F_{\xi}(\cdot)\,\mathrm{d}\mathbb{P}_{N}(\xi)$. Fix
$h\in X$, let $\bar{x}\in\mathrm{dom}(\psi)$ and let
$(x_{N})\subset\mathrm{dom}(\psi)$ with $x_{N}\to\bar{x}$ be arbitrary.
According to 5.1.3, there exists $\varepsilon=\varepsilon(\bar{x})>0$ such
$F_{\xi}$ is Lipschitz continuous on $\mathbb{B}_{X}(\bar{x},\varepsilon)$
with a Lipschitz constant $L_{\bar{x}}(\xi)$ for each $\xi\in\Xi$,
$L_{\bar{x}}$ is continuous, $\mathbb{P}$\- and $\mathbb{P}_{N}$-integrable,
and
$\limsup_{N\to\infty}\,\int_{\Xi}L_{\bar{x}}(\xi)\,\mathrm{d}\mathbb{P}_{N}(\xi)\leq\int_{\Xi}L_{\bar{x}}(\xi)\,\mathrm{d}\mathbb{P}(\xi)$.
The mean value theorem yields
$|\langle\mathrm{D}F_{\xi}(x),h\rangle_{{X}^{*}\\!,X}|\leq
L_{\bar{x}}(\xi)\|h\|_{X}$ for all $x\in\mathbb{B}_{X}(\bar{x},\varepsilon)$,
$h\in X$, and $\xi\in\Xi$. Combined with 5.1.1 and 5.1.2, [56, Cor. 3.4]
yields
$\displaystyle\limsup_{N\to\infty}\,\int_{\Xi}\langle\mathrm{D}F_{\xi}(x_{N}),h\rangle_{{X}^{*}\\!,X}\,\mathrm{d}\mathbb{P}_{N}(\xi)\leq\int_{\Xi}\langle\mathrm{D}F_{\xi}(\bar{x}),h\rangle_{{X}^{*}\\!,X}\,\mathrm{d}\mathbb{P}(\xi).$
Using the chain rule [24, Lem. C.3], we can interchange derivatives and
integrals in the above equation. Hence 3.1.6 holds true.
Both assertions are now implied by Theorem 3.4. ∎
## 6 Application to risk-averse semilinear PDE-constrained optimization
We show that our main result, Theorem 4.3, applies to risk-averse PDE-
constrained optimization using the average value-at-risk. For $\beta\in(0,1)$,
and a $\mathbb{P}$-integrable random variable $Z\colon\Xi\to$, the average
value-at-risk $\mathrm{AVaR}_{\beta}$ evaluated at $Z$ is defined by
$\displaystyle\mathrm{AVaR}_{\beta}[Z]=\inf_{t\in}\,\big{\\{}\,t+\tfrac{1}{1-\beta}\mathbb{E}_{\mathbb{P}}[(Z-t)_{+}]\,\big{\\}},$
(6.1)
where $(s)_{+}=\max\\{0,s\\}$ for $s\in$ [73, eq. (6.23)]. We consider the
risk-averse semilinear PDE-constrained problem
$\displaystyle\min_{u\in
U_{\text{ad}}}\,(1/2)\mathrm{AVaR}_{\beta}[\|\iota_{0}S(u,\cdot)-y_{d}\|_{L^{2}(D)}^{2}]+\wp(\|u\|_{L^{2}(D)}),$
(6.2)
where $\wp\colon[0,\infty)\to[0,\infty)$, $y_{d}\in L^{2}(D)$,
$U_{\text{ad}}\subset L^{2}(D)$, and for each $(u,\xi)\in L^{2}(D)\times\Xi$,
$y=S(u,\xi)\in H_{0}^{1}(D)$ solves the semilinear PDE
$\displaystyle(\kappa(\xi)\nabla y,\nabla
v)_{L^{2}(D)^{d}}+(y^{3},v)_{L^{2}(D)}=(u,v)_{L^{2}(D)}+(b(\xi),v)_{L^{2}(D)}\>\>\text{for
all}\>\>v\in H_{0}^{1}(D).$
We impose mild conditions on the semilinear PDE, the feasible set
$U_{\text{ad}}$, and $\wp$.
###### Assumption 6.1 (semilinear control problem).
1. 1.
$D\subset^{d}$ is a bounded, convex, polygonal/polyhedral domain with
$d\in\\{2,3\\}$.
2. 2.
$\kappa:\Xi\to L^{\infty}(D)$ is strongly measurable and there exists
$0<\kappa_{\min}\leq\kappa_{\max}<\infty$ such that for all $\xi\in\Xi$,
$\kappa_{\min}\leq\kappa(\xi)(x)\leq\kappa_{\max}$ for a.e. $x\in D$.
3. 3.
$b:\Xi\to L^{2}(D)$ is measurable and there exists $b_{\max}\geq 0$ with
$\|b(\xi)\|_{L^{2}(D)}\leq b_{\max}$ for all $\xi\in\Xi$.
4. 4.
$U_{\text{ad}}\subset L^{2}(D)$ is closed, convex, nonempty, and there exists
$r_{\text{ad}}\in(0,\infty)$ such that $\|u\|_{L^{2}(D)}\leq r_{\text{ad}}$
for all $u\in U_{\text{ad}}$.
5. 5.
$\wp\colon[0,\infty)\to[0,\infty)$ is convex and strictly increasing.
If not stated otherwise, let Assumption 6.1 hold throughout the section. Let
us define $G\colon\times\to$ and $\widehat{J}\colon
L^{2}(D)\times\Xi\to[0,\infty)$ by
$\displaystyle
G(s,t)=t+\tfrac{1}{1-\beta}(s-t)_{+}\quad\text{and}\quad\widehat{J}(u,\xi)=(1/2)\|\iota_{0}S(u,\xi)-y_{d}\|_{L^{2}(D)}^{2}.$
Moreover, we define $L^{2}(D)\times\times\Xi\ni(u,t,\xi)\mapsto
F_{\xi}(u,t)\in[0,\infty)$ by
$\displaystyle F_{\xi}(u,t)=G(\widehat{J}(u,\xi),t).$
To study consistency of SAA C-stationary points, we reformulate (6.2)
equivalently as
$\displaystyle\min_{(u,t)\in
U_{\text{ad}}\times}\,\int_{\Xi}G(\widehat{J}(u,\xi),t)\,\mathrm{d}\mathbb{P}(\xi)+\wp(\|u\|_{L^{2}(D)}).$
(6.3)
Let $\Xi$, let $\mathbb{P}$, and let $\xi^{1}$, $\xi^{2},\ldots$ be as in
section 4. The SAA problem of (6.3) is given by
$\displaystyle\min_{(u,t)\in
U_{\text{ad}}\times}\,\frac{1}{N}\sum_{i=1}^{N}G(\widehat{J}(u,\xi^{i}),t)+\wp(\|u\|_{L^{2}(D)}).$
(6.4)
For the remainder of the section, we verify Assumption 4.1. For the spaces
$L^{2}(D)\times$, $V=H_{0}^{1}(D)\times$, and the open, convex, and bounded
set $U_{0}=U_{\text{ad}}+\mathbb{B}_{L^{2}(D)}(0;1)$, 4.1.1 holds true. Let
$\psi$ be the sum of the indicator function of $U_{\text{ad}}$ and the
regularizer $\wp\circ\|\cdot\|_{L^{2}(D)}$. Since $\wp$ is convex and strictly
increasing, and $L^{2}(D)$ is a Hilbert space, $\psi$ has the Kadec property;
see [62, Lem. 1] and [8, Thm. 6.5]. Hence 4.1.2 holds true.
For each $(u,\xi)\in L^{2}(D)\times\Xi$, the PDE solution $S(u,\xi)$ is well-
defined [85, Thm. 26.A], and we have for all $(u,\xi)\in L^{2}(D)\times\Xi$,
the stability estimate
$\displaystyle\|S(u,\xi)\|_{H_{0}^{1}(D)}\leq(C_{D}/\kappa_{\min})\|u\|_{L^{2}(D)}+C_{D}(b_{\max}/\kappa_{\min}).$
(6.5)
Defining
$\displaystyle\mathfrak{c}_{S}=(C_{D}/\kappa_{\min})(r_{\text{ad}}+1)+C_{D}(b_{\max}/\kappa_{\min}),$
we obtain for all $(u,\xi)\in U_{0}\times\Xi$, the stability estimate
$\displaystyle\widehat{J}(u,\xi)\leq\|\iota_{0}S(u,\xi)\|_{L^{2}(D)}^{2}+\|y_{d}\|_{L^{2}(D)}^{2}\leq
C_{D}^{2}\mathfrak{c}_{S}^{2}+\|y_{d}\|_{L^{2}(D)}^{2}.$ (6.6)
The adjoint approach [37, sect. 1.6.2] ensures that for each $\xi\in\Xi$,
$\widehat{J}(\cdot,\xi)$ is continuously differentiable and
$\displaystyle\nabla_{u}\widehat{J}(\cdot,\xi)=-\iota_{0}z(u,\xi)\quad\text{for
all}\quad(u,\xi)\in L^{2}(D)\times\Xi,$
where for each $(u,\xi)\in L^{2}(D)\times\Xi$, $z=z(u,\xi)\in H_{0}^{1}(D)$
solves the adjoint equation
$\displaystyle(\kappa(\xi)\nabla z,\nabla
v)_{L^{2}(D)^{d}}+3(S(u,\xi)^{2}z,v)_{L^{2}(D)}=-(\iota_{0}S(u,\xi)-y_{d},v)_{L^{2}(D)}$
for all $v\in H_{0}^{1}(D)$. For all $(u,\xi)\in L^{2}(D)\times\Xi$, we have
$\displaystyle\|z(u,\xi)\|_{H_{0}^{1}(D)}\leq(C_{D}/\kappa_{\min})\|\iota_{0}S(u,\xi)\|_{L^{2}(D)}+(C_{D}/\kappa_{\min})\|y_{d}\|_{L^{2}(D)}.$
Combined with the stability estimate (6.5), we have for each $(u,\xi)\in
U_{0}\times\Xi$,
$\displaystyle\|z(u,\xi)\|_{H_{0}^{1}(D)}\leq(C_{D}^{2}/\kappa_{\min})\mathfrak{c}_{S}+(C_{D}/\kappa_{\min})\|y_{d}\|_{L^{2}(D)}.$
(6.7)
Let $\mathfrak{c}_{z}$ be the right-hand side in (6.7). Since $G$ is Lipschitz
continuous and the bound $\|\nabla_{u}\widehat{J}(\cdot,\xi)\|_{L^{2}(D)}\leq
C_{D}\mathfrak{c}_{z}$ is valid for all $(u,\xi)\in U_{0}\times\Xi$, 4.1.3 is
satisfied.
Fix $\xi\in\Xi$. Since the mapping $(u,t)\mapsto(\widehat{J}(u,\xi),t)\in^{2}$
is continuously differentiable and $g$ is finite-valued and convex, $F_{\xi}$
is Clarke regular on $L^{2}(D)\times$ and
$\displaystyle\partial_{C}F_{\xi}(u,t)=(\nabla_{u}\widehat{J}(u,\xi),-1)^{*}\partial
G(\widehat{J}(u,\xi),t)=\begin{bmatrix}\frac{1}{1-\beta}\partial(\widehat{J}(u,\xi)-t)_{+}\nabla_{u}\widehat{J}(u,\xi)\\\
1-\frac{1}{1-\beta}\partial(\widehat{J}(u,\xi)-t)_{+}\end{bmatrix};$
see [18, Prop. 2.3.6 and Thm. 2.3.10]. The Clarke regularity ensures 4.1.4.
Using [4, Thm. 8.2.9], we can show that $S(u,\cdot)$ is measurable for each
$u\in L^{2}(D)$. Combined with (6.6), we find that 4.1.5 holds true.
We define $B\colon H_{0}^{1}(D)\times\to L^{2}(D)\times$ and $M_{\xi}\colon
L^{2}(D)\times\rightrightarrows H_{0}^{1}(D)\times$ by
$\displaystyle B(v,s)=(-\iota_{0}v,s)\quad\text{and}\quad
M_{\xi}(u,t)=\begin{bmatrix}\frac{1}{1-\beta}\partial(\widehat{J}(u,\xi)-t)_{+}z(u,\xi)\\\
1-\frac{1}{1-\beta}\partial(\widehat{J}(u,\xi)-t)_{+}\end{bmatrix}.$ (6.8)
We obtain
$\displaystyle\partial_{C}F_{\xi}(u,t)=BM_{\xi}(u,t)\quad\text{for
all}\quad(u,t,\xi)\in L^{2}(D)\times\times\Xi.$
Combined with the fact that $\iota_{0}$ is linear and compact [37, Thm. 1.14],
we conclude that 4.1.6 is fulfilled.
Next we verify 4.1.7 using Lemma 4.2. For each $s\in$,
$\partial(s)_{+}\subset[0,1]$. Fix $(u,t,\xi)\in
U_{\text{ad}}\times\times\Xi$. For all $m=(m_{1},m_{2})\in M_{\xi}(u,t)$,
(6.8) yields
$\displaystyle\|m\|_{H_{0}^{1}(D)\times}=\|m_{1}\|_{H_{0}^{1}(D)}+|m_{2}|\leq\tfrac{1}{1-\beta}\mathfrak{c}_{z}+1+\tfrac{1}{1-\beta}.$
Hence Lemma 4.2 implies 4.1.7.
To summarize, we have shown that Assumption 4.1 holds true. In order to
demonstrate the convergence of SAA C-stationary points of (6.4) towards those
of (6.3) via Theorem 4.3, we need to show that the SAA C-stationary points are
contained in a bounded, deterministic set. Let $(u_{N},t_{N})\in
U_{\text{ad}}\times$ be a C-stationary point of (6.3). We have $u_{N}\in
U_{\text{ad}}$, which is by assumption a bounded set. Next we show that
$\displaystyle
t_{N}\in[0,C_{D}^{2}\mathfrak{c}_{S}^{2}+\|y_{d}\|_{L^{2}(D)}^{2}].$
Since $(u_{N},t_{N})$ is a C-stationary point of (6.3), the second formula in
(6.8) implies that
$1\in\tfrac{1}{N(1-\beta)}\sum_{i=1}^{N}\partial(\widehat{J}(u_{N},\xi^{i})-t_{N})_{+}$.
Combined with $\beta\in(0,1)$ and $\widehat{J}\geq 0$, we have $t_{N}\geq 0$.
Let $\widehat{\mathrm{AVaR}}_{\beta,N}$ be the empirical estimate of
$\mathrm{AVaR}_{\beta}$ with $\mathbb{P}$ replaced by the empirical
distribution of the sample $\xi^{1},\ldots,\xi^{N}$ in (6.1). The minimization
problem in (6.1) with $\mathbb{P}$ replaced by the empirical distribution is
convex. Hence $t_{N}$ solves (6.1) with $\mathbb{P}$ replaced by the empirical
distribution. As with $\mathrm{AVaR}_{\beta}$ [73, p. 231], the empirical
average value-at-risk $\widehat{\mathrm{AVaR}}_{\beta,N}$ is a coherent risk
measure. Since $(s)_{+}\geq 0$ for all $s\in$, $\beta\in(0,1)$, and coherent
risk measures are monotone, translation equivariant, and positively
homogeneous [73, Def. 6.4], the upper bound on $\widehat{J}$ in (6.6) yields
$\displaystyle
t_{N}\leq\widehat{\mathrm{AVaR}}_{\beta,N}[\widehat{J}(u_{N},\xi)]\leq
C_{D}^{2}\mathfrak{c}_{S}^{2}+\|y_{d}\|_{L^{2}(D)}^{2}.$
## 7 Application to risk-neutral bilinear PDE-constrained optimization
Motivated by the deterministic bilinear control problems studied in [14, 27,
41, 52], we consider risk-neutral optimization problems governed by a bilinear
elliptic PDEs with random inputs:
$\displaystyle\min_{u\in
U_{\text{ad}}}\,\int_{\Xi}J(S(u,\xi))\,\mathrm{d}\mathbb{P}(\xi)+(\alpha/2)\|u\|_{L^{2}(D)}^{2},$
(7.1)
where $\alpha>0$, and for each $(u,\xi)\in U_{\text{ad}}\times\Xi$,
$y=S(u,\xi)\in H_{0}^{1}(D)$ is the solution to
$\displaystyle(\kappa(\xi)\nabla y,\nabla
v)_{L^{2}(D)^{d}}+(g(\xi)uy,v)_{L^{2}(D)}=(b(\xi),v)_{L^{2}(D)}\quad\text{for
all}\quad v\in H_{0}^{1}(D).$ (7.2)
The following assumptions ensure existence and regularity of the solutions to
(7.2), and impose conditions on the function $J$ and the set $U_{\text{ad}}$.
These conditions allow us to Assumption 4.1 and apply Theorem 4.3 to study the
consistency of SAA C-stationary points corresponding to (7.1). Conditions
beyond those formulated in Assumption 7.1 may be required to allow for
applications of Theorem 5.2.
###### Assumption 7.1 (bilinear control problem).
1. 1.
$D\subset^{d}$ is a bounded, convex, polygonal/polyhedral domain with
$d\in\\{2,3\\}$.
2. 2.
$\kappa:\Xi\to C^{1}(\bar{D})$ is measurable and there exists
$\kappa_{\min}>0$ such that $\kappa_{\min}\leq\kappa(\xi)(x)$ for all
$(\xi,x)\in\Xi\times\bar{D}$. Moreover
$\mathbb{E}_{\mathbb{P}}[\|\kappa\|_{C^{1}(\bar{D})}^{p}]<\infty$ for all
$p\in[1,\infty)$.
3. 3.
$b:\Xi\to L^{2}(D)$ and $g:\Xi\to C^{0,1}(\bar{D})$ are measurable and there
exist $b_{\max}$, $g_{\max}>0$ with $\|b(\xi)\|_{L^{2}(D)}\leq b_{\max}$ and
$\|g(\xi)\|_{C^{0,1}(\bar{D})}\leq g_{\max}$ for all $\xi\in\Xi$. Moreover,
$g(\xi)(x)\geq 0$ for all $(\xi,x)\in\Xi\times\bar{D}$.
4. 4.
$J:H_{0}^{1}(D)\to[0,\infty)$ is convex, continuously differentiable and its
derivative is Lipschitz continuous with Lipschitz constant $\ell>0$. For some
nondecreasing function $\varrho:[0,\infty)\to[0,\infty)$, it holds that
$\displaystyle J(y)\leq\varrho(\|y\|_{H_{0}^{1}(D)})\quad\text{for all}\quad
y\in H_{0}^{1}(D).$
5. 5.
$U_{\text{ad}}=\\{\,u\in L^{2}(D):\,0\leq
u(x)\leq\mathfrak{u}(x)\;\text{a.e.}\;x\in D\,\\}$ with $\mathfrak{u}\in
L^{\infty}(D)$ and $\mathfrak{u}(x)\geq 0$ a.e. $x\in D$.
If not stated otherwise, let Assumption 7.1 hold throughout the section. We
introduce notation used throughout the section. Let
$C_{H_{0}^{1};L^{4}}\in(0,\infty)$ be the embedding constant of the compact
embedding $H_{0}^{1}(D)\xhookrightarrow{}L^{4}(D)$ and let
$C_{H^{2};L^{\infty}}>0$ be that of the continuous embedding
$H^{2}(D)\xhookrightarrow{}L^{\infty}(D)$ [37, Thm. 1.14]. We define
$C_{\mathfrak{u}}=\|\mathfrak{u}\|_{L^{\infty}(D)}$ and
$|D|=\int_{D}1\,\mathrm{d}x$.
Let $\psi$ be the sum of the indicator function of $U_{\text{ad}}$ and
$(\alpha/2)\|u\|_{L^{2}(D)}^{2}$. The function $\psi$ has the Kadec property
[8, Thm. 6.5], and 4.1.2 holds true.
Next we provide an example function satisfying 7.1.4. Let $y_{d}\in L^{2}(D)$.
We define $J:H_{0}^{1}(D)\to[0,\infty)$ by
$J(y)=(1/2)\|\iota_{0}y-y_{d}\|_{L^{2}(D)}^{2}$. The function $J$ satisfies
7.1.4 with $\ell=C_{D}^{2}$ and
$\varrho(t)=C_{D}^{2}t^{2}+\|y_{d}\|_{L^{2}(D)}^{2}$.
We use two technical facts for our analysis. Lemma 7.2 establishes a bound on
the derivatives of the objective functions fulfilling 7.1.4.
###### Lemma 7.2.
If $D\subset^{d}$ is a bounded domain and 7.1.4 holds, then
$\|\mathrm{D}J(y)\|_{H^{-1}(D)}^{2}\leq 2\ell J(y)$ for all $y\in
H_{0}^{1}(D)$.
###### Proof.
Fix $\tilde{y}$, $y\in H_{0}^{1}(D)$. Let $\nabla J(y)\in H_{0}^{1}(D)$ be the
Riesz representation of $\mathrm{D}J(y)$. Using 7.1.4 and [86, Cor. 3.5.7 and
Rem. 3.5.2], we have $0\leq J(\tilde{y})\leq J(y)+(\nabla
J(y),\tilde{y}-y)_{H_{0}^{1}(D)}+(\ell/2)\|\tilde{y}-y\|_{H_{0}^{1}(D)}^{2}$.
Minimizing over $\tilde{y}\in H_{0}^{1}(D)$ yields $0\leq
J(y)-(1/(2\ell))\|\nabla J(y)\|_{H_{0}^{1}(D)}^{2}$. ∎
Using 7.1.1 and theorems on the multiplication of Sobolev functions (see [6,
Thm. 7.4] and [29, Thm. 1.4.1.1]), we can show that the trilinear (pointwise
multiplication) operator
$\displaystyle C^{0,1}(\bar{D})\times H^{1}(D)\times H^{2}(D)\ni(g,v,w)\mapsto
gvw\in H^{1}(D)$ (7.3)
is continuous. Hence it is bounded [55, p. 68]. Therefore, we obtain the
existence of a constant $C_{\mathrm{tri}}\in(0,\infty)$ such that for all
$(g,v,w)\in C^{0,1}(\bar{D})\times H^{1}(D)\times H^{2}(D)$,
$\displaystyle\|gvw\|_{H^{1}(D)}\leq
C_{\mathrm{tri}}\|g\|_{C^{0,1}(\bar{D})}\|v\|_{H^{1}(D)}\|w\|_{H^{2}(D)}.$
(7.4)
### 7.1 Existence and uniqueness of PDE solutions
We show that the PDE (7.2) has a unique solution $S(u,\xi)\in H_{0}^{1}(D)$
for each $\xi\in\Xi$ and $u$ in an open neighborhood of the set
$U_{\text{ad}}$ in $L^{2}(D)$. This allows us to verify 4.1.3, 4.1.4, 4.1.5,
and 4.1.1. The existence results established in [14, sect. 3.1] are not
applicable to our setting, as we consider $L^{2}(D)$ instead of
$L^{\infty}(D)$ the control space. An approach different from ours to
constructing such a neighborhood is developed in [82, p. 158].
We define
$\displaystyle U_{0}=\bigcup_{u_{0}\in
U_{\text{ad}}}\,\mathbb{B}_{L^{2}(D)}(u_{0};\delta),\quad\text{where}\quad\delta=\tfrac{\kappa_{\min}}{2g_{\max}C_{H_{0}^{1};L^{4}}^{2}}>0.$
(7.5)
We have the identity
$U_{0}=U_{\text{ad}}+\mathbb{B}_{L^{2}(D)}(u_{0};\delta)$.
Fix $(u,\xi)\in U_{0}\times\Xi$ and $y\in H_{0}^{1}(D)$. Since $u\in U_{0}$,
there exists $u_{0}\in U_{\text{ad}}$ with
$u\in\mathbb{B}_{L^{2}(D)}(u_{0};\delta)$. Combining Hölder’s inequality and
the continuity of $H_{0}^{1}(D)\xhookrightarrow{}L^{4}(D)$ with the definition
of $\delta$ provided in (7.5) and $\|u-u_{0}\|_{L^{2}(D)}\leq\delta$, we
obtain
$\displaystyle|(g(\xi)(u-u_{0})y,y)_{L^{2}(D)}|\leq
C_{H_{0}^{1};L^{4}}^{2}g_{\max}\|u-u_{0}\|_{L^{2}(D)}\|y\|_{H_{0}^{1}(D)}^{2}\leq(\kappa_{\min}/2)\|y\|_{H_{0}^{1}(D)}^{2}.$
Using $u_{0}(x)\geq 0$ and $g(\xi)(x)\geq 0$ for a.e. $x\in D$,
$(g(\xi)u_{0}y,y)_{L^{2}(D)}\geq 0$. It follows that
$\displaystyle\begin{aligned} &(\kappa(\xi)\nabla y,\nabla
y)_{L^{2}(D)^{d}}+(g(\xi)uy,y)_{L^{2}(D)}\\\ &\quad=(\kappa(\xi)\nabla
y,\nabla
y)_{L^{2}(D)^{d}}+(g(\xi)(u-u_{0})y,y)_{L^{2}(D)}+(g(\xi)u_{0}y,y)_{L^{2}(D)}\\\
&\quad\geq\kappa_{\min}\|y\|_{H_{0}^{1}(D)}^{2}-(\kappa_{\min}/2)\|y\|_{H_{0}^{1}(D)}^{2}=(\kappa_{\min}/2)\|y\|_{H_{0}^{1}(D)}^{2}.\end{aligned}$
(7.6)
The right-hand side in (7.2) defines a continuous bilinear form. Hence the
Lax–Milgram lemma ensures that the state equation (7.2) has a unique solution
$S(u,\xi)\in H_{0}^{1}(D)$ and yields the stability estimate
$\displaystyle\|S(u,\xi)\|_{H_{0}^{1}(D)}\leq(2/\kappa_{\min})C_{D}\|b(\xi)\|_{L^{2}(D)}\quad\text{for
all}\quad(u,\xi)\in U_{0}\times\Xi.$ (7.7)
We define the parameterized operator $E:H_{0}^{1}(D)\times
L^{2}(D)\times\Xi\to H^{-1}(D)$ by
$\displaystyle\langle
E(y,u,\xi),v\rangle_{H^{-1}(D),H_{0}^{1}(D)}=(\kappa(\xi)\nabla y,\nabla
v)_{L^{2}(D)^{d}}+(g(\xi)uy,v)_{L^{2}(D)}-(b(\xi),v)_{L^{2}(D)}.$
The mapping $E$ is well-defined and $E(\cdot,\cdot,\xi)$ is infinitely many
times continuously differentiable. For each $(y,u,\xi)\in H_{0}^{1}(D)\times
L^{2}(D)\times\Xi$ and $h$, $v\in H_{0}^{1}(D)$, we have
$\displaystyle\begin{aligned} \langle
E_{y}(y,u,\xi)h,v\rangle_{H^{-1}(D),H_{0}^{1}(D)}&=(\kappa(\xi)\nabla h,\nabla
v)_{L^{2}(D)^{d}}+(g(\xi)uh,v)_{L^{2}(D)}.\end{aligned}$
Together with (7.6), we obtain for all $(u,\xi)\in U_{0}\times\Xi$,
$\displaystyle\langle
E_{y}(S(u,\xi),u,\xi)h,h\rangle_{H^{-1}(D),H_{0}^{1}(D)}\geq(\kappa_{\min}/2)\|h\|_{H_{0}^{1}(D)}^{2}\quad\text{for
all}\quad h\in H_{0}^{1}(D).$
The Lax–Milgram lemma ensures that $E_{y}(S(u,\xi),u,\xi)$ has a bounded
inverse and
$\displaystyle\|E_{y}(S(u,\xi),u,\xi)^{-1}\|_{\mathscr{L}(H_{0}^{1}(D),H^{-1}(D))}\leq(2/\kappa_{\min})\quad\text{for
all}\quad(u,\xi)\in U_{0}\times\Xi.$ (7.8)
For each $y$, $v\in H_{0}^{1}(D)$ and $u\in L^{2}(D)$, we can show that
$\langle E(y,u,\cdot),v\rangle_{H^{-1}(D),H_{0}^{1}(D)}$ is measurable, as
7.1.1, 7.1.2, and 7.1.3 hold. Since $H^{-1}(D)$ is separable, $E(y,u,\cdot)$
is measurable [39, Thm. 1.1.6]. Hence $S(u,\cdot)$ is measurable [4, Thm.
8.2.9]. Combined with 7.1.4, we find that $J(S(u,\cdot))$ is a random variable
for each $u\in U_{0}$.
### 7.2 PDE regularity
We show that the solution $S(u,\xi)$ to (7.2) is contained in $H^{2}(D)$ and
derive a stability estimate for each $(u,\xi)\in U_{\text{ad}}\times\Xi$. The
computations performed here form the basis of verifying 4.1.6.
Fix $(u,\xi)\in U_{\text{ad}}\times\Xi$. Hence $u\in L^{\infty}(D)$. To
establish an $H^{2}(D)$-stability estimate, we apply [77, Thm. 3.1] to the
solution $\widetilde{y}=\widetilde{y}(u,\xi)\in H_{0}^{1}(D)$ to
$\displaystyle(\kappa(\xi)\nabla\widetilde{y},\nabla
v)_{L^{2}(D)^{d}}=(b(\xi),v)_{L^{2}(D)}-(g(\xi)uS(u,\xi),v)_{L^{2}(D)}\quad\text{for
all}\quad v\in H_{0}^{1}(D).$
Since this equation has a unique solution $\widetilde{y}(u,\xi)$ and
$S(u,\xi)$ is a solution, we have $\widetilde{y}(u,\xi)=S(u,\xi)$. Defining
the random variable
$\displaystyle
C_{\kappa}(\xi)=\frac{\|\kappa(\xi)\|_{C^{0}(\bar{D})}\|\kappa(\xi)\|_{C^{1}(\bar{D})}^{2}}{\kappa_{\min}^{4}},$
(7.9)
and applying [77, Thm. 2.1], we find that $S(u,\xi)\in H^{2}(D)$ and
$\displaystyle\|S(u,\xi)\|_{H^{2}(D)}\leq
C_{H^{2}}C_{\kappa}(\xi)\Big{(}\|g(\xi)uS(u,\xi)\|_{L^{2}(D)}+\|b(\xi)\|_{L^{2}(D)}\Big{)},$
where $C_{H^{2}}=C_{H^{2}}(D)>0$ is a deterministic constant. Combined with
Hölder’s and Friedrichsínequalities, we obtain for all $(u,\xi)\in
U_{\text{ad}}\times\Xi$,
$\displaystyle\|S(u,\xi)\|_{H^{2}(D)}\leq
C_{H^{2}}C_{\kappa}(\xi)\big{(}C_{D}C_{\mathfrak{u}}g_{\max}\|S(u,\xi)\|_{H_{0}^{1}(D)}+\|b(\xi)\|_{L^{2}(D)}\big{)}.$
(7.10)
### 7.3 Gradient regularity
In this section, we show that 4.1.3, 4.1.4, 4.1.6, and 4.1.7 hold true. Using
the construction of $U_{0}$, we can define the objective function
$U_{0}\times\Xi\ni(u,\xi)\mapsto F_{\xi}(u)\in[0,\infty)$ by
$\displaystyle F_{\xi}(u)=J(S(u,\xi)).$
Assumption 7.1 ensures that $F_{\xi}$ is continuously differentiable for every
$\xi\in\Xi$. Hence 4.1.4 holds true [18, Prop. 2.3.6]. The adjoint approach
[37, sect. 1.6.2] yields for each $(u,\xi)\in U_{0}\times\Xi$,
$\displaystyle(\nabla
F_{\xi}(u),s)_{L^{2}(D)}=(g(\xi)S(u,\xi)z(u,\xi),s)_{L^{2}(D)}\quad\text{for
all}\quad s\in L^{2}(D),$ (7.11)
where for each $(u,\xi)\in U_{0}\times\Xi$, the adjoint state $z=z(u,\xi)\in
H_{0}^{1}(D)$ solves
$\displaystyle(\kappa(\xi)\nabla z,\nabla
v)_{L^{2}(D)^{d}}+(g(\xi)uz,v)_{L^{2}(D)}=-\langle\mathrm{D}J(S(u,\xi)),v\rangle_{H^{-1}(D),H_{0}^{1}(D)}$
(7.12)
for all $v\in H_{0}^{1}(D)$. Using (7.8) and (7.12), we obtain the stability
estimate
$\displaystyle\begin{aligned}
\|z(u,\xi)\|_{H_{0}^{1}(D)}&\leq(2/\kappa_{\min})\|\mathrm{D}J(S(u,\xi))\|_{H^{-1}(D)}\;\;\text{for
all}\;\;(u,\xi)\in U_{0}\times\Xi.\end{aligned}$
Combined with the stability estimate (7.7), 7.1.4, and Lemma 7.2, we have for
all $(u,\xi)\in U_{0}\times\Xi$,
$\displaystyle\begin{aligned}
\|z(u,\xi)\|_{H_{0}^{1}(D)}&\leq(2/\kappa_{\min})\big{(}2\ell\varrho((2/\kappa_{\min})C_{D}b_{\max})\big{)}^{1/2}.\end{aligned}$
(7.13)
We show that 4.1.3 is satisfied. Using the Hölder and Friedrichs inequalities,
the continuity of $H_{0}^{1}(D)\xhookrightarrow{}L^{4}(D)$, the stability
estimates (7.7) and (7.13), and the gradient formula (7.11), we have
$(u,\xi)\in U_{0}\times\Xi$,
$\displaystyle\|\nabla F_{\xi}(u)\|_{L^{2}(D)}$ $\displaystyle\leq
C_{H_{0}^{1};L^{4}}^{2}g_{\max}\|S(u,\xi)\|_{H_{0}^{1}(D)}\|z(u,\xi)\|_{H_{0}^{1}(D)}$
$\displaystyle\leq
C_{H_{0}^{1};L^{4}}^{2}g_{\max}(2/\kappa_{\min})^{2}C_{D}b_{\max}\big{(}2\ell\varrho((2/\kappa_{\min})C_{D}b_{\max})\big{)}^{1/2}.$
Right-hand side in the above equation is a deterministic constant. Hence 4.1.3
holds true.
The stability estimate (7.7) and 7.1.4 yield for all $(u,\xi)\in
U_{0}\times\Xi$,
$\displaystyle F_{\xi}(u)\leq\varrho((2/\kappa_{\min})C_{D}b_{\max}).$ (7.14)
Combined with the measurability of $J(S(u,\cdot))$ for each $u\in U_{0}$, we
find that 4.1.5 holds true.
We verify 4.1.6 by showing that for each $(u,\xi)\in U_{\text{ad}}\times\Xi$,
$\displaystyle\nabla F_{\xi}(u)=\iota_{1}m_{\xi}(u),\quad\text{where}\quad
m_{\xi}(u)=g(\xi)S(u,\xi)z(u,\xi).$ (7.15)
We define $M_{\xi}(u)=\\{\,m_{\xi}(u)\,\\}$, $V=H^{1}(D)$, and $B=\iota_{1}$.
Fix $(u,\xi)\in U_{\text{ad}}\times\Xi$. We have $S(u,\xi)\in H^{2}(D)$ (see
section 7.2 and (7.10)) and $z(u,\xi)\in H^{1}(D)$ (see (7.13)). Combined with
$g(\xi)\in C^{0,1}(\bar{D})$ and (7.4), we find that $m_{\xi}(u)\in H^{1}(D)$.
Hence (7.11) implies (7.15).
We show that 4.1.7 holds true using Lemma 4.2. Fix $(u,\xi)\in
U_{\text{ad}}\times\Xi$. Using (7.4) and (7.15), we have
$\displaystyle\|m_{\xi}(u)\|_{H^{1}(D)}\leq
C_{\mathrm{tri}}g_{\max}\|S(u,\xi)\|_{H^{2}(D)}\|z(u,\xi)\|_{H^{1}(D)}.$
Combined with the stability estimates (7.7), (7.10), and (7.13), and
$\|y\|_{H^{1}(D)}\leq(1+C_{D})\|y\|_{H_{0}^{1}(D)}$ valid for all $y\in
H_{0}^{1}(D)$, we find that
$\displaystyle\|m_{\xi}(u)\|_{H^{1}(D)}$ $\displaystyle\leq
C_{\mathrm{tri}}g_{\max}C_{H^{2}}C_{\kappa}(\xi)\big{(}C_{D}^{2}C_{\mathfrak{u}}g_{\max}(2/\kappa_{\min})b_{\max}+b_{\max}\big{)}$
$\displaystyle\quad\cdot(2/\kappa_{\min})C_{D}(1+C_{D})\big{(}2\ell\varrho((2/\kappa_{\min})C_{D}b_{\max})\big{)}^{1/2}.$
We define $\zeta$ as the right-hand side in the above equation. Using 7.1.2
and (7.9), we find that $\zeta$ is integrable. Hence Lemma 4.2 yields 4.1.7.
## 8 Discussion
A multitude of applications in science and engineering yield PDE-constrained
optimization problems under uncertainty. Motivated by such applications, we
have considered infinite-dimensional stochastic optimization problems. We
approximated the expectations in the stochastic programs using two approaches:
the SAA approach and using probability measures weakly converging to the
random element’s probability distribution. For both approximation approaches,
we established asymptotic consistency statements for C-stationary points under
Clarke regularity of integrands. We applied our framework to a risk-averse
semilinear PDE-constrained optimization problem with the average value-at-risk
as a risk measure, and to risk-neutral bilinear PDE-constrained optimization.
Our consistency analysis requires integrands be Clarke regular. For many
applications in PDE-constrained optimization, integrands are continuously
differentiable and hence Clarke regular. For nonregular integrands,
consistency analysis of SAA first-order stationary points is more demanding.
Using smoothing functions of integrands, the SAA approach has been analyzed in
[12, 83] as the sample size approaches infinity and smoothing parameters
converge to zero. We refer the reader to [69] for recent contributions to the
consistency analysis of SAA stationary points as applied to risk-neutral
optimization problems with integrands that lack Clarke regularity.
Our main result, Theorem 4.3, establishes the consistency of SAA C-stationary
points of infinite-dimensional risk-neutral optimization problems. While
applicable to a large set of applications, it may be desirable to establish
consistency results for sample-based approximations of chance-constrained
problems governed by PDEs with random inputs [15, 21, 79], and PDE-constrained
problems under uncertainty with state constraints [25, 47]. Further future
work includes extending the consistency analysis to a broader class of risk-
averse PDE-constrained optimization problems.
## Acknowledgments
JM is very grateful to Professor Alexander Shapiro for taking the time to
address several of my questions related to the SAA approach.
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# Caught in the Act: A Metal-Rich High-Velocity Cloud in the Inner Galaxy
Frances H. Cashman Space Telescope Science Institute, 3700 San Martin Drive,
Baltimore, MD 21218, USA Andrew J. Fox AURA for ESA, Space Telescope Science
Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA Bart P. Wakker
Department of Astronomy, University of Wisconsin-Madison, 475 North Charter
Street, Madison, WI 53706, USA Trisha Ashley Space Telescope Science
Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA Derck Massa Space
Science Institute, 4750 Walnut Street Suite 205 Boulder, Colorado 80301, USA
Edward B. Jenkins Department of Astrophysical Sciences, Princeton University,
Princeton, NJ 08544-1001, USA Dhanesh Krishnarao NSF Astronomy & Astrophysics
Postdoctoral Fellow, Johns Hopkins University, 3400 N. Charles Street,
Baltimore, MD 21218, USA Department of Physics, Colorado College, 14 East
Cache La Poudre Street, Colorado Springs, CO 80903, USA Space Telescope
Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA Felix J.
Lockman Green Bank Observatory, P.O. Box 2, Rt. 28/92, Green Bank, WV 24944,
USA Robert A. Benjamin Department of Physics, University of Wisconsin-
Whitewater, 800 West Main Street, Whitewater, WI 53190, USA Rongmon Bordoloi
Department of Physics, North Carolina State University, 421 Riddick Hall,
Raleigh, NC 27695-8202, USA Tae-Sun Kim Department of Astronomy, University
of Wisconsin-Madison, 475 North Charter Street, Madison, WI 53706, USA
(Received September 28, 2022)
###### Abstract
We characterize the chemical and physical conditions in an outflowing high-
velocity cloud in the inner Galaxy. We report a super-solar metallicity of
[O/H] = $+0.36\pm 0.12$ for the high-velocity cloud at $v_{\mathrm{LSR}}$ =
125.6 km s-1 toward the star HD 156359 (catalog ) ($l$ = 328.$\degree$7, $b$ =
$-$14.$\degree$5, $d$ = 9 kpc, $z$ = $-$2.3 kpc). Using archival observations
from FUSE, HST STIS, and ESO FEROS we measure high-velocity absorption in H I,
O I, C II, N II, Si II, Ca II, Si III, Fe III, C IV, Si IV, N V, and O VI. We
measure a low H I column density of log $N$(H I) = $15.54\pm 0.05$ in the HVC
from multiple unsaturated H I Lyman series lines in the FUSE data. We
determine a low dust depletion level in the HVC from the relative strength of
silicon, iron, and calcium absorption relative to oxygen, with
[Si/O]=$-0.33\pm 0.14$, [Fe/O]=$-0.30\pm 0.20$, and [Ca/O] =$-0.56\pm 0.16$.
Analysis of the high-ion absorption using collisional ionization models
indicates that the hot plasma is multi-phase, with the C IV and Si IV tracing
104.9 K gas and N V and O VI tracing 105.4 K gas. The cloud’s metallicity,
dust content, kinematics, and close proximity to the disk are all consistent
with a Galactic wind origin. As the HD 156359 (catalog ) line of sight probes
the inner Galaxy, the HVC appears to be a young cloud caught in the act of
being entrained in a multi-phase Galactic outflow and driven out into the
halo.
Galactic Center (565) — Ultraviolet astronomy (1736) — High-velocity clouds
(735) – Chemical abundances (224) — Galactic winds (572)
††journal: ApJ††facilities: FUSE, HST(STIS), ESO(FEROS)††software: astropy
(Astropy Collaboration et al., 2018), Cloudy (Ferland et al., 2017), linetools
(Prochaska et al., 2017), VPFIT (Carswell & Webb, 2014), CalFUSE (Dixon &
Kruk, 2009), calSTIS (Dressel et al., 2007), FEROS-DRS (Kaufer et al., 1999)
## 1 Introduction
Figure 1: Left panel: The location of HD 156359 (catalog ) with reference to
the Fermi and eROSITA bubbles. The composite Fermi–eROSITA image from Predehl
et al. (2020), where the softer X-ray emission (0.6–1 keV, in cyan) envelopes
the harder component of the extended GeV emission of the Fermi bubbles (in
red; adapted from Selig et al. 2015). Right panel: H I column density map from
the 21 cm HI4PI survey showing the distribution of H I in the region from
100–150 km s-1 HI4PI Collaboration et al. (2016) in a section of “Complex WE”
by Wakker & van Woerden (1991). The location of HD 156359 (catalog ) is marked
with a white cross. The contours correspond to log $N_{\mathrm{HI}}$ = 2.08,
6.23, 10.4, 14.5, 18.7 $\times$ 1018 cm-2.
The supermassive black hole, Sagittarius A∗, and surrounding regions of active
star formation power an outflowing multiphase wind from the center of the
Milky Way (MW). Outflows play a critical role in the baryon cycle, the cycling
of gas into and out of galaxies, which helps regulate the evolution of
galaxies. The MW offers us a front-row seat view to study outflows over a
range of wavelengths and phases and understand their impact on galaxy
evolution.
Presently, the farthest-reaching known outflows in the MW are the Fermi (Su et
al., 2010; Ackermann et al., 2014) and eROSITA (Bland-Hawthorn & Cohen, 2003;
Predehl et al., 2020) Bubbles, which are giant gamma- and X-ray emission
extending $\sim$10 kpc and 14 kpc above and below the Galactic disk,
respectively (see Figure 1). The origin of the Fermi Bubbles is believed to be
an energetic GC Seyfert flare event $\sim$3.5 Myr ago (Bland-Hawthorn et al.,
2019).
Ultraviolet (UV) absorption-line studies have found gas outflowing at high-
velocity in a number of sight lines passing through or near the Fermi Bubbles
(see Keeney et al. 2006; Zech et al. 2008; Fox et al. 2015; Bordoloi et al.
2017; Savage et al. 2017; Karim et al. 2018; Ashley et al. 2020, 2022). These
high-velocity clouds (HVCs) consist of neutral (e.g., C I, O I), singly
ionized (S II, Si II, Fe II), and highly ionized (C IV, Si IV, O VI) gas. Cold
gas has also been detected in the nuclear outflow via emission in neutral
hydrogen and molecules. Several hundred H I 21 cm clouds have been detected at
low latitudes in the Fermi Bubbles outflowing from the Galactic Center (GC)
(McClure-Griffiths et al., 2013; Di Teodoro et al., 2018; Lockman et al.,
2020). Carbon monoxide (CO) has been detected in submillimeter emission in two
dense molecular clouds comoving within the H I 21 cm outflow (Di Teodoro et
al., 2020). More recently, molecular hydrogen was discovered in the Galactic
nuclear outflow $\sim$1 kpc below the GC (Cashman et al., 2021). Outflowing
gas associated with the Fermi Bubbles has also been seen in optical emission
(Krishnarao et al., 2020a). Together, these observations of HVCs near the
Galactic Center provide observational constraints on the properties of the MW
nuclear wind.
The outflowing wind interacts with gas in the disk in multiple ways. First, it
can disrupt the position and velocity of the disk (Krishnarao et al., 2020b),
leading to warps and perturbations. Second, it can accelerate and entrain cool
gas into the halo. In some instances, this high-velocity gas may cool, lose
buoyancy, and fall back onto the disk as a “Galactic fountain”, supplying new
fuel for further star formation (see Shapiro & Field 1976; Bregman 1980; Kahn
1981; de Avillez 1999). In other cases, the high-velocity gas may survive
being accelerated into the halo, and even escape. Cloud survival is the
subject of recent theoretical work focusing on how cool gas clouds develop and
grow in the hot wind (see Gronke & Oh 2020; Sparre et al. 2020).
Metallicities of HVCs are important to determine because they provide direct
evidence for the origin of the clouds. Fully UV-based metallicities (with the
metal and H I column densities measured along the same UV sight lines) are
ideal to ensure that the metal absorption and hydrogen absorption are tracing
the same gas, but they are rare because of the difficulty in isolating
unsaturated high-velocity H I components in UV absorption. These high-velocity
H I components are often saturated or blended, even for high-order Lyman
series lines, thus there is only a narrow range of $N$(H I) values for which
precise measurements are possible (French et al., 2021). Therefore HVC
metallicities are frequently determined from a combination of a UV measurement
along an infinitesimal sight line with a H I measurement made using a much
larger 21 cm beam. However, this combination introduces beam-smearing
uncertainties on the metallicity. Currently, the only fully-UV-based
metallicities are for two HVCs toward M5-ZNG1 (Zech et al., 2008), but this is
a high-latitude direction ($b\approx 50\degree$) far from the GC.
At lower latitudes, we can target UV-bright stars as background sources, but
very few UV spectra of distant GC stars ($d\gtrsim 8$ kpc) exist. Therefore,
it is crucial to fully analyze the few spectra available, including LS 4825
(Savage et al., 2017; Cashman et al., 2021) and HD 156359 (this paper). It is
precisely these low-latitude sight lines that are likely to harbor young
clouds that have only recently been entrained in the Galactic outflow. Finding
an outflowing cloud at low latitude means catching it close to its origin,
thus offering a rare opportunity to study the cloud before it has undergone
significant mixing as it begins its journey into the halo. Observing recently
entrained clouds therefore gives a snapshot of outflowing gas when it first
exits the disk.
In this paper we present a detailed spectroscopic analysis of an HVC detected
toward an inner Galaxy sight line, HD 156359 (catalog ). By studying the
properties of this cloud, we characterize the physical and chemical conditions
of the gas in the inner Galaxy, in a region likely influenced by the nuclear
wind.
## 2 Data
### 2.1 The HD 156359 Sight line
HD 156359 (at $l,b,d$ = 328.68°, $-$14.52°, 9 kpc) is one of the best-studied
GC sight lines (Sembach et al., 1991, 1995). The sight line lies in the inner
Galaxy at the boundary of the southern Fermi Bubble and within the eROSITA
X-ray bubble (Predehl et al., 2020), a region of enhanced X-ray emission (see
Figure 1). This sight line also passes close to a complex of small HVCs dubbed
“Complex WE” by Wakker & van Woerden (1991), less than 1$\degree$ from one of
the densest cores. Sembach et al. (1991) classified HD 156359 (catalog ) as a
O9.7 Ib–II star on the basis of stellar photospheric lines and wind profiles
in its UV spectrum. The spectral type and apparent magnitude of the star yield
a spectroscopic distance of 11.1$\pm$2.8 kpc. However, recent Gaia EDR3
parallax measurements (Bailer-Jones et al., 2021) place HD 156359 (catalog )
at a distance of 9.0${}^{+3.1}_{-2.1}$ kpc, which implies a $z$-distance below
the plane of 2.3 kpc. We adopt the Gaia distance in our analysis.
The sight line toward HD 156359 (catalog ) intersects at least three spiral
arms, the Sagittarius, Scutum, and Norma arms. Spiral-arm models from Vallée
(2017) give the expected velocities of the Sagittarius, Scutum, and Norma arms
at approximately $-$10, $-$55, and $-$103 km s-1, respectively. Thus,
absorption components observed at these velocities can be attributed to gas in
(or associated with) the spiral arms. However, the spiral arms cannot explain
the HVC observed at $v_{\rm LSR}$=125 km s-1.
High-ion absorption toward HD 156359 (catalog ) was first observed with the
International Ultraviolet Explorer (IUE; Sembach et al., 1991) and later by
the Goddard High Resolution Spectrograph (GHRS; Sembach et al., 1995). The
high-ion information in the FUSE and HST STIS spectra of this sight line has
not been previously published – the sight line is not included in the FUSE O
VI survey of Galactic disk sight lines by Bowen et al. (2008), as that survey
was restricted to $|b|<10\degree$. We also include analysis of a single
archival optical spectrum of HD 156359 (catalog ) taken with FEROS at the
European Southern Observatory (ESO) at La Silla. Details of all observations
of spectra used in this paper are described below.
### 2.2 UV Observations
HD 156359 (catalog ) was observed by FUSE (Moos et al., 2000) under programs
P101, S701, and U109 between 2000 April 12 and 2006 April 25 (PIs Sembach,
Andersson, and Blair, respectively). The raw spectra were downloaded from the
MAST FUSE archive, and the CalFUSE pipeline (v3.2.2; Dixon & Kruk 2009) was
used to reduce and extract the spectra. Data from the SiC, LiF1, and LiF2
channels were used for the analysis. A detailed explanation of the data
reduction, as well as refinements to the CalFUSE data reduction procedures can
be found in Wakker et al. (2003) and Wakker (2006). The spectra have a signal-
to-noise ratio (S/N) $\sim$12–26 per resolution element and a velocity
resolution of 20 km s-1 (FWHM). The data were binned by three pixels for the
fitting analysis. The FUSE wavelength coverage is $\sim$912–1180Å.
A single exposure of HD 156359 was obtained with HST/STIS on 2003 March 26
under program 9434 (PI Lauroesch) using the E140M grating. The data were
downloaded from the MAST HST archive and reduced using calstis (v.3.4.2,
Dressel et al. 2007). The echelle orders were combined to create a single
continuous spectrum, and in regions of order-overlap spectral counts were
combined to increase the S/N ratio. The data have S/N $\sim$10–38 per
resolution element and a FWHM velocity resolution of 6.5 km s-1, i.e. spectral
resolution ($\lambda$/$\Delta\lambda$) $\sim$45,800. The STIS E140M wavelength
coverage is $\sim$1160–1725Å. Wavelengths and velocities for the absorption-
line features are given in the local standard of rest (LSR) reference frame,
where the correction factor $v_{\mathrm{LSR}}$ $-$ $v_{\mathrm{Helio}}$ =
$-0.35$ km s-1 for the direction toward HD 156359 (catalog ).
### 2.3 Optical Observations
A spectrum of HD 156359 (catalog ) was captured using the Fibre-fed Extended
Range Optical Spectrograph (FEROS) at the European Southern Observatory (ESO)
La Silla 2.2 meter telescope on 2006 April 30 (PI: Bouret, PID: 077.D-0635).
The retrieved archival product covers the wavelength range $\sim$3565–9214 Å
with R = 48000 and FWHM = 6.25 km s-1. The primary data product was reduced
automatically using the FEROS Data Reduction Software (DRS) pipeline version
fern/1.0. In the automated reduction process, the bias is subtracted using
overscan regions and bad columns are replaced by the mean values of the
neighboring columns. The orders are rectified and then extracted using the
standard method. Next, the extracted spectra are flat-fielded and wavelength
calibrated, then rebinned to a constant dispersion of 0.03 Å. Finally, the
individual orders are combined into a single 1D spectrum. As the archival
reduced data are not flux calibrated, the continuum of the stellar spectrum
was normalized using the linetools software package.
## 3 Measurements
In this section we describe our absorption-line measurement processes,
including stellar continuum fitting and procedures for measuring lines of
different ionization states. Our measurements are based on the VPFIT software
program (v12.2; Carswell & Webb 2014), which we used to conduct a set of
Voigt-profile fits to the absorption-line profiles, with wavelengths and
oscillator strengths taken from the compilations of Morton (2003) and Cashman
et al. (2017). The measured lines are listed in Table 1.
### 3.1 Stellar continuum modeling
Given the spectral type of HD 156359 (catalog ) (O9.7 Ib–II; Sembach et al.,
1991), the stellar continuum placement requires careful consideration. In
addition to estimating the stellar continuum through comparison with FUSE
spectra of stars of similar spectral type, we also constructed a TLUSTY model
(Lanz & Hubeny, 2003) of this type of star as a continuum placement guide.
This was particularly useful in regions where the interstellar absorption was
stronger.
The adopted TLUSTY model has $T_{\mathrm{eff}}$ = 26000 K, log $g$ = 3.0,
$v_{\mathrm{turb}}$ = 10.0 km s-1 and is rotationally broadened by 90 km s-1
in order to match the Si III 1300 Å triplets. We matched the observed UV wind
signatures found in the observed FUSE and STIS spectra, where all observed
spectra are adjusted for the stellar radial velocity $v_{\mathrm{rad}}=-82$ km
s-1 (Gontcharov, 2006). The model is reddened by an $E(B-V)=0.09$, using the
mean Fitzpatrick & Massa (2007) extinction curve (extrapolated to 912 Å), and
Lyman series absorption for a foreground H I column of $N$(H I) = 6.0 $\times$
1020 cm-2 is also included (Sembach et al., 1991). The model is then scaled by
1.25 $\times$ 10-20 in order to match observations to $\sim$10%. Finally, all
spectra were binned to 0.1 Å, $\sim$15 to 30 km s-1, depending on wavelength,
see Figure 2. When consulting the model as a guide for interstellar line
continua, additional tweaks of about 5% were needed to match local continua.
Our subsequent column density measurements account for the uncertainty
inherent in the continuum-placement process.
Figure 2: A comparison of a section of the FUSE FUV spectrum with a TLUSTY
stellar model for the same spectral type, O9.7, which aided in the continuum
placement. The FUSE SiC2a spectrum from 914–944 Å is shown in black with a
stellar TLUSTY model shown in red. The TLUSTY model has $T_{\mathrm{eff}}$ =
26000 K, log $g$ = 3.0, $v_{\mathrm{turb}}$ = 10.0 km s-1, and is rotationally
broadened by 90 km s-1 in order to match the observed Si III 1300 Å triplets.
The model is reddened by an $E(B-V)$ = 0.09 using the mean Fitzpatrick & Massa
(2007) extinction curve and Lyman line absorption for log $N$(H I) = 20.78
(Sembach et al., 1991) is also applied. The flux was scaled by
1.25$\times$10-20 to provide an overall agreement with observations to
$\sim$10%.
### 3.2 H I absorption
Figure 3: Velocity profiles of the neutral, low-, and intermediate-ion
absorption lines detected toward HD 156359. The normalized flux is shown in
black, the continuum level is in red, and the 1$\sigma$ error in the
normalized flux is in blue. The vertical line at 0 km s-1 marks the region
associated with the Milky Way. Each of these profiles shows a Voigt fit to the
data except for O I for which we provide an AOD measurement, and for Fe II,
which is a non-detection. The solid green curve is the overall Voigt profile
fit to the absorption. The magenta curve is the Voigt profile fit to the HVC
absorption feature near $+$125 km s-1 and the vertical dashed magenta line
marks the velocity centroid of the fitted component. Figure 4: A comparison
of the normalized apparent column density plots between O I 1302 in violet and
Si II 1526 in green. The profiles have similar shapes although the O I 1302
profile shows a lower signal to noise. The inset panel shows the apparent
column density ratio of Si II 1526 to O I 1302. A linear fit to 5 pixels in
the region from $+110\lesssim v\lesssim 125$ km s-1 has a slope near zero
within the margin of error, supporting the reality of the detection in both
ions. Figure 5: Velocity profiles of the Ca II $\lambda\lambda$3934, 3969 and
Na I $\lambda\lambda$5891, 5897 absorption lines detected toward HD 156359 in
the archival FEROS spectrum. The normalized flux is shown in black, the
continuum level is in red. The gray vertical line at 0 km s-1 marks the
absorption region associated with the Milky Way. The green curves in the top
two panels are the Voigt profile fits to the data, where the green vertical
line at 123.7 km s-1 marks the location of the HVC. There is a non-detection
of high-velocity Na I, although telluric H2O absorption lines are present in
this region. The lower end of the y-axis begins at $+0.2$ in all panels to
improve the visibility of the absorption features.
The FUSE spectrum of HD 156359 (catalog ) covers almost the entire H I Lyman
series, from Lyman-$\beta$ at 1025 Å down to the Lyman limit at 912 Å.
However, we limited our fitting to H I lines in regions where the stellar
continuum was better-defined and showed the least contamination from stellar
features and/or interstellar molecular lines, which are prolific in the FUSE
bandpass.
We selected H I $\lambda\lambda$923, 926, 930, and 937 (see Figure 3) to
derive the H I column density. We fit a continuum to each of these lines
locally. In order to account for sensitivity to continuum placement, we
consider a high and low continuum placement for our column density measurement
across the wavelength range $\lambda$917–944. H I $\lambda$923 lies in a noisy
region of the upper Lyman lines frequently associated with the Inglis-Teller
effect (Inglis & Teller, 1939), in which overlapping stellar absorption lines
have merged to produce the appearance of a depressed continuum. We perform a
simultaneous Voigt profile fit to $\lambda\lambda$926, 930, 937 with the
positions of the low-velocity absorption initially based on weak ISM lines. We
derive log $N_{\mathrm{HI}}$ = 15.54 $\pm$ 0.02 $\pm$ 0.05 in the HVC at
$+125.6$ km s-1, where the first error is the statistical error due to photon
noise and the second is the systematic continuum-placement error. The result
of these fits is shown Table 2 and in Figure 3, where we show that the
simultaneous fit is also consistent with the noisier H I $\lambda$923 profile.
We include a detailed explanation of our continuum placement procedures and
how we considered contamination from stellar features in Appendix A.
An important step in measuring H I absorption in FUSE spectra is to
decontaminate H2 absorption. Fortunately, the FUSE LiF1 and LiF2 channels
provide us with an opportunity to model isolated H2 lines. This allows us to
account for the Milky Way’s molecular contribution to the interstellar
absorption lines blended with H I in the SiC2A spectrum. We conducted an H2
decontamination and describe and illustrate this process in detail in Appendix
B.
### 3.3 O I absorption
After placing a smooth continuum through the O I line at 1302 Å in the STIS
E140M spectrum, we noticed a small absorption feature near $+125$ km s-1 (see
Figure 3). This feature spans five pixels and has a significance of
3.2$\sigma$ ($EW$/$\sigma_{EW}$=3.2). Since only one STIS exposure exists, we
cannot confirm the O I detection in another dataset, and this feature is not
seen in the much weaker O I line at 1039 Å in the FUSE spectrum. To explore
whether this feature is real, we compared the absorption profile to another
low-ion line. Figure 4 shows an apparent column density profile comparison of
the O I 1302 Å line to the weak Si II 1526 Å line. We chose $\lambda$1526
because it is unsaturated, unblended, and seen in a region with high signal to
noise. Although noisier, the profile of O I $\lambda$1302 is very similar to
Si II $\lambda$1526 in the velocity region near $+125$ km s-1, and the ratio
of their apparent column densities is flat with velocity. To confirm this
similarity, the inset plot in Figure 4 shows a linear fit to the ratio over
five pixels (over 2 resolution elements) with a slope equal to zero within the
margin of error, as expected for a genuine O I detection. Therefore, we
proceed with treating the O I $\lambda$1302 feature as real on the basis that:
1) the line is detected at 3.2$\sigma$ significance, 2) there is close
kinematic consistency with Si II, and 3) the feature is centered at the same
velocity as multiple other ions, e.g., H I, C II, N II, and Ca II. After
applying a low and high continuum to this region, we measure log
$N_{\mathrm{OI}}$ = 12.43 $\pm$ 0.08 $\pm$ 0.07, which includes the
statistical error and a continuum-placement (systematic) error. This
measurement is the foundation of our metallicity measurement in the HVC, which
we discuss in Section 4.
### 3.4 Low and intermediate ion absorption
We detected C II $\lambda$1334, Si II $\lambda\lambda$1190, 1193, 1304, 1526,
N II $\lambda$1083, Si III $\lambda$1206, and Fe III $\lambda$1122 in the HVC
near $+125$ km s-1 in the FUSE and STIS spectra (see Figure 3). Our
absorption-line measurements of these lines are given in Table 2. Achieving a
simultaneous Voigt profile fit using all Si lines is hindered by difficulties
in continuum placement for the stronger lines since large spans of the
variable stellar continuum are absorbed. Instead, we adopt the Voigt profile
measurement log $N_{\mathrm{SiII}}$ = 12.94$\pm$0.07 for the weakest unblended
line available at 1526 Å. We show that it is a good fit for
$\lambda\lambda$1190, 1193, 1304 in Figure 3.
Although detected, N II 1083 lies in a portion of the FUSE spectrum where the
stellar continuum is highly variable over a short range in wavelength.
Comparing the neighboring low-velocity H2 $J$3 1084 Å line to other $J$3 lines
in regions with smooth continua reveals that the continuum must drop
significantly in this region, making it difficult to estimate log
$N_{\mathrm{NII}}$. For this reason, our measurement for N II is an upper
limit. Fe II 1144 has a low absorption profile in the vicinity of the HVC. We
applied a high, middle, and low continuum across the absorbing region of
$\lambda$1144 and calculate a significance of 2.2$\sigma$ after including a
continuum placement error. We find a 3$\sigma$ limiting column density of log
$N_{\mathrm{FeII}}$ $\leq$ 12.99, however, given the Fe III detection in the
HVC, we use the Fe III column density in metallicity calculations going
forward. In addition to the high-velocity absorption we observe for Fe III
$\lambda$1122, we detect an intermediate-velocity cloud (IVC) centered near
$+$86 km s-1 with log $N$ = 12.98$\pm$0.13.
The archival ESO FEROS optical spectrum covers the Ca II K and H and Na I D
lines, see Figure 5. Telluric H2O absorption lines are present in the velocity
range of the HVC, however, there is a non-detection of high-velocity Na I and
we find a 3$\sigma$ limiting column density limit of log
$N_{\mathrm{NaI}}<9.45$. We see absorption in Ca II across 6 pixels at 123.7
km s-1 and measure an AOD column density of log $N_{\mathrm{CaII}}$ =
10.68$\pm$0.01 for Ca II 3934. We also performed a simultaneous Voigt profile
fit to Ca II $\lambda$3934, 3969 and find log $N$ = 10.63$\pm$0.10. The
resulting $b$-value for this small component has a high error. However, since
the log $N$ value from Voigt profile fitting agrees with the AOD measurement
within the margin of error, we adopt its value.
### 3.5 High ion absorption
Figure 6: Velocity profiles of the high-ion absorption lines detected toward
HD 156359. The flux is shown in black and the 1$\sigma$ error in the flux is
in blue. The dashed black vertical line marks 0 km s-1. The solid green curve
is the overall Voigt profile fit to the absorption. The solid magenta curve is
the Voigt profile fit to the HVC absorption near $+125$ km s-1 and the
vertical dashed magenta line marks the velocity centroid of the fitted
component. The pink and yellow curves are fits to components at intermediate
velocities.
We detected C IV $\lambda\lambda$1548, 1550, Si IV $\lambda\lambda$1393, 1402,
and N V $\lambda\lambda$1238, 1242 absorption near $+$130 km s-1 in our STIS
E140M spectrum. The C IV and Si IV profiles are complex, showing multiple
distinct low and intermediate-velocity components, in addition to the HVC. The
high-ion absorption features lie on well-defined continua due to their
corresponding broad and smooth stellar P Cygni wind profiles, which gradually
elevates their stellar continua in the region from $-100$ to $+200$ km s-1 for
this star, as shown in Figure 6. The high-ion continuum fitting was also
guided by fits to the GHRS data previously published by Sembach et al. (1991,
1995). We performed Voigt-profile fitting to all high ions with the initial
positions of the components and $b$-values determined from the C IV 1548 Å
line. The position and $b$-values were allowed to vary freely and the results
for the fits are given in Table 2. We note the detection of an IVC centered
near $+$78 km s-1 in both C IV and Si IV with log $N$ = $13.57\pm 0.08$ and
$13.09\pm 0.10$, respectively.
We observe high-velocity absorption in O VI 1031 and 1037 in the LiF1 channel
of the FUSE spectrum at $v_{\mathrm{LSR}}$ = 141 km s-1. However, we only
include $\lambda$1031 in our analysis, because $\lambda$1037 lies on the steep
blueward side of the O VI P Cygni profile and suffers from significant
blending with C II∗ 1037 and H2 $J$1 1038 Å.
The wavelength region around O VI 1031 Å contains several absorption lines
which can serve as contaminants, most notably HD 6–0 $R$(0) $\lambda$1031.915,
Cl I $\lambda$1031.507, and several lines of molecular hydrogen including H2
6–0 $P$(3) $\lambda$1031.192 and H2 6–0 $R$(4) $\lambda$1032.351. To gauge the
effect of contamination by HD 6–0 $R$(0) $\lambda$1031.915, we examined other
HD lines of similar oscillator strength that are isolated from interstellar
absorption, such as 5–0 $R$(0) $\lambda$1042.850, 7–0 $R$(0)
$\lambda$1021.460, and 8–0 $R$(0) $\lambda$1011.461. We detect no HD molecular
absorption in these lines and conclude that no subtraction of the HD 6–0
$R$(0) line at $\lambda$1031.915 from the O VI profile is necessary. For Cl I,
a small amount of absorption in Cl I $\lambda$1347 is present near 0 km s-1 in
the STIS spectrum and we simultaneously fit Cl I $\lambda$1031 to derive its
contribution to the O VI profile. For molecular hydrogen, H2 6–0 $R$(4)
$\lambda$1032.351 is the most relevant potential contaminant as it overlaps
with the high-velocity component in O VI $\lambda$ 1031\. We modeled other
isolated H2 $J4$ lines in the FUSE spectrum, e.g. 5–0 $R$(4) $\lambda$1044.543
and 4–0 $R$(4) $\lambda$1057.381 (see description in Appendix B). Through
simultaneous fitting of those lines with H2 6–0 $R$(4) $\lambda$1032 , we
account for the small amount of H2 present in the O VI high-velocity
absorption. A similar procedure was followed to account for contamination from
H2 6–0 $P$(3) $\lambda$1031.192 near $v_{\mathrm{LSR}}$ = $-$200 km s-1 using
the isolated H2 $J3$ lines 6–0 $R$(3) $\lambda$1028.986 and 5–0 $P$(3)
$\lambda$1043.503. The resulting Voigt profile fit and O VI column density are
shown in Figure 6 and Table 2.
## 4 Results: The Cool Low-ion Gas
The neutral and low ionization species including H I, O I, C II, N II, Si II,
Ca II, Fe II, Si III, and Fe III trace the cool photoionized phase of the HVC
detected at $v_{\mathrm{LSR}}$ = +125 km s-1 toward HD 156359 (catalog ). In
this section we use the ratios of metal column densities to H I column
densities to derive the ion abundances for each observed species in the HVC,
as shown in Table 3. The elemental abundances are then derived from the ion
abundances using ionization corrections derived from custom photoionization
modeling. Finally a comparison of the relative abundances of different
elements is used to derive the dust depletion pattern in the HVC.
Table 1: Measured Absorption Lines
Ion | $\lambda^{a}$ | $f$ | Dataset
---|---|---|---
| (Å) | |
H I | 923.150 | 0.0022b | FUSE SiC2A
| 926.226 | 0.0032b | FUSE SiC2A
| 930.748 | 0.0048b | FUSE SiC2A
| 937.804 | 0.0078b | FUSE SiC2A
C II | 1334.532 | 0.1290c | STIS E140M
C IV | 1548.202 | 0.1900d | STIS E140M
| 1550.774 | 0.0948d | STIS E140M
N II | 1083.990 | 0.1110c | FUSE SiC2B
N V | 1238.821 | 0.1560e | STIS E140M
| 1242.804 | 0.0777e | STIS E140M
O I | 1302.168 | 0.0480c | STIS E140M
O VI | 1031.912 | 0.1330e | FUSE LiF1
Na I | 5891.583 | 0.6500g | FEROS
Si II | 1190.420 | 0.2560f | STIS E140M
| 1193.280 | 0.5440f | STIS E140M
| 1304.370 | 0.0910f | STIS E140M
| 1526.720 | 0.1440f | STIS E140M
Si III | 1206.510 | 1.6100g | STIS E140M
Si IV | 1393.760 | 0.5130g | STIS E140M
| 1402.770 | 0.2540g | STIS E140M
Ca II | 3934.770 | 0.6490h | FEROS
| 3969.590 | 0.3210h | FEROS
Fe II | 1144.926 | 0.0830i | FUSE LiF2
Fe III | 1122.524 | 0.0642j | FUSE LiF2
Note. — a All wavelengths are taken from Morton (2003). b Pal’chikov (1998), c
Froese Fischer & Tachiev (2004), d Yan et al. (1998), e Peach et al. (1988), f
Bautista et al. (2009), g Froese Fischer et al. (2006), h Safronova &
Safronova (2011) i Donnelly & Hibbert (2001), j Deb & Hibbert (2009)
Table 2: HD 156359 High-velocity absorption
Ion | $v_{\mathrm{LSR}}$ | $b$ | log $N$
---|---|---|---
| (km s-1) | (km s-1) | ($N$ in cm-2)
H I | 125.6 $\pm$ 0.9 | 18.9 $\pm$ 1.2 | 15.54 $\pm$ 0.05
O I | 125.0 $\pm$ 4.8 | … | 12.43 $\pm$ 0.11a
Na I | … | … | $<$ 9.45b
C II | 120.9 $\pm$ 0.6 | 9.8 $\pm$ 1.4 | 13.83 $\pm$ 0.10
N II | … | … | $<$ 13.98b
Si II | 118.5 $\pm$ 1.0 | 7.0 $\pm$ 1.6 | 12.94 $\pm$ 0.07
Ca II | 123.7 $\pm$ 1.0 | 2.3 $\pm$ 2.7 | 10.63 $\pm$ 0.10
Fe II | … | … | $<$ 12.99b
Si III | 117.6 $\pm$ 1.4 | 6.9 $\pm$ 2.8 | 12.32 $\pm$ 0.16
Fe III | 129.3 $\pm$ 2.3 | 11.0 $\pm$ 3.6 | 12.82 $\pm$ 0.14
C IV | 133.3 $\pm$ 2.3 | 8.6 $\pm$ 3.6 | 12.60 $\pm$ 0.13
Si IV | 129.5 $\pm$ 1.5 | 5.7 $\pm$ 2.9 | 11.93 $\pm$ 0.15
N V | 130.2 $\pm$ 2.7 | 12.2 $\pm$ 4.0 | 12.64 $\pm$ 0.11
O VI | 141.3 $\pm$ 1.2 | 24.0 $\pm$ 1.8 | 13.65 $\pm$ 0.03
Note. — a Except where noted, all measurements presented in this table are
derived from Voigt-profile fitting, with the exception of O I, for which we
derive an apparent column density from the profile in the range $+110\lesssim
v\lesssim+125$ km s-1, see Figure 4. The error listed for log $N$(O I) is the
combination of the statistical error and a continuum placement error in
quadrature, and is $\pm$ 0.08 $\pm$ 0.07 separately.
b This measurement is a non-detection and the 3$\sigma$ limiting column
density is given.
### 4.1 Photoionization Modeling
To model the ionization breakdown in the HVC and characterize its physical
conditions, we ran a multi-dimensional grid of Cloudy (v. 17.02, Ferland et
al. 2017) photoionization models for the low- and intermediate-ions, assuming
they arise in the same gas phase as the H I and are photoionized by the same
incident radiation field. Our model assumes a plane-parallel slab of uniform
density exposed to the magnitude of the escaping UV ionizing flux of the Milky
Way (Fox et al., 2005; Barger et al., 2013; Fox et al., 2014) at the location
of HD 156359 (catalog ) and includes the extragalactic UV background radiation
field from Khaire & Srianand (2019).
We ran our models for a grid of metallicity values with [Z/H] varying from
$-0.5$ to $+1.5$ in steps of 0.1 dex, each over a range of hydrogen number
densities log ($n_{\mathrm{H}}$/cm-3) from $-3$ to 0, in order to explore
possible metallicities and densities. We determined the best-fit log
$n_{\mathrm{H}}$ for each metallicity model by matching the observed column
density ratio of Si III/Si II to the model value (see the top panel of Figure
7). The Si III/Si II ratio was chosen because both ions have unsaturated
detections in the HVC, and using a ratio of adjacent ions from the same
element (Si) minimizes metallicity or depletion effects from different metal
ions. Using the pairs of metallicity and log $n_{\mathrm{H}}$ determined from
the Si III/Si II ratio, we narrow the range of metallicities by identifying
which models are consistent with the observed log $N_{\mathrm{OI}}$ = 12.43
$\pm$ 0.11, as illustrated in the bottom left panel of Figure 7. We then run
an even finer grid of metallicity values in increments of 0.02 dex over the
narrowed metallicity and number density region to determine the range of
metallicities and densities allowed by the data.
We find that metallicities in the range $0.18\leq\mathrm{[O/H]}\leq 0.51$ and
densities from $-1.69\leq$ log ($n_{\mathrm{H}}$/cm-3) $\leq-1.37$ are
consistent with the data, giving a model best-fit of [O/H] =
0.36${}^{+0.15}_{-0.18}$ at log $n_{\mathrm{H}}$ = $-$1.53$\pm$-0.16. We
repeated this procedure for C II (see bottom-right panel of Figure 7), which
is expected to be weakly depleted (Jenkins, 2009), and find that the range
0.21 $\leq$ [C/H] $\leq$ 0.42 overlaps with the observed data, giving a model
best-fit of [C/H] = 0.30${}^{+0.12}_{-0.09}$ at log $n_{\mathrm{H}}$ =
$-$1.49${}^{+0.09}_{-0.10}$. The agreement between the oxygen-based
metallicity and the carbon-based metallicity lends support to our methodology
and to the robustness of the super-solar metallicity we infer.
The metallicity for this cloud ranks among the highest UV-based metallicities
of any HVC observed thus far, along with the HVC at $-$125 km s-1 toward
M5-ZNG ($l$ = 3.$\degree$9, $b$ = $+$47.$\degree$7 at $z$ = $+$5.3 kpc) with
[O/H] = $+$0.22 $\pm$ 0.10 (Zech et al., 2008). M5-ZNG was also observed with
FUSE and STIS E140M spectra. Their observed metallicity is not corrected for
ionization effects, but is likely higher than reported, given the measured low
log $N_{\mathrm{HI}}$ = 16.50 $\pm$ 0.06 in the cloud and that oxygen shows a
positive ionization correction when log $N$(H I) $<$ 18.5 (Bordoloi et al.,
2017). The metallicity of the HD 156359 (catalog ) HVC is also on the high end
of the range of $<$20% to 3 times solar reported by Ashley et al. (2022) in
their survey of metallicities of Fermi Bubble HVCs. We note that a super-solar
abundance in the inner Galaxy may not be unexpected, given the oxygen
abundance gradients reported from emission line measurements in H II regions
(see Wenger et al. 2019; Arellano-Córdova et al. 2021).
Table 3: HVC elemental abundances and depletions
Ion | [Xi/H I]a | IC(Xi)b | [X/H]c | $\delta_{\mathrm{O}}$(X)d
---|---|---|---|---
O I | 0.20$\pm$0.10 | $+$0.16${}^{+0.04}_{-0.07}$ | 0.36${}^{+0.11}_{-0.12}$ | 0
C II | 1.86$\pm$0.12 | $-$1.59$\pm$0.04 | 0.27$\pm$0.13 | $-$0.09$\pm$0.17
N II | $<$ 2.60 | $<-$1.67 | $<$ 0.93 | $<$ 0.57
Si II | 1.89$\pm$0.08 | $-$1.86$\pm$0.03 | 0.03$\pm$0.09 | $-$0.33$\pm$0.14
Fe II | $<$ 1.95 | $-$1.17${}^{+0.20}_{-0.12}$ | $<$ 0.79 | $<$ 0.43
Fe III | 1.78$\pm$0.15 | $-$1.73$\pm$0.08 | 0.06$\pm$0.17 | $-$0.30$\pm$0.20
Ca II | 0.75$\pm$0.11 | $-$0.95$\pm$0.01 | $-$0.20$\pm$0.11 | $-$0.56$\pm$0.16
Note. —
a Ion abundance [Xi/H I] = log (Xi/H I)HVC $-$ log (X/H)⊙ where Xi is the
observed ion of element X. These are not corrected for ionization effects. The
error includes both systematic and continuum placement uncertainties.
b Ionization correction IC(Xi) = [X/H]model $-$ [Xi/H I], derived from the
difference between the model elemental abundance and the observed ion
abundance.
c Ionization-corrected elemental abundance [X/H] = [Xi/H I] + IC(Xi). Also
referred to as gas-phase abundance.
d $\delta_{\mathrm{O}}$(X) $\equiv$ [X/O] = [X/H] $-$ [O/H] is the depletion
of element X relative to oxygen.
Figure 7: Cloudy photoionization models of the HVC toward HD 156359 (catalog )
that explore a parameter space in metallicity and number density, $-0.50\leq$
[Z/H] $\leq+1.50$ and $-3\leq$ log ($n_{\mathrm{H}}$/cm-3) $\leq 0$,
respectively. Top panel: Model results for log
$N_{\mathrm{SiIII}}$/$N_{\mathrm{SiII}}$ for the set metallicities are shown
as solid colored curves. The dashed horizontal turquoise line is the observed
ratio of $-$0.62$\pm$0.17. The best-fit log $n_{\mathrm{H}}$ for each
metallicity model is determined by the intersection of the model and observed
ratio and is marked by vertical colored lines and circle markers. Bottom left
panel: The model curves for log $N_{\mathrm{OI}}$ for each metallicity are
plotted against log $n_{\mathrm{H}}$, where the values derived from log
$N_{\mathrm{SiIII}}$/$N_{\mathrm{SiII}}$ are marked with vertical lines and
markers. The horizontal turquoise line and bar show the observed log
$N_{\mathrm{OI}}$ = 12.43 $\pm$ 0.11. The model data corresponding to the
metallicity region from $+0.18\leq$[Z/H] $\leq+0.51$ and $-1.69\leq$ log
$n_{\mathrm{H}}$ $\leq-1.37$ overlap with the observed data. The model best-
fit values are [Z/H] = $+$0.36${}^{+0.15}_{-0.18}$ and log $n_{\mathrm{H}}$ =
$-1.53\pm 0.16$. Bottom right panel: Same as for the bottom left panel, but
for C II. The horizontal turquoise line and bar show observed log
$N_{\mathrm{CII}}$ = 13.83 $\pm$ 0.10. The model data corresponding to the
metallicity region from $+0.21\leq$ [Z/H] $\leq+0.42$ and $-1.59\leq$ log
$n_{\mathrm{H}}$ $\leq-1.40$ overlap with the observed data. The model best
fit values are [Z/H]= $+$0.30${}^{+0.12}_{-0.09}$ and log $n_{\mathrm{H}}$ =
$-1.49^{+0.09}_{-0.10}$. Figure 8: Top panel: ionization corrections for the
low ions O I, C II, N II, Si II, Fe III, and Ca II (colored curves) plotted
against log $n_{\mathrm{H}}$ from our Cloudy model to the HD 156359 (catalog )
HVC. The model uses our best-fit [Z/H]=$+$0.36. The vertical line marks the
best-fit log ($n_{\rm H}$/cm-3)=$-$1.53, determined from the Cloudy models for
O I (see Figure 7). The ionization correction calculated at $-$1.53 is added
to the observed ion abundance to determine the elemental abundance. Middle
panel: comparison of the ionization-corrected abundances of the low ions
detected in the HVC, where the solar abundance is plotted as a horizontal
line. Bottom panel: Comparison of the dust depletion levels $\delta$(X) =
[X/O] in the HVC with the depletion pattern for [X/O] measured for sight lines
with the lowest and highest collective depletions ($F_{*}$=0 and $F_{*}$=1,
respectively) from Jenkins (2009). A slight offset is applied in the
$x$-direction of each element for distinction. Figure 9: Our final Cloudy
photoionization model of the low- and intermediate ions detected in the HVC
toward HD 156359 near $+125$ km s-1. The model uses the best-fit [Z/H] =
$+$0.36 determined from our O I analysis. These model predictions for each ion
(colored dotted curves) have been scaled by the dust depletions required to
match the observed values, where the observed column densities are indicated
by circle markers at the best-fit log ($n_{\mathrm{H}}$/cm-3)=$-$1.53 (black
vertical line).
### 4.2 Ionization corrections
In ISM abundance work, [O I/H I] is often considered a robust indicator of
elemental abundance [O/H], since (1) charge-exchange reactions closely couple
the ionization state of hydrogen and oxygen together Field & Steigman (1971),
and (2) oxygen is only lightly depleted onto dust grains (Jenkins, 2009).
However, the assumption that [O I/H I]=[O/H] breaks down when $N$(H I) is so
low that the gas is optically thin or when the ionizing photon flux is
extremely high (Viegas, 1995). In the HVC at +125 km s-1 toward HD 156359
(catalog ), log $N$(H I) is only 15.54$\pm$0.05, so an ionization correction
must be made to account for the column densities of all observed neutral and
ion stages.
Our Cloudy models directly provide the ionization corrections for all our
observed ion stages. We define the ionization correction as the difference
between the model (true) elemental abundance and the observed ion abundance,
i.e.,
$IC(X^{i})=[\text{X/H}]-[\text{X}^{i}/\textsc{H i}].$ (1)
The resulting ion abundances, ionization corrections, and ionization-corrected
elemental abundances are given in Table 3. The relationships between the
ionization corrections and the hydrogen number density for [Z/H] = 0.36 are
shown in the top panel of Figure 8 and the ionization-corrected abundances are
shown in the middle panel of Figure 8.
The positive ionization correction of $IC$(O I) = $0.16^{+0.04}_{-0.07}$ from
the photoionization modeling results in a high gas-phase oxygen abundance of
$+0.36$. We calculate gas-phase abundances of [C/H] = 0.27$\pm$0.13, [N/H] =
$\leq$ 0.93, [Si/H] = 0.03$\pm$0.09, [Fe/H] = 0.06$\pm$0.17, and [Ca/H] =
$-$0.20$\pm$0.11 (see Table 3).
### 4.3 Dust depletion effects
Following convention, we define the depletion $\delta_{\mathrm{O}}$(X) of each
refractory element X as the ionization-corrected abundances of that element
compared to the ionization-corrected oxygen abundance, i.e.,
$\delta_{\mathrm{O}}(\text{X})\equiv[\text{X/O}]=[\text{X/H}]-[\text{O/H}],$
(2)
where oxygen represents an undepleted (or lightly depleted) volatile element.
In this formalism, a negative value of $\delta_{\mathrm{O}}$(X) means that
element X is depleted relative to oxygen. This method assumes that the total
(gas+dust) abundances are solar, which is often assumed to apply to the
Galactic ISM (Savage & Sembach, 1996), though local ISM abundance variations
cannot be ruled out in some sight lines (De Cia et al., 2021). We show
$\delta_{\mathrm{O}}$(X) for the low ions in the bottom panel of Figure 8
compared to $F_{*}$, the line-of-sight depletion strength factor, for [X/O]
determined for low-depletion ($F_{*}$=0) and high-depletion ($F_{*}$=1) gas
from the comprehensive ISM gas-phase element depletions study of Jenkins
(2009).
For the HVC toward HD 156359 (catalog ), we find a low value of
$\delta_{\mathrm{O}}$(C)=$-$0.09$\pm$0.17, i.e. carbon shows no significant
depletion. This is consistent with the low values of [C/O] measured in
Galactic ISM gas (Jenkins, 2009). For the other low ions, we find a 3$\sigma$
upper limit for the nitrogen depletion $\delta_{\mathrm{O}}$(N)$\leq+0.57$ and
low depletions for silicon, iron, and calcium of
$\delta_{\mathrm{O}}$(Si)=$-$0.33$\pm$0.14$,\delta_{\mathrm{O}}$(Fe)=$-$0.30$\pm$0.20,
and $\delta_{\mathrm{O}}$(Ca)=$-$0.56$\pm$0.16, respectively. To summarize the
final results of the CLOUDY modeling after the dust depletion levels have been
derived, Figure 9 shows the model curves of the detected ions shifted by their
respective depletions at [X/H] = $+$0.36 and log $n_{\mathrm{H}}$= $-$1.53.
The ability of this model to match the observed column densities shows that
all low-ion measurements can be explained by photoionization once dust
depletion effects are accounted for.
A low level of dust depletion for the HVC is consistent with the well-known
Routly-Spitzer effect (RS effect), in which the amount of dust observed in
high-velocity clouds decreases significantly at higher LSR velocity (see
Routly & Spitzer 1952; Siluk & Silk 1974; Vallerga et al. 1993; Smoker et al.
2011; Ben Bekhti et al. 2012; Murga et al. 2015). The RS effect is typically
traced by the $N$(Na I)/$N$(Ca II) ratio, which has been found to
significantly decrease with increasing LSR velocity. Although the RS effect
was historically interpreted as a dust depletion effect, it may also be
related to ionization effects, and likely differs depending on the environment
of the cloud. We measure an 3$\sigma$ upper limit of log $N$(Na I)/$N$(Ca II)
$\leq$ $-$1.2 in the $+124$ km s-1 HVC toward HD 156359 (catalog ) from the
optical FEROS data, which is on the low end of the observed ratios for HVCs
(Smoker et al., 2011; Ben Bekhti et al., 2012; Murga et al., 2015), and is
also lower than is typically observed at low velocities. In summary, the dust
depletion pattern we derive in the HVC is consistent with other HVCs and with
the Routly-Spitzer effect, even though the cloud has high metallicity.
## 5 Results: the Highly Ionized Gas
The HVC toward HD 156359 (catalog ) shows high-ion absorption in the resonance
doublets of Si IV, C IV, N V, and O VI at a velocity range of 130–140 km s-1
(Figure 6; Table 2), slightly higher than the velocity range of the low ions,
which are centered near 125 km s-1. To compare the high-ion absorption between
different ions, we show in Figure 10 a comparison of the normalized optical
depth profiles of each high ion. These plots provide a way to inter-compare
the profiles of different ions in a linear manner (e.g., Fox et al. 2003). The
HVC line profiles of C IV and Si IV are well aligned in velocity, but are
offset from the centroid of O VI absorption by $\approx$8 km s-1. This
suggests that the O VI exists in a separate (likely hotter) gas phase. The
STIS profile of N V is too noisy to draw any conclusions about line centroid.
However, the GHRS N V profiles (Sembach et al., 1995) are less noisy and have
a similar profile shape to the O VI seen in the FUSE data, supporting the
placement of O VI and N V in the same phase.
### 5.1 Collisional Ionization Modeling
Figure 10: Optical depth profiles of the high ions, where each profile is
arbitrarily normalized to its peak value for the velocity region shown in
order to facilitate comparison. Top panel: C IV $\lambda$1548 and Si IV
$\lambda$1393 display similar line shapes in the velocity region of the HVC
from $112\lesssim v\lesssim 140$ km s-1, indicated with cyan shading. Middle
panel: C IV $\lambda$1548 and O VI $\lambda$1031 are compared, where the
center of the HVC absorption for O VI is shifted to a higher velocity at
$v\approx 141$ km s-1. Bottom panel: N V $\lambda$1238 and O VI $\lambda$1031
are compared, where the range of velocities attribute to absorption by O VI
are indicated in light pink. Figure 11: Comparison of observed high-ion
column-density ratios with predictions from the collisional ionization models
of Gnat & Sternberg (2007). The observed ratios in the HD 156359 (catalog )
HVC are shown as dashed orange horizontal lines. The top and bottom panels
show the C IV/Si IV and O VI/N V ratios, respectively. In both panels, the
solid black curve is the collisional ionization equilibrium model ion
fraction. The solid green and violet curves are the time-dependent isochoric
and isobaric collisional ionization models for a solar metallicity,
respectively. Their dashed counterparts are the same but for a metallicity
twice the solar value. The vertical dashed blue line is the temperature of the
gas determined by the models. The vertical blue band in the top panel
represents a second time-dependent solution for a range of temperatures from
log $T$ = 4.2–4.4 K.
The observed high ions have column densities that are too high to be explained
by the Cloudy photoionization models described in Section 4.1. Where the Si IV
and C IV column densities differ by $\sim$0.7 and 1.0 dex respectively, the N
V and O VI differ by orders of magnitudes. These discrepancies imply that a
separate ionization mechanism is required for the high ions. We do not account
for energetic photons from the radiation field intrinsic to the Fermi Bubbles,
however, we explore collisional ionization as a separate mechanism for gas
with temperatures of $T\gtrsim$ 105 K. We consult the collisional ionization
models from Gnat & Sternberg (2007) to determine the exact range of
temperatures allowed by the high-ion column densities and their ratios. While
we considered the more recent models from Gnat 2017, which include
photoionization effects from the extragalactic UV background, we do not adopt
them because HD 156359 lies in close proximity to the radiation field of the
Galactic plane which is much stronger. We considered both the collisional
ionization equilibrium and non-equilibrium regimes.
A single-temperature solution explaining all four high ions (Si IV, C IV, N V,
O VI) in the HVC is ruled out, as no such solution exists to the observed
high-ion column densities. Instead, we find that a two-phase solution is
needed, with one temperature explaining the
$N_{\mathrm{CIV}}$/$N_{\mathrm{SiIV}}$ ratio and another explaining
$N_{\mathrm{OVI}}$/$N_{\mathrm{NV}}$. This two-phase model is consistent with
the kinematic information in the UV spectra, where Si IV and C IV show very
similar line profiles but O VI is offset in velocity centroid. The two-phase
high-ion model is illustrated in Figure 11, which shows the observed high-ion
ratios of $N_{\mathrm{CIV}}$/$N_{\mathrm{SiIV}}$ and
$N_{\mathrm{OVI}}$/$N_{\mathrm{NV}}$ compared to model predictions from Gnat &
Sternberg (2007) for solar ([Z/H]=0) and super-solar ([Z/H]=$+$0.3)
metallicities.
For the C IV/Si IV phase, we find two possible solutions for the temperature.
First, the non-equilibrium isochoric (constant volume) and isobaric (constant
pressure) models give a low-temperature solution at $T$ = $10^{4.2-4.4}$ K,
where the lower end corresponds to the solar-metallicity isochoric model and
the higher end with the super-solar isobaric model. Second, the collisional
ionization equilibrium (CIE) model returns a higher temperature $T$=104.9 K.
We are inclined to adopt the non-equilibrium (lower-temperature) solution, as
plasma near $T$=105 K is at the peak of the cooling curve, where oxygen
dominates the radiative cooling and the gas can cool faster than it
recombines, reaching a non-equilibrium state. For the O VI/N V phase, a single
temperature solution for the observed log
$N_{\mathrm{OVI}}$/$N_{\mathrm{NV}}$=1.01 is found at $T$=105.4 K for all
models (see bottom panel of Figure 11).
In conclusion, we can successfully model the high-ion plasma in the HVC toward
HD 156359 (catalog ) as containing collisionally-ionized gas at two
temperatures: a cooler phase seen in C IV and Si IV at $T$ = $10^{4.2-4.4}$ K,
and a hotter phase seen in N V and O VI at $T$=105.4 K.
## 6 Summary
Using archival FUSE, HST STIS, and ESO FEROS spectra, we have analyzed the
chemical composition of the HVC near $+$125 km s-1 toward HD 156359 (catalog
), a massive star lying 9 kpc away toward the Galactic Center. The sight line
passes less than 1$\degree$ from one of the densest cores of a complex of
small HVCs dubbed “Complex WE” by Wakker & van Woerden (1991), as shown in
Figure 1. Furthermore, the sight line passes through a region of enhanced
X-ray emission (the southern eROSITA Bubble; Predehl et al., 2020); this
region indicates energetic feedback from the Galactic Center. Our main results
are as follows.
1. 1.
We determined an H I column density measurement of log $N$(H I) = $15.54\pm
0.05$ in the HVC using unsaturated Lyman series absorption lines.
2. 2.
We measured a super-solar O I abundance of [O I/H I] = 0.20$\pm$0.11 in the
HVC. After applying an ionization correction, we derive an oxygen abundance of
[O/H] = 0.36$\pm$0.12, indicating the cloud is enriched to 2.3 times the solar
level. This abundance is consistent with super solar O abundances of H II
regions measured in the inner Galaxy (Wenger et al., 2019; Arellano-Córdova et
al., 2021).
3. 3.
The ionization-corrected carbon abundance [C/H] = 0.27$\pm$0.13 is consistent
with the oxygen abundance. This indicates that carbon shows low dust depletion
relative to oxygen, consistent with the pattern seen by Jenkins (2009), as
shown in Figure 8.
4. 4.
A low level of depletion is inferred from the silicon, iron, and calcium
lines, with [Si/O] = $-$0.33 $\pm$ 0.14, [Fe/O] = $-$0.30 $\pm$ 0.20, and
[Ca/O] = $-$0.56 $\pm$ 0.16. The Na I/Ca II ratio in the HVC is $\leq-1.2$,
which is lower than is typically observed at low velocities, consistent with
the trend known as the Routly-Spitzer effect.
5. 5.
We detect high ion species C IV, Si IV, N V, and O VI near $+$130 km s-1, at
slightly higher velocities than the velocity range of the lower ions. We find
that an ionization mechanism separate from photoionization, such as
collisional ionization, is required to explain the column densities of the
high ions. We determine that a two-phase temperature solution best explains
the observed $N_{\mathrm{CIV}}$/$N_{\mathrm{SiIV}}$ and
$N_{\mathrm{OVI}}$/$N_{\mathrm{NV}}$ ratios, with a cooler phase seen in C IV
and Si IV at $T$ = $10^{4.2-4.4}$ K, and a hotter phase seen in N V and O VI
at $T$=105.4 K.
The high metallicity, low depletion, complex high-ion absorption, and high
positive velocity of the HVC toward HD 156359 (catalog ) are all consistent
with a wind origin, in which a swept-up clump of material is being carried out
from the Galactic disk into the halo. As such, this HVC may represent a
freshly entrained cool clump of gas caught in the act of being accelerated
into the nuclear wind. While we cannot rule out a foreground origin for the
HVC, in which the cloud exists at an anomalous velocity somewhere between the
Sun and the Galactic Center, we can exclude a spiral-arm explanation for the
HVC on kinematic grounds, because the cloud’s $+125$ km s-1 velocity lies over
100 km s-1 away from the nearest spiral arm, the Sagittarius Arm at $\sim-10$
km s-1. Under a biconical outflow model (Fox et al. 2015; Bordoloi et al.
2017; Di Teodoro et al. 2018), the cloud’s radial velocity and its location
only 2.3 kpc below the disk imply a short timescale of $\sim$5 Myr for the HVC
to have reached its current position, which is much shorter than the
timescales on which chemical mixing is expected to be significant (10s to 100s
of Myr; Gritton et al., 2014; Heitsch et al., 2022). Our observations
therefore provide a snapshot into the chemical and physical conditions
prevailing in this early stage of a nuclear outflow, before chemical mixing
has diluted or enriched the gas from its initial state.
We gratefully acknowledge the invaluable contributions to early versions of
this manuscript from the late Blair Savage, who passed away during the
manuscript’s preparation. This paper would not have been possible without
Blair’s foundational work on the HD 156359 (catalog ) sight line, the chemical
abundances in the ISM, apparent optical depth analysis, and the inner Galaxy.
We gratefully acknowledge support from the NASA Astrophysics Data Analysis
Program (ADAP) under grant 80NSSC20K0435, 3D Structure of the ISM toward the
Galactic Center. The FUSE data were obtained under program P101. FUSE was
operated for NASA by the Department of Physics and Astronomy at the Johns
Hopkins University. D.K. is supported by an NSF Astronomy and Astrophysics
Postdoctoral Fellowship under award AST-2102490. We thank Isabel Rebollido for
valuable conversations on the FEROS spectrograph. The FUSE and HST STIS data
presented in this paper were obtained from the Mikulski Archive for Space
Telescopes (MAST) at the Space Telescope Science Institute. The ESO FEROS
spectrum was obtained from the ESO Archive Science Portal. The FUSE and HST
STIS data presented in this paper were obtained from the Mikulski Archive for
Space Telescopes (MAST) at the Space Telescope Science Institute. The specific
observations analyzed can be accessed via http://dx.doi.org/10.17909/rrnk-3e58
(catalog 10.17909/rrnk-3e58).
## Appendix A Global Continuum Fitting
Stellar absorption lines present a continuum-fitting challenge, particularly
in the complex far-UV FUSE spectra used to derive the H I column density in
the HVC under study. Our approach followed in the analysis is to fit the
continuum _locally_ around each H I Lyman series line of interest, since this
allows us to account for stellar absorption lines, which are present in the
continuum when the radiation field encounters the HVC. However, for
completeness here we consider a _global_ continuum fitting process, which fits
the H I continuum over a larger range (916–944 Å) in the SiC2 channel. This
approach neglects stellar absorption features but ensures the continua are
continuous between adjacent lines in the Lyman series. Wavelength regions free
from stellar emission and absorption lines defined the flux of the global
continuum. We show a global fit to the stellar flux in the vicinity of the
higher order H I Lyman series lines in Figure 12, in which areas of emission
and absorption due to both stellar and interstellar features can be seen. We
performed a simultaneous Voigt profile fit on the same H I lines modeled with
the global continua, using the mid-, high-, and low-continuum fits shown in
Figure 12. A separate local continuum was applied to the depressed region
resulting from stellar line blanketing and which contains H I $\lambda$923\.
Using the global continuum fit (which has a slightly higher continuum
placement than the local fits) results in log $N_{\mathrm{HI}}$ =
15.69$\pm$0.03$\pm$ 0.03 for the HVC, where the first error is the statistical
error due to photon noise and the second error is the systematic continuum-
placement error. This corresponds to a moderate difference of $\Delta$log
$N_{\mathrm{HI}}$ = 0.15 dex between the H I column densities derived by the
global and local continuum methods.
Figure 12: A portion of the FUSE SiC2 spectrum from 916–944 Å in which several
higher order H I Lyman series lines are identified. The flux and 1$\sigma$
error in the flux are shown in black and blue, respectively. A global fit to
the stellar continuum is traced by a solid red curve, with a high and low
stellar continuum placement marked by dashed red lines. The orange box shows a
local fit to a depressed portion of the stellar continuum created by stellar
line blanketing, from $\sim$920–925 Å.
## Appendix B Decontamination of Molecular Hydrogen
The FUSE LiF1 and LiF2 channels provide us with an opportunity to model
isolated H2 lines. This effort is important because it enables us to account
for the Milky Way’s molecular contribution to the interstellar absorption
lines. In particular, contamination from Milky Way H2 affected the higher
order Lyman series H I in the SiC2A channel and O VI in the LiF1 channel. We
performed simultaneous single-component Voigt profile fits to multiple H2
transitions from rotational levels $J$ = 0–6, allowing both the velocity and
$b$-value to vary freely for the stronger $J$ = 0–4 transitions. The
$b$-values of the significantly weaker $J$ = 5,6 transitions were fixed to the
$b$-value of the $J$ = 4 transition in order to deter unrealistic column
densities for these fainter lines. These results of these fittings are shown
in Figure 13.
Figure 13: We performed Voigt profile fits to vibrationally and rotationally
excited H2 lines from the FUSE LiF1 and LiF2 channels in order to account for
molecular blending with H I. In each panel, the flux and 1$\sigma$ error in
the flux are shown in black and blue, respectively. Voigt profile fits to the
data are shown in green. Green vertical lines mark the positions of the H2
absorption lines. Top panel: LiF1 channel H2 transitions from upper
vibrational levels 4 and 5, spanning rotational levels $J=0-6$. Fits to Ar I
$\lambda$1048 and Fe II $\lambda$1055 are included. Bottom left panel: LiF1
channel H2 transitions from upper vibrational levels 2 and 3, including
rotational levels $J=0,1,2,5,6$. Bottom right panel: LiF2 channel H2
transitions from upper vibrational levels 1 and 2, including rotational levels
$J=0,1,2,5$.
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|
# Robust Implicit Adaptive Low Rank Time-Stepping Methods for Matrix
Differential Equations
Daniel Appelö Department of Mathematics, Virginia Tech, Blacksburg, VA 24061
U.S.A<EMAIL_ADDRESS>Research supported by DOE Office of Advanced Scientific
Computing Research under the Advanced Research in Quantum Computing program,
subcontracted from award 2019-LLNL-SCW-1683, NSF DMS-2208164, and Virginia
Tech. Yingda Cheng Department of Mathematics, Virginia Tech, Blacksburg, VA
24061 U.S.A<EMAIL_ADDRESS>Research supported by NSF DMS-2011838, DOE grant
DE-SC0023164 and Virginia Tech.
###### Abstract
In this work, we develop implicit rank-adaptive schemes for time-dependent
matrix differential equations. The dynamic low rank approximation (DLRA) is a
well-known technique to capture the dynamic low rank structure based on
Dirac–Frenkel time-dependent variational principle. In recent years, it has
attracted a lot of attention due to its wide applicability. Our schemes are
inspired by the three-step procedure used in the rank adaptive version of the
unconventional robust integrator (the so called BUG integrator) [2] for DLRA.
First, a prediction (basis update) step is made computing the approximate
column and row spaces at the next time level. Second, a Galerkin evolution
step is invoked using a base implicit solve for the small core matrix.
Finally, a truncation is made according to a prescribed error threshold. Since
the DLRA is evolving the differential equation projected on to the tangent
space of the low rank manifold, the error estimate of the BUG integrator
contains the tangent projection (modeling) error which cannot be easily
controlled by mesh refinement. This can cause convergence issue for equations
with cross terms.
To address this issue, we propose a simple modification, consisting of merging
the row and column spaces from the explicit step truncation method together
with the BUG spaces in the prediction step. In addition, we propose an
adaptive strategy where the BUG spaces are only computed if the residual for
the solution obtained from the prediction space by explicit step truncation
method, is too large. We prove stability and estimate the local truncation
error of the schemes under assumptions. We benchmark the schemes in several
tests, such as anisotropic diffusion, solid body rotation and the combination
of the two, to show robust convergence properties.
## 1 Introduction
In this work, we are interested in solving linear matrix differential
equations in the following form
$\frac{d}{dt}X(t)=F(X(t),t),\quad X(t)\in\mathbb{R}^{m_{1}\times m_{2}},\quad
X(0)=X_{0},$ (1)
where
$F(X(t),t)=\sum_{j=1}^{s}A_{j}X(t)B_{j}^{T}+G(t),$
and $A_{j}\in\mathbb{R}^{m_{1}\times m_{1}},B_{j}\in\mathbb{R}^{m_{2}\times
m_{2}}$ are sparse or structured matrices for which a fast matrix-vector
product is assumed to be known. Further, $G(t)\in\mathbb{R}^{m_{1}\times
m_{2}}$ is a given function that is assumed to have a known low rank
decomposition. The integer $s,$ which denotes the separation rank of
$F(\cdot),$ is assumed not to be too large. We focus on implicit methods,
which are particularly needed for computing solutions to stiff equations.
Equation (1) arises in many applications governed by partial differential
equations (PDE). For example, using the method of lines approach, we can cast
numerical discretizations of two-dimensional (or even higher dimensional)
linear time-dependent convection-diffusion-reaction equations in the form of
(1). If we represent the unknowns on a Cartesian grid, the elements of $X(t)$
will represent the unknowns (e.g. point values of the solution) at a given
time $t.$ The matrices $A_{j},B_{j}$ will, e.g., correspond to difference
operators or variable coefficients, and $G(t)$ encodes the given source terms
and boundary conditions. Compared to the traditional approach of numerical PDE
solvers, the matrix approach does not vectorize the unknowns into a single
vector at each time step, but rather we represent the unknowns in a matrix
format, retaining the relative relations in each spatial dimension.
In recent years, the matrix (or tensor in higher dimensions) approach received
increased attention in numerical analysis because it offers a way to tame _the
curse of dimensionality_ , enabling approximations to solutions of high
dimensional PDEs [10, 13]. The overarching idea is to expose low rank
structure in the solution manifold in order to drastically reduce storage and
computation. This is the idea of low rank tensor methods for PDEs [7, 1]. In
two dimensions, this approach reduces to the low rank matrix method, where it
is assumed that the matrix $X(t)$ has a small rank $r.$ As is well known, a
rank $r$ matrix of size $m_{1}\times m_{2}$ allows for a SVD decomposition
with only $(m_{1}+m_{2}-r)r$ degrees of freedom. This is much smaller than
$m_{1}m_{2}$, the degrees of freedom of the full matrix, when $r$ is small
compared to $m_{1}$ and $m_{2}$.
If the rank of $X(t)$ is fixed, i.e. we constrain the solution $X(t)$ to live
on the rank $r$ matrix manifold
$\mathcal{M}_{r}=\\{X\in\mathbb{R}^{m_{1}\times m_{2}},{\sf rank}(X)=r\\},$
then an effective numerical solution can be computed by the dynamic low rank
approximation (DLRA) [15, 16]. The DLRA solves
$\frac{d}{dt}X(t)=\Pi_{X(t)}F(X(t),t),$ where $\Pi_{X(t)}$ is the orthogonal
projection onto the tangent space $T_{X(t)}\mathcal{M}_{r}$ of the manifold
$\mathcal{M}_{r}$ at $X(t).$ The DLRA has found success in many applications,
including extensions to various tensor formats [18, 17]. However, the
constraint that $X(t)\in\mathcal{M}_{r}$ requires that the rank $r$ is
estimated a-priori and this is in contrast with the rank adaptive schemes.
A rank adaptive method adaptively choose the rank at each time step according
to a prescribed error tolerance. Several approaches to rank adaptivity can be
found in the literature. For example, a straightforward idea is the so-called
step truncation method [4] which evolves the low rank solution for one time
step by a traditional time stepping method in an ambient space of higher rank,
then performs a truncation (by SVD with given tolerance). This method is
intuitive and effective for explicit time stepping methods. For implicit
schemes [21], implementation of step truncation methods is nontrivial and
requires an effective iterative solver in adaptive low rank format [7, 1].
Another approach that has been suggested by multiple groups, and that is
similar to what we propose, is to modify the DLRA method by using rank
increase and decrease indicators and augment or truncate the spaces at each
time step [12, 11, 24]. In particular the ideas in [24], while being
specialized to time dependent Schrödinger, are similar to the approach we take
here. A promising idea in [2] uses the so called unconventional (BUG)
integrator [3] to achieve rank adaptivity. This approach was recently extended
to higher order in time in [19].
In the BUG integrator [3], two subproblems that come from the projector
splitting of DLRA [16] are solved. Then, the solutions to the two subproblems
are used to update the column and row spaces of the matrix; and a Galerkin
evolution step is performed in the resulting space, searching for the
coefficients of a small matrix. For the rank adaptive version of the
unconventional integrator [2], this translates to solving two matrix
differential equation of sizes $m_{1}\times r,m_{2}\times r$ in the first step
and one matrix differential equation of size $2r\times 2r$ in the Galerkin
step, followed by a truncated SVD step. The ideas in [3, 2] can be implemented
for implicit methods because the matrix size is fixed in each subproblems, so
a standard linear solver will suffice. However, the errors of both methods in
[3, 2] are subject to the tangent projection error or the so called _modeling
error_ [14].
It is important to understand that this modeling error is present in many
problems and can, as we will see below, result in non convergent numerical
methods. We now describe a trivial but prototypical example that will make the
DLRA/BUG method fail. Assume that $X(t)=U\Sigma V^{T}$ and that the column and
row space of the matrix $F(X(t),t)$ are both (respectively) orthogonal to $U$
and $V$, then $\Pi_{X(t)}F(X(t),t)=0,$ which means the solution to the DLRA
description will remain stationary. For example, this will happen for the
numerical discretization when $X(t)$ is a rank 1 even function in both
variables (say $X(t,x_{1},x_{2})=\exp(-x_{1}^{2})\exp(-x_{2}^{2})$) on a
square domain with center $(0,0)$, and $F(X(t),t)=AXB^{T}$, with $A={\sf
diag}(x_{1})$ and $B={\sf diag}(x_{2})$.
The main contribution of this paper is to propose a simple improvement to the
BUG integrator that enhances the robustness with respect to convergence. In
our method, we merge the row and column spaces from the explicit step
truncation method with the BUG spaces in the prediction step to alleviate the
tangent projection error associated with DLRA. As the BUG spaces can be
expensive to compute when implicit schemes are used, we propose an adaptive
acceleration strategy based on the residual of the underlying classic scheme.
The strategy is straightforward, if the residual obtained when only the step
truncation spaces are used, is larger than the truncation threshold then we
add in the BUG spaces. We observe that this strategy is robust in convergence
and can reduce the computational time for moderately stiff problems.
The rest of the paper is organized as follows. Section 2 reviews the
background on low rank time integrators. In Section 3, we describe the
proposed schemes and perform numerical analysis of their properties. Section 4
provides numerical experiments and in Section 5 we draw conclusions.
## 2 Background review
In this section, we will review basic concepts of low rank time stepping
methods for (1). In Section 2.1, we gather the notations used in the paper.
Section 2.2 reviews an explicit scheme truncation method based on the forward
Euler method, while Section 2.3 reviews the rank adaptive BUG scheme.
### 2.1 Notations and preliminaries
In this paper, we use $\|\cdot\|$ to denote the matrix Frobenius norm. The
Frobenius norm is a natural choice for matrix functions as $\|A\|$ coincides
with the $L^{2}$ vector norm of the vectorized matrix ${\sf vec}(A),$ which is
commonly used in the analysis of the standard ODE/PDE solvers. We also use
matrix inner product defined as follows, for $A,B\in\mathbb{R}^{m_{1}\times
m_{2}},$ we define $\langle
A,B\rangle=\sum_{i,j=1}^{m_{1},m_{2}}A_{ij}B_{i,j}.$ Then $\langle
A,A\rangle=\|A\|^{2}.$ The following property,
$\langle A,USV^{T}\rangle=\langle U^{T}AV,S\rangle,$ (2)
holds for any $U\in\mathbb{R}^{m_{1}\times r},S\in\mathbb{R}^{r\times
s},V\in\mathbb{R}^{m_{2}\times s}$. Here we stop to caution the reader that
throughout this paper a matrix $\Sigma,S,$ or $\mathcal{S}$ does not always
represent the diagonal matrix holding the singular values of another matrix.
The numerical solution at $t^{n}$ is denoted by
$\hat{X}^{n}\in\mathbb{R}^{m_{1}\times m_{2}}.$ In particular, this is a rank
$r_{n}$ matrix with singular value decomposition (SVD)
$\hat{X}^{n}=U^{n}\Sigma^{n}(V^{n})^{T},$ where
$U^{n}\in\mathbb{R}^{m_{1}\times r_{n}}$, $V^{n}\in\mathbb{R}^{m_{2}\times
r_{n}}$ have orthogonal columns, and $\Sigma^{n}={\sf
diag}(\sigma_{1},\cdots,\sigma_{r_{n}})$ is a $r_{n}\times r_{n}$ diagonal
matrix with diagonal entries $\sigma_{1}\geq\cdots\geq\sigma_{r_{n}}>0.$ Here,
the rank $r_{n}$ will be chosen adaptively by the numerical scheme so that
$\hat{X}^{n}$ approximate the true solution $X(t^{n})$ with a prescribed
accuracy.
We use $\mathcal{T}_{\epsilon}$ to denote a generic matrix approximation
operator with accuracy $\epsilon$ in the Frobenius norm, i.e.
$\|A-\mathcal{T}_{\epsilon}(A)\|\leq\epsilon.$ A prominent example is the
truncated SVD of a matrix. Namely, given a generic rank $r$ matrix
$A\in\mathbb{R}^{m_{1}\times m_{2}}$ with reduced SVD: $A=U\Sigma V^{T},$ then
$\mathcal{T}^{svd}_{\epsilon}(A)=U[:,1:s]\,{\sf
diag}(\sigma_{1},\cdots,\sigma_{r})\,V[:,1:s]^{T},$ where we have used the
standard MATLAB notation for submatrices and $r$ is chosen to be the smallest
integer so that $(\sum_{j=s+1}^{r}\sigma_{j}^{2})^{1/2}\leq\epsilon.$
In this paper, we will also encounter low rank matrix sum operations. This
frequently appearing operation in low rank methods can be efficiently computed
by Algorithm 1. The inputs to this algorithm are the SVD representations of
the matrices $X_{j}=U_{j}\Sigma_{j}(V_{j})^{T},j=1\ldots m,$ and the output is
the truncated SVD of their sum
$\mathcal{T}^{sum}_{\epsilon}(\sum_{j=1}^{m}X_{j})=\mathcal{U}\mathcal{S}\mathcal{V}^{T}.$
We first write $U=[U_{1},\ldots,U_{m}],$ $\Sigma={\sf
diag}(\Sigma_{1},\ldots,\Sigma_{m}),$ $V=[V_{1},\ldots,V_{m}]$. Then use the
column-pivoted QR decomposition, [6], denoted by $[Q,R,\Pi]={\sf qr}(A)$,
which computes $QR\Pi=A.$ In this factorization the columns of $Q$ are
orthogonal as usual, but the introduction of the permutation matrix $\Pi$,
makes it possible to guarantee that the diagonal entries of the upper
triangular matrix $R$ are strictly decreasing in magnitude. Assume that the
column pivoted QR procedure applied to to $U$ and $V$ yields the
factorizations $Q_{1}R_{1}\Pi_{1}$, and $Q_{2}R_{2}\Pi_{2}$. Then the
truncated matrix sum
$\mathcal{T}^{sum}_{\epsilon}(\sum_{j=1}^{m}X_{j})=Q_{1}\mathcal{U}\mathcal{S}(Q_{2}\mathcal{V})^{T}.$
is obtained by a truncated SVD on the small matrix
$\mathcal{T}^{svd}_{\epsilon}(R_{1}\Pi_{1}\,\Sigma\,\Pi_{2}^{T}R_{2}^{T})=\mathcal{U}\mathcal{S}\mathcal{V}^{T}$.
1
Input : low rank matrices $X_{j},j=1\ldots m$ or their SVD
$U_{j}\Sigma_{j}(V_{j})^{T},j=1\ldots m$
Output : truncated SVD of their sum
$\mathcal{T}^{sum}_{\epsilon}(\sum_{j=1}^{m}X_{j})=\mathcal{U}\mathcal{S}\mathcal{V}^{T}$
Parameter : tolerance $\epsilon$
2 Form $U=[U_{1},\ldots,U_{m}],\,\Sigma={\sf
diag}(\Sigma_{1},\ldots,\Sigma_{m}),\,V=[V_{1},\ldots,V_{m}]$.
3Perform column pivoted QR: $[Q_{1},R_{1},\Pi_{1}]={\sf qr}(U),$
$[Q_{2},R_{2},\Pi_{2}]={\sf qr}(V).$
4Compute the truncated SVD:
$\mathcal{T}^{svd}_{\epsilon}(R_{1}\Pi_{1}\,\Sigma\,\Pi_{2}^{T}R_{2}^{T})=\mathcal{U}\mathcal{S}\mathcal{V}^{T}$.
5Form $\mathcal{U}\leftarrow Q_{1}\mathcal{U},\mathcal{V}\leftarrow
Q_{2}\mathcal{V}$.
Return
$[\mathcal{U},\mathcal{S},\mathcal{V}]=\mathcal{T}^{sum}_{\epsilon}(\sum_{j=1}^{m}X_{j})$
Algorithm 1 Sum of low rank matrices
### 2.2 Explicit step truncation schemes
The primary assumption to guarantee the efficiency of any low rank solver is
that $r_{n}\ll m_{1},r_{n}\ll m_{2},$ i.e. the solution can be well
approximated by a low rank matrix. When such assumptions hold, the main steps
of the numerical algorithm should specify the evolution of
$U^{n},\Sigma^{n},V^{n}$ based on the matrix differential equation (1). For
explicit schemes, a simple approach is to numerically integrate $\hat{X}^{n}$
according to a standard time integrator to $t^{n+1}$ and then perform a
truncated SVD according to the error threshold to obtain $\hat{X}^{n+1}.$ This
is the so-called step truncation method. Methods of this type and their
variations have been discussed in [14, 4]. The main steps of the rank adaptive
forward Euler time scheme are highlighted in Algorithm 2. The first step of
Algorithm 2 is a standard forward Euler step for (1) with the truncated right
hand side in Line 2. This will result in a numerical solution
$\hat{X}^{n+1,pre}$ with higher rank than needed. Then, the second step in
Line 2 is to truncate the solution to a lower rank matrix. In both steps, we
use Algorithm 1 to perform the sum of the low rank matrices involved.
1
Input : numerical solution at $t^{n}:$ rank $r_{n}$ matrix $\hat{X}^{n}$ in
its SVD form $U^{n}\Sigma^{n}(V^{n})^{T}.$
Output : numerical solution at $t^{n+1}:$ rank $r_{n+1}$ matrix
$\hat{X}^{n+1}$ in its SVD form $U^{n+1}\Sigma^{n+1}(V^{n+1})^{T}.$
Parameter : time step $\Delta t$, error tolerance $\epsilon_{1},\epsilon_{2}$
2 (Evolution). $\hat{X}^{n+1,pre}=\hat{X}^{n}+\Delta
t\mathcal{T}^{sum}_{\epsilon_{1}}(F(\hat{X}^{n},t^{n})).$
(Truncation).
$\hat{X}^{n+1}=\mathcal{T}^{sum}_{\epsilon_{2}}(\hat{X}^{n+1,pre}).$
Algorithm 2 Forward Euler scheme $t^{n}\rightarrow t^{n+1}$
The convergence of this algorithm is well understood [21]. The local
truncation error is on the order of $O(\Delta t^{2}+\epsilon_{1}\Delta
t+\epsilon_{2}).$ If one choose $\epsilon_{1}=O(\Delta
t),\epsilon_{2}=O(\Delta t^{2}),$ we will obtain a first order accurate
solution. As for computational cost, we can see that the procedure involves QR
factorization of tall matrices and SVD for a small matrix. If we assume
$r_{n}=O(r),$ $s$ is small (i.e. $s=O(1)$) and $G$ has low rank (i.e. ${\sf
rank}(G)=O(r)$), the computational complexity is on the order of
$O(m_{1}r^{2}+m_{2}r^{2}+r^{3}).$ Algorithm 2 can be readily extended to
higher order by embedding in Runge-Kutta, multistep methods, and was also
considered with tangent projection in the projected Runge-Kutta schemes [14,
4, 8].
### 2.3 Dynamic low rank approximation
The DLRA modifies the equation, and solves
$\frac{d}{dt}X(t)=\Pi_{X(t)}F(X(t),t),$ (3)
where $\Pi_{X(t)}$ is the orthogonal projection onto the tangent space
$T_{X(t)}\mathcal{M}_{r}$ of the the rank $r$ matrix manifold
$\mathcal{M}_{r}=\\{X\in\mathbb{R}^{m_{1}\times m_{2}},{\sf rank}(X)=r\\}$ at
$X(t).$ The DLRA is particularly suited for a fixed rank calculation, if the
computation is constrained on $\mathcal{M}_{r}.$ It has found success in many
applications, including extensions to various tensor formats [18, 17] and
particularly in quantum mechanics [20].
We reminder the reader that if the SVD of $X\in\mathcal{M}_{r}$ is given by
$U\Sigma V^{T},$ then the tangent space of $\mathcal{M}_{r}$ at $X$ is given
by
$T_{X}\mathcal{M}_{r}=\left\\{[U\quad
U_{\perp}]\begin{bmatrix}\mathbb{R}^{r\times
r}&\mathbb{R}^{r\times(m_{2}-r)}\\\ \mathbb{R}^{(m_{1}-r)\times
r}&0^{(m_{1}-r)\times(m_{2}-r)}\end{bmatrix}[V\quad V_{\perp}]^{T}\right\\},$
where $U_{\perp},V_{\perp}$ are orthogonal complements of $U,V$ in
$\mathbb{R}^{m_{1}},\mathbb{R}^{m_{2}},$ respectively. As we can see later in
the PDE examples, typically if there are cross terms like $F(X)=AXB$ (e.g.
cross derivatives or variable coefficient problem with operators acting on
both left and right side of the matrix), then in general $AXB\notin
T_{X}\mathcal{M}_{r}.$ In this case, the _modeling error_ [14] (i.e. the error
from the tangent projection, which is $\|(I-\Pi_{X(t)})F(X,t)\|$) will be
evident in the numerical approximation, see for example error estimates in [3,
2].
Below we will review the rank adaptive BUG integrator in [2] for the
discretization of (3). First, two subproblems that come from the projector
splitting of DLRA [16] are solved in the $K-$ and $L-$step. Then, the
solutions to the two subproblems are used to update the column and row spaces
of the matrix; and a Galerkin evolution step is performed in the resulting
space, searching for the optimal solution with the Galerkin condition. The
idea of finding the appropriate subspace and computing the solution in that
space is similar to a widely used technique in numerical linear algebra called
projection methods [22].
For completeness, we describe the rank adaptive BUG integrator in Algorithm 3.
The original algorithm is written in a time continuous format, but here we
present it using an implicit Euler discretization of the $K-,L-,S-$steps to
facilitate the discussion in the remaining part of the paper.
Input : numerical solution at $t^{n}:$ rank $r_{n}$ matrix $\hat{X}^{n}$ in
its SVD form $U^{n}\Sigma^{n}(V^{n})^{T}.$
Output : numerical solution at $t^{n+1}:$ rank $r_{n+1}$ matrix
$\hat{X}^{n+1}$ in its SVD form $U^{n+1}\Sigma^{n+1}(V^{n+1})^{T}.$
Parameter : time step $\Delta t$, error tolerance $\epsilon_{2}$
1 (Prediction). $K-$step and $L-$step integrating from $t^{n}$ to $t^{n+1}$.
2Solve $K^{n+1}-K^{n}=\Delta tF(K^{n+1}(V^{n})^{T},t)V^{n},\quad
K^{n}=U^{n}\Sigma^{n},$ to obtain $K^{n+1},$ and $[\tilde{U},\sim,\sim]={\sf
qr}([U^{n},K^{n+1}]).$
3Solve $L^{n+1}-L^{n}=\Delta tF(U^{n}(L^{n+1})^{T},t)^{T}U^{n},\quad
L^{n}=V^{n}\Sigma^{n},$ to obtain $L^{n+1},$ and $[\tilde{V},\sim,\sim]={\sf
qr}([V^{n},L^{n+1}]).$
4(Galerkin Evolution). $S-$step: solve for $S^{n+1}$ from
$S^{n+1}-S^{n}=\Delta
t\tilde{U}^{T}F(\tilde{U}S^{n+1}\tilde{V}^{T},t)\tilde{V},\quad
S^{n}=\tilde{U}^{T}U^{n}\Sigma^{n}(V^{n})^{T}\tilde{V},$ to obtain $S^{n+1}.$
5(Truncation).
$\hat{X}^{n+1}=\tilde{U}\mathcal{T}_{\epsilon_{2}}^{svd}(S^{n+1})\tilde{V}^{T}=U^{n+1}\Sigma^{n+1}(V^{n+1})^{T}.$
Algorithm 3 Rank adaptive BUG integrator using implicit Euler [2].
## 3 Numerical method
We now present the proposed numerical methods, the Merge method and its
adapted version. We first describe the algorithms, and then discuss the
properties and the rationale behind the design of the schemes.
### 3.1 The Merge method
In the Merge method, we merge the column and row spaces from the explicit step
truncation method and the spaces from the $K-$ and $L-$ steps in the BUG
method. As mentioned above the method consists of three stages.
Prediction: Our first ingredient in the method is to predict the column and
row space using the explicit Euler scheme in combination with the BUG
prediction spaces from Algorithm 3. Precisely, the predicted column and row
spaces are defined as the column and row spaces of the collection of matrices
$[\hat{X}^{n},\,\mathcal{T}_{\epsilon_{1}}(F(\hat{X}^{n},t^{n})),\,K^{n+1}(L^{n+1})^{T}].$
Here the truncation level $\epsilon_{1}$ can be set to zero (no truncation
used) or chosen as $\epsilon_{1}=C_{1}\Delta t$. Both choices gives first
order accurate solutions but depending on the form of (1) the added cost from
the $\epsilon_{1}=C_{1}\Delta t$ truncation may be offset by the reduction in
the dimension of the predicted row and column spaces. To complete the next
step we first orthogonalize the predicted column and row spaces. That is, we
use column pivoted QR to find orthogonal matrices
$\tilde{U}\in\mathbb{R}^{m_{1}\times
s_{1}},\tilde{V}\in\mathbb{R}^{m_{2}\times s_{2}}$ spanning these spaces. Here
$s_{1},s_{2}$ is bounded from above by $2r_{n}(s+1)+r_{G},$ where $r_{G}$ is
the rank of the source $G(t^{n}).$
Galerkin evolution: The Galerkin evolution step can be understood as to find
an approximation $\hat{X}^{n+1,pre}\in W_{\tilde{U},\tilde{V}},$ such that
$\langle\hat{X}^{n+1,pre},A\rangle=\langle\hat{X}^{n}+\Delta
tF(\hat{X}^{n+1,pre},t^{n+1}),A\rangle,\qquad\forall A\in
W_{\tilde{U},\tilde{V}},$ (4)
where $W_{\tilde{U},\tilde{V}}=\\{A\in\mathbb{R}^{m_{1}\times
m_{2}}:A=\tilde{U}\Sigma\tilde{V}^{T},\textrm{with
}\Sigma\in\mathbb{R}^{s_{1}\times s_{2}}\\}$ denote all size $m_{1}\times
m_{2}$ spaces with column and row spaces as $\tilde{U}$ and $\tilde{V}.$
To solve this problem we let
$\hat{X}^{n+1,pre}=\tilde{U}\tilde{\Sigma}\tilde{V}^{T},A=\tilde{U}\Sigma^{*}\tilde{V}^{T}.$
Then by (2), we get
$\langle\tilde{\Sigma},\Sigma^{*}\rangle=\langle\tilde{U}^{T}(\hat{X}^{n}+\Delta
tF(\tilde{U}\tilde{\Sigma}\tilde{V}^{T},t^{n+1})))\tilde{V},\Sigma^{*}\rangle,\qquad\forall\Sigma^{*}\in\mathbb{R}^{s_{1}\times
s_{2}}.$
This means
$\displaystyle\tilde{\Sigma}$ $\displaystyle=$
$\displaystyle\tilde{U}^{T}(\hat{X}^{n}+\Delta
tF(\tilde{U}\tilde{\Sigma}\tilde{V}^{T},t^{n+1})))\tilde{V}$ (5)
$\displaystyle=$
$\displaystyle\tilde{U}^{T}U^{n}\Sigma^{n}(V^{n})^{T}\tilde{V}^{T}+\Delta
t\sum_{j=1}^{s}\tilde{U}^{T}A_{j}\tilde{U}\tilde{\Sigma}\tilde{V}^{T}B_{j}^{T}\tilde{V}+\Delta
t\tilde{U}^{T}U^{n}_{G}\Sigma^{n}_{G}(V^{n}_{G})^{T}\tilde{V}.$
The equation (5) is a linear matrix equation (generalized Sylvester equation)
for the unknown $\tilde{\Sigma}.$ Since the problems we are considering have
low rank solutions, we expect the dimensions of the matrix $\tilde{\Sigma}$ to
be small. To solve this equation one can either use a direct solves or an
iterative solver (e.g. GMRES) [23].
Truncation: The truncation step uses the truncated SVD of
$\hat{X}^{n+1}=\mathcal{T}_{\epsilon_{2}}^{svd}(\tilde{U}\tilde{\Sigma}\tilde{V}^{T})=\tilde{U}\mathcal{T}_{\epsilon_{2}}^{svd}(\tilde{\Sigma})\tilde{V}^{T}.$
Here the truncation level is chosen as $\epsilon_{2}=C_{2}\Delta t^{2}$ to
ensure accuracy.
Algorithm 4 summarizes the Merge method.
1
Input : numerical solution at $t^{n}:$ rank $r_{n}$ matrix $\hat{X}^{n}$ in
its SVD form $\hat{U}^{n}\hat{\Sigma}^{n}(\hat{V}^{n})^{T}$.
Output : numerical solution at $t^{n+1}:$ rank $r_{n+1}$ matrix
$\hat{X}^{n+1}$ in its SVD form $U^{n+1}\Sigma^{n+1}(V^{n+1})^{T}.$
Parameters : truncation tolerances $\epsilon_{1},\epsilon_{2}$
2 (Merge Prediction). Compute the truncated SVD of the right hand side:
$[\mathcal{U},\mathcal{S},\mathcal{V}]=\mathcal{T}^{sum}_{\epsilon_{1}}(F(\hat{X}^{n},t^{n})),$.
3Compute the BUG prediction spaces $K^{n+1},L^{n+1}$ according to Algorithm 3.
4Orthogonalize the merged spaces by column pivoted QR to get the prediction
spaces $[\tilde{U},\sim,\sim]={\sf qr}([[\hat{U}^{n},\mathcal{U},K^{n+1}]),$
$[\tilde{V},\sim,\sim]={\sf qr}([\hat{V}^{n},\mathcal{V}^{n},L^{n+1}]).$
5(Galerkin Evolution). Find $\tilde{\Sigma}$ by solving
$\tilde{\Sigma}=\tilde{U}^{T}(\hat{X}^{n}+\Delta
tF(\tilde{U}\tilde{\Sigma}\tilde{V}^{T},t^{n+1}))\tilde{V}.$
(Truncation).
$\hat{X}^{n+1}=\hat{U}^{n+1}\hat{\Sigma}^{n+1}(\hat{V}^{n+1})^{T}=\tilde{U}\mathcal{T}_{\epsilon_{2}}^{svd}(\tilde{\Sigma})\tilde{V}^{T}.$
Algorithm 4 Merge method $t^{n}\rightarrow t^{n+1}$
We now provide estimates of the computational cost of the different parts of
the computation. Here we assume $m_{1}=m_{2}=m$, $s$ terms in differential
equation and rank $r$.
* •
An evaluation of $AU,(BV)^{T}$ costs $\mathcal{O}(mr)$ for each of $AU$, $BV$,
assuming $A$ and $B$ are sparse.
* •
The truncation of the right hand side $\mathcal{T}^{sum}_{\epsilon_{1}}(F)$
has two components. Two QR orthogonalizations, each costs
$\mathcal{O}(2(m(rs)^{2}-\frac{(rs)^{2}}{3})$ and one SVD of the core dense
matrix, this costs $\mathcal{O}((rs)^{3})$.
* •
The $K$ and $L$ solves used to compute the BUG spaces also require the
solution of matrix equations but the dimension of these matrix solves are for
a $m\times r$ matrix rather than a $sr\times sr$ matrix. For these equations
we have found that applying GMRES in vectorized form works well. It should be
noted that this is typically the most expensive part of the solvers.
* •
The cost of solving for the Galerkin core matrix $\tilde{\Sigma}$ in
$\tilde{\Sigma}-\Delta
t\tilde{U}^{T}F(\tilde{U}\tilde{\Sigma}\tilde{V}^{T},t^{n+1})\tilde{V}=\tilde{U}^{T}\hat{X}^{n}\tilde{V},$
depends on the algorithm used. For example, if GMRES is applied on the
vectorized version of this equation each evaluation of the left hand side
costs $\mathcal{O}(m(rs)^{2})$ if $F$ is a general function. If $F$ is in the
form described in equation (1) it is possible to pre-compute dense $rs\times
rs$ matrices (at a cost of $\mathcal{O}(ms(rs)^{2})$) and bring down the cost
of the evaluation of the left hand side to $\mathcal{O}(s(rs)^{3})$ per
evaluation.
Neglecting the cost associated with the growth of the GMRES Krylov subspace
this cost would then be multiplied with the number of iterations needed to
converge (which is difficult to estimate). We have found that when the
governing equation has elliptic terms it also works well to reformulate the
Galerkin equation into a fixed point iteration
$A_{2}^{-1}(I-\tilde{U}^{T}\Delta
tA_{1}\tilde{U})\tilde{\Sigma}^{k}-\tilde{\Sigma}^{k}(\Delta
tB_{2}\tilde{V})^{T}\tilde{V}B^{-1}_{1}=P(\tilde{\Sigma}^{k-1},\hat{X}^{n},\tilde{U},\tilde{V}).$
In the iteration we use a dense Sylvester solve with cost
$\mathcal{O}((rs)^{3})$ for each iteration. When the matrices $A_{1}$ and
$B_{2}$ correspond to approximations to second derivatives in the 1 and 2
direction and $A_{2}$ and $B_{1}$ are identity or diagonal positive definite
matrices we find that this iteration converges to machine precision in a
handful of iterations.
Finally, we would like to comment on the rank evolution of the numerical
solution. We find in our numerical experiments that the low rank method
generally track the rank growth of the implicit Euler scheme well, i.e. the
rank of $\hat{X}^{n}$ is on par with the rank of the implicit Euler solution
at $t^{n}.$ However, due to the merge in the prediction step, the predicted
rank is larger than the rank of the BUG space (which is equal to $2r$). This
do translate to larger computational cost in the Galerkin step compared to the
BUG solver. However we note that the upper bound in the BUG space $2r$ can be
insufficient to capture rapidly changing initial layers in the PDEs as pointed
out in [19] (where it was observed a larger than actual rank needs to be
imposed for the numerical initial solution). For the numerical experiments we
performed in this paper, we find that by merging with the spaces generated by
the explicit schemes, this issue seems to be addressed and we do not need to
impose large artificial rank for the numerical initial condition.
### 3.2 The Merge-adapt method
As will be shown in Lemma 3.3, the column and row spaces generated from the
current solution $\hat{U}^{n},\hat{V}^{n}$ combined with $\mathcal{U}$ and
$\mathcal{V}$ in
$[\mathcal{U},\mathcal{S},\mathcal{V}]=\mathcal{T}^{sum}_{\epsilon_{1}}(F(\hat{X}^{n},t^{n})),$
will be sufficiently accurate to yield a first order accurate solution as long
as the time step is sufficiently small compared to the Lipschitz constant $L$
of $\|F\|$. This time step restriction can be overcome by merging with the BUG
spaces $K^{n+1},L^{n+1}$, which are computed useing an implicit solver and can
handle the stiffness as shown in the Merge method in Algorithm 4. However,
since the computation of $K^{n+1}$ and $L^{n+1}$ requires a linear solve, it
is preferable to only add those when necessary. This motivates the design of
the Merge-adapt method, which is described in Algorithm 5. In Steps 1-4 in
Algorithm 5, we perform a calculation based purely on column and row spaces
generated by the explicit scheme for the term
$\mathcal{T}^{sum}_{\epsilon_{1}}(F(\hat{X}^{n},t^{n})).$ In Step 5, we do a
residual check, and if this fails, the method will fall back to the Merge
method as shown in Steps 6-8. Otherwise, we proceed to the next time step.
We, heuristically, argue that for this residual check can be used to robustly
maintain first order accuracy. For simplicity, assume that the differential
equation we would like to solve is
$\frac{d}{dt}X(t)=AX(t)B^{T}+G(t).$
Let $X^{n+1}$ be the solution obtained by the classic implicit Euler method,
i.e. it solves
${\sf vec}(X^{n+1})=\underbrace{(I-\Delta t(B\otimes A))^{-1}}_{C}{\sf
vec}(X^{n})+\Delta t\,{\sf vec}(G(t_{n+1})).$
The low rank solution is not, in general, expected to solve this equation but
we may introduce the residual $\hat{R}^{n+1}$ so that
${\sf vec}(\hat{X}^{n+1})=\underbrace{(I-\Delta t(B\otimes A))^{-1}}_{C}{\sf
vec}(\hat{X}^{n})+\Delta t\,{\sf vec}(G(t_{n+1}))+{\sf vec}(\hat{R}^{n+1}).$
Combining the equations we find
${\sf vec}(\hat{X}^{n+1})-{\sf vec}(X^{n+1})=C[{\sf vec}(\hat{X}^{n})-{\sf
vec}(X^{n})]+{\sf vec}(\hat{R}^{n+1}).$
Now if we assume $\|C\|\leq 1$ and the residual check pass, then the error
between the implicit Euler solution and the implicit adaptive low rank
solution $e^{n}={\sf vec}(\hat{X}^{n})-{\sf vec}(X^{n})$ satisfies
$\|e^{n+1}\|\leq\|e^{n}\|+C_{2}\Delta t^{2}.$ This is enough to guarantee that
the low rank solution is convergent because the implicit Euler solution is
first order in time. It is reasonable to assume that the condition $\|C\|\leq
1$ is satisfied for most discretizations of diffusion or advection-diffusion
equations [9].
Input : numerical solution at $t^{n}:$ rank $r_{n}$ matrix $\hat{X}^{n}$ in
its SVD form $\hat{U}^{n}\hat{\Sigma}^{n}(\hat{V}^{n})^{T}$.
Output : numerical solution at $t^{n+1}:$ rank $r_{n+1}$ matrix
$\hat{X}^{n+1}$ in its SVD form $U^{n+1}\Sigma^{n+1}(V^{n+1})^{T}.$
Parameters : truncation tolerances $\epsilon_{1},\epsilon_{2}$
1 (Cheap Prediction). Compute a first order prediction of the column and row
spaces of $\hat{X}^{n+1}$. Compute the truncated SVD of the right hand side:
$[\mathcal{U},\mathcal{S},\mathcal{V}]=\mathcal{T}^{sum}_{\epsilon_{1}}(F(\hat{X}^{n},t^{n}))$.
2Orthogonalize by column pivoted QR to get the prediction spaces
$[\tilde{U},\sim,\sim]={\sf qr}([[\hat{U}^{n},\mathcal{U}]),$
$[\tilde{V},\sim,\sim]={\sf qr}([\hat{V}^{n},\mathcal{V}^{n}]).$
3(Galerkin Evolution). Find $\tilde{\Sigma}$ by solving
$\tilde{\Sigma}=\tilde{U}^{T}(\hat{X}^{n}+\Delta
tF(\tilde{U}\tilde{\Sigma}\tilde{V}^{T},t^{n+1}))\tilde{V}.$
4(Truncation).
$\hat{X}^{n+1}=\hat{U}^{n+1}\hat{\Sigma}^{n+1}(\hat{V}^{n+1})^{T}=\tilde{U}\mathcal{T}_{\epsilon_{2}}^{svd}(\tilde{\Sigma})\tilde{V}^{T}.$
5(Residual Check). Compute $\hat{R}^{n+1}=\hat{X}^{n+1}-\hat{X}^{n}-\Delta
tF(\hat{X}^{n+1},t^{n+1})$. If $\|\hat{R}^{n+1}\|<\epsilon_{2}$ return the
solution $\hat{X}^{n+1}$. If not
6(Merge Predicion). Compute the BUG prediction spaces $K^{n+1},L^{n+1}$
according to Algorithm 3. Orthogonalize the merged spaces by column pivoted
QR: $[\tilde{U},\sim,\sim]={\sf qr}([[\hat{U}^{n},\mathcal{U},K^{n+1}]),$
$[\tilde{V},\sim,\sim]={\sf qr}([\hat{V}^{n},\mathcal{V}^{n},L^{n+1}]).$
7(Galerkin Evolution). Find $\tilde{\Sigma}$ by solving
$\tilde{\Sigma}=\tilde{U}^{T}(\hat{X}^{n}+\Delta
tF(\tilde{U}\tilde{\Sigma}\tilde{V}^{T},t^{n+1}))\tilde{V}.$
8(Truncation).
$\hat{X}^{n+1}=\hat{U}^{n+1}\hat{\Sigma}^{n+1}(\hat{V}^{n+1})^{T}=\tilde{U}\mathcal{T}_{\epsilon_{2}}^{svd}(\tilde{\Sigma})\tilde{V}^{T}.$
Algorithm 5 Merge-adapt method $t^{n}\rightarrow t^{n+1}$
We note that the acceleration of Algorithm 5 over Algorithm 4 is based on the
assumption that residual check passes in Step 5 and the BUG spaces are not
computed. If the residual check fails, extra computation is performed in Steps
3-5 which may incur more computational cost. We observe that, for moderately
stiff problems with small $s$, the Merge-adapt method has computational
advantages. More discussion is provided in the numerical experiment section.
### 3.3 Analysis
In this section, we analyze properties of the proposed methods.
#### 3.3.1 Stability
First, we consider conservative or dissipative systems for which $\langle
F(X,t),X\rangle\leq 0,\forall t,X.$ Then it follows that
$\frac{d}{dt}\|X\|^{2}=\langle F(X,t),X\rangle\leq 0,$
that is, the energy ($L^{2}$ norm) is monotonically decreasing.
###### Theorem 3.1.
If we have $\langle F(X,t),X\rangle\leq 0,\forall t,X,$ then the numerical
solutions from Algorithm 4 or 5 satisfy
$\|\hat{X}^{n+1}\|\leq\|\hat{X}^{n}\|.$
###### Proof.
For Algorithm 4, first note that
$\|X^{n+1,pre}\|=\|\tilde{U}\tilde{\Sigma}\tilde{V}^{T}\|=\|\tilde{\Sigma}\|.$
By (5) and (2),
$\displaystyle\|\tilde{\Sigma}\|^{2}=\langle\tilde{\Sigma},\tilde{\Sigma}\rangle=\langle\tilde{U}^{T}(\hat{X}^{n}+\Delta
tF(\tilde{U}\tilde{\Sigma}\tilde{V}^{T},t^{n+1}))\tilde{V},\tilde{\Sigma}\rangle$
$\displaystyle=\langle\hat{X}^{n}+\Delta
tF(\tilde{U}\tilde{\Sigma}\tilde{V}^{T},t^{n+1}),\tilde{U}\tilde{\Sigma}\tilde{V}^{T}\rangle$
$\displaystyle\leq\langle\hat{X}^{n},\tilde{U}\tilde{\Sigma}\tilde{V}^{T}\rangle=\langle\hat{X}^{n},X^{n+1,pre}\rangle.$
Therefore, by Cauchy-Schwarz inequality,
$\|X^{n+1,pre}\|^{2}\leq\|\hat{X}^{n}\|\|X^{n+1,pre}\|,$ which gives
$\|X^{n+1,pre}\|\leq\|\hat{X}^{n}\|.$ Finally, because of the property of the
truncated SVD,
$\|\hat{X}^{n+1}\|=\|\mathcal{T}_{\epsilon_{2}}^{svd}(X^{n+1,pre})\|\leq\|X^{n+1,pre}\|\leq\|\hat{X}^{n}\|.$
The proof for Algorithm 5 is the same and is omitted. ∎
We can easily generalize this result to semi-bounded operator [9].
Specifically, if $\langle F(X,t),X\rangle\leq\alpha\|X\|^{2},\forall\,t,X,$
then
$\frac{d}{dt}\|X\|^{2}=\langle F(X,t),X\rangle\leq\alpha\|X\|^{2},$
which implies $\|X(t)\|\leq e^{\alpha t}\|X(0)\|.$
###### Theorem 3.2 (Stability for semi-bounded operator).
If we have $\langle F(X,t),X\rangle\leq\alpha\|X\|^{2},\forall\,t,X$ then the
numerical solutions from Algorithm 4 or 5 satisfy
$\|\hat{X}^{n+1}\|\leq e^{\alpha\Delta
t}\|\hat{X}^{n}\|,\quad\|\hat{X}^{n}\|\leq
e^{\alpha\,t^{n}}\|\hat{X}^{0}\|,\quad\textrm{if}\,\,\alpha\Delta t\leq 1.$
###### Proof.
The proof is similar to the proof of Theorem 3.1, so we only highlight the
difference. We have
$\langle\hat{X}^{n+1},\hat{X}^{n+1}\rangle\leq\langle\hat{X}^{n},\hat{X}^{n+1}\rangle+\Delta
t\alpha\|\hat{X}^{n+1}\|^{2},$
which implies
$\|\hat{X}^{n+1}\|\leq\frac{1}{1-\alpha\Delta t}\|\hat{X}^{n}\|,$
if $\alpha\Delta t\leq 1.$ The theorem follows using a similar arguments as in
Theorem 3.1. ∎
#### 3.3.2 Convergence
A convergence estimate has been shown in Theorem 2 of [2] for the rank
adaptive BUG scheme. Assuming the Lipschitz continuity and boundedness of the
operator $F,$ the authors show that the BUG scheme has numerical error bounded
by the sum of the initial numerical error, the tangent projection error,
$\epsilon_{2}$ and first order in time error. The proof is based on the time
continuous version of the BUG schemes. We want to point out that for PDE
applications, in general the (differential) operators are not Lipschitz
bounded. Nevertheless, it is still of theoretical interests to investigate the
convergence properties of the schemes under such assumptions.
For simplicity, below we will estimate the local truncation error of the first
order implicit schemes with column and row space spanned by the cheap
prediction space, i.e. we are investigating the Steps 1-4 in Algorithm 5.
Since the Merge method uses a space that is union of the cheap prediction (as
shown below) and the BUG space, the local truncation error of Algorithm 4 will
be also be upper bounded by the estimate below.
###### Lemma 3.3 (Local truncation error with cheap prediction space).
Suppose $F$ is Lipschitz-continuous in both variables and bounded, i.e. there
exists a constant $L$ such that $\|F(Y,s)-F(Z,t)\|\leq L\|Y-Z\|+L|s-t|$ and
$\|F(X,t)\|\leq B$. If we denote $\mathcal{X}$ as the exact solution to (1)
with initial condition $\hat{X}^{n}=U^{n}\Sigma^{n}(V^{n})^{T}$ from $t^{n}$
to $t^{n+1}$, $\hat{X}^{n+1}$ the numerical solution obtained from from
Algorithm 5, Steps 1-4 with the same initial condition $\hat{X}^{n},$ then for
sufficiently small $\Delta t,$ we have the following error bound
$\|\mathcal{X}-\hat{X}^{n+1}\|\leq C\Delta t^{2}+2\Delta
t\epsilon_{1}+\epsilon_{2}.$ (6)
In (6), $C$ is a constant that only depends $L$ and $B$.
###### Proof.
First we prove (6) for $\epsilon_{1}=0$. Then
$\mathcal{T}^{sum}_{\epsilon_{1}=0}(F(\hat{X}^{n},t^{n}))=F(\hat{X}^{n},t^{n})$.
From (5), and left multiplying by $\tilde{U}$ and right multiplying by
$\tilde{V}^{T},$ we obtain
$\hat{X}^{n+1,pre}=\tilde{U}\tilde{\Sigma}\tilde{V}^{T}=\tilde{U}\tilde{U}^{T}(\hat{X}^{n}+\Delta
tF(\hat{X}^{n+1,pre},t^{n+1})))\tilde{V}\tilde{V}^{T}.$
Since the column space of $\hat{X}^{n}$ is a subset of the column space of
$\tilde{U},$ i.e. $R(\hat{X}^{n})\subseteq R(\tilde{U}),$ we have
$\tilde{U}\tilde{U}^{T}\hat{X}^{n}=\hat{X}^{n}.$ Similarly, because
$R((\hat{X}^{n})^{T})\subseteq R(\tilde{V}),$
$\hat{X}^{n}\tilde{V}\tilde{V^{T}}=\hat{X}^{n}.$ Therefore,
$\tilde{U}\tilde{U}^{T}\hat{X}^{n}\tilde{V}\tilde{V^{T}}=\hat{X}^{n}.$ By a
similar argument, $R(F(\hat{X}^{n},t^{n}))\subseteq R(\tilde{U})$ and
$R(F(\hat{X}^{n},t^{n})^{T})\subseteq R(\tilde{V})$ implies
$\tilde{U}\tilde{U}^{T}F(\hat{X}^{n},t^{n})\tilde{V}\tilde{V}^{T}=F(\hat{X}^{n},t^{n}).$
This gives
$\displaystyle\hat{X}^{n+1,pre}=\hat{X}^{n}+\Delta
t\tilde{U}\tilde{U}^{T}F(\hat{X}^{n+1,pre},t^{n+1})\tilde{V}\tilde{V}^{T}$
$\displaystyle=\hat{X}^{n}+\Delta
t\tilde{U}\tilde{U}^{T}F(\hat{X}^{n},t^{n})\tilde{V}\tilde{V}^{T}+\Delta
t\tilde{U}\tilde{U}^{T}(F(\hat{X}^{n+1,pre},t^{n+1})-F(\hat{X}^{n},t^{n}))\tilde{V}\tilde{V}^{T}$
$\displaystyle=\hat{X}^{n}+\Delta tF(\hat{X}^{n},t^{n})+\Delta
t\tilde{U}\tilde{U}^{T}(F(\hat{X}^{n+1,pre},t^{n+1})-F(\hat{X}^{n},t^{n}))\tilde{V}\tilde{V}^{T}.$
(7)
Therefore,
$\|\hat{X}^{n+1,pre}-\mathcal{X}\|\\\ =\|\hat{X}^{n}-\mathcal{X}+\Delta
tF(\hat{X}^{n},t^{n})+\Delta
t\tilde{U}\tilde{U}^{T}(F(\hat{X}^{n+1,pre},t^{n+1})-F(\hat{X}^{n},t^{n}))\tilde{V}\tilde{V}^{T}\|.$
(8)
We have $\hat{X}^{n}-\mathcal{X}=-\Delta tF(X(t^{*}),t^{*}),$ where $t^{*}$ is
a point on the interval $[t^{n},t^{n+1}].$ Therefore,
$\displaystyle\|\hat{X}^{n}-\mathcal{X}+\Delta tF(\hat{X}^{n},t^{n})\|=\Delta
t\|-F(X(t^{*}),t^{*})+F(\hat{X}^{n},t^{n})\|$ $\displaystyle\leq\Delta
tL(\Delta t+\|X(t^{*})-\hat{X}^{n}\|)\leq\Delta t^{2}L(B+1).$ (9)
On the other hand,
$\displaystyle\|\tilde{U}\tilde{U}^{T}(F(\hat{X}^{n+1,pre},t^{n+1})-F(\hat{X}^{n},t^{n}))\tilde{V}\tilde{V}^{T}\|$
$\displaystyle\leq\|\tilde{U}\tilde{U}^{T}\|_{2}\|F(\hat{X}^{n+1,pre},t^{n+1})-F(\hat{X}^{n},t^{n})\|\|\tilde{V}\tilde{V}^{T}\|_{2},$
where $\|\cdot\|_{2}$ is the matrix 2-norm. Since
$\|\tilde{U}\tilde{U}^{T}\|_{2}=\|\tilde{V}\tilde{V}^{T}\|_{2}=1$ and
$\|F(\hat{X}^{n+1,pre},t^{n+1})-F(\hat{X}^{n},t^{n})\|\leq L(\Delta
t+\|\hat{X}^{n}-\mathcal{X}\|+\|\hat{X}^{n+1,pre}-\mathcal{X}\|)\leq L(\Delta
t+B\Delta t+\|\hat{X}^{n+1,pre}-\mathcal{X}\|),$ we get
$\displaystyle\|\tilde{U}\tilde{U}^{T}(F(\hat{X}^{n+1,pre},t^{n+1})-F(\hat{X}^{n},t^{n}))\tilde{V}\tilde{V}^{T}\|\leq
L(\Delta t+B\Delta t+\|\hat{X}^{n+1,pre}-\mathcal{X}\|).$ (10)
Combining (8),(3.3.2),(10), we get
$\|\hat{X}^{n+1,pre}-\mathcal{X}\|\leq\Delta t^{2}L(2B+2)+L\Delta
t\|\hat{X}^{n+1,pre}-\mathcal{X}\|,$
Choosing $\Delta t$ small enough (i.e. $\Delta t\leq\frac{1}{2L}$), we proved
$\displaystyle\|\hat{X}^{n+1,pre}-\mathcal{X}\|\leq 4\Delta t^{2}L(B+1).$ (11)
Therefore, (6) follows by using
$\|\hat{X}^{n+1,pre}-\hat{X}^{n+1}\|=\|\hat{X}^{n+1,pre}-\mathcal{T}_{\epsilon_{2}}^{svd}(\hat{X}^{n+1,pre})\|\leq\epsilon_{2}.$
The proof for (6) with $\epsilon_{1}\neq 0$ largely follows the same process.
The only difference is that we no longer have
$\tilde{U}\tilde{U}^{T}F(\hat{X}^{n},t^{n})\tilde{V}\tilde{V}^{T}=F(\hat{X}^{n},t^{n})$
due to the truncation in the prediction step. Instead, we have
$\tilde{U}\tilde{U}^{T}\mathcal{T}_{\epsilon_{1}}(F(\hat{X}^{n},t^{n}))\tilde{V}\tilde{V}^{T}=\mathcal{T}_{\epsilon_{1}}(F(\hat{X}^{n},t^{n})).$
In this case,
$\tilde{U}\tilde{U}^{T}F(\hat{X}^{n},t^{n})\tilde{V}\tilde{V}^{T}=\mathcal{T}_{\epsilon_{1}}(F(\hat{X}^{n},t^{n}))+\tilde{U}\tilde{U}^{T}(F(\hat{X}^{n},t^{n})-\mathcal{T}_{\epsilon_{1}}(F(\hat{X}^{n},t^{n})))\tilde{V}\tilde{V}^{T},$
and because $\|\tilde{U}\tilde{U}^{T}\|_{2}=\|\tilde{V}\tilde{V}^{T}\|_{2}=1,$
this leads to
$\|\tilde{U}\tilde{U}^{T}F(\hat{X}^{n},t^{n})\tilde{V}\tilde{V}^{T}-F(\hat{X}^{n},t^{n})\|\leq
2\|F(\hat{X}^{n},t^{n})-\mathcal{T}_{\epsilon_{1}}(F(\hat{X}^{n},t^{n}))\|=2\epsilon_{1}.$
By a similar argument, (6) follows. The details are omitted for brevity. ∎
The numerical convergence follows from Lemma 3.3 by a standard result for the
convergence of one-step method for ODE and is skipped for brevity. Now we
would like to comment on the results in this Lemma to understand property of
the scheme. As we can see from the results in Section 3.3.1, the schemes are
unconditionally stable due to the Galerkin step. Also, in Lemma 3.3, the
tangent projection error is absent because the DLRA approximation is not used.
However, the time step is limited by the Lipschitz constant and also the
constant in the estimate $C$ depends on the Lipschitz constant. For standard
error estimates for implicit schemes for stiff problems, the concept of
B-convergence [5] can remove such restrictions with the help of one-sided
Lipschitz continuity. Unfortunately, such results are not available in our
case.
## 4 Numerical results
We now present numerical results that illustrate the features of the methods
described above. In this section we will refer to the different methods in
figures and tables according to the following naming convention. We will
denote the Merge method by M, and the Merge-adapt method by MA. For many
problems we will also compare to the classic implicit Euler discretization as
IE or Implicit Euler. When computing errors we will compare to a solution
computed using Matlab’s built in ODE15s or ODE45 solvers with relative and
absolute tolerance set to $10^{-12}$. We always report relative errors in the
Frobenius norm. We note that in all of the examples below, the BUG method does
not converge since all or part of the right hand side of (1) is outside the
tangent projection space. We also note that in all the examples below we use
$\epsilon_{1}=0$ but that the results are very similar for
$\epsilon_{1}=\Delta t$.
The main PDE examples used in this section are simulations of solutions to the
equation
$\frac{\partial\rho}{\partial t}+r_{1}(x_{1})\frac{\partial\rho}{\partial
x_{2}}+r_{2}(x_{2})\frac{\partial\rho}{\partial x_{1}}=\\\
b_{1}(x_{2})\frac{\partial}{\partial
x_{1}}\left[a_{1}(x_{1})\frac{\partial\rho}{\partial
x_{1}}\right]+b_{2}(x_{2})\frac{\partial^{2}\left[a_{2}(x_{1})\rho\right]}{\partial
x_{1}\partial
x_{2}}+a_{3}(x_{1})\frac{\partial^{2}\left[b_{3}(x_{2})\rho\right]}{\partial
x_{1}\partial x_{2}}+a_{4}(x)\frac{\partial}{\partial
y}\left[b_{4}(y)\frac{\partial\rho}{\partial y}\right].$ (12)
Here $\rho=\rho(t,x_{1},x_{2})$ and we always use homogenous Dirichlet
boundary conditions. Equation (12) is discretized on the domain
$(x_{1},x_{2})\in[-1,1]^{2},$ using second order accurate finite differences.
Consider the grid $(x_{1,i},x_{2,j})=(-1+ih_{1},-1+jh_{2})$, with
$h_{1}=2/(m_{1}+1),$ $h_{2}=2/(m_{2}+1).$ The solution
$\rho(t_{n},x_{1,i},x_{2,j})$ is then approximated by the grid function
(matrix) $\hat{X}^{n}_{i,j}$ and derivatives are approximated as follow:
$\frac{\partial}{\partial
x_{1}}\left[a_{1}(x_{1})\frac{\partial\rho(t_{n},x_{1},x_{2})}{\partial
x_{1}}\right]\Bigg{|}_{x_{1,i},x_{2,j}}\approx
D_{+}^{1}\left[\frac{a_{1}(x_{1,i})+a_{1}(x_{1,i-1})}{2}\right]D_{-}^{1}\,X_{i,j}^{n},$
$\frac{\partial}{\partial
x_{1}}\left[a_{2}(x_{1})\frac{\partial\rho(t_{n},x_{1},x_{1})}{\partial
x_{2}}\right]\Bigg{|}_{x_{1,i},x_{2,j}}\approx
D_{0}^{1}a_{1}(x_{1,i})D_{0}^{2}\,X_{i,j}^{n}.$
with the remaining two terms discretized in the same way. Here the difference
operators in the 1-direction are defined
$2h_{1}D_{0}^{1}w_{i,j}\equiv w_{i+1,j}-w_{i-1,j},\ \
h_{1}D_{+}^{1}w_{i,j}\equiv w_{i+1,j}-w_{i,j},\ \ h_{1}D_{-}^{1}w_{i,j}\equiv
w_{i,j}-w_{i-1,j},$
with the operators in the 2-direction defined analogously.
### 4.1 Rotation with anisotropic diffusion
In this experiment we consider solid body rotation, $r_{1}(x_{1})=x_{1}$
$r_{2}(x_{2})=-x_{2}$, with anisotropic diffusion. The diffusion coefficients
are:
$\displaystyle a_{1}(x_{1})=a_{4}(x_{1})=\sqrt{\mu}(1+0.1\sin(\pi x_{1})),$
$\displaystyle a_{2}(x_{1})=\sqrt{\mu}(0.15+0.1\sin(\pi x_{1})),$
$\displaystyle a_{3}(x_{1})=\sqrt{\mu}(0.15+0.1\cos(\pi x_{1})),$
$\displaystyle b_{1}(x_{2})=b_{4}(x_{2})=\sqrt{\mu}(1+0.1\cos(\pi x_{2})),$
$\displaystyle b_{2}(x_{2})=\sqrt{\mu}(0.15+0.1\cos(\pi x_{2})),$
$\displaystyle b_{3}(x_{2})=\sqrt{\mu}(0.15+0.1\sin(\pi x_{2})),$
and $\mu=10^{-3}$. We start from the rank-1 initial data
$\rho(0,x_{1},x_{2})=e^{-\left(\frac{x_{1}}{0.3}\right)^{2}}e^{-\left(\frac{x_{2}}{0.1}\right)^{2}}$
and evolve the solution until time $t=\pi$ using a timestep $\Delta
t=\pi/n_{T}$, with $40,80,160,320$. We carry out the computation on three
different grids with $m_{1}=m_{2}=99,199,799$.
In Figure 1 we display 10 equidistant contours between 0.1 and 0.9 of the
solution. Results using the Merge method (left), the classic implicit Euler
(middle) and the BUG method (right) are presented. The solutions are displayed
at time $0.5\pi$ when they have rotated from being horizontal to vertical. The
BUG method does not see the rotation (it is outside the tangent space) and
remains stationary. The solution computed using the Merge method rotates as
expected and is almost identical to the classical implicit Euler solution.
Figure 1: Displayed are 10 equidistant contours between 0.1 and 0.9 of the solution computed using the Merge method (left), the classic implicit Euler (middle) and the BUG method (right). The solutions are displayed at time $0.5\pi$ when they have rotated from being horizontal to vertical. The BUG method does not see the rotation (it is outside the tangent space) and remains stationary. The solution computed using the Merge method rotates as expected and is almost identical to the classical implicit Euler solution. $n_{T}$ | M | MA | IE | $\Delta t/h$ : $\mu\Delta t/h^{2}$
---|---|---|---|---
40 | 1.60(-1) | 1.60(-1) F = 5 | 1.60(-1) | 3.9 : 0.2
80 | 1.01(-1) [0.65] | 1.01(-1) [0.65] F = 3 | 1.01(-1) [0.65] | 2 : 0.098
160 | 6.01(-2) [0.75] | 6.01(-2) [0.75] F = 4 | 6.01(-2) [0.75] | 0.98 : 0.049
320 | 3.36(-2) [0.83] | 3.36(-2) [0.83] F = 0 | 3.36(-2) [0.83] | 0.49 : 0.025
40 | 1.60(-1) | 1.60(-1) F = 8 | 1.60(-1) | 7.9 : 0.79
80 | 1.02(-1) [0.65] | 1.02(-1) [0.65] F = 6 | 1.02(-1) [0.65] | 3.9 : 0.39
160 | 6.07(-2) [0.75] | 6.07(-2) [0.75] F = 4 | 6.07(-2) [0.75] | 2 : 0.2
320 | 3.40(-2) [0.83] | 3.40(-2) [0.83] F = 1 | 3.40(-2) [0.83] | 0.98 : 0.098
40 | 1.61(-1) | 1.61(-1) F = 13 | 1.61(-1) | 31 : 13
80 | 1.02(-1) [0.65] | 1.02(-1) [0.65] F = 11 | 1.02(-1) [0.65] | 16 : 6.3
160 | 6.09(-2) [0.75] | 6.09(-2) [0.75] F = 11 | 6.09(-2) [0.75] | 7.9 : 3.1
320 | 3.41(-2) [0.83] | 3.41(-2) [0.83] F = 9 | 3.41(-2) [0.83] | 3.9 : 1.6
Table 1: Solid body rotation with anisotropic diffusion. Displayed are the
errors (here 1.3(-1) means $1.3\cdot 10^{-1}$) for different timesteps along
with estimated rates of convergence (in brackets). The numbers on the far
right are the “hyperbolic and parabolic CFL numbers”. The top box is for
$m_{1}=m_{2}=99$, the middle for $m_{1}=m_{2}=199$ and the bottom for
$m_{1}=m_{2}=799$. The F indicates how many times the Merge-adapt needed to
add the BUG spaces.
In Table 1 we display the errors and rates of convergence for the Merge,
Merge-adapt and the implicit Euler method. All three methods behave almost
identically (the errors and rates of convergence only differ in the third or
fourth digit). However, the time it takes to compute the solution is quite
different. In Table 2 we display the CPU times for the different methods. The
computations were performed in Matlab on a MacBook Pro with an M2 chip and
16GB RAM. Reasonable effort went into optimizing the codes but the timings
should be seen as indicative rather than decisive. As can be seen the growth
in CPU time as a function of number of degrees of freedom is very mild for the
Merge and Merge-adapt methods in comparison to the classic implicit Euler
method. We further note that the Merge-adapt method has comparable cost with
the Merge method in this example because the equation involves relatively
complex dynamics, which means the explicit prediction is quite costly because
of the large number $s.$
$m_{1}=m_{2}$ | M time$[s]$ | MA time$[s]$ | IE time$[s]$
---|---|---|---
99 | 0.9 | 0.9 | 5.0
199 | 1.4 | 1.5 | 27.1
799 | 12.8 | 11.4 | 926.2
Table 2: Displayed are the CPU times in seconds for the rotation with
anisotropic diffusion example. The results are all for $n_{T}=320$. As can be
seen the growth in CPU time as a function of number of degrees of freedom is
very mild for the Merge and Merge-adapt methods in comparison to the classic
implicit Euler method.
We also track the rank of the solution as a function of time. In Figure 2 we
display the rank for the different methods as a function of time. Once the
solution becomes more diagonal at time $\pi/4$ the rank is decreased and then
increased again as the rotation continues. For the Merge and Merge-adapt
method we use the truncation tolerance $\epsilon_{2}=\Delta t^{2}$ and in
order to compare with the classic implicit Euler we compute the SVD and count
the rank of that solution using the same threshold. We observe the ranks from
the low rank methods are slightly smaller than the implicit Euler method.
Figure 2: Displayed is the rank for the Merge, Merge-adapt and Implicit Euler
method for the problem with rotation and anisotropic diffusion. Here “Fail for
MA” is the indicator where the BUG space is needed for the Merge-adapt method.
For the Implicit Euler method we constantly use the threshold $\Delta t^{2}$
when computing the rank via the truncated SVD. All the computations are done
with $m_{1}=m_{2}=799$ and $n_{T}=320.$
### 4.2 Anisotropic diffusion
In this experiment we consider anisotropic diffusion without rotation
($r_{1}=r_{2}=0$). We take the diffusion coefficients to be constants with
$a_{1}=a_{4}=b_{1}=b_{4}=1$ and $a_{2}=a_{3}=b_{2}=b_{3}=0.3$. Starting from
the rank-1 initial data
$\rho(0,x_{1},x_{2})=\sin(\pi x_{1})\sin(\pi x_{2}),$
we evolve the solution until time $t=0.5$ using a base timestep of $\Delta
t=0.5/n_{T}$, with $40,80,160,320,640,1280$. We carry out the computation on
three different grids with $m_{1}=m_{2}=99,399,799$.
$n_{T}$ | M | MA | IE | $\Delta t/h$ : $\Delta t/h^{2}$
---|---|---|---|---
40 | 8.65(-2) | 8.65(-2) F = 37 | 9.31(-2) | 0.62 : 31
80 | 2.84(-2) [1.60] | 2.84(-2) [1.60] F = 70 | 4.39(-2) [1.08] | 0.31 : 16
160 | 9.94(-3) [1.51] | 9.94(-3) [1.51] F = 160 | 2.13(-2) [1.03] | 0.16 : 7.8
320 | 5.89(-3) [0.75] | 5.89(-3) [0.75] F = 296 | 1.05(-2) [1.01] | 0.078 : 3.9
640 | 3.78(-3) [0.63] | 3.78(-3) [0.63] F = 595 | 5.22(-3) [1.01] | 0.039 : 2
1280 | 2.19(-3) [0.78] | 2.19(-3) [0.78] F = 989 | 2.58(-3) [1.01] | 0.02 : 0.98
40 | 1.13(-1) | 1.13(-1) F = 40 | 9.30(-2) | 1.2 : 120
80 | 4.23(-2) [1.42] | 4.23(-2) [1.42] F = 80 | 4.38(-2) [1.08] | 0.62 : 62
160 | 1.33(-2) [1.66] | 1.33(-2) [1.66] F = 160 | 2.13(-2) [1.03] | 0.31 : 31
320 | 4.64(-3) [1.52] | 4.64(-3) [1.52] F = 320 | 1.05(-2) [1.01] | 0.16 : 16
640 | 2.88(-3) [0.69] | 2.88(-3) [0.69] F = 640 | 5.22(-3) [1.01] | 0.078 : 7.8
1280 | 1.87(-3) [0.61] | 1.87(-3) [0.61] F = 1280 | 2.58(-3) [1.01] | 0.039 : 3.9
40 | 1.03(-1) | 1.03(-1) F = 40 | 9.30(-2) | 5 : 2000
80 | 6.52(-2) [0.66] | 6.52(-2) [0.66] F = 80 | 4.38(-2) [1.08] | 2.5 : 1000
160 | 2.70(-2) [1.27] | 2.70(-2) [1.27] F = 160 | 2.13(-2) [1.03] | 1.2 : 500
320 | 9.69(-3) [1.48] | 9.69(-3) [1.48] F = 320 | 1.05(-2) [1.02] | 0.62 : 250
640 | 2.97(-3) [1.70] | 2.97(-3) [1.70] F = 640 | 5.21(-3) [1.01] | 0.31 : 1.25
1280 | 1.01(-3) [1.55] | 1.01(-3) [1.55] F = 1280 | 2.57(-3) [1.01] | 0.16 : 62
Table 3: Anisotropic diffusion. Displayed are the errors for different
timesteps along with estimated rates of convergence (in brackets). The numbers
on the far right are the “hyperbolic and parabolic CFL numbers”. The top box
is for $m_{1}=m_{2}=99$, the middle for $m_{1}=m_{2}=199$ and the bottom for
$m_{1}=m_{2}=799$. The F indicates how many times the Merge-adapt needed to
add the BUG spaces.
Figure 3: Displayed is the rank for the Merge, Merge-adapt and Implicit Euler
method for the problem with anisotropic diffusion initial data $\sin(\pi
x_{1})\sin(\pi x_{2}),$ (left) and $\sin(2\pi x_{1})\sin(2\pi x_{2})$ (right)
and the solid body rotation problem (right). Here “Fail for MA” is the
indicator where the BUG space is needed for the Merge-adapt method. For the
Implicit Euler method we constantly use the threshold $\Delta t^{2}$ when
computing the rank via the truncated SVD. All the computations are done with
$m_{1}=m_{2}=799$ and $n_{T}=320.$
In Table 3 we display the errors and rates of convergence. Again, the methods
are comparable although it should be noted that the rates of convergence for
the classic implicit Euler method is more uniform than for the low rank
methods. Curiously the error levels are smaller for the low rank method. It
should also be noted that for this example the Merge-adapt method requires the
addition of the BUG spaces almost every time step due to the stiffness of the
problem. We also track the rank of the solution as a function of time, the
results can be found in Figure 3. The rank evolution is very similar for all
methods.
We also consider the more oscillatory initial data
$\rho(0,x_{1},x_{2})=\sin(2\pi x_{1})\sin(2\pi x_{2}),$
which we again evolve until time $t=0.5$ using a base timestep of $\Delta
t=0.5/n_{T}$, with $40,80,160,320,640,1280$. Again, we carry out the
computation on three different grids with $m_{1}=m_{2}=99,399,799$.
This initial data decays more rapidly in time and as a consequence the
residual norm becomes smaller and the Merge-adapt method does not have to add
the BUG spaces as often. The errors and rates of convergence can be found in
Table 4, and the rank of the solution as a function of time, the results can
be found in Figure 2. It can be observed that for this example the classic
implicit Euler method at first is a bit more accurate than the low rank
methods but they catch up as the timestep decreases.
$n_{T}$ | M | MA | IE | $\Delta t/h$ : $\Delta t/h^{2}$
---|---|---|---|---
40 | 4.26(-1) | 4.30(-1) F = 18 | 7.58(-2) | 0.62 : 31
80 | 2.07(-1) [1.03] | 2.08(-1) [1.04] F = 35 | 3.80(-2) [0.99] | 0.31 : 16
160 | 7.90(-2) [1.39] | 7.96(-2) [1.38] F = 61 | 1.91(-2) [0.99] | 0.16 : 7.8
320 | 2.36(-2) [1.74] | 2.42(-2) [1.71] F = 104 | 9.67(-3) [0.98] | 0.078 : 3.9
640 | 5.14(-3) [2.19] | 5.15(-3) [2.23] F = 188 | 4.97(-3) [0.95] | 0.039 : 2
1280 | 1.00(-3) [2.35] | 1.01(-3) [2.34] F = 334 | 2.68(-3) [0.89] | 0.02 : 0.98
40 | 4.87(-1) | 4.89(-1) F = 20 | 7.57(-2) | 1.2 : 125
80 | 2.62(-1) [0.89] | 2.63(-1) [0.89] F = 40 | 3.80(-2) [0.99] | 0.62 : 62
160 | 1.13(-1) [1.20] | 1.13(-1) [1.21] F = 81 | 1.90(-2) [0.99] | 0.31 : 31
320 | 3.89(-2) [1.54] | 3.89(-2) [1.54] F = 134 | 9.60(-3) [0.98] | 0.16 : 16
640 | 1.05(-2) [1.87] | 1.06(-2) [1.87] F = 277 | 4.87(-3) [0.97] | 0.078 : 7.8
1280 | 2.01(-3) [2.39] | 2.01(-3) [2.39] F = 521 | 2.52(-3) [0.95] | 0.039 : 3.9
40 | 5.23(-1) | 5.23(-1) F = 39 | 7.58(-2) | 5 : 2000
80 | 3.49(-1) [0.58] | 3.49(-1) [0.58] F = 75 | 3.80(-2) [0.99] | 2.5 : 1000
160 | 1.80(-1) [0.95] | 1.80(-1) [0.95] F = 132 | 1.91(-2) [0.99] | 1.2 : 500
320 | 7.71(-2) [1.22] | 7.71(-2) [1.22] F = 242 | 9.71(-3) [0.97] | 0.62 : 250
640 | 2.89(-2) [1.41] | 2.89(-2) [1.41] F = 479 | 5.04(-3) [0.94] | 0.31 : 125
1280 | 9.17(-3) [1.65] | 9.17(-3) [1.65] F = 1162 | 2.78(-3) [0.85] | 0.16 : 62
Table 4: Anisotropic diffusion for a higher frequency initial data. Displayed
are the errors for different timesteps along with estimated rates of
convergence (in brackets). The numbers on the far right are the “hyperbolic
and parabolic CFL numbers”. The top box is for $m_{1}=m_{2}=99$, the middle
for $m_{1}=m_{2}=199$ and the bottom for $m_{1}=m_{2}=799$. The F indicates
how many times the Merge-adapt needed to add the BUG spaces.
### 4.3 Solid body rotation with isotropic diffusion
$m_{1}=m_{2}$ | time MA [s] | time M [s] | $\Delta t/h$ : $\mu\Delta t/h^{2}$
---|---|---|---
999 | 1.9 [0.1%] | 3.1 | 0.39 : 0.020
1999 | 3.5 [0.1%] | 6.0 | 0.79 : 0.79
3999 | 7.1 [0.4%] | 15.5 | 1.57 : 0.31
7999 | 52.8 [72%] | 46.2 | 3.14 : 1.26
Table 5: CPU-times for the problem with solid body rotation with isotropic
diffusion. The numbers in brackets are the percentages for how often MA has a
residual that necessitates the addition of the BUG space.
In this experiment we time a problem with solid body rotation and isotropic
diffusion, corresponding to, $r_{1}(x_{1})=x_{1}$ $r_{2}(x_{2})=-x_{2}$,
$a_{1}=b_{1}=a_{4}=b_{4}=10^{-4}$, and $a_{j}=b_{j}=0,\,j=2,\ldots,3$. We
focus on the comparison of the computational cost of Merge and Merge-adapt
schemes for large mesh size. We start from the rank-1 initial data
$\rho(0,x_{1},x_{2})=e^{-\left(\frac{x_{1}}{0.3}\right)^{2}}e^{-\left(\frac{x_{2}}{0.1}\right)^{2}}$
and evolve the solution until time $t=0.25\pi$ using a base timestep of
$\Delta t=0.25\pi/n_{T}$, with $n_{T}=1000$. We carry out the computation on
very fine grids with $m_{1}=m_{2}=999,1999,3999,7999$. Here the size of the
diffusion is chosen so that the ratios $\Delta t/h$ : $\mu\Delta t/h^{2}$
change from smaller than one to larger than one. We report the CPU times for
the M and MA methods in Table 5. As can be seen the MA is faster than the M
method when the number of timesteps where the BUG space is used is small. The
more intensive use of the BUG space appears to happen when the ratio
$\mu\Delta t/h^{2}$ increases above one.
### 4.4 Solid body rotation
In this experiment we consider a hyperbolic problem with pure solid body
rotation, corresponding to, $r_{1}(x_{1})=x_{1}$ $r_{2}(x_{2})=-x_{2}$ and
$a_{j}=b_{j}=0,\,j=1,\ldots,4$. We start from the rank-1 initial data
$\rho(0,x_{1},x_{2})=e^{-\left(\frac{x_{1}}{0.3}\right)^{2}}e^{-\left(\frac{x_{2}}{0.1}\right)^{2}}$
and evolve the solution until time $t=\pi$ using a base timestep of $\Delta
t=\pi/n_{T}$, with $n_{T}=40,80,160,320$. We carry out the computation on
three different grids with $m_{1}=m_{2}=99,199,799$. The errors are listed in
Table 6. Again similar conclusions can be drawn, namely all three methods give
comparable errors. The rank evolution is reported in Figure 4. In this
problem, the rank evolution exhibits periodic pattern in time due to the
solution structure. The rank of the low rank methods are lower than the
implicit Euler method.
$n_{T}$ | M | MA | IE | $\Delta t/h$
---|---|---|---|---
40 | 2.47(-1) | 2.47(-1) F = 11 | 2.51(-1) | 3.9
80 | 1.71(-1) [0.53] | 1.71(-1) [0.53] F = 19 | 1.73(-1) [0.53] | 2
160 | 1.10(-1) [0.63] | 1.10(-1) [0.63] F = 4 | 1.10(-1) [0.64] | 0.98
320 | 6.57(-2) [0.74] | 6.57(-2) [0.74] F = 4 | 6.60(-2) [0.74] | 0.49
40 | 2.50(-1) | 2.50(-1) F = 14 | 2.54(-1) | 7.9
80 | 1.74(-1) [0.52] | 1.74(-1) [0.52] F = 27 | 1.76(-1) [0.53] | 3.9
160 | 1.12(-1) [0.62] | 1.12(-1) [0.62] F = 32 | 1.13(-1) [0.63] | 2
320 | 6.79(-2) [0.73] | 6.79(-2) [0.73] F = 12 | 6.81(-2) [0.73] | 0.98
40 | 2.51(-1) | 2.51(-1) F = 24 | 2.55(-1) | 31
80 | 1.75(-1) [0.52] | 1.75(-1) [0.52] F = 41 | 1.77(-1) [0.52] | 16
160 | 1.13(-1) [0.62] | 1.13(-1) [0.62] F = 61 | 1.14(-1) [0.63] | 7.9
320 | 6.85(-2) [0.72] | 6.85(-2) [0.72] F = 117 | 6.88(-2) [0.73] | 3.9
Table 6: Solid body rotation. Displayed are the errors for different timesteps
along with estimated rates of convergence (in brackets). The top box is for
$m_{1}=m_{2}=99$, the middle for $m_{1}=m_{2}=199$ and the bottom for
$m_{1}=m_{2}=799$. The F indicates how many times the Merge-adapt needed to
add the BUG spaces. Figure 4: Displayed is the rank for the Merge, Merge-
adapt and Implicit Euler method for the problem with solid body rotation
problem. Here “Fail for MA” is the indicator where the BUG space is needed for
the Merge-adapt method. For the Implicit Euler method we constantly use the
threshold $\Delta t^{2}$ when computing the rank via the truncated SVD. All
the computations are done with $m_{1}=m_{2}=799$ and $n_{T}=320.$
## 5 Conclusions and future work
This work prototypes a class of implicit adaptive low rank time-stepping
schemes. By simply merging the explicit step truncation and the BUG spaces,
the modeling error is removed and the numerical schemes achieve robust
convergence upon mesh refinement. An adaptive strategy is proposed for the
prediction of row and column spaces, which is computationally advantageous for
moderately stiff problems. The immediate future work is the generalization to
higher order in time. More importantly, the ideas can be applied to tensor
differential equations, which will be investigated in the future.
## 6 Acknowledgement
We would like to thank Steven R. White for pointing out the reference [24].
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|
University at Albany, State University of New York, USA
jmcurry@albany.eduhttps://orcid.org/ 0000-0003-2504-8388Supported by NSF
CCF-1850052 and NASA 80GRC020C0016 Florida State University, Tallahassee,
<EMAIL_ADDRESS>by NSF grant DMS-1722995 Florida State
University, Tallahassee,
Floridatneedham@fsu.eduhttps://orcid.org/0000-0001-6165-3433Supported by NSF
DMS-2107808 Max Planck Institute for Mathematics in the Sciences, Leipzig,
<EMAIL_ADDRESS>Graz University of Technology,
<EMAIL_ADDRESS>by the Austrian Science Fund (FWF): W1230
Justin Curry, Washington Mio, Tom Needham, Osman Okutan, Florian
Russold[300]Mathematics of computing Algebraic topology [300]Theory of
computation Computational geometry
# Convergence of Leray Cosheaves for Decorated Mapper Graphs
Justin Curry Washington Mio Tom Needham Osman Berat Okutan Florian
Russold111Corresponding Author
###### Abstract
We introduce decorated mapper graphs as a generalization of mapper graphs
capable of capturing more topological information of a data set. A decorated
mapper graph can be viewed as a discrete approximation of the cellular Leray
cosheaf over the Reeb graph. We establish a theoretical foundation for this
construction by showing that the cellular Leray cosheaf with respect to a
sequence of covers converges to the actual Leray cosheaf as the resolution of
the covers goes to zero.
###### keywords:
Leray cosheaves, Reeb graphs, Mapper, convergence
## 1 Introduction
Reeb11footnotetext: This is an abstract of a presentation given at CG:YRF
2023. It has been made public for the benefit of the community and should be
considered a preprint rather than a formally reviewed paper. Thus, this work
is expected to appear in a conference with formal proceedings and/or in a
journal. graphs and their discrete analogs—mapper graphs—are important tools
in computational topology [3, 14, 8, 2]. They are used for data visualization
[12, 10], for comparing scalar fields via distances between Reeb graphs [7, 1,
6], and data skeletonization [9], among other things. Given a continuous map
$f\colon X\rightarrow\mathbb{R}$, the Reeb graph $R_{f}$ summarizes the zero-
dimensional connectedness of $X$. In practice, we deal with maps $f\colon
P\rightarrow\mathbb{R}$ on discrete data sets $P$, where Reeb graphs are
replaced by mapper graphs. A mapper graph is constructed by choosing a cover
$\mathcal{U}=(U_{i})_{i\in I}$ of $f(X)$ or $f(P)$, taking the components or
clustering the data points of $f^{\minus 1}(U_{i})$, collapsing each of the
obtained components to a single vertex and connecting two vertices if their
corresponding components have common points. In [6, 11] it is shown that Reeb
graphs can be viewed as cosheaves and that the cellular Reeb cosheaf w.r.t. a
cover converges to the Reeb cosheaf if the resolution of the cover goes to
zero, establishing a theoretical justification for working with finite covers.
A limitation of this approach is that Reeb/mapper graphs fail to capture
higher-dimensional topological features. To overcome these limitations we
introduce decorated mapper graphs.
Figure 1: Pulling back the cover $\mathcal{U}$ of $f(X)$ along the induced map
$\hat{f}$ on the Reeb graph and refining it into components yields a cover
$\hat{\mathcal{U}}$ of $R_{f}$. The decorated mapper graph is the nerve
complex of this cover decorated by the homology
$H_{\bullet}\big{(}\pi_{f}^{\minus 1}(-)\big{)}$ of the preimages of
intersections in $\hat{\mathcal{U}}$ under the quotient map $\pi_{f}$. Since
$f=\hat{f}\circ\pi_{f}$, this is analogous to using components of preimages
under $f$.
A decorated mapper graph collects the homology (possibly inferred from a point
sample) in all degrees $H_{\bullet}(-)\coloneqq\bigoplus_{n\geq 0}H_{n}(-)$ of
the components of $f^{\minus 1}(U_{i})$ and $f^{\minus 1}(U_{i}\cap U_{j})$ on
the corresponding vertices and edges of the mapper graph; see Figure 1. It can
be viewed as a discrete approximation of the Leray cosheaf over the Reeb
graph. In this abstract we show that the cellular Leray cosheaf w.r.t. a cover
converges to the actual Leray cosheaf for any finite 1D cover $\mathcal{U}$ as
the resolution of the cover goes to zero.
## 2 Leray Cosheaves
The Leray (pre)cosheaf [4, 5, 15] parameterizes the homology of a space $X$
when viewed along a continuous map $f\colon X\rightarrow Y$. It does this by
recording for each pair of open subsets $V\subseteq W\subseteq Y$ their
homology and the induced map $H_{\bullet}\big{(}f^{\minus
1}(V)\big{)}\xrightarrow{H_{\bullet}(f^{\minus 1}(V)\subseteq f^{\minus
1}(W))}H_{\bullet}\big{(}f^{\minus 1}(W)\big{)}$. We further assume that $X$
is compact and $H_{\bullet}\big{(}f^{\minus 1}(V)\big{)}$ is finite-
dimensional for every $V\subseteq Y$.
###### Definition 2.1 (Leray (pre)cosheaf).
We define the graded Leray (pre)cosheaf $\mathcal{L}^{f}$ of a continuous map
$f\colon X\rightarrow Y$ by the following assignments: For all $V\subseteq
W\subseteq Y$ open
$\displaystyle\mathcal{L}^{f}(V)=\bigoplus_{n\in\mathbb{N}_{0}}\mathcal{L}^{f}_{n}(V)$
$\displaystyle\coloneqq\bigoplus_{n\in\mathbb{N}_{0}}H_{n}\big{(}f^{\minus
1}(V)\big{)}$ $\displaystyle\mathcal{L}^{f}(V\subseteq
W)=\bigoplus_{n\in\mathbb{N}_{0}}\mathcal{L}^{f}_{n}(V\subseteq W)$
$\displaystyle\coloneqq\bigoplus_{n\in\mathbb{N}_{0}}H_{n}\big{(}f^{\minus
1}(V)\subseteq f^{\minus 1}(W)\big{)}\hskip 2.0pt.$
In practice, we can not access the whole Leray cosheaf. We can only get a
cellular version [4, 4.1.6] given by its values on a finite cover of $f(X)$.
An open cover $\mathcal{U}=(U_{i})_{i\in I}$ of $f(X)$ defines a simplicial
complex
$\mathcal{N}_{\mathcal{U}}\coloneqq\\{\sigma=(i_{0},\ldots,i_{k})\text{}|\text{
}\mathbf{U}_{\sigma}\coloneqq U_{i_{0}}\cap\ldots\cap
U_{i_{k}}\neq\emptyset\\}$, called the nerve complex of $\mathcal{U}$. We call
$\mathcal{U}$ a finite 1D cover if it is finite and has a one-dimensional
nerve complex and denote by $\leq$ the face-relation of
$\mathcal{N}_{\mathcal{U}}$, i.e. $\sigma\leq\tau\iff\sigma\text{ is a face of
}\tau$ .
###### Definition 2.2 (Cellular Leray cosheaf).
We define the cellular Leray cosheaf $D_{\mathcal{U}}\mathcal{L}^{f}$ w.r.t. a
cover $\mathcal{U}$ as the cosheaf on $\mathcal{N}_{\mathcal{U}}$ given by the
following assignments: For all $\sigma\leq\tau\in\mathcal{N}_{\mathcal{U}}$
$\displaystyle D_{\mathcal{U}}\mathcal{L}^{f}(\sigma)$
$\displaystyle\coloneqq\mathcal{L}^{f}(U_{\sigma})$ $\displaystyle
D_{\mathcal{U}}\mathcal{L}^{f}(\sigma\leq\tau)$
$\displaystyle\coloneqq\mathcal{L}^{f}(U_{\tau}\subseteq U_{\sigma})\hskip
2.0pt.$
If $\pi_{f}\colon X\rightarrow R_{f}$ is the quotient map from $X$ to the Reeb
graph and $\hat{\mathcal{U}}$ a cover of $R_{f}$, then we define a decorated
mapper graph as $D_{\hat{\mathcal{U}}}\mathcal{L}^{\pi_{f}}$; see Figure 1.
## 3 Convergence
It is obvious that this process of discretization can lose information. This
raises the question: How well is $\mathcal{L}^{f}$ represented by
$D_{\mathcal{U}}\mathcal{L}^{f}$? To compare $\mathcal{L}^{f}$ and
$D_{\mathcal{U}}\mathcal{L}^{f}$, we define a process of transforming
$D_{\mathcal{U}}\mathcal{L}^{f}$ into a (pre)cosheaf on $Y$. We want to
approximate the value of $\mathcal{L}^{f}$ on any open set $V\subseteq Y$
given only the information in $D_{\mathcal{U}}\mathcal{L}^{f}$. To this end,
we define the subcomplex
$K_{V}\coloneqq\\{\sigma\in\mathcal{N}_{\mathcal{U}}\text{ }|\text{
}U_{\sigma}\cap V\neq\emptyset\\}\leq\mathcal{N}_{\mathcal{U}}$, which can be
viewed as a simplicial approximation of $V\subseteq\underset{\sigma\in
K_{V}}{\bigcup}U_{\sigma}$; see Figure 2. Moreover, if $V\subseteq W$, we
obtain an inclusion $\iota\colon K_{V}\xhookrightarrow{}K_{W}$.
###### Definition 3.1 (Continuous extension).
Let $\mathcal{U}$ be a finite 1D cover of $Y$. We define the continuous
extension $C_{\mathcal{U}}D_{\mathcal{U}}\mathcal{L}^{f}$ of
$D_{\mathcal{U}}\mathcal{L}^{f}$ as a (pre)cosheaf on $Y$ via the following
assignments: For all $V\subseteq W\subseteq Y$ open
$\displaystyle C_{\mathcal{U}}D_{\mathcal{U}}\mathcal{L}^{f}(V)$
$\displaystyle\coloneqq\bigoplus_{n\in\mathbb{N}_{0}}\Big{(}H_{0}\big{(}K_{V};D_{\mathcal{U}}\mathcal{L}^{f}_{n}|_{K_{V}}\big{)}\oplus
H_{1}\big{(}K_{V};D_{\mathcal{U}}\mathcal{L}^{f}_{n\minus
1}|_{K_{V}}\big{)}\Big{)}$ $\displaystyle
C_{\mathcal{U}}D_{\mathcal{U}}\mathcal{L}^{f}(V\subseteq W)$
$\displaystyle\coloneqq\bigoplus_{n\in\mathbb{N}_{0}}\Big{(}H_{0}\big{(}\iota\big{)}_{n}\oplus
H_{1}\big{(}\iota\big{)}_{n\minus 1}\Big{)}$
where $H_{i}\big{(}K_{V};D_{\mathcal{U}}\mathcal{L}^{f}_{n}|_{K_{V}}\big{)}$
is the degree $i$ homology [4, 6.2.2] of the cosheaf
$D_{\mathcal{U}}\mathcal{L}^{f}_{n}$ restricted to $K_{V}$ and
$H_{i}(\iota)_{n}$ is the map induced on cosheaf homology by $\iota\colon
K_{V}\xhookrightarrow{}K_{W}$ cf. [13, A.17].
As shown in [4, 5, 15], if $\mathcal{U}$ is a finite 1D cover of $f(X)$ the
continuous extension yields
$C_{\mathcal{U}}D_{\mathcal{U}}\mathcal{L}^{f}(V)\cong\underset{n\in\mathbb{N}_{0}}{\bigoplus}H_{n}\Big{(}f^{\minus
1}\big{(}\underset{\sigma\in K_{V}}{\bigcup}U_{\sigma}\big{)}\Big{)}$
approximating $H_{\bullet}\big{(}f^{\minus 1}(V)\big{)}$; see Figure 2.
Figure 2: The figure shows $D_{\mathcal{U}}\mathcal{L}^{f}_{0}\cong
D_{\mathcal{U}}\mathcal{L}^{f}$ on $\mathcal{N}_{\mathcal{U}}$ w.r.t. two
covers with different resolution as well as the restriction
$D_{\mathcal{U}}\mathcal{L}^{f}_{0}|_{K_{V}}$ to $K_{V}$ (in orange). Homology
is taken with coefficients in a field $k$ and unlabeled arrows represent
identity maps. For the coarse cover we get
$C_{\mathcal{U}}D_{\mathcal{U}}\mathcal{L}^{f}(V)\cong k\ncong
H_{0}\big{(}f^{\minus 1}(V)\big{)}$ but for the finer one we get
$C_{\mathcal{U}}D_{\mathcal{U}}\mathcal{L}^{f}(V)\cong k^{2}\cong
H_{0}\big{(}f^{\minus 1}(V)\big{)}$.
The following proposition shows that
$C_{\mathcal{U}}D_{\mathcal{U}}\mathcal{L}^{f}(V\subseteq W)$ also gives us
the correct induced map.
###### Proposition 3.2.
Let $\mathcal{U}$ be a finite 1D cover of $f(X)$. Then, for all $V\subseteq
W\subseteq Y$ open, the following diagram commutes and the horizontal arrows
are isomorphisms:
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To talk about convergence, we have to define a distance on (pre)cosheaves.
Assume $Y$ is a metric space and that all the involved (pre)cosheaves are
constructible [4, 11.0.10]. For every $\epsilon>0$ define
$V^{\epsilon}\coloneqq\\{y\in Y\text{ }|\text{ }d(y,V)<\epsilon\\}$. To a
(pre)cosheaf $F$ on $Y$ we now associate
$F^{\bullet}\coloneqq\\{(F^{\epsilon})_{\epsilon\geq 0}\hskip 2.0pt,\hskip
2.0pt(F_{\epsilon}^{\delta}\colon F^{\epsilon}\rightarrow
F^{\delta})_{\epsilon\leq\delta}\\}$, a parameterized family of (pre)cosheaves
on $Y$, by setting $F^{\epsilon}(V)\coloneqq F(V^{\epsilon})$. This allows us
to define the distance of two (pre)cosheaves $F$ and $G$ as the interleaving
distance of $F^{\bullet}$ and $G^{\bullet}$, i.e. $d(F,G)\coloneqq
d_{I}(F^{\bullet},G^{\bullet})$. Denote by
$\text{res}(\mathcal{U})\coloneqq\underset{U\in\mathcal{U}}{\text{sup}}\hskip
2.0pt\underset{x,y\in U}{\text{sup}}d(x,y)$ the resolution of an open cover
$\mathcal{U}$. We are now able to show that if the resolution of the cover
$\mathcal{U}$ goes to zero, $C_{\mathcal{U}}D_{\mathcal{U}}\mathcal{L}^{f}$
converges to $\mathcal{L}^{f}$.
###### Theorem 3.3.
If $f\colon X\rightarrow Y$ is continuous and $\mathcal{U}$ is a finite 1D
cover of $f(X)$, then
$d\big{(}C_{\mathcal{U}}D_{\mathcal{U}}\mathcal{L}^{f},\mathcal{L}^{f}\big{)}\leq\text{res}(\mathcal{U})\hskip
2.0pt.$
If we only consider $\mathcal{L}^{f}_{0}$, the degree-zero part of
$\mathcal{L}^{f}$, the construction in Definition 3.1 and Theorem 3.3
specializes to an abelianization of the convergence result in [11].
## 4 Future Work
Theorem 3.3 guarantees that, if we choose a cover of resolution $\leq\delta$,
the cellular Leray cosheaf is a $\delta$-approximation of the continuous one.
This result establishes a theoretical justification for working with finite
covers. Since we have to deal with maps from finite point sets in practice, in
future work we plan to investigate under what conditions we can infer the
cellular Leray cosheaf of a map $f\colon X\rightarrow Y$ from a finite sample
of $X$; cf. [2].
## References
* [1] Ulrich Bauer, Xiaoyin Ge, and Yusu Wang. Measuring distance between reeb graphs. In Proceedings of the Thirtieth Annual Symposium on Computational Geometry, SOCG’14, page 464–473, New York, NY, USA, 2014. Association for Computing Machinery. doi:10.1145/2582112.2582169.
* [2] Adam Brown, Omer Bobrowski, Elizabeth Munch, and Bei Wang. Probabilistic convergence and stability of random mapper graphs. Journal of Applied and Computational Topology, 5(1):99–140, 2021\. doi:10.1007/s41468-020-00063-x.
* [3] Hamish Carr, Jack Snoeyink, and Ulrike Axen. Computing contour trees in all dimensions. Computational Geometry, 24(2):75–94, 2003. Special Issue on the Fourth CGC Workshop on Computational Geometry. doi:https://doi.org/10.1016/S0925-7721(02)00093-7.
* [4] Justin Curry. Sheaves, Cosheaves and Applications. PhD thesis, University of Pennsylvania, 2014. arXiv:1303.3255.
* [5] Justin Curry, Robert Ghrist, and Vidit Nanda. Discrete morse theory for computing cellular sheaf cohomology. Foundations of Computational Mathematics, 16, 2013. doi:10.1007/s10208-015-9266-8.
* [6] Vin de Silva, Elizabeth Munch, and Amit Patel. Categorified reeb graphs. Discrete & Computational Geometry, 55(4):854–906, 2016. doi:10.1007/s00454-016-9763-9.
* [7] Barbara Di Fabio and Claudia Landi. The edit distance for reeb graphs of surfaces. Discrete & Computational Geometry, 55(2):423–461, 2016. doi:10.1007/s00454-016-9758-6.
* [8] Herbert Edelsbrunner, John Harer, and Amit K. Patel. Reeb spaces of piecewise linear mappings. In Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry, SCG ’08, page 242–250, New York, NY, USA, 2008. Association for Computing Machinery. doi:10.1145/1377676.1377720.
* [9] Xiaoyin Ge, Issam Safa, Mikhail Belkin, and Yusu Wang. Data skeletonization via reeb graphs. Advances in Neural Information Processing Systems, 24, 2011.
* [10] P. Y. Lum, G. Singh, A. Lehman, T. Ishkanov, M. Vejdemo-Johansson, M. Alagappan, J. Carlsson, and G. Carlsson. Extracting insights from the shape of complex data using topology. Scientific Reports, 3(1):1236, 2013. doi:10.1038/srep01236.
* [11] Elizabeth Munch and Bei Wang. Convergence between Categorical Representations of Reeb Space and Mapper. In 32nd International Symposium on Computational Geometry (SoCG 2016), volume 51 of Leibniz International Proceedings in Informatics (LIPIcs), pages 53:1–53:16. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2016. doi:10.4230/LIPIcs.SoCG.2016.53.
* [12] Monica Nicolau, Arnold Levine, and Gunnar Carlsson. Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival. Proceedings of the National Academy of Sciences of the United States of America, 108:7265–70, 04 2011. doi:10.1073/pnas.1102826108.
* [13] Florian Russold. Persistent sheaf cohomology, 2022. arXiv:2204.13446.
* [14] Gurjeet Singh, Facundo Memoli, and Gunnar Carlsson. Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition. In Eurographics Symposium on Point-Based Graphics. The Eurographics Association, 2007. doi:10.2312/SPBG/SPBG07/091-100.
* [15] Hee Rhang Yoon and Robert Ghrist. Persistence by parts: Multiscale feature detection via distributed persistent homology, 2020. arXiv:2001.01623.
|
TextBox
# An Exploratory Study on Refactoring Documentation in Issues Handling
Eman Abdullah AlOmar Rochester Institute of TechnologyRochester, New YorkUSA
<EMAIL_ADDRESS>, Anthony Peruma Rochester Institute of
TechnologyRochester, New YorkUSA<EMAIL_ADDRESS>, Mohamed Wiem Mkaouer
Rochester Institute of TechnologyRochester, New YorkUSA<EMAIL_ADDRESS>,
Christian D. Newman Rochester Institute of TechnologyRochester, New YorkUSA
<EMAIL_ADDRESS>and Ali Ouni ETS Montreal, University of QuebecMontreal,
QuebecCanada<EMAIL_ADDRESS>
###### Abstract.
Understanding the practice of refactoring documentation is of paramount
importance in academia and industry. Issue tracking systems are used by most
software projects enabling developers, quality assurance, managers, and users
to submit feature requests and other tasks such as bug fixing and code review.
Although recent studies explored how to document refactoring in commit
messages, little is known about how developers describe their refactoring
needs in issues. In this study, we aim at exploring developer-reported
refactoring changes in issues to better understand what developers consider to
be problematic in their code and how they handle it. Our approach relies on
text mining 45,477 refactoring-related issues and identifying refactoring
patterns from a diverse corpus of 77 Java projects by investigating issues
associated with 15,833 refactoring operations and developers’ explicit
refactoring intention. Our results show that (1) developers mostly use move
refactoring related terms/phrases to target refactoring-related issues; and
(2) developers tend to explicitly mention the improvement of specific quality
attributes and focus on duplicate code removal. We envision our findings
enabling tool builders to support developers with automated documentation of
refactoring changes in issues.
Refactoring documentation, issues, software quality, mining software
repositories
††copyright: none††conference: MSR ’22: Proceedings of the 19th International
Conference on Mining Software Repositories; May 23–24, 2022; Pittsburgh, PA,
USA††booktitle: MSR ’22: Proceedings of the 19th International Conference on
Mining Software Repositories, May 23–24, 2022, Pittsburgh, PA, USA††price:
15.00††isbn: 978-1-4503-XXXX-X/18/06††ccs: Software Engineering Software
Quality††ccs: Software Engineering Refactoring
## 1\. Introduction
Code refactoring is a disciplined software engineering practice that is known
as the process of changing a software system in such a way that it does not
alter the external behavior of the code yet improves its internal structure
(Fowler et al., 1999; AlOmar et al., 2021c). Refactoring is commonly used in
different development and maintenance tasks (AlOmar et al., 2021e). It
supports developers in revolving submitted issues such as feature requests
(Nyamawe et al., 2019) or bug reports (Di Penta et al., 2020). Issue tracking
systems are used by most contemporary software projects enabling developers,
quality assurance, managers, and users to submit feature or enhancement
requests, as well as other tasks such as bug fixing and code review.
Previous studies have focused on recommending refactorings through the
detection of refactoring opportunities, either by identifying code anti-
patterns that need correction (Brito et al., 2020; Oizumi et al., 2020;
Lenarduzzi et al., 2020), or by optimizing code quality metrics (Mkaouer et
al., 2015; Anwer et al., 2017; Sellitto et al., [n. d.]). Yet, recent studies
have shown that there is a gap between automated refactoring tools, and what
developers consider to a need-to-refactor situation in code (Cedrim et al.,
2016; AlOmar et al., 2019b; Fernandes et al., 2020). To bridge this gap, it is
important to understand what triggers developers to refactor their code, and
what do developers care about when it comes to code improvement. Such
information provides insights, to software practitioners and researchers,
about the developer’s perception of refactoring. This can question whether
developers do care about structural metrics and code smells when refactoring
their code, or if there are other factors that are of direct influence on
these non-functional changes.
In this paper, we focus on investigating issues that are written to express a
need for refactoring. We extract patterns differentiating refactoring issues.
These patterns represent what developers consider to be worth refactoring. We
are also interested in the solutions, i.e., refactoring operations, being
proposed as correction measures. Our investigation is driven by answering the
following research questions:
* •
RQ1: What textual patterns do developers use to describe their refactoring
needs in issues? This RQ explores the existence of refactoring documentation
in issues containing refactorings. This RQ aims to identify developers’ common
phrases when describing their refactoring problem/challenge.
* •
RQ2: What are the quality attributes developers care about when documenting in
issues? In this RQ, we investigate whether developers explicitly indicate the
purpose of their refactoring activity applied in issues, e.g., improving
structural metrics of fixing code smells.
The results of this exploratory study strengthen our understanding of what
circumstances cause the need for refactorings. Using the evolution history of
77 open-source projects exhibiting a total of 45,477 refactoring commits with
issues, our study reveals that developers are mainly driven by reducing
complexity and increasing comprehension and performance. While various studies
associate refactoring tightly with fixing code smells, the only anti-pattern
that was highlighted is duplicate code. Furthermore, various studies have
shown that the rename category is the most frequent in terms of refactoring
operations (e.g., rename method, rename attribute, etc.) (Peruma et al., 2018;
Negara et al., 2013), but our study reveals that developers tend also to
discuss more complex refactorings, including refactorings dealing with extract
and move code fragments, due to their impact on the design and code semantics
preservation. We have also prepared a replication package of issues and their
corresponding fixes (refactorings), to support the reproducibility and
extension of our work (AlOmar, 2022).
## 2\. Study Design
Figure 1. Overview of our experiment design.
Table 1. Dataset overview of refactoring-related issues.
Item Count Refactoring commits with issues 45,477 Refactoring commits with
issues having keyword ‘refactor*’ in title 835 Refactoring operations
associated with issues 15,833 Issue Status Item Count Closed 603 In progress 8
Open 28 Resolved 196 Issue Resolution Item Count Done 4 Fixed 780 Implemented
8 Resolved 4 Won’t fix 3 Issue Types Item Count Bug 95 Improvement 390 New
Feature 2 Story 1 Sub-task 71 Task 274 Test 2
Figure 1 depicts a general overview of our experimental setup. In the
following subsections, we elaborate on the activities involved in the process.
We provide the dataset that we generate available in our replication package
(AlOmar, 2022).
### 2.1. Source Dataset
Our study utilizes the SmartSHARK MongoDB Release 2.1 dataset (Trautsch et
al., 2021). This dataset contains a wide range of information for 77 open-
source Java projects, such as commit history, issues, refactorings, code
metrics, mailing lists, and continuous integration data, among others. All 77
Java projects are part of the Apache ecosystem and utilize GitHub as their
version control repository and JIRA for issue tracking. Furthermore,
SmartSHARK utilizes RefDiff (Silva and Valente, 2017) and RefactoringMiner
(Tsantalis et al., 2018) to mine refactoring operations. Finally, the
SmartSHARK dataset schema provides the necessary relationship attributes
between data collections to join two or more related collection types.
### 2.2. Refactoring Documentation Detection
To identify refactoring documentation patterns in issues, we perform a series
of manual and automated activities, as follows:
Step #1: Issues associated with a refactoring activity. As our study focuses
on issues and refactorings, our analysis is limited to issues where one or
more refactoring operations were performed as part of the issue resolution.
Hence, we first extracted all different refactorings from the source dataset.
Next, we identify all commits containing the refactoring operations. Finally,
we extracted issues that was addressed using the identified commits.
Step #2: Issues associated with developer intention about refactoring. To
ensure that the selected issues are about refactoring, we focused on a subset
of 835 issues that reported developers’ intention about the application of
refactoring (i.e., having the keyword ‘refactor’). The choice of ‘refactor’,
besides being used by all related studies, is intuitively the first term to
identify ideal refactoring-related issues (Murphy-Hill et al., 2008; Kim et
al., 2014; AlOmar et al., 2019a; AlOmar et al., 2021a). Finally, to reduce the
occurrence of false positives, we limited our analysis to only the occurrence
of the term ‘refactor’ in the title of the issue as the title is a concise
description of the problem (Peruma et al., 2022; Rosen and Shihab, 2016).
Step #3: Annotation of issues. When creating issues, developers use natural
language to describe the issues. Hence, given the diverse nature of developers
describing the problem, an automated approach to analyzing the issue text is
not feasible. Therefore, we performed a manual analysis of the issue title and
body to identify refactoring documentation patterns. Next, we grouped this
subset of issues based on specific patterns. Further, to avoid redundancy of
any pattern, we only considered one phrase if we found different patterns with
the same meaning. For example, if we find patterns such as ‘simplifying the
code’, ‘code simplification’, and ‘simplify code’, we add only one of these
similar phrases in the list of patterns. This enables having a list of the
most insightful and unique patterns, and it also helps in making more concise
patterns that are usable for readers.
## 3\. Experimental Results
### 3.1. RQ1: What textual patterns do developers use to describe their
refactoring needs in issues?
Methodology. To identify refactoring documentation patterns, we manually
inspect a subset of issues. These patterns are represented in the form of a
keyword or phrase that frequently occurs in the issues associated with
refactoring-related commits.
Table 2. List of refactoring documentation in issues (‘*’ captures the
extension of the keyword) .
Patterns Add* Chang* Chang* the name Clean* up code Cleanup Code clean* Code
optimization Creat* Extend* Extract* Fix* Fix* code style Improv* Improv* code
quality Inlin* Introduc* Merg* Mov* Pull* up Push* down Repackag* Redesign*
Reduc* Refactor* Refin* Remov* Renam* Reorganiz* Replac* Restructur* Rewrit*
Simplify* code Split*
Results. Our in-depth inspection of the issues results in a list of 33
refactoring documentation patterns, as shown in Table 2. Our findings show
that the names of refactoring operations (e.g., ‘extract*’, ‘mov*’, ‘renam*’)
occur in the top frequently occurring patterns, and these patterns are mainly
linked to code elements at different levels of granularity such as classes,
methods, and variables. These specific terms are well-known software
refactoring operations and indicate developers’ knowledge of the catalog of
refactoring operations. We also observe that the top-ranked refactoring
operation-related keywords include ‘extract*’ and ‘mov*’. ‘pull up’ and ‘push
down’ operations are among the least discussed refactoring operations (similar
to findings in (Silva et al., 2016; Peruma et al., 2022)). Moreover, we
observe the occurrences of issue-fixing specific terms such as ‘fix*’,
‘remov*’, and ‘reduc*’. Next, we examine the most common keywords that
developers use when expressing refactoring documentation in issues. Figure 3
shows the top keywords used to identify refactoring documentations across the
examined projects, that are ranked according to their number of occurrences.
Extract 37.2% Move 24.6% Rename 21.5% Change 14.7% Inline 2.0% Figure 2. Percentage of refactorings, clustered by operation class. Table 3. Summary of refactoring patterns, clustered by refactoring related categories. Internal QA (%) | External QA (%) | Code Smell (%)
---|---|---
Complexity (0.26 %) | Readability (0.23 %) | Duplicate code (0.73 %)
Design Size (0.25 %) | Performance (0.11 %) |
Encapsulation (0.18 %) | Usability (0.08 %) |
Dependency (0.16 %) | Extensibility (0.07 %) |
Inheritance (0.08 %) | Compatibility (0.06 %) |
Coupling (0.02 %) | Accuracy (0.05 %) |
Abstraction (0.02 %) | Modularity (0.05 %) |
| Flexibility (0.04 %) |
| Understandability (0.04 %) |
| Reusability (0.04 %) |
| Testability (0.03 %) |
| Maintainability (0.03 %) |
| Manageability (0.02 %) |
| Stability (0.006 %) |
| Accessibility (0.01 %) |
| Configurability (0.01 %) |
| Robustness (0.006 %) |
| Repeatability (0.006 %) |
| Effectiveness (0.006 %) |
To better understand the nature of refactoring documentation, we have
classified the associated refactoring operations into 5 classes, namely,
‘changing’, ‘extracting’, ‘inlining’, ‘moving’, and ‘renaming’. Depicted in
Figure 2, we cluster these operations associated with issues using a list of
refactoring keywords defined in a previous work (AlOmar et al., 2022). The
changing of the types belongs to the ‘changing’ class, whereas the extraction
of classes and methods are included in the ‘extracting’ class. As for,
‘moving’, it gathers all the movement of code elements, e.g., moving methods,
or pushing code elements across hierarchies. Merging-related activities are
included in the ‘inlining’ class. Finally, the ‘renaming’ class contains all
refactorings that rename a given code element such as a class, a package or an
attribute. As shown in Figure 2, the ‘extracting’ operations are highly
documented in issues across the projects, and it reached the percentage of
37.2%, higher than ‘moving’, ‘renaming’, and ‘changing’ whose percentage is
respectively 24.6%, 21.5%, and 14.7%. The ‘inlining’ operations, however, is
the least documented refactoring which had a ratio of only 2 %.
Figure 3. Popular refactoring textual patterns in issues.
### 3.2. RQ2: What are the quality attributes developers care about when
documenting in issues?
Methodology. After identifying the different refactoring documentation
patterns, we identify and categorize the patterns into three main categories
(similar to (AlOmar et al., 2019a; AlOmar et al., 2021e; AlOmar et al., 2021b;
AlOmar et al., 2021d)): (1) internal quality attributes, (2) external quality
attributes, and (3) code smells.
Results. Table 3 provides the list of refactoring documentation patterns,
ranked based on their frequency, we identify in refactoring-related issues. We
observe that developers frequently mention key internal quality attributes
(such as inheritance, complexity, etc.), a wide range of external quality
attributes (such as readability and performance), and code duplication code
smell that might impact code quality. To improve the internal design, the
system structure optimization regarding its complexity and design size seems
to be the dominant focus that is consistently mentioned in issues (0.26% and
0.25%, respectively). Concerning external quality attribute-related issues, we
observe the mention of refactorings to enhance nonfunctional attributes.
Patterns such as ‘readability’, ‘performance’, and ‘usability’ represent the
developers’ main focus, with 0.23%, 0.11%, and 0.08%, respectively. Finally,
for code smell-focused refactoring issues, duplicate code represents the most
popular anti-pattern developers intend to refactor (0.73%).
## 4\. Discussion
Our research aims to explore refactoring documentation in issues to provide
future research directions that support developers in understanding
refactoring applied in issues.
RQ1 indicates that developers tend to use a variety of textual patterns to
document their refactorings in issues. These patterns can provide either a (1)
generic description of problems developers encounter or (2) a specific
refactoring operation name following Fowler’s names (Fowler et al., 1999).
Although previous studies show that rename refactorings are a common type of
refactoring, e.g., (Peruma et al., 2018), we notice that ‘mov*’ and ‘extract*’
are the topmost documented refactorings in issues. This can be explained by
the fact that developers tend to make many design improvement decisions that
include remodularizing packages by moving classes, reducing class-level
coupling, and increasing cohesion by moving methods. Additionally, developers
might use similar terminology when performing move-related refactoring
operations, i.e., Extract/Inline/Pull-up/Push-down (AlOmar et al., 2022). As
shown in Figure 2, ‘extracting’ is the most documented refactorings. An
interpretation for this comes from the nature of the debugging process that
may include the separation of concerns which helps in reducing the core
complexity of a larger module and reduce its proneness to errors (Tsantalis
and Chatzigeorgiou, 2011). In other words, developers tend to discuss more
complex refactorings in issues, including refactorings from the extract and
move categories, due to their impact on the design and code semantic
preservation. This information can provide valuable references for refactoring
documentation practice in issues. For example, whether refactoring-related
issue descriptions have the relevant information is a critical indicator for
reproducing refactoring-related issues.
From RQ2, we observe that developers discuss quality concerns when documenting
refactorings in issues that can be related to: (1) internal quality
attributes, (2) external quality attributes, or (3) code smells. When
analyzing these quality concerns per issue types reported in Table 1, we
notice that complexity and duplicate code are mostly documented with issue
type named ‘bug’, whereas the duplicate code and readability were the popular
sub-categories for refactoring-related issues type named ‘improvement’. For
instance, the developer discussed fixing design issues by putting common
functionalities into a superclass to eliminate duplicate code, breaking up
lengthier methods to make the code more readable, and avoiding nested complex
data structure to reduce code complexity. Moreover, we observe that code smell
is rarely documented in issues. As shown in Table 3, developers only focused
on duplicate code removal. Conversely, developers tend to report a variety of
external quality attributes, focusing mainly on improving readability of the
code. This corroborates the finding by Palomba et al. (Palomba et al., 2017),
where refactoring targeting program comprehension was mostly applied during
bug fixing activities. As developers discussed functional and non-functional
aspects of source code, future research can further investigate the intent as
to why and how developers perform refactoring in issues. With a better
understanding of this phenomenon, researchers and tool builders can support
developers with automatically documenting refactorings in issues.
One of the main purposes of exploring refactoring documentation in issues is
to better understand how developers cope with their software decay by
extracting any refactoring strategies that can be associated with removing
code smells (Tsantalis et al., 2008; Bavota et al., 2013), or improving the
design structural measurements (Mkaouer et al., 2014; Bavota et al., 2014).
However, these techniques only analyze the changes at the source code level,
and provide the operations performed, without associating it with any textual
description, which may infer the rationale behind the refactoring application.
Our proposal, of textual patterns, is the first step towards complementing the
existing effort in detecting refactorings, by augmenting it with any
description that was intended to describe the refactoring activity. As
previously shown in Tables 2 and 3, developers tend to add a high-level
description of their refactoring activity, and mention their intention behind
refactoring (remove duplicate code, improve readability, etc.), along with
mentioning the refactoring operations they apply (type migration, inline
methods, etc.).
Overall, the documentation of refactoring in issues is an important research
direction that requires further attention. It has been known that there is a
general shortage of refactoring documentation, and there is no consensus about
how refactoring should be documented, which makes it subjective and developer-
specific. Lack of design documentation forced developers to rely on the source
code to identify design problems (Sousa et al., 2018). Moreover, the fine-
grained description of refactoring can be time-consuming, as a typical
description should contain an indication about the operations performed,
refactored code elements, and a hint about the intention behind the
refactoring. In addition, the developer specification can be ambiguous as it
reflects the developer’s understanding of what has been improved in the source
code, which can be different in reality, as the developer may not necessarily
adequately estimate the refactoring impact on the quality improvement.
## 5\. Threats To Validity
The first threat relates to the analysis of open-source Java projects. Our
results may not generalize to systems written in other languages. Another
potential threat to validity relates to our findings regarding counting the
reported quality attributes and code smells. Due to the large number of commit
messages, we have not performed a manual validation to remove false positive
commit messages. Thus, this may have an impact on our findings. Finally, we
constructed our dataset by extracting issues containing the term ‘refactor’ in
the title. There is the possibility that we may have excluded synonymous
terms/phrases. However, even though this approach reduces the number of issues
in our dataset, it also decreases false-positives, and ensures that we analyze
issues that are explicitly focused on refactorings.
## 6\. Conclusion & Future Work
In this study, we performed an exploratory study to understand how developers
document refactorings in issues. Specifically, we identify refactoring
terms/phrases patterns, study possible refactoring documentation types, and
determine how many refactoring terms/phrases exist in issues. Our results show
that (1) developers mostly use move refactoring related terms/phrases to
target refactoring-related issues; and (2) developers tend to explicitly
mention the improvement of specific quality attributes and focus on duplicate
code removal. We envision our findings enabling tool builders to support
developers with automatically document refactoring in issues. Future work in
this area includes investigating which refactoring operation is more
problematic in issues.
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|
$\lim_{k\rightarrow\infty}\operatorname*{p-lim}_{n\rightarrow\infty}(\lambda^{2}/n)(Q({\bm{m}}^{k-1})-q_{\infty})\|{\bm{m}}^{k-1}\|=0.$
(123)
Because $\operatorname*{p-lim}_{n\rightarrow\infty}Q({\bm{m}}^{k-1})=q_{k}$,
and $q_{k}\rightarrow q_{\infty}$ as $k\rightarrow\infty$, the previous
display holds, and the result follows. |
# HyperDeepONet: learning operator with complex target function space using
the limited resources via hypernetwork
Jae Yong Lee
Center for Artificial Intelligence and Natural Sciences
Korea Institute for Advanced Study
Seoul, 02455, Republic of Korea
<EMAIL_ADDRESS>
&Sung Woong Cho11footnotemark: 1 & Hyung Ju Hwang
Department of Mathematics
Pohang University of Science and Technology
Pohang, 37673, Republic of Korea
<EMAIL_ADDRESS>These authors contributed
equally.corresponding author
###### Abstract
Fast and accurate predictions for complex physical dynamics are a significant
challenge across various applications. Real-time prediction on resource-
constrained hardware is even more crucial in real-world problems. The deep
operator network (DeepONet) has recently been proposed as a framework for
learning nonlinear mappings between function spaces. However, the DeepONet
requires many parameters and has a high computational cost when learning
operators, particularly those with complex (discontinuous or non-smooth)
target functions. This study proposes HyperDeepONet, which uses the expressive
power of the hypernetwork to enable the learning of a complex operator with a
smaller set of parameters. The DeepONet and its variant models can be thought
of as a method of injecting the input function information into the target
function. From this perspective, these models can be viewed as a particular
case of HyperDeepONet. We analyze the complexity of DeepONet and conclude that
HyperDeepONet needs relatively lower complexity to obtain the desired accuracy
for operator learning. HyperDeepONet successfully learned various operators
with fewer computational resources compared to other benchmarks.
## 1 Introduction
Operator learning for mapping between infinite-dimensional function spaces is
a challenging problem. It has been used in many applications, such as climate
prediction (Kurth et al., 2022) and fluid dynamics (Guo et al., 2016). The
computational efficiency of learning the mapping is still important in real-
world problems. The target function of the operator can be discontinuous or
sharp for complicated dynamic systems. In this case, balancing model
complexity and cost for computational time is a core problem for the real-time
prediction on resource-constrained hardware (Choudhary et al., 2020; Murshed
et al., 2021).
Many machine learning methods and deep learning-based architectures have been
successfully developed to learn a nonlinear mapping from an infinite-
dimensional Banach space to another. They focus on learning the solution
operator of some partial differential equations (PDEs), e.g., the initial or
boundary condition of PDE to the corresponding solution. Anandkumar et al.
(2019) proposed an iterative neural operator scheme to learn the solution
operator of PDEs.
Simultaneously, Lu et al. (2019; 2021) proposed a deep operator network
(DeepONet) architecture based on the universal operator approximation theorem
of Chen & Chen (1995). The DeepONet consists of two networks: branch net
taking an input function at fixed finite locations, and trunk net taking a
query location of the output function domain. Each network provides the $p$
outputs. The two $p$-outputs are combined as a linear combination (inner-
product) to approximate the underlying operator, where the branch net produces
the coefficients ($p$-coefficients) and the trunk net produces the basis
functions ($p$-basis) of the target function.
While variant models of DeepONet have been developed to improve the vanilla
DeepONet, they still have difficulty approximating the operator for a
complicated target function with limited computational resources. Lanthaler et
al. (2022) and Kovachki et al. (2021b) pointed out the limitation of linear
approximation in DeepONet. Some operators have a slow spectral decay rate of
the Kolmogorov $n$-width, which defines the error of the best possible linear
approximation using an $n-$dimensional space. A large $n$ is required to learn
such operators accurately, which implies that the DeepONet requires a large
number of basis $p$ and network parameters for them.
Hadorn (2022) investigated the behavior of DeepONet, to find what makes it
challenging to detect the sharp features in the target function when the
number of basis $p$ is small. They proposed a Shift-DeepONet by adding two
neural networks to shift and scale the input function. Venturi & Casey (2023)
also analyzed the limitation of DeepONet via singular value decomposition
(SVD) and proposed a flexible DeepONet (flexDeepONet), adding a pre-net and an
additional output in the branch net. Recently, to overcome the limitation of
the linear approximation, Seidman et al. (2022) proposed a nonlinear manifold
decoder (NOMAD) framework by using a neural network that takes the output of
the branch net as the input along with the query location. Even though these
methods reduce the number of basis functions, the total number of parameters
in the model cannot be decreased. The trunk net still requires many parameters
to learn the complex operators, especially with the complicated (discontinuous
or non-smooth) target functions.
In this study, we propose a new architecture, HyperDeepONet, which enables
operator learning, and involves a complex target function space even with
limited resources. The HyperDeepONet uses a hypernetwork, as proposed by Ha et
al. (2017), which produces parameters for the target network. Wang et al.
(2022) pointed out that the final inner product in DeepONet may be inefficient
if the information of the input function fails to propagate through a branch
net. The hypernetwork in HyperDeepONet transmits the information of the input
function to each target network’s parameters. Furthermore, the expressivity of
the hypernetwork reduces the neural network complexity by sharing the
parameters (Galanti & Wolf, 2020). Our main contributions are as follows.
* •
We propose a novel HyperDeepONet using a hypernetwork to overcome the
limitations of DeepONet and learn the operators with a complicated target
function space. The DeepONet and its variant models are analyzed primarily in
terms of expressing the target function as a neural network (Figure 4). These
models can be simplified versions of our general HyperDeepONet model (Figure
5).
* •
We analyze the complexity of DeepONet (Theorem 2) and prove that the
complexity of the HyperDeepONet is lower than that of the DeepONet. We have
identified that the DeepONet should employ a large number of basis to obtain
the desired accuracy, so it requires numerous parameters. For variants of
DeepONet combined with nonlinear reconstructors, we also present a lower bound
for the number of parameters in the target network.
* •
The experiments show that the HyperDeepONet facilitates learning an operator
with a small number of parameters in the target network even when the target
function space is complicated with discontinuity and sharpness, which the
DeepONet and its variants suffer from. The HyperDeepONet learns the operator
more accurately even when the total number of parameters in the overall model
is the same.
## 2 Related work
Many machine learning methods and deep learning-based architectures have been
successfully developed to solve PDEs with several advantages. One research
direction is to use the neural network directly to represent the solution of
PDE (E & Yu, 2018; Sirignano & Spiliopoulos, 2018). The physics-informed
neural network (PINN), introduced by Raissi et al. (2019), minimized the
residual of PDEs by using automatic differentiation instead of numerical
approximations.
There is another approach to solve PDEs, called operator learning. Operator
learning aims to learn a nonlinear mapping from an infinite-dimensional Banach
space to another. Many studies utilize the convolutional neural network to
parameterize the solution operator of PDEs in various applications (Guo et
al., 2016; Bhatnagar et al., 2019; Khoo et al., 2021; Zhu et al., 2019; Hwang
et al., 2021). The neural operator (Kovachki et al., 2021b) is proposed to
approximate the nonlinear operator inspired by Green’s function. Li et al.
(2021) extend the neural operator structure to the Fourier Neural Operator
(FNO) to approximate the integral operator effectively using the fast Fourier
transform (FFT). Kovachki et al. (2021a) proved the universality of FNO and
identified the size of the network.
The DeepONet (Lu et al., 2019; 2021) has also been proposed as another
framework for operator learning. The DeepONet has significantly been applied
to various problems, such as bubble growth dynamics (Lin et al., 2021),
hypersonic flows (Mao et al., 2021), and fluid flow (Cai et al., 2021).
Figure 1: Example of operator learning: the input function and the output
function for the solution operator of shallow water equation
Lanthaler et al. (2022) provided the universal approximation property of
DeepONet. Wang et al. (2021) proposed physics-informed DeepONet by adding a
residual of PDE as a loss function, and Ryck & Mishra (2022) demonstrated the
generic bounds on the approximation error for it. Prasthofer et al. (2022)
considered the case where the discretization grid of the input function in
DeepONet changes by employing the coordinate encoder. Lu et al. (2022)
compared the FNO with DeepONet in different benchmarks to demonstrate the
relative performance. FNO can only infer the output function of an operator as
the input function in the same grid as it needs to discretize the output
function to use Fast Fourier Transform(FFT). In contrast, the DeepONet can
predict from any location.
Ha et al. (2017) first proposed hypernetwork, a network that creates a weight
of the primary network. Because the hypernetwork can achieve weight sharing
and model compression, it requires a relatively small number of parameters
even as the dataset grows. Galanti & Wolf (2020) proved that a hypernetwork
provides higher expressivity with low-complexity target networks. Sitzmann et
al. (2020) and Klocek et al. (2019) employed this approach to restoring images
with insufficient pixel observations or resolutions. de Avila Belbute-Peres et
al. (2021) investigated the relationship between the coefficients of PDEs and
the corresponding solutions. They combined the hypernetwork with the PINN’s
residual loss. For time-dependent PDE, Pan et al. (2022) designated the time
$t$ as the input of the hypernetwork so that the target network indicates the
solution at $t$. von Oswald et al. (2020) devised a chunk embedding method
that partitions the parameters of the target network; this is because the
output dimension of the hypernetwork can be large.
## 3 Operator learning
### 3.1 Problem setting
The goal of operator learning is to learn a mapping from infinite-dimensional
function space to the others using a finite pair of functions. Let
$\mathcal{G}:\mathcal{U}\to\mathcal{S}$ be a nonlinear operator, where
$\mathcal{U}$ and $\mathcal{S}$ are compact subsets of infinite-dimensional
function spaces $\mathcal{U}\subset C(\mathcal{X};\mathbb{R}^{d_{u}})$ and
$\mathcal{S}\subset C(\mathcal{Y};\mathbb{R}^{d_{s}})$ with compact domains
$\mathcal{X}\subset\mathbb{R}^{d_{x}}$ and
$\mathcal{Y}\subset\mathbb{R}^{d_{y}}$. For simplicity, we focus on the case
$d_{u}=d_{s}=1$, and all the results could be extended to a more general case
for arbitrary $d_{u}$ and $d_{s}$. Suppose we have observations
$\\{u_{i},\mathcal{G}(u_{i})\\}_{i=1}^{N}$ where $u_{i}\in\mathcal{U}$ and
$\mathcal{G}(u_{i})\in\mathcal{S}$. We aim to find an approximation
$\mathcal{G}_{\theta}:\mathcal{U}\to\mathcal{S}$ with parameter $\theta$ using
the $N$ observations so that $\mathcal{G}_{\theta}\approx\mathcal{G}$. For
example, in a dam break scenario, it is an important to predict the fluid flow
over time according to a random initial height of the fluid.
Figure 2: Diagram for the three components for operator learning.
To this end, we want to find an operator $\mathcal{G}_{\theta}$, which takes
an initial fluid height as an input function and produces the fluid height
over time at any location as the output function (Figure 1).
As explained in Lanthaler et al. (2022), the approximator
$\mathcal{G}_{\theta}$ can be decomposed into the three components (Figure 2)
as
$\mathcal{G}_{\theta}:=\mathcal{R}\circ\mathcal{A}\circ\mathcal{E}.$ (1)
First, the encoder $\mathcal{E}$ takes an input function $u$ from
$\mathcal{U}$ to generate the finite-dimensional encoded data in
$\mathbb{R}^{m}$. Then, the approximator $\mathcal{A}$ is an operator
approximator from the encoded data in finite dimension space $\mathbb{R}^{m}$
to the other finite-dimensional space $\mathbb{R}^{p}$. Finally, the
reconstructor $\mathcal{R}$ reconstructs the output function
$s(y)=\mathcal{G}(u)(y)$ with $y\in\mathcal{Y}$ using the approximated data in
$\mathbb{R}^{p}$.
### 3.2 DeepONet and its limitation
DeepONet can be analyzed using the above three decompositions. Assume that all
the input functions $u$ are evaluated at fixed locations
$\\{x_{j}\\}_{j=1}^{m}\subset\mathcal{X}$; they are called ”sensor points.”
DeepONet uses an encoder as the pointwise projection
$\mathcal{E}(u)=(u(x_{1}),u(x_{2}),...,u(x_{m}))$ of the continuous function
$u$, the so-called ”sensor values” of the input function $u$. An intuitive
idea is to employ a neural network that simply concatenates these $m$ sensor
values and a query point $y$ as an input to approximate the target function
$\mathcal{G}(u)(y)$. DeepONet, in contrast, handles $m$ sensor values and a
query point $y$ separately into two subnetworks based on the universal
approximation theorem for the operator (Lu et al., 2021). See Appendix B for
more details. Lu et al. (2021) use the fully connected neural network for the
approximator $\mathcal{A}:\mathbb{R}^{m}\to\mathbb{R}^{p}$. They referred to
the composition of these two maps as branch net
$\beta:\mathcal{U}\to\mathbb{R}^{p},\beta(u):=\mathcal{A}\circ\mathcal{E}(u)$
(2)
for any $u\in\mathcal{U}$. The role of branch net can be interpreted as
learning the coefficient of the target function $\mathcal{G}(u)(y)$. They use
one additional neural network, called trunk net $\tau$ as shown below.
$\tau:\mathcal{Y}\to\mathbb{R}^{p+1},\tau(y):=\\{\tau_{k}(y)\\}_{k=0}^{p}$ (3)
for any $y\in\mathcal{Y}$. The role of trunk net can be interpreted as
learning an affine space $V$ that can efficiently approximate output function
space $C(\mathcal{Y};\mathbb{R}^{d_{s}})$. The functions $\tau_{1}(y)$,
…,$\tau_{p}(y)$ become the $p$-basis of vector space associated with $V$ and
$\tau_{0}(y)$ becomes a point of $V$. By using the trunk net $\tau$, the
$\tau$-induced reconstructor $\mathcal{R}$ is defined as
$\mathcal{R}_{\tau}:\mathbb{R}^{p}\to
C(\mathcal{Y};\mathbb{R}^{d_{s}}),\mathcal{R}_{\tau}(\beta)(y):=\tau_{0}(y)+\sum_{k=1}^{p}\beta_{k}\tau_{k}(y)$
(4)
where $\beta=(\beta_{1},\beta_{2},...,\beta_{p})\in\mathbb{R}^{p}$. In
DeepONet, $\tau_{0}(y)$ is restricted to be a constant $\tau_{0}\in\mathbb{R}$
that is contained in a reconstructor $\mathcal{R}$. The architecture of
DeepONet is described in Figure 4 (b).
Here, the $\tau$-induced reconstructor $\mathcal{R}_{\tau}$ is the linear
approximation of the output function space. Because the linear approximation
$\mathcal{R}$ cannot consider the elements in its orthogonal complement, a
priori limitation on the best error of DeepONet is explained in Lanthaler et
al. (2022) as
$\left(\int_{\mathcal{U}}\int_{\mathcal{Y}}|\mathcal{G}(u)(y)-\mathcal{R}_{\tau}\circ\mathcal{A}\circ\mathcal{E}(u)(y)|^{2}dyd\mu(u)\right)^{\frac{1}{2}}\geq\sqrt{\sum_{k>p}\lambda_{k}},$
(5)
where $\lambda_{1}\geq\lambda_{2}\geq...$ are the eigenvalues of the
covariance operator $\Gamma_{\mathcal{G}_{\\#\mu}}$ of the push-forward
measure $\mathcal{G}_{\\#\mu}$. Several studies point out that the slow decay
rate of the lower bound leads to inaccurate approximation operator learning
using DeepONet (Kovachki et al., 2021b; Hadorn, 2022; Lanthaler et al., 2022).
For example, the solution operator of the advection PDEs (Seidman et al.,
2022; Venturi & Casey, 2023) and of the Burgers’ equation (Hadorn, 2022) are
difficult to approximate when we are using the DeepONet with the small number
of basis $p$.
Figure 3: The perspective target network parametrization for operator
learning.
### 3.3 Variant models of DeepONet
Several variants of DeepONet have been developed to overcome its limitation.
All these models can be viewed from the perspective of parametrizing the
target function as a neural network. When we think of the target network that
receives $y$ as an input and generates an output $\mathcal{G}_{\theta}(u)(y)$,
the DeepONet and its variant model can be distinguished by how information
from the input function $u$ is injected into this target network
$\mathcal{G}_{\theta}(u)$, as described in Figure 3. From this perspective,
the trunk net in the DeepONet can be considered as the target network except
for the final output, as shown in Figure 4 (a). The output of the branch net
gives the weight between the last hidden layer and the final output.
Hadorn (2022) proposed Shift-DeepONet. The main idea is that a scale net and a
shift net are used to shift and scale the input query position $y$. Therefore,
it can be considered that the information of input function $u$ generates the
weights and bias between the input layer and the first hidden layer, as
explained in Figure 4 (b).
Venturi & Casey (2023) proposed FlexDeepONet, explained in Figure 4 (c). They
used the additional network, pre-net, to give the bias between the input layer
and the first hidden layer. Additionally, the output of the branch net also
admits the additional output $\tau_{0}$ to provide more information on input
function $u$ at the last inner product layer.
NOMAD is recently developed by Seidman et al. (2022) to overcome the
limitation of DeepONet. They devise a nonlinear output manifold using a neural
network that takes the output of branch net $\\{\beta_{i}\\}_{i=1}^{p}$ and
the query location $y$. As explained in Figure 4 (d), the target network
receives information about the function $u$ as an additional input, similar to
other conventional neural embedding methods (Park et al., 2019; Chen & Zhang,
2019; Mescheder et al., 2019).
Figure 4: DeepONet and its variant models for operator learning.
These methods provide information on the input function $u$ to only a part of
the target network. It is a natural idea to use a hypernetwork to share the
information of input function $u$ to all parameters of the target network. We
propose a general model HyperDeepONet (Figure 5), which contains the vanilla
DeepONet, FlexDeepONet, and Shift-DeepONet, as a special case of the
HyperDeepONet.
Figure 5: The proposed HyperDeepONet structure
## 4 Proposed model: HyperDeepONet
### 4.1 Architecture of HyperDeepONet
The HyperDeepONet structure is described in Figure 5. The encoder
$\mathcal{E}$ and the approximator $\mathcal{A}$ are used, similar to the
vanilla DeepONet. The proposed structure replaces the branch net with the
hypernetwork. The hypernetwork generate all parameters of the target network.
More precisely, we define the hypernetwork $h$ as
$h_{\theta}:\mathcal{U}\to\mathbb{R}^{p},\;h_{\theta}(u):=\mathcal{A}\circ\mathcal{E}(u)$
(6)
for any $u\in\mathcal{U}$. Then, $h(u)=\Theta\in\mathbb{R}^{p}$ is a network
parameter of the target network, which is used in reconstructor for the
HyperDeepONet. We define the reconstructor $\mathcal{R}$ as
$\mathcal{R}:\mathbb{R}^{p}\to
C(\mathcal{Y};\mathbb{R}^{d_{s}}),\;\mathcal{R}(\Theta)(y):=\text{NN}(y;\Theta)$
(7)
where $\Theta=[W,b]\in\mathbb{R}^{p}$, and NN denotes the target network. Two
fully connected neural networks are employed for the hypernetwork and target
network.
Therefore, the main idea is to use the hypernetwork, which takes an input
function $u$ and produces the weights of the target network. It can be thought
of as a weight generator for the target network. The hypernetwork determines
the all parameters of the target network containing the weights between the
final hidden layer and the output layer. It implies that the structure of
HyperDeepONet contains the entire structure of DeepONet. As shown in Figure 4
(b) and (c), Shift-DeepONet and FlexDeepONet can also be viewed as special
cases of the HyperDeepONet, where the output of the hypernetwork determines
the weights or biases of some layers of the target network. The outputs of the
hypernetwork determine the biases for the first hidden layer in the target
network for NOMAD in Figure 4 (d).
### 4.2 Comparison on complexity of DeepONet and HyperDeepONet
In this section, we would like to clarify the complexity of the DeepONet
required for the approximation $\mathcal{A}$ and reconstruction $\mathcal{R}$
based on the theory in Galanti & Wolf (2020). Furthermore, we will show that
the HyperDeepONet entails a relatively lower complexity than the DeepONet
using the results on the upper bound for the complexity of hypernetwork
(Galanti & Wolf, 2020).
#### 4.2.1 Notations and definitions
Suppose that the pointwise projection values (sensor values) of the input
function $u$ is given as $\mathcal{E}(u)=(u(x_{1}),u(x_{2}),...,u(x_{m}))$.
For simplicity, we consider the case $\mathcal{Y}=[-1,1]^{d_{y}}$ and
$\mathcal{E}(u)\in[-1,1]^{m}$. For the composition
$\mathcal{R}\circ\mathcal{A}:\mathbb{R}^{m}\rightarrow
C(\mathcal{Y};\mathbb{R})$, we focus on approximating the mapping
$\mathcal{O}:\mathbb{R}^{m+d_{y}}\rightarrow\mathbb{R}$, which is defined as
follows:
$\mathcal{O}(\mathcal{E}(u),y):=(\mathcal{R}\circ\mathcal{A}(\mathcal{E}(u)))(y),\quad\text{for
$y\in[-1,1]^{d_{y}},\mathcal{E}(u)\in[-1,1]^{m}$}.$ (8)
The supremum norm $\|h\|_{\infty}$ is defined as
$\max_{y\in\mathcal{Y}}\|h(y)\|$. Now, we introduce the Sobolev space
$\mathcal{W}_{r,n}$, which is a subset of $C^{r}([-1,1]^{n};\mathbb{R})$. For
$r,n\in\mathbb{N}$,
$\mathcal{W}_{r,n}:=\bigg{\\{}h:[-1,1]^{n}\rightarrow\mathbb{R}\quad\Big{|}\|h\|_{r}^{s}:=\|h\|_{\infty}+\sum_{1\leq|\mathbf{k}|\leq
r}\|D^{\mathbf{k}}h\|_{\infty}\leq 1\bigg{\\}},$
where $D^{\mathbf{k}}h$ denotes the partial derivative of $h$ with respect to
multi-index
$\mathbf{k}\in\left\\{\mathbb{N}\cup\left\\{0\right\\}\right\\}^{d_{y}}$. We
assume that the mapping $\mathcal{O}$ lies in the Sobolev space
$\mathcal{W}_{r,m+d_{y}}$.
For the nonlinear activation $\sigma$, the class of neural network
$\mathcal{F}$ represents the fully connected neural network with depth $k$ and
corresponding width $(h_{1}=n,h_{2},\cdots,h_{k+1})$, where
$W^{i}\in\mathbb{R}^{h_{i}}\times\mathbb{R}^{h_{i+1}}$ and
$b_{i}\in\mathbb{R}^{h_{i+1}}$ denote the weights and bias of the $i$-th layer
respectively.
$\displaystyle\mathcal{F}:=\left\\{f:[-1,1]^{n}\rightarrow\mathbb{R}|f(y;[\mathbf{W},\mathbf{b}])=W^{k}\cdot\sigma(W^{k-1}\cdots\sigma(W^{1}\cdot
y+b^{1})+b^{k-1})+b^{k}\right\\}$
Some activation functions facilitate an approximation for the Sobolev space
and curtail the complexity. We will refer to these functions as universal
activation functions. The formal definition can be found below, where the
distance between the class of neural network $\mathcal{F}$ and the Sobolev
space $\mathcal{W}_{r,n}$ is defined by
$d(\mathcal{F};\mathcal{W}_{r,n}):=\sup_{\psi\in\mathcal{W}_{r,n}}\inf_{f\in\mathcal{F}}\|f-\psi\|_{\infty}$.
Most well-known activation functions are universal activations that are
infinitely differentiable and non-polynomial in any interval (Mhaskar, 1996).
Furthermore, Hanin & Sellke (2017) state that the ReLU activation is also
universal.
###### Definition 1.
(Galanti & Wolf, 2020) (Universal activation). The activation function
$\sigma$ is called universal if there exists a class of neural network
$\mathcal{F}$ with activation function $\sigma$ such that the number of
parameters of $\mathcal{F}$ is $O(\epsilon^{-n/r})$ with
$d(\mathcal{F};\mathcal{W}_{r,n})\leq\epsilon$ for all $r,n\in\mathbb{N}$.
We now introduce the theorem, which offers a guideline on the neural network
architecture for operator learning. It suggests that if the entire
architecture can be replaced with a fully connected neural network, large
complexity should be required for approximating the target function. It also
verifies that the lower bound for a universal activation function is a sharp
bound on the number of parameters. First, we give an assumption to obtain the
theorem.
###### Assumption 1.
Suppose that $\mathcal{F}$ and $\mathcal{W}_{r,n}$ represent the class of
neural network and the target function space to approximate, respectively. Let
$\mathcal{F}^{\prime}$ be a neural network class representing a structure in
which one neuron is added rather than $\mathcal{F}$. Then, the followings
holds for all $\psi\in\mathcal{W}_{r,n}$ not contained in $\mathcal{F}$.
$\inf_{f\in\mathcal{F}}\|f-\psi\|_{\infty}>\inf_{f\in\mathcal{F}^{\prime}}\|f-\psi\|_{\infty}.$
For $r=0$, Galanti & Wolf (2020) remark that the assumption is valid for
2-layered neural networks with respect to the $L^{2}$ norm when an activation
function $\sigma$ is either a hyperbolic tangent or sigmoid function. With
Assumption 1, the following theorem holds, which is a fundamental approach to
identifying the complexity of DeepONet and its variants. Note that a real-
valued function $g\in L^{1}(\mathbb{R})$ is called a bounded variation if its
total variation $\sup_{\phi\in C_{c}^{1}(\mathbb{R}),\|\phi\|_{\infty}\leq
1}\int_{\mathbb{R}}g(x)\phi^{\prime}(x)dx$ is finite.
###### Theorem 1.
(Galanti & Wolf, 2020). Suppose that $\mathcal{F}$ is a class of neural
networks with a piecewise $C^{1}(\mathbb{R})$ activation function
$\sigma:\mathbb{R}\rightarrow\mathbb{R}$ of which derivative $\sigma^{\prime}$
is bounded variation. If any non-constant $\psi\in\mathcal{W}_{r,n}$ does not
belong to $\mathcal{F}$, then $d(\mathcal{F};W_{r,n})\leq\epsilon$ implies the
number of parameters in $\mathcal{F}$ should be $\Omega(\epsilon^{-n/r})$.
#### 4.2.2 Lower bound for the complexity of the DeepONet
Now, we provide the minimum number of parameters in DeepONet. The following
theorem presents a criterion on the DeepONet’s complexity to get the desired
error. It states that the number of required parameters increases when the
target functions are irregular, corresponding to a small $r$.
$\mathcal{F}_{\text{DeepONet}}(\mathcal{B},\mathcal{T})$ denotes the class of
function in DeepONet, induced by the class of branch net $\mathcal{B}$ and the
class of trunk net $\mathcal{T}$.
###### Theorem 2.
(Complexity of DeepONet) Let $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ be a
universal activation function in $C^{r}(\mathbb{R})$ such that $\sigma$ and
$\sigma^{\prime}$ are bounded. Suppose that the class of branch net
$\mathcal{B}$ has a bounded Sobolev norm (i.e., $\|\beta\|_{r}^{s}\leq
l_{1},\forall\beta\in\mathcal{B}$). Suppose any non-constant
$\psi\in\mathcal{W}_{r,n}$ does not belong to any class of neural network. In
that case, the number of parameters in the class of trunk net $\mathcal{T}$ is
$\Omega(\epsilon^{-d_{y}/r})$ when
$d(\mathcal{F}_{\text{DeepONet}}(\mathcal{B},\mathcal{T});\mathcal{W}_{r,d_{y}+m})\leq\epsilon$.
The core of this proof is showing that the inner product between the branch
net and the trunk net could be replaced with a neural network that has a low
complexity (Lemma 1). Therefore, the entire structure of DeepONet could be
replaced with a neural network that receives
$[\mathcal{E}(u),y]\in\mathbb{R}^{d_{y}+m}$ as input. It gives the lower bound
for the number of parameters in DeepONet based on Theorem 1. The proof can be
found in Appendix C.1.
The analogous results holds for variant models of DeepONet. Models such as
Shift-DeepONet and flexDeepONet could achieve the desired accuracy with a
small number of basis. Still, there was a trade-off in which the first hidden
layer of the target network required numerous units. There was no restriction
on the dimension of the last hidden layer in the target network for NOMAD,
which uses a fully nonlinear reconstruction. However, the first hidden layer
of the target network also had to be wide enough, increasing the number of
parameters. Details can be found in Appendix C.2.
For the proposed HyperDeepONet, the sensor values $\mathcal{E}(u)$ determine
the weight and bias of all other layers as well as the weight of the last
layer of the target network. Due to the nonlinear activation functions between
linear matrix multiplication, it is difficult to replace HyperDeepONet with a
single neural network that receives
$[\mathcal{E}(u),y]\in\mathbb{R}^{d_{y}+m}$ as input. Galanti & Wolf (2020)
state that there exists a hypernetwork structure (HyperDeepONet) such that the
number of parameters in the target network is $O(\epsilon^{-d_{y}/r})$. It
implies that the HyperDeepONet reduces the complexity compared to all the
variants of DeepONet.
## 5 Experiments
In this section, we verify the effectiveness of the proposed model
HyperDeepONet to learn the operators with a complicated target function space.
To be more specific, we focus on operator learning problems in which the space
of output function space is complicated. Each input function $u_{i}$ generates
multiple triplet data points $(u_{i},y,\mathcal{G}(u)(y))$ for different
values of $y$. Except for the shallow water problem, which uses 100 training
function pairs and 20 test pairs, we use 1,000 training input-output function
pairs and 200 test pairs for all experiments.
For the toy example, we first consider the identity operator
$\mathcal{G}:u_{i}\mapsto u_{i}$. The Chebyshev polynomial is used as the
input (=output) for the identity operator problem. The Chebyshev polynomials
of the first kind $T_{l}$ of degree 20 can be written as
$u_{i}\in\\{\sum_{l=0}^{19}c_{l}T_{l}(x)|c_{l}\in[-1/4,1/4]\\}$ with random
sampling $c_{l}$ from uniform distribution $U[-1/4,1/4]$. The differentiation
operator $\mathcal{G}:u_{i}\mapsto\frac{d}{dx}u_{i}$ is considered for the
second problem. Previous works handled the anti-derivative operator, which
makes the output function smoother by averaging (Lu et al., 2019; 2022). Here,
we choose the differentiation operator instead of the anti-derivative operator
to focus on operator learning when the operator’s output function space is
complicated. We first sample the output function $\mathcal{G}(u)$ from the
above Chebyshev polynomial of degree 20. The input function is generated using
the numerical method that integrates the output function.
Finally, the solution operators of PDEs are considered. We deal with two
problems with the complex target function in previous works (Lu et al., 2022;
Hadorn, 2022). The solution operator of the advection equation is considered a
mapping from the rectangle shape initial input function to the solution
$w(t,x)$ at $t=0.5$, i.e., $\mathcal{G}:w(0,x)\mapsto w(0.5,x)$. We also
consider the solution operator of Burgers’ equation which maps the random
initial condition to the solution $w(t,x)$ at $t=1$, i.e.,
$\mathcal{G}:w(0,x)\mapsto w(1,x)$. The solution of the Burgers’ equation has
a discontinuity in a short time, although the initial input function is
smooth. For a challenging benchmark, we consider the solution operator of the
shallow water equation that aims to predict the fluid height
$h(t,x_{1},x_{2})$ from the initial condition $h(0,x_{1},x_{2})$, i.e.,
$\mathcal{G}:h(0,x_{1},x_{2})\mapsto h(t,x_{1},x_{2})$ (Figure 1). In this
case, the input of the target network is three dimension ($t,x_{1},x_{2}$),
which makes the solution operator complex. Detail explanation is provided in
Appendix E.
Model | DeepONet | Shift | Flex | NOMAD | Hyper(ours)
---|---|---|---|---|---
Identity | 0.578$\pm$0.003 | 0.777$\pm$0.018 | 0.678$\pm$0.062 | 0.578$\pm$0.020 | 0.036$\pm$0.005
Differentiation | 0.559$\pm$0.001 | 0.624$\pm$0.015 | 0.562$\pm$0.016 | 0.558$\pm$0.003 | 0.127$\pm$0.043
Table 1: The mean relative $L^{2}$ test error with standard deviation for the
identity operator and the differentiation operator. The DeepONet, its
variants, and the HyperDeepONet use the target network $d_{y}$-20-20-10-1 with
tanh activation function. Five training trials are performed independently.
Figure 6: One test data example of differentiation operator problem.
Expressivity of target network. We have compared the expressivity of the small
target network using different models. We focus on the identity and
differentiation operators in this experiment. All models employ the small
target network $d_{y}$-20-20-10-1 with the hyperbolic tangent activation
function. The branch net and the additional networks (scale net, shift net,
pre-net, and hypernetwork) also use the same network size as the target
network for all five models.
| Branch net (Hypernetwork) | Target | $\\#$Param | Rel error
---|---|---|---|---
Advection | DeepONet | $m$-256-256 | $d_{y}$-256-256-256-256-1 | 274K | 0.0046$\pm$0.0017
Hyper(ours) | $m$-70-70-70-70-70-$N_{\theta}$ | $d_{y}$-33-33-33-33-1 | 268K | 0.0048$\pm$0.0009
c-Hyper(ours) | $m$-128-128-128-128-128-$1024$ | $d_{y}$-256-256-256-256-1 | 208K | 0.0043$\pm$0.0004
Burgers | DeepONet | $m$-128-128-128-128 | $d_{y}$-128-128-128-128-1 | 115K | 0.0391$\pm$0.0040
Hyper(ours) | $m$-66-66-66-66-66-$N_{\theta}$ | $d_{y}$-20-20-20-20-1 | 114K | 0.0196$\pm$0.0044
c-Hyper(ours) | $m$-66-66-66-66-66-$512$ | $d_{y}$-128-128-128-128-1 | 115K | 0.0066$\pm$0.0009
Shallow | DeepONet | $m$-100-100-100-100 | $d_{y}$-100-100-100-100-1 | 107K | 0.0279 $\pm$ 0.0042
Hyper(ours) | $m$-30-30-30-30-$N_{\theta}$ | $d_{y}$-30-30-30-30-1 | 101K | 0.0148 $\pm$ 0.0002
Shallow w/ small param | DeepONet | $m$-20-20-10 | $d_{y}$-20-20-10-1 | 6.5K | 0.0391 $\pm$ 0.0066
Hyper(ours) | $m$ -10-10-10-$N_{\theta}$ | $d_{y}$-10-10-10-1 | 5.7K | 0.0209 $\pm$ 0.0013
Table 2: The mean relative $L^{2}$ test error with standard deviation for
solution operator learning problems. $N_{\theta}$ and $\\#$Param denote the
number of parameters in the target network and the number of learnable
parameters, respectively. Five training trials are performed independently.
Table 1 shows that the DeepONet and its variant models have high errors in
learning complex operators when the small target network is used. In contrast,
the HyperDeepONet has lower errors than the other models. This is consistent
with the theorem in the previous section that HyperDeepONet can achieve
improved approximations than the DeepONet when the complexity of the target
network is the same. Figure 6 shows a prediction on the differentiation
operator, which has a highly complex target function. The same trends are
observed when the activation function or the number of sensor points changes
(Table 5) and the number of layers in the branch net and the hypernetwork vary
(Figure 11).
Same number of learnable parameters. The previous experiments compare the
models using the same target network structure. In this section, the
comparison between the DeepONet and the HyperDeepONet is considered when using
the same number of learnable parameters. We focus on the solution operators of
the PDEs.
For the three solution operator learning problems, we use the same
hyperparameters proposed in Lu et al. (2022) and Seidman et al. (2022) for
DeepONet. First, we use the smaller target network with the larger
hypernetwork for the HyperDeepONet to compare the DeepONet. Note that the
vanilla DeepONet is used without the output normalization or the boundary
condition enforcing techniques explained in Lu et al. (2022) to focus on the
primary limitation of the DeepONet. More Details are in Appendix E. Table 2
shows that the HyperDeepONet achieves a similar or better performance than the
DeepONet when the two models use the same number of learnable parameters. The
HyperDeepONet has a slightly higher error for advection equation problem, but
this error is close to perfect operator prediction. It shows that the
complexity of target network and the number of learnable parameters can be
reduced to obtain the desired accuracy using the HyperDeepONet. The fourth row
of Table 2 shows that HyperDeepONet is much more effective than DeepONet in
approximating the solution operator of the shallow water equation when the
number of parameters is limited. Figure 7 and Figure 12 show that the
HyperDeepONet learns the complex target functions in fewer epochs for the
desired accuracy than the DeepONet although the HyperDeepONet requires more
time to train for one epoch (Table 8).
Figure 7: One test data example of prediction on the advection equation (First
row) and Burgers’ equation (Second row) using the DeepONet and the
HyperDeepONet.
Scalability. When the size of the target network for the HyperDeepONet is
large, the output of the hypernetwork would be high-dimensional (Ha et al.,
2017; Pawlowski et al., 2017) so that its complexity increases. In this case,
the chunked HyperDeepONet (c-HyperDeepONet) can be used with a trade-off
between accuracy and memory based on the chunk embedding method developed by
von Oswald et al. (2020). It generates the subset of target network parameters
multiple times iteratively reusing the smaller chunked hypernetwork. The
c-HyperDeepONet shows a better accuracy than the DeepONet and the
HyperDeepONet using an almost similar number of parameters, as shown in Table
2. However, it takes almost 2x training time and 2$\sim$30x memory usage than
the HyperDeepOnet. More details on the chunked hypernetwork are in Appendix D.
## 6 Conclusion and discussion
In this work, the HyperDeepONet is developed to overcome the expressivity
limitation of DeepONet. The method of incorporating an additional network and
a nonlinear reconstructor could not thoroughly solve this limitation. The
hypernetwork, which involves multiple weights simultaneously, had a desired
complexity-reducing structure based on theory and experiments.
We only focused on when the hypernetwork and the target network is fully
connected neural networks. In the future, the structure of the two networks
can be replaced with a CNN or ResNet, as the structure of the branch net and
trunk net of DeepONet can be changed to another network (Lu et al., 2022).
Additionally, it seems interesting to research a simplified modulation network
proposed by Mehta et al. (2021), which still has the same expressivity as
HyperDeepONet.
Some techniques from implicit neural representation can improve the
expressivity of the target network (Sitzmann et al., 2020). Using a sine
function as an activation function with preprocessing will promote the
expressivity of the target network. We also leave the research on the class of
activation functions satisfying the assumption except for hyperbolic tangent
or sigmoid functions as a future work.
#### Acknowledgments
J. Y. Lee was supported by a KIAS Individual Grant (AP086901) via the Center
for AI and Natural Sciences at Korea Institute for Advanced Study and by the
Center for Advanced Computation at Korea Institute for Advanced Study. H. J.
Hwang and S. W. Cho were supported by the National Research Foundation of
Korea (NRF) grant funded by the Korea government (MSIT) (RS-2022-00165268) and
by Institute for Information & Communications Technology Promotion (IITP)
grant funded by the Korea government(MSIP) (No.2019-0-01906, Artificial
Intelligence Graduate School Program (POSTECH)).
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## Appendix A Notations
We list the main notations in Table 3 which is not concretely described in
this paper.
$\mathcal{U}$ | Domain of input function
---|---
$\mathcal{Y}$ | Domain of output function
$d_{u}$ | Dimension of $\mathcal{U}$
$d_{y}$ | Dimension of $\mathcal{Y}$
$d_{x}$ | Dimension of the codomain of input function
$d_{s}$ | Dimension of the codomain of output function
$\left\\{x_{1},\cdots,x_{m}\right\\}$ | Sensor points
$m$ | Number of sensor points
$\mathbb{R}^{d}$ | Euclidean space of dimension $d$
$C^{r}(\mathbb{R})$ | Set of functions that has continuous $r$-th derivative.
$C(\mathcal{Y};\mathbb{R}^{d_{s}})$ | Set of continuous function from $\mathcal{Y}$ to $\mathbb{R}^{d_{s}}$
$C^{r}([-1,1]^{n};\mathbb{R})$ | Set of functions from $[-1,1]^{n}$ to $\mathbb{R}$ whose $r$-th partial derivatives are continuous.
$n=O(\epsilon)$ | There exists a constant $C$ such that $n\leq C\cdot\epsilon,\forall\epsilon>0$
$n=\Omega(\epsilon)$ | There exists a constant $C$ such that $n\geq C\cdot\epsilon,\forall\epsilon>0$
$n=o(\epsilon)$ | $n/\epsilon$ converges to 0 as $\epsilon$ approaches to 0.
Table 3: Notation
## Appendix B Original structure of DeepONet
Figure 8: The structure of DeepONet
For a clear understanding of previous works, we briefly leave a description of
DeepONet. In particular, we explain the structure of the unstacked DeepONet in
Lu et al. (2019) which is being widely used in various experiments of the
papers. Note that Figure 4(a) represents the corresponding model which is
called simply DeepONet throughout this paper. The overall architecture of the
model is formulated as
$\mathcal{R}_{\tau}\circ\mathcal{A}_{\beta}(u(x_{1}),\cdots,u(x_{m}))(y):=\langle\beta(u(x_{1}),\cdots,u(x_{m});\theta_{\beta}),\tau(y;\theta_{\tau})\rangle,$
where $\tau$ and $\beta$ are referred to as the trunk net and the branch net
respectively. Note that $R_{\tau}$ and $A_{\beta}$ denote the reconstructor
and the operator in Section 3.1. For $m$ fixed observation points
$(x_{1},\cdots,x_{m})\in\mathcal{X}^{m}$, the unstacked DeepONet consists of
an inner product of branch Net and trunk Net, which are fully connected neural
networks. For a function $u$, the branch net receives pointwise projection
values $(u(x_{1}),\cdots,u(x_{m}))$ as inputs to detect which function needs
to be transformed. The trunk net queries a location $y\in\mathcal{Y}$ of
interest where $\mathcal{Y}$ denotes the domain of output functions.
It was revealed that the stacked DeepONet, the simplified version of the
unstacked DeepONet, is a universal approximator in the set of continuous
functions. Therefore, the general structure also becomes a universal
approximator which enables close approximation by using a sufficient number of
parameters. Motivated by the property, we focus on how large complexity should
be required for DeepONet and its variants to achieve the desired error.
## Appendix C On complexity of DeepONet and its variants
### C.1 Proof of Theorem 2 on DeepONet complexity
The following lemma implies that the class of neural networks is sufficiently
efficient to approximate the inner product.
###### Lemma 1.
For the number of basis $p\in\mathbb{N}$, consider the inner product function
$\pi_{p}:[-1,1]^{2p}\rightarrow\mathbb{R}$ defined by
$\pi_{p}(a_{1},\cdots,a_{p},b_{1},\cdots,b_{p}):=\sum_{i=1}^{p}a_{i}b_{i}=\langle(a_{1},\cdots,a_{p}),(b_{1},\cdots,b_{p})\rangle.$
For an arbitrary positive $t$, there exists a class of neural network
$\mathcal{F}$ with universal activation
$\sigma:\mathbb{R}\rightarrow\mathbb{R}$ such that the number of parameters of
$\mathcal{F}$ is $O(p^{1+1/t}\epsilon^{-1/t})$ with
$\inf_{f\in\mathcal{F}}\|f-\pi_{p}\|_{\infty}\leq\epsilon$.
###### Proof.
Suppose that $t$ is a positive integer and the Sobolev space $W_{2t,2}$ is
well defined. First, we would like to approximate the product function
$\pi_{1}:[-1,1]^{2}\rightarrow\mathbb{R}$ which is defined as
$\pi_{1}(a,b)=ab.$
Note that partial derivatives $D^{\mathbf{k}}\pi_{1}=0$ for all multi-index
$\mathbf{k}\in\left\\{\mathbb{N}\bigcup\left\\{0\right\\}\right\\}^{2}$ such
that $|\mathbf{k}|\geq 2$. For a multi-index $\mathbf{k}$ with
$|\mathbf{k}|=1$, $D^{\mathbf{k}}\pi_{1}$ contains only one term which is
either $a$ or $b$. In this case, we can simply observe that
$\sum_{|\mathbf{k}|=1}\|D^{\mathbf{k}}\pi_{1}\|_{\infty}\leq 2\cdot 1=2$ by
the construction of the domain $[-1,1]^{2}$ for $\pi_{1}$. And finally,
$\displaystyle\|\pi_{1}\|_{\infty}\leq\|ab\|_{\infty}\leq\|a\|_{\infty}\|b\|_{\infty}\leq
1\cdot 1=1,$
so that a function $\pi_{1}/3$ should be contained in $\mathcal{W}_{r,2}$ for
any $r\in\mathbb{N}$. In particular, $\pi_{1}/3$ lies in $\mathcal{W}_{2t,2}$
so that there exists a neural network approximation $f_{nn}$ in some class of
neural network $\mathcal{F}^{*}$ with an universal activation function
$\sigma$ such that the number of parameters of $\mathcal{F}^{*}$ is
$O((\epsilon/3p)^{-2/2t})=O(p^{1/t}\epsilon^{-1/t})$, and
$\displaystyle\|\pi_{1}/3-f_{nn}\|_{\infty}\leq\epsilon/3p,$
by Definition 1. Then the neural network $3f_{nn}$ approximates the function
$\pi_{1}$ by an error $\epsilon/p$ which can be constructed by adjusting the
last weight values directly involved in the output layer of neural network
$f_{nn}$.
Finally, we construct a neural network approximation for the inner product
function $\pi_{p}$. Decompose the $2p-$dimensional inner product function
$\pi_{p}$ into $p$ product functions
$\left\\{\text{Proj}_{i}(\pi_{p})\right\\}_{i=1}^{p}$ which are defined as
$\displaystyle\text{Proj}_{i}(\pi_{p}):\mathbb{R}^{2p}\rightarrow\mathbb{R},\quad\text{Proj}_{i}(\pi_{p})(a_{1},\cdots,a_{p},b_{1},\cdots,b_{p}):=\pi_{1}(a_{i},b_{i})=a_{i}b_{i},$
for $\forall i\in\left\\{1,\cdots,p\right\\}$. Then each function
$\text{Proj}_{i}(\pi_{p})$ could be approximated within an error $\epsilon/p$
by neural network $NN_{i}$ which has $O(p^{1/t}\epsilon^{-1/t})$ parameters by
the above discussion. Finally, by adding the last weight
$[1,1,\cdots,1]\in\mathbb{R}^{1\times p}$ which has input as the outputs of
$p$ neural networks $\left\\{NN_{i}\right\\}_{i=1}^{p}$, we can construct the
neural network approximation $NN$ of
$\pi_{p}=\sum_{i=1}^{p}\text{Proj}_{i}(\pi_{p})$ such that the number of
parameters is $O(1+p+p\cdot
p^{1/t}\epsilon^{-1/t})=O(p^{1+1/t}\epsilon^{-1/t})$. Class of neural network
$\mathcal{F}$, which represents the structure of $NN$, satisfies the desired
property.
Obviously, the statement holds for an arbitrary real $t$ which is not an
integer.
∎
Now we assume that $\mathcal{O}$ (defined in Eq. (8)) lies in the Sobolev
space $\mathcal{W}_{r,d_{y}+m}$. Then, we can obtain the following lemma which
presents the lower bound on the number of basis $p$ in DeepONet structure.
Note that we apply $L_{\infty}$-norm for the outputs of branch net and trunk
net which are multi-dimensional vectors.
###### Lemma 2.
Let $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ be a universal activation
function in $C^{r}(\mathbb{R})$ such that $\sigma^{\prime}$ is a bounded
variation. Suppose that the class of branch net $\mathcal{B}$ has a bounded
Sobolev norm (i.e., $\|\beta\|_{r}^{s}\leq l_{1},\forall\beta\in\mathcal{B}$).
Assume that supremum norm $\|\cdot\|_{\infty}$ of the class of trunk net
$\mathcal{T}$ is bounded by $l_{2}$ and the number of parameters in
$\mathcal{T}$ is $o(\epsilon^{-(d_{y}+m)/r})$. If any non-constant
$\psi\in\mathcal{W}_{r,d_{y}+m}$ does not belong to any class of neural
network, then the number of basis $p$ in $\mathcal{T}$ is
$\Omega(\epsilon^{-d_{y}/r})$ when
$d(\mathcal{F}_{\text{DeepONet}}(\mathcal{B},\mathcal{T});\mathcal{W}_{r,d_{y}+m})\leq\epsilon$.
###### Proof.
To prove the above lemma by contradiction, we assume the opposite of the
conclusion. Suppose that there is no constant $C$ that satisfies the
inequality $p\geq C(\epsilon^{-d_{y}/r})$ for $\forall\epsilon>0$. In other
words, there exists a sequence of DeepONet which has
$\left\\{p_{n}\right\\}_{n=1}^{\infty}$ as the number of basis with a sequence
of error $\left\\{\epsilon_{n}\right\\}_{n=1}^{\infty}$ in $\mathbb{R}$ such
that $\epsilon_{n}\rightarrow 0$, and satisfies
$\displaystyle
p_{n}\leq\frac{1}{n}\epsilon_{n}^{-d_{y}/r}\quad\left(\text{i.e.,}\quad
p_{n}=o(\epsilon_{n}^{-d_{y}/r})\quad\text{with respect to }n\right),$ (9)
and
$\displaystyle
d(\mathcal{F}_{\text{DeepONet}}(\mathcal{B}_{n},\mathcal{T}_{n});\mathcal{W}_{r,d_{y}+m})\leq\epsilon_{n},$
where $\mathcal{B}_{n}$ and $\mathcal{T}_{n}$ denote the corresponding class
sequence of branch net and trunk net respectively. Then, we can choose the
sequence of branch net
$\left\\{\beta_{n}:\mathbb{R}^{m}\rightarrow\mathbb{R}^{p_{n}}\right\\}_{n=1}^{\infty}$
and trunk net
$\left\\{\tau_{n}:\mathbb{R}^{d_{y}}\rightarrow\mathbb{R}^{p_{n}}\right\\}_{n=1}^{\infty}$
satisfying
$\displaystyle\|\mathcal{O}(\mathcal{E}(u),y)-\pi_{p_{n}}(\beta_{n}(\mathcal{E}(u)),\tau_{n}(y))\|_{\infty}\leq
2\epsilon_{n},\forall[\mathcal{E}(u),y]\in[-1,1]^{d_{y}+m},\mathcal{O}\in\mathcal{W}_{r,d_{y}+m}$
for the above sequence of DeepONet by the definition of
$d(\mathcal{F}_{\text{DeepONet}}(\mathcal{B}_{n},\mathcal{T}_{n});\mathcal{W}_{r,d_{y}+m})$.
Now, we would like to construct neural network approximations $f_{n}$ for the
branch net $\beta_{n}$. By the assumption on the boundedness of $\mathcal{B}$,
the $i$-th component $[\beta_{n}]_{i}$ of $\beta_{n}$ has a Sobolev norm
bounded by $l_{1}$. In other words, $\|[\beta_{n}]_{i}/l_{1}\|_{r}^{s}\leq 1$
and therefore, $[\beta_{n}]_{i}/l_{1}$ is contained in $W_{1,m}$. Since
$\sigma$ is a universal activation function, we can choose a neural network
approximation $[f_{n}]_{i}$ of $[\beta_{n}]_{i}$ such that the number of
parameters is $O((\epsilon_{n}/l_{1})^{-m/r})=O(\epsilon_{n}^{-m/r})$, and
$\displaystyle\|[f_{n}]_{i}-[\beta_{n}]_{i}\|_{\infty}\leq\epsilon_{n}/l_{1}.$
Then, $f_{n}=(l_{1}[f_{n}]_{1},l_{1}[f_{n}]_{2},\cdots,l_{1}[f_{n}]_{p_{n}})$
becomes neural network approximation of $\beta_{n}$ which has
$O(p_{n}\epsilon_{n}^{-m/r})$ parameters within an error $\epsilon_{n}$.
Recall the target function corresponding to $m$ observation
$\mathcal{E}(u)\in[-1,1]^{m}$ by
$\mathcal{O}(\mathcal{E}(u),\cdot):\mathbb{R}^{d_{y}}\rightarrow\mathbb{R}$
which is defined in Eq. (8). Then, for $\forall\mathcal{E}(u)$, we can observe
the following inequalities:
$\|\mathcal{O}(\mathcal{E}(u),y)-\pi_{p_{n}}(f_{n}(\mathcal{E}(u)),\tau_{n}(y))\|_{\infty}\\\
\leq\|\mathcal{O}(\mathcal{E}(u),y)-\pi_{p_{n}}\left(\beta_{n}\left(\mathcal{E}(u)\right),\tau_{n}(y)\right)\|_{\infty}+\|\pi_{p_{n}}\left(\beta_{n}\left(\mathcal{E}\left(u\right)\right)-f_{n}(\mathcal{E}(u)\right),\tau_{n}(y))\|_{\infty}\\\
\leq
2\epsilon_{n}+\epsilon_{n}\|\tau_{n}(y)\|_{\infty}\leq\epsilon_{n}(2+l_{2}),$
(10)
by the assumption on the boundedness of $\mathcal{T}$. Now we would like to
consider the sequence of neural network which is an approximation of inner
product between $p_{n}-$dimensional vector in $[-1,1]^{p_{n}}$. Note the
following inequality
$\displaystyle\|f_{n}(\mathcal{E}(u))\|_{\infty}$
$\displaystyle\leq\|f_{n}(\mathcal{E}(u))-\beta_{n}(\mathcal{E}(u))\|_{\infty}+\|\beta_{n}(\mathcal{E}(u))\|_{\infty}$
$\displaystyle\leq\epsilon_{n}+\|\beta_{n}(\mathcal{E}(u))\|_{r}^{s}$
$\displaystyle\leq\epsilon_{n}+l_{1}\leq 2l_{1},$
with $\|\tau_{n}\|_{\infty}\leq l_{2}$ for large $n$. It implies that
$f_{n}(\mathcal{E}(u))/2l_{1}$ and $\tau_{n}(x)/l_{2}$ lie in
$[-1,1]^{p_{n}}$. By Lemma 1, there exists a class of neural network
$\mathcal{H}_{n}$ such that the number of parameters is
$O(p_{n}^{1+1/2d_{y}r}\epsilon_{n}^{-1/2d_{y}r})$ and,
$\displaystyle\inf_{h\in\mathcal{H}_{n}}\|h-\pi_{p_{n}}\|_{\infty}\leq\epsilon_{n}$
where $\pi_{p_{n}}$ is the inner product corresponding to $p_{n}-$dimensional
vector. Choose a neural network $h_{n}:[-1,1]^{2p_{n}}\rightarrow\mathbb{R}$
such that $\|h_{n}-\pi_{p_{n}}\|_{\infty}\leq 2\epsilon_{n}$. Then, by the
triangular inequality,
$\|\mathcal{O}(\mathcal{E}(u),y)-2l_{1}l_{2}h_{n}(f_{n}(\mathcal{E}(u))/2l_{1},\tau_{n}(y)/l_{2})\|_{\infty}\\\
\leq\|\mathcal{O}(\mathcal{E}(u),y)-\pi_{p_{n}}(f_{n}(\mathcal{E}(u)),\tau_{n}(y))\|_{\infty}\\\
+2l_{1}l_{2}\|\pi_{p_{n}}(f_{n}(\mathcal{E}(u))/2l_{1},\tau_{n}(y)/l_{2})-h_{n}(f_{n}(\mathcal{E}(u))/2l_{1},\tau_{n}(y)/l_{2})\|_{\infty}\\\
\leq\epsilon_{n}(2+l_{2})+2l_{1}l_{2}(2\epsilon_{n})=\epsilon_{n}(2+l_{2}+4l_{1}l_{2}).$
(11)
Finally, we compute the number of parameters which is required to implement
the function
$2l_{1}l_{2}h_{n}(f_{n}(\mathcal{E}(u))/2l_{1},\tau_{n}(y)/l_{2})$. The only
part that needs further consideration is scalar multiplication. Since we need
one weight to multiply a constant with one real value, three scalar
multiplications
$\displaystyle h_{n}(f_{n}(\mathcal{E}(u))/2l_{1},\tau_{n}(y)/l_{2})\mapsto
2l_{1}l_{2}h_{n}(f_{n}(\mathcal{E}(u))/2l_{1},\tau_{n}(y)/l_{2}),$
$\displaystyle f_{n}(\mathcal{E}(u))\mapsto
f_{n}(\mathcal{E}(u))/2l_{1},\quad\text{and}\quad\tau_{n}(x)\mapsto\tau_{n}(y)/l_{2},$
require $1,p_{n},p_{n}$-parameters respectively. Combining all the previous
results with the size of trunk net, the total number of parameters is obtained
in the form of
$\displaystyle
O(1+2p_{n}+p_{n}^{1+1/2d_{y}r}\epsilon_{n}^{-1/2d_{y}r}+p_{n}\epsilon_{n}^{-m/r})+o(\epsilon_{n}^{-(d_{y}+m)/r})=o(\epsilon_{n}^{-(d_{y}+m)/r}),$
since the initial assumption (9) on the number of basis gives the following
inequality.
$\displaystyle
p_{n}^{1+1/2d_{y}r}\epsilon_{n}^{-1/2d_{y}r}+p_{n}\epsilon_{n}^{-m/r}$
$\displaystyle\leq
p_{n}(p_{n}^{1/2d_{y}r}\epsilon_{n}^{-1/2d_{y}r}+\epsilon_{n}^{-m/r})$
$\displaystyle\leq\frac{1}{n}\epsilon_{n}^{-d_{y}/r}(\epsilon_{n}^{-1/2r^{2}-1/2d_{y}r}+\epsilon_{n}^{-m/r})$
$\displaystyle\leq\frac{1}{n}\epsilon_{n}^{-d_{y}/r}2\epsilon_{n}^{-m/r}=\frac{2}{n}\epsilon_{n}^{-(d_{y}+m)/r}.$
On the one hand, the sequence of function
$\left\\{2l_{1}l_{2}h_{n}(f_{n}(\mathcal{E}(u))/2l_{1},\tau_{n}(y)/l_{2})\right\\}_{n=1}^{\infty}$
is an sequence of approximation for $\mathcal{O}(\mathcal{E}(u),y)$ within a
corresponding sequence of error
$\left\\{\epsilon_{n}(2+l_{2}+4l_{1}l_{2})\right\\}_{n=1}^{\infty}$. Denote
the sequence of the class of neural networks corresponding to the sequence of
the function $\left\\{2l_{1}l_{2}h_{n}(f_{n}(\mathcal{E}(u))/2l_{1},\\\
\tau_{n}(y)/l_{2})\right\\}_{n=1}^{\infty}$ by
$\left\\{\mathcal{F}_{n}\right\\}_{n=1}^{\infty}$. By the assumption, Theorem
1 implies the number of parameters in
$\left\\{\mathcal{F}_{n}\right\\}_{n=1}^{\infty}$ is
$\Omega((\epsilon_{n}(2+l_{2}+4l_{1}l_{2}))^{-(d_{y}+m)/r})=\Omega(\epsilon_{n}^{-(d_{y}+m)/r})$.
Therefore, the initial assumption (9) would result in a contradiction so the
desired property is valid. ∎
Note that the assumption on the boundedness of the trunk net could be valid if
we use the bounded universal activation function $\sigma$. Using the above
results, we can prove our main theorem, Theorem 2.
###### Proof of Theorem 2.
Denote the number of parameters in $\mathcal{T}$ by $N_{\mathcal{T}}$. Suppose
that there is no constant $C$ satisfies the inequality $N_{\mathcal{T}}\geq
C\epsilon^{-d_{y}/r},\forall\epsilon>0$. That is, there exists a sequence of
DeepONet with the corresponding sequence of trunk net class
$\left\\{\mathcal{T}_{n}\right\\}_{n=1}^{\infty}$ and sequence of error
$\left\\{\epsilon_{n}\right\\}_{n=1}^{\infty}$ such that
$\epsilon_{n}\rightarrow 0$, and it satisfies
$\displaystyle
N_{\mathcal{T}_{n}}<\frac{1}{n}\epsilon_{n}^{-d_{y}/r}\left(\text{i.e.,}N_{\mathcal{T}_{n}}=o(\epsilon_{n}^{-d_{y}/r})\text{
with respect to }n\right).$
Note that the above implies
$N_{\mathcal{T}_{n}}=o(\epsilon_{n}^{-(d_{y}+m)/r})$. On the one hand, the
following inequality holds where $\mathcal{B}_{n}$ denotes the corresponding
class sequence of branch net.
$\displaystyle
d(\mathcal{F}_{\text{DeepONet}}(\mathcal{B}_{n},\mathcal{T}_{n});\mathcal{W}_{r,d_{y}+m})\leq\epsilon_{n}.$
Since $\sigma$ is bounded, $\mathcal{T}_{n}$ consists of bounded functions
with respect to the supremum norm. Therefore, if we apply the Lemma 2 with
respect to $n$, the number of basis $p_{n}$ should be
$\Omega(\epsilon_{n}^{-d_{y}/r})$. Since $p_{n}$ is also the number of output
dimensions for the class of trunk net $\mathcal{T}_{n}$, the number of
parameters in $\mathcal{T}_{n}$ should be larger than
$p_{n}=\Omega(\epsilon_{n}^{-d_{y}/r})$. This leads to a contradiction. ∎
Finally, we present a lower bound on the total number of parameters of
DeepONet, considering the size of the branch net. Keep in mind that the proof
of this theorem can be applied to other variants of DeepONet.
###### Theorem 3.
(Total Complexity of DeepONet) Let $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ be
a universal activation function in $C^{r}(\mathbb{R})$ such that $\sigma$ and
$\sigma^{\prime}$ are bounded. Suppose that the class of branch net
$\mathcal{B}$ has a bounded Sobolev norm (i.e., $\|\beta\|_{r}^{s}\leq
l_{1},\forall\beta\in\mathcal{B}$). If any non-constant
$\psi\in\mathcal{W}_{r,n}$ does not belong to any class of neural network,
then the number of parameters in DeepONet is $\Omega(\epsilon^{-(d_{y}+m)/R})$
for any $R>r$ when
$d(\mathcal{F}_{\text{DeepONet}}(\mathcal{B},\mathcal{T});\mathcal{W}_{r,d_{y}+m})\leq\epsilon$.
###### Proof.
For a positive $\epsilon<l_{1}$, suppose that there exists a class of branch
net $\mathcal{B}$ and trunk net $\mathcal{T}$ such that
$d(\mathcal{F}_{\text{DeepONet}}(\mathcal{B},\mathcal{T});\mathcal{W}_{r,d_{y}+m})\leq\epsilon$.
By the boundedness of $\sigma$, there exists a constant $l_{2}$ which is the
upper bound on the supremum norm $\|\cdot\|_{\infty}$ of the trunk net class
$\mathcal{T}$. Let us denote the number of parameters in
$\mathcal{F}_{\text{DeepONet}}$ by $N_{\mathcal{F}_{\text{DeepONet}}}$. Using
the Lemma 1 to replace DeepONet’s inner products with neural networks as in
the inequality (11), we can construct a class of neural network $\mathcal{F}$
such that the number of parameters $\mathcal{F}$ is
$O(N_{\mathcal{F}_{\text{DeepONet}}}+p^{1+1/t}\epsilon^{-1/t})$ and,
$\displaystyle
d(\mathcal{F};\mathcal{W}_{r,d_{y}+m})\leq(1+l_{1}l_{2})\epsilon.$
Suppose that $N_{\mathcal{F}_{\text{DeepONet}}}=o(\epsilon^{-(d_{y}+m)/r})$.
Then, by Theorem 1, $p^{1+1/t}\epsilon^{-1/t}$ should be
$\Omega(\epsilon^{-(d_{y}+m)/r})$. Since $t$ can be arbitrary large, the
number of basis $p$ should be $\Omega(\epsilon^{-(d_{y}+m)/R})$ for any $R>r$.
∎
### C.2 Complexity analysis on variants of DeepONet.
We would like to verify that variant models of DeepONet require numerous units
in the first hidden layer of the target network. Now we denote the class of
Pre-net in Shift-DeepONet and flexDeepONet by $\mathcal{P}$. The class of
Shift-DeepONet and flexDeepONet will be written as $\mathcal{F}_{\text{shift-
DeepONet}}(\mathcal{P},\mathcal{B},\mathcal{T})$ and
$\mathcal{F}_{\text{flexDeepONet}}(\mathcal{P},\mathcal{B},\mathcal{T})$
respectively. The structure of Shift-DeepONet can be summarized as follows.
Denote the width of the first hidden layer of the target network by $w$. We
define the pre-net as
$\rho=[\rho_{1},\rho_{2}]:\mathbb{R}^{m}\rightarrow\mathbb{R}^{w\times({d_{y}+1})}$
where $\rho_{1}:\mathbb{R}^{m}\rightarrow\mathbb{R}^{w\times d_{y}}$ and
$\rho_{2}:\mathbb{R}^{m}\rightarrow\mathbb{R}^{w}$, the branch net as
$\beta:\mathbb{R}^{m}\rightarrow\mathbb{R}^{p}$, and the trunk net as
$\tau:\mathbb{R}^{w}\rightarrow\mathbb{R}^{p}$. The Shift-DeepONet
$f_{\text{Shift-DeepONet}}(\rho,\beta,\tau)$ is defined as
$\displaystyle f_{\text{Shift-
DeepONet}}(\rho,\beta,\tau)(\mathcal{E}(u),y):=\pi_{p}(\beta(\mathcal{E}(u)),\tau(\Phi(\rho_{1}(\mathcal{E}(u)),y)+\rho_{2}(\mathcal{E}(u))))$
where $\Phi$ is defined in Eq. (12).
We claim that it does not improve performance for the branch net to
additionally output the weights on the first layer of a target network. The
following lemma shows that the procedure can be replaced by a small neural
network structure.
###### Lemma 3.
Consider a function $\Phi:\mathbb{R}^{d_{y}(w+1)}\rightarrow\mathbb{R}^{w}$
which is defined below.
$\Phi(x_{1},\cdots,x_{d_{y}},\cdots,x_{(d_{y}-1)w+1}\cdots,x_{d_{y}w},y_{1},\cdots,y_{d_{y}})\\\
:=\left(\sum_{i=1}^{d_{y}}x_{i}y_{i},\cdots,\sum_{i=1}^{d_{y}}x_{(d_{y}-1)w+i}y_{i}\right).$
(12)
For any arbitrary positive $t$, there exists a class of neural network
$\mathcal{F}$ with universal activation
$\sigma:\mathbb{R}\rightarrow\mathbb{R}$ such that the number of parameters of
$\mathcal{F}$ is $O(wd_{y}^{1+1/t}\epsilon^{-1/t})$ with
$\inf_{f\in\mathcal{F}}\|f-\Phi\|_{\infty}\leq\epsilon$.
###### Proof.
Using the Lemma 1, we can construct a sequence of neural network
$\left\\{f_{i}\right\\}_{i=1}^{w}$ which is an $\epsilon$-approximation of the
inner product with $O(d_{y}^{1+{1/t}}\epsilon^{-1/t})$ parameters. If we
combine all of the $w$ approximations, we get the desired neural network. ∎
Now we present the lower bound on the number of parameters for Shift-DeepONet.
We derive the following theorem with an additional assumption that the class
of trunk net is Lipschitz continuous. The function
$\tau:\mathbb{R}^{d_{y}}\rightarrow\mathbb{R}^{p}$ is called Lipschitz
continuous if there exists a constant $C$ such that
$\|\tau(y_{1})-\tau(y_{2})\|_{1}\leq C\|y_{1}-y_{2}\|_{1}.$
For the neural network $f$, the upper bound of the Lipschitz constant for $f$
could be obtained as $L^{k-1}\Pi_{i=1}^{k}\|W^{i}\|_{1}$, where $L$ is the
Lipschitz constant of $\sigma$ and the norm $\|\cdot\|_{1}$ denotes the matrix
norm induced by vector $1$-norms. We can impose constraints on the upper bound
of the weights, which consequently enforce affine transformation $W^{i}$ to be
bounded with respect to the $L^{1}$ norm. Therefore, we can guarantee the
Lipschitz continuity of the entire neural network in this way.
We would like to remark on the validity of the weight assumptions in the
theorem since the bounded assumptions of the weight may be a possible reason
for increasing the number of parameters. However, the definition of Sobolev
space forces all elements to have the supremum norm $\|\cdot\|_{\infty}$ less
than 1. It may be somewhat inefficient to insist on large weights for
approximating functions with a limited range.
###### Theorem 4.
Let $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ be a universal activation
function in $C^{r}(\mathbb{R})$ such that $\sigma$ and $\sigma^{\prime}$ are
bounded. Suppose that the class of branch net $\mathcal{B}$ and pre-net
$\mathcal{P}$ has a bounded Sobolev norm (i.e., $\|\beta\|_{r}^{s}\leq
l_{1},\forall\beta\in\mathcal{B}$, and $\|\rho\|_{r}^{s}\leq
l_{3},\forall\rho\in\mathcal{P}$) and any neural network in the class of trunk
net $\mathcal{T}$ is Lipschitz continuous with constant $l_{2}$. If any non-
constant $\psi\in\mathcal{W}_{r,n}$ does not belong to any class of neural
network, then the number of parameters in $\mathcal{T}$ is
$\Omega(\epsilon^{-d_{y}/r})$ when $d(\mathcal{F}_{\text{shift-
DeepONet}}(\mathcal{P},\mathcal{B},\mathcal{T});\mathcal{W}_{r,d_{y}+m})\leq\epsilon$.
###### Proof.
Denote the number of parameters in $\mathcal{T}$ by $N_{\mathcal{T}}$. Suppose
that there exists a sequence of pre-net
$\left\\{\rho_{n}\right\\}_{n=1}^{\infty}$, branch net
$\left\\{\beta_{n}\right\\}_{n=1}^{\infty}$ and trunk net
$\left\\{\tau_{n}\right\\}_{n=1}^{\infty}$ with the corresponding sequence of
error $\left\\{\epsilon_{n}\right\\}_{n=1}^{\infty}$ such that
$\epsilon_{n}\rightarrow 0$ and,
$\displaystyle
N_{\mathcal{T}}=o(\epsilon_{n}^{-d_{y}/r}),\quad\text{and}\quad\sup_{\psi\in\mathcal{W}_{r,d_{y}+m}}\|f_{\text{shift-
DeepONet}}(\rho_{n},{\beta_{n}},\tau_{n})-\psi\|_{\infty}\leq\epsilon_{n}.$
The proof can be divided into three parts. Firstly, we come up with a neural
network approximation $\rho^{NN}_{n}$ of $\rho_{n}$ of which size is
$O(w\cdot\epsilon_{n}^{-m/r})$ within an error $\epsilon_{n}$. Next, construct
a neural network approximation of $\Phi$ using the Lemma 3. Finally, the inner
product $\pi_{p_{n}}(\beta_{n},\tau_{n})$ is replaced with a neural network as
in (11) of Lemma 2.
Since all techniques such as triangular inequality are consistent with the
previous discussion, we will briefly explain why additional Lipschitz
continuity is required for the trunk network, and omit the details.
Approximating the Pre-Net of Shift DeepOnet, which is not in DeepOnet,
inevitably results in an error in the input of the trunk net. We are reluctant
to allow this error to change the output of the trunk net significantly. In
this situation, the Lipschitz continuity provides the desired result. ∎
For $d_{y}=1$, the additional rotation is only multiplying by $1$ or $-1$.
Since the weight and bias of the first layer alone can cover the scalar
multiplication, flexDeepONet has the same properties as Shift-DeepONet in the
above theorem.
###### Theorem 5.
Consider the case $d_{y}=1$. Let $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ be a
universal activation function in $C^{r}(\mathbb{R})$ such that $\sigma$ and
$\sigma^{\prime}$ are bounded. Suppose that the class of branch net
$\mathcal{B}$ and pre-net $\mathcal{P}$ has a bounded Sobolev norm(i.e.,
$\|\beta\|_{r}^{s}\leq l_{1},\forall\beta\in\mathcal{B}$, and
$\|\rho\|_{r}^{s}\leq l_{3},\forall\rho\in\mathcal{P}$), and any neural
network in the class of trunk net $\mathcal{T}$ is Lipschitz continuous with
constant $l_{2}$. If any non-constant $\psi\in\mathcal{W}_{r,n}$ does not
belong to any class of neural network, then the number of parameters in
$\mathcal{T}$ is $\Omega(\epsilon^{-d_{y}/r})$ when
$d(\mathcal{F}_{\text{flexDeepONet}}(\mathcal{B},\mathcal{T});\mathcal{W}_{r,d_{y}+m})\leq\epsilon$.
###### Proof.
The main difference between flexDeepONet and Shift-DeepONet, which is not
mentioned earlier, is that the branch net affects the bias of the output
layer. However, adding the values of the two neurons can be implemented in a
neural network by adding only one weight of value 1 for each neuron, so all of
the previous discussion are valid. ∎
In fact, NOMAD can be substituted with the embedding method handled by Galanti
& Wolf (2020). Suppose that the branch net of NOMAD is continuously
differentiable. Let’s also assume that the Lipschitz constant of branch net
and trunk net is bounded. We would like to briefly cite the relevant theorem
here.
###### Theorem 6.
(Galanti & Wolf (2020)) Suppose that $\sigma$ is a universal activation in
$C^{1}(\mathbb{R})$ such that $\sigma^{\prime}$ is a bounded variation on
$\mathbb{R}$. Additionally, suppose that there is no class of neural network
that can represent any function in $\mathcal{W}_{1,d_{y}+m}$ other than a
constant function. If the weight on the first layer of target network in NOMAD
is bounded with respect to $L^{1}$-norm, then
$d(\mathcal{N};\mathcal{W}_{1,d_{y}+m})\leq\epsilon$ implies the number of
parameters in $\mathcal{N}$ is $\Omega(\epsilon^{-min(d_{y}+m,2\cdot m_{y})})$
where $\mathcal{N}$ denotes the class of function contained as a target
network of NOMAD.
## Appendix D Chunked embedding method
The HyperDeepONet may suffer from the large complexity of the hypernetwork
when the size of the target network increases. Although even a small target
network can learn various operators with proper performance, a larger target
network will be required for more accurate training. To take into account this
case, we employ a chunk embedding method which is developed by von Oswald et
al. (2020). The original hypernetwork was designed to generate all of the
target network’s weights so that the complexity of hypernetwork could be
larger than the complexity of the target network. Such a problem can be
overcome by using a hypernetwork with smaller outputs.
Figure 9: Chunk Embedding Method
More specifically, Figure 9 describes how the chunk embedding method reduces
the number of learnable parameters. First, they partition the weights and
biases of the target network. The hypernetwork then creates the weights of
each parameter group by using the sensor values
$\left\\{u(x_{i})\right\\}_{i=1}^{m}$ with a latent vector $z_{j}$. All groups
share the hypernetwork so that the complexity decreases by a factor of the
number of groups. Since the latent vectors $\\{z_{j}\\}_{j=1}^{N_{c}}$ learn
the characteristics of each group during the training period, the chunked
embedding method preserves the expressivity of the hypernetwork. The chunked
architecture is a universal approximator for the set of continuous functions
with the existence of proper partitions (Proposition 1 in von Oswald et al.
(2020)). We remark that the method can also generate the additional weights
and discard the unnecessary ones when the number of the target network’s
parameters is not multiple of $N_{C}$, which is the number of group.
## Appendix E Experimental details
| DeepONet | Shift | Flex | NOMAD | Hyper(ours)
---|---|---|---|---|---
Identity | 0.0005 | 0.0002 | 0.0005 | 0.0001 | 0.0001
Differentiation | 0.0005 | 0.0002 | 0.0005 | 0.0001 | 0.0001
Advection | 0.0005 | 0.0001 | 0.0001 | 0.0002 | 0.0005
Burgers | 0.0001 | 0.0005 | 0.0002 | 0.0001 | 0.0001
Shallow | 0.0001 | 0.0005 | - | 0.0001 | 0.0005
Table 4: Setting of the decay rate for each operator problem
In most experiments, we follow the hyperparameter setting in Lu et al. (2019;
2021; 2022). We use ADAM in Kingma & Ba (2015) as an optimizer with a learning
rate of $1e-3$ and zero weight decay. In all experiments, an InverseTimeDecay
scheduler was used, and the step size was fixed to 1. In the experiments of
identity and differentiation operators, grid search was performed using the
sets 0.0001, 0.0002, 0.0005, 0.001, 0.002, 0.005 for decay rates. The selected
values of the decay rate for each model can be found in Table 4.
### E.1 Identity
As in the text, we developed an experiment to learn the identity operator for
the 20th-order Chebyshev polynomials (Figure 10). Note that the absolute value
of the coefficients of all orders is less than or equal to $1/4$. We
discretize the domain $[-1,1]$ with a spatial resolution of 50. For the
experiments described in the text, we construct all of the neural networks
with tanh. We use 1,000 training pairs and 200 pairs to validate our
experiments. The batch size during the training is determined to be 5,000,
which is one-tenth the size of the entire dataset.
### E.2 Differentiation
In this experiment, we set functions whose derivatives are 20th-order
Chebyshev polynomials as input functions. As mentioned above, all coefficients
of the Chebyshev polynomial are between $-1/4$ and $1/4$. We use the 100
uniform grid on the domain [-1, 1]. The number of training and test samples is
1,000 and 200, respectively. We use a batch size of 10,000, which is one-tenth
the size($100,000=100\cdot 1,000$) of the entire dataset.
Figure 10: One test data example of differentiation operator problem.
### E.3 Advection equation
We consider the linear advection equation on the torus
$\mathbb{T}:=\mathbb{R}/\mathbb{Z}$ as follows:
$\begin{cases}\frac{\partial w}{\partial t}+c\cdot\frac{\partial w}{\partial
x}=0.\\\ w(x,0)=w_{0}(x),&x\in\mathbb{T},\end{cases}$ (13)
where $c$ is a constant which denotes the propagation speed of $w$. By
constructing of the domain as $\mathbb{T}$, we implicitly assume the periodic
boundary condition. In this paper, we consider the case when $c=1$. Our goal
is to learn the operator which maps $w_{0}(x)(=w(0,x))$ to $w(0.5,x)$. We use
the same data as in Lu et al. (2022). We discretize the domain $[0,1]$ with a
spatial resolution of 40. The number of training samples and test samples is
1,000 and 200, respectively. We use the full batch for training so that the
batch size is $40\cdot 1,000=40,000$.
### E.4 Burgers’ equation
We consider the 1D Burgers’ equation which describes the movement of the
viscous fluid
$\begin{cases}\frac{\partial w}{\partial t}=-u\cdot\frac{\partial w}{\partial
x}+\nu\frac{\partial^{2}w}{\partial x^{2}},&(x,t)\in(0,1)\times(0,1],\\\
w(x,0)=w_{0}(x),&x\in(0,1),\end{cases}$ (14)
where $w_{0}$ is the initial state and $\nu$ is the viscosity. Our goal is to
learn the nonlinear solution operator of the Burgers’ equation, which is a
mapping from the initial state $w_{0}(x)(=w(x,0))$ to the solution $w(x,1)$ at
$t=1$. We use the same data of Burgers’ equation provided in Li et al. (2021).
The initial state $w_{0}(x)$ is generated from the Gaussian random field
$\mathcal{N}(0,5^{4}(-\Delta+25I)^{-2})$ with the periodic boundary
conditions. The split step and the fine forward Euler methods were employed to
generate a solution at $t=1$. We set viscosity $\nu$ and a spatial resolution
to 0.1 and $2^{7}=128$, respectively. The size of the training sample and test
sample we used are 1,000 and 200, respectively.
We take the ReLU activation function and the InverseTimeDecay scheduler to
experiment with the same setting as in Lu et al. (2022). For a fair
comparison, all experiments on DeepONet retained the hyperparameter values
used in Lu et al. (2022). We use the full batch so that the batch size is
1,280,000.
### E.5 Shallow Water equation
The shallow water equations are hyperbolic PDEs which describe the free-
surface fluid flow problems. They are derived from the compressible Navier-
Stokes equations. The physical conservation laws of the mass and the momentum
holds in the shock of the solution. The specific form of the equation can be
written as
$\begin{cases}\frac{\partial h}{\partial t}+\frac{\partial}{\partial
x}(hu)+\frac{\partial}{\partial y}(hv)=0,\\\ \frac{\partial(hu)}{\partial
t}+\frac{\partial}{\partial
x}(u^{2}h+\frac{1}{2}gh^{2})+\frac{\partial}{\partial y}(huv)=0,\\\
\frac{\partial(hv)}{\partial t}+\frac{\partial}{\partial
y}(v^{2}h+\frac{1}{2}gh^{2})+\frac{\partial}{\partial x}(huv)=0,\\\
h(0,x_{1},x_{2})=h_{0}(x_{1},x_{2}),\end{cases}$ (15)
for $t\in[0,1]$ and $x_{1},x_{2}\in[-2.5,2.5]$ where $h(t,x_{1},x_{2})$
denotes the height of water with horizontal and vertical velocity $(u,v)$. $g$
denotes the gravitational acceleration. In this paper, we aim to learn the
operator
$h_{0}(x_{1},x_{2})\mapsto\left\\{h(t,x_{1},x_{2})\right\\}_{t\in[1/4,1]}$
without the information of $(u,v)$. For the sampling of initial conditions and
the corresponding solutions, we directly followed the setting of Takamoto et
al. (2022). The 2D radial dam break scenario is considered so that the
initialization of the water height is generated as a circular bump in the
center of the domain. The initial condition is generated by
$h(t=0,x_{1},x_{2})=\begin{cases}2.0,&\text{for
$r<\sqrt{x_{1}^{2}+x_{2}^{2}}$}\\\ 1.0,&\text{for
$r\geq\sqrt{x_{1}^{2}+x_{2}^{2}}$}\end{cases}$ (16)
with the radius $r$ randomly sampled from $U[0.3,0.7]$. The spatial domain is
determined to be a 2-dimensional rectangle $[-2.5,2.5]\times[-2.5,2.5]$. We
use $256=16^{2}$ grids for the spatial domain. We train the models with three
snapshots at $t=0.25,0.5,0.75$, and predict the solution $h(t,x_{1},x_{2})$
for four snapshots $t=0.25,0.5,0.75,1$ on the same grid. We use 100 training
samples and the batch size is determined to be 25,600.
## Appendix F Additional experiments
### F.1 Comparison under various conditions
The experimental results under various conditions are included in Table 5 by
modifying the network structure, activation function, and number of sensor
points. Although the DeepONet shows good performance in certain settings, the
proposed HyperDeepONet shows good performance without dependency on the
various conditions.
Activation: ReLU, $M=30$ | DeepONet | Shift | Flex | NOMAD | Hyper(Ours)
---|---|---|---|---|---
Target network | $d_{y}$-30-30-30-1 | 0.16797 | 1.30852 | 1.04292 | 0.27209 | 0.02059
$d_{y}$-50-50-1 | 0.04822 | 1.08760 | 1.11957 | 0.21391 | 0.05562
Activation: ReLU, $M=100$ | DeepONet | Shift | Flex | NOMAD | Hyper(Ours)
---|---|---|---|---|---
Target network | $d_{y}$-30-30-30-1 | 0.02234 | 1.08310 | 1.03741 | 0.19089 | 0.01743
$d_{y}$-50-50-1 | 0.07255 | 1.47373 | 1.13217 | 0.14020 | 0.04645
Activation: PReLU, $M=30$ | DeepONet | Shift | Flex | NOMAD | Hyper(Ours)
---|---|---|---|---|---
Target network | $d_{y}$-30-30-30-1 | 0.11354 | 1.09395 | 1.03502 | 0.25651 | 0.02844
$d_{y}$-50-50-1 | 0.00873 | 1.14073 | 1.06947 | 0.04054 | 0.04302
Activation: PReLU, $M=100$ | DeepONet | Shift | Flex | NOMAD | Hyper(Ours)
---|---|---|---|---|---
Target network | $d_{y}$-30-30-30-1 | 0.01035 | 1.05080 | 1.07791 | 0.16592 | 0.01083
$d_{y}$-50-50-1 | 0.07255 | 1.47373 | 1.13217 | 0.14020 | 0.04645
Table 5: The relative $L^{2}$ test errors for experiments on training the
identity operator under various conditions
### F.2 Varying the number of layers in branch net and hypernetwork
Figure 11 compares the relative $L^{2}$ error of the training data and test
data for the DeepONet and the HyperDeepONet by varying the number of layers in
the branch net and the hypernetwork while maintaining the same small target
network. Note that the bottom 150 training and test data with lower errors are
selected to observe trends cleary. The training and test error for the
DeepONet is not reduced despite the depth of the branch net becoming larger.
This is a limitation of DeepONet’s linear approximation. DeepONet approximates
the operator with the dot product of the trunk net’s output that approximates
the basis of the target function and the branch net’s output that approximates
the target function’s coefficient. Even if a more accurate coefficient is
predicted by increasing the depth of the branch net, the error does not
decrease because there is a limit to approximating the operator with a linear
approximation using the already fixed trunk net.
The HyperDeepONet approximates the operator with a low test error in all cases
with a different number of layers. Figure 11 shows that the training error of
the HyperDeepONet remains small as the depth of the hypernetwork increases,
while the test error increases. The increasing gap between the training and
test errors is because of overfitting. HyperDeepONet overfits the training
data because the learnable parameters of the model are more than necessary to
approximate the target operator.
Figure 11: Varying the number of layers of branch net and hypernetwork in
DeepONet and HyperDeepONet for identity operator problem (left) and
differentiation operator problem (right).
### F.3 Comparison of HyperDeepONet with Fourier Neural Operator
The Fourier Neural Operator (FNO) (Li et al., 2021) is a well-known method for
operator learning. Lu et al. (2022) consider 16 different tasks to explain the
relative performance of the DeepONet and the FNO. They show that each method
has its advantages and limitations. In particular, DeepONet has a great
advantage over FNO when the input function domain is complicated, or the
position of the sensor points is not uniform. Moreover, the DeepONet and the
HyperDeepONet enable the inference of the solution of time-dependent PDE even
in a finer time grid than a time grid used for training, e.g.the continuous-
in-time solution operator of the shallow water equation in our experiment.
Since the FNO is image-to-image based operator learning model, it cannot
obtain a continuous solution operator over time $t$ and position
$x_{1},x_{2}$. In this paper, while retaining these advantages of DeepONets,
we focused on overcoming the difficulties of DeepONets learning complex target
functions because of linear approximation. Therefore, we mainly compared the
vanilla DeepONet and its variants models to learn the complex target function
without the result of the FNO.
Model | HyperDeepONet (ours) | Fourier Neural Operator
---|---|---
Mode 2 | Mode 4 | Mode 8 | Mode 16
Identity | 0.0358 | 0.0005 | 0.0004 | 0.0003 | 0.0004
Differentiation | 0.1268 | 0.8256 | 0.6084 | 0.3437 | 0.0118
$\\#$Param | | 15741(or 16741)
---
20993 | 29185 | 45569 | 78337
Table 6: The relative $L^{2}$ test errors and the number of parameters for the
identity and differentiation operator problems using HyperDeepONet and FNO
with different number of modes. $\\#$Param denote the number of learnable
parameters.
Table 6 shows the simple comparison of the HyperDeepONet with the FNO for the
identity operator and differentiation operator problems. Although the FNO
structure has four Fourier layers, we use only one Fourier layer with 2,4,8,
and 16 modes for fair comparison using similar number of parameters. The FNO
shows a better performance than the HyperDeepONet for the identity operator
problem. Because the FNO has a linear transform structure with a Fourier
layer, the identity operator is easily approximated even with the 2 modes. In
contrast, the differentiation operator is hard to approximate using the FNO
with 2, 4, and 8 modes. Although the FNO with mode 16 can approximate the
differentiation operator with better performance than the HyperDeepONet, it
requires approximately 4.7 times as many parameters as the HyperDeepONet.
Model | DeepONet | Shift | Flex | NOMAD | Hyper(Ours)
---|---|---|---|---|---
Advection | $\\#$Param | 274K | 281K | 282K | 270K | 268K
Rel error | 0.0046 | 0.0095 | 0.0391 | 0.0083 | 0.0048
Burgers | $\\#$Param | 115K | 122K | 122K | 117K | 114K
Rel error | 0.0391 | 0.1570 | 0.1277 | 0.0160 | 0.0196
Shallow | $\\#$Param | 107K | 111K | - | 117K | 101K
Rel error | 0.0279 | 0.0299 | - | 0.0167 | 0.0148
Shallow w/ small param | $\\#$Param | 6.5K | 8.5K | - | 6.4K | 5.6K
Rel error | 0.0391 | 0.0380 | - | 0.0216 | 0.0209
Table 7: The relative $L^{2}$ test errors and the number of parameters for the
solution operators of PDEs experiments. $\\#$Param denote the number of
learnable parameters. Note that the all five models use the similar number of
parameters for each problem.
### F.4 Performance of other baselines with the same number of learnable
parameters
For three different PDEs with complicated target functions, we compare all the
baseline methods in Table 7 to evaluate the performances. We analyze the
model’s computation efficiency based on the number of parameters and fix the
model’s complexity for each equation. All five models demonstrated their
prediction abilities for the advection equation. DeepONet shows the greatest
performance in this case, and other variants can no longer improve the
performance. For the Burgers’ equation, NOMAD and HyperDeepONet are the two
outstanding algorithms from the perspective of relative test error. NOMAD
seems slightly dominant to our architectures, but the two models compete
within the margin of error. Furthermore, HyperDeepONet improves its accuracy
using the chunk embedding method, which enlarge the target network’s size
while maintaining the complexity. Finally, HyperDeepONet and NOMAD outperform
the other models for 2-dimensional shallow water equations. The HyperDeepONet
still succeeds in accurate prediction even with a few parameters. It can be
observed from Table 7 that NOMAD is slightly more sensitive to an extreme case
using a low-complexity model. Because of the limitation in computing
3-dimensional rotation, FlexDeepONet cannot be applied to this problem.
Figure 13 shows the results on prediction of shallow water equations’ solution
operator using the DeepONet and the HyperDeepONet. The overall performance of
the DeepONet is inferior to that of the HyperDeepONet, which is consistent
with the result in Figure 12. In particular, the DeepONet has difficulty
matching the overall circular shape of the solution when the number of
parameters is small. This demonstrates the advantages of the HyperDeepONet
when the computational resource is limited.
Figure 12: The test $L^{2}$ relative errors of four methods during training for the solution operator of shallow water equations. | | Training time (s)
---
(per 1 epoch)
Inference time (ms)
Same target (Differentiation) | DeepONet | 1.018 | 0.883
HyperDeepONet | 1.097 | 1.389
Same $\\#$param (Advection) | DeepONet | 0.466 | 0.921
HyperDeepONet | 0.500 | 1.912
Table 8: The training time and inference time for the differentiation operator
problem and the solution operator of advection equation problem using DeepONet
and HyperDeepONet.
### F.5 Comparison of training time and inference time
Table 8 shows the training time and the inference time for the DeepONet and
the HyperDeepONet for two different operator problems. When the same small
target network is employed for the DeepONet and the HyperDeepONet, the
training time and inference time for the HyperDeepONet are larger than for the
DeepONet. However, in this case, the time is meaningless because DeepONet does
not learn the operator with the desired accuracy at all (Table 1 and Figure
6).
Even when both models use the same number of training parameters,
HyperDeepONet takes slightly longer to train for one epoch than the DeepONet.
However, the training complex operator using the HyperDeepONet takes fewer
epochs to get the desired accuracy than DeepONet, as seen in Figure 7. This
phenomenon can also be observed for the shallow water problem in Figure 12. It
shows that the HyperDeepONet converges to the desired accuracy faster than any
other variants of DeepONet.
The HyperDeepONet also requires a larger inference time because it can infer
the target network after the hypernetwork is used to generate the target
network’s parameters. However, when the input function’s sensor values are
already fixed, the inference time to predict the output of the target function
for various query points is faster than that of the DeepONet. This is because
the size of the target network for HyperDeepONet is smaller than that of the
DeepONet, although the total number of parameters is the same.
Figure 13: The examples of predictions on the solution operator of shallow
water equations using the DeepONet and the HyperDeepONet. The first column
represents the exact solution generated in Takamoto et al. (2022), and the
other four columns denote the predicted solutions using the corresponding
methods. The four rows shows the predictions $h(t,x_{1},x_{2})$ at four
snapshots $t=[0.25,0.5,0.75,1]$.
|
# Influence of Crystalline Nanoprecipitates on Shear-Band Propagation in Cu-
Zr-Based Metallic Glasses
Tobias Brink<EMAIL_ADDRESS>Fachgebiet Materialmodellierung,
Institut für Materialwissenschaft, Technische Universität Darmstadt, Jovanka-
Bontschits-Straße 2, D-64287 Darmstadt, Germany Martin Peterlechner Institut
für Materialphysik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-
Straße 10, D-48149 Münster, Germany Harald Rösner Institut für
Materialphysik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-
Straße 10, D-48149 Münster, Germany Karsten Albe Fachgebiet
Materialmodellierung, Institut für Materialwissenschaft, Technische
Universität Darmstadt, Jovanka-Bontschits-Straße 2, D-64287 Darmstadt, Germany
Gerhard Wilde Institut für Materialphysik, Westfälische Wilhelms-Universität
Münster, Wilhelm-Klemm-Straße 10, D-48149 Münster, Germany
(May 6, 2016)
###### Abstract
The interaction of shear bands with crystalline nanoprecipitates in Cu-Zr-
based metallic glasses is investigated by a combination of high-resolution TEM
imaging and molecular-dynamics computer simulations. Our results reveal
different interaction mechanisms: Shear bands can dissolve precipitates, can
wrap around crystalline obstacles, or can be blocked depending on the size and
density of the precipitates. If the crystalline phase has a low yield
strength, we also observe slip transfer through the precipitate. Based on the
computational results and experimental findings, a qualitative mechanism map
is proposed that categorizes the various processes as a function of the
critical stress for dislocation nucleation, precipitate size, and distance.
Published in:
T. Brink et al., Phys. Rev. Applied 5, 054005 (2016) DOI:
10.1103/PhysRevApplied.5.054005
This article is available under the terms of the Creative Commons Attribution
3.0 License. Further distribution of this work must maintain attribution to
the authors and the published article’s title, journal citation, and DOI.
Materials Science, Mechanics
## I Introduction
Metallic glasses (MGs) have advantageous mechanical properties, such as a high
yield strength and a large elastic limit, but suffer from brittle failure,
especially under tension, at temperatures significantly below the glass
transition.Inoue (2000); Ashby and Greer (2006) Under compression, improved
ductility is found for composites of MGs and crystalline secondary phases,
namely, for Cu-Zr-based,Fan and Inoue (1997); Lee _et al._ (2006); Hajlaoui
_et al._ (2007); Fornell _et al._ (2010); Li _et al._ (2011) Cu-Ti-
based,Calin _et al._ (2003) and Zr-Ti-based MGs.Hays _et al._ (2000, 2001);
Hofmann _et al._ (2008) Under tension, a small ductility with 1% to 2% strain
is observed for Cu-Zr composites containing nanocrystals.Pauly _et al._
(2010a); Barekar _et al._ (2010); Pauly _et al._ (2010b) With a higher
volume fraction of the crystalline phase, not only compressive but also
significant tensile ductility is reported for Zr-Ti-based,Hofmann _et al._
(2008); Qiao _et al._ (2013) Ti-based,Kim _et al._ (2011) and Cu-Zr-based
MGs.Liu _et al._ (2012); Wu _et al._ (2014)
For dendritic precipitates, there is a correlation between the location of
dendrites and the occurrence of shear-band patterns.Hays _et al._ (2000,
2001) The improved ductility is generally ascribed to the increased number of
shear bands and their limited length given by the constraints of the
crystalline phase.Hofmann _et al._ (2008) Thus, a high volume fraction of
ductile crystalline phase improves the ductility in compression and
microindentation tests, while a brittle secondary phase does not.Fu _et al._
(2007) This is confirmed by Song et al., who suggest a crystalline volume
fraction between $40\%$ and $80\%$ in Cu-Zr-based MGs for obtaining good
mechanical properties,Song _et al._ (2013) which is consistent with the fact
that tensile ductility is observed only in glasses with high volume fractions
of ductile crystalline phases.Hofmann _et al._ (2008); Kim _et al._ (2011);
Liu _et al._ (2012); Wu _et al._ (2014)
While the enhancement of macroscopic ductility of MGs with high volume
fractions of a ductile crystalline phase can be explained by simple composite
models, the influence of nanoprecipitates on the mechanical properties of MGs
containing a much lower volume fraction of crystalline matter is still not
clear. Similar to the case of dendritic precipitates, shear-band patterns were
also observed in glasses containing small spherical crystallites with sizes
around $2\,\mathrm{nm}$.Hajlaoui _et al._ (2007) These nanoprecipitates can
grow during deformation in certain metallic glasses.Lee _et al._ (2006); Cao
_et al._ (2007); Pauly _et al._ (2010a); Barekar _et al._ (2010); Pauly _et
al._ (2010b); Fornell _et al._ (2010) An increased growth rate of
nanocrystallites in shear bands is observed,Chen _et al._ (2006) which has
been related to enhanced atomic mobility inside shear bands.Wilde and Rösner
(2011); Bokeloh _et al._ (2011) Deformation-grown nanocrystallites are
observed to contain twins,Cao _et al._ (2007); Pauly _et al._ (2010a, b)
which occur only in larger crystallites, e.g., with a size greater than
$20\,\mathrm{nm}$ in a Cu-Zr-Al MG.Pauly _et al._ (2010b) These deformation-
grown precipitates are the possible reason for strain hardening during
nanoindentation, Fornell _et al._ (2010) as well as increased plastic strain
during compression.Lee _et al._ (2006) It is proposed that the participation
of the crystallites in the plastic deformation is the reason for the enhanced
ductility: Wu et al. demonstrate that reducing the stacking-fault energy of B2
CuZr by alloying leads to increased twinning and higher ductility under
tension.Wu _et al._ (2012) Pauly et al. propose that a martensitic
transformation from the B2 phase to the B$19^{\prime}$ phase with a subsequent
volume change is responsible for toughening in Cu-Zr-based metallic
glasses.Pauly _et al._ (2010a) This interpretation, however, is not generally
accepted. Corteen et al., in contrast, note that the volume change of the
martensitic transformation is very small and cannot contribute significantly
to toughening.Corteen _et al._ (2011) They instead suggest that precipitates
increase plasticity by favoring the nucleation of new shear bands over the
growth of critical shear bands. Indeed, recent simulation and experimental
results provide evidence for the fact that crystal–glass interfaces serve as
nucleation sites for shear bands and are therefore responsible for the
simultaneous nucleation of multiple shear bands.Albe _et al._ (2013); Zaheri
_et al._ (2014); Wang _et al._ (2014a) In tensile tests and corresponding
molecular dynamics (MD) simulations of nanolaminates of copper nanocrystals
separated by thin Cu-Zr glass layers, the crystal–glass interface acts as a
source or sink for dislocations. Shear transformation zones (STZs) are
activated by interactions with dislocations.Wang _et al._ (2007); Arman _et
al._ (2011); Brandl _et al._ (2013)
Figure 1: An undeformed Zr53.8Cu31.6Ag7.0Al7.6 sample after annealing in a TEM
bright-field image. An overview of the sample is shown in (a), with clearly
visible crystalline precipitates. The inset (b) shows a magnified view,
indicating the transition from globular to dendritic morphologies during
precipitate growth. The different brightness of parts of one precipitate in
(b) is most likely due to the different geometrical orientations of different
parts of the same precipitate with respect to the incident electron beam. The
strong black-white contrast change observed for some nanocrystals in (a) is
most likely due to twinning.
Computational studies on the interaction of crystalline precipitates with
shear bands provide further insights into nanoscale mechanisms. Lund and Schuh
conduct quasi-2D molecular-statics simulations of a binary Lennard-Jones
system with a nanocrystal inclusion.Lund and Schuh (2007) They identify three
mechanisms of deformation, depending on the ratio of shear-band thickness to
crystal size. For small crystals, the deformation is accommodated either in
the interface (for example, by rotation) or by dissolution of the crystal. For
wide shear bands and intermediate crystal sizes, dislocations in the crystal
nucleate at the interface. Finally, for crystals larger than the shear band,
they observe homogeneous dislocation nucleation due to stress building in the
nanocrystal center. However, it is somewhat unclear how the observed
homogeneous dislocation nucleation depends on the artificially induced shear
band and the resulting stress state in the system. Shi and Falk conduct
molecular-dynamics simulations on a monoatomic amorphous model system with a
high fraction of bcc nanocrystallites.Shi and Falk (2008) They find that
deformation is induced at the interfaces and that shear bands bend around
crystallites away from a direction of maximum resolved shear stress. They also
observe blocking of shear bands by crystallites. Because of the high fraction
of crystalline phase, the system more closely resembles a nanocrystalline
structure. The observation of the initiation of plastic deformation at
interfaces still matches the simulations by Albe et al. Albe _et al._ (2013)
and underlines the importance of the crystal–glass interface in these
composite systems.
While it has been shown that interfaces promote shear-band nucleation and that
precipitates can act as obstacles or can deform together with the matrix,
there is no comprehensive study that investigates the influence of the size
and number density of the precipitates. Furthermore, some mechanisms governing
the interaction between a propagating shear band and a preexisting precipitate
have been observed but not investigated and discussed in detail. Therefore the
goal of this study is to investigate the interaction of a shear band with
preexisting precipitates in Cu-Zr-based MGs. In the experimental part of the
study, we anneal Zr-Cu-Ag-Al melt-spun ribbons to induce the formation of
nanocrystalline precipitates. We present transmission electron microscopy
(TEM) images of the samples before and after deformation by cold rolling and
identify the effects of crystalline precipitates on the shear-band
propagation. Using molecular-dynamics computer simulations, we model composite
systems with a metallic-glass matrix and crystalline precipitates. We control
the initiation of a shear band using a stress concentrator and put
precipitates in its propagation path. The focus of the simulations is on the
size effects of “hard” precipitates that do not partake in the plastic
deformation. Additionally, we study the shear-band interaction with “soft,”
plastically deformable precipitates. Finally, we derive a deformation map from
the combined observations of simulations and experiments that classifies the
observed mechanisms.
## II Experiment
### II.1 Experimental setup
We prepare metallic-glass samples of nominal composition
Zr53.8Cu31.6Ag7.0Al7.6 from pure components (Cu: 99.999%, Zr: 99.998%, Ag:
99.999%, Al: 99.999%; all in at.%) by prealloying using arc melting. After
repeated arc melting with intermittent turning of the specimen to enhance
homogenization, the entire ingots are inserted into quartz-glass crucibles for
melt spinning. The weight loss during alloying is minimal and subsequent
composition analyses by energy-dispersive x-ray diffraction (EDX) confirm that
the composition of the material is equal to the nominal composition within the
accuracy of the measurement. For melt spinning, the ingots are inductively
melted under an Ar atmosphere and the melt is ejected onto a rotating Cu wheel
(tangential wheel velocity: $30\,\mathrm{m/s}$), resulting in completely
amorphous thin ribbon samples of approximately $80$-$\mathrm{\upmu{}m}$
thickness. X-ray diffraction on the as-quenched ribbon samples does not
indicate the presence of any crystalline fraction exceeding the sensitivity
threshold of this method. Parts of the ribbon samples are cut to perform
differential scanning calorimetry (DSC) measurements in a Perkin Elmer Diamond
DSC device. Both isochronal and isothermal measurements under a purified Ar
gas flow are conducted and in conjunction with microstructure analyses the
time and temperature dependence of the evolving crystalline fraction is
determined. On the basis of these results, the samples for deformation
processing are annealed in the DSC device at $410\,^{\circ}\mathrm{C}$ for
$3\,\mathrm{h}$. This thermal treatment results in the formation of
nanocrystalline precipitates with an average diameter of about
$70\,\mathrm{nm}$ and a number density on the order of
$10^{20}\,\mathrm{m^{-3}}$. The resulting microstructure is shown in Fig. 1.
Given that the width of shear bands is on the order of
$10\,\mathrm{nm}$,Rösner _et al._ (2014) there is no straight path for
propagating shear bands to avoid the interaction with nanoprecipitates in
these samples.
Figure 2: TEM image of a shear band in a Zr-Cu-Ag-Al ribbon after deformation.
The green arrows mark the positions of crystalline precipitates.
After the DSC heat treatment, the partially crystalline material is deformed
by cold rolling at room temperature in one step to true strain values of about
$\varepsilon=5\%$ at room temperature, applying a strain rate of the order of
$\dot{\varepsilon}=1\,\mathrm{s^{-1}}$. Subsequently, specimens for TEM
investigations are prepared by grinding, dimpling, and finally precision ion
polishing (PIPS, Gatan), using a low acceleration voltage of
$2.5\,\mathrm{kV}$ and low incidence angles ($<4^{\circ}$) to minimize damage
by the preparation process. The electron transparent samples are then analyzed
in dark-field, bright-field, and high-resolution transmission electron
microscopy modes in a Zeiss Libra 200FE TEM operated at an acceleration
voltage of $200\,\mathrm{kV}$. Since intrinsic shear bands formed upon cold
rolling could not be found during the TEM inspections of the thin foil
regions, the shear bands generated as a result of sample preparation and/or
subsequent sample manipulation are studied instead. Thus, the final sample
state resembles the state in an in situ TEM experiment comparable to the work
in Ref. Wilde and Rösner, 2011, where the shear bands are generated at crack
tips in the thin TEM foil.
### II.2 Experimental results
Figure 3: TEM and HRTEM images of a shear band. In the upper left corner, the
electron transparent hole is visible, which stems from the sample preparation
for TEM. (a–b) Bright-field images of the same area, using a different tilt
angle. The shear band appears as a white stripe, with part of it already
cracked (bright white). (c) Dark-field image of the same area: The
nanoprecipitates are visible and marked by circles. The crack stops at
precipitate I; the shear band continues from there to precipitate II. (d) The
power spectrum of precipitate II along the $\langle\overline{1}10\rangle$ zone
axis revealing superlattice reflections of the martensitic B$19^{\prime}$
structure. (e) HRTEM image of precipitate II, showing the shear band stopping
at the precipitate. The contrast in (c) and (e) is enhanced to improve the
visibility of the precipitates and the shear band.
Figure 2 shows a shear band in a sample after TEM preparation. It is
noticeable that the shear band switches propagation directions in the vicinity
of precipitates. Between precipitates 2 and 3, the shear band has an
additional bend. The propagation path, in general, suggests that the shear
band is “attracted” to the precipitates, possibly due to a stress field
resulting from the density change on crystallization, which explains the
change of direction between precipitates 2 and 3. Because of the processing of
the samples, it can be excluded that the repeated bending of the shear band
results from a change of the external stress state: The deformation by cold
rolling is performed in a single step, and the observed shear bands were
created during TEM preparation, resembling an in situ experiment. As the path
change in the presence of precipitates is rather large (in Fig. 2 around
$45^{\circ}$) and correlated to the position of the precipitates, it is most
likely induced by the presence of the precipitates. The literature supports
this, as shear bands in homogeneous metallic glasses (Cu-Zr-based or
otherwise) are straight on the length scale presented here.Kumar _et al._
(2007); Wang _et al._ (2008); Rösner _et al._ (2014); Schmidt _et al._
(2015) It is therefore clear that the precipitates play a major role in
influencing shear-band propagation and thereby the macroscopic plastic
deformation of the material. TEM images of a second shear band, shown in Fig.
3, shed more light on this interaction. The shear band interacts with two
precipitates: A crack follows the path of the shear band up to the first
precipitate and the shear band continues from the first to the second
precipitate. Because of the visible crack opening near the first precipitate,
it can be ruled out that the shear band originates from the second
precipitate. While the first precipitate is passed, the second precipitate,
which is encountered centrally, stops the shear band. This can also be
observed at the end of the shear band in Fig. 2. A detailed analysis of the
gray-scale intensity distribution of the high-resolution TEM (HRTEM) image in
Fig. 3(e) indicates that the shear band changes its path slightly near the
crystalline precipitate but does not proceed further or shows slip transfer
into the precipitate. Additionally, a smaller, shear-band-like region emerges
almost perpendicular to the previous propagation direction (yellow arrow).
This indicates either a shear-band deflection, or a nucleation of a new,
perpendicular shear band. Based on the highly local nature of the intensity
distribution and based on the comparison of the contrast of other
precipitates, preparation artifacts can be excluded. Thus, this type of
interaction observed here is part of the intrinsic interaction mechanism
between precipitates and advancing shear bands. The power spectrum in Fig.
3(d) indicates a precipitate with B$19^{\prime}$ crystal structure,Schryvers
_et al._ (1997) which is consistent with prior observations in Cu-Zr-based
MGs.Pauly _et al._ (2010a)
It is clear that the interaction of a propagating shear band with a
distribution of crystalline precipitates depends on the actual stress state
near the shear-band tip, the already-accommodated stress by the shear-band
propagation and the details of the local distribution of crystalline
precipitates, their sizes, and the residual stresses in the glass matrix due
to the formation of the precipitates. Depending on these factors, there are
various explanations for the observed paths: The path changes are caused
either by a deflection of the shear band, by the blocking and subsequent
renucleation of a new shear band, or by several nascent shear bands growing
together. The unhindered “passing” of a precipitate can be explained either by
a temporary path change of the shear band or by the participation of the
precipitate in the plastic deformation. Because of the necessarily limited
amount of data that can be obtained in the experiment, we undertake MD
simulations to test the aforementioned hypotheses and investigate their
relation to the sample geometry.
## III Simulation
### III.1 Simulation setup and analysis
Figure 4: Schematic simulation setup. The picture shows a cut through the
three-dimensional simulation box in the $xz$ plane at $l/2$. A notch is
inserted to control the origin of the shear band (yellow). The spherical
precipitate is shown in blue. The box has open boundaries in the $x$ direction
and is otherwise periodic. A constant strain rate is applied in the $z$
direction. Figure 5: Snapshots of a simulation with a $30$-$\mathrm{nm}$ CuZr
precipitate. The shear band wraps around the precipitate and continues
unhindered. The glass matrix is colored according to the atomic strain. The
precipitate atoms are shown in blue if they appear in the B2 structure; no
defects are visible. The left column shows a cut through the middle of the
precipitate. On the right, all atoms with $\eta_{i}<0.3$ are deleted. A video
version is provided in Video 1.
List of myvideos 1 Simulation of a shear band wrapping around a
$30$-$\mathrm{nm}$ CuZr precipitate as shown in Fig. 5.
In order to gain more insights into the nanoscale mechanisms of shear-band
interaction with precipitates, we perform a number of MD computer simulations,
which allow for an “in situ” observation of shear-band propagation. We use the
software lammps Plimpton (1995) to quench metallic-glass samples, insert
precipitates, and perform mechanical testing. The simulated metallic glass is
a $\mathrm{Cu_{64}Zr_{36}}$ alloy modeled with a Finnis–Sinclair-type
potential by Mendelev et al.Mendelev _et al._ (2009) Metallic-glass samples
with dimensions $10\times 10\times 10\,\mathrm{nm}^{3}$ and $20\times 20\times
20\,\mathrm{nm}^{3}$ are prepared by melting the material at
$2000\,\mathrm{K}$ and subsequent quenching to $50\,\mathrm{K}$ with a cooling
rate of $0.01\,\mathrm{K/ps}$.
We use the sample geometry illustrated in Fig. 4 to investigate the influence
of preexisting precipitates on an approaching shear band. Size effects are
studied by varying the diameter $d$ of the precipitates and by controlling the
distance between periodic precipitate images by varying the box width $l$.
Sample sizes are $120\,\mathrm{nm}\times l\times 60\,\mathrm{nm}$, with
$l=10\,\mathrm{nm}$, $20\,\mathrm{nm}$, $30\,\mathrm{nm}$, and
$40\,\mathrm{nm}$. With a single precipitate per simulation box, this
corresponds to number densities for the precipitates of $1.4\times
10^{22}/\mathrm{m^{3}}$, $6.9\times 10^{21}/\mathrm{m^{3}}$, $4.6\times
10^{21}/\mathrm{m^{3}}$, and $3.5\times 10^{21}/\mathrm{m^{3}}$, respectively.
The Cu-Zr glass samples are replicated to reach the desired box dimensions,
and spherical precipitates with diameters from $3\,\mathrm{nm}$ to
$40\,\mathrm{nm}$ were inserted. For this, a hole is cut into the glass matrix
with a size chosen to accommodate the precipitate without overlapping atoms.
For the CuZr precipitate, we use the experimentally observed B2
structure.Pauly _et al._ (2007); Sun _et al._ (2005); Das _et al._ (2007);
Jiang _et al._ (2007) The B$19^{\prime}$ structure, which was found in the
precipitates in the experimental part of this paper, is a distortion of the B2
structure.Schryvers _et al._ (1997) A notch controls the origin of the shear
band and makes sure that it always hits the precipitate. Periodic boundary
conditions are applied in the $y$ and $z$ directions and open boundaries in
the $x$ direction. The resulting structure is equilibrated at $50\,\mathrm{K}$
for $2\,\mathrm{ns}$ with a barostat at ambient pressure in periodic
directions. After equilibration, no long ranging stress field around the
precipitate is left; any mismatches are accommodated by the glass during the
interface creation.
The resulting composite samples are deformed at $50\,\mathrm{K}$ under a
constant tensile strain rate of $\dot{\varepsilon}=4\times 10^{7}/\mathrm{s}$
in the $z$ direction up to a total strain of at least $10\%$. The trajectories
from equilibrated to fully deformed samples are analyzed to observe the shear-
band propagation path. The shear band is identified using the von Mises local
shear invariant $\eta_{i}$ Shimizu _et al._ (2007) as implemented in the
visualization tool Ovito.Stukowski (2010) Atoms with a local shear greater
than around $0.2$ are assigned to the shear band. To observe plastic
deformation events in the crystalline phase, we perform atomic structure
identification, which can identify crystal structures, stacking faults, and
other defects.Stukowski (2012)
### III.2 Wrapping and blocking
Figure 6: Comparison of the Orowan mechanism (a) with a shear band wrapping
around precipitates (b). The slip plane is shown in gray, the precipitates in
blue, the dislocation as a black line, and the shear band as a red plane. The
precipitates can stop a dislocation because it must remain in a defined slip
plane, while the shear band can temporarily leave its slip plane and continue
unhindered afterwards.
The first observed interaction mechanism between a propagating shear band and
a preexisting precipitate is shown in Fig. 5 and Video 1. Here, the shear band
wraps around the precipitate like a carpet moving over a small obstacle. The
particle does not deform and simply moves along with one half of the glass
matrix. We call this mechanism the _wrapping mechanism_. This kind of athermal
mechanism is virtually unknown in crystalline materials. The closest analog in
a crystal—dislocation climb—is purely thermally activated. It is therefore
instructive to compare the two material classes, as shown in Fig. 6. In a
crystalline material, a dislocation moves on defined slip planes. A change of
slip plane is possible only for the screw components of the
dislocation,Cottrell (1953) is connected with a high energy barrier, and is
usually observed only in stage III work hardening.Hirth and Lothe (1982)
Therefore, the Orowan mechanism applies: The dislocation is bent around the
obstacle and finally forms dislocation rings [Fig. 6(a)].Gottstein (2004)
These rings may pile up, thereby hardening the material. In a metallic glass,
as in any isotropic material, all slip directions are equivalent. Only an
applied external stress differentiates the directions. Under tensile stress,
the planes oriented in $45^{\circ}$ angles towards the tensile axis experience
the highest resolved stress. An obstacle can be avoided simply by temporarily
and locally changing the slip path, thereby wrapping around the obstacle [Fig.
6(b)]. Depending on the precipitate distance, this wrapping mechanism is
observed for precipitates with diameters smaller than $25\,\mathrm{nm}$ to
$35\,\mathrm{nm}$ in our simulations.
Figure 7: Snapshots of a simulation with a $37.5$-$\mathrm{nm}$ CuZr-
precipitate. The shear band is blocked by the precipitate, while a second
shear band is immediately nucleated in another plane of high resolved shear
stress. The glass matrix is colored according to the atomic strain. The
precipitate atoms are shown in blue if they appear in the B2 structure; no
defects are visible. The left column shows a cut through the middle of the
precipitate. On the right, all atoms with $\eta_{i}<0.3$ are deleted. A video
version is provided in Video 2.
List of myvideos 2 Simulation of a shear band being blocked by a
$37.5\,\mathrm{nm}$ CuZr precipitate as shown in Fig. 7.
An alternative mechanism appears for increasing diameters and decreasing
distances between precipitates: The shear band is _blocked_ by the
precipitate. This causes the simultaneous nucleation of a second shear band
perpendicular to the first one on the opposite side of the precipitate as
shown in Fig. 7 and in Video 2. Again, there is no slip transfer to the
crystalline phase. The paths of the shear bands in this simulation and in the
HRTEM image of the experimental sample [Fig. 3(e)] are comparable. Both show
what looks like a shear band that starts to wrap around the precipitate but
does not propagate further. In the experiment, though, no fully formed shear
band appears perpendicular to the original one. This may be a result of the
more complex stress state or the fact that a new shear band can nucleate at
another precipitate that is not visible in the images. Still, a region
resembling a nascent shear band appears perpendicular to the original
propagation direction, strengthening the agreement with the simulation
results.
A systematic investigation of the parameters of precipitate distance and size
reveal a clear correlation with the mechanism. For a more quantitative
analysis, we define an empirical parameter $\Lambda$, which is given by the
ratio of the cross-sectional area $A$ of the precipitate divided by the
distance of precipitate centers $l$ as shown in Fig. 8(a). Similar to the
derivation of the Orowan stress, the crystalline volume fraction
$f=V_{\mathrm{precipitates}}/V$ can be estimated from the average crystallite
distance by the relationGottstein (2004)
$l=\frac{d/2}{\sqrt{f}}\quad\Leftrightarrow\quad f=\frac{d^{2}}{4l^{2}}.$ (1)
Using that, we can express $\Lambda$ only in terms of volume fraction and
precipitate geometry:
$\Lambda=\frac{A}{l}=\frac{A\sqrt{f}}{d/2}.$ (2)
For nonoverlapping precipitates ($d<l$), it is $A=\pi d^{2}/4$ and $\Lambda$
reduces to
$\Lambda=\pi\frac{d}{2}\sqrt{f}.$ (3)
When the precipitates overlap ($d>l$), we reduce the area $A$ to remove the
overlapping circle segments. In the limit $l\rightarrow 0$, $\Lambda$
corresponds to $A/l$ of an infinite cylinder parallel to the shear-band front.
Figure 8(b) shows a contour plot of the $\Lambda$ parameter as a function of
the precipitate diameter and distance. The data points in the plot represent
results from our MD simulations, divided into those showing the wrapping and
those showing the blocking mechanism. For the simulation geometry used in this
work, there exists a given $\Lambda$ that clearly separates the two
mechanisms:
$\Lambda_{\mathrm{crit}}\approx 12.65\,\mathrm{nm}$ (4)
This $\Lambda_{\mathrm{crit}}$ is not universal, as, for example, the distance
between precipitate and notch is not varied. A test simulation finds that
increasing the distance from the notch and therefore increasing shear-band
length favors the wrapping mechanism.
Figure 8: Influence of the precipitate size and distance on the mechanism. (a)
Explanation of the parameter $l$, distance of the precipitates, and $A$, the
area of a cut through the precipitate. (b) shows a contour plot of the
empirical geometry factor $\Lambda=A/l$. The dashed line shows the critical
value of $\Lambda$ for a transition from the wrapping to the blocking
mechanism. Data points are MD simulations. The data points at
$l=0\,\mathrm{nm}$ are infinite cylinders along the $y$ axis, representing the
case of overlapping spheres with infinitesimally small distances between their
centers. Figure 9: Stress-strain curves of samples which exhibit the wrapping
or blocking mechanisms. The arrows indicate when the shear band hits the
precipitate. Figure 10: Mechanical dissolution of a $3$-$\mathrm{nm}$ copper
particle in a shear band. The gray-scale color coding shows atomic strain
$\eta_{i}$ from $0.0$ (black) to $1.0$ (white). The precipitate is shown in
color: Light blue atoms are in the fcc structure, green atoms are in a
stacking fault, and dark blue atoms are disordered. The arrows indicate the
shear direction. Subfigure (a) shows a cut through the middle of the
nanoprecipitate, while (b) shows only the atoms that initially belonged to the
nanoprecipitate.
Using these formulas, we can also estimate $\Lambda$ for the experimental
results. Given a number density of precipitates $n=10^{20}\,\mathrm{m^{-3}}$
and particle diameters of around $70\,\mathrm{nm}$, we obtain
$\displaystyle f$
$\displaystyle=\frac{V_{\mathrm{precipitates}}}{V}=n\frac{4}{3}\pi(35\,\mathrm{nm})^{3}\approx
1.8\%$ (5) $\displaystyle\Lambda_{\mathrm{exp}}$ $\displaystyle=\pi\times
35\,\mathrm{nm}\times\sqrt{1.8\%}\approx
14.7\,\mathrm{nm}>\Lambda_{\mathrm{crit}}.$ (6)
This is consistent with the fact that the shear bands that are observed in the
glass samples are blocked by the precipitates. Still, the value is close to
$\Lambda_{\mathrm{crit}}$, which means that slightly smaller precipitates (or
precipitates that are not hit centrally, virtually decreasing the prefactor
$d$) may be susceptible to wrapping.
Figure 11: Stress-strain curves for the composites with fcc copper
precipitates and for a $10$-$\mathrm{nm}$ copper nanowire. (a) Curves from
simulations with the Mendelev potential. The copper precipitates do not deform
plastically and show similar behavior to the CuZr precipitates discussed
earlier. The curve of the copper nanowire explains this: The yield stress, and
therefore the critical shear stress, is higher than the highest resolved shear
stress in the steady state in the composite. To be able to compare the tensile
stress in the composite with the shear stress in the nanowire, the tensile
stress axis is scaled with a factor $0.5$ corresponding to the Schmid factor
for the plane of highest resolved shear stress. (b) Curves from simulations
with the Ward potential. Now the critical stress for heterogeneous dislocation
nucleation is lower than the steady-state stress in the composite and the
precipitate deforms plastically. (c) The simulation setup for the nanowire.
The wire is sheared in the $[110]$ direction of the fcc crystal structure on
the $(111)$ plane. The red atoms are fixed and the atoms on the top are
shifted with a constant velocity to shear the nanowire.
Figure 9 shows examples of the stress-strain curves of samples that exhibit
either the wrapping or the blocking mechanism. Neither a pronounced ductility
nor significant strain hardening can be observed. This lack of strain
hardening is in accordance with experimental data for tensile tests on Cu-Zr-
based metallic glasses with crystalline precipitates.Pauly _et al._ (2010a);
Barekar _et al._ (2010); Pauly _et al._ (2010b) It also fits with a recent
study of compression tests of Cu-Zr-based metallic glass, that finds an effect
of particle size on mechanical parameters but no particle hardening.Wang _et
al._ (2014b)
### III.3 Plastic deformation of the crystalline phase
Figure 12: Snapshots of a simulation with a $30$-$\mathrm{nm}$ copper
precipitate. This simulation uses the Ward potential, in which the critical
stress for dislocation nucleation is realistic. As the precipitate is
sufficiently soft, the shear band can cut through it. The glass matrix is
colored according to the atomic strain (the same scale as Figs. 5 and 7).
Defects in the fcc crystal structure are colored according to the legend. The
left column shows a cut through the middle of the precipitate. On the right,
all atoms with $\eta_{i}<0.3$ and all fcc-coordinated atoms are deleted. A
video version is provided in Video 3.
List of myvideos 3 Simulation of a shear band cutting a $30$-$\mathrm{nm}$
copper precipitate as shown in Fig. 12.
The precipitates discussed until now were all “hard,” i.e., not susceptible to
plastic deformation under the simulation conditions: Because of the high
antiphase-boundary energy in the B2 structure, superdislocations or twinning
with respect to a martensitic transformation would be needed for a plastic
deformation of the precipitates. The stress available at the shear-band tip is
not sufficient to nucleate these defects, which explains why no plastic
deformation of the crystalline phase can be observed in our simulations. In
experiments, deformation-grown precipitates show twinning defects which may
cause softening effects and make the precipitate susceptible to plastic
deformation.Cao _et al._ (2007) As a model for a softer precipitate, we thus
exchange the B2 crystal phase for fcc copper.
For precipitates that are small relative to the shear-band width, the
nanocrystals undergo mechanical dissolution.Lund and Schuh (2007); Albe _et
al._ (2013) This is also observed in our setup with $3$-$\mathrm{nm}$
particles as shown in Fig. 10. In samples with larger diameters, the
precipitates do not deform plastically but instead show the same wrapping and
blocking interactions as described before for the “hard” precipitates. Even if
the $(111)$ glide plane is oriented parallel to the shear-band direction to
maximize resolved shear stress on the preferred fcc slip plane, we do not
observe slip transfer into the nanoprecipitate. The corresponding stress-
strain curves are shown in Fig. 11(a). To explain this, we estimate the
critical stress for heterogeneous dislocation nucleation in fcc copper by
shearing a nanowire on the $(111)$ plane in the $[110]$ direction [see Fig.
11(c) for the simulation setup]. For this, we hold the lower layer of atoms
fixed and move the top layer of atoms with a constant velocity in the $[110]$
direction to achieve volume-conserving shear. The diameter of the nanowire is
$10\,\mathrm{nm}$, i.e., on the order of the smaller precipitates to maximize
surface effects. The resulting shear stress over shear curve is also plotted
in Fig. 11(a). The yield stress of the nanowire, $\tau_{\mathrm{crit}}$, is an
estimate for the stress needed for heterogeneous dislocation nucleation at the
glass-precipitate interface. For comparing the shear stress in the nanowire
with the tensile stress in the composite, we assume a Schmid factor of $0.5$
and scale the tensile axis by a factor $0.5$ compared to the shear stress
axis. This graphically estimates an upper bound of the shear stress $\tau_{m}$
that arrives at the precipitate, corresponding values are given in Table 1.
The upper bound of the estimated maximum resolved shear stress in the
composite is comparable to the lower bound for heterogeneous dislocation
nucleation. While this suggests that dislocation nucleation may be possible,
it is important to keep in mind that the nanowire shear test provides only a
lower bound in an idealized case and that the actual shear stress available to
nucleate a dislocation may be lower than $0.5\sigma_{z}$, which is why no
plastic deformation of the precipitate is observed. Evidently, the value of
$\tau_{\mathrm{crit}}$ in the Mendelev potential is much too high, and
therefore the Cu precipitates are “harder” than expected, which is a
deficiency of the potential model for pure Cu.
Table 1: Comparison of maximum stress $\sigma_{m}$ and the corresponding resolved shear stress $\tau_{m}$ with the critical shear stress $\tau_{\text{crit}}$ for heterogeneous dislocation nucleation in copper in the Mendelev and Ward potentials. The calculation of the resolved shear stress assumes a Schmid factor of $0.5$. | $\sigma_{m}$ (GPa) | $\tau_{m}$ (GPa) | $\tau_{\text{crit}}$ (GPa)
---|---|---|---
Mendelev | $3.2$ | $1.6$ | $1.55$
Ward | $2.9$ | $1.45$ | $0.75$
Because of this, we switch to a different potential, which provides a better
description of crystalline Cu. The Finnis–Sinclair-type potential by Ward et
al. Ward _et al._ (2012) is created by using preexisting potentials for the
elemental phases Zhou _et al._ (2004) and fitting the cross terms to the
intermetallic phases. As shown in the Appendix, this potential has a more
realistic unstable stacking-fault energy and critical stress for homogeneous
dislocation nucleation than the Mendelev potential. As shown in Fig. 11(b) and
Table 1, the critical stress for heterogeneous nucleation is much lower than
even the steady-state stress in the composite, easily allowing plastic
deformation of the particle.
The results for a $30$-$\mathrm{nm}$ copper precipitate are shown in Fig. 12
and in Video 3. The precipitate is _cut_ by the shear band, and slip transfer
through the particle can be observed. This mechanism replaces the previously
discussed blocking of the shear band if the nanoprecipitates are “soft”: For
the crystal to partake in the plastic deformation, dislocation nucleation must
be possible at shear stresses below the highest resolved shear stress in the
metallic glass at yield. Despite the participation of the crystalline phase in
the plastic deformation, the stress-strain curve in Fig. 11(b) shows the
distinctive stress drop connected with a single critical shear band and no
strain hardening. The reason is that in this setup the crystalline phase
accounts only for roughly $5\,\mathrm{vol\%}$ of the sample. This means that
the macroscopic mechanical properties are still dominated by the metallic
glass. As the shear band can simply cut through the crystal, the precipitate
poses no obstacle to the percolation of the critical shear band. For a larger
crystalline volume, a ductile crystalline phase could possibly also constrain
the shear bands.Hofmann _et al._ (2008)
## IV Discussion
Using TEM imaging, we observe shear-band bending around or close to
precipitates, an attraction of shear bands to the precipitates, and shear
bands being blocked by precipitates. In our MD simulations, we find four
mechanisms of interaction between shear bands and precipitates:
1. (i)
precipitates that are small relative to the shear-band width dissolve
mechanically,
2. (ii)
shear bands can wrap around precipitates,
3. (iii)
shear bands are blocked by precipitates, and
4. (iv)
shear bands cut through precipitates, and slip transfer into the crystalline
phase takes place.
Which of these mechanisms is active for a given precipitate depends on the
competition between the propagation of the existing shear band, the
heterogeneous nucleation of a new shear band, and the heterogeneous
dislocation nucleation in the precipitate. The wrapping-to-blocking transition
can be quantified by the parameter $\Lambda=A/l\propto A\sqrt{f}/d$. Below
$\Lambda=\Lambda_{\mathrm{crit}}$, the wrapping mechanism is favored. This
value can be be explained by the following simple argument. When the shear
band reaches a precipitate which does not deform, the shearing of the sample
momentarily stops. The stress $\tau_{\mathrm{SB}}$ in the shear band resulting
from the externally applied tensile stress $\sigma_{\mathrm{ext}}$ amounts to
$\tau_{\mathrm{SB}}=\frac{1}{2}\sigma_{\mathrm{ext}}.$ (7)
This stress acts mainly on the shear-band front, allowing us to write
$F_{\mathrm{ext}}\approx\tau_{\mathrm{SB}}\times l\times
h_{\mathrm{SB}}=\frac{\sigma_{\mathrm{ext}}}{2}\,l\,h_{\mathrm{SB}},$ (8)
where $h_{\mathrm{SB}}$ is the width of the shear band. At the moment that the
shear band hits the precipitate, the force $F_{\mathrm{ext}}$ must be equal to
a reaction force $F_{\mathrm{back}}$ from the precipitate (actio est reactio).
Using the projected precipitate area $A$ (cf. Fig. 13), we can convert that
force into a normal stress:
$\sigma_{A}^{n}=\frac{F_{\mathrm{back}}}{A/2}.$ (9)
Figure 13: Projection of the forces and stresses acting around the precipitate
(blue circle) onto the $xz$ plane. Shear bands are shown as hatched areas,
where yellow signifies the arriving shear band, green the path for wrapping,
and red the site for the nucleation of a new shear band. Figure 14: Schematic
view of different mechanisms for the interaction of a shear band with a
crystalline precipitate. With increasing precipitate sizes, the dissolution of
the precipitate is first replaced by the wrapping mechanism. Depending on the
critical stress for dislocation nucleation, wrapping is replaced by blocking
or cutting. Wrapping can be favored by increasing the precipitate distance.
We assume that $F_{\mathrm{ext}}$ predominantly acts on one half of the
obstacle (area $A/2$), which is supported by the deformation pattern of the
plastically deformed particle (Fig. 12 and Video 12). This back stress results
in a shear stress $\tau_{\mathrm{wrap}}\approx 0.5\sigma_{A}^{n}$ in the plane
of the wrapping shear band (green shear band in Fig. 13). With
$F_{\mathrm{back}}=F_{\mathrm{ext}}$, it is
$\sigma_{A}^{n}=\frac{F_{\mathrm{ext}}}{A/2}=\frac{\sigma_{\mathrm{ext}}\,l\,h_{\mathrm{SB}}}{A}=2\tau_{\mathrm{wrap}}.$
(10)
$\tau_{\mathrm{wrap}}$ must surpass a critical value
$\tilde{\tau}_{\mathrm{wrap}}$ to allow the initiation of the wrapping
mechanism; otherwise the shear band simply stops propagating. This is not
observed in our simulations, suggesting that $\sigma_{\mathrm{ext}}$ at yield
is greater than
$\tilde{\sigma}_{\mathrm{ext}}=\frac{2A\tilde{\tau}_{\mathrm{wrap}}}{l\,h_{\mathrm{SB}}}.$
(11)
The competing mechanism, blocking the shear band and nucleating a new one (red
shear band in Fig. 13), can simply be expressed by a critical shear stress
$\tilde{\tau}_{\mathrm{nucl}}$. Because of the low temperature in the
simulation and a stress close to the yield stress, we consider only the
athermal case and do not invoke a nucleation term which takes into account the
relative volume of the interface. With
$\tau_{\mathrm{nucl}}=0.5\sigma_{\mathrm{ext}}$, the transition from wrapping
to blocking takes place where
$\displaystyle
2\tilde{\tau}_{\mathrm{nucl}}=\tilde{\sigma}_{\mathrm{ext}}=\frac{2A\tilde{\tau}_{\mathrm{wrap}}}{l\,h_{\mathrm{SB}}},\quad\text{giving}$
(12)
$\displaystyle\frac{A}{l}=\Lambda_{\mathrm{crit}}=\frac{\tilde{\tau}_{\mathrm{nucl}}}{\tilde{\tau}_{\mathrm{wrap}}}h_{\mathrm{SB}}.$
(13)
This derivation also works in the case of externally applied shear stress, by
replacing $\sigma_{\mathrm{ext}}$ with $2\tau_{\mathrm{ext}}$.
Assuming that nucleating a new shear band at the interface and propagating the
wrapping shear band along the interface have similar critical stresses, we can
simplify Eq. 13 to
$\Lambda_{\mathrm{crit}}\approx h_{\mathrm{SB}}.$ (14)
Shear bands in Cu-Zr-based glasses have widths of around
$10\,\mathrm{nm}$,Ritter and Albe (2011) which fits to the
$\Lambda_{\mathrm{crit}}=12.65\,\mathrm{nm}$ observed in our model systems. As
stated earlier, the wrapping mechanism becomes more favorable again if the
shear band is longer before it hits the precipitate. The reason is that this
gives the shear band time to deviate slightly from its path, so that it does
not hit the precipitate centrally, thus effectively reducing $A$. In practice,
this is not a big problem, as the free shear-band length is constrained to
approximately $l$ anyway due to the distribution of precipitates in the
sample. While this derivation is only approximate, it seems to be sufficient
to explain the observed phenomena and guide future efforts in tuning the
mechanical behavior of crystal–glass composites. Furthermore, it is easily
possible to explain the fourth mechanism, a slip transfer into the crystalline
phase. The critical stress for heterogeneous nucleation of a dislocation in
the precipitate $\tilde{\tau}_{\mathrm{disl}}$ must be provided by the shear
band via $\tau_{\mathrm{SB}}=0.5\sigma_{\mathrm{ext}}$. If
$\tilde{\tau}_{\mathrm{disl}}<\tilde{\tau}_{\mathrm{nucl}}$, we can simply
replace $\tilde{\tau}_{\mathrm{nucl}}$ by $\tilde{\tau}_{\mathrm{disl}}$ in
Eqs. 12 and 13, thereby replacing the blocking mechanism with the plastic
deformation of the precipitate. A lowered $\tilde{\tau}_{\mathrm{disl}}$ also
lowers $\Lambda_{\mathrm{crit}}$.
A simple deflection of the shear band is not observed and seems unlikely, as
any deviation of the shear band from its path leads to a reduced resolved
shear stress and thereby to a driving force to put it back “on track.”
Contrary to the experiment, a change of shear-band path towards the
precipitate was also not observed. Because the precipitates in our simulations
are inserted artificially and do not have a large stress field around them,
this seems reasonable. In thermally grown precipitates, a stress field due to
density mismatch between glass and crystal seems likely. Still, due to the
geometry of the MD simulations, the shear band has two equivalent propagation
pathways from the notch but always chooses the one leading towards the
precipitate. This seems to be a weaker form of the attraction observed
experimentally.
With these results, we can attempt an explanation of the experimentally
observed phenomena. First of all, the blocking of the shear band is a one-to-
one correspondence between simulation and experiment. Comparing Fig. 3(e) with
Fig. 7, we can see that the path of the shear band looks identical. The shear
band wraps partly around the precipitate but is then blocked and does not
propagate. Contrary to the simulation, no fully formed shear band but only a
small shear-band-like region appears at the opposite crystal–glass interface.
This may be a result either of the more complex stress state in the experiment
or the fact that other precipitates are available at which the new shear band
may nucleate. For the winding shear-band path, we can now exclude a simple
deflection as discussed above. A possible explanation would be the concurrent
nucleation of nascent shear bands at the crystal–glass interfaces which grow
together into a single mature shear band. While the interfaces are known to be
nucleation sources for shear bands,Albe _et al._ (2013); Zaheri _et al._
(2014); Wang _et al._ (2014a) our simulations show that the nucleation of a
shear band at a stress concentrator like a notch or a crack always takes
precedence to nucleation at interfaces or surfaces. The shear bands shown in
the TEM images all originate from crack tips, making it unlikely that the
shear band shown in Fig. 2 consists of several concurrently nucleated shear
bands. This leaves the explanation that this winding path is a series of
subsequent blocking and renucleation events.
The observed $\Lambda_{\mathrm{crit}}$ corresponds to precipitate diameters
somewhere between $20\,\mathrm{nm}$ and $40\,\mathrm{nm}$, depending on the
interparticle distance. This critical diameter is on the order of magnitude
reported in several experimental studies for twinning in B2 crystals in Cu-Zr-
Al-based metallic glasses of $20\pm 5\,\mathrm{nm}$.Pauly _et al._ (2010b);
Kuo _et al._ (2014) It also fits an experimental work on Al-based glasses,
where crystallites growing during deformation are sheared apart when they
reach a critical size of about $10\,\mathrm{nm}$.Hebert _et al._ (2006)
Cu50Zr45Ti5 metallic glasses exhibit a critical size of about $9\,\mathrm{nm}$
for twinning of B2 precipitates.Wang _et al._ (2014b) The interparticle
distances in these experiments are on the same order of magnitude as for our
simulations. This supports an explanation of a transition from wrapping to
slip transfer in these systems.
Figure 14 summarizes the competition between the different mechanisms.
Mechanical dissolution of the crystalline particles occurs only if their size
is comparable to the shear-band size.Lund and Schuh (2007) With further
increasing precipitate sizes, the shear band can still wrap around the
obstacle until the size reaches a threshold value. This critical size also
depends on the precipitate distance, as discussed before, expressed in the
parameter $\Lambda_{\mathrm{crit}}(d,l)$. If wrapping is no longer possible, a
precipitate which reacts only elastically to the applied stress will block the
shear band. If the precipitate is susceptible to plastic deformation, slip
transfer into the precipitate will take place.
Concerning the mechanical performance of such in situ composites with
crystalline precipitates that originate from nucleation and growth within the
glass, the current results suggest that the discussed geometrical effects
serve to improve the macroscopic mechanical performance. None of the presented
mechanisms seem to inhibit the percolation of a critical shear band, yet,
catastrophic slip along a shear band leading to complete failure is delayed in
the case of winding shear bands due to the increase of the shear-band path
length as well as the raised activation barriers for slip along shear bands
that have a more complex topology. The wrapping mechanism does not pose an
obstacle to shear-band propagation, but can be avoided by appropriate
adjustment of the crystalline volume fraction and precipitate diameter. “Soft”
precipitates additionally open possibilities to adjust the plastic deformation
by participating in it. Consistently, by increasing the volume fraction of the
ductile crystalline phase, the constraints on shear-band propagation can be
increased, immediate failure can be prevented (cf. Ref. Hofmann _et al._ ,
2008), and the composite displays macroscopic mechanical behavior according to
a mixing rule, further allowing one to tailor the properties.
## V Conclusions
Using TEM imaging, we observe shear-band bending around or close to
precipitates, an attraction of shear bands to the precipitates, and shear
bands being blocked by precipitates. MD simulations reveal that the shear-band
bending is most likely the result of the subsequent blocking and renucleation
of shear bands. Moreover, we identify shear bands wrapping around precipitates
and slip transfer into the crystalline phase. By describing the competition
between the critical stress for wrapping, the nucleation of a new shear band,
and the nucleation of dislocations in the crystal, we could derive a mechanism
map for metallic glasses with nanocrystalline precipitates. This detailed
description of shear-band propagation not only helps to understand the
mechanical failure of these composites but also aids in tuning them.
Figure 15: Generalized stacking-fault energies (top) and the resulting shear
stresses (bottom) for fcc copper. The DFT values for the stress curve are from
Ogata et al.,Ogata _et al._ (2002) and the corresponding stacking-fault
energies are approximated by using a numerical integration of the stress data.
###### Acknowledgements.
The authors thank J. Bünz and M. Gerlitz for help with sample preparation and
DSC characterization. Financial support by the Deutsche Forschungsgemeinschaft
(DFG) through project grants nos. AL 578/13-1, AL 578/6-2, and WI 1899/12-1 is
gratefully acknowledged. Computing time was granted by the John von Neumann
Institute for Computing (NIC) and provided on the supercomputer JUROPA at
Jülich Supercomputing Centre (JSC), as well as by the Gauss Centre for
Supercomputing (GCS) through the NIC on the GCS share of the supercomputer
JUQUEEN at JSC. GCS is the alliance of the three national supercomputing
centers HLRS (Universität Stuttgart), JSC (Forschungszentrum Jülich), and LRZ
(Bayerische Akademie der Wissenschaften), funded by the German Federal
Ministry of Education and Research and the German State Ministries for
Research of Baden-Württemberg, Bayern, and Nordrhein-Westfalen. Additional
computing time was made available by the Technische Universität Darmstadt on
the Lichtenberg cluster.
*
## Appendix A Generalized stacking-fault energy in fcc copper
The generalized stacking-fault energy in fcc copper is calculated by using
both Mendelev Mendelev _et al._ (2009) and Ward Zhou _et al._ (2004); Ward
_et al._ (2012) potentials. Density-functional theory (DFT) calculations from
Ogata et al. Ogata _et al._ (2002) using the same method are used for
comparison. This is plotted in Fig. 15. The Ward potential much more
accurately describes the stacking-fault energy than the Mendelev potential,
which has a critical stress for homogeneous dislocation nucleation which is
more than two times too high.
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|
Staggered Fermions
Maarten Golterman
Department of Physics and Astronomy, San Francisco State University,
San Francisco, CA 94132, USA
> These notes are based on a series of lectures on staggered fermions given at
> the Centre de Physique Théorique, Luminy, in Marseille, France, January
> 17-25, 2024.
## I Introduction
These lecture notes present the lattice formulation of vectorlike gauge
theories such as QED and QCD employing staggered fermions, which are
extensively used in numerical simulations of QCD. While many aspects of the
use of staggered fermions are discussed, these notes are not intended as a
comprehensive review. Several topics and alternative formulations are only
briefly mentioned, or not covered at all (see Sec. IX for pointers to
additional reading). Likewise, while important references on staggered
fermions and related topics have been included in the bibliography, the list
of references is not complete. Staggered fermions were first introduced in a
hamiltonian formalism [2, 3], but here we will only be concerned with the
euclidean theory. These notes complement the review of Ref. [4].
## II Staggered fermion action
In this section we begin with the naive fermion lattice action, from which we
derive the staggered fermion action through spin diagonalization. We then
derive the Feynman rules, show that the classical continuum limit reproduces
the expected gauge theory, and we list and discuss the symmetries of staggered
fermions. We end with the formulation of the staggered theory on the so-called
“taste basis.”
### II.1 Naive fermions
The naive fermion action is given by
$S={1\over
2}\sum_{x\mu}\left(\overline{\psi}(x)\gamma_{\mu}U_{\mu}(x)\psi(x+\mu)-\overline{\psi}(x+\mu)\gamma_{\mu}U^{\dagger}_{\mu}(x)\psi(x)\right)+m\sum_{x}\overline{\psi}(x)\psi(x)\
.$ (1)
Points on the hypercubic lattice are labeled by $x$, and $x+\mu$ is the point
one lattice spacing away from $x$ in the $+\mu$ direction. Most of the time we
will set the lattice spacing $a=1$; if not, this point will be denoted as
$x+a\mu$. $\psi(x)$ is a Dirac spinor transforming in the fundamental
irreducible representation (irrep) of $SU(3)$, gauged by the link variables
$U_{\mu}(x)$. We are in four-dimensional euclidean space, and the Dirac
matrices $\gamma_{\mu}$ are hermitian. The parameter $m$ is the bare mass;
later on, we will generalize this to a mass matrix $M$ when we consider
$N_{f}>1$ flavors of staggered fermions. For now, we take $N_{f}=1$.
Let us obtain the free fermion propagator in momentum space. We define the
Fourier transform of $\psi$ and $\overline{\psi}$ by
$\psi(x)=\int_{p}e^{ipx}\tilde{\psi}(p)\
,\qquad\overline{\psi}(x)=\int_{p}e^{-ipx}\tilde{\overline{\psi}}(p)\ ,$ (2)
where
$\int_{p}\equiv\int_{-\pi}^{\pi}\frac{d^{4}p}{(2\pi)^{4}}\ .$ (3)
In momentum space, the free action, obtained by setting $U_{\mu}(x)=1$, is
$S_{\rm
free}=\int_{p}\sum_{\mu}\tilde{\overline{\psi}}(p)\,i\gamma_{\mu}\sin(p_{\mu})\tilde{\psi}(p)+m\int_{p}\tilde{\overline{\psi}}(p)\tilde{\psi}(p)\
.$ (4)
We see that the inverse propagator (at $m=0$) not only has a zero at $p=0$,
but at all momenta
$p=\pi_{A}\ ,$ (5)
where $\pi_{A}$ has components $0$ or $\pi$. There are 16 such vectors, and
the index $A$ thus runs from 1 to 16. For $p_{\mu}=\tilde{p}_{\mu}+\pi_{A}$,
$\sin(p_{\mu})=\sin(\tilde{p}_{\mu}+\pi_{A\mu})=S^{A}_{\mu}\sin(\tilde{p}_{\mu})\
,$ (6)
which defines the signs $S^{A}_{\mu}=e^{i\pi^{A}_{\mu}}$. Therefore, the
inverse propagator near $p=\pi_{A}$ for small $\tilde{p}$ equals
$S^{-1}(p)=\sum_{\mu}i\gamma_{\mu}S^{A}_{\mu}\tilde{p}_{\mu}+m\ ,$ (7)
showing that the continuum limit of the naive fermion action contains 16
relativistic fermions. These are the so-called fermion “species doublers.” In
particular, the matrices
$\gamma^{A}_{\mu}=\gamma_{\mu}S^{A}_{\mu}$ (8)
satisfy the Dirac algebra
$\\{\gamma_{\mu}^{A},\gamma_{\nu}^{A}\\}=2\delta_{\mu\nu}$, and are unitarily
equivalent to the original set. For the relativistic fermion near $p=\pi_{A}$
we can define
$\gamma_{5}^{A}=\gamma^{A}_{1}\gamma^{A}_{2}\gamma^{A}_{3}\gamma^{A}_{4}=S_{1}^{A}S_{2}^{A}S_{3}^{A}S_{4}^{A}\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}=\pm\gamma_{5}\
,$ (9)
with the plus (minus) sign if the number of components of $\pi_{A}$ equal to
$\pi$ is even (odd). This implies that half of the doublers have axial charge
$+1$, and half of them have axial charge $-1$. The multiplet formed by the 16
species is thus anomaly free. This explains how the naive fermion action can
regulate a Dirac fermion while maintaining exact chiral symmetry (for $m=0$)
[5].
We note that the action (1) is invariant under hypercubic rotations in the
$\mu\nu$ plane with the matrix
$R_{1\over 2}=e^{{1\over 2}\omega\gamma_{\mu}\gamma_{\nu}}\ ,\qquad\mu<\nu$
(10)
with $\omega=\pi/2$ acting on the Dirac spinor $\psi$.111Note that the
generators ${1\over 2}\gamma_{\mu}\gamma_{\nu}$, $\mu<\nu$, are anti-
hermitian.
### II.2 Spin diagonalization
We now carry out the unitary basis transformation
$\psi(x)=\gamma_{1}^{x_{1}}\gamma_{2}^{x_{2}}\gamma_{3}^{x_{3}}\gamma_{4}^{x_{4}}\chi(x)\
,\qquad\overline{\psi}(x)=\bar{\chi}(x)\gamma_{4}^{x_{4}}\gamma_{3}^{x_{3}}\gamma_{2}^{x_{2}}\gamma_{1}^{x_{1}}\
.$ (11)
This brings the action (1) into the form
$S={1\over
2}\sum_{x\mu}\eta_{\mu}(x)\left(\bar{\chi}(x)U_{\mu}(x)\chi(x+\mu)-\bar{\chi}(x+\mu)U^{\dagger}_{\mu}(x)\chi(x)\right)+m\sum_{x}\bar{\chi}(x)\chi(x)\
,$ (12)
with the phase factors $\eta_{\mu}(x)$ defined by
$\eta_{\mu}(x)=(-1)^{x_{1}+\dots+x_{\mu-1}}\ .$ (13)
We can make different choices in Eq. (11) by reordering the $\gamma$ matrices,
and this will lead to different results for the phase factors $\eta_{\mu}(x)$.
However, all essential properties of these phase factors that we will use
below are independent of this choice.
In Eq. (12) the spin matrices $\gamma_{\mu}$ have been diagonalized. We now
change the theory by dropping the spin index on the fields $\chi$ and
$\bar{\chi}$. This defines the staggered fermion action, with the “one-
component” fields222Of course, these fields still carry a color index and
possibly a flavor index, so “one-component” here only refers to spin. $\chi$
and $\bar{\chi}$ (for a discussion of the euclidean action and earlier
references, see Ref. [6]).
The reduction from the naive to the staggered theory reduces the number of
degrees of freedom by a factor 4, so one might expect the staggered theory to
only have 4 species doublers in the continuum limit. However, as this
reduction was obtained by spin diagonalization, it is not obvious that this is
the correct interpretation of the staggered theory. In order to demonstrate
this, one first should show that the action (12) has the correct classical
continuum limit, and then that quantum corrections do not change this
conclusion. If the theory is asymptotically free, so that the continuum limit
is taken at the gaussian fixed point, the latter question can be studied in
weak-coupling perturbation theory (WCPT), and we will do so in the next
chapter.
In order to see that the staggered action (12) may have a sensible continuum
limit, we will define several operators on the field $\chi$. First, we define
$T_{\pm\mu}:\ \chi(x)\to\eta_{\mu}(x)\chi(x\pm\mu)\ ,$ (14)
from which it follows that
$T_{\mu}T_{\nu}:\
\chi(x)\to\eta_{\mu}(x)\eta_{\nu}(x+\mu)\chi(x+\mu+\nu)=\varepsilon_{\mu\nu}(-1)^{x_{1}+\dots+x_{\mu-1}+x_{1}+\dots+x_{\nu-1}}\chi(x+\mu+\nu)\
,$ (15)
with
$\varepsilon_{\mu\nu}=\Biggl{\\{}\begin{array}[]{cc}+1\ ,&\mu\geq\nu\\\ -1\
,&\mu<\nu\end{array}\ .$ (16)
From this, it follows that
$\\{T_{\mu},T_{\nu}\\}:\ \chi(x)\to 2\delta_{\mu\nu}\chi(x+2\mu)\ .$ (17)
We see that the square of $T_{\mu}$ is a normal translation over two lattice
spacings in the $\mu$ direction. Moreover the $T_{\mu}$ themselves satisfy the
Dirac algebra modulo these translations by $2\mu$. The irreps of the group
generated by the $T_{\mu}$ live in momentum space, as $T_{\mu}$ involves a
translation. If we define $\hat{T}_{\mu}$ as the translation (14) modding out
the translations over $2\mu$, the $\hat{T}_{\mu}$ generate a 32-element finite
group isomorphic with that generated by the Dirac matrices $\gamma_{\mu}$. We
will denote this group as $\Gamma_{4}$.333Not to be confused with the matrix
$\Gamma_{4}$ that will be introduced later on. The context should help avoid
confusion.
This group has 16 one-dimensional irreps (obtained by mapping the matrices
$\gamma_{\mu}$ onto $\pm 1$ in all possible ways), and one four-dimensional
irrep generated by the $\gamma_{\mu}$. Since the 16 fields $\chi(x)$ within a
hypercube form a 16-dimensional representation of $\Gamma_{4}$, this
representation is reducible (but it will be irreducible under the full lattice
symmetry group, see Sec. II.4). We thus expect that a basis transformation
exists such that the $\hat{T}_{\mu}$ can be written as
$\hat{T}_{\mu}=\gamma_{\mu}\times 1_{4}\ ,$ (18)
with $1_{4}$ the $4\times 4$ unit matrix.
We expect similar transformations to exist of the form $1_{4}\times\xi_{\nu}$,
with $\xi_{\nu}$ also satisfying the Dirac algebra. Indeed, we can define
shifts [7, 8]
$S_{\pm\mu}:\ \chi\to\zeta_{\mu}(x)\chi(x\pm\mu)\ ,$ (19)
and require the new phase factors $\zeta_{\mu}(x)$ to be such that
$[S_{\mu},T_{\nu}]=0\ .$ (20)
This implies that
$\zeta_{\mu}(x)\eta_{\nu}(x+\mu)=\eta_{\nu}(x)\zeta_{\mu}(x+\nu)\ ,$ (21)
which can be rewritten as
$\zeta_{\mu}(x+\nu)=\eta_{\nu}(x)\eta_{\nu}(x+\mu)\zeta_{\mu}(x)=\varepsilon_{\mu\nu}\zeta_{\mu}(x)\
,$ (22)
and thus
$\zeta_{\mu}(x)=(-1)^{x_{\mu+1}+\dots+x_{4}}\ .$ (23)
With $S_{\mu}$ acting on the gauge fields as
$S_{\mu}:\ U_{\nu}(x)\to U_{\nu}(x+\mu)\ ,$ (24)
$S_{\mu}$ is a symmetry of the action (12), and
$\\{S_{\mu},S_{\nu}\\}:\ \chi(x)\to 2\delta_{\mu\nu}\chi(x+2\mu)\ .$ (25)
We note the similarity to Eq. (17). We will refer to this symmetry as shift
symmetry. It is a crucial symmetry for staggered fermions, as we will see
below. In contrast, the $T_{\mu}$ are not a symmetry of the theory, but they
are helpful for the interpretation of the theory: as we will see in more
detail below, they “generate” the spin matrices $\gamma_{\mu}$ needed to
interpret the continuum limit as a theory of four Dirac fermions.
It is instructive to recast the operators $T_{\mu}$ and $S_{\mu}$ in momentum
space. We first define a subset of the vectors $\pi_{A}$:
$\displaystyle\pi_{\eta_{1}}$ $\displaystyle=$ $\displaystyle(0,0,0,0)\ ,$
(26) $\displaystyle\pi_{\eta_{2}}$ $\displaystyle=$ $\displaystyle(\pi,0,0,0)\
,$ $\displaystyle\pi_{\eta_{3}}$ $\displaystyle=$ $\displaystyle(\pi,\pi,0,0)\
,$ $\displaystyle\pi_{\eta_{4}}$ $\displaystyle=$
$\displaystyle(\pi,\pi,\pi,0)\ ,$
in terms of which
$\eta_{\mu}(x)=e^{i\pi_{\eta_{\mu}}x}\ .$ (27)
We then have
$\eta_{\mu}(x)\chi(x+\mu)=\int_{p}e^{ipx+i\pi_{\eta_{\mu}}x+ip_{\mu}}\tilde{\chi}(p)=\int_{p}e^{ipx}e^{ip_{\mu}}\tilde{\chi}(p+\pi_{\eta_{\mu}})\
,$ (28)
where we used that $(\pi_{\eta_{\mu}})_{\mu}=0$ and that the exponential has a
periodicity $2\pi$ in each direction because $x_{\mu}$ is integer on the
lattice. We now define the full Brillouin zone (BZ) as
$-\frac{\pi}{2}<p_{\mu}\leq\frac{3\pi}{2}\ ,$ (29)
and the reduced BZ as
$-\frac{\pi}{2}<\tilde{p}_{\mu}\leq\frac{\pi}{2}\ .$ (30)
We write $p=\tilde{p}+\pi_{A}$ and define
$\phi_{A}(\tilde{p})=\chi(\tilde{p}+\pi_{A})\ .$ (31)
It follows that
$T_{\mu}:\ \phi_{A}(\tilde{p})=\chi(\tilde{p}+\pi_{A})\to
S^{A}_{\mu}e^{i\tilde{p}_{\mu}}\chi(\tilde{p}+\pi_{A}+\pi_{\eta_{\mu}})=S_{\mu}^{A}(\hat{\pi}_{\eta_{\mu}})_{AB}e^{i\tilde{p}_{\mu}}\phi_{B}(\tilde{p})\
,$ (32)
with
$(\hat{\pi}_{\eta_{\mu}})_{AB}=\overline{\delta}(\pi_{\eta_{\mu}}+\pi_{A}+\pi_{B})\
,$ (33)
and $\overline{\delta}(\pi_{A})=1$ for $\pi_{A}=0$ (mod $2\pi$) and zero
otherwise. Defining also
$(S_{\mu})_{AB}=S_{\mu}^{A}\delta_{AB}$ (34)
and
$\Gamma_{\mu}=S_{\mu}\hat{\pi}_{\eta_{\mu}}\ ,$ (35)
Eq. (32) can be rewritten as
$T_{\mu}:\ \phi(\tilde{p})\to e^{i\tilde{p}_{\mu}}\Gamma_{\mu}\phi(\tilde{p})\
.$ (36)
From Eq. (17) it then follows that
$\\{\Gamma_{\mu},\Gamma_{\nu}\\}=2\delta_{\mu\nu}\ ,$ (37)
i.e., the $16\times 16$ matrices $\Gamma_{\mu}$ obey the Dirac algebra.
Analogously, we can define
$\displaystyle\pi_{\zeta_{1}}$ $\displaystyle=$ $\displaystyle(0,\pi,\pi,\pi)\
,$ (38) $\displaystyle\pi_{\zeta_{2}}$ $\displaystyle=$
$\displaystyle(0,0,\pi,\pi)\ ,$ $\displaystyle\pi_{\zeta_{3}}$
$\displaystyle=$ $\displaystyle(0,0,0,\pi)\ ,$ $\displaystyle\pi_{\zeta_{4}}$
$\displaystyle=$ $\displaystyle(0,0,0,0)\ ,$
and
$\displaystyle(\hat{\pi}_{\zeta_{\mu}})_{AB}$ $\displaystyle=$
$\displaystyle\overline{\delta}(\pi_{\zeta_{\mu}}+\pi_{A}+\pi_{B})\ ,$ (39)
$\displaystyle\Xi_{\mu}$ $\displaystyle=$ $\displaystyle
S_{\mu}\hat{\pi}_{\zeta_{\mu}}\ ,$
to rewrite Eq. (19) in momentum space as
$S_{\mu}:\ \phi(\tilde{p})\to e^{i\tilde{p}_{\mu}}\Xi_{\mu}\phi(\tilde{p})\ ,$
(40)
with $\Xi_{\mu}$ another set of $16\times 16$ matrices. From Eqs. (20) and
(25), it follows that
$\\{\Xi_{\mu},\Xi_{\nu}\\}=2\delta_{\mu\nu}\
,\qquad[\Gamma_{\mu},\Xi_{\nu}]=0\ .$ (41)
As in Eq. (18), we can find a basis transformation such that
$\Gamma_{\mu}\Xi_{\nu}\to\gamma_{\mu}\times\xi_{\nu}\ ,$ (42)
with $\gamma_{\mu}$ and $\xi_{\nu}$ both sets of $4\times 4$ matrices
satisfying the Dirac algebra. We note that
$\displaystyle[S_{\mu},\hat{\pi}_{\eta_{\mu}}]=[S_{\mu},\hat{\pi}_{\zeta_{\mu}}]=0\
,$ $\displaystyle S_{\mu}=S_{\mu}^{T}\
,\quad\hat{\pi}_{\eta_{\mu}}=\hat{\pi}_{\eta_{\mu}}^{T}\
,\quad\hat{\pi}_{\zeta_{\mu}}=\hat{\pi}_{\zeta_{\mu}}^{T}\quad\Rightarrow\quad\Gamma_{\mu}=\Gamma_{\mu}^{T}\
,\quad\Xi_{\mu}=\Xi_{\mu}^{T}\ .$
For an explicit representation, see Ref. [8].
### II.3 Feynman rules and classical continuum limit
Using Eq. (27), the free staggered action can be written in momentum space as
$S_{\rm free}={1\over
2}\int_{p}\int_{q}\tilde{\overline{\chi}}(q)\left(\sum_{\mu}\delta(p-q+\pi_{\eta_{\mu}})(e^{ip_{\mu}}-e^{-iq_{\mu}})+m\delta(p-q)\right)\tilde{\chi}(p)\
,$ (44)
from which we obtain the inverse propagator (using
$(\pi_{\eta_{\mu}})_{\mu}=0$ again)
$G^{-1}(p,-q)=\sum_{\mu}\delta(p-q+\pi_{\eta_{\mu}})i\sin(p_{\mu})+m\delta(p-q)\
,$ (45)
where $\delta$ is the periodic delta function with period $2\pi$ in each
direction. We now set
$p=\tilde{p}+\pi_{A}\ ,\qquad q=\tilde{q}+\pi_{B}\ ,$ (46)
to obtain
$\displaystyle G^{-1}(\tilde{p}+\pi_{A},-\tilde{q}-\pi_{B})$ $\displaystyle=$
$\displaystyle\sum_{\mu}\delta(\tilde{p}-\tilde{q}+\pi_{A}+\pi_{B}+\pi_{\eta_{\mu}})\,i\sin(p_{\mu})$
$\displaystyle\hskip 28.45274pt+m\delta(\tilde{p}-\tilde{q}+\pi_{A}+\pi_{B})$
$\displaystyle=$
$\displaystyle\delta(\tilde{p}-\tilde{q})\left(iS^{A}_{\mu}(\hat{\pi}_{\eta_{\mu}})_{AB}\sin(\tilde{p}_{\mu})+m\delta_{AB}\right)$
$\displaystyle\equiv$
$\displaystyle\delta(\tilde{p}-\tilde{q})S^{-1}_{AB}(\tilde{p})\ .$
The second line follows, because, from Eq. (30),
$-\frac{\pi}{2}<\tilde{p}_{\mu}-\tilde{q}_{\mu}<\frac{\pi}{2}$, with strict
inequalities. We thus find for the free propagator
$S(\tilde{p})=\left(\sum_{\mu}i\Gamma_{\mu}\sin(\tilde{p}_{\mu})+m\right)^{-1}\
,$ (48)
and the free action becomes
$S_{\rm
free}=\int_{\tilde{p}}\sum_{AB}\overline{\phi}_{A}(\tilde{p})\left(\sum_{\mu}i(\Gamma_{\mu})_{AB}\sin(\tilde{p}_{\mu})+m\delta_{AB}\right)\phi_{B}(\tilde{p})\
.$ (49)
This action is invariant under a group $U(4)$ generated by the $\Xi_{\mu}$
(all products of these matrices precisely give the generators of the group
$U(4)$), because this group commutes with $\Gamma_{\mu}$. Free staggered
fermions contain 4 degenerate species of Dirac spinors in the continuum limit.
We will refer to these 4 species as “tastes,” and we find that the free theory
has an exact $U(4)$ taste symmetry on the lattice. In the interacting theory,
the product of this $U(4)$ and lattice translations is broken to the discrete
symmetries $S_{\mu}$, cf. Eqs. (19), (24) and (40).
To obtain the one-gluon vertex, we expand
$U_{\mu}(x)=e^{igA_{\mu}(x)}=1+igA_{\mu}(x)-{1\over
2}\,g^{2}A_{\mu}^{2}(x)+\dots\ ,$ (50)
and Fourier transform
$A_{\mu}(x)=\int_{k}e^{ikx}\tilde{A}_{\mu}(k)\ .$ (51)
The one-gluon part of the action is thus
$S_{\rm
1-gluon}=ig\int_{p}\int_{q}\int_{k}\sum_{\mu}\tilde{\overline{\chi}}(q)\delta(p+k-q+\pi_{\eta_{\mu}}){1\over
2}(e^{ip_{\mu}}+e^{-iq_{\mu}})\tilde{A}_{\mu}(k)\tilde{\chi}(p)\ ,$ (52)
from which we read off the one-gluon vertex
$V^{m}_{\mu}(p,-q,k)=-ig\,\delta(p+k-q+\pi_{\eta_{\mu}})\,{1\over
2}(e^{ip_{\mu}}+e^{-iq_{\mu}})\,t_{m}\ ,$ (53)
where $\tilde{A}_{\mu}(k)=\sum_{m}\tilde{A}_{\mu}^{m}(k)t_{m}$, with $t_{m}$
the gauge group generators.
We can take the classical continuum limit by setting $p=\tilde{p}+\pi_{A}$,
$q=\tilde{q}+\pi_{B}$ with $\tilde{p}$, $\tilde{q}$ and $k$ small. Using
${1\over 2}(e^{ip_{\mu}}+e^{-iq_{\mu}})={1\over
2}(S_{\mu}^{A}e^{i\tilde{p}_{\mu}}+S_{\mu}^{B}e^{-i\tilde{q}_{\mu}})\approx{1\over
2}(S_{\mu}^{A}+S_{\mu}^{B})$ (54)
and
$\delta(\tilde{p}-\tilde{q}+k+\pi_{A}+\pi_{B}+\pi_{\eta_{\mu}})=\delta(\tilde{p}-\tilde{q}+k)(\hat{\pi}_{\eta_{\mu}})_{AB}$
(55)
for $\tilde{p}$, $\tilde{q}$ and $k$ small, the one-gluon vertex becomes
$V^{m}_{\mu}(\tilde{p}+\pi_{A},-\tilde{q}-\pi_{B},k)\approx-
ig\,\delta(\tilde{p}-\tilde{q}+k)(\Gamma_{\mu})_{AB}t_{m}\ ,$ (56)
which is the correct classical continuum limit. In particular, the classical
continuum limit is invariant under $U(4)_{\rm taste}$.
The two-gluon vertex can be read off using the $O(g^{2})$ part of the
expansion (50), with $\ell$ the momentum of the second gluon
$V^{m_{1}m_{2}}_{\mu_{1}\mu_{2}}(p,-q,k,\ell)={1\over
2}\,g^{2}\sum_{\mu}\delta(p+k+\ell-q+\pi_{\eta_{\mu}})\,{1\over
2}(e^{ip_{\mu}}-e^{-iq_{\mu}})\,\delta_{\mu\mu_{1}}\delta_{\mu\mu_{2}}\,{1\over
2}\\{t_{m_{1}},t_{m_{2}}\\}\ .$ (57)
Because of the minus sign between the momentum-dependent phase factors, it is
straightforward to show that this vertex vanishes in the classical continuum
limit.
We end this subsection with the observation that if we take $k_{\mu}=\pi$ in
Eq. (53), and use
${1\over 2}(e^{ip_{\mu}}+e^{-iq_{\mu}})={1\over
2}(e^{ip_{\mu}}-e^{-ip_{\mu}})=i\sin(p_{\mu})=iS_{\mu}^{A}\sin(\tilde{p}_{\mu})$
(58)
for this choice of $k$, we find that the one-gluon vertex for a gluon with
momentum $k_{\mu}=\pi$ is suppressed for small $\tilde{p}_{\mu}$. We will
return to this point in Sec. VIII. For the complete Feynman rules, we refer to
Ref. [8].
### II.4 More symmetries
We have already identified one symmetry of the staggered action, the shifts
defined by Eqs. (19) and (24). There are more symmetries of course, which we
will discuss in this subsection. For more detail, see Ref. [8].
#### II.4.1 Hypercubic rotations
The action is invariant under hypercubic rotations, which are rotations over
an angle $\pi/2$ in each plane. Clearly, the action (49) is invariant under
rotations $R_{\rho\sigma}$ in the $\rho\sigma$ plane over an angle $\pi/2$
which takes $\tilde{p}_{\rho}\to\tilde{p}_{\sigma}$ and
$\tilde{p}_{\sigma}\to-\tilde{p}_{\rho}$, and
$\phi(\tilde{p})\to
e^{\frac{\pi}{4}\Gamma_{\rho}\Gamma_{\sigma}}\phi(R_{\rho\sigma}^{-1}\tilde{p})\qquad(\mbox{tentative})\
.$ (59)
However, this is in the free theory, which has more symmetry than the
interacting theory. Noting that the operators $T_{\mu}$ and $S_{\mu}$ are
expected to transform similarly under hypercubic rotations, we replace Eq.
(59) by
$\phi(\tilde{p})\to
e^{\frac{\pi}{4}\Gamma_{\rho}\Gamma_{\sigma}}e^{\frac{\pi}{4}\Xi_{\rho}\Xi_{\sigma}}\phi(R_{\rho\sigma}^{-1}\tilde{p})\
.$ (60)
Of course, the gauge fields have to be rotated accordingly, and Eq. (60) can
be translated back to position space. The form of rotations on the original
staggered fields $\chi(x)$ and $\bar{\chi}(x)$ as well as the links
$U_{\mu}(x)$ is given in Ref. [8]. Equation (60), along with the
transformation on the gauge fields, constitutes the hypercubic rotational
symmetry of the staggered action. These rotations form the discrete hypercubic
subgroup of a group $SO(4)$, which is the diagonal subgroup of $SO(4)_{\rm
spin}\times SO(4)_{\rm taste}$, with $SO(4)_{\rm taste}$ the unique $SO(4)$
subgroup of $SU(4)_{\rm taste}$.
The field $\phi$ transforms in a bosonic irrep of Eq. (60). However, we can
define a different group of lattice symmetries under which it transforms as a
fermion field. Since we have shift and rotational symmetry, we can define a
composite symmetry
$\tilde{R}_{\rho\sigma}=R_{\rho\sigma}S_{\rho}\ ,$ (61)
which can be visualized as rotations around the center of the hypercube based
at the origin. Using that
$R_{\rho\sigma}^{-1}S_{\rho}R_{\rho\sigma}=S_{\sigma}\ ,\qquad
R_{\rho\sigma}^{-1}S_{\sigma}R_{\rho\sigma}=-S_{\rho}\ ,$ (62)
one can prove that, modulo lattice translations over an even number of lattice
spacings in any direction,
$(\tilde{R}_{\rho\sigma})^{4}=-1\ ,$ (63)
i.e., that a rotation over $2\pi$ exists which is equal to $-1$, as one would
expect for a fermionic theory. In contrast, the fourth power of
$R_{\rho\sigma}$ equals $+1$. In the construction of the symmetry (61) shift
symmetry plays a key role. In momentum space, we find that
$\tilde{R}_{\rho\sigma}=e^{i\tilde{p}_{\rho}}e^{\frac{\pi}{4}\Gamma_{\rho}\Gamma_{\sigma}}\frac{1}{\sqrt{2}}(\Xi_{\rho}-\Xi_{\sigma})=-ie^{i\tilde{p}_{\rho}}e^{\frac{\pi}{4}\Gamma_{\rho}\Gamma_{\sigma}}e^{i\frac{\pi}{2}\frac{1}{\sqrt{2}}(\Xi_{\rho}-\Xi_{\sigma})}\
,$ (64)
using
$\left(\frac{1}{\sqrt{2}}(\Xi_{\rho}-\Xi_{\sigma})\right)^{2}=1\ .$ (65)
#### II.4.2 Axis reversal
Axis reversal $I_{\rho}$ in the $\rho$ direction takes $x_{\rho}\to-x_{\rho}$,
or $\tilde{p}_{\rho}\to-\tilde{p}_{\rho}$ in momentum space. Again, we expect
symmetry under the interchange $\Gamma_{\mu}\leftrightarrow\Xi_{\mu}$, and in
momentum space,
$\phi(\tilde{p})\to\Gamma_{\rho}\Gamma_{5}\Xi_{\rho}\Xi_{5}\phi(I_{\rho}\tilde{p})\
,$ (66)
where
$\Gamma_{5}=\Gamma_{1}\Gamma_{2}\Gamma_{3}\Gamma_{4}\
,\qquad\Xi_{5}=\Xi_{1}\Xi_{2}\Xi_{3}\Xi_{4}\ .$ (67)
This form follows because we need the matrix acting on $\phi$ to commute with
$\Gamma_{\mu}$ for $\mu\neq\rho$ and anticommute with $\Gamma_{\rho}$, because
$\sin(I_{\rho}\tilde{p}_{\rho})=-\sin(\tilde{p}_{\rho})$ (cf. Eq. (49)). In
position space, we have that [8]
$I_{\rho}:\quad\chi(x)\to(-1)^{x_{\rho}}\chi(I_{\rho}x)\ .$ (68)
We can define a parity transformation,
$I_{s}=I_{1}I_{2}I_{3}\ ,$ (69)
but we note that this involves a taste transformation:
$I_{s}:\ \phi(\tilde{p})\to\Gamma_{4}\Xi_{4}\phi(I_{s}\tilde{p})\ .$ (70)
Thus, “tasteless” parity $P$ can be defined by combining this with $S_{4}$,
$P:\ \phi(\tilde{p})\to e^{i\tilde{p}_{4}}\Gamma_{4}\phi(I_{s}\tilde{p})\ ,$
(71)
or, in position space,
$P:\
\chi(x)\to(-1)^{x_{1}+x_{2}+x_{3}}\chi(I_{s}x+4)=\eta_{4}(x)\chi(I_{s}x+4)\ .$
(72)
Of course, the gauge field also transforms non-trivially under axis reversal
[8].
#### II.4.3 Fermion number
Because the action is bilinear in $\chi$ and $\bar{\chi}$, it is invariant
under
$\chi\to e^{i\alpha}\chi\ ,\qquad\bar{\chi}\to\bar{\chi}e^{-i\alpha}\ .$ (73)
#### II.4.4 $U(1)_{\epsilon}$ symmetry
For $m=0$, the action is invariant under [5]
$\chi(x)\to e^{i\beta\epsilon(x)}\chi(x)\
,\qquad\bar{\chi}(x)\to\bar{\chi}(x)e^{i\beta\epsilon(x)}\ ,$ (74)
in which
$\epsilon(x)=(-1)^{x_{1}+x_{2}+x_{3}+x_{4}}=e^{i\pi_{\epsilon}x}\
,\qquad\pi_{\epsilon}=(\pi,\pi,\pi,\pi)\ .$ (75)
Since this is only a symmetry of the action for $m=0$, we can interpret this
symmetry as a chiral symmetry. (As always, this interpretation depends on the
form of the mass term. For instance, for a one-link mass term,
$U(1)_{\epsilon}$ is a vectorlike symmetry [8]!)
Using that
$T_{4}T_{3}T_{2}T_{1}S_{-1}S_{-2}S_{-3}S_{-4}:\ \chi(x)\to\epsilon(x)\chi(x)\
,$ (76)
and using the fact that $T_{\pm\mu}$ ($S_{\pm\mu}$) generates a $\Gamma_{\mu}$
($\Xi_{\mu}$), we find that
$\hat{\pi}_{\epsilon}=\Gamma_{5}\Xi_{5}\ ,$ (77)
and thus that in momentum space $U(1)_{\epsilon}$ symmetry takes the form
$\phi(\tilde{p})\to e^{i\beta\Gamma_{5}\Xi_{5}}\phi(\tilde{p})\
,\qquad\overline{\phi}(\tilde{p})\to\overline{\phi}(\tilde{p})e^{i\beta\Gamma_{5}\Xi_{5}}\
.$ (78)
Indeed, this looks like an non-singlet axial transformation.
#### II.4.5 Charge conjugation
The staggered action is invariant under charge conjugation $C_{0}$:
$\displaystyle\chi(x)\to i\epsilon(x)\bar{\chi}^{T}(x)\
,\qquad\bar{\chi}(x)\to i\epsilon(x)\chi^{T}(x)\ ,$ (79) $\displaystyle
U_{\mu}(x)\to U^{*}_{\mu}(x)\ ,$
where the transpose acts on the color (and possible flavor) indices. The
factors $i\epsilon(x)$ are needed to make both the mass term and the kinetic
term invariant under this symmetry. This implies that charge conjugation
involves a taste transformation; in momentum space it looks like
$\phi(\tilde{p})\to i\Gamma_{5}\Xi_{5}\overline{\phi}^{T}(-\tilde{p})\
,\qquad\overline{\phi}(\tilde{p})\to i\Gamma_{5}\Xi_{5}\phi^{T}(-\tilde{p})\
.$ (80)
The hope is, of course, that the lattice symmetries are sufficient to restore
the full $SO(4)_{\rm rot}\,\times\,SU(4)_{\rm taste}$ in the continuum limit
without any fine tuning of counter terms. We have noted already that the
hypercubic rotations (60) are a discrete subgroup of the diagonal $SO(4)_{\rm
diag}$ of $SO(4)_{\rm rot}\,\times\,SO(4)_{\rm taste}$, with $SO(4)_{\rm
taste}\,\subset\,SU(4)_{\rm taste}$. Likewise, the group $\Gamma_{4}$
generated by shifts (modulo even translations) is a discrete subgroup of
$SU(4)_{\rm taste}$. Finally, $U(1)_{\epsilon}$ provides one exact (non-
singlet, in taste space) chiral symmetry. At the classical level, we have seen
that indeed the continuum limit coincides with QCD. We will revisit this
question at the one-loop level in the next chapter, and in terms of the
Symanzik effective theory in Sec. IV.
### II.5 Taste basis
Another basis that can sometimes be useful has been introduced a long time
ago, the so-called taste basis [9, 10]. We will review the construction here
for the free theory.
We define a unitary transformation
$\psi_{\alpha a}(y)=\frac{1}{2^{3/2}}\sum_{A}(\gamma_{A})_{\alpha
a}\chi(2y+A)\
,\qquad\overline{\psi}_{a\alpha}(y)=\frac{1}{2^{3/2}}\sum_{A}\bar{\chi}(2y+A)(\gamma_{A})^{\dagger}_{a\alpha}\
.$ (81)
Here $A$ runs over the 16 vectors with components $0$ or $1$. The $\psi$
fields live on a “coarse” lattice, labeled by lattice sites $y$, and
$\gamma_{A}=\gamma_{1}^{A_{1}}\gamma_{2}^{A_{2}}\gamma_{3}^{A_{3}}\gamma_{4}^{A_{4}}\
.$ (82)
The index $\alpha$ will be interpreted as a spin index, and the index $a$ as a
taste index below. In order to write the (free) one-component staggered
fermion action in terms of the fields $\psi$ and $\overline{\psi}$, we need to
invert Eq. (82). Using
${\rm tr}\,(\gamma_{B}^{\dagger}\gamma_{A})=4\delta_{AB}\ ,$ (83)
we find that
$\chi(2y+A)=\frac{1}{2^{5/2}}{\rm tr}\,(\gamma_{A}^{\dagger}\psi(y))\
,\qquad\bar{\chi}(2y+A)=\frac{1}{2^{5/2}}{\rm
tr}\,(\gamma_{A}\overline{\psi}(y))\ .$ (84)
We will also need the completeness relation
$\sum_{A}(\gamma_{A})_{\alpha
a}(\gamma_{A})^{\dagger}_{b\beta}=4\delta_{\alpha\beta}\delta_{ab}\ .$ (85)
Using this, it is straightforward to show that
$m\sum_{x}\bar{\chi}(x)\chi(x)=\frac{1}{8}m\sum_{y}\overline{\psi}(y)\psi(y)\
.$ (86)
The kinetic term is a little more involved:
$\displaystyle{1\over
2}\sum_{x\mu}\eta_{\mu}(x)(\bar{\chi}(x)\chi(x+\mu)-\bar{\chi}(x+\mu)\chi(x))$
$\displaystyle=$ $\displaystyle{1\over
2}\sum_{yA\mu}\eta_{\mu}(A)(\bar{\chi}(2y+A)\chi(2y+A+\mu)-\bar{\chi}(2y+A+\mu)\chi(2y+A))$
$\displaystyle=$ $\displaystyle{1\over
2}\sum_{yA\mu,A_{\mu}=0}\eta_{\mu}(A)(\bar{\chi}(2y+A)\chi(2y+A+\mu)-\bar{\chi}(2y+A+\mu)\chi(2y+A))$
$\displaystyle+{1\over
2}\sum_{yA\mu,A_{\mu}=1}\eta_{\mu}(A)(\bar{\chi}(2y+A)\chi(2y+A+\mu)-\bar{\chi}(2y+A+\mu)\chi(2y+A))$
$\displaystyle=$
$\displaystyle\frac{1}{2^{6}}\sum_{yA\mu,A_{\mu}=0}\eta_{\mu}(A)({\rm
tr}\,(\gamma_{A}\overline{\psi}(y)){\rm
tr}\,(\gamma_{A+\mu}^{\dagger}\psi(y))-{\rm
tr}\,(\gamma_{A+\mu}\overline{\psi}(y)){\rm
tr}\,(\gamma_{A}^{\dagger}\psi(y)))$
$\displaystyle+\frac{1}{2^{6}}\sum_{yA\mu,A_{\mu}=0}\eta_{\mu}(A)({\rm
tr}\,(\gamma_{A+\mu}\overline{\psi}(y)){\rm
tr}\,(\gamma_{A}^{\dagger}\psi(y+\mu))-{\rm
tr}\,(\gamma_{A}\overline{\psi}(y+\mu)){\rm
tr}\,(\gamma_{A+\mu}^{\dagger}\psi(y)))\ .$
We now use that
$\displaystyle\sum_{A,A_{\mu}=0}\eta_{\mu}(A)(\gamma_{A+\mu})_{\alpha
a}(\gamma_{A}^{\dagger})_{b\beta}$ $\displaystyle=$
$\displaystyle\sum_{A}\eta_{\mu}(A)(\gamma_{A+\mu})_{\alpha
a}(\gamma_{A}^{\dagger})_{b\beta}{1\over 2}((1+(-1)^{A_{\mu}})$
$\displaystyle={1\over
2}\sum_{A}(\gamma_{\mu})_{\alpha\gamma}(\gamma_{A})_{\gamma
a}(\gamma^{\dagger}_{A})_{b\beta}+{1\over
2}\sum_{A}(-1)^{A_{\mu}}\eta_{\mu}(A)\zeta_{\mu}(A)(\gamma_{A})_{\alpha
c}(\gamma_{\mu})_{ca}(\gamma^{\dagger}_{A})_{b\beta}$ $\displaystyle={1\over
2}\sum_{A}(\gamma_{\mu})_{\alpha\gamma}(\gamma_{A})_{\gamma
a}(\gamma^{\dagger}_{A})_{b\beta}+{1\over
2}\sum_{A}\epsilon(A)(\gamma_{A})_{\alpha
c}(\gamma_{\mu})_{ca}(\gamma^{\dagger}_{A})_{b\beta}$ $\displaystyle={1\over
2}\sum_{A}(\gamma_{\mu})_{\alpha\gamma}(\gamma_{A})_{\gamma
a}(\gamma^{\dagger}_{A})_{b\beta}+{1\over
2}\sum_{A}(\gamma_{5})_{\alpha\gamma}(\gamma_{A})_{\gamma
d}(\gamma_{5})_{dc}(\gamma_{\mu})_{ca}(\gamma^{\dagger}_{A})_{b\beta}$
$\displaystyle=2(\gamma_{\mu})_{\alpha\beta}\delta_{ab}+2(\gamma_{5})_{\alpha\beta}(\gamma_{5}\gamma_{\mu})_{ba}\
,$
and, likewise,
$\sum_{A,A_{\mu}=0}\eta_{\mu}(A)(\gamma_{A})_{\alpha
a}(\gamma_{A+\mu}^{\dagger})_{b\beta}=2(\gamma_{\mu})_{\alpha\beta}\delta_{ab}-2(\gamma_{5})_{\alpha\beta}(\gamma_{5}\gamma_{\mu})_{ba}\
.$ (89)
This allows us to write Eq. (II.5) as
$\displaystyle\frac{1}{2^{4}}{1\over 2}\sum_{y\mu}\Biggl{(}{\rm
tr}\,(\overline{\psi}(y)\gamma_{\mu}\psi(y+\mu)-\overline{\psi}(y+\mu)\gamma_{\mu}\psi(y))$
(90) $\displaystyle\hskip 36.98866pt+{\rm
tr}\,(\overline{\psi}(y)\gamma_{5}\psi(y+\mu)\gamma_{5}\gamma_{\mu}+\overline{\psi}(y+\mu)\gamma_{5}\psi(x)\gamma_{5}\gamma_{\mu}-2\overline{\psi}(y)\gamma_{5}\psi(y)\gamma_{5}\gamma_{\mu})\Biggr{)}\
.$
We see that in this form, the free action is that of a naive fermion, with the
addition of a spin ($\gamma_{5}$) and taste ($\gamma_{5}\gamma_{\mu}$) non-
diagonal Wilson-like mass term, which removes the doublers. The fermion field
$\psi$ is thus undoubled, and shows the explicit four-fold degeneracy of
staggered fermions in the continuum limit.
One can also translate the shift symmetry to the new basis. One obtains, in
the free theory, that
$S_{\mu}:\ \psi(y)\to{1\over 2}(\psi(x)+\psi(x+\mu))\gamma_{\mu}+{1\over
2}\gamma_{5}\gamma_{\mu}(\psi(x)-\psi(x+\mu))\gamma_{5}\approx\psi(y)\gamma_{\mu}\
,$ (91)
where the approximate equality on the right applies in the continuum limit. We
can identify a taste matrix $\xi_{\mu}=\gamma_{\mu}^{T}$ acting on the taste
index of the field $\psi$.
It is tempting to gauge Eq. (90) on the coarse lattice, instead of starting
from the one-component staggered action. However, it is clear that doing this
breaks shift symmetry,444Gauging also Eq. (91) does not make it into a
symmetry of the gauged version of Eq. (90). and we would thus end up with a
different interacting theory, with fewer lattice symmetries. We will see in
the next chapter that this would undo a key property of staggered fermions,
which is that the mass renormalizes multiplicatively.
## III One loop calculations
In this section, we calculate the one-loop vacuum polarization, and
demonstrate that it satisfies the Ward–Takahashi identity (WTI), which follows
from the gauge invariance of the lattice theory. We also consider the one-loop
fermion self energy, and show how staggered symmetries limit possible counter
terms. These examples illustrate the conjecture that lattice QCD (or QED) with
staggered fermions has a well-defined relativistic continuum limit, with no
need for additional counter terms in the lattice action.
### III.1 Vacuum polarization
In Fig. 1 we show the two diagrams contributing at one loop to the vacuum
polarization. For simplicity, we consider the case of QED with gauge coupling
$g$. Using the Feynman rules of Sec. II.3, the tadpole diagram is equal to
$\Pi^{\rm
tadpole}_{\mu\nu}(k,\ell)=-g^{2}\,\delta_{\mu\nu}\int_{pq}\delta(q-p+k-\ell+\pi_{\eta_{\mu}})\,{1\over
2}\left(e^{i(q+k-\ell)_{\mu}}-e^{-iq_{\mu}}\right)G(p,-q)\ .$ (92)
Setting $p=\tilde{p}+\pi_{A}$ and $q=\tilde{q}+\pi_{B}$ this becomes
$-g^{2}\delta_{\mu\nu}e^{\frac{i}{2}(k-\ell)_{\mu}}\\!\\!\int_{\tilde{p}\tilde{q}}\sum_{AB}\delta(\tilde{q}-\tilde{p}+k-\ell+\pi_{A}+\pi_{B}+\pi_{\eta_{\mu}})i\sin(\tilde{q}_{\mu}+\mbox{$\small{{1\over
2}}$}(k_{\mu}-\ell_{\mu}))S_{\mu}^{B}G(\tilde{p}+\pi_{A},-\tilde{q}-\pi_{B})\
.$ (93)
We now use that (cf. Eq. (II.3))
$G(\tilde{p}+\pi_{A},-\tilde{q}-\pi_{B})=\delta(\tilde{p}-\tilde{q})S_{AB}(\tilde{p})\
,$ (94)
which sets the combination $\tilde{q}-\tilde{p}$ in
$\delta(\tilde{q}-\tilde{p}+k-\ell+\pi_{A}+\pi_{B}+\pi_{\eta_{\mu}})$ equal to
zero. Then, we use that $k$ and $\ell$, being external momenta, are taken
small, so that we can factorize
$\delta(k-\ell+\pi_{A}+\pi_{B}+\pi_{\eta_{\mu}})=\delta(k-\ell)(\hat{\pi}_{\eta_{\mu}})_{AB}\
,$ (95)
and use Eq. (35) to arrive at
$\displaystyle\Pi^{\rm tadpole}_{\mu\nu}(k)$ $\displaystyle=$
$\displaystyle-g^{2}\delta_{\mu\nu}\int_{\tilde{q}}{\rm
tr}\,\left(\Gamma_{\mu}\frac{-i\sum_{\kappa}\Gamma_{\kappa}\sin(\tilde{q}_{\kappa})+m}{\sum_{\kappa}\sin^{2}(\tilde{q}_{\kappa})+m^{2}}\right)i\sin(\tilde{q}_{\mu})$
$\displaystyle=$
$\displaystyle-16g^{2}\delta_{\mu\nu}\int_{\tilde{q}}\frac{\sin^{2}(\tilde{q}_{\mu})}{\sum_{\kappa}\sin^{2}(\tilde{q}_{\kappa})+m^{2}}\
,$
where we omitted $\delta(k-\ell)$. We see that $\Pi^{\rm tadpole}_{\mu\nu}(k)$
does not actually depend on $k$, as one would expect for a tadpole diagram.
$\mu$$\nu$$\to k$$\to\ell$$p$$q$$\mu$$\nu$$q^{\prime}$$p^{\prime}$$p$$q$$\to
k$$\to\ell$ Figure 1: The two diagrams contributing to the one-loop vacuum
polarization. The left panel shows the tadpole diagram, the right panel the
sunset diagram.
The other diagram in Fig. 1 is
$\displaystyle\Pi^{\rm sunset}_{\mu\nu}(k,\ell)$ $\displaystyle=$
$\displaystyle
g^{2}\int_{pqp^{\prime}q^{\prime}}\delta(q^{\prime}-p+k+\pi_{\eta_{\mu}})\,{1\over
2}\left(e^{i(q^{\prime}+k)_{\mu}}+e^{-iq^{\prime}_{\mu}}\right)G(p,-q)$
$\displaystyle\hskip
34.14322pt\times\delta(q-p^{\prime}-\ell+\pi_{\eta_{\nu}})\,{1\over
2}\left(e^{i(q-\ell)_{\nu}}+e^{-iq_{\nu}}\right)G(p^{\prime},-q^{\prime})\ .$
Integrating over $p^{\prime}$ and $q^{\prime}$ and writing
$p=\tilde{p}+\pi_{A}$, $q=\tilde{q}+\pi_{B}$ makes this equal to
$\displaystyle\Pi^{\rm sunset}_{\mu\nu}(k,\ell)$ $\displaystyle=$
$\displaystyle g^{2}\int_{\tilde{p}\tilde{q}}\sum_{AB}{1\over
2}S_{\mu}^{A}\left(e^{i\tilde{p}_{\mu}}+e^{-i(\tilde{p}-k)_{\mu}}\right)\delta(\tilde{p}-\tilde{q})S_{AB}(\tilde{p})$
$\displaystyle\times{1\over
2}S_{\nu}^{B}\left(e^{i(\tilde{q}-\ell)_{\nu}}+e^{-i\tilde{q}_{\nu}}\right)G(\tilde{q}-\ell+\pi_{\eta_{\nu}}+\pi_{B},-\tilde{p}+k+\pi_{\eta_{\mu}}+\pi_{A})\
,$
where we used the inverse of Eq. (II.3). Using that we can set
$\tilde{q}=\tilde{p}$ because of the $\delta(\tilde{p}-\tilde{q})$ inside the
integrand, we use Eqs. (45) to write
$\displaystyle
G^{-1}(\tilde{p}-\ell+\pi_{\eta_{\nu}}+\pi_{B},-\tilde{p}+k+\pi_{\eta_{\mu}}+\pi_{A})=$
(99) $\displaystyle\hskip
28.45274pt\sum_{\kappa}\delta(-\ell+k+\pi_{\eta_{\kappa}}+\pi_{\eta_{\nu}}+\pi_{B}+\pi_{\eta_{\mu}}+\pi_{A})iS_{\kappa}^{\eta_{\nu}}S_{\kappa}^{B}\sin(\tilde{p}_{\kappa}-\ell_{\kappa})$
$\displaystyle\hskip
28.45274pt+m\delta(-\ell+k+\pi_{\eta_{\nu}}+\pi_{B}+\pi_{\eta_{\mu}}+\pi_{A})\
.$
Now, using that $k$ and $\ell$ are small,
$\displaystyle\delta(-\ell+k+\pi_{\eta_{\nu}}+\pi_{B}+\pi_{\eta_{\mu}}+\pi_{A})$
$\displaystyle=$
$\displaystyle\delta(k-\ell)\overline{\delta}(\pi_{\eta_{\nu}}+\pi_{B}+\pi_{\eta_{\mu}}+\pi_{A})$
$\displaystyle=$
$\displaystyle\delta(k-\ell)\sum_{C}\overline{\delta}(\pi_{\eta_{\nu}}+\pi_{B}+\pi_{C})\overline{\delta}(\pi_{\eta_{\mu}}+\pi_{A}+\pi_{C})$
$\displaystyle=$
$\displaystyle\delta(k-\ell)(\hat{\pi}_{\eta_{\nu}}\hat{\pi}_{\eta_{\mu}})_{BA}\
,$
and, likewise,
$\displaystyle\delta(-\ell+k+\pi_{\eta_{\kappa}}+\pi_{\eta_{\nu}}+\pi_{B}+\pi_{\eta_{\mu}}+\pi_{A})S_{\kappa}^{\eta_{\nu}}S_{\kappa}^{B}=$
(101) $\displaystyle\hskip
28.45274pt=\delta(k-\ell)\overline{\delta}(\pi_{\eta_{\nu}}+\pi_{B}+\pi_{\eta_{\kappa}}+\pi_{\eta_{\mu}}+\pi_{A})S_{\kappa}^{\eta_{\nu}}S_{\kappa}^{B}$
$\displaystyle\hskip
28.45274pt=\delta(k-\ell)\sum_{CD}\overline{\delta}(\pi_{\eta_{\nu}}+\pi_{B}+\pi_{D})\overline{\delta}(\pi_{\eta_{\kappa}}+\pi_{D}+\pi_{C})\overline{\delta}(\pi_{\eta_{\mu}}+\pi_{C}+\pi_{A})S_{\kappa}^{D}$
$\displaystyle\hskip
28.45274pt=\delta(k-\ell)(\hat{\pi}_{\eta_{\nu}}\Gamma_{\kappa}\hat{\pi}_{\eta_{\mu}})_{BA}\
.$
We thus obtain
$G(\tilde{p}-\ell+\pi_{\eta_{\nu}}+\pi_{B},-\tilde{p}+k+\pi_{\eta_{\mu}}+\pi_{A})=\delta(k-\ell)(\hat{\pi}_{\eta_{\mu}}S(\tilde{p}-\ell)\hat{\pi}_{\eta_{\nu}})_{BA}\
.$ (102)
Combining with the factors $S_{\nu}^{B}$ and $S_{\mu}^{A}$ in Eq. (III.1), we
arrive at
$\Pi^{\rm
sunset}_{\mu\nu}(k,\ell)=g^{2}e^{\frac{i}{2}(k_{\mu}-\ell_{\nu})}\delta(k-\ell)\int_{\tilde{p}}{\rm
tr}\,\left(S(\tilde{p})\Gamma_{\mu}S(\tilde{p}-k)\Gamma_{\nu}\right)\cos(\tilde{p}_{\mu}-\mbox{$\small{{1\over
2}}$}k_{\mu})\cos(\tilde{p}_{\nu}-\mbox{$\small{{1\over 2}}$}k_{\nu})\ .$
(103)
Using
${\rm
tr}\,(\Gamma_{\kappa}\Gamma_{\nu}\Gamma_{\lambda}\Gamma_{\mu})=16(\delta_{\kappa\nu}\delta_{\lambda\mu}-\delta_{\kappa\lambda}\delta_{\mu\nu}+\delta_{\kappa\mu}\delta_{\lambda\nu})\
,$ (104)
this equals
$\displaystyle\Pi^{\rm
sunset}_{\mu\nu}(k)=16g^{2}e^{\frac{i}{2}(k_{\mu}-k_{\nu})}$ (105)
$\displaystyle\int_{\tilde{p}}\frac{\delta_{\mu\nu}(\sum_{\kappa}\sin(\tilde{p}_{\kappa})\sin(\tilde{p}_{\kappa}-k_{\kappa})+m^{2})-\sin(\tilde{p}_{\mu})\sin(\tilde{p}_{\nu}-k_{\nu})-\sin(\tilde{p}_{\nu})\sin(\tilde{p}_{\mu}-k_{\mu})}{(\sum_{\kappa}\sin^{2}(\tilde{p}_{\kappa})+m^{2})(\sum_{\lambda}\sin^{2}(\tilde{p}_{\lambda}-k_{\lambda})+m^{2})}$
$\displaystyle\phantom{\int_{\tilde{p}}}\times\cos(\tilde{p}_{\mu}-\mbox{$\small{{1\over
2}}$}k_{\mu})\cos(\tilde{p}_{\nu}-\mbox{$\small{{1\over 2}}$}k_{\nu})\ ,$
where again we omitted $\delta(k-\ell)$. Formally, in the continuum we would
have obtained one quarter of this result (with
$\sin(\tilde{p}_{\mu})\to\tilde{p}_{\mu}$, $\cos(\tilde{p}_{\mu}-{1\over
2}k_{\mu})\to 1$, etc.), confirming that there are four species doublers
(tastes) in the staggered theory.
Defining
$\Pi_{\mu\nu}(k)=\Pi^{\rm sunset}_{\mu\nu}(k)+\Pi^{\rm tadpole}_{\mu\nu}(k)\
,$ (106)
we first show that this vanishes at $k=0$, thus demonstrating that the tadpole
diagram removes the quadratic divergence present in the sunset diagram. At
$k=0$,
$\Pi^{\rm
sunset}_{\mu\nu}(0)=16g^{2}\int_{\tilde{p}}\left(-2\frac{\sin(\tilde{p}_{\mu})\sin(\tilde{p}_{\nu})\cos(\tilde{p}_{\mu})\cos(\tilde{p}_{\nu})}{(\sum_{\kappa}\sin^{2}(\tilde{p}_{\kappa})+m^{2})^{2}}+\delta_{\mu\nu}\frac{\cos^{2}(\tilde{p}_{\mu})}{\sum_{\kappa}\sin^{2}(\tilde{p}_{\kappa})+m^{2}}\right)\
.$ (107)
Using
$-2\,\frac{\sin(\tilde{p}_{\mu})\cos(\tilde{p}_{\mu})}{(\sum_{\kappa}\sin^{2}(\tilde{p}_{\kappa})+m^{2})^{2}}=\frac{\partial}{\partial\tilde{p}_{\mu}}\frac{1}{\sum_{\kappa}\sin^{2}(\tilde{p}_{\kappa})+m^{2}}$
(108)
and partially integrating the first term, one can show that $\Pi^{\rm
sunset}_{\mu\nu}(0)$ is equal to minus $\Pi^{\rm tadpole}_{\mu\nu}(0)$. There
are no boundary terms, because the integrand is periodic in every direction
with period $\pi$.
Next, we demonstrate that the WTI is satisfied, which on the lattice takes the
form
$\sum_{\mu}(1-e^{-ik_{\mu}})\Pi_{\mu\nu}(k)=0\ .$ (109)
Using that
$(1-e^{-ik_{\mu}})e^{\frac{i}{2}k_{\mu}}\Gamma_{\mu}\cos(\tilde{p}_{\mu}-{1\over
2}k_{\mu})=S^{-1}(\tilde{p})-S^{-1}(\tilde{p}-k)\ ,$ (110)
one can simplify
$\displaystyle\sum_{\mu}(1-e^{-ik_{\mu}})\Pi^{\rm sunset}_{\mu\nu}(k)$
$\displaystyle\\!\\!=\\!\\!$ $\displaystyle
16g^{2}e^{-\frac{i}{2}k_{\nu}}\int_{\tilde{p}}{\rm
tr}\,(\Gamma_{\nu}(S(\tilde{p}-k)-S(\tilde{p}))\cos(\tilde{p}_{\nu}-\mbox{$\small{{1\over
2}}$}k_{\nu})$
$\displaystyle=-16ig^{2}e^{-\frac{i}{2}}\int_{\tilde{p}}\left(\frac{\sin(\tilde{p}_{\nu}-k_{\nu})}{\sum_{\kappa}\sin^{2}(\tilde{p}_{\kappa}-k_{\kappa})+m^{2}}-\frac{\sin(\tilde{p}_{\nu})}{\sum_{\kappa}\sin^{2}(\tilde{p}_{\kappa})+m^{2}}\right)\cos(\tilde{p}_{\nu}-\mbox{$\small{{1\over
2}}$}k_{\nu})$
$\displaystyle=-16ig^{2}e^{-\frac{i}{2}k_{\nu}}\int_{\tilde{p}}\frac{\sin(\tilde{p}_{\nu})}{\sum_{\kappa}\sin^{2}(\tilde{p}_{\kappa})+m^{2}}(\cos(\tilde{p}_{\nu}+\mbox{$\small{{1\over
2}}$}k_{\nu})-\cos(\tilde{p}_{\nu}-\mbox{$\small{{1\over 2}}$}k_{\nu})$
$\displaystyle=16g^{2}(1-e^{-ik_{\nu}})\int_{\tilde{p}}\frac{\sin^{2}(\tilde{p}_{\nu})}{\sum_{\kappa}\sin^{2}(\tilde{p}_{\kappa})+m^{2}}\
,$
where we again used that the integrand is periodic in $\tilde{p}_{\mu}$ with
period $\pi$. This equals $-\sum_{\mu}(1-e^{-ik_{\mu}})\Pi^{\rm
tadpole}_{\mu\nu}(k)$, thus proving Eq. (109).
Finally, in this subsection, we will indicate how one may actually go about
completing the calculation of the vacuuum polarization at one-loop. We start
from the expression
$\displaystyle\Pi_{\mu\nu}(k)=16g^{2}e^{\frac{i}{2}(k_{\mu}-k_{\nu})}$ (112)
$\displaystyle\int_{\tilde{p}}\Biggl{(}\frac{\delta_{\mu\nu}(\sum_{\kappa}\sin(\tilde{p}_{\kappa})\sin(\tilde{p}_{\kappa}-k_{\kappa})+m^{2})-\sin(\tilde{p}_{\mu})\sin(\tilde{p}_{\nu}-k_{\nu})-\sin(\tilde{p}_{\nu})\sin(\tilde{p}_{\mu}-k_{\mu})}{(\sum_{\kappa}\sin^{2}(\tilde{p}_{\kappa})+m^{2})(\sum_{\lambda}\sin^{2}(\tilde{p}_{\lambda}-k_{\lambda})+m^{2})}$
$\displaystyle\phantom{\int_{\tilde{p}}}\times\cos(\tilde{p}_{\mu}-\mbox{$\small{{1\over
2}}$}k_{\mu})\cos(\tilde{p}_{\nu}-\mbox{$\small{{1\over 2}}$}k_{\nu})$
$\displaystyle\phantom{\int_{\tilde{p}}}-\frac{\delta_{\mu\nu}(\sum_{\kappa}\sin^{2}(\tilde{p}_{\kappa})+m^{2})-2\sin(\tilde{p}_{\mu})\sin(\tilde{p}_{\nu})}{(\sum_{\kappa}\sin^{2}(\tilde{p}_{\kappa})+m^{2})^{2}}\cos(\tilde{p}_{\mu})\cos(\tilde{p}_{\nu})\Biggr{)}\
,$
where the last line, obtained by setting $k=0$ in the middle two lines, is
equal to the tadpole contribution because $\Pi_{\mu\nu}(0)=0$. This form is
useful because it shows explicitly that $\Pi_{\mu\nu}(k)$ is logarithmically
divergent; the last line explicitly subtracts the quadratic divergence present
in $\Pi^{\rm sunset}_{\mu\nu}(k)$. Moreover, by shifting $\tilde{p}-{1\over
2}k\to\tilde{p}$, which is allowed in an integral that is only logarithmically
divergent, one can show that the integral is quadratic in $k$ in the continuum
limit (modulo, of course, a logarithmic dependence on $k$).
Returning to Eq. (112), we split the integral into two regions:
$\int_{\tilde{p}}=\int_{|\tilde{p}|<\delta}+\int_{|\tilde{p}|>\delta}\ ,$
(113)
where we take
$|ak|\sim|am|\ll\delta\ll 1\ ,$ (114)
and we take the limit $a\to 0$ first, followed by $\delta\to 0$. (In Eq. (114)
we wrote $k$ and $m$ in physical units, temporarily restoring the lattice
spacing $a$.) We will refer to these two regions as the “inner” and “outer”
regions.
In the inner region, we may replace the integrand by its covariant form:
$\displaystyle\Pi^{\rm inner}_{\mu\nu}(k)$ $\displaystyle=$ $\displaystyle
16g^{2}\int_{|\tilde{p}|<\delta}\Biggl{(}\frac{\delta_{\mu\nu}(\tilde{p}(\tilde{p}-k)+m^{2})-\tilde{p}_{\mu}(\tilde{p}_{\nu}-k_{\nu})-\tilde{p}_{\nu}(\tilde{p}_{\mu}-k_{\mu})}{(\tilde{p}^{2}+m^{2})((\tilde{p}-k)^{2}+m^{2})}$
$\displaystyle\phantom{16g^{2}\int_{|\tilde{p}|<\delta}\Biggl{(}}-\frac{\delta_{\mu\nu}(\tilde{p}^{2}+m^{2})-2\tilde{p}_{\mu}\tilde{p}_{\nu}}{(\tilde{p}^{2}+m^{2})^{2}}\Biggr{)}$
$\displaystyle=$
$\displaystyle\frac{1}{3\pi^{2}}g^{2}(k_{\mu}k_{\nu}-\delta_{\mu\nu}k^{2})\log(\delta^{2})+\mbox{finite\
terms}\ .$
Since $\Pi_{\mu\nu}(k)$ does not depend on $\delta$, the same logarithmic
divergence (up to a sign) also has to be present in the outer region integral.
We handle this by adding and subtracting
$\frac{1}{3}g^{2}(k_{\mu}k_{\nu}-\delta_{\mu\nu}k^{2})\int_{|p|>\delta}\frac{1}{(\sum_{\mu}\sin^{2}({1\over
2}p_{\mu}))^{2}}\ ,$ (116)
because, as we will show,
$I\equiv\lim_{\delta\to
0}\left(\int_{|p|>\delta}\frac{1}{(\sum_{\mu}\sin^{2}({1\over
2}p_{\mu}))^{2}}+\frac{1}{\pi^{2}}\log(\delta^{2})\right)$ (117)
is finite and can be explicitly calculated. This means that the sum of Eq.
(116) and the inner region is finite, and thus that the difference of the
outer region integral and Eq. (116) is finite as well. Since, in the outer
region, $|ak|\sim|am|\ll\delta<|\tilde{p}|$, we can expand the subtracted
outer-region integral in powers of $ak_{\mu}$ and $am$, keep those powers that
survive in the continuum limit, and then take $\delta$ to zero, leading to
purely numerical integrals that can be computed numerically. All non-analytic
dependence on $k$ and $m$ resides in the inner-region integral.
What remains to be done is to calculate Eq. (117). We begin rewriting this as
$\lim_{\delta\to 0}\left(\lim_{m\to
0}\left(\int_{p}\frac{1}{(\sum_{\mu}\sin^{2}({1\over
2}p_{\mu})+\frac{1}{4}m^{2})^{2}}-\int_{|p|<\delta}\frac{16}{(p^{2}+m^{2})^{2}}\right)+\frac{1}{\pi^{2}}\log(\delta^{2})\right)\
.$ (118)
The integral over the components $p_{\mu}$ of $p$ in the first integral runs
from $-\pi\to\pi$. We use
$\displaystyle\int_{p}\frac{1}{\sum_{\mu}\sin^{2}({1\over
2}p_{\mu})+\frac{1}{4}m^{2}}$ $\displaystyle=$ $\displaystyle
2\int_{0}^{\infty}dx\int_{p}e^{-{1\over 2}xm^{2}-2x\sum_{\mu}\sin^{2}({1\over
2}p_{\mu})}$ $\displaystyle=$ $\displaystyle 2\int_{0}^{\infty}dx\,e^{-{1\over
2}xm^{2}}e^{-4x}I_{0}^{4}(x)\ ,$
where $I_{0}(x)$ is the modified Bessel function, and hence
$\displaystyle\int_{p}\frac{1}{(\sum_{\mu}\sin^{2}({1\over
2}p_{\mu})+\frac{1}{4}m^{2})^{2}}$ $\displaystyle=$
$\displaystyle-4\,\frac{\partial}{\partial
m^{2}}\int_{p}\frac{1}{\sum_{\mu}\sin^{2}({1\over
2}p_{\mu})+\frac{1}{4}m^{2}}$ $\displaystyle=$ $\displaystyle
4\int_{0}^{\infty}dx\,x\,e^{-{1\over 2}xm^{2}}e^{-4x}I_{0}^{4}(x)\ .$
Furthermore,
$-16\int_{|p|<\delta}\frac{1}{(p^{2}+m^{2})^{2}}=-16\int_{|p|<1}\frac{1}{(p^{2}+\frac{m^{2}}{\delta^{2}})^{2}}=-\frac{1}{\pi^{2}}\log\frac{\delta^{2}}{m^{2}}+O\left(\frac{m^{2}}{\delta^{2}}\right)\
,$ (121)
showing that the double limit in Eq. (118) is finite. We then calculate the
integral on the left of Eq. (121) in a way similar to the calculation in Eq.
(III.1), obtaining
$-16\int_{|p|<\delta}\frac{1}{(p^{2}+m^{2})^{2}}=-\frac{1}{\pi^{2}}\int_{0}^{\infty}\frac{dx}{x}\,e^{-{1\over
2}x\frac{m^{2}}{\delta^{2}}}\left(1-(1+{1\over 2}x)e^{-{1\over 2}x}\right)\ .$
(122)
We can now combine all terms in Eq. (118), take first $m$ and then $\delta$ to
zero, obtaining
$I=4\int_{0}^{\infty}dx\,x\left(e^{-4x}I_{0}^{4}(x)-\frac{1}{4\pi^{2}x^{2}}\left(1-(1+{1\over
2}x)e^{-{1\over 2}x}\right)\right)=0.4855321\ .$ (123)
### III.2 Fermion self energy
The one-loop fermion self energy can be calculated similarly. The steps are
similar to those followed in Sec. III.1 (for details, see Ref. [8]), and here
we just give the result (in Feynman gauge):
$\displaystyle\Sigma(\tilde{p})$ $\displaystyle=$ $\displaystyle
g^{2}\sum_{\mu}\int_{k}{1\over
2}(1+\cos(k_{\mu}+2\tilde{p}_{\mu}))\Gamma_{\mu}S(k+\tilde{p})\Gamma_{\mu}D(k)$
$\displaystyle-g^{2}\sum_{\mu}\Gamma_{\mu}\,i\sin(\tilde{p}_{\mu})\int_{k}D(k)\
,$ $\displaystyle D(k)$ $\displaystyle=$
$\displaystyle\frac{1}{4\sum_{\mu}\sin^{2}({1\over 2}k_{\mu})}\ .$
The term on the second line comes from a gauge-field tadpole diagram. This
result can be further evaluated by splitting into inner and outer regions,
etc., as we did for the vacuum polarization. $\Sigma(\tilde{p})$ is a
$16\times 16$ matrix, and thus a linear combination of products of the
matrices $\Gamma_{\mu}$ and $\Xi_{\nu}$. In the continuum limit, non-analytic
dependence on $\tilde{p}$ and $m$ is the same as we would obtain in any other
regularization, and comes from the inner-region integral.
By dimensional analysis, the most general contact term one can find in the
continuum limit has the form
$C\equiv\tilde{p}_{\mu}X_{\mu}+\frac{1}{a}\,Y_{0}+mY\ ,$ (125)
where again we restored the lattice spacing, and where $X_{\mu}$, $Y_{0}$ and
$Y$ are $16\times 16$ matrices. We will now see how lattice symmetries
restrict the form of $C$. First, shift symmetry implies that $C$ should
commute with all $\Xi_{\nu}$, and therefore $X_{\mu}$, $Y_{0}$ and $Y$ cannot
contain any $\Xi_{A}$.
Then (hypercubic) rotational symmetry implies that $X_{\mu}$ has to be a
vector, which implies that $X_{\mu}\propto\Gamma_{\mu}$ or
$X_{\mu}\propto\Gamma_{\mu}\Gamma_{5}$, while both $Y_{0}$ and $Y$ have to be
proportional to the $16\times 16$ unit matrix or $\Gamma_{5}$. Reflection
symmetry $I_{s}$ of Eq. (70) excludes $\Gamma_{5}$ from appearing in any of
these, and thus we find that
$C=\alpha\,\tilde{p}_{\mu}\Gamma_{\mu}+\beta\,\frac{1}{a}+\gamma\,m\ ,$ (126)
with $\alpha$, $\beta$ and $\gamma$ constants. Finally, we use that for $m=0$
we have $U(1)_{\epsilon}$ symmetry, which implies that, for $m=0$, $C$ should
anti-commute with $\Gamma_{5}\Xi_{5}$. This sets $\beta=0$, and we thus see
that $m$ renormalizes multiplicatively, while the contact terms take on the
form expected in the continuum theory [8] (with coefficients specific to the
lattice regularization, of course).
We note that, in this argument, shift symmetry played a crucial role. Suppose
we were to gauge the staggered theory on the taste basis, with action (90)
plus (86) on the coarse lattice, losing shift symmetry in the process because
the gauge fields $U_{\mu}(y)$ would now live on the coarse lattice. We would
expect a counter term of the form
$\sum_{\mu}{\rm
tr}\,(\overline{\psi}(y)\gamma_{5}\psi(y)\gamma_{5}\gamma_{\mu})$ (127)
because of the form of the “Wilson” term in Eq. (90). On our momentum basis,
this translates into a counter term
$Y_{0}\propto\sum_{\mu}\Gamma_{5}\Xi_{5}\Xi_{\mu}$. Indeed, this anti-commutes
with $\Gamma_{5}\Xi_{5}$, so it is not excluded by $U(1)_{\epsilon}$ symmetry!
And, indeed, such a counter term appears at one loop in the theory on the
taste basis if it is gauged on the coarse lattice [11]. One can verify that
the action on the taste basis is invariant under hypercubic rotations of the
form (64). In particular,
$\tilde{R}^{-1}_{\rho\sigma}\left(\sum_{\mu}\Gamma_{5}\Xi_{5}\Xi_{\mu}\right)\tilde{R}_{\rho\sigma}=\sum_{\mu}\Gamma_{5}\Xi_{5}\Xi_{\mu}\
,$ (128)
showing that a counter term of the form (127) is consistent with rotational
symmetry.
## IV Symanzik effective theory for staggered fermions
The Symanzik effective theory (SET) is a continuum effective theory that
reproduces correlation functions of the underlying lattice theory expanded in
powers of the lattice spacing, for momenta $\Lambda_{\rm QCD}\ll|p|\ll 1/a$,
where $p$ is the typical momentum of these correlation functions.555In these
notes we ignore anomalous dimensions. However, these could become important
with high precision computations near the continuum limit [12]. In this
chapter, we discuss the SET for QCD with $N_{f}$ flavors of staggered
fermions, for $N_{f}>1$. The SET is formulated in terms of continuum quark
fields $q_{i}$ ($i$ is the flavor index; we leave spin and color indices
implicit) and gluons $A_{\mu}$, with field strength $G_{\mu\nu}$ [13, 14],
because at these momenta quarks and gluons are still the relevant degrees of
freedom.
At leading order, the SET consists of all 4-dimensional operators consistent
with the lattice symmetries. As we have seen in the previous chapter, the
fermion mass renormalizes multiplicatively, and there are no dimension-3
operators.666In the previous chapter, we demonstrated this only to one loop,
but the symmetry arguments we used generalize to higher loops. As we have
seen, a crucial role in arriving at this conclusion is played by shift and
$U(1)_{\epsilon}$ symmetries. Dimensional analysis, gauge invariance, and
staggered symmetries imply that the dimension-4 operators in the SET are just
those defining continuum QCD.
Before we go on to discuss operators with dimension $d$ larger than 4 (which
are thus multiplied by $a^{d-4}$, with $a$ the lattice spacing), we make a
general observation on how shift symmetry works in the SET, using the fact
that this is a continuum theory, which, by assumption, is invariant under
continuous translation symmetry [15]. In momentum space, a shift symmetry
takes the form
$\phi(p)\to S_{\mu}\phi(p)=e^{iap_{\mu}}\Xi_{\mu}\phi(p)\ ,$ (129)
where $-\pi/(2a)<p\leq\pi/(2a)$ is the physical momentum of the field $\phi$
on which $S_{\mu}$ acts, and we made the lattice spacing dependence explicit.
The SET is invariant under shift symmetry because the underlying lattice
theory is. At the same time, it is also invariant under continuous
translations of the form
$\phi(p)\to e^{ipr}\phi(p)\ ,$ (130)
where $r$ is the vector over which we translate the field. Combining these two
symmetries, choosing $r_{\mu}=-a$, $r_{\nu\neq\mu}=0$, we find that the SET is
invariant under
$\phi(p)\to\Xi_{\mu}\phi(p)\ .$ (131)
This is convenient, as this symmetry does not mix operators of different
dimensionality in the SET. Below we will make extensive use of Eq. (131), and
we will still refer to this symmetry as “shift” symmetry. For additional
arguments supporting this claim, diagramatically or using the taste basis, see
Ref. [15].
In the discussion below, we will assume that a basis transformation has been
carried out such that
$\Gamma_{\mu}\to\gamma_{\mu}1_{4}\ ,\qquad\Xi_{\mu}\to 1_{4}\xi_{\mu}\ ,$
(132)
where we omit the direct-product sign (i.e.,
$\gamma_{\mu}1_{4}=\gamma_{\mu}\times 1_{4}$, etc.), and we will work with the
spin matrices $\gamma_{\mu}$ and the taste matrices $\xi_{\mu}$, both in the
4-dimensional irrep of the Dirac algebra in four dimensions. We will also drop
the index $4$ on the $4\times 4$ unit matrix; in fact, we will mostly not
write the unit matrix at all, when no confusion is possible.
### IV.1 Dimension-5 operators
We begin with operators of dimension 5, which would be of order $a$ in the
Symanzik expansion [16]. These operators can at most contain one bilinear in
$q_{i}$ and $\overline{q}_{i}$. They cannot contain any non-trivial taste
matrix, because of the symmetry (131). Bilinears of the form
$\overline{q}_{i}\sigma_{\mu\nu}G_{\mu\nu}q_{i}\
,\qquad\overline{q}_{i}D_{\mu}D_{\mu}q_{i}$ (133)
(where $D_{\mu}$ is the color-covariant derivative) are ruled out immediately
by $U(1)_{\epsilon}$ symmetry. In order to consider other dimension-5
operators, we first define
$\displaystyle q_{Li}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!{1\over 2}(1-\gamma_{5}\xi_{5})q_{i}\ ,\qquad
q_{Ri}={1\over 2}(1+\gamma_{5}\xi_{5})q_{i}\ ,$ (134)
$\displaystyle\overline{q}_{Li}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\overline{q}_{i}{1\over 2}(1+\gamma_{5}\xi_{5})\
,\qquad\overline{q}_{Ri}=\overline{q}_{i}{1\over 2}(1-\gamma_{5}\xi_{5})\ .$
The lattice theory has a $U(N_{f})_{L}\times U(N_{f})_{R}$ symmetry,777Note
that this symmetry is smaller than the $U(4N_{f})_{L}\times U(4N_{f})_{R}$
symmetry of the continuum limit. under which these fields transform as
$\displaystyle q_{L}\to V_{L}q_{L}\ ,\qquad$ $\displaystyle q_{R}\to
V_{R}q_{R}\ ,$
$\displaystyle\overline{q}_{L}\to\overline{q}_{L}V_{L}^{\dagger}\ ,\qquad$
$\displaystyle\overline{q}_{R}\to\overline{q}_{R}V_{R}^{\dagger}\ ,$
with
$V_{L}=e^{\frac{i}{2}\alpha^{a}T^{a}(1-\gamma_{5}\xi_{5})}\ ,\qquad
V_{R}=e^{\frac{i}{2}\beta^{a}T^{a}(1+\gamma_{5}\xi_{5})}\ ,$ (136)
in which $T^{a}$ are the $U(N_{f})$ generators. This symmetry is a
generalization of $U(1)\times U(1)_{\epsilon}$ symmetry for one flavor to the
case of $N_{f}$ flavors. Note that other symmetries, such as shift and
hypercubic symmetry, do not enlarge with the presence of more flavors, because
the gauge field transforms non-trivially under these symmetries.
Mass terms are not invariant under this symmetry, but can be made invariant by
making the $N_{f}\times N_{f}$ mass matrix ${\cal M}$ into a spurion field
(i.e., a non-dynamical field) that transforms as
${\cal M}\to V_{R}{\cal M}V_{L}^{\dagger}\ ,$ (137)
so that the 4-dimensional staggered mass term
$\bar{\chi}_{R}(x){\cal M}\chi_{L}(x)+\bar{\chi}_{L}(x){\cal
M}^{\dagger}\chi_{R}(x)$ (138)
is invariant (with the symmetry $U(N_{f})_{L}\times U(N_{f})_{R}$ acting on
the staggered fields $\chi$ and $\bar{\chi}$ as in Eq. (136) with
$\gamma_{5}\xi_{5}$ replaced by $\epsilon(x)$). The idea is to use the spurion
form of ${\cal M}$ to see what terms can show up in the SET, and then, once we
have found all such terms, set ${\cal M}$ equal to the physical mass matrix
$M$. The dimension-4 mass term in the SET,
$\overline{q}_{R}{\cal M}q_{L}+\overline{q}_{L}{\cal M}^{\dagger}q_{R}$ (139)
is also invariant. We are now ready to return to dimension-5 quark bilinears
containing one or more masses. Using the transformations (IV.1) and (137), the
symmetry $U(N_{f})_{L}\times U(N_{f})_{R}$ rules out terms of the form
$\overline{q}M\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}\\!Dq\
,\qquad\overline{q}M^{2}q\ ,\qquad{\rm tr}\,(M)G_{\mu\nu}G_{\mu\nu}\ .$ (140)
This exhausts all possible dimension-5 contributions to the SET, and we thus
conclude that dependence on the lattice spacing starts at order $a^{2}$, i.e.,
at dimension 6. We note that no use of equations of motion or field
redefinitions was made to rule out all dimension-5 operators, and our
conclusion thus holds for off-shell correlation functions as well.
We end this subsection with a comment on the taste basis of Sec. II.5. The
action (90) has a Wilson term, which is, in fact, a dimension-5 operator.
However, this action is invariant under shift symmetry in the form (91), but
not under the symmetry $\psi\to\psi\gamma_{\mu}$, which is the continuum form
equivalent to Eq. (131). As it is the latter symmetry which is relevant for
restricting the form of the SET, there is thus no contradiction.
### IV.2 Dimension-6 operators
In this section, we will discuss the order-$a^{2}$ part of the SET, i.e., all
dimension-6 operators that contribute, with the focus on operators that break
$U(4)_{\rm taste}$ (which is, of course, only a symmetry of the continuum
limit). These operators were constructed in Refs. [16, 17] (see also Ref.
[18]).
There are no taste-breaking bilinears. This follows from the fact that any
bilinear has to be invariant under the discrete group $\Gamma_{4}$, generated
by the $\xi_{\mu}$. (There exist taste-invariant bilinear operators, such as
$\overline{q}\,\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}\\!D^{3}q$, see the
end of this subsection.)
Many 4-quark operators can be constructed, and if we require them to be
invariant under $SU(N_{f})$, they take the form888They can be color-mixed or
-unmixed.
$(\overline{q}_{i}Xq_{i})(\overline{q}_{j}Yq_{j})\ ,$ (141)
in which sums over $i$ and $j$ are implied, and where $X=\gamma_{A}\xi_{B}$
and $Y=\gamma_{C}\xi_{D}$, with
$\displaystyle\gamma_{A}\ ,\gamma_{C}$ $\displaystyle\in$ $\displaystyle\\{1,\
\gamma_{\mu},\ i\gamma_{\mu}\gamma_{\nu},\ i\gamma_{5}\gamma_{\mu},\
\gamma_{5}\\}\ ,$ (142) $\displaystyle\xi_{B}\ ,\xi_{D}$ $\displaystyle\in$
$\displaystyle\\{1,\ \xi_{\mu},\ i\xi_{\mu}\xi_{\nu},\ i\xi_{5}\xi_{\mu},\
\xi_{5}\\}\ ,$
in which $\mu<\nu$. Since no mass matrix can appear in these operators,
$U(N_{f})_{L}\times U(N_{f})_{R}$ symmetry implies that both
$\gamma_{A}\xi_{B}$ and $\gamma_{C}\xi_{D}$ have to anti-commute with
$\gamma_{5}\xi_{5}$, which, in turn, implies that
$\displaystyle\gamma_{A}\in\\{1\ \mbox{(S)},\ i\gamma_{\mu}\gamma_{\nu}\
\mbox{(T)},\ \gamma_{5}\ \mbox{(P)}\\}\quad$
$\displaystyle\Rightarrow\quad\xi_{B}\in\\{\xi_{\mu}\ \mbox{(V)},\
i\xi_{5}\xi_{\mu}\ \mbox{(A)}\\}\ ,$
$\displaystyle\gamma_{A}\in\\{\gamma_{\mu}\ \mbox{(V)},\
i\gamma_{5}\gamma_{\mu}\ \mbox{(A)}\\}\quad$
$\displaystyle\Rightarrow\quad\xi_{B}\in\\{1\ \mbox{(S)},\
i\xi_{\mu}\xi_{\nu}\ \mbox{(T)},\ \xi_{5}\ \mbox{(P)}\\}\ .$
Here we denoted the various tensor structures with S (scalar), T (tensor), P
(pseudo-scalar), V (vector) and A (axial-vector), for later convenience. Both
bilinears thus have to be “odd,” which means that the spin-taste matrix has to
have an odd number of Lorentz indices. Translating this back to lattice
operators implies that the corresponding lattice bilinears have to be odd-link
operators, since each $\gamma_{\mu}$ or $\xi_{\mu}$ is generated by a shift
over one lattice spacing in the $\mu$ direction.
Next, $\Gamma_{4}$ symmetry implies that $\xi_{B}=\xi_{D}$, because, if
$\xi_{B}$ were not equal to $\xi_{D}$, we could find a $\xi_{X}$ such that
$\xi_{B}\xi_{X}=-\xi_{X}\xi_{B}$ but $\xi_{D}\xi_{X}=+\xi_{X}\xi_{D}$, and the
operator would not be invariant under $q\to\xi_{X}q$,
$\overline{q}\to\overline{q}\xi_{X}$. Our operators thus have the form
$(\overline{q}_{i}\gamma_{A}\xi_{B}q_{i})(\overline{q}_{j}\gamma_{C}\xi_{B}q_{j})\
,$ (144)
with the constraint (IV.2). A similar argument sets $\gamma_{A}=\gamma_{C}$.
Rotations over 180 degrees take the form (cf. Eq. (60))
$q\to\gamma_{\rho}\gamma_{\sigma}\xi_{\rho}\xi_{\sigma}q\
,\quad\rho\neq\sigma\ ,$ (145)
and if we combine these with shifts in the $\rho$ and $\sigma$ directions, it
follows that the SET has to be invariant under
$q(x)\to\gamma_{\rho}\gamma_{\sigma}q(R^{-1}_{\rho\sigma}(\pi)x),\quad\rho\neq\sigma\
.$ (146)
Likewise, we can combine $I_{s}$ of Eq. (70) with a shift in the 4th
direction, resulting in a SET symmetry
$q(x)\to\gamma_{4}q(I_{s}x)\ .$ (147)
Together, these imply that $\gamma_{A}$ has to be equal to $\gamma_{C}$ (with
Eq. (147) excluding $\gamma_{C}=\gamma_{A}\gamma_{5}$). Our 4-fermion
operators thus take the form
$(\overline{q}_{i}\gamma_{A}\xi_{B}q_{i})(\overline{q}_{j}\gamma_{A}\xi_{B}q_{j})\
,$ (148)
with the proviso that $\gamma_{A}\xi_{B}$ has to be odd. It remains to
consider rotations over 90 degrees. In order to make sure our 4-fermion
operators are invariant under these, we need to sum over Lorentz indices, to
make all operators into scalars under 90-degree rotations. There will be two
types of operators [16]: those operators which are invariant not only under
hypercubic rotations, but also under $SO(4)$, which we will refer to as “type
A,” and those which are not invariant under $SO(4)$, but only under hypercubic
rotations, “type B.” To show how this works, it is easiest to give some
examples. Examples of type-A operators are
$\displaystyle\sum_{\mu}(\bar{\gamma}_{i}\xi_{\mu}q_{i})(\overline{q}_{j}\xi_{\mu}q_{j})\
,$ (149)
$\displaystyle\sum_{\mu,\nu\neq\rho}(\overline{q}_{i}\gamma_{\mu}\,i\xi_{\nu}\xi_{\rho}q_{i})(\overline{q}_{j}\gamma_{\mu}\,i\xi_{\nu}\xi_{\rho}q_{j})\
.$
These operators are both invariant under $SO(4)$. The first of these two
operators we refer to as $S\times V$, since $\gamma_{A}$ is scalar and
$\xi_{B}$ is vector; the second we refer to as $V\times T$, because
$\gamma_{A}$ is vector and $\xi_{B}$ is tensor. An example of a type-B
operator is
$\sum_{\mu\neq\nu}(\overline{q}_{i}\gamma_{\mu}\,i\xi_{\mu}\xi_{\nu}q_{i})(\overline{q}_{j}\gamma_{\mu}\,i\xi_{\mu}\xi_{\nu}q_{j})\
.$ (150)
Because the index $\mu$ is repeated four times, this operator is invariant
under hypercubic rotations, but not under $SO(4)$. We may refer to this
operator as $V_{\mu}\times T_{\mu}$, where the repeated $\mu$ reminds us of
the four times repeated index $\mu$ in Eq. (150). This type can only occur if
either $\gamma_{A}$ or $\xi_{B}$ is tensor (and thus the other is vector or
axial-vector).
In fact, the type-B operator of our example can be taken to be
$\sum_{\mu\neq\nu}\left((\overline{q}_{i}\gamma_{\mu}\,i\xi_{\mu}\xi_{\nu}q_{i})(\overline{q}_{j}\gamma_{\mu}\,i\xi_{\mu}\xi_{\nu}q_{j})-(\overline{q}_{i}\gamma_{\mu}\,i\xi_{\mu}\xi_{\nu}\xi_{5}q_{i})(\overline{q}_{j}\gamma_{\mu}\,i\xi_{\mu}\xi_{\nu}\xi_{5}q_{j})\right)\
,$ (151)
because the orthogonal combination
$\displaystyle\sum_{\mu\neq\nu}\left((\overline{q}_{i}\gamma_{\mu}\,i\xi_{\mu}\xi_{\nu}q_{i})(\overline{q}_{j}\gamma_{\mu}\,i\xi_{\mu}\xi_{\nu}q_{j})+(\overline{q}_{i}\gamma_{\mu}\,i\xi_{\mu}\xi_{\nu}\xi_{5}q_{i})(\overline{q}_{j}\gamma_{\mu}\,i\xi_{\mu}\xi_{\nu}\xi_{5}q_{j})\right)$
$\displaystyle=$
$\displaystyle\sum_{\mu\neq\nu}\left((\overline{q}_{i}\gamma_{\mu}\,i\xi_{\mu}\xi_{\nu}q_{i})(\overline{q}_{j}\gamma_{\mu}\,i\xi_{\mu}\xi_{\nu}q_{j})+(\overline{q}_{i}\gamma_{\mu}\,i\xi_{\rho}\xi_{\sigma}q_{i})(\overline{q}_{j}\gamma_{\mu}\,i\xi_{\rho}\xi_{\sigma}q_{j})_{\rho\sigma\neq\mu\nu}\right)$
$\displaystyle=$
$\displaystyle\sum_{\mu,\rho\neq\sigma}(\overline{q}_{i}\gamma_{\mu}\,i\xi_{\rho}\xi_{\sigma}q_{i})(\overline{q}_{j}\gamma_{\mu}\,i\xi_{\rho}\xi_{\sigma}q_{j})\
,$
which is type A. We see that the requirement that the 4-quark operators are
invariant under the staggered symmetry group leads to a huge reduction in the
number of 4-quark operators. With $\gamma_{A}$, $\xi_{B}$, $\gamma_{C}$ and
$\xi_{D}$ unrestricted, there are $16^{4}=65536$ choices!
For $N_{f}=1$ there is a subtlety, based on the fact that the flavor symmetry
group is just $U(1)\times U(1)_{\epsilon}$. It is straightforward to verify
that, e.g.,
$(\overline{q}q)(\overline{q}q)-(\overline{q}\gamma_{5}\xi_{5}q)(\overline{q}\gamma_{5}\xi_{5}q)$
(153)
is invariant under this group. Using our shorthand notation, we can label this
operator as $S\times S-P\times P$. The fermion bilinears in this combination
are even, and this operator thus does not fit into the general pattern we
established above. However, it can be made to fit, by a Fierz transformation
[16], as we will now show.
A Fierz transformation is based on identities of the form
$(\gamma_{A})_{\alpha\beta}(\gamma_{A})_{\gamma\delta}=\sum_{B}c(A,B)(\gamma_{B})_{\alpha\delta}(\gamma_{B})_{\gamma\beta}\
,$ (154)
with numerical coefficients $c(A,B)$ that depend on $A$ and $B$. For instance,
$1_{\alpha\beta}1_{\gamma\delta}=\frac{1}{4}\sum_{B}(\gamma_{B})_{\alpha\delta}(\gamma_{B})_{\gamma\beta}\
.$ (155)
Similar identities hold, of course, for the $\xi_{A}$. Using the abbreviations
$S$, $V$, $T$, $A$ and $P$ introduced above, we can use Eq. (155) for both the
$\gamma$ and $\xi$ matrices to write
$\displaystyle S\times S$ $\displaystyle=$
$\displaystyle\frac{1}{16}(S+V+T+A+P)\times(S+V+T+A+P)\ ,$ (156)
$\displaystyle P\times P$ $\displaystyle=$
$\displaystyle\frac{1}{16}(S-V+T-A+P)\times(S-V+T-A+P)\ .$
Subtracting these two, we find that
$\displaystyle S\times S-P\times P=\frac{1}{8}\Bigl{(}S\times V+V\times
S+T\times V+V\times T+P\times V+V\times P$ $\displaystyle\hskip
56.9055pt+S\times A+A\times S+T\times A+A\times T+P\times A+A\times P\Bigr{)}\
,$ (157)
which allows us to write the operator (153) in the general form we derived
above for arbitrary $N_{f}$. Of course, Fierzing interchanges color-mixed and
color-unmixed operators, but both types occur in the general analysis as well.
There are also dimension-6 operators with just one quark bilinear. Staggered
and flavor symmetries restrict these to the list999The operator
$\sum_{\mu}\overline{q}_{i}(D_{\mu}^{2}\,\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\\!D-\,\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\\!DD_{\mu}^{2})q_{i}$ has negative charge parity [16].
$\displaystyle\overline{q}_{i}\,\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\\!D^{3}q_{i}\
,\quad\sum_{\mu}\overline{q}_{i}(D_{\mu}^{2}\,\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\\!D+\,\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\\!DD_{\mu}^{2})q_{i}\
,\quad\sum_{\mu}\overline{q}_{i}D_{\mu}\,\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\\!DD_{\mu}q_{i}\
,\quad\sum_{\mu}\overline{q}_{i}\gamma_{\mu}D_{\mu}^{3}q_{i}\ ,$
$\displaystyle\overline{q}_{i}M\,\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\\!D^{2}q_{i}\ ,\quad\overline{q}_{i}MD_{\mu}^{2}q_{i}\
,\quad\overline{q}_{i}M^{2}\,\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\\!Dq_{i}\ ,\quad\overline{q}_{i}M^{3}q_{i}\ .$ (158)
In particular, the fermion bilinear cannot contain any taste matrix, because
of shift symmetry. The operators containing the mass matrix $M$ can be
constructed using Eqs. (IV.1) and (137). For example, the last operator is
obtained from
$\overline{q}_{Ri}{\cal M}{\cal M}^{\dagger}{\cal
M}q_{Li}+\overline{q}_{Li}{\cal M}^{\dagger}{\cal M}{\cal M}^{\dagger}q_{Ri}\
,$ (159)
and setting ${\cal M}={\cal M}^{\dagger}=M$. Finally, there are purely gluonic
dimension-6 operators, shared with other discretizations of QCD, which we do
not list here.
### IV.3 Operators with dimension larger than 6
We briefly discuss what happens beyond dimension 6, i.e., at order $a^{3}$ and
higher.
#### IV.3.1 Dimension 7
Let us start with dimension-7 operators. We again restrict ourselves to
operators containing quark fields (no purely gluonic operators with odd
dimension exist). Dimension-7 operators, if they exist, can contain one or two
quark bilinears.
First, consider operators with only one quark bilinear. Starting from the list
in Eq. (IV.2), we need to add one mass dimension, which can be done by adding
an extra covariant derivative $D_{\mu}$, or an extra mass matrix $M$. However,
if we insert a $D_{\mu}$, Lorentz indices still have to balance, and the only
way to do that is to remove or add a $\gamma_{\mu}$ inside the bilinear. But
this would change the properties of this bilinear under $U(N_{f})_{L}\times
U(N_{f})_{R}$, so this does not work. An analogous argument excludes adding a
mass matrix $M$ (using again a spurion-based argument).
Essentially the same argument applies to operators with two quark bilinears.
Inserting a $D_{\mu}$ leads to an unmatched Lorentz index, or a change in the
symmetry properties of one fermion bilinear, making it inconsistent with
$U(N_{f})_{L}\times U(N_{f})_{R}$. A similar argument applies with inserting a
mass matrix $M$. If we replace one of the two bilinears
$\overline{q}_{i}Xq_{i}$ with $\overline{q}_{i}MYq_{i}$, we need to choose $Y$
such that $\overline{q}_{i}MYq_{i}$ is invariant under $U(N_{f})_{L}\times
U(N_{f})_{R}$ (after promoting $M$ to the spurions ${\cal M}$ and ${\cal
M}^{\dagger}$). But this implies that $Y\neq X$, and thus shift and/or lattice
rotation/reflection symmetries would be broken. We conclude that no
dimension-7 operators exist that are part of the SET.
#### IV.3.2 Dimension 8
In contrast, many operators of dimension 8 exist, and they are easily
constructed from dimension-6 operators, e.g., by inserting $D_{\mu}^{2}$ or
$\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}\\!D^{2}$, or by multiplying with
${\rm tr}\,(M^{2})$. However, all continuum symmetries are already broken down
to the staggered symmetry group at the level of dimension-6 operators (because
of type-B operators!), so no new symmetry breaking occurs at order $a^{4}$
[19].101010In particular, no new taste spurions occur, because the number of
quark bilinears is still restriced to two. See next section.
#### IV.3.3 Odd dimension $\geq 9$
At dimension 9, we should consider operators with three quark bilinears.
Because of dimensional reasons, these cannot contain any derivatives or
masses. Each of the bilinears needs to be invariant under $U(N_{f})_{L}\times
U(N_{f})_{R}$, which implies that each bilinear has to be odd—it has an odd
number of Lorentz indices. But this is impossible, because all Lorentz indices
need to be contracted because of hypercubic invariance. In general, any
operator, at any dimension, has to have an even number of Lorentz indices.
What about dimension-9 operators with only one quark bilinear? Either the
bilinear is odd, which means it has 1 or 3 Lorentz indices, or, if it contains
a mass matrix $M$, it carries an even number of Lorentz indices, because of
$U(N_{f})_{L}\times U(N_{f})_{R}$ symmetry. (If there are two mass matrix
insertions, it has to be odd again; if there are three mass matrices, it has
to be even again, etc.)
The rest of the operator has to be made out of $k$ appearances of $D_{\mu}$
and/or $\ell$ appearances of $G_{\kappa\lambda}$. This part of the operator
thus has dimension $d=k+2\ell$, and it has $k+2\ell$ mod 2 Lorentz indices. If
the quark bilinear has dimension 3 (no $M$), the complete operator thus has
dimension $d=3+k+2\ell$, and it has $1+k+2\ell$ mod 2 Lorentz indices. For odd
$d$, we thus need $k+2\ell$ even, but that implies that the number of Lorentz
indices is odd. Likewise, if the bilinear contains one factor of $M$, we need
$d=4+k+2\ell$ to be odd, and thus that $k+2\ell$ is odd. But now the quark
bilinear needs to be even, so again the number of Lorentz indices is odd. This
argument generalizes to operators with more insertions of $M$.
A very similar argument takes care of dimension-9 operators with two quark
bilinears. The general pattern is that the dimension of the gluonic part to be
added is always such that the number of Lorentz indices is odd. The conclusion
appears to be that no operators with odd dimension, and thus an odd power of
the lattice spacing, can appear in the SET.
### IV.4 What causes taste breaking?
Finally, in this chapter, let us have a closer look at what causes taste-
breaking operators to appear in the SET. As they first appear at dimension 6
through 4-quark operators, we expect them to come from quark-quark scattering,
as depicted in Fig. 2. Indeed, when the gluon momentum exchanged between the
two quarks has momentum $ak=\pi_{E}\neq 0$, with $\pi_{E}$ one of the momentum
vectors with components equal to 0 or $\pi$, this short-distance gluon
exchange generates taste-breaking 4-quark operators, as we will show below.
Note that in this figure, we restored the lattice spacing $a$.
$ak=\pi_{E}$$a\tilde{p}^{\prime}+\pi_{C}$$a\tilde{q}^{\prime}+\pi_{D}$$a\tilde{p}+\pi_{A}$$a\tilde{q}+\pi_{B}$
Figure 2: Exchange of a gluon with momentum equal to $\pi_{E}$ between two
staggered quarks, in momentum space.
In Feynman gauge, the gluon propagator $D(k)$ in Eq. (III.2), for such
momenta, is equal to
$D_{\mu\nu}(k=\pi_{E}/a)=\frac{\delta_{\mu\nu}}{\sum_{\mu}\frac{4}{a^{2}}\sin^{2}({1\over
2}ak_{\mu})}\Bigg{|}_{k=\pi_{E}/a}=\frac{a^{2}}{4n}\,\delta_{\mu\nu}\ ,$ (160)
where $n$ is the number of components of $\pi_{E}$ equal to $\pi$. Using Eq.
(53), and working in Feynman gauge, the diagram is thus equal to (retaining
the explicit lattice spacing $a$ only in the prefactor and omitting color
generators)
$\displaystyle-g^{2}\frac{a^{2}}{4n}\sum_{\mu}\delta(\tilde{p}+\pi_{A}-\tilde{p}^{\prime}+\pi_{C}+\pi_{E}+\pi_{\eta_{\mu}}){1\over
2}(S_{\mu}^{A}e^{i\tilde{p}_{\mu}}+S_{\mu}^{C}e^{-i\tilde{p}^{\prime}_{\mu}})$
$\displaystyle\phantom{-g^{2}\frac{a^{2}}{4n}\sum_{\mu}}\times\delta(\tilde{q}+\pi_{B}-\tilde{q}^{\prime}+\pi_{D}+\pi_{E}+\pi_{\eta_{\mu}}){1\over
2}(S_{\mu}^{B}e^{i\tilde{q}_{\mu}}+S_{\mu}^{D}e^{-i\tilde{q}^{\prime}_{\mu}})$
$\displaystyle=$
$\displaystyle-g^{2}\frac{a^{2}}{4n}\sum_{\mu}\delta(\tilde{p}-\tilde{p}^{\prime})\delta(\tilde{q}-\tilde{q}^{\prime}){1\over
2}(S_{\mu}^{A}e^{i\tilde{p}_{\mu}}+S_{\mu}^{C}e^{-i\tilde{p}_{\mu}}){1\over
2}(S_{\mu}^{B}e^{i\tilde{q}_{\mu}}+S_{\mu}^{D}e^{-i\tilde{q}_{\mu}})$
$\displaystyle\phantom{-g^{2}\frac{a^{2}}{4n}\sum_{\mu}}\times\overline{\delta}(\pi_{A}+\pi_{C}+\pi_{E}+\pi_{\eta_{\mu}})\overline{\delta}(\pi_{B}+\pi_{D}+\pi_{E}+\pi_{\eta_{\mu}})$
$\displaystyle\approx$
$\displaystyle-g^{2}\frac{a^{2}}{4n}\frac{1}{4}\sum_{\mu}\delta(\tilde{p}-\tilde{p}^{\prime})\delta(\tilde{q}-\tilde{q}^{\prime})\Biggl{(}(\hat{\pi}_{E}\Gamma_{\mu})_{CA}(\hat{\pi}_{E}\Gamma_{\mu})_{DB}+(\Gamma_{\mu}\hat{\pi}_{E})_{CA}(\Gamma_{\mu}\hat{\pi}_{E})_{DB}$
$\displaystyle\phantom{-g^{2}\frac{a^{2}}{4n}\frac{1}{4}\sum_{\mu}}+(\hat{\pi}_{E}\Gamma_{\mu})_{CA}(\Gamma_{\mu}\hat{\pi}_{E})_{DB}+(\Gamma_{\mu}\hat{\pi}_{E})_{CA}(\hat{\pi}_{E}\Gamma_{\mu})_{DB}\Biggr{)}\
,$
where in the last expression we approximated $e^{i\tilde{p}_{\mu}}\approx 1$,
etc., and we used
$\overline{\delta}(\pi_{A}+\pi_{C}+\pi_{E}+\pi_{\eta_{\mu}})=\sum_{F}\overline{\delta}(\pi_{E}+\pi_{C}+\pi_{F})\overline{\delta}(\pi_{\eta_{\mu}}+\pi_{F}+\pi_{A})=(\hat{\pi}_{E}\hat{\pi}_{\eta_{\mu}})_{CA}=(\hat{\pi}_{\eta_{\mu}}\hat{\pi}_{E})_{CA}$
(162)
and likewise for
$\overline{\delta}(\pi_{B}+\pi_{D}+\pi_{E}+\pi_{\eta_{\mu}})$. (We recall that
$(\pi_{\eta_{\mu}})_{\mu}=0$.)
We now use that
$\hat{\pi}_{E}\in\\{\Gamma_{\mu}\Xi_{\mu},\
\Gamma_{\mu}\Gamma_{\nu}\Xi_{\mu}\Xi_{\nu},\
\Gamma_{\mu}\Gamma_{5}\Xi_{\mu}\Xi_{5},\ \Gamma_{5}\Xi_{5}\\}\ .$ (163)
This is easy to see, because in these combinations, the factors $S_{\mu}$ in
Eqs. (35) and (39) cancel, so
$\Gamma_{\mu}\Xi_{\mu}=\hat{\pi}_{\eta_{\mu}}\hat{\pi}_{\zeta_{\mu}}$, etc.,
and hence the matrices in Eq. (163) generate “pure” $\hat{\pi}$ matrices, a
total of 15 of them. Equation (IV.4) thus generates the 4-quark operators with
spin-taste structure
$\displaystyle\hat{\pi}_{E}=$ $\displaystyle\Gamma_{\mu}\Xi_{\mu}:\qquad$
$\displaystyle S\times V\ ,$
$\displaystyle\Gamma_{\nu}\Xi_{\nu}|_{\nu\neq\mu}:\qquad$ $\displaystyle
T_{\nu}\times V_{\nu}\ ,$
$\displaystyle\Gamma_{\mu}\Gamma_{\nu}\Xi_{\mu}\Xi_{\nu}:\qquad$
$\displaystyle V_{\nu}\times T_{\nu}\ ,$
$\displaystyle\Gamma_{\rho}\Gamma_{\sigma}\Xi_{\rho}\Xi_{\sigma}|_{\rho,\sigma\neq\mu}:\qquad$
$\displaystyle A_{\nu}\times T_{\rho\sigma}\ (\nu\neq\rho,\sigma)\ ,$
$\displaystyle\Gamma_{\mu}\Gamma_{5}\Xi_{\mu}\Xi_{5}:\qquad$ $\displaystyle
P\times A\ ,$
$\displaystyle\Gamma_{\nu}\Gamma_{5}\Xi_{\nu}\Xi_{5}|_{\nu\neq\mu}:\qquad$
$\displaystyle T_{\rho\sigma}\times A_{\nu}\ (\nu\neq\rho,\sigma)\ ,$
$\displaystyle\Gamma_{5}\Xi_{5}:\qquad$ $\displaystyle A\times P\ .$
As expected, these are all odd operators, because the $U(N_{f})_{L}\times
U(N_{f})_{R}$ symmetry is respected by the kinetic term in the staggered
action from which the Feynman rule used above derives. We note that we do not
obtain all 4-quark operators from this diagram, but other processes, for
instance with more than one gluon exchanged, can occur. The scale of the
coupling at the quark-gluon vertex in Fig. 2 is that of the exchanged gluon,
and we thus expect taste breaking to be suppressed by a factor of order
$\alpha_{s}(\pi/a)$ (with $\alpha_{s}$ the strong coupling). This already
follows from the fact that the free theory has exact $U(4)$ taste symmetry, so
internal gluons are needed to produce taste-breaking operators.
## V Staggered chiral perturbation theory
The continuum limit of QCD with $N_{f}$ staggered quarks has $4N_{f}$ quarks
in the continuum limit, with flavor symmetry group $U(1)\times
SU(4N_{f})_{L}\times SU(4N_{f})_{R}$. It follows that this theory is expected
to have $(4N_{f})^{2}-1$ Nambu–Goldstone bosons (NGBs). Of course, the number
of NGBs is too large, and this is why, in practical numerical computations,
the 4th root of the staggered determinant is taken, in order to reduce the
number of quark degrees of freedom by a factor of 4.111111The staggered tastes
can also be interpreted as physical flavors, and an action can be constructed
that completely breaks the taste degeneracy in the continuum limit [8].
However, this construction leads to a practical problems, and has not been
used in practice. While we will not review the arguments supporting the “4th-
root trick,” we can accommodate this trick in both WCPT and chiral
perturbation theory (ChPT) by employing the replica trick. (For a review of
the rooting trick and further references, see Ref. [20]. See also Ref. [15].)
We introduce $n_{r}$ replicas of each staggered fermion, so that now we have
$N=4n_{r}N_{f}$ quarks in the continuum limit, and we thus expect
$(4n_{r}N_{f})^{2}-1$ NGBs. Setting, at the end of any calculation (in WCPT or
ChPT) $n_{r}=\frac{1}{4}$ thus yields effectively $N_{f}^{2}-1$ NGBs, i.e.,
the desired number. We will thus develop staggered ChPT (SChPT) for $N$
continuum quarks, and see in an example how the replica trick works. Of
course, if we wish to consider unrooted staggered quarks, we would set
$n_{r}=1$.
### V.1 Construction of the SChPT lagrangian
The chiral lagrangian is, like the SET, a continuum effective theory, with
lattice spacing dependence taken into account in terms of an expansion in
powers of $a^{2}$. It is formulated in terms of a non-linear field
$\Sigma=e^{\frac{i}{f}\pi}\ ,$ (165)
where $\pi$ is a traceless, hermitian $4n_{r}N_{f}\times 4n_{r}N_{f}$ matrix
describing the $(4n_{r}N_{f})^{2}-1$ NGBs. The field $\Sigma$ transforms as
$\Sigma_{ia,jb}\sim q_{Lia}\overline{q}_{Rjb}\ ,$ (166)
with $i,\ j=1,\dots,n_{r}N_{f}$ replica-flavor indices, and $a,\ b=1,\dots,4$
taste indices. In this chapter, the left-handed and right-handed projections
are defined as in continuum QCD:
$\displaystyle q_{L}\\!\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!\\!{1\over
2}(1-\gamma_{5})q\ ,\qquad q_{R}={1\over 2}(1+\gamma_{5})q\ ,$ (167)
$\displaystyle\overline{q}_{L}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\overline{q}{1\over 2}(1+\gamma_{5})\
,\qquad\overline{q}_{R}=\overline{q}{1\over 2}(1-\gamma_{5})\ .$
We embed the taste matrices $\xi_{A}$ into $4n_{r}N_{f}\times 4n_{r}N_{f}$
matrices by simply repeating the $4\times 4$ matrices $\xi_{A}$ $n_{r}N_{f}$
times along the main diagonal, thus constructing a block-diagonal matrix
$\xi_{A}^{(n_{r}N_{F})}$. However, we will drop the superscript $(n_{r}N_{F})$
below, when no confusion is possible. For instance, since $\Sigma$ is a
$4n_{r}N_{f}\times 4n_{r}N_{f}$ matrix, we have that
$\xi_{A}\Sigma\equiv\xi_{A}^{(n_{r}N_{F})}\Sigma\ .$ (168)
For reviews of continuum ChPT and applications to various discretizations of
QCD, see Refs. [21, 22].
At leading order (LO) the chiral lagrangian is given by
${\cal L}_{LO}=\frac{1}{8}f^{2}\,{\rm
tr}\,(\partial_{\mu}\Sigma^{\dagger}\partial_{\mu}\Sigma)-\frac{1}{4}Bf^{2}\,{\rm
tr}\,({\cal M}^{\dagger}\Sigma+{\cal M}\Sigma^{\dagger})\ .$ (169)
The trace is over replica/flavor and taste indices (cf. Eq. (166)). This
lagrangian is invariant under
$\Sigma\to V_{L}\Sigma V^{\dagger}_{R}\ ,\qquad{\cal M}\to V_{L}{\cal
M}V_{R}^{\dagger}\ ,$ (170)
with now $V_{L}\in SU(N)_{L}$ and $V_{R}\in SU(N)_{R}$, with $N=4n_{r}N_{f}$.
As before, ${\cal M}$ is a mass spurion, to be set equal to the quark mass
matrix $M$ after the construction of the chiral lagrangian. $f$ and $B$ are
low-energy constants (LECs), to be determined by matching with the underlying
theory, QCD.121212$f$, of course, is the pion decay constant, normalized such
that at LO, $f_{\pi}=f=130.4$ MeV. We will choose the quark mass matrix $M$ to
be diagonal and positive. The first four diagonal elements will be equal to
the up quark mass $m_{u}$, the next four will be equal to the down quark mass
$m_{d}$, etc. (As before, the four tastes produced by each staggered fermion
are degenerate in the continuum limit.)
To incorporate terms of order $a^{2}$ into ChPT, we use a spurion trick just
like for the mass term in Eq. (169). It is easiest to explain this through
some examples. Our first example starts from the following operator appearing
in the SET lagrangian:
$a^{2}(\overline{q}\xi_{5\nu}q)(\overline{q}\xi_{5\nu}q)=a^{2}(\overline{q}_{R}\xi_{5\nu}q_{L}+\overline{q}_{L}\xi_{5\nu}q_{R})^{2}\to(\overline{q}_{R}X_{R}q_{L}+\overline{q}_{L}X_{L}q_{R})^{2}\
,$ (171)
where $\xi_{5\nu}=i\xi_{5}\xi_{\nu}$ and $X_{L,R}$ are new spurions
transforming as
$X_{L}\to V_{L}X_{L}V^{\dagger}_{R}\ ,\qquad X_{R}\to
V_{R}X_{R}V^{\dagger}_{L}\ .$ (172)
We can take $X_{R}=X_{L}^{\dagger}$, but this is not necessary. To recover the
4-quark operator we started from, we have to set
$X_{L}=X_{R}=a\xi_{5\nu}\ .$ (173)
The next step is simply to write down all operators involving the non-linear
field $\Sigma$ and two spurions $X_{L}$ and/or $X_{R}$, such that these
operators are invariant under Eqs. (170) and (172). The possibilities are
$\displaystyle{\rm tr}\,(X_{L}\Sigma^{\dagger})\,{\rm tr}\,(X_{R}\Sigma)\ ,$
(174) $\displaystyle\left({\rm
tr}\,(X_{L}\Sigma^{\dagger})\right)^{2}+\left({\rm
tr}\,(X_{R}\Sigma)\right)^{2}\ ,$ $\displaystyle{\rm
tr}\,(X_{L}\Sigma^{\dagger}X_{L}\Sigma^{\dagger})+{\rm tr}\,(X_{R}\Sigma
X_{R}\Sigma)\ .$
The plus signs follow from parity symmetry (Eq. (173) requires parity to
exchange $X_{L}$ and $X_{R}$).131313We note that parity
$q(x)\to\gamma_{4}\xi_{4}q(I_{s}x)$ is a symmetry of the SET. Setting the
spurions equal to their physical values, Eq. (173), yields the operators
$\displaystyle a^{2}{\rm tr}\,(\xi_{5\nu}\Sigma^{\dagger})\,{\rm
tr}\,(\xi_{5\nu}\Sigma)\ ,$ (175) $\displaystyle a^{2}\left({\rm
tr}\,(\xi_{5\nu}\Sigma^{\dagger})\right)^{2}+a^{2}\left({\rm
tr}\,(\xi_{5\nu}\Sigma)\right)^{2}\ ,$ $\displaystyle a^{2}{\rm
tr}\,(\xi_{5\nu}\Sigma^{\dagger}\xi_{5\nu}\Sigma^{\dagger})+a^{2}{\rm
tr}\,(\xi_{5\nu}\Sigma\xi_{5\nu}\Sigma)\ .$
Each of these operators should be added to Eq. (169) with an arbitary new
coefficient, thus introducing new LECs into the chiral lagrangian.
Before we go on to our next example, we note that we could proceed slightly
differently. Instead of the spurions $X_{L,R}$ we could first write
$a^{2}(\overline{q}\xi_{5\nu}q)(\overline{q}\xi_{5\nu}q)=a^{2}\left((\overline{q}_{R}\xi_{5\nu}q_{L})^{2}+(\overline{q}_{R}\xi_{5\nu}q_{L})(\overline{q}_{L}\xi_{5\nu}q_{R})+(\overline{q}_{L}\xi_{5\nu}q_{R})^{2}\right)\
,$ (176)
and introduce ${\cal O}(a^{2})$ spurions
$\displaystyle X_{LL}$ $\displaystyle\to$ $\displaystyle
a^{2}\xi_{5\nu}\times\xi_{5\nu}\ ,$ (177) $\displaystyle X_{LR}$
$\displaystyle\to$ $\displaystyle a^{2}\xi_{5\nu}\times\xi_{5\nu}\ ,$
$\displaystyle X_{RR}$ $\displaystyle\to$ $\displaystyle
a^{2}\xi_{5\nu}\times\xi_{5\nu}\ ,$
for each of the three terms on the right-hand side of Eq. (176), thus working
only with spurions of order $a^{2}$. The resulting terms in the chiral
lagrangian will be the same, so instead we will use the simpler ${\cal O}(a)$
spurions above, with the extra rule that all terms need to contain two of the
spurions $X_{L}$ and $X_{R}$.
Our second example starts from
$a^{2}(\overline{q}\gamma_{\mu}\xi_{5}q)(\overline{q}\gamma_{\mu}\xi_{5}q)=a^{2}(\overline{q}_{L}\gamma_{\mu}\xi_{5}q_{L}+\overline{q}_{R}\gamma_{\mu}\xi_{5}q_{R})^{2}\to((\overline{q}_{L}\gamma_{\mu}Y_{L}q_{L}+\overline{q}_{R}\gamma_{\mu}Y_{R}q_{R})^{2}\
,$ (178)
where we introduced two new spurions $Y_{L}$ and $Y_{R}$ transforming as
$Y_{L}\to V_{L}Y_{L}V_{L}^{\dagger}\ ,\qquad Y_{R}\to
V_{R}Y_{R}V_{R}^{\dagger}\ .$ (179)
Their physical values are
$Y_{L}=Y_{R}=a\xi_{5}\ .$ (180)
Again, we need the rule that any term in the chiral lagrangian should contain
two $Y$ spurions, conforming with Eq. (178). The only operator that can be
constructed from $\Sigma$ and two $Y$-spurions is
${\rm tr}\,(Y_{L}\Sigma Y_{R}\Sigma^{\dagger})\to a^{2}{\rm
tr}\,(\xi_{5}\Sigma\xi_{5}\Sigma^{\dagger})\ ,$ (181)
which thus introduces one new LEC into the chiral lagrangian.
Performing this procedure for all dimension-6 operators we constructed in Sec.
IV.2 leads to an order-$a^{2}$ “potential”
$a^{2}{\cal V}=a^{2}({\cal U}+{\cal U}^{\prime})$ (182)
to be added to Eq. (169), with [16, 18]
$\displaystyle-{\cal U}$ $\displaystyle=$ $\displaystyle C_{1}\,{\rm
tr}\,(\xi_{5}\Sigma\xi_{5}\Sigma^{\dagger})+{1\over
2}C_{3}\sum_{\nu}\left({\rm
tr}\,(\xi_{\nu}\Sigma\xi_{\nu}\Sigma)+\mbox{h.c.}\right)$
$\displaystyle+{1\over 2}C_{4}\sum_{\nu}\left({\rm
tr}\,(\xi_{5\nu}\Sigma\xi_{5\nu}\Sigma)+\mbox{h.c.}\right)+C_{6}\sum_{\mu<\nu}{\rm
tr}\,(\xi_{\mu\nu}\Sigma\xi_{\mu\nu}\Sigma^{\dagger})\ ,$ $\displaystyle-{\cal
U}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{4}C_{2V}\sum_{\nu}\left(({\rm
tr}\,(\xi_{\nu}\Sigma))^{2}+\mbox{h.c.}\right)+\frac{1}{4}C_{2A}\sum_{\nu}\left(({\rm
tr}\,(\xi_{5\nu}\Sigma))^{2}++\mbox{h.c.}\right)$ $\displaystyle+{1\over
2}C_{5V}\sum_{\nu}{\rm tr}\,(\xi_{\nu}\Sigma)\,{\rm
tr}\,(\xi_{\nu}\Sigma^{\dagger})+{1\over 2}C_{5A}\sum_{\nu}{\rm
tr}\,(\xi_{5\nu}\Sigma)\,{\rm tr}\,(\xi_{\nu}\Sigma^{\dagger})\ ,$
in which $\xi_{\mu\nu}=i\xi_{\mu}\xi_{\nu}$. We find that the operators
$S\times V$, $S\times A$, $P\times V$, $P\times A$, $T\times V$, and $T\times
A$ are like example 1 above, and contribute to the LECs $C_{3}$, $C_{4}$,
$C_{2V,A}$ and $C_{5V,A}$. The operators $V\times P$, $A\times P$, $V\times
T$, and $A\times T$ are like example 2 above, and contribute to $C_{1}$ and
$C_{6}$. The operators $V\times S$ and $A\times S$ do not break taste
symmetry, and thus do not contribute any new operators to the chiral
lagrangian. However, they do lead to order-$a^{2}$ corrections to LECs already
present in the continuum lagrangian. The lagrangian obtained by adding ${\cal
V}$ to Eq. (169) defines staggered ChPT (SChPT) at lowest-order.
Operators like $V_{\mu}\times T_{\mu}$, i.e., operators of type B, do
contribute at order $a^{2}$ to the SET, but not at this order to SChPT. The
reason is that operators that break $SO(4)$ (while preserving hypercubic
rotations) can occur only at higher orders in SChPT. For example one can make
a hypercubic invariant $\sim\sum_{\mu}\partial_{\mu}^{4}$ which is not an
$SO(4)$ invariant. Such an operator is $O(a^{2}p^{4})$ in ChPT power counting
(the $a^{2}$ has to be present because the continuum theory respects $SO(4)$
rotational invariance), and thus always of higher order than the LO
lagrangian. This has an important consequence: To order $a^{2}$, the physics
of NGBs is invariant under the $SO(4)_{\rm taste}$ subgroup of $SU(4)_{\rm
taste}$, and not just under the smaller staggered symmetry group [16]! Other
examples are operators of the schematic form
$\sim\sum_{\mu}\partial_{\mu}^{2}\xi_{\mu}\xi_{\mu}$ or
$\sum_{\mu}\xi_{\mu}\xi_{\mu}\xi_{\mu}\xi_{\mu}$, which also respect only the
staggered hypercubic invariance. We will discuss such examples toward the end
of this subsection; such operators are also higher order than the LO
lagrangian.
This point brings us to the issue of power counting. SChPT has three small
parameters, the typical momentum of physical low-energy pions, the quark
masses in $M_{ia,jb}=m_{i}\delta_{ij}\delta_{ab}$, and the parameter $a^{2}$.
Since for physical momenta, $p^{2}=-m_{\pi}^{2}$, and, as we will see shortly,
$m_{\pi}^{2}\sim m_{i}$, the first two expansion parameters are
related.141414However, for an alternative scenario, which appears not to be
realized in nature, see Ref. [23]. But $a^{2}$ and the $m_{i}$ are in
principle independent, and it depends on the actual numerical computations
carried out in lattice QCD which power counting is most appropriate. Most
staggered computations have NGB (squared-mass) taste splittings comparable in
size to the NGB (squared) masses. We will thus assume a power counting
$a^{2}\Lambda_{\rm QCD}^{2}\sim\frac{m_{i}}{\Lambda_{\rm
QCD}}\sim\frac{p^{2}}{\Lambda_{\rm QCD}^{2}}\ .$ (184)
Of course, if the lattice spacing is small enough, it may be possible to
ignore lattice spacing artifacts, in which case one can use continuum ChPT. On
the other hand, if the lattice spacing is too large, SChPT may break down,
even if the masses $m_{i}$ are small enough.
We can now look at the NGB masses predicted by LO SChPT. We write the pion
matrix in Eq. (165) as
$\pi_{ij}=\sum_{A}\xi_{A}\pi^{A}_{ij}\ ,\qquad\xi_{A}\in\\{\xi_{5},\
\xi_{5\mu},\ \xi_{\mu\nu},\ \xi_{\mu},\ 1\\}$ (185)
and expand the LO lagrangian to second order in the pion fields. This allows
us to read off the masses at lowest order. For $i\neq j$ we find
$m_{Aij}^{2}=B(m_{i}+m_{j})+a^{2}\Delta(\xi_{A})\ ,$ (186)
with
$\displaystyle\Delta(\xi_{5})$ $\displaystyle=$ $\displaystyle 0\ ,$ (187)
$\displaystyle\Delta(\xi_{5\mu})$ $\displaystyle=$
$\displaystyle\frac{16}{f^{2}}(C_{1}+3C_{3}+C_{4}+3C_{6})\ ,$
$\displaystyle\Delta(\xi_{\mu\nu})$ $\displaystyle=$
$\displaystyle\frac{16}{f^{2}}(2C_{3}+2C_{4}+4C_{6})\ ,$
$\displaystyle\Delta(\xi_{\mu})$ $\displaystyle=$
$\displaystyle\frac{16}{f^{2}}(C_{1}+C_{3}+3C_{4}+3C_{6})\ ,$
$\displaystyle\Delta(1)$ $\displaystyle=$
$\displaystyle\frac{16}{f^{2}}(4C_{3}+4C_{4})\ .$
We see that there is a degeneracy predicted by the accidental $SO(4)$ symmetry
that is present in the chiral lagrangian at LO, i.e., to order $a^{2}$. In the
next section, we will see that the flavor non-diagonal NGBs form eight
different multiplets (instead of the five seen in Eq. (187)) if taste symmetry
is fully broken. We also note that the $\xi_{5}$ NGBs are massless when all
quark masses vanish. This is the NGB associated with the exactly conserved
axial currents for the $U(N_{f})_{L}\times U(N_{f})_{R}$ symmetry group of
Eqs. (IV.1) and (136).
The LECs in the double-trace part of the potential, ${\cal U}^{\prime}$,
contribute only to flavor-diagonal NGB masses. To quadratic order,
${\cal U}^{\prime}_{\rm
quad}=\frac{32}{f^{2}}(C_{2V}-C_{5V})\sum_{ij}\pi^{V}_{ii}\pi^{V}_{jj}+\frac{32}{f^{2}}(C_{2A}-C_{5A})\sum_{ij}\pi^{A}_{ii}\pi^{A}_{jj}\
,$ (188)
leading to corresponding contributions to (the squares of) their masses. It is
important to note that, even if one restricts oneself to external pions that
are flavor off-diagonal, flavor diagonal pions still contribute to loops. For
examples, see Ref. [18].
If any of the $\Delta(\xi_{A})$ would be negative, this could induce
flavor/taste symmetry breaking when the quark masses $m_{i}$ are small enough.
However, QCD inequalities (see e.g. Ref. [24]) prevent this from happening.
There is, however, no argument that the combinations $C_{2V,A}-C_{5V,A}$
cannot be negative. For further discussion, see Ref. [25]. In practice no
spontaneous breaking of flavor/taste symmetries has been observed.
At the next order in SChPT, $SO(4)_{\rm taste}$ breaks down to $\Gamma_{4}$,
and rotational symmetry breaks down to the hypercubic group. We give a few
examples [19]. A single insertion of a 4-quark operator can lead to a ChPT
operator of the form
$a^{2}\sum_{\mu}{\rm
tr}\,(\Sigma\partial_{\mu}\Sigma^{\dagger}\xi_{\mu}\Sigma^{\dagger}\partial_{\mu}\Sigma\xi_{\mu})=\frac{4a^{2}}{f^{2}}\sum_{\mu}{\rm
tr}\,(\partial_{\mu}\pi\xi_{\mu}\partial_{\mu}\pi\xi_{\mu})+\dots\ ,$ (189)
in which the first $\xi_{\mu}$ comes from a spurion like $X_{L}$ and the
second from a spurion like $X_{R}$. This operator leads to direction
dependence of the dispersion relation. This contribution appears at order
$a^{2}p^{2}$, i.e., at next-to-leading order (NLO).
Another example comes from a double insertion of two 4-quark operators, and
takes the form
$a^{4}\sum_{\mu}\sum_{\nu\neq\mu}{\rm
tr}\,(\xi_{\mu\nu}\Sigma\xi_{\mu}\Sigma\xi_{\mu\nu}\Sigma^{\dagger}\xi_{\mu}\Sigma^{\dagger})\
,$ (190)
with the taste matrices coming from spurions $Y_{L}$, $X_{R}$, etc. Because of
the four spurions, this contribution appears at order $a^{4}$, which in our
power counting is NLO. At tree level, it only contributes to processes with
four or more pions. This is easy to see: if we set the last $\Sigma^{\dagger}$
equal to one, we can use $\xi_{\mu}\xi_{\mu\nu}=i\xi_{\nu}$ to write the
operator as
$a^{4}\sum_{\mu}\sum_{\nu\neq\mu}i\,{\rm
tr}\,(\xi_{\nu}\Sigma\xi_{\mu}\Sigma\xi_{\mu\nu}\Sigma^{\dagger})\ ,$ (191)
which has $SO(4)$ taste symmetry. Similar simplifications happen when one of
the other non-linear fields is set equal to one. At next-to-next-to-leading
order (NNLO) this operator does contribute to two-point functions through
loops. For a complete construction of the NLO SChPT lagrangian, see Ref. [19].
Finally, in this section, we discuss whether the accidental $SO(4)$ symmetry
at order $a^{2}$ also occurs for other hadrons than pions. As an example, we
consider taste breaking for the $\rho$ meson [16], introducing a “heavy”
$\rho$ field $\rho_{\mu}$ [26]. In the infinite-mass limit, its 4-velocity
$v_{\mu}$ is fixed. We first choose $v_{\mu}$ to be a spurion, transforming
like a vector under hypercubic rotations, and once the effective lagrangian
for the $\rho$ mass has been constructed, we set $v_{4}=1$ and $\vec{v}=0$,
choosing the $\rho$’s restframe. Using this and staggered symmetries, we can
write down the most general quadratic term in the field $\rho_{\mu
ij}=\sum_{A}\xi_{A}\rho^{A}_{\mu ij}$:
$\displaystyle{\cal L}_{\rho\rm-mass}$ $\displaystyle=$ $\displaystyle
R_{1}\sum_{\mu}{\rm tr}\,(\rho_{\mu}^{\dagger}\rho_{\mu})+R_{2}\sum_{\mu}{\rm
tr}\,(\rho_{\mu}^{\dagger}\xi_{5}\rho_{\mu}\xi_{5})+R_{3}\sum_{\mu\nu}{\rm
tr}\,(\rho_{\mu}^{\dagger}\xi_{\nu}\rho_{\mu}\xi_{\nu})$ $\displaystyle+$
$\displaystyle R_{4}\sum_{\mu\nu}{\rm
tr}\,(\rho_{\mu}^{\dagger}\xi_{5\nu}\rho_{\mu}\xi_{5\nu})+R_{5}\sum_{\mu\
\nu<\kappa}{\rm
tr}\,(\rho_{\mu}^{\dagger}\xi_{\nu\kappa}\rho_{\mu}\xi_{\nu\kappa})+R_{6}\sum_{\mu}{\rm
tr}\,(\rho_{\mu}^{\dagger}\xi_{\mu}\rho_{\mu}\xi_{\mu})$ $\displaystyle+$
$\displaystyle R_{7}\sum_{\mu}{\rm
tr}\,(\rho_{\mu}^{\dagger}\xi_{5\mu}\rho_{\mu}\xi_{5\mu})+R_{8}\sum_{\mu\neq\nu}{\rm
tr}\,(\rho_{\mu}^{\dagger}\xi_{\mu\nu}\rho_{\mu}\xi_{\mu\nu})+R_{9}\sum_{\mu}{\rm
tr}\,(\rho_{\mu}^{\dagger}\xi_{4}\rho_{\mu}\xi_{4})$ $\displaystyle+$
$\displaystyle R_{10}\sum_{\mu}{\rm
tr}\,(\rho_{\mu}^{\dagger}\xi_{45}\rho_{\mu}\xi_{45})+R_{11}\sum_{\mu\nu}{\rm
tr}\,(\rho_{\mu}^{\dagger}\xi_{4\nu}\rho_{\mu}\xi_{4\nu})+R_{12}\sum_{\mu}{\rm
tr}\,(\rho_{\mu}^{\dagger}\xi_{4\mu}\rho_{\mu}\xi_{4\mu})\ .$
For example, the term with $R_{9}$ is obtained from
$\sum_{\mu\nu}{\rm
tr}\,(\rho_{\mu}^{\dagger}\xi_{\nu}\rho_{\mu}\xi_{\nu})v_{\nu}v_{\nu}\big{|}_{v=(0,0,0,1)}=\sum_{\mu}{\rm
tr}\,(\rho_{\mu}^{\dagger}\xi_{4}\rho_{\mu}\xi_{4})\ .$ (193)
There are 11 different quark bilinear operators that create a $\rho$ with the
form
$\displaystyle\overline{q}\gamma_{k}\xi_{A}q\ ,$ (194)
$\displaystyle\xi_{A}=(\xi_{5},\xi_{m},\xi_{4k},\xi_{m5},\xi_{45},\xi_{k5},\xi_{m4},\xi_{k4},\xi_{km},\xi_{\ell
m})\ ,$
where $k\neq\ell\neq m\neq k$ [27]. Each of these transforms in an irrep of
the relevant staggered symmetry group (see next section). Thus, the 11
operators with LECs $R_{2}$ to $R_{12}$ in Eq. (V.1) completely break the
degeneracy (the operator with LEC $R_{1}$ gives the same mass to all
tastes).151515It appears that the term with LEC $R_{12}$ was missed in Ref.
[16].
### V.2 Example: Electromagnetic current correlator at NLO
As an example of SChPT at work, we will calculate the electromagnetic (EM)
current two-point function to one loop, which is the leading order for this
correlation function in ChPT. We usually refer to this as NLO, since it
involves a loop, but there is no tree-level contribution, as we will see.
We will use SChPT with two flavors, and $n_{r}$ replicas. This theory has
$4n_{r}$ up quarks, and $4n_{r}$ down quarks in the continuum limit; we will
consider the isospin-symmetric case with $m_{u}=m_{d}$. At the end of the
calculation, we will set $n_{r}=1/4$, thus implementing the rooting trick. In
the continuum limit, this will yield the expected result, which, at this
order, is that one would obtain in scalar QED (where the complex scalar is the
charged pion). This validates, in this example, the rooting trick.
In two-flavor QCD, the continuum EM current for $n_{r}$ replicas of two light
flavors is
$j_{\mu}=\sum_{i}\left(\frac{2}{3}\,\overline{u}_{i}\gamma_{\mu}u_{i}-\frac{1}{3}\,\overline{d}_{i}\gamma_{\mu}d_{i}\right)\
,$ (195)
where the index $i$ is now the replica index.
To couple SChPT to electromagnetism, we introduce the covariant derivative
$D_{\mu}\Sigma=\partial_{\mu}\Sigma-i\ell_{\mu}\Sigma+i\Sigma r_{\mu}\ ,$
(196)
where $\ell_{\mu}$ and $r_{\mu}$ are external vector fields gauging the groups
$U(N)_{L}$ and $U(N)_{R}$, respectively, with $N=8n_{r}$ (since have now
specialized to $N_{f}=2$). To obtain the coupling of the pions to the EM
field, we take $\ell_{\mu}=r_{\mu}=Qv_{\mu}$, with
$Q=\left(\begin{array}[]{cc}\frac{2}{3}I&0\\\
0&-\frac{1}{3}I\end{array}\right)\ ,$ (197)
with $I$ the $4n_{r}\times 4n_{r}$ unit matrix. We thus have
$\displaystyle D_{\mu}\Sigma$ $\displaystyle=$
$\displaystyle\partial_{\mu}\Sigma-iv_{\mu}[Q,\Sigma]\ ,$ (198)
$\displaystyle(D_{\mu}\Sigma)^{\dagger}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\Sigma^{\dagger}-iv_{\mu}[Q,\Sigma^{\dagger}]\ .$
The kinetic term in the chiral lagrangian becomes
${\cal L}_{\rm kin}=\frac{f^{2}}{8}{\rm
tr}\,\left((D_{\mu}\Sigma)^{\dagger}D_{\mu}\Sigma\right),$ (199)
and the ChPT form of the EM current (to lowest order) is obtained from the
terms linear in $v_{\mu}$:
$\displaystyle j_{\mu}$ $\displaystyle=$ $\displaystyle\frac{f^{2}}{8}\,{\rm
tr}\,\left(-i[Q,\Sigma^{\dagger}]\partial_{\mu}\Sigma-i\partial_{\mu}\Sigma^{\dagger}[Q,\Sigma]\right)$
$\displaystyle=$ $\displaystyle-\frac{i}{4}\,{\rm
tr}\,\left(Q(\pi\partial_{\mu}\pi-\partial_{\mu}\pi\pi)\right)+\dots\ ,$
where in the second line we expanded $\Sigma=1+\frac{i}{f}\pi+\dots$. Writing
the pion fields as
$\pi=\left(\begin{array}[]{cc}\frac{1}{\sqrt{2}}\pi^{0}&\pi^{+}\\\
\pi^{-}&-\frac{1}{\sqrt{2}}\pi^{0}\end{array}\right)\ ,$ (201)
and using $\pi^{\pm}_{ij}=\sum_{A}\xi_{A}\pi^{\pm A}_{ij}$, this can be
written as
$j_{\mu}=i\left(\pi^{-A}_{ij}\partial_{\mu}\pi^{+A}_{ji}-\pi^{+A}_{ij}\partial_{\mu}\pi^{-A}_{ji}\right)\
.$ (202)
It is straightforward to check that this current is conserved by using the
equation of motion $\raisebox{-1.72218pt}{\large$\Box$}\pi^{\pm
A}_{ij}=m_{A}^{2}\pi^{\pm A}_{ij}$. This current corresponds to the conserved
staggered current
$\displaystyle J_{\mu}^{\rm cons}$ $\displaystyle=$ $\displaystyle{1\over
2}\frac{2}{3}\left(\bar{\chi}_{u}(x)\eta_{\mu}(x)U_{\mu}(x)\chi_{u}(x+\mu)+\bar{\chi}_{u}(x+\mu)\eta_{\mu}(x)U^{\dagger}_{\mu}(x)\chi_{u}(x)\right)$
$\displaystyle-{1\over
2}\frac{1}{3}\left(\bar{\chi}_{d}(x)\eta_{\mu}(x)U_{\mu}(x)\chi_{d}(x+\mu)+\bar{\chi}_{d}(x+\mu)\eta_{\mu}(x)U^{\dagger}_{\mu}(x)\chi_{d}(x)\right)\
.$
The conserved staggered current is not the only discretization of the EM
current we may consider. Another example, which is not conserved on the
lattice, is the local current
$J_{\mu}^{\rm
local}=\frac{2}{3}\,\bar{\chi}_{u}(x)\eta_{\mu}(x)\zeta_{\mu}(x)\chi_{u}(x)-\frac{1}{3}\,\bar{\chi}_{d}(x)\eta_{\mu}(x)\zeta_{\mu}(x)\chi_{d}(x)\
.$ (204)
In Eqs. (V.2) and (204) we left the replica index implicit. The continuum
limit of $J_{\mu}^{\rm local}$ is161616We use $J$ for lattice currents, and
$j$ for continuum currents.
$j_{\mu}^{\rm
local}=\frac{2}{3}\,\overline{u}_{i}\gamma_{\mu}\xi_{\mu}u_{i}-\frac{1}{3}\,\overline{d}_{i}\gamma_{\mu}\xi_{\mu}d_{i}=\overline{q}\gamma_{\mu}Q\xi_{\mu}^{(2n_{r})}q\
,$ (205)
where in the second step we grouped the $u_{i}$ and $d_{i}$ quarks together
into the field $q$. To translate this current into ChPT, we set
$\ell_{\mu}=r_{\mu}=Q\xi_{\mu}^{(2n_{r})}v_{\mu}$. In general, we can define
“tasteful” currents (as opposed to the conserved current, which is tasteless)
by generalizing this to
$j_{\mu}^{A}=\frac{2}{3}\,\overline{u}_{i}\gamma_{\mu}\xi_{A}u_{i}-\frac{1}{3}\,\overline{d}_{i}\gamma_{\mu}\xi_{A}d_{i}=\overline{q}\gamma_{\mu}Q\xi_{A}^{(2n_{r})}q\
,$ (206)
with the local current corresponding to $\xi_{A}=\xi_{\mu}$, and the conserved
current corresponding to $\xi_{A}=1$. In SChPT, the corresponding currents are
$\displaystyle j_{\mu}^{A}$ $\displaystyle=$
$\displaystyle-i\,\frac{f^{2}}{4}\,{\rm
tr}\,\left(Q\xi_{A}^{(2n_{r})}[\Sigma^{\dagger},\partial_{\mu}\Sigma]\right)$
$\displaystyle=$ $\displaystyle-\frac{i}{4}\,{\rm
tr}\,\left(Q\xi_{A}^{(2n_{r})}[\pi,\partial_{\mu}\pi]\right)+\dots\ ,$
where in the second line we again expanded in the pion fields. Instead of Eq.
(201), we will now write
$\pi=\left(\begin{array}[]{cc}U&\pi^{+}\\\ \pi^{-}&D\end{array}\right)\ ,$
(208)
which, as we will see, is convenient. In terms of the fields introduced in Eq.
(208),
$j_{\mu}^{A}=-\frac{i}{6}\,{\rm
tr}\,\left(\xi_{A}\left(U{\overleftrightarrow{\partial}_{\\!\\!\\!\mu}}U+\pi^{+}{\overleftrightarrow{\partial}_{\\!\\!\\!\mu}}\pi^{-}\right)\right)+\frac{i}{12}\,{\rm
tr}\,\left(\xi_{A}\left(\pi^{-}{\overleftrightarrow{\partial}_{\\!\\!\\!\mu}}\pi^{+}+D{\overleftrightarrow{\partial}_{\\!\\!\\!\mu}}D\right)\right)\
.$ (209)
We used the notation
$f{\overleftrightarrow{\partial}_{\\!\\!\\!\mu}}g=f\partial_{\mu}g-(\partial_{\mu}f)g\
.$ (210)
Note that in these expressions, we have replaced $\xi_{A}^{(n_{r})}$ by
$\xi_{A}$, for simplicity. If $\xi_{A}=1$, we have that
${\rm tr}\,(U{\overleftrightarrow{\partial}_{\\!\\!\\!\mu}}U)=0\ ,\qquad{\rm
tr}\,(D{\overleftrightarrow{\partial}_{\\!\\!\\!\mu}}D)=0\ ,\qquad{\rm
tr}\,(\pi^{-}{\overleftrightarrow{\partial}_{\\!\\!\\!\mu}}\pi^{+})=-{\rm
tr}\,(\pi^{+}{\overleftrightarrow{\partial}_{\\!\\!\\!\mu}}\pi^{-})\ ,$ (211)
and we recover Eq. (202). If $\xi_{A}\neq 1$,
${\rm
tr}\,(\xi_{A}U{\overleftrightarrow{\partial}_{\\!\\!\\!\mu}}U)=\sum_{BC}{\rm
tr}\,(\xi_{A}[U^{B}\xi_{B},\partial_{\mu}U^{C}\xi_{C}]=\sum_{BC}{\rm
tr}\,(\xi_{A}[\xi_{B},\xi_{C}])U^{B}_{ij}\partial_{\mu}U^{C}_{ji}\neq 0$ (212)
in general, so the terms with $U$ and $D$ in Eq. (209) do not vanish. However,
if $\xi_{B}=1$, the commutator $[\xi_{B},\xi_{C}]=[1,\xi_{C}]=0$, and thus the
contribution with $U^{B}=U^{S}$ (with $U^{S}$ the coefficient of $1$ in
$U=\sum_{B}\xi_{B}U^{B}$) does in fact vanish, and hence the neutral singlet
fields $U^{S}$ and $D^{S}$ do not contribute to the current (209). Since there
is no mixing between any of the $U^{A}$ and any of the $D^{B}$ for $A,\ B\neq
S$, we can work on the basis of Eq. (208). We have that
$\displaystyle\partial_{\mu}j_{\mu}^{A}$ $\displaystyle=$
$\displaystyle-\frac{i}{6}\sum_{BC}{\rm
tr}\,(\xi_{A}[\xi_{B},\xi_{C}])U^{B}_{ij}m_{\pi C}^{2}U^{C}_{ji}+\dots$
$\displaystyle=$ $\displaystyle-\frac{i}{12}\sum_{BC}{\rm
tr}\,(\xi_{A}[\xi_{B},\xi_{C}])U^{B}_{ij}(m_{\pi C}^{2}-m_{\pi
B}^{2})U^{C}_{ji}+\dots\neq 0\ ,$
we see that indeed the current $j_{\mu}^{A}$ is also not conserved in SChPT,
unless $\xi_{A}=1$. Of course, in the continuum limit, where all pions become
degenerate become equal, all currents are conserved.
We note that
$Q={1\over 2}\tau_{3}+\frac{1}{6}Y={1\over 2}\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)+\frac{1}{6}\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right)\ .$ (214)
Since only the commutator $[Q,\Sigma]$ appears in the conserved current, the
hypercharge part $Y$ drops out, and thus in SChPT, the conserved EM current is
the isospin-1 current. For the other currents, with a non-trivial $\xi_{A}$,
the situation is different, as we will discuss shortly.
First, let us return to the $\xi_{A}=1$ current $j_{\mu}^{S}=j_{\mu}$, in the
continuum, at the quark level. With $\langle\dots\rangle_{U_{\mu}}$ denoting
the average over the gauge fields
$\langle(\overline{q}(x)\gamma_{\mu}{1\over
2}\tau_{3}q(x))(\overline{q}(y)\gamma_{\nu}{1\over
2}\tau_{3}q(y))\rangle=-{1\over 2}\langle{\rm
tr}\,(\gamma_{\mu}S(x,y;U_{\mu})\gamma_{\nu}S(y,x;U_{\mu})\rangle_{U_{\mu}}\
,$ (215)
when $m_{u}=m_{d}$, because any quark-disconnected parts cancel. Here
$S(x,y;U_{\mu})$ is the quark propagator (up or down) in a gauge-field
background. The trace is over spin, taste and replica indices. We also have
$\langle(\overline{q}(x)\gamma_{\mu}{1\over
2}\tau_{3}q(x))(\overline{q}(y)\gamma_{\nu}\frac{1}{6}Yq(y))\rangle=0\ ,$
(216)
and, finally,
$\displaystyle\langle(\overline{q}(x)\gamma_{\mu}\frac{1}{6}Yq(x))(\overline{q}(y)\gamma_{\nu}\frac{1}{6}Yq(y))\rangle$
$\displaystyle=$ $\displaystyle-\frac{1}{18}\langle{\rm
tr}\,(\gamma_{\mu}S(x,y;U_{\mu})\gamma_{\nu}S(y,x;U_{\mu})\rangle_{U_{\mu}}$
$\displaystyle+\frac{1}{9}\langle{\rm tr}\,(\gamma_{\mu}S(x,x;U_{\mu}))\,{\rm
tr}\,(\gamma_{\nu}S(y,y;U_{\mu}))\rangle_{U_{\mu}}\ .$
The hypercharge current two-point function is the sum of a quark-connected
(first line) and quark-disconnected (second line) part.
It follows that the light-quark connected part of the EM current two-point
function is equal to
$-\frac{5}{9}\langle{\rm
tr}\,(\gamma_{\mu}S(x,y;U)\gamma_{\nu}S(y,x;U)\rangle_{U}=\frac{10}{9}\times\langle(\overline{q}(x)\gamma_{\mu}{1\over
2}\tau_{3}q(x))(\overline{q}(y)\gamma_{\nu}{1\over 2}\tau_{3}q(y))\rangle\ ,$
(218)
i.e., the light-quark connected part of the EM current two-point function is
equal to $10/9$ times the $I=1$ current two-point function.
Now consider a similar exercise for the current $j_{\mu}^{A}$ with
$\xi_{A}\neq 1$. In this case, the relations (215) and (V.2) hold, with a
$\xi_{A}$ inserted next to every $\gamma_{\mu}$ and $\gamma_{\nu}$. For this
current, there is no quark-disconnected part, because ${\rm tr}\,(\xi_{A})=0$!
Therefore the $j_{\mu}^{A}$ two-point function is equal to its light-quark
connected part, and, in the continuum limit, the light-quark connected parts
of all currents, including the one with $\xi_{A}$, are equal. Therefore, in
the continuum limit,
$\langle j_{\mu}^{A}(x)j_{\nu}^{A}(y)\rangle\big{|}_{A\neq
S}=\frac{10}{9}\langle j_{\mu}^{S}(x)j_{\nu}^{S}(y)\rangle\ .$ (219)
We thus expect to reproduce this relation in the continuum limit of SChPT as
well.
Next, we carry out the actual calculation, beginning with the $U$ part of the
current (209). As we will be interested in
$C^{A}(t)=\frac{1}{3}\sum_{i=1}^{3}\sum_{\vec{x}}\langle
j_{i}^{A}(\vec{x},t)j_{i}^{A}(0)\rangle\ ,$ (220)
we calculate, using $U_{k\ell}(x)=\sum_{X}\xi_{X}U^{X}_{k\ell}(x)$ (where $k$
and $\ell$ are replica indices),
$\displaystyle\langle j_{\mu}^{A}(x)j_{\nu}^{A}(y)\rangle_{U\mbox{\tiny-
part}}$ $\displaystyle=$
$\displaystyle-\frac{1}{36}\sum_{XX^{\prime}YY^{\prime}}\langle(U^{X}_{mn}(x){\overleftrightarrow{\partial}_{\\!\\!\\!\mu}}U^{X^{\prime}}_{nm}(x))(U^{Y}_{k\ell}(y){\overleftrightarrow{\partial}_{\\!\\!\\!\mu}}U^{Y^{\prime}}_{\ell
k}(y))\rangle$
$\displaystyle\phantom{-\frac{1}{36}\sum_{XX^{\prime}YY^{\prime}}}\times\,{\rm
tr}\,(\xi_{A}\xi_{X}\xi_{X^{\prime}})\,{\rm
tr}\,(\xi_{A}\xi_{Y}\xi_{Y^{\prime}})$ $\displaystyle=$
$\displaystyle-\frac{n_{r}^{2}}{36}\int\frac{dp_{4}}{2\pi}\int\frac{dq_{4}}{2\pi}\frac{1}{V}^{2}\sum_{\vec{p},\vec{q}}\sum_{XX^{\prime}}\frac{e^{ip(x-y)}e^{iq(y-x)}}{(p^{2}+m_{X}^{2})(q^{2}+m_{X^{\prime}}^{2})}(p_{\mu}+q_{\mu})^{2}$
$\displaystyle\phantom{-}\times\Bigl{(}{\rm
tr}\,(\xi_{A}\xi_{X}\xi_{X^{\prime}})\,{\rm
tr}\,(\xi_{A}\xi_{X}\xi_{X^{\prime}})-{\rm
tr}\,(\xi_{A}\xi_{X}\xi_{X^{\prime}})\,{\rm
tr}\,(\xi_{A}\xi_{X^{\prime}}\xi_{X})\Bigr{)}\ ,$
where $V$ is the spatial volume (we take the time direction infinite), and
$\langle
U^{X}_{mn}(x)U_{k\ell}^{Y}(y)\rangle=\delta_{XY}\delta_{m\ell}\delta_{nk}\int\frac{dp_{4}}{2\pi}\frac{1}{V}\sum_{\vec{p}}\frac{e^{ip(x-y)}}{p^{2}+m_{X}^{2}}\
.$ (222)
Setting $\mu=i$, summing over $\vec{x}$, $\vec{y}$ and $i$, and using
$\displaystyle\int\frac{dp_{4}}{2\pi}\frac{e^{ip_{4}t}}{p^{2}+m_{X}^{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{2E_{X}(\vec{p})}\,e^{-E_{X}(\vec{p})|t|}\ ,$ (223)
$\displaystyle E_{X}(\vec{p})$ $\displaystyle=$
$\displaystyle\sqrt{{\vec{p}}^{2}+m_{X}^{2}}\ ,$
we obtain
$\displaystyle 3C^{A}(t)_{U\mbox{\tiny-part}}$ $\displaystyle=$
$\displaystyle-\frac{n_{r}^{2}}{36V}\sum_{\vec{p}}\sum_{XX^{\prime}}\frac{e^{-(E_{X}(\vec{p})+E_{X^{\prime}}(\vec{p}))|t|}}{E_{X}(\vec{p})E_{X^{\prime}}(\vec{p})}\,{\vec{p}}^{2}$
$\displaystyle\phantom{-}\times\Bigl{(}{\rm
tr}\,(\xi_{A}\xi_{X}\xi_{X^{\prime}})\,{\rm
tr}\,(\xi_{A}\xi_{X}\xi_{X^{\prime}})-{\rm
tr}\,(\xi_{A}\xi_{X}\xi_{X^{\prime}})\,{\rm
tr}\,(\xi_{A}\xi_{X^{\prime}}\xi_{X})\Bigr{)}\ .$
We will refer to the first double trace over taste matrices as the “unmixed”
part, and to the second double trace as the “mixed” part. We note that for
$\xi_{A}=1$ this vanishes, consistent with the fact that there is no $U$
contribution to the conserved current. For the $D$ contribution, we get $1/4$
of this, and thus the sum of the $U$ and $D$ contributions is equal to
$36\times\left(1/36+1/144\right)=5/4$ times Eq. (V.2).
It remains to add the contribution from the charged-pion part of the current.
One can verify that choosing the term
$\pi^{+}{\overleftrightarrow{\partial}_{\\!\\!\\!\mu}}\pi^{-}$ in both
currents yields 1 times the mixed term in Eq. (V.2), choosing the term
$\pi^{-}{\overleftrightarrow{\partial}_{\\!\\!\\!\mu}}\pi^{+}$ in both
currents yields $1/4$ times the mixed term in Eq. (V.2), while the cross terms
yields $-1$ times the unmixed term in Eq. (V.2). Adding all contributions
leads to
$\displaystyle 3C^{A}(t)$ $\displaystyle=$
$\displaystyle\frac{n_{r}^{2}}{144V}\sum_{\vec{p}}\sum_{XX^{\prime}}\frac{e^{-(E_{X}(\vec{p})+E_{X^{\prime}}(\vec{p}))|t|}}{E_{X}(\vec{p})E_{X^{\prime}}(\vec{p})}\,{\vec{p}}^{2}$
$\displaystyle\times\Bigl{(}10{\rm tr}\,(\xi_{A}\xi_{X}\xi_{X^{\prime}})\,{\rm
tr}\,(\xi_{A}\xi_{X^{\prime}}\xi_{X})-{\rm
tr}\,(\xi_{A}\xi_{X}\xi_{X^{\prime}})\,{\rm
tr}\,(\xi_{A}\xi_{X}\xi_{X^{\prime}})\Bigr{)}\ .$
For $\xi_{A}=1$ the traces set $X^{\prime}=X$, and setting all masses $m_{X}$
equal (i.e., taking the continuum limit), the factor on the second line of
this result is equal to $9\times 4^{2}\times 16=144\times 16$ (a 4 for each
trace; the 16 comes from the $\sum_{X}$). With $n_{r}=1/4$, the total
prefactor becomes equal to one, which is what one would have obtained directly
in the continuum. For $a>0$, the taste pion $\pi_{X}$ runs around the loop,
and these loops are averaged over all tastes.
When $\xi_{A}\neq 1$, a little more work is needed. First, we have that
${\rm tr}\,(\xi_{A}\xi_{X}\xi_{X^{\prime}})={\rm
tr}\,((\xi_{A}\xi_{X}\xi_{X^{\prime}})^{T})={\rm
tr}\,(\xi_{X^{\prime}}^{T}\xi_{X}^{T}\xi_{A}^{T})={\rm
tr}\,(\xi_{A}\xi_{X^{\prime}}\xi_{X}^{\prime})^{*}\ ,$ (226)
and thus
${\rm tr}\,(\xi_{A}\xi_{X}\xi_{X^{\prime}})\,{\rm
tr}\,(\xi_{A}\xi_{X^{\prime}}\xi_{X})=16\delta_{X^{\prime},X+A}\ .$ (227)
This takes care of the mixed trace in Eq. (V.2). We still get the factor 16,
and the current inserts taste $A$ into the pion loop. For the unmixed trace,
we use that
$\xi_{A}\xi_{X}=(-1)^{|A||X|+A\cdot X}\xi_{X}\xi_{A}\ ,$ (228)
with $|X|=\sum_{\mu}X_{\mu}$ and $A\cdot X=\sum_{\mu}A_{\mu}X_{\mu}$. Thus
$({\rm tr}\,(\xi_{A}\xi_{X}\xi_{X^{\prime}}))^{2}=16(-1)^{|A||X|+A\cdot
X}\delta_{X^{\prime},X+A}\ .$ (229)
For $\xi_{A}=1$ the right-hand side is equal to $\delta_{X,X^{\prime}}$, as
before. For any $\xi_{A}\neq 1$, $(-1)^{|A||X|+A\cdot X}=+1$ for half of the
tastes $X$, and $-1$ for the other half. Thus, when all masses become
degenerate in the continuum limit, the trace term on the second line of Eq.
(V.2) becomes equal to 11 when $(-1)^{|A||X|+A\cdot X}=-1$, and 9 when
$(-1)^{|A||X|+A\cdot X}=+1$. The total prefactor thus becomes
$\frac{n_{r}^{2}}{144}\times 4^{2}\times 8\times(11+9)=\frac{10}{9}\times
16n_{r}^{2}\ ,$ (230)
which is $10/9$ times the continuum limit for $\xi_{A}=1$, as we expect from
Eq. (219). Here the factors 4 again come from the traces, and the factor
$8\times(11+9)$ from the $\sum_{X}$. Again, the current $j_{\mu}^{A}$ inserts
a taste $A$ into the pion loop. This verifies Eq. (219) at this order SChPT,
in the continuum limit. For $a>0$, the sums in Eq. (V.2) have to be carried
out to obtain the complete result in explicit form. The Kronecker delta in Eq.
(227) reduces the sum over $X$ and $X^{\prime}$ to a single sum.
## VI Staggered states and operators
In this chapter, we first consider the spectrum of QCD with staggered fermions
and general properties of staggered correlation functions, after which we
discuss the construction of operators that create and annihilate states in the
staggered Hilbert space. Most of the chapter will deal with mesons, but in the
final section we will briefly discuss aspects of baryonic states in QCD with
staggered fermions. The material of the first three sections can be found in
Refs. [27, 28, 29], while the material of the final section can be found in
Refs. [28, 30]. Another useful reference is Ref. [31].
### VI.1 Staggered spectrum: states and operators
We will consider states in momentum space, because the irreducible
representations of a theory with staggered fermions live in momentum space.
Operators acting on the staggered Hilbert space will be denoted with a hat.
Thus, $\hat{S}_{\mu}$ is the shift operator acting on the Hilbert space.
Defining $\hat{T}_{\mu}=\hat{S}_{\mu}^{2}$, $\hat{T}_{\mu}$ is a normal
translation over two lattice spacings in the $\mu$ direction. In momentum
space, the eigenvalues are
$\hat{T}_{k}:\quad e^{2ip_{k}}\ ,\qquad\hat{T}_{4}:\quad e^{-2E}\ ,$ (231)
where $p_{k}$ with $-\pi/2<p_{k}\leq\pi/2$ are the components of the physical
spatial momentum of a state, and $E$ is the energy. The operator $\hat{T}_{4}$
can be taken as the transfer matrix of the theory. With Eq. (231) we can
define the operators $\hat{T}_{\mu}^{-{1\over 2}}$ as having eigenvalues
$e^{-ip_{k}}$ and $e^{E}$, and thus we can define
$\hat{\Xi}_{\mu}=\hat{S}_{\mu}\hat{T}_{\mu}^{-{1\over 2}}\ ,$ (232)
with
$\hat{\Xi}_{\mu}^{2}=1\ .$ (233)
We will consider states with $\vec{p}=0$ (except in Sec. VI.3). Such states
transform in representations of the geometrical restframe group ($GRF$)
$GRF=G(\Xi_{\mu},R^{(k\ell)},I_{s})\ ,$ (234)
where the spatial rotations $R^{(k\ell)}=R^{(k\ell)}(\pi/2)$ generate the
little group of $\vec{p}=0$ states, and we recall that
$I_{s}=I_{1}I_{2}I_{3}=\Gamma_{4}\Xi_{4}$ in the defining representation. The
notation $G(X)$ denotes the finite group generated by the elements $X$. Since
$\Xi_{4}\in GRF$, also parity $P=I_{s}\Xi_{4}=\Gamma_{4}\in GRF$, and, with
$P$ commuting with all other elements of the group,
$GRF=G(\Xi_{\mu},R^{(k\ell)})\times\\{1,\ P\\}\ .$ (235)
A maximally commuting set of elements of $GRF$ is the set $\\{\Xi_{4},\
\Xi_{1}\Xi_{2},\ R^{(12)},\ P\\}$.
We will label states in the Hilbert space by the eigenvalues $E$, the
eigenvalue $\sigma$ of $P$, and the eigenvalue $\sigma_{t}$ of $\Xi_{4}$:
$\displaystyle\hat{S}_{4}|Er\sigma;\sigma_{t}\rangle$ $\displaystyle=$
$\displaystyle\sigma_{t}e^{-E}|Er\sigma;\sigma_{t}\rangle\ ,$ (236)
$\displaystyle\Xi_{4}|Er\sigma;\sigma_{t}\rangle$ $\displaystyle=$
$\displaystyle\sigma_{t}|Er\sigma;\sigma_{t}\rangle\ ,$ $\displaystyle
P|Er\sigma;\sigma_{t}\rangle$ $\displaystyle=$
$\displaystyle\sigma|Er\sigma;\sigma_{t}\rangle\ ,$ $\displaystyle
I_{s}|Er\sigma;\sigma_{t}\rangle$ $\displaystyle=$
$\displaystyle\sigma\sigma_{t}|Er\sigma;\sigma_{t}\rangle\equiv\sigma_{s}|Er\sigma;\sigma_{t}\rangle\
.$
The label $r$ denotes an irrep of $G(\Xi_{\mu},R^{(k\ell)})$; within such an
irrep, we can label the state with the eigenvalue $\sigma_{t}$ of $\Xi_{4}$.
The final equation defines the parity $\sigma_{s}$ of $I_{s}$.
Next, we introduce local operators $\hat{\phi}$ and $\hat{\overline{\phi}}$ at
fixed time $t$, which thus transform in representations of a smaller group,
the geometric time-slice group
$GTS=G(\Xi_{m},R^{(k\ell)},I_{s})\ ,$ (237)
which does not contain $\Xi_{4}$. We define time-dependent operators (i.e., we
use the Heisenberg picture) through
$\hat{\phi}(t)=\hat{S}_{4}^{-t}\hat{\phi}\hat{S}_{4}^{t}\
,\qquad\hat{\overline{\phi}}(t)=\hat{S}_{4}^{-t}\hat{\overline{\phi}}\hat{S}_{4}^{t}\
.$ (238)
There is a $U(1)$ charge operator $\hat{Q}$ (quark number) with
$[\hat{Q},\hat{\phi}]=q\hat{\phi}\
,\qquad[\hat{Q},\hat{\overline{\phi}}]=-q\hat{\overline{\phi}}\ ,$ (239)
with $q$ the charge of $\hat{\phi}$ (we can take $q=1$ for the operator
$\hat{\chi}$), and a charge conjugation operator $\hat{C}_{0}$ with
$\hat{C}_{0}^{-1}\hat{\phi}\,\hat{C}_{0}=\tau\hat{\overline{\phi}}\
,\qquad\hat{C}_{0}^{-1}\hat{\overline{\phi}}\,\hat{C}_{0}=\overline{\tau}\hat{\phi}\
.$ (240)
The phases $\tau$ and $\overline{\tau}$ depend on convention, but the phase
$\sigma_{c}=\tau\overline{\tau}$ defined by
$\hat{C}_{0}^{-2}\hat{\phi}\,\hat{C}_{0}^{2}=\sigma_{c}\hat{\phi}$ (241)
is not. Using the definition of $C_{0}$ in Sec. II.4 we find that
$\sigma_{c}=-1$ for the staggered quark field $\chi$, and thus that it is
equal to $+1$ for mesonic operators, and $-1$ for baryonic operators. The
fields $\hat{\phi}$ and $\hat{\overline{\phi}}$ transform in a representation
$r$ of $GTS$, which is isomorphic to $G(\Xi_{\mu},R^{(k\ell)})$ and can thus
be taken to be the same irrep as in Eq. (236).
We note that $GRF=GTS\times\\{1,\ P\\}$, but the fields $\hat{\phi}$ and
$\hat{\overline{\phi}}$ do not have $P$ quantum numbers, since the definition
of $P$ involves a translation in time. We can consider operators $\phi$ and
$\overline{\phi}$ with a well-defined value $\sigma_{s}$ of $I_{s}$:
$\hat{I}_{s}^{-1}\hat{\phi}\hat{I}_{s}=\sigma_{s}\hat{\phi}\
,\qquad\hat{I}_{s}^{-1}\hat{\overline{\phi}}\hat{I}_{s}=\sigma_{s}\hat{\overline{\phi}}\
.$ (242)
($\hat{\phi}$ and $\hat{\overline{\phi}}$ have the same $I_{s}$ quantum number
because $\hat{I}_{s}$ commutes with $\hat{C}_{0}$.) If $\hat{\phi}$,
$\hat{\overline{\phi}}$ have the $I_{s}$ quantum number $\sigma_{s}$, we have
that
$\langle
Er\sigma;\sigma_{t}|\hat{\overline{\phi}}|0\rangle=0\qquad\mbox{unless}\
\sigma_{s}=\sigma\sigma_{t}\ ,$ (243)
because $\hat{I}_{s}=\hat{P}\hat{\Xi}_{4}$.
An example of $\hat{\phi}$ is
$\sum_{\vec{m}}\hat{\chi}(\vec{x}+2\vec{m})\ ,$ (244)
which transforms in an 8-dimensional irrep of $GTS$. This irrep decomposes as
${\bf 8}\to A_{1}^{+}+A_{1}^{-}+F_{1}^{+}+F_{1}^{-}\ ,$ (245)
where $A_{1}$ is the trivial irrep of the cubic group and $F_{1}$ the vector
irrep. The superscript denotes the value of $I_{s}$. These irreps of the cubic
group can be projected out as [28]
$\displaystyle A_{1}^{+}:\qquad$ $\displaystyle\sum_{\vec{x}\ {\rm
even}}\hat{\chi}(\vec{x})\ ,$ $\displaystyle A_{1}^{-}:\qquad$
$\displaystyle\sum_{\vec{x}\ {\rm even}}\hat{\chi}(\vec{x}+1+2+3)\ ,$
$\displaystyle F_{1}^{-}:\qquad$ $\displaystyle\sum_{\vec{x}\ {\rm
even}}\left(\begin{array}[]{c}\hat{\chi}(\vec{x}+1)\\\
\hat{\chi}(\vec{x}+2)\\\ \hat{\chi}(\vec{x}+3)\end{array}\right)\ ,$ (250)
$\displaystyle F_{1}^{+}:\qquad$ $\displaystyle\sum_{\vec{x}\ {\rm
even}}\left(\begin{array}[]{c}\hat{\chi}(\vec{x}+2+3)\\\
-\hat{\chi}(\vec{x}+1+3)\\\ \hat{\chi}(\vec{x}+1+2)\end{array}\right)\ .$
(254)
The appearance of the irreps $A_{1}$ and $F_{1}$ for a fermionic field can be
understood as follows. In the rest frame, the relevant continuum group is
$SU(2)_{\rm spin}\times SU(2)_{\rm taste}$. The group $G(R^{(k\ell)})$ is a
discrete subgroup of the diagonal $SU(2)$ subgroup of the continuum symmetry
group, and the relevant decomposition is, for the groups $SU(2)_{\rm
spin}\times SU(2)_{\rm taste}\supset SU(2)_{\rm diag}\supset G(R^{(k\ell)})$,
$\left({\scriptstyle{{1\over 2}}}_{\rm spin},\ {\scriptstyle{{1\over 2}}}_{\rm
taste}\right)\to{\bf 0}+{\bf 1}\to A_{1}+F_{1}\ .$ (255)
Now, let us consider a correlation function of the form
$C(t)=\langle 0|\hat{\phi}(t_{1})\hat{\overline{\phi}}(t_{2})|0\rangle\ ,$
(256)
with $t=t_{1}-t_{2}>0$. Using Eqs. (236), (238) and (243), this can be written
as (note that $\hat{S}_{4}|0\rangle=|0\rangle$)
$\displaystyle C(t)$ $\displaystyle=$ $\displaystyle\sum_{E\sigma}\langle
0|\hat{\phi}(t_{1})|Er\sigma;\sigma_{t}\rangle\langle
Er\sigma;\sigma_{t}|\hat{\overline{\phi}}(t_{2})|0\rangle$ $\displaystyle=$
$\displaystyle\sum_{E\sigma}\langle
0|\hat{\phi}\hat{S}_{4}^{t_{1}}|Er\sigma;\sigma_{t}\rangle\langle
Er\sigma;\sigma_{t}|\hat{S}_{4}^{-t_{2}}\hat{\overline{\phi}}|0\rangle$
$\displaystyle=$ $\displaystyle\sum_{E\sigma}\sigma_{t}^{t}e^{-Et}\langle
0|\hat{\phi}|Er\sigma;\sigma_{t}\rangle\langle
Er\sigma;\sigma_{t}|\hat{\overline{\phi}}|0\rangle$ $\displaystyle=$
$\displaystyle\sum_{E_{+}}|\langle
0|\hat{\phi}|E_{+}r;\sigma_{t}=\sigma_{s}\rangle|^{2}\sigma_{s}^{t}e^{-E_{+}t}$
$\displaystyle+\sum_{E_{-}}|\langle
0|\hat{\phi}|E_{-}r;\sigma_{t}=-\sigma_{s}\rangle|^{2}(-\sigma_{s})^{t}e^{-E_{-}t}\
,$
where in the first term the sum is over states with positive parity
$\sigma=\sigma_{t}\sigma_{s}$ and energies $E_{+}$, and in the second term the
sum is over states with negative parity and energies $E_{-}$. The subscript on
$E_{\pm}$ indicates the parity $\sigma$. If, for example, the operator
$\hat{\phi}$ has $I_{s}$ parity $\sigma_{s}=+1$, the negative $P$ parity
states yield a rapidly oscillating contribution to $C(t)$. The correlator gets
contributions from both parities because $P$ is not well-defined for operators
localized in time.
The result (VI.1) is valid when the time extent of the lattice is infinite.
If, instead, it is $2T$ (it is convenient to take the lattice even in all
directions for staggered fermions), then, instead of considering $\langle
0|\hat{\phi}(t_{1})\hat{\overline{\phi}}(t_{2})|0\rangle$, we should consider
${\rm
Tr}\,(\hat{S}_{4}^{2T-t}\hat{\phi}\hat{S}_{4}^{t}\hat{\overline{\phi}})$, with
${\rm Tr}\,$ the trace in Hilbert space. Let us instead derive the correction
to Eq. (VI.1) more intuitively.
If a particle can travel from $t_{2}$ to $t_{1}$, an anti-particle can travel
from $t_{1}$ to $t_{2}^{\prime}=t_{2}+2T$. This leads to the replacement
$e^{-Et}\to e^{-E(t_{2}^{\prime}-t_{1})}=e^{-E(2T-t)}$. With anti-
periodic/periodic boundary conditions for the staggered quarks, we pick up a
phase factor $\kappa=+1$, respectively, $\kappa=-1$ for fermionic operators,
while always $\kappa=+1$ for bosonic operators. Furthermore,
$\langle 0|\hat{\phi}|Er\sigma;\sigma_{t}\rangle\langle
Er\sigma;\sigma_{t}|\hat{\overline{\phi}}|0\rangle=\sigma_{c}\langle
0|\hat{C}_{0}^{-1}\hat{\overline{\phi}}\hat{C}_{0}|Er\sigma;\sigma_{t}\rangle\langle
Er\sigma;\sigma_{t}|\hat{C}_{0}^{-1}\hat{\phi}\hat{C}_{0}|0\rangle\ .$ (258)
Using invariance under charge-conjugation, Eq. (VI.1) gets replaced by
$\displaystyle C(t)$ $\displaystyle=$ $\displaystyle\sum_{E_{+}}|\langle
0|\hat{\phi}|E_{+}r;\sigma_{t}=\sigma_{s}\rangle|^{2}\sigma_{s}^{t}\left(e^{-E_{+}t}+\sigma_{c}\kappa\,e^{-(2T-t)E_{+}}\right)$
$\displaystyle+\sum_{E_{-}}|\langle
0|\hat{\phi}|E_{-}r;\sigma_{t}=-\sigma_{s}\rangle|^{2}(-\sigma_{s})^{t}\left(e^{-E_{-}t}+\sigma_{c}\kappa\,e^{-(2T-t)E_{-}}\right)+\dots\
.$
Contributions proportional to $|\langle n|\hat{\phi}|m\rangle|^{2}$ with both
$|n\rangle$ and $|m\rangle$ not equal to $|0\rangle$ are not shown. Note that
if $\langle 0|\hat{\phi}|0\rangle\neq 0$, we need to subtract $\langle
0|\hat{\phi}|0\rangle\langle 0|\hat{\overline{\phi}}|0\rangle$ to get the
connected part of $C(t)$.
### VI.2 Mesons
In this subsection, we will specialize to meson operators, in particular,
operators bilinear in $\chi$ and $\bar{\chi}$ [27]. If $X_{\mu}=D(\Xi_{\mu})$
is the representation of $\Xi_{\mu}$ on a mesonic operator, they commute,
because
$D(\Xi_{\mu})D(\Xi_{\nu})=e^{iq\pi}D(\Xi_{\nu})D(\Xi_{\mu})\ ,$ (259)
and $q=0$ for mesonic operators. (This can be easily checked on the explicit
operators we will construct below.) Defining
$\tilde{X}_{m}=X_{1}X_{2}X_{3}X_{m}\ ,$ (260)
the restframe symmetry group for mesonic states is
$(RF)_{\rm mesons}=G(\tilde{X}_{m},R^{(k\ell)})\times\\{1,\
X_{4}\\}\times\\{1,\ X_{1}X_{2}X_{3}\\}\times\\{1,\ P\\}\times\\{1,\ C_{0}\\}\
.$ (261)
Mesonic states can thus be labeled by
$|E,\overline{r}^{\sigma_{t}\sigma_{123}},\sigma,\tau_{0}\rangle\ ,$ (262)
with $\overline{r}$ an irrep of $G(\tilde{X}_{m},R^{(k\ell)})$, and
$\sigma_{t}$, $\sigma_{123}$, $\sigma$ and $\tau_{0}$ the $X_{4}$, $X_{123}$,
$P$ and $C_{0}$ parities, respectively.
Defining171717After constructing meson operators, they can be made gauge
covariant by inserting averages over the shortest Wilson lines between $\chi$
and $\bar{\chi}$.
$D_{k}\chi(x)={1\over 2}\left(\chi(x+k)+\chi(x-k)\right)\ ,$ (263)
we define classes of bilinear operators (dropping hats on $\chi$ and
$\bar{\chi}$) on a fixed time slice $t$:
$\displaystyle M_{0}$ $\displaystyle=$ $\displaystyle\sum_{\vec{x}\ {\rm
even}}\bar{\chi}(x)\chi(x)\ ,$ (264) $\displaystyle M_{1}$ $\displaystyle=$
$\displaystyle\sum_{\vec{x}\ {\rm even}}\bar{\chi}(x)D_{1}\chi(x)\ ,$
$\displaystyle M_{12}$ $\displaystyle=$ $\displaystyle\sum_{\vec{x}\ {\rm
even}}\bar{\chi}(x)D_{1}D_{2}\chi(x)\ ,$ $\displaystyle M_{123}$
$\displaystyle=$ $\displaystyle\sum_{\vec{x}\ {\rm
even}}\bar{\chi}(x)D_{1}D_{2}D_{3}\chi(x)\ ,$
where all members of each class are obtained by taking the operator explicitly |
# Performance Bounds for Policy-Based Average Reward Reinforcement Learning
Algorithms
Yashaswini Murthy
Electrical and Computer Engineering
University of Illinois Urbana-Champaign
Urbana, IL 61801
<EMAIL_ADDRESS>
&Mehrdad Moharrami
Electrical and Computer Engineering
University of Illinois Urbana-Champaign
Urbana, IL 61801
<EMAIL_ADDRESS>&R. Srikant
Electrical and Computer Engineering
University of Illinois Urbana-Champaign
Urbana, IL 61801
<EMAIL_ADDRESS>
###### Abstract
Many policy-based reinforcement learning (RL) algorithms can be viewed as
instantiations of approximate policy iteration (PI), i.e., where policy
improvement and policy evaluation are both performed approximately. In
applications where the average reward objective is the meaningful performance
metric, discounted reward formulations are often used with the discount factor
being close to $1,$ which is equivalent to making the expected horizon very
large. However, the corresponding theoretical bounds for error performance
scale with the square of the horizon. Thus, even after dividing the total
reward by the length of the horizon, the corresponding performance bounds for
average reward problems go to infinity. Therefore, an open problem has been to
obtain meaningful performance bounds for approximate PI and RL algorithms for
the average-reward setting. In this paper, we solve this open problem by
obtaining the first finite-time error bounds for average-reward MDPs, and show
that the asymptotic error goes to zero in the limit as policy evaluation and
policy improvement errors go to zero.
## 1 Introduction
Reinforcement Learning algorithms can be broadly classified into value-based
methods and policy-based methods. In the case of discounted-reward Markov
Decision Processes (MDPs), value-based methods such as Q-learning [WD92,
Tsi94, JJS93, SB18, BT96], fitted value iteration [MS08] and target network
Q-learning [CCM22] can be viewed as approximately solving the fixed point of
the Bellman optimality equation to find the value function and an
approximately optimal control policy. In other words value iteration is
approximately implemented for discounted reward MDPs [Ber11, Ber12, Put14]. In
policy-gradient methods [AKLM21], a gradient step is used to improve the
policy, somewhat similar to the policy improvement step in policy iteration
for discounted-reward MDPs. In a recent paper [CM22], it has been shown that
many policy-based methods for discounted-reward MDPs can be viewed as special
cases of approximate policy iteration. The classical results on approximate
policy iteration [BT96] assume that the error in the policy evaluation and
improvement steps are constant, independent of the iteration. The key idea in
[CM22] is to show that a simple modification of the proof in [BT96] can be
used to allow for iteration-dependent policy evaluation and improvement
errors, which is then used to make the connection to policy-based methods. Our
goal in this paper is to derive similar results for average-reward problems.
Average-reward problems [ABB01, WNS21, ZWSW21, Mah96, YB09, Gos04, Sin94] are,
in some sense, harder to study than their discounted-reward counterparts. We
now discuss why this is so by recalling the error bounds for discounted-reward
problems and examining them in an appropriate limit to study their
applicability to average-reward problems.
The fundamental result on approximate policy iteration for discounted reward
MDPs which allows the connection to policy-based methods is the following
[BT96]:
$\limsup_{k\to\infty}\|J_{\mu_{k}}-J_{*}^{\alpha}\|_{\infty}\leq
H_{\alpha}^{2}(\epsilon+2\alpha\delta),$ (1)
where $\alpha$ is the discount factor, $H_{\alpha}=1/(1-\alpha),$
$J_{*}^{\alpha}$ is the optimal value function, $J_{\mu_{k}}$ is the value
function associated with the policy obtained after $k$ iterations of
approximate policy iteration, $\epsilon$ is the policy improvement error, and
$\delta$ is the policy evaluation error. Thus, as $\epsilon,\delta\rightarrow
0,$ we recover the result that standard policy iteration converges. On the
other hand, to understand whether the above bound is useful to study average-
reward problems, we write Equation 1 as
$\limsup_{k\to\infty}\frac{1}{H_{\alpha}}\|J_{\mu_{k}}-J_{*}^{\alpha}\|_{\infty}\leq
H_{\alpha}(\epsilon+2\alpha\delta).$
Under mild conditions [Ber11, Ros14], it is well known that each element of
the value function vector approaches the average-reward for each fixed policy
and for the optimal policy, in the limit $\alpha\rightarrow 1$. Hence, the
left-hand side of the above equation is an approximation to the error in
approximate policy iteration for average-reward MDPs; the right-hand side
which gives an upper bound on this error blows up to infinity, i.e., when
$\alpha\rightarrow 1.$ Thus, unlike in the discounted-reward case, the above
bound fails to even recover the well-known convergence of standard policy
iteration in average-reward case in the limit $\epsilon,\delta\rightarrow 0$
(since we have to let $\alpha\rightarrow 1$ before letting
$\epsilon,\delta\rightarrow 0$ ). However, as mentioned in [BT96], it is also
well known that approximate policy iteration performs much better than the
bound suggests. The main goal of our paper is to resolve this discrepancy
between theory and practice.
### 1.1 Contributions
It is well known that the error bound for discounted-reward, approximate
policy iteration is tight for unichain MDPs [BT96]. Thus, it is impossible to
improve upon the bound in general. However, as will be explained in a later
section, it is easy to convert most reasonable unichain MDPs to MDPs where
every stationary policy results in an irreducible Markov chain with an
arbitrarily small loss of reward. So the natural question to ask is whether
the bound can be dramatically improved for MDPs where every stationary policy
results in an irreducible Markov chain. To the best of our knowledge, no such
bound was available in the prior literature.
Our main contributions are as follows:
* •
Under the assumption that every stationary policy induces an irreducible
Markov chain, we first perform a Schweitzer transformation of the MDP [Sch71]
and obtain finite-time error bounds for average-reward approximate policy
iteration.
* •
Using the above finite-time error bounds, we prove an error bound of the form
$\limsup_{k\to\infty}J^{*}-J_{\mu_{k}}\leq f(\epsilon,\delta),$
for average-reward approximate policy iteration for a function $f$ such that
$f(\epsilon,\delta)\rightarrow 0$ in the limit as $\epsilon,\delta\rightarrow
0.$ Note that this is in sharp contrast to the result that one can obtain from
the discounted-reward analysis where the error bound blows up to infinity when
applied to discount factors approaching $1,$ which is the appropriate limit to
convert discounted-reward problems to average-reward problems.
* •
The main difficulty in obtaining the above bound when compared to the
discounted-reward case is the lack of infinity-norm contraction property for
the Bellman operator in the average-reward case. In most analysis of average-
reward problems, this difficulty is circumvented by the use of a span-
contraction property [Put14]. However, the span contraction property is
insufficient for our purpose and therefore, we use a technique due to [VdW80]
for the study of a different algorithm called modified policy iteration in
[Put14].
* •
Next, we extend the above analysis to obtain finite-iteration error bounds for
the case where the policy evaluation and improvement errors can potentially
vary from iteration to iteration. Further, we allow these errors to be random
(as would be the case, for example, when one uses TD learning for policy
evaluation and soft policy improvement) and obtain expected error bounds on
approximate policy iteration after a finite number of iterations.
* •
Our error bounds are in a form such that one can then apply them to many
policy-based RL algorithms, i.e., we can simply plug in error bounds for
specific policy evaluation or policy improvement algorithms. We illustrate the
applicability of the results to several different policy-based RL methods
(softmax update, greedy update and mirror descent update) studied in the
literature [MXSS20, BT96, AYBB+19, AKLM21].
* •
While the main focus of our paper is on offline learning, we provide
connections to regret bounds for online learning in average-reward MDPS as in
[AYBB+19, LYAYS21].
### 1.2 Related Work
Our work in this paper is most closely related to the work in [CM22]. The main
contribution there was to recognize that an extension of the approximate
policy iteration results in [BT96] can be used to derive finite-time error
bounds for many RL-based algorithms. The idea of considering iteration-
dependent policy evaluation and improvement bounds was also used to study
policy iteration with function approximation in discounted-reward MDPs in
[WLLS21]. Our contribution here is to derive the first error bound for
approximate policy iteration for average-reward MDPs and then write it in a
form which can be applied to policy-based RL methods for average-reward TD
learning. To the best of our knowledge, no known error bounds existed in prior
literature for average-reward approximate policy iteration due to the fact
that the corresponding bounds for discounted MDPs increase proportional to the
square of length of the horizon. Thus, in the limit as the horizon increases
to infinity, we have essentially reduced the dependence of the error from the
square of the length of the horizon to linear in the length of the horizon.
RL algorithms have not been studied as extensively for the average-reward case
as they are for discounted-reward MDPs. There is recent work [ZZM21] on
average-reward TD-learning which we leverage to illustrate the applicability
of our results. There are also asymptotic convergence results for policy
gradient and actor-critic methods for average-reward MDPs [TVR99, KT99,
BSGL09, MMT00]. However, these results only prove convergence to a stationary
point and there are no global convergence results. The original paper on
natural policy gradient (NPG) is written for average-reward MDPs, but the
extensive performance analysis of NPG in subsequent work such as [AKLM21]
seems to only consider the discounted-reward case. Recent work [AYBB+19,
LYAYS21] considers mirror descent in average reward RL but do not provide
performance results for other RL algorithms. In a later section, we compare
our results to [AYBB+19, LYAYS21] in the special case of mirror descent-type
policy updates
## 2 Model and Preliminaries
### 2.1 Average Reward Formulation
We consider the class of infinite horizon MDPs with finite state space
$\mathcal{S}$, finite action space $\mathcal{A}$, and transition kernel
$\mathbb{P}$, where $|\mathcal{S}|=n$ and $|\mathcal{A}|=m$. Let
$\Delta\mathcal{A}$ denote the probability simplex over actions. We consider
the class of randomized policies
$\Pi=\\{\mu:\mathcal{S}\to\Delta\mathcal{A}\\}$, so that a policy $\mu$
assigns a probability vector over actions to each state. Given a policy $\mu$,
the transition kernel for the underlying Markov process is denoted by
$\mathbb{P}_{\mu}:\mathcal{S}\to\mathcal{S}$, where
$\mathbb{P}_{\mu}(s^{\prime}|s):=\sum_{a\in\mathcal{A}}\mu(a|s)\mathbb{P}(s^{\prime}|s,a)$
is the probability of moving to state $s^{\prime}\in\mathcal{S}$ from
$s\in\mathcal{S}$ upon taking action $\mu(s)\in\mathcal{A}$. Associated with
each state-action pair $(s,a)$, is a one-step reward which is denoted by
$r_{\mu}(s):=\mathbb{E}_{a\sim\mu(s)}r(s,a)$.
Let $J_{\mu}\in\mathbb{R}$ be the average reward associated with the policy
$\mu\in\Pi$, i.e. $J_{\mu}$ is defined as:
$J_{\mu}=\lim_{T\to\infty}\frac{\mathbb{E}_{\mu}\left(\sum_{i=0}^{T-1}r_{\mu}(s_{i})\right)}{T}.$
Here the expectation is taken with respect to the measure $\mathbb{P}_{\mu}$
associated with the policy $\mu$. Let $h_{\mu}\in\mathbb{R}^{n}$ be the
relative value function associated with the policy $\mu$. Defining
$\boldsymbol{1}\in\mathbb{R}^{n}$ to be the vector of all ones, the pair
$(J_{\mu},h_{\mu})$ satisfies the following average reward Bellman equation:
$J_{\mu}\boldsymbol{1}+h_{\mu}=r_{\mu}+\mathbb{P}_{\mu}h_{\mu}.$ (2)
Let $\pi_{\mu}\in\mathbb{R}^{n}$ denote the stationary distribution associated
with the kernel $\mathbb{P}_{\mu}$. Let
$\mathbb{P}^{*}_{\mu}=\boldsymbol{1}\pi_{\mu}^{\top}\in\mathbb{R}^{n\times n}$
denote the matrix whose rows are $\pi_{\mu}^{\top}$. Using
$\mathbb{P}^{*}_{\mu}=\mathbb{P}_{\mu}^{*}\mathbb{P}_{\mu}$, we get the
following characterization of the average reward by multiplying both sides of
Equation 2 with $\mathbb{P}^{*}_{\mu}$:
$J_{\mu}\boldsymbol{1}=\mathbb{P}^{*}_{\mu}r_{\mu}$.
Let $J^{*}:=\max_{\mu\in\Pi}J_{\mu}$ be the optimal average reward. From
standard MDP theory, there exists $h^{*}\in\mathbb{R}^{n}$, for which the pair
($J^{*}$,$h^{*}$) satisfies the following Bellman optimality equation:
$J^{*}\boldsymbol{1}+h^{*}=\max_{\mu\in\Pi}r_{\mu}+\mathbb{P}_{\mu}h^{*}.$ (3)
Let $\mu^{*}$ be the optimizing policy in Equation 3. Then, similar reasoning
shows that $J^{*}\boldsymbol{1}=\mathbb{P}^{*}_{\mu^{*}}r_{\mu^{*}}.$
The goal of dynamic programming and reinforcement learning is to determine
this optimal average reward $J^{*}$ and its corresponding optimal policy
$\mu^{*}$. Policy iteration is one of the most widely used dynamic programming
algorithms for this purpose. It consists of two steps: (i) evaluation of the
value function associated with a policy, and (ii) determining a greedy policy
with respect to this evaluation. There are primarily two challenges that arise
when applying such an algorithm: (i) high memory and time complexity when the
state space is too large, and (ii) the unknown transition probability kernel
that governs the dynamics of the MDP.
When the state space is too large, it may be computationally infeasible to
perform policy iteration exactly. Existing literature suggests that
approximate policy iteration in the context of discounted reward MDPs yields
good performance. However, such an algorithm has not been studied in the
context of average reward MDPs. In fact, the results obtained for the
discounted reward MDPs yield vacuous performance bounds for average reward
MDPs given the standard relationship between discounted reward and average
reward MDPs. We explore this further in Section 2.2.
### 2.2 Relationship to the Discounted Reward MDP
To provide some background, we first discuss the approximate policy iteration
algorithm in the context of discounted reward MDPs. Let $\alpha\in[0,1)$
denote the discount factor. The discounted reward associated with a
deterministic policy $\mu\in\Pi$ starting from some state $s\in\mathcal{S}$ is
denoted $J_{\mu}^{\alpha}(s)$ and is defined as:
$J_{\mu}^{\alpha}(s)=\mathbb{E}_{\mu}\left[\sum_{i=0}^{\infty}\alpha^{i}r(s_{i},\mu(s_{i}))\Big{|}s_{0}=s\right].$
The value function $J_{\mu}^{\alpha}\in\mathbb{R}^{n}$ is the unique fixed
point of the Bellman operator
$\mathsf{T}^{\alpha}_{\mu}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ associated with
the policy $\mu$, which is defined as
$\mathsf{T}^{\alpha}_{\mu}J=r_{\mu}+\alpha\mathbb{P}_{\mu}J$. For each state
$s\in\mathcal{S}$, let $J_{*}^{\alpha}(s)$ denote the optimal discounted
reward, i.e., $J_{*}^{\alpha}(s)=\max_{\mu\in\Pi}J_{\mu}^{\alpha}(s).$
Similarly, $J_{*}^{\alpha}\in\mathbb{R}^{n}$ is the unique fixed point of the
optimality Bellman operator
$\mathsf{T}^{\alpha}:\mathbb{R}^{n}\to\mathbb{R}^{n}$, which is defined as
$\mathsf{T}^{\alpha}J=\max_{\mu\in\Pi}\left(r_{\mu}+\alpha\mathbb{P}_{\mu}J\right)$.
Algorithm 1 is Approximate Policy Iteration for the discounted reward MDP:
Algorithm 1 Approximate Policy Iteration: Discounted Reward
Require $J_{0}\in\mathbb{R}^{n}$
for $k=0,1,2,\ldots$ do
1\. Compute $\mu_{k+\\!1}\\!\\!\in\\!\Pi$ such that
$\|\mathsf{T}^{\alpha}J_{k}\\!-\\!\mathsf{T}^{\alpha}_{\mu_{k+1}}\\!J_{k}\|_{\infty}\\!\leq\\!\epsilon$
$\triangleright$ Approximate Policy Improvement
2\. Choose $J_{k+1}$ such that $\|J_{k+1}-J_{\mu_{k+1}}\|_{\infty}\leq\delta$
$\triangleright$ Approximate Policy Evaluation
where $J_{\mu_{k+1}}=\mathsf{T}^{\alpha}_{\mu_{k+1}}J_{\mu_{k+1}}$
end for
###### Theorem 2.1.
Let $\mu_{k}$ be the sequence of policies generated from the approximate
policy iteration algorithm (Algorithm 1). Then the performance error is
bounded as:
$\limsup_{k\to\infty}\|J_{\mu_{k}}-J_{*}^{\alpha}\|_{\infty}\leq\frac{\epsilon+2\alpha\delta}{(1-\alpha)^{2}}.$
(4)
###### Proof.
The proof of this theorem can be found in [Ber11] ∎
From literature [Ros14, Ber11], we know that the average reward $J_{\mu}$
associated with any policy $\mu$ is related to its discounted reward
$J^{\alpha}_{\mu}(s)$ counterpart as follows:
$J_{\mu}=\lim_{\alpha\to 1}(1-\alpha)J^{\alpha}_{\mu}(s).$
Note that the above relation is independent of the state $s\in\mathcal{S}$, as
the average reward does not depend on the initial state. Multiplying Equation
4 with $(1-\alpha)$, and letting $\alpha\to 1$, still yields an approximation
performance bound with $(1-\alpha)$ in the denominator. This term blows up to
infinity as $\alpha\to 1$. Note that this bound is known to be tight under
unichain Markov structure of the probability transition kernel [Ber11].
However, in practice it is observed that approximate policy iteration works
well in the average reward MDP case although these theoretical bounds are not
representative of this performance. To the best of our knowledge, we are
unaware of an average reward approximate policy iteration performance bound,
when the policies induce an irreducible Markov chain. We bridge this gap
between theory and practice by providing a theoretical analysis of approximate
policy iteration in the average reward MDP setting, with non trivial
performance bounds.
## 3 Approximate Policy Iteration for Average Reward
A crucial component of the proof of convergence of approximate policy
iteration in the context of discounted reward MDPs is the contraction due to
the discount factor $\alpha$ which is absent in the average reward setting
(since $\alpha=1$). To get some source of contraction, we have to make some
assumptions on the MDP.
###### Assumption 3.1.
We make the following assumptions:
1. (a)
Every deterministic policy $\mu\in\Pi$ induces an irreducible Markov Chain
$\mathbb{P}_{\mu}$.
2. (b)
For all policies, the diagonal elements of the probability transition matrix
are positive, i.e., the Markov chain stays in the same state with non-zero
probability.
Assumption (a) need not be satisfied by all MDPs. However, in order to satisfy
this assumption, we consider a modified MDP where at every time step with
probability $\varepsilon,$ an action is chosen from the set of all possible
actions with equal probability. Simultaneously, with probability
$1-\varepsilon$, we choose an action dictated by some policy. The problem then
is to choose this policy optimally. For most MDPs of interest, this small
modification will satisfy our assumption with a small $O(\varepsilon)$ loss in
performance. It is straightforward to show the $O(\varepsilon)$ loss, but we
include the proof in the appendix for completeness. Assumption (b) is without
loss of generality in the following sense: there exists a simple
transformation which essentially leaves the MDP unchanged but ensures that
this assumption is satisfied. This transformation known as the aperiodicity
transformation or Schweitzer transformation was introduced in [Sch71]. For
more details, the reader is referred to the appendix. In the remainder of this
paper, we assume that all MDPs are transformed accordingly to ensure the
assumptions are satisfied. One consequence of Assumption (b) is the following
lemma which will be useful to us later.
###### Lemma 3.2.
There exists a $\gamma>0$ such that, for any policy $\mu\in\Pi$,
$\min_{i\in\mathcal{S}}\pi_{\mu}(i)\geq\gamma,$
where $\pi_{\mu}$ is stationary distribution over states associated with the
policy $\mu$.
###### Proof.
This lemma has been proved in a more general sense in the context of
deterministic policies in [VdW80]. Since the aperiodicity transformation
ensures a non-zero probability of staying in the same state, the expected
return times are bounded and the lemma holds true for randomized policies as
well. ∎
Prior to presenting the algorithm, consider the following definitions. The
average-reward Bellman operator
$\mathsf{T}_{\mu}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ corresponding to a policy
$\mu$ is defined as $\mathsf{T}_{\mu}h={r}_{\mu}+{\mathbb{P}}_{\mu}h.$ The
average-reward optimal Bellman operator
$\mathsf{T}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is defined as
$\mathsf{T}h=\max_{\mu\in\Pi}{r}_{\mu}+{\mathbb{P}}_{\mu}h.$ The value
function $h_{\mu}$ associated with a policy $\mu$ satisfies the Bellman
Equation 2 with single step reward ${r}_{\mu}$ and the transition kernel
${\mathbb{P}}_{\mu}$. Note that $h_{\mu}$ is unique up to an additive
constant.
Solving for $\left({J}_{\mu},{h}_{\mu}\right)$ using Equation 2 involves
setting the value function for some fixed state $x^{*}$ as $0$ (since this
reduces the system from being underdetermined to one with a unique solution).
Further, the value function $h_{\mu}$ can be alternatively expressed as the
$\lim_{m\to\infty}\mathsf{T}^{m}_{\mu}h_{0}$ for any $h_{0}\in\mathbb{R}^{n}$
as is the case with discounted reward MDPs. However, unlike the discounted
reward Bellman Equation, there is no discount factor $\alpha<1$ preventing the
value function from exploding to infinity. Hence, we consider value function
computed using the relative Bellman operator $\widetilde{\mathsf{T}}_{\mu}$
defined as
$\widetilde{\mathsf{T}}_{\mu}h={r}_{\mu}+{\mathbb{P}}_{\mu}h-{r}_{\mu}(x^{*})\boldsymbol{1}-\left({\mathbb{P}}_{\mu}h\right)(x^{*})\boldsymbol{1}.$
We now state the algorithm for Approximate Policy Iteration for Average Reward
MDPs.
### 3.1 Approximate Policy Iteration Algorithm for Average Reward MDPs
Algorithm 2 Approximate Policy Iteration: Average Reward
1:Require $h_{0}\in\mathbb{R}^{n}$
2:for $k=0,1,2,\ldots$ do
3: 1\. Compute $\mu_{k+1}\\!\in\\!\Pi$ such that
$\|\mathsf{T}h_{k}\\!-\\!\mathsf{T}_{\mu_{k+1}}h_{k}\|_{\infty}\leq\epsilon$
$\triangleright$ Approx. Policy Improvement
4: 2\. Compute $h_{k+1}$ such that
$\|h_{k+1}-h_{\mu_{k+1}}\|_{\infty}\leq\delta$ $\triangleright$ Approx. Policy
Evaluation
5: where
$h_{\mu_{k+1}}=\lim_{m\to\infty}\widetilde{\mathsf{T}}^{m}_{\mu_{k+1}}h_{k}$
6:end for
### 3.2 Performance bounds for Average Reward Approximate Policy Iteration
We are now ready to present the main result of this work.
###### Theorem 3.3.
Let ${J}^{*}$ be the optimal average reward of an MDP that satisfies 3.1. The
sequence of policies $\mu_{k}$ generated by Algorithm 2 and their associated
average rewards ${J}_{\mu_{k}}$ satisfy the following bound:
$\displaystyle\Big{(}{J}^{*}-$
$\displaystyle{J}_{\mu_{k+1}}\Big{)}\leq\underbrace{\frac{\left(1-\left(1-\gamma\right)^{k}\right)}{\gamma}\left(\epsilon\left(\gamma+1\right)+2\delta\right)}_{\text{approximation
error}}+\underbrace{\left(1-\gamma\right)^{k}\left({J}^{*}-{\min_{i}\left(\mathsf{T}h_{0}-h_{0}\right)(i)+\epsilon}\right)}_{\text{initial
condition error}}.$
Interpretation of the Bound: Since $0<\gamma<1$, as $k\to\infty$, the error
due to initial condition $h_{0}$ drops to zero. The approximation error
consists of two components, $\epsilon$ and $\delta$. Here, $\epsilon$
represents the approximation error due to a suboptimal policy and $\delta$
represents the approximation error associated with evaluating the value
function corresponding to a policy. The limiting behavior of the average
reward associated with the policies obtained as a consequence of the algorithm
is captured in the following corollary.
###### Corollary 3.4.
The asymptotic performance of the policies obtained from Algorithm 2
satisfies:
$\limsup_{k\to\infty}{\left({J}^{*}-{J}_{\mu_{k}}\right)}\leq\frac{(1+\gamma)\epsilon+2\delta}{\gamma}.$
(5)
Note that the asymptotic bound in Equation 5 is in a very similar form as the
asymptotic performance bound for the discounted reward MDP in Equation 4 where
$\gamma$ plays the role of $(1-\alpha)^{2}$. However, when $\epsilon$ and
$\delta$ got to zero, this bound also goes to zero which is not the case if we
try to approximate average-reward problems by using the results for
discounted-reward problems in the limit where the horizon goes to infinity.
## 4 Application to Reinforcement Learning
Approximate policy iteration is closely related to RL. In general, approximate
policy improvement and approximate policy evaluation in Algorithm 2 are
executed through approximations to policy improvement (such as greedy update,
mirror descent, softmax) and TD learning for value function approximation.
First, we present a generic framework to analyze policy-based RL algorithms by
building upon the theory presented in the last section. For this purpose, we
define the state-action relative value function $Q$ (instead of the relative
state value function $h$ as before) to evaluate a policy. Let
$\mathsf{T}^{\mathsf{Q}}_{\mu}Q$ denote the Bellman operator with respect to
$Q$ and policy $\mu$ where
$(\mathsf{T}^{\mathsf{Q}}_{\mu}Q)(s,a)=r(s,a)+\left(\mathbb{Q}_{\mu}Q\right)(s,a),$
where
$\mathbb{Q}_{\mu}(s^{\prime},a^{\prime}|s,a)=\mu(a^{\prime}|s^{\prime})\mathbb{P}(s^{\prime}|s,a).$
The relative state action value function corresponding to policy $\mu$ is
represented by $Q_{\mu}$ and is the solution to the following Bellman
equation:
$J_{\mu}+Q_{\mu}(s,a)=r(s,a)+\sum_{s^{\prime}\in\mathcal{S},a^{\prime}\in\mathcal{A}}\mathbb{Q}_{\mu}(s^{\prime},a^{\prime}|s,a)Q_{\mu}(s^{\prime},a^{\prime})$,
for all state action pairs $(s,a)\in(\mathcal{S},\mathcal{A})$. We present a
generic policy-based algorithm for average-reward problem below.
Algorithm 3 Generic Policy Based Algorithm: $Q$-function Average Reward
Require $Q_{0}\in\mathbb{R}^{|\mathcal{S}||\mathcal{A}|}$
for $k=0,1,2,\ldots,T$ do
1\. Determine $\mu_{k+1}\\!\in\\!\Pi$ as a function of $Q_{k}$ using a
possibly random policy improvement step
2\. Compute $Q_{k+1}$ as an approximation to $Q_{\mu_{k+1}}$ using (state,
action, reward) samples from a trajectory generated by policy $\mu_{k+1}$
end for
Note that the error in Steps 1 (policy improvement) and 2 (policy evaluation)
of the above algorithm could be random. Thus, the analysis for Theorem 3.3 has
to be adapted to Algorithm 3. The resulting expected deviation from optimality
is characterized in the lemma below.
###### Lemma 4.1.
Let $\mu_{k}$ and $Q_{k}$ be the sequence of policies and relative state-
action value function iterates generated by Algorithm 3. For all $k\in
0,\ldots,T$, we have:
$\displaystyle\\!\\!\Big{(}{J}^{*}-\left(\min_{(s,a)}\left(\mathsf{T}^{\mathsf{Q}}Q_{k}-Q_{k}\right)(s,a)\right)\\!\Big{)}$
$\displaystyle\leq\left(1-\gamma\right)\left({J}^{*}-\left(\min_{(s,a)}\left(\mathsf{T}^{\mathsf{Q}}Q_{k-1}-Q_{k-1}\right)(s,a)\right)\right)$
$\displaystyle+\left\|\mathsf{T}^{\mathsf{Q}}Q_{k-\\!1}\\!-\\!\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}Q_{k-\\!1}\\!\right\|_{\infty}\\!\\!\\!+\\!2\\!\left\|Q_{k}\\!-\\!Q_{\mu_{k}}\\!\right\|_{\infty}\\!,$
where
$\gamma=\min_{\begin{subarray}{c}\mu\in{\mu_{1},\mu_{2},\ldots,\mu_{T}}\\\
(s,a)\in\mathcal{S}\times\mathcal{A}\end{subarray}}\mathbb{Q}_{\mu}^{*}(s,a)>0,$
and $\mathbb{Q}_{\mu}^{*}$ is a matrix with identical rows corresponding to
the invariant distribution of $\mathbb{Q}_{\mu}$.
Iteratively applying Lemma 4.1 yields the following proposition for a generic
policy-based algorithm.
###### Proposition 4.2.
For any $T>0$, the iterates of Algorithm 3 satisfy
$\displaystyle\mathbb{E}\big{[}{J}^{*}\\!-\\!{J}_{\mu_{T+1}}\big{]}\\!$
$\displaystyle\leq\\!\underbrace{\left(1\\!-\\!\gamma\right)^{T}\mathbb{E}\\!\left[{J}^{*}\\!-\\!\min_{i}\left(\mathsf{T}^{\mathsf{Q}}Q_{0}\\!-\\!Q_{0}\right)(i)\right]}_{\text{Initial
condition
error}}\\!+\\!\underbrace{2\sum_{\ell=0}^{T-1}\left(1\\!-\\!\gamma\right)^{\ell}\mathbb{E}\\!\left[\left\|Q_{T-\ell}\\!-\\!Q_{\mu_{T-\ell}}\right\|_{\infty}\right]}_{\text{Policy
evaluation error}}$
$\displaystyle\\!+\\!\underbrace{\\!\sum_{\ell=1}^{T-1}\\!\left(1-\gamma\right)^{\ell-1}\mathbb{E}\\!\left[\left\|\mathsf{T}^{\mathsf{Q}}_{\mu_{T+1-\ell}}\\!Q_{T-\ell}\\!-\\!\mathsf{T}^{\mathsf{Q}}Q_{T-\ell}\right\|_{\infty}\right]+\mathbb{E}\left[{\left\|\mathsf{T}^{\mathsf{Q}}_{\mu_{T+1}}Q_{T}-\mathsf{T}^{\mathsf{Q}}Q_{T}\right\|_{\infty}}\right]}_{\text{Policy
improvement error}}.$
We now illustrate the applicability of the above result to specific RL
algorithms. More specifically, we characterize finite time performance bounds
when state-action relative value functions are learnt through TD-Learning
[ZZM21], while several different update rules are employed for policy
improvement.
### 4.1 Performance bounds for RL algorithms
For every iteration of Algorithm 3, it is necessary to observe trajectories of
length $\tau$ in order to learn the state action value function associated
with any policy. Using linear function approximations for value functions, the
error in policy evaluation at iteration $k$ in Algorithm 3 can be expressed in
the following form:
$\|Q_{k}-Q_{\mu_{k}}\|_{\infty}\leq\|\Phi(\theta_{k}-\theta^{*}_{\mu_{k}})\|_{\infty}+\|Q_{\mu_{k}}-\Phi\theta^{*}_{\mu_{k}}\|_{\infty},$
(6)
where $\Phi\in\mathbb{R}^{|\mathcal{S}||\mathcal{A}|\times d}$ corresponds to
the feature vector matrix and $\theta\in\mathbb{R}^{d}$ is the parameter
vector. Further, $\theta_{k}$ is the parameter vector obtained as a result of
the TD learning algorithm. The projection of $Q_{\mu_{k}}$ onto the span of
$\Phi$ is given by $\Phi\theta^{*}_{\mu_{k}}$. The last term in Equation 6
corresponds to the function approximation error which depends on the span of
the feature matrix $\Phi$. We denote this error term by $\delta_{0,k}.$ The
error due to TD learning (first term in Equation 6) is denoted by
$\delta_{\text{TD},k}$. Assuming
$\max_{(s,a)\in\mathcal{S}\times\mathcal{A}}\|\phi(s,a)\|_{2}\leq 1$, and
$\|\theta_{\mu_{k}}^{*}\|_{2}$ is bounded uniformly in $k$, [ZZM21] shows that
there exists $C>0$ such that the length of the sample trajectory $\tau$ and
$\delta_{\text{TD},k}$ are related as follows:
$\mathbb{E}\left[\delta_{\text{TD},k}\right]=\frac{C}{\sqrt{\tau}}.$ (7)
Let
$c_{0}=\left(J^{*}-\min_{i\in\mathcal{S}}\left(\mathsf{T}^{\mathsf{Q}}Q_{0}-Q_{0}\right)(i)\right)$
be the error due to initial condition and
$\overline{\delta}_{0}=\max_{t}\delta_{0,t}$ be the maximum function
approximation error across all iterations. Then we obtain the following
performance bounds when TD learning is used for policy evaluation.
###### Corollary 4.3.
(Greedy policy update) Let
$\tilde{a}(s)=\operatorname*{argmax}_{a\in\mathcal{A}}Q_{k}(s,a^{\prime}).$
Let $\beta>0$ be such that the sequence of policies in Algorithm 3
corresponding to all state action pairs $(s,a)$ are obtained using the
following algorithm, which we call the greedy policy update.
$\displaystyle\mu_{k+1}(a|s)=\begin{cases}\frac{1}{\beta|\mathcal{A}|},&\text{if
}a\neq\tilde{a}(s)\\\ \frac{1}{\beta|\mathcal{A}|}+1-\frac{1}{\beta},&\text{if
}a=\tilde{a}(s),\end{cases}$ (8)
Let $\eta=\max_{\begin{subarray}{c}s\in\mathcal{S},a\in\mathcal{A}\\\ k\in
1\ldots T\end{subarray}}|Q_{k}(s,a)|.$ Using TD Learning with linear value
function approximation for policy evaluation, we get the following finite time
performance bound:
$\displaystyle\mathbb{E}\left[J^{*}-J_{\mu_{T+1}}\right]$
$\displaystyle\leq(1-\gamma)^{T}c_{0}+\left(\frac{1+\gamma}{\gamma}\right)\frac{2\eta}{\beta}+\frac{2}{\gamma}\left(\overline{\delta}_{0}+\frac{C}{\sqrt{\tau}}\right).$
(9)
###### Corollary 4.4.
(Softmax policy update) Let $\beta>0$ be such that the sequences of policies
in Algorithm 3 corresponding to all state action pairs $(s,a)$ are obtained
using the following algorithm, which we call the softmax policy update.
$\mu_{k+1}(a|s)=\frac{\exp\left(\beta
Q_{k}(s,a)\right)}{\sum_{a^{\prime}\in\mathcal{A}}\exp\left(\beta
Q_{k}(s,a^{\prime})\right)}.$ (10)
Using TD Learning with linear value function approximation for policy
evaluation, we get the following finite time performance bound:
$\displaystyle\mathbb{E}\left[J^{*}-J_{\mu_{T+1}}\right]$
$\displaystyle\leq(1-\gamma)^{T}c_{0}+\left(\frac{1+\gamma}{\gamma}\right)\frac{\log\left(|\mathcal{A}|\right)}{\beta}+\frac{2}{\gamma}\left(\overline{\delta}_{0}+\frac{C}{\sqrt{\tau}}\right).$
(11)
###### Corollary 4.5.
(Mirror descent update) Let $\beta>0$ be such that the sequences of policies
in Algorithm 3 corresponding to all state action pairs $(s,a)$ are obtained
using the following algorithm, which we call the mirror descent policy update.
$\mu_{k+1}(a|s)\propto\mu_{k}(a|s)\exp\\{\beta Q_{k}(s,a)\\}.$ (12)
Let $\omega=\min_{\begin{subarray}{c}k\in 1\ldots T\\\
s\in\mathcal{S}\end{subarray}}\mu_{k}(a^{*}(s)|s)$ where $a^{*}(s)$ is the
optimal action at state $s.$ Using TD Learning with linear value function
approximation for policy evaluation, we get the following finite time
performance bound:
$\displaystyle\mathbb{E}\left[J^{*}-J_{\mu_{T+1}}\right]$
$\displaystyle\leq(1-\gamma)^{T}c_{0}+\left(\frac{1+\gamma}{\gamma\beta}\right)\log\left(\frac{1}{\omega}\right)+\frac{2}{\gamma}\left(\overline{\delta}_{0}+\frac{C}{\sqrt{\tau}}\right).$
(13)
###### Remark 4.6.
The following remarks are in order:
* •
In each of the above performance bounds, there is an inevitable constant error
$(\overline{\delta}_{0})$ due to function approximation. If the value function
lies in the class of functions chosen, then this error will be zero.
* •
The error due to policy update (characterized by $\beta$) can be made
arbitrarily small by choosing $\beta$ large. However, choosing a large $\beta$
may increase $C$ due to the fact that $C$ is a function of the mixing time of
each policy, which is affected by the choice of $\beta.$ However, $C$ cannot
be arbitrarily large due to Assumption (a).
* •
The error due to the initial condition decays exponentially fast but the rate
of decay can be slow because $\gamma$ can be very small. In this regard, it is
interesting to compare our work with the results in [AYBB+19, LYAYS21]. We do
so in the next subsection.
### 4.2 Comparison with [AYBB+19, LYAYS21]
Our model for approximate policy iteration is intended to be a general
framework to study policy-based RL algorithms. A specific class of such
algorithms has been studied in [AYBB+19, LYAYS21], where regret bounds are
presented for a mirror descent like algorithm referred to as Politex. Although
our model is intended for offline algorithms, it is possible to adapt the
techniques in [AYBB+19, LYAYS21] to obtain regret bounds for our model as
well.
Let $r(s_{t},a_{t})$ be the reward obtained at time $t.$ Let $K$ be the total
number of time units for which Algorithm 3 is executed. Hence $K=T\tau$, where
$\tau$ is the length of the sample trajectory for every iteration of policy
evaluation. The regret, defined as
$R(K)=\sum_{t=1}^{K}\left(J^{*}-r(s_{t},a_{t})\right)$ can be further
decomposed as:
$R(K)=\sum_{t=1}^{K}\left(J^{*}-J_{\mu_{t}}\right)+\left(J_{\mu_{t}}-r(s_{t},a_{t})\right).$
(14)
The first term i.e., $\sum_{t=1}^{K}\left(J^{*}-J_{\mu_{t}}\right)$ is
referred to as the pseudo regret $R_{\text{PS}}(K)$. The second term is
essentially bounding the average value from its estimate and can be bounded
using a martingale analysis identical to [AYBB+19, LYAYS21]. Here, we focus on
obtaining an upper bound on the pseudo regret with mirror descent update and
TD learning. The pseudo regret can be further decomposed as:
$R_{\text{PS}}(K)=\tau\sum_{t=1}^{T}\left(J^{*}-J_{\mu_{t}}\right).$
For the mirror descent policy update in Corollary 4.5, we get the following
expected regret bound
$R_{\text{PS}}(K)\leq\tau\sum_{t=1}^{T}(1-\gamma)^{T}c_{0}+\frac{1+\gamma}{\gamma\beta}\log\left(\frac{1}{\omega}\right)+\frac{2}{\gamma}\left(\overline{\delta}_{0}+\frac{c_{3}}{\sqrt{\tau}}\right).$
(15)
Let $\beta=\sqrt{\tau}$, then the above regret can be expressed as:
$R_{\text{PS}}(K)\leq\frac{1}{\gamma}\left({\tau
c_{0}}+{2K\overline{\delta}_{0}}+\frac{Kc_{5}}{\sqrt{\tau}}\right).$
Optimizing for the regret yields $\tau=O(K^{\frac{2}{3}})$ and the following
pseudo regret upper bound,
$R_{\text{PS}}(K)\leq\frac{1}{\gamma}\left(2K\overline{\delta}_{0}+c_{6}K^{\frac{2}{3}}\right).$
(16)
We now compare our regret bounds to the ones in [AYBB+19, LYAYS21]. Since all
of the bounds will have an inevitable error $(\overline{\delta}_{0})$ due to
function approximation, the comparison is with respect to the other terms in
the regret bound.
* •
We get a better order-wise bound on the regret compare to [AYBB+19] which
obtains a bound of $O(K^{3/4}).$ However, they have better constants which are
in terms of mixing-time coefficients rather than in terms of $\gamma.$
* •
[LYAYS21] gets a better regret bound $(O(\sqrt{K}))$ than us.
The results in [AYBB+19, LYAYS21] use Monte Carlo policy evaluation, which
allows for a closed-form expression for the value function estimate. The
structure of the closed-form expressions allows them to exploit the fact that
the policy varies slowly from one iteration to the next (which is specific to
the mirror descent policy update) and hence $\gamma$ does not appear in their
analysis. However, the Monte Carlo policy evaluation algorithms use matrix
inversion which is often too expensive to implement in practice unlike the TD
learning algorithm in our analysis which is widely used. Nevertheless, the
ideas in [AYBB+19, LYAYS21] provide a motivation to investigate whether one
can extend their analysis to more general policy evaluation and policy
improvement algorithms. This is an interesting topic for future research.
## 5 Conclusions
We present a general framework for analyzing policy-based reinforcement
learning algorithms for average-reward MDPs. Our main contribution is to
obtain performance bounds without using the natural source of contraction
available in discounted reward formulations. We apply our general framework to
several well-known RL algorithms to obtain finite-time error bounds on the
average reward. We conclude with a comparative regret analysis and an outline
of possible future work.
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## 6 Appendix
### 6.1 Discussion on Assumption 3.1
#### 6.1.1 Assumption 3.1(a)
In order to ensure every policy induces an irreducible Markov chain, we modify
the MDP where at every time step with probability $\epsilon$, an action is
chosen from the set of all possible actions with equal probability.
Simultaneously, with probability $1-\epsilon$, we choose an action dictated by
some policy. Let the transition kernel of the MDP associated with policy $\mu$
be denoted as $\mathbb{P}_{\mu}$. We denote the modified kernel by
$\widehat{\mathbb{P}}_{\mu}$, where the transition probability from state $i$
to $j$ is given by:
$\widehat{\mathbb{P}}_{\mu}(j|i)=(1-\epsilon)\mathbb{P}_{\mu}(j|i)+\epsilon\left(\frac{1}{|\mathcal{A}|}\sum_{a\in\mathcal{A}}\mathbb{P}(j|i,a)\right)$
Let
$\mathbb{P}_{\rho}(j|i)=\frac{1}{|\mathcal{A}|}\sum_{a\in\mathcal{A}}\mathbb{P}(j|i,a),$
where $\rho$ represents randomized policy, that is
$\rho(a|i)=\frac{1}{|\mathcal{A}|.}$ Let $(J_{\mu},V_{\mu})$ be the average
reward and state value function vector associated with the policy $\mu.$ Then
they the satisfy the following Bellman Equation:
$J_{\mu}+V_{\mu}=r_{\mu}+\mathbb{P}_{\mu}V_{\mu},$
where $r_{\mu}(i)=\sum_{a\in\mathcal{A}}\mu(a|i)r(i,a).$ Similarly
$(\widehat{J}_{\mu},\widehat{V}_{\mu})$ be the average reward and state value
function vector that satisfy the following Bellman Equation,
$\widehat{J}_{\mu}+\widehat{V}_{\mu}=\widehat{r}_{\mu}+\widehat{\mathbb{P}}_{\mu}\widehat{V}_{\mu}$
where $\widehat{r}_{\mu}=(1-\epsilon)r_{\mu}+\epsilon r_{\rho}$. Since
$\widehat{\mathbb{P}}_{\mu}=(1-\epsilon)\mathbb{P}_{\mu}+\epsilon\mathbb{P}_{\rho},$
the above equation can be rewritten as:
$\widehat{J}_{\mu}=\left(1-\epsilon\right)\left(r_{\mu}+\mathbb{P}_{\mu}\widehat{V}_{\mu}-\widehat{V}_{\mu}\right)+\epsilon\left(r_{\rho}+\mathbb{P}_{\rho}\widehat{V}_{\mu}-\widehat{V}_{\mu}\right)$
Multiplying the above equation by the vector $\mathbb{P}^{*}_{\mu}$, which is
the invariant distribution over the state space due to policy $\mu$, we
obtain,
$\widehat{J}_{\mu}=\left(1-\epsilon\right)\left(\mathbb{P}^{*}_{\mu}r_{\mu}\right)+\epsilon\mathbb{P}^{*}_{\mu}\left(r_{\rho}+\mathbb{P}_{\rho}\widehat{V}_{\mu}-\widehat{V}_{\mu}\right)$
since $\mathbb{P}^{*}_{\mu}\mathbb{P}_{\mu}=\mathbb{P}^{*}_{\mu}$. We also
know that $J_{\mu}=\mathbb{P}^{*}_{\mu}r_{\mu}.$. Hence we obtain,
$\widehat{J}_{\mu}=\left(1-\epsilon\right)J_{\mu}+\epsilon\mathbb{P}^{*}_{\mu}\left(r_{\rho}+\mathbb{P}_{\rho}\widehat{V}_{\mu}-\widehat{V}_{\mu}\right)$
Therefor we obtain the following result,
$J_{\mu}-\widehat{J}_{\mu}=\epsilon\left(J_{\mu}-\mathbb{P}^{*}_{\mu}\left(r_{\rho}+\mathbb{P}_{\rho}\widehat{V}_{\mu}-\widehat{V}_{\mu}\right)\right)$
Since the rewards are bounded, so are $J_{\mu},r_{\rho}$ and
$\mathbb{P}_{\rho}\widehat{V}_{\mu}-\widehat{V}_{\mu}$ for all $\mu$. Hence we
obtain,
$J_{\mu}-\widehat{J}_{\mu}=O(\epsilon)$
The difference in the average reward associated with the original MDP and the
modified MDP vary by $O(\epsilon).$ Let the optimal policy corresponding to
original MDP be $\mu^{*}$ and the optimal policy corresponding to modified MDP
be $\widehat{\mu}^{*}$. Then,
$J_{\mu^{*}}-\widehat{J}_{\widehat{\mu}^{*}}=J_{\mu^{*}}-\widehat{J}_{\mu^{*}}+\widehat{J}_{\mu^{*}}-\widehat{J}_{\widehat{\mu}^{*}}$
Since $\widehat{\mu}^{*}$ is the optimizing policy for the modified MDP, we
have $\widehat{J}_{\mu^{*}}-\widehat{J}_{\widehat{\mu}^{*}}\leq 0$. Hence, we
obtain,
$J_{\mu^{*}}-\widehat{J}_{\widehat{\mu}^{*}}\leq
J_{\mu^{*}}-\widehat{J}_{\mu^{*}}=O(\epsilon).$
Similarly,
$J_{\mu^{*}}-\widehat{J}_{\widehat{\mu}^{*}}=J_{\mu^{*}}-J_{\widehat{\mu}^{*}}+J_{\widehat{\mu}^{*}}-\widehat{J}_{\widehat{\mu}^{*}}$
Since $\mu^{*}$ is the optimizing policy for the original MDP, we have
$J_{\mu^{*}}-J_{\widehat{\mu}^{*}}\geq 0$. Hence we obtain,
$J_{\mu^{*}}-\widehat{J}_{\widehat{\mu}^{*}}\geq
J_{\widehat{\mu}^{*}}-\widehat{J}_{\widehat{\mu}^{*}}=O(\epsilon).$
Thus the optimal average reward of the original MDP and modified MDP differ by
$O(\epsilon).$ That is,
$|J_{\mu^{*}}-\widehat{J}_{\widehat{\mu}^{*}}|\leq O(\epsilon)$
#### 6.1.2 Assumption 3.1 (b)
To ensure Assumption 3.1 (b) is satisfied, an aperiodicity transformation can
be implemented. Under this transformation, the transition probabilities of the
original MDP are modified such that the probability of staying in any state
under any policy is non-zero. In order to compensate for the change in
transition kernel, the single step rewards are analogously modified such that
the Bellman equation corresponding to the original and transformed dynamics
are scaled versions of one another. This thus ensures that the optimality of a
policy remains the same irrespective of this transformation, along with
yielding quantifiable convergence properties. Mathematically, this
transformation is described below.
###### Definition 6.1 (Aperiodicity Transformation).
Let $\kappa\in\left(0,1\right)$. For every policy $\mu\in\Pi$ and for all
states $i,j\in\mathcal{S}$, consider the following transformation:
$\displaystyle\widehat{\mathbb{P}}_{\mu}\left(i|i\right)$
$\displaystyle=\kappa+(1-\kappa)\mathbb{P}_{\mu}\left(i|i\right)$ (17)
$\displaystyle\widehat{\mathbb{P}}_{\mu}\left(j|i\right)$
$\displaystyle=(1-\kappa)\mathbb{P}_{\mu}\left(j|i\right),\qquad j\neq i$ (18)
$\displaystyle\widehat{r}(i,\mu(i))$ $\displaystyle=(1-\kappa)r(i,\mu(i))$
(19)
###### Theorem 6.2.
Given the transformation in Equation 17-Equation 19, for every $\mu\in\Pi$,
let $(J_{\mu},h_{\mu})$ and $(\widehat{J}_{\mu},\widehat{h}_{\mu})$ be the
solution to the Bellman Equation (Equation 2) corresponding to the original
MDP $(\mathbb{P}_{\mu},r_{\mu})$ and the transformed MDP
$(\widehat{\mathbb{P}}_{\mu},\widehat{r}_{\mu})$. Then,
$\left(\widehat{J}_{\mu},\widehat{h}_{\mu}\right)=\left((1-\kappa)J_{\mu},h_{\mu}\right)$
###### Proof.
The proof of this theorem can be found in [Sch71]. ∎
###### Remark 6.3.
Due to this bijective relationship between the original and the transformed
problem, it suffices to solve the transformed problem. Given that the
trajectory used for algorithms such as TD Learning corresponds to the original
system, such a transformation necessitates a mild change in how samples from a
given policy are utilized in TD learning. More specifically, for each (state,
action) sample, with probability $\kappa,$ we have to add the same sample to
the data set for the next time instant. With probability $1-\kappa$ choose the
next state in the trajectory. Hence, this transformation is easy to
incorporate in RL algorithms.
### 6.2 Proof of Theorem 3.3
Prior to presenting the proof, we define
$\displaystyle u_{k}=\max_{i}(\mathsf{T}h_{k}-h_{k})(i),$ (20) $\displaystyle
l_{k}=\min_{i}(\mathsf{T}h_{k}-h_{k})(i).$
A key lemma in the proof of convergence of approximate policy iteration in the
context of average reward is:
###### Lemma 6.4.
Let $J^{*}$ be the optimal average reward associated with the transformed MDP.
For all $k\in\mathbb{N}$:
$l_{k}-\epsilon\leq J_{\mu_{k+1}}\leq{J}^{*}\leq u_{k}$
###### Proof.
From definition,
$\displaystyle l_{k}\boldsymbol{1}$ $\displaystyle\leq\mathsf{T}h_{k}-h_{k}$
Since $\|\mathsf{T}h_{k}-\mathsf{T}_{\mu_{k+1}}h_{k}\|_{\infty}\leq\epsilon$,
$\displaystyle l_{k}\boldsymbol{1}$
$\displaystyle\leq\mathsf{T}_{\mu_{k+1}}h_{k}-h_{k}+\epsilon\boldsymbol{1}$
$\displaystyle=r_{\mu_{k+1}}+\mathbb{P}_{\mu_{k+1}}h_{k}-h_{k}+\epsilon\boldsymbol{1}$
$\displaystyle\mathbb{P}_{\mu_{k+1}}^{*}l_{k}\boldsymbol{1}$
$\displaystyle\leq\mathbb{P}_{\mu_{k+1}}^{*}r_{\mu_{k+1}}+\mathbb{P}_{\mu_{k+1}}^{*}\mathbb{P}_{\mu_{k+1}}h_{k}-\mathbb{P}_{\mu_{k+1}}^{*}h_{k}+\mathbb{P}_{\mu_{k+1}}^{*}\epsilon\boldsymbol{1}$
where $\mathbb{P}_{\mu_{k+1}}^{*}$ is a matrix whose rows are identical and
are the invariant distribution associated with the probability transition
kernel $\mathbb{P}_{\mu_{k+1}}$. Since
$\mathbb{P}_{\mu_{k+1}}^{*}\mathbb{P}_{\mu_{k+1}}=\mathbb{P}_{\mu_{k+1}}^{*}$,
$\displaystyle l_{k}\boldsymbol{1}$
$\displaystyle\leq\mathbb{P}^{*}_{\mu_{k+1}}r_{\mu_{k+1}}+\epsilon\boldsymbol{1}$
$\displaystyle l_{k}\boldsymbol{1}-\epsilon\boldsymbol{1}$
$\displaystyle\leq{J}_{\mu_{k+1}}\boldsymbol{1}$ $\displaystyle
l_{k}-\epsilon$ $\displaystyle\leq{J}_{\mu_{k+1}}$
From definition,
$\displaystyle{J}^{*}=\max_{\mu\in\Pi}{J}_{\mu}\geq{J}_{\mu_{k+1}}$
Let $\mu^{*}$ be the policy corresponding to the optimal average reward
${J}^{*}$ and value function $h^{*}$, ie.,
$\displaystyle{J}^{*}+h^{*}$
$\displaystyle=\max_{\mu}r_{\mu}+\mathbb{P}_{\mu}h^{*}$
$\displaystyle=r_{\mu^{*}}+\mathbb{P}_{\mu^{*}}h^{*}$
Then,
$\displaystyle u_{k}\boldsymbol{1}$ $\displaystyle\geq\mathsf{T}h_{k}-h_{k}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\geq}}\mathsf{T}_{\mu^{*}}h_{k}-h_{k}$
$\displaystyle=r_{\mu^{*}}+\mathbb{P}_{\mu^{*}}h_{k}-h_{k}$
$\displaystyle\mathbb{P}_{\mu^{*}}^{*}u_{k}\boldsymbol{1}$
$\displaystyle\geq\mathbb{P}_{\mu^{*}}^{*}r_{\mu^{*}}+\mathbb{P}_{\mu^{*}}^{*}\mathbb{P}_{\mu^{*}}h_{k}-\mathbb{P}_{\mu^{*}}^{*}h_{k}$
$\displaystyle u_{k}\boldsymbol{1}$
$\displaystyle\geq\mathbb{P}^{*}_{\mu^{*}}r_{\mu^{*}}$ $\displaystyle u_{k}$
$\displaystyle\geq{J}^{*}$
where $(a)$ is due to the fact that $\mathsf{T}$ is the maximising Bellman
operator.
Hence $\forall k\in\mathbb{N}$,
$\displaystyle l_{k}-\epsilon\leq{J}_{\mu_{k+1}}\leq{J}^{*}\leq u_{k}$
∎
Without any approximation, the above lemma indicates that there exists a state
whose value function changes by an amount less than the average reward
associated with the optimizing policy at that iteration, and also that there
exists a state whose value function increases by an amount larger than the
optimal average reward.
The proof proceeds by showing a geometric convergence rate for $l_{k}$. More
precisely, the following lemma is proved.
###### Lemma 6.5.
For all $k\in\mathbb{N}$, it is true that:
$\left({J}^{*}-l_{k}\right)\leq\left(1-\gamma\right)\left({J}^{*}-l_{k-1}\right)+\epsilon+2\delta$
###### Proof.
Define:
$g_{k}(i)=\left(\mathsf{T}h_{k}-h_{k}\right)(i)$ $\displaystyle g_{k}(i)$
$\displaystyle\geq\left(\mathsf{T}_{\mu_{k}}h_{k}-h_{k}\right)(i)$
Since $\mathsf{T}$ is the maximizing Bellman operator.
We know that:
1. 1.
$\left\|h_{k}-h_{\mu_{k}}\right\|_{\infty}\leq\delta$
2. 2.
For all constant $p\in\mathbb{R}$:
$\displaystyle\mathsf{T}_{\mu_{k}}\left(h_{k}+p\boldsymbol{1}\right)$
$\displaystyle=r_{\mu_{k}}+\mathbb{P}_{\mu_{k}}\left(h_{k}+p\boldsymbol{1}\right)$
$\displaystyle=r_{\mu_{k}}+\mathbb{P}_{\mu_{k}}h_{k}+p\boldsymbol{1}$
$\displaystyle=\mathsf{T}_{\mu_{k}}h_{k}+p\boldsymbol{1}$
We thus obtain,
$\displaystyle g_{k}(i)$
$\displaystyle\geq\mathsf{T}_{\mu_{k}}(h_{\mu_{k}}-\delta)-(h_{\mu_{k}}+\delta)$
(21)
$\displaystyle\geq\left(\mathsf{T}_{\mu_{k}}h_{\mu_{k}}-h_{\mu_{k}}\right)(i)-2\delta$
(22)
Recall that $h_{\mu_{k}}=\lim_{m\to\infty}\widetilde{T}^{m}_{\mu_{k+1}}h_{k}$
$\displaystyle\widetilde{\mathsf{T}}_{\mu_{k}}^{m}h_{k-1}$
$\displaystyle=\widetilde{\mathsf{T}}_{\mu_{k}}^{m-1}\widetilde{\mathsf{T}}_{\mu_{k}}h_{k-1}$
$\displaystyle=\widetilde{\mathsf{T}}_{\mu_{k}}^{m-1}\left(r_{\mu_{k}}+\mathbb{P}_{\mu_{k}}h_{k-1}-r_{\mu_{k}}(x^{*})\boldsymbol{1}-\left(\mathbb{P}_{\mu_{k}}h_{k-1}\right)(x^{*})\boldsymbol{1}\right)$
$\displaystyle=\widetilde{\mathsf{T}}_{\mu_{k}}^{m-1}\left(\mathsf{T}_{\mu_{k}}h_{k-1}-\left(\mathsf{T}_{\mu_{k}}h_{k-1}\right)(x^{*})\boldsymbol{1}\right)$
$\displaystyle=\widetilde{\mathsf{T}}_{\mu_{k}}^{m-2}\widetilde{\mathsf{T}}_{\mu_{k}}\left(\mathsf{T}_{\mu_{k}}h_{k-1}-\left(\mathsf{T}_{\mu_{k}}h_{k-1}\right)(x^{*})\boldsymbol{1}\right)$
$\displaystyle=\widetilde{\mathsf{T}}_{\mu_{k}}^{m-2}\left(r_{\mu_{k}}+\mathbb{P}_{\mu_{k}}\left(\mathsf{T}_{\mu_{k}}h_{k-1}-\mathsf{T}_{\mu_{k}}h_{k-1}(x^{*})\right)-r_{\mu_{k}}(x^{*})\boldsymbol{1}-\left(\mathbb{P}_{\mu_{k}}\left(\mathsf{T}_{\mu_{k}}h_{k-1}-\mathsf{T}_{\mu_{k}}h_{k-1}(x^{*})\right)\right)(x^{*})\boldsymbol{1}\right)$
$\displaystyle=\widetilde{\mathsf{T}}_{\mu_{k}}^{m-2}\left(r_{\mu_{k}}+\mathbb{P}_{\mu_{k}}\left(\mathsf{T}_{\mu_{k}}h_{k-1}\right)-\mathsf{T}_{\mu_{k}}h_{k-1}(x^{*})-r_{\mu_{k}}(x^{*})\boldsymbol{1}-\mathbb{P}_{\mu_{k}}\left(\mathsf{T}_{\mu_{k}}h_{k-1}\right)(x^{*})\boldsymbol{1}+\mathsf{T}_{\mu_{k}}h_{k-1}(x^{*})\right)$
$\displaystyle=\widetilde{\mathsf{T}}_{\mu_{k}}^{m-2}\left(\mathsf{T}^{2}_{\mu_{k}}h_{k-1}-\mathsf{T}^{2}_{\mu_{k}}h_{k-1}(x^{*})\boldsymbol{1}\right)$
Iterating, we thus obtain,
$h_{\mu_{k}}=\lim_{m\to\infty}\mathsf{T}^{m}_{\mu_{k}}h_{k-1}-\mathsf{T}^{m}_{\mu_{k}}h_{k-1}(x^{*})\boldsymbol{1}$
Substituting back in Equation 21,
$\displaystyle g_{k}(i)$
$\displaystyle\geq\left(\mathsf{T}_{\mu_{k}}\left(\lim_{m\to\infty}\mathsf{T}^{m}_{\mu_{k}}h_{k-1}-\mathsf{T}^{m}_{\mu_{k}}h_{k-1}(x^{*})\boldsymbol{1}\right)-\lim_{m\to\infty}\mathsf{T}^{m}_{\mu_{k}}h_{k-1}-\mathsf{T}^{m}_{\mu_{k}}h_{k-1}(x^{*})\boldsymbol{1}\right)(i)-2\delta$
$\displaystyle=\left(\mathsf{T}_{\mu_{k}}\left(\lim_{m\to\infty}\mathsf{T}^{m}_{\mu_{k}}h_{k-1}\right)-\lim_{m\to\infty}\mathsf{T}^{m}_{\mu_{k}}h_{k-1}(x^{*})\boldsymbol{1}-\lim_{m\to\infty}\mathsf{T}^{m}_{\mu_{k}}h_{k-1}-\lim_{m\to\infty}\mathsf{T}^{m}_{\mu_{k}}h_{k-1}(x^{*})\boldsymbol{1}\right)(i)-2\delta$
$\displaystyle=\left(\mathsf{T}_{\mu_{k}}\left(\lim_{m\to\infty}\mathsf{T}^{m}_{\mu_{k}}h_{k-1}\right)-\lim_{m\to\infty}\mathsf{T}^{m}_{\mu_{k}}h_{k-1}\right)(i)-2\delta$
Since $\mathsf{T}_{\mu_{k}}$ is a continuous operator, it is possible to take
the limit outside. We thus obtain,
$\displaystyle g_{k}(i)$
$\displaystyle\geq\lim_{m\to\infty}\left(\mathsf{T}^{m+1}_{\mu_{k}}h_{k-1}-\mathsf{T}^{m}_{\mu_{k}}h_{k-1}\right)(i)-2\delta$
$\displaystyle=\lim_{m\to\infty}\left(\mathsf{T}^{m}_{\mu_{k}}\mathsf{T}_{\mu_{k}}h_{k-1}-\mathsf{T}^{m}_{\mu_{k}}h_{k-1}\right)(i)-2\delta$
$\displaystyle=\lim_{m\to\infty}\left(\mathbb{P}^{m}_{\mu_{k}}\left(\mathsf{T}_{\mu_{k}}h_{k-1}-h_{k-1}\right)\right)(i)-2\delta$
$\displaystyle=\mathbb{P}^{*}_{\mu_{k}}\left(\mathsf{T}_{\mu_{k}}h_{k-1}-h_{k-1}\right)(i)-2\delta$
Since $\left\|T_{\mu_{k}}h_{k-1}-Th_{k-1}\right\|_{\infty}\leq\epsilon$,
$\displaystyle g_{k}(i)$
$\displaystyle\geq\mathbb{P}^{*}_{\mu_{k}}\left(\mathsf{T}h_{k-1}-h_{k-1}\right)(i)-2\delta-\epsilon$
$\displaystyle=\left(\mathbb{P}^{*}_{\mu_{k}}g_{k-1}\right)(i)-2\delta-\epsilon$
Note that from Lemma 3.2 in the mainpaper, we know that
$\min_{i,j\in\mathcal{S}}{\mathbb{P}}^{*}_{\mu_{k}}(j|i)>\gamma>0$
Also, since every policy is assumed to induce an irreducible Markov Process,
we have a limiting distribution with all positive entries. We thus obtain the
following crucial relationship,
$\displaystyle g_{k}(i)\geq(1-\gamma)l_{k-1}+\gamma u_{k-1}-2\delta-\epsilon$
(23)
Since the above inequation is true for all states $i$, it has to also be true
for $\operatorname*{argmin}_{i}\left(\mathsf{T}h_{k}-h_{k}\right)(i)$.
Hence, we obtain,
$\displaystyle l_{k}\geq(1-\gamma)l_{k-1}+\gamma u_{k-1}-2\delta-\epsilon$
Note that from Lemma 6.4, we know that $u_{k}\geq{J}^{*}$ for all
$k\in\mathbb{N}$.
Hence we obtain,
$\displaystyle l_{k}\geq(1-\gamma)l_{k-1}+\gamma{J}^{*}-2\delta-\epsilon$
Upon rearranging we obtain the result, that is,
$\displaystyle\left({J}^{*}-l_{k}\right)\leq(1-\gamma)\left({J}^{*}-l_{k-1}\right)+2\delta+\epsilon$
∎
We now present the proof of Theorem 3.3.
From Lemma 6.5, we thus have,
$\left({J}^{*}-l_{k}\right)\leq(1-\gamma)\left({J}^{*}-l_{k-1}\right)+2\delta+\epsilon$
(24)
However, we know that,
$l_{k}-\epsilon\leq{J}^{*}$
In order to iterate Equation 24, need to ensure the terms are non-negative.
Rearranging the terms thus yields,
$\left({J}^{*}-l_{k}+\epsilon\right)\leq(1-\gamma)\left({J}^{*}-l_{k-1}+\epsilon\right)+2\delta+(1+\gamma)\epsilon$
Let
$\omega=2\delta+(1+\gamma)\epsilon$
and
$a_{k}={J}^{*}-l_{k}+\epsilon.$
Then upon iterating, we obtain,
$\displaystyle a_{k}$ $\displaystyle\leq(1-\gamma)a_{k-1}+\omega$
$\displaystyle\leq(1-\gamma)((1-\gamma)a_{k-2}+\omega)+\omega$
$\displaystyle\leq\omega\frac{1-(1-\gamma)^{k}}{\gamma}+(1-\gamma)^{k}a_{0}$
Substituting back, we obtain,
$\displaystyle\left({J}^{*}-l_{k}+\epsilon\right)\leq\left(\frac{1-(1-\gamma)^{k}}{\gamma}\right)\left(2\delta+(1+\gamma)\epsilon\right)+(1-\gamma)^{k}({J}^{*}-l_{0}+\epsilon)$
(25)
$\displaystyle\left({J}^{*}-l_{k}\right)\leq\left(\frac{1-(1-\gamma)^{k}}{\gamma}\right)\left(2\delta+(1+\gamma)\epsilon\right)-\epsilon+(1-\gamma)^{k}({J}^{*}-l_{0}+\epsilon)$
Note that from Lemma 6.4 we know that
$l_{k}\leq{J}_{\mu_{k+1}}+\epsilon\leq{J}^{*}+\epsilon$
Hence we get,
$\left({J}^{*}-{J}_{\mu_{k+1}}\right)\leq\left({J}^{*}-l_{k}+\epsilon\right)$
Thus we obtain,
$\left({J}^{*}-{J}_{\mu_{k+1}}\right)\leq\left(\frac{1-(1-\gamma)^{k}}{\gamma}\right)\left(2\delta+(1+\gamma)\epsilon\right)+(1-\gamma)^{k}({J}^{*}-l_{0}+\epsilon)$
Theorem 3.3 presents an upper bound on the error in terms of the average
reward. Recall that, for the optimal relative value function $h$, the
following relationship holds: $Th-h=J^{*}\boldsymbol{1}.$ Thus, it is also
interesting to understand how $Th_{k}-h_{k}$ behaves under approximate policy
iteration. The following proposition characterizes the behavior of this term.
###### Proposition 6.6.
The iterates generated from approximate policy iteration algorithm 2 satisfy
the following bound:
$\displaystyle\left(u_{k-1}-l_{k-1}\right)$
$\displaystyle\leq\underbrace{\left(\frac{1-(1-\gamma)^{k}}{\gamma^{2}}\right)\left(2\delta+(1+\gamma)\epsilon\right)+\frac{2\delta+\epsilon}{\gamma}}_{\text{approximation
error}}+\underbrace{\frac{(1-\gamma)^{k}({J}^{*}-l_{0}+\epsilon)}{\gamma}}_{\text{initial
condition error}},$
where $u_{k},l_{k}$ are defined in Equation 20.
###### Proof.
It is known from Equation 23 that,
$g_{k}(i)\geq(1-\gamma)l_{k-1}+\gamma u_{k-1}-2\delta-\epsilon$
This further yields,
$l_{k}\geq(1-\gamma)l_{k-1}+\gamma u_{k-1}-2\delta-\epsilon$
From Lemma 6.4, we know that,
$l_{k}\leq{J}^{*}+\epsilon$ ${J}^{*}+\epsilon\geq(1-\gamma)l_{k-1}+\gamma
u_{k-1}-2\delta-\epsilon$
${J}^{*}-l_{k}+\epsilon\geq\gamma(u_{k-1}-l_{k-1})-2\delta-\epsilon$
$\gamma(u_{k-1}-l_{k-1})\leq\left({J}^{*}-l_{k}+\epsilon\right)+2\delta+\epsilon$
From Equation 25, we thus obtain,
$\gamma(u_{k-1}-l_{k-1})\leq\left(\frac{1-(1-\gamma)^{k}}{\gamma}\right)\left(2\delta+(1+\gamma)\epsilon\right)+(1-\gamma)^{k}({J}^{*}-l_{0}+\epsilon)+2\delta+\epsilon$
We thus obtain the result in Proposition 3.11,
$\left(u_{k-1}-l_{k-1}\right)\leq\left(\frac{1-(1-\gamma)^{k}}{\gamma^{2}}\right)\left(2\delta+(1+\gamma)\epsilon\right)+\frac{2\delta+\epsilon}{\gamma}+\frac{(1-\gamma)^{k}({J}^{*}-l_{0}+\epsilon)}{\gamma}$
∎
###### Corollary 6.7.
The asymptotic behavior of the relative value function iterates is given by
$\limsup_{k\to\infty}\left(u_{k}-l_{k}\right)\leq\frac{\epsilon\left(1+2\gamma\right)+2\delta\left(1+\gamma\right)}{\gamma^{2}}$
##### Comment of the Novelty of our Proof Technique:
As mentioned in the main body of the paper, our proof is inspired by the proof
of convergence of modified policy iteration in [VdW80]. However, since our
algorithm is quite different from modified policy iteration due to the
presence of errors in each step of the algorithm, special care is needed to
obtain useful performance bounds. Specifically, the impact of the errors at
each step have to be carefully bounded to ensure that the performance bounds
do not blow up to infinity as it does when we obtain a similar result for the
discounted-reward case and let the discount factor go to one.
### 6.3 Proofs from Section 4
Algorithm 2 and its analysis, adapted to the context of $Q$ function with time
dependent approximation errors is presented in this section. Most RL
algorithms use the state-action relative value function $Q$ instead of the
relative state value function $h$ to evaluate a policy. The corresponding
Bellman Equation in terms of $Q$ associated with any state-action pair $(s,a)$
is given by
${J}_{\mu}+Q_{\mu}(s,a)=r(s,a)+\left(\mathbb{Q}_{\mu}Q_{\mu}\right)(s,a)$
where
$\mathbb{Q}_{\mu}(s^{\prime},a^{\prime}|s,a)=\mu(a^{\prime}|s^{\prime})\mathbb{P}(s^{\prime}|s,a).$
Since we are interested in randomized policies for exploration reasons, the
irreducibility assumptions imposed on $\mathbb{P}_{\mu}$ also hold true for
the transition kernel $\mathbb{Q}_{\mu}$. The state relative value function
$h_{\mu}$ and state-action relative value function $Q_{\mu}$ are related as
$h_{\mu}(s)=\sum_{a}\mu(a|s)Q_{\mu}(s,a)$. Consider the following similar
definitions of the Bellman operators corresponding to the state-action value
function:
$(\mathsf{T}^{\mathsf{Q}}_{\mu}Q)(s,a)=r(s,a)+\left(\mathbb{Q}_{\mu}Q\right)(s,a)$
and
$(\mathsf{T}^{\mathsf{Q}}Q)(s,a)=r(s,a)+\max_{\mu\in\Pi}\left({\mathbb{Q}}_{\mu}Q\right)(s,a).$
Let $(s^{*},a^{*})$ represent some fixed state. Then the relative Bellman
operator, relative to $(s^{*},a^{*})$ is defined as:
$\left(\widetilde{\mathsf{T}}^{\mathsf{Q}}_{\mu}Q\right)(s,a)=r(s,a)+\left(\mathbb{Q}_{\mu}Q\right)(s,a)-r(s^{*},a^{*})-\left(\mathbb{Q}_{\mu}Q\right)(s^{*},a^{*}).$
The algorithm, thus adapted to state action relative value function is given
below:
Algorithm 4 Approximate Policy Iteration: Average Reward
1:Require $Q_{0}\in\mathbb{R}^{n}$
2:for $k=0,1,2,\ldots$ do
3: 1\. Compute $\mu_{k+1}\in\Pi$ such that
$\|\mathsf{T}Q_{k}-\mathsf{T}_{\mu_{k+1}}Q_{k}\|_{\infty}\leq\epsilon_{k}$
4: 2\. Compute $Q_{k+1}$ such that
$\|Q_{k+1}-Q_{\mu_{k+1}}\|_{\infty}\leq\delta_{k+1}$
5: where
$Q_{\mu_{k+1}}=\lim_{m\to\infty}\left({\widetilde{\mathsf{T}}^{\mathsf{Q}}_{\mu_{k+1}}}\right)^{m}Q_{k}$
6:end for
Define
$\displaystyle
g_{k}(s,a)=\left(\mathsf{T}^{\mathsf{Q}}Q_{k}-Q_{k}\right)(s,a),\qquad\forall(s,a)\in\mathcal{S}\times\mathcal{A},$
and set
$\displaystyle
l_{k}=\min_{(s,a)\in\mathcal{S}\times\mathcal{A}}g_{k}(s,a),\qquad
u_{k}=\max_{(s,a)\in\mathcal{S}\times\mathcal{A}}g_{k}(s,a).$
We prove the convergence in three parts:
1. 1.
We prove that $l_{k}-\epsilon_{k}\leq{J}_{\mu_{k+1}}\leq{J}^{*}\leq u_{k}$.
2. 2.
We use the result from 1 to prove Lemma 4.1.
3. 3.
We iteratively employ the result in Lemma 4.1 to prove Proposition 4.2.
#### 6.3.1 Proof of part 1
By definition, we have
$\displaystyle l_{k}\boldsymbol{1}$
$\displaystyle\leq\mathsf{T}^{\mathsf{Q}}Q_{k}-Q_{k}.$
Since
$\|\mathsf{T}^{\mathsf{Q}}Q_{k}-\mathsf{T}^{\mathsf{Q}}_{\mu_{k+1}}Q_{k}\|_{\infty}\leq\epsilon_{k}$,
it follows that
$\displaystyle l_{k}\boldsymbol{1}$
$\displaystyle\leq\mathsf{T}^{\mathsf{Q}}_{\mu_{k+1}}Q_{k}-Q_{k}+\epsilon_{k}\boldsymbol{1}$
$\displaystyle=r+\mathbb{Q}_{\mu_{k+1}}Q_{k}-Q_{k}+\epsilon_{k}\boldsymbol{1}.$
Multiplying by $\mathbb{Q}^{*}_{\mu_{k+1}}$, we get
$\displaystyle\mathbb{Q}_{\mu_{k+1}}^{*}l_{k}\boldsymbol{1}$
$\displaystyle\leq\mathbb{Q}_{\mu_{k+1}}^{*}r+\mathbb{Q}_{\mu_{k+1}}^{*}\mathbb{Q}_{\mu_{k+1}}Q_{k}-\mathbb{Q}_{\mu_{k+1}}^{*}Q_{k}+\mathbb{Q}_{\mu_{k+1}}^{*}\epsilon_{k}\boldsymbol{1}$
$\displaystyle=\mathbb{Q}_{\mu_{k+1}}^{*}r+\epsilon_{k}\boldsymbol{1}$ (26)
where $\mathbb{Q}_{\mu_{k+1}}^{*}$ is a matrix whose rows are identical and
are the invariant distribution associated with the probability transition
kernel $\mathbb{Q}_{\mu_{k+1}}$. Note that
$\mathbb{Q}_{\mu_{k+1}}^{*}\boldsymbol{1}=\boldsymbol{1}$, and that
$\mathbb{Q}_{\mu_{k+1}}^{*}\mathbb{Q}_{\mu_{k+1}}=\mathbb{Q}_{\mu_{k+1}}^{*}$.
Consider the Bellman Equation corresponding to the state-action relative value
function associated with the policy $\mu_{k+1}$:
$\displaystyle{J}_{\mu_{k+1}}+Q_{\mu_{k+1}}=r+\mathbb{Q}_{\mu_{k+1}}Q_{\mu_{k+1}}$
It follows that
$\displaystyle\mathbb{Q}^{*}_{\mu_{k+1}}{J}_{\mu_{k+1}}+\mathbb{Q}^{*}_{\mu_{k+1}}Q_{\mu_{k+1}}=\mathbb{Q}^{*}_{\mu_{k+1}}r+\mathbb{Q}^{*}_{\mu_{k+1}}\mathbb{Q}_{\mu_{k+1}}Q_{\mu_{k+1}}.$
Since
$\mathbb{Q}_{\mu_{k+1}}^{*}\mathbb{Q}_{\mu_{k+1}}=\mathbb{Q}^{*}_{\mu_{k+1}}$,
we have that
$\displaystyle{J}_{\mu_{k+1}}\boldsymbol{1}=\mathbb{Q}_{\mu_{k+1}}^{*}r.$
Hence, Equation 26 yields
$\displaystyle l_{k}\leq{J}_{\mu_{k+1}}+\epsilon_{k}.$
Equivalently, we have
$\displaystyle l_{k}-\epsilon_{k}\leq{J}_{\mu_{k+1}}.$
From definition,
$\displaystyle{J}^{*}=\max_{\mu\in\Pi}{J}_{\mu}\geq{J}_{\mu_{k+1}}.$
Let $\mu^{*}$ be the policy corresponding to the optimal average reward
${J}^{*}$. Let $Q^{*}$ denote the state-action relative value function
associated with the policy $\mu^{*}$. Note that $({J}^{*},Q^{*})$ is the
solution of the Bellman optimality equation, i.e.,
$\displaystyle{J}^{*}+Q^{*}$
$\displaystyle=r+\max_{\mu\in\Pi}\mathbb{Q}_{\mu}Q^{*}$
$\displaystyle=r+\mathbb{Q}_{\mu^{*}}Q^{*}.$
Then,
$\displaystyle u_{k}\boldsymbol{1}$
$\displaystyle\geq\mathsf{T}^{\mathsf{Q}}Q_{k}-Q_{k}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\geq}}\mathsf{T}^{\mathsf{Q}}_{\mu^{*}}Q_{k}-Q_{k}$
$\displaystyle=r+\mathbb{Q}_{\mu^{*}}Q_{k}-Q_{k},$
where (a) is due to the fact that $\mathsf{T}^{\mathsf{Q}}$ is the Bellman
optimality operator. Therefore, we have
$\displaystyle\mathbb{Q}_{\mu^{*}}^{*}u_{k}\boldsymbol{1}$
$\displaystyle\geq\mathbb{Q}_{\mu^{*}}^{*}r+\mathbb{Q}_{\mu^{*}}^{*}\mathbb{Q}_{\mu^{*}}Q_{k}-\mathbb{Q}_{\mu^{*}}^{*}Q_{k}$
Equivalently,
$\displaystyle u_{k}\boldsymbol{1}$
$\displaystyle\geq\mathbb{Q}^{*}_{\mu^{*}}r$
Therefore, we conclude that
$\displaystyle u_{k}$ $\displaystyle\geq{J}^{*}$
Hence, for all $k\in\mathbb{N}$,
$\displaystyle l_{k}-\epsilon_{k}\leq{J}_{\mu_{k+1}}\leq{J}^{*}\leq u_{k}$
(27)
#### 6.3.2 Proof of Lemma 4.1
Recall that
$\displaystyle
g_{k}(s,a)=\left(\mathsf{T}^{\mathsf{Q}}Q_{k}-Q_{k}\right)(s,a)\geq\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}Q_{k}-Q_{k}\right)(s,a),$
where the inequality follows by the fact that $\mathsf{T}^{\mathsf{Q}}$ is the
Bellman optimality operator. Note that the Bellman operator
$\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}$ is shift-invariant, i.e., for all
$p\in\mathbb{R}$:
$\displaystyle\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}\left(Q_{k}+p\boldsymbol{1}\right)$
$\displaystyle=r+\mathbb{Q}_{\mu_{k}}\left(Q_{k}+p\boldsymbol{1}\right)$
$\displaystyle=r+\mathbb{Q}_{\mu_{k}}Q_{k}+p\boldsymbol{1}$
$\displaystyle=\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}Q_{k}+p\boldsymbol{1}$
Recall that $\left\|Q_{k}-Q_{\mu_{k}}\right\|_{\infty}\leq\delta_{k}$. Hence,
we have
$\displaystyle g_{k}(s,a)$
$\displaystyle\geq\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}(Q_{\mu_{k}}-\delta_{k}\boldsymbol{1})(s,a)-(Q_{\mu_{k}}(s,a)+\delta_{k})$
$\displaystyle\geq\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}Q_{\mu_{k}}-Q_{\mu_{k}}\right)(s,a)-2\delta_{k}$
(28)
Recall that
$Q_{\mu_{k}}=\lim_{m\to\infty}\left(\widetilde{\mathsf{T}}^{\mathsf{Q}}_{\mu_{k+1}}\right)^{m}Q_{k}$.
Therefore,
$\displaystyle\left(\widetilde{\mathsf{T}}^{\mathsf{Q}}_{\mu_{k}}\right)^{m}Q_{k-1}$
$\displaystyle=\left(\widetilde{\mathsf{T}}^{\mathsf{Q}}_{\mu_{k}}\right)^{m-1}\left(\widetilde{\mathsf{T}}^{\mathsf{Q}}_{\mu_{k}}\right)Q_{k-1}$
$\displaystyle=\left(\widetilde{\mathsf{T}}^{\mathsf{Q}}_{\mu_{k}}\right)^{m-1}\left(r+\mathbb{Q}_{\mu_{k}}Q_{k-1}-r(s^{*},a^{*})\boldsymbol{1}-\left(\mathbb{Q}_{\mu_{k}}Q_{k-1}\right)(s^{*},a^{*})\boldsymbol{1}\right)$
$\displaystyle=\left(\widetilde{\mathsf{T}}^{\mathsf{Q}}_{\mu_{k}}\right)^{m-1}\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}Q_{k-1}-\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}Q_{k-1}\right)(s^{*},a^{*})\boldsymbol{1}\right)$
$\displaystyle=\left(\widetilde{\mathsf{T}}^{\mathsf{Q}}_{\mu_{k}}\right)^{m-2}\left(\widetilde{\mathsf{T}}^{\mathsf{Q}}_{\mu_{k}}\right)\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}Q_{k-1}-\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}Q_{k-1}\right)(s^{*},a^{*})\boldsymbol{1}\right)$
$\displaystyle=\left(\widetilde{\mathsf{T}}^{\mathsf{Q}}_{\mu_{k}}\right)^{m-2}\bigg{(}r+\mathbb{Q}_{\mu_{k}}\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}Q_{k-1}-\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}Q_{k-1}\right)(s^{*},a^{*})\boldsymbol{1}\right)$
$\displaystyle\qquad\qquad\qquad-r(s^{*},a^{*})\boldsymbol{1}-\left(\mathbb{Q}_{\mu_{k}}\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}Q_{k-1}-\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}Q_{k-1}\right)(s^{*},a^{*})\boldsymbol{1}\right)\right)(s^{*},a^{*})\boldsymbol{1}\bigg{)}$
$\displaystyle=\left(\widetilde{\mathsf{T}}^{\mathsf{Q}}_{\mu_{k}}\right)^{m-2}\bigg{(}r+\mathbb{Q}_{\mu_{k}}\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}Q_{k-1}\right)-\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}Q_{k-1}\right)(s^{*},a^{*})\boldsymbol{1}$
$\displaystyle\qquad\qquad\qquad-r(s^{*},a^{*})\boldsymbol{1}-\mathbb{Q}_{\mu_{k}}\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}Q_{k-1}\right)(s^{*},a^{*})\boldsymbol{1}+\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}Q_{k-1}\right)(s^{*},a^{*})\boldsymbol{1}\bigg{)}$
$\displaystyle=\left(\widetilde{\mathsf{T}}^{\mathsf{Q}}_{\mu_{k}}\right)^{m-2}\left(\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}\right)^{2}Q_{k-1}-\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}\right)^{2}Q_{k-1}(s^{*},a^{*})\boldsymbol{1}\right)$
Iterating, we thus obtain,
$Q_{\mu_{k}}=\lim_{m\to\infty}(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}})^{m}Q_{k-1}-(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}})^{m}Q_{k-1}(s^{*},a^{*})\boldsymbol{1}$
Substituting back in Equation 28,
$\displaystyle g_{k}(s,a)$
$\displaystyle\geq\bigg{(}\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}\left(\lim_{m\to\infty}(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}})^{m}Q_{k-1}-(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}})^{m}Q_{k-1}(s^{*},a^{*})\boldsymbol{1}\right)$
$\displaystyle\qquad\qquad\qquad\qquad-\lim_{m\to\infty}\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}\right)^{m}Q_{k-1}-\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}\right)^{m}Q_{k-1}(s^{*},a^{*})\boldsymbol{1}\bigg{)}(s,a)-2\delta_{k}$
$\displaystyle=\bigg{(}\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}\left(\lim_{m\to\infty}(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}})^{m}Q_{k-1}\right)-\lim_{m\to\infty}(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}})^{m}Q_{k-1}(s^{*},a^{*})\boldsymbol{1}$
$\displaystyle\qquad\qquad\qquad\qquad-\lim_{m\to\infty}\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}\right)^{m}Q_{k-1}-\lim_{m\to\infty}\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}\right)^{m}Q_{k-1}(s^{*},a^{*})\boldsymbol{1}\bigg{)}(s,a)-2\delta_{k}$
$\displaystyle=\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}\left(\lim_{m\to\infty}\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}\right)^{m}Q_{k-1}\right)-\lim_{m\to\infty}\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}\right)^{m}Q_{k-1}\right)(s,a)-2\delta_{k}$
Since $\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}$ is a continuous operator, it is
possible to take the limit outside. We thus obtain,
$\displaystyle g_{k}(s,a)$
$\displaystyle\geq\lim_{m\to\infty}\left(\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}\right)^{m+1}Q_{k-1}-\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}\right)^{m}Q_{k-1}\right)(s,a)-2\delta_{k}$
$\displaystyle=\lim_{m\to\infty}\left(\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}\right)^{m}\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}Q_{k-1}-\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}\right)^{m}Q_{k-1}\right)(s,a)-2\delta_{k}$
$\displaystyle=\lim_{m\to\infty}\left(\mathbb{Q}^{m}_{\mu_{k}}\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}Q_{k-1}-Q_{k-1}\right)\right)(s,a)-2\delta_{k}$
$\displaystyle=\mathbb{Q}^{*}_{\mu_{k}}\left(\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}Q_{k-1}-Q_{k-1}\right)(s,a)-2\delta_{k}.$
Since
$\left\|\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}Q_{k-1}-\mathsf{T}^{\mathsf{Q}}Q_{k-1}\right\|_{\infty}\leq\epsilon_{k-1}$,
$\displaystyle g_{k}(s,a)$
$\displaystyle\geq\mathbb{Q}^{*}_{\mu_{k}}\left(\mathsf{T}^{\mathsf{Q}}Q_{k-1}-Q_{k-1}\right)(s,a)-2\delta_{k}-\epsilon_{k-1}$
$\displaystyle=\left(\mathbb{Q}^{*}_{\mu_{k}}g_{k-1}\right)(s,a)-2\delta_{k}-\epsilon_{k-1}.$
Recall the definition of $\gamma>0$ in Lemma 4.1. Note that all entries of
$\mathbb{Q}^{*}_{\mu_{k}}$ are bounded from below by $\gamma$. Hence, we have
$\displaystyle g_{k}(s,a)\geq(1-\gamma)l_{k-1}+\gamma
u_{k-1}-2\delta_{k}-\epsilon_{k-1}.$
Since the above inequality is true for all states
$(s,a)\in\mathcal{S}\times\mathcal{A}$, we obtain,
$\displaystyle l_{k}\geq(1-\gamma)l_{k-1}+\gamma
u_{k-1}-2\delta_{k}-\epsilon_{k-1}.$
The above inequality combined with Equation 27, yields
$\displaystyle
l_{k}\geq(1-\gamma)l_{k-1}+\gamma{J}^{*}-2\delta_{k}-\epsilon_{k-1}.$
Upon rearranging the above inequality, we obtain the result, that is,
$\displaystyle\left({J}^{*}-l_{k}\right)\leq(1-\gamma)\left({J}^{*}-l_{k-1}\right)+2\delta_{k}+\epsilon_{k-1}.$
Subsequently,
$\displaystyle\left({J}^{*}-\min_{(s,a)}\left(\mathsf{T}^{\mathsf{Q}}Q_{k}-Q_{k}\right)(s,a)\right)$
$\displaystyle\leq(1-\gamma)\left({J}^{*}-\min_{(s,a)}\left(\mathsf{T}^{\mathsf{Q}}Q_{k-1}-Q_{k-1}\right)(s,a)\right)$
$\displaystyle\qquad\qquad\qquad+2\left\|Q_{k}-Q_{\mu_{k}}\right\|_{\infty}+\left\|\mathsf{T}^{\mathsf{Q}}Q_{k-1}-\mathsf{T}^{\mathsf{Q}}_{\mu_{k}}Q_{k-1}\right\|_{\infty}$
#### 6.3.3 Proof of Proposition 4.2
We now prove Proposition 4.2. From Lemma 4.1, we have
$\left({J}^{*}-l_{k}\right)\leq(1-\gamma)\left({J}^{*}-l_{k-1}\right)+2\delta_{k}+\epsilon_{k-1}.$
(29)
To iterate over Equation 29, we need to ensure the terms are non-negative.
Note that by the first part,
$l_{k}-\epsilon_{k}\leq{J}^{*}.$
Hence, rearranging the terms thus yields,
$\left({J}^{*}-l_{k}+\epsilon_{k}\right)\leq(1-\gamma)\left({J}^{*}-l_{k-1}+\epsilon_{k-1}\right)+\epsilon_{k}+2\delta_{k}+\epsilon_{k-1}-(1-\gamma)\epsilon_{k-1}.$
This is equivalent to
$\displaystyle\left({J}^{*}-l_{k}+\epsilon_{k}\right)\leq\left(1-\gamma\right)\left({J}^{*}-l_{k-1}+\epsilon_{k-1}\right)+\left(\epsilon_{k}+\gamma\epsilon_{k-1}\right)+2\delta_{k}.$
Let
$a_{k}={J}^{*}-l_{k}+\epsilon_{k}.$
Then, we obtain,
$\displaystyle a_{k}$
$\displaystyle\leq(1-\gamma)a_{k-1}+\epsilon_{k}+\gamma\epsilon_{k-1}+2\delta_{k}$
$\displaystyle\leq(1-\gamma)\left[(1-\gamma)a_{k-2}+\epsilon_{k-1}+\gamma\epsilon_{k-2}+2\delta_{k-1}\right]+\epsilon_{k}+\gamma\epsilon_{k-1}+2\delta_{k}$
$\displaystyle=(1-\gamma)^{k}a_{0}+\sum_{\ell=0}^{k-1}(1-\gamma)^{\ell}\epsilon_{k-\ell}+\gamma\sum_{\ell=0}^{k-1}(1-\gamma)^{\ell}\epsilon_{k-1-\ell}+2\sum_{\ell=0}^{k-1}(1-\gamma)^{\ell}\delta_{k-\ell}$
Since $\epsilon_{0}=0$,
$\displaystyle
a_{k}\leq\left(1-\gamma\right)^{k}a_{0}+\epsilon_{k}+\sum_{\ell=1}^{k-1}\left(1-\gamma\right)^{\ell-1}\epsilon_{k-\ell}+2\sum_{\ell=0}^{k-1}\left(1-\gamma\right)^{\ell}\delta_{k-\ell}$
Substituting for $a_{k}$, we get
$\displaystyle\left({J}^{*}-l_{k}\right)\leq\left(1-\gamma\right)^{k}\left({J}^{*}-l_{0}\right)+\sum_{\ell=1}^{k-1}\left(1-\gamma\right)^{\ell-1}\epsilon_{k-\ell}+2\sum_{\ell=0}^{k-1}\left(1-\gamma\right)^{\ell}\delta_{k-\ell}.$
Since from part 1, we know that
${J}^{*}-{J}_{\mu_{k+1}}\leq{J}^{*}-l_{k}+\epsilon_{k}$, we have
$\displaystyle{J}^{*}-{J}_{\mu_{k+1}}$
$\displaystyle\leq\left(1-\gamma\right)^{k}\left[{J}^{*}-\min_{i}\left(\mathsf{T}^{\mathsf{Q}}Q_{0}-Q_{0}\right)(i)\right]$
$\displaystyle+\sum_{\ell=1}^{k-1}\left(1-\gamma\right)^{\ell-1}\left[\left\|\mathsf{T}^{\mathsf{Q}}_{\mu_{k+1-\ell}}Q_{k-\ell}-\mathsf{T}^{\mathsf{Q}}Q_{k-\ell}\right\|_{\infty}\right]+{\left\|\mathsf{T}^{\mathsf{Q}}_{\mu_{k+1}}Q_{k}-\mathsf{T}^{\mathsf{Q}}Q_{k}\right\|_{\infty}}$
$\displaystyle+2\sum_{\ell=0}^{k-1}\left(1-\gamma\right)^{\ell}\left[\left\|Q_{k-\ell}-Q_{\mu_{k-\ell}}\right\|_{\infty}\right].$
Taking expectation, we have
$\displaystyle\mathbb{E}\left[{J}^{*}-{J}_{\mu_{k+1}}\right]$
$\displaystyle\leq\left(1-\gamma\right)^{k}\mathbb{E}\left[{J}^{*}-\min_{i}\left(\mathsf{T}^{\mathsf{Q}}Q_{0}-Q_{0}\right)(i)\right]$
$\displaystyle+\sum_{\ell=1}^{k-1}\left(1-\gamma\right)^{\ell-1}\mathbb{E}\left[\left\|\mathsf{T}^{\mathsf{Q}}_{\mu_{k+1-\ell}}Q_{k-\ell}-\mathsf{T}^{\mathsf{Q}}Q_{k-\ell}\right\|_{\infty}\right]+\mathbb{E}\left[\left\|\mathsf{T}^{\mathsf{Q}}_{\mu_{k+1}}Q_{k}-\mathsf{T}^{\mathsf{Q}}Q_{k}\right\|_{\infty}\right]$
$\displaystyle+2\sum_{\ell=0}^{k-1}\left(1-\gamma\right)^{\ell}\mathbb{E}\left[\left\|Q_{k-\ell}-Q_{\mu_{k-\ell}}\right\|_{\infty}\right].$
#### 6.3.4 TD Learning for $Q$ function
One RL algorithm to estimate the state-action relative value function with
linear function approximation, associated with a policy $\mu$ is
TD($\lambda$). Finite-time sample complexity of TD($\lambda$) has been
recently studied in [ZZM21]. Invoking their results, restating it in terms of
the state-action relative value function rather than the state relative value
function, we characterize the sample complexity required to achieve certain
accuracy in the approximation.
We estimate $Q_{\mu}(s,a)$ linearly by $\phi(s,a)^{\top}\theta^{*}_{\mu}$, for
some $\theta^{*}_{\mu}\in\mathbb{R}^{d}$, where $\phi(s,a)=$
$[\phi_{1}(s,a),\cdots,\phi_{d}(s,a)]^{T}\in\mathbb{R}^{d}$ is the feature
vector associated with $(s,a)\in\mathcal{S}\times\mathcal{A}$. Here,
$\theta^{*}_{\mu}$ is a fixed point of
$\Phi\theta=\Pi_{D,W_{\phi}}\mathsf{T}^{\mathsf{Q},\lambda}_{\mu}\Phi\theta$,
where $\Phi$ is an $|\mathcal{S}|\times d$ matrix whose $k$-th column is
$\phi_{k}$,
$\mathsf{T}^{\mathsf{Q},\lambda}_{\mu}=(1-\lambda)\sum_{m=0}^{\infty}\lambda^{m}\left(\widehat{\mathsf{T}}^{\mathsf{Q}}_{\mu}\right)^{m+1}$
where
$\widehat{\mathsf{T}}^{\mathsf{Q}}_{\mu}=\mathsf{T}^{\mathsf{Q}}_{\mu}-{J}_{\mu}\boldsymbol{1}$,
$D$ is the diagonal matrix with diagonal elements given by the stationary
distribution of the policy $\mu$ on $\mathcal{S}\times\mathcal{A}$, and
$\Pi_{D,W_{\phi}}=\Phi(\Phi^{\top}D\Phi)^{-1}\Phi^{\top}D$ is the projection
matrix onto $W_{\Phi}=\\{\Phi\theta|\theta\in\mathbb{R}^{d}\\}$ with respect
to the norm $\|\cdot\|_{D}$. Note that this fixed point equation may have
multiple solutions. In particular, if $\Phi\theta_{e}=\boldsymbol{1}$, then
$\theta^{*}_{\mu}+p\theta_{e}$ is also a solution for any $p\in\mathbb{R}$.
Hence, we focus on the set of equivalent classes $E$ where we say
$\theta_{1}\sim\theta_{2}$ if $\Phi(\theta_{1}-\theta_{2})=\boldsymbol{1}$.
The value of $\mathbb{E}[\|\Pi_{2,E}(\theta-\theta^{*}_{\mu})\|_{2}^{2}]$
characterizes the accuracy of our approximation, where $\Pi_{2,E}$ is the
projection onto $E$ with respect to $\|\cdot\|_{2}$. Suppose that
$\\{\phi_{1},\phi_{2},\cdots,\phi_{d}\\}$ are linearly independent and that
$\max_{(s,a)\in\mathcal{S}\times\mathcal{A}}\|\phi(s,a)\|_{2}\leq 1$.
Upon making a new observation $(s_{t+1},a_{t+1})$ for $t\geq 0$, the average
reward $TD(\lambda)$ uses the following update equations:
$\displaystyle\text{Eligibility trace: }z_{t}=\lambda
z_{t-1}+\phi(s_{t},a_{t})$ (30) $\displaystyle\text{TD error:
}d_{t}=r(s_{t},a_{t})-{J}_{t}+\phi(s_{t+1},a_{t+1})^{\top}\theta_{t}$
$\displaystyle\qquad\qquad\qquad\qquad-\phi(s_{t},a_{t})^{\top}\theta_{t}$
$\displaystyle\text{average-reward update:
}{J}_{t+1}={J}_{t}+c_{\alpha}\beta_{t}(r(s_{t},a_{t})-{J}_{t}),$
$\displaystyle\text{parameter update:
}\theta_{t+1}=\theta_{t}+\beta_{t}\delta_{t}(\theta)z_{t},$
where $\beta_{t}$ is the scalar step size, $c_{\alpha}>0$ is a constant, and
$z_{-1}$ is initialized to be zero. Following the same argument as in [ZZM21],
we get the following theorem which characterizes the number of samples
required to get a certain accuracy in the approximation.
###### Theorem 6.8.
Let $\beta_{t}=\frac{c_{1}}{t+c_{2}}$, and suppose that the positive constants
$c_{1}$, $c_{2}$, and $c_{\alpha}$ are properly chosen. Then, the number of
samples required to ensure an approximation accuracy of
$\mathbb{E}[\|\Pi_{2,E}(\theta-\theta^{*}_{\mu})\|_{2}^{2}]\leq\delta$, is
given by
$\displaystyle\tau=O\left(\frac{K\log(\frac{1}{\Delta})\|\theta^{*}_{\mu}\|_{2}^{2}}{\Delta^{4}(1-\lambda)^{4}\delta^{2}}\right),$
where $K$ is the minimum mixing time constant across all probability
transition matrices induced by the policies,
$\displaystyle\Delta=\min_{\|\theta\|_{2}=1,\theta\in
E}\theta^{T}\Phi^{T}D(I-\mathbb{Q}^{\lambda})\Phi\theta>0,$
and
$\mathbb{Q}^{\lambda}=(1-\lambda)\sum_{m=0}^{\infty}\lambda^{m}\mathbb{Q}^{m+1}$.
###### Proof.
The proof is a simple adaptation of the proof in [ZZM21] for the state
relative value function. ∎
Let $Q_{k}=\Phi\theta_{k}$, where $\theta_{k}$ is the output of TD($\lambda$)
algorithm with $T$ samples. Then, we have
$\displaystyle\|Q_{k}-Q_{\mu_{k}}\|_{\infty}\leq\|\Phi(\theta_{k}-\theta_{\mu_{k}}^{*})\|_{\infty}+\|\Phi\theta_{\mu_{k}}^{*}-Q_{\mu_{k}}\|_{\infty},$
where the second term above is the error associated with the choice of $\Phi$.
Since $\max_{(s,a)\in\mathcal{S}\times\mathcal{A}}\|\phi(s,a)\|_{2}\leq 1$, we
get
$\displaystyle\delta_{k}=d\|\theta_{k}-\theta_{\mu_{k}}^{*}\|_{\infty}+\|\Phi\theta_{\mu_{k}}^{*}-Q_{\mu_{k}}\|_{\infty}.$
Using Theorem 6.8, we get a bound on the value of
$\|\theta_{k}-\theta_{\mu_{k}}^{*}\|_{\infty}$. Note that if
$\Phi\theta_{e}=\boldsymbol{1}$ for some $\theta_{e}\in\mathbb{R}^{d}$, then
we can pick $\theta_{k+1}$ and $\theta_{\mu_{k+1}}^{*}$ so that
$\phi(s^{*},a^{*})^{T}\theta_{k+1}=\phi(s^{*},a^{*})^{T}\theta_{\mu_{k+1}}^{*}=0$.
Otherwise, there is a unique choice for $\theta_{\mu_{k+1}}^{*}$, and we have
$E=\mathbb{R}^{d}$.
The expected value of learning error can thus be expressed as:
$\displaystyle\mathbb{E}\left[\delta_{k}\right]=\mathbb{E}\left[\delta_{\mathsf{TD},k}\right]+\mathbb{E}\left[\delta_{0,k}\right]$
where
${\delta_{\mathsf{TD},k}}=d\|\theta_{k+1}-\theta_{\mu_{k+1}}^{*}\|_{\infty}$
represents the TD learning error of the parameter vector
$\theta_{\mu_{k+1}}^{*}$ and
${\delta_{0,k}}=\|\Phi\theta_{\mu_{k+1}}^{*}-Q_{\mu_{k+1}}\|_{\infty}$
represents the function approximation error associated with the span of the
feature vector matrix $\Phi$.
${\delta_{\mathsf{TD},k}}$ has a direct dependence on the number of samples
utilized for TD learning and the mixing time of the corresponding Markov Chain
induced by the policy $\mu_{k}.$ More precisely, from Theorem 3.1,
$\displaystyle\mathbb{E}\left[\delta_{\mathsf{TD},k}\right]=O\left(\sqrt{\frac{K\log(\frac{1}{\Delta})\|\theta^{*}_{\mu_{k}}\|_{2}^{2}}{\Delta^{4}(1-\lambda)^{4}\tau}}\right),$
Hence as long as the mixing constant $\Delta$ is uniformly greater than zero,
and the feature vectors $\theta^{*}_{\mu_{k}}$ are such that they are
uniformly upper bounded in $k$, it is true that
$\displaystyle\mathbb{E}\left[\delta_{\mathsf{TD},k}\right]=\frac{C}{\sqrt{\tau}},$
for some constant $C>0$.
Next, we prove the corollaries for specific policy improvement algorithms.
Since the policy improvement part does not depend on whether the problem is a
discounted-reward problem or an average reward problem, we can borrow the
results from [CM22] to identify $\epsilon_{k}$ in each of the corollaries.
#### 6.3.5 Proof of Corollary 4.3
Recall the greedy update rule. Let
$a^{*}=\operatorname*{argmax}_{a^{\prime}}Q_{k}(s,a^{\prime})$. Given a
parameter $\beta>0$, at any time instant $k$, the greedy policy update
$\mu_{k+1}$ is given by:
$\displaystyle\mu_{k+1}(a|s)=\begin{cases}\frac{1}{\beta|\mathcal{A}|},&\text{if
}a\neq a^{*}\\\ \frac{1}{\beta|\mathcal{A}|}+1-\frac{1}{\beta},&\text{if
}a=a^{*}\end{cases}$ (31)
Let $\eta_{k}=\max_{\begin{subarray}{c}s^{\prime}\in\mathcal{S}\\\
a^{\prime}\in\mathcal{A}\end{subarray}}|\mathbb{Q}_{k}(s^{\prime},a^{\prime})|$.
The policy improvement approximation can be shown to be the following:
$\displaystyle\epsilon_{k}$
$\displaystyle=\left(\mathsf{T}^{\mathsf{Q}}Q_{k}-\mathsf{T}^{\mathsf{Q}}_{\mu_{k+1}}Q_{k}\right)(s,a)$
$\displaystyle\leq\sum_{s^{\prime}\in\mathcal{S}}\mathbb{P}(s^{\prime}|s,a)\frac{2}{\beta}\max_{a^{\prime}\in\mathcal{A}}|Q_{k}(s^{\prime},a^{\prime})|$
$\displaystyle\leq\frac{2\eta_{k}}{\beta}.$
Recall the error due to TD Learning
$\displaystyle\mathbb{E}\left[\delta_{k}\right]=\mathbb{E}\left[\delta_{\mathsf{TD},k}\right]+\mathbb{E}\left[\delta_{0,k}\right].$
Let $\overline{\delta}_{0}=\max_{t}\delta_{0,t}$. Then we obtain the
following:
$\displaystyle\mathbb{E}\left[\delta_{k}\right]=\overline{\delta}_{0}+\frac{C}{\sqrt{\tau}}.$
Substituting for expressions in Proposition 4.2, we obtain,
$\displaystyle\mathbb{E}\left[J^{*}-J_{\mu_{T+1}}\right]$
$\displaystyle\leq\left(1-\gamma\right)^{T}\mathbb{E}\left[{J}^{*}-\min_{i}\left(\mathsf{T}^{\mathsf{Q}}Q_{0}-Q_{0}\right)(i)\right]$
$\displaystyle+2\sum_{\ell=0}^{T-1}\left(1-\gamma\right)^{\ell}\left(\overline{\delta}_{0}+\frac{C}{\sqrt{\tau}}\right)+\sum_{\ell=1}^{T-1}\left(1-\gamma\right)^{\ell-1}\frac{2\eta_{T-\ell}}{\beta}+\frac{2\eta_{T}}{\beta}.$
Let
$c_{0}=\mathbb{E}\left[{J}^{*}-\min_{i}\left(\mathsf{T}^{\mathsf{Q}}Q_{0}-Q_{0}\right)(i)\right]$
be the error associated with the initial condition. Let
$\overline{\eta}=\max_{t}\eta_{t}$ ($\overline{\eta}$ can be the uniform upper
bound of the estimates $Q_{k}$ of relative state action value function
$Q_{\mu_{k}}$ over $k.$) Then we obtain the result in the corollary,
$\displaystyle\mathbb{E}\left[J^{*}-J_{\mu_{T+1}}\right]$
$\displaystyle\leq(1-\gamma)^{T}c_{0}+\frac{2}{\gamma}\left(\overline{\delta}_{0}+\frac{C}{\sqrt{\tau}}\right)+\left(\frac{1+\gamma}{\gamma}\right)\frac{2\overline{\eta}}{\beta}.$
(32)
#### 6.3.6 Proof of Corollary 4.4
Recall the softmax policy update. Given a parameter $\beta>0$, at any time
instant $k$, the Softmax update policy $\mu_{k+1}$ is:
$\mu_{k+1}(a|s)=\frac{\exp\left(\beta
Q_{k}(s,a)\right)}{\sum_{a^{\prime}\in\mathcal{A}}\exp\left(\beta
Q_{k}(s,a^{\prime})\right)}.$ (33)
The policy improvement approximation for this update rule turns out to be time
independent and is given by:
$\displaystyle\epsilon_{k}=\left(\mathsf{T}^{\mathsf{Q}}Q_{k}-\mathsf{T}^{\mathsf{Q}}_{\mu_{k+1}}Q_{k}\right)(s,a)\leq\frac{\log|\mathcal{A}|}{\beta}.$
Given $\overline{\delta}_{0}=\max_{t}\delta_{0,t}$ the following is true,
$\displaystyle\mathbb{E}\left[\delta_{k}\right]=\overline{\delta}_{0}+\frac{C}{\sqrt{\tau}}.$
Substituting for expressions in Proposition 4.2, we obtain,
$\displaystyle\mathbb{E}\left[J^{*}-J_{\mu_{T+1}}\right]$
$\displaystyle\leq\left(1-\gamma\right)^{T}\mathbb{E}\left[{J}^{*}-\min_{i}\left(\mathsf{T}^{\mathsf{Q}}Q_{0}-Q_{0}\right)(i)\right]$
$\displaystyle+2\sum_{\ell=0}^{T-1}\left(1-\gamma\right)^{\ell}\left(\overline{\delta}_{0}+\frac{C}{\sqrt{\tau}}\right)+\sum_{\ell=1}^{T-1}\left(1-\gamma\right)^{\ell-1}\frac{\log|\mathcal{A}|}{\beta}+\frac{\log|\mathcal{A}|}{\beta}.$
Let
$c_{0}=\mathbb{E}\left[{J}^{*}-\min_{i}\left(\mathsf{T}^{\mathsf{Q}}Q_{0}-Q_{0}\right)(i)\right]$
be the error associated with the initial condition. Then we obtain the result
in the corollary,
$\displaystyle\mathbb{E}\left[J^{*}-J_{\mu_{T+1}}\right]$
$\displaystyle\leq(1-\gamma)^{T}c_{0}+\frac{2}{\gamma}\left(\overline{\delta}_{0}+\frac{C}{\sqrt{\tau}}\right)+\left(\frac{1+\gamma}{\gamma}\right)\frac{\log\left(|\mathcal{A}|\right)}{\beta}.$
(34)
#### 6.3.7 Proof of Corollary 4.5
Recall the mirror descent update. Given $\beta>0$ the mirror descent update is
given by:
$\displaystyle\mu_{k+1}(a|s)=\frac{\mu_{k}(a|s)e^{\beta
Q_{k}(s,a)}}{\sum_{a^{\prime}\in\mathcal{A}}\mu_{k}(a^{\prime}|s)e^{\beta
Q_{k}(s,a^{\prime})}}$ (35)
The policy improvement error for this update rule and is given by:
$\displaystyle\epsilon_{k}$
$\displaystyle=\left(\mathsf{T}^{\mathsf{Q}}Q_{k}-\mathsf{T}^{\mathsf{Q}}_{\mu_{k+1}}Q_{k}\right)(s,a)$
$\displaystyle\leq\frac{1}{\beta}\log\frac{1}{\min_{s\in\mathcal{S}}\mu_{k+1}\left(a^{*}(s)|s\right)}.$
where $a^{*}(s)$ is the optimal action at state $s.$ Let
$\omega_{k+1}=\min_{s\in\mathcal{S}}\mu_{k+1}\left(a^{*}(s)|s\right)$ Given
that the TD learning error is of the form
$\displaystyle\mathbb{E}\left[\delta_{k}\right]=\overline{\delta}_{0}+\frac{C}{\sqrt{\tau}},$
Substituting for expressions in Proposition 4.2, we obtain,
$\displaystyle\mathbb{E}\left[J^{*}-J_{\mu_{T+1}}\right]$
$\displaystyle\leq\left(1-\gamma\right)^{T}\mathbb{E}\left[{J}^{*}-\min_{i}\left(\mathsf{T}^{\mathsf{Q}}Q_{0}-Q_{0}\right)(i)\right]$
$\displaystyle+2\sum_{\ell=0}^{T-1}\left(1-\gamma\right)^{\ell}\left(\overline{\delta}_{0}+\frac{C}{\sqrt{\tau}}\right)+\sum_{\ell=1}^{T-1}\left(1-\gamma\right)^{\ell-1}\frac{2}{\beta}\log{\frac{1}{\omega_{T-\ell}}}+\frac{2}{\beta}\log{\frac{1}{\omega_{T}}}.$
Let
$c_{0}=\mathbb{E}\left[{J}^{*}-\min_{i}\left(\mathsf{T}^{\mathsf{Q}}Q_{0}-Q_{0}\right)(i)\right]$
be the error associated with the initial condition. Let
$\underline{\omega}=\min_{t}\omega_{t}$ Then we obtain the result in the
corollary,
$\displaystyle\mathbb{E}\left[J^{*}-J_{\mu_{T+1}}\right]$
$\displaystyle\leq(1-\gamma)^{T}c_{0}+\frac{2}{\gamma}\left(\overline{\delta}_{0}+\frac{C}{\sqrt{\tau}}\right)+\left(\frac{1+\gamma}{\gamma\beta}\right)\log\left(\frac{1}{\underline{\omega}}\right).$
(36)
#### 6.3.8 Regret Analysis
Recall the pseudo regret defined as follows:
$\displaystyle R_{\text{PS}}(K)=\sum_{t=1}^{K}\left(J^{*}-J_{\mu_{t}}\right),$
(37)
where $\mu_{t}$ us the policy utilized at time $t$ and obtained through mirror
descent, and $K$ is the time horizon. Since $K=\tau T$, we have
$\displaystyle
R_{\text{PS}}(K)=\tau\sum_{t=1}^{T}\left(J^{*}-J_{\mu_{t}}\right).$ (38)
Substituting for $J^{*}-J_{\mu_{t}}$, it is true that
$\displaystyle
R_{\text{PS}}(K)=\tau\left(\sum_{t=1}^{T}(1-\gamma)^{T}c_{0}+\frac{1+\gamma}{\gamma\beta}\log\left(\frac{1}{\omega}\right)+\frac{2}{\gamma}\left(\overline{\delta}_{0}+\frac{c_{3}}{\sqrt{\tau}}\right)\right).$
(39)
Let $\beta=\sqrt{\tau}$. The above regret can be simplified as:
$\displaystyle R_{\text{PS}}(K)=\frac{\tau
c_{0}}{\gamma}+\frac{K}{\gamma\sqrt{\tau}}\left((1+\gamma)\log\left(\frac{1}{\omega}\right)+2c_{3}\right)+\frac{2K\overline{\delta}_{0}}{\gamma}$
(40)
Let $c_{5}=(1+\gamma)\log(1/\omega)+2c_{3}$. We then have
$\displaystyle R_{\text{PS}}(K)=\frac{\tau
c_{0}}{\gamma}+\frac{Kc_{5}}{\gamma\sqrt{\tau}}+\frac{2K\overline{\delta}_{0}}{\gamma}.$
(41)
Optimizing for regret involves equating $\frac{\tau c_{0}}{\gamma}$ and
$\frac{Kc_{5}}{\gamma\sqrt{\tau}}$. This yields
$\tau=\left(\frac{Kc_{5}}{c_{0}}\right)^{2/3}$. Further substituting for
$\tau$ yields
$\displaystyle R_{\text{PS}}(K)$
$\displaystyle=\left(\frac{Kc_{3}}{c_{0}}\right)^{2/3}\cdot\frac{c_{0}}{\gamma}+\frac{Kc_{5}c_{0}^{1/3}}{\gamma(Kc_{5})^{1/3}}+\frac{2K\overline{\delta}_{0}}{\gamma}$
(42) $\displaystyle=\frac{(Kc_{5})^{2/3}\cdot
c_{0}^{1/3}}{\gamma}+\frac{(Kc_{5})^{2/3}\cdot
c_{0}^{1/3}}{\gamma}+\frac{2K\overline{\delta}_{0}}{\gamma}$ (43)
$\displaystyle=\frac{2c_{5}^{2/3}\cdot c_{0}^{1/3}\cdot
K^{2/3}}{\gamma}+\frac{2K\overline{\delta}_{0}}{\gamma}.$ (44)
Let $c_{6}=2c_{5}^{2/3}c_{0}^{1/3}$. We have
$\displaystyle
R_{\text{PS}}(K)=\frac{K^{2/3}c_{6}}{\gamma}+\frac{2K\overline{\delta_{0}}}{\gamma}.$
(45)
|
# Improving Mitosis Detection Via UNet-based Adversarial Domain Homogenizer
††thanks: * Indicates equal contribution.
1st Tirupati Saketh Chandra Electrical Engineering
Indian Institute of Technology Bombay
Mumbai, India
0000-0002-0325-5821 2nd Sahar Almahfouz Nasser Electrical Engineering
Indian Institute of Technology Bombay
Mumbai, India
0000-0002-5063-9211 3rd Nikhil Cherian Kurian Electrical Engineering
Indian Institute of Technology Bombay
Mumbai, India
0000-0003-1713-0736 4th Amit Sethi Electrical Engineering
Indian Institute of Technology Bombay
Mumbai, India
0000-0002-8634-1804
###### Abstract
The effective localization of mitosis is a critical precursory task for
deciding tumor prognosis and grade. Automated mitosis detection through deep
learning-oriented image analysis often fails on unseen patient data due to
inherent domain biases. This paper proposes a domain homogenizer for mitosis
detection that attempts to alleviate domain differences in histology images
via adversarial reconstruction of input images. The proposed homogenizer is
based on a U-Net architecture and can effectively reduce domain differences
commonly seen with histology imaging data. We demonstrate our domain
homogenizer’s effectiveness by observing the reduction in domain differences
between the preprocessed images. Using this homogenizer, along with a
subsequent retina-net object detector, we were able to outperform the
baselines of the 2021 MIDOG challenge in terms of average precision of the
detected mitotic figures.
###### Index Terms:
MIDOG, domain generalization, mitosis detection, domain homogenizer, auto-
encoder.
## I Introduction and Related Work
The demand for robust models which perform well on unseen testing data has
increased rationally in the last two years, especially in the medical domain.
In many practical applications of machine learning models domain shift occurs
after training, wherein the characteristics of the test data are different
from the training data. Particularly in the application of deep neural
networks (DNNs) to pathology images, the test data may have different colors,
stain concentration, and magnification compared to what the DNN was trained on
due to changes in scanner, staining reagents, and sample preparation
protocols. MIDOG2021 [20] was the first MICCAI challenge that addressed the
problem of domain shift as it is one of the reasons behind the failure of
machine learning models after training, including those for mitosis detection
techniques when tested on data from a different domain (scanner) than the one
used during the training. MIDOG2021 was the first MICCAI challenge that
addressed the problem of domain shift due to change in scanner [20].
Domain generalization is the set of techniques that improve the prediction
accuracy of machine learning models on data from new domains without assuming
access to those data during training. Proposing and testing various domain
generalization techniques was the main goal of the MIDOG2021 challenge. In
this section, we introduce the most significant notable solutions for the
MIDOG20201 challenge before introducing our proposed method.
In [1] the authors modified RetinaNet network for mitosis detection [2] for
mitosis detection by adding a domain classification head and a gradient
reversal layer to encourage domain agnosticism. In this work, they used a pre-
trained Resnet18 for the encoder. For their discriminator, it was a simple
sequence of three convolutional blocks and a fully connected layer. The domain
classifier was placed at the bottleneck of the encoder. Breen et al [3]
proposed a U-Net type architecture that outputs the probability map of the
mitotic figures. These probabilities get converted into bounding boxes around
the mitotic figures. They used a neural style transfer (NST) as a domain
adaptation technique. This technique casts the style of one image on the
content of another. The method proposed by [4] consists of two parts, a patch
selection and a style transfer module. To learn the styles of images from
different scanners, they used a StarGAN. A two steps domain-invariant mitotic
detection method was proposed by [5]. This method is based on Fast RCNN [6].
For domain generalization purposes they used StainTools software [17] to
augment the images. StainTools package decomposes the image into two matrices,
a concentration matrix C and a stain matrix S. By combining the C and S
matrices from different images they produced the augmented images. A cascaded
pipeline of a Mask RCNN [9] followed by a classification ensemble was proposed
by [7] to detect mitotic candidates. A Cycle GAN [8] was used to transfer
every scanner domain to every other scanner domain. In [10] the authors used a
stain normalization method proposed by [11] as a preprocessing step for the
images. Others like [12] merged Hard negative mining with immense data
augmentation for domain generalization was proposed by [12]. Stain
normalization techniques such as [14] and [15] were used in [13] to account
for the domain difference between images. Almahfouz Nasser et al., [16]
proposed an autoencoder trained adversarially on the sources of domain
variations. This autoencoder makes the appearance of images uniform across
different domains.
In this paper, we present our work which is an extension of our proposed
method for MIDOG2021 [16]. Our contribution is three-fold and can be
summarised in as follows. Firstly, we modified [16] by shifting the domain
classifier from the latent space to the end of the autoencoder, which improved
the results drastically. Secondly, we showed the importance of perceptual loss
in preserving the semantic information which affects the final accuracy of the
object detection part. Finally, unlike our previous work, training the auto-
encoder along with the object detection network end-to-end improved the
quality of the homogenized outputs substantially.
## II Methodology
### II-A Notations
In domain generalization there are source (seen) domains which are shown to
the model during training, and there are some target (unseen) domains which
are used only during testing. Labelled samples from the source domains are
represented by $D_{ls}$=$\\{(x^{ls}_{i},y^{ls}_{i})\\}^{N_{ls}}_{i=1}$,
unlabelled source domains are represented by
$D_{us}$=$\\{(x^{us}_{i})\\}^{N_{us}}_{i=1}$ and labelled target domains are
represented by $D_{lus}$=$\\{(x^{lus}_{i},y^{lus}_{i})\\}^{N_{lus}}_{i=1}$.
Let the unlabelled images from all subsets be represented by
$D_{all}$=$D_{ls}$ $\cup$ $D_{us}$ $\cup$ $D_{lus}$.
### II-B Adversarial end-to-end trainable architecture
Inspired by the work of [18], we have used an encoder-decorder network to
translate the patches from different domains (scanners) to a common space. The
translated images are then passed through RetinaNnet [2] for object detection.
The architecture also consists of an adversarial head with domain
classification as an auxiliary task. This head encourages the encoder-decoder
network to erase all the domain- specific information. The architecture of our
method is as shown in figure 1.
Figure 1: The pipeline of our proposed method for mitosis detection.
### II-C Training Objectives
The object detection loss consists of bounding box loss ($\mathcal{L}_{bb}$)
and instance classification loss ($\mathcal{L}_{inst}$). The bounding box loss
($\mathcal{L}_{bb}$) is computed as smooth L1 loss and the focal loss function
is used for the instance classification ($\mathcal{L}_{inst}$).
In order to ensure that the images translated by encoder-decoder network
contains the semantic information we have used a perceptual loss
($\mathcal{L}_{percp}$). We have used the perceptual loss based on pretrained
VGG-16, which is proposed in [19].
At the end of the adversarial head we have used standard cross entropy loss
($\mathcal{L}_{CE}$) for domain classification.
The overall loss for the end-to-end training is given by,
$\mathcal{L}=\mathcal{L}_{bb}+\mathcal{L}_{inst}+\lambda_{1}\mathcal{L}_{percep}+\lambda_{2}\mathcal{L}_{CE}$
(1)
## III Data and Experiments
### III-A Dataset
The experiments were conducted on MIDOG 2021 dataset [20] which consists of 50
whole slide images of breast cancer from four scanners namely Hamamatsu XR
NanoZoomer 2.0, Hamamatsu S360, Aperio ScanScope CS2, and Leica GT450 forming
four domains. Two classes of objects are to be detected namely mitotic figures
and hard negatives. The whole slide images from scanners other than the Leica
GT450 areis labelled. Small patches of size 512 x 512 are mined for supervised
end-to-end training such that the cells belonging to at least one of the
mitotic figures or hard negatives are present in the patch.
The seen and unseen domains i.e., the scanners are $D_{ls}$={Hamamatsu XR
NanoZoomer 2.0, the Hamamatsu S360}, $D_{us}$={Leica GT450}, $D_{lus}$={Aperio
ScanScope CS2} (refer II-A for notations.)
### III-B Implementation Details
The model is implemented using Pytorch [21] library [21]. For supervised end-
to-end training a batch size of 12 is used with equal number of patches being
included from each scanner. Here the model is trained using FastAI [22]
library default settings with an initial learning rate of $1e^{-4}$. In the
equation 1 we have set the values of hyperparameters as $\lambda_{1}$=10 and
$\lambda_{2}$=25.
### III-C Results
Two classes of objects – hard negatives, and mitotic figures – are detected.
The models are evaluated on $D_{ls}\cup D_{lus}$. Average precision at IoU
threshold of 0.5 is used as metric for evaluation. End-to-end training with
(AEC_RetinaNet + Pecp) and without using perceptual loss (AEC_RetinaNet) were
tried. The results are compared with the reference algorithm DA_RetinaNet
[23], RetinaNet [24] with and without data augmentation. The results obtained
are as shown in table I.
TABLE I: Results obtained using end -to -end training of models Model | AP-Hard Neg | AP-Mitotic figures | mAP
---|---|---|---
RetinaNet | 0.196 | 0.352 | 0.274
RetinaNet + Aug | 0.238 | 0.619 | 0.429
DA_RetinaNet | 0.347 | 0.655 | 0.501
AEC_RetinaNet | 0.289 | 0.448 | 0.369
AEC_RetinaNet + Pecp | 0.248 | 0.72 | 0.484
The count of mitotic figures is an important clinical goal. So, the
performance on the class of mitotic figures was our focused. The results in
the above table show that the newly designed end-to-end training architectures
performs better than the reference algorithm and the basic Retina-Net based
algorithms. The best performance is in terms of recall metric, which is
evident from the plot shown in figure 2.
The perceptual loss added at the output of the decoder helps in retaining the
semantic information. This information which helps in better object detection.
This is also validated by higher AP score obtained when perceptual loss
component is added.
As shown in figure 3 the modified domain homogenizer produced much more
plausible images than the original domain homogenizer [16].
Besides, figure 4 shows the detection accuracy of our proposed method.
Figure 2: The precision vs recall plot shows that the newly designed end-to-
end model performs better than the baselines especially for the recall values.
## IV Conclusions
In this paper, we proposed a modified version of our previous domain
homogenizer proposed by us and tested it on the data from for the MIDOG 2021
challenge 2021. We showed that the great impact of the position of the domain
classifier has a significant impact on the performance of the homogenizer.
[EXPLAIN HOW AND WHY] Additionally, our experiments revealed that training the
homogenizer along with the object detection network end-to-end improves the
detection accuracy by a significant margin. Finally, we showed that our method
substantially improves upon the baseline of the MIDOG challenge in terms of
mitotic figures detection.
Figure 3: A visual comparison of the performances of the domain homogenizer
and the modified domain homogenizer(proposed method). Figure 4: Two examples
show the detection accuracy of our proposed method.
## References
* [1] Wilm, Frauke, et al. ”Domain Adversarial RetinaNet as a Reference Algorithm for the MItosis DOmain Generalization Challenge.” International Conference on Medical Image Computing and Computer-Assisted Intervention. Springer, Cham, 2021.
* [2] Marzahl, Christian, et al. ”Deep learning-based quantification of pulmonary hemosiderophages in cytology slides.” Scientific reports 10.1 (2020): 1-10.
* [3] Breen, Jack, et al. ”Assessing domain adaptation techniques for mitosis detection in multi-scanner breast cancer histopathology images.” International Conference on Medical Image Computing and Computer-Assisted Intervention. Springer, Cham, 2021.
* [4] Chung, Youjin, Jihoon Cho, and Jinah Park. ”Domain-Robust Mitotic Figure Detection with Style Transfer.” International Conference on Medical Image Computing and Computer-Assisted Intervention. Springer, Cham, 2021.
* [5] Nateghi, Ramin, and Fattaneh Pourakpour. ”Two-Step Domain Adaptation for Mitotic Cell Detection in Histopathology Images.” International Conference on Medical Image Computing and Computer-Assisted Intervention. Springer, Cham, 2021.
* [6] Girshick, Ross. ”Fast r-cnn.” Proceedings of the IEEE international conference on computer vision. 2015.
* [7] Fick, Rutger HJ, et al. ”Domain-Specific Cycle-GAN Augmentation Improves Domain Generalizability for Mitosis Detection.” International Conference on Medical Image Computing and Computer-Assisted Intervention. Springer, Cham, 2021.
* [8] Zhu, Jun-Yan, et al. ”Unpaired image-to-image translation using cycle-consistent adversarial networks.” Proceedings of the IEEE international conference on computer vision. 2017.
* [9] He, Kaiming, et al. ”Mask r-cnn.” Proceedings of the IEEE international conference on computer vision. 2017.
* [10] Jahanifar, Mostafa, et al. ”Stain-robust mitotic figure detection for the Mitosis Domain Generalization Challenge.” International Conference on Medical Image Computing and Computer-Assisted Intervention. Springer, Cham, 2021.
* [11] Vahadane, Abhishek, et al. ”Structure-preserving color normalization and sparse stain separation for histological images.” IEEE transactions on medical imaging 35.8 (2016): 1962-1971.
* [12] Dexl, Jakob, et al. ”MitoDet: Simple and robust mitosis detection.” International Conference on Medical Image Computing and Computer-Assisted Intervention. Springer, Cham, 2021.
* [13] Long, Xi, et al. ”Domain Adaptive Cascade R-CNN for MItosis DOmain Generalization (MIDOG) Challenge.” International Conference on Medical Image Computing and Computer-Assisted Intervention. Springer, Cham, 2021.
* [14] Reinhard, Erik, et al. ”Color transfer between images.” IEEE Computer graphics and applications 21.5 (2001): 34-41.
* [15] Vahadane, Abhishek, et al. ”Structure-preserved color normalization for histological images.” 2015 IEEE 12th International Symposium on Biomedical Imaging (ISBI). IEEE, 2015.
* [16] Almahfouz Nasser, Sahar, Nikhil Cherian Kurian, and Amit Sethi. ”Domain Generalisation for Mitosis Detection Exploting Preprocessing Homogenizers.” International Conference on Medical Image Computing and Computer-Assisted Intervention. Springer, Cham, 2021.
* [17] StainTools Homepage. https://github.com/Peter554/StainTools. Accessed 03 Aug 2022
* [18] Yaroslav Ganin and Victor Lempitsky. “Unsupervised Domain Adaptation by Backpropagation”. In: Proceedings of the 32nd International Conference on Machine Learning. Ed. by Francis Bach and David Blei. Vol. 37. Proceedings of Machine Learning Research. Lille, France: PMLR, July 2015, pp. 1180–1189. url: https://proceedings.mlr.press/v37/ganin15.html
* [19] Justin Johnson, Alexandre Alahi, and Li Fei-Fei. “Perceptual losses for real-time style transfer and super-resolution”. In: European Conference on Computer Vision. 2016
* [20] Marc Aubreville et al. MItosis DOmain Generalization Challenge. Mar. 2021. doi: 10.5281/zenodo.4573978. url: https://doi.org/10.5281/zenodo.4573978
* [21] Adam Paszke et al. “PyTorch: An Imperative Style, High-Performance Deep Learning Library”. In: Advances in Neural Information Processing Systems. Ed. by H. Wallach et al. Vol. 32. Curran Associates, Inc., 2019. url: https://proceedings.neurips.cc/paper/2019/file/bdbca288fee7f92f2bfa9f7012727740-Paper.pdf
* [22] Jeremy Howard et al. fastai. https://github.com/fastai/fastai. 2018
* [23] Frauke Wilm et al. “Domain Adversarial RetinaNet as a Reference Algorithm for the MItosis DOmain Generalization Challenge”. In: Biomedical Image Registration, Domain Generalisation and Out-of-Distribution Analysis. Ed. by Marc Aubreville, David Zimmerer, and Mattias Heinrich. Cham: Springer International Publishing, 2022, pp. 5–13. isbn: 978-3-030-97281-3.
* [24] Tsung-Yi Lin et al. “Focal Loss for Dense Object Detection”. In: 2017 IEEE International Conference on Computer Vision (ICCV). 2017, pp. 2999–3007. doi: 10.1109/ICCV.2017.324.
|
# Progress on Local Properties Problems of Difference Sets††thanks: This
research project was done as part of the 2020 NYC Discrete Math REU, supported
by NSF awards DMS-1802059, DMS-1851420, DMS-1953141, and DMS-2028892.
Anqi Li Massachusetts Institute of Technology, 77 Massachusetts Ave,
Cambridge, MA 02139<EMAIL_ADDRESS>Supported by MIT Department of Science.
###### Abstract
We derive several new bounds for the problem of difference sets with local
properties, such as establishing the super-linear threshold of the problem.
For our proofs, we develop several new tools, including a variant of higher
moment energies and a Ramsey-theoretic approach for the problem.
## 1 Introduction
Erdős and Shelah [3] initiated the study of graphs with local properties. For
parameters $n,k,$ and $\ell$, they considered edge colorings of the complete
graph $K_{n}$, such that every $K_{k}$ subgraph contains at least $\ell$
colors. They defined $f(n,k,\ell)$ as the minimum number of colors that such
an edge coloring can have. For example, $f(n,3,2)$ is the minimum number of
colors that is required to color $K_{n}$, so that it does not contain a
monochromatic triangle.
The value of $f(n,k,2)$ can be thought of as an inverse of Ramsey’s theorem.
In Ramsey’s theorem, we look for the largest $n$ such that there exists a
coloring of the edges of $K_{n}$ with $c$ colors and no monochromatic $K_{k}$.
When studying $f(n,k,2)$, we are given the number of vertices $n$, and instead
look for the number of colors $c$.
By definition, $1\leq f(n,k,\ell)\leq\binom{n}{2}$. We usually consider $n$ to
be asymptotically large, while $k$ and $\ell$ are fixed constants. For
asymptotically large $n$ and a fixed $k$, the _linear threshold_ is the
smallest $\ell$ for which $f(n,k,\ell)=\Omega(n)$. This value of $\ell$ is
expressed as a function of $k$. Similarly, the _quadratic threshold_ is the
smallest $\ell$ that satisfies $f(n,k,\ell)=\Theta(n^{2})$. The _polynomial
threshold_ is the smallest $\ell$ that satisfies
$f(n,k,\ell)=\Omega(n^{\varepsilon})$ for some ${\varepsilon}>0$.
Erdős and Gyárfás [4] identified the linear and quadratic thresholds to be
$\ell=\binom{k}{2}-k+3$ and $\ell=\binom{k}{2}-\lfloor k/2\rfloor+2$,
respectively. In the same paper, they also observed that $f(n,k,k)$ is
polynomial in $n$. Conlon, Fox, Lee and Sudakov [2] proved that $f(n,k,k-1)$
is sub-polynomial in $n$, which established the polynomial threshold $\ell=k$.
Beyond these thresholds, almost tight bounds for $f(n,k,\ell)$ are known in
other regimes. Particularly, Pohoata and Sheffer [9] showed that, for any
integers $k>m\geq 2$,
$f\left(n,k,\binom{k}{2}-m\cdot\left\lfloor\frac{k}{m+1}\right\rfloor+m+1\right)=\Omega\left(n^{1+\frac{1}{m}}\right).$
(1)
Probabilistic constructions of Erdős and Gyárfás [4] show that (1) is
asymptotically tight up to sub-polynomial factors.
Zeev Dvir recently suggested an additive combinatorics variant of the local
properties problem. As stated in [4, Section 9], Erdős and Sós also studied
this problem, but did not publish their results. We define the _difference
set_ of a set $A\subset\mathbb{R}$ as
$A-A=\\{a-a^{\prime}\ :\ a,a^{\prime}\in A\ \text{ and }\ a-a^{\prime}>0\\}.$
The standard definition of a difference set does not include the condition
$a-a^{\prime}>0$. However, this less common definition makes the following
local properties problem nicer to study.
For parameters $n,k,$ and $\ell$, we consider sets $A$ of $n$ real numbers,
such that every subset $A^{\prime}\subset A$ of size $k$ satisfies that
$|A^{\prime}-A^{\prime}|\geq\ell$. Let $g(n,k,\ell)$ be the minimum size of
$\lvert A-A\rvert$, over all such sets $A$. In the current work, we study
$g(n,k,\ell)$. The trivial bounds for this problem are $n-1\leq
g(n,k,\ell)\leq\binom{n}{2}$. As before, for a fixed $k$, the quadratic
threshold is the smallest $\ell$ for which $g(n,k,\ell)=\Theta(n^{2})$. We
define the _super-linear threshold_ as the largest $\ell$ for which
$g(n,k,\ell)=O(n)$.
Previous bounds. Currently, not much is known about the behavior of
$g(n,k,\ell)$. The unpublished work of Erdős and Sós proved that
$g(n,4,5)\geq\binom{n}{2}-n+2$. We also have the bound $g(n,k,\ell)\geq
f(n,k,\ell)$, for every $n,k,$ and $\ell$. For example, the bounds of (1) also
hold after replacing $f(\cdot)$ with $g(\cdot)$. Indeed, we can construct a
complete graph with a vertex for every element of $A$. The color of an edge
corresponds to the difference of the numbers that correspond to the two
vertices. If every subset $A^{\prime}\subset A$ of size $k$ satisfies that
$|A^{\prime}-A^{\prime}|\geq\ell$, then every subgraph $K_{k}$ spans at least
$\ell$ colors. Similarly, the total number of colors is $|A-A|$.
Fish, Pohoata, and Sheffer [6] observed that Roth’s theorem implies the
following bound.
###### Theorem 1.1.
For every $\epsilon>0$, there exists $c$ that satisfies the following. For
every sufficiently large integer $k$, we have that
$g\left(n,k,c\cdot k\cdot\log^{1/4-\epsilon}k\right)=n2^{O(\sqrt{\log n})}.$
Fish, Lund, and Sheffer [5] obtained the following result by projecting a
hypercube onto a line.
###### Theorem 1.2.
For every $n>k\geq 0$ where $n$ is a power of two, we have that
$g\left(n,k,\frac{k^{\log_{2}(3)}-1}{2}\right)=O\left(n^{\log_{2}(3)}\right).$
Our results. We derive several new bounds for $g(n,k,\ell)$. First, we
identify the super-linear threshold of $g(n,k,\ell)$. Recall that
$\omega(\cdot)$ means “asymptotically strictly larger than”.
###### Theorem 1.3.
For every $k$ and sufficiently large $n$, we have that
$\displaystyle g(n,k,k-1)=n-1\qquad\text{ and }\qquad g(n,k,k)=\omega(n).$
After establishing the super-linear threshold, it is natural to ask how
quickly the bound changes as $\ell$ grows. We obtain the following bound via a
probabilistic argument.
###### Theorem 1.4.
For every $c\geq 2$ and integer $k>(c^{2}+1)^{2}$, we have that
$g(n,k,ck+1)=O\Big{(}n^{1+\frac{c^{2}+1}{k}}\Big{)}.$
When fixing $c\geq 2$, the bound of Theorem 1.4 becomes arbitrarily close to
$O(n)$ as $k$ grows. We note that the assumption $k>(c^{2}+1)^{2}$ can be
removed from the statement of this theorem. Indeed, when $k\leq(c^{2}+1)^{2}$,
the bound of the theorem is trivial.
Additive combinatorics provides many tools for studying sets that have a small
difference set. We rely on such tools to study the case of small $\ell$, in
the proofs of Theorems 1.3 and 1.4. To study sets with larger values of
$\ell$, we have to develop new tools. In particular, we obtain the following
result by studying a new variant of higher moment energies. For the definition
of higher moment energies and of our new variant, see Section 2.
###### Theorem 1.5.
Every $k\geq 8$ that is a multiple of 8 satisfies
$g\left(n,k,\frac{9k^{2}}{32}-\frac{9k}{8}+5\right)=\Omega(n^{4/3}).$
By (1), for the graphs local properties problem, we have that
$f\left(n,k,\binom{k}{2}-3\cdot\left\lfloor\frac{k}{4}\right\rfloor+4\right)=\Omega\left(n^{4/3}\right).$
This bound is tight up to subpolynomial factors. Theorem 1.5 shows that, in
the additive combinatorics variant, a much weaker local restriction leads to
$\Omega(n^{4/3})$, up to subpolynomial factors. Unlike the graph variant, in
this case the coefficient of $k^{2}$ is smaller than $1/2$.
Theorem 1.2 provides an upper bound for the local properties problem by
introducing a construction. Surprisingly, we show that the same construction
can be used to obtain a proof for a lower bound.
###### Theorem 1.6.
For every $k\geq 8$ that is a power of two and sufficiently large $n$, we have
that
$g\left(n,k,\frac{k^{\log_{2}(3)}+1}{2}\right)=\Omega\left(n^{1+\frac{2}{k-2}}\right).$
The proof of Theorem 1.6 is in some sense Ramsey theoretic: We show that a set
with many repeated differences must contain a copy of the construction from
Theorem 1.2. Such a subset contradicts the local property, and thus cannot
exist.
Finally, we mention the best bound that we managed to obtain for the quadratic
threshold.
###### Claim 1.7.
When $k$ is a multiple of two, we have that
$g\left(n,k,\frac{3k^{2}}{8}-\frac{3k}{4}+2\right)=\Omega(n^{2}).$
It is not difficult to prove Claim 1.7, and it is possible that the quadratic
threshold is much smaller. We include Claim 1.7 for completeness. That is, to
document all the current best bounds for the arithmetic variant of the local
properties problem.
Lastly, we note that while most results in this paper apply to the setting of
a general Abelian group, we decided to not pursue this direction and leave
this to future work.
Outline. In Section 2, we define our new energy variant and use it to prove
Theorem 1.5. In Section 3, we prove our other results for large values of
$\ell$: Theorem 1.6 and Claim 1.7. In Section 4, we prove our bounds for small
values of $\ell$: Theorem 1.3 and Theorem 1.4. In Section 5, we pose some
questions for future work.
Acknowledgements. We would like to thank MIT Department of Science for the
generous support. We would also like to express our gratitude towards Adam
Sheffer for his mentorship and support throughout the NYC Discrete Math REU,
as well as his invaluable help in the preparation of this paper. In addition,
we would like to thank Sara Fish for many helpful discussions as well as Ilya
Shkredov for his enlightening suggestion that led to an improvement in the
bound of Theorem 1.5. Lastly, we thank the referees for their many useful
comments and suggestions.
## 2 Dumbbell energy and Theorem 1.5
In this section, we prove Theorem 1.5. Our proof relies on a new energy
variant. Before we introduce this energy, we briefly recall the notion of
higher moment energies.
The _additive energy_ of a finite set $A\subset\mathbb{R}$ is
$E^{+}(A)=\lvert\\{(a_{1},a_{2},a_{3},a_{4})\in A^{4}\ :\
a_{1}-a_{2}=a_{3}-a_{4}\\}\rvert.$
Additive energy is a common tool in additive combinatorics. For example, it
provides the following lower bound for the size of the difference set:
$|A-A|\geq\frac{|A|^{4}}{E^{+}(A)}.$
In the past decade, several variants of additive energy led to interesting new
results. One of those variants is _higher moment energies_. For an integer
$\ell>1$, the $\ell$th moment energy of $A$ is
$E_{\ell}^{+}(A)=\lvert\\{(a_{1},\ldots,a_{2\ell})\in A^{2\ell}\ :\
a_{1}-a_{2}=a_{3}-a_{4}=\cdots=a_{2\ell-1}-a_{2\ell}\\}\rvert.$
We note that $E_{2}^{+}(A)$ is the standard additive energy $E^{+}(A)$. For
applications of higher moment energies, see for example [11, 12].
Figure 1: (a) A dumbbell. (b) An arithmetic dumbbell.
We now define a new variant of additive energy, which we call _dumbbell
energy_. We say that a quadruple $(a_{1},a_{2},a_{3},a_{4})\in\mathbb{R}$
forms a _dumbbell_ if $a_{1}<a_{2}<a_{3}<a_{4}$ and $a_{2}-a_{1}=a_{4}-a_{3}$.
See Figure 1. For distances $d,d^{\prime}>0$, we define $D(d,d^{\prime})$ as
the congruence class of all dumbbells $(a_{1},a_{2},a_{3},a_{4})$ that satisfy
$a_{2}-a_{1}=a_{4}-a_{3}=d$ and $a_{3}-a_{2}=d^{\prime}$. That is, two
dumbbells are congruent if there exists a translation of $\mathbb{R}$ that
takes one dumbbell to the other.
We can think of the $\ell$th additive energy as the number of $\ell$-tuples of
congruent intervals. Similarly, we consider an energy variant that counts
$\ell$-tuples of congruent dumbbells. We define the $\ell$th _dumbbell energy_
of $A\subset\mathbb{R}$ as
$\displaystyle E^{D}_{\ell}(A)=\Big{|}\Big{\\{}(a_{1},\ldots,a_{4\ell})\in
A^{4\ell}\ :\ $ $\displaystyle\text{ For }1\leq i\leq\ell,\text{ the
quadruples }(a_{4i-3},a_{4i-2},a_{4i-1},a_{4i})$ $\displaystyle\text{ form
}\ell\text{ distinct dumbbells from the same congruence class
}\Big{\\}}\Big{|}.$
For $d,d^{\prime}\in A-A$, we set
$r^{D}_{A}(d,d^{\prime})=|\\{(a_{1},a_{2},a_{3},a_{4})\in A^{4}\ :\
a_{2}-a_{1}=a_{4}-b_{3}=d\text{ and }a_{3}-a_{2}=d^{\prime}\\}|.$
In other words, $r^{D}_{A}(d,d^{\prime})$ is the number of dumbbells in the
congruence class $D(d,d^{\prime})$ that are spanned by $A$. We note that a
dumbbell $(a_{1},a_{2},a_{3},a_{4})$ of $D(d,d^{\prime})$ is uniquely
determined by $a_{1}$. Thus, $r^{D}_{A}(d,d^{\prime})\leq|A|$ for all
$d,d^{\prime}\in A-A$. We also note that
$E^{D}_{\ell}(A)=\sum_{d,d^{\prime}\in
A-A}\binom{r^{D}_{A}(d,d^{\prime})}{\ell}.$
For a difference $d\in A-A$, we set
$r^{-}_{A}(d)=|\\{(a,a^{\prime})\in A^{2}\ :\ a-a^{\prime}=d\\}|.$
In other words, $r^{-}_{A}(d)$ is the number of representations that the
difference $d$ has in $A$.
The following lemma states a connection between the $\ell$th dumbbell energy
$E^{D}_{\ell}(A)$ and the size of the difference set $|A-A|$.
###### Lemma 2.1.
The following holds for every even constant $\ell\geq 2$. Every set $A$ of $n$
reals satisfies that $|A-A|=\Omega\left(n^{4/3}\right)$ or that
$E^{D}_{\ell}(A)=\Omega\left(\frac{n^{4\ell}}{|A-A|^{3\ell-2}}\right).$
###### Proof.
As a first step, we will prove that
$\sum_{d,d^{\prime}\in
A-A}r^{D}_{A}(d,d^{\prime})^{2}=\Omega\left(\frac{n^{8}}{|A-A|^{4}}\right).$
Let
$\mathbf{1}_{B}(x)=\begin{cases}1&\text{if }x\in B,\\\
0&\text{otherwise},\end{cases}$
be the indicator function for a set $B\subset\mathbb{R}$. We may write
$S:=\sum_{d,d^{\prime}\in A-A}r_{A}^{D}(d,d^{\prime})=\sum_{a\in
A}\sum_{d,d^{\prime}\in
A-A}\mathbf{1}_{A\cap(A-d)}(a)\mathbf{1}_{A\cap(A-d)}(a+d^{\prime})=\sum_{a\in
A}\sum_{d\in A-A}\mathbf{1}_{A\cap(A-d)}(a)r_{A}^{-}(d).$
Note that $\sum_{d\in A-A}r_{A}^{-}(d)=\sum_{a\in A}\sum_{d\in
A-A}\mathbf{1}_{A\cap(A-d)}(a)=\binom{|A|}{2}=\binom{n}{2}$.
Consider $\mathcal{D}=\left\\{d:|r_{A}^{-}(d)|\geq
S/4\cdot\binom{n}{2}^{-1}\right\\}$. It is not difficult to see that
$\mathcal{D}$ captures a large fraction of the above sum: since
$\sum_{a\in
A}\sum_{d\not\in\mathcal{D}}\mathbf{1}_{A\cap(A-d)}(a)r_{A}^{-}(d)<\max_{d\not\in\mathcal{D}}r_{A}^{-}(d)\cdot\binom{n}{2}=\frac{S}{4},$
it follows that
$\sum_{a\in
A}\sum_{d\in\mathcal{D}}\mathbf{1}_{A\cap(A-d)}(a)r_{A}^{-}(d)\geq\frac{3S}{4}.$
We also have the following bound
$|\mathcal{D}|\cdot S/4\cdot\binom{n}{2}^{-1}\leq\sum_{d\in
A-A}r_{A}^{-}(d)=\binom{n}{2}.$
That is, $|\mathcal{D}|\leq 4/S\cdot\binom{n}{2}^{2}$.
Let $\mathcal{B}\subset(A-A)\times(A-A)\subset\mathbb{R}^{2}$ be the set which
projects onto a copy of $\mathcal{D}$ on each coordinate axes. By the symmetry
in $d,d^{\prime}$, the above inequality implies that
$\sum_{d,d^{\prime}\in\mathcal{B}}r_{A}^{D}(d,d^{\prime})\geq\frac{3S}{4}.$
Furthermore, the Loomis-Whitney inequality gives the upper bound
$|\mathcal{B}|\leq|\mathcal{D}|^{2}\leq 16/S^{2}\cdot\binom{n}{2}^{4}$.
Consequently, the Cauchy-Schwarz inequality implies that
$\frac{9S^{2}}{16}\leq\left(\sum_{d,d^{\prime}\in\mathcal{B}}r_{A}^{D}(d,d^{\prime})\right)^{2}\leq|\mathcal{B}|\sum_{d,d^{\prime}\in\mathcal{B}}r_{A}^{D}(d,d^{\prime})^{2}\leq
16/S^{2}\cdot\binom{n}{2}^{4}\sum_{d,d^{\prime}\in
A}r_{A}^{D}(d,d^{\prime})^{2}.$
Equivalently,
$\sum_{d,d^{\prime}\in
A-A}r^{D}_{A}(d,d^{\prime})^{2}\geq\frac{9S^{4}}{256\cdot\binom{n}{2}^{4}}.$
(2)
By definition, we have that
$\displaystyle S=\sum_{d,d^{\prime}\in A-A}r^{D}_{A}(d,d^{\prime})=\sum_{d\in
A-A}\binom{r^{D}_{A}(d)}{2}$ $\displaystyle=\sum_{d\in
A-A}\frac{r^{D}_{A}(d)^{2}}{2}-\sum_{d\in A-A}\frac{r^{D}_{A}(d)}{2}$
$\displaystyle=\sum_{d\in
A-A}\frac{r^{D}_{A}(d)^{2}}{2}-\frac{1}{2}\binom{n}{2}.$
Combining the above with the Cauchy-Schwarz inequality leads to
$\sum_{d,d^{\prime}\in A-A}r^{D}_{A}(d,d^{\prime})\geq\frac{(\sum_{d\in
A-A}r^{D}_{A}(d))^{2}}{2|A-A|}-\frac{1}{2}\binom{n}{2}=\binom{n}{2}^{2}/2|A-A|-\frac{1}{2}\binom{n}{2}.$
If $|A-A|=\Theta(n^{2})$ then we are done. We may thus assume that
$S=\Omega\left(\frac{n^{4}}{|A-A|}\right).$ (3)
Combining this Equation 2, we get the desired lower bound
$\sum_{d,d^{\prime}\in
A-A}r^{D}_{A}(d,d^{\prime})^{2}=\Omega\left(\frac{n^{8}}{|A-A|^{4}}\right).$
Lastly, by Hölder’s inequality and since $\ell$ is constant, we obtain that
$\displaystyle E_{\ell}^{D}(A)$ $\displaystyle=\sum_{d,d^{\prime}\in
A-A}\binom{r_{A}^{D}(d,d^{\prime})}{\ell}\geq\sum_{d,d^{\prime}\in
A-A}\left(\frac{r_{A}^{D}(d,d^{\prime})}{\ell}\right)^{\ell}\geq\frac{(\sum_{d,d^{\prime}\in
A-A}r_{A}^{D}(d,d^{\prime})^{2})^{\ell/2}}{|A-A|^{2(\ell/2-1)}\ell^{\ell}}$
$\displaystyle=\Omega\left(\frac{(n^{8}/|A-A|^{4})^{\ell/2}}{|A-A|^{\ell-2}\ell^{\ell}}\right)=\Omega\left(\frac{n^{4\ell}}{|A-A|^{3\ell-2}}\right).$
∎
We next introduce a variant of the energy graph that is defined in [6]. For a
finite $A\subset\mathbb{R}$ and $\ell$ even, we define the $\ell$th _dumbell
energy graph_ $G$ of $A$, as follows. The graph $G$ has a vertex for every
$2\ell$-tuple from $A^{2\ell}$. There is an edge between the vertices
$(a_{1},\ldots,a_{2\ell})$ and $(b_{1},\ldots,b_{2\ell})$ if the tuple
$(a_{1},\ldots,a_{2\ell},b_{1},\ldots,b_{2\ell})$ contributes to
$E^{D}_{\ell}$. Equivalently, if each vertex consists of $\ell/2$ dumbbells
and all $\ell$ dumbbells are congruent. By definition, the number of edges in
$G$ is $E_{\ell}^{D}(A)$.
We are now ready to prove Theorem 1.5. We first recall the statement of this
theorem.
Theorem 1.5. _Every $k\geq 8$ that is a multiple of 8 satisfies_
$g\left(n,k,\frac{9k^{2}}{32}-\frac{9k}{8}+5\right)=\Omega(n^{4/3}).$
###### Proof.
Let $A$ be a set of $n$ reals, such that every subset $A^{\prime}\subset A$ of
size $k$ satisfies that $|A^{\prime}-A^{\prime}|\geq\frac{5k^{2}}{16}-k+5$.
Let $G$ be the $k/4$ dumbbell energy graph of $A$. Consider three vertices
$v_{1},v_{2},v_{3}$ of $G$, such that there is an edge between $v_{1}$ and
$v_{2}$ and another edge between $v_{2}$ and $v_{3}$. Then each of the three
vertices consists of $k/8$ dumbbells, and all $3k/8$ dumbbells are congruent.
Thus, there is also an edge between $v_{1}$ and $v_{3}$. We conclude that
every connected component of $G$ is a clique.
We may discard connected components with a single vertex without changing the
number of edges. Every remaining connected component corresponds to a distinct
dumbbell congruency class. Since there are $O(|A-A|^{2})$ distinct (non-
congruent) dumbbells, the number of remaining connected components in $G$ is
$O(|A-A|^{2})$.
Assume for contradiction that $A$ spans $k/4$ disjoint congruent dumbbells.
Let $A^{\prime}\subset A$ be the set of $k$ numbers that form these $k/4$
dumbbells. At most four distinct differences are spanned by pairs of numbers
from the same dumbbell. In Figure 1(b), these distances are
$d,d^{\prime},d+d^{\prime}$, and $2d+d^{\prime}$.
Figure 2: The nine distances between the two dumbbells are
$d_{3},d_{1}+d_{3},2d_{1}+d_{3},d_{1}+d_{2}+d_{3},2d_{1}+d_{2}+d_{3},3d_{1}+d_{2}+d_{3},2d_{1}+2d_{2}+d_{3},3d_{1}+2d_{2}+d_{3},4d_{1}+2d_{2}+d_{3}$.
There are $\binom{k/4}{2}$ pairs of distinct dumbbells as described above.
Each such pair spans at most 9 distinct differences with one number from each
dumbbell. See Figure 2. We conclude that
$|A^{\prime}-A^{\prime}|\leq 4+\binom{k/4}{2}\cdot
9=\frac{9k^{2}}{32}-\frac{9k}{8}+4.$
This contradicts the assumption, so $A$ does not span $k/4$ disjoint congruent
dumbbells.
Let $S\subset A$ be a dumbbell. We wish to derive an upper bound for the
number of dumbbells $S^{\prime}$ that are congruent to $S$ but are not
disjoint to it. There are at most four choices for the element of $S$ that
also appears in $S^{\prime}$. Then, there are at most three possible positions
for this element in $S^{\prime}$. Since $S^{\prime}$ is congruent to $S$, the
other three coordinates of $S^{\prime}$ are uniquely determined. Thus, at most
12 dumbbells are congruent to $S$ and not disjoint to it.
By combining the two preceding paragraphs, we conclude that $A$ cannot span
$3k$ congruent dumbbells. For a fixed dumbbell congruence class, every vertex
in the corresponding connected component of $G$ consists of $k/8$ copies of
that dumbbell, possibly with repetitions. Since there are fewer than $3k$ such
dumbbells to choose from, this connected component consists of fewer than
$(3k)^{k/8}$ vertices. The number of edges in the connected component is
smaller than $(3k)^{k/4}$. This holds for every connected component of $G$.
Recalling that the number of remaining connected components in $G$ is at most
$|A-A|^{2}$, we conclude that the number of edges in $G$ is
$O_{k}(|A-A|^{2})$.
Since the number of edge in $G$ is $E_{k/4}^{D}(A)$, we have that
$E_{k/4}^{D}(A)=O_{k}(|A-A|^{2})$. By Lemma 2.1, we have that
$E^{D}_{k/4}(A)=\Omega\left(\frac{n^{k}}{|A-A|^{3k/4-2}}\right).$
Combining the two above bounds for $E^{D}_{k/4}(A)$ implies that
$|A-A|=\Omega(n^{4/3})$. ∎
###### Remark 2.2.
When performing the above analysis with $E_{2}^{D}(A)$ instead of
$E^{D}_{k/4}(A)$, we obtain the weaker bound $|A-A|=\Omega(n^{8/7})$ with the
same local condition.
###### Remark 2.3.
We say that an 8-tuple
$(a_{1},a_{2},a_{3},a_{4},a_{5},a_{6},a_{7},a_{8})\in\mathbb{R}^{8}$ forms a
_double dumbbell_ if $a_{1}<a_{2}<a_{3}<a_{4}<a_{5}<a_{6}<a_{7}<a_{8}$ and
$a_{2}-a_{1}=a_{4}-a_{3}=a_{6}-a_{5}=a_{8}-a_{7}$ and
$a_{3}-a_{2}=a_{7}-a_{6}$. A natural generalization of the dumbbell energy
studied above is the $\ell$th _double dumbbell energy_ , defined for
$A\subset\mathbb{R}$ as
$\displaystyle\mathcal{E}^{D}_{\ell}(A)=\Big{|}\Big{\\{}(a_{1},\ldots,a_{8\ell})\in
A^{8\ell}\ :\ $ $\displaystyle\text{ For }1\leq i\leq\ell,\text{ the
quadruples }(a_{8i-7},a_{8i-6},\ldots,,a_{8i})$ $\displaystyle\text{ form
}\ell\text{ distinct double dumbbells from the same congruence class
}\Big{\\}}\Big{|}.$
As before, two double dumbbells are congruent if there exists a translation of
$\mathbb{R}$ that takes one double dumbbell to the other.
An identical argument as that for Theorem 1.5 gives the bound
$g\left(n,k,\frac{31k^{2}}{128}-\frac{31k}{16}+9\right)=\Omega(n^{8/7})$
for every $k$ a multiple of 16. Here we have slightly weaker local condition
than that in Theorem 1.5 at the expense of a worse exponent in
$\Omega\left(n^{8/7}\right)$. We chose to present Theorem 1.5 because it is
less calculation intensive and better illustrates the main ideas of the proof.
## 3 Additional proofs for large values of $\ell$
In this section, we prove our bounds for large values of $\ell$ that are not
related to dumbbell energy. That is, we prove Theorem Theorem 1.6 and Claim
1.7.
We begin with the proof of Claim 1.7, since this simple proof is a nice warm-
up. The proof relies on the observation that $a-b=c-d$ implies $a-c=b-d$.
Recall that for a set $A$ and a difference $d\in A-A$ we set
$r^{-}_{A}(d)=|\\{(a,a^{\prime})\in A^{2}\ :\ a-a^{\prime}=d\\}|.$
Claim 1.7. _When $k$ is a multiple of two, we have that_
$g\left(n,k,\frac{3k^{2}}{8}-\frac{3k}{4}+2\right)=\Omega(n^{2}).$
###### Proof.
Let $A$ be a set of $n$ reals, such that every subset $A^{\prime}\subset A$ of
size $k$ satisfies that
$|A^{\prime}-A^{\prime}|\geq\frac{3k^{2}}{8}-\frac{3k}{4}+2$. Assume for
contradiction that there exists $d\in A-A$ for which $r_{A}^{-}(d)\geq k$.
Since every number participates in at most two representations of $d$, there
exist distinct $a_{1},\ldots,a_{k/2},b_{1},\ldots,b_{k/2}$ such that
$a_{1}-b_{1}=\cdots=a_{k/2}-b_{k/2}=d$. We set
$S=\\{a_{1},\ldots,a_{k/2},b_{1},\ldots,b_{k/2}\\}$.
We may assume, without loss of generality, that $a_{1}>a_{2}>\cdots>a_{k/2}$.
This implies that $b_{1}>b_{2}>\cdots>b_{k/2}$. For every $1\leq i<j\leq k/2$,
we have that $a_{i}-b_{i}=a_{j}-b_{j}$, which in turn implies that
$a_{i}-a_{j}=b_{i}-b_{j}$. Since there are $\binom{k/2}{2}$ such choices of
$i$ and $j$, there are $\binom{k/2}{2}$ equations of the form
$a_{i}-a_{j}=b_{i}-b_{j}$. We conclude that
$|S-S|\leq\binom{k}{2}-\left(\frac{k}{2}-1\right)-\binom{k/2}{2}=\frac{3k^{2}}{8}-\frac{3k}{4}+1.$
Since the above contradicts the assumption about $A$, we get that every $d\in
A-A$ satisfies $r_{A}^{-}(d)<k$. Since $k$ is constant, we have that
$|A-A|>\frac{1}{k}\binom{n}{2}=\Theta(n^{2}).$
∎
We now move to prove Theorem 1.6. We rely on the following configuration from
[5]. Fish, Lund and Sheffer utilize the configuration to obtain an upper bound
on $g(n,k,k^{2/3})$. We will instead use this construction to prove a lower
bound on $g(n,k,\ell)$ where $\ell=\Theta(k^{2/3})$.
For an integer $i\geq 1$ and positive reals $a,\delta_{1},\ldots,\delta_{i}$,
we define a _projected- $i$-cube_ as
$P(a,\delta_{1},\ldots,\delta_{i})=\\{a+x_{1}\delta_{1}+x_{2}\delta_{2}+\cdots+x_{i}\delta_{i}\
:\ x_{1},\ldots,x_{i}\in\\{0,1\\}\\}.$
We note that a projected-$i$-cube is a set of at most $2^{i}$ real numbers. A
projected-$i$-cube can be thought of as a projection of an $i$-dimensional
hypercube onto $\mathbb{R}$, which explains the name of this object.
For $i>1$, every projected-$i$-cube is the union of two
projected-$(i-1)$-cubes that are identical up to a translation of distance
$p_{i}$. The following lemma states another useful property of
projected-$i$-cubes.
###### Lemma 3.1.
If $A$ is a projected-$i$-cube then
$\lvert A-A\rvert\leq\frac{3^{i}-1}{2}.$
For comparison, a set $A$ of $2^{i}$ random reals is expected to satisfy
$\lvert A-A\rvert=\Theta(4^{i})$.
###### Proof.
The proof is by induction on $i$. For the induction basis, we consider the
case of $i=1$. In this case $A$ is a set of two numbers, so
$|A-A|=1=(3^{1}-1)/2$.
For the induction step, we assume that the claim holds for projected-$i$-cubes
and consider a projected-$(i+1)$-cube $A=P(a,\delta_{1},\ldots,\delta_{i+1})$.
Then $A$ is the union of the two projected-$i$-cubes
$A_{1}=P(a,\delta_{1},\ldots,\delta_{i})$ and
$A_{2}=P(a+\delta_{i+1},\delta_{1},\ldots,\delta_{i})$. These two
projected-$i$-cubes have the same difference set $D=A_{1}-A_{1}=A_{2}-A_{2}$.
The induction hypothesis implies that $|D|\leq(3^{i}-1)/2$.
Next, we note that every difference in $(A-A)\setminus\\{\delta_{i+1}\\}$ is
either formed by two numbers from $A_{1}$ or by one number from $A_{1}$ and
another from $A_{2}$. By the preceding paragraph, there are at most
$(3^{i}-1)/2$ differences of the former type. Every difference of the latter
type is in $D\pm\delta_{i+1}$. The number of such differences is at most
$2\cdot|D|=3^{i}-1$. In total,
$|A-A|\leq(3^{i}-1)/2+(3^{i}-1)+1=(3^{i+1}-1)/2.$
This completes the proof of the induction step and thus the proof of the
theorem. ∎
We define the _left endpoint_ of a projected-$i$-cube as the smallest number
of that cube. The _right endpoint_ is the largest number of the cube.
Theorem 1.6. _For every $k\geq 8$ that is a power of two and sufficiently
large $n$, we have that_
$g\left(n,k,\frac{k^{\log_{2}(3)}+1}{2}\right)=\Omega\left(n^{1+\frac{2}{k-2}}\right).$
###### Proof.
We consider a set $A\subset\mathbb{R}$ such that $|A|=n$ and $|A-A|\leq
n^{1+\frac{2}{k-2}}/9$. We show that such a set must contain a
projected-$(\log_{2}(k)-2)$-cube. Then, Lemma 3.1 implies that $A$ does not
satisfy the local property in the statement of the theorem, which completes
the proof.
For an integer $i$, we set $\eta_{i}=1-(2^{i+1}-2)/(k-2)$. We prove by
induction on $1\leq i\leq\log_{2}(k)-2$ that $A$ contains at least
$n^{\eta_{i}}$ disjoint translated copies of a projected-$i$-cube with exactly
$2^{i}$ elements. Combining this with the preceding paragraph completes the
proof of the theorem.
There are $\binom{n}{2}>n^{2}/3$ pairs of elements from $A$ and $|A-A|\leq
n^{1+\frac{2}{k-2}}/9$. By the pigeonhole principle there exists a distance
that occurs at least $3n^{(k-4)/(k-2)}$ times. We arbitrarily fix one such
distance and denote it as $d_{1}$. We set $A_{1}$ to be the set of elements of
$A$ that are at distance $d_{1}$ from another element of $A$. Let $G_{1}$ be
the graph whose vertices are the elements of $A_{1}$. Two vertices are
connected by an edge if they span the difference $d_{1}$. We note that $G_{1}$
contains at least $3n^{(k-4)/(k-2)}$ edges. Since every vertex is of degree at
most two, we can obtain a matching by repeatedly choosing an arbitrary edge
$e$ and discarding at most two other edges that share a vertex with $e$. This
implies that $G_{1}$ contains a matching of size at least $n^{(k-4)/(k-2)}$.
Since a projected-1-cube consists of two points and $\eta_{1}=(k-4)/(k-2)$, we
obtain the case of $i=1$. This completes the induction basis.
For the induction step, we fix $1\leq i<\log_{2}(k)-2$ and assume that the
claim holds for $i$. That is, $A$ contains at least $n^{\eta_{i}}$ disjoint
translated copies of a projected-$i$-cube. We note that, in this range,
$\eta_{i}>0$. We fix exactly $n^{\eta_{i}}$ translated copies and set
$A_{i+1}$ to be the set of endpoints of these. By definition, we have that
$|A_{i+1}|=2n^{\eta_{i}}$. The number of pairs of endpoints that do not belong
to the same translated copy is
$\frac{1}{2}\cdot
2n^{\eta_{i}}\cdot\left(2n^{\eta_{i}}-2\right)=2n^{2\eta_{i}}-2n^{\eta_{i}}.$
Each difference is spanned by $A_{i+1}$ fewer than $2n^{\eta_{i}}$ times.
Thus, the number of above pairs that span a difference that is also spanned by
the projected-$i$-cube is $O(n^{\eta_{i}})$. When $n$ is sufficiently large,
we may assume that the number of pairs that span a distance that is not in the
projected-$i$-cube is at least $n^{2\eta_{i}}$. Since $A_{i+1}\subset A$ and
$|A-A|\leq n^{1+\frac{2}{k-2}}/9$, there exists a difference that is spanned
by at least
$9n^{2\eta_{i}-1-2/(k-2)}=9n^{\eta_{i+1}}$
of these pairs. We fix such a difference and denote it as $d_{i+1}$. In each
of the corresponding pairs, the difference $d_{i+1}$ occurs between the two
left endpoints, between the two right endpoints, or between one right endpoint
and one left endpoint. By the pigeonhole principle, at least one third of the
pairs are of the same type. Thus, $A_{i+1}$ contains at least
$3n^{\eta_{i+1}}$ translatd copies of the same projected-$(i+1)$-cube.
The above translated copies of a projected-$(i+1)$-cube might not be disjoint.
In particular, a projected projected-$i$-cube from $A_{i}$ might participate
in two projected-$(i+1)$-cubes. By repeating the matching argument from the
induction basis, we obtain that at least one third of the
projected-$(i+1)$-cubes are disjoint. That is $A_{i+1}$ contains at least
$n^{\eta_{i+1}}$ disjoint translated copies of the same
projected-$(i+1)$-cube. This concludes the proof of the induction step and
thus the proof of the theorem. ∎
## 4 Small values of $\ell$
In this section, we prove our bounds for small values of $\ell$: Theorem 1.3
and Theorem 1.4. We first briefly go over the tools that we require from
additive combinatorics.
Tools from additive combinatorics. Szemerédi’s theorem states that every
sufficiently large subset of $\\{1,2,\ldots,n\\}$ contains a long arithmetic
progression. We rely on the following variant of this result, due to Gowers
[7].
###### Theorem 4.1.
For every $k$, there exists $c>0$ that satisfies the following. Every set of
$n/(\log\log n)^{c}$ numbers from $\\{1,2,\ldots,n\\}$ contains a $k$-term
arithmetic progression. Moreover, $c$ can be taken to be $2^{-2^{k+9}}$.
A _generalized arithmetic progression of dimension_ $d$ is defined as
$\Big{\\{}a+\sum_{j=1}^{d}k_{j}b_{j}\ :\
a,b_{1},\ldots,b_{d}\in\mathbb{R}\text{ and with integer }0\leq k_{j}\leq
n_{j}-1\text{ for every }1\leq j\leq d\Big{\\}}.$ (4)
The size of a generalized arithmetic progression is the number of elements in
it. We note that an arithmetic progression is a generalized arithmetic
progression of dimension one.
One of the most common tools for studying sets with a small different set is
_Freiman’s theorem._ The following is a variant of this theorem over the
reals. For example, see Sanders [10, Theorem 1.3].
###### Theorem 4.2.
For every sufficiently large finite set $A\subset\mathbb{R}$ and sufficiently
large $d>0$, if $|A-A|\leq d|A|$ then $A$ is contained in a generalized
arithmetic progression of dimension at most $d\cdot\log^{4}d$ and size at most
$|A|\cdot e^{d\cdot\log^{4}d}$.
Similarly to a difference set $A-A$, we define the _sum set_ of a set
$A\subset\mathbb{R}$ as
$A+A=\\{a+a^{\prime}\ :\ a,a\in A\\}.$
The following _Plünnecke-Ruzsa estimate_ (for example, see [13, Corollary
6.29]) shows one connection between $A+A$ and $A-A$.
###### Lemma 4.3.
If a finite $A\subset\mathbb{R}$ satisfies $|A-A|\leq c|A|$ then
$|A+A|\leq c^{2}|A|.$
The following is a special case of a result of Green and Morris [8] about sum
sets.
###### Theorem 4.4.
For every $c>0$, every sufficiently large $m$ satisfies the following. For
every integer $n$, the number of subsets $A\subset\\{1,2,\ldots,n\\}$ that
satisfy $|A|=m$ and $|A+A|\leq c|A|$ is at most
$2^{m}\cdot n^{c+1}\cdot\binom{cm/2}{m}.$
Our proofs. We now derive the super-linear threshold for our local properties
problem. We first recall the statement of this result.
Theorem 1.3. _For every $k$ and sufficiently large $n$, we have that_
$\displaystyle g(n,k,k-1)=n-1\qquad\text{ and }\qquad g(n,k,k)=\omega(n).$
###### Proof.
To see that $g(n,k,k-1)=n-1$, we set $A=\\{1,2,\ldots,n\\}$ and note that
$|A-A|=n-1$. The local property trivially holds, since every $k$ numbers span
at least $k-1$ differences. It remains to prove that $g(n,k,k)=\omega(n)$.
For a constant $c>0$, we consider a set $A$ such that $|A|=n$ and $|A-A|\leq
cn$. By Theorem 4.2, the set $A$ is contained in a generalized arithmetic
progression $G$ of size at most $Cn$ and dimension at most $D$. The constants
$C$ and $D$ depend on $c$ but not on $n$. We define $G$ as in (4). Without
loss of generality, we may assume that $n_{1}\geq n^{1/D}$.
We may think of $G$ as the union of disjoint arithmetic progressions with step
$b_{1}$, each of size at least $n_{1}\geq n^{1/D}$. Since $|A|/|G|\geq 1/C$,
there exists such an arithmetic progression $S$ such that $|A\cap S|/|S|\geq
1/C$. We translate and scale $\mathbb{R}$ such that $S$ becomes
$\\{1,2,3,...,|S|\\}$. Then, $A$ contains at least $|S|/C$ elements of
$\\{1,2,3,...,|S|\\}$. Theorem 4.1 implies that $A$ contains a $k$-term
arithmetic progression.
We proved that the following holds for every constant $c>0$, when $n$ is
sufficiently large: Every set $A$ of $n$ reals that satisfies $|A-A|\leq cn$
contains $k$ numbers that span $k-1$ differences. Thus, a set that satisfies
the local restrictions of $g(n,k,\ell)$ also satisfies that $|A-A|>cn$ for
every $c$. ∎
###### Remark 4.5.
Quantitatively, the proof above gives the bound $g(n,k,k)\leq
n\cdot\sqrt{\log(\log\log n)^{c}}$, where $c=2^{-2^{k+9}}$.
For our probabilistic arguments, we recall the following Chernoff bounds (for
example, see [1, Corollary A.1.14]).
###### Theorem 4.6.
For every ${\varepsilon}>0$, there exists $c_{\varepsilon}$ that satisfies the
following. Let $Y$ be a sum of independent indicator random variables and let
$\mu$ be the expected value of $Y$. Then
$\Pr[|Y-\mu|>{\varepsilon}\mu]<2e^{-e_{\varepsilon}\mu}.$
We may set
$c_{\varepsilon}=\min\\{\ln(e^{-{\varepsilon}}(1+{\varepsilon})^{1+{\varepsilon}}),{\varepsilon}^{2}/2\\}.$
We now prove our upper bound for small values of $\ell$. We first recall the
statement of this result.
Theorem 1.4. _For every $c\geq 2$ and integer $k>(c^{2}+1)^{2}$, we have that_
$g(n,k,ck+1)=O\Big{(}n^{1+\frac{c^{2}+1}{k}}\Big{)}.$
###### Proof.
Let $N$ be a sufficiently large constant multiple of $n$, which is determined
below. We define the probability $p=3N^{-(c^{2}+1)/k}$ and set
$M=\big{\\{}1,2,\ldots,N^{1+(c^{2}+1)/k}\big{\\}}$. Let $A^{\prime}$ be a set
that is obtained by selecting each element of $M$ independently with
probability $p$. The expected size of $A^{\prime}$ is
$p\cdot N^{1+(c^{2}+1)/k}=3N.$
By applying Theorem 4.6 with $\mu=3N$ and ${\varepsilon}=1/2$, we obtain that
$\Pr\left[\frac{3N}{2}\leq\lvert A\rvert\leq\frac{9N}{2}\right]>1-2e^{-N/4}.$
(5)
By Theorem 4.4, the number of subsets $B\subset M$ that satisfy $|B|=k$ and
$|B+B|\leq c^{2}k$ is at most
$2^{k}\cdot(N^{1+(c^{2}+1)/k})^{c^{2}+1}\cdot\binom{c^{2}k/2}{k}=O_{c,k}(N^{c^{2}+1+(c^{2}+1)^{2}/k}).$
By the contrapositive of Lemma 4.3, the number of subsets $B\subset M$ that
satisfy $|B|=k$ and $|B-B|\leq ck$ is $O_{c,k}(N^{c^{2}+1+(c^{2}+1)^{2}/k})$.
We refer to such a set as a _set with small doubling_.
The probability that a fixed set with small doubling is in $A$ is $p^{k}$.
Thus, the expected number of sets with small doubling in $A$ is
$p^{k}\cdot
O_{c,k}(N^{c^{2}+1+(c^{2}+1)^{2}/k})=O_{c,k}(N^{(c^{2}+1)^{2}/k}).$
By the assumption $k>(c^{2}+1)^{2}$, the expected number of sets with small
doubling in $A$ is $o_{c,k}(N)$. Combining this with (5) implies that, with
positive probability, we have that $\frac{3N}{2}\leq\lvert
A\rvert\leq\frac{9N}{2}$ and that the number of sets with small doubling is
$o_{c,k}(N)$. Thus, there exists $A$ that satisfies both of these properties.
We fix such a set $A$ and arbitrarily remove one number from each of the
remaining sets with small doubling. This leads to $|A|=\Theta(N)$ and no sets
with small doubling in $A$. In other words, every $k$ elements from $A$ span
at least $ck+1$ distinct differences.
Since $A\subset M$, we have that $|A-A|<N^{1+(c^{2}+1)/k}$. By taking $N$ to
be a sufficiently large constant mulitple of $n$, we obtain that $|A|\geq n$.
We arbitrarily remove elements from $A$ until $|A|=n$. This leads to the
assertion of the theorem. ∎
## 5 Future work
We are still far from understanding the behavior of $g(n,k,\ell)$ an have more
questions than answers. One main open problem is to identify the quadratic
threshold of $g(n,k,\ell)$. Since this turned out to be challenging, we
suggest the following problem as a step towards the quadratic threshold.
###### Question 5.1.
For what values of $\ell$ can we prove a bound of the form
$g(n,k,\ell)=\Omega(n^{c})$ for some $4/3<c<2$?
New ideas would be needed for this problem: as mentioned in Remark 2.3, though
the idea of dumbbell energies can be generalized to “higher” dumbbell
energies, the bounds obtained that way would be of the form
$g(n,k,\ell)=\Omega(n^{c})$ for some $c<4/3$.
In terms of quadratic lower bounds, we pose the following question.
###### Question 5.2.
Is it true that when $k$ is a multiple of four, we have that
$g\left(n,k,\frac{k^{2}}{4}\right)=\Omega(n^{2})?$
There is also a natural generalization of the construction of the
projected-$(i+1)$-cube, where instead of projecting an $i$-dimensional Boolean
hypercube onto $\mathbb{R}$, we could instead project an $i$-dimensional
$b$-ary hypercube onto $\mathbb{R}$. Specifically, let
$A_{1}=\\{1,2,\ldots,b\\}$ and $D=\\{0,1,\ldots,b-1\\}$. For every $i>1$, we
let $s_{i}=4b\cdot\max\\{A_{i-1}-A_{i-1}\\}$ and $T_{i}=s_{i}\cdot D$. We then
set
$A_{i}=T_{i}+A_{i-1}.$
We note that $1\in A_{i}$ for every $i\geq 1$. This implies that $s_{i}\geq
4b(\max\\{A_{i}\\}-1)$. Since the nonzero elements of $T_{i}$ are
significantly larger than the elements of $A_{i}$, we get that
$|A_{i}|=b\cdot|A_{i-1}|$. Since $|A_{1}|=b$, we conclude that $|A_{i}|=b^{i}$
for every $i\geq 1$.
By definition, every element of $A_{i}-A_{i}$ can be represented as $x+y$
where $x\in T_{i}-T_{i}$ and $y\in A_{i-1}-A_{i-1}$. By our choice of $s_{i}$,
we have that $x,x^{\prime}\in T_{i}-T_{i}$ and $y,y^{\prime}\in
A_{i-1}-A_{i-1}$ satisfy that $x+y=x^{\prime}+y^{\prime}$ if and only if
$(x,y)=(x^{\prime},y^{\prime})$. Indeed, if $x\neq x^{\prime}$ then $\lvert
x-x^{\prime}\rvert\geq 4b\max\\{|y|,|y^{\prime}|\\}\geq 2b\lvert
y-y^{\prime}\rvert$. In particular, it can be easily checked that as a
consequence we have $\lvert A_{i}-A_{i}\rvert=\frac{(2b-1)^{i}-1}{2}$. We
conjecture that the natural generalization of [5] should hold in this case.
###### Conjecture 5.3.
When $k$ is a power of $b$,
$g\left(n,k,\frac{k^{\log_{b}(2b-1)}+1}{2}\right)=O\left(k^{\log_{b}(2b-1)}\right)$.
In the same flavor as Theorem 1.6, we may ask the following question.
###### Question 5.4.
Can we prove a lower bound for
$g\left(n,k,\frac{k^{\log_{b}(2b-1)}+1}{2}\right)$ when $k$ is a power of $b$?
## References
* [1] Noga Alon and Joel H Spencer. The probabilistic method. John Wiley & Sons, 2004.
* [2] David Conlon, Jacob Fox, Choongbum Lee, and Benny Sudakov. The Erdős–Gyárfás problem on generalized Ramsey numbers. Proceedings of the London Mathematical Society, 110(1):1–18, 2015\.
* [3] Paul Erdos. Problems and results on finite and infinite graphs. In Recent advances in graph theory (Proc. Second Czechoslovak Sympos., Prague, 1974), pages 183–192, 1975.
* [4] Paul Erdős and András Gyárfás. A variant of the classical ramsey problem. Combinatorica, 17(4):459–467, 1997.
* [5] Sara Fish, Ben Lund, and Adam Sheffer. A construction for difference sets with local properties. European Journal of Combinatorics, 79:237–243, 2019.
* [6] Sara Fish, Cosmin Pohoata, and Adam Sheffer. Local properties via color energy graphs and forbidden configurations. SIAM Journal on Discrete Mathematics, 34(1):177–187, 2020.
* [7] William Timothy Gowers. A new proof of Szemerédi’s theorem. Geometric & Functional Analysis GAFA, 11(3):465–588, 2001.
* [8] Ben Green and Robert Morris. Counting sets with small sumset and applications. Combinatorica, 36(2):129–159, 2016.
* [9] Cosmin Pohoata and Adam Sheffer. Local properties in colored graphs, distinct distances, and difference sets. Combinatorica, 39(3):705–714, 2019.
* [10] Tom Sanders. The structure theory of set addition revisited. Bull. Amer. Math. Soc. (N.S.), 50(1):93–127, 2013.
* [11] Tomasz Schoen. New bounds in Balog-Szemerédi-Gowers theorem. Combinatorica, 35(6):695–701, 2015.
* [12] Tomasz Schoen and Ilya D Shkredov. Higher moments of convolutions. Journal of Number Theory, 133(5):1693–1737, 2013.
* [13] Terence Tao and Van H Vu. Additive combinatorics, volume 105. Cambridge University Press, 2006.
|
# Termination Analysis of Programs with Multiphase Control-Flow††thanks: This
paper is a summary of the corresponding invited talk of the second author
given at HCVS 2021, and is based on the PhD thesis of the first author [15]
and on collaborations with Amir M. Ben-Amram [9, 5] and John P. Gallagher
[16]. This work was funded partially by the Spanish MCIU, AEI and FEDER (EU)
project RTI2018-094403-B-C31, by the CM project S2018/TCS-4314 co-funded by
EIE Funds of the European Union, and by the UCM CT42/18-CT43/18 grant.
Jesús J. Domenech1 Samir Genaim1,2
1 Universidad Complutense de Madrid, Spain2 Instituto de Tecnología del
Conocimiento, Madrid, Spain<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
Programs with multiphase control-flow are programs where the execution passes
through several (possibly implicit) phases. Proving termination of such
programs (or inferring corresponding runtime bounds) is often challenging
since it requires reasoning on these phases separately. In this paper we
discuss techniques for proving termination of such programs, in particular:
(1) using multiphase ranking functions, where we will discuss theoretical
aspects of such ranking functions for several kinds of program
representations; and (2) using control-flow refinement, in particular partial
evaluation of Constrained Horn Clauses, to simplify the control-flow allowing,
among other things, to prove termination with simpler ranking functions.
## 1 Introduction
Proving that a program will eventually terminate, i.e., that it does not go
into an infinite loop, is one of the most fundamental tasks of program
verification, and has been the subject of voluminous research. Perhaps the
best known, and often used, technique for proving termination is that of
_ranking functions_ , which has already been used by Alan Turing in his early
work on program verification [34]. This consists of finding a function $\rho$
that maps program states into the elements of a well-founded ordered set, such
that $\rho(s)\succ\rho(s^{\prime})$ holds for any consecutive states $s$ and
$s^{\prime}$. This implies termination since infinite descent is impossible in
a well-founded order. Besides proving termination, Turing [34] mentions that
ranking functions can be used to bound the length computations as well. This
is useful in applications such as cost analysis and loop optimisation [17, 4,
2, 12].
Unlike termination of programs in general, which is undecidable, the
algorithmic problems of detection (deciding the existence) or generation
(synthesis) of a ranking function can well be solvable, given certain choices
of the program representation, and the class of ranking functions. There is a
considerable amount of research in this direction, in which different kinds of
ranking functions for different kinds of program representations were
considered. In some cases, the algorithmic problems have been completely
settled, and efficient algorithms provided, while other cases remain open.
A common program representation in this context is _Single-path Linear-
Constraint_ (SLC) loops, where a state is described by the values of numerical
variables, and the effect of a transition (one iteration) is described by a
conjunction of _linear constraints_. Here is an example of this loop
representation; primed variables $x^{\prime}$ and $y^{\prime}$ refer to the
state following the transition:
$while\leavevmode\nobreak\ (x\leq y)\leavevmode\nobreak\
do\leavevmode\nobreak\ x^{\prime}=x+1,\,y^{\prime}\leq y$ (1)
Note that $x^{\prime}=x+1$ is an equation, not an assignment. The description
of a loop may involve linear inequalities rather than equations, such as
$y^{\prime}\leq y$ above, and consequently, be non-deterministic. Note that
for a SLC loop with $n$ variables, a transition can be seen as a point
$\bigl{(}\begin{smallmatrix}{\mathbf{x}}\hfill\\\
{\mathbf{x}^{\prime}}\hfill\end{smallmatrix}\bigr{)}\in\mathbb{Q}^{2n}$, where
its first $n$ components correspond to $\mathbf{x}$ and its last $n$
components to $\mathbf{x}^{\prime}$. We denote the set of all transitions by
${\mathcal{Q}}$, which is a polyhedron.
A more general program representation is _Transition Systems_ (TSs), which are
defined by _Control-Flow Graphs_ (CFGs) with numerical variables, consisting
of nodes representing program locations and edges annotated with linear
constraints (polyhedra) describing how values of variables change when moving
from one location to another. Figure 1 includes a program and its
corresponding TS ${\cal T}$ to its right. Primed variables in the linear
constraints refer to the state following the transition, exactly as in the
case of SLC loops.
In both program representations mentioned above, the domain of variables is
also important as it typically affects the complexity of the underlying
decision and synthesis problems. Although these program representations allow
only numerical variables and linear constraints, data structures can be
handled using _size abstractions_ , e.g., length of lists, depth of trees,
etc. [27, 24, 13, 33, 28, 20]. In such case, variables represent sizes of
corresponding data structures.
⬇
1void phases1(int x, int y, int z) {
2 while( x >= 1 ){
3 if ( y <= z - 1 ) {
4 y = y + 1;
5 } else {
6 x = x - 1;
7 }
8 }
9}
$\mathtt{n_{\\_}0}$
$\mathtt{n_{\\_}1}$
$\mathtt{n_{\\_}2}$
$\mathtt{n_{\\_}3}$
${\cal T}$
$\scriptscriptstyle{\mathcal{Q}}_{\\_}0$
$\scriptscriptstyle{\mathcal{Q}}_{\\_}1$
$\scriptscriptstyle{\mathcal{Q}}_{\\_}2$
$\scriptscriptstyle{\mathcal{Q}}_{\\_}3$
$\scriptscriptstyle{\mathcal{Q}}_{\\_}4$
$\mathtt{n_{\\_}0}$
$\mathtt{n_{\\_}1^{3}}$
$\mathtt{n_{\\_}3^{2}}$
$\mathtt{n_{\\_}2^{2}}$
$\mathtt{n_{\\_}1^{2}}$
${\cal T}_{\\_}{pe}$
$\mathtt{n_{\\_}2^{1}}$
$\mathtt{n_{\\_}3^{1}}$
$\mathtt{n_{\\_}1^{1}}$
$\scriptscriptstyle{\mathcal{Q}}_{\\_}0$
$\scriptscriptstyle{\mathcal{Q}}_{\\_}1$
$\scriptscriptstyle{\mathcal{Q}}_{\\_}2$
$\scriptscriptstyle{\mathcal{Q}}_{\\_}6$
$\scriptscriptstyle{\mathcal{Q}}_{\\_}5$
$\scriptscriptstyle{\mathcal{Q}}_{\\_}7$
$\scriptscriptstyle{\mathcal{Q}}_{\\_}8$
$\scriptscriptstyle{\mathcal{Q}}_{\\_}{6}$
$\scriptscriptstyle{\mathcal{Q}}_{\\_}{1}$
$\begin{array}[]{r@{}llllll}{\mathcal{Q}}_{\\_}0\equiv&\\{&&x^{\prime}=x,&y^{\prime}=y,&z^{\prime}=z&\\}\\\
{\mathcal{Q}}_{\\_}1\equiv&\\{x\geq
1,&&x^{\prime}=x,&y^{\prime}=y,&z^{\prime}=z&\\}\\\
{\mathcal{Q}}_{\\_}2\equiv&\\{x\leq
0,&&x^{\prime}=x,&y^{\prime}=y,&z^{\prime}=z&\\}\\\
{\mathcal{Q}}_{\\_}3\equiv&\\{y\leq
z-1,&&x^{\prime}=x,&y^{\prime}=y+1,&z^{\prime}=z&\\}\\\
{\mathcal{Q}}_{\\_}4\equiv&\\{y\geq
z,&&x^{\prime}=x-1,&y^{\prime}=y,&z^{\prime}=z&\\}\\\
{\mathcal{Q}}_{\\_}5\equiv&\\{x\geq 1,&y\leq
z-1,&x^{\prime}=x,&y^{\prime}=y+1,&z^{\prime}=z&\\}\\\
{\mathcal{Q}}_{\\_}6\equiv&\\{x\geq 1,&y\geq
z,&x^{\prime}=x-1,&y^{\prime}=y,&z^{\prime}=z&\\}\\\
{\mathcal{Q}}_{\\_}7\equiv&\\{x\geq 1,&y\geq
z,&x^{\prime}=x,&y^{\prime}=y,&z^{\prime}=z&\\}\\\
{\mathcal{Q}}_{\\_}8\equiv&\\{x\leq 0,&y\geq
z,&x^{\prime}=x,&y^{\prime}=y,&z^{\prime}=z&\\}\\\ \end{array}$
Figure 1: A loop with $2$ phases, it corresponding TS ${\cal T}$, and the TS
${\cal T}_{\\_}{pe}$ after applying CFR.
Due to practical considerations, termination analysis tools typically focus on
classes of ranking functions that can be synthesised efficiently. This,
however, does not mean that theoretical aspects of such classes are put aside,
as understanding theoretical limits and properties of the underlying problems
is necessary for developing practical algorithms. The most popular class of
ranking functions in this context is probably that of _Linear Ranking
Functions_ (LRFs). A LRF is a function
$\rho(x_{\\_}1,\dots,x_{\\_}n)=a_{\\_}1x_{\\_}1+\dots+a_{\\_}nx_{\\_}n+a_{\\_}0$
such that any transition from $\mathbf{x}$ to $\mathbf{x}^{\prime}$ satisfies
(i) $\rho(\mathbf{x})\geq 0$; and (ii)
$\rho(\mathbf{x})-\rho(\mathbf{x}^{\prime})\geq 1$. For example,
$\rho(x,y)=y-x$ is a LRF for Loop (1). Several polynomial-time algorithms to
find a LRF using linear programming exist [4, 14, 17, 29, 30, 32]. These
algorithms are complete111Complete means that if there is a LRF, they will
find one. for TSs with rational-valued variables, but not with integer-valued
variables. Ben-Amram and Genaim [6] showed how completeness for the integer
case can be achieved, and also classified the corresponding decision problem
as co-NP complete.
Despite their popularity, LRFs do not suffice for all programs, and a natural
question is what to do when a LRF does not exist; and a natural answer is to
try a richer class of ranking functions. Of particular importance is the class
of _Lexicographic-Linear Ranking Functions_ (LLRFs). These are tuples of
linear functions that decrease lexicographically over a corresponding well-
founded ordered set. For example, the program depicted in Figure 1 does not
have a LRF, but it has the LLRF $\langle z-y,x\rangle$ since: for the _then_
branch $z-y$ decreases and for the _else_ branch $x$ decreases while $z-y$
does not change. LLRFs might be necessary even for SLC loops. For example, the
following SLC loop
$while\leavevmode\nobreak\ (x\geq 0)\leavevmode\nobreak\
do\leavevmode\nobreak\ x^{\prime}=x+y,\,y^{\prime}=y+z,z^{\prime}=z-1$ (2)
does not have a LRF, but can be proved terminating using the LLRF $\langle
z,y,x\rangle$.
There are several definitions for LLRFs in the literature [4, 7, 11, 23] and
they have different power, i.e., some can prove termination of a program while
others fail. They have corresponding polynomial-time synthesis algorithms for
the case of rational variables, and the underlying decision problems are co-NP
complete for the case of integer variables [7, 8]. The definition of Larraz et
al. [23] is the most general in this spectrum of LLRFs, its complexity
classification is not known yet and corresponding complete synthesis
algorithms do not exist.
The LLRF $\langle z,y,x\rangle$ used above for Loop (2) belongs to a class of
LLRFs that is know as _Multiphase-Linear Ranking Functions_ (M$\Phi$RFs).
Ranking functions in this class are characterised by the following behaviour:
the first component _always_ decreases, and when it becomes negative the
second component starts to decrease (and will keep decreasing during the rest
of the execution), and when it becomes negative the third component starts to
decrease, and so on. This behaviour defines phases through which executions
pass. This is different from LLRFs in general since once a component becomes
negative it cannot be used anymore. Also the LLRF $\langle z-y,x\rangle$ for
the program of Figure 1 induces a similar multiphase behaviour, since once
$z-y$ becomes negative it cannot be used anymore. However, it is not a
M$\Phi$RF since $z-y$ stops decreasing once we move to the second phase.
In the rest of this paper we overview works on termination analysis of
multiphase programs, in particular: in Section 2 we discuss techniques for
proving termination using M$\Phi$RFs; and in Section 3 we discuss techniques
for proving termination using _Control-Flow Refinement_ (CFR). Section 4 ends
the paper with some concluding remarks and discusses further research
directions.
## 2 Termination analysis using multiphase ranking functions
A M$\Phi$RF is a tuple $\langle\rho_{\\_}1,\ldots,\rho_{\\_}d\rangle$ of
linear functions that define phases of the program that are linearly ranked,
where $d$ is the _depth_ of the M$\Phi$RF, intuitively the number of phases.
The decision problem _Existence of a M $\Phi$RF_ asks to determine whether a
program has a M$\Phi$RF. The _bounded_ decision problem restricts the search
to M$\Phi$RFs of depth $d$, where $d$ is part of the input.
For the case of SLC loops, the complexity and algorithmic aspects of the
bounded version of the M$\Phi$RF problem were settled by Ben-Amram and Genaim
[9]. The decision problem is $\mathtt{PTIME}$ for SLC loops with rational-
valued variables, and $\mathtt{coNP}$-complete for SLC loops with integer-
valued variables; synthesising M$\Phi$RFs, when they exist, can be performed
in polynomial and exponential time, respectively. We note that the proof of
the rational case is done by showing that M$\Phi$RFs and _nested_ ranking
functions [25] (a strict subclass of M$\Phi$RFs for which a polynomial-time
algorithm exists) have the same power for SLC loops. Besides, they show that
for SLC loops M$\Phi$RFs have the same power as general lexicographic-linear
ranking functions, and that M$\Phi$RFs induce linear iteration bounds. The
problem of deciding if a given SLC loop admits a M$\Phi$RF, without a given
bound on the depth, is still open.
In practice, termination analysis tools search for M$\Phi$RFs incrementally,
starting by depth $1$ and increase the depth until they find one, or reach a
predefined limit, after which the returned answer is _don’t know_. Finding a
theoretical upper-bound on the depth of a M$\Phi$RF, given the loop, would
also settle this problem, however, as shown by Ben-Amram and Genaim [9] such
bound must depend not only on the number of constraints or variables, as for
other classes of LLRFs [4, 7, 11], but also on the coefficients used in the
corresponding constraints. Yuan et al. [35] proposed an incomplete method to
bound the depth of M$\Phi$RFs for SLC loops.
Ben-Amram et al. [5] have done a significant progress towards solving the
problem of _existence of a M $\Phi$RF_ for SLC loop, i.e., seeking a M$\Phi$RF
without a given bound on the depth. In particular, they present an algorithm
for seeking M$\Phi$RFs that reveals novel insights on the structure of these
ranking functions. In a nutshell, the algorithm starts from the set of
transitions of the given SLC loop, which is a polyhedron, and iteratively
removes transitions $\bigl{(}\begin{smallmatrix}{\mathbf{x}}\hfill\\\
{\mathbf{x}^{\prime}}\hfill\end{smallmatrix}\bigr{)}$ such that
$\rho(\mathbf{x})-\rho(\mathbf{x}^{\prime})>0$ for some function
$\rho(\mathbf{x})=\vec{a}\cdot\mathbf{x}+b$ that is _non-negative on all
enabled states_. The process continues iteratively, since after removing some
transitions, more functions $\rho$ may satisfy the non-negativity condition,
and they may eliminate additional transitions in the next iteration. When all
transitions are eliminated in a finite number of iterations, one can construct
a M$\Phi$RF using the $\rho$ functions; and when reaching a situation in which
no transition can be eliminated, the remaining set of transitions, which is a
polyhedron, is actually a recurrent set that witnesses non-termination. The
algorithm always finds a M$\Phi$RF if one exists, and in many cases, it finds
a recurrent set when the loop is non-terminating, however, it is not a
decision procedure as it diverges in some cases. Nonetheless, the algorithm
provides important insights into the structure of M$\Phi$RFs. Apart from
revealing a relation between seeking M$\Phi$RFs and seeking recurrent sets,
these insights are useful for finding classes of SLC loops for which, when
terminating, there is always a M$\Phi$RF and thus have linear run-time bound.
This result was proven for two kinds of SLC loops, both considered in previous
work, namely _octagonal relations_ and _affine relations with the finite-
monoid property_ – for both classes, termination has been proven decidable
[22].
Ben-Amram et al. [5] have also suggested a new representation for SLC loops,
called the _displacement_ representation, that provides new tools for studying
termination of SLC loops in general, and the existence of a M$\Phi$RF in
particular. In this representation, a transition
$\bigl{(}\begin{smallmatrix}{\mathbf{x}}\hfill\\\
{\mathbf{x}^{\prime}}\hfill\end{smallmatrix}\bigr{)}$ is represented as
$\bigl{(}\begin{smallmatrix}{\mathbf{x}}\hfill\\\
{\mathbf{y}}\hfill\end{smallmatrix}\bigr{)}$ where
$\mathbf{y}=\mathbf{x}^{\prime}-\mathbf{x}$. Using this representation the
algorithm described above can be formalised in a simple way that avoids
computing the $\rho$ functions mentioned above, and reduces the existence of a
M$\Phi$RF of depth $d$ to unsatisfiability of a certain linear constraint
system. Moreover, any satisfying assignment for this linear constraint system
is a witness that explains why the loop has no M$\Phi$RF of depth $d$. As an
evidence on the usefulness of this representation in general, they showed that
some non-trivial observations on termination of bounded SLC loops (i.e., the
set of transitions is a bounded polyhedron) are made straightforward in this
representation, while they are not easy to see in the normal representation,
in particular: a bounded SLC loop terminates if and only if it has a LRF, and
it does not terminate if and only if it has a fixpoint transition
$\bigl{(}\begin{smallmatrix}{\mathbf{x}}\hfill\\\
{\mathbf{x}}\hfill\end{smallmatrix}\bigr{)}$.
The works discussed above are limited to the case of SLC loops. The case of
general TSs has been considered by Leike and Heizmann [25] and Li et al. [26],
where both translate the existence of a M$\Phi$RF of a given depth $d$ to
solving a corresponding non-linear constraint problem, which is complete for
the case of rational variables. However, while complete, these approaches do
not provide any insights on the complexity of the underlying decision
problems. The technique of Borralleras et al. [10] can be used to infer, among
other things, M$\Phi$RFs for TSs without a given bound on the depth. It is
based on solving corresponding safety problems using Max-SMT, and it is not
complete.
## 3 Termination analysis using control-flow refinement
As we have seen in Section 1, there are TSs with multiphase behaviour that do
not admit M$\Phi$RFs, but rather a different notion of LLRF that does not
require the components to keep decreasing after turning negative. To prove
termination of such TSs one can use other classes of LLRFs [4, 7, 11, 23],
however, this might not be enough due to complex control-flow where abstract
properties of several implicit execution paths are merged. To overcome this
imprecision, not only for termination but for program analysis in general, one
approach is to simplify the control-flow in order to make implicit execution
paths explicit, and thus makes it possible to infer the desired properties
with a weaker version of the analysis. This is known as _Control-Flow
Refinement_ (CFR).
Let us see how CFR can simplify the kind of ranking functions needed to proved
termination, and thus improve the precision of the underlying termination
analyser. Consider the program depicted Figure 1 again, and recall that we
failed to prove its termination when using only LRFs. Examining this program
carefully, we can see that any execution passes in two phases: in the first
one, $\mathtt{y}$ is incremented until it reaches the value of $\mathtt{z}$,
and in the second phase $\mathtt{x}$ is decremented until it reaches
$\mathtt{0}$. Lets us transform the program into a semantically equivalent one
such that the two phases are separate and explicit:
⬇
1 while (x >= 0 and y <= z-1) y = y + 1;
2 while (x >= 0 and y >= z) x = x - 1;
Now we can prove termination of this program using LRFs only: for the first
loop $z-y$ is a LRF, and for the second loop $x$ is a LRF. Moreover, cost
analysis tools that are based on bounding loop iterations using LRFs [2] would
infer a linear bound for this program while they would fail on the original
one.
Apart from simplifying the termination proof, there are also cases where it is
not possible to prove termination without such transformations even when using
LLRFs. In addition, CFR can help in inferring more precise invariants, without
the need for expensive disjunctive abstract domains, which can benefit any
analysis that uses such invariants, e.g., termination and cost analysis.
CFR has been considered by Gulwani et al. [21] and Flores-Montoya and Hähnle
[18] to improve the precision of cost analysis, and by Sharma et al. [31] to
improve the precision of invariants in order to prove program assertions.
While all these techniques can automatically obtain the transformed program
above, they are developed from scratch and tailored to some analysis of
interest. Recently, CFR has also been considered by Albert et al. [3] to
improve cost analysis as well, but from a different perspective that uses
termination witnesses to guide CFR.
Since CFR is, in principle, a program transformation that specialises programs
to distinguish different execution scenarios, Domenech et al. [16] explored
the use of general-purpose specialisation techniques for CFR, in particular
the techniques of Gallagher [19] for partial evaluation of Constrained Horn
Clauses (CHCs). Basing CFR on partial evaluation has the clear advantage that
soundness comes for free because partial evaluation guarantees semantic
equivalence between the original program and its transformed version.
Moreover, this way we obtain a CFR procedure that is not tailored for a
particular purpose, but rather can be tuned depending on the application
domain.
Domenech et al. [16] developed such a CFR procedure for TSs, by transforming
TSs into CHCs and using the partial evaluator of Gallagher [19], and
integrated it in a termination analysis algorithm in a way that allows
applying CFR at different levels of granularity, and thus controlling the
trade-off between precision and performance. This is done by suggesting
different schemes for applying CFR, not only as a preprocessing step but also
on specific parts of the TS which we could not prove terminating. Moreover,
they developed heuristics for automatically configuring partial evaluation
(i.e., inferring properties to guides specialisation) in order to achieve the
desired CFR. Experimental evaluation provides a clear evidence to that their
CFR procedure significantly improves the precision of termination analysis,
cost analysis, and invariants generation.
Let use demonstrate this CFR procedure on the TS ${\cal T}$ that is depicted
in Figure 1. In a first step, the TS ${\cal T}$ is translated into the
following (semantically) equivalent CHC program:
$\begin{array}[]{rlllll}q_{\\_}{\mathtt{n_{\\_}0}}(x,y,z)\leftarrow&q_{\\_}{\mathtt{n_{\\_}1}}(x,y,z).\\\
q_{\\_}{\mathtt{n_{\\_}1}}(x,y,z)\leftarrow&\\{x\geq
1\\},&q_{\\_}{\mathtt{n_{\\_}2}}(x,y,z).\\\
q_{\\_}{\mathtt{n_{\\_}1}}(x,y,z)\leftarrow&\\{x\leq
0\\},&q_{\\_}{\mathtt{n_{\\_}3}}(x,y,z).\\\
q_{\\_}{\mathtt{n_{\\_}2}}(x,y,z)\leftarrow&\\{y\leq
z-1,&y^{\prime}=y+1\\},&q_{\\_}{\mathtt{n_{\\_}1}}(x,y^{\prime},z).\\\
q_{\\_}{\mathtt{n_{\\_}2}}(x,y,z)\leftarrow&\\{y\geq
z,&x^{\prime}=x-1\\},&q_{\\_}{\mathtt{n_{\\_}1}}(x^{\prime},y,z).\\\
\end{array}$
Then the partial evaluator of Gallagher [19] is applied, using the
(automatically inferred) properties $\\{x\geq 1,y\geq z\\}$ for the loop head
predicate $q_{\\_}{\mathtt{n_{\\_}1}}(x,y,z)$, which results in the following
CHC program:
$\begin{array}[]{rllllll}q_{\\_}{\mathtt{n_{\\_}0}}(x,y,z)\leftarrow&q_{\\_}{\mathtt{n_{\\_}1^{3}}}(x,y,z).\\\
q_{\\_}{\mathtt{n_{\\_}1^{3}}}(x,y,z)\leftarrow&\\{x\leq
0\\},&q_{\\_}{\mathtt{n_{\\_}3^{2}}}(x,y,z).\\\
q_{\\_}{\mathtt{n_{\\_}1^{3}}}(x,y,z)\leftarrow&\\{x\geq
1\\},&q_{\\_}{\mathtt{n_{\\_}2^{2}}}(x,y,z).\\\
q_{\\_}{\mathtt{n_{\\_}2^{2}}}(x,y,z)\leftarrow&\\{x\geq 1,&y\leq
z-1,&y^{\prime}=y+1\\},&q_{\\_}{\mathtt{n_{\\_}1^{2}}}(x,y^{\prime},z).\\\
q_{\\_}{\mathtt{n_{\\_}2^{2}}}(x,y,z)\leftarrow&\\{x\geq 1,&y\geq
z,&x^{\prime}=x-1\\},&q_{\\_}{\mathtt{n_{\\_}1^{1}}}(x^{\prime},y,z).\\\
q_{\\_}{\mathtt{n_{\\_}1^{2}}}(x,y,z)\leftarrow&\\{x\geq
1\\},&q_{\\_}{\mathtt{n_{\\_}2^{2}}}(x,y,z).\\\
q_{\\_}{\mathtt{n_{\\_}1^{1}}}(x,y,z)\leftarrow&\\{x\leq 0,&y\geq
z\\},&q_{\\_}{\mathtt{n_{\\_}3^{1}}}(x,y,z).\\\
q_{\\_}{\mathtt{n_{\\_}1^{1}}}(x,y,z)\leftarrow&\\{x\geq 1,&y\geq
z\\},&q_{\\_}{\mathtt{n_{\\_}2^{1}}}(x,y,z).\\\
q_{\\_}{\mathtt{n_{\\_}2^{1}}}(x,y,z)\leftarrow&\\{x\geq 1,&y\geq
z,&x^{\prime}=x-1\\},&q_{\\_}{\mathtt{n_{\\_}1^{1}}}(x^{\prime},y,z).\\\
\end{array}$
Now translating this CHC program into a TS results in the TS ${\cal
T}_{\\_}{pe}$ that is depicted in Figure 1. Note that the two phases are now
separated into two different strongly connected components, which can be
proven terminating using only LRFs.
## 4 Concluding Remarks
In this paper we have discussed techniques for proving termination of programs
with multiphase control-flow, where the execution passes through several
(possibly implicit) phases. In particular, we have discussed techniques that
correspond to our recent work: (1) the use of multiphase ranking functions [9,
5]; and (2) the use of control-flow refinement [16]. As a byproduct of our
research, we have developed an open-source termination analyser called
iRankFinder 222http://irankfinder.loopkiller.com that implements all these
techniques as well as other state-of-the-art techniques. Some of the
components of iRankFinder can be used independently, in particular the CFR
component that can be used to incorporate CFR in static analysers with little
effort.
### Future Work.
For M$\Phi$RFs, an obvious future direction is to study the problem of
deciding whether a TS has a M$\Phi$RF, both from algorithmic and theoretical
complexity perspectives. In initial unpublished work, we have proven that the
corresponding decision problem is NP-hard for the rational setting, but we
could not obtain a further classification. Further exploration of the
M$\Phi$RF problem for SLC loops is also required since it is not solved for
the general case yet. For CFR, we have developed some heuristics for the
automatic inference of properties, which is crucial for obtaining the desired
transformations, but further research in this direction is required. We
concentrated on CFR of TSs, and in a future direction one could apply our CFR
techniques for program representations that allow recursion as well.
Technically, this would not require much work since the partial evaluation
techniques of Gallagher [19] specialise CHCs that include recursion already.
We also concentrated on numerical programs, and a possible future direction
can concentrate on using CFR for program analysis tools where the data is not
necessarily numerical. Here one should also adapt the partial evaluation
techniques to support such specialisations, which seems doable for the partial
evaluation techniques of Gallagher [19] since it is based on using abstract
properties like those used in abstract domains of program analysis.
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|
# Neutrino meets ultralight dark matter: $\boldsymbol{0\nu\beta\beta}$ decay
and cosmology
Guo-yuan Huang<EMAIL_ADDRESS>Max-Planck-Institut für
Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany Newton Nath
<EMAIL_ADDRESS>Istituto Nazionale di Fisica Nucleare, Via Orabona 4,
70126 Bari, Italy
###### Abstract
We explore the neutrinoless double beta ($0\nu\beta\beta$) decay induced by an
ultralight dark matter field coupled to neutrinos. The effect on
$0\nu\beta\beta$ decay is significant if the coupling violates the lepton
number, for which the $\Delta L=2$ transition is directly driven by the dark
matter field without further suppression of small neutrino masses. As the
ultralight dark matter can be well described by a classical field, the effect
features a periodic modulation pattern in decay events. However, we find that
in the early Universe such coupling will be very likely to alter the standard
cosmological results. In particular, the requirement of neutrino free-
streaming before the matter-radiation equality severely constrains the
parameter space, such that the future $0\nu\beta\beta$ decay experiments can
hardly see any signal even with a meV sensitivity to the effective neutrino
mass.
## I Introduction
In spite of its remarkable success, the Standard Model (SM) of particle
physics fails to address several fundamental issues, e.g., the nature of dark
matter (DM). The mass of viable DM candidates can cover a huge landscape
spanning from the primordial black holes Bird:2016dcv ; Carr:2016drx ;
Chen:2016pud ; Georg:2017mqk to the ultralight regime Hu:2000ke . At present,
the DM direct detection experiments such as Xenon1T XENON:2020gfr , PandaX-II
PandaX-II:2017hlx come out with null signal of weakly interacting massive
particle (WIMP), one of the most promising DM candidates, hence setting
stringent limits on the available WIMP parameter space. Another particular DM
class, the ultralight DM Ferreira:2020fam ; Urena-Lopez:2019kud with a mass
ranging from $\mathcal{O}(10^{-22})~{}{\rm eV}$ 111This is often called fuzzy
dark matter Hu:2000ke ; Hui:2016ltb . The term “fuzzy” corresponds to a huge
Compton wavelength, $\lambda=2\pi/m_{\phi}\simeq 0.4\,{\rm
pc}\times(10^{-22}\,\text{eV}/m_{\phi})$ with $m_{\phi}$ being the DM mass.
to $\mathcal{O}(1)~{}{\rm eV}$, for example the well-motivated quantum
chromodynamics (QCD) axion Kim:2008hd ; Graham:2015ouw ; Irastorza:2018dyq ;
Marsh:2015xka , is receiving renewing attention. Different from WIMP, the
ultralight DM is produced non-thermally in the early Universe and can be well
described by a classical-number field.
Beside DM, the nature of massive neutrinos, i.e., whether they are Majorana or
Dirac fermions, is yet to be answered. At present, the only feasible process
that can uncover the nature of neutrinos is the neutrinoless double beta
($0\nu\beta\beta$) decay, which violates the lepton number by two units
DellOro:2016tmg ; Vergados:2016hso .
There have been many attempts to connect the mentioned two mysterious sectors,
i.e., neutrinos and ultralight DM Berlin:2016woy ; Brdar:2017kbt ;
Krnjaic:2017zlz ; Liao:2018byh ; Capozzi:2018bps ; Reynoso:2016hjr ;
Huang:2018cwo ; Pandey:2018wvh ; Farzan:2018pnk ; Choi:2019ixb ; Baek:2019wdn
; Choi:2019zxy ; Choi:2020ydp ; Dev:2020kgz ; Baek:2020ovw ; Losada:2021bxx ;
Smirnov:2021zgn ; Alonso-Alvarez:2021pgy ; Huang:2021zzz . In this work, we
investigate $0\nu\beta\beta$ decays in the presence of a coupling between
neutrinos and the ultralight DM with a general mass spanning from
$10^{-22}~{}{\rm eV}$ to sub-eV. Among general interaction forms, the scalar
and tensor couplings are able to violate the lepton number (or equivalently,
the DM particle carries two units of lepton number), and hence important for
$0\nu\beta\beta$-decay experiments. The Lagrangian reads
$\displaystyle-\mathcal{L}^{\rm M}_{\rm
int}=\begin{dcases}\frac{g_{\phi}^{\alpha\beta}}{2}\,\phi\,\overline{\nu_{\rm
L\alpha}}\nu^{\rm c}_{\rm L\beta}+{\rm h.c.}\;,\\\\[2.15277pt]
\frac{g_{T}^{\alpha\beta}}{2}T^{\mu\nu}\overline{\nu_{\rm
L\alpha}}\sigma_{\mu\nu}\nu^{\rm c}_{\rm L\beta}+{\rm h.c.}\;,\end{dcases}$
(1)
where $\alpha,\beta=e,\mu,\tau$ are the flavor indices, and $\phi$ and
$T^{\mu\nu}$ represent real scalar and tensor fields with neutrino coupling
constants $g_{\phi}^{\alpha\beta}$ and $g_{T}^{\alpha\beta}$, respectively. We
have assumed that neutrinos are Majorana particles, such that only the field
$\nu_{\rm L}$ is involved at low energies.
In analogy with the electromagnetic force, the tensor interaction can be
generated by a dark photon field $A^{\prime}$ coupled to neutrinos with a dark
magnetic dipole moment Capozzi:2018bps . However, for Majorana neutrinos, only
transition magnetic moment exists and the diagonal part of the tensor coupling
constant $g_{T}^{\alpha\beta}$ is vanishing, thus not contributing to the
$0\nu\beta\beta$ process which exchanges two electron neutrinos. Hence, we
will restrict our following discussions to only the ultralight scalar DM. It
is worth mentioning that the transition magnetic moment can be relevant for
the lepton-number-violating meson decays by exchanging different flavors of
neutrinos.
For Majorana neutrinos, $\nu\equiv\nu_{\rm L}+{\nu^{\rm c}_{\rm L}}$, in the
background of scalar DM, the Hamiltonian density of the system is
$\displaystyle\mathcal{H}=\frac{1}{2}(\partial\phi)^{2}+\frac{1}{2}m^{2}_{\phi}\phi^{2}+\frac{1}{2}\overline{\nu}(\mathrm{i}\not{\partial}+\widehat{M}_{\nu}+g\phi)\nu\;,$
(2)
where $m_{\phi}$ is the DM mass, $\widehat{M}_{\nu}$ is the neutrino mass
matrix in vacuum, and the coupling constant $g$ is a real symmetric matrix in
general. Note that the pseudoscalar coupling has been ignored in the
expression, which will induce a similar effect. The corresponding equation of
motion is given by
$\displaystyle(\partial^{2}+m^{2}_{\phi})\phi=-\frac{1}{2}g\overline{\nu}\nu\;,$
(3) $\displaystyle i\not{\partial}\nu=(\widehat{M}_{\nu}+g\phi)\nu\;.$ (4)
As long as the neutrino number density is not too large in Eq. (3), the
ultralight DM evolves as free classical field in the Universe as
$\phi=\phi_{0}\cos{m_{\phi}t}$ after the scalar mass “freeze-in”, i.e.,
$H<m_{\phi}$, with $H$ being the Hubble expansion rate. The field strength is
given by $\phi_{0}=\sqrt{2\rho}/m_{\phi}$, where $\rho\approx
m_{\phi}n_{\phi}\approx 0.3~{}{\rm GeV\cdot cm^{-3}}$ is the DM energy density
in our local galaxy.
The large occupation number of local ultralight DM, namely
$N=n_{\phi}\lambda^{3}\approx 6\times 10^{36}\,(10^{-10}~{}{\rm
eV}/m_{\phi})^{4}$ within the cube of Compton wavelength, guarantees the
classical-field description as an excellent approximation to the more
fundamental quantum field. The formation of the classical field in our local
galaxy will generate a time-varying effective Majorana mass $m_{\nu,{\rm
eff}}=g\phi$ to neutrinos, which is very similar to the Higgs mechanism. At
the particle level, the $0\nu\beta\beta$ decays induced by ultralight DM are
shown as diagrams in Fig. 1. The illustrated leading-order contribution is
accurate enough since the coupling of our concern is set to be extremely
small. The emission process has been discussed before in the context of
Majoron model Blum:2018ljv ; Brune:2018sab ; Cepedello:2018zvr . However,
different from the “spontaneous” Majoron emission process, in our case both
the absorption and emission of scalars benefit from the large occupation
number of ultralight DM, which gives rise to a dramatic stimulation
enhancement.
Figure 1: Feynman diagrams of $0\nu\beta\beta$ decays induced by the
absorption and emission of ultralight scalar DM. Because these transitions
take place in a background of identical scalar bosons, there will be an
enhancement of stimulation.
In the rest of the work, we first describe how the coupled system of
ultralight DM and neutrinos evolves in the early Universe in Sec. II. Based on
the robust cosmic microwave background (CMB) observations, we confine the
allowed parameter space of the mass and coupling of ultralight DM. In Sec.
III, we explore the possible impacts on $0\nu\beta\beta$ decays. Conclusions
are presented in Sec. IV.
## II Cosmological evolution
A popular way to generate the coherent scalar field in the early Universe is
via the misalignment mechanism, which is well known for the QCD axion
Kim:2008hd ; Graham:2015ouw ; Irastorza:2018dyq ; Marsh:2015xka . The
production is typically associated with a global symmetry, which is broken
spontaneously and results in a Goldstone mode. The later dynamics will break
the shift-symmetry of the Goldstone by selecting a preferred phase. A mass
term will then be generated to the field, which populates the DM energy
density after $H\lesssim m_{\phi}$. The initial condition of a misaligned
scalar in our scenario reads Marsh:2015xka
$\displaystyle\phi(t_{\rm i},x)=f_{\phi}\theta_{\phi}\;,~{}\dot{\phi}(t_{\rm
i},x)=0\;,$ (5)
where $t_{\rm i}$ is the time set by the spontaneous symmetry breaking of the
global symmetry, $f_{\phi}$ represents the energy scale of the symmetry
breaking, $\theta_{\phi}$ is the initial phase, and
$\dot{\phi}\equiv\mathrm{d}\phi/\mathrm{d}t$.
At the quantum field level, the ultralight DM produced by the misalignment
mechanism (from the dynamical symmetry breaking) should be descried by a
coherent state of the form Marsh:2015xka ; Ferreira:2020fam ; Miransky1994 ;
Ferreira:lecture ; Davidson:2014hfa ; Veltmaat2020GalaxyFW ; Guth:2014hsa ;
Zhang:1999is ; Barnett:1997
$\displaystyle\left|\phi\right\rangle=\mathrm{e}^{-\frac{N}{2}}\mathrm{e}^{\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\tilde{\phi}(k)\hat{a}^{\dagger}_{k}}\left|0\right\rangle\;,$
(6)
where $\mathrm{e}^{-\frac{N}{2}}$ is a normalization factor, $\tilde{\phi}(k)$
is a general complex field with the wave-vector $k$, and
$\hat{a}^{\dagger}_{k}$ is the corresponding particle creation operator. The
coherent state $|\phi\rangle$ has been thought of as the most field-like
solution of quantum states. The property of this state can be largely captured
by the expectation value of the field operator $\hat{\phi}$, namely
$\displaystyle\phi(t,x)\equiv\left\langle\phi\right|\hat{\phi}\left|\phi\right\rangle$
$\displaystyle=$
$\displaystyle\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\frac{1}{\sqrt{2E_{k}}}\left[\tilde{\phi}(k)\mathrm{e}^{-i(\omega
t-k\cdot x)}+\right.$ (7)
$\displaystyle\left.+\tilde{\phi}^{*}(k)\mathrm{e}^{i(\omega t-k\cdot
x)}\right]\;,$
which is a classical-number field. This non-vanishing expectation value can be
interpreted in a way similar to the Higgs mechanism. The evolution of the
classical field $\phi(t,x)$ right follows the equation of motion in Eq. (3).
For the isotropic DM field without kinematic energy, i.e., $k=0$, we have
$\left|\phi\right\rangle=\mathrm{e}^{-\frac{N}{2}}\mathrm{e}^{\sqrt{N}\mathrm{e}^{\mathrm{i}\theta}\hat{a}^{\dagger}_{0}}\left|0\right\rangle$.
It should be emphasized that the coherent state $|\phi\rangle$ is an
eigenstate of $\hat{a}_{0}$, i.e.,
$\hat{a}_{0}|\phi\rangle=\sqrt{N}\mathrm{e}^{\mathrm{i}\theta}|\phi\rangle$,
but not that of the particle number $\hat{N}=\hat{a}^{\dagger}_{0}\hat{a}_{0}$
in the Fock space. However, for a large enough ensemble of DM, there is a
well-defined mean particle number
$\langle\hat{a}^{\dagger}_{0}\hat{a}_{0}\rangle=N$ with a quantum fluctuation
proportional to $1/\sqrt{N}$. In fact, for the state with a definitive
particle number, e.g., $\hat{a}^{\dagger}_{0}\hat{a}_{0}|N\rangle=N|N\rangle$,
the field expectation value $\left\langle
N\right|\hat{\phi}\left|N\right\rangle$ is vanishing. The difference between
$|\phi\rangle$ and $|{N}\rangle$ is critical and should be highlighted, as
they will induce very distinct signals experimentally. For instance, for
$|{N}\rangle$ the absolute quantum phase
$\mathrm{exp}(-\mathrm{i}\,m_{\phi}t)$ is meaningless in practice, but the
classical phase of $|\phi\rangle$ is physical and will lead to time-varying
signals DirectMeasurementofLightWaves ; Barnett1996 .
In the early Universe, the problem is ascribed to solving the evolution of
classical field. Because the ultralight DM should be homogeneously produced,
the evolution equation of Eq. (3) can be recast to Esteban:2021ozz
$\displaystyle\ddot{\phi}+3H\dot{\phi}+m^{2}_{\phi}\phi=-g\int\frac{\mathrm{d}^{3}\bm{p}}{(2\pi)^{3}}\frac{g\phi}{\sqrt{\bm{p}^{2}+(g\phi)^{2}}}f_{\nu}(a|\bm{p}|),\hskip
14.22636pt$ (8)
where $a$ is the scale factor, connected to the redshift $z$ via $a\equiv
a_{0}/(z+1)$ with $a_{0}$ being the scale factor at present. Here, the
neutrino distribution function simply follows
$f_{\nu}(a|\bm{p}|)=D_{\nu}/(\mathrm{e}^{|\bm{p}|/T_{\nu}(a)}+1)$ with the
temperature $T_{\nu}(a)\propto 1/a$ and the factor $D_{\nu}=6$ accommodating
three generations of neutrinos and antineutrinos. The source term on the
right-hand side of Eq. (8) modifies the evolution of a free scalar. Note that
we have assumed the vacuum neutrino mass to be vanishing in the above
equation, and a larger effect of neutrino source will be expected if the
vacuum neutrino mass is taken into account.
### II.1 Observable impacts
If $T_{\nu}\gg g\phi$ is satisfied (i.e., neutrinos in the plasma are
relativistic), the source term is equivalent to a screening mass of the
scalar, with Davoudiasl:2018hjw ; Esteban:2021ozz
$\displaystyle m^{2}_{\phi,\rm
s}(a)=\int\frac{\mathrm{d}^{3}\bm{p}}{(2\pi)^{3}}\frac{g^{2}}{|\bm{p}|}f_{\nu}(a|\bm{p}|)\sim
g^{2}T^{2}_{\nu}.\hskip 14.22636pt$ (9)
The screening mass describes how the neutrino plasma should response to the
change of the scalar potential, and its main effect will be altering the
oscillating frequency of scalar field. In this case, we have
$\displaystyle\ddot{\phi}+3H\dot{\phi}+\left[m^{2}_{\phi}+m^{2}_{\phi,\rm
s}(a)\right]\phi=0\;,\hskip 14.22636pt$ (10)
with an effective mass varying as the Universe expands, $m_{\phi,{\rm
eff}}=[m^{2}_{\phi}+m^{2}_{\phi,\rm s}(a)]^{1/2}$. For the numerical
convenience, it is useful to replace the physical time $t$ with the scale
factor $a=m\,x$, where $m$ is an arbitrary scale and $x$ is dimensionless. The
equation of motion can be recast into
$\displaystyle
x^{2}\phi^{\prime\prime}+\left(4x+x^{2}\frac{H^{\prime}}{H}\right)\phi^{\prime}+\frac{m^{2}_{\phi,{\rm
eff}}}{H^{2}}\phi=0\;,$ (11)
where ${\phi}^{\prime}\equiv\mathrm{d}\phi/\mathrm{d}x$.
The scalar field $\phi$ starts to oscillate when $H\lesssim m_{\phi,{\rm
eff}}$. If $\phi$ is not coupled to neutrinos, its amplitude will simply scale
as $\phi_{0}\propto a^{-3/2}$ Marsh:2015xka , and the energy density
$\rho_{\phi}\approx m^{2}_{\phi}\phi^{2}_{0}/2$ will be proportional to
$a^{-3}$, which justifies its capability of being the cold DM. However, if the
scalar mass is significantly screened, this benefit will be lost due to the
behavior $m_{\phi,{\rm s}}\propto a^{-1}$. In the completely screened case,
i.e. $m_{\phi,{\rm s}}\gg m_{\phi}$, we find 222In the fast-oscillating
regime, to obtain the evolution of the amplitude we decompose the scalar field
as $\phi(t)=\phi_{0}(t)\cos{(m_{\phi,{\rm eff}}\cdot t)}$ Marsh:2015xka .
Keeping the leading order in the assumption of $H\sim\dot{\phi}_{0}\ll
m_{\phi,{\rm eff}}$, we obtain
$2\dot{\phi_{0}}=-3H\phi_{0}-\phi_{0}\dot{m}_{\phi,{\rm eff}}/m_{\phi,{\rm
eff}}$. For $m_{\phi,{\rm eff}}\propto a^{-1}$, we have
$\dot{\phi_{0}}=-H\phi_{0}$, and hence $\phi_{0}\propto a^{-1}$.
$\displaystyle\phi_{0}\propto
a^{-1}\;,~{}\rho_{\phi}\sim\frac{m^{2}_{\phi,{\rm s}}\phi^{2}_{0}}{2}\propto
a^{-4}\;,$ (12)
which indicates that $\phi$ evolves more like a radiation instead of cold DM
with the screened mass. The screening effect is thus to be constrained by the
cosmological observations if we require the scalar to be DM.
An example of $\phi^{2}$ ($\geq 0$) evolution with respect to the scale factor
$a$ is given in Fig. 2 by numerically solving Eq. (11). The lighter fast-
oscillating curves stand for the magnitude of $\phi^{2}$ with (red) or without
(blue) the screening effect, and the darker ones are produced by averaging
over fast-oscillating modes to see the overall amplitude. As the screening
mass $m_{\phi,{\rm s}}$ is dominant for $a<500$, the scalar field scales as
$a^{-1}$ instead of $a^{-3/2}$ and contributes to the radiation component.
After $a=500$ the vacuum mass starts to dominate over the screening one, and
the scalar field plays the role of DM as normal.
Figure 2: A demonstration of evolution of ultralight DM as a function of the
scale factor. For demonstration we give $\phi^{2}$ in the logarithmic scale.
The case with the screening effect of neutrinos is shown as the red curve,
while that without the screening effect is given as the blue one. The lighter
curves are the actual solutions, and the darker ones are averaged over certain
scale factor to see the magnitude of the field.
Another effect to avoid is the impact on neutrino velocity $v_{\nu}$, which is
related to the neutrino free-streaming length before the decoupling of CMB. As
has been mentioned, in the background of ultralight DM, neutrinos acquire an
effective mass $m_{\nu,{\rm eff}}=g\phi$. The neutrino velocity in the early
Universe will be given by
$\displaystyle\frac{v_{\nu}}{c}=\frac{|\bm{p}|}{\sqrt{\bm{p}^{2}+m^{2}_{\nu,{\rm
eff}}}}\;,$ (13)
where the average momentum of neutrino plasma reads
$\langle|\bm{p}|\rangle\sim 3T_{\nu}$. Notice that the effective neutrino mass
$m_{\nu,{\rm eff}}\propto a^{-3/2}$ increases faster than $T_{\nu}\propto
a^{-1}$ as we go to higher redshift $z$. This means that neutrinos will become
more and more non-relativistic and hence less and less free-streaming at
higher $z$. In comparison, the CMB data are consistent with neutrino free-
streaming with a speed of light up to a redshift as high as $z=10^{5}$
Hannestad:2004qu ; Hannestad:2005ex ; Bell:2005dr ; Basboll:2008fx ;
Archidiacono:2013dua ; Forastieri:2015paa ; Cyr-Racine:2013jua ;
Oldengott:2014qra ; Forastieri:2017oma ; Oldengott:2017fhy ; Choudhury:2020tka
; Brinckmann:2020bcn ; Das:2020xke ; Mazumdar:2020ibx . For simplicity, in the
following we will consider neutrinos as free-streaming with the speed of light
if $T_{\nu}>g\phi$, and vise versa.
It is worth noting that the CMB constraint on the sum of neutrino masses is
not as competitive as the above two effects. The combined analysis of
$Planck~{}{\rm TT,TE,EE+lowE+lensing+BAO}$ data leads to $\Sigma
m_{\nu}<0.12~{}{\rm eV}$ Planck:2018vyg , which actually takes account of data
at very low redshift long after the recombination $z_{\rm rc}=1100$. In fact,
around the recombination era neutrinos with a mass $\mathcal{O}(0.1)~{}{\rm
eV}$ are still relativistic. Since $m_{\nu,{\rm eff}}\propto a^{-3/2}$, the
impact of effective neutrino masses turns out to be less and less important at
lower $z$. Hence, we do not attempt to set the bound according to $\Sigma
m_{\nu,\rm eff}(z)<0.12~{}{\rm eV}$ at recombination $z_{\rm rc}=1100$, which
has been considered previously.
Figure 3: The evolution of average neutrino velocity in terms of the redshift
in the early Universe. The colorful curves stand for the cases with a coupling
between neutrinos and the ultralight DM, with the vacuum neutrino mass
vanishing and DM-induced neutrino mass $g\phi$ at present indicated close to
the curve. The black curve represents the neutrino velocity of the standard
case with a mass $m_{\nu}=0.04~{}{\rm eV}$.
### II.2 Updated CMB constraints
Depending on the parameter choice of $g$ and $m_{\phi}$, there can be four
different scenarios in the early Universe for the evolution equation of Eq.
(8):
* •
Case I: $g\phi<T_{\nu}$ and $g^{2}T^{2}_{\nu}<m^{2}_{\phi}$. The effective
neutrino mass is negligible compared to the neutrino temperature and the
screening effect is also not important. In this case, the scalar field evolves
like matter with $\phi_{0}\propto a^{-3/2}$ while $T_{\nu}\propto a^{-1}$, and
these two conditions will maintain themselves as the Universe expands.
* •
Case II: $g\phi<T_{\nu}$ and $g^{2}T^{2}_{\nu}>m^{2}_{\phi}$. Neutrinos are
relativistic but the screening scalar mass $m_{\phi,\rm s}\sim gT_{\nu}$
dominates the oscillation of $\phi$. In such scenario, the scalar field
evolves like radiation with $\phi_{0}\propto a^{-1}$. This state is not stable
and will collapse into Case I eventually.
* •
Case III: $g\phi>T_{\nu}$ and $gT^{3}_{\nu}<m^{2}_{\phi}\phi$. The screening
effect is not important compared to the vacuum mass of $\phi$, but neutrinos
are non-relativistic (thus not free-streaming with $c$) due to the large
effective mass. This case will also end up into Case I as the Universe
expands.
* •
Case IV: $g\phi>T_{\nu}$ and $gT^{3}_{\nu}>m^{2}_{\phi}\phi$. Both the
screening effect and the effective neutrino mass are dominant factors, which
is to be avoided over the observable history of CMB.
For Cases I and III, in terms of redshift, the neutrino temperature and the
scalar field strength read as $T_{\nu}\approx 1.68\times 10^{-4}~{}{\rm
eV}(z+1)$ and $\phi\approx 2.15\times 10^{-3}~{}{\rm
eV}^{2}/m_{\phi}\left[(z+1)/45\right]^{3/2}$, respectively, assuming the
ultralight scalar accommodates all the DM abundance.
The evolution of the Universe can be in a sequence of Case III$\to$I or
II$\to$I. In Fig. 3, we illustrate the neutrino velocity evolution $\langle
v_{\nu}\rangle/c$, i.e., Eq. (13), in terms of the redshift $z$ from Case III
to I, where the screening effect is not important but the neutrino effective
mass is severely affected. This can be achieved by properly choosing the DM
parameter. It can be noticed that the neutrino velocity drops rapidly as we go
back to higher $z$, which should be avoided to be in accordance with CMB free-
streaming requirement. In comparison, we also give the neutrino velocity in
the standard case with a constant mass $m_{\nu}=0.04~{}{\rm eV}$ corresponding
to the Planck limit $\Sigma m_{\nu}<0.12~{}{\rm eV}$. In this case, the
neutrino velocity evolves in an opposite manner compared to that with a
coupling to the ultralight DM. As has been mentioned, around the recombination
era $z_{\rm rc}\approx 1100$, neutrinos of the Planck limit are still ultra-
relativistic, and it is the data at lower $z$ that contribute to this limit.
In order not to significantly alter the CMB observations, two criteria should
be satisfied: (1) the ultralight scalar must evolve as matter before the
matter-radiation equality around $z_{\rm eq}=3000$; (2) neutrinos should be
free-streaming up to the redshift $z_{\rm eq}=3000$ (conservative) or $z_{\rm
fs}=10^{5}$ (aggressive). The aggressive redshift for neutrino free-streaming
is inspired by the investigation of neutrino self-interactions
Hannestad:2004qu ; Hannestad:2005ex ; Bell:2005dr ; Basboll:2008fx ;
Archidiacono:2013dua ; Forastieri:2015paa ; Cyr-Racine:2013jua ;
Oldengott:2014qra ; Forastieri:2017oma ; Oldengott:2017fhy ; Choudhury:2020tka
; Brinckmann:2020bcn ; Das:2020xke ; Mazumdar:2020ibx , which similarly
shrinks the neutrino free-streaming length. One may argue that the scalar
becomes a dynamical field (acquiring the shift-symmetry breaking terms) only
after a certain redshift, e.g., $z=10^{4}$, such that the aggressive free-
streaming limit does not apply in general. But there is no way to avoid the
robust limit set at $z_{\rm eq}=3000$, because the CMB data require the
dominance of matter component after that redshift.
If these two criteria are satisfied, during the observable history of CMB the
ultralight scalar will play the role of cold DM and neutrinos will stream as
free particles. We can evolve the current Universe back to higher redshift for
the CMB constraints. To be more specific, we require $g\phi<T_{\nu}$ and
$g^{2}T^{2}_{\nu}<m^{2}_{\phi}$ before $z=3000$ for the conservative limit,
and $g\phi<T_{\nu}$ before $z=10^{5}$ for the aggressive one. We obtain our
CMB limits as
$\displaystyle g$ $\displaystyle<$ $\displaystyle 0.4\cdot\frac{m_{\phi}}{\rm
eV}\hskip 14.22636pt({\rm conservative}),$ (14) $\displaystyle g$
$\displaystyle<$ $\displaystyle 0.09\cdot\frac{m_{\phi}}{\rm eV}\hskip
9.67383pt({\rm aggressive}),$ (15)
which correspond to the limits on the effective neutrino mass $g\phi$ in our
local galaxy
$\displaystyle m_{\nu,{\rm eff}}$ $\displaystyle<$ $\displaystyle 9\times
10^{-4}~{}{\rm eV}\hskip 11.38092pt({\rm conservative}),$ (16) $\displaystyle
m_{\nu,{\rm eff}}$ $\displaystyle<$ $\displaystyle 2\times 10^{-4}~{}{\rm
eV}\hskip 11.38092pt({\rm aggressive}).$ (17)
These limits are dominated by the neutrino free-streaming requirement. We find
that, for the aggressive limit, as we go to higher redshift the state will
transit from Case I to II around $z=6\times 10^{4}$, after which the free-
streaming condition $g\phi<T_{\nu}$ becomes stabilized as $\phi_{0}\propto
a^{-1}$ for Case II. For $g$ larger than the aggressive limit Case I will
transit to III instead before $z_{\rm fs}=10^{5}$, and the free-streaming
requirement is spoiled. We need to point out that a thorough analysis of the
CMB spectrum by simulating the evolution of perturbations in this context will
make the results more robust, and we leave it as a possible future work.
## III Implication for $\boldsymbol{0\nu\beta\beta}$ decays
There are many dedicated experiments that are looking for the signatures of
$0\nu\beta\beta$ decays, e.g., GERDA Phase-II Agostini:2018tnm , CUORE
Alduino:2017ehq , SuperNEMO Barabash:2011aa , KamLAND-Zen KamLAND-Zen:2016pfg
and EXO Agostini:2017jim . The discovery of lepton-number-violating processes
in the future can reveal the Majorana nature of massive neutrinos
Dolinski:2019nrj .
As has been shown in Fig. 1, even without the vacuum Majorana neutrino mass
the $0\nu\beta\beta$ transition can still take place in the background of
ultralight DM. The induced transition is very similar to the Majoron emission
process in vacuum. However, because the vacuum in our case is occupied by
dense coherent scalars, the dominant process will be the “stimulated” emission
and absorption of the scalar particle, in analogy with the atomic transition
in a laser field.
Because the occupation number of scalars $N=n_{\phi}\lambda^{3}$ is very
large, such a transition will leave the DM field almost unaffected. This is
equivalent to simply taking the expectation value for all relevant operators
in the Lagrangian or scattering matrix. In the presence of ultralight DM, the
effective neutrino mass ($m_{ee}$) will hence receive an additional time-
varying contribution
$\displaystyle{m}^{\prime}_{ee}(t)=\langle\phi|m_{ee}+g\hat{\phi}(t,x)|\phi\rangle=m_{ee}+g{\phi}(t,x)\;,$
(18)
The $0\nu\beta\beta$-decay process typically takes place with a very small
uncertainty of time. It is therefore a good approximation to take the factor
$\cos{m_{\phi}t}$ as a constant during the transition. But the long-term
events will feature a time-varying pattern if enough data have been collected.
The above result should be able to be reproduced from the more fundamental
quantum field theory. However, it is not apparent to obtain the time-varying
effect from a view of the Feynman diagrams as illustrated in Fig. 1, where the
emission and absorption processes are themselves time-independent. To
understand this, we note that the mass of the ultralight DM ($<{\rm eV}$),
namely the change in the energy of final electrons, is much smaller than the
energy resolution of $0\nu\beta\beta$-decay experiments ($\sim{\rm keV}$).
Hence, it is not possible at the quantum level to distinguish between the
emission and absorption diagrams, which will lead to an interference. One can
find that it is this interference that gives rise to the time variation. To be
more precise, after the $0\nu\beta\beta$ transition via the vertex
$(m_{ee}+g\hat{\phi})\overline{\nu^{\rm c}_{\rm L}}\nu_{\rm L}/2$ the scalar
state will be taken to
$\displaystyle\left[m_{ee}+g\hat{\phi}(t,x)\right]\left|\phi\right\rangle$
$\displaystyle=\left[m_{ee}+g\sqrt{\frac{2}{m_{\phi}V}}\left(\hat{a}\mathrm{e}^{-\mathrm{i}m_{\phi}t}+\hat{a}^{\dagger}\mathrm{e}^{\mathrm{i}m_{\phi}t}\right)\right]|\phi\rangle\;.$
(19)
Note that $|\phi\rangle$, $\hat{a}|\phi\rangle$ and
$\hat{a}^{\dagger}|\phi\rangle$ almost completely overlap with each other,
e.g.,
$\langle\phi|\hat{a}\,\hat{a}|\phi\rangle=N\mathrm{e}^{2\mathrm{i}\theta}$.
Together with other terms, this feature results in a probability proportional
to $\Sigma_{N}|\left\langle N\right|m_{ee}+g\hat{\phi}(t,x)|\phi\rangle|^{2}$,
namely $\left|m_{ee}+g{\phi}(t,x)\right|^{2}$. The above observations are also
applicable to the impact on neutrino oscillation experiments.
As for our $0\nu\beta\beta$ process, the half-life of a given isotope can be
expressed as Rodejohann:2011mu
$\dfrac{1}{T^{0\nu}_{1/2}}=G_{0\nu}\cdot\left|{M}_{0\nu}\right|^{2}\cdot\frac{\left|m^{\prime}_{ee}(t)\right|^{2}}{M^{2}_{e}}\;,$
(20)
where $G_{0\nu}$ represents the phase-space factor, $M_{0\nu}$ is the nuclear
matrix element (NME), and $M_{e}\approx 0.51~{}{\rm MeV}$ is the electron
mass. The present lower limit on the half-life of $0\nu\beta\beta$ decays from
KamLAND-Zen collaboration KamLAND-Zen:2016pfg is $T^{0\nu}_{1/2}>1.07\times
10^{26}$ yr at 90% C. L. Using the phase-space factor $G_{0\nu}=3.793\times
10^{-14}~{}{\rm yr^{-1}}$ Kotila:2012zza with $g=1.27$ and NME
$M_{0\nu}\in(1.5-4.2)$ Guzowski:2015saa for ${}^{136}{\rm Xe}$, the upper
bound on the effective neutrino mass in standard three-flavor neutrino
scenario is obtained as $|m_{ee}|<(61-169)$ meV. Here, $m_{ee}$ is given by
$\displaystyle{m}_{ee}$ $\displaystyle\equiv$ $\displaystyle
m_{1}\cos^{2}\theta_{13}\cos^{2}\theta_{12}e^{{\rm
i}\rho}+m_{2}\cos^{2}\theta_{13}\sin^{2}\theta_{12}$ (21)
$\displaystyle+m_{3}\sin^{2}\theta_{13}e^{{\rm i}\sigma}\;,$
where $m_{i}$ (for $i=1,2,3$) stand for the absolute masses of three
neutrinos, $\\{\theta_{12},\theta_{13}\\}$ are the leptonic mixing angles, and
$\\{\rho,\sigma\\}$ are the Majorana CP phases.
Figure 4: Exclusion and sensitivities of various experiments on the parameter
space of the ultralight DM mass $m_{\phi}$ and the coupling to neutrinos $g$.
The exclusion using the result of KamLAND-Zen are given by cyan lines, while
the CMB bounds in present work are shown as the dashed brown and solid black
lines. See the text for details of other limits.
To find out how the ultralight scalar exactly modifies the rate, we expand the
vacuum and DM contributions
$\displaystyle|{m}^{\prime}_{ee}|^{2}$ $\displaystyle=$
$\displaystyle\left|m_{ee}\right|^{2}+g^{2}\phi^{2}_{0}\cos^{2}{m_{\phi}t}$
(22) $\displaystyle+2|m_{ee}|g\phi_{0}\cos\eta\cos{m_{\phi}t}\;,$
where $\eta$ is the relative phase between $m_{ee}$ and $g\phi$, the second
term is induced solely by the DM field, and the third term is the interference
with the vacuum neutrino mass. Such time-varying effect is in principle
observable, if enough $0\nu\beta\beta$-decay events with proper time
resolution have been obtained. When the ultralight DM oscillates fast enough,
the long-term decay rate depends on the time average of
$|{m}^{\prime}_{ee}|^{2}$, i.e.,
$\displaystyle\overline{|{m}^{\prime}_{ee}|^{2}}=\left|m_{ee}\right|^{2}+\frac{1}{2}g^{2}\phi^{2}_{0}\;.$
(23)
Note that the interference term has been averaged out in such case. The
coupling constant $g$ is in general not related to $m_{ee}$. Hence, when the
cancellation of effective neutrino mass occurs, i.e., $m_{ee}=0$ for normal
ordering, the DM-induced term can dominate the transition.
We present our noteworthy results as Fig. 4 in the $m_{\phi}$-$g$ parameter
plane. The exclusion regions of $0\nu\beta\beta$-decay experiments and
cosmology are shown along with bounds and sensitivities arising from different
experimental searches. The vertical blue band shows the constraint based on
the results of Lyman-$\alpha$ forest. It has been pointed out (see for
detailed analysis in Ref. Hui:2016ltb ) that fuzzy DM lighter than $\sim
10^{-21}$ eV is in tension with observations of the Lyman-$\alpha$ forest. The
gray shaded region gives the limit from Super-K and Sudbury Neutrino
Observatory (SNO) neutrino oscillation experiments Berlin:2016woy ; Super-
Kamiokande:2003snd ; SNO:2005ftm . The sensitivity from forthcoming neutrino
oscillation experiments Krnjaic:2017zlz ; Dev:2020kgz like DUNE DUNE:2015lol
and JUNO JUNO:2015zny have also been presented using the magenta shaded
region and orange dot-dashed line, respectively.
The constraint from $0\nu\beta\beta$-decay searches (KamLAND-Zen KamLAND-
Zen:2016pfg ) is shown as the solid cyan line. The projection with an ultimate
sensitivity goal of $0\nu\beta\beta$-decay experiments
$|m^{\prime}_{ee}|=1~{}{\rm meV}$ Ge:2016tfx ; Penedo:2018kpc ; Cao:2019hli ;
Huang:2020mkz is shown as the dotted cyan line. In order to translate results
into the $m_{\phi}$-$g$ plane, we have used Eq. (23). We present our CMB
bounds as the dashed brown and solid black lines, respectively. Two different
limits corresponding to conservative and aggressive considerations are shown,
as given by Eqs. (14) and (15), respectively. It can be noticed that JUNO can
cover the limit of $0\nu\beta\beta$-decay experiment for most of the parameter
spcae. However, for the region where the ultralight DM mass is greater than
$10^{-11}~{}{\rm eV}$, $0\nu\beta\beta$-decay experiments set tighter bound
than neutrino oscillation experiments. On the other hand, Super-K and SNO put
the best constraint for ultralight DM below $10^{-17}~{}{\rm eV}$, whereas the
CMB analysis provides the most stringent limit for ultralight DM above
$10^{-17}$ eV.
Figure 5: The effective Majorana neutrino mass as a function of the lightest
active neutrino mass $m_{1}$. The orange region stands for the standard case
with normal mass ordering for neutrinos, while the blue region is for the case
with $g\phi_{0}=0.01~{}{\rm eV}$.
Finally, as a visual guide for the ultralight DM searches at
$0\nu\beta\beta$-decay experiments, we present the effective Majorana neutrino
mass, in presence of ultralight DM, in Fig. 5 Graf:2020cbf . The horizontal
green band indicates the current experimental limit from KamLAND-Zen,
$(61-165)~{}{\rm meV}$ KamLAND-Zen:2016pfg . The dashed lines correspond to
the future sensitivities projected by SNO+ with $(19-46)~{}{\rm meV}$
Andringa:2015tza , LEGEND with $(10.7-22.8)~{}{\rm meV}$ Abgrall:2017syy , and
nEXO with $(5.7-17.7)~{}{\rm meV}$ Albert:2017hjq , respectively. The vertical
gray band represents the current neutrino mass limit of cosmological data from
the Planck collaboration Aghanim:2018eyx . The orange region depicts the
standard three-flavor neutrino scenario. The impact of ultralight DM has been
demonstrated using the blue region for a benchmark choice of
$g\phi_{0}=0.01~{}{\rm eV}$. It turns out that even the averaged effect of
ultralight DM can be tested by the next generation of $0\nu\beta\beta$-decay
experiments.
## IV Conclusion
In this work, we investigate the impact of ultralight DM on
$0\nu\beta\beta$-decay experiments. We find that the scalar DM can directly
induce the $0\nu\beta\beta$-decay transition, even without the vacuum neutrino
mass term. The effect can be significant for very small couplings between the
ultralight DM and neutrinos, due to the stimulated absorption and emission of
scalars in the phase space filled with identical particles. As the
consequence, the vacuum Majorana neutrino mass will be modified with a time-
varying term.
We also explore the consequence of such a scalar field on cosmology. There are
mainly two effects. First, the neutrino plasma will induce a screening mass to
the ultralight DM, such that the DM field evolves as radiation instead of
matter component. Second, the dense ultralight DM in the early Universe will
shrink the neutrino free-streaming length significantly and is in tension with
the CMB observations.
The constraints and sensitivities of different experiments are given in Fig.
4. We conclude that $0\nu\beta\beta$-decay experiments can give stringent
constraint compared to the forthcoming neutrino oscillation experiments like
DUNE and JUNO for the DM mass above $\thicksim 10^{-11}$ eV. In addition, the
CMB observations have the best constraining capability for the mass range
above $\thicksim 10^{-17}$ eV. Furthermore, we observe from Fig. 5 that our
framework of ultralight DM can be tested even without analyzing the time-
varying effect by the next generation of $0\nu\beta\beta$-decay experiments.
###### Acknowledgements.
Authors are thankful to Werner Rodejohann for his valuable comments and
careful reading of the manuscript. Authors would also like to thank Eligio
Lisi, Pablo Martínez-Miravé, Manibrata Sen, and Shun Zhou for fruitful
discussions. GYH is supported by the Alexander von Humboldt Foundation. NN is
supported by the Istituto Nazionale di Fisica Nucleare (INFN) through the
“Theoretical Astroparticle Physics” (TAsP) project.
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|
# CEHR-GPT: Generating Electronic Health Records with Chronological Patient
Timelines
Chao Pang111indicates equal contribution Department of Biomedical Informatics,
Columbia University Irving Medical Center Observational Health Data Sciences
and Informatics Xinzhuo Jiang∗ Department of Biomedical Informatics, Columbia
University Irving Medical Center Observational Health Data Sciences and
Informatics Nishanth Parameshwar Pavinkurve Department of Biomedical
Informatics, Columbia University Irving Medical Center Observational Health
Data Sciences and Informatics Krishna S. Kalluri Department of Biomedical
Informatics, Columbia University Irving Medical Center Observational Health
Data Sciences and Informatics Elise L. Minto Department of Biomedical
Informatics, Columbia University Irving Medical Center Observational Health
Data Sciences and Informatics Jason Patterson Department of Biomedical
Informatics, Columbia University Irving Medical Center Observational Health
Data Sciences and Informatics Linying Zhang Institute for Informatics, Data
Science, and Biostatistics. Washington University in St Louis Observational
Health Data Sciences and Informatics George Hripcsak Department of
Biomedical Informatics, Columbia University Irving Medical Center Medical
Informatics Services, New York-Presbyterian Hospital Observational Health
Data Sciences and Informatics Noémie Elhadad Department of Biomedical
Informatics, Columbia University Irving Medical Center Medical Informatics
Services, New York-Presbyterian Hospital Observational Health Data Sciences
and Informatics Karthik Natarajan Department of Biomedical Informatics,
Columbia University Irving Medical Center Medical Informatics Services, New
York-Presbyterian Hospital Observational Health Data Sciences and Informatics
## Abstract
Synthetic Electronic Health Records (EHR) have emerged as a pivotal tool in
advancing healthcare applications and machine learning models, particularly
for researchers without direct access to healthcare data. Although existing
methods, like rule-based approaches and generative adversarial networks
(GANs), generate synthetic data that resembles real-world EHR data, these
methods often use a tabular format, disregarding temporal dependencies in
patient histories and limiting data replication. Recently, there has been a
growing interest in leveraging Generative Pre-trained Transformers (GPT) for
EHR data. This enables applications like disease progression analysis,
population estimation, counterfactual reasoning, and synthetic data
generation. In this work, we focus on synthetic data generation and
demonstrate the capability of training a GPT model using a particular patient
representation derived from CEHR-BERT, enabling us to generate patient
sequences that can be seamlessly converted to the Observational Medical
Outcomes Partnership (OMOP) data format.
## Keywords
Generative Pre-trained Transformer, Synthetic Electronic Health Records,
Patient Representation, Observational Medical Outcomes Partnership - Common
Data Model, Observational Health Data Sciences and Informatics
## 1 Introduction
Access to electronic health records (EHRs) is fundamental to healthcare
research, drug surveillance, clinical machine learning, and system
development. However, the use of real-world EHR data comes with considerable
challenges such as privacy and security issues, institutional consent, and
restrictions on data sharing. Synthetic data emerges as a promising solution,
offering a more expedient and secure pathway to healthcare information, which
could accelerate progress across various sectors, including academic research,
clinical settings, and the pharmaceutical industry [1].
Synthetic data is not real data, that is, it doesn’t relate to any specific
individual. However, it mimics the statistical characteristics and journeys of
specific patient populations. Synthetic data enables a broader range of
researchers to answer their questions of interest without going through the
cumbersome process of accessing real data and worrying about patient privacy
[2]. In recent years, many machine learning, specifically deep learning and
generative artificial intelligence (AI) models, have been developed to derive
synthetic data from real EHR data. [3] However, most existing methods for
synthetic EHR data generation fail to adequately capture the temporal
dependencies that are often critical in medical scenarios. These temporal
aspects, such as medication schedules, symptom progression, and lab result
timelines, are vital for a comprehensive understanding of patient health
trajectories and for developing effective treatment strategies. An ideal
synthetic dataset derived from institutional data should maintain the inherent
correlations among time-series features, thus enabling researchers to
externally validate machine learning models in different populations.
Crucially, the synthetic dataset must preserve accurate patient timelines, as
predictive tasks are highly susceptible to temporal variations. A synthetic
dataset is considered to exhibit comparable machine learning utility to the
original data if it meets two key criteria: 1) it demonstrates similar outcome
prevalence to the source data; 2) machine learning models trained on the
synthetic data achieve performance metrics akin to those trained with the
original data.
The majority of the existing research focuses on developing new deep learning
models in generative EHR research but without adequate emphasis on retaining
accurate temporal information [4, 5, 6]. Unfortunately, synthetic EHR datasets
developed as such will not support use cases that require the accurate
construction of a patient timeline e.g., 30-day readmission, one-year risk of
heart failure, and disease progression. This limits existing work to only
perform simple code prediction in their evaluations instead of comprehensive
phenotype predictions. Another challenge in using synthetic EHR data in
practice is its difficulty with dissemination due to a lack of standards.
Synthetic patient sequences cannot be widely adopted for analyses without use
of a common data model, however, none of the existing works have included such
a component in their frameworks to present the synthetic data in an easy-to-
consume fashion.
In our view, time-series synthetic data should not only capture the underlying
characteristics of heterogeneous EHRs but also satisfy the following temporal
requirements, 1) a matching distribution of the starting age; 2) a matching
distribution of the starting year; 3) a matching distribution of the inpatient
duration; 4) a matching distribution of time intervals between neighboring
visits. Furthermore, synthetic EHR data should be stored in common data models
such as the Observational Medical Outcomes Partnership (OMOP) Common Data
Model, which is used in many large data networks, [7] for easy dissemination
and consumption. Although creating such a time-series EHR dataset seems to be
a challenging task, we think that this problem can be solved through a patient
representation approach. The key is to focus on designing a good patient
representation rather than creating a sophisticated architecture to model time
and medical events simultaneously. In this paper, we present the CEHR-GPT
framework for building an end-to-end workflow to generate time-series
synthetic EHR data. Our contributions are summarized below,
* •
We design a novel patient representation that not only keeps track of the
visit types and discharge facility for inpatient visits but also preserves all
temporal information including patients’ starting year, starting age, time
intervals between visits, and inpatient visit spans. To the best of our
knowledge, this is the first time that temporal information is fully retained.
* •
We treat patient sequence generation as a language modeling problem, which
allowed us to use the state-of-the-art language model Generative Pre-trained
Transformers (GPT) to learn the distribution of patient sequences to generate
new synthetic sequences [8, 9].
* •
We converted synthetic sequences to the common data format OMOP with almost no
loss of temporal information. Synthetic OMOP can be easily evaluated using the
OHDSI tools and disseminated to others.
* •
We evaluated the synthetic EHR data on three levels, dimension-wise
distribution (marginal distribution), co-occurrence relationship, and machine
learning model performance metrics.
## 2 Related work
With the adoption of Generative Adversarial Networks (GANs) [10], researchers
have found creative ways to generate synthetic EHR data. Since 2017, several
groups have applied GANs to tabular EHRs and developed several evaluation and
privacy metrics to quantify the performance of GANs [3]. Despite the success,
one limitation is that the tabular format fails to capture the temporal nature
of EHR data because they are constructed from patient histories using a bag-
of-words approach. It was not until 2020 that researchers started developing
new GAN architectures to tackle the time series data. Dual adversarial
autoencoder (DAAE) [6] used a combination of a variational autoencoder (VAE)
and two GAN components, where the inner GAN was trained to replicate the
encoded representation generated by the encoder, and the outer GAN was trained
to generate realistic-looking patient sequences in addition to the
reconstruction error. Another model called EHR-M-GAN [5] employed a similar
autoencoder architecture with two main differences 1) they used a dual-VAE
framework to handle the continuous and discrete valued features, where the
continuous and discrete representations were generated by encoders first and
then collectively used for decoding; 2) in the GAN generator, they used two
parallel recurrent neural networks (one for sampling continuous noise and one
for discrete noise) with a so-called Bilateral LSTM cell to allow the
continuous and discrete noise vectors to interact with each other to generate
better sampling vectors. Although these GANs took the temporal order of events
into consideration, they did not generate timestamps for visits or medical
events, therefore limiting the use of the synthetic data.
Improving upon the previous works, a two-stage learning algorithm (dependency
learning and conditional simulation) named SynTEG was proposed to generate
timestamped synthetic data [11]. In dependency learning, transformer encoders
learned visit representations, which were used to feed into an recurrent
neural network (RNN) model to learn the dependencies between visits. Two self-
supervising learning tasks were used for training – prediction of the
timestamp and diagnoses at the next visit. In the conditional simulation, the
learned visit representations generated from the previous step were used to
train a conditional GAN to generate the diagnosis codes at each visit. This
approach achieved superior performance in the time-sensitive evaluations over
the previous methods. However, there are still ongoing challenges that have
not been addressed in their work, 1) other EHR data was not utilized because
only diagnosis codes were included in training; 2) the visits were assumed to
start and end on the same day, which would fail to model in-patient visits
that normally span days; therefore, the constructed patient timeline would be
inaccurate; 3) the synthetic data didn’t include visit types and discharge
facilities.
Until now, almost all the existing approaches used some variation of GAN for
learning the data distribution, but unfortunately, GANs are notoriously
difficult to train and easily subject to mode collapse. Despite the recent
advancements in optimization techniques such as Wasserstein-GAN [12], it would
require a significant amount of time to tune hyperparameters to properly train
GANs. As the previous works have demonstrated [13, 14, 15], patient sequence
generation could be conceptually represented as a language modeling problem.
Foresight [16] adapted GPT to forecast patient trajectories. Their methodused
a name entity recognition (NER) tool to extract medical concepts from
discharge summaries, based on which a patient sequence was constructed
chronologically, and then they trained a standard GPT model using all patient
sequences constructed from the previous step. To forecast future events, they
fed a patient history to prompt the model and employed a Monte Carlo sampling
strategy to calculate the probability of developing certain conditions. One
limitation of this work was that the model could not predict when a certain
condition would happen due to a lack of temporal information in their patient
sequence. Nevertheless, their work demonstrated the potential of using GPT for
modeling patient sequences. We aim to address this limitations in our work.
## 3 Methods
In Figure 1, we present a framework for generating synthetic EHR data from an
OMOP source. To retain the temporal dependencies, we opted to work directly
with time-series patient sequences instead of using a bag-of-words (BOW)
representation by building on our previous work [13]. We first encoded the
OMOP data into patient sequences using a specific patient representation,
described later in this section. Secondly, we trained a generative model on
the converted patient sequences and utilized it to generate new synthetic
patient sequences.
Figure 1: Overall architecture. The OMOP data is first converted to patient
sequences by an OMOP encoder based on the patient representation that
preserves demographics, visit types, and temporal intervals between visits.
Then a generative model is trained to learn the sequence distribution in order
to generate new sequences. Next, the generated sequences are converted back to
the OMOP format using an OMOP decoder.
Finally, we fed the synthetic patient sequences into an OMOP decoder to create
a synthetic OMOP dataset. Furthermore, an evaluation procedure was developed
to assess the similarity between the synthetic OMOP and the source OMOP data.
### 3.1 Patient Representation
We designed a patient representation in CEHR-BERT [13] that captures medically
relevant events and their timelines while exhibiting certain characteristics
of a sentence. In order to fully leverage Large Language Models (LLM) on
patient sequences, we further extended this patient representation to include
demographic information, patient history, and temporal dependencies as shown
in Figure 2. The start of the sequence defines the demographic prompt
containing EHR start year, age, gender, and race. It is followed by visit
blocks separated by artificial time tokens (ATT) representing time intervals
in days, e.g., $D_{1}$ represents an interval of 1 day. For time intervals
surpassing 1080 days, we grouped these into a single Long Term (LT) token, a
decision guided by the low occurrence rate in this time frame. Each visit
block starts with a visit type token (VTT) to signify the type of visit and is
then followed by domain records arranged in chronological order. In the case
of inpatient visits, a distinct inpatient ATT (IATT) was inserted between
neighboring inpatient spans, defined as the groups of records that occurred on
the same day. These are distinct from ATT and are used for capturing the time
between multiple events characterized by concepts within the same visit. In
addition, a discharge facility code (e.g. discharge home and long-term care)
was inserted at the end of the inpatient visit. The use of IATT, which is
distinct from ATT, was necessary since they are attributed to two different
contexts, and resulted in better performance than using the same ATT across
both contexts.
This patient representation allows us to convert from any common data model
(e.g. OMOP) to patient sequences and vice versa without any loss of temporal
information. To formulate this property, let’s denote $D_{i}$ to be data
associated with the $i$th patient in the source format, $P_{i}$ to be the
patient sequence converted from $D_{i}$, and $F$ to be the function that
converts source data to patient sequences, represented as $P_{i}=F(D_{i})$.
Let’s then denote $D_{i}^{\prime}$ to be the reconstructed source data and
$F^{\prime}$ to be the inverse function that converts patient sequences back
to original source format. This indicates $D_{i}^{\prime}=F^{\prime}(P_{i})$.
Finally, let’s denote $T$ to be a function that extracts all the dates for a
set of patient records. The patient representation is said to preserve
temporal information perfectly if and only if the following statement is true
for every single patient, $T(D_{i})=T(D_{i}^{\prime})+C_{i}$, where $C_{i}$ is
a constant that represents a consistent time shift e.g., $C_{i}=4$ days.
Figure 2: The patient representation preserves demographics, visit types, and
temporal intervals between visits and inpatient duration. It’s designed to
have the demographic prompt at the beginning including year at the first
visit, age at the first visit, gender and race tokens, then followed by a
series of visit blocks to represent the complete patient timeline. An
artificial time token (ATT) is inserted between the neighboring visit blocks
to keep track of the time intervals in days. In each visit block, all the
essential information is retained including the visit type and domain records.
In the case of inpatient visits, the inpatient ATT tokens (representing time
intervals in days) are inserted between groups of concepts that occur on the
same day, in addition, a discharge token is provided at the end of the visit
block.
### 3.2 OMOP Encoder
To create a patient sequence, we began by generating a demographic prompt
using data from the OMOP person and visit tables. This prompt included
essential demographic information such as the patient’s age at their initial
visit, the year of their first visit, their gender, and their race.
Subsequently, we constructed a series of visit blocks to represent the
patient’s entire medical history. We inserted an ATT token between these visit
blocks to signify the time intervals between them. Within each visit block, we
gathered all relevant records from OMOP domain tables (e.g., condition) and
arranged them chronologically based on their respective timestamps. In cases
where there are timestamp ties, we sorted the concepts. Additionally, three
artificial tokens (VS, VE, and VTT) were added at the beginning and end of
each visit block to denote the start, end, and type of the visit, as
illustrated in Figure 2. For inpatient visit blocks, extra processing steps
were necessary. Initially, we grouped records by their timestamps to identify
all inpatient spans, arranging them in chronological order. Next, we inserted
IATT between these spans, aligning them with the respective time intervals.
Finally, the discharge facility code was extracted from the OMOP visit table
and appended to the end of the block.
### 3.3 Generative Model
We used a GPT model with standard transformer decoders, where the input layer
utilized concept embedding and trainable positional embedding. The model was
trained using the Next Word Prediction learning objective. When generating a
patient sequence, we randomly sampled a demographic prompt from our source
sequences, which served as the input to the GPT model. Using these prompts,
the entire patient history was generated autoregressively by sampling tokens
from the predictive distribution at the final layer.
### 3.4 OMOP Decoder
The patient sequence was converted back to the OMOP format using the OMOP
decoder. The start-year prompt determined the EHR history’s beginning, using
January 1st as the default. Demographic data was stored in the person table,
while concepts were transformed into condition, drug, and procedure tables. A
date cursor was used to represent the “current time” as we were processing
each patient sequence, it was initially set to the star year and was updated
whenever an ATT token was encountered. We first parsed out the number of days
represented by the ATT token, and then moved the data cursor by the same
number of days to the future. During the sequence processing, the VS token
marked the start of a new visit block. We, therefore, extracted the token
corresponding to the visit type (the token immediately followed by VS) and
created a new visit record with the corresponding type.
All the tokens subsumed by this visit block were converted to condition, drug,
and procedure records and linked to the current visit. For outpatient visits,
we assumed that all data points were generated on the same day; therefore, we
set the start and end dates of the visit to the current value of the date
cursor. Similarly, all the domain records were set to the date cursor as well.
When processing inpatient visits, the date cursor was updated inside the visit
block based on the time intervals represented by IATT tokens between inpatient
spans. Domain records were generated with the current value of the date
cursor. Towards the end of the inpatient visit block, we extracted the last
token (the token right before VE) corresponding to a discharge facility and
updated the visit end date using the current value of the date cursor (as the
date cursor was frequently updated inside an inpatient visit block). This
allowed us to preserve the complete information about inpatient visits. If
generated sequences do not follow patterns presented in the patient
representation, they will be discarded to ensure the quality of the synthetic
data.
## 4 Experiments and Results
### 4.1 Data and Preprocessing
The source patient sequences were generated from the OMOP database derived
from Columbia University Irving Medical Center-New York Presbyterian Hospital
EHR data, which includes 3.7 million unique patients’ medical histories
including condition, medication, and procedure. Unknown concepts (i.e.,
$concept\\_id=0$) were removed from all domains except for the visit type when
constructing the patient sequences using the proposed patient representation.
Patients with less than 20 tokens were removed from the training dataset, and
approximately 2.3 million patients were included for training whereas 75,000
patients were held out for privacy evaluations. For the GPT model, we used a
context window of 512, 16 transformer decoders, 8 attention heads with a
dropout rate of 0.1, and 128 dimensions for both the embedding and hidden
units. All patients with longer than 512 tokens were post-truncated to fit the
context window. The statistics of training data was summarized in Table 1. We
trained the model for 2 epochs on 2 Nvidia 2080 TI GPUs with a batch size of
32 and a learning rate of 0.0002. The model checkpoint was created every
10,000 steps.
| No. of visits per patient | Sequence length per patient
---|---|---
mean | 16 | 148
std | 19 | 154
min | 2 | 20
25% | 4 | 38
50% | 8 | 78
75% | 21 | 198
max | 102 | 512
Table 1: Summary statistics of the CUIMC-NYP OMOP training data
During the first epoch, we used a standard data generator strategy, where
every training example was fed to the model; however, we switched to a random
sampling strategy to draw training examples during the second epoch. For
synthetic data generation, we used the 10th model snapshot because early
experiments showed its superior performance compared to other snapshots.
Furthermore, we used several sampling hyper-parameters including top k=300,
top k=200, top k=100, top p=90%, top=95%, and top=100% to generate different
synthetic OMOP datasets. Using specific top p/k values is a common technique
in language models for data generation. For instance, the top k approach
limits the selection of the k most probable tokens in the prediction
distribution during sampling. On the other hand, the top p method selects a
set of most likely tokens whose combined probability reaches p% (e.g. 90%) in
the predictive distribution. For each sampling strategy, 1M synthetic
sequences were generated and converted to OMOP. On average, 98% of the
generated sequences passed the validation and were converted to OMOP.
### 4.2 Evaluations
We followed the evaluation procedures proposed by [17] to compute the data
utility metrics including dimension-wise distribution, co-occurrence
relationship, and machine learning model performance. Some of the metrics were
originally designed for tabular EHR data; therefore, we adapted them to the
time-series setting. When using Kullback-Leibler (KL) divergence to evaluate
source and synthetic datasets, we use concept probabilities defined as,
$P_{prob}(c)=\frac{\sum_{i}^{n}\mathbbm{1}\Big{[}c\in
h_{i}\Big{]}}{\sum_{i}^{n}\sum_{j}^{m}\mathbbm{1}\Big{[}c_{j}\in
h_{i}\Big{]}}$
where $c$ denotes the target concept, $h_{i}$ denotes the $ith$ patient
history, $n$ and $m$ denote the total number of patients and concepts
respectively. Due to the small probability values, we opted to use the
prevalence instead for data visualization using a slightly modified formula
below,
$P_{prev}(c)=\frac{\sum_{i}^{n}\mathbbm{1}\Big{[}c\in h_{i}\Big{]}}{n}$
The only difference between the prevalence and the probability is the
normalization constant used as the denominator.
For baseline comparison, we added three variants of GPT models trained on
slightly different patient representations. The first baseline model was
trained on the adjusted CEHR-BERT representation, which differs from the
proposed patient representation by 1) CEHR-BERT ATT tokens contain a mix of
day/week/month/year tokens to represent time intervals while CEHR-GPT only
used the day tokens and 2) the IATT tokens and discharge facility tokens were
not used in the CEHR-BERT representation. The second baseline model used the
proposed patient representation with IATT tokens removed. We will refer to
this baseline as GPT-OUTPAT. The last baseline GPT was trained on the patient
representation widely used in time-series EHR research, where the concepts
were simply ordered chronologically and put in a sequence without any
additional artificial tokens. To use such sequences for comparison, we assumed
that all medical events in the patient sequence belonged to a single visit.
This baseline model will be referred to as GPT-Vanilla. We only used the
top_p=95% sampling strategy to generate synthetic OMOPs for these baseline
models. The comparison between these patient representations can be seen in
Supplementary Figure 9.
#### 4.2.1 Dimension-wise Distribution
KL divergence was assessed to compare the concept probability distributions
between synthetic and source datasets among the entire population. In Figure
3, synthetic datasets generated by different patient representations and
sampling strategies were evaluated against real patient data. The results
showed that baseline models CEHR-BERT, GPT-Vanilla and GPT-OUTPAT with
sampling strategy using the threshold of top_p$=95\%$ diverged the least from
real concept probability distributions followed by GPT models trained with
top_p$=100\%$ and top_p$=95\%$. GPT models with threshold of top_k$=300$ and
top_k$=200$ with relatively similar divergence. However, GPT models with
threshold of top_k$=100$ sampling strategy had the largest KL divergence.
Figure 3: KL divergence for comparing concept probability distribution between
synthetic data and real data. The probabilities of concepts were calculated on
the scale of the entire population.
Furthermore, we conducted a qualitative analysis using the synthetic data
top_p=95% to gain a more comprehensive understanding. The dimension-wise
distributions between the synthetic and source datasets were compared at three
distinct levels: the entire population, specific sub-groups (e.g., female
population), and particular cohorts (e.g., hospitalization cases). Figure 4
illustrates the concept prevalence comparison between the original OMOP
dataset and the generated OMOP dataset, using a threshold of top_p$=95\%$. In
the high-frequency regions, most data points cluster closely around the
diagonal line, indicating a strong agreement between the source and synthetic
data. Conversely, data points appear more dispersed for low-frequency
concepts. Notably, in the subplot representing female conditions (located in
the first column, second row), there is an unusual cluster of concepts
positioned above the diagonal line. Further examination revealed that these
concepts were male-specific and should not appear in the female population.
Although there are a few instances of such cases in the source data, GPT
amplified such cases in synthetic data.
Figure 4: Concept prevalence comparison between the source OMOP and generated
OMOP using top $p=95\%$ in the log scale stratified by domain in columns and
by population in rows, where x-axis and y-axis represent the source and the
synthetic data respectively, and each dot represents a concept
In addition, we conducted a detailed comparison of the visit tables between
the two OMOP datasets. Our specific focus was on performing demographic
breakdowns and analyses for gender, race, and age group. The supplementary
Figures 10, 12, 11 highlight the top 10 most prevalent visits for each
demographic breakdown, showcasing that the trends in both the source and
generated visit tables exhibited notable similarities.
#### 4.2.2 Co-occurrence Relationship
To measure how closely the generated datasets resemble the source, we computed
the KL divergence between their co-occurrence matrices. The matrix was
constructed temporally with the following logic: 1) for each concept in the
patient sequence, only the future concepts were taken into consideration for
creating concept pairs; 2) each patient could only contribute to the same
concept pair once; 3) the matrix was normalized into a proper probability
distribution by dividing the occurrences of each concept pair by the overall
number of pairs. Additionally, we set benchmarks for our analysis: a lower-
bound and an upper-bound. The lower bound was determined by applying the KL
divergence method to two random samples from the source data. The upper bound
was established by creating a theoretical co-occurrence matrix under the
assumption that all concepts in the source data were independent. The KL
divergence was then applied to this hypothetical matrix to calculate the upper
bound. Figure 5 illustrates that the datasets with CEHR-BERT and top_k=300
most closely approach the lower bound, with the top_p=95% and GPT-OUTPAT
baselines coming next. Datasets with top_k=100 and top_k=200 had marginally
higher KL divergence values, while the top_p=100%, the GPT-Vanilla datasets
exhibited the largest KL divergence.
To thoroughly examine the similarities between the original and the synthetic
data, we carried out a qualitative analysis using one of the synthetic
datasets. This involved comparing the most frequent pairs of co-occurring
concepts within each category, such as condition-condition (interpreted as one
condition concept followed by another condition).
Figure 5: KL divergence associated with different synthetic data. The closer
to the lower bound in the bottom left corner, the better the synthetic data.
Figure 6: Top 100 co-occurring concept pairs for each co-occurrence category
e.g. condition-condition interpreted as a condition concept followed by
another condition concept. The x and y axes represent the synthetic and source
data respectively
The results, illustrated in Figure 6, display the top 100 pairs for each type
of co-occurrence relationship. The analysis revealed that the synthetic data
accurately mirrored the co-occurrence patterns in most categories, with the
exception of the categories ending with a drug concept (as shown in the second
column of Figure 6), where the data points were more scattered. Notably, a
majority of the points in condition-condition and procedure-procedure pairs
aligned along the diagonal line.
Figure 7: Comparison of temporal co-occurrence networks around Type 2 Diabetes
Mellitus (T2DM) for real (left) and synthetic data (right). Networks were
constructed with T2DM’s (large circle) top 5 co-occurring condition concepts
and their own top 5 co-occurring condition concepts. The metrics Prevalence
(top) and Pointwise Mutual Information-3 (PMI3) (bottom) were used to quantify
co-occurrence. Gold edges indicate edges that are shared by both real and
synthetic data, edge thickness indicates the strength of co-occurrence, and
arrow direction indicates the direction of the temporal association.
For an in-depth analysis, we focused on examining the co-occurrence
relationships associated with Type 2 Diabetes Mellitus (T2DM). Figure 7
qualitatively compares the real and top_p=95% synthetic data networks of
condition concepts that co-occur around T2DM. The degree of co-occurrence is
calculated with prevalence, which evaluates the frequency of concept co-
occurrence, and Pointwise Mutual Information-3 (PMI3), which evaluates the
probabilistic association of concepts. There is much overlap between the real
and synthetic networks when prevalence is used to construct them, but
connections are more disparate when PMI3 is used.
#### 4.2.3 Predictive performance
For this analysis, we constructed five prediction tasks using the method
described in [13] and the Book of OHDSI [18]. Table 2 shows the cohort and the
corresponding definition.
Cohort | Definition
---|---
HF readmission | HF patients who have a 30-day all-cause readmission. Observation window: 360 days, Prediction windows 30 days
Hospitalization | 2-year risk of hospitalization starting from the 3rd year since the initial entry into the EHR system. Observation window: 540 days, hold-off window: 180 days, prediction windows: 720 days
COPD readmission | COPD patients who have a 30-day all-cause readmission. Observation window: 360 days, prediction windows: 30 days
Afib ischemic stroke | Afib patients with 1-year risk since the initial diagnosis of afib ischemic stroke. Observation window: 720 days, prediction windows: 360 days
CAD CABG | Patients initially diagnosed with Coronary Arterial Disease (CAD) without any prior stent graft will receive the Coronary artery bypass surgery (CABG) treatment. Observation window: 720 days, prediction windows: 360 days
Table 2: Cohort definitions
To extract features, we first rolled up the medical concepts using ontological
hierarchies to reduce dimensionality [see supplementary materials] and used
the bag-of-word (BOW) approach, where we counted the frequency of each concept
in a given observation window. For each task, we split the cohort data (both
synthetic and real) into training and testing sets with a split ratio of
85:15. We ran logistic regression using Sklearn’s implementation with the
default configuration. Finally, the area under the receiver operating
characteristics curve (AUC) was calculated using the test set. In addition, we
reported PR-AUC (precision-recall) due to the class imbalance often present in
EHR data. Table 3 shows the prevalence of the positive cases, ROC-AUC, and PR-
AUC for each synthetic data. The metrics associated with the baseline CEHR-
BERT are reported in the Supplementary Materials Table 7. For easy comparison
of synthetic datasets, we defined a consolidated distance metric as the
weighted average of the relative differences of the three aforementioned
metrics,
$dist=\frac{|\delta_{Pre}|\times 0.5}{Pre_{true}}+\frac{|\delta_{AUC}|\times
0.25}{AUC_{true}}+\frac{|\delta_{PR}|\times 0.25}{PR_{true}}$
where $\delta_{Pre}$, $\delta_{AUC}$, and $\delta_{PR}$ represent the
differences in prevalence, ROC-AUC, and PR-AUC between the source and the
synthetic data; $Pre_{true}$, $AUC_{true}$, and $PR_{true}$ denote the ground
truth metrics generated from the source data.
Figure 8 presents the distance metrics for various synthetic datasets across
different cohorts. The dataset created with top_k=300 displayed the best
performance in HF and COPD Readmission but was less effective in other
cohorts. The top_p=95% dataset maintained consistent performance levels,
ranking as the best in Hospitalization and Afib ischemic stroke, and as
second-best in HF and COPD readmission. In comparison, the GPT-OUTPAT baseline
dataset exhibited a slightly higher divergence across the Hospitalization,
Afib Ischemic Stroke, and CAD CABG groups. However, the CEHR-BERT baseline
showed similar patterns in most cohorts, with a notably lower divergence in
CAD CABG. It is noted that both GPT-OUTPAT and CEHR-BERT were not included in
the HF Readmission and COPD Readmission analyses due to the absence of
synthetic patients meeting the selection criteria for these cohorts. Other
synthetic datasets, specifically those generated with top_p=100% and top_k=200
lagged behind in performance. Interestingly, top_k=100 showed a unique
pattern, being the closest in the distance for the CAD CABG cohort but
underperforming in the others.
Figure 8: The consolidated distance metrics for different synthetic datasets
stratified by cohort. GPT-Vanilla and CEHR-BERT were omitted as the cohorts
couldn’t be constructed due to the loss of temporal information. GPT-OUTPAT
was omitted from HF readmission and COPD readmission as the cohorts for the
same reason.
Cohort | Real | p=95% | p=100% | k=100 | k=200 | k=300 | GPT-OUTPAT
---|---|---|---|---|---|---|---
HF readmission | Pre = 25.7 AUC = 65.7 PR = 39.3 | Pre = 27.6 AUC = 69.2 PR = 45.7 | Pre =27.7 AUC = 52.4 PR = 29.0 | Pre = 30.7 AUC = 68.1 PR = 47.8 | Pre = 29.3 AUC = 54.0 PR = 32.9 | Pre = 26.5 AUC = 61.1 PR = 33.8 | Pre =100.0 AUC = NA PR = NA
Hospitalization | Pre = 5.6 AUC = 75.3 PR = 19.5 | Pre = 5.2 AUC = 77.1 PR = 21.4 | Pre = 7.4 AUC = 71.3 PR = 20.2 | Pre = 2.8 AUC = 87.0 PR = 22.1 | Pre = 5.2 AUC = 84.2 PR = 20.8 | Pre = 6.3 AUC = 78.7 PR = 24.6 | Pre = 5.2 AUC = 70.2 PR = 14.3
COPD readmission | Pre = 34.5 AUC = 74.2 PR = 83.8 | Pre = 37.8 AUC = 76.4 PR = 84.4 | Pre = 47.2 AUC = 74.1 PR = 67.2 | Pre = 26.4 AUC = 75.9 PR = 90.3 | Pre = 28.3 AUC = 70.1 PR = 82.8 | Pre = 34.5 AUC = 68.8 PR = 80.2 | Pre = NA AUC = NA PR = NA
Afib ischemic stroke | Pre = 8.7 AUC = 84.0 PR = 48.5 | Pre = 10.2 AUC = 78.9 PR = 41.2 | Pre = 10.4 AUC = 70.7 PR = 39.1 | Pre = 16.6 AUC = 77.1 PR = 50.5 | Pre = 15.8 AUC =68.9 PR = 36.6 | Pre = 10.8 AUC = 76.8 PR = 38.5 | Pre = 9.7 AUC = 67.2 PR = 27.2
CAD CABG | Pre = 7.1 AUC = 88.4 PR = 55.9 | Pre = 4.1 AUC = 81.5 PR = 25.2 | Pre = 4.4 AUC = 52.9 PR = 4.3 | Pre = 7.2 AUC = 84.7 PR = 31.3 | Pre = 4.9 AUC = 73.5 PR = 24.3 | Pre = 4.0 AUC = 79.0 PR = 24.1 | Pre = 3.5 AUC = 81.5 PR = 44.4
Table 3: Logistic regression model performance across different datasets. In
each cell, three numbers were reported including the prevalence of the
positive cases, ROC-AUC, and PR-AUC
### 4.3 Privacy Evaluations
We adopted the privacy evaluation framework outlined in [17], focusing on
quantifying the risk of privacy breaches through attribute and member
inference attacks. The attribute inference risk assesses the likelihood of
inferring sensitive attributes from a synthetic dataset by utilizing targeted
patients’ demographic information and common diagnoses found in real datasets
available to the adversary. In a membership inference attack, the adversary
tries to determine if the target record was in the training set given all or
partial attributes.
#### 4.3.1 Membership Inference Attack
We simulated two different attack scenarios, namely the dataset attack and the
model attack. In a dataset attack, attackers only have access to the synthetic
dataset, whereas they can query the model itself in a model attack. Their goal
is to find a way to infer whether or not a real patient record was used for
training [19]. For example, the model might output a high probability
associated with the training data points and a low probability for the non-
training ones, similarly, the synthetic data might be more similar to the
training data than the non-training ones.
The attack dataset was constructed based on the following procedure 1) we
created the negative examples by taking the 75000 holdout set and labeling
them as negative; 2) we created the positive examples by randomly selecting
75000 patients from the training set and assigning them as positive. In the
dataset attack, for each attack data point, we found the best match from the
synthetic data using a hamming distance metric, which sums up the absolute
differences in age, number of visits, and concepts [19], finally, the median
distance was used as a threshold, below which patients were predicted as
positive and negative otherwise. In the model attack, we computed the average
loss of each patient sequence by feeding the attack dataset to the model
directly, similarly, the median loss was used to infer the positive/negative
cases, where the positive cases were assigned to those below the threshold and
positive otherwise. Finally, we calculated recall/precision/f1/accuracy using
the ground truth labels against those inferred predictions. Table 4 shows that
the accuracy of both attacks is slightly less than 50%, indicating that the
performance of such attacks is worse than a random guess.
| Accuracy | Recall | Precision | F1
---|---|---|---|---
Model Attack | 0.4955 | 0.4955 | 0.4942 | 0.4949
Data Attack | 0.4941 | 0.4996 | 0.4959 | 0.4978
Table 4: Membership Inference Attack metrics
#### 4.3.2 Attribute Inference Attack
In attribute inference attack, we assumed the attacker has access to a group
of target patients along with their demographic data, including age, gender,
race, the year of their initial clinical visit, and prevalent clinical
conditions like hypertension, abdominal pain, chest pain, etc. With these
features, the attacker identifies the patient in the synthetic dataset with
the closest attribute resemblance to the target individual. Then the attacker
aims to use the sensitive attributes of matched patient from synthetic dataset
to infer the corresponding sensitive attributes of the target patient.
To quantitatively evaluate the attribute inference risk, we randomly sampled
150,000 patients from the real patient dataset as a target group. The search
group compromised 1 million generated synthetic patients. To find most similar
patients from the search group, we created a set of common attributes
including the demographic data along with the top 1% of the most prevalent
condition concepts, represented as one-hot encoded features. Then a k-nearest
neighbors (KNN with k = 1) algorithm was applied to each target patient and a
synthetic patient with the smallest euclidean distance was found. Finally we
extracted the sensitive attributes (condition concepts not in the top 1% tier)
from matched target and synthetic patients. F1 scores were computed for the
sensitive attributes of each matched patient pair and aggregated across all
matched patients.
A baseline analysis was performed substituting real patients for synthetic
patients. A result lower than the baseline suggests that the likelihood of
finding a synthetic data similar to a real patient is lower than finding a
real patient similar to a real patient who share sensitive attributes. This
implies a lower attribute inference risk which could be acceptable. Our
training set was randomly divided into two halves, with 1 million real
patients assigned as the target group, and the remaining 1 million as the
search group. The matching process was consistently applied and an aggregated
F1 score was computed. Table 5 shows that the F1 score of synthetic vs real is
less than real vs real scenario. But it’s still higher than other models who
didn’t capture the real patterns as effectively. [19]
| Recall | Precision | F1
---|---|---|---
Synthetic vs Real | 0.0350 | 0.0343 | 0.0271
Real vs Real | 0.0612 | 0.0468 | 0.0421
Table 5: Attribute Inference Attack metrics
## 5 Discussion
To the best of our knowledge, this is the first attempt to utilize GPT for
generating time-series heterogeneous EHR data, while preserving patient
privacy. Our main contribution lies in designing a novel patient
representation that preserves a complete timeline of the patient’s history,
along with crucial visit details, thereby enabling GPT to create realistic
patient sequences. Importantly, this representation facilitates seamless
conversion back to the OMOP format, simplifying dissemination and analysis.
This patient representation could serve as an effective messenger for
transferring information across various standard data models. At present, our
system is tailored to the OMOP format. However, it is designed with
adaptability in mind, enabling us to seamlessly integrate new encoder/decoder
pairs. This flexibility would facilitate the conversion of patient sequences
to other widely-used data models, such as i2b2[20].
The study undertakes a three-tiered evaluation approach, systematically
comparing synthetic and real datasets based on their marginal (column-wise
distribution), conditional (co-occurrence relationship), and joint
distributions (predictive performance). Concurrently, as the evaluation
progresses through these three levels, there is a corresponding escalation in
the complexity and challenge of the tasks involved. The outcomes of the KL
divergence analysis revealed a nuanced relationship between the top k/p
sampling strategies and the performance across the evaluation levels.
Specifically, an increase in top k/p values enhanced performance in level 1
concept prevalence. However, an excessively high or low top k/p value
adversely affects both level 2 co-occurrence metrics and level 3 machine
learning predictions. This pattern suggests that including more tokens in the
predictive distribution introduces greater uncertainty and a wider array of
potential patient trajectory variations in the data generation process,
thereby escalating the difficulty of achieving comparable performance
outcomes.
The sampling strategies of top_p=95% and top_k=300 seem to be most effective
for generating the synthetic data. For instance, the synthetic data created
with top_p=95% demonstrates the second smallest divergence in both dimension-
wise distribution and co-occurrence relationship. Simultaneously, the
corresponding synthetic cohorts successfully replicated the performance
metrics in all predictive tasks, with the exception of CAD CABG. Finally, the
significance of this patient representation transcends synthetic data
generation; we believe it has the potential to establish the groundwork for
integrating time into patient representations across diverse EHR-based deep-
learning models.
### 5.1 Loss of Temporal Information
The reason that CEHR-GPT replicated the performance metrics of the prediction
tasks can be attributed to the use of time tokens in its underlying patient
representation. The majority of the prediction problems are phrased as “For a
group of target patients who share similar characteristics, who would
experience a particular medical event in one year from the index event?” in
EHR research, therefore maintaining a complete patient timeline is crucial for
time-sensitive cohort constructions [18].
We claimed that the proposed patient representation had almost zero loss of
temporal information, although this makes intuitive sense, there does not
exist a formal metric to quantify this. To bridge this gap, we conceived a new
metric named loss of temporal information (LOTI) to estimate the shrinkage of
the patient timeline due to the use of the patient representation in an EHR
dataset. Let’s denote $T$ to be the time interval measured in days, ATT to be
an artificial time token that represents a time interval ($W_{0}$), $F$ to be
a function that maps $T$ to an ATT token (four days to $W_{0}$). In addition,
let $G$ be the inverse function of $F$ that converts an ATT to $T$, moreover,
we impose the constraint on G such that it takes the lower bound of ATT e.g.,
$W_{0}\implies 0$ days. Formally, we define LOTI as the expected difference
between the original time interval $T$ and the reconstructed time interval
$G(F(T))$ as the following,
$LOTI=E_{p(T)}\Big{[}T-G\big{(}F(T)\big{)}\Big{]}$
where $P(T)$ is the probability of $T$ observed in the training data defined
as,
$P(T)=\frac{\textit{Freq of T}}{\Sigma\textsuperscript{T}\textit{Freq of T}}$
We computed LOTI for the patient representations utilized in CEHR-GPT and
baseline models, shown in Figure 9. As Table 6 shows, CEHR-GPT has the least
LOTI compared to the other patient representations while GPT-OUTPAT has a
slightly higher time shrinkage because the inpatient duration was not
retained. CEHR-BERT has a relatively large LOTI compared to the previous two
representations due to the use of coarse ATT tokens. Finally, GPT-Vanilla has
the most LOTI, which is equal to the expected length of the timeline in the
source population due to the complete collapse of the timeline.
Representation | Between visit ATT token | Between inpatient span ATT token | LOTI
---|---|---|---
CEHR-GPT | Day token for $T\leq 1080$ LT token for $T>1080$ | Day token | 7.739
GPT-OUTPAT | Day token for $T\leq 1080$ LT token for $T>1080$ | N/A | 7.962
CEHR-BERT | Day token for $T<7$ Week token for $7\leq T<30$ Month token for $30\leq T<360$ LT token $T\geq 360$ | N/A | 31.482
GPT-Vanilla | N/A | N/A | 111.164
Table 6: Loss of Temporal Information for Different Patient Representations
### 5.2 Time Invariance and Sensitivity
Across all synthetic datasets, the dimension-wise distribution (marginal
distribution) and co-occurrence relationship were well preserved, regardless
of the patient timeline’s integrity within the models used. Even with a high
LOTI, baseline models such as CEHR-BERT accurately mirrored both marginal
distribution and co-occurrence relationship, yielding results similar to those
from CEHR-GPT. This implies that these two measures may be largely time-
invariant, unaffected by any shrinkage in the patient timeline. The rationale
behind this is rooted in their construction methods, which either disregard or
marginalize the temporal factor. Marginal distribution was constructed by
counting the unique number of patients associated with the target concept,
which was then normalized by a constant. The construction disregarded
temporality, as the placement of a concept on the timeline was not a factor of
consideration. The co-occurrence matrix was created using time initially but
was marginalized after all co-occurring pairs were collected from the patient
population.
On the contrary, the predictive performance is extremely sensitive to any
change made to the patient timeline as the cohort construction requires the
integrity of the patient timeline. For example, HF readmission requires a
30-day prediction window from the index event (defined as the hospitalization
episode with a heart failure diagnosis). Any shrinkage to the patient timeline
will disrupt the construction of this cohort. The synthetic HF readmission
cohort produced by CEHR-BERT showed a readmission rate of 100% due to the
shrinkage of the timeline in this patient representation; whereas, the actual
expected rate of readmission should be approximately 25%. Compared to CEHR-
GPT, GPT-OUTPAT encoded time intervals between visits but did not preserve the
duration of inpatient visits, therefore having a slightly higher LOTI. As a
consequence, it showed reasonable performance in the cohorts (Hospitalization
and Afib ischemic stroke), which had a 360-day prediction window and were thus
less impacted by timeline shrinkage. However, in the case of HF Readmission
(where a 100% readmission rate was observed) and COPD Readmission (which
identified no patients), GPT-OUTPAT was less successful. These cohorts used a
short 30-day prediction window, making them highly sensitive to any
distortions in the timeline, which likely led to synthetic patients not
meeting the cohort selection criteria. Interestingly, the CAD CABG cohort
presents a notable deviation from the general trend, where the CEHR-GPT
dataset with top_k=100 outperformed both the top_p=95% and top_k=300
configurations. Additionally, the CEHR-BERT synthetic data accurately
replicated the machine learning performance metrics as well. This indicates
that the CAD CABG cohort was less affected by time shrinkage.
Therefore, selecting the appropriate patient representation is pivotal in
maintaining specific properties of the source data when generating synthetic
data. The choice hinges on the intended application of synthetic data,
ensuring that critical features and patterns inherent to the patient
information are accurately reflected and retained.
### 5.3 Time Sensitive Forecasting
Because the patient representation encodes all the temporal information in the
sequence, the trained GPT model could be used potentially for time-sensitive
forecasting. We could prompt the trained GPT model with a patient history and
estimate the time of the next visit via a Monte Carlo Sampling approach shown
in the following equation,
$P(\delta_{t}|h)\approx\frac{\sum^{n}_{i=1}\mathbbm{1}\Big{[}M_{gpt}(h)=\delta_{t}\Big{]}}{n}$
where $M_{gpt}$ denotes the GPT model, $h$ denotes a patient history,
$\delta_{t}$ denotes any time interval, and $n$ represents the number of
samples. Then we can use the expectation $E\big{(}\delta_{t}\big{)}$ as the
predicted time interval. In addition, we can quantify the confidence by
calculating the standard deviation e.g. $sd(\delta_{t})$. Similarly, we can
predict the visit type ($v$) using the same Monte Carlo approach.
$P(v|E\big{(}\delta_{t}\big{)},h)\approx\frac{\sum^{n}_{i=1}\mathbbm{1}\Big{[}M_{gpt}\Big{(}E\big{(}\delta_{t}\big{)},h\Big{)}=v\Big{]}}{n}$
Finally, we can predict the most likely medical event ($c$) given the
predicted visit type $v$,
$P(c|v,E\big{(}\delta_{t}\big{)},h)\approx\frac{\sum^{n}_{i=1}\mathbbm{1}\Big{[}M_{gpt}\Big{(}v,E\big{(}\delta_{t}\big{)},h\Big{)}=c\Big{]}}{n}$
This approach goes beyond conventional prediction methods by not only
forecasting future medical events but also determining the timing of the next
visit and the specific medical events associated with that visit type. This
predictive model could provide a more detailed and actionable timeline for
patient care.
### 5.4 Limitations
While synthetic datasets demonstrated a high degree of similarity to source
data, they are subject to certain known constraints. Firstly, a selection bias
was present in the training data due to the constraints on sequence length,
ranging from 20 to 512. This limitation resulted in the partial inclusion of
patients with chronic conditions, which typically require a longer context
window for accurate representation. While extending the context window of the
model could potentially address this issue, it may introduce unforeseen
effects. Finding the optimal configuration to accommodate a broader context
window would require comprehensive experiments.
Secondly, identifying an optimal sampling strategy for generating synthetic
data remains a challenge due to the presence of numerous hyperparameters such
as temperature, top_p, and top_k. These parameters, when used in conjunction,
could yield a wide array of configurations. While the top_p=95% strategy
showed the least divergence from the source data, it was unable to accurately
replicate performance metrics for CAD CABG. As an interim solution, we may
publish multiple versions of synthetic data, along with their corresponding
performance metrics. This approach would allow researchers to select the most
suitable dataset for their specific use case.
Thirdly, the GPT model showed a propensity to over-represent prevalent
concepts, skewing towards those with higher frequencies in the dataset. An
illustration of this is seen in the synthetic data, where 78% of patients had
at least one outpatient visit, in contrast to the actual data where this
figure was 73%. This discrepancy indicates a bias in the model towards more
common occurrences. In addition, the co-occurrence analysis also demonstrated
that the synthetic reconstruction faithfully represents the frequent concept
pairs in the original data but may be less effective at recovering the finer
associations between rare concepts as shown in Figure 7. To address the over-
representation of prevalent concepts by the GPT model, future work will look
into implementing regularization techniques. One promising approach is
adaptive regularization, which can be outlined in several steps: 1) Model
Training: Begin by training the GPT model for a predetermined number of steps;
2) Sequence Generation and Analysis: Generate a sample of patient sequences
from the trained model and calculate the marginal distribution of the concepts
within these sequences; 3) Distribution Comparison and Adjustment Score
Calculation: Compare the model-generated distribution to the empirical
distribution derived from the actual data. From this comparison, calculate an
adjustment score for each concept; 4) Logit Adjustment: Modify the logits for
each concept in the model according to the calculated adjustment scores. By
implementing this procedure, the influence of each concept on the model’s
learning process would be adaptively modified during back-propagation,
allowing for an update in the model parameters that takes into account the
disparity between the generated and actual data distributions. This should
help in reducing the bias towards over-represented concepts.
Furthermore, improving patient representation is an area for further
development. Currently, the model’s representation is limited to daily
intervals and does not capture more precise measurements like hours or
minutes. This limitation is particularly relevant for intensive care unit
(ICU) data where time-sensitive decisions are critical. Furthermore, the
current model framework assigns the first visit of every synthetic patient to
the start of the year, as it only includes a year token to denote the
commencement of patient history. To refine the accuracy of patient history
commencement, integrating a month token alongside the year token is being
considered to properly represent seasonality. This would provide a more
accurate reconstruction of the starting point for a patient’s first visit in
the generated data.
Lastly, there is a necessity to incorporate the death event within patient
sequences. Including this event would allow the synthetic data to more
accurately represent mortality, enhancing its utility for predictions related
to patient outcomes and lifespan. These enhancements aim to create a more
precise and clinically relevant synthetic dataset that better mirrors the
complexities of real-world patient trajectories.
## 6 Conclusion
To our knowledge, this is the first attempt to utilize GPT for generating
time-series EHR data. Our main contribution lies in the design of a patient
representation that captures temporal dependencies among token types, enabling
GPT to generate realistic patient sequences. Moreover, this representation
allows for easy conversion back to the OMOP format. Comprehensive evaluations
showed that the synthetic data effectively captures the intricate patterns
present in EHR data.
## References
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## 7 Supplementary Materials
Figure 9: The comparison of the patient representations.
Cohort | Real | GPT-OUTPAT | CEHR-BERT
---|---|---|---
HF readmission | Pre = 25.7 AUC = 65.7 PR = 39.3 | Pre =100.0 AUC = NA PR = NA | Pre =100.0 AUC = NA PR = NA
Hospitalization | Pre = 5.6 AUC = 75.3 PR = 19.5 | Pre = 5.3 AUC = 70.2 PR = 14.3 | Pre = 0.3 AUC = 76.0 PR = 1.0
COPD readmission | Pre = 34.5 AUC = 74.2 PR = 83.8 | Pre = NA AUC = NA PR = NA | Pre = NA AUC = NA PR = NA
Afib ischemic stroke | Pre = 8.7 AUC = 84.0 PR = 48.5 | Pre = 9.7 AUC = 67.2 PR = 27.2 | Pre = 19.5 AUC = 72.4 PR = 60.0
CAD CABG | Pre = 7.1 AUC = 88.4 PR = 55.9 | Pre = 3.5 AUC = 81.5 PR = 44.4 | Pre = 5.4 AUC = 77.2 PR = 44.1
Table 7: Logistic regression (LR) model performance across baseline synthetic
datasets. In each cell, three numbers were reported including the prevalence
of the positive cases, ROC-AUC, and PR-AUC. If the prevalence=NA, this
indicates that $0$ patients were identified in the cohort. AUC=NA and PR=NA
indicate the LR model could not be run successfully due to either 100% or 0%
prevalence.
Figure 10: The visit prevalence stratified by age group
Figure 11: The visit prevalence stratified by race
Figure 12: The visit prevalence stratified by gender
|
Ronald A. Remmerswaal
Arthur E.P. Veldman
Bernoulli Institute, University of Groningen
PO Box 407, 9700 AK Groningen, The Netherlands
two-phase flow volume of fluid method parabolic reconstruction
This work is part of the research programme SLING, which is (partly) financed by the Netherlands Organisation for Scientific Research (NWO).
|
# Formation of moons and equatorial ridge around top-shaped asteroids after
surface landslide
Ryuki Hyodo ISAS/JAXA, Sagamihara, Kanagawa, Japan<EMAIL_ADDRESS>Keisuke Sugiura Earth-Life Science Institute, Tokyo Institute of Technology,
Meguro-ku, Tokyo 152-8550, Japan
###### Abstract
Top-shaped asteroids have been observed among near-Earth asteroids. About half
of them are reported to have moons (on the order of $\sim 1$wt.% of the top-
shaped primary) and many of them have an equatorial ridge. A recent study has
shown that the enigmatic top-shaped figure of asteroids (e.g., Ryugu, Bennu,
and Didymos) could result from an axisymmetric landslide of the primary during
a fast spin-up near the breakup rotation period. Such a landslide would
inevitably form a particulate disk around an asteroid with a short timescale
($\sim 3$ hours). However, the long-term full dynamical evolution is not
investigated. Here, we perform a continuous simulation ($\sim 700$ hours) that
investigates the sequence of events from the surface landslide that forms a
top-shaped asteroid and a particulate disk to disk evolution. We show that the
disk quickly spreads and produces moons (within $\sim 300$ hours). The mass of
the formed moon is consistent with what is observed around the top-shaped
asteroids. We also demonstrate that an equatorial ridge is naturally formed
because a fraction of the disk particles re-accretes selectively onto the
equatorial region of the primary. We envision that Ryugu and Bennu could once
have an ancient moon that was later lost due to a successive moon’s orbital
evolution. Alternatively, at the top-shaped asteroid that has a moon, such as
Didymos, no significant orbital evolution of the moon has occurred that would
result in its loss. Our study would also be qualitatively applicable to any
rubble-pile asteroids near the breakup rotation period.
Asteroids (72); Near-Earth objects (1092)
## 1 Introduction
Figure 1: Schematic summary of our paper. Panel (a): a rubble-pile asteroid
spins up due to, for example, the YORP effect, small impacts, a close
encounter with a planet, or re-accumulation after a catastrophic impact. Panel
(b): a surface landslide occurs when a critical spin-state is realized and a
top-shaped figure is formed (Sugiura et al., 2021). Panel (c): a particulate
disk spreads due to inelastic collisions and gravitational interactions among
particles. Panel (d): a moon is gravitationally accreted outside the Roche
limit of the central top-shaped body and an axisymmetric equatorial ridge is
formed due to the re-accretion of disk particles. Panels (e) and (f): the
formed moon is lost or remains, depending on the long-term orbital evolution
between the moon and the primary.
More and more growing interests are on near-Earth asteroids (NEAs) together
with recent activities of asteroid exploration missions. Top-shaped asteroids
may exist ubiquitously among NEAs (Walsh & Jacobson, 2015; Margot et al.,
2015). These include asteroid 162173 Ryugu (Watanabe et al., 2019), 101955
Bennu (Lauretta et al., 2019), 65803 Didymos, 1999 KW4 (Ostro et al., 2006),
and 2001 SN263 (Becker et al., 2015). Their diameters are less than several
kilometers, and recent in-situ observations by spacecrafts showed that they
might be rubble-pile bodies (e.g., Watanabe et al., 2019). Because of their
small size, Yarkovsky-O’Keefe-Radzievskii-Paddack (YORP) effect (Rubincam,
2000), small impacts (Takeda & Ohtsuki, 2009), and/or a close encounter with a
planet (Hyodo et al., 2016), for example, could efficiently change their spin
states. The acceleration rate of spin-up (or spin-down) depends on each
physical process (e.g., roughly instantaneous by an impact and $\gtrsim 100$
Kyr by YORP effect on a kilometer-sized body).
Interestingly, about half of the reported top-shaped asteroids have moons
around them (e.g., 2001 NE263; Becker et al. 2015, 1999 KW4; Ostro et al.
2006, 1994 CC; Brozović et al. 2011, Didymos; Naidu et al. 2020). The masses
of the moons are roughly on the order of $\sim 1$wt.% of the host top-shaped
asteroids. Furthermore, many of the top-shaped asteroids are reported to have
an equatorial ridge (Benner et al., 2015).
There are several proposed mechanisms to form a top-shaped asteroid. The top-
shaped figure may be formed during re-accretion after catastrophic disruption
of a parent body (Michel et al., 2020). Alternatively, a mass movement with
reshaping or a landslide of surface materials of a rubble-pile asteroid due to
fast spin-up may form the top-shaped figure (Walsh et al., 2008, 2012; Harris
et al., 2009; Hirabayashi et al., 2020; Sugiura et al., 2021).
Recently, Sugiura et al. (2021) used the Smoothed Particle Hydrodynamics (SPH)
method to study rotational deformation during spin-up, which has also been
used to model the shape deformation due to impacts (Jutzi & Asphaug, 2015). In
their study, they included effective bulk friction of rubble pile bodies as a
parameter (i.e., effective friction angle $\phi_{\rm fri}$), which could
effectively include, for example, the effect of cohesion (see Sec. 2.2 in
Sugiura et al. 2021). They demonstrated that, for $\phi_{\rm fri}\geq 70$ deg
with a spin-up timescale of $\lesssim$ a few days (defined as the elapsed time
of spin-up from the rotation period of 3.5 h to 3.0 h), an axisymmetric set of
surface landslides would occur and top-shaped rubble-pile bodies would be
correspondingly formed (panels (a)-(b) in Fig. 1).
The story of the landslide hypothesis, however, should not have ended here,
although Sugiura et al. (2021) did not study the fate of the surface materials
of the landslide. In this study, we first show that the surface materials are
distributed around the newly formed top-shaped central body, forming a
transient particle disk (panel (c) in Fig. 1). Here, the disk mass is
typically found to be about $\sim 10$wt.% of the top-shaped primary body.
In this study, following the numerical approach used in Sugiura et al. (2021),
we investigate a full long-term dynamical evolution – from the landslide to
the end of the disk evolution – to understand the fate of the surface
materials that are distributed around the newly formed top-shaped body (Fig.
1). We show that moons and an axisymmetric equatorial ridge are naturally and
inevitably formed around the top-shaped primary (panel (d) in Fig. 1).
In Sect. 2, we describe our numerical models. In Sect. 3, we show our
numerical results. Section 4 discusses the long-term orbital evolution between
a moon and the central primary due to the binary YORP (BYORP) effect, which
may lead to diverse systems of the top-shaped asteroids with and without a
companion moon(s) (panel (e) and panel (f) in Fig. 1). Finally, Section 5
summarizes this paper.
## 2 Numerical method
Our SPH simulations are the same as those in Sugiura et al. (2021), and we
refer the readers to their Sec. 2 for more details. Our simulations solve
hydrodynamic equations (the equations of continuity and motion) with self-
gravity and yield processes with the aid of a friction model that determines
the shear strength of granular material (Jutzi, 2015).
In a similar context, a Discrete Element Methods (DEMs) with cohesionless
particles were used in Walsh et al. (2008). Shape deformation using this
approach could be numerically affected due to the inconsistency between the
realistic particle size and that used in their simulations. In the SPH
approach, in contrast, a rubble-pile body was constructed by a continuum of
granular material and we explicitly set a specific value of the friction angle
of the material. We expect that the SPH method has the advantage of precise
control of bulk friction of rubble piles (see also Jutzi & Asphaug, 2015). We
also note that there is a difference in the process of the moon formation
between their study and ours; in Walsh et al. (2008), the moon formation
occurred via a step-by-step accumulation of discretely ejected particles,
while, in our study, the moon formation occurs because of particle disk
evolution.
The rubble-pile body initially had a uniform and spherical body with a radius
of $R_{\rm ini}=500$ m, which was then numerically spun up with the angular
acceleration of $8.954\times 10^{-10}$ rad s-2. We stopped the acceleration
when 1wt.% of the primary is ejected; this prevents further artificial
deformation after the landslide, and the arbitrary choice of 1wt.% does not
affect our results as long as the landslide occurs with a shorter timescale
(here $\sim 3$ h) than the spin-up timescale (here $\sim 30$ h). The density
of particles at uncompressed states is $\rho_{\rm ini}=1.19$ g cm-3. Particle
mass is given as $m=M_{\rm tot}/N_{\rm tot}$, where $M_{\rm
tot}\equiv(4/3)\pi\rho_{\rm ini}R_{\rm ini}^{3}$ and $N_{\rm tot}=25470$ is
the total number of the SPH particles.
In this study, following Sugiura et al. (2021), we specifically focused on a
set of parameters (i.e., $\phi_{\rm fri}=80$ deg; see panels (k)-(o) in their
Fig. 1), which successfully resulted in the formation of an axisymmetric top-
shaped figure as seen at asteroids Ryugu, Bennu, and Didymos. Lowering
$\phi_{\rm fri}$, for example, resulted in an elongated lemon-shaped primary
via internal deformation (see panels (a)-(e) and (f)-(j) in Fig. 1 of Sugiura
et al. 2021), which eventually led to distribute particles as disks with a
variety of initial disk mass. The dependences on the disk parameters (e.g.,
initial disk mass) and tidal parameters are studied in detail using an
independent numerical approach (1D fluid simulations) in our companion paper
(Madeira et al. submitted). Here, we used the direct and continuous approach –
from the landslide to the end of the disk evolution – and focused on studying
the details regarding the relevant consequences of the top-shape formation.
Below, we show that the surface materials of the primary are distributed in a
disk-like structure and such a particulate disk eventually forms moon(s)
around the top-shaped primary. We numerically distinguished member particles
of the primary as follows. We start from a randomly chosen particle and then
iteratively detect any particles within a critical distance in a bottom-up
fashion (Hyodo & Genda, 2018). This procedure continues until no more new
particles are detected. As a critical distance, we used 1.5 times the
smoothing length of an SPH particle to be consistent with Sugiura et al.
(2021), although this specific choice does not affect our results.
In this study, we continued our simulation up to $t\sim 25\times 10^{5}$ s
($\sim 700$ hours). This is much after the landslide had occurred; Sugiura et
al. (2021) stopped their simulations at $t\sim 1\times 10^{5}$ s ($\sim 30$
hours) when the top-shaped figure was formed. We may need to be careful about
the angular momentum (AM) conservation in SPH simulations, particularly for a
rotating system (such as disk evolution). In our calculations, after an
artificial spin-up of the primary (i.e., artificial AM increase in the system)
was stopped, the cumulative AM error was $\sim 2.6$% during and after the disk
formation until the end of the simulation (i.e., from $t\sim 1.5\times 10^{5}$
s to $t\sim 25\times 10^{5}$ s).
Although our SPH simulations contain some error in AM conservation,
independent numerical simulations of particle disk evolutions of the Earth’s
Moon formation (e.g., $N$-body simulations; e.g., Kokubo et al. 2000 and 1D
semi-analytical fluid simulations; e.g., Salmon & Canup 2012) used similar
initial disk conditions (e.g., in terms of disk to primary mass ratio) and
their results (e.g., resultant moon to primary mass ratio) were consistent
with those obtained here. This is because the gravitational disk evolution can
be well characterized by the mass ratio between the disk and the primary
(Kokubo et al., 2000; Salmon & Canup, 2012). Furthermore, using $N$-body
simulations, Hyodo et al. (2015) showed a scaling law for the mass of the
largest moon formed via disk spreading as a function of the initial disk mass
(see their Fig. 12) and it is generally consistent with that obtained in this
study. We note that, although these studies generally assumed particulate
disks, a more realistic Moon-forming disk would be vapor-rich and thus the
Earth’s Moon formation process would be more complex (e.g., Thompson &
Stevenson, 1988; Nakajima & Stevenson, 2014; Lock et al., 2018)
Figure 2: Overall disk evolution after the landslide. Gray particles represent
member particles of the top-shaped primary. Dark-gray particles represent disk
particles and member particles of gravitational clumps (including the formed
moon at later epochs). The black circle indicates the Roche limit of the
primary ($r_{\rm R}\sim 2.5R_{\rm c}$ where $R_{\rm c}=500$m is used). Figure
3: Same as Fig. 2, but edge on views.
## 3 Numerical results: moon formation
Figure 4: Evolutions of the disk mass (left panel; not including the mass of
the largest moons) and the masses of the largest three moons (right panel). At
around $t\sim 1\times 10^{5}$s with a time-interval of $\Delta t\sim 10^{4}$s,
the landslide occurs and a particle disk is formed (left panel). The disk
spreads and starts to form gravitational clumps with masses larger than $\sim
0.001M_{\rm c}$ at $t\sim 5.5\times 10^{5}$s, where $M_{\rm c}$ is the mass of
the central primary. Two large moons are formed by the end of our simulations
($t=25\times 10^{5}$s; $\sim 700$h).
Figures 2 and 3 show an overall evolution of a particle disk including the
landslide. The landslide (i.e., disk formation) occurs with a very short
timescale ($\sim 10^{4}$ s $\simeq 3$ hours) compared to the disk evolution
timescale ($\sim 10^{6}$ $\simeq 300$ hours; see Fig. 4). Because of the
landslide, a top-shaped figure of the primary is formed (see more details in
Sugiura et al. 2021 and their Fig.1). The surface materials of the primary are
distributed around the primary, forming a particle disk. The initial mass of
the disk is $M_{\rm disk}\sim 20$wt.% of the top-shaped primary (Fig. 4). Most
of the disk mass is initially within the Roche limit of the primary, defined
as $r_{\rm R}\equiv 2.456(\rho_{\rm m}/\rho_{\rm c})^{-1/3}R_{\rm c}$ where
$\rho_{\rm m}$ and $\rho_{\rm c}$ are densities of the moon and the central
body. $R_{\rm c}$ is the radius of the central body.
The disk mass is large enough to become gravitationally unstable and the
spiral arm structures are formed as a result of gravitational instability.
This leads to an efficient angular momentum transfer and the outer disk
spreads further radially outward (Takeda & Ida, 2001), distributing the disk
materials beyond the Roche limit. Because of the angular momentum
conservation, the inner part of the disk spreads inward, resulting in re-
accretion onto the primary.
The materials scattered beyond the Roche limit start to coagulate via their
own gravity, forming gravitational clumps (Figs. 2 and 3). With time,
accretion among clumps proceeds, forming larger gravitational bodies (see also
the right panel of Fig. 4 for the mass evolution of the largest objects).
After $\sim 300$ hours since the landslide, large moons around the top-shaped
primary asteroid are formed. The total mass of the formed moons is $\sim
4$wt.% of the primary.
Here, two large moons are formed (Figs. 2, 3, and 4). The number of the large
moon is the result of the stochastic nature of the accretion processes that
can be seen in the 3D simulations, especially when the disk is massive to the
central body; the details of the stochastic nature were reported in
independent $N$-body simulations (Hyodo et al., 2015). This means that, if one
runs another simulation with the same parameters but with slightly different
initial positions of particles, the final outcome could be the formation of a
single moon. Thus, the number of large moons is not a decisive parameter that
characterizes the general outcome. More importantly, instead, regardless of
the number of moons, the total mass of the moons rarely changes and it
characterizes the outcome of specific choices of the disk parameters (e.g.,
initial disk mass; see Ida et al. 1997; Kokubo et al. 2000).
## 4 Discussion
### 4.1 Formation of equatorial ridge
Figure 5: Comparison of asteroid shapes between the initial figure ($t=0$h;
left panel) and after the disk evolution ($t\sim 700$h; right panel). A black
circle indicates a circle with a radius of 500 meters.
Our numerical simulations show another interesting feature as a result of disk
evolution around a top-shaped asteroid. Because the disk spreads both inward
and outward to conserve the angular momentum, the disk particles initially at
the inner region re-accrete onto the top-shaped primary. This has led to the
formation of an equatorial ridge.
Other potential physical mechanisms to form an equatorial ridge are discussed
in other literature. These are, for example, rotational reshaping and mass
movement (e.g., Walsh et al., 2012; Hirabayashi & Scheeres, 2015; Hirabayashi
et al., 2020), re-accumulation of debris after a catastrophic disruption of a
parent body (Michel et al., 2020), or selective accumulation of ejecta of
small impacts at rapidly rotating asteroid (Ikeya & Hirata, 2021).
Figure 5 shows an edge on views of the central rubble-pile body at $t=0$h
(left panel) and at the end of our simulation without disk particles and
formed moons ($t\sim 700$h; right panel). A significant shape change can be
seen from the initial spherical shape to a top-shaped figure with a prominent
axisymmetric equatorial ridge. This is a direct consequence of the re-
accretion of equatorial disk particles onto the top-shaped primary.
Many of the top-shaped asteroids and relatively spheroidal bodies in the near-
Earth region are reported to have an equatorial ridge (Benner et al., 2015).
JAXA’s Hayabusa2 mission revealed that asteroid Ryugu is a top-shaped asteroid
with an equatorial ridge (Watanabe et al., 2019). NASA’s OSIRIS-REx mission
also reported that asteroid Bennu has an equatorial ridge (Walsh et al.,
2019). Another top-shaped asteroid, Didymos, has also an equatorial ridge as
well as a moon, Dimorphos (Naidu et al., 2020).
We importantly note that the axis ratio of $c/a$ and the equatorial ridge seen
in Fig. 5 are smaller and much more prominent than the observations of, for
example, Ryugu, Bennu, and Didymos (Watanabe et al., 2019; Lauretta et al.,
2019; Naidu et al., 2020). Our SPH simulations would be limited in their
ability to precisely demonstrate the detailed accumulation processes of
particles in the equatorial ridge; because our SPH simulations assumed and
employed, for example, the same size and simple physical properties among all
particles. We also emphasize the importance of studying the long-term
geological evolution (e.g., including degradation) of the ridge. Continuous
micrometeoroid impacts and/or thermal fatigue would also change the ridge
shape. Therefore, further studies of the detailed ridge formation process as
well as the post-formation geological processes need to be done to fully
validate our ridge formation scenario via the landslides followed by the disk
evolution. We leave these points to later work.
### 4.2 On the diversity of top-shaped asteroids
Some top-shaped asteroids, e.g., Didymos, are found to have a companion
moon(s), while others do not (e.g., Ryugu and Bennu). In the discussion below,
we focus on asteroids that have (1) top-shaped figures and (2) equatorial
ridges, as background considerations. We, then, additionally consider the
existence and non-existence of a companion moon around the top-shaped primary.
Our results together with those of Sugiura et al. (2021) indicate that small
rubble-pile asteroids may inevitably experience a landslide due to spin-up by
the YORP effect or by other physical processes (e.g., re-accumulation, small
impact, and/or a close encounter with a planet). Depending on the acceleration
rate of spin-up as well as the effective friction angle, $\phi_{\rm fri}$, of
constituent particles of an asteroid, the resultant shape of asteroids via a
landslide would change (e.g., a lemon-shape or a top-shape; see Fig.1 of
Sugiura et al. 2021).
When the top-shaped primary was formed with the fast spin-up rate and
$\phi_{\rm fri}\gtrsim 70$ deg, Sugiura et al. (2021) numerically demonstrated
that about $\sim 10$wt.% of the primary is generally ejected. When the initial
disk mass is $\sim 10$wt.% or larger, the mass of the moon would be linearly
scaled with the disk mass (Ida et al., 1997; Kokubo et al., 2000), and thus a
change in the resultant disk mass within the same order of magnitude does not
significantly change the mass of the moon (would be on the order of $\sim
1$wt.% of the primary as observed for the top-shaped binary asteroids). In
contrast, when the effective friction angle is smaller than the above value,
the shape deformation became more gradual, and a lemon-shaped primary was
formed. In this case, the ejected mass tends to be smaller (e.g., $\sim 3$wt.%
and $\sim 5$wt.% for $\phi_{\rm fri}=40$ deg and $\phi_{\rm fri}=60$ deg,
respectively), although more detailed studies on throughout parameters may be
needed (see Fig. 1 of Sugiura et al., 2021). In these cases of smaller initial
disk masses, the mass of the resultant moon depends more strongly on the disk
mass and much smaller moons can be produced from the disk (Hyodo et al.,
2015).
Importantly, as a natural consequence of a landslide, a particle disk is
formed around the primary. Once a particle disk exists, the disk inevitably
spreads and forms moon(s) as long as the disk is massive enough. The final
mass and orbital configurations of the moon systems formed through the disk
spreading depend on the initial disk mass and tidal parameters. Such
dependencies are studied using 1D fluid simulations in our companion paper in
the context of the Didymos-Dimorphos system formation (Madeira et al.
submitted). In short, the direct consequence of the landslide would be the
system of a moon(s) around a central asteroid – regardless of the top-shaped
or the lemon-shaped central primary – with an equatorial ridge on the primary.
Then, the question now is can we remove the formed moon if the top-shaped
figures of Ryugu and/or Benuu are formed via landslides? This is because these
top-shaped asteroids today do not have a moon around them. Ejection via tidal
evolution would not be promising because its efficiency significantly
decreases as the distance between the primary and the moon becomes larger.
A potential dominant dynamical mechanism on the binary system may be the
binary YORP (BYORP) effect (e.g., Ćuk & Burns, 2005; Ćuk & Nesvorný, 2010;
McMahon & Scheeres, 2010; Jacobson & Scheeres, 2011). Although the timescale
of orbital separation to eject a companion moon via the BYORP effect can be as
small as $\sim 10^{5}$ years (see Sec.4.3 in Sugiura et al. 2021 and Ćuk &
Burns, 2005), its timescale and direction of the orbital separation (shrinking
or expanding) strongly depend on the details of the surface properties of the
asteroids and moons (Ćuk & Burns, 2005; McMahon & Scheeres, 2010).
We envision that such a complex dependence on the shapes and surface
properties can potentially lead to a diversity of the top-shaped asteroids
with and without a companion moon. The moons can be ejected by the BYORP
effect in some cases, while the dynamical evolution of the orbital separation
may not be efficient in other cases, remaining the system as binary asteroids
(Figure 1).
## 5 Summary
In this study, using the SPH simulations, we studied a continuous dynamical
sequence of events from the surface landslide that forms a top-shaped asteroid
and a particulate disk to disk evolution (Figure 1). We numerically
demonstrated that the particle disk formed by a surface landslide would
quickly spread and produce moons just outside the Roche limit of the top-
shaped primary (within $\sim 300$ hours). The mass of the moon is consistent
with what is observed around the top-shaped asteroids (on the order of $\sim
1$wt.% of the primary). We also demonstrated that an equatorial ridge would be
naturally formed because a fraction of the disk particles re-accrete
selectively on the equatorial region of the primary.
Tidal interaction as well as the binary YORP (BYORP) effect between moons and
the primary would change the orbital separation between them. The timescale
and direction of the orbital separation (shrinking or expanding) strongly
depend on the details of the surface properties of the moons and the primary.
This indicates that a long-term orbital evolution could produce diverse moon
systems around the top-shaped asteroids.
We envision that top-shaped Ryugu and Bennu could once have an ancient moon
that was later lost as a result of a successive moon’s orbital evolution.
Alternatively, other top-shaped asteroids today, such as Didymos, have a
companion moon. In these cases, no significant orbital evolution of the moon
may have occurred that would result in its loss.
Our study focused on the top-shaped asteroids. However, our results of the
consequences of a surface landslide – the moon formation and the equatorial
ridge formation – can be qualitatively (but not always quantitatively) applied
to any small rubble-pile asteroids near the breakup rotation period. Indeed,
Sugiura et al. (2021) showed that changing the effective friction angle and/or
the spin-up rate results in different patterns of the deformation mode with a
variety of disk formation (see their Fig. 1). Once a particle disk is formed,
the moon and the equatorial ridge would be naturally and inevitably formed.
The long-term evolution, then, would lead to a diverse asteroid system (see
also Ćuk, 2007; Jacobson & Scheeres, 2011; Ćuk et al., 2021).
Further development of the theoretical and modeling research, especially on
the BYORP effect, together with a better understanding of the surface and
particle properties of individual asteroids would be needed to further
constrain and validate the results presented in this study. Further studies on
the accumulation and post-formation geological processes of the equatorial
ridge are also demanded. Analysis of the return samples by JAXA’s Hayabusa2
and NASA’s OSIRIS-REx as well as data that would be obtained by NASA’s DART
and ESA’s HERA missions would help us to better understand the nature and
evolution of the top-shaped asteroids.
R.H. acknowledges the financial support of MEXT/JSPS KAKENHI (Grant Number
JP22K14091). R.H. also acknowledges JAXA’s International Top Young program.
K.S. acknowledges the financial support of JSPS, Japan KAKENHI Grant
(JP20K14536, JP20J01165). Numerical simulations in this work were carried out
on the Cray XC50 supercomputer at the Center for Computational Astrophysics,
National Astronomical Observatory of Japan.
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# Existence and emergent dynamics of quadratically separable states to the
Lohe tensor model
Seung-Yeal Ha
Department of Mathematical Sciences and Research Institute of Mathematics,
Seoul National University, Seoul 08826 and
Korea Institute for Advanced Study, Hoegiro 85, Seoul, 02455, Republic of
Korea<EMAIL_ADDRESS>, Dohyun Kim
School of Mathematics, Statistics and Data Science,
Sungshin Women’s University, Seoul 02844, Republic of Korea
<EMAIL_ADDRESS>and Hansol Park
Department of Mathematical Sciences,
Seoul National University, Seoul 08826, Republic of Korea
<EMAIL_ADDRESS>
###### Abstract.
A tensor is a multi-dimensional array of complex numbers, and the Lohe tensor
model is an aggregation model on the space of tensors with the same rank and
size. It incorporates previously well-studied aggregation models on the space
of low-rank tensors such as the Kuramoto model, Lohe sphere and matrix models
as special cases. Due to its structural complexities in cubic interactions for
the Lohe tensor model, explicit construction of solutions with specific
structures looks daunting. Recently, we obtained completely separable states
by associating rank-1 tensors. In this paper, we further investigate another
type of solutions, namely “quadratically separable states” consisting of
tensor products of matrices and their component rank-2 tensors are solutions
to the double matrix model whose emergent dynamics can be studied using the
same methodology of the Lohe matrix model.
###### Key words and phrases:
Aggregation, double sphere model, gradient flow, Kuramoto model, Lohe matrix
model, Lohe tensor model, synchronization
###### 2010 Mathematics Subject Classification:
82C10, 82C22, 35B37
Acknowledgment. The work of S.-Y. Ha is supported by National Research
Foundation of Korea (NRF-2020R1A2C3A01003881).
## 1\. Introduction
Collective behaviors often appear in large population systems for weakly
coupled oscillators or interacting units [3, 6, 7] in diverse scientific
disciplines including biology, social sciences, engineering with various space
and time scales, for instance, colonies of bacteria [25], school of fish [4,
26], flock of starlings [29], pedestrian dynamics [18], opinion dynamics [17],
power grid networks [22], etc. For a brief introduction to collective
dynamics, we refer the reader to survey articles and book [1, 5, 10, 24].
Mathematical approach toward the understanding of collective motions has been
established in literature by Winfree [28] and Kuramoto [19] in the 1970s, and
by Vicsek [27] in the 1990s. After their remarkable works, the aforementioned
models have been extended in several directions, particularly in high-
dimensional extension to Riemannian manifolds including the hypersurfaces [2,
12] and the matrix Lie group [11] which have attracted lots of interest thanks
to its powerful application, for instance, nonconvex optimization. In this
work, among high-dimensional models, we are concerned with the Lohe tensor
model in [13].
Next, we briefly discuss tensors and an aggregation model on the space of
tensors, namely “the Lohe tensor model”. A rank-$m$ complex valued tensor can
be represented as a multi-dimensional array of complex numbers with multi-
indices. The rank of a tensor is the number of indices, say a rank-$m$ tensor
with size $d_{1}\times\cdots\times d_{m}$ is an element of
${\mathbb{C}}^{d_{1}\times\cdots\times d_{m}}$. For example, scalars, vectors
and matrices correspond to rank-0, 1 and 2 tensors, respectively. Let $T$ be a
rank-$m$ tensor with a size $d_{1}\times\cdots\times d_{m}$. Then, we denote
$(\alpha_{1},\cdots,\alpha_{m})$-th component of the tensor $T$ by
$[T]_{\alpha_{1}\cdots\alpha_{m}}$, and we set $\overline{T}$ by the rank-$m$
tensor whose components are the complex conjugate of the elements in $T$:
$[\overline{T}]_{\alpha_{1}\cdots\alpha_{m}}:=\overline{[T]_{\alpha_{1}\cdots\alpha_{m}}}.$
Let ${\mathcal{T}}_{m}(\mathbb{C};d_{1}\times\cdots\times d_{m})$ be the
collection of all rank-$m$ tensors with size $d_{1}\times\cdots\times d_{m}$.
Then, it is a complex vector space. Several well-known first-order aggregation
models, for instance, the Kuramoto model [19], the swarm sphere model [23] and
matrix models [8, 20] can be regarded as aggregation models on
${\mathcal{T}}_{0}(\mathbb{C};1),{\mathcal{T}}_{1}(\mathbb{R};d)$ and
${\mathcal{T}}_{2}(\mathbb{C};d\times d)$, respectively. Let $A_{j}$ be the
skew-hermitian rank-$2m$ tensor with size $(d_{1}\times\cdots\times
d_{m})\times(d_{1}\times\cdots\times d_{m})$. For simplicity, we introduce
handy notation as follows: for
$T\in{\mathcal{T}}_{m}(\mathbb{C};d_{1}\times\cdots\times d_{m})$ and
$A\in{\mathcal{T}}_{2m}(\mathbb{C};d_{1}\times\cdots\times d_{m}\times
d_{1}\times\cdots\times d_{m})$, we set
$\displaystyle\begin{aligned}
&[T]_{\alpha_{*}}:=[T]_{\alpha_{1}\alpha_{2}\cdots\alpha_{m}},\quad[T]_{\alpha_{*0}}:=[T]_{\alpha_{10}\alpha_{20}\cdots\alpha_{m0}},\quad[T]_{\alpha_{*1}}:=[T]_{\alpha_{11}\alpha_{21}\cdots\alpha_{m1}},\\\
&[T]_{\alpha_{*i_{*}}}:=[T]_{\alpha_{1i_{1}}\alpha_{2i_{2}}\cdots\alpha_{mi_{m}}},\quad[T]_{\alpha_{*(1-i_{*})}}:=[T]_{\alpha_{1(1-i_{1})}\alpha_{2(1-i_{2})}\cdots\alpha_{m(1-i_{m})}},\\\
&[A]_{\alpha_{*}\beta_{*}}:=[A]_{\alpha_{1}\alpha_{2}\cdots\alpha_{m}\beta_{1}\beta_{2}\cdots\beta_{m}}.\end{aligned}$
Then, the Lohe tensor model in a component form reads as follows:
$\begin{cases}\displaystyle\dot{[T_{j}]}_{\alpha_{*0}}=[A_{j}]_{\alpha_{*0}\alpha_{*1}}[T_{j}]_{\alpha_{*1}}\\\
\displaystyle\hskip
42.67912pt+\sum_{i_{*}\in\\{0,1\\}^{m}}\kappa_{i_{*}}\Big{(}[T_{c}]_{\alpha_{*i_{*}}}\bar{[T_{j}]}_{\alpha_{*1}}[T_{j}]_{\alpha_{*(1-i_{*})}}-[T_{j}]_{\alpha_{*i_{*}}}\bar{[T_{c}]}_{\alpha_{*1}}[T_{j}]_{\alpha_{*(1-i_{*})}}\Big{)},\\\
\displaystyle\bar{[A_{j}]}_{\alpha_{*0}\alpha_{*1}}=-[A_{j}]_{\alpha_{*1}\alpha_{*0}},\end{cases}$
(1.1)
where $\kappa_{i_{*}}$’s are nonnegative coupling strengths.
Before we discuss our main issues, we introduce a concept of a “quadratically
separable state” for the Lohe tensor model (1.1).
###### Definition 1.1.
Let $\\{T_{i}\\}$ be a quadratically separable state to (1.1), if it is
decomposed as a tensor product of rank-2 tensors (or matrices):
$T_{i}=U_{i}^{1}\otimes U_{i}^{2}\otimes\cdots\otimes U_{i}^{m},\quad
U_{i}^{k}\in\mathbb{C}^{d_{1}^{k}\times
d_{2}^{k}},\quad\|U_{i}^{k}\|_{\textup{F}}=1,\quad 1\leq i\leq N,\quad 1\leq
k\leq m,$
where $\|\cdot\|_{\textup{F}}$ is the Frobenius norm induced by Frobenius
inner product: for matrices $A$ and $B$,
$\langle
A,B\rangle_{\textup{F}}:=\textup{tr}(A^{\dagger}B),\quad\|A\|_{\textup{F}}:=\sqrt{\textup{tr}(A^{\dagger}A)}.$
In this paper, we are interested in the following simple questions:
* •
(Q1): Are there quadratically separable states for the Lohe tensor model?
* •
(Q2): If so, do they exhibit collective behaviors under which circumstances?
Our main results deal with the raised two questions (Q1) and (Q2). More
precisely, our main results of this paper can be summarized as follows.
First, we introduce the double matrix model induced from the Lohe tensor model
whose elements have rank-4 with a specific condition on natural frequencies
$B_{j}$ and $C_{j}$ (see (3.9)):
$\displaystyle\begin{cases}\dot{U}_{j}=B_{j}U_{j}+\displaystyle\frac{\kappa_{1}}{N}\sum_{k=1}^{N}\left(\langle
V_{j},V_{k}\rangle_{\textup{F}}~{}U_{k}U_{j}^{\dagger}U_{j}-\langle
V_{k},V_{j}\rangle_{\textup{F}}~{}U_{j}U_{k}^{\dagger}U_{j}\right)\\\ \hskip
56.9055pt+\displaystyle\frac{\kappa_{2}}{N}\sum_{k=1}^{N}\left(\langle
V_{j},V_{k}\rangle_{\textup{F}}~{}U_{j}U_{j}^{\dagger}U_{k}-\langle
V_{k},V_{j}\rangle_{\textup{F}}~{}U_{j}U_{k}^{\dagger}U_{j}\right),\\\
\dot{V}_{j}=C_{j}V_{j}+\displaystyle\frac{\kappa_{1}}{N}\sum_{k=1}^{N}\left(\langle
U_{j},U_{k}\rangle_{\textup{F}}~{}V_{k}V_{j}^{\dagger}V_{j}-\langle
U_{k},U_{j}\rangle_{\textup{F}}~{}V_{j}V_{k}^{\dagger}V_{j}\right)\\\ \hskip
56.9055pt+\displaystyle\frac{\kappa_{2}}{N}\sum_{k=1}^{N}\left(\langle
U_{j},U_{k}\rangle_{\textup{F}}~{}V_{j}V_{j}^{\dagger}V_{k}-\langle
U_{k},U_{j}\rangle_{\textup{F}}~{}V_{j}V_{k}^{\dagger}V_{j}\right),\\\
\end{cases}$ (1.2)
where $B_{j}\in\mathbb{C}^{d_{1}\times d_{2}\times d_{1}\times d_{2}}$ and
$C_{j}\in\mathbb{C}^{d_{3}\times d_{4}\times d_{3}\times d_{4}}$ are skew-
symmetric rank-4 tensors. For a solution $\\{(U_{i},V_{i})\\}$ to the double
matrix model (1.2), a special solution $T_{i}$ to (1.1) can be represented as
follows:
$T_{i}(t)\equiv U_{i}(t)\otimes V_{i}(t),\quad t>0.$
Precisely, if $T_{i}$ is initially decomposed into the tensor product of two
matrices $U_{i}$ and $V_{i}$, then its separability is propagated along the
flow for all time. For details, we refer the reader to Section 3.
Second, we study emergent dynamics of the double matrix model by investigating
several aggregation quantities:
$\displaystyle\mathcal{D}(\mathcal{U}(t)):=\max_{1\leq i,j\leq
N}\|U_{i}(t)-U_{j}(t)\|_{\textup{F}},\quad\mathcal{S}(\mathcal{U}(t)):=\max_{1\leq
i,j\leq N}|n-\langle U_{i},U_{j}\rangle_{\textup{F}}(t)|,$
$\displaystyle\mathcal{D}(\mathcal{V}(t)):=\max_{1\leq i,j\leq
N}\|V_{i}(t)-V_{j}(t)\|_{\textup{F}},\quad\mathcal{S}(\mathcal{V}(t)):=\max_{1\leq
i,j\leq N}|m-\langle V_{i},V_{j}\rangle_{\textup{F}}(t)|.$
For a homogeneous ensemble (i.e. $B_{i}=B_{j}$ and $C_{i}=C_{j}$ for all $i$
and $j$.) we show that system (1.2) exhibits complete aggregation in which all
relative distances for $\\{U_{i}\\}$ and $\\{V_{i}\\}$ tend to zero
respectively (see Theorem 4.1). On the other hand, for a heterogeneous
ensemble ($B_{i}\neq B_{j}$ and $C_{i}\neq C_{j}$ in general) complete
aggregation (one-point collapse) would not be expected. Instead, our concern
is dedicated to emergence of locked states in which relative distances
converge to positive definite values (see Theorem 4.2). For our analytical
results, we need to assume that the size of unitary matrices satisfy
$\min(n,m)>4\sqrt{\max(n,m)}$ that requires restriction on $n,m$. In fact,
this technical assumption on the sizes is mainly due to the fact that elements
are complex-valued. Thus, when the unitary groups $\mathbf{U}(n)$ and
$\mathbf{U}(m)$ are replaced by the special orthogonal groups $\textbf{SO}(n)$
and $\textbf{SO}(m)$, such restriction on $n,m$ would be removed (see Theorem
C.1 and Theorem C.2).
The rest of the paper is organized as follows. In Section 2, we begin with
previous results on the relation between the Lohe tensor model and the swarm
double sphere model presented in [9]. As a natural extension, we construct the
double matrix model in Section 3 and study existence and uniqueness of
quadratically separable states. In Section 4, we study emergent dynamics of
the double matrix model for both homogeneous and heterogeneous ensembles.
Next, the double matrix model is further generalized to the multi matrix model
in Section 5. Finally, Section 6 is devoted to a brief summary of the paper
and future work. In Appendix A and Appendix B, we provide proofs of Lemma 4.1
and Lemma 4.2, respectively. In Appendix C, emergent dynamics of the double
matrix model on $\textbf{SO}(n)\times\textbf{SO}(m)$ is provided.
For simplicity of presentation, we use the following abbreviated jargons:
* •
LT model: Lohe tensor model, LM model: Lohe matrix model,
* •
SDS model: swarm double sphere model, SMS model: swarm multi-sphere model,
* •
DM model: double matrix model, DUM model: double unitary matrix model,
* •
DSOM model: double special orthogonal matrix model, QSS: quadratically
separable state,
* •
MM model: multiple matrix model, MUM model: multiple unitary matrix model.
## 2\. Preliminaries
In this section, we review how the SDS model [9, 21] can be related to the LT
model via completely separable states, and discuss extension of the SDS model
to the SMS model leading to the DM model.
### 2.1. From the LT model to the SDS model
In this subsection, we briefly recall the relation between the LT model and
the SDS model which was first observed in [9]. In [21], Lohe introduced a
first-order aggregation model on the product of two unit spheres
$(u_{i},v_{i})\in\mathbb{S}^{d_{1}-1}\times\mathbb{S}^{d_{2}-1}$:
$\displaystyle\begin{cases}\dot{u}_{i}=\Omega_{i}u_{i}+\displaystyle\frac{\kappa}{N}\sum_{j=1}^{N}\langle
v_{i},v_{j}\rangle(u_{j}-\langle u_{i},u_{j}\rangle u_{i}),\quad t>0,\\\
\dot{v}_{i}=\Lambda_{i}v_{i}+\displaystyle\frac{\kappa}{N}\sum_{j=1}^{N}\langle
u_{i},u_{j}\rangle(v_{j}-\langle v_{i},v_{j}\rangle v_{i}),\\\
(u_{i},v_{i})(0)=(u_{i}^{0},v_{i}^{0})\in\mathbb{S}^{d_{1}-1}\times\mathbb{S}^{d_{2}-1},\quad
1\leq i\leq N,\end{cases}$ (2.1)
where $\Omega_{i}$ and $\Lambda_{i}$ are skew-symmetric matrices of sizes
$d_{1}\times d_{1}$ and $d_{2}\times d_{2}$, respectively:
$\Omega_{i}^{\top}=-\Omega_{i},\quad\Lambda_{i}^{\top}=-\Lambda_{i},\quad
1\leq i\leq N,$
and ${\kappa}$ denotes the (uniform) coupling strength.
On the other hand, if we choose the following parameters:
$m=2,\quad{\kappa}_{00}={\kappa}_{11}=0,\quad{\kappa}_{01}={\kappa}_{10}={\kappa},$
system (1.1) reduces to the generalized Lohe matrix model in [15]:
$\begin{cases}\displaystyle{\dot{T}}_{i}=A_{i}T_{i}+\kappa(T_{c}T_{i}^{\dagger}T_{i}-T_{i}T_{c}^{\dagger}T_{i})+\kappa(T_{i}T_{i}^{\dagger}T_{c}-T_{i}T_{c}^{\dagger}T_{i}),\quad
t>0,\\\ \displaystyle
T_{i}(0)=T_{i}^{0},\quad\|T_{i}^{0}\|_{\textup{F}}=1,\quad
T_{c}:=\frac{1}{N}\sum_{k=1}^{N}T_{k},\quad i=1,\cdots,N.\end{cases}$ (2.2)
Next, we present how models (2.1) and (2.2) can be viewed as equivalent
systems under well-prepared natural frequency tensors and initial data in the
following proposition.
###### Proposition 2.1.
[9] Systems (2.1) and (2.2) are equivalent in the following sense.
1. (1)
Suppose $\\{(u_{i},v_{i})\\}$ is a solution to (2.1). Then, rank-2 real
tensors $T_{i}$ defined by $T_{i}:=u_{i}\otimes v_{i}$ is a solution to (2.2)
with initial data $T_{i}^{0}=u_{i}^{0}\otimes v_{i}^{0}$ and well-prepared
free flow tensors $A_{i}$:
$A_{i}T_{i}:=\Omega_{i}T_{i}+T_{i}\Lambda_{i}^{\top}.$ (2.3)
2. (2)
Suppose $T_{i}$ is a solution to (2.2)-(2.3) with completely factorized
initial data:
$T_{i}^{0}=:u_{i}^{0}\otimes v_{i}^{0},\quad 1\leq i,j\leq N,$
for rank-1 real tensors $u_{i}^{0}\in\mathbb{S}^{d_{1}1}$ and
$v_{i}^{0}\in\mathbb{S}^{d_{2}-1}$. Then, there exists a pair of unit vectors
$(u_{i}(t),v_{i}(t))$ such that
$T_{i}(t)=u_{i}(t)\otimes v_{i}(t),\quad t>0,$
where $(u_{i},v_{i})$ is a solution to (2.1) with initial data
$(u_{i},v_{i})(0)=(u_{i}^{0},v_{i}^{0})$.
By applying the completely separability stated in Proposition 2.1, emergent
behaviors for (2.1) and those for (2.2) are exactly the same. Thus, it
suffices to investigate the SDS model (2.1).
###### Proposition 2.2.
_[9]_ Suppose the initial data $\\{(u_{i}^{0},v_{i}^{0})\\}$ satisfy the
following conditions:
$\min_{1\leq i,j\leq N}\langle u_{i}^{0},u_{j}^{0}\rangle>0,\quad\min_{1\leq
i,j\leq N}\langle v_{i}^{0},v_{j}^{0}\rangle>0,$
and let $\\{(U,V)\\}$ be a solution to system (2.1). Then, we have
$\lim_{t\to\infty}\max_{1\leq i,j\leq
N}|u_{i}(t)-u_{j}(t)|=0\quad\textup{and}\quad\lim_{t\to\infty}\max_{1\leq
i,j\leq N}|v_{i}(t)-v_{j}(t)|=0.$
Now, it is worthwhile mentioning that system (2.1) can be represented as a
coupled gradient flow:
$\begin{cases}\displaystyle{\dot{u}}_{i}=-\frac{N\kappa}{2}{\mathbb{P}}_{T_{u_{i}}\mathbb{S}^{d_{1}-1}}\Big{(}\nabla_{u_{i}}\mathcal{E}(U,V)\Big{)},\vspace{0.2cm}\\\
\displaystyle{\dot{v}}_{i}=-\frac{N\kappa}{2}{\mathbb{P}}_{T_{v_{i}}\mathbb{S}^{d_{2}-1}}\Big{(}\nabla_{v_{i}}\mathcal{E}(U,V)\Big{)},\end{cases}$
(2.4)
where the projection operators ${\mathbb{P}}_{T_{u_{i}}\mathbb{S}^{d_{1}-1}}$
and ${\mathbb{P}}_{T_{v_{i}}\mathbb{S}^{d_{2}-1}}$ onto the tangent spaces of
$\mathbb{S}^{d_{1}-1}$ and $\mathbb{S}^{d_{2}-1}$ at $u_{i}$ and $v_{i}$ are
defined by the formulae, respectively: for $w_{1}\in\mathbb{R}^{d_{1}}$ and
$w_{2}\in\mathbb{R}^{d_{2}}$,
$\begin{cases}\displaystyle{\mathbb{P}}_{T_{u_{i}}\mathbb{S}^{d_{1}-1}}(w_{1}):=w_{1}-\langle
w_{1},u_{i}\rangle u_{i},\\\
\displaystyle{\mathbb{P}}_{T_{v_{i}}\mathbb{S}^{d_{2}-1}}(w_{2}):=w_{2}-\langle
w_{2},v_{i}\rangle v_{i},\end{cases}$
and the potential function $\mathcal{E}(U,V)$ is defined as
$\mathcal{E}(U,V):=1-\frac{1}{N^{2}}\sum_{i,j=1}^{N}\langle
u_{i},u_{j}\rangle\langle v_{i},v_{j}\rangle.$ (2.5)
Thanks to the gradient flow formulation (2.4), any solution to system (2.1)
converges to an equilibrium as $t\to\infty$.
### 2.2. From the SDS model to the SMS model
In this subsection, we extend the SDS model (2.1) on the product of two unit
spheres to an aggregation model on the product of multiple unit spheres,
namely, the SMS model. Note that the SDS model can be represented as a
gradient flow with a potential function as can be seen in (2.4)–(2.5). Thus,
we first generalize the potential function (2.5) as follows: for
$u_{i}^{k}\in\mathbb{S}^{d_{k}-1},\quad i=1,\cdots,N,\quad k=1,\cdots,m,$ we
set
$\mathcal{E}(U^{1},U^{2},\cdots,U^{m}):=1-\frac{1}{N^{2}}\sum_{i,j=1}^{N}\prod_{k=1}^{m}\langle
u_{i}^{k},u_{j}^{k}\rangle,\quad
U^{k}:=\\{u_{1}^{k},u_{2}^{k},\cdots,u_{N}^{k}\\}.$ (2.6)
Using the same spirit for a gradient flow with the potential function (2.6),
we propose the SMS model as follows.
$\displaystyle\begin{cases}\displaystyle{\dot{u}}_{i}^{k}=\frac{\kappa}{N}\sum_{j=1}^{N}\left(\prod_{l\neq
k}\langle u_{i}^{l},u_{j}^{l}\rangle\right)\Big{(}u_{j}^{k}-\langle
u_{i}^{k},u_{j}^{k}\rangle u_{i}^{k}\Big{)},\quad t>0,\\\
u_{i}^{k}(0)=u_{i}^{k,0}\in\mathbb{S}^{d_{k}-1}\qquad
i\in\\{1,2,\cdots,N\\},\quad k\in\\{1,2,\cdots,m\\}.\end{cases}$ (2.7)
As in Section 2.1, we set rank-$m$ real tensor $T_{i}$:
$T_{i}:=u_{i}^{1}\otimes\cdots\otimes u_{i}^{m},\quad i=1,\cdots,N.$
Then, it is easy to check that $T_{i}$ satisfies
$[\dot{T}_{p}]_{\alpha_{*0}}=\frac{\kappa}{N}\sum_{k=1}^{m}\sum_{\ell=1}^{N}\left([T_{\ell}]_{\alpha_{*i_{*}^{k}}}[\bar{T}_{p}]_{\alpha_{*1}}[T_{p}]_{\alpha_{*(1-i_{*}^{k})}}-[T_{p}]_{\alpha_{*i_{*}^{k}}}[\bar{T}_{\ell}]_{\alpha_{*1}}[T_{p}]_{\alpha_{*(1-i_{*}^{k})}}\right).$
(2.8)
It should be noted that (2.8) can be derivable from the Lohe tensor model
(1.1) with the following conditions:
$\displaystyle\kappa_{i_{*}}=\begin{cases}\kappa\quad&\text{when
}i_{*}=i_{*}^{k},\quad 1\leq k\leq m,\\\
0&\text{otherwise},\end{cases}\quad\text{where}\quad
i_{*}^{k}:=\underbrace{(0,\cdots,0,1,0,\cdots,0)}_{\text{only $k^{th}$ index
is $1$}}.$
Hence, systems (2.7) and (2.8) can be related in view of a completely
separable state, and since the emergent dynamics of the LT model has been
discussed in literature [13, 14, 15], we conclude that system (2.7) exhibits
complete aggregation under suitable circumstances.
###### Proposition 2.3.
_[9]_ Suppose that initial data $T^{0}=\\{T_{i}^{0}\\}$ are completely
factorized as a tensor product of rank-1 real tensors:
$\displaystyle\begin{aligned} &T_{i}^{0}=u_{i}^{1,0}\otimes
u_{i}^{2,0}\otimes\cdots\otimes u_{i}^{m,0},\quad i=1,\cdots,N,\\\
&{\mathcal{A}}(U^{k,0}):=\min_{1\leq i,j\leq N}\langle
u^{k,0}_{i},u^{k,0}_{j}\rangle>0,\quad k=1,\cdots,m,\end{aligned}$
and let $T=\\{T_{i}\\}$ be a solution to system (2.8) and
$\\{U^{1},\cdots,U^{d}\\}$ be a solution to system (2.7). Then, the following
assertions hold.
1. (1)
$T_{i}=T_{i}(t)$ is completely separable in the sense that
$T_{i}(t)=u_{i}^{1}(t)\otimes u_{i}^{2}(t)\otimes\cdots\otimes
u_{i}^{m}(t),\quad t>0,\quad i=1,\cdots,N.$
2. (2)
The solution exhibits the complete aggregation:
$\lim_{t\to\infty}\max_{1\leq i,j\leq N}\|T_{i}(t)-T_{j}(t)\|_{\textup{F}}=0.$
###### Proof.
For a proof, we refer the reader to Theorem 6.2 and Proposition 7.1 in [9]. ∎
## 3\. Existence of rank-4 quadratically separable states
In this section, we present existence of the QSS for the LT model with rank-4
tensors and introduce the DM model than can be induced from the LT model.
### 3.1. The DM model
In this subsection, we propose the DM model consisting of two generalized Lohe
matrix model on the rectangular matrices with possibly different sizes:
$U_{j}\in\mathbb{C}^{d_{1}\times d_{2}}\quad\mbox{and}\quad
V_{j}\in\mathbb{C}^{d_{3}\times d_{4}},\quad j=1,\cdots,N.$
Below, we sketch our strategy how to derive the DM model from the LT model:
* •
Step A (A homogeneous ensemble): we present a DM model for a homogeneous
ensemble (Section 3.1.1).
* •
Step B (A heterogeneous ensemble): by adding natural frequency tensors with
suitable structure conditions, we derive the DM model from the LT model
(Section 3.1.2).
#### 3.1.1. A homogeneous ensemble
Let $T_{j}\in\mathbb{C}^{d_{1}\times d_{2}\times d_{3}\times d_{4}}$ be a
rank-4 tensor that is a solution to (1.1) with zero natural frequency tensors
$A_{j}\equiv O$:
$[\dot{T}_{j}]_{\alpha_{*0}}=\displaystyle\sum_{i_{*}\in\\{0,1\\}^{4}}\left[\frac{\kappa_{i_{*}}}{N}\sum_{k=1}^{N}\left([T_{k}]_{\alpha_{*i_{*}}}[\bar{T}_{j}]_{\alpha_{*1}}[T_{j}]_{\alpha_{*(1-i_{*})}}-[T_{j}]_{\alpha_{*i_{*}}}[\bar{T}_{k}]_{\alpha_{*1}}[T_{j}]_{\alpha_{*(1-i_{*})}}\right)\right].$
(3.1)
For a given solution $T_{j}$ to (3.1), we assume that there exist two matrices
$U_{j}\in\mathbb{C}^{d_{1}\times d_{2}}$ and $V_{j}\in\mathbb{C}^{d_{3}\times
d_{4}}$ such that
$T_{j}=U_{j}\otimes
V_{j},\quad[T_{j}(t)]_{\alpha\beta\gamma\delta}=[U_{j}(t)]_{\alpha\beta}[V_{j}(t)]_{\gamma\delta}\quad\mbox{in
a component form}.$
Next, we rewrite cubic interaction terms in (3.1) in terms of $U_{j}$ and
$V_{j}$. For this, we decompose the index vectors $i_{*}$ and
$\alpha_{*i_{*}}$ as
$i_{*}:=(i_{1},i_{2},i_{3},i_{4}),\quad\alpha_{*i_{*}}:=(\beta_{*j_{*}},\gamma_{*k_{*}}),\quad
j_{*}:=(i_{1},i_{2}),\quad k_{*}:=(i_{3},i_{4}),$
where $j_{*}$ and $k_{*}$ correspond to the index vectors for $U_{j}$ and
$V_{j}$, respectively. We now observe
$\displaystyle\begin{aligned}
&[T_{i}]_{\alpha_{*i_{*}}}[\bar{T}_{j}]_{\alpha_{*1}}[T_{k}]_{\alpha_{*(1-i_{*})}}\\\
&\hskip
14.22636pt=[T_{i}]_{(\beta_{*j_{*}},\gamma_{*k_{*}}}[\bar{T}_{j}]_{(\beta_{*1},\gamma_{*1})}[T_{k}]_{(\beta_{*(1-j_{*})},\gamma_{*(1-k_{*})})}\\\
&\hskip
14.22636pt=[U_{i}]_{\beta_{*j_{*}}}[V_{i}]_{\gamma_{*k_{*}}}[\bar{U}_{j}]_{\beta_{*1}}[\bar{V}_{j}]_{\gamma_{*1}}[U_{k}]_{\beta_{*(1-j_{*})}}[V_{k}]_{\gamma_{*(1-k_{*})}}\\\
&\hskip
14.22636pt=\left([U_{i}]_{\beta_{*j_{*}}}[\bar{U}_{j}]_{\beta_{*1}}[U_{k}]_{\beta_{*(1-j_{*})}}\right)\left([V_{i}]_{\gamma_{*k_{*}}}[\bar{V}_{j}]_{\gamma_{*1}}[V_{k}]_{\gamma_{*(1-k_{*})}}\right).\end{aligned}$
By interchanging the roles of $j\leftrightarrow k$, the term inside of the
summation in the right-hand side of (3.1) becomes
$\displaystyle[T_{k}]_{\alpha_{*i_{*}}}[\bar{T}_{j}]_{\alpha_{*1}}[T_{j}]_{\alpha_{*(1-i_{*})}}-[T_{j}]_{\alpha_{*i_{*}}}[\bar{T}_{k}]_{\alpha_{*1}}[T_{j}]_{\alpha_{*(1-i_{*})}}$
$\displaystyle\hskip
28.45274pt=\left([U_{k}]_{\beta_{*j_{*}}}[\bar{U}_{j}]_{\beta_{*1}}[U_{j}]_{\beta_{*(1-j_{*})}}\right)\left([V_{k}]_{\gamma_{*k_{*}}}[\bar{V}_{j}]_{\gamma_{*1}}[V_{j}]_{\gamma_{*(1-k_{*})}}\right)$
$\displaystyle\hskip
42.67912pt-\left([U_{j}]_{\beta_{*j_{*}}}[\bar{U}_{k}]_{\beta_{*1}}[U_{j}]_{\beta_{*(1-j_{*})}}\right)\left([V_{j}]_{\gamma_{*k_{*}}}[\bar{V}_{k}]_{\gamma_{*1}}[V_{j}]_{\gamma_{*(1-k_{*})}}\right).$
Since the left-hand side of (3.1) has the form of
$\dot{T}_{j}=\dot{U}_{j}\otimes V_{j}+U_{j}\otimes\dot{V}_{j},$ (3.2)
one should impose either ${\kappa}_{*}=(1,1)$ or $j_{*}=(1,1)$ to derive the
restriction on ${\kappa}_{i_{*}}$:
$\kappa_{i_{*}}=0\quad\textup{for all $i_{*}\in\\{0,1\\}^{4}$ with
$(i_{1},i_{2})\neq(1,1)$ and $(i_{3},i_{4})\neq(1,1)$.}$
Then, the right-hand side of (3.1) further reduces to
$\displaystyle\begin{aligned}
&\displaystyle\sum_{i_{*}\in\\{0,1\\}^{4}}\left[\frac{\kappa_{i_{*}}}{N}\sum_{k=1}^{N}\left([T_{k}]_{\alpha_{*i_{*}}}[\bar{T}_{j}]_{\alpha_{*1}}[T_{j}]_{\alpha_{*(1-i_{*})}}-[T_{j}]_{\alpha_{*i_{*}}}[\bar{T}_{k}]_{\alpha_{*1}}[T_{j}]_{\alpha_{*(1-i_{*})}}\right)\right]\\\
&\hskip
28.45274pt=\displaystyle\sum_{j_{*}\in\\{0,1\\}^{2}}\biggl{[}\frac{\kappa_{(j_{*},1,1)}}{N}\sum_{k=1}^{N}\Big{(}[T_{k}]_{\beta_{*j_{*}}\gamma_{*1}}[\bar{T}_{j}]_{\beta_{*1}\gamma_{*1}}[T_{j}]_{\beta_{*(1-j_{*})}\gamma_{*0}}\\\
&\hskip
142.26378pt-[T_{j}]_{\beta_{*j_{*}}\gamma_{*1}}[\bar{T}_{k}]_{\beta_{*1}\gamma_{*1}}[T_{j}]_{\beta_{*(1-j_{*})}\gamma_{*0}}\Big{)}\biggr{]}\\\
&\hskip
42.67912pt+\displaystyle\sum_{k_{*}\in\\{0,1\\}^{2}}\biggl{[}\frac{\kappa_{(1,1,k_{*})}}{N}\sum_{k=1}^{N}\Big{(}[T_{k}]_{\beta_{*1}\gamma_{*k_{*}}}[\bar{T}_{j}]_{\beta_{*1}\gamma_{*1}}[T_{j}]_{\beta_{*0}\gamma_{*(1-k_{*})}}\\\
&\hskip
142.26378pt-[T_{j}]_{\beta_{*1}\gamma_{*k_{*}}}[\bar{T}_{k}]_{\beta_{*1}\gamma_{*1}}[T_{j}]_{\beta_{*0}\gamma_{*(1-k_{*})}}\Big{)}\biggr{]}\\\
&\hskip
28.45274pt=[V_{j}]_{\gamma_{*0}}\displaystyle\sum_{j_{*}\in\\{0,1\\}^{2}}\biggl{[}\frac{\kappa_{(j_{*},1,1)}}{N}\sum_{k=1}^{N}\Big{(}\langle
V_{j},V_{k}\rangle_{\textup{F}}[U_{k}]_{\beta_{*j_{*}}}[\bar{U}_{j}]_{\beta_{*1}}[U_{j}]_{\beta_{*(1-j_{*})}}\\\
&\hskip 142.26378pt-\langle
V_{k},V_{j}\rangle_{\textup{F}}[U_{j}]_{\beta_{*j_{*}}}[\bar{U}_{k}]_{\beta_{*1}}[U_{j}]_{\beta_{*(1-j_{*})}}\Big{)}\biggr{]}\\\
&\hskip
42.67912pt+[U_{j}]_{\beta_{*0}}\displaystyle\sum_{k_{*}\in\\{0,1\\}^{2}}\biggl{[}\frac{\kappa_{(1,1,k_{*})}}{N}\sum_{k=1}^{N}\Big{(}\langle
U_{j},U_{k}\rangle_{\textup{F}}[V_{k}]_{\gamma_{*k_{*}}}[\bar{V}_{j}]_{\gamma_{*1}}[V_{j}]_{\gamma_{*(1-k_{*})}}\\\
&\hskip 142.26378pt-\langle
U_{k},U_{j}\rangle_{\textup{F}}[V_{j}]_{\gamma_{*k_{*}}}[\bar{V}_{k}]_{\gamma_{*1}}[V_{j}]_{\gamma_{*(1-k_{*})}}\Big{)}\biggr{]}.\end{aligned}$
By comparing $\cdot\otimes V_{j}$ and $U_{j}\otimes\cdot$ in (3.2), one has
$\begin{cases}[\dot{U}_{j}]_{\beta_{*0}}=\displaystyle\sum_{j_{*}\in\\{0,1\\}^{2}}\biggl{(}\frac{\kappa_{(j_{*},1,1)}}{N}\sum_{k=1}^{N}\Big{(}\langle
V_{j},V_{k}\rangle_{\textup{F}}[U_{k}]_{\beta_{*j_{*}}}[\bar{U}_{j}]_{\beta_{*1}}[U_{j}]_{\beta_{*(1-j_{*})}}\\\
\displaystyle\hskip 170.71652pt-\langle
V_{k},V_{j}\rangle_{\textup{F}}[U_{j}]_{\beta_{*j_{*}}}[\bar{U}_{k}]_{\beta_{*1}}[U_{j}]_{\beta_{*(1-j_{*})}}\Big{)}\biggr{)},\\\
[\dot{V}_{j}]_{\gamma_{*0}}=\displaystyle\sum_{k_{*}\in\\{0,1\\}^{2}}\biggl{(}\frac{\kappa_{(1,1,k_{*})}}{N}\sum_{k=1}^{N}\Big{(}\langle
U_{j},U_{k}\rangle_{\textup{F}}[V_{k}]_{\gamma_{*k_{*}}}[\bar{V}_{j}]_{\gamma_{*1}}[V_{j}]_{\gamma_{*(1-k_{*})}}\\\
\displaystyle\hskip 170.71652pt-\langle
U_{k},U_{j}\rangle_{\textup{F}}[V_{j}]_{\gamma_{*k_{*}}}[\bar{V}_{k}]_{\gamma_{*1}}[V_{j}]_{\gamma_{*(1-k_{*})}}\Big{)}\biggr{)}.\\\
\end{cases}$
If we choose the coupling strengths as for remaining $i_{*}$:
$\displaystyle\kappa_{(0,1,1,1)}=\kappa_{(1,1,0,1)}=\kappa_{1},\quad\kappa_{(1,0,1,1)}=\kappa_{(1,1,1,0)}=\kappa_{2},\quad\kappa_{i_{*}}=0,$
(3.3)
we obtain the desired system for $(U_{j},V_{j})$:
$\displaystyle\begin{cases}\dot{U}_{j}=\displaystyle\frac{\kappa_{1}}{N}\sum_{k=1}^{N}\left(\langle
V_{j},V_{k}\rangle_{\textup{F}}~{}U_{k}U_{j}^{\dagger}U_{j}-\langle
V_{k},V_{j}\rangle_{\textup{F}}~{}U_{j}U_{k}^{\dagger}U_{j}\right)\\\ \hskip
28.45274pt+\displaystyle\frac{\kappa_{2}}{N}\sum_{k=1}^{N}\left(\langle
V_{j},V_{k}\rangle_{\textup{F}}~{}U_{j}U_{j}^{\dagger}U_{k}-\langle
V_{k},V_{j}\rangle_{\textup{F}}~{}U_{j}U_{k}^{\dagger}U_{j}\right),\\\
\dot{V}_{j}=\displaystyle\frac{\kappa_{1}}{N}\sum_{k=1}^{N}\left(\langle
U_{j},U_{k}\rangle_{\textup{F}}~{}V_{k}V_{j}^{\dagger}V_{j}-\langle
U_{k},U_{j}\rangle_{\textup{F}}~{}V_{j}V_{k}^{\dagger}V_{j}\right)\\\ \hskip
28.45274pt+\displaystyle\frac{\kappa_{2}}{N}\sum_{k=1}^{N}\left(\langle
U_{j},U_{k}\rangle_{\textup{F}}~{}V_{j}V_{j}^{\dagger}V_{k}-\langle
U_{k},U_{j}\rangle_{\textup{F}}~{}V_{j}V_{k}^{\dagger}V_{j}\right).\\\
\end{cases}$ (3.4)
#### 3.1.2. A heterogeneous ensemble
Similar to several aggregation models such as the Lohe matrix model [20] and
the Lohe tensor model [13], a natural candidate for heterogeneous (or non-
identical) extension of (3.4) would be the model (3.4) together with natural
frequency tensors $B_{j}$ and $C_{j}$ whose ranks are 4. Thus, the DM model
for a heterogeneous ensemble reads as
$\displaystyle\begin{cases}\dot{U}_{j}=B_{j}U_{j}+\displaystyle\frac{\kappa_{1}}{N}\sum_{k=1}^{N}\left(\langle
V_{j},V_{k}\rangle_{\textup{F}}~{}U_{k}U_{j}^{\dagger}U_{j}-\langle
V_{k},V_{j}\rangle_{\textup{F}}~{}U_{j}U_{k}^{\dagger}U_{j}\right)\\\ \hskip
28.45274pt+\displaystyle\frac{\kappa_{2}}{N}\sum_{k=1}^{N}\left(\langle
V_{j},V_{k}\rangle_{\textup{F}}~{}U_{j}U_{j}^{\dagger}U_{k}-\langle
V_{k},V_{j}\rangle_{\textup{F}}~{}U_{j}U_{k}^{\dagger}U_{j}\right),\\\
\dot{V}_{j}=C_{j}V_{j}+\displaystyle\frac{\kappa_{1}}{N}\sum_{k=1}^{N}\left(\langle
U_{j},U_{k}\rangle_{\textup{F}}~{}V_{k}V_{j}^{\dagger}V_{j}-\langle
U_{k},U_{j}\rangle_{\textup{F}}~{}V_{j}V_{k}^{\dagger}V_{j}\right)\\\ \hskip
28.45274pt+\displaystyle\frac{\kappa_{2}}{N}\sum_{k=1}^{N}\left(\langle
U_{j},U_{k}\rangle_{\textup{F}}~{}V_{j}V_{j}^{\dagger}V_{k}-\langle
U_{k},U_{j}\rangle_{\textup{F}}~{}V_{j}V_{k}^{\dagger}V_{j}\right),\\\
\end{cases}$ (3.5)
where $B_{j}\in\mathbb{C}^{d_{1}\times d_{2}\times d_{1}\times d_{2}}$ and
$C_{j}\in\mathbb{C}^{d_{3}\times d_{4}\times d_{3}\times d_{4}}$ are rank-4
tensors satisfying skew-symmetric properties: for suitable indices,
$[B_{j}]_{\alpha_{1}\beta_{1}\alpha_{2}\beta_{2}}=-[\bar{B}_{j}]_{\alpha_{2}\beta_{2}\alpha_{1}\beta_{1}},\quad[C_{j}]_{\gamma_{1}\delta_{1}\gamma_{2}\delta_{2}}=-[\bar{C}_{j}]_{\gamma_{2}\delta_{2}\gamma_{1}\delta_{1}}.$
(3.6)
Moreover, $U_{j}B_{j}$ and $C_{j}V_{j}$ can be understood as tensor
contractions between rank-4 and rank-2 tensors:
$[B_{j}U_{j}]_{\alpha\beta}=[B_{j}]_{\alpha\beta\gamma\delta}[U_{j}]_{\gamma\delta},\quad[C_{j}V_{j}]_{\alpha\beta}=[C_{j}]_{\alpha\beta\gamma\delta}[V_{j}]_{\gamma\delta}.$
Next, we find a condition for $A_{j}$ in (1.1) in terms of $B_{j}$ and $C_{j}$
in (3.5) so that models (1.1) and (3.5) are equivalent. For this, it suffices
to focus on the free flows by setting ${\kappa}_{1}={\kappa}_{2}=0$, i.e.,
$\dot{U}_{j}=B_{j}U_{j},\quad\dot{V}_{j}=C_{j}V_{j},\quad j=1,\cdots,N.$
If we use the relations above and
$(U_{j}\otimes V_{j})^{\prime}=(B_{j}U_{j})\otimes
V_{j}+U_{j}\otimes(C_{j}V_{j}),$
then we can find
$A_{j}(U_{j}\otimes V_{j})=(B_{j}U_{j})\otimes
V_{j}+U_{j}\otimes(C_{j}V_{j}).$ (3.7)
In addition, if we rewrite (3.7) as a component form, then it becomes
$\displaystyle\begin{aligned}
0&=[A_{j}]_{\alpha_{1}\beta_{1}\gamma_{1}\delta_{1}\alpha_{2}\beta_{2}\gamma_{2}\delta_{2}}[U_{j}]_{\alpha_{2}\beta_{2}}[V_{j}]_{\gamma_{2}\delta_{2}}-[B_{j}]_{\alpha_{1}\beta_{1}\alpha_{2}\beta_{2}}[U_{j}]_{\alpha_{2}\beta_{2}}[V_{j}]_{\gamma_{1}\delta_{1}}\\\
&\hskip
14.22636pt-[U_{j}]_{\alpha_{1}\beta_{1}}[C_{j}]_{\gamma_{1}\delta_{1}\gamma_{2}\delta_{2}}[V_{j}]_{\gamma_{2}\delta_{2}}\\\
&=\Big{(}[A_{j}]_{\alpha_{1}\beta_{1}\gamma_{1}\delta_{1}\alpha_{2}\beta_{2}\gamma_{2}\delta_{2}}-[B_{j}]_{\alpha_{1}\beta_{1}\alpha_{2}\beta_{2}}\delta_{\gamma_{1}\gamma_{2}}\delta_{\delta_{1}\delta_{2}}-[C_{j}]_{\gamma_{1}\delta_{1}\gamma_{2}\delta_{2}}\delta_{\alpha_{1}\alpha_{2}}\delta_{\beta_{1}\beta_{2}}\Big{)}\\\
&\hskip
14.22636pt\times[U_{j}]_{\alpha_{2}\beta_{2}}[V_{j}]_{\gamma_{2}\delta_{2}}.\end{aligned}$
(3.8)
Since (3.8) holds for arbitrary $U_{j}$ and $V_{j}$, we can find an explicit
relation between $A_{j}$ and $B_{j}$, $C_{j}$:
$\displaystyle[A_{j}]_{\alpha_{1}\beta_{1}\gamma_{1}\delta_{1}\alpha_{2}\beta_{2}\gamma_{2}\delta_{2}}=[B_{j}]_{\alpha_{1}\beta_{1}\alpha_{2}\beta_{2}}\delta_{\gamma_{1}\gamma_{2}}\delta_{\delta_{1}\delta_{2}}+[C_{j}]_{\gamma_{1}\delta_{1}\gamma_{2}\delta_{2}}\delta_{\alpha_{1}\alpha_{2}}\delta_{\beta_{1}\beta_{2}}.$
(3.9)
So far, we have derived the DM model (3.5) with (3.9) from the LT model (1.1).
However, if we use the same argument reversely and recall that system (3.5)
admits a unique solution, then one can identify the LT model from the DM
model. Thus, we can say that system (3.5)–(3.9) and system (1.1) are
equivalent in some sense. Below, we summarize the argument above in the
following proposition.
###### Proposition 3.1.
1. (1)
Suppose $\\{(U_{i},V_{i})\\}$ is a solution to (3.5). Then, a rank-4 tensor
$T_{i}$ defined by $T_{i}:=U_{i}\otimes V_{i}$ is a QSS to (1.1) with a well-
prepared free flow tensor $A_{i}$ satisfying (3.9).
2. (2)
Suppose a rank-4 tensor $T_{i}$ is a solution to (1.1) with (3.9) and
quadratically separable initial data:
$T_{i}^{0}=:U_{i}^{0}\otimes V_{i}^{0},\quad 1\leq i\leq N,$
for rank-2 tensors $U_{i}^{0}\in\mathbb{C}^{d_{1}\times d_{2}}$ and
$V_{i}^{0}\in\mathbb{C}^{d_{3}\times d_{4}}$ with unit norms. Then, there
exist two matrices with unit norms $U_{i}=U_{i}(t)$ and $V=V_{i}(t)$ such that
$T_{i}(t)=U_{i}(t)\otimes V_{i}(t),\quad t>0,$
where $(U_{i},V_{i})$ is a solution to (3.5) with
$(U_{i},V_{i})(0)=(U_{i}^{0},V_{i}^{0})$.
### 3.2. Gradient flow formulation of the DM model
In this subsection, we show that system (3.5) can be formulated as a gradient
flow with a suitable analytical potential induced from the Lohe tensor model.
Recall from [14] that the following functional can be associated with the LT
model:
$\mathcal{V}(T)=1-\frac{1}{N^{2}}\sum_{i,j=1}^{N}\langle
T_{i},T_{j}\rangle_{\textup{F}}.$ (3.10)
If we use the decomposition $T_{i}=U_{i}\otimes V_{i}$, then we find the
corresponding functional for (3.5):
$\mathcal{E}(U,V):=1-\frac{1}{N^{2}}\sum_{i,j=1}^{N}\langle U_{i}\otimes
V_{i},U_{j}\otimes
V_{j}\rangle_{\textup{F}}=1-\frac{1}{N^{2}}\sum_{i,j=1}^{N}\langle
U_{i},U_{j}\rangle_{\textup{F}}\langle V_{i},V_{j}\rangle_{\textup{F}}.$
(3.11)
In the following lemma, we show that $\mathcal{E}(U,V)$ is non-increasing
along the flow (3.5).
###### Lemma 3.1.
Let $\\{(U_{i},V_{i})\\}$ be a solution to (3.5) with $(B_{j},C_{j})=(O,O)$.
Then, the functional $\mathcal{E}(U,V)$ is non-increasing in time:
$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{E}(U,V)=-\frac{\kappa_{1}}{N}\sum_{j=1}^{N}\left\|\frac{1}{N}\sum_{i=1}^{N}\Big{(}\langle
V_{j},V_{i}\rangle_{\textup{F}}U_{i}U_{j}^{\dagger}-\langle
V_{i},V_{j}\rangle_{\textup{F}}U_{j}U_{i}^{\dagger}\Big{)}\right\|_{\textup{F}}^{2}$
$\displaystyle\hskip
19.91684pt-\frac{\kappa_{1}}{N}\sum_{j=1}^{N}\left\|\frac{1}{N}\sum_{i=1}^{N}\left(\langle
U_{j},U_{i}\rangle_{\textup{F}}V_{i}V_{j}^{\dagger}-\langle
U_{i},U_{j}\rangle_{\textup{F}}V_{j}V_{i}^{\dagger}\right)\right\|_{\textup{F}}^{2}$
$\displaystyle\hskip
19.91684pt-\frac{\kappa_{2}}{N}\sum_{j=1}^{N}\left\|\frac{1}{N}\sum_{i=1}^{N}\Big{(}\langle
V_{j},V_{i}\rangle_{\textup{F}}U_{j}^{\dagger}U_{i}-\langle
V_{i},V_{j}\rangle_{\textup{F}}U_{i}^{\dagger}U_{j}\Big{)}\right\|_{\textup{F}}^{2}$
$\displaystyle\hskip
19.91684pt-\frac{\kappa_{2}}{N}\sum_{j=1}^{N}\left\|\frac{1}{N}\sum_{i=1}^{N}\left(\langle
U_{j},U_{i}\rangle_{\textup{F}}V_{j}^{\dagger}V_{i}-\langle
U_{i},U_{j}\rangle_{\textup{F}}V_{i}^{\dagger}V_{j}\right)\right\|_{\textup{F}}^{2}.$
###### Proof.
It follows from Proposition 4.1 in [14] that $\mathcal{V}(T)$ in (3.10) for
the LT model (1.1) satisfies
$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{V}(T)=-\frac{1}{N}\sum_{j=1}^{N}\sum_{i_{*}\in\\{0,1\\}^{4}}\kappa_{i_{*}}\|M^{i_{*}}(T_{c})M^{i_{*}}(T_{j})^{\dagger}-M^{i_{*}}(T_{j})M^{i_{*}}(T_{c})^{\dagger}\|_{\textup{F}}^{2}.$
(3.12)
Here, we recall from [14, Definition 3.1] that for a rank-4 tensor $T$,
$\mathcal{M}^{i_{*}}(T)$ is defined as a rank-2 tensor reshaped from $T$ or it
can be understood as a bijective linear map which conserves the Frobenius
norm. Since we only consider four types among $i_{*}\in\\{0,1\\}^{4}$ as in
(3.3), we set
$\kappa_{(0,1,1,1)}=\kappa_{(1,1,0,1)}=\kappa_{1},\quad\kappa_{(1,0,1,1)}=\kappa_{(1,1,1,0)}=\kappa_{2},\quad\kappa_{i_{*}}=0\quad\text{for
other }i_{*}.$ (3.13)
On the other hand, in (3.12), we are concerned with the term
$M^{i_{*}}(T_{i})M^{i_{*}}(T_{j})^{\dagger}$ for $T_{i}=U_{i}\otimes V_{i}$
and $i_{*}$ in (3.13), for instance, if $i_{*}=(0,1,1,1)$, then we have
$\displaystyle[M^{i_{*}}(T_{i})M^{i_{*}}(T_{j})^{\dagger}]_{\alpha_{20}\alpha_{21}}$
$\displaystyle=[T_{i}]_{\alpha_{11}\alpha_{20}\alpha_{31}\alpha_{41}}[\bar{T}_{j}]_{\alpha_{11}\alpha_{21}\alpha_{31}\alpha_{41}}$
$\displaystyle=[U_{i}]_{\alpha_{11}\alpha_{20}}[V_{i}]_{\alpha_{31}\alpha_{41}}[\bar{U}_{j}]_{\alpha_{11}\alpha_{21}}[\bar{V}_{j}]_{\alpha_{31}\alpha_{41}}=\langle
V_{j},V_{i}\rangle_{\textup{F}}[U_{j}^{\dagger}U_{i}]_{\alpha_{21}\alpha_{20}}.$
If we repeat similar calculations as above, we obtain the desired dissipative
estimate. ∎
### 3.3. The DM model
In this subsection, we further reduce the DM model (3.5) to the model defined
on the product of two unitary matrices $\mathbf{U}(n)\times\mathbf{U}(m)$
which we call as the DUM model. By the construction of model (3.5), we know
that the Frobenius norms of $U_{j}$ and $V_{j}$ are conserved. In addition to
the conservation of the Frobenius norms, we show that unitarity is also
preserved, when rectangular matrices are replaced by square matrices.
###### Lemma 3.2.
Suppose that the system parameters and initial data satisfy
$d_{1}=d_{2}=n,\quad d_{3}=d_{4}=m,\quad U_{j}^{0}\in\mathbf{U}(n),\quad
V_{j}^{0}\in\mathbf{U}(m),$
and let $\\{(U_{j},V_{j})\\}$ be a solution to (3.5). Then, one has
$\displaystyle U_{j}(t)\in\mathbf{U}(n),\quad V_{j}(t)\in\mathbf{U}(m),\quad
t>0,\quad j=1,\cdots,N.$
###### Proof.
Since $U_{j}$ and $V_{j}$ have the same structure, we focus only on $U_{j}$.
We rewrite the dynamics of $U_{j}$ as a simpler form:
$\dot{U}_{j}=B_{j}U_{j}+(P_{j}-P_{j}^{\dagger})U_{j}+U_{j}(Q_{j}-Q_{j}^{\dagger}),$
where $P_{j}$ and $Q_{j}$ are defined as
$\displaystyle P_{j}:=\frac{{\kappa}_{1}}{N}\sum_{k=1}^{N}\langle
V_{j},V_{k}\rangle_{\textup{F}}U_{k}U_{j}^{\dagger},\quad
Q_{j}:=\frac{{\kappa}_{2}}{N}\sum_{k=1}^{N}\langle
V_{j},V_{k}\rangle_{\textup{F}}U_{j}^{\dagger}U_{k}.$
Then, one has
$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}(U_{j}U_{j}^{\dagger})$
$\displaystyle=B_{j}U_{j}U_{j}^{\dagger}-U_{j}U_{j}^{\dagger}B_{j}+(P_{j}-P_{j}^{\dagger})U_{j}U_{j}^{\dagger}+U_{j}(Q_{j}-Q_{j}^{\dagger})U_{j}^{\dagger}$
$\displaystyle\hskip 14.22636pt-
U_{j}U_{j}^{\dagger}(P_{j}-P_{j}^{\dagger})-U_{j}(Q_{j}-Q_{j}^{\dagger})U_{j}^{\dagger}$
$\displaystyle=(B_{j}+P_{j}-P_{j}^{\dagger})U_{j}U_{j}^{\dagger}-U_{j}U_{j}^{\dagger}(B_{j}+P_{j}-P_{j}^{\dagger}).$
By straightforward calculation, we find
$\frac{{\textup{d}}}{{\textup{d}t}}\|I_{n}-U_{j}U_{j}^{\dagger}\|_{\textup{F}}=0.$
(3.14)
Since we assume $U_{j}^{0}\in\mathbf{U}(n)$, i.e.,
$I_{n}-U_{j}U_{j}^{\dagger}=O$, the relation (3.14) yields the desired result.
∎
Due to the unitarity, system (3.5) can further be reduced to the model on
$\mathbf{U}(n)\times\mathbf{U}(m)$ by using the relations:
$U_{j}U_{j}^{\dagger}=I_{n}=U_{j}^{\dagger}U_{j},\quad
V_{j}V_{j}^{\dagger}=I_{m}=V_{j}^{\dagger}V_{j},\quad j=1,\cdots,N.$
Thus, the DM model reads as
$\displaystyle\begin{cases}\dot{U}_{j}=B_{j}U_{j}+\displaystyle\frac{\kappa}{N}\sum_{k=1}^{N}\left(\langle
V_{j},V_{k}\rangle_{\textup{F}}~{}U_{k}-\langle
V_{k},V_{j}\rangle_{\textup{F}}~{}U_{j}U_{k}^{\dagger}U_{j}\right),\\\
\dot{V}_{j}=C_{j}V_{j}+\displaystyle\frac{\kappa}{N}\sum_{k=1}^{N}\left(\langle
U_{j},U_{k}\rangle_{\textup{F}}~{}V_{k}-\langle
U_{k},U_{j}\rangle_{\textup{F}}~{}V_{j}V_{k}^{\dagger}V_{j}\right),\end{cases}$
where $\kappa=\kappa_{1}+\kappa_{2}$.
Note that natural frequency tensors $B_{j}$ and $C_{j}$ have rank-4 satisfying
skew-symmetric properties as in (3.6). In order to give a meaning of
Hamiltonian, we associate two Hermitian matrices, namely,
$H_{j}\in\mathbb{C}^{n\times n}$ and $G_{j}\in\mathbb{C}^{m\times m}$:
$[B_{j}]_{\alpha_{1}\beta_{1}\alpha_{2}\beta_{2}}=:[-\mathrm{i}H_{j}]_{\alpha_{1}\alpha_{2}}\delta_{\beta_{1}\beta_{2}},\quad[C_{j}]_{\gamma_{1}\delta_{1}\gamma_{2}\delta_{2}}=:[-\mathrm{i}G_{j}]_{\gamma_{1}\gamma_{2}}\delta_{\delta_{1}\delta_{2}}.$
Then, system (3.10) reduces to the model on
$\mathbf{U}(n)\times\mathbf{U}(m)$:
$\displaystyle\begin{cases}\dot{U}_{j}=-\mathrm{i}H_{j}U_{j}+\displaystyle\frac{\kappa}{N}\sum_{k=1}^{N}\left(\langle
V_{j},V_{k}\rangle_{\textup{F}}~{}U_{k}-\langle
V_{k},V_{j}\rangle_{\textup{F}}~{}U_{j}U_{k}^{\dagger}U_{j}\right),\\\
\dot{V}_{j}=-\mathrm{i}G_{j}V_{j}+\displaystyle\frac{\kappa_{1}}{N}\sum_{k=1}^{N}\left(\langle
U_{j},U_{k}\rangle_{\textup{F}}~{}V_{k}-\langle
U_{k},U_{j}\rangle_{\textup{F}}~{}V_{j}V_{k}^{\dagger}V_{j}\right),\end{cases}$
(3.15)
where $H_{j}U_{j}$ and $G_{j}V_{j}$ are now usual matrix products. Then as in
Lemma 3.1, system (3.15) also satisfies the dissipative energy estimate. Since
the proof can be directly obtained from Lemma 3.1, we omit it.
###### Corollary 3.1.
Let $\\{(U_{i},V_{i})\\}$ be a solution to (3.15) with $H_{j}=G_{j}\equiv O$.
Then the Lyapunov functional (5.1) satisfies
$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{E}(U,V)=-\frac{\kappa}{N}\sum_{j=1}^{N}\left(\|\dot{U}_{j}\|_{\textup{F}}^{2}+\|\dot{V}_{j}\|_{\textup{F}}^{2}\right),\quad
t>0.$
## 4\. Emergent dynamics of rank-4 quadratically separable states
In this section, we study the emergent behavior of the QSS for the Lohe tensor
model by investigating the dynamics of the DUM model which reads as follows:
$\begin{cases}\dot{U}_{j}=\displaystyle-\mathrm{i}H_{j}U_{j}+\frac{\kappa}{N}\sum_{k=1}^{N}\left(\langle
V_{j},V_{k}\rangle_{\textup{F}}~{}U_{k}-\langle
V_{k},V_{j}\rangle_{\textup{F}}~{}U_{j}U_{k}^{\dagger}U_{j}\right),\quad
t>0,\\\
\dot{V}_{j}=\displaystyle-\mathrm{i}G_{j}V_{j}+\frac{\kappa}{N}\sum_{k=1}^{N}\left(\langle
U_{j},U_{k}\rangle_{\textup{F}}~{}V_{k}-\langle
U_{k},U_{j}\rangle_{\textup{F}}~{}V_{j}V_{k}^{\dagger}V_{j}\right),\\\
(U_{j},V_{j})(0)=(U_{j}^{0},V_{j}^{0})\in\mathbf{U}(n)\times\mathbf{U}(m),\quad
1\leq j\leq N,\end{cases}$ (4.1)
where $H_{j}\in\mathbb{C}^{n\times n}$ and $G_{j}\in\mathbb{C}^{m\times m}$
are Hermitian matrices which play roles of natural frequencies for each
oscillator.
First, we recall several definitions of emergent behaviors for (4.1).
###### Definition 4.1.
_[16]_ Let $(\mathcal{U},\mathcal{V}):=\\{U_{j},V_{j}\\}_{j=1}^{N}$ be a
solution to (4.1).
1. (1)
System (4.1) exhibits complete aggregation if the following estimate holds:
$\lim_{t\to\infty}\max_{1\leq i,j\leq
N}\Big{(}\|U_{i}(t)-U_{j}(t)\|_{\textup{F}}+\|V_{i}(t)-V_{j}(t)\|_{\textup{F}}\Big{)}=0.$
(4.2)
2. (2)
The state $(\mathcal{U},\mathcal{V})$ tends to a phase-locked state if the
following relation holds:
$\exists~{}\lim_{t\to\infty}U_{i}(t)U_{j}(t)^{\dagger}\quad\textup{and}\quad\exists~{}\lim_{t\to\infty}V_{i}(t)V_{j}(t)^{\dagger}.$
In order to investigate emergent behaviors for (4.1), we denote the following
quantities for notational simplicity:
$\displaystyle\begin{aligned} &X_{ij}:=U_{i}U_{j}^{\dagger},\quad
S_{ij}:=I_{n}-U_{i}U_{j}^{\dagger},\quad d_{ij}:=\langle
U_{i},U_{j}\rangle_{\textup{F}},\\\ &Y_{ij}:=V_{i}V_{j}^{\dagger},\quad
T_{ij}:=I_{m}-V_{i}V_{j}^{\dagger}\quad c_{ij}:=\langle
V_{i},V_{j}\rangle_{\textup{F}}.\end{aligned}$ (4.3)
It follows from simple observations that
$\displaystyle\|U_{j}\|_{\textup{F}}=n,\quad\|V_{j}\|_{\textup{F}}=m,\quad|d_{ij}|\leq
n,\quad|c_{ij}|\leq m,$
$\displaystyle\|S_{ij}\|^{2}_{\textup{F}}=\|U_{i}-U_{j}\|_{\textup{F}}^{2}=2\textup{Re}(n-d_{ij}),\quad\|T_{ij}\|_{\textup{F}}^{2}=\|V_{i}-V_{j}\|_{\textup{F}}^{2}=2\textup{Re}(m-c_{ij}).$
Then, it is easy to see that
$\|U_{i}-U_{j}\|_{\textup{F}}\to 0\Longleftrightarrow|n-d_{ij}|\to
0,\quad\|V_{i}-V_{j}\|_{\textup{F}}\to 0\Longleftrightarrow|m-c_{ij}|\to 0.$
Thus, the complete aggregation in (4.2) can be represented in terms of the
quantities in (4.3):
$\displaystyle\lim_{t\to\infty}\max_{1\leq i,j\leq
N}\Big{(}\|S_{ij}(t)\|_{\textup{F}}+\|T_{ij}(t)\|_{\textup{F}}\Big{)}=0\quad\textup{or}\quad\lim_{t\to\infty}\max_{1\leq
i,j\leq N}\Big{(}|n-d_{ij}(t)|+|m-c_{ij}(t)|\Big{)}=0.$
In this regard, we define aggregation quantities: for $t>0$,
$\displaystyle\mathcal{D}(\mathcal{U}(t)):=\max_{1\leq i,j\leq
N}\|U_{i}(t)-U_{j}(t)\|_{\textup{F}},\quad\mathcal{S}(\mathcal{U}(t)):=\max_{1\leq
i,j\leq N}|n-d_{ij}(t)|,$
$\displaystyle\mathcal{D}(\mathcal{V}(t)):=\max_{1\leq i,j\leq
N}\|V_{i}(t)-V_{j}(t)\|_{\textup{F}},\quad\mathcal{S}(\mathcal{V}(t)):=\max_{1\leq
i,j\leq N}|m-c_{ij}(t)|.$
Note that $\mathcal{S}(\mathcal{U})$ has a second-order with respect to the
state $\mathcal{U}$, whereas $\mathcal{D}(\mathcal{U})$ has a first-order.
In the following two subsections, we present emergent dynamics for homogeneous
and heterogeneous ensembles respectively.
### 4.1. Homogeneous ensemble
In this subsection, we deal with the emergent dynamics of (4.1) with a
homogeneous ensemble:
$H_{j}\equiv H,\quad G_{j}\equiv G,\quad j=1,\cdots,N,$
and by the solution splitting property, we may assume
$H=O,\quad G=O.$
In this setting, system (3.5) becomes
$\displaystyle\begin{cases}\dot{U}_{j}=\displaystyle\frac{\kappa}{N}\sum_{k=1}^{N}\left(\langle
V_{j},V_{k}\rangle_{\textup{F}}~{}U_{k}-\langle
V_{k},V_{j}\rangle_{\textup{F}}~{}U_{j}U_{k}^{\dagger}U_{j}\right),\quad
t>0,\\\ \dot{V}_{j}=\displaystyle\frac{\kappa}{N}\sum_{k=1}^{N}\left(\langle
U_{j},U_{k}\rangle_{\textup{F}}~{}V_{k}-\langle
U_{k},U_{j}\rangle_{\textup{F}}~{}V_{j}V_{k}^{\dagger}V_{j}\right),\\\
(U_{j},V_{j})(0)=(U_{j}^{0},V_{j}^{0})\in\mathbf{U}(n)\times\mathbf{U}(m).\end{cases}$
(4.4)
Without loss of generality, we may assume
$n\geq m.$ (4.5)
Our goal of this subsection is to find a sufficient condition under which
$\lim_{t\to\infty}\mathcal{L}(t)=0,\quad\mathcal{L}(t):=\mathcal{D}(\mathcal{U}(t))+\mathcal{D}(\mathcal{V}(t))+\mathcal{S}(\mathcal{U}(t))+\mathcal{S}(\mathcal{V}(t)),$
where $\mathcal{L}=\mathcal{L}(t)$ is called as a total aggregation
functional.
In [16], it suffices to study the temporal evolutions of
$\mathcal{D}(\mathcal{U})$. However in our case, time evolutions of
$\mathcal{S}(\mathcal{U})$ as well as $\mathcal{S}(\mathcal{V})$ are needed to
achieve complete aggregation estimates. Below, we derive a differential
inequality for $\mathcal{L}$.
###### Lemma 4.1.
Let $\\{(U_{j},V_{j})\\}$ be a solution to (4.4) with (4.5). Then, the total
aggregation functional $\mathcal{L}$ satisfies
$\displaystyle\begin{aligned}
\dot{\mathcal{L}}\leq-2{\kappa}(m-4\sqrt{n})\mathcal{L}+{\kappa}(4n+9)\mathcal{L}^{2}+{\kappa}\left(2n+\frac{8}{3}\right)\mathcal{L}^{3},\quad
t>0.\end{aligned}$ (4.6)
###### Proof.
Since a proof is lengthy, we provide it in Appendix A. ∎
We are now ready to provide a sufficient condition leading to the complete
aggregation for (4.4).
###### Theorem 4.1.
Suppose that the system parameters and initial data satisfy
$\displaystyle\begin{aligned} &\textup{(i)}~{}~{}n\geq m>4\sqrt{n}.\\\
&\textup{(ii)}~{}~{}\mathcal{L}^{0}<\alpha_{n,m}:=\frac{-(12n+27)+\sqrt{(12n+27)^{2}+48(m-4\sqrt{n})(3n+4)}}{4(3n+4)},\end{aligned}$
(4.7)
and let $\\{(U_{j},V_{j})\\}$ be a solution to (4.4). Then, we have
$\lim_{t\to\infty}\mathcal{L}(t)=0.$
Moreover, the convergence rate is at least exponential. In other words, system
(4.4) exhibits complete aggregation with an exponential convergence.
###### Proof.
Consider an auxiliary quadratic polynomial:
$f(s):=\left(2n+\frac{8}{3}\right)s^{2}+(4n+9)s-2(m-4\sqrt{n}).$
Since $m>4\sqrt{n}$, the algebraic relation $f=0$ admits a unique positive
root $\alpha_{n,m}$ defined in (4.7)(ii). Then, the relation (4.6) is
rewritten in terms of $f$:
$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{L}\leq{\kappa}\mathcal{L}f(\mathcal{L}),\quad
t>0.$
Since initial data satisfy (4.7)(ii), the desired result follows from
dynamical systems theory. ∎
###### Remark 4.1.
(i) In (4.7)(i), we have assumed that $n\geq m>4\sqrt{n}$ which imposes a
restriction on the size of $U_{i}$ such as
$n>16.$
It also should be mentioned that such restriction arises, for instance, when
we estimate the terms $\mathcal{I}_{15}$ in (A.7) and $\mathcal{I}_{25}$ in
(A.10). We indeed show in Appendix C that this technical assumption would be
removed.
(ii) Since $m\leq n$, we have
$\limsup_{n\to\infty}\alpha_{n,m}\leq\lim_{n\to\infty}\frac{-(12n+27)+\sqrt{(12n+27)^{2}+48(n-4\sqrt{n})(3n+4)}}{4(3n+4)}=\frac{1}{2}.$
### 4.2. Heterogeneous ensemble
In this subsection, we study the case of heterogeneous Hamiltonians in which
$H_{j}$ and $G_{j}$ in (4.1) are given to be different in general. In order to
establish the emergence of the phase-locked state, we will follow a strategy
developed in [16]. For any two solutions $\\{U_{j},V_{j}\\}$ and
$\\{\tilde{U}_{j},\tilde{V}_{j}\\}$ to (4.1), we define the diameters
measuring the dissimilarity of two configurations:
$\displaystyle\begin{aligned} &d(U,\tilde{U})(t):=\max_{1\leq i,j\leq
N}\|U_{i}(t)U_{j}^{\dagger}(t)-\tilde{U}_{i}(t)\tilde{U}_{j}^{\dagger}(t)\|_{\textup{F}},\\\
&d(V,\tilde{V})(t):=\max_{1\leq i,j\leq
N}\|V_{i}(t)V_{j}^{\dagger}(t)-\tilde{V}_{i}(t)\tilde{V}_{j}^{\dagger}(t)\|_{\textup{F}}.\end{aligned}$
(4.8)
Then, we will show that the diameters above converge to zero:
$\lim_{t\to\infty}\Big{(}d(U,\tilde{U})(t)+d(V,\tilde{V})(t)\Big{)}=0.$ (4.9)
As a next step, since our system is autonomous, for any $T>0$, we choose
$\tilde{U}_{j}$ and $\tilde{V}_{j}$ as
$\tilde{U}_{j}(t)=U_{j}(t+T),\quad\tilde{V}_{j}(t)=V_{j}(t+T).$
By discretizing the time $t\in\mathbb{R}_{+}$ as $n\in\mathbb{Z}_{+}$ and
setting $T=m\in\mathbb{Z}_{+}$, we conclude that
$\\{U_{i}(n)U_{j}^{\dagger}(n)\\}_{n\in\mathbb{Z}_{+}}$ and
$\\{V_{i}(n)V_{j}^{\dagger}(n)\\}_{n\in\mathbb{Z}_{+}}$ are indeed Cauchy
sequences in the complete spaces $\mathbf{U}(n)$ and $\mathbf{U}(m)$,
respectively. Hence, there exist two constant unitary matrices
$U_{ij}^{\infty}\in\mathbf{U}(n)$ and $V_{ij}^{\infty}\in\mathbf{U}(m)$ such
that
$\lim_{t\to\infty}\|U_{i}(t)U_{j}^{\dagger}(t)-U_{ij}^{\infty}\|_{\textup{F}}=0,\quad\lim_{t\to\infty}\|V_{i}(t)V_{j}^{\dagger}(t)-V_{ij}^{\infty}\|_{\textup{F}}=0.$
Hence, we aim to find a sufficient condition under which (4.9) holds. To show
(4.9), we associate another diameters measuring the difference between two
solution configurations $\\{U_{j},V_{j}\\}$ and
$\\{\tilde{U}_{j},\tilde{V}_{j}\\}$:
$\displaystyle\begin{aligned} &\mathcal{S}(U,\tilde{U})(t):=\max_{1\leq
i,j\leq N}|\langle
U_{i},U_{j}\rangle(t)-\langle\tilde{U}_{i},\tilde{U}_{j}\rangle(t)|,\\\
&\mathcal{S}(V,\tilde{V})(t):=\max_{1\leq i,j\leq N}|\langle
V_{i},V_{j}\rangle(t)-\langle\tilde{V}_{i},\tilde{V}_{j}\rangle(t)|.\end{aligned}$
(4.10)
Note that
$\mathcal{S}(U,\tilde{U})\leq\sqrt{n}d(U,\tilde{U}),\quad\mathcal{S}(V,\tilde{V})\leq\sqrt{m}d(V,\tilde{V}).$
To this end, our goal of this subsection is to find a sufficient framework
leading to
$\lim_{t\to\infty}\mathcal{F}(t)=0,\quad\mathcal{F}(t):=d(U,\tilde{U})(t)+d(V,\tilde{V})(t)+\mathcal{S}(U,\tilde{U})(t)+\mathcal{S}(V,\tilde{V})(t).$
(4.11)
Below, we derive a differential inequality for $\mathcal{F}$ in (4.11).
###### Lemma 4.2.
Let $\\{(U_{i},V_{i})\\}$ and $\\{(\tilde{U}_{i},\tilde{V}_{i})\\}$ be any two
solutions to (4.1) with (4.5), respectively. Then, the aggregation functional
$\mathcal{F}$ satisfies
$\frac{{\textup{d}}}{{\textup{d}t}}{\mathcal{F}}\leq-{\kappa}\left(2m-8\sqrt{n}-\frac{\max\\{\mathcal{D}(\mathcal{H}),\mathcal{D}(\mathcal{G})\\}}{{\kappa}}\right)\mathcal{F}+{\kappa}(4n+22)\mathcal{L}\mathcal{F}+20{\kappa}\mathcal{L}^{2}\mathcal{F},$
(4.12)
where $\mathcal{D}(\mathcal{H})$ and $\mathcal{D}(\mathcal{G})$ are defined as
$\mathcal{D}(\mathcal{H}):=\max_{1\leq i,j\leq
N}\|H_{i}-H_{j}\|_{\infty},\quad\mathcal{D}(\mathcal{G}):=\max_{1\leq i,j\leq
N}\|G_{i}-G_{j}\|_{\infty}.$
###### Proof.
We postpone its proof in Appendix B. ∎
In what follows, using the differential inequality (4.12), we provide a
sufficient condition leading to the phase-locked state. First, in order to
make a leading coefficient of (4.12) negative, we assume that a coupling
strength ${\kappa}$ is sufficiently large:
${\kappa}>\frac{\max\\{\mathcal{D}(\mathcal{H}),\mathcal{D}(\mathcal{G})\\}}{2(m-4\sqrt{n})}.$
For a handy notation, we denote
$\Lambda:=2(m-4\sqrt{n})-\frac{\max\\{\mathcal{D}(\mathcal{H}),\mathcal{D}(\mathcal{G})\\}}{{\kappa}}.$
Next, we show that we can make the total aggregation functional $\mathcal{L}$
small as we wish by controlling the coupling strength ${\kappa}$.
###### Proposition 4.1.
Suppose system parameters and initial data satisfy
$\displaystyle\begin{aligned} &\textup{(i)}~{}~{}n\geq
m,\quad\mathcal{D}(\mathcal{H})\geq\mathcal{D}(\mathcal{G}),\qquad\textup{(ii)}~{}~{}{\kappa}\geq{\kappa}_{\textup{c}},\quad\mathcal{L}^{0}\ll\nu_{2},\end{aligned}$
(4.13)
where ${\kappa}_{\textup{c}}$ and $\nu_{2}$ are specified later in (4.18) and
(4.19), respectively, and let $\\{(U_{j},V_{j})\\}$ be a solution to (4.1).
Then, we have
$\lim_{{\kappa}\to\infty}\limsup_{t\to\infty}\mathcal{L}(t)=0.$
###### Proof.
Since we assumed (4.13)(i), it follows from (4.6) that
$\dot{\mathcal{L}}\leq-2{\kappa}(m-4\sqrt{n})\mathcal{L}+{\kappa}(4n+9)\mathcal{L}^{2}+{\kappa}\left(2n+\frac{8}{3}\right)\mathcal{L}^{3}+2(1+3\sqrt{n})\mathcal{D}(\mathcal{H}).$
(4.14)
Now, we introduce an auxiliary cubic polynomial:
$g(s):=2(m-4\sqrt{n})s-(4n+9)s^{2}-\left(2n+\frac{8}{3}\right)s^{3}.$ (4.15)
Then, (4.14) can be rewritten as
$\dot{\mathcal{L}}\leq{\kappa}\left(\frac{2(1+3\sqrt{n})\mathcal{D}(\mathcal{H})}{{\kappa}}-g(\mathcal{L})\right).$
By investigating roots of the polynomial $g$ in (4.15), we deduce that for a
sufficient large ${\kappa}$, the polynomial $g$ admits one negative root, say,
$\nu_{0}<0$, and two positive roots, say, $0<\nu_{1}<\nu_{2}$ with continuous
dependence of ${\kappa}$, i.e.,
$\lim_{{\kappa}\to\infty}\nu_{1}({\kappa})=0,\quad\lim_{{\kappa}\to\infty}\nu_{2}({\kappa})=\alpha_{*},\quad\textup{$\alpha_{*}$:
a unique positive root of $g$}.$ (4.16)
Since we assume (4.13)(ii), there exists a finite entrance time $T_{*}>0$ such
that
$\mathcal{L}(t)<\nu_{1},\quad t>T_{*}.$ (4.17)
Finally, we combine (4.16) and (4.17) to obtain the desired estimate:
$\lim_{{\kappa}\to\infty}\limsup_{t\to\infty}\mathcal{L}(t)=0.$
∎
###### Remark 4.2.
For an explicit value for ${\kappa}_{\textup{c}}$ in (4.13)(ii), by simple
calculus, we know that the polynomial $g=g(s)$ in (4.15) admits a global
maximum in $\mathbb{R}_{+}$ at $s=s_{*}$:
$s_{*}:=\frac{-(4n+9)+\sqrt{(4n+9)^{2}+2(6n+8)(m-4\sqrt{n})}}{6n+8}.$
Thus, in order to guarantee the existence of a positive root for $g$, one
should impose
$g(s_{*})>\frac{2(1+3\sqrt{n})\mathcal{D}(\mathcal{H})}{{\kappa}},\quad\textup{i.e.,}\quad{\kappa}>\frac{2(1+3\sqrt{n})\mathcal{D}(\mathcal{H})}{g(s_{*})}=:{\kappa}_{\textup{c}}.$
(4.18)
For this ${\kappa}_{\textup{c}}$, $\nu_{2}$ in (4.13)(ii) can be explicitly
determined as the largest positive root of
$g(s)=\frac{2(1+3\sqrt{n})\mathcal{D}(\mathcal{H})}{{\kappa}}\quad\textup{when
${\kappa}>{\kappa}_{\textup{c}}$}.$ (4.19)
It follows from Proposition 4.1 that under the assumption on the smallness of
the initial data $\mathcal{L}^{0}$ and largeness of the coupling strength
${\kappa}$, we have
$\lim_{{\kappa}\to\infty}\limsup_{t\to\infty}\mathcal{L}(t)=0.$
Hence, there exists a (large) coupling strength ${\kappa}_{\textup{p}}$ larger
than ${\kappa}_{\textup{c}}$ such that for $t\gg 1$,
$(4n+22)\mathcal{L}(t)+20\mathcal{L}(t)^{2}<\frac{\Lambda}{2},\quad\Lambda=2(m-4\sqrt{n})-\frac{\max\\{\mathcal{D}(\mathcal{H}),\mathcal{D}(\mathcal{G})\\}}{{\kappa}}.$
Then, (4.12) becomes
$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{F}$
$\displaystyle\leq-\Lambda{\kappa}\mathcal{F}+{\kappa}(4n+22)\mathcal{L}\mathcal{F}+20{\kappa}\mathcal{L}^{2}\mathcal{F}\leq-\Lambda{\kappa}\mathcal{F}+\frac{\Lambda{\kappa}}{2}\mathcal{F}=-\frac{\Lambda{\kappa}}{2}\mathcal{F}.$
This yields a desired exponential decay for $\mathcal{F}$. The argument above
can be stated and shown as follows.
###### Theorem 4.2.
Suppose system parameters and initial data satisfy
$\displaystyle\begin{aligned} &\textup{(i)}~{}~{}n\geq
m,\quad\mathcal{D}(\mathcal{H})\geq\mathcal{D}(\mathcal{G}),\qquad&\textup{(ii)}~{}~{}{\kappa}>{\kappa}_{\textup{p}}>{\kappa}_{\textup{c}},\quad\max\\{\mathcal{L}^{0},\tilde{\mathcal{L}}^{0}\\}\leq\nu_{2},\end{aligned}$
and let $\\{(U_{i},V_{i})\\}$ and $\\{(\tilde{U}_{i},\tilde{V}_{i})\\}$ be any
two solutions to (4.1), respectively. Then, the following assertions hold.
1. (1)
The functional $\mathcal{F}$ converges to zero with an exponential rate.
2. (2)
The normalized velocities $\mathrm{i}\dot{U}_{j}U_{j}^{\dagger}$ and
$\mathrm{i}\dot{V}_{j}V_{j}^{\dagger}$ synchronize:
$\lim_{t\to\infty}\|\mathrm{i}\dot{U}_{j}U_{j}^{\dagger}-\mathrm{i}\dot{\tilde{U}}_{j}\tilde{U}_{j}^{\dagger}\|_{\textup{F}}=0,\quad\lim_{t\to\infty}\|\mathrm{i}\dot{V}_{j}V_{j}^{\dagger}-\mathrm{i}\dot{\tilde{V}}_{j}\tilde{V}_{j}^{\dagger}\|_{\textup{F}}=0.$
3. (3)
There exist unitary matrices $X_{\infty}\in\mathbf{U}(n)$ and
$Y_{\infty}\in\mathbf{U}(m)$ such that
$\displaystyle\lim_{t\to\infty}U_{i}^{\dagger}(t)\tilde{U}_{i}(t)=X_{\infty},\quad\lim_{t\to\infty}\|\tilde{U}_{i}(t)-U_{i}(t)X_{\infty}\|_{\textup{F}}=0,$
$\displaystyle\lim_{t\to\infty}V_{i}^{\dagger}(t)\tilde{V}_{i}(t)=Y_{\infty},\quad\lim_{t\to\infty}\|\tilde{V}_{i}(t)-V_{i}(t)Y_{\infty}\|_{\textup{F}}=0.$
4. (4)
Asymptotic phase-locking emerge: for any indices $i$ and $j$,
$\exists~{}\lim_{t\to\infty}U_{i}(t)U_{j}^{\dagger}(t)\quad\textup{and}\quad\exists~{}\lim_{t\to\infty}U_{i}(t)U_{j}^{\dagger}(t).$
Moreover, there exist phase-locked state
$\mathcal{X}^{\infty}:=\\{X_{i}^{\infty}\\}_{i=1}^{N}$ and
$\mathcal{Y}^{\infty}:=\\{Y_{i}^{\infty}\\}_{i=1}^{N}$, and unitary matrices
$P\in\mathbf{U}(n)$ and $Q\in\mathbf{U}(m)$ such that
$\displaystyle\lim_{t\to\infty}\|U_{i}(t)-X_{i}^{\infty}P\|_{\textup{F}}=0=\lim_{t\to\infty}\|V_{i}(t)-Y_{i}^{\infty}Q\|_{\textup{F}}=0,\quad\max\\{\mathcal{D}(\mathcal{X}^{\infty}),\mathcal{D}(\mathcal{Y}^{\infty})\\}<\nu_{2}.$
###### Proof.
(i) It follows from (4.12) in Proposition 4.1 that $\mathcal{F}$ satisfies
$\frac{{\textup{d}}}{{\textup{d}t}}{\mathcal{F}}\leq-{\kappa}\left(2m-8\sqrt{n}-\frac{\max\\{\mathcal{D}(\mathcal{H}),\mathcal{D}(\mathcal{G})\\}}{{\kappa}}\right)\mathcal{F}+{\kappa}(4n+22)\mathcal{L}\mathcal{F}+20{\kappa}\mathcal{L}^{2}\mathcal{F}.$
(4.20)
Since $\mathcal{L}$ can be sufficiently small under (4.13), i.e.,
$\mathcal{L}(t)<\nu_{1},\quad t>T_{*},$
we choose ${\kappa}_{\textup{p}}>{\kappa}_{\textup{c}}$ sufficiently large
such that
$(4n+22)\mathcal{L}+20\mathcal{L}^{2}<\frac{\Lambda}{2},\quad\Lambda=2(m-4\sqrt{n})-\frac{\max\\{\mathcal{D}(\mathcal{H}),\mathcal{D}(\mathcal{G})\\}}{{\kappa}}.$
Thus, (4.20) becomes
$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{F}\leq-\frac{\Lambda{\kappa}}{2}\mathcal{F},\quad
t>T_{*},$ (4.21)
and the relation (4.21) yields the desired exponential convergence of
$\mathcal{F}$ toward zero.
(ii) For the second assertion, we claim:
$\|\mathrm{i}\dot{U}_{j}U_{j}^{\dagger}-\mathrm{i}\dot{\tilde{U}}_{j}\tilde{U}_{j}^{\dagger}\|_{\textup{F}}\leq
2{\kappa}(m+n)\mathcal{F},\qquad\|\mathrm{i}\dot{V}_{j}V_{j}^{\dagger}-\mathrm{i}\dot{\tilde{V}}_{j}\tilde{V}_{j}^{\dagger}\|_{\textup{F}}\leq
2{\kappa}(m+n)\mathcal{F}.$ (4.22)
Once the relations (4.22) hold, it follows from the first assertion to derive
the second assertion. Note that
$\displaystyle\begin{aligned}
&\|\mathrm{i}\dot{U}_{j}U_{j}^{\dagger}-\mathrm{i}\dot{\tilde{U}}_{j}\tilde{U}_{j}^{\dagger}\|_{\textup{F}}=\left\|\frac{{\kappa}}{N}\sum_{k=1}^{N}\Big{(}\langle
V_{j},V_{k}\rangle_{\textup{F}}U_{k}U_{j}^{\dagger}-\langle
V_{k},V_{j}\rangle_{\textup{F}}U_{j}U_{k}^{\dagger}\Big{)}\right.\\\ &\hskip
142.26378pt\left.-\frac{{\kappa}}{N}\sum_{k=1}^{N}\Big{(}\langle\tilde{V}_{j},\tilde{V}_{k}\rangle_{\textup{F}}\tilde{U}_{k}\tilde{U}_{j}^{\dagger}-\langle\tilde{V}_{k},\tilde{V}_{j}\rangle_{\textup{F}}\tilde{U}_{j}\tilde{U}_{k}^{\dagger}\Big{)}\right\|_{\textup{F}}.\end{aligned}$
(4.23)
By algebraic manipulations, one has
$\displaystyle\|\langle
V_{j},V_{k}\rangle_{\textup{F}}U_{k}U_{j}^{\dagger}-\langle\tilde{V}_{j},\tilde{V}_{k}\rangle_{\textup{F}}\tilde{U}_{k}\tilde{U}_{j}^{\dagger}\|_{\textup{F}}$
$\displaystyle\hskip 14.22636pt\leq\|\langle
V_{j},V_{k}\rangle_{\textup{F}}(U_{k}U_{j}^{\dagger}-\tilde{U}_{k}\tilde{U}_{j}^{\dagger})\|_{\textup{F}}+\|(\langle
V_{j},V_{k}\rangle_{\textup{F}}-\langle\tilde{V}_{j},\tilde{V}_{k}\rangle_{\textup{F}})\tilde{U}_{k}\tilde{U}_{j}^{\dagger}\|_{\textup{F}}\leq(m+n)\mathcal{F}.$
Thus, the relation (4.23) yields
$\|\mathrm{i}\dot{U}_{j}U_{j}^{\dagger}-\mathrm{i}\dot{\tilde{U}}_{j}\tilde{U}_{j}^{\dagger}\|_{\textup{F}}\leq
2{\kappa}(m+n)\mathcal{F}.$
This shows the desired synchronization of the normalized velocities and the
exactly same argument is applied to $\mathrm{i}\dot{V}_{j}V_{j}^{\dagger}$.
(iii) Since system (4.1) is autonomous, we directly use Theorem 2(3) of [16].
Hence, we briefly sketch a proof. We observe
$\displaystyle\left\|\frac{{\textup{d}}}{{\textup{d}}s}(U_{i}^{\dagger}\tilde{U}_{i})\right\|_{\textup{F}}=\|\dot{U}_{i}U_{i}^{\dagger}-\dot{\tilde{U}}_{i}\tilde{U}_{i}^{\dagger}\|_{\textup{F}}\leq
2{\kappa}(m+n)\mathcal{F}.$
Since $\mathcal{F}$ tends to zero exponentially,
$\lim_{t\to\infty}(U_{i}^{\dagger}(t)\tilde{U}_{i}(t))=U_{i}^{0,{\dagger}}\tilde{U}_{i}^{0}+\int_{0}^{\infty}\frac{{\textup{d}}}{{\textup{d}}s}(U_{i}^{\dagger}(s)\tilde{U}_{i}(s)){\textup{d}}s.$
(4.24)
In addition, since $d(U,\tilde{U})$ converges to zero, one deduces that the
right-hand side of (4.24) does not depend on the index $i$. This establishes
the desired assertion.
(iv) We first show the existence of the asymptotic limit of
$U_{i}U_{j}^{\dagger}$ and $V_{i}V_{j}^{\dagger}$. For any $T>0$, since system
(4.1) is autonomous, we choose $\tilde{U}_{i}$ and $\tilde{V}_{i}$ as
$\tilde{U}_{i}(t)=U_{i}(t+T),\quad V_{i}(t)=V_{i}(t+T).$
If we discretize the time $t\in\mathbb{R}_{+}$ and $n\in\mathbb{Z}_{+}$ and
choose $T=m\in\mathbb{Z}_{+}$, we use the convergence of $d(U,\tilde{U})$ and
$d(V,\tilde{V})$ to conclude that
$\\{U_{i}(n)U_{j}^{\dagger}(n)\\}_{n\in\mathbb{Z}_{+}}$ and
$\\{V_{i}(n)V_{j}^{\dagger}(n)\\}_{n\in\mathbb{Z}_{+}}$ become Cauchy
sequences in the complete spaces $\mathbf{U}(n)$ and $\mathbf{U}(m)$,
respectively. Thus, the limits of $U_{i}U_{j}^{\dagger}$ and
$V_{i}V_{j}^{\dagger}$ exist. In particular, if we denote
$X_{i}^{\infty}:=\lim_{t\to\infty}U_{i}(t)U_{1}^{\dagger}(t),$
then one has
$\lim_{t\to\infty}U_{i}(t)U_{j}^{\dagger}(t)=\lim_{t\to\infty}U_{i}U_{1}^{\dagger}(t)(U_{j}(t)U_{1}^{\dagger}(t))^{\dagger}=X_{i}^{\infty}X_{j}^{\infty,{\dagger}},\quad\lim_{t\to\infty}\langle
U_{i},U_{j}\rangle_{\textup{F}}=\langle
X_{i}^{\infty},X_{j}^{\infty}\rangle_{\textup{F}}.$
On the other hand, we recall (A.2)
$\displaystyle\begin{aligned}
&\frac{{\textup{d}}}{{\textup{d}t}}(U_{i}U_{j}^{\dagger})=\mathrm{i}(U_{i}U_{j}^{\dagger}H_{j}-H_{i}U_{i}U_{j}^{\dagger})\\\
&+\frac{{\kappa}}{N}\sum_{k=1}^{N}\Big{(}\langle
V_{i},V_{k}\rangle_{\textup{F}}U_{k}U_{j}^{\dagger}-\langle
V_{k},V_{i}\rangle_{\textup{F}}U_{i}U_{k}^{\dagger}U_{i}U_{j}^{\dagger}+\langle
V_{k},V_{j}\rangle_{\textup{F}}U_{i}U_{k}^{\dagger}-\langle
V_{j},V_{k}\rangle_{\textup{F}}U_{i}U_{j}^{\dagger}U_{k}U_{j}^{\dagger}\Big{)}.\end{aligned}$
(4.25)
In (4.25), if we let $t\to\infty$ and apply Barbalat’s lemma, the left-hand
side of (4.25) vanishes and consequently, the relation (4.25) becomes
$\displaystyle\begin{aligned}
O&=\mathrm{i}(X_{i}^{\infty}X_{j}^{\infty,{\dagger}}H_{j}-H_{i}X_{i}^{\infty}X_{j}^{\infty,{\dagger}})\\\
&\hskip 14.22636pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}\Big{(}\langle
Y_{i}^{\infty},Y_{k}^{\infty}\rangle_{\textup{F}}X_{k}^{\infty}X_{j}^{\infty,{\dagger}}-\langle
Y_{k}^{\infty},Y_{i}^{\infty,{\dagger}}\rangle_{\textup{F}}X_{i}^{\infty}X_{k}^{\infty,{\dagger}}X_{i}^{\infty}X_{j}^{\infty,{\dagger}}\\\
&\hskip 14.22636pt+\langle
Y_{k}^{\infty},Y_{j}^{\infty}\rangle_{\textup{F}}X_{i}^{\infty}X_{k}^{\infty,{\dagger}}-\langle
Y_{j}^{\infty},Y_{k}^{\infty,{\dagger}}\rangle_{\textup{F}}X_{i}^{\infty}X_{j}^{\infty,{\dagger}}X_{k}^{\infty}X_{j}^{\infty,{\dagger}}\Big{)}.\end{aligned}$
(4.26)
By performing left-multiplication $X_{i}^{\infty,{\dagger}}$ and right-
multiplication $X_{j}^{\infty}$ with (4.26), we obtain
$\displaystyle\begin{aligned}
&-\mathrm{i}X_{j}^{\infty,{\dagger}}H_{j}X_{j}^{\infty}+\frac{{\kappa}}{N}\sum_{k=1}^{N}\Big{(}\langle
Y_{j}^{\infty},Y_{k}^{\infty}\rangle_{\textup{F}}X_{j}^{\infty,{\dagger}}X_{k}^{\infty}-\langle
Y_{k}^{\infty},Y_{j}^{\infty}\rangle_{\textup{F}}X_{k}^{\infty,{\dagger}}X_{j}^{\infty}\Big{)}\\\
&\hskip
14.22636pt=-\mathrm{i}X_{i}^{\infty,{\dagger}}H_{i}X_{i}^{\infty}+\frac{{\kappa}}{N}\sum_{k=1}^{N}\Big{(}\langle
Y_{i}^{\infty},Y_{k}^{\infty}\rangle_{\textup{F}}X_{i}^{\infty,{\dagger}}X_{k}^{\infty}-\langle
Y_{k}^{\infty},Y_{i}^{\infty}\rangle_{\textup{F}}X_{k}^{\infty,{\dagger}}X_{i}^{\infty}\Big{)}.\end{aligned}$
(4.27)
Since the relation(4.27) does not depend on the index, we can set
$-\mathrm{i}\Theta:=-\mathrm{i}X_{j}^{\infty,{\dagger}}H_{j}X_{j}^{\infty}+\frac{{\kappa}}{N}\sum_{k=1}^{N}\Big{(}\langle
Y_{j}^{\infty},Y_{k}^{\infty}\rangle_{\textup{F}}X_{j}^{\infty,{\dagger}}X_{k}^{\infty}-\langle
Y_{k}^{\infty},Y_{j}^{\infty}\rangle_{\textup{F}}X_{k}^{\infty,{\dagger}}X_{j}^{\infty}\Big{)}.$
This yields
$X_{j}^{\infty}\Theta
X_{j}^{\infty,{\dagger}}=H_{j}+\frac{{\kappa}}{N}\sum_{k=1}^{N}\Big{(}\langle
Y_{k}^{\infty},Y_{j}^{\infty}\rangle_{\textup{F}}X_{k}^{\infty,{\dagger}}X_{j}^{\infty}-\langle
Y_{j}^{\infty},Y_{k}^{\infty}\rangle_{\textup{F}}X_{j}^{\infty,{\dagger}}X_{k}^{\infty}\Big{)}.$
Similarly for $\\{V_{i}\\}$, we can define a matrix $\Gamma$ independent of
the index such that
$Y_{j}^{\infty}\Gamma
Y_{j}^{\infty,{\dagger}}=G_{j}++\frac{{\kappa}}{N}\sum_{k=1}^{N}\Big{(}\langle
X_{k}^{\infty},X_{j}^{\infty}\rangle_{\textup{F}}Y_{k}^{\infty,{\dagger}}Y_{j}^{\infty}-\langle
X_{j}^{\infty},X_{k}^{\infty}\rangle_{\textup{F}}Y_{j}^{\infty,{\dagger}}Y_{k}^{\infty}\Big{)}.$
Then, $\\{\\{X_{i}^{\infty}\\},\Theta\\}$ and
$\\{\\{Y_{i}^{\infty}\\},\Gamma\\}$ indeed consist of the phase-locked state.
Moreover, one has
$\mathcal{D}(\mathcal{X}^{\infty})=\max_{1\leq i,j\leq
N}\|X_{i}^{\infty}X_{j}^{\infty,{\dagger}}-I_{n}\|=\lim_{t\to\infty}\max_{1\leq
i,j\leq
N}\|U_{i}(t)U_{j}^{\dagger}(t)-I_{n}\|_{\textup{F}}=\lim_{t\to\infty}\mathcal{D}(\mathcal{U})<\nu_{2}.$
Exactly the same estimate holds for $\mathcal{D}(\mathcal{Y}^{\infty})$ as
well. ∎
###### Remark 4.3.
In [20], the phase-locked state of (4.1) are defined to be of the following
form:
$U_{i}(t)=U_{i}^{\infty}e^{-\mathrm{i}\Gamma_{U}t},\quad
V_{i}(t)=V_{i}^{\infty}e^{-\mathrm{i}\Gamma_{V}t},$
where $U_{i}^{\infty}\in\mathbf{U}(n)$ and $V_{i}^{\infty}\in\mathbf{U}(m)$
are unitary matrices, and $\Lambda_{U}$ and $\Lambda_{V}$ satisfy
$\displaystyle\begin{aligned}
&U_{i}^{\infty}\Gamma_{U}U_{i}^{\infty,{\dagger}}=H_{i}+\frac{\mathrm{i}{\kappa}}{N}\sum_{k=1}^{N}\Big{(}\langle
V_{k}^{\infty},V_{j}^{\infty}\rangle_{\textup{F}}U_{k}^{\infty}U_{i}^{\infty,{\dagger}}-\langle
V_{j}^{\infty},V_{k}^{\infty}\rangle_{\textup{F}}U_{i}^{\infty}U_{k}^{\infty,{\dagger}}\Big{)},\\\
&V_{i}^{\infty}\Gamma_{V}V_{i}^{\infty,{\dagger}}=G_{i}+\frac{\mathrm{i}{\kappa}}{N}\sum_{k=1}^{N}\Big{(}\langle
U_{k}^{\infty},U_{j}^{\infty}\rangle_{\textup{F}}V_{k}^{\infty}V_{i}^{\infty,{\dagger}}-\langle
U_{j}^{\infty},U_{k}^{\infty}\rangle_{\textup{F}}V_{i}^{\infty}V_{k}^{\infty,{\dagger}}\Big{)}.\end{aligned}$
## 5\. Rank-$2m$ quadratically separable state
In this section, we study a quadratically separable state of the LT model for
rank-$2m$ tensors by introducing the MM model whose solution configuration is
given as follows.
$\\{\mathcal{U}^{1},\mathcal{U}^{2},\cdots,\mathcal{U}^{m}\\},\quad\mathcal{U}^{p}:=(U_{1}^{p},\cdots,U_{N}^{p}),\quad
p=1,\cdots,m.$
In the following two subsections, we introduce extended models for the DM
model (3.5) and the DUM model (3.15). Since the procedures are similar as
those in Section 3, we omit details.
### 5.1. The MM model
In this subsection, we introduce the MM model:
$\displaystyle\begin{cases}\displaystyle\dot{U}_{j}^{p}=B_{j}^{p}U_{j}^{p}+\displaystyle\frac{\kappa_{1}}{N}\sum_{k=1}^{N}\left(\prod_{\begin{subarray}{c}\ell=1\\\
\ell\neq p\end{subarray}}^{m}\langle
U_{j}^{\ell},U_{k}^{\ell}\rangle_{\textup{F}}U_{k}^{p}\big{(}U_{j}^{p}\big{)}^{\dagger}U_{j}^{p}-\prod_{\begin{subarray}{c}\ell=1\\\
\ell\neq p\end{subarray}}^{m}\langle
U_{k}^{l},U_{j}^{l}\rangle_{\textup{F}}U_{j}^{p}\big{(}U_{k}^{p}\big{)}^{\dagger}U_{j}^{p}\right)\\\
\hskip
28.45274pt\displaystyle+\frac{\kappa_{2}}{N}\sum_{k=1}^{N}\left(\prod_{\begin{subarray}{c}\ell=1\\\
\ell\neq p\end{subarray}}^{m}\langle
U_{j}^{\ell},U_{k}^{\ell}\rangle_{\textup{F}}U_{j}^{p}\big{(}U_{j}^{p}\big{)}^{\dagger}U_{k}^{p}-\prod_{\begin{subarray}{c}\ell=1\\\
\ell\neq p\end{subarray}}^{m}\langle
U_{k}^{\ell},U_{j}^{\ell}\rangle_{\textup{F}}U_{j}^{p}\big{(}U_{k}^{p}\big{)}^{\dagger}U_{j}^{p}\right),\end{cases}$
(5.1)
where $B_{j}^{p}$ is a rank-$4m$ tensor satisfying the skew-symmetric
property.
For a solution $\\{\mathcal{U}^{p}\\}_{p=1}^{m}$ to (5.1), we set a rank-$2m$
tensor denoted by $T_{i}$:
$T_{i}(t)=U_{i}^{1}(t)\otimes U_{i}^{2}(t)\otimes\cdots\otimes
U_{i}^{m}(t),\quad U_{i}^{p}\in\mathbb{C}^{d_{1}^{p}\times d_{2}^{p}},\quad
i=1,\cdots,N,\quad p=1,\cdots,m,$ (5.2)
which can be written in a component form:
$[T_{i}]_{\alpha_{1}\beta_{1}\alpha_{2}\beta_{2}\cdots\alpha_{m}\beta_{m}}=[U_{i}^{1}]_{\alpha_{1}\beta_{1}}[U_{i}^{2}]_{\alpha_{2}\beta_{2}}\cdots[U_{i}^{m}]_{\alpha_{m}\beta_{m}}.$
In order to relate the model (5.1) with the LT model (1.1), we consider the
index vector $i_{*}$ in ${\kappa}_{i_{*}}$. Then, we introduce two subsets of
$\\{0,1\\}^{2d}$: for $q=1,\cdots,m$,
$\displaystyle\Lambda_{1}$ $\displaystyle=\\{i_{*}\in\\{0,1\\}^{2d}:\text{0
appears once at $2q-1$-th coordinate}\\},$ $\displaystyle\Lambda_{2}$
$\displaystyle=\\{i_{*}\in\\{0,1\\}^{2d}:\text{0 appears once at $2q$-th
coordinate}\\}.$
Note that $|\Lambda_{1}|=|\Lambda_{2}|=m$ and for instance with $m=2$,
$(0,1,1,1),(1,1,0,1)\in\Lambda_{1},\quad(1,0,1,1),(1,1,1,0)\in\Lambda_{2}.$
In this regard, we choose the index vector as
$\displaystyle\kappa_{i_{*}}=\begin{cases}\kappa_{1}\quad&\text{if
}i_{*}\in\Lambda_{1},\\\ \kappa_{2}\quad&\text{if }i_{*}\in\Lambda_{2},\\\
0&\text{otherwise},\end{cases}$ (5.3)
which generalizes (3.3) for the DM model.
Next, for the natural frequency tensors, we use $\\{B_{j}^{p}\\}_{p=1}^{m}$ to
associate a rank-$4m$ tensor $A_{j}$ as
$\displaystyle[A_{j}]_{\alpha_{1}\alpha_{2}\cdots\alpha_{2m}\beta_{1}\beta_{2}\cdots\beta_{2m}}=\sum_{k=1}^{m}\left([B_{j}^{k}]_{\alpha_{2k-1}\alpha_{2k}\beta_{2k-1}\beta_{2k}}\prod^{d}_{\begin{subarray}{c}\ell=1\\\
\ell\neq
k\end{subarray}}\left(\delta_{\alpha_{2\ell-1}\beta_{2\ell-1}}\delta_{\alpha_{2\ell}\beta_{2\ell}}\right)\right),$
(5.4)
which corresponds to (3.9) for the DM model. Then, it follows from
straightforward calculation that the Lohe tensor model with (5.3) and (5.4)
reduces to (5.1) whose solutions can be related by (5.2). The argument above
is summarized in the following proposition analogous to Proposition 3.1.
###### Proposition 5.1.
The following assertions hold.
1. (1)
Suppose $\\{\mathcal{U}^{p}\\}_{p=1}^{m}$ is a solution to (5.1). Then, a
rank-$2m$ tensor defined by $T_{i}:=U_{i}^{1}\otimes
U_{i}^{2}\otimes\cdots\otimes U_{i}^{m}$ is the QSS to (1.1) with well-
prepared initial data and free flow tensors $A_{i}$ satisfying (5.4).
2. (2)
Suppose a rank-$2m$ tensor $T_{i}$ is a solution to (1.1) with (5.4) and
quadratically separable initial data:
$T_{i}^{0}:=U_{i}^{1,0}\otimes U_{i}^{2,0}\otimes\cdots\otimes
U_{i}^{p,0},\quad 1\leq i\leq N,$
for rank-2 tensors $U_{i}^{p,0}\times\mathbb{C}^{d_{1}^{p}\times d_{2}^{p}}$
with unit norms. Then, there exist matrices $\\{U_{i}^{p}\\}$ with unit norms
such that
$T_{i}(t)=U_{i}^{1}(t)\otimes U_{i}^{2}(t)\otimes\cdots\otimes
U_{i}^{p}(t),\quad t>0,$
where $\\{U_{i}^{p}\\}$ is a solution to (5.1) with
$U_{i}^{p,0}=U_{i}^{p}(0)$.
### 5.2. The MUM model
In this subsection, we further reduce to the MM model to the model on the
product of the unitary groups. Since the unitary group is concerned, we set
$d_{1}^{p}=d_{2}^{p}=:d_{p},\quad 1\leq p\leq m.$ (5.5)
For the modeling of natural frequencies, we also define Hermitian matrices
with a size $d_{p}\times d_{p}$:
$[-\mathrm{i}H_{j}^{p}]_{\alpha_{1}\alpha_{2}}\delta_{\beta_{1}\beta_{2}}:=[B_{j}^{p}]_{\alpha_{1}\beta_{1}\alpha_{2}\beta_{2}}.$
In addition, $\\{H_{j}^{p}\\}$ satisfy
$\displaystyle[A_{j}]_{\alpha_{1}\alpha_{2}\cdots\alpha_{2m}\beta_{1}\beta_{2}\cdots\beta_{2m}}$
$\displaystyle=\sum_{k=1}^{m}\left([-\mathrm{i}H_{j}^{k}]_{\alpha_{2k-1}\beta_{2k-1}}\delta_{\alpha_{2k}\beta_{2k}}\prod^{m}_{\begin{subarray}{c}\ell=1\\\
\ell\neq
k\end{subarray}}\left(\delta_{\alpha_{2\ell-1}\beta_{2\ell-1}}\delta_{\alpha_{2\ell}\beta_{2\ell}}\right)\right).$
(5.6)
By Lemma 3.2, one can verify that system (5.1) with (5.5) conserves the
unitarity of $U_{j}^{p}$. Thus, system (5.1) reduces to the following model on
the unitary group:
$\displaystyle\begin{cases}\displaystyle\dot{U}_{j}^{p}=-\mathrm{i}H_{j}^{p}U_{j}^{p}+\displaystyle\frac{\kappa}{N}\sum_{k=1}^{N}\left(\prod_{\begin{subarray}{c}\ell=1\\\
\ell\neq p\end{subarray}}^{m}\langle
U_{j}^{l},U_{k}^{l}\rangle_{\textup{F}}U_{k}^{p}-\prod_{\begin{subarray}{c}\ell=1\\\
\ell\neq p\end{subarray}}^{m}\langle
U_{k}^{\ell},U_{j}^{\ell}\rangle_{\textup{F}}U_{j}^{p}\big{(}U_{k}^{p}\big{)}^{\dagger}U_{j}^{p}\right),\\\
U_{j}^{p}(0)=U_{j}^{p,0}\in\mathbf{U}(d_{p}),\end{cases}$ (5.7)
where $\kappa=\kappa_{1}+\kappa_{2}$ and $H_{j}^{p}U_{j}^{p}$ is a usual
matrix product. As in Proposition 5.1, existence and uniqueness of the QSS for
(5.7) can be stated as follows.
###### Proposition 5.2.
The following assertions hold.
1. (1)
Suppose $\\{\mathcal{U}^{p}\\}_{p=1}^{m}$ is a solution to (5.7). Then, a
rank-$2m$ tensor defined by $T_{i}:=U_{i}^{1}\otimes
U_{i}^{2}\otimes\cdots\otimes U_{i}^{m}$ is a quadratically separable state to
(1.1) with a well-prepared free flow tensor $A_{i}$ satisfying (5.6).
2. (2)
Suppose a rank-$2m$ tensor $T_{i}$ is a solution to (1.1) with (5.6) and
quadratically separable initial data:
$T_{i}^{0}:=U_{i}^{1,0}\otimes U_{i}^{2,0}\otimes\cdots\otimes
U_{i}^{p,0},\quad 1\leq i\leq N,$
for rank-2 tensors $U_{i}^{p,0}\times\mathbb{C}^{n_{1}^{p}\times n_{2}^{p}}$
with unit norms. Then, there exist matrices $\\{U_{i}^{p}\\}$ with unit norms
such that
$T_{i}(t)=U_{i}^{1}(t)\otimes U_{i}^{2}(t)\otimes\cdots\otimes
U_{i}^{p}(t),\quad t>0,$
where $\\{U_{i}^{p}\\}$ is a solution to (5.7) with
$U_{i}^{p,0}=U_{i}^{p}(0)$.
###### Remark 5.1.
In Section 4, we provided the emergent dynamics of the double unitary matrix
model (4.1). However for its generalized model (5.7), emergent dynamics will
not be studied, since it can be straightforwardly obtained from the results in
Section 4.
## 6\. Conclusion
In this paper, we have studied the existence and emergent dynamics of
quadratically separable states for the Lohe tensor model which incorporates
several well-known low-rank aggregation models such as the Kuramoto model, the
Lohe sphere model and the Lohe matrix model, etc. In our previous work [9], we
obtained completely separable states as special solutions to the Lohe tensor
model defined as tensor products of rank-1 real tensors (or vectors). In
analogy with the previous work, we consider the states in which a solution can
be decomposed as a tensor product of rank-2 tensors (or matrices), namely, a
quadratically separable state. Precisely, if initial data are quadratically
separable, then such separability is preserved along the Lohe tensor flow.
Moreover, by introducing and analyzing double matrix and unitary matrix
models, we are able to study asymptotic behavior of the quadratically
separable states to the Lohe tensor model. Of course, there are several issues
that are not discussed in this work. For example, one can naturally consider
the state consisting of tensors with possibly different ranks and sizes. We
explore this issue in a future work.
## Appendix A Proof of Lemma 4.1
In this appendix, we provide a proof of Lemma 4.1 in which a differential
inequality for the aggregation functional
$\mathcal{L}=\mathcal{D}(\mathcal{U})+\mathcal{D}(\mathcal{V})+\mathcal{S}(\mathcal{U})+\mathcal{S}(\mathcal{V})$
is derived. We divide a proof into two steps:
* •
Step A: we derive differential inequalities for $\mathcal{D}(\mathcal{U})$ and
$\mathcal{D}(\mathcal{V})$ (see Lemma A.1).
* •
Step B: we derive differential inequalities for $\mathcal{S}(\mathcal{U})$ and
$\mathcal{S}(\mathcal{V})$ (see Lemma A.2).
###### Lemma A.1.
Let $\\{(U_{i},V_{i})\\}$ be a solution to (4.4). Then,
$\mathcal{D}(\mathcal{U})$ and $\mathcal{D}(\mathcal{V})$ satisfy
$\displaystyle\begin{aligned}
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{U})\leq-2m{\kappa}\mathcal{D}(\mathcal{U})+m{\kappa}\mathcal{D}(\mathcal{U})^{3}+6{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})+2{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{2}+4{\kappa}\sqrt{n}S(\mathcal{V}),\\\
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{V})\leq-2n{\kappa}\mathcal{D}(\mathcal{V})+n{\kappa}\mathcal{D}(\mathcal{V})^{3}+6{\kappa}\mathcal{S}(\mathcal{U})\mathcal{D}(\mathcal{V})+2{\kappa}\mathcal{S}(\mathcal{U})\mathcal{D}(\mathcal{V})^{2}+4{\kappa}\sqrt{m}S(\mathcal{U}).\end{aligned}$
(A.1)
###### Proof.
By straightforward calculations, one finds a differential equation for
$G_{ij}=U_{i}U_{j}^{\dagger}$:
$\frac{{\textup{d}}}{{\textup{d}t}}G_{ij}=\frac{{\kappa}}{N}\sum_{k=1}^{N}(c_{ik}G_{kj}-c_{ki}G_{ik}G_{ij}+c_{kj}G_{ik}-c_{jk}G_{ij}G_{kj}).$
(A.2)
By using the relation $G_{ij}=I_{n}-S_{ij}$, we see that $S_{ij}$ satisfies
$\displaystyle\begin{aligned}
\frac{{\textup{d}}}{{\textup{d}t}}S_{ij}&=\frac{{\kappa}}{N}\sum_{k=1}^{N}\Big{[}c_{ik}S_{kj}-c_{ki}S_{ik}-c_{ki}S_{ij}+c_{i}S_{k}S_{ij}+c_{kj}S_{ik}-c_{jk}S_{kj}-c_{jk}S_{ij}+c_{jk}S_{ij}S_{kj}\\\
&\hskip 71.13188pt+(c_{ki}-c_{ik}+c_{jk}-c_{kj})I_{n}\Big{]}.\end{aligned}$
(A.3)
After algebraic manipulation, we rewrite (A.3) in terms of $S_{ij}$ and
$c_{ij}-m$ that are expected to converge to zero:
$\displaystyle\begin{aligned}
\frac{{\textup{d}}S_{ij}}{dt}&=-2m{\kappa}S_{ij}+\frac{m{\kappa}}{N}\sum_{k=1}^{N}\Big{(}S_{ik}S_{ij}+S_{ij}S_{kj}\Big{)}\\\
&\hskip
5.69046pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}\Big{[}(c_{ik}-m)S_{kj}-(c_{ki}-m)S_{ik}-(c_{ik}-m)S_{ij}+(c_{ki}-m)S_{ik}S_{ij}\Big{]}\\\
&\hskip
5.69046pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}\Big{[}(c_{kj}-m)S_{ik}-(c_{jk}-m)S_{kj}-(c_{jk}-m)S_{ij}+(c_{jk}-m)S_{ij}S_{kj}\Big{]}\\\
&\hskip
5.69046pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}\Big{[}(c_{ki}-c_{ik}+c_{jk}-c_{kj})I_{n}\Big{]}.\end{aligned}$
(A.4)
On the other hand for an $n\times n$ matrix $A$, one has
$\frac{1}{2}\frac{{\textup{d}}}{{\textup{d}t}}\|A\|_{\textup{F}}^{2}=\frac{1}{2}\frac{{\textup{d}}}{{\textup{d}t}}\textup{tr}(AA^{\dagger})=\frac{1}{2}\textup{tr}(\dot{A}A^{\dagger}+A\dot{A}^{\dagger})=\textup{Re}\textup{tr}(\dot{A}A^{\dagger}).$
We multiply (A.4) with $S_{ij}^{\dagger}$ to find
$\displaystyle\begin{aligned}
&\frac{1}{2}\frac{{\textup{d}}}{{\textup{d}t}}\|S_{ij}\|_{\textup{F}}^{2}=-2m{\kappa}\|S_{ij}\|_{\textup{F}}^{2}+\frac{m{\kappa}}{N}\sum_{k=1}^{N}\textup{Re}\textup{tr}(S_{ik}S_{ij}S_{ij}^{\dagger}+S_{ij}S_{kj}S_{ij}^{\dagger})\\\
&\hskip
14.22636pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}\textup{Re}[(c_{ik}-m)\textup{tr}(S_{kj}S_{ij}^{\dagger})]-\textup{Re}[(c_{ki}-m)\textup{tr}(S_{ik}S_{ij}^{\dagger})]-\textup{Re}(c_{ik}-m)\|S_{ij}\|_{\textup{F}}^{2}\\\
&\hskip
14.22636pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}\textup{Re}[(c_{kj}-m)\textup{tr}(S_{ik}S_{ij}^{\dagger})]-\textup{Re}[(c_{jk}-m)\textup{tr}(S_{kj}S_{ij}^{\dagger})]-\textup{Re}(c_{jk}-m)\|S_{ij}\|_{\textup{F}}^{2}\\\
&\hskip
14.22636pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}\textup{Re}[(c_{ki}-m)\textup{tr}(S_{ik}S_{ij}S_{ij}^{\dagger})]+\textup{Re}[(c_{jk}-m)\textup{tr}(S_{ij}S_{kj}S_{ij}^{\dagger})]\\\
&\hskip
14.22636pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}\textup{Re}[(c_{ki}-c_{ik}+c_{jk}-c_{kj})\textup{tr}(S_{ij}^{\dagger})]\\\
&=:-2m{\kappa}\|S_{ij}\|_{\textup{F}}^{2}+\mathcal{I}_{11}+\mathcal{I}_{12}+\mathcal{I}_{13}+\mathcal{I}_{14}+\mathcal{I}_{15}.\end{aligned}$
(A.5)
Below, we present estimates for $\mathcal{I}_{1k},~{}k=1,\cdots,5$,
respectively.
$\bullet$ (Estimate of $\mathcal{I}_{11}$): We use (A.6) and
$S_{ij}+S_{ji}=S_{ij}S_{ji},\quad S_{ij}^{\dagger}=S_{ji}$ (A.6)
to derive
$\displaystyle\textup{Re}\textup{tr}(S_{ik}S_{ij}S_{ji})$
$\displaystyle=\frac{1}{2}\textup{tr}(S_{ik}S_{ij}S_{ji}+S_{ij}S_{ji}S_{ki})=\frac{1}{2}\textup{tr}(S_{ij}S_{ji}(S_{ki}+S_{ik}))$
$\displaystyle=\frac{1}{2}\textup{tr}(S_{ij}S_{ji}S_{ki}S_{ik})=\frac{1}{2}\|S_{ki}S_{ij}\|_{\textup{F}}^{2}.$
Similarly, one has
$\textup{Re}\textup{tr}(S_{ij}S_{kj}S_{ij}^{\dagger})=\frac{1}{2}\|S_{ij}S_{jk}\|_{\textup{F}}^{2}.$
Hence, $\mathcal{I}_{11}$ satisfies
$\mathcal{I}_{11}=\frac{m{\kappa}}{N}\sum_{k=1}^{N}\textup{Re}\textup{tr}(S_{ik}S_{ij}S_{ij}^{\dagger}+S_{ij}S_{kj}S_{ij}^{\dagger})=\frac{m{\kappa}}{2N}\sum_{k=1}^{N}\big{(}\|S_{ki}S_{ij}\|_{\textup{F}}^{2}+\|S_{ij}S_{jk}\|_{\textup{F}}^{2}\big{)}\leq
m{\kappa}\mathcal{D}(\mathcal{U})^{4}.$
$\bullet$ (Estimates of $\mathcal{I}_{12}$ and $\mathcal{I}_{13}$): By the
maximality of $\mathcal{D}(\mathcal{U})$ and $\mathcal{S}(\mathcal{V})$, we
have
$\displaystyle\mathcal{I}_{12}$
$\displaystyle=\frac{{\kappa}}{N}\sum_{k=1}^{N}\textup{Re}[(c_{ik}-m)\textup{tr}(S_{kj}S_{ij}^{\dagger})]-\textup{Re}[(c_{ki}-m)\textup{tr}(S_{ik}S_{ij}^{\dagger})]-\textup{Re}(c_{ik}-m)\|S_{ij}\|_{\textup{F}}^{2}$
$\displaystyle\leq
3{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{2},$
$\displaystyle\mathcal{I}_{13}$
$\displaystyle=\frac{{\kappa}}{N}\sum_{k=1}^{N}\textup{Re}[(c_{kj}-m)\textup{tr}(S_{ik}S_{ij}^{\dagger})]-\textup{Re}[(c_{jk}-m)\textup{tr}(S_{kj}S_{ij}^{\dagger})]-\textup{Re}(c_{jk}-m)\|S_{ij}\|_{\textup{F}}^{2}$
$\displaystyle\leq
3{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{2}.$
$\bullet$ (Estimate of $\mathcal{I}_{14}$): By straightforward calculation,
one has
$\displaystyle\mathcal{I}_{14}$
$\displaystyle=\frac{{\kappa}}{N}\sum_{k=1}^{N}\textup{Re}[(c_{ki}-m)\textup{tr}(S_{ik}S_{ij}S_{ij}^{\dagger})]+\textup{Re}[(c_{jk}-m)\textup{tr}(S_{ij}S_{kj}S_{ij}^{\dagger})]\leq
2{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{3}.$
$\bullet$ (Estimate of $\mathcal{I}_{15}$): Note that for an $n\times n$
matrix $A$,
$\textup{tr}(A)=\textup{tr}(I_{n}A)\leq\|I_{n}\|_{\textup{F}}\|A\|_{\textup{F}}=\sqrt{n}\|A\|_{\textup{F}}.$
This yields
$\mathcal{I}_{15}=\frac{{\kappa}}{N}\sum_{k=1}^{N}\textup{Re}[(c_{ki}-c_{ik}+c_{jk}-c_{kj})\textup{tr}(S_{ij}^{\dagger})]\leq
4{\kappa}\sqrt{n}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U}).$ (A.7)
In (A.5), we collect all the estimates for $\mathcal{I}_{1k},~{}k=1,\cdots,5$
to obtain
$\frac{1}{2}\frac{{\textup{d}}}{{\textup{d}t}}\|S_{ij}\|_{\textup{F}}^{2}\leq-2m{\kappa}\|S_{ij}\|_{\textup{F}}^{2}+m{\kappa}\mathcal{D}(\mathcal{U})^{4}+6{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{2}+2{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{3}+4{\kappa}\sqrt{n}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U}).$
Hence, $\mathcal{D}(\mathcal{U})$ satisfies
$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{U})\leq-2m{\kappa}\mathcal{D}(\mathcal{U})+m{\kappa}\mathcal{D}(\mathcal{U})^{3}+6{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})+2{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{2}+4{\kappa}\sqrt{n}\mathcal{S}(\mathcal{V}).$
Similarly, one can find a differential inequality for
$\mathcal{D}(\mathcal{V})$ by exchanging the roles of $\mathcal{U}$ and
$\mathcal{V}$:
$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{V})\leq-2n{\kappa}\mathcal{D}(\mathcal{V})+n{\kappa}\mathcal{D}(\mathcal{V})^{3}+6{\kappa}\mathcal{S}(\mathcal{U})\mathcal{D}(\mathcal{V})+2{\kappa}\mathcal{S}(\mathcal{U})\mathcal{D}(\mathcal{V})^{2}+4{\kappa}\sqrt{m}\mathcal{S}(\mathcal{U}).$
∎
In (A.1), note that $\mathcal{S}(\mathcal{U})$ and $\mathcal{S}(\mathcal{V})$
appear in the differential inequalities for $\mathcal{D}(\mathcal{U})$ and
$\mathcal{D}(\mathcal{V})$. Hence, we derive the differential inequalities for
$\mathcal{S}(\mathcal{U})$ and $\mathcal{S}(\mathcal{V})$ below.
###### Lemma A.2.
Let $\\{(U_{i},V_{i})\\}$ be a solution to (4.4). Then,
$\mathcal{S}(\mathcal{U})$ and $\mathcal{S}(\mathcal{V})$ satisfy
$\displaystyle\begin{aligned}
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(\mathcal{U})\leq-2m{\kappa}\mathcal{S}(\mathcal{U})+2m{\kappa}\mathcal{D}(\mathcal{U})^{2}+6{\kappa}\mathcal{S}(\mathcal{U})\mathcal{S}(\mathcal{V})+2{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{2}+4{\kappa}\sqrt{n}\mathcal{S}(\mathcal{V}),\\\
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(\mathcal{V})\leq-2n{\kappa}\mathcal{S}(\mathcal{V})+2n{\kappa}\mathcal{D}(\mathcal{V})^{2}+6{\kappa}\mathcal{S}(\mathcal{U})\mathcal{S}(\mathcal{V})+2{\kappa}\mathcal{S}(\mathcal{U})\mathcal{D}(\mathcal{V})^{2}+4{\kappa}\sqrt{m}\mathcal{S}(\mathcal{U}).\end{aligned}$
(A.8)
###### Proof.
In (A.4), we take trace to obtain
$\displaystyle\begin{aligned}
&\frac{{\textup{d}}}{{\textup{d}t}}(n-d_{ij})=-2m{\kappa}(n-d_{ij})+\frac{m{\kappa}}{N}\sum_{k=1}^{N}\textup{tr}(S_{ik}S_{ij}+S_{ij}S_{kj})\\\
&\hskip
28.45274pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}(c_{ik}-m)(n-d_{kj})-(c_{ki}-m)(n-d_{ik})-(c_{ik}-m)(n-d_{ij})\\\
&\hskip
28.45274pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}(c_{kj}-m)(n-d_{ik})-(c_{jk}-m)(n-d_{kj})-(c_{jk}-m)(n-d_{ij})\\\
&\hskip
28.45274pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}(c_{ki}-m)\textup{tr}(S_{ik}S_{ij})+(c_{jk}-m)\textup{tr}(S_{ij}S_{kj})\\\
&\hskip
28.45274pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}[(c_{ki}-c_{ik})+(c_{jk}-c_{kj})]\sqrt{n}\\\
&\hskip
28.45274pt=:-2m{\kappa}(n-d_{ij})+\mathcal{I}_{21}+\mathcal{I}_{22}+\mathcal{I}_{23}+\mathcal{I}_{24}+\mathcal{I}_{25}.\end{aligned}$
(A.9)
Below, we present estimates of $\mathcal{I}_{2k},~{}k=1,\cdots,5$, separately.
$\bullet$ (Estimate of $\mathcal{I}_{21}$): We use the definition of
$\mathcal{D}(\mathcal{U})$ to find
$\displaystyle\mathcal{I}_{21}=\frac{m{\kappa}}{N}\sum_{k=1}^{N}\textup{tr}(S_{ik}S_{ij}+S_{ij}S_{kj})\leq
2m{\kappa}\mathcal{D}(\mathcal{U})^{2}.$
$\bullet$ (Estimates of $\mathcal{I}_{22}$ and $\mathcal{I}_{23}$): It is easy
to see that
$\displaystyle\mathcal{I}_{22}=\frac{{\kappa}}{N}\sum_{k=1}^{N}(c_{ik}-m)(n-d_{kj})-(c_{ki}-m)(n-d_{ik})-(c_{ik}-m)(n-d_{ij})\leq
3{\kappa}\mathcal{S}(\mathcal{U})\mathcal{S}(\mathcal{V}),$
$\displaystyle\mathcal{I}_{23}=\frac{{\kappa}}{N}\sum_{k=1}^{N}(c_{kj}-m)(n-d_{ik})-(c_{jk}-m)(n-d_{kj})-(c_{jk}-m)(n-d_{ij})\leq
3{\kappa}\mathcal{S}(\mathcal{U})\mathcal{S}(\mathcal{V}).$
$\bullet$ (Estimate of $\mathcal{I}_{24}$): Similar to $\mathcal{I}_{21}$, one
has
$\displaystyle\mathcal{I}_{24}=\frac{{\kappa}}{N}\sum_{k=1}^{N}(c_{ki}-m)\textup{tr}(S_{ik}S_{ij})+(c_{jk}-m)\textup{tr}(S_{ij}S_{kj})\leq
2{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{2}.$
$\bullet$ (Estimate of $\mathcal{I}_{25}$): We find
$\mathcal{I}_{25}=\frac{{\kappa}}{N}\sum_{k=1}^{N}[(c_{ki}-c_{ik})+(c_{jk}-c_{kj})]\sqrt{n}\leq
4{\kappa}\sqrt{n}\mathcal{S}(\mathcal{V}).$ (A.10)
In (A.9), we combine all estimates to obtain
$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(\mathcal{U})\leq-2m{\kappa}\mathcal{S}(\mathcal{U})+2m{\kappa}\mathcal{D}(\mathcal{U})^{2}+6{\kappa}\mathcal{S}(\mathcal{U})\mathcal{S}(\mathcal{V})+2{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{2}+4{\kappa}\sqrt{n}\mathcal{S}(\mathcal{V}).$
By exchanging the roles of $\mathcal{U}$ and $\mathcal{V}$, we derive a
differential inequality for $\mathcal{S}(\mathcal{V})$:
$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(\mathcal{V})\leq-2n{\kappa}\mathcal{S}(\mathcal{V})+2n{\kappa}\mathcal{D}(\mathcal{V})^{2}+6{\kappa}\mathcal{S}(\mathcal{U})\mathcal{S}(\mathcal{V})+2{\kappa}\mathcal{S}(\mathcal{U})\mathcal{D}(\mathcal{V})^{2}+4{\kappa}\sqrt{m}\mathcal{S}(\mathcal{U}).$
∎
###### Remark A.1.
For homogeneous Hamiltonians, we add $\eqref{AA-12}_{1}$ and
$\eqref{AA-12}_{2}$ to find
$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}(\mathcal{S}(\mathcal{U})+\mathcal{S}(\mathcal{V}))$
$\displaystyle\leq-2{\kappa}(m-2\sqrt{m})\mathcal{S}(\mathcal{U})-2{\kappa}(n-2\sqrt{n})\mathcal{S}(\mathcal{V})+\mathcal{O}((\mathcal{S}(\mathcal{U})+\mathcal{S}(\mathcal{V}))^{2}).$
Hence, in order to obtain the desired convergence, we assume
$n>2\sqrt{n},\quad m>2\sqrt{m},\quad\textup{i.e.,}\quad n,m>4,$
which requires the restriction on the size of $U_{i}$ and $V_{j}$. This
technical assumption arises from the estimate of $\mathcal{I}_{25}$ in (A.10).
Now, we are ready to present a proof of Lemma 4.1 using Lemma A.1 and Lemma
A.2.
Proof of Lemma 4.1: Recall the inequalities in (A.1) and (A.8):
$\displaystyle\begin{aligned}
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{U})\leq-2m{\kappa}\mathcal{D}(\mathcal{U})+m{\kappa}\mathcal{D}(\mathcal{U})^{3}+6{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})+2{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{2}+4{\kappa}\sqrt{n}\mathcal{S}(\mathcal{V}),\\\
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{V})\leq-2n{\kappa}\mathcal{D}(\mathcal{V})+n{\kappa}\mathcal{D}(\mathcal{V})^{3}+6{\kappa}\mathcal{S}(\mathcal{U})\mathcal{D}(\mathcal{V})+2{\kappa}\mathcal{S}(\mathcal{U})\mathcal{D}(\mathcal{V})^{2}+4{\kappa}\sqrt{m}\mathcal{S}(\mathcal{U}),\\\
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(\mathcal{U})\leq-2m{\kappa}\mathcal{S}(\mathcal{U})+2m{\kappa}\mathcal{D}(\mathcal{U})^{2}+6{\kappa}\mathcal{S}(\mathcal{U})\mathcal{S}(\mathcal{V})+2{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{2}+4{\kappa}\sqrt{n}\mathcal{S}(\mathcal{V}),\\\
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(\mathcal{V})\leq-2n{\kappa}\mathcal{S}(\mathcal{V})+2n{\kappa}\mathcal{D}(\mathcal{V})^{2}+6{\kappa}\mathcal{S}(\mathcal{U})\mathcal{S}(\mathcal{V})+2{\kappa}\mathcal{S}(\mathcal{U})\mathcal{D}(\mathcal{V})^{2}+4{\kappa}\sqrt{m}\mathcal{S}(\mathcal{U}).\end{aligned}$
(A.11)
Without loss of generality, we may assume $n\geq m$, and add all the
inequalities in (A.11) to find
$\displaystyle\begin{aligned}
\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{L}&\leq-2{\kappa}(m-4\sqrt{n})\mathcal{L}+n{\kappa}(\mathcal{D}(\mathcal{U})^{3}+2\mathcal{D}(\mathcal{U})^{2}+\mathcal{D}(\mathcal{V})^{3}+2\mathcal{D}(\mathcal{V})^{2})\\\
&\hskip
14.22636pt+6{\kappa}(\mathcal{D}(\mathcal{U})\mathcal{S}(\mathcal{V})+\mathcal{D}(\mathcal{V})\mathcal{S}(\mathcal{U})+2\mathcal{S}(\mathcal{U})\mathcal{S}(\mathcal{V}))+4{\kappa}(\mathcal{D}(\mathcal{U})^{2}\mathcal{S}(\mathcal{V})+\mathcal{D}(\mathcal{V})^{2}\mathcal{S}(\mathcal{U}))\\\
&=:-2{\kappa}(m-4\sqrt{n})\mathcal{L}+n{\kappa}\mathcal{I}_{31}+6{\kappa}\mathcal{I}_{32}+4{\kappa}\mathcal{I}_{33}.\end{aligned}$
(A.12)
In the sequel, we provide estimates for $\mathcal{I}_{3k},~{}k=1,2,3$,
respectively.
$\bullet$ (Estimate of $\mathcal{I}_{31}$): We use a rough estimate to find
$\displaystyle\mathcal{I}_{31}=\mathcal{D}(\mathcal{U})^{3}+2\mathcal{D}(\mathcal{U})^{2}+\mathcal{D}(\mathcal{V})^{3}+2\mathcal{D}(\mathcal{V})^{2}\leq\mathcal{L}^{3}+2\mathcal{L}^{2}.$
$\bullet$ (Estimate of $\mathcal{I}_{32}$): By straightforward calculation,
one has
$\displaystyle\mathcal{I}_{32}$
$\displaystyle=\mathcal{D}(\mathcal{U})\mathcal{S}(\mathcal{V})+\mathcal{D}(\mathcal{V})\mathcal{S}(\mathcal{U})+2\mathcal{S}(\mathcal{U})\mathcal{S}(\mathcal{V})$
$\displaystyle\leq\frac{1}{2}\mathcal{D}(\mathcal{U})^{2}+\frac{1}{2}\mathcal{D}(\mathcal{V})^{2}+\frac{3}{2}\mathcal{S}(\mathcal{U})^{2}+\frac{3}{2}\mathcal{S}(\mathcal{V})^{2}\leq\frac{3}{2}\mathcal{L}^{2}.$
$\bullet$ (Estimate of $\mathcal{I}_{33}$): We use Young’s inequality that for
$a,b>0$,
$a^{2}b\leq\frac{2}{3}a^{3}+\frac{1}{3}b^{3}$
to find
$\displaystyle\mathcal{I}_{33}=\mathcal{D}(\mathcal{U})^{2}\mathcal{S}(\mathcal{V})+\mathcal{D}(\mathcal{V})^{2}\mathcal{S}(\mathcal{U})\leq\frac{2}{3}\mathcal{D}(\mathcal{U})^{3}+\frac{1}{3}\mathcal{S}(\mathcal{V})^{3}+\frac{2}{3}\mathcal{D}(\mathcal{V})^{3}+\frac{1}{3}\mathcal{S}(\mathcal{U})^{3}\leq\frac{2}{3}\mathcal{L}^{3}.$
In (A.12), we combine all the estimates to obtain the desired inequality for
$\mathcal{L}$:
$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{L}$
$\displaystyle\leq-2{\kappa}(m-4\sqrt{n})\mathcal{L}+2{\kappa}n(\mathcal{L}^{3}+2\mathcal{L}^{2})+9{\kappa}\mathcal{L}^{2}+\frac{8{\kappa}}{3}\mathcal{L}^{3}$
$\displaystyle=-2{\kappa}(m-4\sqrt{n})\mathcal{L}++{\kappa}(4n+9)\mathcal{L}^{2}+{\kappa}\left(2n+\frac{8}{3}\right)\mathcal{L}^{3}.$
## Appendix B Proof of Lemma 4.2
In this appendix, we present a proof of Lemma 4.2 in which the differential
inequality for the aggregation functional $\mathcal{F}$ in (4.11) will be
derived.
###### Lemma B.1.
Let $\\{(U_{i},V_{i})\\}$ be a solution to (4.1). Then, $d(U,\tilde{U})$ and
$d(V,\tilde{V})$ satisfy
$\displaystyle\begin{aligned}
\frac{{\textup{d}}}{{\textup{d}t}}d(U,\tilde{U})&\leq-2m{\kappa}d(U,\tilde{U})+4m{\kappa}\mathcal{L}d(U,\tilde{U})+6\mathcal{L}(d(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))\\\
&\hskip
14.22636pt+2\mathcal{L}^{2}(4d(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))+4\sqrt{n}\mathcal{S}(V,\tilde{V}).\\\
\frac{{\textup{d}}}{{\textup{d}t}}d(V,\tilde{V})&\leq-2n{\kappa}d(V,\tilde{V})+4n{\kappa}\mathcal{L}d(V,\tilde{V})+6\mathcal{L}(d(V,\tilde{V})+\mathcal{S}(U,\tilde{U}))\\\
&\hskip
14.22636pt+2\mathcal{L}^{2}(4d(V,\tilde{V})+\mathcal{S}(U,\tilde{U}))+4\sqrt{m}\mathcal{S}(U,\tilde{U}).\end{aligned}$
(B.1)
###### Proof.
First, we recall (A.4):
$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}S_{ij}$
$\displaystyle=-2m{\kappa}S_{ij}+\frac{m{\kappa}}{N}\sum_{k=1}^{N}S_{ik}S_{ij}+S_{ij}S_{kj}+\mathrm{i}(H_{i}-H_{j})+\mathrm{i}(S_{ij}H_{j}-H_{i}S_{ij})$
$\displaystyle\hskip
14.22636pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}(c_{ik}-m)S_{kj}-(c_{ki}-m)S_{ik}-(c_{ik}-m)S_{ij}+(c_{ki}-m)S_{ik}S_{ij}$
$\displaystyle\hskip
14.22636pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}(c_{kj}-m)S_{ik}-(c_{jk}-m)S_{kj}-(c_{jk}-m)S_{ij}+(c_{jk}-m)S_{ij}S_{kj}$
$\displaystyle\hskip
14.22636pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}(c_{ki}-c_{ik}+c_{jk}-c_{kj})I_{n}.$
Thus, $S_{ij}-\tilde{S}_{ij}$ satisfies
$\displaystyle\begin{aligned}
&\frac{{\textup{d}}}{{\textup{d}t}}(S_{ij}-\tilde{S}_{ij})=-2m{\kappa}(S_{ij}-\tilde{S}_{ij})+\frac{m{\kappa}}{N}\sum_{k=1}^{N}\underbrace{(S_{ik}S_{ij}+S_{ij}S_{kj}-\tilde{S}_{ik}\tilde{S}_{ij}-\tilde{S}_{ij}\tilde{S}_{kj})}_{=:\mathcal{I}_{41}}\\\
&\hskip
28.45274pt+\underbrace{\mathrm{i}((S_{ij}-\tilde{S}_{ij})H_{j}-H_{i}(S_{ij}-\tilde{S}_{ij}))}_{=:\mathcal{I}_{42}}+\frac{{\kappa}}{N}\sum_{k=1}^{N}(\mathcal{I}_{43}-\tilde{\mathcal{I}}_{43})+\frac{{\kappa}}{N}\sum_{k=1}^{N}(\mathcal{I}_{44}-\tilde{\mathcal{I}}_{44})\\\
&\hskip
28.45274pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}\underbrace{((c_{ki}-c_{ik}+c_{jk}-c_{kj})-(\tilde{c}_{ki}-\tilde{c}_{ik}+\tilde{c}_{jk}-\tilde{c}_{kj}))I_{n}}_{=:\mathcal{I}_{45}},\end{aligned}$
(B.2)
where $\mathcal{I}_{43}$ and $\mathcal{I}_{44}$ are defined as
$\displaystyle\mathcal{I}_{43}:=(c_{ik}-m)S_{kj}-(c_{ki}-m)S_{ik}-(c_{ik}-m)S_{ij}+(c_{ki}-m)S_{ik}S_{ij},$
$\displaystyle\mathcal{I}_{44}:=(c_{kj}-m)S_{ik}-(c_{jk}-m)S_{kj}-(c_{jk}-m)S_{ij}+(c_{jk}-m)S_{ij}S_{kj}.$
Below, we provide the estimates for $\mathcal{I}_{4k},~{}k=1,\cdots,5$,
respectively.
$\bullet$ (Estimate of $\mathcal{I}_{41}$): Note that
$\displaystyle\mathcal{I}_{41}$
$\displaystyle=S_{ik}S_{ij}+S_{ij}S_{kj}-\tilde{S}_{ik}\tilde{S}_{ij}-\tilde{S}_{ij}\tilde{S}_{kj}$
$\displaystyle=S_{ik}(S_{ij}-\tilde{S}_{ij})+(S_{ik}-\tilde{S}_{ik})\tilde{S}_{ij}+S_{ij}(S_{kj}-\tilde{S}_{kj})+(S_{ij}-\tilde{S}_{ij})\tilde{S}_{kj}.$
This yields
$\|\mathcal{I}_{41}\|_{\textup{F}}\leq 4\mathcal{L}d(U,\tilde{U}).$ (B.3)
$\bullet$ (Estimate of $\mathcal{I}_{42}$): Note that for a skew-Hermitian
matrix $\Omega$ and a matrix $A$, one has
$\displaystyle\textup{tr}(\Omega AA^{\dagger})=0.$
This implies
$\displaystyle\textup{Re}\textup{tr}(\mathcal{I}_{42}(S_{ij}-\tilde{S}_{ij})^{\dagger})=\textup{Re}\textup{tr}(\mathrm{i}H_{j}(S_{ij}-\tilde{S}_{ij})(S_{ij}-\tilde{S}_{ij})^{\dagger})=0.$
$\bullet$ (Estimate of $\mathcal{I}_{43}$): We observe
$\displaystyle\begin{aligned}
&\|(c_{ik}-m)S_{kj}-(\tilde{c}_{ik}-m)\tilde{S}_{kj}\|_{\textup{F}}\\\ &\hskip
28.45274pt=\|(c_{ik}-m)(S_{kj}-\tilde{S}_{kj})+(c_{ik}-\tilde{c}_{ik})\tilde{S}_{kj}\|_{\textup{F}}\\\
&\hskip
28.45274pt\leq\mathcal{L}d(U,\tilde{U})+\mathcal{L}\mathcal{S}(V,\tilde{V})=\mathcal{L}(d(U,\tilde{U})+\mathcal{S}(V,\tilde{V})).\end{aligned}$
Moreover, we use (B.3)
$\displaystyle\|(c_{ki}-m)S_{ik}S_{ij}-(\tilde{c}_{ki}-m)\tilde{S}_{ik}\tilde{S}_{ij}\|_{\textup{F}}$
$\displaystyle\hskip
28.45274pt=\|(c_{ki}-m)(S_{ik}S_{ij}-\tilde{S}_{ik}\tilde{S}_{ij})+(c_{ik}-\tilde{c}_{ik})\tilde{S}_{ik}\tilde{S}_{ij}\|_{\textup{F}}$
$\displaystyle\hskip 28.45274pt\leq\mathcal{L}\cdot
4\mathcal{L}d(U,\tilde{U})+\mathcal{S}(V,\tilde{V})\mathcal{L}^{2}=\mathcal{L}^{2}(4d(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))$
to obtain
$\displaystyle\|\mathcal{I}_{43}-\tilde{\mathcal{I}}_{43}\|_{\textup{F}}\leq
3\mathcal{L}(d(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))+\mathcal{L}^{2}(4d(U,\tilde{U})+\mathcal{S}(V,\tilde{V})).$
$\bullet$ (Estimate of $\mathcal{I}_{44}$): Similar to $\mathcal{I}_{43}$, one
finds
$\displaystyle\|\mathcal{I}_{44}-\tilde{\mathcal{I}}_{44}\|_{\textup{F}}\leq
3\mathcal{L}(d(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))+\mathcal{L}^{2}(4d(U,\tilde{U})+\mathcal{S}(V,\tilde{V})).$
$\bullet$ (Estimate of $\mathcal{I}_{45}$): We directly find
$\displaystyle\|\mathcal{I}_{45}\|_{\textup{F}}\leq
4\sqrt{n}\mathcal{S}(V,\tilde{V}).$
In (B.2), we combine all the estimates to obtain
$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}d(U,\tilde{U})$
$\displaystyle\leq-2m{\kappa}d(U,\tilde{U})+4m{\kappa}\mathcal{L}d(U,\tilde{U})+6\mathcal{L}(d(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))$
$\displaystyle\hskip
14.22636pt+2\mathcal{L}^{2}(4d(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))+4\sqrt{n}\mathcal{S}(V,\tilde{V}).$
By exchanging the roles of $\mathcal{U}$ and $\mathcal{V}$, we find the
differential inequality for $d(V,\tilde{V})$:
$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}d(V,\tilde{V})$
$\displaystyle\leq-2n{\kappa}d(V,\tilde{V})+4n{\kappa}\mathcal{L}d(V,\tilde{V})+6\mathcal{L}(d(V,\tilde{V})+\mathcal{S}(U,\tilde{U}))$
$\displaystyle\hskip
14.22636pt+2\mathcal{L}^{2}(4d(V,\tilde{V})+\mathcal{S}(U,\tilde{U}))+4\sqrt{m}\mathcal{S}(U,\tilde{U}).$
∎
Below, we derive differential inequalities for $\mathcal{S}(U,\tilde{U})$ and
$\mathcal{S}(V,\tilde{V})$.
###### Lemma B.2.
Let $\\{(U_{i},V_{i})\\}$ be a solution to (4.1). Then,
$\mathcal{S}(U,\tilde{U})$ and $\mathcal{S}(V,\tilde{V})$ satisfy
$\displaystyle\begin{aligned}
\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(U,\tilde{U})&\leq-2m{\kappa}\mathcal{S}(U,\tilde{U})+4\mathcal{L}d(U,\tilde{U})+\mathcal{D}(\mathcal{H})d(U,\tilde{U})+6\mathcal{L}(\mathcal{S}(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))\\\
&\hskip
14.22636pt+8\mathcal{L}^{2}d(U,\tilde{U})+2\mathcal{L}^{2}\mathcal{S}(V,\tilde{V})+4\sqrt{n}\mathcal{S}(V,\tilde{V}).\\\
\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(V,\tilde{V})&\leq-2n{\kappa}\mathcal{S}(V,\tilde{V})+4\mathcal{L}d(V,\tilde{V})+\mathcal{D}(\mathcal{H})d(V,\tilde{V})+6\mathcal{L}(\mathcal{S}(V,\tilde{V})+\mathcal{S}(U,\tilde{U}))\\\
&\hskip
14.22636pt+8\mathcal{L}^{2}d(V,\tilde{V})+2\mathcal{L}^{2}\mathcal{S}(U,\tilde{U})+4\sqrt{m}\mathcal{S}(U,\tilde{U}).\end{aligned}$
(B.4)
###### Proof.
We recall (A.9):
$\displaystyle\begin{aligned}
\frac{{\textup{d}}}{{\textup{d}t}}(n-d_{ij})&=-2m{\kappa}(n-d_{ij})+\frac{m{\kappa}}{N}\sum_{k=1}^{N}\textup{tr}(S_{ik}S_{ij}+S_{ij}S_{kj})\\\
&\hskip
14.22636pt+\mathrm{i}\textup{tr}(H_{i}-H_{j})+\mathrm{i}\textup{tr}(S_{ij}H_{j}-H_{i}S_{ij})\\\
&\hskip
14.22636pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}(c_{ik}-m)(n-d_{kj})-(c_{ki}-m)(n-d_{ik})-(c_{ik}-m)(n-d_{ij})\\\
&\hskip
14.22636pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}(c_{kj}-m)(n-d_{ik})-(c_{jk}-m)(n-d_{kj})-(c_{jk}-m)(n-d_{ij})\\\
&\hskip
14.22636pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}(c_{ki}-m)\textup{tr}(S_{ik}S_{ij})+(c_{jk}-m)\textup{tr}(S_{ij}S_{kj})\\\
&\hskip
14.22636pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}[(c_{ki}-c_{ik})+(c_{jk}-c_{kj})]\sqrt{n}.\end{aligned}$
We denote
$p_{ij}:=n-d_{ij},\quad q_{ij}:=m-c_{ij}.$
Then, $p_{ij}-\tilde{p}_{ij}$ satisfies
$\displaystyle\begin{aligned}
&\frac{{\textup{d}}}{{\textup{d}t}}(p_{ij}-\tilde{p}_{ij})=-2m{\kappa}(p_{ij}-\tilde{p}_{ij})+\frac{m{\kappa}}{N}\sum_{k=1}^{N}\underbrace{\textup{tr}(S_{ik}S_{ij}+S_{ij}S_{kj}-\tilde{S}_{ik}\tilde{S}_{ij}-\tilde{S}_{ij}\tilde{S}_{kj})}_{=:\mathcal{I}_{51}}\\\
&\hskip
5.69046pt+\underbrace{\mathrm{i}\textup{tr}((S_{ij}-\tilde{S}_{ij})(H_{j}-H_{i}))}_{=:\mathcal{I}_{52}}+\frac{{\kappa}}{N}\sum_{k=1}^{N}(\mathcal{I}_{53}-\tilde{\mathcal{I}}_{53})+\frac{{\kappa}}{N}\sum_{k=1}^{N}(\mathcal{I}_{54}-\tilde{\mathcal{I}}_{54})\\\
&\hskip
5.69046pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}(\mathcal{I}_{55}-\tilde{\mathcal{I}}_{55})+\frac{{\kappa}}{N}\sum_{k=1}^{N}\underbrace{(p_{ik}-\tilde{p}_{ik}+p_{kj}-\tilde{p}_{kj})+(\tilde{p}_{ki}-\tilde{p}_{ki})+(\tilde{p}_{jk}-p_{jk})}_{=:\mathcal{I}_{56}}\sqrt{n}.\end{aligned}$
(B.5)
Below, we present the estimates for $\mathcal{I}_{5k},~{}k=1,\cdots,6$,
separately.
$\bullet$ (Estimate of $\mathcal{I}_{51}$): We recall (B.3) to find
$|\mathcal{I}_{51}|\leq 4\mathcal{L}d(U,\tilde{U}).$
$\bullet$ (Estimate of $\mathcal{I}_{52}$): we set
$\mathcal{D}(\mathcal{H}):=\max_{1\leq i,j\leq N}\|H_{i}-H_{j}\|_{\infty},$
Then, one has
$|\mathcal{I}_{52}|\leq\mathcal{D}(\mathcal{H})d(U,\tilde{U}).$
$\bullet$ (Estimate of $\mathcal{I}_{53}$): Note that
$\displaystyle|(c_{ik}-m)(n-d_{kj})-(\tilde{c}_{ik}-m)(n-\tilde{d}_{kj})|$
$\displaystyle\hskip
28.45274pt\leq|p_{ik}-\tilde{p}_{ik}||q_{ik}|+|\tilde{p}_{ik}||q_{kj}-\tilde{q}_{kj}|\leq\mathcal{L}(\mathcal{S}(U,\tilde{U})+\mathcal{S}(V,\tilde{V})).$
Thus, we find
$\displaystyle\|\mathcal{I}_{53}-\tilde{\mathcal{I}}_{53}\|_{\textup{F}}\leq
3\mathcal{L}(\mathcal{S}(U,\tilde{U})+\mathcal{S}(V,\tilde{V})).$
$\bullet$ (Estimate of $\mathcal{I}_{54}$): We observe
$\displaystyle\|\mathcal{I}_{54}-\tilde{\mathcal{I}}_{54}\|_{\textup{F}}\leq
3\mathcal{L}(\mathcal{S}(U,\tilde{U})+\mathcal{S}(V,\tilde{V})).$
$\bullet$ (Estimate of $\mathcal{I}_{55}$): Note that
$\displaystyle
q_{ki}\textup{tr}(S_{ik}S_{ij}-\tilde{q}_{ki}\textup{tr}(\tilde{S}_{ik}\tilde{S}_{ij})$
$\displaystyle\hskip
14.22636pt=q_{ij}\textup{tr}(S_{ik}S_{ij}-\tilde{S}_{ik}\tilde{S}_{ij})+(q_{ij}-\tilde{q}_{ij})\textup{tr}(\tilde{S}_{ik}\tilde{S}_{ij})\leq
4\mathcal{L}^{2}d(U,\tilde{U})+\mathcal{L}^{2}\mathcal{S}(V,\tilde{V}).$
Thus, we have
$\displaystyle\|\mathcal{I}_{55}-\tilde{\mathcal{I}}_{55}\|_{\textup{F}}\leq
8\mathcal{L}^{2}d(U,\tilde{U})+2\mathcal{L}^{2}\mathcal{S}(V,\tilde{V}).$
$\bullet$ (Estimate of $\mathcal{I}_{56}$): We directly find
$\displaystyle|\mathcal{I}_{56}|\leq 4\sqrt{n}\mathcal{S}(V,\tilde{V}).$
In (B.5), we collect all the estimates to find
$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(U,\tilde{U})$
$\displaystyle\leq-2m{\kappa}\mathcal{S}(U,\tilde{U})+4\mathcal{L}d(U,\tilde{U})+\mathcal{D}(\mathcal{H})d(U,\tilde{U})+6\mathcal{L}(\mathcal{S}(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))$
$\displaystyle\hskip
14.22636pt+8\mathcal{L}^{2}d(U,\tilde{U})+2\mathcal{L}^{2}\mathcal{S}(V,\tilde{V})+4\sqrt{n}\mathcal{S}(V,\tilde{V}).$
By exchanging the roles of $\mathcal{U}$ and $\mathcal{V}$, we can derive a
differential inequality for $\mathcal{S}(V,\tilde{V})$:
$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(V,\tilde{V})$
$\displaystyle\leq-2n{\kappa}\mathcal{S}(V,\tilde{V})+4\mathcal{L}d(V,\tilde{V})+\mathcal{D}(\mathcal{H})d(V,\tilde{V})+6\mathcal{L}(\mathcal{S}(V,\tilde{V})+\mathcal{S}(U,\tilde{U}))$
$\displaystyle\hskip
14.22636pt+8\mathcal{L}^{2}d(V,\tilde{V})+2\mathcal{L}^{2}\mathcal{S}(U,\tilde{U})+4\sqrt{m}\mathcal{S}(U,\tilde{U}).$
∎
Now, we are ready to provide a proof of Lemma 4.2 with the help of Lemma B.1
and Lemma B.2.
Proof of Lemma 4.2: recall the inequalities in (B.1) and (B.4):
$\displaystyle\begin{aligned}
\frac{{\textup{d}}}{{\textup{d}t}}d(U,\tilde{U})&\leq-2m{\kappa}d(U,\tilde{U})+4m{\kappa}\mathcal{L}d(U,\tilde{U})+6\mathcal{L}(d(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))\\\
&\hskip
14.22636pt+2\mathcal{L}^{2}(4d(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))+4\sqrt{n}\mathcal{S}(V,\tilde{V}),\\\
\frac{{\textup{d}}}{{\textup{d}t}}d(V,\tilde{V})&\leq-2n{\kappa}d(V,\tilde{V})+4n{\kappa}\mathcal{L}d(V,\tilde{V})+6\mathcal{L}(d(V,\tilde{V})+\mathcal{S}(U,\tilde{U}))\\\
&\hskip
14.22636pt+2\mathcal{L}^{2}(4d(V,\tilde{V})+\mathcal{S}(U,\tilde{U}))+4\sqrt{m}\mathcal{S}(U,\tilde{U}),\\\
\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(U,\tilde{U})&\leq-2m{\kappa}\mathcal{S}(U,\tilde{U})+4\mathcal{L}d(U,\tilde{U})+\mathcal{D}(\mathcal{H})d(U,\tilde{U})+6\mathcal{L}(\mathcal{S}(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))\\\
&\hskip
14.22636pt+8\mathcal{L}^{2}d(U,\tilde{U})+2\mathcal{L}^{2}\mathcal{S}(V,\tilde{V})+4\sqrt{n}\mathcal{S}(V,\tilde{V}),\\\
\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(V,\tilde{V})&\leq-2n{\kappa}\mathcal{S}(V,\tilde{V})+4\mathcal{L}d(V,\tilde{V})+\mathcal{D}(\mathcal{G})d(V,\tilde{V})+6\mathcal{L}(\mathcal{S}(V,\tilde{V})+\mathcal{S}(U,\tilde{U}))\\\
&\hskip
14.22636pt+8\mathcal{L}^{2}d(V,\tilde{V})+2\mathcal{L}^{2}\mathcal{S}(U,\tilde{U})+4\sqrt{m}\mathcal{S}(U,\tilde{U}).\end{aligned}$
(B.6)
We add all the inequalities in (B.6) to obtain the desired inequality for
$\mathcal{F}$:
$\displaystyle\frac{{\textup{d}}\mathcal{F}}{dt}$
$\displaystyle\leq-{\kappa}\left(2m-8\sqrt{n}-\frac{\mathcal{D}(\mathcal{H})}{{\kappa}}\right)\mathcal{F}+4(n+1){\kappa}\mathcal{L}(d(U,\tilde{U})+d(V,\tilde{V}))+6{\kappa}\mathcal{L}\mathcal{F}$
$\displaystyle\hskip
14.22636pt+12{\kappa}\mathcal{L}(\mathcal{S}(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))+16{\kappa}\mathcal{L}^{2}(d(U,\tilde{U})+d(V,\tilde{V}))+4{\kappa}\mathcal{L}^{2}(\mathcal{S}(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))$
$\displaystyle\leq-{\kappa}\left(2m-8\sqrt{n}-\frac{\max\\{\mathcal{D}(\mathcal{H}),\mathcal{D}(\mathcal{G})\\}}{{\kappa}}\right)\mathcal{F}+{\kappa}(4n+22)\mathcal{L}\mathcal{F}+20{\kappa}\mathcal{L}^{2}\mathcal{F}.$
## Appendix C Emergent dynamics of the DSOM model
In this section, we replace the product of two unitary groups
$\mathbf{U}(n)\times\mathbf{U}(m)$ by the product of two special orthogonal
groups $\mathbf{SO}(n)\times\mathbf{SO}(m)$. Since all elements of special
orthogonal matrices are real-valued, one has for any (real-valued) square
matrices $A$ and $B$,
$\langle
A,B\rangle_{\textup{F}}=\textup{tr}(AB^{\top})=\textup{tr}(BA^{\top})=\langle
B,A\rangle_{\textup{F}}.$ (C.1)
Thus, model (3.5) reduces to
$\begin{cases}\displaystyle\dot{U}_{j}=B_{j}U_{j}+\frac{{\kappa}}{N}\sum_{k=1}^{N}\langle
V_{j},V_{k}\rangle_{\textup{F}}(U_{k}-U_{j}U_{k}^{\top}U_{j}),\quad t>0,\\\
\displaystyle\dot{V}_{j}=C_{j}V_{j}+\frac{{\kappa}}{N}\sum_{k=1}^{N}\langle
U_{j},U_{k}\rangle_{\textup{F}}(V_{k}-V_{j}V_{k}^{\top}V_{j}),\\\
\displaystyle(U_{j},V_{j})(0)=(U_{j}^{0},V_{j}^{0})\in\mathbf{SO}(n)\times\mathbf{SO}(m),\quad
1\leq j\leq N,\end{cases}$ (C.2)
where $B_{j}\in\mathbb{R}^{n\times n\times n\times n}$ and
$C_{j}\in\mathbb{R}^{m\times m\times m\times m}$ are given rank-4 tensors
satisfying skew-symmetric properties:
$[B_{j}]_{\alpha_{1}\beta_{1}\alpha_{2}\beta_{2}}=-[B_{j}]_{\alpha_{2}\beta_{2}\alpha_{1}\beta_{1}},\quad[C_{j}]_{\gamma_{1}\delta_{1}\gamma_{2}\delta_{2}}=-[{C}_{j}]_{\gamma_{2}\delta_{2}\gamma_{1}\delta_{1}}.$
###### Remark C.1.
Due to the symmetric property of the Frobenius inner product on real-valued
matrices, one has
$d_{ij}=\langle U_{i},U_{j}\rangle_{\textup{F}}=\langle
U_{j},U_{i}\rangle_{\textup{F}}=d_{ji},\quad c_{ij}=\langle
V_{i},V_{j}\rangle_{\textup{F}}=\langle
V_{j},V_{i}\rangle_{\textup{F}}=c_{ji}.$ (C.3)
Due to the relation (C.3), for example, the last term
$c_{ki}-c_{ik}+c_{jk}-c_{kj}$ in (A.3) vanishes. Thus, it would be expected
that the relation (C.3) relaxes the condition such as the dimension condition
$n\geq m>4\sqrt{n}$ in (4.7)(i) (see also Remark A.1).
The contents of this appendix are exactly the same as those of Section 4, and
we only provide a sketch of the proof. In Section C.1, we are concerned with
the identical (or homogeneous) Hamiltonian, and in Section C.2, the non-
identical (or heterogeneous) Hamiltonian is considered to exhibit the phase-
locked state.
### C.1. Homogeneous ensemble
In this subsection, thanks to the solution splitting property, we assume that
$B_{j}\equiv O,\quad C_{j}\equiv O,\quad j=1,\cdots,N.$ (C.4)
Below, we state the main theorem for an identical ensemble under which the
complete aggregation occurs. For this, we recall diameters and aggregation
functional:
$\displaystyle\begin{aligned} &\mathcal{D}(\mathcal{U}(t))=\max_{1\leq i,j\leq
N}\|U_{i}(t)-U_{j}(t)\|_{\textup{F}},\quad\mathcal{S}(\mathcal{U}(t))=\max_{1\leq
i,j\leq N}|n-d_{ij}(t)|,\\\ &\mathcal{D}(\mathcal{V}(t))=\max_{1\leq i,j\leq
N}\|V_{i}(t)-V_{j}(t)\|_{\textup{F}},\quad\mathcal{S}(\mathcal{V}(t))=\max_{1\leq
i,j\leq N}|m-c_{ij}(t)|,\\\
&\mathcal{L}=\mathcal{D}(\mathcal{U})+\mathcal{D}(\mathcal{V})+\mathcal{S}(\mathcal{U})+\mathcal{S}(\mathcal{V}).\end{aligned}$
###### Theorem C.1.
Suppose initial data satisfy
$\mathcal{L}^{0}<\alpha_{m,n}=\frac{-(12n+27)+\sqrt{(12n+27)^{2}+24m(3n+4)}}{4(4n+3)},$
(C.5)
and let $\\{(U_{i},V_{i})\\}$ be a solution to (C.2) with (C.4). Then, we have
$\lim_{t\to\infty}\mathcal{L}(t)=0.$
###### Proof.
By recalling the inequalities in (A.1) with the relation (C.1), we see
$\displaystyle\begin{aligned}
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{U})\leq-2m{\kappa}\mathcal{D}(\mathcal{U})+m{\kappa}\mathcal{D}(\mathcal{U})^{3}+6{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})+2{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{2},\\\
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{V})\leq-2n{\kappa}\mathcal{D}(\mathcal{V})+n{\kappa}\mathcal{D}(\mathcal{V})^{3}+6{\kappa}\mathcal{S}(\mathcal{U})\mathcal{D}(\mathcal{V})+2{\kappa}\mathcal{S}(\mathcal{U})\mathcal{D}(\mathcal{V})^{2}.\end{aligned}$
(C.6)
Similarly, we use (A.8) to find
$\displaystyle\begin{aligned}
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(\mathcal{U})\leq-2m{\kappa}\mathcal{S}(\mathcal{U})+2m{\kappa}\mathcal{D}(\mathcal{U})^{2}+6{\kappa}\mathcal{S}(\mathcal{U})\mathcal{S}(\mathcal{V})+2{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{2},\\\
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(\mathcal{V})\leq-2n{\kappa}\mathcal{S}(\mathcal{V})+2n{\kappa}\mathcal{D}(\mathcal{V})^{2}+6{\kappa}\mathcal{S}(\mathcal{U})\mathcal{S}(\mathcal{V})+2{\kappa}\mathcal{S}(\mathcal{U})\mathcal{D}(\mathcal{V})^{2}.\end{aligned}$
(C.7)
Now, we add (C.6) and (C.7) to obtain a differential inequality for
$\mathcal{L}$:
$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{L}\leq-2{\kappa}m\mathcal{L}+{\kappa}(4n+9)\mathcal{L}^{2}+{\kappa}\left(2n+\frac{8}{3}\right)\mathcal{L}^{3}.$
(C.8)
If we introduce an auxiliary polynomial:
$f(s):=\left(2n+\frac{8}{3}\right)s^{2}+(4n+9)s-2m,$ (C.9)
then (C.8) can be written as
$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{L}\leq{\kappa}\mathcal{L}f(\mathcal{L}).$
Furthermore, we notice that $\alpha_{m,n}$ in (C.5) becomes a unique positive
root of the quadratic polynomial $f$ in (C.9). Hence, the desired zero
convergence of $\mathcal{L}$ directly follows from the dynamical systems
theory. ∎
### C.2. Heterogeneous ensemble
In this subsection, we are concerned with the heterogeneous Hamiltonians where
the natural frequency tensors are different in general. Similar to Section
4.2, there exist two skew-symmetric matrices $\Omega_{j}\in\mathbb{R}^{n\times
n}$ and $\Psi_{j}\in\mathbb{R}^{m\times m}$ such that
$B_{j}U_{j}=\Omega_{j}U_{j},\quad C_{j}V_{j}=\Psi_{j}V_{j},\quad 1\leq j\leq
N.$
Thus, our model reads as
$\begin{cases}\displaystyle\dot{U}_{j}=\Omega_{j}U_{j}+\frac{{\kappa}}{N}\sum_{k=1}^{N}\langle
V_{j},V_{k}\rangle_{\textup{F}}(U_{k}-U_{j}U_{k}^{\top}U_{j}),\\\
\displaystyle\dot{V}_{j}=\Psi_{j}V_{j}+\frac{{\kappa}}{N}\sum_{k=1}^{N}\langle
U_{j},U_{k}\rangle_{\textup{F}}(V_{k}-V_{j}V_{k}^{\top}V_{j}),\\\
\displaystyle(U_{j},V_{j})(0)=(U_{j}^{0},V_{j}^{0})\in\mathbf{SO}(n)\times\mathbf{SO}(m),\quad
1\leq j\leq N,\end{cases}$ (C.10)
A crucial estimate used for the emergence of the phase-locked state is (4.12)
stated in Lemma 4.2 where the aggregation functional $\mathcal{F}$ tends to
zero. Here, $\mathcal{F}$ measures the inter-distance between any two solution
configurations $\\{(U_{i},V_{i})\\}$ and
$\\{(\tilde{U}_{i},\tilde{V}_{i})\\}$:
$\mathcal{F}(t)=d(U,\tilde{U})(t)+d(V,\tilde{V})(t)+\mathcal{S}(U,\tilde{U})(t)+\mathcal{S}(V,\tilde{V})(t),$
where the diameters are defined in (4.8) and (4.10). We also denote the
diameters for natural frequencies:
$\mathcal{D}(\Omega):=\max_{1\leq i,j\leq
N}\|\Omega_{i}-\Omega_{j}\|_{\infty},\quad\mathcal{D}(\Psi):=\max_{1\leq
i,j\leq N}\|\Psi_{i}-\Psi_{j}\|_{\infty}.$
Without loss of generality, we may assume
$\mathcal{D}(\Omega)\geq\mathcal{D}(\Psi).$
In the following lemma, we provide a temporal evolution of $\mathcal{F}$.
###### Lemma C.1.
Let $\\{(U_{i},V_{i})\\}$ and $\\{(\tilde{U}_{i},\tilde{V}_{i})\\}$ be any two
solutions to (C.10). Then, the aggregation functionals $\mathcal{L}$ and
$\mathcal{F}$ satisfy
$\displaystyle\begin{aligned} &\frac{{\textup{d}}\mathcal{L}}{dt}\leq
2(1+3\sqrt{n})\mathcal{D}(\Omega)-2{\kappa}m\mathcal{L}+{\kappa}(4n+9)\mathcal{L}^{2}+{\kappa}\left(2n+\frac{8}{3}\right)\mathcal{L}^{3},\quad
t>0,\\\
&\frac{{\textup{d}}\mathcal{F}}{dt}\leq-{\kappa}\left(2m-\frac{\max\\{\mathcal{D}(\Omega),\mathcal{D}(\Psi)\\}}{{\kappa}}\right)\mathcal{F}+{\kappa}(4n+22)\mathcal{L}\mathcal{F}+20{\kappa}\mathcal{L}^{2}\mathcal{F}.\end{aligned}$
(C.11)
###### Proof.
For the inequality for $\mathcal{L}$, it directly follows from (C.8) in
Theorem C.1. Similarly, if we closely follow a proof of Lemma 4.2 presented in
Appendix B, then we find the desired inequality for $\mathcal{F}$. ∎
By applying $\eqref{Y-15}_{2}$, we establish the desired practical aggregation
estimate for (C.10).
###### Proposition C.1.
Suppose that the system parameters and initial data satisfy
$\mathcal{D}(\Omega)\geq\mathcal{D}(\Psi),\quad{\kappa}>{\kappa}_{\textup{c}},\quad\mathcal{L}^{0}<\nu_{2},$
(C.12)
where ${\kappa}_{\textup{c}}$ and $\nu_{2}$ are specified in (C.14) and
(C.15), respectively, and let $\\{(U_{i},V_{i})\\}$ be a solution to (C.10).
Then, one has
$\lim_{{\kappa}\to\infty}\limsup_{t\to\infty}\mathcal{L}(t)=0.$
###### Proof.
We introduce an auxiliary polynomial:
$g(s):=2ms-(4n+9)s^{2}-\left(2n+\frac{8}{3}\right)s^{3}.$ (C.13)
Then, we notice that $g$ has three roots, say, $\alpha_{0}<0<\alpha_{1}$.
Since $\eqref{Y-15}_{2}$ is rewritten as
$\dot{\mathcal{L}}\leq{\kappa}\left(\frac{2(1+3\sqrt{n})\mathcal{D}(\mathcal{H})}{{\kappa}}-g(\mathcal{L})\right)=:{\kappa}p(s),$
for a sufficiently large ${\kappa}>0$, $p$ admits one negative root, say,
$\nu_{0}<0$ and two positive roots, say, $0<\nu_{1}<\nu_{2}$ with continuous
dependence on ${\kappa}$: for $\nu_{1}=\nu_{1}({\kappa})$ and
$\nu_{2}=\nu_{2}({\kappa})$,
$\lim_{{\kappa}\to\infty}\nu_{1}({\kappa})=0,\quad\lim_{{\kappa}\to\infty}\nu_{2}({\kappa})=\alpha_{1}.$
Since we assume $\eqref{Y-18}_{3}$, it follows from dynamical systems theory
that there exists a finite entrance time $T_{*}>0$ such that
$\mathcal{L}(t)<\nu_{1},\quad t>T_{*}.$
Hence, this yields the desired result. ∎
###### Remark C.2.
For the explicit value of ${\kappa}_{\textup{c}}$, we see that the cubic
polynomial $g$ in (C.13) admits the local maximum at $s=s_{*}$:
$s_{*}=\frac{-(4n+9)+\sqrt{(4n+9)^{2}+4m(3n+4)}}{6n+8}.$
Hence, ${\kappa}_{\textup{c}}$ is chosen to be
${\kappa}_{\textup{c}}:=\frac{2(1+3\sqrt{n})\mathcal{D}(\Omega)}{g(s_{*})}\quad\textup{so
that}\quad g(s_{*})>\frac{2(1+3\sqrt{n})}{{\kappa}_{\textup{c}}}.$ (C.14)
For this ${\kappa}_{\textup{c}}$, $\nu_{2}=\nu_{2}({\kappa})$ is completely
determined as the largest positive root of
$p(s)=0\quad\textup{for ${\kappa}>{\kappa}_{\textup{c}}$}.$ (C.15)
It follows from Proposition 5.1 that we can make $\mathcal{L}$ sufficiently
small by increasing the value of ${\kappa}$ for $t>T_{*}$. Thus, there exists
${\kappa}_{\textup{p}}>{\kappa}_{\textup{c}}$ such that for
${\kappa}>{\kappa}_{\textup{p}}$,
$(4n+22)\mathcal{L}+20\mathcal{L}^{2}<\frac{1}{2}\left(2m-\frac{\max\\{\mathcal{D}(\Omega),\mathcal{D}(\Psi)\\}}{{\kappa}}\right)=:\frac{1}{2}\Lambda.$
Hence, $\eqref{Y-15}_{2}$ yields
$\frac{d\mathcal{F}}{dt}\leq-\frac{1}{2}\Lambda\mathcal{F},\quad t>T_{*}.$
Since we have established the zero convergence of $\mathcal{F}$, we can obtain
the same result in Theorem 4.2. Thanks to exactly the same proof, we only
state the results.
###### Theorem C.2.
Suppose that the system parameters and initial data satisfy
$n\geq
m,\quad\mathcal{D}(\Omega)\geq\mathcal{D}(\Psi),\quad{\kappa}>{\kappa}_{\textup{p}},\quad\max\\{\mathcal{L}^{0},\tilde{\mathcal{L}}^{0}\\}\leq\nu_{2},$
and let $\\{(U_{i},V_{i})\\}$ and $\\{(\tilde{U}_{i},\tilde{V}_{i})\\}$ be any
two solutions to (C.10). Then, the following assertions hold.
1. (1)
The aggregation functional $\mathcal{F}$ converges to zero exponentially.
2. (2)
The normalized velocities $\dot{U}_{i}U_{i}^{\top}$ and
$\tilde{\dot{U}}_{i}\tilde{U}_{i}^{\top}$ synchronize:
$\|\dot{U}_{i}U_{i}^{\top}-\tilde{\dot{U}}_{i}\tilde{U}_{i}^{\top}\|_{\textup{F}}\leq
2{\kappa}(m+n)\mathcal{F}.$
3. (3)
There exist special orthogonal matrices $X_{\infty}\in\mathbf{SO}(n)$ and
$Y_{\infty}\in\mathbf{SO}(m)$ such that
$\displaystyle\lim_{t\to\infty}U_{i}^{\top}(t)\tilde{U}_{i}(t)=X_{\infty},\quad\lim_{t\to\infty}\|\tilde{U}_{i}(t)-U_{i}(t)X_{\infty}\|_{\textup{F}}=0,$
$\displaystyle\lim_{t\to\infty}V_{i}^{\top}(t)\tilde{V}_{i}(t)=Y_{\infty},\quad\lim_{t\to\infty}\|\tilde{V}_{i}(t)-V_{i}(t)Y_{\infty}\|_{\textup{F}}=0.$
4. (4)
System (C.10) exhibits asymptotic phase-locking: for any indices $i$ and $j$,
$\exists~{}~{}\lim_{t\to\infty}U_{i}(t)U_{j}^{\top}(t)\quad\textup{and}\quad\exists~{}~{}\lim_{t\to\infty}V_{i}(t)V_{j}^{\top}(t).$
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|
Test of the Orbital-Based LI3 Index as a Predictor of the Height of the 3MLCT
$\rightarrow$ 3MC Transition-State Barrier for Gas-Phase [Ru(N∧N)3]2+
Polypyridine Complexes
Denis Magero
School of Science, Technology and Engineering, Department of Chemistry and
Biochemistry, Alupe University, P.O. Box 845-50400, Busia, Kenya
e-mail<EMAIL_ADDRESS>
Ala Aldin M. H. M. Darghouth
College of Sciences, University of Mosul, Al Majmoaa Street, Mosul, 41002
Iraq.
e-mail<EMAIL_ADDRESS>
Mark E. Casida
Laboratoire de Spectrométrie, Interactions et Chimie théorique (SITh),
Département de Chimie Moléculaire (DCM, UMR CNRS/UGA 5250), Institut de Chimie
Moléculaire de Grenoble (ICMG, FR-2607), Université Grenoble Alpes (UGA) 301
rue de la Chimie, CS40700, F-38058 Grenoble Cedex 9, France
e-mail<EMAIL_ADDRESS>
Abstract
Ruthenium(II) polypyridine compounds often luminesce but the luminescence
lifetime depends upon the precise nature of the ligands. This luminescence
lifetime is thought to be controlled by the barrier to conversion from an
initial phosphorescent triplet metal-ligand charge transfer (3MLCT) state to a
nonluminscent triplet metal-centered (3MC) state which decays nonradiatively.
Earlier work [J. Photochem. Photobiol. A 348, 305 (2017)] took room
temperature and liquid nitrogen (77 K) lifetimes from a large previously-
published database [Coord. Chem. Rev. 84, 85 (1988)] and extracted empirical
average 3MLCT $\rightarrow$ 3MC transition state (TS) barrier heights
($E_{ave}$s) which were not believed to be quantitative but which were
believed to capture the trends in the true barrier heights correctly. These
were then used together with information from partial density of states
calculations [J. Photochem. Photobiol. A 276, 8 (2014)] to derive several
orbital-based luminescence indices of which the third (LI3) was based upon
frontier-molecular-orbital-like ideas and correlated linearly with values of
$E_{ave}$. As it is known that $E_{ave}$ is a large underestimate of the true
3MLCT $\rightarrow$ 3MC TS barrier height in the case of the trisbipyridine
ruthenium(II) cation $\\{$ [Ru(bpy)3]2+ $\\}$, but accurate TS barrier heights
are difficult to obtain experimentally, it was judged useful to verify the
ideas used to derive the LI3 index by calculating the energetics of the gas-
phase 3MLCT $\rightarrow$ 3MC reaction for a series of ruthenium(II) tris
bipyridine complexes using the same density functional and basis sets used in
calculating LI3. Specifically, four closely-related bipyradine complexes $\\{$
[Ru(N∧N)3]2+ with N∧N = bpy (6), 4,4’-dm-bpy (70), 4,4’-dph-bpy (73), and
4,4’-DTB-bpy (74) $\\}$ were used for these calculations. We examine the trans
dissociation mechanism in great detail at the B3LYP/6-31G+LANLDZ(Ru) level of
detail and uncover a two part mechanism. In the first part, the electron is
transferred to a single ligand rather than symmetrically to all three ligands.
It is the two Ru-N bonds to this ligand which are equally elongated in the
transition state. The intrinsic reaction coordinate then continues down a
ridge in hyperspace and bifurcates into one of two symmetry-equivalent 3MC
structures with elongated trans bonds. Interestingly, no significant
difference is found for the TS barriers for the four complexes treated here.
Instead, LI3 is linearly correlated with the energy difference $\Delta
E=E(\mbox{${}^{3}$MLCT})-E(\mbox{${}^{3}$MC})$. While this work shows that LI3
predicts the total energy difference and this does correlate well with the
luminescence lifetime, we do not have a detailed understanding yet of how this
happens and are also wary of oversimplicity as there are likely other excited-
state reactions occurring on similar time scales which may also impact the
luminescence lifetimes.
Graphical Abstract
## 1 Introduction
Luminescent ruthenium(II) complexes elicit immense interest [1, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11, 12] owing to their wide range of applications, including
photochemical molecular devices, biological sensors, organic light emitting
diodes, and biomedical applications, ††margin: LFT among others. The commonly
assumed ligand-field theory (LFT) mechanism explaining this luminescence
begins with the $d^{6}$ ruthenium complex initially in ††margin: GS its
closed-shell ground-state (1GS) configuration,
$\mbox{metal:
}\begin{array}[]{c}\underbrace{[\,\,\,,\,\,\,][\,\,\,,\,\,\,]}_{e_{g}^{*}}\\\
\underbrace{[\uparrow,\downarrow][\uparrow,\downarrow][\uparrow,\downarrow]}_{t_{2g}}\end{array}+\mbox{ligands:
}\underbrace{[\,\,\,,\,\,\,][\,\,\,,\,\,\,]\cdots}_{\pi^{*}}\,.$ (1)
(The 1X and 3X pre-exponents denote respectively that the state X ††margin:
1X , 3X is a singlet or triplet. Note that the $t_{2g}$ orbital is nonbonding
but that the $e_{g}^{*}$ orbitals are antibonding. We have also indicated some
unoccupied $\pi^{*}$ orbitals on the ligands.) Figure 1 emphasizes that these
$\pi^{*}$ ligand orbitals are within the LFT gap of the metal.
Figure 1: Pseudo-octahedral ligand field theory diagram for ruthenium(II)
complexes.
Upon absorption of a photon, an electron is excited from the metal to the
ligand to create a high-lying singlet ††margin: MLCT metal-ligand-charge-
transfer (1MLCT) configuration,
$\mbox{metal:
}\begin{array}[]{c}\underbrace{[\,\,\,,\,\,\,][\,\,\,,\,\,\,]}_{e_{g}^{*}}\\\
\underbrace{[\uparrow,\downarrow][\uparrow,\downarrow][\downarrow,\,\,\,]}_{t_{2g}}\end{array}+\mbox{ligands:
}\underbrace{[\uparrow,\,\,\,][\,\,\,,\,\,\,]\cdots}_{\pi^{*}}\,,$ (2)
which relaxes by non-radiative relaxation to the lowest 3MLCT state,
$\mbox{metal:
}\begin{array}[]{c}\underbrace{[\,\,\,,\,\,\,][\,\,\,,\,\,\,]}_{e_{g}^{*}}\\\
\underbrace{[\uparrow,\downarrow][\uparrow,\downarrow][\uparrow,\,\,\,]}_{t_{2g}}\end{array}+\mbox{ligands:
}\underbrace{[\uparrow,\,\,\,][\,\,\,,\,\,\,]\cdots}_{\pi^{*}}\,.$ (3)
its closed-shell ground-state (1GS) configuration,
$\mbox{metal:
}\begin{array}[]{c}\underbrace{[\,\,\,,\,\,\,][\,\,\,,\,\,\,]}_{e_{g}^{*}}\\\
\underbrace{[\uparrow,\downarrow][\uparrow,\downarrow][\uparrow,\downarrow]}_{t_{2g}}\end{array}+\mbox{ligands:
}\underbrace{[\,\,\,,\,\,\,][\,\,\,,\,\,\,]\cdots}_{\pi^{*}}\,.$ (4)
Energetically close to this state is a triplet metal-centered (3MC) state
which mixes with the 3MLCT state to form an avoided crossing, allowing
repopulation of the $e_{g}^{*}$ orbitals via back transfer of an electron,
$\mbox{metal:
}\begin{array}[]{c}\underbrace{[\uparrow,\,\,\,][\,\,\,,\,\,\,]}_{e_{g}^{*}}\\\
\underbrace{[\uparrow,\downarrow][\uparrow,\downarrow][\uparrow,\,\,\,]}_{t_{2g}}\end{array}+\mbox{ligands:
}\underbrace{[\,\,\,,\,\,\,][\,\,\,,\,\,\,]\cdots}_{\pi^{*}}\,.$ (5)
As this 3MC state does not luminesce, its formation quenches the
phosphoresence of the ruthenium(II) complex leading to shortened luminescence
lifetimes. Here it is important to realize that, while we talk of states, the
picture in most chemical physicists’/physical chemists’ minds is that of LFT
orbitals. However most transition metal complex calculations are carried out
using ††margin: DFT density-functional theory (DFT) where extracting LFT
orbitals is far from obvious. Previous work showed how a LFT-like
understanding of ruthenium(II) polypyridine complexes could be ††margin: PDOS
obtained using a partial density of state (PDOS) method [13], in particular
allowing assignment of the energy of the antibonding $e_{g}^{*}$ state which
had posed a problem in previous theoretical work. This, in turn, allowed the
construction of an orbital-based luminscence ††margin: LI3 index of which the
third try (LI3) correlated well with experimentally-derived 3MLCT
$\rightarrow$ 3MC activation energies $E_{\mbox{ave}}$ for about one hundred
compounds [14]. However we have no reason to believe that $E_{\mbox{ave}}$ is
a quantitative measure of the 3MLCT $\rightarrow$ 3MC barrier height, even
though it is derived from experiment and so reflects general luminescence
trends in ruthenium(II) polypyridine complexes. This is further confirmed by
comparison of $E_{\mbox{ave}}$ with the best available experimental and
theoretical values for the same barrier height [14]. It thus makes sense to
carry out a more stringent test of LI3 using barriers calculated at the same
level as the original PDOS calculations used to calculate LI3, namely gas-
phase B3LYP/6-31G+LANLDZ(Ru) (see Sec. 2 for computational details). Such
calculations are both computer-resource and human-time intensive and so we
restricted our calculations to only a few ruthenium(II) polypyridine
complexes. Several aspects of the results turned out to be rather surprising!
(Vide infra!)
The 3MLCT $\rightarrow$ 3MC process may be thought of as a chemical reaction
involving the metal $d$ orbitals and the ligand $\pi^{*}$ orbitals. Just as
frontier molecular orbital ††margin: FMOT theory (FMOT) has been successful
in explaining many chemical reactions, similar arguments were used to motivate
the LI3 orbital-based luminescence index [14]
$\mbox{LI3}=\frac{\left[\left(\epsilon_{{e}_{g}^{*}}+\epsilon_{\pi^{*}}\right)/2\right]^{2}}{\epsilon_{{e}_{g}^{*}}-\epsilon_{\pi^{*}}}\,.$
(6)
Notice how the form of the LI3 descriptor is typical of FMOT with an orbital
energy difference in the denominator. The numerator represents the square of
the off-diagonal term in the orbital hamiltonian in the Helmholtz
approximation just as in familiar semi-empirical applications of FMOT, but
neglecting the orbital overlap term. As shown in Fig. 2, this descriptor works
remarkably well for predicting $E_{\mbox{ave}}$ for the compounds considered
in the present article. It still works very well when considering a much
larger number of compounds though there is necessarily more scatter in the
corresponding graph and a small number of compounds appear far enough away
from the line that we might start to question whether they follow the same
luminescence mechanism as the other compounds [14]. This is not entirely
surprising because not all bonds are
Figure 2: Performance of LI3 for complexes of formula [RuX3]${}^{2}+$. Numbers
in parentheses designate the compounds shown in Fig. 3 which, in turn, are
named after the ligands shown in Fig. 4.
symmetry equivalent in many of these compounds. Given the difficulty of a
thorough investigation of reaction barriers, we decided to focus first on only
a small series of similar complexes for which results are reported in the
present article. These complexes are shown in Fig. 3 and the Lewis dot
structures of the corresponding ligands are shown in Fig. 4.
|
---|---
(a) N∧N = bpy (6) | (b) N∧N = 4,4’-dm-bpy (70)
|
(c) N∧N = 4,4’-dph-bpy (73) | (d) N∧N = 4,4’-DTB-bpy (74)
Figure 3: Structures of [Ru(N∧N)3]2+. Hydrogen atoms have been suppressed for
clarity. We have chosen to use the $\Delta$ stereoisomers, though similar
results are expected for the corresponding $\Lambda$ stereoisomers.
---
(a) bpy: 2,2’-bipyridine (6)
(b) 4,4’-dm-bpy: 4,4’-dimethyl-2,2’-bipyridine (70)
(c) 4,4’-dph-bpy: 4,4’-diphenyl-2,2’-bipyridine (73)
(d) 4,4’-DTB-bpy: 4,4’-di-tert-butyl-2,2’-bipyridine (74)
Figure 4: Ligand list.
This series of complexes was chosen as small and “simple” as possible to
††margin: TS minimize the problem of multiple transition states (TSs) because
locating TSs and determining their barrier heights can be difficult.
Let us look in more detail at the conventional mechanism for ruthenium(II)
photoluminescence. On the basis of experimental results and LFT group
theoretical arguments, Adamson has given a set of rules for photodissociation
rules for $O_{h}$ complexes [15]:
> Rule 1. Trans dissociation will occur for those opposing ligands with the
> weakest average ligand field strength.
> Rule 2. In the case of nonidentical ligands, the ligand of greater field
> strength aquates first.
As Rule 1 appears to be stronger than Rule 2, ruthenium(II) polypyridine
complexes have been almost universally assumed to undergo trans dissociation.
This coordinate provides a 1D cut allowing us to sketch ††margin: PEC an
approximate potential energy curve (PEC). The luminsence mechanism described
in the first paragraph is then easy to understand in terms of the diagram in
Fig. 5. Note that the 1GS
Figure 5: The diagram shows the principle potential energy curves in our
model. The abscissa corresponds to a reaction pathway involving partial
removal of a ligand while the ordinate represents the state energy. The dashed
lines indicate diabatic states whose avoided crossing leads to the energetic
barrier on the adiabatic surface between the 3MLCT and 3MC minima.
and 3MLCT geometries are represented as very similar, radiationless 1MLCT
$\rightarrow$ 3MLCT intersystem crossing is very easy because the two curves
have very similar energies and because of spin-orbit coupling due to the heavy
ruthenium atom, the 1MC and 3MC geometries are also expected to be similar but
with longer ruthenium-ligand bond lengths because of the occupation of the
antibonding $e^{*}_{g}$ orbital, there is an avoided crossing between the 3MC
and 3MLCT states, and there is easy radiationless de-activation of the 3MC
state back to the 1GS.
Such a picture is subject to experimental verification but obtaining barrier
heights experimentally requires careful rate measurements at a series of
temperatures and extraction using an appropriate model, usually assuming an
Arrhenius rate law. An accurate experimental value for the 3MLCT $\rightarrow$
3MC barrier height of 3800 cm-1 is known for the prototypical trisbipyridine
ruthenium(II) cation complex [Ru(bpy)3]2+ in acetonitrile [16] (3960 cm-1 in
propionitrile/buylronitile (4:5 v/v) is quoted in Ref. [14]), but for very few
other complexes. The problem of obtaining experimental barrier heights has
been discussed in more detail in Ref. [14] where a compromise was made in
calculating ††margin: $E_{\mbox{ave}}$ $E_{\mbox{ave}}$ so as to be able to
use data covering a large number of compounds. The value $E_{\mbox{ave}}$ =
132 cm-1 was found for [Ru(bpy)3]2+ which clearly underestimates the
previously quoted experimental barriers, but the hope was that trends in
barrier heights would be accurately captured by the experimentally-derived
$E_{\mbox{ave}}$. This hypothesis is exactly what we wish to test in the
present work.
It might seem that determining reaction paths and barrier heights might be
easier than determining them experimentally. However such complexes are
expected to have a plethera of TSs. Adamson’s trans dissociation of $O_{h}$
complexes with six identical monodentate ligands is completely consistent with
the expected Jahn-Teller distortion of an open-shell $O_{h}$ complex. However
there are three symmetry-equivalent trans dissociation pathways, each of which
should be associated with three ††margin: PES different TSs on a Mexican hat
potential energy (hyper)surface (PES) [17]. This has also been discussed in
the context of cobalt(II) imine complexes having a single type of bidentate
ligand (e.g., Fig. 7 of Ref. [18]). In addition to the TSs of each of the
three symmetry-equivalent trans dissociation products, we may also expect TSs
between the symmetry-equivalent 3MLCT geometries and between the symmetry-
equivalent 3MC geometries. In the case of three identical bidentatate ligands,
our $O_{h}$ complex has $\Lambda$ and $\Delta$ enantiomers. These also
interconvert via TSs. Interconversion between the isomers has been proposed to
happen either via Bailar or by Ray-Dutt twists [19], each of which has
associated TSs. And there is the question of whether the excited electron in
the 3MLCT is transferred equally to all three bidentate ligands or
preferentially to a single bidentate ligand? Certainly photodissociation
experiments which observe the replacement of a bpy ligand with water in
[Ru(bpy)3]2+ to form [Ru(bpy)2(H2O)2]2+ [20] seem consistent with the idea
that symmetry breaking is occurring in such a way that one of the bpy is
treated differently than the others. And this reaction also has its own TS.
All of which means that we must expect many probably close-lying TSs to
complicate our investigation.
Of course we are not the first to carryout a theoretical investigation of
3MCLT $\rightarrow$ 3MC PECs, but the number seems to still be rather small
[21, 22, 23, 24] and relatively recent. In their 2015 paper [21], Yoshikawa et
al. report the energies and geometries of 3MCLT $\rightarrow$ 3MC transition
states for nine [Ru(bpy)2(phen derivative)]2+, where “phen derivative” refers
to either (1) phen, (2) 4-phenyl-phen, (3) 4,7-diphenyl-phen, (4)
2,9-dimethyl-4,7-diphenyl-phen, (5) 2,9-dimethyl-phen,(6) 3,4,7,8-tetramethyl-
phen, (7) 5-amino-phen, (8) 5-methyl-phen, and (9) 3-phenylethynyl-phen.
However no PECs were reported. In their 2016 paper [22, 23], Zhou et al.
report 3MCLT $\rightarrow$ 3MC PECs for four derivatives of fac-
tris(1-methyl-5-phenyl-3-n-propyl-[1,2,4]-triazolyl)iridium(III). In their
2017 paper [25], Dixon et al. report 3MCLT $\rightarrow$ 3MC PECs for
[Ru(bpy)(btz)2]2+. In their 2017 paper [24], Sun et al. report 3MCLT
$\rightarrow$ 3MC PECs for [Ru(bpy)3]2+ and for [Ru(mphen)3]2+. Finally, in
their 2018 papers [26, 25], Soupart et al. report the 3MCLT $\rightarrow$ 3MC
potential energy curves for trans [25] and for cis [26] dissociation. Fumanal
et al. report transition states for [Mn(im)(CO)3(phen)]+ but without detailed
potential energy curves [27].
Some of the most extensive theoretical work has been done studying
[Ru(bpy)3]2+ [28, 29, 30, 25, 26] where it is clearly seen that the PEC of
Fig. 5 is an oversimplification which sould be replaced with a possibly
complex set of competing processes. To begin with, the 3MLCT state of
[Ru(bpy)3]2+ has more than one minimum, including a minimum with $D_{3}$
symmetry where the ruthenium electron has been transferred equally to the
three ligands and a minimum with $C_{3v}$ symmetry where the ruthenium
electron has been transferred to a single bpy ligand [28]. The same reference
reports a 3MC in order to remove the spatial degeneracy of the 3MLCT state.
Spin-orbit effects can explain radiationless relaxation to the ground state
via S${}_{0}/^{3}$MC mixing as in shown in Fig. 5 [29]. However LFT
considerations [15] suggest that cis dissociation might also occur. Indeed
theoretical calculations show that cis triplet dissociation has a comparable
but higher barrier than does trans triplet dissociation [30, 25]. The trans
3MCLT $\rightarrow$ 3MC pathway has been mapped out in gas phase and solvent
for [Ru(bpy)3]2+ [26]. While the overall conclusion is that complex processes
are going on, the dominant one still seems to be trans dissociation. We
emphasize that we have consulted with the authors of Ref. [26] and use the
same techniques, but have pushed the methodology a little further in the
direction of answering our driving question about LI3 and the energetics of
trans dissociation.
Although the present work is restricted to a small family of closely related
molecules, we have nevertheless realized that great care was needed in our
calculations because something strange seemed to be happening. Complex 6
([Ru(bpy)3]2+) has $N=61$ atoms and therefore $F=3N-6=177$ internal degrees of
freedom. Even though some of these degrees of freedom are fairly rigid, this
still leaves a lot of possibilities for processes to occur which are different
from those initially imagined! So we have made a very careful effort to be
systematic:
1. 1.
Our first step was to make 2D scans of the PES by fixing two trans bond
lengths at different fixed values and minimizing all the other degrees of
freedom. This gives us our first crude view of the reaction mechanism, but
scans are well-known to be misleading when it comes to finding TSs and
reaction paths (Ref. [31] gives a particularly clear explanation of why this
is so.) It does, however, provide us with first guesses that are subsequently
used to optimize the 3MC and 3MCLT geometries for trans dissociation.
††margin: NEB
2. 2.
We then used the nudged elastic band (NEB) method [32, 33] to start from an
interpolated path for trans dissociation and find a first approximation to the
reaction path. This is important as only ††margin: MEP the NEB, and not the
2D scan, was able to give us a maximum energy point (MEP) close enough to a TS
to be useful.
3. 3.
The MEP geometry still has to be optimized using a TS optimizer. Special
methods must then be used to locate the TS and to determine the reaction
pathways [34, 35]. These are typically computationally resource intensive both
because they require multiple single point quantum chemical calculations but
also because of the need to calculate first and second derivatives of the PESs
[34, 35, 36, 37, 38, 39, 40].
4. 4.
The optimized TS still has to be confirmed as linking the 3MC and 3MLCT minima
and this is done by following the intrinsic reaction ††margin: IRC coordinate
(IRC) in both directions along the direction of the imaginary vibrational
coordinate to verify that the TS does indeed correspond to the desired
reaction path.
It is also possible to follow the change in character of the molecule along
the IRC by using Mulliken population analysis to follow the spin- and charge-
density along the reaction path. Given the surprising nature and complexity of
what we found, we prefer to give a schematic of our main conclusions and then
later justify them. This schematic PES is shown in Fig. 6. The details of the
notation will be explained later. Briefly the initial 3MLCT state is quite
symmetric so that it is difficult to distinguish three symmetry equivalent
Jahn-Teller distorted geometries. The IRC is consistent with an electron being
transferred to a particular bidentate ligand whose two bonds to the central
ruthenium atom elongate simultaneously so as to maintain $C_{2}$ symmetry at
the TS and beyond. The reaction path is then on a ridge and can bifurcate in
either of two directions to find one of two symmetry-equivalent minima. This
accounts for small indications here and there in the literature (and our own
experience) that, and explains why, the IRC is unusually difficult to
calculate for this reaction.
(a) |
---|---
(b) |
|
Figure 6: Schematic PES representing our best understanding of the 3MLCT
$\rightarrow$ 3MC reaction path: A represents three nearly identical 3MLCT
minima, the Bi,j represent three TS which lead to a ridge descending to three
TSs (Ci,j) representing the barrier to interconversion between symmetry-
equivalent 3MC minima (Di and Dj). Somewhere along the ridge between Bi,j and
Ci,j is a valley-ridge inflection (VLI) point defined mathematically as the
point at which the eigenvalue of the Hessian matrix is zero and the gradient
vector is perpendicular to the corresponding eigenvector) [41, 42]. The VLI
may be thought of as (close to) where the 3MLCT $\rightarrow$ 3MC reaction
path bifurcates [part (a) of this figure].
The rest of this paper is organized as follows: Our computational methods are
described in the next section. Section 3 justifies the general picture seen in
Fig. 6 and then goes on to compare calculated barrier and reaction energies
with LI3. Section 4 summarizes our conclusions. Additional information is
available as Supplementary Information ††margin: SI (SI) (Sec. Supplementary
Information).
## 2 Computational Details
The four-step procedure that was used in the present work has already been
described in the introduction. This section goes into greater technical
detail.
We used two programs to carryout this procedure, namely version 09 revision
D.01 of the Gaussian code [43] and version 5.0.2 of the ORCA code [44]. All
calculations were gas phase calculations with the same basis sets and the same
functionals.
††margin: 6-31G
The basis consisted of the valence double-$\zeta$ quality 6-31G basis set for
H [45], N [46], and C [46]. The Ru atom requires an effective core potential
(ECP) in order to include relativistic effects. We used ††margin: LANL2DZ
ECP the LANL2DZ ECP and the corresponding double-$\zeta$ basis set [47].
effective core potential (ECP) [47] was used to characterize Ru. H, N and C
atoms were described by 6-31G basis set.
††margin: B3LYP
The density functional approximation used is the B3LYP (Becke exchange, 3
parameter, Lee-Yang-Parr correlation) functional described in Ref. [48] and
initially programmed in Gaussian. Unfortunately this functional has a history
of confusion which can ††margin: VWN be traced back to the implementation of
the Vosko-Wilke-Nusair (VWN) ††margin: LDA parameterization of the local
density approximation (LDA) in Gaussian. Their parameterization of Ceperley
and Alder’s quantum Monte Carlo results (found in the caption of Fig. 5 of
Ref. [49]) should have been used as this is what had been referred to as VWN
in works by all previous authors. However the Vosko-Wilke-Nusair
parameterization ††margin: RPA of a random phase approximation (RPA) result
for the electron gas result was called VWN in Gaussian. Instead of correcting
the error, Gaussian refers to the original parameterization of the Ceperley-
Alder ††margin: VWN5 results as VWN5. The implementation of B3LYP in Gaussian
uses VWN for the LDA. However B3LYP in ORCA uses the orginal VWN (i.e.,
††margin: B3LYP/G VWN5) in its implementation of B3LYP and requires the
specification of B3LYP/G in order to get the original B3LYP. We used the
original B3LYP in our calculations througout this paper.
While there may be some value in using other functionals and better basis sets
in implicit solvent, such changes would defeat our goal of calculating
barriers to compare with the LI3 indices already calculated at the
B3LYP/6-31G+LANLDZ(Ru) level used in Ref. [14]. So we decided to stick with
the original level of calculation.
In Gaussian, all calculations were carried out without any symmetry
contraints, using an ultrafine grid, and the spin-unrestricted formalism.
††margin: SCF
DIIS Self-consistent field (SCF) convergience was achieved using the direct
inversion of iterative subspace (DIIS) extrapolation algorithm. Frequency
calculations were performed on the optimized geometries so as to check whether
they were the true minima (with no imaginary frequencies) or a transition
state (with one imaginary frequency). For all the calculations, (pop=full) was
set to full so as to extract all the information. In ORCA, keywords used
included NORI (no approximation is used), TightSCF, TightOpt, SlowConv,
NumFreq and an ultra-fine grid. We explicitly confirmed that Gaussian and Orca
gave the same results for the same basis sets and functionals in single-point
calculations.
Figure 7: The $m$t$n$ labeling system for the N atoms around the central Ru.
We also need some system of N atom numbering in order to be able to discuss
the photochemical reaction pathways. One possibility is to just use the
numbers shown in Fig. 3. However we found that we could be a little less
arbitrary by using a system where each of the N∧N ligands with the numbers
1,2, and 3. Each of the N in a given N∧N is ††margin: $n$t$m$ trans to a
different N∧N, so we may label as $n$t$m$ the $N$ in the $n$th N∧N that is
trans to the $m$th N∧N. This is a great improvement over using the atom
numbers in Fig. 3, but still is complicated by the fact that there are still
six different ways to permute the three numbers (1,2,3). The possible
alternative numberings are shown in Table 1. With a little practice and an
occasional sketch (Fig. 7), it is relatively easy to master. Usually it is the
2t1-1t2 or the 3t1-1t3 trans stretch which interests us in our calculations.
We will now take a step-by-step look at how we carried out our calculations.
Many of these steps could have been carried out using either the Gaussian or
the ORCA codes, but we chose to allow different people use the codes with
which they were most familiar once we had assured ourselves that the two codes
did indeed give the same results.
Complex | Ru-N Bond | (1,2,3) | (1,3,2) | (2,1,3) | (2,3,1) | (3,1,2) | (3,2,1)
---|---|---|---|---|---|---|---
[Ru(bpy)3]2+ | 1-2 | 3t2 | 2t3 | 3t1 | 1t3 | 2t1 | 1t2
(6) | 1-3 | 3t1 | 2t1 | 3t2 | 1t2 | 2t3 | 1t3
| 1-22 | 2t1 | 3t1 | 1t2 | 3t2 | 1t3 | 2t3
| 1-23 | 2t3 | 3t2 | 1t3 | 3t1 | 1t2 | 2t1
| 1-42 | 1t3 | 1t2 | 2t3 | 2t1 | 3t2 | 3t1
| 1-43 | 1t2 | 1t3 | 2t1 | 2t3 | 3t1 | 3t2
[Ru(4,4’-dm-bpy)3]2+ | 1-2 | 3t2 | 2t3 | 3t1 | 1t3 | 2t1 | 1t2
(70) | 1-3 | 2t1 | 3t1 | 1t2 | 3t2 | 1t3 | 2t3
| 1-4 | 1t3 | 1t2 | 2t3 | 2t1 | 3t1 | 3t1
| 1-29 | 3t1 | 2t1 | 3t2 | 3t2 | 2t3 | 1t3
| 1-30 | 1t2 | 1t3 | 2t1 | 2t3 | 2t1 | 3t2
| 1-31 | 2t3 | 1t3 | 1t3 | 3t1 | 1t2 | 2t1
[Ru(4,4’-dph-bpy)3]2+ | 1-2 | 1t2 | 1t3 | 2t1 | 2t3 | 3t1 | 3t2
(73) | 1-3 | 1t3 | 1t2 | 2t3 | 2t1 | 3t2 | 3t1
| 1-4 | 3t2 | 2t3 | 3t1 | 1t3 | 2t1 | 1t2
| 1-29 | 2t1 | 3t1 | 1t2 | 3t2 | 1t3 | 2t3
| 1-39 | 2t3 | 3t2 | 1t3 | 3t1 | 2t1 | 2t1
| 1-41 | 3t1 | 2t1 | 3t2 | 1t2 | 2t3 | 1t3
[Ru(4,4’-DTB-bpy)3]2+ | 1-2 | 3t1 | 2t1 | 3t2 | 1t2 | 2t3 | 1t3
(74) | 1-3 | 1t2 | 1t3 | 2t1 | 2t3 | 3t1 | 3t2
| 1-4 | 2t3 | 3t2 | 1t3 | 3t1 | 2t1 | 2t1
| 1-68 | 3t2 | 2t3 | 3t1 | 1t3 | 2t1 | 1t2
| 1-69 | 2t1 | 3t1 | 1t2 | 3t2 | 1t3 | 2t3
| 1-70 | 1t3 | 1t2 | 2t3 | 2t1 | 3t2 | 3t1
Table 1: The six different ways to relabel N atoms as in the $n$th N∧N unit
trans to the $m$th N∧N unit. With a little practice, this notation becomes
fairly easy to use.
#### Step 1
consists of making a 2D scan of the PES. This was done using Gaussian with the
objective of finding a suitable 3MLCT minimum energy geometry. This was
achieved by first doing a vertical excitation ††margin: FC from the 1GS
minimum to obtain the Franck-Condon (FC) 3MLCT. This FC 3MLCT geometry was
then optimized to get the 3MLCT local minimum. (It is not a global minimum!)
This minimum was confirmed by checking for the absence of imaginary
frequencies. We then made a simultaneous 2D-scan by independently stretching
two trans Ru-N bonds and made a contour plot. This strategy was adopted from
[28] which suggested that the ground state molecular orbitals are of the
$d_{z^{2}}$-type and the population of this $d_{z^{2}}$-like orbital in the
triplet excited state would thus result in the bond elongations (i.e., the
$e^{*}_{g}$ LFT orbital is antibonding). Of course all three trans distortions
should give the same reaction path. Two axial bonds that were trans to each
other were independently elongated from the initial bond length in the
optimized geometry of the 3MLCT state to 2.500 Å in steps of 0.002 Å, letting
all other geometric parameters relax to give the lowest energy with only the
two trans Ru-N distances constrained. The resulting energies had to be
resorted before plotting because Gaussian uses a boustrophedon (as an ox plows
a field) order when it does a 2D-scan in order to minimize changes in
geometries during the scan. The lowest energy point with long trans bonds is a
first guess at the 3MC geometry. This was then optimized once again to obtain
the minimum energy geometry and the minima was confirmed by checking for the
absence of any imaginary frequencies. Mulliken spin density analysis was used
to confirm of the nature of the excited-state {0.9 excess spin-up electron on
Ru [Eq. (2)] for the 3MLCT state and 1.8 excess spin-up electron on Ru for a
3MC state [Eq. (3)]}.
#### Step 2
Having found our 3MLCT and 3MC end points, we then carried out an NEB
calculation [32, 33] with ORCA to find a best first guess at the IRC.
#### Step 3
The MEP from the NEB calculation in ORCA was further optimized using ORCA’s
NEB-TS algorithm. The resultant TS was reoptimized with a different algorithm
with Gaussian.
#### Step 4
Finally Gaussian was used to follow the IRC from the calculated TS and insure
that it is indeed connected to the originally input 3MLCT and 3MC input
geometries.
## 3 Results and Discussion
We are now prepared to investigate the question initially posed, namely how
well the orbital-based luminescence index LI3 predicts the 3MLCT $\rightarrow$
3MC barrier height. A first subsection will focus on the mechanism and
energetics of the 3MLCT $\rightarrow$ 3MC reaction. The method of calculation
of the barrier height was explained earlier as was the reason for the choice
of the level of our calculations to be consistent with our earlier
calculations of LI3. While we will focus on results for complex 70 (the
results for the other three complexes are in the SI), we also want to reveal
the correctness of the schematic PES of Fig. 6 as we believe this to be a new
and novel contribution to the literature of this well-studied and important
family of molecules. A second, and final, subsection will then examine the
question of what we can say about the predictive value of LI3.
### 3.1 3MLCT $\rightarrow$ 3MC Mechanism and Energetics
In the first instance, we focus on complex 70 and show the results for the
various steps of our reaction mechanism investigation.
Figure 8: Contour plot of 2D-Scan of Ru1-N3 (2t1) and Ru1-N30 (1t2) for
complex 70.
#### Step 1
The calculated contour plot is shown in Fig. 8. Inspection of a model shows
that this plot should be symmetric about the 45∘ line representing interchange
between the 2t1 and 1t2 N atoms. It is approximately but not exactly symmetric
because of the existance of multiple internal degrees of freedom in this
system which may be trapped in different local minima due to the boustrophedon
nature of Gaussian’s 2D scan. This phenomenon is well understood [31].
Nevertheless the contour plot is nearly symmetric. It allows us to see that
there is an energetically high-lying 3MLCT minimum and an energetically lower-
lying 3MC minimum, separated by a barrier. Importantly, this allows us to find
a first guess of the 3MC structure which is subsequently optimized. Key
optimized Ru-N bond lengths are collected in Table 2.
Ru-N | Ground State | 3MLCT | TS | 3MC
---|---|---|---|---
Bond | Gaussian | Orca | Gaussian | Orca | Gaussian | Orca | Gaussian | Orca
[Ru(bpy)3]2+ (6)
3t2 | 2.110 (s) | 2.111 (s) | 2.096 (s) | 2.098 (s) | 2.101 (s) | 2.100 (s) | 2.115 (s) | 2.115 (s)
3t1 | 2.110 (s) | 2.111 (s) | 2.097 (s) | 2.098 (s) | 2.102 (s) | 2.102 (s) | 2.112 (s) | 2.115 (s)
2t1 | 2.110 (s) | 2.110 (s) | 2.118 (s) | 2.119 (s) | 2.211 (m) | 2.214 (m) | 2.463 (l) | 2.464 (l)
2t3 | 2.110 (s) | 2.110 (s) | 2.118 (s) | 2.119 (s) | 2.215 (m) | 2.212 (m) | 2.176 (m) | 2.170 (m)
1t3 | 2.110 (s) | 2.110 (s) | 2.097 (s) | 2.099 (s) | 2.103 (s) | 2.102 (s) | 2.164 (m) | 2.170 (m)
1t2 | 2.110 (s) | 2.110 (s) | 2.095 (s) | 2.097 (s) | 2.098 (s) | 2.101 (s) | 2.454 (l) | 2.464 (l)
[Ru(4,4’-dm-bpy)3]2+ (70)
3t2 | 2.109 (s) | 2.109 (s) | 2.097 (s) | 2.097 (s) | 2.098 (s) | 2.096 (s) | 2.129 (s) | 2.112 (s)
2t1 | 2.109 (s) | 2.109 (s) | 2.097 (s) | 2.097 (s) | 2.227 (m) | 2.225 (m) | 2.437 (l) | 2.465 (l)
1t3 | 2.109 (s) | 2.109 (s) | 2.097 (s) | 2.097 (s) | 2.103 (s) | 2.103 (s) | 2.139 (m) | 2.186 (m)
3t1 | 2.109 (s) | 2.109 (s) | 2.097 (s) | 2.097 (s) | 2.103 (s) | 2.104 (s) | 2.104 (s) | 2.112 (s)
1t2 | 2.109 (s) | 2.109 (s) | 2.097 (s) | 2.097 (s) | 2.098 (s) | 2.097 (s) | 2.387 (l) | 2.465 (l)
2t3 | 2.109 (s) | 2.109 (s) | 2.097 (s) | 2.097 (s) | 2.226 (m) | 2.225 (m) | 2.222 (m) | 2.168 (m)
[Ru(4,4’-dph-bpy)3]2+ (73)
1t2 | 2.104 (s) | 2.104 | 2.114 (s) | 2.094 (s) | 2.251 (m) | 2.251 (m) | 2.165 (m) | 2.459 (l)
1t3 | 2.104 (s) | 2.104 | 2.115 (s) | 2.091 (s) | 2.258 (m) | 2.258 (m) | 2.458 (l) | 2.165 (m)
3t2 | 2.104 (s) | 2.104 | 2.091 (s) | 2.098 (s) | 2.103 (s) | 2.103 (s) | 2.165 (m) | 2.108 (s)
2t1 | 2.104 (s) | 2.104 | 2.087 (s) | 2.094 (s) | 2.099 (s) | 2.099 (s) | 2.108 (s) | 2.457 (l)
2t3 | 2.104 (s) | 2.104 | 2.094 (s) | 2.091 (s) | 2.106 (s) | 2.106 (s) | 2.108 (s) | 2.165 (m)
3t1 | 2.104 (s) | 2.104 | 2.086 (s) | 2.098 (s) | 2.101 (s) | 2.101 (s) | 2.456 (l) | 2.108 (s)
[Ru(4,4’-DTB-bpy)3]2+ (74)
3t1 | 2.107 (s) | 2.107 | 2.130 (s) | 2.130 (s) | 2.229 (m) | 2.230 (m) | 2.357 (l) | 2.465 (l)
1t2 | 2.107 (s) | 2.107 | 2.092 (s) | 2.092 (s) | 2.102 (s) | 2.102 (s) | 2.106 (s) | 2.166 (s)
2t3 | 2.107 (s) | 2.107 | 2.075 (s) | 2.073 (s) | 2.090 (s) | 2.090 (s) | 2.210 (m) | 2.110 (m)
3t2 | 2.107 (s) | 2.107 | 2.130 (s) | 2.130 (s) | 2.231 (m) | 2.231 (m) | 2.357 (l) | 2.167 (m)
2t1 | 2.107 (s) | 2.107 | 2.092 (s) | 2.092 (s) | 2.102 (s) | 2.102 (s) | 2.105 (s) | 2.109 (s)
1t3 | 2.107 (s) | 2.107 | 2.075 (s) | 2.073 (s) | 2.090 (s) | 2.090 (s) | 2.210 (m) | 2.459 (l)
Table 2: Key Ru-N bond lengths (Å) for different compounds as computed with
Gaussian and with ORCA. In parentheses: “s” stands for “short” ($\sim$ 2.0 Å),
“m” for “medium length” ($\sim$ 2.1 Å), and “l” for “long” ($\sim$ 2.4 Å).
Figure 9: NEB-TS optimization for complex 70. The initial guess IRC is in
black, the final converged IRC is in red, and the intermediate IRCs obtained
during the NEB procedure are shown in grey.
#### Step 2
Figure 9 shows the iterative convergence of the NEB IRC from its initial guess
to its final result. This is an expensive calculation involving the
simultaneous displacement of 24 points (the end points are fixed) in a
parallelized calculation. It is clear that its convergence is approximate.
Figure 10: Plot of the NEB IRC for complex 70. The red circles are on top of
the black squares. The blue point is the energy of the optimized TS.
#### Step 3
Figure 10 shows the IRC obtained from the NEB calculation. In this particular
case, the MEP optimizes to a point which gives a considerably lower barrier.
The geometries of the optimized TSs obtained with Gaussian and Orca may
compared in Table 2. Agreement is to within 0.002 Å.
Figure 11: Plot of the Gaussian total energy in Hartree against the IRC for
complex 70.
#### Step 4
Figure 11 shows the more accurate IRC obtained using Gaussian. Note that this
figure evolves from left to right in the 3MLCT $\rightarrow$ 3MC direction
while Fig 10 evolves from left to right in the 3MC $\rightarrow$ 3MLCT
direction. These calculations confirmed that the TS is connected by the IRC to
the originally input 3MC and 3MLCT structures.
Figure 12: Plot of bond lengths along the IRC for complex 70 as calculated
with Orca. Figure 13: Plot of bond lengths along the IRC for complex 70 as
calculated with Gaussian.
#### Reaction Mechanism
We have also looked at spin densities, but we have found the evolution of Ru-N
bond lengths along the IRC to be the most valuable indicator of how the
reaction mechanism proceeds. Figures 12 and 13 show how these bond lengths
vary along the IRC during the photoreaction. The NEB calculation (Fig. 12)
from left to right follows the 3MC $\rightarrow$ 3MLCT reaction. It is less
accurate than the IRC calculation carried out with Gaussian (Fig. 13) which
from left to right follows the 3MLCT $\rightarrow$ 3MC direction. Only the
Gaussian IRC is accurate enough to understand what is going on. From right to
left: The initial 3MLCT Ru-N bond lengths are nearly equal until the IRC is
reduced to about 3, at which point the 2t1 and 2t3 bond lengths remain equal
but increase while the other bond lengths continue to stay about the same.
This means that ligand 2 is moving away from the metal in a way that might be
expected if the $e_{g}^{*}$ electron on Ru had been transferred to a $\pi^{*}$
orbital on ligand 2. This continues beyond the TS (at IRC coordinate 0 in Fig.
13) until we get to an IRC coordinate of about -5. So, up to this point, one
of the ligands is coming off by elongating both of its Ru-N bonds equally.
When the IRC is less than -5, there is a sudden change in the bond lengths, so
that the two trans Ru-N pairs (2t1 and 1t2) become longest while two other
pairs (3t2 and 2t2) reduce to a medium bond length. Two Ru-N pairs (3t2 and
3t1) remain short. Such behavior is consistent with the PES scheme shown in
Fig. 6. Such a PES should show a bifurcation such that different trans bond
lengths could lengthen in different calculations. This is seen in Table 2 for
complexes 73 and 74, confirming our overall interpretation of the trans
dissociation reaction mechanism.
### 3.2 What Does LI3 Predict?
We established in our earlier work that the orbital-based luminescence index
LI3 correlates well with $E_{\mbox{ave}}$ values of the 3MLCT $\rightarrow$
3MC barrier derived from experimental luminescence data obtained at room
temperature and at the boiling point of liquid nitrogen [14]. This has been
illustrated in Fig. 2. The question is not whether LI3 works (it does!) but
how it is able to do this. We will address this question more fully in this
subsection.
Complex | $\tau$/$\mu$s (77K) | $\tau$/$\mu$s (RT) | $\Delta E_{\mbox{ave}}$/cm-1
---|---|---|---
(6) | 5.23 | 0.845 | 132.
(70) | 4.6 | 0.525 | 157.
(73) | 4.79 | 1.31 | 94.
(74) | 9.93 | 0.673 | 194.
Table 3: Luminescence lifetimes at both room temperature (RT) and liquid
nitrogen temperature (77K) along with the empirically-derived $\Delta
E_{\mbox{ave}}$ as reported in Tables 10 and 11 of Ref. [14].
It is perhaps appropriate to recall that the fundamental 3MLCT $\rightarrow$
3MC reaction involves an electron transfer. One famous theory of electron
transfer is Marcus theory [50] and Marcus theory has indeed been used in
connection with studying electrogenerated chemiluminescence in
tris(polypyridine) ruthenium complexes [51]. Marcus theory is valid when the
Massey parameter,
$\Gamma=\frac{2\pi|H_{1,2}|^{2}}{|d(E_{1}-E_{2})/dt|}\,,$ (7)
is small, where $E_{1}$ and $E_{2}$ are the energies of the two diabatic
curves (3MLCT and 3MC here) and $H_{1,2}$ is the adiabatic coupling.
Diagonalizing the corresponding $2\times 2$ matrix gives the adiabatic curves,
corresponding to the PESs treated in the present work. Marcus theory assumes
rapid crossing of the avoided crossing region in the weak coupling case (i.e.,
when $E_{1}\approx E_{2}$). A detailed derivation [52] shows that Marcus
theory assumes rapid oscillation between the reactants and products. The final
formula for the reaction rate constant, expressed in terms of Gibb’s free
energies $G$ is,
$k=|G_{R,P}|^{2}\sqrt{\frac{\pi}{\lambda
k_{B}T}}e^{-\frac{\left(\lambda+\Delta G^{0}\right)^{2}}{4\lambda k_{B}T}}\,,$
(8)
where the relaxation parameter $\lambda$ is the difference between the product
diabatic curve at the reactant minimum and the same curve at the product
minimum. In contrast, the strong coupling limit combined with a no-recrossing
rule, leads to Eyring’s transition state theory [53] and the equally well-
known formula,
$k=\frac{\kappa k_{B}T}{2\pi}e^{-{\frac{\Delta G^{\ddagger}}{RT}}}\,,$ (9)
which explains the famous Arrhenius equation,
$k=Ae^{-\frac{E_{a}}{RT}}\,,$ (10)
familiar from first-year University chemistry courses. Marcus theory and
Eyring transition state theory are, in fact, just two limiting cases of the
more general theory of charge transfer reactions [54]. We have not defined
every parameter in these well-known equations because we do not need them in
the present work and because we are sure that the interested reader can easily
find suitable references to fill in any missing details. But it is evident
that temperature $T$ is in these equations which may at first seem a little
strange for a photochemical reaction. A justification is that the luminescence
lifetimes of our complexes are on the order of microseconds (Table 3) whereas
a typical vibrational time is on the order of picoseconds, which suggests that
some degree of thermodynamic equilibration may occur, even if it is not
necessarily complete. In turn, this explains why the usual analysis of
barriers for these complexes is carried out in terms of the Arrhenius equation
with some additional terms involving equilibrium between near-lying electronic
states and a melting term (see Ref. [14] and references therein). In our case,
we are only interested in the energetics of the 3MLCT $\rightarrow$ 3MC
reaction and the extent to which it correlates with LI3. We will assume the
Arrhenius picture with the same freqency factor $A$ for all our complexes so
that only the activation energy $E_{a}$ is important. Of course, we are also
limited by the fact that we are using the same level of calculation of the
PESs that we used to calculate our LI3s. We have to accept in advance that
this level of calculation may or may not be adequate for directly explaining
the experimental observations, even if we expect it to shed light upon our
problem.
Energy\Complex | 6 | 70 | 73 | 74
---|---|---|---|---
LI3 | 16.78 eV | 13.78 eV | 9.68 eV | 11.97 eV
3MLCT | -1579.29913 Ha | -1815.18022 Ha | -2965.37532 Ha | -2522.64515 Ha
3MC | -1579.30474 Ha | -1815.18504 Ha | -2965.37816 Ha | -2522.64934 Ha
3MEP | -1579.29661 Ha | -1815.17613 Ha | -2965.37120 Ha | -2522.64201 Ha
3TS | -1579.29769 Ha | -1815.17880 Ha | -2965.37392 Ha | -2522.64328 Ha
3MEP-3MLCT | 0.00252 Ha | 0.00409 Ha | 0.00412 Ha | 0.00314 Ha
| 553.08 cm-1 | 897.65 cm-1 | 904.24 cm-1 | 689.15 cm-1
3TS-3MLCT | 0.00144 Ha | 0.00142 Ha | 0.0014 Ha | 0.00187 Ha
| 316.04 cm-1 | 311.65 cm-1 | 307.26 cm-1 | 410.42 cm-1
3MEP-3MC | 0.00813 Ha | 0.00891 Ha | 0.00696 Ha | 0.00733 Ha
| 1784.33 cm-1 | 1955.52 cm-1 | 1527.54 cm-1 | 1608.75 cm-1
3TS-3MC | 0.00705 Ha | 0.00624 Ha | 0.00424 Ha | 0.00606 Ha
| 1547.52 cm-1 | 1369.52 cm-1 | 930.57 cm-1 | 1330.02 cm-1
3MC-3MLCT | 0.005610 Ha | 0.00482 Ha | 0.00284 Ha | 0.004190 Ha
| 1231.25 cm-1 | 1057.87 cm-1 | 623.31 cm-1 | 919.60 cm-1
Table 4: Energies for 3MLCT, 3MC and TS of 4 compounds obtained from NEB
calculations as implemented in the ORCA code.
Table 4 gives energies for the 3MLCT states, 3MC states, and TSs for the four
compounds obtained from NEB calculations as implemented in the Orca code.
Values in parenthesis are in cm-1. Except for the 3MEP values which come from
NEB calculations, the same numbers are obtained with Gaussian. It is
immediately clear that the 3MLCT $\rightarrow$ 3MC barrier (3MEP-3MLCT or, for
more accuracy, 3TS-3MC) is really very small and on the order of about 300-400
cm-1 (0.86-1.14 kcal/mol). Chemical accuracy is usually cited as 1 kcal/mol
(349.757 cm-1) and is very hard to achieve even with the best quantum chemical
methods.
Figure 14: Graph of the 3MLCT-3MC energy difference versus LI3.
Figure 14 shows that the 3MLCT-3MC energy difference correlates very well with
L3—in fact even better than the correlation of $\Delta E_{\mbox{ave}}$ with
LI3 (Fig. 2). This is consistent with the origin of the FMOT origin of the
orbital-based luminescence index LI3.
Figure 15: Explanation of the Bell-Evans-Polanyi model [55, 56] in the present
context: RE, reactant PEC, $E_{R}(x)=(k_{R}/2)(x-x_{R})^{2}+E_{R}^{0}$; LRE,
linearized RE, $E_{R}(x)=m_{R}(x-x_{R})+E_{R}^{0}$; PE, product PEC,
$E_{P}(x)=(k_{P}/2)(x-x_{P})^{2}+E_{P}^{0}$; LPE, linearized PE,
$E_{P}(x)=m_{P}(x-x_{P})+E_{P}^{0}$. Here $x$ is the reaction coordinate.
versus
There is an argument that the 3TS-3MLCT energy barrier should vary linearly
with the 3MLCT-3MC energy difference,
$E[\mbox{${}^{3}$TS}]-E[\mbox{${}^{3}$MLCT}]=\alpha\left(E[\mbox{${}^{3}$MC}]-E[\mbox{${}^{3}$MLCT}]\right)+E_{0}\,,$
(11)
and hence that the 3TS-3MLCT energy barrier should also correlate quite nicely
with L3. This can be understood in terms of two parabollic intersecting
diabatic curves as shown in Fig. 15. The intersection point
$(x_{a},E_{a}+E_{R}^{0})$ defines the activation energy ($E_{a}$). The
corresponding reaction coordinate may be found by solving the equation,
$x_{a}=\frac{x_{P}+x_{R}}{2}+\frac{\Delta E^{0}+\left(\frac{\Delta
k}{2}\right)(x_{a}-x_{R})^{2}}{k_{P}(x_{P}-x_{R})}\,,$ (12)
where $\Delta E^{0}=E_{P}^{0}-E_{R}^{0}$ and $\Delta k=k_{P}-k_{R}$. Equation
(12) may be solved for $x_{a}$ in many different ways, but taking $\Delta k=0$
as an initial guess and then interating leads to rapid convergence in the case
we tried. Back in 1936, Bell, Evans, and Polanyi [55, 56] presented an
argument which is easily used in the present context. It requires
linearlization of the parabolas (dotted lines in Fig. 15) and then solving,
which gives,
$E_{a}=\underbrace{\frac{m_{R}}{m_{R}-m_{P}}}_{\alpha}\Delta
E^{0}+\underbrace{E_{R}^{0}+\frac{m_{R}m_{P}}{m_{P}-m_{R}}(x_{P}-x_{R})}_{E_{0}}\,.$
(13)
Identification of the reactant as the 3MLCT state and the product as the 3MC
state, then gives Eq. (11). Solving with the full parabolas (which is a step
in the derivation of Marcus theory) would lead to an additional quadratic
term, which we will ignore here.
But this derivation was important here because it underlines that we need a
PEC that resembles two intersecting parabolas, which is evidently not the case
in Fig. 11 because of the plateau in the IRC curve caused by the presence of a
bifurcation. For this reason, we do not expect the 3TS-3MLCT energy to follow
the Bell-Evans-Polanyi postulate, even approximately. However the first guess
IRC of the NEB does not go through the bifurcation region and so in in
approximate agreement with the Bell-Evans-Polanyi postulate as shown in Fig.
16. As might be expected, after a little reflection, the slope of the first
guess NEB MEPs opposite would be opposite to that seen in Fig. 2 for
$E_{\mbox{ave}}$. Also, as previously mentioned our best results (namely
3TS-3MLCT) are actually essentially constant.
Figure 16: 3MLCT $\rightarrow$ 3MC barrier heights: black square, accurate TS;
black diamond, converged NEB MEP; black downward facing triangle, initial NEB
guess MEP.
We are now left with a final conundrum. All our calculations for the four
complexes studied in this article suggest that trans dissociation kinetics is
actually unable to explain the observed correlation between the
experimentally-derived $E_{\mbox{ave}}$ and LI3, because our theoretically-
calculated 3TS-3MLCT is complex-independent to within the expected accuracy of
our methodology. Nevertheless LI3 is positively correlated with the 3MLCT-3MC
energy difference as expected from FMOT and this leads to a strong correlation
between $E_{\mbox{ave}}$ and the 3MLCT-3MC energy difference $\Delta E^{0}$.
Why? Could it be that our experimentally-derived $E_{\mbox{ave}}$ reflects
something other than kinetics? Afterall, the measured luminescence lifetimes
are orders of magnitude longer than typical vibrational lifetimes. This might
allow some sort of quasiequilibrium to take place. Indeed, our observations up
to this point are more consistent with the hypothesis that 3MLCT states are
being removed via a fast 3MLCT = 3MC equilibrium process whose equilibrium
constant is roughly,
$\frac{[\mbox{${}^{3}$MC}]}{[\mbox{${}^{3}$MLCT}]}=K_{eq}=e^{-\Delta
G^{0}/RT}\propto e^{-\Delta E^{0}/RT}\,.$ (14)
Hence, all other things being equal, a larger 3MLCT-3MC means a more negative
value of $\Delta E^{0}$ and hence a larger ratio of 3MC product in comparison
to the 3MLCT reactant, effectively removing the luminescent 3MLCT state in
favor of the nonluminescent 3MC state and leading to shorter observed
luminescence lifetimes which were previously interpretted in terms of larger
values of $E_{\mbox{ave}}$. Of course, we cannot rule out the presence of
other competing pathways leading to different products and transition states
and hence the possibility of completing kinetics and quasiequilibria.
## 4 Conclusion
This may be regarded as the third in a series of articles seeking an orbital-
based explanation of luminescence life-times in ruthenium(II) polypyridine
complexes. Article I [13] introduced the use of the partial density of states
(PDOS) for the extraction of a ligand-field theory (LFT) like picture from
density-functional theory (DFT) calculations. Direct visualization of
molecular orbitals (MOs) is insufficient for this purpose because the
$e_{g}^{*}$ metal MOs mix heavily with ligand MOs. However the PDOS analysis
allows the assignment of an energy to these orbitals in a quantitative, albeit
fuzzy and both basis set and functional dependent, fashion. This allowed the
investigation of on the order of 100 molecules in Article II [14] to develop
an orbital-based luminescence index, called LI3, which correlated strongly
with an average “activation energy” $E_{\mbox{ave}}$ calculated from
experimental luminescence lifetimes measured in solution at room temperature
and liquid nitrogen temperature. The work in Articles I and II is based upon
the widely assumed 3MLCT $\rightarrow$ 3MC trans distortion mechanism and made
use of ideas from frontier molecular orbital theory (FMOT). However it was
made clear in Article II that $E_{\mbox{ave}}$ is quite different from an
accurately determined experimental 3MLCT $\rightarrow$ 3MC activation energy
for [Ru(bpy)3]2+, complex 6 in the present article, for which $E_{\mbox{ave}}$
= 132 cm-1 [14], but for which experimental studies in solution gave 3800 cm-1
[16] and 3960 cm-1 depending upon the solvent and details of the analysis. The
present article corrects this somewhat unsatisfactory state of affairs by
carrying out calculations of the transition states (TSs) and intrinsic
reaction coordinates (IRCs) for four closely related ruthenium(II) bipyridine
complexes under the same conditions as used to calculate LI3 in Article II,
namely gas phase calculations with the B3LYP functional and the
6-31G+LANLDZ(Ru) basis set. Our calculated 3MLCT $\rightarrow$ 3MC TS barrier
is 316.04 cm-1 which may be compared with the previously calculated
theoretical gas phase value of 700 cm-1 [11].
It is important to note that we used the same nudged elastic band (NEB)
methodology as in Ref. [11], though the basis set and functional differ, but
made a more thorough investigation of our four molecules using more accurate
TS and IRC searches. The result carries some surprises. For example, our NEB
maximum energy point (MEP) gives a TS barrier of 553 cm-1 (closer to the 700
cm-1 value quoted in Ref. [11], particularly when considering that chemical
accuracy is often quoted as 1 kcal/mol = 350 cm-1). But we realized that the
NEB MEP is not sufficiently accurate for our purposes and so used specific TS
optimizers and proved the correctness of our optimized TS by calculation of
the IRC. Examination of Ru-N bond lengths along the IRC showed some surprising
results which can only be explained by a TS corresponding to one of the
ligands symmetrically lengthening its Ru-N bonds, hence going through a sort
of cis TS consistent with electron transfer to a single ligand, rather than
symmetrically to all three ligands. The trans dissociation then continues
along a ridge and then bifurcates into one of two symmetry-equivalent minima.
Once this is taken into account, not only is a lower TS barrier obtained, but
a much richer and more complex description is obtained of the trans
dissociation mechanism.
Returning to LI3 and the energetics of trans dissociation, we find that the
3MLCT-3MC energy correlates linearly with LI3 but that the 3TS-3MLCT is, to
within the accuracy of our calculation, molecule independent. Attempts to make
alternative kinetic arguments, such as by using the Bell-Evans-Polanyi
postulate, only lead to ideas in contradiction with the observed relation
between $E_{\mbox{ave}}$ and LI3. This suggests that it is the total energy
difference of the reaction, rather than the barrier height, which determines
the luminescence lifetime. That is, we are looking at a quasielquilibrium
property rather than a kinetic property. Given that the measured luminescence
lifetimes are on the order of $\mu$s which is much several orders of magnitude
longer than a typical vibrational time, then such an assumption is not
unreasonable.
Our picture of what is going on in our study is limited by it being a gas-
phase study with a particular basis set and a particular functional. It is
also limited by the fact that only one luminescence quenching mechanism has
been studied in a family of molecules which is rich in degrees of freedom and
hence also rich in dissociation mechanisms as shown by the brief review of
related work given in the introduction of this article. Nevertheless we think
that our work brings a new and more detailed picture of part of what could be
going on in the luminescence mechanism for this important family of molecules.
## Acknowledgement
DM and MEC gratefully acknowledge helpful funding from the African School on
Electronic Structure Methods and Applications (ASESMA), ASESMANet, and the US-
Africa Initiative. We thank the following people for insightful discussions:
Ragnar Björnsson, Isabelle M. Dixon, Jean Louis Heully, Valid Mwatati, Max
Daku Latévi Lawson, Sandeep Sharma. In particular, Isabelle M. Dixon is
gratefully acknowledged for sharing unpublished computational results with us
early on in our project. We would like to thank Pierre Girard and Sébastien
Morin for technical support in the context of the Grenoble Centre
d’Experimentation du Calcul Intensif en Chimie (CECIC) computers used for the
calculations reported here.
## Supplementary Information
Only plots for complex 70 have been used in the main article. The
corresponding plots for complexes 6, 73, and 74 are available on-line:
1. 1.
Contour plots
2. 2.
NEB optimizations
3. 3.
Converged NEB Minimum Energy Paths
4. 4.
IRC Energy Profiles
5. 5.
Variation of Metal-Ligand Bond Lengths Along the NEB Reaction Path
6. 6.
Variation of Metal-Ligand Bond Lengths Along the IRC
7. 7.
Author contributions
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|
# Efficiency of Nonthermal Particle Acceleration in Magnetic Reconnection
Masahiro Hoshino Department of Earth and Planetary Science, The University of
Tokyo, Tokyo 113-0033, Japan<EMAIL_ADDRESS>
###### Abstract
The nonthermal particle acceleration during magnetic reconnection remains a
fundamental topic in several astrophysical phenomena, such as solar flares,
pulsar wind, magnetars, etc, for more than half a century, and one of the
unresolved questions is its efficiency. Recently, nonthermal particle
acceleration mechanisms during reconnection have been extensively studied by
particle-in-cell simulations, yet it is an intriguing enigma as to how the
magnetic field energy is divided into thermally heated plasmas and nonthermal
particles. Here we study both non-relativistic and relativistic magnetic
reconnections using large-scale particle-in-cell simulation for a pair plasma,
and indicate that the production of the nonthermal particle becomes efficient
with increasing the plasma temperature. In the relativistic hot plasma case,
we determine that the heated plasmas by reconnection can be approximated by a
kappa distribution function with the kappa index of approximately $3$ or less
(equivalent to $2$ or less for the power-law index), and the nonthermal energy
density of reconnection is approximately over $95\%$ of the total internal
energy in the downstream exhaust.
magnetic reconnection — plasmas — particle acceleration
## I Introduction
Recently, the rapid magnetic energy release by magnetic reconnection, as well
as the supersonic bulk flow energy release by collisionless shock waves, is
regarded as an important particle acceleration process, and reconnection has
garnered significant attention in various plasma environments such as the
solar flares, Earth’s magnetosphere, pulsar magnetospheres, and magnetars
[e.g., 1, 2, 3, 4, 5]. In these hot and rarefied magneto-active plasmas, the
magnetic energy stored in a plasma sheet with an anti-parallel magnetic field
component can be efficiently converted into not only the Alfvénic bulk flow
energy, but also the energies of plasma heating and nonthermal particle
acceleration by magnetic reconnection. While the reconnection paradigm has
been established as a powerful magnetic energy dissipation process in the
plasma universe [e.g., 6, 7, 8, 9, 10], it is recently being recognized that
the reconnection plays a much more important role than before in the
astrophysical phenomena, because it can efficiently generate the nonthermal
particles, and the nonthermal energy spectra observed in various astrophysical
objects often indicate a hard power-law spectrum, which cannot be easily
explained by the conventional acceleration processes such as the diffusive
Fermi shock acceleration [e.g. 11, 12].
From the satellite observations of the solar flares and substorms in the
Earth’s magnetotail, it is recognized that not only the plasma heating, but
also the nonthermal particle acceleration is an ubiquitous process of
reconnection, and it is suggested that the reconnection process with a
relativistic hot gas temperature such as gamma-ray flares in the Crab nebula
can generate nonthermal particles as well. In fact, by employing particle-in-
cell (PIC) simulation studies, it is discussed that a hard non-thermal energy
spectrum $N(\varepsilon)\propto\varepsilon^{-s}$ with its power-law index
$s\leq 2$ can be formed for a relativistic reconnection using 2-dimensional
PIC simulations [e.g. 13, 14]. It is revealed in three-dimensional PIC
simulations [e.g. 15, 16, 18, 17] that such a hard power-spectrum can be
formed for a large magnetization parameter $\sigma=B^{2}/(4\pi\rho c^{2})$,
whereas the power-law slope becomes softer as the magnetization parameter
$\sigma$ decreases, where $B$ and $\rho$ are the magnetic field and mass
density, respectively.
While the magnetic reconnection is understood as an efficient particle
accelerator in plasma universe, one of the remaining challenges is to
quantitatively understand the energy conversion rate into nonthermal
particles. The efficiency of nonthermal particle production against the total
internal plasma energy during the magnetic energy dissipation is not
necessarily understood in a consistent way. Ball et al. [18] discussed the
nonthermal particle efficiency in trans-relativistic reconnection using PIC
simulations, and determined that the efficiency increases with increasing
magnetization parameter $\sigma$. However, it is interesting to know the
efficiency of the nonthermal production for a wider range of $\sigma$ values,
from non-relativistic cold plasmas to relativistic hot plasma reconnections.
It is also important to carefully discuss the spectral feature from the low-
energy to high-energy nonthermal range, and to understand the energy partition
between the thermal and nonthermal high energies by analyzing the reconnection
energy spectrum. The energy partition between the thermal plasma and
nonthermal particles is still an important open question.
To understand the energy partition between the thermal and nonthermal
energies, we investigate magnetic reconnection using particle-in-cell
simulations by focusing on the time and spatial evolutions of the energy
spectrum. The energy spectrum obtained by the reconnection simulation, in
general, comprises the thermal and non-thermal populations, and a model
fitting of the energy spectrum by the combination of Maxwellian and kappa
distribution functions is made as well. [e.g. 19]. We quantify the thermal
plasma and nonthermal populations as the function of the initial plasma
temperature, both in non-relativistic and relativistic regimes, and prove that
the production of the nonthermal particle becomes efficient with an increase
in plasma temperature. In the relativistic hot plasma case, we determine that
the nonthermal power-law index $s$ can be close to $2$, and the nonthermal
energy density of reconnection reaches over approximately $95\%$ of the total
heated plasma in the exhaust.
## II Overview of Simulation Study
We study the plasma heating and particle acceleration of magnetic reconnection
using two-dimensional PIC simulation [20, 21, 22] with a periodic boundary in
the $x$-direction for $x=-215\lambda$ and $x=215\lambda$, and the conducting
walls for the upper and lower boundaries at $y=\pm 215\lambda$, where
$\lambda$ is the thickness of the initial plasma sheet. The total system size
for our fiducial runs is $L_{x}\times L_{y}=430\lambda\times 430\lambda$ and
the computational grid size is $5376\times 5376$. The total number of
particles is $8.4\times 10^{10}$, to calculate the high-$\beta$ plasma sheet
as accurately as possible.
For simplicity, we adopt the Harris solution [23] for the pair plasma with the
same mass $m$ and temperature $T$ uniform in space, and focus on the energy
partition of an idealized magnetic reconnection in a collisionless plasma
system. The magnetic field ${\bf B}=B_{x}(y){\bf e_{x}}$ and plasma density
$N(y)$ are given by
$B_{x}(y)=B_{0}\tanh(y/\lambda),$ (1)
and
$N(y)=N_{0}\cosh^{-2}(y/\lambda)+N_{\rm b},$ (2)
respectively, where $N_{\rm b}$ is the background uniform plasma density
adopted to demonstrate continuous plasma injection from the outside plasma
sheet to the reconnection exhaust. The background plasma density $N_{\rm b}$
is set to be $5\%$ of the maximum Harris plasma density $N_{0}$. Note that the
relationships of the pressure balance $B_{0}^{2}/8\pi=2N_{0}T$ and force
balance $2T/\lambda=|e|u_{d}B_{0}/c$ are satisfied, where $u_{d}$ is the drift
velocity , and the initial electric field ${\bf E}$ is zero.
We study eight different cases of plasma temperature from a cold non-
relativistic to hot relativistic temperature of
$T/mc^{2}=10^{n/2-2}\quad{\rm with}\quad n=0,1,2,..,7.$ (3)
The above temperatures correspond to the magnetization parameter
$\sigma=B_{0}^{2}/(8\pi N_{0}mc^{2})=2\times 10^{n/2-2}$, if we use the number
density for the value of the plasma sheet $N_{0}$, owing to the pressure
balance between the gas pressure inside the plasma sheet and magnetic pressure
outside the plasma sheet. Note that the Alfvén speed $v_{A}$ is given by
$v_{A}=c\sqrt{\sigma/(1+\sigma)}$ in a cold plasma limit, where the density is
used for the value of the central plasma sheet and the magnetic field is the
value of the outside plasma sheets. We keep the ratio of the gyro-radius
$r_{g}$ and thickness of the plasma sheet $\lambda$ for all simulation runs,
and set $r_{g}/\lambda=0.45$, where
$r_{g}=mc^{2}\sqrt{\gamma_{th}^{2}-1}/(eB_{0})$, because a thin plasma sheet
whose thickness is almost equal to the gyro-radius has been formed before the
explosive growth of magnetic reconnection in the pair plasma [22] and has been
observed before the onset of substorms in the Earth’s magnetotail [e.g. 24,
25]. For simplicity, we have assumed $(\gamma_{th}-1)mc^{2}/T=1$, and then the
drift speed $u_{d}$ can be calculated by
$u_{d}/c=2(r_{g}/\lambda)/\sqrt{1+2mc^{2}/T}=0.9/\sqrt{1+2\times 10^{2-n/2}}$
.
We add a small initial perturbation for the vector potential $\delta A_{z}$ to
the Harris equilibrium, which is given by
$\delta
A_{z}(x,y)\propto\exp(-(\frac{x}{\lambda_{x}})^{2}-(\frac{y}{\lambda_{y}})^{2})\cosh(\frac{y}{\lambda_{y}}),$
where $\lambda_{x}=2\lambda$, $\lambda_{y}=\lambda/2$, and the initial
amplitude of the reconnected magnetic field is set to be
$\max(B_{y}/B_{0})=10^{-2}$ in the neutral sheet.
Figure 1: Time evolution of the collisionless magnetic reconnection for a
Harris current sheet obtained by PIC simulation for $T/mc^{2}=10^{-2}$. The
top panel ($t/\tau_{A}=119$) shows the early stage of reconnection, and the
X-type neutral point is formed at the center of the simulation box. The middle
panel ($t/\tau_{A}=176$) is the nonlinear stage where several plasmoids are
coalescing and forming the larger plasmoids, and the bottom panel
($t/\tau_{A}=308$) is the almost final stage when two large plasmoids are just
coalescing into one large plasmoid in the system. The thick white lines
indicate the magnetic field lines that pass through the X-type neutral line,
i.e., the separatrix between the downstream exhaust and the upstream plasma
sheet. The color bars in the right-hand side indicate that the logarithmic
plasma density normalized the initial plasma sheet density. Figure 2: Energy
spectra for the downstream exhaust region for $T/mc^{2}=10^{-2}$ (left),
$T/mc^{2}=1/\sqrt{10}$ (middle) and $T/mc^{2}=10$ (right), respectively. The
color lines indicate the time evolution of spectra, whose time stages are
indicated in the right-hand side bar. The dark blue line is the initial state,
and the red one is the final stage when a large plasmoid is formed in the
simulation box. The thick solid lines are the model fitting for the composed
function of the Maxwellian and kappa distributions $N_{{\rm
M}+\kappa}(\gamma)$. The black dashed lines in the lower energy regime
represent the Maxwellian part of the fitting result $N_{\rm M}(\gamma)$, while
the other dashed lines in the higher energy regime are the kappa distribution
parts $N_{\kappa}(\gamma)$. The bottom panels indicate the errors of the model
fitting, which show the difference between the fitting model function and
simulation result. The dashed lines in the bottom plot are depicted as
reference of an acceptable level of error.
Illustrated in Figure 1 are three-time stages of the reconnection structure
obtained by the particle-in-cell simulation for a non-relativistic
reconnection with the temperature of $T/mc^{2}=0.01$. The plasma density and
magnetic field lines are indicated by the color contour and white lines,
respectively. The thick white lines indicate the separatrix of the magnetic
reconnection for the X-type neutral point formed around the center
$x/\lambda\sim 0$, i.e., the most recently reconnected magnetic field lines.
The downstream of the separatrix is occupied by the reconnection heated
plasmas. These lines of the separatrix can be obtained by the contour lines of
the vector potential $A_{z}$ that has the minimum value in the neutral line of
$y=0$. A part of the simulation system is illustrated as well.
The top panel is the early time stage at $t/\tau_{A}=119$ when the
reconnection just started at the center, where $\tau_{A}=\lambda/v_{A}$ is the
Alfvén transit time, and several small scale plasmoids have been formed along
$y=0$. The middle panel indicates the time stage when several plasmoids grow
rapidly and coalesce with each other. As we imposed the initial small
perturbation $\delta A_{z}$ at $x/\lambda=0$, the most prominent reconnection
occurs at the center, but in addition to that, we can observe several other
small size plasmoids embedded in the downstream domain of the main
reconnection. The bottom panel is the approximate final stage of the active
reconnection in our periodic system after the two large plasmoids have merged
into one large magnetic island.
## III Model Fitting of Energy Spectrum
As our objective is to understand the energy partition between the thermal and
nonthermal populations during reconnection, we study the energy spectrum in
the downstream of the separatrix lines in details. Figure 2 illustrates the
time evolutions of three energy spectra for $T/mc^{2}=10^{-2}$ (left),
$1/\sqrt{10}$ (middle), and $10$ (right), integrated over the reconnection
region sandwiched by the separatrix, i.e., the thick white lines defined in
Figure 1. Note that if the width of the separatrix is smaller than the
thickness of the plasma sheet $\lambda$, the integration is conducted for the
plasma sheet with the size $\lambda$. The horizontal axis is the particle
energy $(\gamma-1)mc^{2}$, and the vertical axis is the number density
$N(\gamma)d\gamma$, where $\gamma=1/\sqrt{1-(v/c)^{2}}$. The color lines
indicate the time evolution of the energy spectra, whose time stages are
indicated in the right-hand color bars. The blueish and reddish colors
correspond to the earlier and later time stages, respectively. As we assumed a
uniform plasma temperature for both the $\cosh$-type Harris density population
and background populations in Equation (2), the dark blue color spectra in
Figure 2 illustrate approximately the initial plasma state with the initial
Maxwellian distribution functions. For the relativistic case in Figure 2c,
however, one-dimensional energy spectrum is modified by the effect of a high-
speed drift velocity with $u_{d}/c\sim 0.82$.
As time goes on, we can clearly observe those Maxwellian plasmas are gradually
heated owing to the reconnection heating process, and the high-energy
components are further accelerated and form the nonthermal population. During
the time evolution, by looking at the energy interval of the time evolution of
the energy spectra, it is determined that the energy intervals are wider in
the early acceleration stage compared with those in the later stage,
suggesting that the rapid energy gain happens in the early stage.
As the spectra with the red curve is approximately stable and at the final
stage, we discuss the spectra behavior for the red curve by making a model
function fitting in detail. It is evident that the energy spectra cannot be
simply approximated by a Maxwell distribution function given by,
$N_{M}(\gamma)\propto\gamma\sqrt{\gamma^{2}-1}\exp(-\frac{\gamma-1}{T_{M}/mc^{2}}),$
(4)
where, for simplicity, a three-dimensional Maxwell distribution function has
been projected into a one-dimensional distribution in the simulation frame as
the function of the particle energy $\gamma$.
To make a better model fitting of the observed distribution, we first tested a
kappa distribution function given by,
$N_{\kappa}(\gamma)\propto\gamma\sqrt{\gamma^{2}-1}\left(1+\frac{\gamma-1}{\kappa
T_{\kappa}/mc^{2}}\right)^{-(1+\kappa)}f_{cut}(\gamma).$ (5)
The kappa distribution comprises a thermal Maxwellian at low energies and a
nonthermal population approximated by a power-law function at high energies
[e.g. 19]. It is known that the observed solar wind distribution function can
be well modeled by a kappa ($\kappa$) distribution function [e.g 19, 26], and
the kappa modeling is widely applied for the space plasma investigation [e.g
27, 28]. In our analysis, we added a high energy cutoff function $f_{cut}$ to
represent a possible high energy cutoff in the simulation data [29, 30], which
is given by,
$f_{cut}(\gamma)=\left\\{\begin{array}[]{@{\,}ll}1&\mbox{for
$\gamma\leq\gamma_{cut}$ }\\\
\exp\left(-(\gamma-\gamma_{cut})/\gamma_{cut}\right)&\mbox{for
$\gamma>\gamma_{cut}$ }\end{array}\right.$
The high energy cutoff may come from the finite time evolution of reconnection
under the limited system size [30]. Note that the high energies of
$N_{\kappa}$ can be approximated by a power-law function of
$N_{\kappa}\propto\gamma^{-\kappa+1}$ with the power-law index $s=\kappa-1$
for $\gamma\gg T_{\kappa}/mc^{2}$.
Although the model function was not necessarily a good approximation, we
determined that this kappa model function is better than the Maxwellian
fitting. As the spectra for the red lines in Figure 2 can be observed, the
contributions of the cold plasma freshly transported from the outside plasma
sheet is not necessarily negligible. Note that the energetic particles
generated in the X-type region/magnetic diffusion region can escape along the
magnetic field line, and simultaneously, the pre-heated plasma is transported
from the outside plasma sheet by $E\times B$ drift motion in association with
the magnetic field lines. Therefore, the separatrix region contains both the
cold and hot plasmas [e.g. 31].
Having the above plasma transport, we finally determined that the best-fitted
model function $N_{M+\kappa}$ is a combination of the Maxwellian and kappa
distribution functions, i.e.,
$N_{M+\kappa}(\gamma)=N_{M}(\gamma)+N_{\kappa}(\gamma).$ (6)
In the top three panels in Figure 2, the sold lines are the model fitting
curves for $N_{M+\kappa}$ at the final time stage, while the two dashed lines
are the fitting curve of $N_{M}$ at lower energies and $N_{\kappa}$ for higher
energies, respectively. It is determined that three simulation spectra in the
downstream exhaust for $T/mc^{2}=10^{-2}$ (left), $1/\sqrt{10}$ (middle), and
$10$ (right) can be approximated by the composed model function of Maxwellian
and kappa distributions $N_{{\rm M}+\kappa}(\gamma)$. We have also confirmed
that five other simulation cases of $T/mc^{2}=10^{n/2-2}$ with $n=1,2,4,5$ and
$7$ express the similar good model fitting.
The bottom three panels indicate the error of the model fitting given by,
$\displaystyle{\rm Error}(\gamma)=|N_{data}(\gamma)-N_{M+\kappa}(\gamma)|/$
$\displaystyle{\rm min}(N_{data}(\gamma),N_{M+\kappa}(\gamma)),$ (7)
where ${\rm min()}$ means a function to return the minimum element from
$N_{data}$ and $N_{M+\kappa}$. If
$\frac{1}{2}N_{data}<N_{M+\kappa}<2N_{data}$, then Error becomes less than
$1$, as depicted by the dashed line in the bottom panel of Figure 2. We can
observe that the errors in the wide energy ranges are less than $10^{-1}$, and
the model fitting of $N_{M+\kappa}$ can well describe the energy spectra
obtained in the simulation runs. Note that a small bump seen around
$\gamma\sim 8$ for $T/mc^{2}=10^{-2}$ is the contribution from the small-scale
plasmoids behind the main large-scale plasmoid, but the total energy
contribution is not necessarily large. Note that the bumpy structure in the
energy spectrum is not a permanent structure and disappears after the small
scale plasmoid is absorbed into the large plasmoid. For the model fitting for
$T/mc^{2}=10$ for the right-hand panel in Figure 2, we can observe a large
discrepancy between the simulation data and model fitting at high energies. By
choosing another cutoff function given by
$f_{cut}(\gamma)=\exp(-((\gamma-\gamma_{cut})/\gamma_{cut})^{2})$, we obtain
the better fitting result (not indicated here).
Figure 3: The model fitting results for eight different initial plasma
temperatures of $T/mc^{2}=10^{-2}\sim 10^{3/2}$. (Left) The temperatures of
the Maxwellian part $T_{M}$ (red dots) and kappa distribution part
$T_{\kappa}$ (blue dots) as the function of the initial plasma temperature.
The kappa index $\kappa$ (middle) and cutoff energy $\gamma_{\rm cut}$ (right)
as the function of the initial plasma temperature.
We quantitatively discuss the energy partition between the thermal and
nonthermal plasmas. Figure 3 illustrates the model fitting results of (a) the
temperatures $T_{M}$ (red dots) and $T_{\kappa}$ (blue dots), (b) the $\kappa$
index with a power-law tail, and (c) the cutoff energy $\gamma_{cut}$ as the
function of the initial plasma temperature $T/mc^{2}$, respectively. We
determine that the temperature of $T_{M}$ remains approximately the same as
the initial background temperature $T$, suggesting that the exhaust region
sandwiched by the separatrix, as indicated by the thick white lines in Figure
1, still contains a part of non-heated plasmas, and that those temperatures
may represent the flux tubes just transported into the reconnection exhaust,
and are not heated strongly [31]. Note that during the transport from the
outside plasma sheet into the plasma sheet, the plasmas are slightly cooled
down owing to the adiabatic expansion of the magnetic flux tube, because the
slow mode expansion waves are emitted from the X-type reconnection region.
However, the temperatures of the kappa distribution function $T_{\kappa}$ are
higher than that of the Maxwellian function $T_{M}$. The ratio of
$T_{\kappa}/T_{M}$ is large for the non-relativistic reconnection cases for
$T/mc^{2}<1$, while those for the relativistic reconnection regime of
$T/mc^{2}>1$ become less than 2. $\kappa$ index, which is equivalent to the
power-law index $s=\kappa-1$ is illustrated in Figure 3b. For the non-
relativistic case of $T/mc^{2}=10^{-2}$, $\kappa$ indicates a large value,
suggesting that the spectrum is close to a Maxwell distribution and the
efficiency of the nonthermal particle production is very low, whereas for the
relativistic regime of $T/mc^{2}>1$, the $\kappa$ values are approximately $3$
and the nonthermal tails are well developed. Note that $\kappa=3$ can be
rephrased as the power-law index of $2$. Figure 3c is the cutoff energy
$\gamma_{cut}$. Except for the case of $T/mc^{2}=10^{-2}$, we determine that
the ratio of $\gamma_{cut}/(T_{\kappa}/mc^{2})$ is approximately $50\sim
10^{2}$, suggesting that the nonthermal tails are well extended above the
thermal population of the kappa distribution function.
It would be interesting to mention the relationship between the gyro-radius of
the accelerated particle and the size of the magnetic island. The initial
gyro-radius was set to be 0.45 $\lambda$ for all runs, and we can find in
Figure 2 that the highest energies after reconnection attained to several
$10^{2}$ \- $10^{3}$ times larger than the initial thermal energies.
Therefore, we can estimate that the gyro-radius of the highest energy
particles is about $100-200\lambda$, which is almost same as the size of the
magnetic island.
It would be better to make a short comment on the model fitting error to
estimate the fitting parameters. As can be observed in Figure 2, the spectral
profile in the final stages of the simulation have approximately the same
spectral structure in the logarithmic scale, but the fitting parameters may
have approximately 10% variations depending on the time stages.
## IV Efficiency of Nonthermal Density and Energy Production
Figure 4: The model fitting results for the same data set as Figure 3. (Left)
The percentage in the number density of the kappa distribution part $\xi_{\rm
den}$ (red) and percentage in the energy density $\xi_{\rm ene}$ (blue).
(Right) The percentage in the number density of the nonthermal population of
the kappa distribution against the total distribution $\varepsilon_{\rm den}$
(red) and percentage in the energy density of the nonthermal population
$\varepsilon_{\rm ene}$ (blue).
Based on the above model fitting, we can study the efficiency of the
nonthermal particle acceleration in reconnection. In the previous section, we
discovered that the energy spectrum in the exhaust region can be approximated
by the combination of Maxwellian and kappa distribution functions. To quantify
the efficiency of the production of the nonthermal particles, we calculate the
percentage of the nonthermal population from the fitted model function.
As illustrated in Figure 4, the left-hand panel a is the percentage of the
number density and the energy density for the population of the $\kappa$
distribution against the entire population of the energy spectrum as the
function of the initial plasma temperature $T/mc^{2}$. The solid red and
dashed blue lines are the number and energy densities defined by,
$\displaystyle\xi_{\rm
den}=\int_{1}^{\infty}N_{\kappa}(\gamma)d\gamma/\int_{1}^{\infty}N_{{\rm
M}+\kappa}(\gamma)d\gamma,$
and
$\displaystyle\xi_{\rm
ene}=\int_{1}^{\infty}(\gamma-1)N_{\kappa}(\gamma)d\gamma/\int_{1}^{\infty}(\gamma-1)N_{{\rm
M}+\kappa}(\gamma)d\gamma,$
respectively. We discussed that the energy spectrum can be approximated by the
combination of Maxwellian and kappa distribution functions in Figure 2, and
the above denominator indicates the total contribution to the energy spectrum
coming from both the Maxwellian and $\kappa$ portions. The percentages of the
number and energy densities of the population of the kappa distribution
function weakly depend on the plasma temperature $T/mc^{2}$, and we determine
that the kappa distributions occupy more than $95\%$ and $70\%$ the total
energy and total number densities, respectively. The exhaust region contains
the freshly injected plasma from the outside plasma sheet, but the
contribution of the cold Maxwellian population is less than $5\%$ in the total
energy density.
Figure 4b illustrates the ratios of the nonthermal to total number densities
(solid red line), and the nonthermal energy to total energy densities (dashed
blue line) as the function of the initial plasma temperature $T/mc^{2}$,
respectively. We estimate the percentage of the nonthermal number density by
the following definition of
$\displaystyle\varepsilon_{\rm
den}=\int_{1}^{\infty}(N_{\kappa}(\gamma)-N_{\kappa}^{\rm
M}(\gamma))d\gamma/\int_{1}^{\infty}N_{{\rm M}+\kappa}(\gamma)d\gamma,$
where $N_{\kappa}^{\rm M}(\gamma)$ represents the portion of Maxwellian
distribution function out of the $\kappa$ distribution function, i.e., the
$\kappa$ value is replaced by $\kappa=\infty$ by keeping other fitted
parameters. Therefore, the numerator of $N_{\kappa}(\gamma)-N_{\kappa}^{\rm
M}$ corresponds to only the nonthermal power-law component. Note that a kappa
distribution function approaches a Maxwellian distribution as
$\kappa\to\infty$, as follows,
$\displaystyle\lim_{\kappa\to\infty}\left(1+\frac{\gamma-1}{\kappa
T_{\kappa}}\right)^{-(\kappa+1)}\simeq\exp\left(-\frac{\gamma-1}{T_{\kappa}}\right).$
(8)
Similarly, the nonthermal energy density can be calculated by,
$\displaystyle\varepsilon_{\rm
ene}=\int_{1}^{\infty}(\gamma-1)(N_{\kappa}(\gamma)-N_{\kappa}^{\rm
M}(\gamma))d\gamma/$ $\displaystyle\int_{1}^{\infty}(\gamma-1)N_{{\rm
M}+\kappa}(\gamma)d\gamma.$
From Figure 4b, it is determined that the nonthermal number and energy
densities increase with an increase in the plasma temperature in the non-
relativistic regime of $T/mc^{2}<1$, while the number and energy densities are
approximately constant in the relativistic regime of $T/mc^{2}>1$. We
determine that the relativistic reconnection can efficiently generate a lot of
nonthermal particles compared with the non-relativistic reconnection. The
percentage of the nonthermal number density saturates at $60\%$, while that of
the energy density is approximately $95\%$ of the total internal energy in the
downstream exhaust. This result suggests that the energy density of the
nonthermal population dominates for the relativistic reconnection with a
relativistic temperature, and the thermal energy density population is
negligible. The similar nonthermal efficiency has been discussed for trans-
relativistic reconnection by Ball et al. [18].
## V Evolution of Nonthermal Plasma in Association with Magnetic Flux Tube
Transport
So far, we discussed the nonthermal energy spectra integrated all particles in
the exhaust plasma sheet, which is sandwiched by the last reconnected magnetic
field line; however, it would be interesting to discuss where and how those
nonthermal particles are accelerated in the plasma sheet. It is known that the
exhaust plasma sheet has several distinct structures responsible for particle
acceleration: (i) the magnetic diffusion region with weak magnetic fields,
where the fresh particles transported from the outside plasma sheet are picked
up and accelerated by the reconnection electric field, (ii) the plasma sheet
boundary layer between the outside and exhaust plasma sheets where the plasma
gas pressure suddenly increases, (iii) the plasmoids where the high-
temperature and high-density plasmas are confined by the closed magnetic flux
rope, and (iv) the magnetic field pile-up region where the Alfvénic jet from
the diffusion region collides with the pre-existing plasma sheet/plasmoid etc.
In association with those distinct regions, their roles on plasma heating and
acceleration processes have been discussed by paying attention to the
individual particle motion [e.g 34, 35, 36, 31, 37, 38, 39].
Figure 5: The colour contour of the plasma density structure of magnetic
reconnection for $T/mc^{2}=10^{-2}$. The logarithmic density contour is
indicated in the right-hand side bar. The white lines indicate the magnetic
field lines, i.e., the contour lines of the vector potential $A_{z}(x,y)$. The
interval of the contour line is chosen in such a way that the downstream
exhaust region is divided by ten magnetic field lines. The thick white lines
are depicted with the flux tube number for every five lines, which corresponds
to the spatial evolution of the energy spectrum indicated by the right-hand
color box in Figure 6. Figure 6: Energy spectra for $T/mc^{2}=10^{-2}$ (left),
$T/mc^{2}=1/\sqrt{10}$ (middle) and $T/mc^{2}=10$ (right), respectively. The
same data set as Figure 2. The color lines correspond to the energy spectra
for the 19 divided regions illustrated in Figure 5. The dark blue color
spectrum is taken for the outermost region, while the red one is the innermost
region. The right-hand side color bar indicates the region number. The solid
black lines are the energy spectra for the downstream exhaust regions, which
are the same spectra as illustrated in Figure 2. The black dashed lines are
the downstream exhaust spectra in which the contributions in the inner most
flux tube are subtracted.
The relative importance of the efficiency of nonthermal particle production in
those different regions, however, has not necessarily been discussed in a
systematic manner. To understand the relative importance, we focus on the
spatial evolution of the nonthermal energy spectrum in association with the
magnetic flux transportation. We divide the exhaust plasma sheet with 10
magnetic flux tubes by the equal interval of the vector potential $\Delta
A_{z}=(A_{z}^{\rm max}-A_{z}^{\rm min})/10$, where $A_{z}^{\rm max}$ and
$A_{z}^{\rm min}$ correspond to the vector potential for the last reconnected
magnetic field line at the X-type neutral point, and that for the minimum
bottom of the O-type point in the plasmoid, respectively. The reconnection
structure divided by the 19 magnetic flux tubes for $T/mc^{2}=10^{-2}$ is
illustrated in Figure 5, and we mark labels on the magnetic field lines from 1
– 19 from inside to outside. The last reconnected magnetic field line
corresponds to the label number of 10. The underlying assumption of this
analysis is that the thermal and energetic particles are transported together
with the magnetic flux, because we have confirmed that the frozen-in condition
is approximately satisfied [32].
Figure 6 illustrates the energy spectra for those 19 divided regions labeled
by the rainbow colors indicated by the right-hand side color bar. To focus on
the nonthermal tail, the energy range in the horizontal axis is chosen to be
three orders of magnitudes. The red line is the energy spectrum for the flux
tube number 1, which corresponds to the innermost region almost occupied by
the yellow color in Figure 5. The green line of the label number 10
corresponds to the flux tube just attached to the X-type neutral point, and
the dark blue is that of outside the plasma sheet. Evidently, the blueish
lines are basically the thermal Maxwellian spectrum with the initial plasma
temperature almost maintained, thus suggesting that no plasma heating and
acceleration occurs during the plasma transport from the outside plasma sheet
toward the separatrix region. (Note that this statement is not necessarily
correct if the evolution of the velocity distribution function is discussed in
detail, because the adiabatic plasma process can be observed in association
with the change of flux tube, which is initiated by the slow mode expansion
waves generated around the magnetic diffusion region [32]).
However, when the magnetic flux tube is attached to the separatrix, the rapid
increase in the nonthermal particle population can be observed, regardless of
the initial plasma temperature. The flux numbers of 10 and 11 with the
greenish colors corresponds to this region, and the flux number 11 has been
highlighted by the thick green color. This behavior suggests that the initial
significant particle acceleration can happen during the reconnection of two
different magnetic flux tube. The flux tube number 11 is just situated outside
the separatrix, but the magnetic diffusion region at the X-type neutral point
has a finite width in $y$-direction, and the flux tube is contaminated by
those accelerated particles around the diffusion region.
The main energy gain in the flux tube numbers 10 and 11 comes from the
meandering particles in the diffusion region [35, 37] where the meandering
particles are accelerated by the inductive electric field $E_{z}$. Note that
the resonant acceleration in the diffusion region can be enhanced in a
relativistic plasma due to the relativistic inertia effect [13]. In addition
to the acceleration in the diffusion region, the beam plasma heating can occur
in the plasma sheet boundary, because a part of the accelerated particles
around the diffusion region may have a small pitch-angle against the local
magnetic field, and those particles can escape outward along the magnetic
field line. The interaction of the field-aligned beam component and cold
plasma transported by $E\times B$ motion from the outside plasma sheet may
lead to the plasma heating together with the generation of Alfvénic waves
[33].
After rapid energization in the magnetic diffusion region and along the
separatrix magnetic field lines, those flux tubes are convected downward, and
further gradual energization can be observed, as indicated by the yellowish
and reddish lines. However, we determine that those energy gains are not
necessarily strong, compared with those in the last reconnecting magnetic flux
tube. Several bumpy structures in the spectra may arise from the dynamical
evolution of plasmoids.
The thick black solid curves are the energy spectra for the downstream exhaust
region integrated from the flux tube number 1 – 10, which is the same as the
final stage illustrated in Figure 2. By looking at the spatial evolution of
the energy spectra from No.10 (flux tube of the separatrix) to No.2 (inner
flux tube), we more or less observe the gradual heating and particle
acceleration. However, we determine that the spectra for the innermost flux
tube No. 1 has a slightly different behavior, because of the pre-existed and
pre-heated plasma before the onset of reconnection. To distinguish the plasma
heating between the pre-existed plasma and freshly injected particles, we
indicate that the energy spectra integrated the region from the flux tube of
No.2 – No.10 as the black dashed line, i.e., we excluded the innermost flux
tube. As can be observed, the difference appears only around the middle energy
range between the thermal Maxwellian part and nonthermal power-law part, and
the kappa index becomes slightly harder as well. Regarding the case of the
relativistic reconnection, we obtain the kappa index of $\kappa<2$, and the
result is roughly consistent with the previous studies [13, 14, 15, 16]. The
efficiency of nonthermal density discussed in the previous section, however,
does not change so much.
Figure 7: Time evolution of energy spectra for fixed magnetic flux tubes for
$T/mc^{2}=10^{-2}$ (left), $T/mc^{2}=1/\sqrt{10}$ (middle) and $T/mc^{2}=10$
(right), respectively. The same data set as Figure 2. The color lines indicate
the time evolution of spectra, whose time stages are indicated in the right-
hand side bar. The dark blue line is the initial state when the magnetic flux
tube is situated in the upstream. On the other hand, the red one is the final
stage when the flux tube is transported into the downstream exhaust. The thick
solid lines are the model fitting for the composed function of the Maxwellian
and kappa distributions $N_{{\rm M}+\kappa}(\gamma)$. The black dashed lines
in the lower energy regime represent the Maxwellian part of the fitting result
$N_{\rm M}(\gamma)$, while the other dashed lines in the higher energy regime
are the kappa distribution parts $N_{\kappa}(\gamma)$. The bottom panels
indicate the errors of the model fitting in the same format as Figure 2.
Although we discussed the spatial evolution of the energy spectra in Figure 6,
it is also interesting to mention the time evolution of the energy spectrum
for a fixed magnetic flux tube, which is transported from the outside plasma
sheet into the downstream exhaust region during reconnection. Figure 7
illustrates the time evolution of the energy spectra for the fix magnetic flux
tubes sandwiched between $\left|y\right|/\lambda=9$ and $12$ at the initial
state $t=0$. As could be easily expected, the behavior of the energization is
basically same as Figure 6. The dark blue color spectra are taken when the
flux tubes are situated in the upstream, and those temperatures almost remain
the initial Maxwellian. As time goes on, we observed the rapid energization
and formation of the nonthermal tail when two flux tubes just reconnect at the
X-type neutral point at $t/\tau_{A}\sim 120$. The timing of the attachment of
two flux tubes normalized by the Alfvén transit time $\tau_{A}$ was almost
same for three cases.
After the attachment of the two magnetic flux tubes, those reconnected
magnetic flux tubes are further transported into the downstream, and the high
energy particles are gradually produced. The red lines are the almost final
stages of the magnetic reconnection. The thick solid lines are the model
fitting for the composed function of the Maxwellian and kappa distributions
$N_{{\rm M}+\kappa}(\gamma)$. The black dashed lines in the lower energy
regime represent the Maxwellian part of the fitting result $N_{\rm
M}(\gamma)$, while the other dashed lines in the higher energy regime are the
kappa distribution parts $N_{\kappa}(\gamma)$. The fitted parameters of
($T_{M}/mc^{2}$, $T_{\kappa}/mc^{2}$, $\kappa$, $\gamma_{cut}$) are ($0.013$,
$0.078$, $6.1$, $6.2\times 10^{2}$), ($0.79$, $0.44$, $3.0$, $2.2\times
10^{1}$), and ($15.$, $9.0$, $2.6$, $4.7\times 10^{2}$) for
$T/mc^{2}=10^{-2}$, $1/\sqrt{10}$, and $10$, respectively. These fitted
parameters are basically same as those obtained in Figures 2 and 3. The
contributions of the Maxwellian part, however, are smaller than the kappa
distribution part compared with Figure 2, because the magnetic flux tubes are
convected into the deep downstream and the upstream cold plasma are heated.
The bottom panels indicate the errors of the model fitting in the same format
as Figure 2. We find that the model fittings are very good for a wide energy
ranges.
## VI Discussions and Summary
During the last two decades, it has been discussed that magnetic reconnection
can produce not only the hot thermal plasma, but also the high-energy supra-
thermal particles. Specifically, relativistic reconnection can efficiently
produce nonthermal particles, whose energy spectrum is approximated by a hard
power-law function. However, how the efficiency changes between the non-
relativistic and relativistic reconnection has remained an open question.
Meanwhile, the detailed acceleration process focusing on the individual
particle motion has been well discussed by several people [e.g. 31, 35, 36,
37, 38, 39].
The interaction between the reconnection electric field $E_{z}$ and meandering
motion under the anti-parallel magnetic field $B_{x}$ in the neutral sheet is
known to play an important role as the primary acceleration process, and the
accelerated particles in and around the X-type neutral line can be ejected
outward by the Lorentz force because of the reconnecting magnetic field [34,
37]. Zenitani and Hoshino [13] investigated this process to the relativistic
reconnection, and argued that the interaction efficiency between the
reconnection electric field and meandering particle can be enhanced in the
relativistic reconnection, because of the relativistic inertia effect during
the Speiser motion [34].
In addition to the particle acceleration in and around the magnetic diffusion
region, several other possible mechanisms have been proposed. The collision
between the Alfvénic jet ejected from the magnetic diffusion region and
magnetic field pile-up region ahead of the plasmoid can provide the betatron-
like acceleration owing to the gradient-B drift motion [e.g 35]. Drake et al.
[36] proposed that the particles trapped by the plasmoid can be energized by a
Fermi process, which is the curvature drift motion in association of the time
evolution of the plasmoid. Drake et al. [36], Dahlin et al. [38] investigated
the above mechanisms responsible for energy gain using the guiding center
formalism. Oka et al. [39] argued that the particles can be accelerated during
coalescence of two plasmoids with the similar process that occurs in the
magnetic diffusion region, but with a stronger inductive electric field in
association with the dynamic coalescence.
We did not analyze such detail acceleration mechanisms in this study, rather
focused on the evolution of energy spectrum during the plasma transport in
association with the magnetic flux tube. We not only observed the primary
particle acceleration around the magnetic diffusion region, but also the
additional acceleration during the plasma transport in the exhaust. These
heating and acceleration are basically consistent with the previous studies.
In addition, we determined that the energy spectra obtained by PIC simulations
can be well approximated by a kappa distribution function for the high energy
population. By examining the model fitting results, we determined that the
efficiency of the nonthermal particle acceleration increases with increasing
plasma temperature, and the nonthermal energy density reaches over $95\%$ of
the total internal energy in the relativistic temperature of the exhaust,
whereas the nonthermal number density remains approximately $40\%\sim 60\%$
for the total number density in the exhaust.
Our simulation study in a two-dimensional system with the anti-parallel
magnetic field topology is only a first step toward understanding the energy
partition between thermal and nonthermal plasmas. We investigated the
efficiency of the nonthermal particle production for a pair plasma, but it is
important to understand the acceleration efficiency for an ion-electron
plasma. Recently, several people discussed the energy partition between ion
and electron, and it is revealed that ions are preferentially heated, compared
to electrons during magnetic reconnection by PIC simulations [40, 41, 42]. It
has been discussed that the ratio of the ion heating to the electron
$T_{i}/T_{e}$ can be approximated by $(m_{i}/m_{e})^{1/4}$ for the magnetic
reconnection without guide field, where $m_{i}$ and $m_{e}$ are the ion and
electron mass, respectively [32], whereas the electrons receive larger
energies than the ions for guide field reconnection [43]. Although ions seem
to receive much thermal energies compared with electrons, suprathermal
electrons seem to be efficiently generated, while the proton nonthermal
spectrum seems not to be necessarily well developed in particle simulations.
Significant attention has been paid to electron nonthermal particle
acceleration, but the study of ion nonthermal production is limited. We are
probably required to study a longer time evolution of reconnection for a large
system size.
It is important to check the dependence of the energy spectrum on the system
size of our PIC simulation. So far, we fixed the system size with (a)
$430\lambda\times 430\lambda$, but we studied other cases with (b)
$215\lambda\times 215\lambda$, (c) $860\lambda\times 215\lambda$, and (d)
$1720\lambda\times 430\lambda$ for $T/mc^{2}=0.1$. The fitting results for
$T_{M}$, $T_{\kappa}$, $\kappa$, $\varepsilon_{\rm den}$ and $\varepsilon_{\rm
ene}$ does not strongly depend on the system size, and we found that those
differences of $T_{M}$, $T_{\kappa}$, $\kappa$ on the system size were less
than $15\%$. However, the cutoff energy seems to weakly depend on the system
size, except for case (b). Regarding case (b), $\gamma_{\rm cut}$ was
approximately 6.7, but other cases of (a),(c) and (d) were $\gamma_{\rm
cut}\sim 9-12$.
It would be important to mention the effect of the plasma temperature in the
outside plasma sheet, which is denoted by $N_{b}$ in Equation (2). So far, we
assumed that the background temperature is the same as that for the Harris
plasma component $N_{0}$, but we can choose any temperature to keep an
equilibrium state. In astrophysical settings, it is likely that the plasma
temperature outside the plasma sheet is colder than the Harris plasma
temperature. In fact, for the case of the Earth’s magnetotail, the temperature
in the lobe is known to be colder than that in the plasma sheet [e.g. 24, 25].
We have studied the cold background plasma temperature $T_{b}/mc^{2}=10^{-4}$
by keeping other plasma parameters the same. We determine that the efficiency
of the nonthermal energy density is almost the same as the case of the uniform
background plasma temperature discussed in this study. We will discuss this
behavior in a separate paper.
Our final comment is on the effect of the background density $N_{b}$. It is
well known that the reconnection rate increases with an increase in the Alfvén
speed in the inflow region. Furthermore, the reconnection electric field
$E_{z}$ becomes larger with a decrease in the background density, and the
magnetic energy dissipation rate being converted into the thermal and
nonthermal plasmas would be enhanced as well. Although the nonthermal particle
production would be expected to increase with a decrease in the background
plasma density, the energy partition between the thermal and nonthermal plasma
production should be carefully studied. In our preliminary study for the low
background density with $N_{b}/N_{0}=1/80$, we obtained the nonthermal
$\kappa$ values of $2.2$, $2.6$, $2.7$, and $4.3$ for $T/mc^{2}=10$, $1$,
$10^{-1}$ and $10^{-2}$, respectively. We find that these $\kappa$ values are
about $30\%$ smaller than those of $N_{b}/N_{0}=1/20$ shown in Figure 3b. It
would be also interesting to study the effects of background temperature and
density for the generalized magnetization parameter $\sigma=B^{2}/(4\pi w)$
and Alfvén speed $v_{A}=c/\sqrt{1+4\pi(e+p)/B^{2}}$, where $w$, $e$ and $p$
are the enthalpy density, the total energy density and the gas pressure,
respectively. We will discuss these effects in a separate paper together with
the effect of the guide field.
###### Acknowledgements.
This study was supported by JSPS KAKENHI, Grant Number 20K20908.
## DATA AVAILABILITY
The data support the findings of this study are available from the
corresponding author upon reasonable request.
## Appendix A Nonthermal contribution in $\kappa$ distribution function
We shortly discuss the basic behavior of $\kappa$ distribution function by
focusing on the thermal and non-thermal populations. As the $\kappa$
distribution function are given by two parameters of the temperature $T$ and
$\kappa$ index, the nonthermal contribution can be given by two parameters. By
assuming that the thermal population of the $\kappa$ distribution function is
given by keeping the same temperature but by setting $\kappa=\infty$, we can
easily estimate the percentages of the nonthermal number and nonthermal energy
densities. The top and bottom panels in Figure 8 illustrate the nonthermal
number density $\varepsilon_{\rm den}$ and the nonthermal energy density
$\varepsilon_{\rm ene}$, respectively. The horizontal axes are the $\kappa$
index and plasma temperature normalized by the rest mass energy $T/mc^{2}$. We
can observe that the nonthermal contributions become small for a large
$\kappa$ value. For the case of a small $\kappa$ and relativistic hot plasma,
we determine that the nonthermal contributions become significant. Regarding
$\kappa=3$, the nonthermal number density contribution $\varepsilon_{\rm den}$
becomes $\varepsilon_{\rm den}\rightarrow 7/9$ for
$T/mc^{2}\rightarrow\infty$, whereas the nonthermal energy density
contribution is $\varepsilon_{\rm ene}\rightarrow 1$ for
$T/mc^{2}\rightarrow\infty$. For $\kappa=4$ and $T/mc^{2}\rightarrow\infty$,
$\varepsilon_{\rm den}\rightarrow 5/8$ and $\varepsilon_{\rm ene}\rightarrow
29/32$.
Figure 8: The percentage of the nonthermal number density $\varepsilon_{\rm
den}$ (top) and nonthermal energy density $\varepsilon_{\rm ene}$ (bottom) as
the functions of the temperature $T/mc^{2}$ and kappa index $\kappa$.
It is also interesting to note the effect of the cutoff energy function
introduced in the simulation analysis. In our simulation results, the cutoff
energies were much larger than the heated plasma temperature, i.e.,
$\gamma_{cut}-1\sim 10^{2}\times T/mc^{2}$. By assuming that
$\gamma_{cut}-1=10^{2}\times T/mc^{2}$, we determined that the difference
between the nonthermal contribution with and without the cut-off energy is
less than several $\%$ for $\kappa>3$.
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|
11institutetext: Istanbul Technical University, Istanbul, Turkey
22institutetext: Mehmet Akif Ersoy Thoracic and Cardiovascular Surgery
Training and Research Hospital, Istanbul, Turkey 33institutetext: Acibadem
University, Istanbul, Turkey 44institutetext: King’s College London
∗Corresponding author: 44email<EMAIL_ADDRESS>
# Shifted Windows Transformers for Medical Image Quality Assessment
Caner Özer ** 1 1 Arda Güler 22 Aysel Türkvatan Cansever 22 Deniz Alis 33
Ercan Karaarslan 33 İlkay Öksüz 1144
###### Abstract
To maintain a standard in a medical imaging study, images should have
necessary image quality for potential diagnostic use. Although CNN-based
approaches are used to assess the image quality, their performance can still
be improved in terms of accuracy. In this work, we approach this problem by
using Swin Transformer, which improves the poor-quality image classification
performance that causes the degradation in medical image quality. We test our
approach on Foreign Object Classification problem on Chest X-Rays (Object-CXR)
and Left Ventricular Outflow Tract Classification problem on Cardiac MRI with
a four-chamber view (LVOT). While we obtain a classification accuracy of 87.1%
and 95.48% on the Object-CXR and LVOT datasets, our experimental results
suggest that the use of Swin Transformer improves the Object-CXR
classification performance while obtaining a comparable performance for the
LVOT dataset. To the best of our knowledge, our study is the first vision
transformer application for medical image quality assessment.
###### Keywords:
Visual Transformers Medical Image Quality Assessment Chest X-Rays Cardiac MRI.
## 1 Introduction
It is the responsibility of technicians to ensure that the acquired medical
scan should be of good quality so that the physician, who asked for the
medical scan, will be able to diagnose accurately. However, this condition may
not be satisfied due to the medical image quality problems such as patient
breathing or arrhythmia in Cardiac Magnetic Resonance Imaging (MRI), ringing
artifact presence in Computed Tomography (CT), and low dose or foreign object
appearance on the Chest X-Rays (CXR). These image quality issues would also
constitute a critical problem for automated processing of medical scans within
a clinical decision support system consisting of big data.
Maintaining a high standard in image quality is imperative also for a machine
learning-guided medical image analysis downstream tasks. More specifically,
existence of poor quality images may lead a machine learning methodology to
learn spurious set of features which are not clinically useful. Hence,
identifying, eliminating or correcting the samples with poor image quality is
necessary for downstream tasks such as segmentation and disease diagnosis. The
reasons for an image that should be eliminated from the pipeline include but
are not limited to artefacts in MR Imaging [11], foreign object existence
[15], or undesirable region appearance [12] within the scans. For this
purpose, several datasets on a variety of modalities have been developed to
overcome such problems. Ma et. al. [11] has developed a brain MRI dataset in
which the patient data has been labelled according to the level of motion
artefact. The authors have also developed a 4-layer convolutional neural
network model to classify the samples according to their motion degradation
level. JFHealthcare [8] proposed a MIDL 2020 challenge which comes up with a
dataset that aims to localize the foreign objects within chest X-Rays.
Swin-Transformers are becoming more common in medical image analysis,
especially in the field of diagnosis. For instance, Zhang et al. [18] uses a
cascaded mechanism for chest CT COVID-19 diagnosis which uses a U-Net to
extract lungs as a foreground and background segmentation mechanism, and Swin-
Transformer uses these segmented lungs to find out whether the patient has
COVID-19 or not. Chen et al. [2] combines a statistical significance-based
slice sampling scheme and Swin-Transformers to propose a framework that can
detect COVID-19 from chest CT scans by using a single slice. In addition to
diagnosis, Swin-Transformers are becoming more common to be used for other
problems, such as brain tumor segmentation [1] [5], and cardiac CT
reconstruction [13]. However, despite being promising for these problems in
medical imaging, Swin-Transformers’ have yet to be studied for medical image
quality assessment tasks.
In this work, we propose a SwinTransformer-based [10] approach for medical
image quality analysis where our contributions are as follows:
* •
We train a variety of SwinTransformer models in order to classify X-Ray images
with foreign objects and Cardiac MRI images with a 4-chamber view with Left
Ventricular Outflow Tract (LVOT).
* •
We compare our approach with deep neural network (DNN) based approaches, such
as ResNet [6] and EfficientNet [14].
* •
To our best knowledge, this is the first paper to perform medical image
quality assessment through SwinTransformers.
## 2 Methodology
Figure 1: Swin Transformer pipeline for medical image quality analysis.
### 2.1 Pre-processing
In order to train our model, we apply a number of preprocessing procedures
prior to feeding the input images to the network. In this regard, instead of
manually defining the data augmentations and their corresponding
hyperparameters, we use RandAugment [3] which uses a previously found
automated data augmentation strategy to train a DNN. After that, we resize
these images to a pre-determined resolution of $H\times W$. Finally, we
normalize the images by using ImageNet statistics that they become ready-to-
use in a pre-trained model.
### 2.2 Swin Transformer
Following the pre-processing procedure, we used the Swin Transformer model
that is being demonstrated in Fig. 1. Given an input image, Swin Transformer
first divides the image into patches using a Patch Partition layer that
generates $48$-dimensional representations for each of the $4\times 4$ patches
on the image. After that, as shown in Stage 1, the Linear Embedding layer
transforms $48$-d representations into $C$ dimensions.
The second component of Stage 1 is the Swin Transformer Blocks which has been
inherited from the general Visual Transformer (ViT) [4] architecture. Despite
having the same high-level block structure as the one in ViT, multiheaded
self-attention (MSA) blocks are replaced with an alternate ordering of window-
based MSA (W-MSA) and shifted window-based MSA (SW-MSA), sequentially. In
W-MSA, the feature map is divided into windows and MSA computes the output on
each of them. However, this brings a cost of missing global connectivity,
since this was not an issue for MSA in ViT that operates globally. For this
reason, cyclic shifting operation is used to overcome this problem with SW-MSA
that, the feature map is circularly shifted on the height and width axes by
the half-length of the patch size. Then, W-MSA is executed on the resulting
feature map, and a reverse circular shift operation is used to obtain a
reverted feature map without any shifts.
In all of Stages 2, 3, and 4, we have a Patch Merging layer and a varying
number of Swin Transformer blocks. In the Patch Merging layer, each of the
pixels located on the $2\times 2$ patches on the feature map are concatenated
across the channel axis, i.e., $\frac{H}{2}\times\frac{W}{2}\times C$ to
$\frac{H}{4}\times\frac{W}{4}\times 4C$. Then, after applying the layer
normalization layer, a linear layer is used to reduce the dimensionality to
half, i.e., to $\frac{H}{4}\times\frac{W}{4}\times 2C$. The output is then
transferred to the Swin Transformer blocks for further processing. Lastly,
after using layer normalization followed by an average pooling layer to
calculate the features of the Swin Transformer model, the output is used for
classification in a single Linear layer. This model classifies if there is
anything present in the input image that can affect the image quality
adversely. The hyperparameter settings can be found in Table 1. Stochastic
depth refers to the probability of disregarding the transformation that is
applied on a sample within a batch during training, while being processed
within a Swin Transformer block, embedding dims $C$ is the dimensionality of
the patch-wise embeddings, depths and MSA heads correspond to the number of
repeated Swin Transformer blocks and head count of the MSA layer for each
stage, respectively.
Table 1: Hyperparameter settings for different Swin Transformer models. Model Name | | Stochastic Depth
---
| Embedding Dims $(C)$
---
Depths | MSA Heads
Swin-Tiny | 0.2 | 96 | 2, 2, 6, 2 | 3, 6, 12, 24
Swin-Small | 0.3 | 96 | 2, 2, 18, 2 | 3, 6, 12, 24
Swin-Base | 0.2 | 128 | 2, 2, 18, 2 | 4, 8, 16, 32
### 2.3 Training the Model
We adopt the official code of Swin
Transformers111https://github.com/microsoft/Swin-Transformer in which we
performed some modifications to run it on PyTorch 1.10.2. We also used
TorchVision 0.11.3 to inherit the pre-trained models for EfficientNet and
ResNet pre-trained architectures. We apply the learning rate warm-up scheme
and learning rate scheduling and leverage color jittering, stochastic depth
(only for Swin Transformer) [7], Random Erasing [19] MixUp [17], CutMix [16]
and RandAugment [3] data augmentation methods for promoting model
generalization. For the foreign object classification model, we set the batch
size to 16, gradient accumulation steps to 8 and window size to 8. Instead for
the LVOT classification model, we set the batch size to 64, gradient
accumulation steps to 2 and window size to 7, due to lower input image
resolution comparing chest X-Rays. We employ an AdamW [9] optimizer with
weight-decay of $10^{-8}$ to fine-tune both of these models by using an
ImageNet pretrained model, as we set the initial learning rate to $0.00006$
and use a linear learning-rate scheduler. The fine-tuning procedure takes $60$
epochs with $5$ epochs of a warm-up period. We also trained the models from
scratch for $300$ epochs that, the learning rate is set to $0.0005$ and the
objective function is Cross Entropy. You can access to our repository by using
the following link: https://github.com/canerozer/qct.
## 3 Experimental Results
### 3.1 Datasets
We demonstrate our results on the Object-CXR [8], a challenge dataset where
the aim is to classify the Chest X-Ray scans with foreign objects and to
localize the foreign objects, and the LVOT classification dataset, which is
constructed by selecting a subset of the Cardiac MR (CMR) scans with 4-chamber
view that were acquired in the Mehmet Akif Ersoy Thoracic and Cardiovascular
Surgery Training and Research Hospital between the years 2016 and 2019.
Object-CXR dataset contains 8,000 samples for training and 1,000 samples for
each of the validation and testing splits. There is an equal number of
positive and negative samples for each split which does and does not contain
foreign objects. We only use the class label, but not the localization ground
truth for this dataset during training. We set the fixed input size to
$H=W=1024$ since the images can reach a spatial resolution of $4048\times
4932$. On the other hand, LVOT classification dataset contains 272 positive
and 278 negative samples that are used for training, 34 positive and 35
negative samples for validation, and 35 positive and 35 negative samples for
testing. These images have relatively way less spatial resolution than the
ones in Object-CXR, hence, we set $H=W=224$. In addition, since our pipeline
is capable of processing 2D images, we slice the 3D CMR patient scans into 2D
images, which is also helpful in terms of increasing the number of samples for
each split by a factor of 25.
### 3.2 Left Ventricular Outflow Tract Classification Results
First, we start by comparing the performances of different Swin Transformer
models that are named as Swin-Tiny, Swin-Small, and Swin-Base in Table 2. We
additionally measured the differences between using a pretrained model and
training from scratch. We notice that, the accuracy is the highest for the
Swin-Tiny model with an accuracy of $95.59\%$. Nevertheless, Swin-Small’s AUC
(Area Under Curve) score is performing the best among these three models with
a score of $0.980$. For this reason, we decided to include both of the Swin-
Small and Swin-Tiny models to compare them with the other baselines.
Meanwhile, switching from pre-trained model to training from scratch
significantly decreases the accuracy of at least around $12.35\%$ and AUC
score of $0.059$. We relate this to the data-hungry regime of Swin
Transformers, as they need abundant data samples to capture the inductive bias
during training. Hence, it becomes necessary to use the pretrained models or
to increase the number of samples to the order of tens of thousands.
Table 2: Comparison among Swin Transformer models with different capacity for the LVOT classification task. Model | Pretrained? | Acc | AUC | | # of params
---
GFLOPs | | Throughput (img/sec)
---
Swin-Tiny | Yes | 95.59 | 0.969 | 27,520,892 | 4.5 | 515
No | 72.93 | 0.757
Swin-Small | Yes | 95.48 | 0.980 | 48,838,796 | 8.7 | 313
No | 83.13 | 0.921
Swin-Base | Yes | 93.86 | 0.976 | 86,745,274 | 15.4 | 222
No | 67.01 | 0.738
Table 3: Our results on the LVOT dataset indicate that the performance is comparable to the other DNN-based methods. Model | Acc | AUC | | # of params
---
GFLOPs | Throughput
EfficientNet-B0 | 93.74 | 0.979 | 4,010,110 | 0.4 | 2274
EfficientNet-B1 | 93.68 | 0.971 | 6,515,746 | 0.6 | 1565
EfficientNet-B2 | 94.61 | 0.971 | 7,703,812 | 0.7 | 1480
ResNet-34 | 95.19 | 0.974 | 21,285,698 | 3.7 | 1963
ResNet-50 | 95.77 | 0.979 | 23,512,130 | 4.1 | 1005
ResNet-101 | 95.59 | 0.969 | 42,504,258 | 7.8 | 587
ResNet-152 | 96.00 | 0.988 | 58,147,906 | 11.6 | 409
Swin-Tiny | 95.59 | 0.969 | 27,520,892 | 4.5 | 515
Swin-Small | 95.48 | 0.980 | 48,838,796 | 8.7 | 313
Then, we compare our approach with other deep learning baselines as ResNet [6]
and EfficientNet [14] as shown in Table 3. We see that, despite the
differences in the number of parameters and GFLOPs, the performance varies
insignificantly ($p>0.05$) with outstanding accuracy and AUC metrics. We
relate this to the triviality of the LVOT classification task, where the goal
is to check if fifth chamber exists on the Cardiac MRI scan - generally
located at the centre. As a result, it becomes possible to obtain testing
accuracy of over $95\%$ and AUC scores exceeding $0.969$ for all ResNet
architectures and Swin-Small without overfitting. Still, these models are
ready to be deployed for a real-world scenario for alerting LVOT regions on
4-chamber Cardiac MRIs almost instantaneously.
### 3.3 Foreign Object Classification Results
Table 4: Results for the Object-CXR dataset. Model | Acc | AUC | | # of params
---
GFLOPs | Throughput
EfficientNet-B0 | 82.1 | 0.873 | 4,010,110 | 8.3 | 119
EfficientNet-B1 | 83.0 | 0.882 | 6,515,746 | 12.3 | 83
EfficientNet-B2 | 83.2 | 0.890 | 7,703,812 | 14.2 | 78
ResNet-34 | 82.9 | 0.895 | 21,285,698 | 91.8 | 94
ResNet-50 | 82.8 | 0.897 | 23,512,130 | 102.7 | 42
ResNet-101 | 84.1 | 0.906 | 42,504,258 | 195.8 | 26
ResNet-152 | 85.1 | 0.909 | 58,147,906 | 241.5 | 22
Swin-Base | 87.1 | 0.922 | 86,766,330 | 324.6 | 11
The results show Swin Transformer’s strength on a higher dimensional input,
namely, on the Chest X-Ray data for foreign object classification. The model
complexity is highly correlated with the model performance that, the best
performance is obtained by the Swin-Base model with $87.1\%$ testing accuracy
and $0.922$ testing AUC score. The closest performance to the best model is
the ResNet-152 model, which has $1.5\times$ less number of parameters and
$25\%$ less floating point operations, at a cost of $2\%$ less testing
accuracy and $0.013$ less AUC score. Despite Swin-Base is working slightly
slower than real-time due to its huge input size, its significant performance
gains ($p<0.005$) makes it a promising approach.
### 3.4 Qualitative Analysis
Prior to utilizing our model in a clinical study, we further examine our model
under extreme conditions. Firstly, we start with the Object-CXR classification
task as some cases are shown in Fig. 2. Regardless of the orientation and
contrast factors, our model can correctly classify the X-Ray images which does
and does not contain foreign objects as demonstrated in Fig. 2a and 2b,
respectively. What is interesting especially for the rightmost image of Fig.
2b is that, even though the foreign objects may not be visible to human eye
without altering the contrast, our model can still successfully classify the
image as it contains foreign objects. Foreign objects are marked with a blue
bounding box in Fig. 2b.
Figure 2: Some true challenging samples due to their medical image quality
factors, from Object-CXR dataset.
We also perform a qualitative analysis on the classification results as some
of the edge cases are shown in Fig. 3 for the LVOT classification task. In
Fig. 3a, where all 3 images are labelled as good quality images without LVOT
appearance, the model can correctly classify even when the images are subject
to motion artefacts and local noise upto some level. Also, as shown in Fig.
3b, our model can correctly classify the images containing LVOT regions under
low contrast. We show the regions with LVOT with a blue bounding box in Fig.
3b.
Figure 3: Some true challenging samples due to their medical image quality
factors, from LVOT dataset.
## 4 Discussion & Conclusion
In this work, we propose to use Swin Transformers for medical image quality
analysis, where we validated our approach on two datasets working on two
different modalities. We outperform a variety of ResNet and EfficientNet
baselines on the Object-CXR dataset by obtaining a testing accuracy of
$87.1\%$ while obtaining a comparable performance for the LVOT classification
objective. Comparing Swin Transformer to the other baselines and performing a
qualitative analysis demonstrate its potential for utilizing it in a clinical
setting.
Although the problem setting has been restricted to foreign object
classification in Chest X-Rays and LVOT classification for Cardiac MRI’s, it
is possible to generalize the approach for other medical image quality
problems such as motion artefact grade classification. In addition, the
interpretability aspect of the Swin Transformers is a critical avenue of
research for the safe transition of an automated medical image analysis
pipeline.
## Acknowledgments
This paper has been produced benefiting from the 2232 International Fellowship
for Outstanding Researchers Program of TUBITAK (Project No: 118C353). However,
the entire responsibility of the publication/paper belongs to the owner of the
paper. The financial support received from TUBITAK does not mean that the
content of the publication is approved in a scientific sense by TUBITAK.
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# Field-free anomalous junction and superconducting diode effect in spin split
superconductor/topological insulator junctions
T.H. Kokkeler<EMAIL_ADDRESS>Donostia International Physics Center
(DIPC), 20018 Donostia–San Sebastián, Spain
Interfaces and Correlated Electron Systems, Faculty of Science and Technology,
University of Twente, Enschede, The Netherlands
A.A. Golubov Interfaces and Correlated Electron Systems, Faculty of Science
and Technology, University of Twente, Enschede, The Netherlands
F. S. Bergeret Centro de Física de Materiales (CFM-MPC) Centro Mixto CSIC-
UPV/EHU, E-20018 Donostia-San Sebastián, Spain
Donostia International Physics Center (DIPC), 20018 Donostia–San Sebastián,
Spain
###### Abstract
We study the transport properties of a diffusive Josephson junction between
two spin-split superconductors made of superconductor-ferromagnetic insulator
bilayers (FIS) on top of a 3D topological insulator (TI). We derive the
corresponding Usadel equation describing the quasiclassical Green’s functions
in these systems and first solve the equation analytically in the weak-
proximity case. We demonstrate the appearance of an anomalous phase in the
absence of an external magnetic field. We also explore non-reciprocal
electronic transport. Specifically, we calculate the junction’s diode
efficiency $\eta$ by solving the Usadel equation. We obtain a sizable diode
effect even at zero applied magnetic field. We discuss how the diode
efficiency $\eta$ depends on the different parameters and find a non-monotonic
behavior of $\eta$ with temperature.
††preprint: APS/123-QED
## I Introduction
Recent advances attracting attention in superconductivity research are effects
related to non-reciprocal charge transport [1, 2, 3, 4, 5, 6, 7, 8, 9, 10],
particularly the superconducting diode effect [11, 12, 13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 23, 24, 25, 26, 27]. A mesoscopic superconducting junction is
called a superconducting diode if the critical current is different for
opposite current directions. That is, for a superconducting diode the minimum,
$I_{-}=\min_{\phi}{I_{s}}(\phi)$, and maximum,
$I_{+}=\max_{\phi}{I_{s}}(\phi)$, of the current phase relation (CPR) are
unequal in magnitude. If a current $I$ flows in such a junction, with
$\min(|I_{-}|,I_{+})<I<\max(|I_{-}|,I_{+})$, in one direction it is a
supercurrent, whereas in the other direction the current dissipates. In a
conventional Josephson junction the critical current is the same in both
directions, the CPR has the following symmetry: $I(-\phi)=-I(\phi)$ [28]. This
symmetry holds if either time reversal symmetry or inversion symmetry is
present in the system [29, 30].
The superconducting diode effect can thus only be obtained if both time
reversal symmetry and inversion symmetry are broken [11, 12]. Time reversal
symmetry breaking can be achieved by a magnetic field. On the other hand
inversion symmetry can be broken intrinsically, such as in topological
insulators [31, 32, 33] or superconductors with Rashba spin orbit coupling
[34, 35, 36, 37, 38, 39, 40, 41]. Inversion symmetry can also be broken by
using an asymmetric junction geometry [42] or asymmetry of the device
originated in the fabrication [16].
If both time-reversal symmetry and inversion-symmetry are broken, in general
$I(-\phi)\neq-I(\phi)$ and thus possibly $I(\phi=0)\neq 0$. Such junctions,
for which the current vanishes at a nonzero phase difference, are called
$\phi_{0}$-junctions [35]. In weak coupling Josephson junctions, the CPR is
proportional to $\sin(\phi+\phi_{0})$; hence, in this regime, one cannot
observe the diode effect. However, if higher harmonics contribute to the
current, in general $I_{+}\neq|I_{-}|$, and the diode effect can be observed
[15, 13, 17, 18].
One way of breaking time-reversal symmetry without an external magnetic field
is to attach a ferromagnetic insulator (FI) to the superconductor. FI-S
systems have been discussed extensively in the literature [43, 44, 45, 46].
The exchange interaction between the localized magnetic moments of the FI and
the itinerant electrons in the SC leads to a spin split in the density of
states of the latter. Spin split superconductors form an active field of
research with varying directions [47, 48, 49, 50, 51, 52, 53].
In this work, we investigate the diode effect in a Josephson junction made of
spin-split superconducting electrodes on the 2D surface of a disordered 3D
topological insulator (TI). Specifically, our setup consists of two spin-split
superconductors (FIS) placed on top of a topological insulator, see Fig. 1a.
The spin-splitting in the superconductor breaks time-reversal symmetry,
whereas inversion symmetry is broken because we consider only the top surface
of the topological insulator. Thus, the conditions to have a $\phi_{0}$-effect
are fulfilled even without an external magnetic field. Using the linearized
Usadel equation, we first show analytically that such CPR exhibits the
$\phi_{0}$-effect. Going beyond the linear regime we compute the diode
efficiency. Even in the case of low transmission FIS/TI interfaces we obtain
an efficiency of 1%. By increasing the interface transmission the efficiency
can reach values larger than 7%. We also find that for short junctions the
efficiency is maximized at a finite temperature independent of the strength of
the exchange field.
The article is structured as follows. In section II we introduce the setup and
the basic equations. We derive the Usadel equation for a diffusive topological
insulator in proximity with a spin-split superconductor. In section III, we
focus on analytical results that can be obtained by linearizing the Usadel
equation, which is valid under the assumption that the proximity effect is
small. Even though in this limit the CPR contains only the first harmonic, and
thus has no diode effect, we can determine the condition for observation of
the anomalous Josephson currents. The latter is the precursor of a diode
effect in the non-linearized equation. We also show that the $\phi_{0}$\-
effect is suppressed by impurity scattering. In section IV we go beyond the
linear regime and present our numerical results for the nonlinear equation and
the diode efficiency as a function of the temperature for different values of
the exchange field, length of the junction, and transparency of the
interfaces. Finally in section V we conclude and propose real material
combinations to test our predictions. Throughout the paper it is assumed that
$\hbar=k_{B}=1$.
## II The system and basic equations
We consider the FIS-TI-FIS system shown in Fig. 1a. A FIS can be realized by
placing a ferromagnetic insulator on top of a superconducting film (SC) with a
conventional s-wave pair potential. The thickness of the latter is assumed to
be small compared to its coherence length, so that we can assume a homogeneous
splitting field in the SC induced by the magnetic proximity effect [54]. We
neglect the inverse proximity effect of the topological insulator on the
superconductor. In this setup a current can flow through the top surface of
the TI from one FIS to the other. Because the system is finite in $x$
direction, no current can flow at $x=\pm L/2$, where $L$ is the length of the
TI. We denote by $L_{1}$ the length of the TI between the two FIS electrodes.
(a)
(b)
Figure 1: (a) Sketch of the junction under consideration. (b) The CPR of the
FIS-TI-FIS junction for different temperatures. The parameters chosen are
$\frac{\gamma_{0}}{E_{\text{Th}}}=\frac{\Delta_{0}}{E_{\text{Th}}}=\frac{5h}{2E_{\text{Th}}}=25$,
$\frac{L_{1}}{L}=\frac{1}{10}$ and $\frac{l}{L}=0.08\ll 1$. Inset: zoom in of
the CPR around $\phi=0$.
We assume that the transport at the TI surface is diffusive and can be
described by the Usadel equation: the derivation of this equation for our
system closely resembles the derivation of the Usadel equation for a TI in an
exchange field as presented in Refs. [55, 56]. However, whereas
superconductivity in the systems discussed in these papers is introduced as an
effective pair potential, for the spin split superconductor a different
approach is taken.
We incorporate the effect of the spin-split superconductor as a self-energy
term $\bar{\Sigma}_{s}$. For the self-energy we follow the approach similar to
[57, 58], in which the self-energy, up to second order in the tunneling
parameter $T_{1}$ between the TI and the superconductor, is introduced as
$\displaystyle\bar{\Sigma}_{s}=T_{1}^{2}\rho_{S}\check{\tau}_{3}\sigma_{0}\bar{G^{\prime}}_{S}\sigma_{0}\check{\tau}_{3}=T_{1}^{2}\rho_{S}\bar{G}_{S},$
(1)
where $\rho_{S}$ is the density of states in the superconductor, $\sigma_{0}$
is the identity matrix in spin space and $\check{\tau}_{3}$ is the third Pauli
matrix in electron-hole space. $\bar{G^{\prime}}_{S}$ is the momentum
integrated Green’s function in the spin-split superconductor and
$\bar{G}_{S}=\check{\tau}_{3}\sigma_{0}\bar{G^{\prime}}_{S}\sigma_{0}\check{\tau}_{3}$
is introduced to shorten notation. Note that the only effect of this
transformation by $\check{\tau}_{3}\sigma_{0}$ is to negate the pair
amplitudes. The self-energy term appears as an added term in the commutator on
the left hand side of the Eilenberger equation. The Eilenberger equation thus
reads, using the same presentation in spin-Nambu space as in [55]:
$\displaystyle\frac{v_{F}}{2}\\{\\{\eta_{j},\nabla_{j}\check{g}(1+\vec{n}_{F}\cdot\vec{\eta})\\}\\}$
$\displaystyle=[\check{g}(1+\vec{n}_{F}\cdot\eta),\omega_{n}\check{\tau}_{3}+\Gamma(x)\bar{G}_{S}+\frac{\langle\check{g}(1+\vec{n}_{F}\cdot\vec{\eta})\rangle}{2\tau}],$
(2)
where $\\{\cdot,\cdot\\}$ denotes the anticommutator and $[\cdot,\cdot]$ the
commutator. In our notation $\check{g}$ is the quasiclassical Green’s
function, $\omega_{n}$ is the nth Matsubara frequency, $\vec{n}_{F}$ is the
direction of the momentum at the Fermi surface, $v_{F}$ is the magnitude of
the Fermi velocity, $\tau$ is the collision time, $\check{\tau}_{3}$ is the
third Pauli matrix in Nambu space, $\mu$ is the Fermi energy and
$\vec{\eta}=(-\sigma_{2},\sigma_{1},0)$. The tunneling parameter $T_{1}$ is
nonzero only in FIS regions. To reflect this we introduce the boundary
parameter $\Gamma(x)$:
$\Gamma(x)=\gamma_{0}\Theta(|x|-\frac{L_{1}}{2})\Theta(\frac{L}{2}-|x|),$ (3)
where $\Theta$ denotes the Heaviside function and
$\gamma_{0}=T_{1}^{2}\rho_{S}$.
The Green’s function is written as
$\check{g}\frac{1}{2}(1+\vec{n}_{F}\cdot\vec{\eta})$ to reflect the strong
coupling between spin and direction of momentum in a topological insulator. In
this work a 2D surface of a 3D TI is studied. Therefore scattering is not
prohibited, unlike in a 1D edge of a 2D TI. We assume the junction is in the
dirty limit, that is, the inverse scattering time $\frac{1}{\tau}$ is much
larger than any energy scale other than the chemical potential $\mu$. In that
case the Green’s function $g$ is almost isotropic. Thus, it is a good
approximation to keep only the zeroth and first term in the expansion in
angular momentum, that is,
$\check{g}\approx\check{g}_{s}+\vec{n}_{F}\cdot\vec{\check{g}}_{a},$ (4)
where the zeroth order $\check{g}_{s}$ and the first order angular momentum
$\vec{\check{g}}_{a}$ satisfy $\check{g}_{s}^{2}=\mathbf{1}$ and
$\check{g}_{s}\vec{\check{g}}_{a}+\vec{\check{g}}_{a}\check{g}_{s}=\vec{0}$ in
order to satisfy the normalisation condition $\bar{g}^{2}=\mathbf{1}$ up to
second order in $\tau$. The Green’s functions $\check{g}_{s}$ and
$\vec{\check{g}}_{a}$ do not have any degrees of freedom in momentum space nor
in spin space and are thus functions which map position into the space of 2 by
2 matrices. Using the expansion in angular momentum, the Usadel equation can
be derived. The strategy followed to derive the Usadel equation for our
structure is very similar to the strategy used in [55], [56]. To this end, we
first write
$\bar{G}_{S}=\check{G}_{S0}\sigma_{0}+\vec{\check{G}}_{S1}\cdot\vec{\sigma},$
(5)
where $\check{G}_{S0}$ and $\vec{\check{G}}_{S1}$ are matrix functions without
spin degrees of freedom, and $\vec{\sigma}$ is the vector of Pauli matrices in
spin space. It is assumed that the exchange field in the FI always points in
the same direction, so that there are no domain walls which may affect the
density of states significantly [53]. The position independent Green’s
function in the spin-split superconductor is
$\displaystyle\bar{G}_{S}$
$\displaystyle=\frac{1}{2}(1+\vec{b}\cdot\vec{\sigma})\check{G}_{\uparrow}+\frac{1}{2}(1-\vec{b}\cdot\vec{\sigma})\check{G}_{\downarrow}$
(6) $\displaystyle\check{G}_{\uparrow,\downarrow}$
$\displaystyle=g_{\uparrow,\downarrow}\tau_{3}+f_{\uparrow,\downarrow}(\cos{\phi}\tau_{1}+\sin\phi\tau_{2}),$
(7)
where $g_{\uparrow,\downarrow}=(\omega_{n}\pm ih)/{\sqrt{(\omega_{n}\pm
ih)^{2}+\Delta^{2}}}$ are the normal parts and
$f_{\uparrow,\downarrow}=\Delta/{\sqrt{(\omega_{n}\pm ih)^{2}+\Delta^{2}}}$
are the anomalous parts [59]. The + sign is used for the spin-up component and
the - sign for the spin down component, $\tau_{1,2,3}$ are the Pauli matrices
in Nambu space, $h$ is the magnitude of the exchange field $\vec{h}=h\vec{b}$,
$\Delta$ is the superconducting potential calculated self-consistently and
$\phi$ is the phase of the superconductor. Combining Eq. (5) with Eqs. (6) and
(7), we may write
$\displaystyle\check{G}_{S0,1}$
$\displaystyle=g_{S0,1}\tau_{3}+f_{S0,1}(\cos{\phi}\tau_{1}+\sin{\phi}\tau_{2}),$
(8) $\displaystyle\vec{\check{G}}_{S1}$
$\displaystyle=\check{G}_{S1}\vec{b}\cdot\vec{\sigma},$ (9)
where $g_{s0,1}=(g_{\uparrow}\pm g_{\downarrow})/2$ and
$f_{S0,1}=(f_{\uparrow}\pm f_{\downarrow})/2$. The component $f_{S0}$ of the
condensate is the usual singlet component, whereas $f_{S1}$ is the odd-
frequency triplet component [59].
The Usadel equation for the angular averaged Green’s function $g_{s}$ without
spin degrees of freedom in the TI is obtained analogous to the approach laid
out in [55]. We combine the spin-trace of the equation obtained by angular
averaging and the equation obtained by angular averaging after multiplication
by $\vec{n}_{F}$. The resulting Usadel equation is
$D\hat{\nabla}(\check{g}_{s}\hat{\nabla}\check{g}_{s})=[\omega_{n}\check{\tau}_{3}+\frac{\Gamma(x)}{2}\check{G}_{S0},\check{g}_{s}],$
(10)
with $D=v_{F}^{2}\tau$. Eq. (10) is similar to the Usadel equation in normal
metals. However, the derivative is replaced by a generalized derivative:
$\displaystyle\hat{\nabla}=\nabla+\frac{\Gamma(x)}{2v_{F}}[\cdot,\check{G}_{S1y}]e_{x}-\frac{\Gamma(x)}{2v_{F}}[\cdot,\check{G}_{S1x}]e_{y},$
(11)
For a spin split superconductor this becomes
$\displaystyle\hat{\nabla}=\nabla+\frac{\Gamma(x)}{2v_{F}}(b_{y}e_{x}-b_{x}e_{y})[\cdot,\check{G}_{S1}].$
(12)
This derivative is similar to the derivative presented in [56] for a
topological insulator with an exchange field, in fact, Eq. (10) reduces to
this expression if $\check{G}_{S1}=h\check{\tau_{3}}$. Throughout this work we
will assume that the magnetic field is oriented perpendicular to the current
direction, so that $b_{y}=1$ and $b_{x}=0$. The equation is accompanied by the
boundary conditions
$\displaystyle\hat{\nabla}G_{S1}(x=\pm\frac{L}{2})=0.$ (13)
In this paper it is assumed that the ferromagnetic insulator is either very
thin or very thick, so that there is no y nor z-dependence in the problem. As
a consequence the equation becomes effectively one-dimensional. From the
solutions of Eqs. (10),(12) and (13) one can determine the current:
$\displaystyle
I=\frac{\sigma_{N}}{2}T\sum_{n}\text{Tr}\Big{(}\tau_{3}\bar{G}(x^{*},\omega_{n})\hat{\nabla}\bar{G}(x^{*},\omega_{n})\Big{)}\;,$
(14)
where $\sigma_{N}$ is the normal state conductance, $T$ is the temperature
entering the Matsubara frequencies $\omega_{n}=(2n+1)\pi T$, and $x^{*}$ is
any position for which $\Gamma(x^{*})=0$. The quantities of interest in this
article are the maximum supercurrents $I_{c}^{+}$ and $|I_{c}^{-}|$ in both
directions, and the diode efficiency, defined by
$\displaystyle\eta=\frac{I_{c}^{+}-|I_{c}^{-}|}{I_{c}^{+}+|I_{c}^{-}|}.$ (15)
Before showing the numerical solution of the above boundary problem, in the
next section, we study the linearized equation in the limit of a small
proximity effect. As discussed in the introduction, in this limiting case the
diode effect vanishes, but the anomalous phase $\phi_{0}$ , can be studied
analytically. The anomalous current is a strong indication for the diode
effect to appear.
## III Linearized case: The $\phi_{0}$-junction
To get an understanding of the physics behind the new Usadel equation, Eq.
(10), we focus first on the case of a weak proximity effect. In this case the
anomalous parts of the GF are much smaller than the normal one, that is,
$\text{Tr}(\tau_{1,2}G)\ll\text{Tr}(\tau_{3}G)$, and $G$ can be approximated
by $G(i\omega_{n})\approx\begin{bmatrix}\text{sgn}(\omega_{n})&F\\\
\tilde{F}&-\text{sgn}(\omega_{n})\end{bmatrix}$, where $|F|,|\tilde{F}|\ll 1$.
Using this approximation the Usadel equation reduces to a linear equation. We
assume here that the system in Fig. 1a is infinite in $x$ direction. The
superconductor is only absent in the region
$(-\frac{L_{1}}{2},\frac{L_{1}}{2})$ and present everywhere outside this
region. In this case Eq. (10) can be written in the three separate spatial
regions:
$\displaystyle\begin{cases}D\partial_{xx}F=2|\omega_{n}|F-\gamma_{0}f_{S0}e^{-i\frac{\phi}{2}}&x<-\frac{L_{1}}{2}\\\
D\partial_{xx}F=2|\omega_{n}|F&|x|<\frac{L_{1}}{2}\\\ \end{cases}$ (16)
For $x\xrightarrow{}\pm\infty$ the Green’s function should commute with
$\omega\tau_{3}+\check{G}_{S0}$. This implies that the pair potential is given
by
$\displaystyle\lim_{x\xrightarrow{}\pm\infty}F=\frac{\gamma_{0}f_{S0}}{2|\omega_{n}|}e^{\pm
i\frac{\phi}{2}}.$ (17)
These are exactly the same equations as for the conventional SNS junction.
However, the equations which joint the solutions at $x=\pm\frac{L_{1}}{2}$,
are different compared to the conventional SNS junction. Requiring continuity
of both the Green’s function and the current through the junction yields
$\displaystyle
F\left(\pm\frac{L_{1}}{2}+0^{+}\right)=F\left(\pm\frac{L_{1}}{2}+0^{-}\right)$
(18)
$\displaystyle\derivative{F}{x}\left(\pm\frac{L_{1}}{2}+0^{\pm}\right)+\frac{\gamma_{0}}{v_{F}}f_{S1}\text{sgn}(\omega_{n})e^{\pm
i\frac{\phi}{2}}=\derivative{F}{x}\left(\pm\frac{L_{1}}{2}+0^{\mp}\right).$
(19)
This expression differs from the expression for the SNS junction in the
appearance of the $f_{S1}$-term on the right hand side of Eq. (19). The CPR
following from these equations is
$\displaystyle I(\phi)$
$\displaystyle=\sum_{n}\frac{1}{4}e^{-2\sqrt{\frac{2|\omega_{n}|}{D}}}\text{Im}((A_{n}-iB_{n})e^{i\frac{\phi}{2}})^{2}$
(20)
$\displaystyle=\frac{1}{4}\sum_{n}e^{-2\sqrt{\frac{2|\omega_{n}|}{D}}}\left(A_{n}^{2}-B_{n}^{2}\sin{\phi}+2A_{n}B_{n}\cos{\phi}\right)$
(21)
where $A_{n}$ and $B_{n}$ are real coefficients given by
$A_{n}=\frac{\gamma_{0}f_{S0}}{2|\omega_{n}|}$ and
$B_{n}=-i\gamma_{0}f_{S1}\sqrt{\frac{D\omega_{n}}{2}}\frac{1}{|\omega_{n}|}$.
This implies that
$\displaystyle\phi_{0}=\arctan
2\frac{\sum_{n=-\infty}^{\infty}A_{n}B_{n}e^{-2\sqrt{\frac{2|\omega_{n}|}{D}}}}{\sum_{n=-\infty}^{\infty}(A_{n}^{2}-B_{n}^{2})e^{-2\sqrt{\frac{2|\omega_{n}|}{D}}}}$
(22)
The $\phi_{0}$-effect is largest if $A_{n}=B_{n}$ and small if
$|\frac{B_{n}}{A_{n}}|$ is not close to 1. This ratio is given by
$\displaystyle|\frac{B_{n}}{A_{n}}|$
$\displaystyle=\frac{\gamma_{0}|f_{S1}|\sqrt{\frac{D}{2|\omega_{n}|}}2|\omega_{n}|}{\gamma_{0}f_{S0}v_{F}}=\frac{|f_{S1}|}{f_{S0}}\frac{1}{v_{F}}\sqrt{2|\omega_{n}|D}$
(23)
Recall that the diffusion constant is given by $D=v_{F}^{2}\tau$. This means
that $\frac{1}{v_{F}}\sqrt{2|\omega_{n}|D}=O(\sqrt{|\omega_{n}|\tau})$, which
is small in the diffusive regime. In fact, in the derivation of the Usadel
equation, it is assumed that $\frac{1}{\tau}\gg|\Delta|$ and thus
$\frac{1}{\tau}\gg|\omega_{n}|$ for every $n$ that contributes significantly
to the current. This means the effect can only be large if $f_{S1}\gg f_{S0}$.
This constraint can only be satisfied if,
$h\gg\sqrt{\omega_{n}^{2}+\Delta^{2}}$ for all Matsubara frequencies that have
a significant contribution to the critical current. However, in the setup
discussed in this paper the condition $h\gg\Delta$ can not be realized, since
the magnetisation is induced via the superconductor and a high magnetisation
destroys the superconductivity. The $\phi_{0}$-effect is suppressed by a
factor $\sqrt{|\omega_{n}|\tau}$ in the linearized case.
Next the temperature dependence is discussed. Because the $\phi_{0}$-effect is
small we can simplify the equation for $\phi_{0}$ to
$\displaystyle\phi_{0}\approx
2\frac{\sum_{n=-\infty}^{\infty}A_{n}B_{n}e^{-2\sqrt{\frac{2|\omega_{n}|}{D}}}}{\sum_{n=-\infty}^{\infty}(A_{n}^{2})e^{-2\sqrt{\frac{2|\omega_{n}|}{D}}}}.$
(24)
Now, consider the behaviour at low temperatures. Since the triplet component
$f_{S1}$ is odd in frequency, whereas the singlet component $f_{S0}$ is even
in frequency, $|\frac{B_{n}}{A_{n}}|$ is small for small Matsubara
frequencies. As the temperature is decreased these terms become more and more
dominant in the sum. Thus, at low temperatures, the $\phi_{0}$-effect is
increases with increasing temperature. On the other hand, for large Matsubara
frequencies the ratio between triplet and singlet components
$|\frac{f_{s1}}{f_{s0}}|=O(\frac{h}{|\omega_{n}|})$. This means that at high
temperatures the $\phi_{0}$-effect decreases. Therefore, the $\phi_{0}$-effect
must be non-monotonic as a function of temperature, it attains a maximum.
Moreover, since $\sqrt{\tau}$ is an $\omega_{n}$-independent prefactor it can
not determine the maximum. The temperature at which the maximum is attained is
determined by only two dimensionless quantities,
$\frac{\Delta}{E_{\text{Th}}}$ and $\frac{h}{\Delta}$.
An interesting limit is the limit in which $\sqrt{\frac{2\pi T}{D}}L_{1}\ll 1$
and $h\ll\Delta$ so that the exponential suppression can be ignored to first
order in $\sqrt{\frac{2\pi T}{D}}L_{1}$. The following expression is obtained
111Strictly speaking, the linearized case leads to a divergency as T goes to
0. We therefore replace $\Delta/|\omega_{n}|$ by
$\Delta/\sqrt{\omega^{2}+\gamma_{0}^{2}}$, based on the non-linearized case:
$\displaystyle\phi_{0}\approx$
$\displaystyle\frac{\sum_{n=0}^{\infty}A_{n}B_{n}}{\sum_{n=0}^{\infty}A_{n}^{2}}=h\sqrt{\tau}\frac{\sum_{n=0}^{\infty}\frac{1}{(\omega_{n}^{2}+\Delta^{2})^{2}\sqrt{\omega_{n}}}}{\sum_{n=0}^{\infty}\frac{1}{\omega_{n}^{4}+\Delta^{2}\omega_{n}^{2}}}$
(25)
The multiplication of the sum with $\sqrt{\tau}$ signals the dirty limit
suppression of the $\phi_{0}$-effect. The resulting expression is evaluated
numerically as a function of temperature. Numerical evaluation confirmed the
non-monotonicity, see Fig. 2.
Figure 2: The $\phi_{0}$-effect as a function of temperature as calculated
using Eq. (25). The $\phi_{0}$-effect is suppressed at low temperatures. The
$\phi_{0}$-effect is given in units of the small quantity
$h\sqrt{\tau/\Delta}$.
In the following sections we discuss the full non-linear equation, and we show
that the diode-effect is non-monotonic with temperature for short junctions.
## IV Non-linearized case: The superconducting diode effect
To investigate the diode effect one needs to go beyond the linear approach and
solve numerically the Usadel equation. In this section we present our
numerical results for the supercurrent in the FIS-TI-FIS junction. As a first
step it is convenient to write the Usadel equation, Eq. (10), in dimensionless
form, normalising $x$ by the total length of the junction. The obtained
equation is
$\displaystyle\hat{\nabla}(\check{G}\hat{\nabla}\check{G})=[\frac{\omega_{n}}{E_{\text{Th}}}\check{\tau}_{3}+\frac{\Gamma(x)}{2E_{\text{Th}}}\check{G}_{S0},\check{G}],$
(26)
$\displaystyle\hat{\nabla}=\derivative{x}+\frac{\Gamma(x)L}{2v_{F}}\tilde{b}_{y}[\check{G}_{S1},\cdot]$
(27)
Thus, all energies are given in units of the Thouless energy
$E_{\text{Th}}=\frac{D}{L^{2}}$, whereas lengths are given in units of $L$.
The strength of the proximity effect is described by the dimensionless
parameter
$\frac{\gamma_{0}L}{v_{F}}=\frac{\gamma_{0}}{E_{\text{Th}}}\frac{l}{L}$, where
$l=v_{F}\tau$ is the mean free path, which in the diffusive limit must be the
shortest length involved in the problem. This puts a constraint on the
magnitude of the new dimensionless quantity, it must be much smaller than
$\frac{\gamma_{0}}{E_{\text{Th}}}$. However, without this term the equations
reduce to the equations for the SNS junction, which is known not to have a
diode effect [28], it has no time-reversal symmetry breaking. Thus, if the new
quantity is very small, the diode effect is very small. Therefore,
$\frac{\gamma_{0}}{E_{\text{Th}}}$ must be chosen large, that is, the contact
between the superconductor and the TI must be good to have a large
$\gamma_{0}$.
To solve the non-linearized Usadel equation, Eq. (10), the so-called Riccati
parametrisation is used,
$\check{G}=\frac{1}{1+\bar{\gamma}\tilde{\gamma}}\begin{bmatrix}1-\bar{\gamma}\tilde{\gamma}&2\gamma\\\
2\tilde{\gamma}&-1+\bar{\gamma}\tilde{\gamma}\end{bmatrix},$ (28)
where $\bar{\gamma}$ and $\tilde{\gamma}$ are the Riccati parameters.
In principle, the pair potential $\Delta$ has to be determined self-
consistently since it is suppressed by the exchange field [61]. In the
numerical calculations we choose values of the exchange field smaller than
$\frac{h}{\Delta_{0}}=\frac{2}{5}$. Other parameters are set as follows:
$\frac{\gamma_{0}}{E_{\text{Th}}}=25$, whereas $\frac{L_{1}}{L}$ is chosen to
be $\frac{1}{10}$ and $\frac{l}{L}=0.08\ll 1$.
Fig. 1b shows our numerical results for the CPR obtained from the non-
linearized Eq. (10). One can see a finite current value at $\phi=0$ associated
with the appearance of the anomalous phase $\phi_{0}$. Moreover, even though
small, there is a difference in the absolute value of the maximum and minimum
of the current. This asymmetry reflects the diode effect. By increasing the
temperature both the current at zero phase and the critical current decrease.
We now study the temperature dependence of the diode effect for different
exchange fields and sizes of the junction. The numerical results for the diode
efficiency are shown in Fig. 3. Interestingly, we find a non-monotonic
behaviour with a maximum efficiency at a finite temperature, $T_{d}$. It is
important to notice, that by computing $\eta$ in Fig. 3, the self-consistency
of the pair potential is ignored, because of the reduction of computational
costs. However, we verify that for all values of $h$ considered here, the
self-consistency leaves the magnitude of the gap almost unchanged for
temperatures of the order of $T=T_{d}$. The critical temperatures for all
cases shown in Fig. 3a lie right of the dashed vertical line.
If the exchange field is increased, Fig. 3a, the diode efficiency becomes
larger. $\eta$ increases approximately linearly with $h$. The temperature at
which the diode efficiency is maximal is almost independent of the exchange
field, with $T_{d}\approx 0.18$.
We also investigate the influence of the distance between the leads, $L_{1}$
on $\eta$, see Fig. 3b. Specifically, $E_{Th1}=D/L_{1}^{2}$ is varied, whereas
the quantities $\frac{l}{L}$ and $\frac{L_{1}}{L}$ are held constant. As the
Thouless energy is decreased, the diode efficiency decreases. Moreover, the
temperature $T_{d}$ at which the diode effect is maximal decreases with the
length of the junction. For enough long junctions the dependence of $\eta$ on
temperature becomes monotonic.
(a)
(b)
Figure 3: The temperature dependence of the diode efficiency $\eta$ for
different values of the exchange field strength $h$ (a), and Thouless energy
$E_{\text{Th}1}=\frac{D}{L_{1}^{2}}$ of the TI part (b). The critical
temperature is for all magnitudes of the exchange field considered here larger
than $0.9T_{c}$, highlighted by the black dotted line.
So far, we have considered disordered systems with low transparent S/FI
interfaces. One can, though, increase the $\phi_{0}$ and diode effects by
relaxing these conditions. On the one hand, our analytical results in section
III indicate that the $\phi_{0}$-effect can be increased by increasing $\tau$,
see for example Eq. (25). We also verified numerically that the diode effect
increases if the degree of disorder decreases.
On the other hand, we also investigated the effect of increasing the ratio
$\frac{L_{1}}{L}$. Our numerical calculations demonstrate that whereas
$I(\phi=0)/I_{c}$ increases, the diode effect decreases with increasing
$\frac{L_{1}}{L}$. This can be explained as follows. By increasing the
distance $L_{1}$ between the electrodes the CPR becomes more sinus-like. To be
precise, when increasing $\frac{L_{1}}{L}$ from $\frac{1}{10}$ to
$\frac{1}{2}$, the ratio between the magnitudes of the second and first
harmonics decreases from $\approx\frac{1}{6}$ to $\approx\frac{1}{10}$. As we
discussed before, besides the breaking of time-reversal and inversion
symmetries, the diode effect relies crucially on the contribution of higher
harmonics to the CPR.
A way to increase the higher harmonics contribution, is to increase the
coupling between the superconducting correlations from the left and right
electrodes. This can be achieved by increasing the transparency of the FIS/TI
interfaces, as shown in Fig. 4.
Figure 4: The diode efficiency as a function of temperature for different
values of $\gamma_{0}$.
Finally, another way to increase the diode effect is by increasing the
exchange field, as shown in Fig. 3. In our junction, however, the value of $h$
is limited by the critical field of the superconductor. To increase the
strength of the exchange field without suppressing superconductivity in the S
electrodes one could add an additional ferromagnetic insulating layers
directly on top of the TI between the two superconductors, similar to the
situation investigated in Refs. [55, 62, 63]. In that case the exchange field
can be larger than the superconducting gap and the diode effect may increase.
## V Conclusions
We present a study of the $\phi_{0}$ and diode effects in a FIS-TI-FIS
Josephson junction. Though disorder tend to suppress them [40], we found, even
in the diffusive limit, sizable effects without applying any external field.
We found that by increasing the FIS/TI interface transparency and the magnetic
field one can increase the diode effect. For short junctions the diode effect
is non-monotonic as a function of temperature. By increasing the distance
between the electrodes the $\phi_{0}$-effect is enhanced, however the diode
effect is suppressed due to the loss of higher harmonics.
From the point of view of materials the proposed structure can be fabricated
with well studied material combinations. On the one hand the use of
topological insulators in Josephson junctions is well understood [64, 65, 66,
67, 68, 69, 70, 71, 72]. On the other hand, spin-split superconductivity has
been measured in several experiments on ferromagnetic insulator/superconductor
bilayers, as for example EuS/Al structures [73, 74, 51, 54]. Moreover, good
interfaces between TI and FI has been reported in Ref. [75].
## Acknowledgements
We thank Stefan Ilic for useful discussions. We acknowledge financial support
from Spanish AEI through project PID2020-114252GB-I00 (SPIRIT). FSB
acknowledges financial support from the European Union’s Horizon 2020 Research
and Innovation Framework Programme under Grant No. 800923 (SUPERTED), the A.
v. Humboldt Foundation, and the Basque Government through grant IT-1591-22.
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|
# One-class Recommendation Systems with the Hinge Pairwise Distance Loss and
Orthogonal Representations
Ramin Raziperchikolaei Young-joo Chung
###### Abstract
In one-class recommendation systems, the goal is to learn a model from a small
set of interacted users and items and then identify the positively-related
user-item pairs among a large number of pairs with unknown interactions. Most
previous loss functions rely on dissimilar pairs of users and items, which are
selected from the ones with unknown interactions, to obtain better prediction
performance. This strategy introduces several challenges such as increasing
training time and hurting the performance by picking ”similar pairs with the
unknown interactions” as dissimilar pairs. In this paper, the goal is to only
use the similar set to train the models. We point out three trivial solutions
that the models converge to when they are trained only on similar pairs:
collapsed, partially collapsed, and shrinking solutions. We propose two terms
that can be added to the objective functions in the literature to avoid these
solutions. The first one is a hinge pairwise distance loss that avoids the
shrinking and collapsed solutions by keeping the average pairwise distance of
all the representations greater than a margin. The second one is an
orthogonality term that minimizes the correlation between the dimensions of
the representations and avoids the partially collapsed solution. We conduct
experiments on a variety of tasks on public and real-world datasets. The
results show that our approach using only similar pairs outperforms state-of-
the-art methods using similar pairs and a large number of dissimilar pairs.
## 1 Introduction
Feedback in recommendation systems (RSs) can be explicit or implicit. In the
case of explicit feedback [20, 24, 35, 12], users provide rating values after
interacting with the items, which shows how satisfied they are. In the case of
implicit feedback [20, 11, 15], which is a common scenario in the real world,
we only know if the user interacted with an item or not (such as clicked,
purchased, etc.). The goal of one-class recommendation systems is to solve the
implicit feedback prediction problem. It’s called a one-class problem since a
”no interaction” between a user and an item in the training set does not
necessarily mean the user does not like that item. It just means we do not
have enough information about their interaction. That’s because the set of the
items in RSs is huge and the users can’t see all the items, i.e., users can
only see a small subset of the items and then interact with a few of them.
There are three main steps in learning a RS model to predict the implicit
feedback. The first step is to learn user and item representations. This can
be done by learning user and item embedding matrices from their IDs [24, 35,
12, 8, 9], learning user and item multi-layer perceptrons (MLPs) from the
interaction vectors [21, 30, 5] or side information [23, 34, 4], or learning
graph neural networks from the bipartite user-item graph [25, 31, 32, 10].
The second step is to model the interaction score from the user and item
representations. The common functions are dot product [11, 15, 24, 12, 4, 32,
10, 18], cosine similarity [30, 31], and neural networks over the concatenated
user and item representations [8, 9, 5, 17, 16].
The third step is to optimize a loss function, which gets smaller values as
the model gives a larger score to the similar pairs of the user and items
compared to the dissimilar ones. Different types of loss functions have been
used in the literature. Mean squared error (MSE) loss [11, 15, 24, 12, 4, 32,
17, 18] and Binary cross-entropy (BCE) [30, 8, 5] directly minimize the
difference between the predicted and the actual scores. The Bayesian
personalized rank (BPR) loss [19, 9, 10] tries to make the interaction score
of the similar pairs greater than the dissimilar ones, instead of directly
mapping them to the actual scores. The contrastive learning loss tries to put
representations of the similar pairs close to each other and put the
dissimilar ones far away [18, 14, 29].
All the above loss functions need both similar and dissimilar pairs of the
users and items to learn a model. If we train these loss functions only using
the similar pairs, we get a collapsed solution: all representations will be
mapped to the same point in the latent space and the model predicts the same
interaction score for all the pairs. The performance of the collapsed solution
is as bad as assigning random representations to the users and items. So the
dissimilar sets are essential in training RS models to avoid the collapsed
solution.
In one-class recommendation systems, we only have access to the implicit
(known) interactions, and the rest of the interactions are unknown. To create
a dissimilar set of users and items, the common approach is to randomly select
a set of user and item pairs with the unknown interactions and consider them
dissimilar [30, 8, 5, 4, 32, 18]. Another strategy is to find out the hard-
negatives: the pairs with the unknown interactions that the model has
difficulty with classifying as dissimilar [33, 2, 3].
Creating dissimilar pairs from unknown ones is problematic for two main
reasons. First, as we show in experiments, we might need a large number of
dissimilar pairs to achieve reasonable results, which makes the training slow.
Second, using a large number of dissimilar pairs increases the chance of
converting a ”similar pair with an unknown interaction” to a dissimilar pair,
which hurts the performance. This issue is more severe in the case of the hard
negative approach, since ”similar pairs with unknown interactions” are by
definition difficult to be classified as dissimilar, and will be mistakenly
taken as hard negatives.
The main goal of this paper is to propose a new objective function that avoids
the collapsed solution without using the dissimilar pairs. As we define it
mathematically in Section 3, at the collapsed solution, the average pairwise
distance between all the user-user, item-item, and user-item representations
is $0$. To avoid the collapsed solution, we propose a hinge pairwise distance
loss which keeps the pairwise distance above a margin. We show that this new
loss function alone is not enough to generate high-quality representations, as
the solution might converge to a ”partially collapsed solution”, where all
users and items converge to exactly two unique representations. To avoid this
solution, which performs as poorly as the collapsed solution, we propose to
minimize the correlation between the dimensions of representations by making
them orthogonal.
We conduct extensive experiments on several real-world and public datasets in
both warm-start and cold-start settings. The experimental results in Section 4
show that our approach, which only uses similar pairs, has a better prediction
performance than state-of-the-art methods, which use both similar and
dissimilar pairs.
#### Notations.
We denote the sparse interaction matrix by $\mathbf{R}\in\mathbb{R}^{m\times
n}$, where $m$ and $n$ are the number of users and items, respectively,
$R_{jk}>0$ is the interaction value of the user $j$ on the item $k$, and
$R_{jk}=0$ means the interaction is unknown. The goal is to predict the
unknown interactions in $\mathbf{R}$. The $i$th row of a matrix $\mathbf{H}$
is shown by $\mathbf{H}_{i,:}$ and the $j$th column is shown by
$\mathbf{H}_{:,j}$. The $d$-dimensional representations of the all users and
all items are denoted by $\mathbf{Z}^{u}\in\mathbb{R}^{m\times d}$ and
$\mathbf{Z}^{i}\in\mathbb{R}^{n\times d}$, respectively. The representation of
the $j$th user and $k$th item are denoted by
$\mathbf{z}_{j}^{u}=\mathbf{Z}_{j,:}^{u}$ and
$\mathbf{z}_{k}^{i}=\mathbf{Z}_{k,:}^{i}$, respectively.
## 2 Related works
Since our main contribution is proposing a new objective function, we review
the different loss functions used in the RS literature. Most of these loss
functions are defined based on the predicted user-item interactions from the
representations. The most common mappings from representations to predicted
interactions are dot product [11, 15, 24, 12, 4, 32, 10, 18], cosine
similarity [30, 31], and neural networks [8, 9, 5, 17, 16]:
$\text{dot
product:}\hat{R}_{jk}=(\mathbf{z}_{j}^{u})^{T}\mathbf{z}_{k}^{i},\quad\text{cosine:}\hat{R}_{jk}=\frac{(\mathbf{z}_{j}^{u})^{T}\mathbf{z}_{k}^{i}}{||\mathbf{z}_{j}^{u}||||\mathbf{z}_{k}^{i}||},\quad\text{NNs:}\hat{R}_{jk}=h([\mathbf{z}_{j}^{u},\mathbf{z}_{k}^{i}])$
(1)
where $[\cdot,\cdot]$ merges the two vectors and $h()$ is a neural network. In
[8, 5],$[\cdot,\cdot]$ is concatenation and $h()$ is an MLP. In [9],
$[\cdot,\cdot]$ is outer product and $h()$ is a convolutional network.
To define the loss functions in RSs, previous works use sets of similar and
dissimilar pairs of the users and items. The similar set $S^{+}$ contains all
the users and items that interacted with each other, i.e., $(j,k)\in S^{+}$ if
$R_{jk}=1$. The dissimilar set $S^{-}$ contains a subset of the users and
items with unknown interactions, i.e., $R_{jk}=0$.
Two popular loss functions in the literature are mean squared error (MSE) [11,
15, 24, 12, 4, 32, 17, 18] and binary cross-entropy (BCE) [30, 8, 5], which
directly minimize the difference between the predicted and actual
interactions:
$\displaystyle
l_{\text{BCE}}(\mathbf{z}_{j}^{u},\mathbf{z}_{k}^{i})=-(R_{jk}\log\hat{R}_{jk}+(1-R_{jk})\log(1-\hat{R}_{jk})),$
$\displaystyle
l_{\text{MSE}}(\mathbf{z}_{j}^{u},\mathbf{z}_{k}^{i})=(\hat{R}_{jk}-R_{jk})^{2},$
(2)
where $j,k\in S^{+}\cup S^{-}$. On the other hand, the BPR loss [19, 9, 10] is
defined based on the difference between the predicted interactions of the
similar and dissimilar pairs:
$l_{\text{BPR}}(\mathbf{z}_{j}^{u},\mathbf{z}_{k}^{i})=-\ln{\sigma(\hat{R}_{jk}-\hat{R}_{jl})},$
(3)
where $j,k\in S^{+}$ and $j,l\in S^{-}$, i.e., user $j$ is similar to item $k$
and dissimilar to item $l$. Note that the above loss functions are optimized
over the user and item representations, $\mathbf{z}_{j}^{u}$ and
$\mathbf{z}_{k}^{i}$, which are used to generate the predicted interaction.
Contrastive loss functions [6] are also used in recommendation systems [18,
14, 29]. The idea is to map representations of similar pairs of the users and
items close to each other and the dissimilar ones far away:
$l_{\text{contrastive}}(\mathbf{z}_{j}^{u},\mathbf{z}_{k}^{i})=R_{jk}||\mathbf{z}^{u}_{j}-\mathbf{z}^{i}_{k}||^{2}+(1-R_{jk})\lambda\max{(0,m_{d}-||\mathbf{z}^{u}_{j}-\mathbf{z}^{i}_{k})||)}^{2},$
(4)
where $m_{d}$ is a margin and $j,k\in S^{+}\cup S^{-}$.
As we explain in the next section, without the dissimilar sets, all the above
loss functions will converge to a collapsed solution, which performs as bad as
a random assignment of the representations. The goal of this paper is to
introduce new terms to let the above loss functions be trained only on similar
sets and still avoid the collapse solution.
## 3 Proposed method
As mentioned in the introduction, there are several issues in using the
dissimilar pairs in one-class RSs. First, we are never sure whether the
selected pairs with the unknown interactions are dissimilar or not. A ”no
interaction” does not necessarily mean that the user did not like the item.
Selecting similar pairs and using them as dissimilar pairs can hurt the
performance. Second, as we show in experiments, previous works use a large
number of dissimilar pairs to get reasonable performance, which will increase
the training time. Our goal is to learn a recommendation system model by
discarding the dissimilar set of users and items, training only on the similar
pairs, avoiding the collapsed solution, and achieving state-of-the-art
results.
In this section, we first explain how the previous objective functions
converge to a collapsed solution when they only use similar pairs. Then, we
explain our proposed method by considering the
$E_{\text{cont}}(\mathbf{Z}^{u},\mathbf{Z}^{i})$ in Eq. (LABEL:e:con_reg) as
the main term of the objective function. We show how we avoid the collapsed,
partially collapsed, and shrinking solutions by proposing new terms. Then, we
will explain how our objective function works with other loss functions.
Finally, we go over the computational complexity of the proposed objective
function.
### 3.1 The collapsed solution without the dissimilar set
The loss functions introduced in Section 2 use both similar and dissimilar
pairs to predict the actual representations. Let us explain what happens if we
only use the similar pairs and discard the dissimilar ones in two cases: 1)
using the loss function $l_{\text{MSE}}$ in Eq. (2) and dot product mapping
function in Eq. (1), and 2) using the constrastive loss defined in Eq. (4):
$\begin{split}&E_{\text{MSE}}(\mathbf{Z}^{u},\mathbf{Z}^{i})=\sum_{j,k\in
S^{+}}((\mathbf{z}^{u}_{j})^{T}\mathbf{z}^{i}_{k}-1)^{2},\\\
&E_{\text{cont}}(\mathbf{Z}^{u},\mathbf{Z}^{i})=\sum_{j,k\in
S^{+}}||\mathbf{z}^{u}_{j}-\mathbf{z}^{i}_{k}||^{2}.\end{split}$ (5)
Both objective functions above are defined on similar pairs $S^{+}$ and
optimizing them will lead to a collapsed solution, where all the users and
item representations are mapped to a certain point in the latent space. The
optimal solution of the $E_{\text{MSE}}$ is achieved by mapping all the user
and item representations to any $d$-dimensional vector with a unit L2 norm.
That’s because the dot product of all pairs becomes $1$, i.e.,
$\hat{R}_{jk}=(\mathbf{z}^{u}_{j})^{T}\mathbf{z}^{i}=1$, which makes the loss
value $0$ for all the terms. In the case of $E_{\text{cont}}$, the optimal
(and collapsed) solution is achieved by mapping all the user and item
representations to any $d$-dimensional vector.
At the collapsed solution, which is the result of removing dissimilar pairs
from the loss functions, the model always returns the same prediction, no
matter what the input pairs are. This works as poorly as a random model.
Note that different combinations of mapping functions and loss functions can
give us different objective functions. In all cases, without dissimilar pairs,
the result is a collapsed solution.
### 3.2 Avoiding the collapsed solution: hinge pairwise distance loss
Let us assume the $d$-dimensional representations of the users and items is
denoted by $\mathbf{Z}^{u}\in\mathbb{R}^{m\times d}$ and
$\mathbf{Z}^{i}\in\mathbb{R}^{n\times d}$, respectively. The joint user-item
representation is achieved by vertically concatenating the user and item
representations,
$\mathbf{Z}=[\mathbf{Z}^{u},\mathbf{Z}^{i}]\in\mathbb{R}^{(m+n)\times d}$. The
pairwise distance between all the representations in $\mathbf{Z}$ is computed
as:
$d_{p}=E_{\text{cont}}(\mathbf{Z},\mathbf{Z})=\frac{1}{(m+n)^{2}}\sum_{l,s=1}^{m+n}||\mathbf{z}_{l}-\mathbf{z}_{s}||^{2}.$
(6)
Note that $d_{p}$ computes distance between all the user-user, item-item, and
user-item representations, which is different from
$E_{\text{cont}}(\mathbf{Z}^{u},\mathbf{Z}^{i})$ that computes the distance
between similar pairs of the users and items.
Mathematically, at the collapsed solution, we have $d_{p}=0$. To avoid the
collapsed solution, we propose a hinge pairwise distance loss that keeps the
average pairwise distance $d_{p}$ greater than a margin $m_{p}$. The new
objective function can be written as:
$E=E_{\text{cont}}(\mathbf{Z}^{u},\mathbf{Z}^{i})+E_{d_{p}}(\mathbf{Z})=\sum_{j,k\in
S^{+}}||\mathbf{z}^{u}_{j}-\mathbf{z}^{i}_{k}||^{2}+\max{(0,m_{p}-d_{p})}^{2}.$
(7)
Note that $d_{p}$ involves computing the distances between all the pairs,
which could be very time-consuming. As we show below, $d_{p}$ is equivalent to
the summation of the variance of each dimension, which can be computed way
faster.
Let us denote the $q$th dimension of the $l$th representation as $z_{l,q}$,
and the pairwise distance of the $q$th dimension as $d_{p}^{q}$. Then, $d_{p}$
in Eq. (6) can be separated over the $d$ dimensions:
$d_{p}=\sum_{q=1}^{d}d_{p}^{q}=\frac{1}{(m+n)^{2}}\sum_{q=1}^{d}\sum_{l,s=1}^{m+n}(z_{lq}-z_{sq})^{2}.$
(8)
We can rewrite $d_{p}^{q}$ as:
$d_{p}^{q}=\frac{1}{(m+n)^{2}}\sum_{l,s=1}^{m+n}(z_{lq}-z_{sq})^{2}=\frac{1}{(m+n)^{2}}\sum_{l,s=1}^{m+n}z_{lq}{{}^{2}}+z_{sq}{{}^{2}}-2z_{lq}z_{sq}=\\\
\frac{2}{(m+n)^{2}}\sum_{l,s=1}^{m+n}z_{lq}{{}^{2}}-\frac{2}{(m+n)^{2}}\sum_{l=1}^{m+n}z_{lq}\sum_{s=1}^{m+n}z_{sq}=\frac{2}{(m+n)}\sum_{l=1}^{m+n}z_{lq}{{}^{2}}-2\bar{\mathbf{Z}}_{:,q}^{2}=2\text{var}(\mathbf{Z}_{:,q}).$
(9)
So twice the variance of a dimension is equivalent to the pairwise distances
of the representations in that dimension. This means that at the collapse
solution, the variance of each dimension is $0$, and to avoid this solution,
we just need the summation of the variance of the dimensions to be greater
than a margin. At the end of this section, we explain how the new formulation
helps the computational complexity.
### 3.3 Avoiding the partially collapsed solution: orthogonality term
While the objective function of Eq. (7) avoids the collapsed solution, it
might give us low-quality representations by converging to a ”partially
collapsed solution”. This solution returns only two representations for the
whole set of users and items: a portion of the users and items are mapped to
one representation and the rest of them are mapped to a second representation.
Below, we explain when the objective function converges to this solution and
how we avoid it by proposing an orthogonality term.
To explain the partially collapsed solution, we consider the user-item
bipartite graph. This graph contains two disjoint sets of nodes: set $I$
contains the items and set $U$ contains the users. The edges are determined by
$S^{+}$, i.e., there is an edge between nodes of user $j$ and item $k$ if
$(j,k)\in S^{+}$. Now, let us assume the graph can be partitioned into $V>1$
disjoint components $\\{C_{v}\\}_{v=1}^{V}$, where the edges in each component
are denoted by $S_{v}^{+}$. Since it’s a disjoint partition, each node from
$U$ and $I$ appears in exactly one component, $\cup\\{S_{v}^{+}\\}=S^{+}$, and
there is no edge across the nodes of the two different components. The
interaction matrix is more than $99\%$ sparse in real world scenarios, which
can easily lead to having multiple disjoint partitions.
The optimal solution of our objective function in the setting above can be
achieved by assigning representation $\mathbf{p}_{1}$ to all the nodes in some
of the components and $\mathbf{p}_{2}$ to all the nodes in the rest of the
components.
First, let us show how the above solution makes makes $E_{\text{cont}}=0$.
Here, for simplicity, let us assign $\mathbf{p}_{1}$ to the components
$1,\dots,\frac{v}{2}$ and assign $\mathbf{p}_{2}$ to the components
$\frac{v}{2}+1,\dots,V$. Then we have:
$E_{\text{con}}=\sum_{j,k\in
S^{+}}||\mathbf{z}^{u}_{j}-\mathbf{z}^{i}_{k}||^{2}=\sum_{v=1}^{V}\sum_{j,k\in
S_{v}^{+}}||\mathbf{z}^{u}_{j}-\mathbf{z}^{i}_{k}||^{2}=\\\
\sum_{v=1}^{\frac{V}{2}}\sum_{j,k\in
S_{v}^{+}}||\mathbf{z}^{u}_{j}-\mathbf{z}^{i}_{k}||^{2}+\sum_{v={\frac{V}{2}+1}}^{V}\sum_{j,k\in
S_{v}^{+}}||\mathbf{z}^{u}_{j}-\mathbf{z}^{i}_{k}||^{2}=\\\
\sum_{v=1}^{\frac{V}{2}}\sum_{j,k\in
S_{v}^{+}}||\mathbf{p}_{1}-\mathbf{p}_{1}||^{2}+\sum_{v={\frac{V}{2}+1}}^{V}\sum_{j,k\in
S_{v}^{+}}||\mathbf{p}_{2}-\mathbf{p}_{2}||^{2}=0.$ (10)
We can easily prove that $E_{\text{con}}$ becomes $0$ for the other ways of
assigning the two representations to the components.
To make the hinge pairwise distance loss $E_{d_{p}}=0$, we just need to put
the two points $\mathbf{p}_{1}$ and $\mathbf{p}_{2}$ far away from each other,
which makes the average variance of the dimensions greater than the margin
$m_{d}$. In the extreme case, consider each dimension of $\mathbf{p}_{1}$ goes
to $+\infty$ and each dimension of $\mathbf{p}_{2}$ goes to$-\infty$.
The partially collapsed solution is (almost) as bad as the collapsed solution.
It assigns the same representation to all the nodes in the same component
which makes them inseparable, while the nodes in each component should be
separated based on their interactions with the other nodes.
To avoid the partially collapsed solution, we look at the correlation
coefficients between the dimensions of the representations. If we have only
two unique representations in the user-item representations matrix
$\mathbf{Z}$, then there is a linear relationship between the dimension of the
$\mathbf{Z}$, and by definition the correlation between the dimensions becomes
maximum ($-1$ or $+1$). To avoid this, we add an orthogonality term to the
objective function to make the representations orthogonal:
$E_{ours}=\lambda_{1}E_{\text{cont}}(\mathbf{Z}^{u},\mathbf{Z}^{i})+\lambda_{2}E_{d_{p}}(\mathbf{Z})+\lambda_{3}E_{\text{orth}}(\mathbf{Z}),$
(11)
where $E_{\text{cont}}(\mathbf{Z}^{u},\mathbf{Z}^{i})$ and
$E_{d_{p}}(\mathbf{Z})$ are defined in Eq. (7) and we defined
$E_{\text{orth}}(\mathbf{Z})$ as follows:
$E_{\text{orth}}(\mathbf{Z})=\sum_{q=1}^{d}\sum_{s=q+1}^{d}\hat{\mathbf{Z}}_{:,q}^{T}\hat{\mathbf{Z}}_{:,s},$
(12)
where $\hat{\mathbf{Z}}$ is achieved by subtracting the mean of each dimension
from $\mathbf{Z}$, and $\hat{\mathbf{Z}}_{:,q}$ is the $q$th column of
$\hat{\mathbf{Z}}$. The orthogonality term let the objective avoid the
partially collapsed solution by making the off-diagonal values of the
covariance matrix small, which consequently makes the correlation of the
dimensions of the representations small.
### 3.4 Shrinking solution: why we need both terms
As explained above, if we only optimize the hinge pairwise distance loss
$E_{d_{p}}(\mathbf{Z})$ and $E_{\text{cont}}(\mathbf{Z}^{u},\mathbf{Z}^{i})$
together, without using the orthogonality term $E_{\text{orth}}(\mathbf{Z})$,
it will converge to a partially collapsed solution. On the other hand,
optimizing the orthogonality term $E_{\text{orth}}(\mathbf{Z})$ and
$E_{\text{cont}}(\mathbf{Z}^{u},\mathbf{Z}^{i})$, without
$E_{d_{p}}(\mathbf{Z})$, is enough to avoid both the collapsed and partially
collapsed solutions.
Then, a question arises here: do we need both hinge pairwise distance loss and
the orthogonality term in the same objective function? Or is it enough to only
enforce representation orthogonality to make the correlation small? In other
words, where does the solution converge if we set $\lambda_{2}=0$ in Eq. (11)?
To answer this question, let us consider a trivial (but not collapsed)
solution, where all users’ and items’ d-dimensional representations are
generated from a multivariate uniform distribution. More specifically, each of
the $d$ dimensions is drawn from the interval $[-r,r]$, independently from
other dimensions. In this scenario, the dimensions are uncorrelated, so the
orthogonality term $E_{\text{orth}}(\mathbf{Z})$ is almost $0$ in practice.
The maximum distance between any two representations is $4dr^{2}$, which is
the upper-bound for $E_{cont}(\mathbf{Z}^{u},\mathbf{Z}^{i})$. So by setting
$r$ to an extremely small number (but larger than $0$), we can make
$E_{cont}(\mathbf{Z}^{u},\mathbf{Z}^{i})$ extremely small. So this solution,
which generates low-quality random representations from a multivariate uniform
distribution, can make the objective function of Eq. (11) very small if we
don’t include the hinge pairwise distance loss. That’s because as we make $r$
smaller, the space of the representation shrinks, and the variance of each
dimension becomes very small. The hinge pairwise distance loss makes sure that
the average variance is larger than a margin and avoids this shrinking
solution.
### 3.5 Other objective functions in RSs
In this section, we explore the impact of the proposed terms on the different
objective functions. We explain what happens if we change the objective from
$E_{\text{cont}}$ to another one, achieved by combining loss functions in Eq.
(2) and mappings in Eq. (1). Is it necessary to use both $E_{d_{p}}$ and
$E_{\text{orth}}$ with the objective functions other than $E_{\text{cont}}$?
Let us explain what happens if we replace $E_{\text{cont}}$ by
$E_{\text{MSE}}$, defined in Eq. (LABEL:e:con_reg). By optimizing
$E_{\text{MSE}}$ on the similar pairs, the collapsed solution is achieved by
assigning all users and items to any point $\mathbf{p}\in\mathbb{R}^{d}$ such
that $||\mathbf{p}||_{2}=1$. Let us assume the user-item graph can be
partitioned into $V$ components. The partially collapsed solution is achieved
by assigning all representations to any two points
$\mathbf{p}_{1}\in\mathbb{R}^{d}$ and $\mathbf{p}_{2}\in\mathbb{R}^{d}$, where
$||\mathbf{p}_{1}||_{2}=1$, $||\mathbf{p}_{2}||_{2}=1$, such that all the
users and items in the same component get the same representation.
Unlike the previous case where we used $E_{\text{cont}}$, we do not get the
shrinking solution if we draw each dimension of each representation from a
multivariate uniform distribution in $[-r,r]$. That’s because as we make $r$
smaller, the absolute value of the dimensions of the representations
decreases, the dot product of the representations becomes smaller and further
away from $1$, and the loss function $E_{\text{MSE}}$ increases. So the loss
function $E_{\text{MSE}}$ itself avoids the shrinking solution.
Since we can prevent the collapsed and partially collapsed solutions using the
orthogonality term, without converging to a shrinking solution, we cannot gain
much by adding $E_{d_{p}}$. The results on CiteULike dataset in our
experimental results section confirm this.
For other objective functions in the literature, one needs to first verify the
existence of the collapsed, partially collapsed, and shrinking solutions and
then decide which terms to use to prevent them.
### 3.6 Computational complexity and batch-wise training
Here, we analyze the computational complexity of each term in our objective
function in Eq. (11). The time complexity of $E_{\text{cont}}$ and
$E_{\text{orth}}$ are $O(|S^{+}|d)$ and $O((m+n)d)$, respectively. For
$E_{d_{p}}$, if we use Eq. (6), the complexity is $O((m+n)^{2}d)$. If we use
Eq. (9) to compute the variance, then the time complexity of computing
$E_{d_{p}}$ will decrease to $O((m+n)d)$, which is linear in the total number
of users and items.
In our training, we use neural networks of the previous works to extract user
and item representations. Since we use batches to train the model and update
the representations, the three terms are computed on a batch and will have a
much smaller time complexity, which depends on the number of users and items
in the batch.
## 4 Experiments
### 4.1 Experimental setup
We conduct experiments in both warm-start and item cold-start settings. In the
warm-start setting, the interaction feedback of the test users and items is
available in training data and no new user or item will be added to the system
at test time. In the item cold-start setting, there is no feedback history for
the test items and we only have access to their side information.
In the cold-start setting, we use the following two datasets:
* •
CiteULike [27]. In this dataset, each user has a set of saved articles and the
goal is to recommend new (cold-start) articles to the users. We use the subset
provided by [26], which contains $5\,551$ users, $16\,980$ articles, and
$204\,986$ user-article pairs. The interaction matrix $\mathbf{R}$ is $99.8$%
sparse and $R_{jk}=1$ means user $j$ saved article $k$. A set of $3\,396$
articles are removed from the training data and used as the test cold-start
articles.
* •
Ichiba10m. This dataset contains $10$ million purchases from a specific
category of the Rakuten Ichiba 111www.rakuten.co.jp. There are around
$844\,000$ items, $1.4$M users, and $2\,000$ cold-start items. The dataset is
$99.9991$% sparse. The model uses the user interaction vector and item side
information, which includes item’s category, price, and title. The train/test
split is done based on the time period. The cold start items are the items
that didn’t appear during the training period.
In Ichiba10m dataset, the goal is to find interested users for a newly
released (cold-start) item. In CiteULike dataset, the goal is to assign a set
of cold-start articles to each user. In both datasets, we report recall to
evaluate the performance of the methods, which shows what portion of the
retrieved items/articles is relevant to the users. Since ranking all users is
time-consuming in Ichiba, we randomly sample 1 000 unobserved interactions for
each item and report recall by returning top 50 users with highest score.
We use the following two datasets in the warm-start setting:
* •
Lastfm222https://www.last.fm/. The
dataset333http://ocelma.net/MusicRecommendationDataset/lastfm-1K.html contains
$69\,149$ interactions from $1\,741$ users on $2\,665$ songs. The dataset is
$98.5$% sparse.
* •
AMusic [7]. The dataset contains $1\,700$ users, $13\,000$ musics, and
$46\,000$ interactions. The dataset is $99.8$% sparse.
We use the pre-processed datasets provided by Dong et al. [5], which is
publicly available for non-commercial
use444https://github.com/familyld/DeepCF.
We report Hit Ratio (HR) to evaluate the implicit feedback prediction
performance. We follow the same protocols as [5, 8] and truncate the rank list
at $10$ for both metrics. HR measures whether the actual test item exists in
the top-ranked list.
#### Baselines.
We compare our method with the state-of-the-art methods in both cold-start and
warm-start settings:
* •
Shared [18]. It uses a shared item network to learn both user and item
representations. The item network utilizes side information to create item
representations. The user representations are generated from the
representations of the items purchased by the user.
* •
Shared attention [18]. Same the the Shared model above, except that it uses
attention mechanism to combine the item representations.
* •
DropoutNet-WMF and DropoutNet-CDL [26]. The key idea is to apply input dropout
to the neural network model such that it can handle the missing input data in
the cold-start setting.
* •
ACCM [22]. It applies the attention mechanism to hybrid RSs to adjust the
importance of the input sources.
* •
BUIR-ID [13]. This is the first work to utilize the stop-gradient approach [1]
in RSs. Similar to SimPDO, BUIR-ID only uses positive pairs. The difference
from SimPDO is that BUIR-ID has two encoders and uses the momentum mechanism
to update the parameters of one encoder and to avoid the collapse solution.
* •
CFNet [5]. It uses interaction vector as the input and learns two user and two
item representations by two neural network models, which are then combined and
mapped to the final interaction by an MLP.
* •
NeuMF [8]. It uses IDs as the input and learns two user and two item
representations, which are then combined by an MLP to predict the final
interaction.
* •
DMF [30]. It learns user and item representations by training a neural
networks which uses the explicit feedback as the input and the dot-product as
the mapping function.
Ichiba10m
---
|
CiteULike
|
Figure 1: Impact of each term of our objective function on the results in
Ichiba10m (top row) and CiteULike (bottom row) datasets. We report the average
variance of each dimension, the average correlation between the dimensions,
and recall, as we train the models. The model that uses all three terms
achieves the best results.
#### RSs models to learn the user and item representations.
The specific models to map users and items to the low-dimensional
representations are taken from previous works since the novelty of our work is
in designing a new objective function, which works no matter what the model
is. In the cold-start setting, we use the Shared model proposed by
Raziperchikolaei et al. [18] and in the warm-start setting we use the neural
collaborative framework proposed by Wang et al. [28].
#### Implementation details.
We implemented our method using Keras with TensorFlow 2 backend. We ran all
the experiments on one Nvidia Tesla V100-SXM2 32GB GPU in the internal
cluster. We use grid search to set the hyper-parameters using a subset of
training set and a small validation set. We set the maximum number of epochs
to $50$. We use SGD with the learning rate of $0.5$ for our method in all
experiments. We set the batch size to $32$ in CiteULike and $128$ in other
datasets. We set $\lambda_{1}=0.01$, $\lambda_{1}=1$, and $\lambda_{3}=1$ in
all datasets. The margin $m_{p}$ is $0.1$ in the CiteULike and $0.01$ in all
other datasets. The dimension of the user and item embeddings are $100$ in the
CiteULike and Ichiba10m dataset and $1\,000$ in AMusic and Lastfm.
Figure 2: SimPDO vs Shared [18] as we change the number of training pairs from
$1$M to $10$M and report recall@50 in Ichiba10m dataset. SimPDO achieves
significantly better results when we use a smaller number of training pairs.
### 4.2 Experimental results
Our proposed objective function uses Similar pairs, Pairwise Distance loss,
and Orthogonality loss, and is denoted by SimPDO.
#### Impact of each term of SimPDO.
Our proposed objective function has three terms. The first term,
$E_{\text{cont}}$ or $E_{\text{MSE}}$, tries to generate a high interaction
score for the similar pairs, either by putting their representations close to
each other or by maximizing the dot product of the representations. The second
term, $E_{d_{p}}$ defined in Eq. (6), tries to maximize the pairwise distance
of all the pairs (or equivalently maximize the variance) and avoid the
collapsed and shrinking solutions. The third term, $E_{\text{orth}}$ in Eq.
(11), avoid the partially collapsed solution by making the representations
orthogonal.
In Fig. 1, we investigate the impact of each term by reporting: 1) the average
variance of the dimensions of all the representations, 2) the average
correlation between the dimensions of the representations, and 3) the recall,
as we train the models.
In the top row, we report the results on Ichiba10m dataset. We can see that
the method with all three terms achieves the maximum recall because: 1)
without $E_{\text{cont}}$, the similarity between the similar pairs will not
be preserved, 2) without $E_{d_{p}}$, as we can see in the second column, the
variance drops significantly towards $0$, which is a sign of the shrinking or
collapsed solution, and 3) without $E_{\text{orth}}$, as we can see in the
third column, the average correlation increase, which is a sign of the
partially collapsed solution.
In the bottom row of the Fig. 1, we show the results on CiteULike, where we
use $E_{\text{MSE}}$ instead of the $E_{\text{cont}}$. The results on the
CiteULike confirm that the model with all three terms performs best. Note that
in CiteULike the curve without $E_{d_{p}}$ performs almost as well as using
all the terms. In addition, note that even without $E_{d_{p}}$, the variance
does not drop significantly and it is a bit smaller compared to the other
curves. That’s because $E_{\text{orth}}$ avoids the collapsed and partially
collapsed solution and the shrinking solution does not happen when we use
$E_{\text{MSE}}$.
#### Training only on similar pairs: better results with fewer training
points.
In Ichiba10m dataset, we fix the test set to $2\,000$ cold-start items, change
the number of training pairs from $1$M to $10$M, and report the recall of our
method (SimPDO) and Shared model [18] in Fig. 2. SimPDO only uses similar
pairs in training and optimizes the objective function of Eq. (11). Shared
model [18] uses both similar and dissimilar pairs and optimizes the objective
function of Eq. (4).
Table 1: Comparison with the state-of-the-art cold-start methods on CiteULike dataset. The * in front of the methods’ name means we report the results from the Shared model paper [18]. Our method outperforms the competitors. We run SimPDO three times and report the mean and standard deviation of the results. method | recall@100
---|---
SimPDO | $\mathbf{66.8}\pm 0.11$%
Shared* [18] | $65.7$%
Shared attention* [18] | $66.4$%
DropoutNet-WMF* [26] | $65.2$%
DropoutNet-CDL* [26] | $62.9$%
ACCM [22] | $63.1$%
In SimPDO, all training pairs are similar pairs so we have no dissimilar
pairs. In Shared, the training pairs are divided between the similar and
dissimilar pairs in different portions. In Fig. 2, we considered two cases,
where the number of dissimilar pairs is equal to (first case) and three times
more than (second case) the number of similar pairs.
There are three remarkable points about the results of Fig. 2. First, SimPDO
performs better than the Shared model no matter how many training pairs are
used, which shows the advantage of training only on similar pairs. Second, the
gap between the Shared and SimPDO becomes smaller as we increase the portion
of the dissimilar pairs compared to the similar ones, which shows the
importance of using a large number of dissimilar pairs. Third, SimPDO is
significantly better than the Shared model using a smaller number of training
pairs. This is a big advantage of SimPDO when the datasets have billions of
pairs and training on all of them is time-consuming: SimPDO can be trained on
a smaller training set and still achieve reasonable results.
Table 2: Comparison with the state-of-the-art warm-start methods on AMusic and Lastfm datasets. The * in front of the methods’ name means we report the results from the DeepCF paper [5]. Our method achieves competitive results using smaller number of training pairs. We put $-$ for HR@5 of some of the methods since only HR@10 is reported in [5]. We run SimPDO, BUIR-ID, and CFNet three times and report the mean and standard deviation of the results. | AMusic | Lastfm
---|---|---
method | HR@5 | HR@10 | HR@5 | HR@10
SimPDO | $\mathbf{32.66\pm 0.4}$ | $\mathbf{43.47\pm 0.09}$ | $75.4\pm 0.1$ | $\mathbf{89.6\pm 0.18}$
BUIR-ID [13] | $25.76\pm 0.2$ | $36.5\pm 0.7$ | $70.16\pm 0.6$ | $84.15\pm 0.9$
CFNet [5] | $29.7\pm 0.08$ | $38.4\pm 0.003$ | $\mathbf{77.1\pm 0.17}$ | $88.6\pm 0.02$
CFNet* [5] | - | $41.2$ | - | $89.9$
CFNet-ml* [5] | - | $40.7$ | - | $88.3$
CFNet-rl* [5] | - | $39.4$ | - | $88.4$
NeuMF* [8] | - | $38.9$ | - | $88.6$
DMF* [30] | - | $37.4$ | - | $88.4$
Table 3: We report number of similar and dissimilar pairs to train different methods and training time per epoch of different methods. SimPDO and BUIR-ID use a smaller number of pairs and can be trained much faster than the other methods. | AMusic | Lastfm | CiteULike
---|---|---|---
method | numS | numD | training time | numS | numD | training time | numS | numD | training time
SimPDO | $46$K | $0$ | $13$s | $69$K | $0$ | $15$s | $200$K | $0$M | $123$s
BUIR-ID | $46$K | $0$ | $8$s | $69$K | $0$ | $11$s | $200$K | $0$M | $123$s
CFNet | $46$K | $184$K | $36$s | $69$K | $276$K | $36$s | - | - | -
Shared | - | - | - | - | - | - | $200$K | $2$M | $1871$s
Shared_attention | - | - | - | - | - | - | $200$K | $2$M | $1963$s
#### Comparison with the state-of-the-art methods.
In Table 1, we compare the methods on CiteULike dataset in the cold-start
setting. The dataset and the Shared code are available
online555https://github.com/rakutentech/shared_rep. We use the code and
processed data provided in this repository to train our model. Note that our
method uses the same user and item models as the shared model, but uses the
new objective function. This makes it clear that the better performance of the
SimPDO comes from the proposed objective function that is only using similar
pairs.
In Table 2, we compare the methods on AMusic and Lastfm datasets in the warm-
start setting. The dataset and the DeepCF code are available
online666https://github.com/familyld/DeepCF. We use the code and processed
data provided in this repository to train our model. We also trained CFNet and
BUIR-ID from the publicly available code and put the results in the second and
third row of Table 2.
Results of Table 1 and Table 2 show that SimPDO outperforms most state-of-the-
art methods and achieves very competitive results. Note that SimPDO and BIUR
use a significantly smaller number of training pairs compared to the other
methods, such as Shared and CFNet. We have put the number of similar and
dissimilar pairs used in training the methods with the best results in Table
3. SimPDO and BUIR-ID use a smaller number of pairs and can be trained much
faster than the other works.
## 5 Conclusion
In this paper, we proposed SimPDO, a new objective function that enables
training of one-class recommendation system models without dissimilar pairs.
We showed that by only using similar pairs, the optimal solution of existing
objective functions becomes a collapsed solution, where every representation
is mapped to the same point in the latent space. We avoided the collapsed
solution by providing a hinge loss for pairwise distance which is equivalent
to the average variance of each dimension of representations. We also showed
that we need an orthogonality term to avoid collapsed and partially collapsed
solutions, where the optimal solution under the pairwise loss returns only two
representations. The orthogonality term minimizes the correlation between each
dimension of representations and forces the objective function to return
various representations. Finally, we showed that both terms are necessary to
learn meaningful representations. The results demonstrated that SimPDO
outperformed the existing RS objective functions without using dissimilar
pairs. We also showed that SimPDO can be trained more efficiently with a
smaller number of training pairs.
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|
††thanks: M. A. N., H. K. and W. B. C. contributed equally to this
work.††thanks: M. A. N., H. K. and W. B. C. contributed equally to this
work.††thanks: M. A. N., H. K. and W. B. C. contributed equally to this work.
# Iterative assembly of 171Yb atom arrays in cavity-enhanced optical lattices
M. A. Norcia H. Kim W. B. Cairncross M. Stone A. Ryou M. Jaffe M. O.
Brown K. Barnes P. Battaglino T. C. Bohdanowicz A. Brown K. Cassella
C.-A. Chen R. Coxe D. Crow J. Epstein C. Griger E. Halperin F. Hummel
A. M. W. Jones J. M. Kindem J. King K. Kotru J. Lauigan M. Li M. Lu E.
Megidish J. Marjanovic M. McDonald T. Mittiga J. A. Muniz S.
Narayanaswami C. Nishiguchi T. Paule K. A. Pawlak L. S. Peng K. L. Pudenz
D. Rodríguez Pérez A. Smull D. Stack M. Urbanek R. J. M. van de Veerdonk
Z. Vendeiro L. Wadleigh T. Wilkason T.-Y. Wu X. Xie E. Zalys-Geller X.
Zhang B. J. Bloom
Atom Computing, Inc
###### Abstract
Assembling and maintaining large arrays of individually addressable atoms is a
key requirement for continued scaling of neutral-atom-based quantum computers
and simulators. In this work, we demonstrate a new paradigm for assembly of
atomic arrays, based on a synergistic combination of optical tweezers and
cavity-enhanced optical lattices, and the incremental filling of a target
array from a repetitively filled reservoir. In this protocol, the tweezers
provide microscopic rearrangement of atoms, while the cavity-enhanced lattices
enable the creation of large numbers of deep optical potentials that allow for
rapid low-loss imaging of atoms. We apply this protocol to demonstrate
deterministic filling (99% per-site occupancy) of 1225-site arrays. Because
the reservoir is repeatedly filled with fresh atoms, the array can be
maintained in a filled state indefinitely. We anticipate that this protocol
will be compatible with mid-circuit reloading, which will be a key capability
for running large-scale error-corrected quantum computations whose durations
exceed the lifetime of a single atom in the system.
## I Introduction
Individually controlled neutral atoms provide a promising platform for quantum
information processing and simulation, and expanding the size and control of
these systems represent key challenges in the ongoing push to access regimes
beyond the capabilities of classical simulation. Tweezer arrays and optical
lattices have emerged as core technologies for optically trapping and
manipulating cold atoms, each with complementary capabilities. Optical
lattices provide a well-defined potential landscape with features on the sub-
wavelength scale, which can enable tight confinement for imaging and a clean
optical potential for simulations involving itinerant atoms [1]. Optical
tweezers provide a means of selectively manipulating individual atoms [2].
Notably, tweezer arrays have gained prominence for deterministic assembly of
atomic arrays [3, 4, 5] with up to hundreds of atoms [6, 7] and inter-atomic
spacing suitable for enabling interactions between atoms through excitation to
Rydberg states [8, 9, 10, 6, 7, 11, 12]. Recently, optical tweezers and
lattices have been combined to create programmable initial conditions in a
Hubbard-regime system [13, 14], and to enable enhanced scaling of atom number
for metrology [15] and quantum simulation[16].
The merits of optical lattices can be further enhanced through the use of
optical buildup cavities. Optical cavities enhance the lattice depth by
enabling the constructive interference of many overlapped reflections of laser
light, in turn enabling the creation of arrays of many deep traps [17, 18].
Figure 1: Conceptual diagram of repeated loading sequence. A “reservoir”
optical tweezer array (smaller left grid) is repeatedly filled with 171Yb
atoms transported from a spatially separated magneto-optical trap (MOT), and
ultimately transferred into a “target” tweezer array (larger right grid) using
a “rearrangement” tweezer. The rearrangement moves required for the transfer
are determined from low-loss images obtained by transferring atoms from the
tweezers into a cavity-enhanced optical lattice and performing site-resolved
fluorescence detection. The cavity-enhanced lattice allows for the scalable
generation of large numbers of deep traps. The call-out shows example images
prior to the 20th cycle of rearrangement, and after the 70th cycle, with the
final reservoir reloading step omitted. This protocol allows us to load over
1200 atoms into 1225 target sites.
In this work, we combine the capabilities of tweezers and cavity-enhanced
optical lattices to demonstrate an iterative approach to creating large arrays
of individually-controlled atoms. Typically, tweezer rearrangement is
performed by stochastically loading up to a single atom into each trap within
an array [19], imaging the atoms to determine trap occupancy, and then
rearranging atoms within the array to create a deterministically occupied sub-
array [4, 5]. Crucially, the number of atoms contained in the final array with
this approach is no greater than the number initially loaded. Further, because
the initial loading is stochastic, the number of sites in the array must
generally be substantially larger than desired final sub-array (though under
certain conditions, near-deterministic loading can be achieved [20, 21], and
several initial arrays can be used to increase the number of available atoms
[22]). Recently, repeated loading of a “buffer” array from an optical dipole
trap “reservoir” has demonstrated that one can decouple the filling of a six-
site target array from a single loading of a cold reservoir [23]. In this
work, we extend this concept to repeated loading of a reservoir array, from
which we create a deterministically filled target array (typically 99%
occupancy) of over 1200 171Yb atoms in 1225 sites. This is made possible by
combining optical tweezer arrays with a cavity-enhanced optical lattice to
provide both microscopic control and the large number of deep traps required
for rapid, high-fidelity, low-loss imaging of large numbers of atoms.
A single cycle of loading proceeds as follows (See figure 1): Atoms are
collected and cooled in a magneto-optical trap 30 cm below a “science region”
where we create a cavity-enhanced optical lattice, as well as tweezer arrays
projected through a high-numerical-aperture imaging. The atoms are then
transported into the science region, where they are loaded into a set of
tweezers that form the reservoir. The reservoir sits adjacent to the target
tweezer array, which on all loading cycles but the first is already partially
occupied with atoms loaded on previous cycles. The atoms from both reservoir
and target tweezer arrays are then transferred into the trapping potential of
the cavity-enhanced optical lattice, where we perform site-resolved
nondestructive fluorescence imaging to determine which sites of each tweezer
array were occupied. We then transfer the atoms back from the cavity lattice
to the tweezers, and use a single separate “rearrangement” tweezer to move
atoms from the reservoir into empty target tweezers. This defines one loading
cycle. Over the course of multiple cycles, the number of atoms in the target
array increases until the target array is filled. At this point, further
operations may be performed on the atoms, and subsequent loading cycles are
applied to maintain a filled array.
Because we reload the reservoir on each loading cycle from a fresh magneto-
optical trap, we can continue loading the target array indefinitely. In the
near-term, this allows for relatively high data-rates for quantum simulation
and computation with large system sizes, and could also be of benefit for
optical clocks for high statistical precision with low dead-time [24].
Ultimately, the ability to reload new atoms while maintaining both the
presence and coherence of existing atoms will be a key capability for
performing error-corrected quantum computations, where execution of an
algorithm may take much longer than the lifetime of any given atom in the
system. When combined with site-selective hiding [25, 26] and mid-circuit
rearrangement techniques already demonstrated in a similar system [26], we
anticipate that this protocol will be compatible with mid-circuit reloading of
atoms into the array. As a means of maintaining a fully filled array, our
approach represents an alternative to the interleaved use of two atomic
species [27, 28]. Unlike the two-species approach, the method presented here
does not require simultaneous replacement of the entire array, and so may be
expected to ease requirements for scaling to large arrays, and would not
require inter-species gates [29] in order to replenish the array during a
quantum computation.
## II Trapping and imaging atoms in a 3D cavity enhanced lattice
Figure 2: Cavity-enhanced optical lattices for rapid low-loss imaging. (a) The
intersection of two cavity modes provide three-dimensional confinement for
atoms within the field of view of our high-numerical-aperture imaging system.
Confinement in the $x$ and $y$ directions is provided by a self-intersecting
standing-wave cavity, while confinement in the $z$ direction is provided by a
separate self-intersecting running-wave cavity mode. Imaging light (not shown
for clarity) is incident along the mode of the running-wave lattice. (b)
Observed counts during a 7 ms long image of single atoms within the optical
lattice, showing well-resolved peaks for occupied and unoccupied sites. For
these conditions, per-image survival is 99.8(1)%. (c) Lattice homogeneity
across the target array as characterized by light shifts on the 1S0
$m_{f}=1/2$ to 3P1, $F=3/2$ $m_{f}=-1/2$ transition. Atoms within 13.4 $\mu$m
squares are averaged together for statistics and display clarity. (d, e) Light
shift contributions from each cavity averaged over rows (d) and columns (e).
Red and orange markers and lines represent the measured and predicted profiles
of the XY (red) and Z (orange) cavities, with the prediction centered on the
array.
Rapid low-loss imaging of the locations of atoms is essential to our repeated
loading protocol. Because atoms are heated by the photon scattering required
for imaging, deep optical traps and imaging protocols with low equilibrium
temperatures are required to prevent atom loss. Optical tweezers represent a
common method for achieving deep optical traps – the tight foci can provide
high intensities for moderate powers. However, the total power requirement for
a large array of tweezers presents scaling limitations due to limited
available laser power. Cavity-enhanced optical lattices offer an alternative
path to achieving many deep traps. While the laser intensity in a lattice is
spread over a larger cross-sectional area than in optical tweezers both
because it spans the regions between utilized sites, and because the Gaussian
mode profile must be large compared to the size of the array to achieve
uniform trapping, interference of many reflected paths in a cavity-supported
lattice can enable more power-efficient generation of traps.
We employ two intersecting optical cavities to provide tight confinement in
three dimensions, as shown in figure 2a. The first cavity – which we call the
XY cavity – is a retro-reflected four-mirror cavity whose mode intersects
itself at the location of the atoms. The polarization is perpendicular to the
plane of propagation, providing interference between the two crossed portions
of the cavity mode [30]. Compared to using two crossed non-interfering
lattices [31, 18], and assuming equal cavity finesse, this configuration
provides an eight-fold enhancement in trap depth (defined as the potential
barrier between adjacent lattice sites) per unit of total laser power. The XY
cavity provides tight confinement in the directions perpendicular to the high
numerical aperture optical axis used for the tweezers and imaging. The second
cavity, which we call the Z cavity, is a four-mirror running wave cavity whose
mode intersects itself at the location of the atoms [32] at 15 degree angles
from the XY plane. This provides confinement along the tweezer axis. The
polarization of the Z cavity is also oriented perpendicular to its
intersection plane, enabling complete interference of the crossing modes. Each
cavity has two highly reflecting mirrors, and two matched partially
transmitting mirrors. The finesse of the XY and Z cavities are 2900(100) and
3000(100) respectively, and the design mode waists at the location of the
atoms are 268 $\mu$m and 183 $\mu$m respectively. For imaging, we typically
operate with XY (Z) trap frequencies near 160 kHz (50 kHz), corresponding to
330 $\mu$K (260 $\mu$K) deep traps.
We image atoms using the the narrow-linewidth (180 kHz) 1S0 to 3P1 transition,
which simultaneously provides cooling and scattering of photons for detection
[33, 34, 26]. We use a single beam, detuned several hundred kHz from the
$F=3/2$, $m_{f}=3/2$ state, with projection onto both the X and Z directions.
For applications like array assembly where we wish to determine the presence
of an atom but not the state of its nuclear spin (the qubit we use for quantum
information applications), we simultaneously apply optical pumping on the 1S0
$m_{f}=-1/2$ to 3P1 $F=1/2$, $m_{f}=1/2$ transition with light incident along
the magnetic field ($x$ direction). At our benchmark imaging parameters, we
collect 50 photons in 7 ms, and distinguish occupied from unoccupied sites
with a fidelity of approximately 99.95%, as determined from the overlap error
in a double-Gaussian fit to the bimodal distribution of counts. Atom loss
during such an image is $2(1)\times 10^{-3}$. Spin-selective imaging requires
us to omit the optical pumping, and rely on the combination of frequency and
polarization selectivity within the 3P1 manifold to create a large imbalance
of scattering between the two nuclear spin states, as demonstrated in optical
tweezers in refs. [34, 26]. The omission of the optical pumping does not
modify the photon scattering rate appreciably, but does introduce a finite
spin flip probability of of $4(1)\times 10^{-3}$ for our typical imaging
parameters, measured as an additional apparent loss in repeated images of 1S0
$m_{f}=1/2$. These spin flips are likely due to the fact that the lattice
polarization is perpendicular to the magnetic field, and so can induce spin-
changing transitions within the 3P1 manifold [34].
In order to ensure uniform detuning of the imaging transition across the array
and between different motional states, we operate with a trapping wavelength
that has equal polarizability for the ground and excited states of the imaging
transition – a so-called magic wavelength. For the 1S0 $m_{f}=1/2$ to 3P1
$m_{f}=3/2$ transition, we find a magic condition near 783.8 nm.
We characterize the homogeneity of the optical lattice potential by probing
the 1S0 $m_{f}=1/2$ to 3P1 $F=3/2$, $m_{f}=-1/2$ transition, for which the
excited state has approximately 40% higher polarizability than the ground
state. We infer the trap depth from the transition light shifts, measured
using optical pumping from the 1S0 $m_{f}=1/2$ state, followed by spin-
selective imaging (fig. 2c). By varying the power in each cavity, we can
extract their independent contributions, displayed averaged over rows and
columns in figure 2d,e. Over our 115 $\mu$m square array, peak deviations are
within 20% for the Z lattice and 10% for the XY lattice. Because we operate
with a magic wavelength condition for critical operations, these deviations do
not present a major limitation.
## III Repeated loading to build up a large array
Figure 3: (a) Timing diagram for our continuous loading sequence. See main
text for a detailed description of the procedure. The beam paths referenced in
the call-out are illustrated in figure 4. (b) Number of atoms and fill
fraction as a function of loading iteration number during initial loading
(left panel) and while maintaining a filled array (right panel). Black points
represent atom number inferred from images preceding the rearrangement step.
Red points represent atom number inferred from additional images inserted
after rearrangement, representing atoms available for computation.
In this section, we describe our continuous loading protocol in detail, and
characterize its performance for loading large arrays.
Our loading cycle (illustrated in figure 3a) begins with collecting atoms from
a pre-cooled atomic beam into a magneto-optical trap (MOT) using a “core-
shell” configuration [35]. In this MOT, each beam consists of a ring of 399 nm
light near resonance with the broad linewidth 1S0 to 1P1 transition
surrounding a central region with only 556 nm light near resonance with the
narrow linewidth 1S0 to 3P1 transition. The shell is primarily responsible for
capturing atoms from the incident beam, while the low Doppler temperature of
the narrow transition used in the core enables cooling to several 10 $\mu$K.
Because atoms do not scatter light from the broad transition except during
initial capture, the core-shell design poses less risk of generating scattered
photons that may be absorbed by atoms in the target array – the total
scattering rate on the narrow transition is much lower, and the large magnetic
field present in the science region dramatically suppresses reabsorption of
photons via the narrow-line transition. In our operating condition, the 399 nm
photon scattering rate is about 2 orders of magnitude lower than the broad-
line stage of a sequential MOT. We find that the core-shell MOT configuration
increases the MOT loading rate by roughly a factor of two, compared to a more
standard sequential MOT configuration in a static magnetic field.
From the core-shell MOT, atoms are loaded into a vertically oriented standing-
wave lattice formed by counterpropagating 532 nm laser beams focused at the
location of the atoms. Loading is achieved by overlapping the lattice with the
center of the MOT, followed by reducing the detuning and intensity of the 556
nm light. We then extinguish the MOT light and transport the atoms vertically
by synchronously applying a frequency offset to the upward-going and downward-
going lattice beams, and translating their foci [26]. This brings the atoms 30
cm vertically into the vicinity of the target array, at which point we
transfer the atoms into a reservoir array formed by 483 nm tweezers (another
magic wavelength for the 1S0 to 3P1 transition [26]). To avoid disturbing
atoms already loaded into the target array, this transfer takes place at a
location displaced 170 $\mu$m horizontally from the target array. No
dissipation is required to transfer atoms from the transport lattice to the
tweezers – loading is achieved by simply increasing the intensity of the
tweezers while the lattice is overlapped and then decreasing the intensity of
the lattice. Once loaded, the reservoir array is translated to be directly
adjacent to the target array by changing the angle of a galvo mirror that is
imaged onto the entrance pupil of the objective.
The target array consists of a rectangular array of tweezer spots formed with
light at 459.5960(5) nm wavelength, a magic wavelength for the 1S0 to 3P0
clock transition [26, 36], and a set of tweezer spots formed by light at
423.31 nm, which is near resonance with a transition between 3P1 to a higher-
lying 6S8S 3S1 state. Both wavelengths provide large light-shifts to atoms in
3P1, which prevents unwanted scattering from atoms loaded in these tweezer
arrays, and can be used interchangeably for this work. For this work, we
operate with a 35x35-site target array with 3.3 $\mu$m spacing, comprised of
two separate rectangular arrays each using one of the two previously mentioned
wavelengths. We have also operated with overlapped full-size arrays to achieve
sufficient trap depth, and obtained similar results. However, the overlapped
condition is more sensitive to alignments, and so we use spatially separated
arrays for all results presented here.
While translating the reservoir array adjacent to the target array, both
arrays are illuminated with light resonant with the 1S0 to 3P1 $F=3/2$,
$m_{f}=3/2$ transition. This induces light-assisted collisions between atoms
in the reservoir tweezers, resulting in either zero or one atom in each
reservoir tweezer with approximately equal probability [19]. Importantly, the
tweezer light illuminating the atoms in the target array prevents them from
scattering the light used to induce collisions, which might otherwise cause
loss of existing atoms in the target array. Next, atoms from both tweezer
arrays are transferred into the optical lattice and imaged, as described
above. The imaging leaves the atoms at a temperature of 20 $\mu$K, which is
too warm to efficiently transfer into the relatively shallow tweezers of the
target array. Following each image, we apply Doppler and Raman sideband
cooling to reach an average of $\sim$ 0.1 motional quanta in each direction
(Appendix A). Once cooled, the atoms are transferred back into the tweezer
arrays, and a single optical tweezer derived from the same laser as the
reservoir light is used to transfer atoms from filled sites of the reservoir
array into empty sites of the target array. This completes a single loading
cycle.
In order to shorten the time required to build up an atomic array, and to
limit the losses associated with background gas collisions, we load the MOT at
the same time as performing operations in the science chamber. This is made
possible by the spatial separation between MOT and science regions, and by the
fact that we operate with static magnetic fields: we can simultaneously
operate with the magnetic field gradients required by the MOT and the large
500 Gauss homogeneous bias field that is desirable for imaging and other
operations required in the science regions. The low scattering rate of 399 nm
photons, combined with the large magnetic field and spatial separation between
the MOT and science regions prevents light scattered in the MOT from being
reabsorbed by atoms in the science region, which could cause heating or
decoherence.
Figure 3b shows examples of the atom number increasing as a function of load
cycles, with 105 sites in the reservoir and 1225 sites in the target array. At
first, the atom number increases approximately linearly as roughly 45 atoms
are transferred into the target array per cycle. The loading rate decreases
slightly as the array fills, as some atoms from the reservoir are used to
counteract loss. Once the number of vacancies in the target array drops below
the typical number of atoms in the reservoir, an equilibrium is reached with
the vacancy fraction set by the per cycle probability of atom loss.
In figure 3b, the atom number is inferred from the image taken before
rearrangement, so the measured fill fraction is sensitive to all losses that
occur between such images. This fill fraction is typically 98%, with the
largest single loss contribution coming from vacuum loss during the typical
300 ms time between images due to our approximately 30 s vacuum lifetime. We
can add additional diagnostic images directly after rearrangement to determine
the atom number available for computation, and observe typical filling
fractions of 99%, again dominated by vacuum loss during the 150 ms typically
allocated for rearrangement. Our experimental control allocates a variable
duration for rearrangement that depends on the number and length of moves to
be performed, as well as the time required to compute the moves. The durations
above are typical for the 1225-site arrays studied in this work. Other sources
of loss or potential loss are described in more detail in Appendix B, and
include the handoff between the tweezer arrays and the lattice, and imaging
loss and infidelity.
## IV Conclusions and outlooks
We have demonstrated an iterative method for assembling large arrays of
individually controlled neutral atoms, suitable for quantum computation,
simulation, and metrology. Through this approach, we decouple the final size
of the array from the number of atoms that can be loaded at once, and the need
to create many tweezer traps with sufficient depth for loading and imaging
atoms. In particular, our reservoir and lattice traps are much deeper than the
target tweezer traps, allowing for the use of large numbers of sites with
moderate laser power. Our current arrays have percent-level defects, which
could be straightforwardly reduced with improved vacuum.
For quantum computing applications, our iterative loading may be combined with
recently demonstrated mid-circuit measurement techniques [25, 26] in order to
achieve continuous mid-circuit refilling of array defects. This capability
will be critical for mitigating the effect of atom loss during execution of
complex error-corrected circuits that may extend beyond the lifetime of an
individual atom in the array.
## V Appendix A: Raman Sideband Cooling
Figure 4: Raman sideband cooling (RSC). (a) Diagram of relevant directions and
polarizations. RSC beams RSC1 and RSC2 are combined to cool the $z$ direction,
while RSC1 and RSC3 are combined to cool the $x$ and $y$ directions. Optical
pumping (beam OP) is applied along the $x$ direction, which is along the
applied magnetic field. (b) Level diagram for Raman sideband cooling. Raman
transitions detuned by 35 MHz from 3P1 $F=3/2$ $m_{f}=1/2$ are driven from 1S0
$m_{f}=1/2$ to 1S0 $m_{f}=-1/2$, while reducing the number of motional quanta
along the addressed direction(s) from $n$ to $n-1$. Optical pumping through
3P1 $F=1/2$ $m_{f}=1/2$ returns atoms to 1S0 $m_{f}=1/2$ to enable further
cycles. (c) Sideband spectra in the $x$, $y$ plane. The sideband imbalance
indicates an average number of motional quanta along the differential momentum
vector of the RSC1 and RSC3 beams of $\bar{n}_{xy}=0.08(1)$. (d) Sideband
spectra in the $z$ direction. The sideband imbalance indicates an average
number of motional quanta along the $z$ direction of $\bar{n}_{z}=0.12(1)$.
(e) Heating in the lattice, as measured from the sideband imbalance in the
$x$, $y$ plane, with a fit to an exponential growth profile, returning a time-
constant of 190(40) ms.
Transferring atoms from the lattice into shallow tweezer traps with low loss
requires that the atoms be much colder than the depth of the tweezer traps. By
operating with colder atoms, we can use shallower tweezers, and in turn scale
to larger array sizes. To accomplish this, we use Raman sideband cooling (RSC)
between the two ground nuclear spin states [21, 25]. For the motional Raman
transitions, we use two pairs of beams oriented along the $x$ and $y$
directions, and along the $x$ and $z$ directions to provide cooling along all
three directions (fig. 4a). The Raman transitions transfer atoms from 1S0
$m_{f}=1/2$ to 1S0 $m_{f}=-1/2$, state, and are detuned from the motional
carrier transition by the appropriate trap oscillation frequency in order to
reduce the motional state by one quanta (fig. 4b). Optical pumping is provided
by a beam oriented along $x$ and the magnetic field that addresses the 1S0
$m_{f}=-1/2$ to 1P1 $F=1/2$, $m_{f}=1/2$ transition.
We perform 20 iterations of cooling, with each iteration consisting of a $\pi$
pulse on the red motional sideband for each pair, followed by optical pumping.
The pulse durations are 200 $\mu$s for the $xy$ pair and 100 $\mu$s for the
$xz$ pair (tuned to the $z$ direction sidband), and 50 $\mu$s for optical
pumping. The total cooling sequence lasts 8 ms. In typical sequences, we apply
a 2 ms Doppler cooling pulse prior to the RSC sequence using the same beam as
for the imaging pulse, but with lower intensity and farther red-detuning. The
Doppler cooling results in 10 $\mu$K temperature in the cavity as measured by
a release and recapture protocol [37]. This roughly corresponds to 1 motional
quanta occupancy in the cavity. Subsequent RSC reduces the average motional
quanta to in each direction to $\sim$ 0.1, as evident from the sideband
imbalance following cooling (fig. 4c, d).
We measure heating in the $x$, $y$ plane while holding atoms in the lattice by
performing Raman sideband spectroscopy to extract the average motional quantum
number $\bar{n}_{xy}$ as a function of time. We fit this quantity to a
function representing exponential growth, as expected for parametric heating
due to intensity fluctuations of the lattice [38], and extract an exponential
time-constant of 190(40) ms. This is likely due to conversion of laser
frequency noise to amplitude by the cavity resonance, and could be improved by
optimizing the lock of lasers to the cavity.
## VI Appendix B: Sources of Per-Cycle Loss
Figure 5: Characterization of per-cycle loss mechanisms. (a), (b). Loss per
pair of handoffs between the 459 nm (a) and 423 nm (b) tweezer arrays and
lattice, measured by performing 25 handoffs with cooling in between. The black
line represents the estimated contribution from vacuum loss, with the red band
representing day-to-day drifts in this value. Tweezer depths are normalized to
their default values, which correspond to roughly 50 $\mu$K. (c) Final vacancy
fraction versus rearrangment tweezer power, normalized to the default tweezer
depth of approximately 150 $\mu$K. For low powers, atoms are not efficiently
transferred from the reservoir to target array. For too high of powers, we
observe increased loss of loaded atoms due to the rearrangement tweezer
passing nearby.
Vacuum loss due to the finite duration of loading cycles is the single largest
source of loss in our system, contributing percent-level per-cycle loss.
Because our loading cycle is of variable length, and because the vacuum level
in our system can fluctuate from day to day, it is difficult to estimate its
exact contribution. However, typical values for the loading cycle duration and
vacuum lifetime are 300 ms and 30 s respectively, so 1% represents a typical
value for this loss. In this section, we describe other loss sources that
contribute to a lesser degree, especially if parameters are not carefully
optimized. In general, the loss mechanisms described below both limit the size
of array we can ultimately load (as the reservoir must be large enough to
replace lost atoms), and limit the final fill fraction of the array.
As described in the main text, typical imaging loss is $2(1)\times 10^{-3}$
where roughly 30 % is the vacuum loss and the rest is the Raman scattering out
of 3P1 due to the trap light , and the discrimination infidelity is typically
at or below the $10^{-3}$ level. If an atom within the target array is lost
during the image after scattering enough photons to be identified as present,
the defect will not be filled during the subsequent rearrangement step,
leading to a defect in the array. If a site within the array is incorrectly
identified as empty, an atom will likely be added to that site from the
reservoir, and subsequently undergo lossy collisions with the original
occupant. Empty sites within the target or reservoir arrays that are
mistakenly identified as full will lead to a previous defect not being
repaired. However, because defects are rare, this mechanism is less
problematic. We currently operate with a discrimination threshold that
balances the correct identification of empty and full sites. In a system where
imaging losses and infidelity become dominant loss sources (a system with
better vacuum), it may be advantageous to bias the threshold to minimize the
more problematic forms of imaging error.
Each image requires handing off atoms from the tweezer arrays into the lattice
and back. We find that this process is sensitive to the alignment of the two
arrays on the scale of a single lattice site, to the depth of the tweezer
arrays, and to the temperature of the atoms. Atom temperature and alignment
are described in Appendix A and C, respectively. We isolate the effect of trap
depth on handoff for cold atoms and well-aligned arrays by performing 25
subsequent handoff pairs between tweezers and lattice in figure 5a, b. We
perform our usual Raman sideband cooling (RSC) each time the atoms are in the
lattice between handoffs. For 459 nm tweezer depths below 75% of our typical
operating conditions, we observe significant loss. Above this power level, we
observe a constant loss rate of 0.0006(1), which is consistent with data taken
with the handoff omitted, and with our typical vacuum losses. The 423 nm
tweezers show a similar behavior, though with a slightly higher loss rate of
0.0015(2) at high powers, likely due to worse alignment or optical
aberrations. If the tweezers are aligned to the trough of the XY cavity
intensity, the loss can be as high as 0.02.
In principle, moving new atoms in the target array may cause loss of existing
atoms in the array, as the rearrangement tweezer moves near the occupied
target sites. We assess this possibility in figure 5c by measuring the final
fill fraction versus the depth of the rearrangement tweezer. For a
rearrangement tweezer that is too shallow, we observe a lower loading rate and
so a lower final fill fraction. For too deep a tweezer, we also see a
reduction in the final fill fraction, as the rearrangement leads to loss of
existing atoms. Near our typical operating conditions, we observe a region
with weak dependency of the fill fraction on the tweezer power.
## VII Appendix C: Alignment of arrays
Figure 6: Alignment of the tweezer arrays to the cavity lattice is performed
by monitoring atomic survival after repeated handoffs, while scanning the
position offset of the tweezer array. For the representative alignment scan of
the 459 nm tweezer array shown here, we perform 25 handoff pairs between the
tweezers and lattice, with no cooling in between. Data is averaged over the
full utilized field of view – the presence of a visible fringe in the average
image indicates that the sites across the array can be aligned simultaneously.
Precise alignment of the different optical potentials is critical for the
performance of our repeated loading protocol. The alignment of the target
tweezer arrays to the optical lattice is particularly sensitive, as
misalignment here can lead to increased handoff loss that can limit both the
largest array that can be loaded, and the final fill fraction of the array.
Alignment of the arrays involves first matching the spacing and tilt of the
lattice – quantities which do not drift appreciably over time – and then at a
higher frequency aligning the $x$ and $y$ offsets. We obtain the correct
spacings and tilts by populating the lattice with atoms transferred from an
expanded but sparse version of the reservoir array. We image the atoms in the
lattice to determine the lattice grid, and then in the reservoir tweezers to
determine required corrections to the tweezer array. We then use a camera
whose imaging system is corrected for chromatic shifts between the different
tweezer wavelengths to register the target array tweezers to the reservoir
tweezers.
We then optimize the translation alignment by performing repeated handoffs
between the tweezers and the lattice and scanning the position offset of the
tweezer arrays. We observe a periodic modulation in the atomic survival (fig.
6), which we fit to determine the optimal alignment. This process is repeated
as necessary to ensure alignment of the arrays.
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|
# Inference for a Large Directed Acyclic Graph with Unspecified Interventions
Chunlin Li<EMAIL_ADDRESS>
School of Statistics,
University of Minnesota,
Minneapolis, MN 55455, USA Xiaotong Shen<EMAIL_ADDRESS>
School of Statistics,
University of Minnesota,
Minneapolis, MN 55455, USA Wei Pan<EMAIL_ADDRESS>
Division of Biostatistics,
University of Minnesota,
Minneapolis, MN 55455, USA
###### Abstract
Inference of directed relations given some unspecified interventions, that is,
the target of each intervention is unknown, is challenging. In this article,
we test hypothesized directed relations with unspecified interventions. First,
we derive conditions to yield an identifiable model. Unlike classical
inference, testing directed relations requires identifying ancestors and
relevant interventions of hypothesis-specific primary variables. Towards this
end, we propose a peeling algorithm based on nodewise regressions to establish
a topological order of primary variables. Moreover, we prove that the peeling
algorithm yields a consistent estimator in low order polynomial time. Second,
we propose a likelihood ratio test integrated with a data perturbation scheme
to account for the uncertainty of identifying ancestors and interventions.
Also, we show that the distribution of a data perturbation test statistic
converges to the target distribution. Numerical examples demonstrate the
utility and effectiveness of the proposed methods, including an application to
infer gene regulatory networks. The implementation of the proposed methods is
available at https://github.com/chunlinli/intdag.
Keywords: High-dimensional inference, data perturbation, structure learning,
identifiability, peeling algorithm
## 1 Introduction
Directed relations are essential to explaining pairwise dependencies among
multiple interacting units. In gene network analysis, regulatory gene-to-gene
relations are a focus of biological investigation (Sachs et al., 2005), while
in a human brain network, scientists investigate causal influences among
regions of interest to understand how the brain functions through the regional
effects of neurons (Liu et al., 2017). In such a situation, inferring directed
effects without other information is generally impossible because of the lack
of identifiability in a Gaussian directed acyclic graph (DAG) model, and hence
external interventions are introduced to treat a non-identifiable situation
(Heinze-Deml et al., 2018). For instance, the genetic variants such as single-
nucleotide polymorphisms (SNPs) can be, and indeed are increasingly, treated
as external interventions to infer inter-trait causal relations in a
quantitative trait network (Brown and Knowles, 2020) and gene interactions in
a gene regulatory network (Teumer, 2018; Molstad et al., 2021). In
neuroimaging analysis, scientists use randomized experimental stimuli as
interventions to identify causal relations in a functional brain network
(Grosse-Wentrup et al., 2016; Bergmann and Hartwigsen, 2021). However, the
interventions in these studies often have unknown targets and off-target
effects (Jackson et al., 2003; Eaton and Murphy, 2007). Consequently,
inferring directed relations while identifying useful interventions for
inference is critical. This paper focuses on the simultaneous inference of
directed relations subject to unspecified interventions, that is, without
known targets.
In a DAG model, the research has been centered on the reconstruction of
directed relations in observational and interventional studies (van de Geer
and Bühlmann, 2013; Oates et al., 2016; Heinze-Deml et al., 2018; Zheng et
al., 2018; Yuan et al., 2019). For inference, Bayesian methods (Friedman and
Koller, 2003; Luo and Zhao, 2011; Viinikka et al., 2020) have been popular.
Yet, statistical inference remains under-studied, especially for
interventional models in high dimensions (Peters et al., 2016; Rothenhäusler
et al., 2019). Recently, for observational data, Janková and van de Geer
(2018) propose a debiased test of the strength of a single directed relation
and Li et al. (2019) derives a constrained likelihood ratio test for multiple
directed relations.
Despite progress, challenges remain. First, inferring directed relations
requires identifying a certain DAG topological order (van de Geer and
Bühlmann, 2013), while the identifiability in a Gaussian DAG with unspecified
interventions remains under-explored. Second, the inferential results should
agree with the acyclicity requirement for a DAG. As a result, degenerate and
intractable situations can occur, making inference greatly different from the
classical ones. Third, likelihood-based methods for learning the DAG
topological order often use permutation search (van de Geer and Bühlmann,
2013) or continuous optimization subject to the acyclicity constraint (Zheng
et al., 2018; Yuan et al., 2019), where a theoretical guarantee of the actual
estimate (instead of the global optimum) has not been established for these
approaches. Recently, an important line of work (Ghoshal and Honorio, 2018;
Rajendran et al., 2021; Rolland et al., 2022) has focused on order-based
algorithms with computational and statistical guarantees. However, in Gaussian
DAGs, existing methods often rely on some error scale assumptions (Peters and
Bühlmann, 2014), which is sensitive to variable scaling like the common
practice of standardizing variables. This drawback could limit their
applications, especially in causal inference, as causal relations are
typically invariant to scaling.
To address the above issues, we develop structure learning and inference
methods for a Gaussian DAG with unspecified additive interventions. Unlike the
existing approach where structure discovery and subsequent inference are
treated separately, our proposal integrates DAG structure learning and testing
directed relations (also known as the natural direct effect in causal
inference), accounting for the uncertainty of structure learning for
inference. With suitable interventions called instrumental variables (IVs),
the proposed approach removes the restrictive error scale assumptions and
delivers creditable outcomes with theoretical guarantees in low order
polynomial time. This result indicates IVs, a well-known tool in causal
inference, can play important roles in structure learning even if some
interventions do not meet the IV criterion. More specifically, our
contributions are summarized as the following aspects.
* •
For modeling, we establish the identifiability conditions for a Gaussian DAG
with unspecified interventions. In particular, the conditions allow
interventions on more than one target, which is suitable for multivariate
causal analysis (Murray, 2006). Moreover, we introduce the concepts of
nondegeneracy and regularity for hypothesis testing under a DAG model to
characterize the behavior of a test.
* •
For methodology, we develop likelihood ratio tests for directed edges and
pathways in a super-graph of the true DAG, where the super-graph is formed by
ancestral relations and candidate interventional relations, offering the
topological order for inference. Furthermore, we reconstruct the super-graph
by the peeling algorithm, which automatically meets the acyclicity
requirement. By integrating structure learning with inference, we account for
the uncertainty of the super-graph estimation for the proposed tests via a
novel data perturbation (DP) scheme, which effectively controls the type-I
error while enjoying high statistical power.
* •
For theory, we prove that the proposed peeling algorithm based on nodewise
regressions yields consistent results in
$O(p\times\log\kappa_{\max}^{\circ}\times(q^{3}+nq^{2}))$ operations almost
surely, where $p,q$ are the numbers of primary and intervention variables, $n$
is the sample size, and $\kappa_{\max}^{\circ}$ is the sparsity. On this
basis, we justify the proposed DP inference method by establishing the
convergence of the DP likelihood ratio to the null distribution and desired
power properties.
* •
The numerical studies and real data analysis demonstrate the utility and
effectiveness of the proposed methods. The implementation of the proposed
tests and structure learning method is available at
https://github.com/chunlinli/intdag.
The rest of the article is structured as follows. Section 2 establishes model
identifiability and introduces the concepts of nondegeneracy and regularity.
Section 3 develops the proposed methods for structure learning and statistical
inference. Section 4 presents statistical theory to justify the proposed
methods. Section 5 performs simulation studies, followed by an application to
infer gene pathways with gene expression and SNP data in Section 6. Section 7
concludes the article. The Appendix contains illustrative examples and
technical proofs.
## 2 Gaussian directed acyclic graph with additive interventions
To infer directed relations among primary variables (variables of primary
interest) $\bm{Y}=(Y_{1},\ldots,Y_{p})^{\top}$, consider a structural equation
model with additive interventions:
$\displaystyle\bm{Y}=\bm{U}^{\top}\bm{Y}+\bm{W}^{\top}\bm{X}+\bm{\varepsilon},\quad\bm{\varepsilon}\sim
N(\bm{0},\bm{\Sigma}),\quad\bm{\Sigma}=\text{Diag}(\sigma^{2}_{1},\ldots,\sigma^{2}_{p}),$
(1)
where $\bm{X}=(X_{1},\ldots,X_{q})^{\top}$ is a vector of additive
intervention variables, $\bm{U}\in\mathbb{R}^{p\times p}$ and
$\bm{W}\in\mathbb{R}^{q\times p}$ are unknown coefficient matrices, and
$\bm{\varepsilon}=(\varepsilon_{1},\ldots,\varepsilon_{p})^{\top}$ is a vector
of random errors with $\sigma_{j}^{2}>0$; $j=1,\ldots,p$. In (1),
$\bm{\varepsilon}$ is independent of $\bm{X}$ but components of $\bm{X}$ can
be dependent. The matrix $\bm{U}$ specifies the directed relations among
$\bm{Y}$, where $U_{kj}\neq 0$ if $Y_{k}$ is a direct cause of $Y_{j}$,
denoted by $Y_{k}\rightarrow Y_{j}$, and $Y_{k}$ is called a parent of $Y_{j}$
or $Y_{j}$ a child of $Y_{k}$. Thus, $\bm{U}$ represents a directed graph,
which is required to be acyclic to ensure the validity of the local Markov
property (Spirtes et al., 2000). The matrix $\bm{W}$ specifies the targets and
strengths of interventions, where $W_{lj}\neq 0$ indicates $X_{l}$ intervenes
on $Y_{j}$, denoted by $X_{l}\to Y_{j}$. In (1), no directed edge from a
primary variable $Y_{j}$ to an intervention variable $X_{l}$ is permissible.
In what follows, we will focus on the DAG $G$ with nodes
$\bm{Y}=(Y_{1},\ldots,Y_{p})$, primary variable edges
$\mathcal{E}=\\{(k,j):U_{kj}\neq 0\\}$, and intervention edges
$\mathcal{I}=\\{(l,j):W_{lj}\neq 0\\}$.
### 2.1 Identifiability
Model (1) is generally non-identifiable without interventions
($\bm{W}=\bm{0}$) when errors do not meet some requirements such as the equal-
variance assumption (Peters and Bühlmann, 2014) and its variants (Ghoshal and
Honorio, 2018; Rajendran et al., 2021). Moreover, a model can be identified
when $\bm{\varepsilon}$ in (1) is replaced by non-Gaussian errors (Shimizu et
al., 2006) or linear relations are replaced by nonlinear ones (Hoyer et al.,
2009). Regardless, suitable interventions can make (1) identifiable. When
intervention targets are known, the identifiability issue has been studied
(Oates et al., 2016; Chen et al., 2018). However, it is less so when the exact
targets and strengths of interventions are unknown, as in many biological
applications (Jackson et al., 2003; Kulkarni et al., 2006), referring to the
case of _unspecified_ or _uncertain_ interventions (Heinze-Deml et al., 2018;
Eaton and Murphy, 2007; Squires et al., 2020).
We now categorize interventions as instruments and invalid instruments.
###### Definition 1 (Instrument)
An intervention is an instrument in a DAG if it satisfies:
1. (A)
(Relevance) it intervenes on at least one primary variable;
2. (B)
(Exclusion) it does not intervene on more than one primary variable.
Otherwise, it is an invalid instrument.
Here, (A) requires an intervention to be active, while (B) prevents
simultaneous interventions of a single intervention variable on multiple
primary variables. This is critical to identifiability because in this
situation an instrument on a cause variable $Y_{1}$ helps reveal its directed
effect on an outcome variable $Y_{2}$, which would otherwise be impossible.
Thus, the instruments are introduced to break the symmetry that results in
non-identifiability of directed relations $Y_{1}\rightarrow Y_{2}$ and
$Y_{2}\rightarrow Y_{1}$.
Next, we make some assumptions on intervention variables to yield an
identifiable model, where dependencies among intervention variables are
permissible.
###### Assumption 1
Assume that model (1) satisfies the following conditions.
1. (1A)
(Positive definiteness) $\operatorname{\mathbb{E}}\bm{X}\bm{X}^{\top}$ is
positive definite, where $\operatorname{\mathbb{E}}$ is the expectation.
2. (1B)
(Local faithfulness)
$\operatorname{Cov}(Y_{j},X_{l}\mid\bm{X}_{\\{1,\ldots,q\\}\setminus\\{l\\}})\neq
0$ when $X_{l}$ intervenes on $Y_{j}$, where $\operatorname{Cov}$ denotes the
covariance.
3. (1C)
(Instrumental sufficiency) Each primary variable is intervened by at least one
instrument.
Assumption 1A imposes mild distributional restrictions on $\bm{X}$, permitting
discrete variables such as SNPs. Assumption 1B requires interventional effects
not to cancel out as multiple targets from an invalid instrument are
permitted. Importantly, if either Assumption 1B or Assumption 1C fails, model
(1) is generally not identifiable, as shown in Example 2 of Appendix A.1. In
Section 5, we empirically examine the situation when Assumption 1C is not met.
###### Proposition 1 (Identifiability)
Under Assumption 1, the parameters $(\bm{U},\bm{W},\bm{\Sigma})$ in model (1)
are identifiable.
Proposition 1 (proved in Appendix B.1) is derived for a DAG model with
unspecified interventions. This is in contrast to Proposition 1 of Chen et al.
(2018), which proves the identifiability of the parameters in a directed graph
with target-known instruments on each primary variable. Moreover, it is worth
noting that the estimated graph in Chen et al. (2018) may be cyclic and lacks
the local Markov property for causal interpretation (Spirtes et al., 2000).
### 2.2 Likelihood-based inference for a DAG
Our primary goal is to perform statistical inference of directed edges and
pathways. Let $\mathcal{H}\subseteq\\{(k,j):k\neq j,1\leq k,j\leq p\\}$ be an
edge set among primary variables $\\{Y_{1},\ldots,Y_{p}\\}$, where
$(k,j)\in\mathcal{H}$ specifies a (hypothesized) directed edge $Y_{k}\to
Y_{j}$ in (1). We shall focus on two types of testing with null and
alternative hypotheses $H_{0}$ and $H_{a}$. For simultaneous testing of
directed edges,
$\displaystyle H_{0}:U_{kj}=0;\text{ for all }(k,j)\in\mathcal{H}\quad\mbox{
versus }\quad H_{a}:U_{kj}\neq 0\text{ for some }(k,j)\in\mathcal{H};$ (2)
for simultaneous testing of pathways,
$\displaystyle H_{0}:U_{kj}=0;\text{ for some }(k,j)\in\mathcal{H}\quad\mbox{
versus }\quad H_{a}:U_{kj}\neq 0\text{ for all }(k,j)\in\mathcal{H},$ (3)
where $(\bm{U}_{\mathcal{H}^{c}},\bm{W},\bm{\Sigma})$ are unspecified nuisance
parameters and c is the set complement.
Note that $H_{0}$ in (3) is a composite hypothesis that can be decomposed into
sub-hypotheses
$H_{0,\nu}:U_{k_{\nu},j_{\nu}}=0,\quad\mbox{ versus
}\quad{H}_{a,\nu}:U_{k_{\nu},j_{\nu}}\neq 0;\quad\nu=1,\ldots,|\mathcal{H}|,$
where
$\mathcal{H}=\\{(k_{1},j_{1}),\ldots,(k_{|\mathcal{H}|},j_{|\mathcal{H}|})\\}$
and each sub-hypothesis is a directed edge test.
Now consider likelihood inference for (2). Given an independent and
identically distributed sample $(\bm{Y}_{i},\bm{X}_{i})_{i=1}^{n}$ from (1),
the log-likelihood is
$L(\bm{\theta},\bm{\Sigma})=-\frac{1}{2}\sum_{i=1}^{n}\|\bm{\Sigma}^{-1/2}((\bm{I}-\bm{U}^{\top})\bm{Y}_{i}-\bm{W}^{\top}\bm{X}_{i})\|^{2}_{2}-n\log\sqrt{\det(\bm{\Sigma})},$
(4)
where $\bm{\theta}=(\bm{U},\bm{W})$,
$\bm{\Sigma}=\text{Diag}(\sigma_{1}^{2},\ldots,\sigma_{p}^{2})$, and $\bm{U}$
is subject to the acyclicity constraint (Zheng et al., 2018; Yuan et al.,
2019) that no directed cycle is permissible in a DAG. The likelihood ratio is
defined as
$\text{Lr}=L(\widehat{\bm{\theta}}^{(1)},\widehat{\bm{\Sigma}})-L(\widehat{\bm{\theta}}^{(0)},\widehat{\bm{\Sigma}})$,
where $\widehat{\bm{\theta}}^{(1)}$ is a maximum likelihood estimate (MLE) in
(4) subject to the acyclicity constraint, $\widehat{\bm{\theta}}^{(0)}$ is an
MLE subject to an additional requirement $\bm{U}_{\mathcal{H}}=\bm{0}$, and
$\widehat{\bm{\Sigma}}$ is an estimator of $\bm{\Sigma}$.
In many statistical models, a likelihood ratio has a nondegenerate and
tractable distribution, for instance, a chi-squared distribution with degrees
of freedom $|\mathcal{H}|$. However, in (1), because of the acyclicity
constraint, some hypothesized edges in $\mathcal{H}$ can be absent given the
edge subset $\mathcal{E}\cap\mathcal{H}^{c}$ defined by the nonzero elements
of nuisance parameter $\bm{U}_{\mathcal{H}^{c}}$, where $\mathcal{E}$ denotes
the edge set of the whole DAG. As a result, Lr may converge to a distribution
with degrees of freedom less than $|\mathcal{H}|$ and the distribution may be
even intractable, making inference for a DAG greatly different from the
classical ones, as illustrated by Example 1.
Figure 1: A true DAG structure of five primary variables $Y_{1},\ldots,Y_{5}$
and five intervention variables $X_{1},\ldots,X_{5}$, where directed edges are
represented by solid arrows while dependencies among $\bm{X}$ are not
displayed.
###### Example 1
Consider the likelihood ratio test under null $H_{0}$ and alternative $H_{a}$
for the DAG displayed in Figure 1. Let $\mathcal{E}$ be the edge set of the
DAG.
* •
$H_{0}:U_{21}=0$ versus $H_{a}:U_{21}\neq 0$, where $\mathcal{H}=\\{(2,1)\\}$.
Here, $(2,1)$ forms a cycle together with the edges in
$\mathcal{E}\cap\mathcal{H}^{c}$ (namely the edges not considered by the
hypothesis), violating the acyclicity constraint. Note that the maximum
likelihood value $L(\widehat{\bm{\theta}},\widehat{\bm{\Sigma}})$ with
$U_{21}\neq 0$ tends to be smaller than that with $U_{21}=0$ when a random
sample is obtained under $H_{0}$, especially so when the asymptotics kicks in
as the sample size increases. Consequently, the likelihood ratio Lr becomes
zero, constituting a degenerate situation.
* •
$H_{0}:U_{45}=U_{53}=0$ versus $H_{a}:U_{45}\neq 0$ or $U_{53}\neq 0$, where
$\mathcal{H}=\\{(4,5),(5,3)\\}$. In this case, $\\{(4,5),(5,3)\\}$ forms a
cycle with the edges in $\mathcal{E}\cap\mathcal{H}^{c}$, violating the
acyclicity constraint. Since the likelihood value
$L({\bm{\theta}},\bm{\Sigma})$ tends to be maximized under the true graph when
data is sampled under $H_{0}$, we have
$L(\widehat{\bm{\theta}}^{(1)},\widehat{\bm{\Sigma}})=\max(L(\widehat{U}_{45},\widehat{U}_{53}=0,\widehat{\bm{U}}_{\mathcal{H}^{c}},\widehat{\bm{W}},\widehat{\bm{\Sigma}}),L(\widehat{U}_{45}=0,\widehat{U}_{53},\widehat{\bm{U}}_{\mathcal{H}^{c}},\widehat{\bm{W}},\widehat{\bm{\Sigma}}))$.
As a result, the likelihood ratio distribution becomes complicated in this
situation due to the dependence between the two components in
$L(\widehat{\bm{\theta}}^{(1)},\widehat{\bm{\Sigma}})$.
Motivated by Example 1, we introduce the concepts of nondegeneracy and
regularity.
###### Definition 2 (Nondegeneracy and regularity with respect to DAG)
1. (A)
An edge $(k,j)\in\mathcal{H}$ is nondegenerate with respect to DAG $G$ if
$\\{(k,j)\\}\cup\mathcal{E}$ contains no directed cycle, where $\mathcal{E}$
denotes the edge set of $G$. Otherwise, $(k,j)$ is degenerate. Let
$\mathcal{D}\subseteq\mathcal{H}$ be the set of all nondegenerate edges with
respect to $G$. A null hypothesis $H_{0}$ is nondegenerate with respect to DAG
$G$ if $\mathcal{D}\neq\emptyset$. Otherwise, $H_{0}$ is degenerate.
2. (B)
A null hypothesis $H_{0}$ is said to be regular with respect to DAG $G$ if
$\mathcal{D}\cup\mathcal{E}$ contains no directed cycle, where $\mathcal{E}$
denotes the edge set of $G$. Otherwise, $H_{0}$ is called irregular.
###### Remark 1
In practice, $\mathcal{D}$ is unknown and needs to be estimated from data.
Nondegeneracy ensures nonnegativity of the likelihood ratio. In testing (2),
regularity excludes intractable situations for the null distribution. In
testing (3), if $H_{0}$ is irregular, then $\mathcal{D}\cup\mathcal{E}$ has a
directed cycle, which means the hypothesized directed pathway cannot exist due
to the acyclicity constraint. Thus, regularity excludes the degenerate
situations in testing (3).
In what follows, we mainly focus on nondegenerate and regular hypotheses. For
the degenerate case, the p-value is defined to be one. For the irregular case
of edge test (2), we decompose the hypothesis into regular sub-hypotheses and
conduct multiple testing. For the irregular case of pathway test (3), the
p-value is defined to be one. More discussions on the implementation in
irregular cases are provided in Appendix A.5.
Finally, we introduce some notations for a DAG $G$ with primary variables
$\\{Y_{1},\ldots,Y_{p}\\}$, intervention variables $\\{X_{1},\ldots,X_{q}\\}$,
a directed edge set $\mathcal{E}\subseteq\\{(k,j):k\neq j,1\leq k,j\leq p\\}$,
and an intervention edge set $\mathcal{I}\subseteq\\{(l,j):1\leq l\leq q,1\leq
j\leq p\\}$. If there is a directed path $Y_{k}\to\cdots\to Y_{j}$ in $G$,
$Y_{k}$ is called an ancestor of $Y_{j}$ or $Y_{j}$ a descendant of $Y_{k}$.
Let $\mathcal{A}=\\{(k,j):Y_{k}\to\cdots\to Y_{j}\\}$ be the set of all
ancestral relations. In a DAG $G$, let
$\textnormal{{pa}}_{G}(j)=\\{k:(k,j)\in\mathcal{E}\\}$,
$\textnormal{{an}}_{G}(j)=\\{k:(k,j)\in\mathcal{A}\\}$,
$\textnormal{{d}}_{G}(j)=\\{k:(k,j)\in\mathcal{D}\\}$, and
$\textnormal{{in}}_{G}(j)=\\{l:(l,j)\in\mathcal{I}\\}$ be the parent,
ancestor, nondegenerate hypothesized parent, and intervention sets of $Y_{j}$,
respectively.
## 3 Methodology
This section develops the main methodology, including the peeling algorithm
for structure learning and the data perturbation inference for simultaneous
testing of directed edges (2) and pathways (3).
One major challenge to the likelihood approach is optimization subject to the
acyclicity requirement. This constraint imposes difficulty on not only
computation but also asymptotic theory. As a result, there is a gap between
the asymptotic distribution based on a global maximum and that on the actual
estimate which could be a local maximum (Janková and van de Geer, 2018; Li et
al., 2019). Moreover, the actual estimate may give an imprecise topological
order, tending to impact adversely on inference.
To circumvent the acyclicity requirement, we propose using a super-graph $S$
of the true DAG $G$, where $S$ is constructed by a novel peeling method
without explicitly imposing the acyclicity constraint. As a result, our method
is scalable with a statistical guarantee of the actual estimate. We will focus
on a specific super-graph $S$ consisting of primary variable edges
$\mathcal{E}_{S}:=\mathcal{A}\supseteq\mathcal{E}$ and candidate intervention
edges $\mathcal{I}_{S}:=\mathcal{C}\supseteq\mathcal{I}$, where $\mathcal{E}$
and $\mathcal{I}$ are sets of primary variable edges and intervention edges in
the original graph $G$, and set $\mathcal{C}$ is to be defined shortly after
(6). Note that $S$, whose primary variable edges consist of all ancestral
relations of $G$, is also a DAG. Given the super-graph $S$, the true DAG $G$
can be reconstructed by nodewise constrained regressions.
Let $\bm{\theta}_{S}=\big{(}\bm{U}_{\mathcal{A}},\bm{W}_{\mathcal{C}}\big{)}$
be $\bm{\theta}$ restricted to $S$. Note that
$\bm{\theta}_{S^{c}}=(\bm{U}_{\mathcal{A}^{c}},\bm{W}_{\mathcal{C}^{c}})=\bm{0}$
in the original graph $G$. Given $S$, the log-likelihood
$L(\bm{\theta},\bm{\Sigma})$ in (4) can be rewritten as
$\begin{split}L_{S}(\bm{\theta}_{S},\bm{\Sigma})&=-\sum_{j=1}^{p}\Big{(}\text{RSS}_{j}(\bm{\theta}_{S})/2\sigma_{j}^{2}+\log\sqrt{2\pi\sigma_{j}^{2}}\Big{)},\\\
\text{RSS}_{j}(\bm{\theta}_{S})&=\sum_{i=1}^{n}\Big{(}Y_{ij}-\sum_{(k,j)\in\mathcal{A}}U_{kj}Y_{ik}-\sum_{(l,j)\in\mathcal{C}}W_{lj}X_{il}\Big{)}^{2},\\\
\end{split}$ (5)
which involves $|\mathcal{A}|+|\mathcal{C}|$ parameters for $\bm{\theta}_{S}$.
Our plan is as follows. In Section 3.1, we hierarchically construct ${S}$ via
nodewise regressions of $Y_{j}$ over $\bm{X}$; $j=1,\ldots,p$ without the
acyclicity constraint for $\bm{U}$. Then in Section 3.2, on the ground of
constructed super-graph $S$ and (5), we develop likelihood ratio tests for
hypotheses (2) and (3).
### 3.1 Structure learning via peeling
This section develops a novel structure learning method based on a peeling
algorithm and nodewise constrained regressions to construct
$S=(\mathcal{A},\mathcal{C})$ in a hierarchical manner.
First, we observe an important connection between primary variables and
intervention variables. Rewrite (1) as
$\bm{Y}=\bm{V}^{\top}\bm{X}+\bm{\varepsilon}_{V},\quad\bm{\varepsilon}_{V}=(\bm{I}-\bm{U}^{\top})^{-1}\bm{\varepsilon}\sim
N(\bm{0},\bm{\Omega}^{-1}),$ (6)
where $\bm{\Omega}=(\bm{I}-\bm{U})\bm{\Sigma}^{-1}(\bm{I}-\bm{U}^{\top})$ is a
precision matrix and $\bm{V}=\bm{W}(\bm{I}-\bm{U})^{-1}$. Now, define the set
of candidate interventional relations $\mathcal{C}:=\\{(l,j):V_{lj}\neq 0\\}$.
###### Proposition 2
Suppose Assumption 1 is satisfied. Then $\mathcal{C}\supseteq\mathcal{I}$.
Moreover,
1. (A)
if $V_{lj}\neq 0$, then $X_{l}$ intervenes on $Y_{j}$ or an ancestor of
$Y_{j}$;
2. (B)
$Y_{j}$ is a leaf variable (having no child) if and only if there is an
instrument $X_{l}$ such that $V_{lj}\neq 0$ and $V_{lj^{\prime}}=0$ for
$j^{\prime}\neq j$.
The proof of Proposition 2 is deferred to Appendix B.2. Intuitively,
$V_{lj}\neq 0$ implies the dependence of $Y_{j}$ on $X_{l}$ through a directed
path $X_{l}\to\cdots\to Y_{j}$, and hence that $X_{l}$ intervenes on $Y_{j}$
or an ancestor of $Y_{j}$. Thus, the instruments on a leaf variable are
independent of the other primary variables conditional on the rest of
interventions. This observation suggests a method to reconstruct the DAG
topological order by sequentially identifying the leaves in the DAG and
removing the identified leaf variables.
Next, we discuss the estimation of $\bm{V}$ and construction of $S$.
#### 3.1.1 Nodewise constrained regressions
We propose nodewise $\ell_{0}$-constrained regressions to estimate nonzero
elements of $\bm{V}$:
$\widehat{\bm{V}}_{\cdot j}=\mathop{\rm arg\,min\,}_{\bm{V}_{\cdot j}}\
\sum_{i=1}^{n}(Y_{ij}-\bm{V}_{\cdot
j}^{\top}\bm{X}_{i})^{2}\quad\text{s.t.}\quad\sum_{l=1}^{q}\operatorname{I}(V_{lj}\neq
0)\leq\kappa_{j};\quad j=1,\ldots,p,$ (7)
where $1\leq\kappa_{j}\leq q$ is an integer-valued tuning parameter
controlling the sparsity of $\bm{V}_{\cdot j}$ and can be chosen by BIC or
cross-validation.
To solve (7), we use $J_{\tau_{j}}(z)=\min(|z|/\tau_{j},1)$ as a surrogate of
the $\ell_{0}$-function $\operatorname{I}(z\neq 0)$ (Shen et al., 2012) and
develop a difference-of-convex (DC) algorithm with the $\ell_{0}$-projection
to improve globality of the solution of (7). At $(t+1)$th iteration, we solve
a weighted Lasso problem,
$\displaystyle\widetilde{\bm{V}}_{\cdot j}^{[t+1]}=\mathop{\rm
arg\,min\,}_{\bm{V}_{\cdot j}}\ \sum_{i=1}^{n}(Y_{ij}-\bm{V}_{\cdot
j}^{\top}\bm{X}_{i})^{2}+2n\gamma_{j}\tau_{j}\sum_{l=1}^{q}\operatorname{I}\big{(}|\widetilde{V}^{[t]}_{lj}|\leq\tau_{j}\big{)}|V_{lj}|;\quad
j=1,\ldots,p,$ (8)
where $\gamma_{j}>0$ is a tuning parameter and
$\widetilde{\bm{V}}^{[t]}_{\cdot j}$ is the solution of (8) at the $t$-th
iteration. The DC algorithm terminates at $\widetilde{\bm{V}}_{\cdot
j}=\widetilde{\bm{V}}_{\cdot j}^{[t]}$ such that $\|\widetilde{\bm{V}}_{\cdot
j}^{[t+1]}-\widetilde{\bm{V}}_{\cdot
j}^{[t]}\|_{\infty}\leq\sqrt{\text{tol}}$, where tol is the machine precision.
Then, we obtain the solution $\widehat{\bm{V}}_{\cdot j}$ of (7) by projecting
$\widetilde{\bm{V}}_{\cdot j}$ onto the $\ell_{0}$-constrained set
$\\{\|\bm{V}_{\cdot j}\|_{0}\leq\kappa_{j}\\}$.
1
2Specify $\gamma_{j}>0$, $\tau_{j}>0$, and $1\leq\kappa_{j}\leq q$. Initialize
$\widetilde{\bm{V}}_{\cdot j}^{[0]}$ with
$|\\{l:\widetilde{V}^{[0]}_{lj}>\tau_{j}\\}|\leq\kappa_{j}$.
3Compute $\widetilde{\bm{V}}_{\cdot j}=\widetilde{\bm{V}}^{[t]}_{\cdot j}$
such that $\|\widetilde{\bm{V}}_{\cdot j}^{[t+1]}-\widetilde{\bm{V}}_{\cdot
j}^{[t]}\|_{\infty}\leq\sqrt{\text{tol}}$.
4($\ell_{0}$-projection) Let
$B_{j}=\\{l:\sum_{l^{\prime}=1}^{q}\operatorname{I}(|\widetilde{V}_{l^{\prime}j}|\geq|\widetilde{V}_{lj}|)\leq\kappa_{j}\\}$.
Set $\widehat{\bm{V}}_{\cdot j}=\mathop{\rm arg\,min\,}_{\bm{V}_{\cdot j}}\
\|\bm{\mathsf{Y}}_{j}-\bm{V}_{\cdot
j}^{\top}\bm{\mathsf{X}}\|_{2}^{2}\quad\text{s.t.}\quad V_{lj}=0\text{ for
}l\notin B_{j}.$
Algorithm 1 Constrained minimization via DC programming + $\ell_{0}$
projection
1
2Let $\widehat{\bm{V}}^{[0]}=\widehat{\bm{V}}$. Begin iteration with
$h=0,\ldots$: at iteration $h$,
3In the current sub-graph identify all instrument-leaf pairs $X_{l}\to Y_{j}$
satisfying $l=\mathop{\rm
arg\,min\,}_{l^{\prime}:\|\widehat{\bm{V}}^{[h]}_{l^{\prime}\cdot}\|_{0}\neq
0}\|\widehat{\bm{V}}^{[h]}_{l^{\prime}\cdot}\|_{0}$ and $j=\mathop{\rm
arg\,max\,}_{j^{\prime}}|\widehat{V}^{[h]}_{lj^{\prime}}|$. Let
$h_{\widehat{G}}(j)=h$.
4For a leaf $Y_{k}$, identify all directed relations $Y_{k}\to Y_{j}$ with
$h_{\widehat{G}}(j)=h-1$ when $\widehat{V}_{lj}\neq 0$ for each instrument
$X_{l}$ of $Y_{k}$ in the current sub-graph.
5Peeling-off all the leaves $Y_{j}$ by deleting $j$th columns from
$\widehat{\bm{V}}^{[h]}$ to yield $\widehat{\bm{V}}^{[h+1]}$.
6Repeat Steps 2-4 until all primary variables are removed.
7Compute $\widehat{S}=(\widehat{\mathcal{A}},\widehat{\mathcal{C}})$ and
$\widehat{\mathcal{D}}$:
$\begin{split}\widehat{\mathcal{A}}&=\Big{\\{}(k,j):Y_{k}\to\cdots\to
Y_{j}\Big{\\}},\quad\widehat{\mathcal{D}}=\Big{\\{}(k,j)\in\mathcal{H}:(j,k)\notin\widehat{\mathcal{A}}\Big{\\}},\\\
\widehat{\mathcal{C}}&=\Big{\\{}(l,j):|\widehat{V}_{lj}|>\tau_{j}\Big{\\}},\text{
where $\tau_{j}$ is specified in Algorithm \ref{algorithm:nodewise-
regression}.}\end{split}$
Algorithm 2 Reconstruction of topological layers by peeling
#### 3.1.2 Peeling
Now, we describe a _peeling_ algorithm to estimate
$S=(\mathcal{A},\mathcal{C})$ based on $\bm{V}$.
Before proceeding, we introduce the notion of _topological height_ to
characterize the DAG topological order. Given a DAG $G$, the height $h_{G}(j)$
of $Y_{j}$ is defined as the maximal length of a directed path from $Y_{j}$ to
a leaf variable, where $0\leq h_{G}(j)\leq p-1$ with a leaf having zero
height. Then the primary variables of the same height form a topological layer
and the layers constitute a partition of primary variables in $G$. For
instance, in Figure 1, $\\{Y_{4},Y_{5}\\},\\{Y_{3}\\},\\{Y_{2}\\},\\{Y_{1}\\}$
are topological layers with height $0,1,2,3$, respectively.
Observe that the topological layer of height $h$ are leaves in the sub-graph
with primary variables of height no less than $h$. Based on Proposition 2, the
peeling algorithm sequentially identifies and peels off the primary variables
of height $h$; $h=0,1,2,\ldots$. Note that instruments in the sub-graph may
not be necessarily the instruments in the graph $G$ after some primary
variables are removed. However, the next proposition indicates that the
instruments in the corresponding sub-graph remain useful to reconstruct
ancestral relations.
###### Proposition 3
Let $Y_{k}$ have height $h_{G}(k)$. Let $G^{\prime}$ be the sub-graph with
primary variables of height $\geq h_{G}(k)$. Under Assumption 1, for a pair
$(k,j)$ with $h_{G}(j)=h_{G}(k)-1$, we have $Y_{k}\to Y_{j}$ if and only if
$V_{lj}\neq 0$ for each instrument $X_{l}$ of $Y_{k}$ in $G^{\prime}$.
Proposition 3 (proved in Appendix B.3) permits the construction of directed
relations between the topological layers of heights $h$ and $h-1$; $h\geq 1$
and thus ancestral relations.
The peeling algorithm is summarized in Algorithm 2 and a detailed illustration
is presented in Example 3 of Appendix A.2.
###### Remark 2 (Recovery of true graph $G$)
$\widehat{\mathcal{A}}$ estimates a superset of directed relations of $\bm{U}$
and $\widehat{\mathcal{C}}$ a superset of interventional relations of
$\bm{W}$. Therefore, under some technical conditions, with suitable tuning
parameters, the true graph $G$ can be reconstructed by estimating
$(\bm{U},\bm{W})$ through $\ell_{0}$-constrained regressions given the
estimated super-graph
$\widehat{S}=(\widehat{\mathcal{A}},\widehat{\mathcal{C}})$,
$\begin{split}\min_{(\bm{U}_{\textnormal{{an}}_{\widehat{S}}(j),j},\bm{W}_{\textnormal{{in}}_{\widehat{S}}(j),j})}\sum_{i=1}^{n}\left(Y_{ij}-\bm{U}_{\textnormal{{an}}_{\widehat{S}}(j),j}^{\top}\bm{Y}_{i,\textnormal{{an}}_{\widehat{S}}(j)}-\bm{W}_{\textnormal{{in}}_{\widehat{S}}(j),j}^{\top}\bm{X}_{i,\textnormal{{in}}_{\widehat{S}}(j)}\right)^{2}\\\
\text{s.t.}\quad\sum_{k\in\textnormal{{an}}_{\widehat{S}}(j)}\operatorname{I}(U_{kj}\neq
0)+\sum_{l\in\textnormal{{in}}_{\widehat{S}}(j)}\operatorname{I}(W_{lj}\neq
0)\leq\kappa_{j}^{\prime}.\end{split}$ (9)
The final estimates $\widehat{\bm{U}}_{\cdot
j}=\Big{(}\widehat{\bm{U}}_{\textnormal{{an}}_{\widehat{S}}(j),j},\bm{0}_{\textnormal{{an}}_{\widehat{S}}(j)^{c}}\Big{)}$
and $\widehat{\bm{W}}_{\cdot
j}=\Big{(}\widehat{\bm{W}}_{{\textnormal{{in}}}_{\widehat{S}}(j),j},\bm{0}_{{\textnormal{{in}}}_{\widehat{S}}(j)^{c}}\Big{)}$;
$j=1,\ldots,p$.
### 3.2 Likelihood ratio test via data perturbation
This subsection proposes an inference method for testing (2) and (3). We first
construct a likelihood ratio $\text{Lr}(S,\bm{\Sigma})$ in given
$(S,\bm{\Sigma})$ and then plug-in an estimate
$(\widehat{S},\widehat{\bm{\Sigma}})$. Next we perform a test via data
perturbation, accounting for the uncertainty of constructing $S$.
#### 3.2.1 Likelihood ratio given $S$
Consider the test in (2). From (5), the likelihood ratio given
$(S,\bm{\Sigma})$ is
$\text{Lr}(S,\bm{\Sigma})=L_{S}(\widehat{\bm{\theta}}_{S}^{(1)},\bm{\Sigma})-L_{S}(\widehat{\bm{\theta}}_{S}^{(0)},\bm{\Sigma})=\sum_{j=1}^{p}\frac{\text{RSS}_{j}(\widehat{\bm{\theta}}^{(0)}_{S})-\text{RSS}_{j}(\widehat{\bm{\theta}}^{(1)}_{S})}{2\sigma_{j}^{2}}$
where $\widehat{\bm{\theta}}_{S}^{(1)}$ is the MLE given $(S,\bm{\Sigma})$,
and $\widehat{\bm{\theta}}_{S}^{(0)}$ is the MLE given $(S,\bm{\Sigma})$
subject to $\bm{U}_{\mathcal{H}}=0$.
Now, recall that $\textnormal{{d}}_{G}(j)=\\{k:(k,j)\in\mathcal{D}\\}$, where
$\mathcal{D}$ is the set of nondegenerate edges of $H_{0}$ with respect to $G$
(Definition 2). Of note, $\textnormal{{d}}_{S}(j)=\textnormal{{d}}_{G}(j)$
since an edge is nondegenerate with respect to $S$ if and only if it is
nondegenerate with respect to $G$. To eliminate the nuisance parameters,
observe that if $\textnormal{{d}}_{S}(j)=\emptyset$, then
${\text{RSS}}_{j}(\widehat{\bm{\theta}}^{(0)}_{S})-{\text{RSS}}_{j}(\widehat{\bm{\theta}}^{(1)}_{S})=0$,
because no $H_{0}$ constraint is imposed for $Y_{j}$. Hence,
$\text{Lr}(S,\bm{\Sigma})$ only summarizes the contributions from the primary
variables with nondegenerate hypothesized edges,
$\text{Lr}(S,\bm{\Sigma})=\sum_{j:\textnormal{{d}}_{S}(j)\neq\emptyset}\frac{{\text{RSS}}_{j}(\widehat{\bm{\theta}}^{(0)}_{S})-{\text{RSS}}_{j}(\widehat{\bm{\theta}}^{(1)}_{S})}{2\sigma_{j}^{2}},$
(10)
where the nuisance parameters for $Y_{j}$ with
$\textnormal{{d}}_{S}(j)=\emptyset$ are eliminated.
The likelihood ratio test statistic becomes
$\text{Lr}=\text{Lr}(\widehat{S},\widehat{\bm{\Sigma}})$. Here, $\widehat{S}$
is constructed by Algorithm 2 in Section 3.1 and
$\widehat{\bm{\Sigma}}=\text{Diag}(\widehat{\sigma}^{2}_{1},\ldots,\widehat{\sigma}^{2}_{p})$,
$\displaystyle\widehat{\sigma}^{2}_{j}=\frac{\text{RSS}_{j}(\widehat{\bm{\theta}}^{(1)}_{\widehat{S}})}{n-|{\textnormal{{an}}}_{\widehat{S}}(j)|-|{\textnormal{{in}}}_{\widehat{S}}(j)|},\quad
j=1,\ldots,p,$ (11)
where ${\textnormal{{an}}}_{\widehat{S}}(j)$ and
${\textnormal{{in}}}_{\widehat{S}}(j)$ are the estimated ancestor and
candidate intervention sets of $Y_{j}$.
#### 3.2.2 Testing directed edges (2) via data perturbation
The likelihood ratio requires an estimation of $S$, where we must account for
the uncertainty of $\widehat{S}$ for accurate finite-sample inference.
To proceed, we consider the test statistic Lr based on a “correct” super-graph
$S\supseteq S^{\circ}$, where $S=(\mathcal{A},\mathcal{C})\supseteq
S^{\circ}=(\mathcal{A}^{\circ},\mathcal{C}^{\circ})$ means that
$\mathcal{A}\supseteq\mathcal{A}^{\circ}$ and
$\mathcal{C}\supseteq\mathcal{C}^{\circ}$ and ∘ denotes the truth.
Intuitively, a “correct” super-graph distinguishes descendants and
nondescendants and thus can help to infer the true directed relations defined
by the local Markov property (Spirtes et al., 2000) without introducing model
errors, yet may lead to a less powerful test when $S$ is much larger than
$S^{\circ}$. By comparison, a “wrong” graph $S\not\supseteq S^{\circ}$
provides an incorrect topological order, and a test based on a “wrong” graph
may be biased, accompanied by an inflated type-I error.
Now, let $\bm{\mathsf{Z}}_{n\times(p+q)}=(\bm{\mathsf{X}},\bm{\mathsf{Y}})$ be
the data matrix, where $\bm{\mathsf{Y}}_{n\times
p}=(\bm{Y}_{1},\ldots,\bm{Y}_{n})^{\top}$ and $\bm{\mathsf{X}}_{n\times
q}=(\bm{X}_{1},\ldots,\bm{X}_{n})^{\top}$. Moreover, let
$\bm{\mathsf{e}}_{j}\in\mathbb{R}^{n}$ be the $j$th column of error matrix
$\bm{\mathsf{e}}_{n\times
p}=(\bm{\varepsilon}_{1},\ldots,\bm{\varepsilon}_{n})^{\top}$. From (10),
assuming $\widehat{S}\supseteq S^{\circ}$ is a “correct” super-graph, the
likelihood ratio becomes
$\begin{split}\text{Lr}&=\sum_{j:{\textnormal{{d}}}_{\widehat{S}}(j)\neq\emptyset}\frac{\bm{\mathsf{Y}}_{j}^{\top}(\bm{\mathsf{P}}_{\widehat{A}_{j}}-\bm{\mathsf{P}}_{\widehat{B}_{j}})\bm{\mathsf{Y}}_{j}}{2\widehat{\sigma}_{j}}\\\
&=\sum_{j:{\textnormal{{d}}}_{\widehat{S}}(j)\neq\emptyset}\frac{(\bm{\mathsf{Y}}_{{\textnormal{{d}}}_{\widehat{S}}(j)}\bm{U}_{{\textnormal{{d}}}_{\widehat{S}}(j),j}+\bm{\mathsf{e}}_{j})^{\top}(\bm{\mathsf{P}}_{\widehat{A}_{j}}-\bm{\mathsf{P}}_{\widehat{B}_{j}})(\bm{\mathsf{Y}}_{{\textnormal{{d}}}_{\widehat{S}}(j)}\bm{U}_{{\textnormal{{d}}}_{\widehat{S}}(j),j}+\bm{\mathsf{e}}_{j})}{2\widehat{\sigma}_{j}},\\\
\widehat{\sigma}_{j}&=\frac{\bm{\mathsf{Y}}_{j}^{\top}(\bm{I}-\bm{\mathsf{P}}_{\widehat{A}_{j}})\bm{\mathsf{Y}}_{j}}{n-|\widehat{A}_{j}|}=\frac{\bm{\mathsf{e}}_{j}^{\top}(\bm{I}-\bm{\mathsf{P}}_{\widehat{A}_{j}})\bm{\mathsf{e}}_{j}}{n-|\widehat{A}_{j}|};\quad
1\leq j\leq p,\end{split}$ (12)
where
$\bm{\mathsf{P}}_{A}=\bm{\mathsf{Z}}_{A}(\bm{\mathsf{Z}}_{A}^{\top}\bm{\mathsf{Z}}_{A})^{-1}\bm{\mathsf{Z}}_{A}^{\top}$
is the projection matrix onto the span of columns $A$ in $\bm{\mathsf{Z}}$,
$\widehat{A}_{j}={\textnormal{{an}}}_{\widehat{S}}(j)\cup{\textnormal{{in}}}_{\widehat{S}}(j)\cup{\textnormal{{d}}}_{\widehat{S}}(j)$,
and
$\widehat{B}_{j}=({\textnormal{{an}}}_{\widehat{S}}(j)\cup{\textnormal{{in}}}_{\widehat{S}}(j))\setminus{\textnormal{{d}}}_{\widehat{S}}(j)$.
Of note,
$\bm{\mathsf{Y}}_{{\textnormal{{d}}}_{S}(j)}\bm{U}_{{\textnormal{{d}}}_{S}(j),j}=\bm{0}$
for all $j$ under $H_{0}$, while
$\bm{\mathsf{Y}}_{{\textnormal{{d}}}_{S}(j)}\bm{U}_{{\textnormal{{d}}}_{S}(j),j}\neq\bm{0}$
for some $j$ under $H_{a}$.
We propose a data perturbation (DP) method (Shen and Ye, 2002; Breiman, 1992)
to approximate the null distribution of Lr in (12). The idea behind DP is to
assess the sensitivity of the estimates through perturbed data
$\bm{Y}^{*}_{i}=\bm{Y}_{i}+\bm{\varepsilon}^{*}_{i}$, where
$\bm{\varepsilon}_{i}^{*}\sim N(0,\widehat{\bm{\Sigma}})$; $i=1,\ldots,n$.
Let
$(\bm{\mathsf{Z}}^{*},\bm{\mathsf{e}}^{*})=(\bm{\mathsf{X}},\bm{\mathsf{Y}}^{*},\bm{\mathsf{e}}^{*})$
be perturbed data, where $\bm{\mathsf{e}}^{*}_{n\times
p}=(\bm{\mathsf{e}}^{*}_{1},\ldots,\bm{\mathsf{e}}^{*}_{p})=(\bm{\varepsilon}^{*}_{1},\ldots,\bm{\varepsilon}^{*}_{n})^{\top}$
denotes perturbation errors. Given
$(\bm{\mathsf{Z}}^{*},\bm{\mathsf{e}}^{*})$, we obtain a perturbation estimate
$\widehat{S}^{*}$. In (12), under $H_{0}$,
$\text{Lr}=f(\bm{\mathsf{X}},\bm{\mu},\bm{\mathsf{e}})$ is a function of
intervention data $\bm{\mathsf{X}}$, conditional mean
$\bm{\mu}=\operatorname{\mathbb{E}}(\bm{\mathsf{Y}}\mid\bm{\mathsf{X}})\in\mathbb{R}^{n\times
p}$ and unobserved error $\bm{\mathsf{e}}$. However, the perturbation error
$\bm{\mathsf{e}}^{*}$ is _known_ given the DP sample
$(\bm{\mathsf{Z}}^{*},\bm{\mathsf{e}}^{*})$, suggesting a form $\text{Lr}^{*}$
based on Lr:
$\begin{split}\text{Lr}^{*}&\equiv
f(\bm{\mathsf{X}},\bm{\mathsf{Y}},\bm{\mathsf{e}}^{*})=\sum_{j:{\textnormal{{d}}}_{\widehat{S}}(j)\neq\emptyset}\frac{(\bm{\mathsf{e}}_{j}^{*})^{\top}(\bm{\mathsf{P}}_{\widehat{A}^{*}_{j}}-\bm{\mathsf{P}}_{\widehat{B}^{*}_{j}})\bm{\mathsf{e}}^{*}_{j}}{2\widetilde{\sigma}_{j}^{*}},\quad\widetilde{\sigma}^{*}_{j}=\frac{(\bm{\mathsf{e}}_{j}^{*})^{\top}(\bm{I}-\bm{\mathsf{P}}_{\widehat{A}^{*}_{j}})\bm{\mathsf{e}}^{*}_{j}}{n-|\widehat{A}^{*}_{j}|},\quad
1\leq j\leq p,\end{split}$ (13)
which mimics (12) when
$\bm{\mathsf{Y}}_{{\textnormal{{d}}}_{S}(j)}\bm{U}_{{\textnormal{{d}}}_{S}(j)}=\bm{0}$
under $H_{0}$, with $\bm{\mathsf{e}}_{j}$ replaced by
$\bm{\mathsf{e}}^{*}_{j}$. As a result, when $\widehat{S}^{*}\supseteq
S^{\circ}$, the conditional distribution of $\text{Lr}^{*}$ given
$\bm{\mathsf{Z}}$ well approximates the null distribution of Lr, where the
model selection effect is accounted for by assessing the variability of
$\widehat{A}^{*}_{j}$ and $\widehat{B}^{*}_{j}$ over different realizations of
$(\bm{\mathsf{Z}}^{*},\bm{\mathsf{e}}^{*})$.
In practice, we use Monte-Carlo to approximate the distribution of
$\text{Lr}^{*}$ given $\bm{\mathsf{Z}}$ in (13). In particular, we generate
$M$ perturbed samples
$(\bm{\mathsf{Z}}^{*}_{m},\bm{\mathsf{e}}_{m}^{*})_{m=1}^{M}$ independently
and compute $\text{Lr}^{*}_{m}$; $m=1,\ldots,M$. Then, we examine the
condition $\widehat{S}^{*}_{m}\supseteq S^{\circ}$ by checking its empirical
counterpart $\widehat{S}^{*}_{m}\supseteq\widehat{S}$. The DP p-value of edge
test in (2) is defined as
$\text{Pval}=\left(\sum_{m=1}^{M}\operatorname{I}(\text{Lr}_{m}^{*}\geq\text{Lr},\widehat{S}^{*}_{m}\supseteq\widehat{S})\right)/\left(\sum_{m=1}^{M}\operatorname{I}(\widehat{S}^{*}_{m}\supseteq\widehat{S})\right),$
(14)
where $\operatorname{I}(\cdot)$ is the indicator function.
###### Remark 3
Assume $\widehat{S}^{*}\supseteq S^{\circ}$ is a “correct” super-graph. A
naive likelihood ratio based on perturbed data
$(\bm{\mathsf{Z}}^{*},\bm{\mathsf{e}}^{*})$ is
$\text{Lr}(\widehat{S}^{*},\widehat{\bm{\Sigma}}^{*})=\sum_{j:{\textnormal{{d}}}_{\widehat{S}}(j)\neq\emptyset}\frac{(\bm{\mathsf{Y}}_{{\textnormal{{d}}}_{\widehat{S}}(j)}\bm{U}_{{\textnormal{{d}}}_{\widehat{S}}(j),j}+\bm{\mathsf{e}}_{j}+\bm{\mathsf{e}}^{*}_{j})^{\top}(\bm{\mathsf{P}}_{\widehat{A}^{*}_{j}}-\bm{\mathsf{P}}_{\widehat{B}^{*}_{j}})(\bm{\mathsf{Y}}_{{\textnormal{{d}}}_{\widehat{S}}(j)}\bm{U}_{{\textnormal{{d}}}_{\widehat{S}}(j),j}+\bm{\mathsf{e}}_{j}+\bm{\mathsf{e}}^{*}_{j})}{2\widehat{\sigma}_{j}^{*}}.$
Note that the error $\bm{\mathsf{e}}_{j}$ given $\bm{\mathsf{Z}}$ is
deterministic and does not vanish under either $H_{0}$ or $H_{a}$;
$j=1,\ldots,p$. Thus, the conditional distribution of
$\text{Lr}(\widehat{S}^{*},\widehat{\bm{\Sigma}}^{*})$ given $\bm{\mathsf{Z}}$
does not approximate the null distribution of Lr in (12), in contrast to the
DP likelihood ratio $\text{Lr}^{*}$ in (13).
#### 3.2.3 Extension to hypothesis testing for a pathway
Next, we extend the DP inference for (3). Denote
$\mathcal{H}=\\{(k_{1},j_{1}),\ldots,(k_{|\mathcal{H}|},j_{|\mathcal{H}|})\\}$.
Then the test of pathways in (3) can be reduced to testing sub-hypotheses
$H_{0,\nu}:U_{k_{\nu},j_{\nu}}=0,\quad\mbox{ versus
}\quad{H}_{a,\nu}:U_{k_{\nu},j_{\nu}}\neq 0;\quad\nu=1,\ldots,|\mathcal{H}|,$
where each sub-hypothesis is a directed edge test. Given $(S,\bm{\Sigma})$,
the likelihood ratio for $\text{H}_{0,\nu}$ is
$\text{Lr}_{\nu}(S,\bm{\Sigma})=L_{S}(\widehat{\bm{\theta}}_{S}^{(1)},\bm{\Sigma})-L_{S}(\widehat{\bm{\theta}}_{S}^{(0,\nu)},\bm{\Sigma})$,
where $\widehat{\bm{\theta}}_{S}^{(0,\nu)}$ is the MLE under the constraint
that $U_{k_{\nu},j_{\nu}}=0$. Then for $\widehat{S}\supseteq S^{\circ}$, we
have
$\begin{split}\text{Lr}_{\nu}&=\frac{1}{2}\frac{(\bm{\mathsf{Y}}_{k_{\nu}}U_{k_{\nu},j_{\nu}}+\bm{\mathsf{e}}_{j_{\nu}})^{\top}(\bm{\mathsf{P}}_{\widehat{A}_{j_{\nu}}}-\bm{\mathsf{P}}_{\widehat{B}_{j_{\nu}}})(\bm{\mathsf{Y}}_{k_{\nu}}U_{k_{\nu},j_{\nu}}+\bm{\mathsf{e}}_{j_{\nu}})}{(\bm{\mathsf{e}}_{j_{\nu}})^{\top}(\bm{I}-\bm{\mathsf{P}}_{\widehat{A}_{j_{\nu}}})\bm{\mathsf{e}}_{j_{\nu}}/(n-|\widehat{A}_{j_{\nu}}|)},\\\
\text{Lr}_{\nu}^{*}&=\frac{1}{2}\frac{(\bm{\mathsf{e}}^{*}_{j_{\nu}})^{\top}(\bm{\mathsf{P}}_{\widehat{A}^{*}_{j_{\nu}}}-\bm{\mathsf{P}}_{\widehat{B}^{*}_{j_{\nu}}})\bm{\mathsf{e}}^{*}_{j_{\nu}}}{(\bm{\mathsf{e}}^{*}_{j_{\nu}})^{\top}(\bm{I}-\bm{\mathsf{P}}_{\widehat{A}^{*}_{j_{\nu}}})\bm{\mathsf{e}}^{*}_{j_{\nu}}/(n-|\widehat{A}^{*}_{j_{\nu}}|)},\end{split}$
(15)
where the distributions of $\text{Lr}_{\nu}^{*}$ given $\bm{\mathsf{Z}}$
approximates the null distributions of $\text{Lr}_{\nu}$. Finally, define the
p-value of pathway test in (3) as
$\text{Pval}=\max_{1\leq\nu\leq|\mathcal{H}|}\left(\sum_{m=1}^{M}\operatorname{I}(\text{Lr}_{\nu,m}^{*}\geq\text{Lr}_{\nu},\widehat{S}^{*}_{m}\supseteq\widehat{S})\right)/\left(\sum_{i=1}^{M}\operatorname{I}(\widehat{S}^{*}_{m}\supseteq\widehat{S})\right).$
(16)
Note that if any $\text{H}_{0,\nu}$ is degenerate, then $\text{Pval}=1$.
Algorithm 3 summarizes the DP method for hypothesis testing.
1Specify the Monte Carlo size $M$.
2Compute Lr and
$\widehat{\bm{\Sigma}}=\text{Diag}(\widehat{\sigma}^{2}_{1},\ldots,\widehat{\sigma}^{2}_{p})$
via Algorithm 2 with original data $\bm{\mathsf{Z}}$.
3(Parallel DP) Generate perturbed data
$(\bm{\mathsf{Z}}^{*}_{m},\bm{\mathsf{e}}_{m}^{*})=(\bm{\mathsf{X}},\bm{\mathsf{Y}}_{m}^{*},\bm{\mathsf{e}}_{m}^{*})$
in parallel, where $\bm{\varepsilon}_{m,i}^{*}\sim
N(\bm{0},\widehat{\bm{\Sigma}})$. For (2), compute $\text{Lr}^{*}_{m}$ based
on $(\bm{\mathsf{Z}}^{*}_{m},\bm{\mathsf{e}}_{m}^{*})$, while, for (3),
compute $\text{Lr}^{*}_{\nu,m}$; $\nu=1,\ldots,|\mathcal{H}|$ based on
$(\bm{\mathsf{Z}}^{*}_{m},\bm{\mathsf{e}}_{m}^{*})$; $m=1,\ldots,M$.
4Compute the p-value of a test as (14) or (16) accordingly.
Algorithm 3 DP likelihood ratio test
###### Remark 4 (Computation)
To accelerate the computation, we consider a parallelizable procedure in
Algorithm 3. In addition, we use the original estimate $\widehat{\bm{\theta}}$
as a warm-start initialization for the DP estimates, which effectively reduces
the computing time.
###### Remark 5 (Connection with bootstrap)
One may consider parametric or nonparametric bootstrap for Lr. The parametric
bootstrap requires a good initial estimate of $(\bm{U},\bm{W})$. Yet, it is
rather challenging to correct the bias of this estimate because of the
acyclicity constraint. By comparison, DP does not rely on such an estimate. On
the other hand, nonparametric bootstrap resamples the original data with
replacement. In a bootstrap sample, only about 63% distinct observations in
the original data are used in model selection and fitting, leading to
deteriorating performance (Kleiner et al., 2012), especially in a small
sample. As a result, nonparametric bootstrap may not well-approximate the
distribution of Lr, while DP provides a better approximation of Lr, taking
advantage of a full sample.
## 4 Theory
This section provides the theoretical justification for the proposed methods.
### 4.1 Convergence and consistency of structure learning
First, we introduce some technical assumptions to derive statistical and
computational properties of Algorithms 1 and 2. Let
$(\bm{\mathsf{Y}},\bm{\mathsf{X}})\in\mathbb{R}^{n\times(p+q)}$ be the data
matrix. Denote by $\bm{\zeta}$ a generic vector and
$\bm{\zeta}_{A}\in\mathbb{R}^{|A|}$ be the projection of $\bm{\zeta}$ onto
coordinates in $A$. Let $\kappa^{\circ}_{j}=\|\bm{V}^{\circ}_{\cdot j}\|_{0}$
and $\kappa^{\circ}_{\max}=\max_{1\leq j\leq p}\kappa_{j}^{\circ}$.
###### Assumption 2 (Restricted eigenvalues)
For a constant $c_{1}>0$,
$\min_{A:|A|\leq
2\kappa^{\circ}_{\max}}\min_{\\{\bm{\zeta}:\|\bm{\zeta}_{A^{c}}\|_{1}\leq
3\|\bm{\zeta}_{A}\|_{1}\\}}\frac{\|\bm{\mathsf{X}}\bm{\zeta}\|_{2}^{2}}{n\|\bm{\zeta}\|_{2}^{2}}\geq
c_{1}.$
###### Assumption 3 (Interventions)
For a constant $c_{2}>0$,
$\max\Big{(}\max_{1\leq l\leq
q}n^{-1}(\bm{\mathsf{X}}^{\top}\bm{\mathsf{X}})_{ll},\max_{A:|A|\leq
2\kappa^{\circ}_{\max}}\max_{1\leq
l\leq|A|}n((\bm{\mathsf{X}}_{A}^{\top}\bm{\mathsf{X}}_{A})^{-1})_{ll}\Big{)}\leq
c_{2}^{2}.$
###### Assumption 4 (Nuisance signals)
$\min_{V^{0}_{lj}\neq 0}|V^{0}_{lj}|\geq 100\frac{c_{2}}{c_{1}}\max_{1\leq
j\leq
p}\sqrt{\Omega_{jj}^{-1}\Big{(}\frac{\log(q)}{n}+\frac{\log(n)}{n}\Big{)}}.$
(17)
Assumptions 2-3, as a replacement of Assumption 1A, are satisfied with a
probability tending to one for isotropic subgaussian or bounded $\bm{X}$
(Rudelson and Zhou, 2012). Assumption 2 is a common condition for proving the
convergence rate of Lasso regression (Bickel et al., 2009; Zhang et al.,
2014). Assumption 4, as an alternative to Assumption 1B, specifies the minimal
signal strength over candidate interventions. A signal strength requirement
like Assumption 4 is a condition for establishing selection consistency in
high dimensional variable selection (Fan et al., 2014; Loh et al., 2017; Zhao
et al., 2018). Moreover, Assumption 4 can be further relaxed to a less
intuitive condition, Assumption 6; see Appendix A.3 for details.
###### Theorem 4 (Finite termination and consistency)
Under Assumptions 1-4 with constants $c_{1}<6c_{2}$, if the machine precision
$\text{tol}\ll 1/n$ is negligible, and tuning parameters of Algorithm 1
satisfy:
1. (1)
$\gamma_{j}\leq c_{1}/6$,
2. (2)
$\sqrt{32c_{2}^{2}\Omega_{jj}^{-1}(\log(q)+\log(n))/n}/\gamma_{j}\leq\tau_{j}\leq
0.4\min_{V^{0}_{lj}\neq 0}|V^{0}_{lj}|$,
3. (3)
$\kappa_{j}=\kappa_{j}^{\circ}$,
for $j=1,\ldots,p$, then Algorithm 1 yields the global minimizer
$\widehat{\bm{V}}_{\cdot j}$ of (7) in at most
$1+\lceil\log(\kappa_{\max}^{\circ})/\log 4\rceil$ DC iterations almost
surely, where $\lceil\cdot\rceil$ is the ceiling function. Moreover, Algorithm
2 recovers $\mathcal{A}^{\circ}$,
$\mathcal{C}^{\circ}\equiv\\{l:V^{\circ}_{lj}\neq
0\\}\supseteq\mathcal{I}^{\circ}$, and $\mathcal{D}^{\circ}$ almost surely as
$n\to\infty$.
By Theorem 4 (proved in Appendix B.4), the computational complexity of
Algorithm 1 is the number of iterations multiplied by that of solving a
weighted Lasso regression, $O(\log\kappa^{\circ}_{\max})\times
O(q^{3}+nq^{2})$ (Efron et al., 2004). Also note that in Steps 2-6 of
Algorithm 2, no heavy computation is involved. Thus, the overall time
complexity of Algorithm 2 is bounded by
$O(p\times\log\kappa^{\circ}_{\max}\times(q^{3}+nq^{2}))$ for computing
$\widehat{\bm{V}}$ matrix. Finally, the peeling algorithm does not apply to
observational data ($\bm{W}=\bm{0}$). In a sense, interventions play an
essential role.
Although Theorem 4 establishes the consistent reconstruction of the peeling
algorithm for the super-graph $S$, it does not provide any uncertainty measure
of the presence of some directed edges of the true DAG. In what follows, we
will develop a theory for hypothesis tests to make valid inference concerning
directed edges of interest.
### 4.2 Inferential theory
###### Assumption 5 (Hypothesis-specific dimension restriction)
For a constant $\rho$,
$\max_{j:\textnormal{{d}}_{G^{\circ}}(j)\neq\emptyset}\frac{|\textnormal{{an}}_{G^{\circ}}(j)|+|\textnormal{{d}}_{G^{\circ}}(j)|+{\kappa_{j}^{\circ}}}{n}\leq\rho<1,\quad\text{
as }n\rightarrow\infty.$
Assumption 5 is a hypothesis-specific condition restricting the underlying
dimension of the testing problem. Usually,
$|\textnormal{{an}}_{G^{\circ}}(j)|\asymp\kappa_{j}^{\circ}\ll p$;
$j=1,\ldots,p$, which dramatically relaxes the condition
$n\gg{|\mathcal{D}|^{1/2}p\log(p)}$ for the constrained likelihood ratio test
(Li et al., 2019).
###### Theorem 5 (Empirical p-values)
Suppose Assumptions 1-5 are met and $H_{0}$ is regular. If the tuning
parameters in Algorithm 1 satisfy (1)-(3) in Theorem 4, then,
1. (A)
(Test of directed edges (2))
$\begin{split}\lim_{\begin{subarray}{c}n\to\infty\\\
\bm{\theta}^{\circ}\textnormal{ satisfies H}_{0}\textnormal{ in
}\eqref{equation:link-
test}\end{subarray}}\lim_{M\rightarrow\infty}\operatorname{\mathbb{P}}_{\bm{\theta}^{\circ}}(\textnormal{Pval}<\alpha)=\alpha,&\quad\text{
if }H_{0}\text{ is nondegenerate};\\\ \lim_{\begin{subarray}{c}n\to\infty\\\
\bm{\theta}^{\circ}\textnormal{ satisfies H}_{0}\textnormal{ in
}\eqref{equation:link-
test}\end{subarray}}\lim_{M\rightarrow\infty}\operatorname{\mathbb{P}}_{\bm{\theta}^{\circ}}(\textnormal{Pval}=1)=1,&\quad\text{
if }H_{0}\text{ is degenerate}.\end{split}$
2. (B)
(Test of directed pathways (3))
$\begin{split}\limsup_{\begin{subarray}{c}n\to\infty\\\
\bm{\theta}^{\circ}\textnormal{ satisfies H}_{0}\textnormal{ in
}\eqref{equation:pathway-
test}\end{subarray}}\lim_{M\rightarrow\infty}\operatorname{\mathbb{P}}_{\bm{\theta}^{\circ}}(\textnormal{Pval}<\alpha)=\alpha,&\quad\text{
if }H_{0}\text{ is nondegenerate with }|\mathcal{D}|=|\mathcal{H}|;\\\
\limsup_{\begin{subarray}{c}n\to\infty\\\ \bm{\theta}^{\circ}\textnormal{
satisfies H}_{0}\textnormal{ in }\eqref{equation:pathway-
test}\end{subarray}}\lim_{M\rightarrow\infty}\operatorname{\mathbb{P}}_{\bm{\theta}^{\circ}}(\textnormal{Pval}=1)=1,&\quad\text{
if }|\mathcal{D}|<|\mathcal{H}|.\end{split}$
By Theorem 5, the DP likelihood ratio test yields a valid p-value for (2) and
(3). Note that $|\mathcal{D}|$ is permitted to depend on $n$. Moreover,
Proposition 6 summarizes the asymptotics for directed edge test (2).
###### Proposition 6 (Asymptotic distribution of Lr of (2))
Assume that the assumptions of Theorem 5 are met. Under a regular $H_{0}$, as
$n\to\infty$,
$\begin{split}2\textnormal{Lr}\overset{d}{\longrightarrow}\chi^{2}_{|\mathcal{D}|},&\text{
if }H_{0}\text{ is nondegenerate with }|\mathcal{D}|>0\text{ being fixed};\\\
\frac{2\textnormal{Lr}-|\mathcal{D}|}{\sqrt{2|\mathcal{D}|}}\overset{d}{\longrightarrow}N(0,1),&\text{
if }H_{0}\text{ is nondegenerate with }|\mathcal{D}|\rightarrow\infty,\
\frac{|\mathcal{D}|\log|\mathcal{D}|}{n}\rightarrow 0.\end{split}$
The proofs of Theorem 5 and Proposition 6 are given in Appendix B.5 and B.6.
###### Remark 6 (Local structure for hypothesis testing)
Theorem 5 or Proposition 6 only requires correct identification of the local
structures of the DAG such as,
$\textnormal{{an}}_{S}(j)=\textnormal{{an}}_{G}(j)$,
$\textnormal{{d}}_{S}(j)=\textnormal{{d}}_{G}(j)$, and
$\textnormal{{in}}_{S}(j)$, for variables $Y_{j}$ with
$\textnormal{{d}}_{S}(j)\neq\emptyset$, as opposed to those of entire $S$.
Next, we analyze the local limit power of the proposed tests for (2) and (3).
Assume $\bm{\theta}^{\circ}=(\bm{U}^{\circ},\bm{W}^{\circ})$ satisfies
$H_{0}$. Let $\bm{\Delta}\in\mathbb{R}^{p\times p}$ satisfy
$\bm{\Delta}_{\mathcal{D}^{c}}=\bm{0}$ so that $\bm{U}^{\circ}+\bm{\Delta}$
represents a DAG. For nondegenerate and regular $H_{0}$, consider an
alternative $H_{a}$:
$\bm{U}_{\mathcal{H}}=\bm{U}_{\mathcal{H}}^{\circ}+\bm{\Delta}_{\mathcal{H}}$,
and define the power function as
$\beta(\bm{\theta}^{\circ},\bm{\Delta})=\operatorname{\mathbb{P}}_{H_{a}}(\text{Pval}<\alpha).$
(18)
###### Proposition 7 (Local power of edge test)
Assume that $H_{0}$ is nondegenerate and regular. Let
$\|\bm{\Delta}\|_{F}=\|\bm{\Delta}_{\mathcal{H}}\|_{F}=n^{-1/2}\delta$ when
$|\mathcal{D}|>0$ is fixed and
$\|\bm{\Delta}\|_{F}=|\mathcal{D}|^{1/4}n^{-1/2}h$ when
$|\mathcal{D}|\to\infty$, where $\delta>0$ and $\|\cdot\|_{F}$ is the matrix
Frobenius norm. If the assumptions of Theorem 5 are met, then under $H_{a}$,
as $n,M\to\infty$,
$\beta(\bm{\theta}^{\circ},\bm{\Delta})\geq\begin{cases}\operatorname{\mathbb{P}}\Big{(}\|\bm{Z}+c_{l}\sqrt{n}\bm{\Delta}\|_{2}^{2}>\chi^{2}_{|\mathcal{D}|,1-\alpha}\Big{)}&\text{if
}|\mathcal{D}|>0\text{ is fixed};\\\
\operatorname{\mathbb{P}}\Big{(}Z>z_{1-\alpha}-{c_{l}\|\bm{\Delta}\|^{2}_{2}}/{\sqrt{2|\mathcal{D}|}}\Big{)}&\text{if
}|\mathcal{D}|\to\infty,\frac{|\mathcal{D}|\log|\mathcal{D}|}{n}\to
0,\end{cases}$
where $\bm{Z}\sim N(\bm{0},\bm{I}_{|\mathcal{D}|})$, $Z\sim N(0,1)$, and
$\chi^{2}_{|\mathcal{D}|,1-\alpha}$ and $z_{1-\alpha}$ are the $(1-\alpha)$th
quantile of distributions $\chi^{2}_{|\mathcal{D}|}$ and $N(0,1)$,
respectively. In particular,
$\lim_{\delta\to\infty}\lim_{n\to\infty}\beta(\bm{\theta}^{\circ},\bm{\Delta})=1$.
###### Proposition 8 (Local power of pathway test)
Assume that $H_{0}$ is nondegenerate and regular with
$|\mathcal{D}|=|\mathcal{H}|$. Let
$\min_{(k,j)\in\mathcal{H}}|U^{\circ}_{kj}+\Delta_{kj}|=n^{-1/2}\delta$ when
$|\mathcal{H}|>0$ is fixed and
$\min_{(k,j)\in\mathcal{H}}|U^{\circ}_{kj}+\Delta_{kj}|=n^{-1/2}\delta\sqrt{\log|\mathcal{H}|}$
when $|\mathcal{H}|\to\infty$. If the assumptions of Theorem 5 are met, then
under $H_{a}$, as $n,M\to\infty$,
$\beta(\bm{\theta}^{\circ},\bm{\Delta})\geq
1-\frac{|\mathcal{H}|}{\sqrt{2\pi}}\exp\Big{(}-\Big{(}\delta\sqrt{\log|\mathcal{H}|}/\max_{1\leq
j\leq p}\Omega_{jj}-\sqrt{\chi^{2}_{1,1-\alpha}}\Big{)}^{2}/2\Big{)},$
where $Z\sim N\Big{(}\delta^{2}/\max_{1\leq j\leq p}\Omega_{jj},1\Big{)}$ and
$\chi^{2}_{1,1-\alpha}$ is the $(1-\alpha)$th quantile of distribution
$\chi^{2}_{1}$. Then,
$\lim_{\delta\to\infty}\lim_{n\to\infty}\beta(\bm{\theta}^{\circ},\bm{\Delta})=1$.
The proofs of Propositions 7 and 8 are deferred to Appendix B.7 and B.8.
## 5 Simulations
This section investigates the operating characteristics of the proposed tests
and the peeling algorithm via simulations. In simulations, we consider two
setups for generating $\bm{U}\in\mathbb{R}^{p\times p}$, representing random
and hub DAGs, respectively.
* •
Random graph. The upper off-diagonal entries $U_{kj}$; $k<j$ are sampled
independently from $\\{0,1\\}$ according to Bernoulli$(1/p)$, while other
entries are zero. This generates a random graph with a sparse neighborhood.
* •
Hub graph. Set $U_{1,2j+1}=1$ and $U_{2,2j+2}=1$ for $j=1,\ldots,\lfloor
p/2\rfloor-2$, while other entries are zero. This generates a hub graph, where
nodes 1 and 2 are hub nodes with a dense neighborhood.
Moreover, we consider three setups for intervention matrix
$\bm{W}\in\mathbb{R}^{q\times p}$, representing different scenarios. Setups A
and B are designed for inference, whereas Setup C in Section 5.2 is designed
to compare with the method of Chen et al. (2018) for structure learning. Let
$\bm{W}=(\bm{A}^{\top},\bm{B}^{\top},\bm{0}^{\top})^{\top}$, where
$\bm{A},\bm{B}\in\mathbb{R}^{p\times p}$ and
$\bm{0}\in\mathbb{R}^{(q-2p)\times p}$.
* •
Setup A. Set $A_{jj}=B_{jj}=B_{j,j+1}=1$; $j=1,\ldots,p-1$, $A_{pp}=1$, while
other entries of $\bm{A}$, $\bm{B}$ are zero. Then, $X_{1},\ldots,X_{p}$ are
instruments for $Y_{1},\ldots,Y_{p}$, respectively, $X_{p+1},\ldots,X_{2p-1}$
are invalid instruments with two targets, and $X_{2p},\ldots,X_{q}$ represent
inactive interventions.
* •
Setup B. Set $A_{jj}=A_{j,j+1}=B_{jj}=B_{j,j+1}=1$; $j=1,\ldots,p-1$,
$A_{pp}=1$, while other entries of $\bm{A}$, $\bm{B}$ are zero. Here, the only
valid instrument is $X_{p}$ on $Y_{p}$, and the other intervention variables
either have two targets or are inactive. Importantly, the instrumental
sufficiency (Assumption 1C) is not met.
To generate $(\bm{X},\bm{Y})$ for each setup, we sample $\bm{X}\sim
N(\bm{0},\bm{\Sigma}_{X})$ with
$(\Sigma_{X})_{ll^{\prime}}=0.5^{|l-l^{\prime}|}$; $1\leq l,l^{\prime}\leq q$
and sample $\bm{Y}$ according to (1) with
$(\bm{U},\bm{W},\sigma_{1}^{2},\ldots,\sigma_{p}^{2})$, where
$\sigma_{1}^{2},\ldots,\sigma_{p}^{2}$ are set to be equally spaced from $0.5$
to $1$.
### 5.1 Inference
We compare three tests in empirical type-I errors and powers in simulated
examples, namely, the DP likelihood ratio test (DP-LR) in Algorithm 3, the
asymptotic likelihood ratio test (LR), and the oracle likelihood ratio test
(OLR). Here LR uses $\text{Lr}(\widehat{S},\widehat{\bm{\Sigma}})$, while OLR
uses $\text{Lr}(S^{\circ},\widehat{\bm{\Sigma}})$ assuming that the super-
graph $S^{\circ}$ were known in advance. The p-values of LR and OLR are
computed via Proposition 6. The implementation details of these tests are in
Appendix C.
For the empirical type-I error of a test, we compute the percentage of times
rejecting $H_{0}$ out of 500 simulations when $H_{0}$ is true. For the
empirical power of a test, we report the percentage of times rejecting $H_{0}$
out of 100 simulations when $H_{a}$ is true under alternative hypotheses
$H_{a}$.
* •
Test of directed edges. For (2), we examine two different hypotheses:
(i) $H_{0}:U_{1,20}=0$ versus $H_{a}:U_{1,20}\neq 0$. In this case,
$|\mathcal{D}|=1$.
(ii) $H_{0}:\bm{U}_{\mathcal{H}}=\bm{0}$ versus
$H_{a}:\bm{U}_{\mathcal{H}}\neq\bm{0}$, where
$\mathcal{H}=\\{(k,20):k=1,\ldots,15\\}$. In this case, $|\mathcal{D}|=15$.
Moreover, five alternatives $H_{a}:U_{1,20}=0.1l$ and
$\bm{U}_{\mathcal{H}\setminus\\{(1,20)\\}}=\bm{0}$; $l=1,2,3,4,5$, are used
for the power analysis in (i) and (ii). The data are generated by modifying
$\bm{U}$ accordingly.
* •
Test of directed pathways. We examine the test of a directed path $Y_{1}\to
Y_{5}\to Y_{10}\to Y_{15}\to Y_{20}$, namely
$\mathcal{H}=\\{(1,5),(5,10),(10,15),(15,20)\\}$ in (3). Since (3) is a test
of composite null hypothesis, the data are generated under a graph with
parameters $(\bm{U},\bm{W},\bm{\Sigma})$ satisfying $H_{0}$, where
$\bm{U}_{\mathcal{H}}=\bm{0}$. Five alternatives
$H_{a}:\bm{U}_{\mathcal{H}}=\bm{0.1}l$; $l=1,2,3,4,5$ are used for the power
analysis.
Figure 2: Empirical type-I errors and powers of tests of directed edges. The
black dotted line marks the nominal level of significance $\alpha=0.05$.
Figure 3: Empirical type-I errors and powers of tests of a directed pathway.
The black dotted line marks the nominal level of significance $\alpha=0.05$.
For testing directed edges, as displayed in Figure 2, DP-LR and LR perform
well compared to the ideal test OLR in Setup A with Assumption 1 satisfied. In
setup B with Assumption 1C not fulfilled, DP-LR appears to have control of
type-I error, whereas LR has an inflated empirical type-I error compared to
the nominal level $\alpha=0.05$. This discrepancy is attributed to the data
perturbation scheme accounting for the uncertainty of identifying $S$.
However, both suffer a loss of power compared to the oracle test OLR in this
setup. This observation suggests that without Assumption 1C, the peeling
algorithm tends to yield an estimate $\widehat{S}\supseteq S^{\circ}$, which
overestimates $S^{\circ}$, resulting in a power loss.
For testing directed pathways, as indicated in Figure 3, we observe similar
phenomena as in the previous directed edge tests. Of note, both LR and DP-LR
are capable in controlling type-I error of directed path tests.
In summary, DP-LR has a suitable control of type-I error when there are
invalid instruments and Assumption 1C is violated. Concerning the power, DP-LR
and LR are comparable in all scenarios and their powers tend to one as the
sample size $n$ or the signal strength of tested edges increases. Moreover,
DP-LR and LR perform nearly as well as the oracle test OLR when Assumption 1
is satisfied. These empirical findings agree with our theoretical results.
### 5.2 Structure learning
This subsection compares the peeling algorithm with the Two-Stage Penalized
Least Squares (2SPLS, Chen et al. (2018)) in terms of the structure learning
accuracy. For peeling, we consider Algorithm 2 with an additional step (9) for
structure learning of $\bm{U}$. For 2SPLS, we use the R package BigSEM.
Figure 4: Structural Hamming distances (SHDs) for the peeling algorithm and
2SPLS, where a smaller value of SHD indicates a better result.
2SPLS requires that all the intervention variables to be target-known
instruments in addition to Assumption 1C. Thus, we consider an additional
Setup C.
* •
Setup C. Let $\bm{W}=(\bm{I}_{p\times p},\bm{0})^{\top}\in\mathbb{R}^{q\times
p}$. Then $X_{1},\ldots,X_{p}$ are valid instruments for $Y_{1},\ldots,Y_{p}$,
respectively, and other intervention variables are inactive.
For 2SPLS, we assign each active intervention variable to its most correlated
primary variable in Setups A-C. In Setup C, this assignment yields a correct
identification of valid instruments, meeting all the requirements of 2SPLS.
For each scenario, we compute the average structural Hamming distance
$\text{SHD}(\widehat{\bm{U}},\bm{U}^{\circ})=\sum_{k,j}|\operatorname{I}(\widehat{U}_{kj}\neq
0)-\operatorname{I}(U^{\circ}_{kj}\neq 0)|$ over 100 runs. As shown in Figure
4, the peeling algorithm outperforms 2SPLS, especially when there are invalid
instruments and Assumption 1C is violated.
### 5.3 Comparison of inference and structure learning
Figure 5: Rejection rates for $H_{0}:U_{1,20}=0$ versus
$H_{a}:U_{1,20}=1/\sqrt{n}$ by DP inference and structure learning. For
structure learning, $H_{0}$ is rejected if $\widehat{U}_{1,20}\neq 0$. The red
lines indicate the results of DP inference using the significance level
$\alpha=0.05$. The other colored lines display the results of structure
learning using different sparsity parameter values $\kappa=1,2,3,4,5$. The
simulation is repeated for 500 times and $\kappa=2$ is chosen by BIC in over
90% cases.
This subsection compares the proposed DP testing method against the proposed
structure learning method in (9) in terms of inferring the true graph
structure. To this end, we consider Setup A in Section 5.1 with $p=30,q=100$,
and the hypotheses
$H_{0}:U_{1,20}=0$ versus $H_{a}:U_{1,20}=1/\sqrt{n}$.
For DP inference, we use $\alpha=0.05$ and choose the tuning parameters by BIC
as in previous experiments; see Appendix C for details. For structure
learning, we reject the null hypothesis when $\widehat{U}_{1,20}\neq 0$.
As displayed in Figure 5, when the null hypothesis $H_{0}$ is true, the DP
testing method controls type-I error very close to the nominal level of 0.05,
whereas the type-I error of the structure learning varies greatly depending on
the tuning parameter selection. Under the alternative hypothesis $H_{a}$, the
DP inference enjoys high statistical power than structure learning methods
when $n\geq 200$. Interestingly, the power of structure learning diminishes as
$n$ increases. This observation is in agreement with our theoretical results
in Theorem 4, suggesting that consistent reconstruction requires the smallest
size of nonzero coefficients to be of order $\sqrt{\log(n)/n}$ with the tuning
parameter $\tau$ of the same order (fixing $p,q$). In this case, the edge
$U_{1,20}$ is of order $1/\sqrt{n}$, which is less likely to be reconstructed
as $n$ increases. In contrast, Proposition 7 indicates that a DP test has a
non-vanishing power when the hypothesized edges are of order $1/\sqrt{n}$.
Figure 5 demonstrates some important distinctions between inference and
structure learning. When different tuning parameters are used, the structure
learning results correspond to different points on an ROC curve. Although it
is asymptotically consistent when optimal tuning parameters are used,
structure learning lacks an uncertainty measure of graph structure
identification. As a result, it is nontrivial for structure learning methods
to trade-off the false discovery rate and detection power in practice. This
makes the interpretation of such results hard, especially when they heavily
rely on hyperparameters as in Figure 5. By comparsion, DP inference aims to
maximize statistical power while controlling type-I error at a given level,
offering a clear interpretation of its result. This observation agrees with
the discussions in the literature on variable selection and inference
(Wasserman and Roeder, 2009; Meinshausen and Bühlmann, 2010; Lockhart et al.,
2014; Candes et al., 2018) and it justifies the demand for inferential tools
for directed graphical models.
## 6 ADNI data analysis
This section applies the proposed tests to analyze an Alzheimer’s Disease
Neuroimaging Initiative (ADNI) dataset. In particular, we infer gene pathways
related to Alzheimer’s Disease (AD) to highlight some gene-gene interactions
differentiating patients with AD/cognitive impairments and healthy
individuals.
The raw data are available in the ADNI database (https://adni.loni.usc.edu),
including gene expression, whole-genome sequencing, and phenotypic data. After
cleaning and merging, we have a sample size of 712 subjects. From the KEGG
database (Kanehisa and Goto, 2000), we extract the AD reference pathway
(hsa05010, https://www.genome.jp/pathway/hsa05010), including 146 genes in the
ADNI data.
For data analysis, we first regress the gene expression levels on five
covariates – gender, handedness, education level, age, and intracranial
volume, and then use the residuals as gene expressions in the following
analysis. Next, we extract the genes with at least one SNP at a marginal
significance level below $10^{-3}$, yielding $p=63$ genes as primary
variables. For these genes, we further extract their marginally most
correlated two SNPs, resulting in $q=63\times 2=126$ SNPs as unspecified
intervention variables, in subsequent data analysis. All gene expression
levels are normalized.
The data contains individuals in four groups, namely, Alzheimer’s Disease
(AD), Early Mild Cognitive Impairment (EMCI), Late Mild Cognitive Impairment
(LMCI), and Cognitive Normal (CN). For our purpose, we treat 247 CN
individuals as controls while the remaining 465 individuals as cases (AD-MCI).
Then, we use the gene expressions and the SNPs to reconstruct the ancestral
relations and infer gene pathways for 465 AD-MCI and 247 CN control cases,
respectively.
(a) AD-MCI
(b) CN
(c) AD-MCI
(d) CN
Figure 6: Display of the subnetworks associated with genes APP and CASP3. (a)
and (b): Solid/dashed arrows indicate significant/insignificant edges at
$\alpha=0.05$ after adjustment for multiplicity by the Bonferroni-Holm
correction. (c) and (d): Solid arrows indicate the reconstructed edges using
2SPLS (Chen et al., 2018).
(a) AD-MCI
(b) CN
Figure 7: The p-values of pathway tests (3) by the proposed tests for the AD-
MCI and CN groups, where p-values are adjusted for multiplicity by the
Bonferroni-Holm correction and solid/dashed arrows indicate
significant/insignificant pathways at $\alpha=0.05$.
In the literature, genes APP, CASP3, and PSEN1 are well-known to be associated
with AD, reported to play different roles in AD patients and healthy subjects
(Julia and Goate, 2017; Su et al., 2001; Kelleher and Shen, 2017). For this
dataset, we conduct hypothesis testing on edges and pathways related to genes
APP, CASP3, and PSEN1 in the KEGG AD reference (hsa05010) to evaluate the
proposed DP inference by checking if DP inference can discover the differences
that are reported in the biomedical literature. First, we consider testing
$H_{0}:U_{kj}=0$ versus $H_{a}:U_{kj}\neq 0$, for each edge $(k,j)$ as shown
in Figure 6 (a) and (b). Moreover, we consider two hypothesis tests of
pathways $H_{0}:U_{kj}=0$ for some $(k,j)\in\mathcal{P}_{\ell}$ versus
$H_{a}:U_{kj}\neq 0$ for all $(k,j)\in\mathcal{P}_{\ell}$; $\ell=1,2$, where
the two pathways are specified by
$\mathcal{P}_{1}=\\{\text{PSEN1}\to\text{CAPN1}\to\text{CDK5R1}\\}$, and
$\mathcal{P}_{2}=\\{\text{PSEN1}\to\text{CAPN2}\to\text{CDK5R1}\\}$. See
Figure 7. Of note, for clear visualization, Figure 6 (a), (b) and Figure 7
only display the edges related to hypothesis testing, whereas the ancestral
relations are reconstructed using $p=63$ genes and $q=126$ SNPs for AD-MCI and
CN groups separately.
In Figures 6-7, the significant results under the level $\alpha=0.05$ after
the Holm-Bonferroni adjustment for $2\times(9+2)=22$ tests are displayed. In
Figures 6, the edge test in (2) exhibits a strong evidence for the presence of
directed connectivity $\\{$APP $\to$ APBB1, APP $\to$ GSK3B, FADD $\to$
CASP3$\\}$ in the AD-MCI group, but no evidence in the CN group. Meanwhile,
this test suggests the presence of connections $\\{$TNFRSF1A $\to$ CASP3, FADD
$\to$ CASP8$\\}$ in the CN group but not so in the AD-MCI group. In both
groups, we identify directed connections $\\{$TNFRSF1A $\to$ FADD, TNFRSF1A
$\to$ CASP8$\\}$. In Figure 7, the pathway test (3) supports the presence of a
pathway PSEN1 $\to$ CAPN1 $\to$ CDK5R1 in the AD-MCI group with a p-value of
$0.044$ but not in the CN group with a p-value of $0.33$. The pathway PSEN1
$\to$ CAPN2 $\to$ CDK5R1 appears insignificant at $\alpha=0.05$ for both
groups. Also noted is that some of our discoveries agree with the literature
according to the AlzGene database (alzgene.org) and the AlzNet database
(https://mips.helmholtz-muenchen.de/AlzNet-DB). Specifically, GSK3B
differentiates AD patients from normal subjects; as shown in Figures 6, our
result indicates the presence of connection APP $\to$ GSK3B for the AD-MCI
group, but not for the CN group, the former of which is confirmed by Figure 1
of Vanleuven (2011). The connection APP $\to$ APBB1 also differs in AD-MCI and
CN groups, which appears consistent with Figure 3 of Bu (2009). Moreover, the
connection CAPN1 $\to$ CDK5R1, in the pathway PSEN1 $\to$ CAPN1 $\to$ CDK5R1
discovered in AD-MCI group, is found in the AlzNet database (interaction-ID
24614, https://mips.helmholtz-muenchen.de/AlzNet-DB/entry/show/1870). Finally,
as suggested by Figure 8, the normality assumption in (1) is adequate for both
groups.
By comparison, as shown in Figure 6 (c) and (d), gene APP in the reconstructed
networks by 2SPLS (Chen et al., 2018) is not connected with other genes,
indicating no regulatory relation of APP with other genes in the AD-MCI and CN
groups. However, as a well-known gene associated with AD, APP is reported to
play different roles in controlling the expressions of other genes for AD
patients and healthy people (Matsui et al., 2007; Julia and Goate, 2017). Our
results in Figure 6 (a) and (b) are congruous with the studies: the
connections of APP with other genes are different in our estimated networks
for AD-MCI and CN groups.
Figure 8: Normal quantile-quantile plots of studentized residuals of the AD-
MCI and CN groups.
In summary, our findings seem to agree with those in the literature (Julia and
Goate, 2017; Su et al., 2001; Kelleher and Shen, 2017), where the subnetworks
of genes APP, CASP3 in Figure 6 and PSEN1 in Figure 7 differentiate the AD-MCI
from the CN groups. Furthermore, the pathway PSEN1 $\to$ CAPN1 $\to$ CDK5R1 in
Figure 7 seems to differentiate these groups, which, however, requires
validation in biological experiments.
## 7 Summary
This article proposes structure learning and inference methods for a Gaussian
DAG with interventions, where the targets and strengths of interventions are
unknown. A likelihood ratio test is derived based on an estimated super-graph
formed by ancestral relations and candidate interventional relations. This
test accounts for the statistical uncertainty of the construction of the
super-graph based on a novel data perturbation scheme. Moreover, we develop a
peeling algorithm for the super-graph construction. The peeling algorithm
allows scalable computing and yields a consistent estimator. The numerical
studies justify our theory and demonstrate the utility of our methods.
The proposed methods can be extended to many practical situations beyond
biological applications with independent and identically distributed data. An
instance is to infer directed relations between multiple autoregressive time
series (Pamfil et al., 2020), where the lagged variables and covariates can
serve as interventions for each time series.
Acknowledgments
The authors would like to thank the action editor and anonymous referees for
helpful comments and suggestions. The research is supported by NSF grants
DMS-1712564, DMS-1952539 and NIH grants R01GM126002, R01HL116720, R01HL105397,
R01AG069895, R01AG065636, R01AG074858, and U01AG073079.
## A Illustrative examples and discussions
### A.1 Identifiability of model (1) and Assumption 1
The parameter space for model (1) is
$\\{(\bm{U},\bm{W},\bm{\Sigma}):\bm{U}\in\mathbb{R}^{p\times p}\text{
represents a DAG},\ \bm{W}\in\mathbb{R}^{p\times q},\
\bm{\Sigma}=\text{diag}(\sigma_{1}^{2},\ldots,\sigma^{2}_{p})\\}.$
As suggested by Proposition 1, Assumption 1 (1A-1C) suffices for
identification of every parameter value in the parameter space. Next, we show
by examples that if Assumption 1B or 1C is violated then model (1) is no
longer identifiable.
Before proceeding, we rewrite (1) as
$\bm{Y}=\bm{V}^{\top}\bm{X}+\bm{\varepsilon}_{V},\quad\bm{\varepsilon}_{V}=(\bm{I}-\bm{U}^{\top})^{-1}\bm{\varepsilon}\sim
N(\bm{0},\bm{\Omega}^{-1}),$
where $\bm{\Omega}=(\bm{I}-\bm{U})\bm{\Sigma}^{-1}(\bm{I}-\bm{U}^{\top})$ is a
precision matrix and $\bm{V}=\bm{W}(\bm{I}-\bm{U})^{-1}$.
###### Example 2 (Identifiability)
In model (1), consider two non-identifiable bivariate situations: (1) $p=2$
and $q=4$ and (2) $p=2$ and $q=2$.
1. (1)
Model (1) is non-identifiable when Assumption 1B breaks down. Consider two
different models with different parameter values:
$\displaystyle\bm{\theta}:$ $\displaystyle
Y_{1}=X_{1}+X_{2}+X_{3}+\varepsilon_{1},\quad$ $\displaystyle
Y_{2}=Y_{1}-X_{2}+X_{3}+X_{4}+\varepsilon_{2},$ (19)
$\displaystyle\widetilde{\bm{\theta}}:$ $\displaystyle
Y_{1}=0.5Y_{2}+0.5X_{1}+X_{2}-0.5X_{4}+\widetilde{\varepsilon}_{1},\quad$
$\displaystyle Y_{2}=X_{1}+2X_{3}+X_{4}+\widetilde{\varepsilon}_{2},$ (20)
where $\varepsilon_{1},\varepsilon_{2}\sim N(0,1)$ are independent, and
$\widetilde{\varepsilon}_{1}\sim N(0,0.5)$, $\widetilde{\varepsilon}_{2}\sim
N(0,2)$ are independent. As depicted in Figure 9, (19) satisfies Assumption
1C. However, Assumption 1B is violated given that
$\operatorname{Cov}(Y_{2},X_{2}\mid\bm{X}_{\\{1,3,4\\}})=0$ and $X_{2}$ is an
intervention variable of $Y_{2}$. This is because the direct interventional
effect of $X_{2}$ on $Y_{2}$ are canceled out by its indirect interventional
effect through $Y_{1}$. Similarly, (20) satisfies Assumption 1C but
$\operatorname{Cov}(Y_{1},X_{4}\mid\bm{X}_{\\{1,2,3\\}})=0$ violating
Assumption 1B. In this case, it can be verified that $\bm{\theta}$ and
$\widetilde{\bm{\theta}}$ correspond to the same distribution
$\operatorname{\mathbb{P}}(\bm{Y}\mid\bm{X})$, because they share the same
$(\bm{V},\bm{\Omega})$ even with different values of
$(\bm{U},\bm{W},\bm{\Sigma})$. Hence, it is impossible to infer the directed
relation between $Y_{1}$ and $Y_{2}$.
2. (2)
Model (1) is non-identifiable when Assumption 1C breaks down. Consider two
different models with different parameter values:
$\displaystyle\bm{\theta}:$ $\displaystyle
Y_{1}=X_{1}+X_{2}+\varepsilon_{1},\quad$ $\displaystyle
Y_{2}=Y_{1}+X_{2}+\varepsilon_{2},$ (21)
$\displaystyle\widetilde{\bm{\theta}}:$ $\displaystyle
Y_{1}=0.5Y_{2}+0.5X_{1}+\widetilde{\varepsilon}_{1},\quad$ $\displaystyle
Y_{2}=X_{1}+2X_{2}+\widetilde{\varepsilon}_{2},$ (22)
where $\varepsilon_{1},\varepsilon_{2}\sim N(0,1)$ are independent, and
$\widetilde{\varepsilon}_{1}\sim N(0,0.5)$, $\widetilde{\varepsilon}_{2}\sim
N(0,2)$ are independent. Note that (21) and (22) satisfy Assumption 1B. In
(21), $Y_{2}$ does not have any instrumental intervention although it has an
invalid instrument $X_{2}$. Similarly, in (22), neither does $Y_{1}$ have any
instrumental intervention while having an invalid instrument $X_{1}$. As in
the previous case, $\bm{\theta}$ and $\widetilde{\bm{\theta}}$ yield the same
distribution $\operatorname{\mathbb{P}}(\bm{Y}\mid\bm{X})$ because they share
the same $(\bm{V},\bm{\Omega})$ even with different values of
$(\bm{U},\bm{W},\bm{\Sigma})$. In this case, it is impossible to infer the
directed relation between $Y_{1}$ and $Y_{2}$.
Figure 9: (a) Display of DAG defined by (19). (b) Display of DAG defined by
(20). Figure 10: (a) Display of DAG defined by (21). (b) Display of DAG
defined by (22).
### A.2 Illustration of Algorithm 2
We now illustrate Algorithm 2 by Example 3.
###### Example 3
Consider model (1) with $p=q=5$,
$\begin{aligned} Y_{1}&=X_{1}+\varepsilon_{1},\\\
Y_{3}&=0.5Y_{2}+X_{5}+\varepsilon_{3},\\\ \end{aligned}\qquad\begin{aligned}
Y_{2}&=0.5Y_{1}+X_{3}+\varepsilon_{2},\\\
Y_{4}&=0.5Y_{3}-0.1Y_{1}+X_{2}+\varepsilon_{4},\\\
\end{aligned}\qquad\begin{aligned} Y_{5}&=X_{4}+\varepsilon_{5},\\\
&\end{aligned}$ (23)
where $\varepsilon_{1},\ldots,\varepsilon_{5}\sim N(0,1)$ are independent.
Then (23) defines a DAG as displayed in Figure 1(a). For illustration, we
generate a random sample of size $n=40$ and compute $\widehat{\bm{V}}$ by
Algorithm 1. In particular,
$\bm{V}^{\circ}=\begin{pmatrix}1&0.5&0.25&0.025&0\\\ 0&0&0&1&0\\\
0&1&0.5&0.25&0\\\ 0&0&0&0&1\\\
0&0&1&0.5&0\end{pmatrix},\quad\widehat{\bm{V}}=\begin{pmatrix}0.92&0.48&0.27&0&0\\\
0&0&0&1.08&0\\\ 0&1.03&0.52&0.21&0\\\ 0&0&0&0&1.06\\\
0&0&0.98&0.55&0\end{pmatrix}.\quad$
Algorithm 2 proceeds as follows.
* •
$h=0$: We have the set $\\{X_{2},X_{4}\\}$ of instruments on leaves.
* –
$X_{2}$ is identified as an instrument of leaf node $Y_{4}$ ($X_{2}\to Y_{4}$)
as $\widehat{V}_{24}\neq 0$ is the only nonzero in the row 2 with the smallest
(positive) row $\ell_{0}$-norm.
* –
$X_{4}$ is identified as an instrument of leaf node $Y_{5}$ ($X_{4}\to Y_{5}$)
as $\widehat{V}_{45}\neq 0$ is the only nonzero in row 4 with the smallest
(positive) row $\ell_{0}$-norm.
Then $h_{G}(4)=h_{G}(5)=0$ and $\\{Y_{4},Y_{5}\\}$ are removed.
* •
$h=1$: We have the set $\\{X_{5}\\}$ of instrument on leaf.
* –
$X_{5}$ is identified as an instrument of a leaf node $Y_{3}$ ($X_{5}\to
Y_{3}$) in the subgraph for $Y_{1},Y_{2},Y_{3}$ given that
$\widehat{V}_{53}\neq 0$ is the only nonzero element in the row with the
smallest (positive) row $\ell_{0}$-norm of the submatrix for
$Y_{1},Y_{2},Y_{3}$. Since $\widehat{V}_{54}\neq 0$ and $h_{G}(4)=0$, we have
$Y_{3}\to Y_{4}$ by Proposition 3.
Then $h_{G}(3)=1$ and $\\{Y_{3}\\}$ is removed.
* •
$h=2$: We have the set $\\{X_{3}\\}$ of instrument on leaf.
* –
$X_{3}$ is identified as an instrument of a leaf node $Y_{2}$ ($X_{3}\to
Y_{2}$) similarly in the subgraph for $Y_{1},Y_{2}$ given that
$\widehat{V}_{32}\neq 0$ is the largest nonzero element in its row of the
submatrix. Since $\widehat{V}_{33}\neq 0$ and $h_{G}(3)=1$, we have $Y_{2}\to
Y_{3}$.
Then $h_{G}(2)=2$ and $\\{Y_{2}\\}$ is removed.
* •
$h=3$: We have the set $\\{X_{1}\\}$ of instrument on leaf.
* –
$X_{1}$ is an instrument of $Y_{1}$ ($X_{1}\to Y_{1}$). Since
$\widehat{V}_{12}\neq 0$ and $h_{G}(2)=2$, we have $Y_{1}\to Y_{2}$.
Then $h_{G}(1)=3$, $\\{Y_{1}\\}$ is removed, and the peeling process is
terminated.
Finally, Step 6 of Algorithm 2 identifies the ancestral relations
$\widehat{\mathcal{A}}=\\{(1,2),(2,3),(3,4),(1,3),(1,4),(2,4)\\}$
and the candidate interventional relations
$\widehat{\mathcal{I}}=\\{(1,1),(1,2),(1,3),(2,4),(3,2),(3,3),(3,4),(4,5),(5,3),(5,4)\\}.$
In Example 3, $\widehat{\bm{V}}\neq\bm{V}^{\circ}$, suggesting that the
selection consistency of matrix $\bm{V}^{\circ}$ is unnecessary for Algorithm
2 to correctly reconstruct ancestral relations $\mathcal{A}^{\circ}$; see also
Section A.3.
### A.3 Relaxation of Assumption 4
Assumption 4 in Theorem 4 leads to consistent identification for
$\bm{V}^{\circ}$. Now, we discuss when Algorithm 2 correctly reconstruct
ancestral relations $\mathcal{A}^{\circ}$ without requiring Assumption 4.
###### Assumption 6
There exists $(\kappa_{j}^{*},\tau^{*}_{j})$ such that
$\kappa_{j}^{*}=|\\{l:|V^{\circ}_{lj}|\geq\tau_{j}^{*}\\}|$ and
$\begin{split}&\min\Big{(}\min_{l:X_{l}\to
Y_{j}}|V^{\circ}_{lj}|,\min_{\begin{subarray}{c}l:X_{l}\to Y_{k}\to Y_{j}\\\
\text{ with }h(k)=h(j)+1\end{subarray}}|V^{\circ}_{lj}|\Big{)}\\\
&\geq\tau^{*}_{j}\geq
100\frac{c_{2}}{c_{1}}\sqrt{\Omega_{jj}^{-1}(\kappa^{\circ}_{j}-\kappa^{*}_{j}+1)\Big{(}\frac{\log(q)}{n}+\frac{\log(n)}{n}\Big{)}};\quad
1\leq j\leq p.\end{split}$ (24)
Assumption 6 requires a subset of nonzero entries in $\bm{V}^{\circ}$ to
exceed a certain signal strength. These signals enable us to reconstruct the
topological layers and $\mathcal{A}^{\circ}$. Note that Assumption 6 reduces
to Assumption 4 when $\kappa^{*}_{j}=\kappa^{\circ}_{j}$.
###### Theorem 9
Under Assumptions 1-3 and 6 with constants $c_{1}<6c_{2}$, if the machine
precision $\text{tol}\ll 1/n$ is negligible and tuning parameters of Algorithm
1 satisfy:
1. (1)
$\gamma_{j}\leq c_{1}/6\sqrt{\kappa^{\circ}_{j}-\kappa^{*}_{j}+1}$,
2. (2)
$\sqrt{32c_{2}^{2}\Omega_{jj}^{-1}(\log(q)+\log(n))/n}/\gamma_{j}\leq\tau_{j}\leq
0.4\tau^{*}_{j}$,
3. (3)
$\kappa_{j}=\kappa_{j}^{\circ}$,
for $j=1,\ldots,p$, then Algorithm 1 terminates in at most
$1+\lceil\log(\kappa_{\max}^{\circ})/\log 4\rceil$ iterations almost surely.
Moreover, $\widehat{\mathcal{A}}=\mathcal{A}^{\circ}$,
$\widehat{\mathcal{C}}\supseteq\\{(l,j):|V_{lj}^{\circ}|\geq\tau^{*}_{j}\\}\supseteq\mathcal{I}^{\circ}$,
and $\widehat{\mathcal{D}}=\mathcal{D}^{\circ}$ almost surely as $n\to\infty$.
The proof of Theorem 9 is given in Appendix B.9.
### A.4 Comparison of strong faithfulness and Assumption 4 (or 6)
In the literature, a faithfulness condition is usually assumed for
identifiability up to Markov equivalence classes (Spirtes et al., 2000). For
discussion, we formally introduce the concepts of faithfulness and strong
faithfulness.
Consider a DAG $G$ with node variables $(Z_{1},\ldots,Z_{p+q})^{\top}$. Nodes
$Z_{i}$ and $Z_{j}$ are adjacent if $Z_{i}\to Z_{j}$ or $Z_{j}\to Z_{i}$. A
_path_ (_undirected_) between $Z_{i}$ and $Z_{j}$ in $G$ is a sequence of
distinct nodes $(Z_{i},\ldots,Z_{j})$ such that all pairs of successive nodes
in the sequence are adjacent. A nonendpoint node $Z_{k}$ on a path
$(Z_{i},\ldots,Z_{k-1},Z_{k},Z_{k+1},\ldots,Z_{j})$ is a _collider_ if
$Z_{k-1}\to Z_{k}\leftarrow Z_{k+1}$. Otherwise it is a _noncollider_. Let
$A\subseteq\\{1,\ldots,p+q\\}$, where $A$ does not contain $i$ and $j$. Then
$\bm{Z}_{A}$ _blocks_ a path $(Z_{i},\ldots,Z_{j})$ if at least one of the
following holds: (i) the path contains a noncollider that is in $\bm{Z}_{A}$,
or (ii) the path contains a collider that is not in $\bm{Z}_{A}$ and has no
descendant in $\bm{Z}_{A}$. A node $Z_{i}$ is d-separated from $Z_{j}$ given
$\bm{Z}_{A}$ if $\bm{Z}_{A}$ block every path between $Z_{i}$ and $Z_{j}$;
$i\neq j$ (Pearl, 2000).
According to Uhler et al. (2013), a multivariate Gaussian distribution of
$(Z_{1},\ldots,Z_{p+q})^{\top}$ is said to be $\varsigma$-strong faithful to a
DAG with node set $\mathcal{V}=\\{1,\ldots,p+q\\}$ if
$\min_{A\subseteq\mathcal{V}\backslash\\{i,j\\}}\Big{\\{}|{\operatorname{Corr}}(Z_{i},Z_{j}\mid\bm{Z}_{A})|:Z_{i}\text{
is not d-separated from }Z_{j}\text{ given }\bm{Z}_{A}\Big{\\}}>\varsigma,$
(25)
for $1\leq i\neq j\leq p+q$, where $\varsigma\in[0,1)$,
${\operatorname{Corr}}$ denotes the correlation. When $\varsigma=0$, (25) is
equivalent to faithfulness. For consistent structure learning (up to Markov
equivalence classes), it often requires that
$\varsigma\gtrsim\sqrt{s_{0}\log(p+q)/n}$, where $s_{0}$ is a sparsity
measure; see Uhler et al. (2013) for a survey. For a pair $(i,j)$, the number
of possible sets for $A$ is $2^{(p+q-2)}$. If $Z_{i}\to Z_{j}$, then
$\text{Corr}(Z_{i},Z_{j}\mid\bm{Z}_{A})\neq 0$ for any $A$. Therefore, for
this $(i,j)$ pair alone, (25) could require exponentially many conditions.
Indeed, (25) is very restrictive in high-dimensional situation (Uhler et al.,
2013).
By comparison, Algorithm 2 yields consistent structure learning based on
Assumption 4 or 6 instead of strong faithfulness. In some sense, Assumption 4
or 6 requires the sufficient signal strength that is analogous to the
condition for consistent feature selection (Shen et al., 2012). This
assumption may be thought of as an alternative to strong faithfulness. As
illustrated in Example 4, Assumption 4 or 6 is less stringent than strong
faithfulness.
###### Example 4 (Faithfulness)
For simplicity, assume $\bm{X}\sim N(\bm{0},\bm{I})$. Consider model (1) with
$p=q=3$,
$\displaystyle Y_{1}=W_{11}X_{1}+\varepsilon_{1},\quad
Y_{2}=U_{12}Y_{1}+W_{22}X_{2}+\varepsilon_{2},\quad
Y_{3}=U_{13}Y_{1}+U_{23}Y_{2}+W_{33}X_{3}+\varepsilon_{3},$
where $\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}\sim N(0,1)$ are
independent and $U_{12},U_{13},U_{23},W_{11},W_{22},W_{33}\neq 0$. Let
$\bm{Z}=(Y_{1},Y_{2},Y_{3},X_{1},X_{2},X_{3})^{\top}$. Since directed
relations among $\bm{X}$ are not of interest, (25) becomes
$\begin{split}\min_{\begin{subarray}{c}\bm{Z}_{A}=(\bm{Y}_{A_{1}},\bm{X}_{A_{2}}):\\\
A_{1}\subseteq\\{i,j\\}^{c},A_{2}\end{subarray}}&\Big{\\{}|{\operatorname{Corr}}(Y_{i},Y_{j}\mid\bm{Z}_{A})|:Y_{i},Y_{j}\text{
are not d-separated given }\bm{Z}_{A}\Big{\\}}>\varsigma,\\\
\min_{\begin{subarray}{c}\bm{Z}_{A}=(\bm{Y}_{A_{1}},\bm{X}_{A_{2}}):\\\
A_{1}\subseteq\\{j\\}^{c},A_{2}\subseteq\\{l\\}^{c}\end{subarray}}&\Big{\\{}|{\operatorname{Corr}}(Y_{j},X_{l}\mid\bm{Z}_{A})|:Y_{j},X_{l}\text{
are not d-separated given }\bm{Z}_{A}\Big{\\}}>\varsigma,\end{split}$ (26)
for each pair $(i,j)$ with $i\neq j$ and each pair $(l,j)$. Then strong-
faithfulness in (26) assumes 152 conditions for the correlations. By
comparison, (17) requires the absolute values of
$V_{11},V_{12},V_{13},V_{22},V_{23},V_{33}\gtrsim\sqrt{\log(q)/n}$, which in
turn requires the minimal absolute value of six correlations
$\gtrsim\sqrt{\log(q)/n}$,
(i) ${\operatorname{Corr}}(Y_{1},X_{1}\mid X_{2},X_{3})$, (ii)
${\operatorname{Corr}}(Y_{2},X_{1}\mid X_{2},X_{3})$, (iii)
${\operatorname{Corr}}(Y_{3},X_{1}\mid X_{2},X_{3})$,
(iv) ${\operatorname{Corr}}(Y_{2},X_{2}\mid X_{1},X_{3})$, (v)
${\operatorname{Corr}}(Y_{3},X_{2}\mid X_{1},X_{3})$, (vi)
${\operatorname{Corr}}(Y_{3},X_{3}\mid X_{1},X_{2})$.
Importantly, (i)-(vi) are required in (26), suggesting that strong-
faithfulness is more stringent than (17).
### A.5 Irregular hypothesis
Assume, without loss of generality, that
$\widehat{\mathcal{D}}=\mathcal{D}^{\circ}$ and
$\widehat{\mathcal{A}}=\mathcal{A}^{\circ}$ are correctly reconstructed in the
following discussion.
* •
For testing of directed edges (2), suppose $H_{0}$ is irregular, namely,
$\mathcal{D}^{\circ}\cup\mathcal{E}^{\circ}$ contains a directed cycle. This
implies that a directed cycle exists in
$\widehat{\mathcal{D}}\cup\widehat{\mathcal{A}}$. In this situation, we
decompose $H_{0}$ into sub-hypotheses $H_{0}^{(1)},\ldots,H_{0}^{(\nu)}$, each
of which is regular. Then testing $H_{0}$ is equivalent to multiple testing
for $H_{0}^{(1)},\ldots,H_{0}^{(\nu)}$. For instance, in Example 1,
$H_{0}:U_{45}=U_{53}=0$ is irregular, and $H_{0}$ can be decomposed into
$H_{0}^{(1)}:U_{45}=0$ and $H_{0}^{(2)}:U_{53}=0$.
* •
For testing of directed pathways (3), if $H_{0}$ is irregular, then
$\widehat{\mathcal{D}}\cup\widehat{\mathcal{A}}$ has a directed cycle. The
p-value is defined to be one in this situation since no evidence supports the
presence of the pathway.
## B Technical proofs
### B.1 Proof of Proposition 1
Suppose that $\bm{\theta}=(\bm{U},\bm{W},\bm{\Sigma})$ and
$\widetilde{\bm{\theta}}=(\widetilde{\bm{U}},\widetilde{\bm{W}},\widetilde{\bm{\Sigma}})$
render the same distribution of $(\bm{X},\bm{Y})$. We will prove that
$\bm{\theta}=\widetilde{\bm{\theta}}$.
Denote by $G(\bm{\theta})$ and $G(\widetilde{\bm{\theta}})$ the DAGs
corresponding to $\bm{\theta}$ and $\widetilde{\bm{\theta}}$, respectively.
First, consider $G(\bm{\theta})$. Without loss of generality, assume $Y_{1}$
is a leaf node. By Assumption 1C, there exists an instrumental intervention
with respect to $G(\bm{\theta})$, say $X_{1}$. Then,
$\displaystyle\operatorname{Cov}(Y_{j},X_{1}\mid\bm{X}_{\\{2,\ldots,q\\}})=0,$
$\displaystyle\quad j=2,\ldots,p,$ (27)
$\displaystyle\operatorname{Cov}(Y_{1},X_{1}\mid\bm{Y}_{A},\bm{X}_{\\{2,\ldots,q\\}})\neq
0,$ $\displaystyle\quad\text{for any }A\subseteq\\{2,\ldots,p\\}.$ (28)
By the local Markov property (Spirtes et al., 2000), (28) implies that
$X_{1}\to Y_{1}$ in $G(\widetilde{\bm{\theta}})$. Suppose $Y_{1}$ is not a
leaf node in $G(\widetilde{\bm{\theta}})$. Assume, without loss of generality,
that $Y_{1}\to Y_{2}$ with
$h_{G(\widetilde{\bm{\theta}})}(1)=h_{G(\widetilde{\bm{\theta}})}(2)+1$. Then
$\operatorname{Cov}(Y_{2},X_{1}\mid\bm{X}_{\\{2,\ldots,q\\}})=0$ but $X_{1}\to
Y_{1}$ and $Y_{1}\to Y_{2}$ with
$h_{G(\widetilde{\bm{\theta}})}(1)=h_{G(\widetilde{\bm{\theta}})}(2)+1$, which
contradicts to Assumption 1B. This implies that if $Y_{1}$ is a leaf node in
$G(\bm{\theta})$ then it must be a leaf node in $G(\widetilde{\bm{\theta}})$.
In both $G(\bm{\theta})$ and $G(\widetilde{\bm{\theta}})$, the parents and
interventions of $Y_{1}$ can be identified by
$\begin{split}\operatorname{\mathbb{E}}(Y_{1}\mid\bm{Y}_{\\{2,\ldots,p\\}},\bm{X})&=\operatorname{\mathbb{E}}(Y_{1}\mid\bm{Y}_{\textnormal{{pa}}_{G(\bm{\theta})}(1)},\bm{X})=\operatorname{\mathbb{E}}(Y_{1}\mid\bm{Y}_{\textnormal{{pa}}_{G(\bm{\theta})}(1)},\bm{X}_{\textnormal{{in}}_{G(\bm{\theta})}(1)}),\\\
\operatorname{\mathbb{E}}(Y_{1}\mid\bm{Y}_{\\{2,\ldots,p\\}},\bm{X})&=\operatorname{\mathbb{E}}(Y_{1}\mid\bm{Y}_{\textnormal{{pa}}_{G(\widetilde{\bm{\theta}})}(1)},\bm{X})=\operatorname{\mathbb{E}}(Y_{1}\mid\bm{Y}_{\textnormal{{pa}}_{G(\widetilde{\bm{\theta}})}(1)},\bm{X}_{\textnormal{{in}}_{G(\widetilde{\bm{\theta}})}(1)}).\end{split}$
Consequently, $Y_{1}$ has the same parents and interventions in
$G(\bm{\theta})$ and $G(\widetilde{\bm{\theta}})$.
The forgoing argument is applied to other nodes sequentially. First, we remove
$Y_{1}$ with any directed edges to $Y_{1}$, which does not alter the joint
distribution of $(\bm{Y}_{\\{2,\ldots,p\\}},\bm{X})$ and the subgraph of nodes
$Y_{2},\ldots,Y_{p}$. By induction, we remove the leafs in $G(\bm{\theta})$
until it is empty, leading to $G(\bm{\theta})=G(\widetilde{\bm{\theta}})$.
Finally, $\bm{\theta}=\widetilde{\bm{\theta}}$ because they have the same
locations for nonzero elements and these model parameters (or regression
coefficients) are uniquely determined under Assumption 1A (Shojaie and
Michailidis, 2010). This completes the proof.
### B.2 Proof of Proposition 2
It follows from Assumption 1B that $\mathcal{C}=\\{(l,j):V_{lj}\neq
0\\}\supseteq\mathcal{I}$.
#### Proof of (A)
Note that the maximal length of a path in a DAG of $p$ nodes is at most $p-1$.
Then it can be verified that $\bm{U}$ is nilpotent in that
$\bm{U}^{p}=\bm{0}$. An application of the matrix series expansion yields that
$(\bm{I}-\bm{U})^{-1}=\bm{I}+\bm{U}+\ldots+\bm{U}^{p-1}$. Using the fact that
$\bm{V}=\bm{W}(\bm{I}-\bm{U})^{-1}$ from (6), we have, for any $1\leq l,j\leq
p$,
$V_{lj}=\sum_{k=1}^{p}W_{lk}\Big{(}I_{kj}+U_{kj}+\ldots+(\bm{U}^{p-1})_{kj}\Big{)},$
where $U_{kj}$ is the $(k,j)$th entry of $\bm{U}$. If $V_{lj}\neq 0$, then
there exists $k$ such that $W_{lk}\neq 0$ and $(\bm{U}^{r})_{kj}\neq 0$ for
some $0\leq r\leq p-1$. If $r=0$, then we must have $k=j$, and $X_{l}\to
Y_{j}$. If $r>0$, then $X_{l}\to Y_{k}$ and $Y_{k}$ is an ancestor of $Y_{j}$.
#### Proof of (B)
First, for any leaf node variable $Y_{j}$, by Assumption 1, there exists an
instrument $X_{l}\to Y_{j}$. If $V_{lj^{\prime}}\neq 0$ for some
$j^{\prime}\neq j$, then $Y_{j}$ must be an ancestor of $Y_{j^{\prime}}$,
which contradicts the fact that $Y_{j}$ is a leaf node variable.
Conversely, suppose that $V_{lj}\neq 0$ and $V_{lj^{\prime}}=0$ for
$j^{\prime}\neq j$. If $Y_{j}$ is not a leaf node variable, then there exists
a variable $Y_{j^{\prime}}$ such that $Y_{j}\to Y_{j^{\prime}}$ with
$h_{G}(j)=h_{G}(j^{\prime})+1$, that is $U_{jj^{\prime}}\neq 0$ and
$(\bm{U}^{r})_{jj^{\prime}}=0$ for $r>1$. Then
$V_{lj^{\prime}}=W_{lj}U_{jj^{\prime}}\neq 0$, a contradiction.
### B.3 Proof of Proposition 3
Suppose $Y_{k}\to Y_{j}$ and $h_{G}(j)=h_{G}(k)-1$. Let $X_{l}$ be an
instrument of $Y_{k}$ in $G^{\prime}$. Then there are two cases: (1) $X_{l}$
intervenes on $Y_{k}$ but does not intervene on $Y_{j}$, namely $X_{l}\to
Y_{k}$ but $X_{l}\not\to Y_{j}$; (2) $X_{l}$ intervenes on $Y_{k}$ and $Y_{j}$
simultaneously, namely $X_{l}\to Y_{k}$ and $X_{l}\to Y_{j}$. For (1),
$V_{lj}=W_{lk}U_{kj}\neq 0$. For (2), Assumption 1B implies that $V_{lj}\neq
0$. This holds for every instrument of $Y_{k}$ in $G^{\prime}$, and the
desired result follows.
Conversely, suppose for each instrument $X_{l}$ of $Y_{k}$ in $G^{\prime}$, we
have $V_{lj}\neq 0$. Let $X_{l^{\prime}}$ be an instrument of $Y_{k}$ in $G$,
which is also an instrument in $G^{\prime}$. Then $0\neq
V_{l^{\prime}j}=W_{l^{\prime}k}U_{kj}$, which implies $U_{kj}\neq 0$. This
completes the proof.
### B.4 Proof of Theorem 4
The proof proceeds in two steps: (A) Show that $\\{l:\widehat{V}_{lj}\neq
0\\}=\\{l:V_{lj}^{\circ}\neq 0\\}$ almost surely for $j=1,\ldots,p$; and (B)
Show that $\widehat{S}=S^{\circ}$ if $\widehat{\bm{V}}$ satisfies the property
in (A). As a result, $\widehat{\mathcal{D}}=\mathcal{D}^{\circ}$ follows
immediately.
#### Proof of (A)
Let $A_{j}^{\circ}=\\{l:V^{\circ}_{lj}\neq 0\\}$ and
$A^{[t]}_{j}=\\{l:|\widetilde{V}^{[t]}_{lj}|\geq\tau_{j}\\}$ be the estimated
support of penalized solution at the $t$-th iteration of Algorithm 1. For the
penalized solution, define the false negative set
$\text{FN}^{[t]}_{j}=A^{\circ}_{j}\setminus A_{j}^{[t]}$ and the false
positive set $\text{FP}^{[t]}_{j}=A_{j}^{[t]}\setminus A_{j}^{\circ}$; $t\geq
0$. Consider a “good” event
$\mathscr{E}_{j}=\\{\|\bm{\mathsf{X}}^{\top}\widehat{\bm{\xi}}_{j}/n\|_{\infty}\leq
0.5\gamma_{j}\tau_{j}\\}\cap\\{\|\widehat{\bm{V}}^{\circ}_{\cdot
j}-\bm{V}^{\circ}_{\cdot j}\|_{\infty}\leq 0.5\tau_{j}\\}$, where
$\widehat{\bm{\xi}}_{j}=\bm{\mathsf{Y}}_{j}-\bm{\mathsf{X}}\widehat{\bm{V}}^{\circ}_{\cdot
j}$ is the residual of the least squares estimate
$\widehat{\bm{V}}^{\circ}_{\cdot j}$ such that
$A_{j}^{\circ}=\\{l:\widehat{V}^{\circ}_{lj}\neq 0\\}$; $j=1,\ldots,p$. We
shall show that $\text{FN}^{[t]}_{j}$ and $\text{FP}^{[t]}_{j}$ are eventually
empty sets on event $\mathscr{E}_{j}$ which has a probability tending to one.
First, we show that if $|A_{j}^{\circ}\cup A^{[t-1]}_{j}|\leq
2\kappa_{\max}^{\circ}$ on $\mathscr{E}_{j}$ for $t\geq 1$, then
$|A_{j}^{\circ}\cup A^{[t]}_{j}|\leq 2\kappa^{\circ}_{\max}$, to be used in
Assumption 2. Now suppose $|A_{j}^{\circ}\cup A^{[t-1]}_{j}|\leq
2\kappa^{\circ}_{\max}$ on $\mathscr{E}_{j}$ for $t\geq 1$. By the optimality
condition of (8),
$\begin{split}\Big{(}\widehat{\bm{V}}^{o}_{\cdot
j}-\widetilde{\bm{V}}^{[t]}_{\cdot
j}\Big{)}^{\top}\Big{(}-\bm{\mathsf{X}}^{\top}(\bm{\mathsf{Y}}_{j}-\bm{\mathsf{X}}\widetilde{\bm{V}}^{[t]}_{\cdot
j})/n+\gamma_{j}\tau_{j}\nabla\|\widetilde{\bm{V}}_{(A^{[t-1]}_{j})^{c},j}^{[t]}\|_{1}\Big{)}\geq
0,\end{split}$
where $\widetilde{\bm{V}}^{[t]}$ is defined in (8). Plugging
$\widehat{\bm{\xi}}_{j}=\bm{\mathsf{Y}}_{j}-\bm{\mathsf{X}}\widehat{\bm{V}}^{\circ}_{\cdot
j}$ into the inequality and rearranging it, we have
$\|\bm{\mathsf{X}}(\widetilde{\bm{V}}_{\cdot j}^{[t]}-\widehat{\bm{V}}_{\cdot
j}^{\circ})\|_{2}^{2}/n$ is no greater than
$\begin{split}&\Big{(}\widetilde{\bm{V}}_{\cdot
j}^{[t]}-\widehat{\bm{V}}_{\cdot
j}^{\circ}\Big{)}^{\top}\Big{(}\bm{\mathsf{X}}^{\top}\widehat{\bm{\xi}}_{j}/n-\gamma_{j}\tau_{j}\nabla\|\widetilde{\bm{V}}^{[t]}_{(A_{j}^{[t-1]})^{c},j}\|_{1}\Big{)}\\\
&=\Big{(}\widetilde{\bm{V}}_{A_{j}^{\circ}\setminus
A_{j}^{[t-1]},j}^{[t]}-\widehat{\bm{V}}_{A_{j}^{\circ}\setminus
A_{j}^{[t-1]},j}^{\circ}\Big{)}^{\top}\Big{(}\bm{\mathsf{X}}^{\top}_{A_{j}^{\circ}\setminus
A^{[t-1]}_{j}}\widehat{\bm{\xi}}_{j}/n-\gamma_{j}\tau_{j}\nabla\|\widetilde{\bm{V}}^{[t]}_{A_{j}^{\circ}\setminus
A^{[t-1]}_{j},j}\|_{1}\Big{)}\\\
&\quad+\Big{(}\widetilde{\bm{V}}_{(A_{j}^{\circ}\cup
A^{[t-1]}_{j})^{c},j}^{[t]}-\widehat{\bm{V}}_{(A_{j}^{\circ}\cup
A^{[t-1]}_{j})^{c},j}^{\circ}\Big{)}^{\top}\Big{(}\bm{\mathsf{X}}^{\top}\widehat{\bm{\xi}}/n-\gamma_{j}\tau_{j}\nabla\|\widetilde{\bm{V}}^{[t]}_{(A_{j}^{\circ}\cup
A^{[t-1]}_{j})^{c},j}\|_{1}\Big{)}\\\
&\quad+\Big{(}\widetilde{\bm{V}}_{A^{[t-1]}_{j}\setminus
A_{j}^{\circ},j}^{[t]}-\widehat{\bm{V}}_{A^{[t-1]}_{j}\setminus
A_{j}^{\circ},j}^{\circ}\Big{)}^{\top}\bm{\mathsf{X}}^{\top}_{A_{j}^{[t-1]}\setminus
A_{j}^{\circ}}\widehat{\bm{\xi}}/n.\end{split}$ (29)
Note that
$\Big{(}\widetilde{\bm{V}}_{(A_{j}^{\circ}\cup
A_{j}^{[t-1]})^{c},j}^{[t]}-\widehat{\bm{V}}_{(A_{j}^{\circ}\cup
A_{j}^{[t-1]})^{c},j}^{\circ}\Big{)}^{\top}\nabla\Big{\|}\widetilde{\bm{V}}^{[t]}_{(A_{j}^{\circ}\cup
A_{j}^{[t-1]})^{c},j}\Big{\|}_{1}=\Big{\|}\widetilde{\bm{V}}_{(A_{j}^{\circ}\cup
A_{j}^{[t-1]})^{c},j}^{[t]}-\widehat{\bm{V}}_{(A_{j}^{\circ}\cup
A_{j}^{[t-1]})^{c},j}^{\circ}\Big{\|}_{1}.$
Then (29) is no greater than
$\begin{split}&\Big{\|}\widetilde{\bm{V}}_{A_{j}^{\circ}\triangle
A^{[t-1]}_{j},j}^{[t]}-\widehat{\bm{V}}_{A_{j}^{\circ}\triangle
A^{[t-1]}_{j},j}^{\circ}\Big{\|}_{1}\Big{(}\|\bm{\mathsf{X}}^{\top}\widehat{\bm{\xi}}_{j}/n\|_{\infty}+\gamma_{j}\tau_{j}\Big{)}\\\
&+\Big{\|}\widetilde{\bm{V}}_{(A_{j}^{\circ}\cup
A_{j}^{[t-1]})^{c},j}^{[t]}-\widehat{\bm{V}}_{(A_{j}^{\circ}\cup
A_{j}^{[t-1]})^{c},j}^{\circ}\Big{\|}_{1}\Big{(}\|\bm{\mathsf{X}}^{\top}\widehat{\bm{\xi}}_{j}/n\|_{\infty}-\gamma_{j}\tau_{j}\Big{)},\end{split}$
(30)
where $\triangle$ denotes the symmetric difference. Note that
$\|\bm{\mathsf{X}}(\widetilde{\bm{V}}_{\cdot j}^{[t]}-\widehat{\bm{V}}_{\cdot
j}^{\circ})\|_{2}^{2}/n\geq 0$. Rearranging the inequality yields that
$\begin{split}&\Big{(}\gamma_{j}\tau_{j}-\|\bm{\mathsf{X}}^{\top}\widehat{\bm{\xi}}_{j}/n\|_{\infty}\Big{)}\Big{\|}\widetilde{\bm{V}}_{(A_{j}^{\circ}\cup
A_{j}^{[t-1]})^{c},j}^{[t]}-\widehat{\bm{V}}_{(A_{j}^{\circ}\cup
A_{j}^{[t-1]})^{c},j}^{\circ}\Big{\|}_{1}\\\
&\leq\Big{(}\|\bm{\mathsf{X}}^{\top}\widehat{\bm{\xi}}_{j}/n\|_{\infty}+\gamma_{j}\tau_{j}\Big{)}\Big{\|}\widetilde{\bm{V}}_{A_{j}^{\circ}\triangle
A^{[t-1]}_{j},j}^{[t]}-\widehat{\bm{V}}_{A_{j}^{\circ}\triangle
A^{[t-1]}_{j},j}^{\circ}\Big{\|}_{1}.\end{split}$
On event $\mathscr{E}_{j}$,
$\|\bm{\mathsf{X}}^{\top}\widehat{\bm{\xi}}_{j}/n\|_{\infty}\leq\gamma_{j}\tau_{j}/2$,
implying that $\|\widetilde{\bm{V}}_{(A_{j}^{\circ}\cup
A_{j}^{[t-1]})^{c},j}^{[t]}-\widehat{\bm{V}}_{(A_{j}^{\circ}\cup
A_{j}^{[t-1]})^{c},j}^{\circ}\|_{1}\leq
3\|\widetilde{\bm{V}}_{A_{j}^{\circ}\triangle
A^{[t-1]}_{j},j}^{[t]}-\widehat{\bm{V}}_{A_{j}^{\circ}\triangle
A^{[t-1]}_{j},j}^{\circ}\|_{1}\leq 3\|\widetilde{\bm{V}}_{A_{j}^{\circ}\cup
A^{[t-1]}_{j},j}^{[t]}-\widehat{\bm{V}}_{A_{j}^{\circ}\cup
A^{[t-1]}_{j},j}^{\circ}\|_{1}$. Note that $|A_{j}^{\circ}\cup
A^{[t-1]}_{j}|\leq 2\kappa^{\circ}_{\max}$. By Assumption 2 and (30),
$\begin{split}c_{1}\|\widetilde{\bm{V}}_{\cdot
j}^{[t]}-\widehat{\bm{V}}_{\cdot
j}^{\circ}\|_{2}^{2}\leq&\Big{(}\|\bm{\mathsf{X}}^{\top}\widehat{\bm{\xi}}_{j}/n\|_{\infty}+\gamma_{j}\tau_{j}\Big{)}\Big{\|}\widetilde{\bm{V}}_{A_{j}^{\circ}\triangle
A_{j}^{[t-1]},j}^{[t]}-\widehat{\bm{V}}_{A_{j}^{\circ}\triangle
A_{j}^{[t-1]},j}^{\circ}\Big{\|}_{1}\\\
&+\Big{(}\|\bm{\mathsf{X}}^{\top}\widehat{\bm{\xi}}_{j}/n\|_{\infty}-\gamma_{j}\tau_{j}\Big{)}\Big{\|}\widetilde{\bm{V}}_{(A_{j}^{\circ}\cup
A_{j}^{[t-1]})^{c},j}^{[t]}-\widehat{\bm{V}}_{(A_{j}^{\circ}\cup
A_{j}^{[t-1]})^{c},j}^{\circ}\Big{\|}_{1}\\\
\leq&\Big{(}\|\bm{\mathsf{X}}^{\top}\widehat{\bm{\xi}}_{j}/n\|_{\infty}+\gamma_{j}\tau_{j}\Big{)}\Big{\|}\widetilde{\bm{V}}_{A_{j}^{\circ}\triangle
A_{j}^{[t-1]},j}^{[t]}-\widehat{\bm{V}}_{A_{j}^{\circ}\triangle
A_{j}^{[t-1]},j}^{\circ}\Big{\|}_{1}.\end{split}$ (31)
By the Cauchy-Schwarz inequality,
$\Big{\|}\widetilde{\bm{V}}_{A_{j}^{\circ}\triangle
A_{j}^{[t-1]},j}^{[t]}-\widehat{\bm{V}}_{A_{j}^{\circ}\triangle
A_{j}^{[t-1]},j}^{\circ}\Big{\|}_{1}\leq\sqrt{A_{j}^{\circ}\triangle
A^{[t-1]}_{j}}\Big{\|}\widetilde{\bm{V}}_{\cdot
j}^{[t]}-\widehat{\bm{V}}^{\circ}_{\cdot j}\Big{\|}_{2}$. Thus,
$c_{1}\|\widetilde{\bm{V}}_{\cdot j}^{[t]}-\widehat{\bm{V}}_{\cdot
j}^{\circ}\|_{2}\leq 1.5\gamma_{j}\tau_{j}\sqrt{2\kappa^{\circ}_{\max}}$,
since $|A_{j}^{\circ}\triangle A_{j}^{[t-1]}|\leq|A_{j}^{\circ}\cup
A_{j}^{[t-1]}|\leq 2\kappa^{\circ}_{\max}$ and
$\|\bm{\mathsf{X}}^{\top}\widehat{\bm{\xi}}_{j}/n\|_{\infty}\leq
0.5\gamma_{j}\tau_{j}$. By the condition (1) of Theorem 4, $\gamma_{j}\leq
c_{1}/6$ and $\|\widetilde{\bm{V}}_{\cdot j}^{[t]}-\widehat{\bm{V}}_{\cdot
j}^{\circ}\|_{2}/\tau_{j}\leq\sqrt{\kappa^{\circ}_{\max}}$. On the other hand,
for any $l\in\text{FP}^{[t]}_{j}=A^{[t]}_{j}\setminus A_{j}^{\circ}$, we have
$|\widetilde{V}_{lj}^{[t]}-\widehat{V}_{lj}^{\circ}|=|\widetilde{V}_{lj}^{[t]}|>\tau_{j}$.
Thus, $\sqrt{|\text{FP}^{[t]}_{j}|}\leq\|\widetilde{\bm{V}}_{\cdot
j}^{[t]}-\widehat{\bm{V}}_{\cdot
j}^{\circ}\|_{2}/\tau_{j}\leq\sqrt{\kappa^{\circ}_{\max}}$. Consequently,
$|A_{j}^{\circ}\cup A^{[t]}_{j}|=|A_{j}^{\circ}|+|\text{FP}^{[t]}_{j}|\leq
2\kappa^{\circ}_{\max}$ on $\mathscr{E}_{j}$ for $t\geq 1$.
Next, we estimate the number of iterations required for termination. Note that
the machine precision is negligible. The termination criterion is met when
$A^{[t]}_{j}=A^{[t-1]}_{j}$ since the weighted Lasso problem (8) remains same
at the $t$th and $(t-1)$th iterations. To show that
$A^{[t]}_{j}=A^{\circ}_{j}$ eventually, we prove that
$|\text{FN}^{[t]}_{j}|+|\text{FP}^{[t]}_{j}|<1$ eventually.
Now, suppose $|\text{FN}^{[t]}_{j}|+|\text{FP}_{j}^{[t]}|\geq 1$. For any
$l\in\text{FN}^{[t]}_{j}\cup\text{FP}_{j}^{[t]}$, by Assumption 4,
$|\widetilde{V}^{[t]}_{lj}-\widehat{V}^{\circ}_{lj}|\geq|\widetilde{V}^{[t]}_{lj}-V^{\circ}_{lj}|-|\widehat{V}^{\circ}_{lj}-V^{\circ}_{lj}|\geq\tau_{j}-0.5\tau_{j}$,
so
$\sqrt{|\text{FN}^{[t]}_{j}|+|\text{FP}^{[t]}_{j}|}\leq\|\widetilde{\bm{V}}_{\cdot
j}^{[t]}-\widehat{\bm{V}}_{\cdot j}^{\circ}\|_{2}/0.5\tau_{j}$. By (31) and
the Cauchy-Schwarz inequality, $c_{1}\|\widetilde{\bm{V}}_{\cdot
j}^{[t]}-\widehat{\bm{V}}_{\cdot j}^{\circ}\|_{2}\leq
1.5\gamma_{j}\tau_{j}\sqrt{|\text{FN}_{j}^{[t-1]}|+|\text{FP}^{[t-1]}_{j}|}.$
By conditions (1) and (2) for $(\tau_{j},\gamma_{j})$ in Theorem 4, we have
$\begin{split}\sqrt{|\text{FN}^{[t]}_{j}|+|\text{FP}^{[t]}_{j}|}\leq\frac{\|\widetilde{\bm{V}}_{\cdot
j}^{[t]}-\widehat{\bm{V}}_{\cdot
j}^{\circ}\|_{2}}{0.5\tau_{j}}&\leq\frac{3\gamma_{j}}{c_{1}}\sqrt{|\text{FN}_{j}^{[t-1]}|+|\text{FP}_{j}^{[t-1]}|}\\\
&\leq 0.5\sqrt{|\text{FN}_{j}^{[t-1]}|+|\text{FP}_{j}^{[t-1]}|}.\end{split}$
Hence,
$\sqrt{|\text{FN}^{[t]}_{j}|+|\text{FP}^{[t]}_{j}|}\leq(1/2)^{t}\sqrt{|A^{\circ}_{j}|+|A^{[0]}_{j}|}$.
In particular, for $t\geq 1+\lceil\log\kappa^{\circ}_{j}/\log 4\rceil$,
$|\text{FN}_{j}^{[t]}|+|\text{FP}^{[t]}_{j}|<1$ implying that
$\text{FN}_{j}^{[t]}=\emptyset$ and $\text{FP}_{j}^{[t]}=\emptyset$.
Let $t_{\max}=1+\lceil\log\kappa^{\circ}_{j}/\log 4\rceil$. Then
$\text{FN}_{j}^{[t_{\max}]}=\text{FP}^{[t_{\max}]}_{j}=\emptyset$ on event
$\mathscr{E}_{j}$. Under condition (3), we have
$\\{l:\widetilde{V}^{[t_{\max}]}_{lj}\neq 0\\}=\\{l:V_{lj}^{\circ}\neq 0\\}$.
To bound $\operatorname{\mathbb{P}}(\bigcup_{j=1}^{p}\mathscr{E}_{j}^{c})$,
let
$\bm{\eta}=\bm{\mathsf{X}}^{\top}(\bm{I}-\bm{\mathsf{P}}_{A^{\circ}_{j}}){\bm{\xi}}_{j}$
and
$\bm{\eta}^{\prime}=n(\bm{\mathsf{X}}^{\top}_{A^{\circ}_{j}}\bm{\mathsf{X}}_{A^{\circ}_{j}})^{-1}\bm{\mathsf{X}}^{\top}_{A_{j}^{\circ}}{\bm{\xi}}_{j}$,
where
$\bm{\xi}_{j}=((\varepsilon_{V})_{1j},\ldots,(\varepsilon_{V})_{nj})^{\top}\in\mathbb{R}^{n}$.
Then $\bm{\eta}\in\mathbb{R}^{q}$ and
$\bm{\eta}^{\prime}\in\mathbb{R}^{|\kappa^{\circ}_{j}|}$ are Gaussian random
vectors, where $\mathop{\rm
Var}(\eta_{l})\leq\Omega_{jj}^{-1}(\bm{\mathsf{X}}^{\top}\bm{\mathsf{X}})_{ll}\leq
n\Omega_{jj}^{-1}c_{2}^{2}$ and $\mathop{\rm
Var}(\eta^{\prime}_{l})\leq\Omega_{jj}^{-1}((\bm{\mathsf{X}}^{\top}_{A^{\circ}_{j}}\bm{\mathsf{X}}_{A^{\circ}_{j}})^{-1})_{ll}\leq
n\Omega^{-1}_{jj}c_{2}^{2}$. Then
$\begin{split}\operatorname{\mathbb{P}}\Big{(}\|\bm{\mathsf{X}}^{\top}\widehat{\bm{\xi}}_{j}/n\|_{\infty}>0.5\gamma_{j}\tau_{j}\Big{)}&=\operatorname{\mathbb{P}}\Big{(}\|\bm{\eta}/n\|_{\infty}>0.5\gamma_{j}\tau_{j}\Big{)}\\\
&\leq\sum_{l=1}^{q}\operatorname{\mathbb{P}}\Big{(}|\eta_{l}|\geq
0.5n\gamma_{j}\tau_{j}\Big{)}\\\ &\leq
q\int_{\frac{0.5\sqrt{n\Omega_{jj}}\gamma_{j}\tau_{j}}{c_{2}}}^{\infty}\frac{e^{-t^{2}/2}}{\sqrt{2\pi}}dt\leq
q\sqrt{\frac{2}{\pi}}\exp\Big{(}-\frac{n\Omega_{jj}\gamma^{2}_{j}\tau^{2}_{j}}{8c^{2}_{2}}\Big{)},\end{split}$
and similarly,
$\begin{split}\operatorname{\mathbb{P}}\Big{(}\|\widehat{\bm{V}}_{\cdot
j}^{\circ}-\bm{V}_{\cdot
j}^{\circ}\|_{\infty}>0.5\tau_{j}\Big{)}\leq\sum_{l\in
A^{\circ}_{j}}\operatorname{\mathbb{P}}\Big{(}|\eta^{\prime}_{l}|>0.5\tau_{j}\Big{)}\leq\kappa^{\circ}_{\max}\sqrt{\frac{2}{\pi}}\exp\Big{(}-\frac{n\Omega_{jj}\tau^{2}_{j}}{8c^{2}_{2}}\Big{)}.\end{split}$
Under conditions (1) and (2) for $(\tau_{j},\gamma_{j})$ in Theorem 4,
$\begin{split}&\operatorname{\mathbb{P}}\Big{(}\Big{\\{}l:\widetilde{V}^{[t_{\max}]}_{lj}\neq
0\Big{\\}}=\Big{\\{}l:V_{lj}^{\circ}\neq 0\Big{\\}};\ 1\leq j\leq p\Big{)}\\\
&\geq
1-\operatorname{\mathbb{P}}\Big{(}\bigcup_{j=1}^{p}\mathscr{E}_{j}^{c}\Big{)}\\\
&\geq
1-p\sqrt{\frac{2}{\pi}}\Big{(}e^{-4(\log(q)+\log(n))+\log(q)}+e^{-144c_{1}^{-2}c_{2}^{2}(\log(q)+\log(n))+\log(\kappa^{\circ}_{j})}\Big{)}\geq
1-\sqrt{\frac{2}{\pi}}pq^{-3}n^{-4}.\end{split}$
By Borel-Cantelli lemma, $\\{l:\widetilde{V}^{[t_{\max}]}_{lj}\neq
0\\}=\\{l:V_{lj}^{\circ}\neq 0\\}$; $j=1,\ldots,p$ almost surely as
$n\to\infty$.
So far, we have $\widetilde{\bm{V}}_{\cdot
j}=\widetilde{\bm{V}}^{[t_{\max}]}_{\cdot j}=\widehat{\bm{V}}^{\circ}_{\cdot
j}$; $j=1,\ldots,p$, almost surely. It remains to show that
$\widehat{\bm{V}}^{\circ}_{\cdot j}$ is a global minimizer of (7);
$j=1,\ldots,p$, with a high probability. Note that Assumptions 2 and 4 imply
the degree of separation condition (Shen et al., 2013). By Theorem 2 of Shen
et al. (2013),
$\operatorname{\mathbb{P}}\Big{(}\widehat{\bm{V}}_{\cdot j}^{\circ}\text{ is
not a global minimizer of }\eqref{pseudo-likelihood};1\leq j\leq p\Big{)}\leq
3p\exp\Big{(}-2(\log(q)+\log(n))\Big{)}.$
implying that $\widehat{\bm{V}}_{\cdot
j}=\widetilde{\bm{V}}^{[t_{\max}]}_{\cdot j}=\widehat{\bm{V}}^{\circ}_{\cdot
j}$ is a global minimizer of (7) almost surely, as $n\to\infty$. This
completes the proof.
#### Proof of (B)
Assume $\widehat{\bm{V}}$ satisfies the properties in (A). Then Propositions 2
and 3 holds true for $\widehat{\bm{V}}$. Clearly, we have
$\widehat{\mathcal{C}}=\mathcal{C}^{\circ}$. Thus, we need to show
$\widehat{\mathcal{A}}=\mathcal{A}^{\circ}$ and
$\widehat{\mathcal{D}}=\mathcal{D}^{\circ}$. We shall prove this by induction.
For $h\geq 0$, the set of instruments on leaves in the graph $G^{[h]}$ (with
primary variables $\bm{Y}_{\\{j:h_{G}(j)\geq h\\}}$ and intervention variables
$\bm{X}$) is
$\mathscr{I}^{[h]}=\Big{\\{}l:l=\mathop{\rm
arg\,min\,}_{l^{\prime}:\|\widehat{\bm{V}}^{[h]}_{l^{\prime}\cdot}\|_{0}\neq
0}\|\widehat{\bm{V}}^{[h]}_{l^{\prime}\cdot}\|_{0}\Big{\\}}=\Big{\\{}l:\|\bm{V}^{[h]}_{l\cdot}\|_{0}=1\Big{\\}}.$
Hence, $X_{l}$ is an instrument of leaf variable $Y_{j}$ in $G^{[h]}$, when
$l\in\mathscr{I}^{[h]}$ and $j=\mathop{\rm
arg\,max\,}_{j^{\prime}}|V_{lj^{\prime}}|$. Moreover, $h_{G}(j)=h$. By
Proposition 3 it also recovers the edges $Y_{k}\to Y_{j}$ such that
$h_{G}(k)=h_{G}(j)+1$. In model (1), after removing $\\{Y_{j}:h_{G}(j)\leq
h\\}$, the local Markov property among $(\bm{Y}_{\\{j:h_{G}(j)\geq
h+1\\}},\bm{X})$ remain intact provided that $\\{j:h_{G}(j)\geq
h+1\\}\neq\emptyset$. Then we increment $h$ and repeat the procedure until all
primary variables are removed, namely, $\\{j:h_{G}(j)\geq h+1\\}=\emptyset$.
As a result, Algorithm 2 correctly identifies the topological height
$h_{G}(j)$ for each primary variable $Y_{j}$ and the edges $Y_{k}\to Y_{j}$
such that $h_{G}(k)=h_{G}(j)+1$. Consequently, the ancestral relations can be
reconstructed by
$\mathcal{A}=\Big{\\{}(k_{0},k_{d}):Y_{k_{0}}\to\cdots\to Y_{k_{d}},\
h_{G}(k_{\nu})=h_{G}(k_{\nu+1})+1,\ \nu=0,\ldots,d-1\Big{\\}}.$
Therefore, Step 6 of Algorithm 2 recovers
$\widehat{\mathcal{A}}=\mathcal{A}^{\circ}$ and
$\widehat{\mathcal{D}}=\mathcal{D}^{\circ}$ almost surely as $n\to\infty$.
This completes the proof of (B).
###### Lemma 10
Let
$T_{j}=\frac{\bm{\mathsf{e}}_{j}^{\top}(\bm{\mathsf{P}}_{A^{\circ}_{j}}-\bm{\mathsf{P}}_{B^{\circ}_{j}})\bm{\mathsf{e}}_{j}}{\bm{\mathsf{e}}_{j}^{\top}(\bm{I}-\bm{\mathsf{P}}_{A^{\circ}_{j}})\bm{\mathsf{e}}_{j}/(n-|A^{\circ}_{j}|)},\quad
T_{j}^{*}=\frac{(\bm{\mathsf{e}}^{*}_{j})^{\top}(\bm{\mathsf{P}}_{A^{\circ}_{j}}-\bm{\mathsf{P}}_{B^{\circ}_{j}})\bm{\mathsf{e}}^{*}_{j}}{(\bm{\mathsf{e}}^{*}_{j})^{\top}(\bm{I}-\bm{\mathsf{P}}_{A^{\circ}_{j}})\bm{\mathsf{e}}^{*}_{j}/(n-|A^{\circ}_{j}|)},$
(32)
where
$A^{\circ}_{j}=\textnormal{{an}}_{S^{\circ}}(j)\cup\textnormal{{in}}_{S^{\circ}}(j)\cup\textnormal{{d}}_{S^{\circ}}(j)$,
$B^{\circ}_{j}=(\textnormal{{an}}_{S^{\circ}}(j)\cup\textnormal{{in}}_{S^{\circ}}(j))\setminus\textnormal{{d}}_{S^{\circ}}(j)$.
Then $(T_{1},\ldots,T_{p})$ are independent and $(T_{1}^{*},\ldots,T_{p}^{*})$
are conditionally independent given $\bm{\mathsf{Z}}$. Moreover,
$T^{*}_{j}/(|A^{\circ}_{j}|-|B^{\circ}_{j}|)\mid\bm{\mathsf{Z}}\overset{d}{=}T_{j}/(|A^{\circ}_{j}|-|B^{\circ}_{j}|)\sim
F_{|A^{\circ}_{j}|-|B^{\circ}_{j}|,n-|A^{\circ}_{j}|};\quad 1\leq j\leq p,$
where $F_{d_{1},d_{2}}$ denotes the F-distribution with $d_{1}$ and $d_{2}$
degrees of freedom.
### Proof of Lemma 10
Let $\bm{\mathsf{Z}}=(\bm{\mathsf{Y}},\bm{\mathsf{X}})$ as in (12). Given data
submatrix $\bm{\mathsf{Z}}_{A^{\circ}_{j}}$,
$\frac{\bm{\mathsf{e}}_{j}^{\top}(\bm{\mathsf{P}}_{A^{\circ}_{j}}-\bm{\mathsf{P}}_{B^{\circ}_{j}})\bm{\mathsf{e}}_{j}}{\sigma^{2}_{j}}\mid\bm{\mathsf{Z}}_{A^{\circ}_{j}}\sim\chi^{2}_{|A^{\circ}_{j}|-|B^{\circ}_{j}|},\quad\frac{\bm{\mathsf{e}}_{j}^{\top}(\bm{I}-\bm{\mathsf{P}}_{A^{\circ}_{j}})\bm{\mathsf{e}}_{j}}{\sigma^{2}_{j}}\mid\bm{\mathsf{Z}}_{A^{\circ}_{j}}\sim\chi^{2}_{n-|A^{\circ}_{j}|},$
and they are independent, because
$\bm{\mathsf{P}}_{A^{\circ}_{j}}-\bm{\mathsf{P}}_{B^{\circ}_{j}}$ and
$\bm{I}-\bm{\mathsf{P}}_{A^{\circ}_{j}}$ are orthonormal projection matrices,
$\bm{\mathsf{e}}_{j}\mid\bm{\mathsf{Z}}_{A^{\circ}_{j}}\sim
N(\bm{0},\sigma_{j}^{2}\bm{I}_{n})$, and
$(\bm{\mathsf{P}}_{A^{\circ}_{j}}-\bm{\mathsf{P}}_{B^{\circ}_{j}})(\bm{I}-\bm{\mathsf{P}}_{A^{\circ}_{j}})=\bm{0}$.
Then, for any $t\in\mathbb{R}$, the characteristic function
$\operatorname{\mathbb{E}}\exp(\iota tT_{j}/(|A^{0}_{j}|-|B^{0}_{j}|))$ can be
written as ($\iota$ is the imaginary unit)
$\operatorname{\mathbb{E}}\left(\operatorname{\mathbb{E}}\left(\exp(\iota
tT_{j}/(|A^{\circ}_{j}|-|B^{\circ}_{j}|))\mid\bm{\mathsf{Z}}_{A^{\circ}_{j}}\right)\right)=\operatorname{\mathbb{E}}\psi_{|A^{\circ}_{j}|-|B^{\circ}_{j}|,n-|A^{\circ}_{j}|}(t)=\psi_{|A^{\circ}_{j}|-|B^{\circ}_{j}|,n-|A^{\circ}_{j}|}(t),$
where $\psi_{|A^{\circ}_{j}|-|B^{\circ}_{j}|,n-|A^{\circ}_{j}|}$ is the
characteristic function of F-distribution with degrees of freedom
$|A^{\circ}_{j}|-|B^{\circ}_{j}|,n-|A^{\circ}_{j}|$. Hence,
$T_{j}/(|A^{\circ}_{j}|-|B^{\circ}_{j}|)\sim
F_{|A^{\circ}_{j}|-|B^{\circ}_{j}|,n-|A^{\circ}_{j}|}$. Similarly,
$T_{j}^{*}/(|A^{\circ}_{j}|-|B^{\circ}_{j}|)\sim
F_{|A^{\circ}_{j}|-|B^{\circ}_{j}|,n-|A^{\circ}_{j}|}$; $j=1,\ldots,p$.
Next, we prove independence for $\bm{T}$ and $\bm{T}^{*}$ via a peeling
argument. Let $\bm{t}=(t_{1},\ldots,t_{p})$. Let $j$ be a leaf node of the
graph $G^{\circ}$. Then $\bm{T}_{-j}\mid\bm{\mathsf{Y}}_{-j},\bm{\mathsf{X}}$
is deterministic, where $\bm{T}_{-j}$ is the subvector of $\bm{T}$ with the
$j$th component removed. The characteristic function of $\bm{t}^{\top}\bm{T}$
is
$\begin{split}\operatorname{\mathbb{E}}\exp(\iota\bm{t}^{\top}\bm{T})&=\operatorname{\mathbb{E}}\Big{(}\operatorname{\mathbb{E}}\Big{(}\exp(\iota
t_{j}T_{j})\mid\bm{\mathsf{Y}}_{-j},\bm{\mathsf{X}}\Big{)}\exp(\iota\bm{t}^{\top}_{-j}\bm{T}_{-j})\Big{)}\\\
&=\psi_{|A^{\circ}_{j}|-|B^{\circ}_{j}|,n-|A^{\circ}_{j}|}(t_{j})\operatorname{\mathbb{E}}\exp(\iota\bm{t}_{-j}^{\top}\bm{T}_{-j}),\end{split}$
(33)
where $\psi_{|A^{\circ}_{j}|-|B^{\circ}_{j}|,n-|A^{\circ}_{j}|}$ is the
characteristic function of the F-distribution with degrees of freedom
$(|A^{\circ}_{j}|-|B^{\circ}_{j}|,n-|A^{\circ}_{j}|)$. Next, let $j^{\prime}$
be a leaf node of the graph $G^{\prime}$ and apply the law of iterated
expectation again, where $G^{\prime}$ is the subgraph of $G^{\circ}$ without
node $j$. Repeat this procedure and after $p$ steps
$\operatorname{\mathbb{E}}\exp(\iota\bm{t}^{\top}\bm{T})=\prod_{j=1}^{p}\psi_{|A^{\circ}_{j}|-|B^{\circ}_{j}|,n-|A^{\circ}_{j}|}(t_{j})$,
which implies $(T_{1},\ldots,T_{p})$ are independent. Similarly, $\bm{T}^{*}$
also has independent components given $\bm{\mathsf{Z}}$ and has the same
distribution as $\bm{T}$. This completes the proof.
### B.5 Proof of Theorem 5
#### Proof of (A)
Without loss of generality, assume $M$ is sufficiently large. For any
$x\in\mathbb{R}$,
$\begin{split}&|\operatorname{\mathbb{P}}(\textnormal{Lr}\leq
x)-\operatorname{\mathbb{P}}(\textnormal{Lr}(S^{\circ},\widehat{\bm{\Sigma}})\leq
x)|\\\ &\leq\operatorname{\mathbb{E}}|\operatorname{I}(\textnormal{Lr}\leq
x)-\operatorname{I}(\textnormal{Lr}(S^{\circ},\widehat{\bm{\Sigma}})\leq
x)|\\\ &=\operatorname{\mathbb{E}}|\operatorname{I}(\textnormal{Lr}\leq
x,\widehat{S}\neq
S^{\circ})+\operatorname{I}(\textnormal{Lr}(S^{\circ},\widehat{\bm{\Sigma}})\leq
x)(\operatorname{I}(\widehat{S}=S^{\circ})-1)|\\\ &\leq
2\operatorname{\mathbb{P}}(\widehat{S}\neq S^{\circ}).\end{split}$ (34)
From (13),
$\textnormal{Lr}^{*}(S^{\circ},\widehat{\bm{\Sigma}}^{*})=\sum_{j:\textnormal{{d}}_{S^{\circ}}(j)\neq\emptyset}T_{j}^{*}$.
By Lemma 10,
$\operatorname{\mathbb{P}}(\textnormal{Lr}(S^{\circ},\widehat{\bm{\Sigma}})\leq
x)=\operatorname{\mathbb{P}}(\textnormal{Lr}^{*}(S^{\circ},\widehat{\bm{\Sigma}}^{*})\leq
x\mid\bm{\mathsf{Z}})$. Note that
$\textnormal{Lr}^{*}=\textnormal{Lr}^{*}(\widehat{S}^{*},\widehat{\bm{\Sigma}}^{*})$.
Then for any $x\in\mathbb{R}$,
$\begin{split}&|\operatorname{\mathbb{P}}(\textnormal{Lr}^{*}\leq
x\mid\bm{\mathsf{Z}})-\operatorname{\mathbb{P}}(\textnormal{Lr}(S^{\circ},\widehat{\bm{\Sigma}})\leq
x)|\\\ &=|\operatorname{\mathbb{P}}(\textnormal{Lr}^{*}\leq
x,\widehat{S}^{*}\neq
S^{\circ}\mid\bm{\mathsf{Z}})+\operatorname{\mathbb{P}}(\textnormal{Lr}^{*}\leq
x,\widehat{S}^{*}=S^{\circ}\mid\bm{\mathsf{Z}})-\operatorname{\mathbb{P}}(\textnormal{Lr}^{*}(S^{\circ},\widehat{\bm{\Sigma}}^{*})\leq
x\mid\bm{\mathsf{Z}})|\\\
&\leq|\operatorname{\mathbb{P}}(\textnormal{Lr}^{*}\leq x,\widehat{S}^{*}\neq
S^{\circ}\mid\bm{\mathsf{Z}})|+|\operatorname{\mathbb{P}}(\textnormal{Lr}^{*}(S^{\circ},\widehat{\bm{\Sigma}}^{*})\leq
x,\widehat{S}^{*}\neq S^{\circ}\mid\bm{\mathsf{Z}})|\\\ &\leq
2\operatorname{\mathbb{P}}(\widehat{S}^{*}\neq
S^{\circ}\mid\bm{\mathsf{Z}}).\end{split}$ (35)
Since (34) and (35) hold uniformly in $x$, we have
$\sup_{x\in\mathbb{R}}|\operatorname{\mathbb{P}}(\textnormal{Lr}\leq
x)-\operatorname{\mathbb{P}}(\textnormal{Lr}^{*}\leq
x\mid\bm{\mathsf{Z}})|\leq 2\operatorname{\mathbb{P}}(\widehat{S}\neq
S^{\circ})+2\operatorname{\mathbb{P}}(\widehat{S}^{*}\neq
S^{\circ}\mid\bm{\mathsf{Z}}).$
By Theorem 4, we have $\operatorname{\mathbb{P}}(\widehat{S}\neq S^{\circ})\to
0$, which holds uniformly for all $\bm{\theta}$ which satisfy the Assumptions
1C-5. For $\operatorname{\mathbb{P}}(\widehat{S}^{*}\neq
S^{\circ}\mid\bm{\mathsf{Z}})$, the error terms of (7) in the perturbed data
$\bm{\mathsf{Z}}^{*}$ are rescaled with
$\Omega_{jj}^{-1}+\widehat{\sigma}^{2}_{j}\leq 2\Omega_{jj}^{-1}$. By Theorem
4, $\operatorname{\mathbb{P}}(\widehat{S}^{*}\neq
S^{\circ})=\operatorname{\mathbb{E}}\operatorname{\mathbb{P}}(\widehat{S}^{*}\neq
S^{\circ}\mid\bm{\mathsf{Z}})\to 0$, which implies
$\operatorname{\mathbb{P}}(\widehat{S}^{*}\neq
S^{\circ}\mid\bm{\mathsf{Z}})\overset{p}{\longrightarrow}0$ as
$n\rightarrow\infty$ by the Markov inequality. Consequently,
$\sup_{x\in\mathbb{R}}|\operatorname{\mathbb{P}}(\textnormal{Lr}\leq
x)-\operatorname{\mathbb{P}}(\textnormal{Lr}^{*}\leq x\mid\bm{\mathsf{Z}})|\to
0$. For $|\mathcal{D}|=0$, $\operatorname{\mathbb{P}}(\textnormal{Lr}=0)\to
1$, $\operatorname{\mathbb{P}}(\textnormal{Lr}^{*}=0\mid\bm{\mathsf{Z}})\to
1$, and $\operatorname{\mathbb{P}}(\textnormal{Pval}=1)\to 1$. For
$|\mathcal{D}|>0$,
$P(\textnormal{Lr}^{*}\geq\textnormal{Lr}\mid\bm{\mathsf{Z}})\to\textnormal{Unif}(0,1)$
and $\operatorname{\mathbb{P}}(\textnormal{Pval}<\alpha)\to\alpha$. This
completes the proof of (A).
#### Proof of (B)
Let
$\textnormal{Pval}_{k}=M^{-1}\sum_{m=1}^{M}\operatorname{I}(\text{Lr}^{*}_{k,m}\geq\text{Lr})$,
the p-value of sub-hypothesis $\text{H}_{0,k}$. For
$|\mathcal{D}|<|\mathcal{H}|$, there exists an edge
$(i_{k},j_{k})\in\mathcal{H}$ but $(i_{k},j_{k})\notin\mathcal{D}$. Then by
(A), $\operatorname{\mathbb{P}}(\textnormal{Pval}=\textnormal{Pval}_{k}=1)\to
1$. For $|\mathcal{D}|=|\mathcal{H}|$, note that as $n,M\to\infty$,
$\operatorname{\mathbb{P}}\Big{(}\textnormal{Pval}<\alpha\Big{)}=\operatorname{\mathbb{P}}\Big{(}\textnormal{Pval}_{1}<\alpha,\ldots,\textnormal{Pval}_{|\mathcal{H}|}<\alpha\Big{)}\leq\operatorname{\mathbb{P}}\Big{(}\textnormal{Pval}_{1}<\alpha\Big{)}\to\alpha.$
Now, define a sequence $\bm{U}_{\mathcal{H}}^{(r)}$; $r=1,2,\ldots$ such that
$\bm{U}_{(i_{1},j_{1})}^{(r)}=0$ and
$\min_{k=2}^{|\mathcal{H}|}|\bm{U}_{(i_{k},j_{k})}^{(r)}|\geq c>0$. Thus,
$\bm{U}_{\mathcal{H}}^{(r)}$; $r=1,2,\ldots$ satisfies $H_{0}$. By Proposition
7, $\text{Pval}_{k}\overset{p}{\longrightarrow}0$ for $k\geq 2$ as
$r\to\infty$ . Hence,
$\limsup_{\begin{subarray}{c}n\to\infty\\\ \bm{U}_{\mathcal{H}}^{(r)}\text{
satisfies
H}_{0}\end{subarray}}\operatorname{\mathbb{P}}_{\bm{\theta}^{(r)}}\Big{(}\text{Pval}<\alpha\Big{)}\geq\sup_{r}\lim_{n,p,q\to\infty}\operatorname{\mathbb{P}}_{\bm{\theta}^{(r)}}\Big{(}\text{Pval}<\alpha\Big{)}=\alpha.$
This completes the proof.
### B.6 Proof of Proposition 6
Since $\operatorname{\mathbb{P}}(\widehat{S}=S^{\circ})\to 1$, it suffices to
consider $\text{Lr}(S^{\circ},\widehat{\bm{\Sigma}})$. For $|\mathcal{D}|=0$,
we have $\operatorname{\mathbb{P}}(\text{Lr}=0)\to 1$. Now, assume
$|\mathcal{D}|>0$. Then
$\begin{split}2\text{Lr}(S^{\circ},\widehat{\bm{\Sigma}})=\underbrace{\sum_{j:\textnormal{{d}}_{S^{\circ}}(j)\neq\emptyset}\frac{\bm{\mathsf{e}}_{j}^{\top}(\bm{\mathsf{P}}_{A^{\circ}_{j}}-\bm{\mathsf{P}}_{B^{\circ}_{j}})\bm{\mathsf{e}}_{j}}{\sigma^{2}_{j}}}_{R_{1}}+\underbrace{\sum_{j:\textnormal{{d}}_{S^{\circ}}(j)\neq\emptyset}\left(\frac{\sigma^{2}_{j}}{\widehat{\sigma}^{2}_{j}}-1\right)\frac{\bm{\mathsf{e}}_{j}^{\top}(\bm{\mathsf{P}}_{A^{\circ}_{j}}-\bm{\mathsf{P}}_{B^{\circ}_{j}})\bm{\mathsf{e}}_{j}}{\sigma_{j}^{2}}}_{R_{2}}.\end{split}$
To derive the asymptotic distribution of $R_{1}$, we apply the peeling
strategy with law of iterated expectation as in the proof of Lemma 10. Then we
have
$\bm{\mathsf{e}}_{j}^{\top}(\bm{\mathsf{P}}_{A^{0}_{j}}-\bm{\mathsf{P}}_{B^{0}_{j}})\bm{\mathsf{e}}_{j}/\sigma^{2}_{j}$;
$\textnormal{{d}}_{S^{\circ}}(j)\neq\emptyset$ are independent. Therefore,
$R_{1}\sim\chi^{2}_{|\mathcal{D}|}$.
To bound $R_{2}$, we apply Lemma 1 of Laurent and Massart (2000). By
Assumption 5,
$\begin{split}\operatorname{\mathbb{P}}\left(\max_{j:\textnormal{{d}}_{S^{\circ}}(j)\neq\emptyset}\left|\frac{\widehat{\sigma}^{2}_{j}}{\sigma^{2}_{j}}-1\right|\geq
4\sqrt{\frac{\log|\mathcal{D}|}{(1-\rho)n}}+8\frac{\log|\mathcal{D}|}{(1-\rho)n}\right)&\leq
2\exp(-\log|\mathcal{D}|).\end{split}$
Hence,
$\max_{j:\textnormal{{d}}_{S^{\circ}}(j)\neq\emptyset}\left|\frac{\sigma^{2}_{j}}{\widehat{\sigma}_{j}^{2}}-1\right|\leq
8\sqrt{\frac{\log|\mathcal{D}|}{(1-\rho)n}}$ with probability tending one as
$n,p\rightarrow\infty$. Thus
$|R_{2}|\leq|R_{1}|\max_{j:\textnormal{{d}}_{S^{\circ}}(j)\neq\emptyset}\left|\frac{\sigma^{2}_{j}}{\widehat{\sigma}_{j}^{2}}-1\right|\leq
O_{\operatorname{\mathbb{P}}}\left(|\mathcal{D}|\sqrt{\frac{\log|\mathcal{D}|}{n}}\right).$
Consequently, as $n,p,q\rightarrow\infty$, if $|\mathcal{D}|>0$ is fixed, then
$2\text{Lr}\overset{d}{\longrightarrow}\chi^{2}_{|\mathcal{D}|}$; if
$|\mathcal{D}|\to\infty$, then
$(2|\mathcal{D}|)^{-1/2}(2\text{Lr}-|\mathcal{D}|)\overset{d}{\longrightarrow}N(0,1)$.
This completes the proof.
### B.7 Proof of Proposition 7
Let $\bm{\theta}^{(n)}=(\bm{U}^{\circ}+\bm{\Delta},\bm{W}^{\circ})$. Then
$L(\bm{\theta}^{(n)},\bm{\Sigma}^{\circ})-L(\bm{\theta}^{\circ},\bm{\Sigma}^{\circ})=\sum_{j:\textnormal{{d}}^{\circ}(j)\neq\emptyset}\Big{(}\sqrt{n}\bm{\eta}_{j}^{\top}\bm{\Delta}_{\cdot
j}-\frac{1}{2}\sqrt{n}\bm{\Delta}_{\cdot
j}\Big{(}n^{-1}\bm{\mathsf{Y}}^{\top}_{\textnormal{{d}}^{\circ}(j)}\bm{\mathsf{Y}}_{\textnormal{{d}}^{\circ}(j)}\Big{)}\sqrt{n}\bm{\Delta}_{\cdot
j}\Big{)},\\\ $
where
$\bm{\eta}_{j}=(n^{-1/2}\bm{\mathsf{Y}}_{\textnormal{{d}}^{\circ}(j)}^{\top}\bm{\mathsf{e}}_{j},\bm{0}_{|\textnormal{{d}}^{\circ}(j)^{c}|})\in\mathbb{R}^{p}$.
For the likelihood ratio, it suffices to consider
$\textnormal{Lr}(S^{\circ},\widehat{\bm{\Sigma}})$, since
$P(\widehat{S}=S^{\circ})\to 1$. Under $H_{0}$, we have
$2\text{Lr}=\sum_{j:\textnormal{{d}}^{\circ}(j)\neq\emptyset}T_{j}$. Note that
$T_{j}=\sum_{r=1}^{|\textnormal{{d}}^{\circ}(j)|}(\bm{\mathsf{q}}_{j,r}^{\top}\bm{\mathsf{e}}_{j})^{2}+o_{\operatorname{\mathbb{P}}}(1)$
by Proposition 6, where
$\bm{\mathsf{P}}_{A^{\circ}_{j}}-\bm{\mathsf{P}}_{B^{\circ}_{j}}=\sum_{r=1}^{|\textnormal{{d}}^{\circ}(j)|}\bm{\mathsf{q}}_{j,r}\bm{\mathsf{q}}_{j,r}^{\top}$.
Let
$\bm{\mathsf{Q}}_{j}=(\bm{\mathsf{q}}_{j,1},\ldots,\bm{\mathsf{q}}_{j,|\textnormal{{d}}^{\circ}(j)|})\in\mathbb{R}^{n\times|\textnormal{{d}}^{\circ}(j)|}$.
Then
$\begin{pmatrix}\bm{\mathsf{Q}}_{j}^{\top}\bm{\mathsf{e}}_{j}\\\
\bm{\eta}_{j}\end{pmatrix}\ \vline\
\bm{\mathsf{Y}}_{\textnormal{{an}}(j)},\bm{\mathsf{X}}\sim
N\left(\bm{0},\begin{pmatrix}\bm{I}_{|\textnormal{{d}}^{\circ}(j)|}&n^{-1/2}\bm{\mathsf{Q}}_{j}^{\top}\bm{\mathsf{Y}}_{\textnormal{{d}}^{\circ}(j)}\\\
n^{-1/2}\bm{\mathsf{Y}}_{\textnormal{{d}}^{\circ}(j)}^{\top}\bm{\mathsf{Q}}_{j}&n^{-1}\bm{\mathsf{Y}}^{\top}_{\textnormal{{d}}^{\circ}(j)}\bm{\mathsf{Y}}_{\textnormal{{d}}^{\circ}(j)}\end{pmatrix}\right),$
$\bm{\mathsf{Q}}^{\top}_{j}\bm{\mathsf{e}}_{j}\mid\bm{\eta}_{j},\bm{\mathsf{Y}}_{\textnormal{{an}}(j)},\bm{\mathsf{X}}\sim
N\Big{(}n^{-1/2}\bm{\mathsf{Y}}_{\textnormal{{d}}^{\circ}(j)}^{\top}\bm{\mathsf{Q}}_{j}\bm{\eta}_{j},\bm{\mathsf{Y}}_{\textnormal{{d}}^{\circ}(j)}^{\top}(\bm{I}_{n}-\bm{\mathsf{Q}}_{j}\bm{\mathsf{Q}}_{j}^{\top})\bm{\mathsf{Y}}_{\textnormal{{d}}^{\circ}(j)}\Big{)}.$
Next, let $k$ be a leaf node of the graph $G^{\circ}$. For fixed
$|\mathcal{D}|>0$, after change of measure,
$\begin{split}&\beta(\bm{\theta}^{\circ},\bm{\Delta})\\\
&\geq\liminf_{n\to\infty}\operatorname{\mathbb{E}}_{\bm{\theta}^{\circ}}\left(\operatorname{I}\left(\sum_{j=1}^{p}\|\bm{\mathsf{Q}}^{\top}\bm{\mathsf{e}}_{j}\|^{2}_{2}>\chi^{2}_{|\mathcal{D}|,1-\alpha}\right)\exp(L(\bm{\theta}^{(n)},\bm{\Sigma}^{\circ})-L(\bm{\theta}^{\circ},\bm{\Sigma}^{\circ}))\right)\\\ |
# Deployment of Energy-Efficient Deep Learning Models on Cortex-M based
Microcontrollers using Deep Compression
Mark Deutel1, Philipp Woller2, Christopher Mutschler2, and Jürgen Teich4 1
Mark Deutel is with Fraunhofer IIS, Fraunhofer Institute for Integrated
Circuits IIS, Nuremberg, Germany and Friedrich-Alexander-Universität Erlangen-
Nürnberg (FAU), Erlangen, Germany<EMAIL_ADDRESS>Phillip Woller and
Christopher Mutschler are with the Fraunhofer IIS, Fraunhofer Institute for
Integrated Circuits IIS, Nuremberg, Germany
<EMAIL_ADDRESS>Jürgen Teich is with Friedrich-
Alexander-Universität Erlangen-Nürnberg (FAU), Erlangen, Germany
<EMAIL_ADDRESS>
###### Abstract
Large Deep Neural Networks (DNNs) are the backbone of today’s artificial
intelligence due to their ability to make accurate predictions when being
trained on huge datasets. With advancing technologies, such as the Internet of
Things, interpreting large quantities of data generated by sensors is becoming
an increasingly important task. However, in many applications not only the
predictive performance but also the energy consumption of deep learning models
is of major interest.
This paper investigates the efficient deployment of deep learning models on
resource-constrained microcontroller architectures via network compression. We
present a methodology for the systematic exploration of different DNN pruning,
quantization, and deployment strategies, targeting different ARM Cortex-M
based low-power systems. The exploration allows to analyze trade-offs between
key metrics such as accuracy, memory consumption, execution time, and power
consumption. We discuss experimental results on three different DNN
architectures and show that we can compress them to below 10% of their
original parameter count before their predictive quality decreases. This also
allows us to deploy and evaluate them on Cortex-M based microcontrollers.
###### Index Terms:
Deep Neural Networks, Deep Compression, Network Pruning, Weight Quantization,
Microcontrollers
## I Introduction
Deep Neural Networks (DNNs) became predominant in many applications that
require autonomous decision-making based on environmental information,
including audio recognition [1, 2], image classification [3, 4], or human
activity monitoring [5]. DNNs are beneficial as they are easy to set up and as
they can be trained to detect correlations even when they are confronted with
high-dimensional data.
However, the execution of DNNs is energy-, resource-, and time-expensive [6,
7]. In situations where the trade-off between resource constraints, execution
time, and predictive quality is key, DNNs often struggle to compete with
classical machine learning approaches [8]. However, with trends like smart
devices and the internet of things (IoT), the demand and interest in deploying
DNNs on microcontrollers grows.
Figure 1: Overview of our methodology.
Deep compression is a relatively young research area that deals with the
compression of DNNs. Prominent techniques include DNN graph pruning [9],
weight quantization [10, 11], and subspace methods [12]. Their goal is to
reduce the resource footprint of a DNN by reducing the number of trainable
weights and computational complexity while preserving the original predictive
performance.
Based on these principles different DNN compression pipelines have been
proposed. Most noticeably, Han et al. [13] who proposed a pipeline combining
network pruning, integer quantization, and Huffman encoding. Others focus on
quantization during network training [10] or on structure-based pruning. This
allows for an immediate removal of pruned weights [14, 15]. However, such
well-established frameworks only trade compression over predictive accuracy
but do not explicitly target energy-efficiency and architecture-specific
constraints like memory availability and processing speed that play an
important role in many embedded applications.
This paper proposes a methodology to systematically train and deploy DNN
architectures on Cortex-M-based microcontrollers. We introduce an automated
pipeline that covers application-specific DNN training and compression, and
that combines it with a target architecture-specific deployment, see Fig. 1.
Our proposed pipeline is composed of two major building blocks. First, from an
application-specific viewpoint we systematically explore compression
techniques, i.e., network pruning and weight quantization, and configurations
during the training of DNNs. Second, incorporating an architecture-specific
view, we realize the mapping from a high-level graph-based DNN representation
to low level code. This step involves an offline code generator and a runtime
library. The former takes care of the data layout, plans memory allocation and
emits executable code while the latter provides implementations of common DNN
layers used by the generated code. Novel methods implemented in our proposed
pipeline include ahead-of-time code generation and memory allocation
scheduling, which eliminates the need for any form of network interpretation
or dynamic memory management at runtime, and the exploitation of sparse
matrices generated as part of our pruning techniques using the _Compressed
Column Storage (CCS)_ format [16].
In our experiments we evaluate both compression and deployment for three
common DNN architectures, i.e., AlexNet [17], ResNet [3] and LeNet [18]. Our
objective is a thorough evaluation of the relation between compressed DNNs and
their predictive quality. In contrast to previous work our results do not only
focus on deployment or compression alone, but provide detailed insight into
the relationship between different pruning, quantization, and deployment
strategies when applied in combination. Furthermore, we deployed the
compressed models on three target systems and discuss their memory
consumption, execution time, and power consumption.
The rest of this paper is structured as follows. Sec. II discusses related
work. Secs. III and IV provide details about our compression and deployment
pipeline. Sec. V discusses our experimental results. Sec. VII concludes.
## II Related Work
Existing research predominately compresses DNNs via network pruning and weight
quantization. These techniques are well understood as research has been
conducted exploring the effects of pruning and quantization on a network’s
predictive performance [19, 20]. However, when deploying applications for
embedded targets, they are defined by the constraints imposed by the platforms
they use. As a result, the suitability of DNN models for deployment on
microcontrollers is not only determined by their accuracy but also by their
memory footprint and inference time. Therefore, this work extends existing
findings by analysing the effects of DNN compression not only on accuracy but
also on relevant deployment metrics, i.e. memory consumption, latency, and
energy consumption.
Furthermore, research that focuses on the deployment of DNNs to
microcontrollers is often published in an application-oriented way, e.g., to
realize real-time drowsiness detection [21] or to perform motor fault
diagnosis [22]. Those platforms do not serve as general purpose frameworks as
they are tightly optimized to the particular application requirement and as
they do not generalize to a broader set of target architectures.
Nevertheless, recent scientific work has provided some insight into
generalized deployment of DNNs on microcontrollers. The approach most closely
related to our proposed methodology is MCUNet [23]. Similar to our pipeline
the authors describe a two stage process to seamlessly combine model design
(TinyNAS) with an inference engine (TinyEngine). Still MCUNet differs from our
approach in the way it generates suitable DNN candidates for deployment. To
find networks that meet target platform constraints, MCUNet focuses on neural
architecture search (NAS) [24] while our framework starts from well-known
existing DNN architectures and then dynamically scales them down during their
training using pruning and quantization techniques.
A more general approach to a deployment framework for microcontrollers is
_tfl-micro_ [25], which supports the execution of quantized tensorflow lite
models on ARM Cortex-M-based hardware using ARM’s _CMSIS_ library. However,
this also limits the framework as it requires the usage of _tensorflow (TF)_
for model training and also only supports a subset of features implemented in
TF.
Besides that there are also commercial frameworks focusing on embedded
platforms. Noticeable examples are STM’s _X-CUBE-AI_
111https://www.st.com/en/embedded-software/x-cube-ai.html, that allows for the
automatic conversion of pre-trained AI algorithms to deployable code as well
as web-service based end-to-end solutions like _Edge Impulse_
222https://www.edgeimpulse.com/ or _SensiML_ 333https://sensiml.com/. However,
such commercial frameworks are often either black boxes (e.g., X-CUBE-AI) or
they base themselves on already existing underlying solutions (e.g., _Edge
Impulse_ uses _tfl-micro_) and their limitations.
## III Compression and Deployment Pipeline
Our pipeline is fully integrated and seamlessly covers the complete DNN
training and deployment process. Our methodology uses both network pruning
(Sec. III-A) and weight quantization (Sec. III-B) which can both be controlled
via a set of additional _hyperparameters_. Furthermore, the trained and
compressed DNNs are directly converted from their graph-based representation
to executable architecture-specific program code (see Sec. IV). As a result,
our pipeline can easily be integrated with existing meta-heuristic
optimization frameworks (e.g. Optuna [26]) to conduct automated design space
exploration.
### III-A Network Pruning
Our pipeline implements configurable elements for network pruning, i.e., (1)
pruning techniques, (2) pruning heuristics, and (3) pruning schedule. which we
describe in the following.
Pruning techniques. Pruning DNNs by removing parameters has been proposed
early [27, 9]. While initially being introduced to improve generalization and
convergence it recently became a standard size reduction technique with no or
low cost of accuracy. Our pipeline implements _element-wise pruning_ and
_structural pruning_. Element-wise pruning removes connections from a DNN’s
compute graph, i.e., parameters of the network are set to zero. Hence, these
parameters do no longer influence the training error and are removed from the
scope of the optimizer that trains the network. Structural pruning sets whole
structures of parameters to zero. This has shown to be very efficient for
pruning filters [14] or channels [15] of 2D-convolutional layers but it can
analogously also be applied to rows and columns of linear layers. Its major
benefit is the removal of complete structures from the weight tensors at once,
which results in a considerable immediate reductions of parameters (which is
in contrast to element-wise pruning that only creates sparse weight tensors).
Pruning heuristics. A critical aspect of pruning is the selection of elements
or structures that, when removed, have the least impact on predictive
performance. Oracle pruning [28] finds an optimal selection by removing every
single structure and element of a network before evaluating its impact on the
prediction quality. In practical applications this approach cannot be applied
as it is too resource- and time-consuming. Fortunately, there have been
proposed a number of heuristics that approximate optimal element-wise or
structural pruning. In our framework we implemented many popular approaches
that are based on different criteria such as magnitude [29], L-norm [14],
gradient [28] or percentage of zeros found in activations [30] to rank
parameters or parameter structures by their approximated importance.
Pruning schedules. The pruning schedule determines when, how often, and to
what extent a DNN will be pruned during training. We implement two well-known
approaches: _One-Shot Pruning_ [9] and _Iterative Pruning_. One-shot pruning
first trains a DNN until it achieves a reasonable accuracy on the test
dataset, and then prunes the DNN (optionally followed by a few epochs of re-
training). Iterative pruning prunes a DNN over the course of training, which
allows for an interleaved re-training. Hence, not all weights are removed at
the same time but step by step over several pruning iterations (finally
enforcing maximal sparsity). We implemented _Automated Gradual Pruning (AGP)_
[31], which gradually increases the number of pruned weights $s_{t}$ starting
at $t_{0}$ from an initial sparsity $s_{i}$ to a final sparsity $s_{f}$ over
$n$ steps:
$s_{t}=s_{f}+(s_{i}-s_{f})\left(1-\frac{t-t_{0}}{n\Delta t}\right)^{3},\
t\in\left\\{t_{0},\dots,t_{0}+n\Delta t\right\\}.$ (1)
### III-B Weight Quantization
Quantization reduces the numerical resolution of the parameters and their
computation. This not only cuts down the memory footprint but also the
computational complexity of a DNN. However, as parameter quantization comes
with an additional error on the predictive performance a major challenge lies
in the selection of a good trade-off between predictive quality and parameter
resolution.
Our framework uses an _affine mapping_ scheme that transforms an original
floating-point parameter into an 8-bit unsigned integer444See also [32] and
https://onnxruntime.ai/docs/how-to/quantization.html.. We apply a function
$f(x)$ in combination with additional sets of trainable scale and zero point
parameters:
$\begin{split}f(x)&=g\left(\left\lfloor\frac{x}{s}\right\rceil+zp\right),\\\
s&=\frac{max_{data}-min_{data}}{255},\ 0\leq zp\leq 255,\end{split}$ (2)
where $g(x)$ is the clamp-function to avoid data type overflows:
$g(x)=\left\\{\begin{array}[]{rcl}x&\mbox{if}&0\leq x\leq 255\\\
255&\mbox{if}&x>255\\\ 0&\mbox{if}&x<0.\end{array}\right.$ (3)
The scale parameter $s$ defines the step size of the quantization, which is
calculated as the ratio between the span in which values are distributed in
the original floating-point space and the span covered by the quantized
integer space. The zero point parameter $zp$ denotes an exact point in
quantization space that represents zero in the original floating-point space.
The two parameters can be defined either per tensor or per structure.
Quantization can not only be applied to weight tensors but also to activation
tensors. We refer to this as _full integer quantization_. During execution
most computations can then be performed in integer instead of floating-point
space which is beneficial for target systems that are not equipped with
floating-point units. We give an example for applying full-integer
quantization to matrix-multiplications. The general form is defined as:
$\begin{split}c_{ij}=\sum_{k=0}^{n}a_{ik}\cdot b_{ki},\forall
i\in\\{0,\dots,m\\},\forall j\in\\{0,\dots,p\\},\end{split}$ (4)
where the first line describes how the elements of a matrix $C$ are calculated
from the elements of a $m\times n$ matrix $A$ and a $n\times p$ matrix $B$. In
a fully-quantized DNN, both matrices $A$ and $B$ contain integer values and we
first must de-quantize them by rearranging Equation 2 before we multiply them.
As the resulting matrix $C$ is represented in the un-quantized space, we have
to quantize it by applying Eq. 2 again. By substituting and rearranging the
previous computations we obtain
$c_{ij}=g\left(zp_{c}+\left\lfloor{\left(\frac{s_{a}\cdot
s_{b}}{s_{c}}\right)\sum_{k=0}^{n}(a_{ik}-zp_{a})(b_{ik}-zp_{b})}\right\rceil\right).$
(5)
Note that only the scale parameters $\\{s_{a},s_{b},s_{c}\\}\in\mathbb{R}$
while all other parameters $\in\mathbb{N}_{0}$.
Our pipeline implements two popular ways of determining at which point
quantization is applied to a DNN. The first method quantizes as a post process
(PPQ) [11, 33], i.e., after training has finished, and the second method
integrates quantization into the training loop. The latter is denoted by
_Quantization-aware Training (QAT)_ [10]. Both techniques come with their
advantages and disadvantages: PPQ is extremely easy to integrate as it can be
performed completely decoupled from a DNN’s training process and does not
requires any invasive changes to a DNN’s architecture (i.e., no re-training to
fine-tune quantization parameters). However, this usually comes at the cost of
a larger error introduced by quantization as the required scale and zero point
parameters are only roughly approximated. In contrast, QAT adapts quantization
parameters as part of a DNN’s training process and can hence yield better
results. However, QAT only works properly with extensive network augmentation,
which leads to a more complex and computationally expensive training process.
## IV Architecture-Specific Deployment
Our pipeline provides a deployment framework for targeting microcontrollers,
see Fig. 2. We call this framework _dnnruntime_. It uses a platform-
independent, offline, and ahead of time conversion tool together with a
runtime library. The conversion tool maps pre-trained DNNs stored in the ONNX
format to C code (Sec. IV-A), while the runtime library implements platform-
specific DNN operators that are subsequently used by the code emitted from the
conversion tool (Sec. IV-B). Our implementation is novel in the way that it
exploits static properties of trained DNNs (i.e. fixed layer configurations
and parameters). and therefore removes the necessity of interpreting the DNN
at runtime. This includes dynamic allocation of memory for intermediate
tensors which can be simulated offline allowing heap allocation to be
conducted at compile time. This not only decreases the computational overhead
at runtime but also allows metrics like simulated memory consumption to be
directly fed back into the overall optimization process without having to
evaluate the model on the target system.
Figure 2: Overview of the process implemented by our conversion tool, mapping
trained ONNX DNN models to C-Code and finally compiling them to deployable
binaries.
### IV-A Conversion Tool
The main functionality of the conversion tool is to generate ANSI C code based
on a given ONNX model of the DNN to be deployed. This involves two steps: (1)
parsing and converting the model to an intermediate representation, and (2)
using this representation to determine a suitable data format, simulate memory
allocation and generate an implementation describing the model’s structure in
code.
The ONNX format stores a DNN’s compute graph as a directed, acyclic graph with
a set of fixed inputs and outputs. Every node in the graph represents an
operation and can have multiple incoming and outgoing edges assigned to it.
Edges describe the dataflow during the DNN’s execution. Besides that, based on
the type of operation a node represents, additional static parameters tensors
can also be assigned to it.
#### IV-A1 Mapping ONNX to target-specific intermediate format
We first map a given ONNX representation to an architecture-specific
intermediate format that can be used to emit program code later on. This
involves three consecutive steps.
First, we concatenate the static tensors of all nodes into a byte-stream. The
single elements of each tensor are stored in the stream by using a little-
endian byte order as this is the default memory format on ARM architectures
(of course, this can be modified easily). Additionally, we add padding bytes
where necessary to avoid triggering the memory protection unit (MPU) when
accessing tensor data at runtime. Afterwards, we generate descriptor
structures containing the location of each tensor in the byte-stream and
additional metadata such as data types and tensor shapes. Sparse tensors are
handled as edge cases as they are generated by element pruning during our
pipeline’s compression stage. To reduce memory usage, our tool applies a
conversion from the original full-sized layout of the tensors to a more
compact _Compressed Row Storage (CRS)_ [16] layout, see Fig. 3 for an example.
CRS reduces the memory footprint, allows for an optimized implementation of
matrix-vector products, and does not pose any requirements on how sparsity is
distributed inside a tensor (hence, element pruning during compression can
ignore the subsequent space-saving storage of pruned tensors). A disadvantage
of CRS is that it can only be applied to 2D-tensors (matrices), which means
that we need to map higher dimensional weight tensors of convolutions to 2D
space before processing them.
$\displaystyle A=\begin{bmatrix}10&0&0&0&1\\\ 0&7&0&2&0\\\ 0&0&8&0&0\\\
14&0&0&0&6\end{bmatrix}$
values | 10 | 1 | 7 | 2 | 8 | 14 | 6
---|---|---|---|---|---|---|---
col_ind | 0 | 4 | 1 | 3 | 2 | 0 | 4
row_ptr | 0 | 2 | 4 | 5 | 7 | |
Figure 3: Conversion of an asymmetric matrix $A$ (left) into its CRS
representation (right): we emit three smaller arrays to the bytes-stream
instead of one. The first one contains all non-zero values, the second one
contains column indices, and the third one contains row pointers.
Second, the conversion tool generates descriptor structures for all dynamic
activation tensors. This is more compilicated than it is for static parameter
tensors as activation tensors are only represented in the ONNX model’s compute
graph as edges. Edges are not required to provide any meta information like
data types or shapes. However, this information is mandatory for our
conversion tool. Hence, we implement a process called _shape inference_. The
idea is to trace the execution of a DNN through its compute graph from input
to output nodes, and to use these traces to infer the shapes and types of
intermediate tensors assigned to inner edges.
Third, we parse and interpret all operator nodes in the ONNX compute graph and
bring them into a topological and serialized order. This information is then
used during code generation to determine the execution order of operations.
#### IV-A2 Code Generation
Using the intermediate representation generated by the first step the
conversion tool can emit code. We start by estimating the minimal heap size
required for storing activation tensors. This information can be queried
offline as once the training of a DNN is complete, its structure and
dimensionality remains unchanged throughout its lifetime. Using the minimal
heap size, we define a fixed-size memory balloon at compile time (eliminating
the need for dynamic memory management at runtime). A naive approach that
estimates the size of this balloon calculates the product of the shapes of all
activation tensors and multiplies them with the byte sizes of their respective
data types. However, this is not space-efficient as usually the lifetimes of
these tensors are rather short. Hence, parts of the heap memory can be reused
for multiple tensors during an inference pass. This may have a big impact on
the amount of memory required.
We take advantage of this by implementing an offline _memory planning
algorithm_ based on graph tracing and using a first-fit allocation strategy
therein. We estimate optimal heap re-usage in two steps: First, based on
incoming and outgoing edges of nodes (i.e., operators) in the input model, the
algorithm creates two lists per operator: The first list contains all tensors
that have to be allocated for that operator (i.e., allocation list) and the
second list contains all previously allocated tensors that can be discarded
(i.e., release list). Second, the algorithm proceeds iterating through the
sequence of operators starting with an empty, infinitely sized memory balloon.
For each operator, it first iterates the tensors in the corresponding free
list and marks their space in the memory balloon as unoccupied. After that, it
iterates the allocate list and tries to reserve pieces of memory based on the
shapes and data types of the tensors. To find suitable locations in the
balloon, the algorithm compares the required sizes with available segments of
free memory starting from the beginning (i.e., first fit). Once it found
suitable pieces of memory, it marks them as allocated in the balloon and stops
searching. During all steps, the algorithm keeps track of the maximum size of
the memory balloon.
The emitted code implements a two-function API: The first function allows to
setup the converted DNN and the second function executes an inference given an
input sample. The latter is implemented based on the list of topologically
sorted ONNX operator nodes stored in the previously generated intermediate
representation. For each operator a function call is emitted. These functions
are implemented by the _runtime library_. To give context to these function
and pass intermediate results between them, we provide references to constant
tensor descriptor structures generated as part of the intermediate
representation. All static data associated with weight tensor descriptors is
stored in a byte-array in the intermediate representation. Therefore, a
constant C-array (i.e. flash memory) containing all the data is emitted,
accordingly.
The amount of random access memory required for intermediate activation
tensors is based on the minimal memory balloon previously estimated by our
_memory planning algorithm_. Hence our tool emits another accordingly sized
zero initialized non-constant C-array (i.e. heap memory).
### IV-B Runtime Library
To perform the operations described by the input ONNX model, the code emitted
by our conversion tool relies on additional DNN operator functionality that we
implement by a runtime library. Currently, this includes operators such as
convolutions, linear transformations, batch-normalization, pooling operations,
and activation functions. All our implementations follow the ONNX operator
specification555https://github.com/onnx/onnx/blob/master/docs/Operators.md.
Where required, we also implement quantized versions of these operators.
Based on the target platform there are different possibilities to optimize the
execution of DNNs. During profile tests we found that most execution time is
spent on computing convolutions or matrix-vector products. Hence, an optimal
implementation of these operator types yields significant improvements in both
resource consumption and execution time. Less crucial but still significantly,
some operations can be removed from a DNN’s compute graph by applying graph
optimization, which we apply after DNN training and compression. Notable
optimization techniques include _batch normalization folding_ [10] and the
fusing of ReLU activation functions into preceding quantized linear or
convolutional operations [10].
For our experiments we focus on Cortex-M0+ and Cortex-M4 processors. This is
why our implementation heavily makes use of the information of these processor
architectures. A major algorithmic optimization that we apply is to unroll 2D
convolutions into more CPU- and memory-friendly matrix-vector products
(_im2col mapping_). Hence, during mapping we rearrange both the input tensors
and the parameter tensors of convolutions. This is a common approach used in
digital signal processing.666e.g., Matlab® provides convmtx() and convmtx2()
for this task.. In addition, this mapping also enables our conversion tool to
apply CRS to convolutional layers.
Since DNNs also use matrix-vector products in linear transformations, a nice
additional side-effect is that by using the im2col mapping complete inference
passes can be described by matrix-vector products and non-linearities alone.
Moreover, ARM provides optimized open-source implementations for matrix
products in their _CMSIS_ library777https://github.com/ARM-software/CMSIS_5.
Using them is especially beneficial on architectures like the Cortex-M4 as it
allows to use SIMD instructions provided by ARM’s _Digital Signal Processing
(DSP)_ extension, see Sec. V.
## V Evaluation
TABLE I: Parameters of the DNN architectures used in our experiments. AlexNet (44.7M) | ResNet (9.4M) | LeNet (1.2M)
---|---|---
Input: [3, 32, 32] | Input: [1, 28, 28]
[3, 64, 2, 2] | [64, 64, 3, 1] | [1, 32, 3, 1]
[64, 192, 3, 1] | [64, 128, 3, 1] | [32, 64, 3, 1]
[192, 384, 3, 1] | [128, 256, 3, 1] |
[384, 256, 3, 1] | [256, 512, 3, 1] |
MaxPool: [2]
[6400, 4096] | [25088,10] | [9216, 128]
[4096, 4096] | | [128, 10]
[4096, 10] | |
Output: [10]
SoftMax
Tuples describe linear layers in the form of [num. inputs, num. outputs].
Every layer uses ReLU as their non-linearity except the last ones which are
followed by SoftMax; AlexNet/LeNet: quadruples describe 2D-convolutions by
[channels, filters, kernel size, stride]; ResNet: quadruples describe residual
blocks each with two 2D-convolutions in the form of [block in. channels, block
out. filters, conv. kernel size, conv. stride]; AlexNet/ResNet: every
2D-convolution is followed by batch normalization.
TABLE II: Microcontrollers considered in our evaluation. | Raspberry Pi Pico | Arduino Nano 33 BLE Sense | Raspberry Pi 4B
---|---|---|---
Processor | RP2040, Cortex-M0+ (Armv6-M) | nrf52840, Cortex-M4 (Armv7-M) | BCM2711 SoC, Cortex-A72 (ARMv8-A)
Clock | 133 MHz | 64 MHz | 1.5 GHz
Flash | 2 MB | 1 MB | 16 GB (SD-Card)
RAM | 256 KB (SRAM) | 256 KB (SRAM) | 8 GB (SDRAM)
SIMD | No | Yes (ARM DSP) | Yes (ARM Neon)
(a) AlexNet on CIFAR-10.
(b) ResNet on CIFAR-10.
(c) LeNet on MNIST.
Figure 4: Element-wise and structural pruning applied to the DNN
architectures. The curves describe the relation between theoretical model size
and accuracy of compressed models relative to their uncompressed baselines
when using different pruning techniques in combination with an iterative AGP
pruning schedule. We define theoretical model size to be the number of weights
a model features excluding all weights that have been set to zero by pruning.
To evaluate our pipeline we selected three popular DNN architectures: (1) a
convolutional network similar to the one proposed by Krizhevsky et al. [17] to
classify CIFAR-10 images (AlexNet), (2) a residual network [3] (ResNet), and
(3) a smaller network architecture initially proposed by LeCun et al. [18]
(LeNet) for classifying the MNIST handwritten digit database. See Table I for
more details.
We trained AlexNet and ResNet on the CIFAR10 [4] dataset for 100 epochs with
mini-batches of size 80 and LeNet on the MNIST handwritten digit datasets [34]
for 20 epochs with mini-batches of size 48 (as training converges on MNIST
considerably faster). On all the models we used stochastic gradient descent
(SGD) with a momentum of 0.9 and a learning rate of $1e-3$. We achieved a
maximum accuracy of 86.51% for AlexNet, 85.97% for ResNet, and 98.46% for
LeNet that serve as baselines for our experiments.
To evaluate our deployment pipeline we selected three target systems: (1) a
Raspberry Pi Pico, (2) an Arduino Nano 33 BLE Sense, and (3) a Raspberry Pi 4B
(that serves as a larger reference system), see Table II. As our runtime
library is mainly optimized for Cortex-M architectures, we instead use
onnxruntime [32] for the deployment to the Raspberry Pi 4B.
We compare the performance of different pruning techniques in Sec. V-A and
discuss their combination with quantization in Sec. V-B. We analyze the memory
footprint of our compressed DNNs in Sec. V-C. In Sec. V-D we discuss the
execution time and power/energy consumption w.r.t. the predictive accuracy
from a real-world execution of the compressed DNNs on the target platforms.
### V-A Comparison of Pruning Techniques
First, we present results of pruning experiments conducted for each of our
three test DNN architectures. We repeated model training from scratch and
increased pruning target rates starting with 0% as the un-pruned baseline and
ending with 99% (i.e., a relative theoretical model size of 1%) as the most
aggressively pruned configuration. For each of the configurations we repeated
the experiment five times and report their means and standard deviations.
Figure 5: Comparison of pruning schedules for both structural and element-wise
pruning with AlexNet on CIFAR-10 (L1: pruning selected parameters based on
$\ell_{1}$-norm; Level: selects parameters based on magnitude level).
Figs. 4 and 5 show the predictive accuracy of the models (relative to the un-
pruned baseline) over percentages of theoretically remaining parameters on the
validation dataset. We first investigate the influence of pruning schedules on
the achievable pruning rate, see AlexNet on CIFAR-10 in Fig. 5. While we
cannot observe major differences between iterative (i.e., Automated Gradual
Pruning, AGP) and one-shot schedules for final parameter counts $>10\%$, we
see that iterative schedules perform slightly better both for element-wise and
structural pruning for parameter counts below $10\%$. We argue that this is
because it is more difficult for the network to retrain when a large number of
parameters are removed at once than when parameters are removed gradually and
retraining is possible in between. Hence, we decided to focus on iterative
pruning schedules for all further experiments.
With the iterative AGP schedule on the same experimental setup we tested
different pruning heuristics for structural and element-wise pruning on all
architectures, see Figs. 4(a) to 4(c). We report four different heuristics for
structural pruning and one for element-wise pruning alongside a random
selection (’Random’) approach as a baseline for the more elaborate heuristics.
For structural pruning, we use both the $\ell_{1}$\- and the $\ell_{2}$-norm
of parameter structures (’L1’ and ’L2’) as well as their gradient size
(’Gradient’) and the average percentage of zeros in their corresponding
activation (’Activation’) as heuristics. For element-wise pruning we use a
magnitude level to decide which elements to prune (’Level’).
While we see a significant improvement of the level-based heuristic over the
corresponding random selection for element-wise pruning on all our three DNN
architectures, we cannot observe a similar behaviour for structural pruning.
Instead, none of the more complex structural pruning heuristics managed to
significantly improve over the random selection approach. There is also only
little variation in the results of the heuristics. We believe that the main
reason for this is the iterative re-training between pruning steps: while
introducing pruning during DNN training can cause degradation in the
predictive quality of a model, it was very often regained in a short number of
epochs when re-training. This is in line with results reported in previous
work [29].
In all our experiments we used the same target compression rates for both
element-wise and structural pruning. However, we see that the structural
pruning experiments resulted in models that exceed their selected targeted
compression rate. In some cases this reduces the parameter count to almost
99.9%. For element-wise pruning, we do not see such an effect. The reason for
this is related to the removal of structures from DNN models during structural
pruning: due to the existence of data dependencies between layers, removing
structures from their parameter tensors also influences the shapes of tensors
in surrounding layers. For element-wise pruning, parameters are not completely
eliminated from the DNN but are instead just set to zero. Therefore, all data
dependencies remain in the network, no tensor shape changes occur, and the
pruning target rate is more precisely met.
### V-B Combining Pruning and Quantization
We present experimental results for weight quantization in combination with
pruning for our three test DNN architectures. Fig. 6(a) shows the results for
quantization in combination with structural pruning and Fig. 6(b) shows the
results for element-wise pruning. The different colors refer to the models we
trained and quantization strategies are differentiated with the line and
marker style. We aim to give an understanding on how much the additional
quantization error alone influences the prediction quality of quantized
models. Hence, as before, the $x$-axes show the relative theoretical model
size reduction while this time the $y$-axes reports the accuracy decrease of
each pruned and quantized model in relation to its pruned but not quantized
version.888We want to point out that quantization does not change the number
of parameters in a model, only their resolution, so the application of
quantization does not affect the relative theoretical model size, although the
model effectively becomes smaller. We present our findings on actual size
reduction achievable with quantization in Sec. V-C. This allows us to focus on
the additional error that is introduced through quantization alone.
When looking at the results of structural pruning in combination with both
Quantization as a post process after training (PPQ) and Quantization Aware
Training (QAT), we see that the techniques work well together with pruning for
all our three architectures. In Fig. 6(a) we see that the accuracy decrease
consistently is $<5\%$ even when using quantization in combination with
aggressive pruning regimes. The only outliers we monitored were part of our
experiments on the LeNet architecture. Here, for the two most aggressive
pruning configurations, the accuracy decrease between the non-quantized and
quantized models went to around 40% for both tested quantization strategies.
Moreover, we observed an increase in standard deviation as accuracy decreased,
which we believe is related to an increase in variance in the trained weight
values that we observed for LeNet at higher compression rates. The higher the
variance of the values in a weight tensor, the worse quantization can
represent these values in integer space.
(a) L1-norm AGP structural pruning
(b) Level AGP element-wise pruning
Figure 6: Decrease in accuracy on the evaluation dataset observed for our
three DNN architectures when combining pruning and quantization techniques.
We also tested element-wise pruning in combination with PPQ and QAT, see Fig.
6(b). Other than for structural pruning, where PPQ performed consistently well
even in combination with aggressive pruning configurations, for element-wise
pruning we observed accuracy decreases of over 70% in comparison to the un-
quantized versions. In particular, we observe that PPQ noticeably failed for
models that have been compressed by element-wise pruning to 10% or less of
their original parameter count. This is despite the fact that the technique
performed well for pruning configurations that target compression rates above
10%. In contrast, QAT performed significantly better than PPQ even when used
together with aggressive pruning configurations. The technique was able to
keep the accuracy decrease very close to 0% during all conducted experiments.
Hence, we conclude that PPQ generally seems to perform better when being
applied in combination with structural pruning than when used with element
pruning. QAT on the other hand performed consistently well, both in
combination with structural and element-wise pruning.
(a) Flash consumption of the compressed models (left: AlexNet/CIFAR-10;
middle: ResNet/CIFAR-10; right: LeNet/MNIST).
(b) SRAM consumption of the compressed models (left: AlexNet/CIFAR-10; middle:
ResNet/CIFAR-10; right: LeNet/MNIST).
Figure 7: Memory required for the deployment of our three DNN architectures
(AlexNet, ResNet, LeNet) when applying the proposed runtime environment. (a)
shows the required amounts of read-only ”Flash” memory to store the DNNs’
trained weights, while (b) shows the required amounts of random access memory
”SRAM” to store the dynamic activation tensors. The dashed horizontal red and
green lines mark the memory limits of our two considered embedded target
platforms, the Raspberry Pi Pico and the Arduino Nano 33 BLE Sense.
### V-C From Size Reduction to Memory Consumption
As described in Sec. IV the execution of DNNs requires both static read-only
and dynamic random access memory. We now discuss the memory footprint of the
models when they are being deployed. Fig. 7(a) shows the relation between
relative model size (on the $x$-axes) and ROM/flash consumption in Kibibyte
(on the $y$-axes) and Fig. 7(b) shows the same relation for SRAM. The curves
for the pruned models are drawn with solid lines while the curves for the
models where additional quantization was applied are dashed. To also highlight
the importance of model compression we added the Flash and SRAM limits of out
two microcontroller target platforms (red and green dashed horizontal lines).
On both of these platforms flash and SRAM availability are two of the main
bottlenecks for deployment.
We see a linear correlation between flash consumption and relative theoretical
model size for structural pruning in Fig. 7(a) (note that both axes are in
logarithmic scale). The reason for this is that pruned structures can be
completely removed from the model which immediately decreases its memory
consumption in the process. For element-wise pruning, there is no such direct
correlation. Instead, when looking at the orange curves in Fig. 7(a) we
observe plateaus for higher theoretical model sizes. This is because pruning
only generates sparse weight matrices. Our runtime environment utilizes
compressed row storage (CRS) as a special decoding technique to store sparse
weight matrices space-efficiently. However, a characteristic of the technique
is that decoding a tensor will only result in memory savings after a certain
percentage of sparsity has been reached. For any amount of sparsity below this
threshold, it is better to just use the default memory layout where all
elements are saved sequentially.
An additional observation we made is that for models that are compressed using
quantization and element-wise pruning the threshold at which CRS becomes
feasible is much higher than for just pruned models. Again, this is related to
the properties of CRS decoding. Instead of all values, only values unequal to
zero (or a zero point) are stored. To preserve their position in the original
un-decoded matrix, the row and column indices of the values must be stored as
well. For larger matrices, like they exist in DNNs, these indices usually
require 16- or 32-bit integers to be stored correctly. Therefore, memory
savings made by CRS can be considered as a trade-off between storing some
elements and their indices versus storing all elements without any indices.
When introducing sparsity into matrices with 32-bit floating point values,
this quickly becomes a good trade-off. However, since quantized values require
only 8 bits, while the index values introduced by CRS are still usually at
least 16 bits long, the amount of sparsity that has to be introduced before
memory can be saved is higher.
Fig. 7(b) shows the relationship between relative theoretical model size and
required SRAM. Similar to flash consumption, a correlation between model size
and SRAM consumption can be observed for structural pruning. When structures
are removed from parameter tensors during pruning, data dependencies between
layers are also removed. Therefore, the shapes of the dynamic intermediate
activation tensors stored in RAM and shared between succeeding layers reduce
as well. The reason why the relation is not perfectly linear is because our
runtime library tries to re-use heap memory for several activation tensors.
How good this strategy works and how much memory it can save depends on the
topology of the DNN that it is applied to. We cannot observe a reduction in
SRAM consumption for element-wise pruning. This is because we do not remove
any elements during element-wise pruning. While CRS can compress the sparse
parameter tensors, it cannot change their original shapes. Therefore, as
expected, no data dependencies are removed during decoding and as a result all
intermediate activation tensors retain their original sizes.
TABLE III: Average ($n=8$) execution time, power and energy consumption observed per inference for compressed version of LeNet targeting two relative accuracy levels. The baseline model achieved an accuracy of $\approx$98% for both float and uint8. rel. accuracy | compression | type | system | execution time (ms) | power (mW) | energy (mWs)
---|---|---|---|---|---|---
$>$99.00% | 99.18% | structure, float | arduino | $44.15$ p m 0.03 | $141.14$ p m 0.07 | $6.23$ p m 0.01
| | | pico | $184.28$ p m 0.17 | $81.42$ p m 0.42 | $15.00$ p m 0.07
| | | pi4b | $1.81$ p m 0.55 | $3754.70$ p m 284.92 | $6.65$ p m 1.35
| | structure, uint8 | arduino | $18.84$ p m 0.04 | $101.06$ p m 1.80 | $1.90$ p m 0.03
| | | pico | $50.06$ p m 0.03 | $79.37$ p m 1.03 | $3.97$ p m 0.05
| | | pi4b | $2.16$ p m 0.60 | $3624.90$ p m 272.88 | $7.68$ p m 1.53
| 95.00% | element, float | arduino | — | — | —
| | | pico | $1004.77$ p m 0.07 | $81.44$ p m 0.10 | $81.82$ p m 0.11
| | | pi4b | $5.66$ p m 1.39 | $3772.18$ p m 263.01 | $21.00$ p m 3.34
| | element, uint8 | arduino | $360.95$ p m 0.09 | $142.28$ p m 0.10 | $51.36$ p m 0.04
| | | pico | $392.23$ p m 0.07 | $84.82$ p m 0.37 | $33.27$ p m 0.15
| | | pi4b | $5.10$ p m 1.27 | $3673.70$ p m 332.91 | $18.31$ p m 3.04
$>$97.00% | 99.7% | structure, float | arduino | $22.48$ p m 0.01 | $105.32$ p m 0.27 | $2.37$ p m 0.01
| | | pico | $73.26$ p m 0.33 | $81.72$ p m 0.48 | $5.99$ p m 0.03
| | | pi4b | $1.91$ p m 0.56 | $3554.33$ p m 232.34 | $6.67$ p m 1.48
| | structure, uint8 | arduino | $18.79$ p m 0.03 | $82.54$ p m 0.80 | $1.55$ p m 0.02
| | | pico | $24.97$ p m 0.44 | $79.92$ p m 0.88 | $2.00$ p m 0.04
| | | pi4b | $2.01$ p m 0.53 | $3581.79$ p m 240.22 | $7.11$ p m 1.41
| 97.00% | element, float | arduino | — | — | —
| | | pico | $658.70$ p m 0.10 | $82.44$ p m 0.68 | $54.30$ p m 0.45
| | | pi4b | $5.35$ p m 1.23 | $3816.72$ p m 246.08 | $20.12$ p m 2.79
| | element, uint8 | arduino | $303.15$ p m 0.09 | $141.58$ p m 0.05 | $42.92$ p m 0.01
| | | pico | $392.27$ p m 0.10 | $84.67$ p m 0.45 | $33.22$ p m 0.18
| | | pi4b | $4.64$ p m 1.39 | $3787.84$ p m 395.19 | $17.05$ p m 3.16
### V-D Deployment Results
As the last step of evaluating our pipeline, we deployed several of the pruned
and quantized models from our previous experiments on our target systems and
monitored key runtime metrics. We especially focused on execution time per
inference, power, and energy consumption. For measuring these metrics on our
test systems, we used an Agilent N6705A DC Power Analyzer to provide them with
a regulated power supply. Furthermore, the Power Analyzer allowed us to
measure the current and power drawn by the systems. To measure the execution
time required for calculating the energy consumption of our DNN models, we
used a GPIO signal. We toggled the signal every time an inference started and
finished and monitored it using an oscilloscope. We present the results of our
measurements for LeNet in Table III and the results for our other two DNN
architectures in Table IV.
TABLE IV: Average ($n=8$) execution time, power and energy consumption
observed per inference for AlexNet and ResNet. To achieve different
compression rates and thus different accuracy scores, we used L1-norm
structural pruning. In addition, we used post training quantization for all
non floating-point measurements.
compression | type | rel. accuracy | system | execution time (ms) | power (mW) | energy (mWs)
---|---|---|---|---|---|---
95.9% | float | 97.79% | arduino | — | — | —
| | | pico | — | — | —
| | | pi4b | $6.60$ p m 1.85 | $3795.81$ p m 334.67 | $24.43$ p m 4.10
| uint8 | 97.56% | arduino | — | — | —
| | | pico | $1766.58$ p m 0.25 | $92.40$ p m 0.03 | $163.23$ p m 0.07
| | | pi4b | $3.96$ p m 1.12 | $3809.41$ p m 250.59 | $14.82$ p m 2.71
98.2% | float | 94.88% | arduino | — | — | —
| | | pico | — | — | —
| | | pi4b | $2.68$ p m 0.70 | $3687.37$ p m 221.25 | $9.71$ p m 1.76
| uint8 | 94.65% | arduino | $403.11$ p m 0.16 | $139.96$ p m 0.17 | $56.42$ p m 0.07
| | | pico | $669.97$ p m 0.19 | $86.41$ p m 0.02 | $57.89$ p m 0.03
| | | pi4b | $2.93$ p m 0.41 | $4187.75$ p m 89.73 | $12.28$ p m 2.05
99.6% | float | 83.26% | arduino | $252.40$ p m 0.40 | $107.83$ p m 0.42 | $27.22$ p m 0.10
| | | pico | $1435.42$ p m 0.34 | $81.55$ p m 0.07 | $117.06$ p m 0.09
| | | pi4b | $2.45$ p m 0.09 | $4182.59$ p m 66.35 | $10.25$ p m 0.35
| uint8 | 82.33% | arduino | $110.00$ p m 0.01 | $141.17$ p m 0.04 | $15.53$ p m 0.01
| | | pico | $186.57$ p m 0.12 | $86.86$ p m 0.21 | $16.21$ p m 0.04
| | | pi4b | $2.10$ p m 0.14 | $4134.21$ p m 52.03 | $8.68$ p m 0.60
(a) AlexNet, CIFAR10, baseline accuracy $\approx$86% for both float and uint8
compression | type | rel. accuracy | system | execution time (ms) | power (mW) | energy (mWs)
---|---|---|---|---|---|---
93.2% | float | 96.25% | arduino | — | — | —
| | | pico | — | — | —
| | | pi4b | $29.41$ p m 2.62 | $4234.81$ p m 121.13 | $124.24$ p m 7.11
| uint8 | 96.43% | arduino | $11\,983.40$ p m 2.05 | $143.74$ p m 0.49 | $1722.48$ p m 5.69
| | | pico | $25\,997.20$ p m 4.00 | $91.94$ p m 0.01 | $2390.24$ p m 0.41
| | | pi4b | $21.32$ p m 3.32 | $4085.92$ p m 213.90 | $86.54$ p m 9.62
98.7% | float | 92.50% | arduino | $6105.20$ p m 1.60 | $105.76$ p m 1.85 | $645.69$ p m 11.16
| | | pico | $46\,266.50$ p m 16.82 | $85.78$ p m 0.02 | $3968.83$ p m 1.49
| | | pi4b | $9.41$ p m 4.27 | $4536.99$ p m 183.46 | $42.06$ p m 16.53
| uint8 | 88.52% | arduino | $2425.50$ p m 0.92 | $102.38$ p m 1.10 | $248.31$ p m 2.70
| | | pico | $4269.78$ p m 2.58 | $87.81$ p m 0.04 | $374.92$ p m 0.36
| | | pi4b | $7.74$ p m 2.78 | $3863.07$ p m 427.70 | $28.82$ p m 6.98
99.6% | float | 83.48% | arduino | $1754.75$ p m 0.31 | $106.57$ p m 2.64 | $187.00$ p m 4.63
| | | pico | $11\,140.25$ p m 1.85 | $83.13$ p m 0.24 | $926.12$ p m 2.61
| | | pi4b | $5.42$ p m 1.66 | $3560.66$ p m 421.12 | $18.70$ p m 4.01
| uint8 | 78.28% | arduino | $806.80$ p m 0.60 | $104.48$ p m 0.09 | $84.29$ p m 0.13
| | | pico | $1214.53$ p m 1.00 | $84.13$ p m 0.06 | $102.18$ p m 0.10
| | | pi4b | $3.45$ p m 0.22 | $3881.06$ p m 72.77 | $13.37$ p m 0.60
(b) ResNet, CIFAR10, baseline accuracy $\approx$85% for float and $\approx$84%
for uint8
For the LeNet architecture, we used the models’ accuracy on its evaluation
dataset to compare the deployment of different pruning and quantization
techniques. The logic behind this was that if models that were compressed
using different approaches can achieve a similar accuracy, then they can be
seen as direct alternatives and are therefore comparable. In our experimental
setup we defined two relative accuracy boundaries for which we selected the
smallest compressed models from our previous tests that meet them: $>$99% and
$>$97%. In the second column of Table III we can see that LeNet can be
compressed to well over 5% of its original parameter count while still passing
both accuracy thresholds. We tested all selected models on our three target
systems, including not only the pruned models but also their quantized
counterparts, see the third and fourth columns. Note that using element
pruning we were not able to deploy all selected models on the Arduino.
For all deployed LeNet models, we monitored execution time, power and energy
consumption over a span of 8 different inferences. Their resulting averages
are presented in the remaining columns of Table III. First, we see that the
execution time per inference on the Pi 4B is significantly lower than on the
Arduino or the Pi Pico. This is expected as the Pi 4B runs between 1.0 to 1.5
GHz while both the Arduino and the Pico run in a lower MHz range. When
comparing the Raspberry Pi Pico and the Arduino Nano we observed a higher
execution time on the Pico than on the Arduino (consistently). This is even
though the Arduino runs with a clock speed only around half as fast as the
Pico. On both systems the quantized models always outperformed their floating-
point counter parts although much more pronounced on the Pico. This can be
explained by the features present on both systems. First, the Arduino’s
Cortex-M4 processor implements a real floating point unit while on the Pi
Pico’s Cortex-M0+ processor floating point arithmetic has to be simulated.
Second, the Arduino’s Cortex-M4 processor supports ARM’s Digital Signal
Processing (DSP) extension giving it access to a subset of SIMD instructions
to accelerate its integer operations. The Pi Pico does not implement the DSP
extension.
Considering the power measured during inference for all deployed models, we
see that all our tested systems on average draw a constant amount of power
while we see a more significant variation in the different samples taken for
each model on the Pi 4B (note that the power consumption is much higher than
on the other two systems). On the Arduino, the measured power consumption was
between 100 to 150 mW on average while on the Pi Pico it was around 80 mW. In
contrast to that, the Pi 4B generally consumed around 4W. However, in addition
to power, execution time is the second factor in calculating a system’s energy
consumption. Looking at the results, we see that for some cases the Pi 4B
scored the best energy consumption per inference. It is often followed by the
Arduino and then the Pi Pico. This is in reverse to the power consumption and
shows that having an excellent runtime can compensate for high power
consumption.
Besides LeNet, we also deployed our AlexNet and ResNet architectures, see
Table IV. For both architectures we evaluated models which were compressed
using structural pruning. This is different from our LeNet experiments where
we tested both structural and element-wise pruning. The reason we did not do
this for AlexNet and ResNet is due to memory limitations on our Arduino and Pi
Pico target systems, which made it impossible for us to deploy any element
pruned models. For structural pruning the situation is different and we were
able to feasibly deploy models. However, we were still forced to select models
trained with aggressive compression rates that removed well over 90% of the
original parameters. As a consequence we had to accept decreases in accuracy
to be able to deploy models, see the third column of Tables 8(a) and 8(b).
To measure execution time, power, and energy consumption, we used the same
approach as before. We again monitored all three metrics over 8 different
inferences and calculated their averages and respective standard deviations.
When looking at the results for AlexNet and ResNet, we see the same patterns
we discussed for LeNet in Table III.
Yet we evaluated the two architectures as a way to explore upper boundaries of
feasible DNN deployment. AlexNet features a very high number of trainable
parameters while ResNet contains a high number of large and computationally
expensive 2D-convolutions. This affects the deployment of the two
architectures differently. While for AlexNet, we need to apply high
compression rates to shrink the model size far enough to fit the it into the
memory of our target microcontrollers (see Sec. V-C) for ResNet execution time
is the primary bottleneck. Even after pruning almost 99% of ResNet’s
parameters we still measure inference times of around 4 seconds on the Pi
Pico, and of around 6 seconds on the Arduino Nano even when applying
quantization, see the fifth column of Table 8(b). Only after removing $>$99%
of all parameters the execution time ended up in an acceptable range (of
around 1 second) on both microcontroller targets. However, these extremely
high compression rates were accompanied by a high loss in accuracy, see the
first column. This may make the usage of such a complex DNN architecture
impractical for microcontroller targets. Therefore, we conclude that not only
a model’s parameter count but also its topology decides if it is deployable on
a target system from a performance standpoint.
## VI Discussion of Results
By using our compression and deployment pipeline, we were able to
automatically and feasibly deploy DNNs to microcontrollers. Compressing DNNs
allowed us to achieve significant savings in memory consumption, execution
time, and energy consumption at runtime, without sacrificing model accuracy
for it. For pruning, we achieved the best results and most savings with
structural pruning. When comparing different pruning strategies, our
experiments indicate that structural pruning offers better opportunities for
saving memory and execution time than element-wise pruning. In addition, the
execution of DNNs that were compressed using this technique did not require
any special support for sparse matrix formats as it is required for element
pruning. Furthermore, we observed that for structural pruning, using different
state-of-the-art heuristics did not have that much of an impact. Choosing a
reasonable pruning schedule and allowing for retraining has proven to be more
effective. Additionally, applying weight quantization together with structural
pruning resulted in even more savings not only in memory consumption, but also
in execution time. This is due to the fact that our target systems were able
to process much more efficient in integer- than in floating-point space.
Besides that, we noticed that our different compression and deployment
strategies had almost no influence on the power drawn by both the Pi Pico and
the Arduino during inference. This means that the observed energy savings were
mainly the result of execution times.
We come to the conclusion that a DNN model is deployed optimally on a
microcontroller, if it runs on a system where it fits into the available
memory and draws the least amount of power under load while still being able
to run inferences in a reasonable time frame. Furthermore, we argue that a
DNN’s execution time has to be seen in relation to the frequency at which
input samples are generated by connected sensors.
## VII Conclusion
In this work we presented a configurable pipeline for compressing DNNs and
deploying them on Cortex-M based microcontrollers. To achieve compression, our
pipeline utilizes network pruning and weight quantization techniques. The
deployment is handled by using a proposed runtime environment, which consists
of a code generator for mapping trained networks to executable C-code and a
runtime library which provides optimized implementations of common DNN layers.
We used the introduced pipeline to compare DNNs compressed with different
pruning and quantization techniques. Furthermore, we tested how compression
influences runtime performance on multiple target systems. We were able to
show that even larger DNN architectures like AlexNet or ResNet can be feasibly
deployed on microcontrollers featuring memory footprints of as little as 1-2
MB Flash and 256 Kb SRAM while still achieving good execution time and
accuracy results.
## Acknowledgement
This work was supported by the Bavarian Ministry for Economic Affairs,
Infrastructure, Transport and Technology through the Center for Analytics-
Data-Applications (ADA- Center) within the framework of “BAYERN DIGITAL II”.
## References
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|
# SEA++: Multi-Graph-based High-Order Sensor Alignment for Multivariate Time-
Series Unsupervised Domain Adaptation
Yucheng Wang, Yuecong Xu, Jianfei Yang, Min Wu, Xiaoli Li, Lihua Xie, Fellow,
IEEE, Zhenghua Chen111Corresponding Author Yucheng Wang and Yuecong Xu are
with Institute for Infocomm Research, A∗STAR, Singapore and the School of
Electrical and Electronic Engineering, Nanyang Technological University,
Singapore (Email<EMAIL_ADDRESS>xuyu0014@e.ntu.edu.sg).Zhenghua
Chen and Min Wu are with Institute for Infocomm Research, A∗STAR, Singapore
(Email<EMAIL_ADDRESS>wumin@i2r.a-star.edu.sg).Xiaoli Li is with
Institute for Infocomm Research, A∗STAR, Singapore and the School of Computer
Science and Engineering, Nanyang Technological University, Singapore (Email:
xlli@i2r.a-star.edu.sg).Jianfei Yang and Lihua Xie are with the School of
Electrical and Electronic Engineering, Nanyang Technological University,
Singapore (Email<EMAIL_ADDRESS>elhxie@ntu.edu.sg).
###### Abstract
Unsupervised Domain Adaptation (UDA) methods have been successful in reducing
label dependency by minimizing the domain discrepancy between a labeled source
domain and an unlabeled target domain. However, these methods face challenges
when dealing with Multivariate Time-Series (MTS) data. MTS data typically
consist of multiple sensors, each with its own unique distribution. This
characteristic makes it hard to adapt existing UDA methods, which mainly focus
on aligning global features while overlooking the distribution discrepancies
at the sensor level, to reduce domain discrepancies for MTS data. To address
this issue, a practical domain adaptation scenario is formulated as
Multivariate Time-Series Unsupervised Domain Adaptation (MTS-UDA). In this
paper, we propose SEnsor Alignment (SEA) for MTS-UDA, aiming to reduce domain
discrepancy at both the local and global sensor levels. At the local sensor
level, we design endo-feature alignment, which aligns sensor features and
their correlations across domains. To reduce domain discrepancy at the global
sensor level, we design exo-feature alignment that enforces restrictions on
global sensor features. We further extend SEA to SEA++ by enhancing the endo-
feature alignment. Particularly, we incorporate multi-graph-based high-order
alignment for both sensor features and their correlations. High-order
statistics are employed to achieve comprehensive alignment by capturing
complex data distributions across domains. Meanwhile, a multi-graph alignment
technique is introduced to effectively align the evolving distributions of MTS
data. Extensive empirical results have demonstrated the state-of-the-art
performance of our SEA and SEA++ on public MTS datasets for MTS-UDA.
Unsupervised Domain Adaptation (UDA) methods have been successful in reducing
label dependency by minimizing the domain discrepancy between a labeled source
domain and an unlabeled target domain. However, these methods face challenges
when dealing with Multivariate Time-Series (MTS) data. MTS data typically
consist of multiple sensors, each with its own unique distribution. This
characteristic makes it hard to adapt existing UDA methods, which mainly focus
on aligning global features while overlooking the distribution discrepancies
at the sensor level, to reduce domain discrepancies for MTS data. To address
this issue, a practical domain adaptation scenario is formulated as
Multivariate Time-Series Unsupervised Domain Adaptation (MTS-UDA). In this
paper, we propose SEnsor Alignment (SEA) for MTS-UDA, aiming to reduce domain
discrepancy at both the local and global sensor levels. At the local sensor
level, we design endo-feature alignment, which aligns sensor features and
their correlations across domains. To reduce domain discrepancy at the global
sensor level, we design exo-feature alignment that enforces restrictions on
global sensor features. We further extend SEA to SEA++ by enhancing the endo-
feature alignment. Particularly, we incorporate multi-graph-based high-order
alignment for both sensor features and their correlations. Extensive empirical
results have demonstrated the state-of-the-art performance of our SEA and
SEA++ on public MTS datasets for MTS-UDA.
###### Index Terms:
Multivariate Time-Series Data; Unsupervised Domain Adaptation; Graph Neural
Network
## I Introduction
Time-Series (TS) data have gained significant attention in various fields due
to their wide range of applications. Deep Learning (DL) methods have played a
pivotal role in addressing TS-related problems by leveraging their ability to
effectively capture latent dependencies within the data [1, 2, 3, 4, 5, 6, 7].
However, the effectiveness of DL models heavily relies on the availability of
large-scale labeled TS data, which is often a challenge in real-world
applications due to the high cost associated with data labeling.
To reduce the cost of labeling, Unsupervised Domain Adaptation (UDA) methods
have been proposed to transfer the knowledge from a labeled source domain to
an unlabeled target domain, enabling effective learning in the absence of
target domain labels [8]. Due to domain shifts, existing UDA methods focused
on reducing discrepancy across domains by learning domain-invariant features
[9, 10, 11, 12, 13, 14]. These methods have achieved decent performance,
showing the effectiveness of UDA in reducing label dependency. Subsequently,
researchers applied UDA to TS data, aiming to reduce domain discrepancy by
learning domain-invariant temporal features extracted by Recurrent Neural
Network (RNN) [15], Long Short-Term Memory (LSTM) [16], and Convolutional
Neural Network (CNN) [17].
Figure 1: Comparisons between existing UDA methods and ours, where before and
after represent before and after alignment respectively. Left: Only global
sensor features are aligned, resulting in the poor alignment of the red
sensor. Right: SEA aligns the sensor information between domains, so the
feature discrepancy of each sensor can be reduced.
While substantial progress has been made, existing TS UDA methods may be
inapplicable for real-world applications where multiple sensors are deployed
simultaneously. For instance, in Remaining Useful Life (RUL) prediction,
various types of sensors are employed to measure different physical
parameters. On the other hand, in Human Activity Recognition (HAR), multiple
sensors are positioned at various locations on the human body to capture
activity data. To address these application scenarios, we formulate a more
challenging yet practical problem as Multivariate Time-Series Unsupervised
Domain Adaptation (MTS-UDA). Applying current TS UDA methods directly to MTS-
UDA poses two key limitations. First, MTS data typically consist of signals
from multiple sensors, where each sensor collects data with distinct
distributions due to variations in sensor types or deployment locations [18].
When adapting existing TS UDA methods to MTS-UDA, they mainly focus on
learning global features to facilitate alignment across domains by treating
the signals from various sensors as a whole [16, 19]. However, these methods
ignore the diverse distributions across sensors, resulting in misalignment at
the sensor level and thus leading to suboptimal solutions for MTS-UDA. As
presented in Fig. 1 (Left), existing UDA methods focused on aligning the
global features between domains, resulting in the poor alignment of the
features from the red sensor and limiting the transferability of the model.
Second, MTS data contain essential spatial-temporal dependency information due
to its multi-source nature. Specifically, the spatial dependency refers to the
correlations between sensors, which capture the critical interactive
relationships among them. For instance, the spatial dependency in the context
of machine RUL prediction corresponds to the correlation between a temperature
sensor and a fan speed sensor, as they exhibit strong correlations. Besides,
the temporal dependency refers to the temporal relationships between
consecutive timestamps. Existing UDA methods are limited in effectively
modelling and transferring both dependencies across domains, thus leading to
suboptimal solutions for MTS-UDA. These characteristics within MTS-UDA render
it a more complex and challenging task compared to standard TS UDA.
To address the above challenges, we propose SEnsor Alignment (SEA) for MTS-
UDA. We first aim to reduce domain discrepancy at both the local and global
sensor levels. At the local sensor level, we propose endo-feature alignment to
align sensor-level information between domains. This alignment process
includes aligning both the sensor features and sensor correlations, which
capture the characteristics of individual sensors and their interactions,
respectively. To achieve this alignment, we introduce contrastive sensor
alignment and sensor correlation alignment. Further, at the global sensor
level, we design exo-feature alignment to reduce global feature discrepancy.
This is achieved by enforcing restrictions on the global features derived from
stacked sensor features. Moreover, to capture the spatial-temporal
dependencies inherent in MTS data and enable their effective transfer across
domains, we introduce a graph-based encoder that is able to capture the
dependency information. Given that MTS data may not come with pre-defined
graphs that represent the relationships between sensors, we propose a multi-
branch self-attention mechanism to model these dependencies.
Although SEA presents a feasible solution for sensor alignment in MTS-UDA, it
employs a low-order statistics-based scheme for achieving endo-feature
alignment. This scheme may be insufficient to achieve comprehensive alignment
when dealing with complex data distributions in certain scenarios.
Furthermore, SEA can benefit from leveraging the dynamically changing
distributions within MTS data to enhance its performance. For instance, as a
machine’s health deteriorates, the distribution of a fan speed sensor
undergoes dynamic changes due to increased friction. Taking these issues into
consideration, we develop SEA++ as an extension of SEA by introducing multi-
graph-based high-order alignment for enhancing our endo-feature alignment. It
incorporates improved sensor correlation alignment and enhanced contrastive
sensor alignment, utilizing high-order statistics to achieve comprehensive
sensor-level alignment by capturing complex data distributions across domains.
Additionally, SEA++ introduces a multi-graph alignment technique specifically
tailored for aligning the evolving distributions within MTS data. This
involves the construction of sequential graphs that represent the
distributions within local temporal intervals, followed by the alignment of
corresponding sequential graphs between domains. To optimize the alignment
process, weights for balancing sequential graphs are learned based on
distribution discrepancies, allowing us to prioritize alignment for
challenging graphs. This module effectively aligns the evolving distributions,
enabling adaptation to the changing nature of the data. With the two types of
enhancements in the improved version, we represent the name with two plus
signs as SEA++.
Our contributions can be summarized as following.
* •
We formulate a challenging scenario of Multivariate Time-Series Unsupervised
Domain Adaptation (MTS-UDA) in accordance with the characteristics of MTS
data. To our best knowledge, this is the first work to design a UDA method
specifically for MTS data.
* •
We analyze the problems underlying MTS-UDA and propose SEnsor Alignment (SEA)
that reduces domain discrepancy at both the local and global sensor levels.
Meanwhile, SEA captures the spatial-temporal dependencies within MTS data,
enabling effective transfer of the dependency information across domains.
* •
To cope with scenarios with complex data distributions, we introduce SEA++ by
enhancing our endo-feature alignment. High-order statistics alignment methods
are employed to achieve comprehensive sensor correlation and sensor feature
alignment.
* •
Considering the dynamically changing distributions within MTS data, SEA++
further incorporates a multi-graph alignment technique. It aligns sequential
graphs representing local distributions, effectively aligning evolving
distributions between domains.
* •
We evaluate the effectiveness of SEA and SEA++ with various real-world MTS
datasets through extensive experiments, demonstrating that the both methods
achieve state-of-the-art performance for MTS-UDA.
This journal paper presents an extended version of our previous work [20] by
introducing multi-graph-based high-order alignment techniques. First, we
introduce high-order statistic alignment to improve endo-feature alignment,
aiming to achieve more comprehensive sensor-level alignment. Second, we
analyze the properties of MTS data and emphasize the significance of
dynamically changing distributions inherent within MTS data. Thus, we
introduce a multi-graph alignment technique to align the evolving
distributions across domains. Third, we have conducted additional experiments,
including more compared methods, comprehensive ablation studies, and extensive
sensitivity analysis, to fully assess SEA and SEA++.
## II Related Work
### II-A Unsupervised domain adaptation
UDA aims to mitigate the need for labeled data by transferring knowledge from
a labeled source domain to an unlabeled target domain. To achieve good
performance on the target domain, existing UDA methods have mainly focused on
minimizing domain discrepancies by utilizing two categories of approaches. The
first category comprises adversarial-based methods, which utilize domain
discriminator networks to encourage a feature extractor to learn domain-
invariant representations. Prominent models in this category include Domain
Adversarial Neural Network (DANN) [10] and Adversarial Discriminative Domain
Adaptation (ADDA) [11]. These models have served as foundations for various
improved variants that address specific challenges [12, 21, 22, 23, 24, 25].
For instance, CDAN [12] and MADA [21] were both extensions of DANN that
incorporate multi-mode structure information for classification tasks. DAAN
[22] introduced class-aware adversarial loss to align the conditional
distribution using DANN. The second category encompasses metric-based methods,
which enforce metric restrictions to enable the learning of invariant
features. Commonly employed metrics include Maximum Mean Discrepancy (MMD) [9]
and deep CORrelation ALignment (CORAL) [13]. Within this category, several
variants have been proposed to tackle specific challenges [14, 26, 27, 28].
For example, JAN [14] was developed based on MMD to align the joint
distribution of feature and label distributions. MK-MMD [26] employed MMD to
align features in the final layers, enhancing their transferability.
Furthermore, high-order moment matching called HoMM [27] was developed as a
high-order moment matching technique to align feature distributions in hidden
layers, addressing the limitations of low-order alignment methods. In
addition, central moment discrepancy [29] and maximum density divergence [30]
are another two criterias to align feature distributions in hidden layers.
Overall, these methods have contributed to the advancement of UDA by
effectively reducing the domain discrepancy and enabling knowledge transfer
between different domains.
### II-B Unsupervised domain adaptation for time-series data
In recent years, there has been a growing trend of exploring existing UDA
methods in various domains, showing their effectiveness in different
applications [31, 32]. In the context of TS data, researchers have primarily
focused on reducing domain discrepancy by aligning temporal features across
different domains [16, 17, 15, 19, 33]. Several notable works have made
significant contributions to this area. For instance, VRADA [15] leveraged a
variational RNN to learn temporal features and employs adversarial-based
methods to reduce the domain discrepancy. Similarly, Zhao et al. [34]
introduced a UDA method combining DANN and CNN for EEG classification, which
incorporated a center loss to minimize intra-class variation and maximize
inter-class distance. CoDATS [35] and CLADA [36], as multi-source UDA methods
for TS data, applied DANN and 1D-CNN to handle the scenarios where multiple
source domains are available. Moreover, some researchers have specifically
designed UDA methods tailored to TS data. ADATIME [19] considered the temporal
properties of TS data and evaluated various temporal encoders for TS UDA.
Besides, AdvSKM [37] mapped features into a high-dimensional space by
proposing adversarial Spectral Kernels and then aligned them using MMD. To
consider the invariant associative structure between variables exhibited by
MTS data, SASA [16] aligned this structural information for TS UDA.
These works are great pioneers for TS UDA, but they may not perform well for
the scenarios requiring multiple sensors, i.e., MTS data. In MTS data, signals
from different sensors often follow various distributions due to their varied
physical locations and measurements. Existing TS UDA methods can be adapted to
MTS data by treating all sensors as a whole [38, 19, 15, 34], facilitate the
learning of global features for alignment, but they are limited in accounting
for the distinct distributions of sensors within MTS data. Meanwhile, they
ignored the multi-source nature of MTS data, restricting their ability to
model and transfer spatial-temporal dependency information across domains. To
address these limitations, we propose our SEA and SEA++ that accounts for
aligning sensor-level information and modelling dependency information within
MTS data to achieve better MTS-UDA.
Figure 2: The overall structure. (1) To transfer spatial-temporal dependencies
across domains, source and target domains share the same graph-based encoder,
including MSGC, GNN, and LSTM. Each sample is segmented as multiple patches,
which are constructed as sequential graphs. With the combination of GNN and
LSTM, decent sensor information is learned, including sensor features and
correlations. (2) In addition to supervised learning on the source domain,
endo-feature alignment and exo-feature alignment are designed to reduce domain
discrepancy at the local and global sensor levels. The endo-feature alignment
includes aligning sensor features and their correlations between domains by
SCA and SFA, both of which are enhanced by incorporating multi-graph alignment
and high-order statistic alignment. Exo-feature alignment aligns global
features mapped by sensor features.
## III SEA Model
### III-A Problem Definition
For MTS-UDA, we are given a source domain with $N_{s}$ labeled samples
$\mathcal{D}_{S}=\\{(x_{i}^{s},y_{i}^{s})\\}_{i=1}^{N_{s}}$ and a target
domain with $N_{t}$ unlabeled samples
$\mathcal{D}_{T}=\\{x_{i}^{t}\\}_{i=1}^{N_{t}}$. Each MTS sample $x_{i}$
(either $x_{i}^{s}$ or $x_{i}^{t}$) originates from $N$ sensors with different
distributions, i.e., $x_{i}=\\{x_{im}\\}_{m=1}^{N}\in\mathbb{R}^{N\times{L}}$,
where $L$ represents the time length. The goal of SEA is to train an encoder
by transferring the knowledge from the source domain to the target domain,
enabling the effective learning of features $h_{i}\in\mathbb{R}^{F}$ from the
MTS data $x_{i}^{t}$ in the target domain. The features can then be used for
downstream tasks such as RUL prediction and HAR. Notably, the SEA framework
consists of two stages. In the first stage, we focus on feature extraction
from MTS data. To simplify the notation and enhance clarity, we omit the index
$i$ in this stage. Thus, each sample is represented as $x^{s}$ and $x^{t}$,
and the data from the $m$-th sensor is denoted as $x_{m}^{s}$ and $x_{m}^{t}$.
While in the second stage that involves domain alignment, we retain the use of
the index to facilitate a clear description of the alignment process.
Furthermore, to model the evolving distributions within MTS data, we construct
sequential graphs
$\\{\mathcal{G}_{T}\\}_{T=1}^{\hat{L}},\mathcal{G}_{T}=(Z_{T},E_{T})$ from the
sample $x$, where $Z_{T}=\\{z_{m,T}\\}_{m=1}^{N}$ and
$E_{T}=\\{e_{mn,T}\\}_{m,n=1}^{N}$ represent the sensor features and
correlations, respectively, in the $T$-th graph.
$\\{\mathcal{G}^{s}_{T}\\}_{T=1}^{\hat{L}}$ and
$\\{\mathcal{G}^{t}_{T}\\}_{T=1}^{\hat{L}}$ represent the sequential graphs in
the source and target domains respectively.
### III-B Preliminary
#### III-B1 Graph Neural Network
Several previous studies have demonstrated the effectiveness of Graph Neural
Network (GNN) in capturing spatial dependencies [39, 40]. Motivated by this
capability, we employ GNN to capture the spatial dependencies, i.e., sensor
correlations, within our constructed sequential graphs. Specifically, we
utilize a variant of GNN known as Message Passing Neural Network (MPNN) [41]
to process each graph. The MPNN framework consists of two stages: propagation
and updating. During the propagation stage, the features of neighboring nodes
are propagated to the central node. Subsequently, the aggregated features are
updated using a non-linear function, such as a Multi-Layer Perceptron (MLP)
network. With MPNN, we can capture the spatial dependencies between sensors,
enabling the effective transfer of the dependency information across domains
in the subsequent alignment processes. The details of MPNN are shown in Eq.
(1), where sensor $m$ is the central node, and $\mathcal{N}(m)$ is the set of
its neighboring nodes.
$\begin{split}{h_{m,T}}&=\sum_{j\in\mathcal{N}(m)}{e_{mj,T}}{z_{j,T}},\\\
{z_{m,T}}&=ReLU({h_{m,T}}W_{G}).\end{split}$ (1)
#### III-B2 Deep Coral
Deep Coral [13] is one of the most typical metric-based methods for UDA.
Different from $\mathcal{L}_{2}$ or cosine distances which mainly employ
first-order statistics, Deep Coral focuses on minimizing the discrepancy in
second-order statistics between domains [42], making it possible to capture
complex data distributions for alignment. Given training batches with $n^{s}$
and $n^{t}$ samples in source and target domains respectively, we denote the
extracted features from these domains as
$\mathcal{H}^{s}\in\mathbb{R}^{n^{s}\times{f}}$ and
$\mathcal{H}^{t}\in\mathbb{R}^{n^{t}\times{f}}$, where $f$ represents the
feature dimension. With the features, we can compute the discrepancy of
second-order statistics with Deep Coral
$M_{c}(\mathcal{H}^{s},\mathcal{H}^{t})$ as Eq. (2), where $||\cdot||_{F}^{2}$
is the squared matrix Frobenius norm. By minimizing the Deep Coral discrepancy
$\mathcal{L}_{Coral}$, the second-order statistics of the features are aligned
between the source and target domains, leading to comprehensive alignment
across domains.
$\begin{split}&C^{s}=\frac{(\mathcal{H}^{s})^{T}\mathcal{H}^{s}-\frac{1}{n^{s}}(\textbf{1}^{T}\mathcal{H}^{s})(\textbf{1}^{T}\mathcal{H}^{s})}{n^{s}-1},\\\
&C^{t}=\frac{(\mathcal{H}^{t})^{T}\mathcal{H}^{t}-\frac{1}{n^{t}}(\textbf{1}^{T}\mathcal{H}^{t})(\textbf{1}^{T}\mathcal{H}^{t})}{n^{t}-1},\\\
&M_{c}(\mathcal{H}^{s},\mathcal{H}^{t})=\frac{1}{4f^{2}}||C^{s}-C^{t}||_{F}^{2}.\end{split}$
(2)
### III-C Overall Structure
The overall structure of SEA is shown in Fig. 2. The superscripts $s$ and $t$
are omitted in Section III-C and III-D, as the forward processes in source and
target domains share the same graph-based encoder, including Multi-branch
Self-attention based Graph Construction (MSGC), GNN, and LSTM. First, we model
the spatial-temporal dependencies within MTS data for effective transfer
across domains. Given the sample $x$, we segment it into multiple patches,
which are constructed as sequential graphs by MSGC. Then, with the sequential
graphs, GNN and LSTM are leveraged to capture the spatial-temporal information
for learning sensor-level information, i.e., sensor features and correlations.
Second, in addition to supervised learning on the source domain, we
incorporate endo-feature alignment and exo-feature alignment techniques to
reduce domain discrepancy at both the local and global sensor levels. At the
local sensor level, we design endo-feature alignment, which involves aligning
sensor features and sensor correlations by proposed Sensor Feature Alignment
(SFA) and Sensor Correlation Alignment (SCA) respectively. The two components
are further improved by both the high-order statistic alignment and multi-
graph alignment to form iSFA and iSCA. The high-order statistic alignment aims
to achieve comprehensive alignment by capturing complex data distributions
while the multi-graph alignment aims to align evolving distributions at the
sensor level across domains. At the global sensor level, we design exo-feature
alignment to align global features by enforcing restrictions. We introduce the
details of each module in subsequent sections.
### III-D Graph-based Encoder
#### III-D1 Segmentation
To capture the evolving distributions within MTS data, we employ a
segmentation approach. In this approach, a sample
$x\in\mathbb{R}^{N\times{L}}$ is segmented into multiple patches, each with a
size of $d$, yielding sequential features denoted as
$\\{Z_{T}\\}_{T=1}^{\hat{L}}\in\mathbb{R}^{N\times{\hat{L}\times{d}}}$. Here,
$\hat{L}=\lfloor{\frac{L}{d}}\rfloor$ represents the number of patches, and
$d$ represents the feature dimension. The features within the $T$-th patch
$Z_{T}\in\mathbb{R}^{N\times{d}}$ capture the distributions in local temporal
intervals. Within each patch, the features of the $m$-th sensor are denoted as
$z_{m,T}\in\mathbb{R}^{d}$. Additionally, we incorporate a nonlinear function
$f_{L}(\cdot)$ to enhance the nonlinear expressiveness of the model.
#### III-D2 MSGC
We construct sequential graphs with the sequential features. Traditionally,
various methods have been employed to construct graphs for MTS data, such as
dot-product[43], Euclidean distance[40], and concatenation [44]. However, the
traditional methods are insufficient to model comprehensive relationships of
sensors due to the complex topological structures presented within MTS data.
To solve the limitation, we propose MSGC, which constructs a graph by
combining multiple graphs generated from various weight initializations across
multiple branches. Each branch employs differently weighted matrices to model
sensor correlations from distinct perspectives, allowing us to capture
comprehensive sensor correlations. Specifically, we adopt query and key to
learn the edge weights between sensors. Eq. (3) describes the edge between
sensors $m$ and $n$ in the $i$-th branch.
$\begin{split}q_{m,T}^{i}&=z_{m,T}W_{Q}^{i},k_{n,T}^{i}=z_{n,T}W_{K}^{i},\\\
e_{mn,T}^{i}&=\frac{\frac{q_{m,T}^{i}(k_{n,T}^{i})^{T}}{\sqrt{d}}}{\sum_{j=1}^{N}\frac{q_{m,T}^{i}(k_{j,T}^{i})^{T}}{\sqrt{d}}},\\\
\end{split}$ (3)
where we treat the $m$-th sensor $z_{m,T}$ as the query sensor and the $n$-th
sensor $z_{n,T}$ as the key sensor, and use this information to learn the edge
weight between them for the $i$-th branch, i.e., $e_{mn,T}^{i}$. The edge
weights are then normalized using the softmax function to ensure they fall
within the range of [0, 1].
To combine the graphs from multiple branches, we calculate the average of the
corresponding elements, i.e.,
$e_{mn,T}=\sum_{i=1}^{n_{b}}\frac{e_{mn,T}^{i}}{n_{b}}$, where $n_{b}$
represents the number of branches. This operation allows us to obtain the
sensor correlations
$E_{T}=\\{e_{mn,T}\\}_{m,n=1}^{N}\in\mathbb{R}^{N\times{N}}$ for the $T$-th
graph. By conducting similar operations in other patches, we can finally
obtain the sequential correlations
$\\{E_{T}\\}_{T=1}^{\hat{L}}\in\mathbb{R}^{\hat{L}\times{N}\times{N}}$.
Figure 3: Left: Sensor correlation alignment is to make the correlations
between domains identical. Middle: Sensor feature alignment is to make the
features from the corresponding sensors in two domains similar and the data
from the different sensors in two domains different. Right: Exo-feature
alignment reduces the discrepancy between the global features in different
domains.
#### III-D3 Capturing Spatial-Temporal Information
With the constructed sequential graphs, we employ two models, MPNN and LSTM,
to capture the spatial-temporal dependencies within MTS data. First, we
utilize MPNN (Eq. (1)) to capture the sensor correlations within each
sequential graph. In this process, the graphs are processed with shared
weights to save training costs and trainable parameters. Once the features of
each sensor in the sequential graphs have been updated using MPNN, our focus
shifts to capture temporal dependencies. It is noted that features from
corresponding sensors across the sequential graphs show temporal dependencies.
To capture these dependencies, we employ LSTM, which captures the dependency
information for each sensor by taking its features
$z_{m}\in\mathbb{R}^{{\hat{L}}\times{d}}$ across sequential graphs. After GNN
and LSTM, we finally obtain updated sequential features
$\\{Z_{T}\\}_{T=1}^{\hat{L}}\in\mathbb{R}^{\hat{L}\times{N}\times{d}}$. By
combining the capabilities of the two networks, we can effectively capture the
spatial-temporal dependencies within MTS data, allowing us to transfer the
dependency information across domains through the alignment at the local and
global sensor levels in the subsequent stages.
### III-E SEnsor Alignment
To reduce the discrepancy between domains for MTS-UDA, we design endo-feature
alignment and exo-feature alignment at the local and global sensor levels
respectively.
#### III-E1 Endo-feature alignment
Existing UDA methods reduce domain discrepancy by aligning global features
across domains, which is applicable for the data coming from a single source.
However, MTS data originate from multiple sensors, with each sensor following
various distributions. When adapting existing UDA methods to MTS-UDA, they can
consider global distributions by treating all sensors as a whole yet ignore
the distinct distributions of each sensor. This results in the misalignment at
the sensor level, restricting the performance of the learned model
transferring from the source domain to the target domain. To address this
limitation, we propose an endo-feature alignment method designed to align
sensor-level information and avoid misalignment for each sensor when
transferring knowledge across domains.
To make better endo-feature alignment, it is necessary to note that the
sensor-level information consists of both the sensor features and sensor
correlations. Sensor features represent the properties of each sensor, while
sensor correlations represent the interactive information between sensors. For
illustration, we present a scenario with temperature and fan speed sensors
deployed together to monitor the status of a machine. It is desirable for the
data from temperature sensors in two domains to follow similar distributions.
Additionally, the correlation between the sensors should also demonstrate
consistent patterns across domains, such as the fan speed consistently
increasing with temperature rises. Based on these considerations, aligning
sensor features and sensor correlations across domains simultaneously is
important to ensure comprehensive alignment at the sensor level.
##### Sensor Correlation Alignment
To explore consistent patterns within the interactions of sensors across
domains, we propose to align the sensor correlations. Specifically, our aim is
to align the distribution of correlations between any two sensors across
domains. To achieve this, we propose sensor correlation alignment (SCA), which
involves individually aligning each edge of the graphs as shown in Fig. 3
Left.
Given training batches with $n^{s}$ and $n^{t}$ samples in source and target
domains, we represent sensor correlations across sequnetial graphs as
$\mathcal{E}^{s}\in\mathbb{R}^{n^{s}\times{\hat{L}}\times{N}\times{N}}$ and
$\mathcal{E}^{t}\in\mathbb{R}^{n^{t}\times{\hat{L}}\times{N}\times{N}}$
respectively, with
$\mathcal{E}_{T}^{s}\in\mathbb{R}^{n^{s}\times{N}\times{N}}$ and
$\mathcal{E}_{T}^{t}\in\mathbb{R}^{n^{t}\times{N}\times{N}}$ signifying sensor
correlations for the $T$-th graph. Here,
$\mathcal{E}_{T}^{s}=\\{\mathcal{E}_{mn,T}^{s}\\}_{m,n=1}^{N}$, wherein
$\mathcal{E}_{mn,T}^{s}=\\{e^{s}_{i,mn,T}\\}_{i}^{n^{s}}$. For SCA, alignment
of sensor correlations across domains is achieved based on averaged sensor
correlations. Specifically, for the edge connecting sensors $m$ and $n$, its
mean value across the training batch is computed as
$e^{s}_{mn,T}=\sum_{i}^{n^{s}}e^{s}_{i,mn,T}/n^{s}$. Subsequently, the
averaged sensor correlations over sequential graphs are calculated as
$e^{s}_{mn}=\sum_{T}^{\hat{L}}e^{s}_{mn,T}/{\hat{L}}$. Next, we minimize the
averaged sensor correlations across domains as shown in Eq. (4), where
$|\cdot|$ is absolute value. This involves minimizing the expected discrepancy
of sensor correlations between the source and target domains. By doing so, we
can reduce distribution discrepancies for sensor correlations in source and
target domains.
$\begin{split}e_{mn}^{st}&=e_{mn}^{s}-e_{mn}^{t},\\\
\mathcal{L}_{SCA}&=\mathbb{E}(|e_{mn}^{st}|).\end{split}$ (4)
##### Sensor Feature Alignment
Next, we focus on aligning sensor features between domains. Our assumption is
that features originating from corresponding sensors in two domains should
exhibit similar distributions. Meanwhile, compared with features from
corresponding sensors, features from different sensors in two domains should
have distinct properties. For example, as shown in Fig. 3 Middle, the
distribution of sensor 1 in the source domain should be more similar to the
distribution of its corresponding sensor in the target domain than to the
distributions of other sensors (e.g., sensors 2, 3, and 4) in the target
domain. To achieve this goal, we propose a sensor contrasting mechanism that
facilitates the alignment of sensor features between domains.
Given training batches with $n^{s}$ and $n^{t}$ samples, we define sensor
features as
$\mathcal{Z}^{s}\in\mathbb{R}^{n^{s}\times{\hat{L}}\times{N}\times{d}}$ and
$\mathcal{Z}^{t}\in\mathbb{R}^{n^{t}\times{\hat{L}}\times{N}\times{d}}$, with
$\mathcal{Z}_{T}^{s}\in\mathbb{R}^{n^{s}\times{N}\times{d}}$ and
$\mathcal{Z}_{T}^{t}\in\mathbb{R}^{n^{t}\times{N}\times{d}}$ denoting sensor
features for the $T$-th graph. Here,
$\mathcal{Z}_{T}^{s}=\\{\mathcal{Z}_{n,T}^{s}\\}_{n=1}^{N}$, wherein
$\mathcal{Z}_{n,T}^{s}=\\{z^{s}_{i,n,T}\\}_{i}^{n^{s}}$. In SFA, domain
contrasting is realized based on averaged sensor features. In Eq. (5),
$p^{s}_{m}$ represents the averaged sensor features over multiple graphs
within a training batch, calulated as
$p^{s}_{m}=\sum_{T}^{\hat{L}}\sum_{i}^{n^{s}}z^{s}_{i,n,T}/n^{s}\hat{L}$.
$\begin{split}\mathcal{L}_{SFA}&=-\frac{1}{N}\sum_{n}^{N}log\frac{e^{\varphi(p_{m}^{s},p_{m}^{t})}}{\sum_{p_{j}^{t}\in{P^{t}}}e^{\varphi(p_{m}^{s},p_{j}^{t})}},\\\
\varphi(p_{m}^{s},p_{m}^{t})&=p_{m}^{s}(p_{m}^{t})^{T}.\end{split}$ (5)
##### Exo-feature alignment.
To reduce the domain discrepancy at the global level, we enforce restrictions
on the global features mapped by sensor features between domains.
Specifically, we first need to learn the global features of each sample. This
involves stacking all sensor features across the sequential graphs and
subsequently employing an MLP network to map the stacked features as global
features $\mathcal{P}^{s}$ and $\mathcal{P}^{t}$ in source and target domains,
as shown in Fig. 3 Right. With the global features, we can then align them
with Deep Coral, expressed as
$\mathcal{L}=\mathcal{M}_{c}(\mathcal{P}^{s},\mathcal{P}^{t})$. Finally, the
overall loss function (6) is minimized:
$\begin{split}\min\mathcal{L}=\mathcal{L}_{C}+\mathcal{L}_{EXO}+\mathcal{L}_{Endo},\\\
\mathcal{L}_{Endo}=\lambda_{SCA}\mathcal{L}_{SCA}+\lambda_{SFA}\mathcal{L}_{SFA}.\end{split}$
(6)
where $\mathcal{L}_{C}$ is the cost function (such as mean square error loss
or cross-entropy loss, depending on specific tasks) computed by the source
data and labels, and $\lambda_{SCA}$ and $\lambda_{SFA}$ are hyperparameters
to balance the effect of SCA and SFA respectively.
## IV SEA++ Model
While the method described above offers a viable approach for achieving sensor
correlation and sensor feature alignment, there is still room for improvement
to enhance endo-feature alignment. Two key directions warrant consideration.
First, SEA employs first-order statistics for alignment, such as averaged
sensor features over a training batch used in the sensor contrastive
mechanism. However, the first-order statistical alignment method is limited in
dealing with scenarios with complex data distributions. This limitation may
restrict the ability to achieve comprehensive alignment across domains,
leading to suboptimal performance in certain scenarios. To address that, we
propose introducing second-order statistical metrics for alignment to achieve
more comprehensive endo-feature alignment.
Second, the constructed sequential graphs, comprising sequential features and
correlations, capture evolving distributions within local temporal intervals
and should be aligned across domains. We consider the example of monitoring
machine health status for illustration. As the machine’s health deteriorates,
the distribution of a fan speed sensor may change due to increased friction or
other factors. Similarly, the interaction between the fan speed sensor and a
temperature sensor will also be affected and undergo changes. Specifically,
the same fan speeds might lead to higher temperature increases due to a
machine in poor health, e.g., friction increasing. Thus, we emphasize the
importance of aligning the evolving distributions at the sensor level between
domains. However, SEA employs averaged sensor correlations and sensor features
over the sequential graphs, limiting its ability to capture the evolving
distributions for alignment. To improve this, we introduce a multi-graph
alignment technique, aiming to effectively align evolving distributions
between domains.
Next, we begin by discussing SCA and SFA enhanced by high-order statistical
alignment within each sequential graph. Subsequently, we focus on combining
the alignment across all sequential graphs by employing our multi-graph
alignment.
### IV-A Sensor Correlation Alignment
Given training batches with $n^{s}$ and $n^{t}$ samples in source and target
domains, we have sensor correlations for the $T$-th graph, denoted as
$\mathcal{E}_{T}^{s}\in\mathbb{R}^{n^{s}\times{N}\times{N}}$ and
$\mathcal{E}_{T}^{t}\in\mathbb{R}^{n^{t}\times{N}\times{N}}$ respectively.
Instead of leveraging averaged sensor correlations within batches, we
introduce second-order statistics with Deep Coral, i.e., $M_{c}$ in Eq. (2),
to align the sensor correlations between the source and target domains.
Building upon this idea, we enhance SCA and introduce improved SCA (iSCA) as
shown in Eq. (7).
$\begin{split}d_{mn,T}^{st}&=M_{c}(\mathcal{E}_{mn,T}^{s},\mathcal{E}_{mn,T}^{t}),\\\
\mathcal{L}_{iSCA,T}&=\mathbb{E}(d_{mn,T}^{st})=\frac{\sum_{m}^{N}\sum_{n}^{N}d_{mn,T}^{st}}{mn}.\end{split}$
(7)
By aligning each edge between domains with the second-order statistical metric
and minimizing the distribution discrepancy of sensor correlations, we can
achieve more comprehensive alignment for sensor correlations.
### IV-B Sensor Feature Alignment
Next, we focus on improving sensor feature alignment. Similar to SCA, the
sensor contrasting mechanism in SFA is realized with the averaged sensor
features over the training batches in source and target domains. Thus, SFA
encounters a challenge similar to SCA. Specifically, the contrasting-based
alignment procedure in Eq. (5) relies on first-order statistics, which might
not achieve comprehensive alignment when dealing with complex distributions.
To overcome this limitation, we enhance SFA and introduce improved SFA (iSFA)
by incorporating Deep Coral into the contrasting process.
In Eq. (8), iSFA begins by calculating sensor-level distribution discrepancies
using Deep Coral, which takes into account second-order statistics.
Subsequently, iSFA leverages the sensor-level discrepancies to perform
contrasting between domains. This involves minimizing the distribution
discrepancies between corresponding sensors across domains while maximizing
the distribution discrepancies between different sensors across domains. This
approach allows for more comprehensive alignment of sensor features.
$\begin{split}\mathcal{L}_{iSFA,T}=-\frac{1}{N}\sum_{n}^{N}log\frac{e^{M_{c}(\mathcal{Z}_{n,T}^{s},\mathcal{Z}_{n,T}^{t}})}{\sum^{N}_{j}e^{M_{c}(\mathcal{Z}_{n,T}^{s},\mathcal{Z}_{j,T}^{t})}}.\end{split}$
(8)
### IV-C Multi-Graph Alignment
To combine the alignment for sensor correlations and sensor features across
all sequential graph, a straightforward approach would involve computing the
average alignment for these graphs. However, the distributions between domains
for different graphs might exhibit various discrepancies. To account for this
and enhance domain adaptation, we propose a method to learn adaptive weights
for adjustment, as shown in Eq. (9).
$\begin{split}\mathcal{L}_{iEndo}&=\sum_{T}\mathcal{W}_{T}(\lambda_{SCA}\mathcal{L}_{iSCA,T}+\lambda_{SFA}\mathcal{L}_{iSFA,T}),\\\
\mathcal{W}_{T}&=M_{c}([\mathcal{Z}_{1,T}^{s},...,\mathcal{Z}_{\hat{L},T}^{s}],[\mathcal{Z}_{1,T}^{t},...,\mathcal{Z}_{\hat{L},T}^{t}]),\end{split}$
(9)
where $[a,...,b]$ represents concatenating the elements from $a$ to $b$. In
Eq. (9), the learnable weights $\mathcal{W}_{T}$ are determined by evaluating
the distribution discrepancy of the $T$-th graph between domains.
Specifically, $\mathcal{W}_{T}$ will be large when there is a substantial
discrepancy in the distributions of the $T$-th graph between the source and
target domains. By assigning larger weights to the graphs with larger
distribution discrepancies between domains, we prioritize alignment in those
graphs. This adaptive weighting scheme allows us to better adapt the model to
the evolving distributions between domains. Furthermore, we also introduce
$\lambda_{SCA}$ and $\lambda_{SFA}$ to balance the effect of iSCA and iSFA. By
combining the exo-feature alignment and the improved endo-feature alignment,
the improved overall loss function (10) can be obtained as:
$\begin{split}\min\mathcal{L}&=\mathcal{L}_{C}+\mathcal{L}_{EXO}+\mathcal{L}_{iEndo}.\end{split}$
(10)
## V Experiment
### V-A Datasets and Setup
To evaluate the performance of SEA and SEA++ under various MTS scenarios, we
conducted experiments on two public datasets: C-MAPSS for remaining useful
life prediction [45], which collects MTS data from different sensor types, and
Opportunity HAR for human activity recognition [46], which collects MTS data
from different locations.
C-MAPSS describes the degradation process of aircraft engines. The MTS data in
C-MAPSS were originated from 14 sensors to measure various physical
parameters, such as temperature, fan speed, and pressure. The dataset includes
four sub-datasets collected under different working conditions and fault
modes, where each sub-dataset represents one domain. We processed the sub-
datasets following the data preparation in the previous work [47], and data
annotations represent the remaining useful life circle of engines.
Opportunity HAR describes human activities. The MTS data in Opportunity HAR
were originated from 113 sensors deployed to various locations of the human
body, such as hand and leg. After removing the sensors with constant values,
110 sensors were used for evaluation. The dataset includes the data collected
from four subjects, each representing one domain. The data annotations include
two levels: 1). locomotion representing the low-level tasks including four
categories, sitting, standing, walking, and lying down; 2). gestures
representing the high-level tasks including 17 various actions. Following the
experimental settings in the previous work [48], we adopted the low-level
tasks. As some values in the data were missing, we adopted the linear
interpolation approach to fill in the missing positions. To construct the
training dataset, we adopted a sliding window with a size of 128 and an
overlapping of 50% following the previous work [48].
The experiments include three parts, comparisons with state-of-the-art
methods, the ablation study, and sensitivity analysis. All experiments were
run ten times with average results reported to remove the effect of random
initialization. Besides, we set the batch size as 50 and employed the
optimizer as Adam with a learning rate of 0.001. Meanwhile, we adopted 20
training epochs for training our model. Furthermore, we built and trained our
model based on Pytorch 1.9 and NVIDIA GeForce RTX 3080Ti GPU.
We employed different evaluation indicators to assess the performance of our
method on the two distinct datasets. For C-MAPSS experiments, which involve
predicting the RUL of an engine, we utilized Root Mean Square Error (RMSE) and
the Score function [47, 49, 50, 51] as evaluation metrics. Lower values for
both indicators indicate better model performance. For Opportunity HAR
experiments, which entail a classification task, we employed accuracy as the
evaluation metric. High accuracy values signify superior model performance.
### V-B Comparisons with State-of-the-Art
TABLE I: The Comparisons with SOTAs in C-MAPSS (R: RMSE; S: Score) Models (R) | 1$\to$2 | 1$\to$3 | 1$\to$4 | 2$\to$1 | 2$\to$3 | 2$\to$4 | 3$\to$1 | 3$\to$2 | 3$\to$4 | 4$\to$1 | 4$\to$2 | 4$\to$3 | Avg.
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Target | 12.93 | 13.33 | 14.66 | 12.36 | 13.33 | 14.66 | 12.36 | 12.93 | 14.66 | 12.36 | 12.93 | 13.33 | 13.32
Source | 28.76 | 35.28 | 27.21 | 21.86 | 39.80 | 33.50 | 37.90 | 30.33 | 24.10 | 33.76 | 25.31 | 21.75 | 29.96
DDC | 43.25 | 39.48 | 42.99 | 40.07 | 39.46 | 43.01 | 40.93 | 43.43 | 43.72 | 41.58 | 43.38 | 39.61 | 41.74
Coral | 16.72 | 26.49 | 25.03 | 14.47 | 37.41 | 31.13 | 32.82 | 23.40 | 19.84 | 28.37 | 32.00 | 22.47 | 25.85
DANN | 17.02 | 23.33 | 24.20 | 14.16 | 30.00 | 25.91 | 19.89 | 20.31 | 19.95 | 31.74 | 24.68 | 17.94 | 22.43
BNM | 48.18 | 41.61 | 46.45 | 56.66 | 48.40 | 62.02 | 49.39 | 48.89 | 42.98 | 40.76 | 43.71 | 41.89 | 47.58
SDAT | 16.50 | 24.11 | 23.07 | 13.57 | 29.76 | 26.38 | 25.34 | 20.29 | 18.63 | 20.13 | 23.23 | 17.65 | 21.55
AdvSKM | 31.09 | 34.16 | 40.74 | 23.93 | 32.41 | 35.73 | 28.47 | 22.17 | 26.45 | 28.56 | 22.75 | 23.51 | 29.16
CoDATs | 15.91 | 26.67 | 25.24 | 15.59 | 24.52 | 25.64 | 21.25 | 21.04 | 19.77 | 19.41 | 18.99 | 17.85 | 20.99
CtsADA | 18.31 | 29.23 | 29.49 | 23.20 | 54.69 | 41.60 | 38.14 | 24.67 | 19.86 | 31.63 | 25.71 | 23.97 | 30.04
CLUDA | 25.31 | 36.80 | 33.24 | 22.51 | 33.82 | 34.63 | 26.48 | 24.09 | 26.29 | 24.33 | 25.14 | 22.18 | 27.90
SEA | 15.63 | 21.09 | 23.21 | 13.29 | 24.62 | 25.32 | 19.69 | 19.44 | 19.37 | 18.92 | 17.70 | 16.45 | 19.56
SEA++ | 15.41 | 19.38 | 22.12 | 13.00 | 24.60 | 24.90 | 19.19 | 19.25 | 17.75 | 18.43 | 18.16 | 16.79 | 19.08
Models (S) | 1$\to$2 | 1$\to$3 | 1$\to$4 | 2$\to$1 | 2$\to$3 | 2$\to$4 | 3$\to$1 | 3$\to$2 | 3$\to$4 | 4$\to$1 | 4$\to$2 | 4$\to$3 | Avg.
Target | 678 | 330 | 1027 | 241 | 330 | 1027 | 241 | 678 | 1027 | 241 | 678 | 330 | 569
Source | 9341 | 4751 | 4699 | 2340 | 8335 | 11018 | 24892 | 26878 | 11006 | 16318 | 8771 | 2047 | 10866
DDC | 23323 | 6164 | 34891 | 11581 | 6578 | 35122 | 22236 | 59400 | 80774 | 29592 | 58029 | 15239 | 31911
Coral | 1158 | 1468 | 3613 | 406 | 6241 | 8492 | 15298 | 12661 | 3048 | 6742 | 25763 | 4758 | 7471
DANN | 1073 | 1612 | 3340 | 392 | 3160 | 4667 | 1666 | 3896 | 2381 | 22225 | 4620 | 762 | 4150
BNM | 24265 | 44650 | 19017 | 526203 | 56769 | 29689 | 46262 | 29188 | 39559 | 11618 | 20566 | 65807 | 76133
SDAT | 974 | 1403 | 2729 | 300 | 4360 | 3859 | 3756 | 3134 | 2289 | 966 | 2369 | 678 | 2235
AdvSKM | 13313 | 5012 | 18444 | 2344 | 4406 | 14624 | 6601 | 4516 | 11857 | 6332 | 4474 | 5218 | 8095
CoDATs | 1043 | 2854 | 5074 | 836 | 1533 | 5094 | 1300 | 3993 | 2352 | 1065 | 2205 | 697 | 2337
CtsADA | 1467 | 1965 | 6832 | 4136 | 29153 | 22515 | 47682 | 17418 | 2138 | 13019 | 15436 | 1593 | 13613
CLUDA | 2686 | 4754 | 6337 | 873 | 3501 | 7159 | 2389 | 3232 | 3078 | 1658 | 5566 | 1093 | 3527
SEA | 922 | 1302 | 3786 | 287 | 2090 | 3782 | 1163 | 2990 | 2278 | 939 | 1930 | 635 | 1842
SEA++ | 881 | 803 | 2669 | 295 | 1510 | 3220 | 1117 | 3045 | 1558 | 859 | 2072 | 572 | 1550
TABLE II: The Comparisons with SOTAs in Opportunity HAR Models | 1$\to$2 | 1$\to$3 | 1$\to$4 | 2$\to$1 | 2$\to$3 | 2$\to$4 | 3$\to$1 | 3$\to$2 | 3$\to$4 | 4$\to$1 | 4$\to$2 | 4$\to$3 | Avg.
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Target | 91.90 | 97.95 | 95.60 | 98.40 | 97.95 | 95.60 | 98.40 | 91.90 | 95.60 | 98.40 | 91.90 | 97.95 | 95.96
Source | 47.50 | 64.50 | 60.83 | 73.00 | 71.00 | 65.67 | 64.83 | 56.67 | 72.17 | 66.67 | 65.50 | 74.67 | 65.25
DDC | 40.50 | 58.50 | 55.50 | 43.35 | 58.30 | 55.50 | 45.00 | 40.50 | 55.50 | 45.00 | 40.50 | 58.50 | 49.72
Coral | 78.55 | 82.71 | 78.17 | 81.58 | 79.93 | 79.07 | 78.50 | 75.40 | 84.88 | 78.56 | 80.38 | 87.88 | 80.47
DANN | 69.62 | 85.12 | 79.00 | 85.62 | 81.62 | 74.88 | 76.50 | 75.75 | 86.12 | 84.25 | 74.12 | 86.62 | 79.94
BNM | 44.00 | 61.00 | 52.00 | 45.00 | 61.00 | 52.00 | 45.00 | 44.00 | 52.00 | 45.50 | 44.00 | 61.00 | 50.54
LMMD | 56.25 | 61.17 | 59.08 | 45.33 | 58.50 | 55.50 | 49.42 | 46.83 | 57.33 | 59.00 | 44.75 | 61.75 | 54.58
SDAT | 84.92 | 72.42 | 77.00 | 87.25 | 83.00 | 76.17 | 87.75 | 84.25 | 87.25 | 74.58 | 66.75 | 91.50 | 81.07
AdvSKM | 58.10 | 58.25 | 51.75 | 45.40 | 40.95 | 48.40 | 69.60 | 72.20 | 70.60 | 68.75 | 55.50 | 58.20 | 58.14
CoDATs | 80.17 | 83.33 | 82.33 | 88.67 | 80.50 | 81.00 | 89.25 | 79.12 | 85.75 | 84.67 | 79.17 | 86.17 | 83.34
CtsADA | 80.00 | 73.67 | 76.08 | 69.75 | 78.17 | 79.08 | 67.75 | 77.33 | 64.75 | 74.58 | 52.17 | 75.00 | 72.36
SLARDA | 80.38 | 80.50 | 79.50 | 82.75 | 78.88 | 79.25 | 75.88 | 78.62 | 70.12 | 80.62 | 84.00 | 77.25 | 78.98
CLUDA | 70.45 | 73.75 | 74.40 | 73.20 | 74.25 | 73.95 | 61.65 | 55.10 | 64.30 | 65.75 | 61.05 | 71.55 | 68.28
SEA | 86.35 | 86.15 | 86.00 | 86.35 | 84.35 | 81.75 | 87.25 | 83.05 | 88.70 | 84.90 | 84.25 | 92.25 | 85.42
SEA++ | 86.85 | 90.30 | 90.85 | 90.25 | 83.85 | 85.40 | 89.80 | 83.10 | 88.10 | 91.70 | 86.95 | 92.55 | 88.31
We compared SEA and SEA++ with State-Of-The-Art methods (SOTAs), including
general UDA methods and TS UDA methods. General UDA methods were designed to
be applicable to various tasks and domains, regardless of the specific data
type, including DDC [9], Deep Coral [52], DANN [53], BNM [54], LMMD [55], and
SDAT [56]. TS UDA methods were specifically designed for TS data, including
AdvSKM [37], CoDATs [35], CtsADA [47], SLARDA [48], and CLUDA [57]. For fair
comparisons, all methods were run ten times based on the same feature
extractor (i.e., GNN-LSTM). LMMD and SLARDA were designed specifically for
multi-class classification UDA, so they only had results on Opportunity HAR.
Notably, in the previous SEA paper, different cross-domain scenarios were
based on various sample lengths. For example, the cross-domain scenario
1$\to$2 was based on a time length of 72 while 3$\to$4 was based on a time
length of 45. However, this training setting may be inapplicable in real-world
systems where obtaining samples with a fixed time length is desired. Thus, all
results in this paper, particularly in the context of the C-MAPSS dataset,
have been re-conducted based on a consistent time length of 60 timestamps.
Additionally, we have included the results of Target-only (Target) and Source-
only (Source) baselines for comparison. The target-only baseline represents
the upper bound of UDA, where training and testing were performed on the
target domain only. The source-only baseline represents the lower bound of
UDA, where only the source data were used for training without any UDA methods
applied.
TABLE I shows the RMSE and Score results for 12 cross-domain scenarios in
C-MAPSS. From the results, we observe that both SEA and SEA++ achieve better
performance than SOTAs in most cross-domain scenarios. Specifically, compared
to the methods excluding SEA++, SEA performs the best performance in 9 out of
12 scenarios and the second-best performance in the remaining scenarios with
very marginal gaps compared to the best methods. We utilize the RMSE results
in C-MAPSS for illustration. In the cross-domain scenario 1$\to$3, SEA
outperforms the second-best method by a significant 9.6%. In scenario 1$\to$4,
SEA is the second-best method and is only 0.6% weaker than the top-performing
method. These results demonstrate the effectiveness of SEA compared to SOTAs.
However, SEA exhibits minor limitations that make it slightly less effective
in specific cases, motivating the development of SEA++.
Regarding the results of SEA++, it not only achieves better performance in
scenarios where SEA performs the best but also achieves the best performance
in 2 out of 3 scenarios where SEA is the second-best. For example, in the
cross-domain scenario 1$\to$4 where SEA is not the top-performing, SEA++
improves by 4.1% compared to the best conventional method. Although SEA++ is
slightly weak in 2$\to$3, the gap is further reduced, being only 0.3% weaker
than the best method. In terms of average RMSE results, SEA outperforms the
best conventional methods by 9.2%. With improved endo-feature alignment and
multi-graph alignment, SEA++ further enhances the performance, surpassing SEA
by 2.45% and outperforming the best conventional method by 11.4%. Similar
improvements can be observed in the Score results of C-MAPSS and Opportunity
HAR in TABLE II. Taking the average accuracy in Opportunity HAR for
illustration, SEA improves by 2.08% compared to the best conventional method.
Additionally, SEA++ outperforms SEA by 2.89% and outperforms the best
conventional method by 4.97%.
These results indicate the effectiveness of SEA, showing the necessity of
aligning sensor-level information. Furthermore, SEA++ addresses the minor
limitations of SEA, resulting in improved performance that establishes its
state-of-the-art status compared to traditional methods.
### V-C Ablation Study
TABLE III: The Ablation Study in C-MAPSS (R: RMSE; S: Score)
Variants (R) | 1$\to$2 | 1$\to$3 | 1$\to$4 | 2$\to$1 | 2$\to$3 | 2$\to$4 | 3$\to$1 | 3$\to$2 | 3$\to$4 | 4$\to$1 | 4$\to$2 | 4$\to$3 | Avg.
---|---|---|---|---|---|---|---|---|---|---|---|---|---
w/ Exo (w/o endo) | 16.96 | 27.74 | 28.28 | 14.47 | 33.70 | 28.18 | 32.88 | 25.45 | 19.98 | 30.75 | 23.19 | 20.03 | 25.13
w/ Exo + SCA | 15.96 | 23.83 | 25.40 | 14.12 | 27.32 | 27.13 | 26.89 | 23.97 | 19.87 | 25.94 | 20.62 | 17.97 | 22.42
w/ Exo + SCA + MGA | 15.96 | 21.95 | 22.68 | 13.65 | 28.93 | 26.31 | 27.98 | 21.64 | 18.70 | 26.15 | 20.31 | 17.33 | 21.80
w/ Exo + SCA + 2-order | 15.94 | 21.61 | 23.05 | 13.32 | 28.76 | 26.94 | 23.60 | 20.73 | 18.41 | 22.40 | 19.85 | 17.25 | 20.99
w/ Exo + iSCA | 15.90 | 21.39 | 22.62 | 13.28 | 27.91 | 26.08 | 22.01 | 19.86 | 18.30 | 22.37 | 19.55 | 17.18 | 20.54
w/ Exo + SFA | 16.71 | 22.03 | 24.96 | 14.03 | 29.01 | 26.15 | 26.71 | 21.28 | 19.71 | 25.97 | 19.63 | 17.30 | 21.96
w/ Exo + SFA + MGA | 16.11 | 21.61 | 22.47 | 13.75 | 28.97 | 26.05 | 27.35 | 21.20 | 18.97 | 23.67 | 19.58 | 17.08 | 21.40
w/ Exo + SFA + 2-order | 16.15 | 22.08 | 22.57 | 13.77 | 28.51 | 26.45 | 23.99 | 20.67 | 18.19 | 22.57 | 19.87 | 17.85 | 21.06
w/ Exo + iSFA | 15.90 | 20.37 | 22.47 | 13.59 | 27.80 | 25.34 | 21.44 | 19.95 | 18.10 | 20.54 | 19.30 | 17.28 | 20.17
SEA++ | 15.41 | 19.38 | 22.12 | 13.00 | 24.60 | 24.90 | 19.19 | 19.25 | 17.75 | 18.43 | 18.16 | 16.79 | 19.08
Variants (S) | 1$\to$2 | 1$\to$3 | 1$\to$4 | 2$\to$1 | 2$\to$3 | 2$\to$4 | 3$\to$1 | 3$\to$2 | 3$\to$4 | 4$\to$1 | 4$\to$2 | 4$\to$3 | Avg.
w/ Exo (w/o endo) | 1156 | 1863 | 7246 | 406 | 3705 | 5160 | 17374 | 17854 | 2240 | 10247 | 6635 | 1224 | 6259
w/ Exo + SCA | 1101 | 1537 | 6392 | 360 | 2358 | 4657 | 11756 | 7351 | 2458 | 4082 | 3925 | 735 | 3893
w/ Exo + SCA + MGA | 980 | 979 | 2936 | 356 | 2255 | 3893 | 9340 | 7284 | 1974 | 4052 | 3652 | 664 | 3197
w/ Exo + SCA + 2-order | 995 | 1070 | 2838 | 328 | 2517 | 4287 | 3489 | 5954 | 1820 | 1976 | 3207 | 696 | 2431
w/ Exo + iSCA | 978 | 1045 | 2674 | 306 | 2076 | 3704 | 2270 | 3906 | 1571 | 1815 | 3084 | 679 | 2009
w/ Exo + SFA | 1052 | 1445 | 4347 | 368 | 2152 | 3855 | 6638 | 7266 | 2111 | 4406 | 3082 | 654 | 3115
w/ Exo + SFA + MGA | 949 | 1194 | 2634 | 364 | 1873 | 3697 | 6193 | 6023 | 1915 | 2443 | 3048 | 627 | 2580
w/ Exo + SFA + 2-order | 987 | 1099 | 2674 | 367 | 2233 | 4214 | 3654 | 7640 | 1621 | 2212 | 3208 | 707 | 2551
w/ Exo + iSFA | 946 | 1083 | 2634 | 342 | 1873 | 3598 | 1902 | 4515 | 1694 | 1395 | 2814 | 630 | 1952
SEA++ | 881 | 803 | 2669 | 295 | 1510 | 3220 | 1117 | 3045 | 1558 | 859 | 2072 | 572 | 1550
* •
**MGA represents Multi-Graph Alignment; 2-order represents high-order
statistical alignment.
TABLE IV: The Ablation Study in Opportunity HAR Variants | 1$\to$2 | 1$\to$3 | 1$\to$4 | 2$\to$1 | 2$\to$3 | 2$\to$4 | 3$\to$1 | 3$\to$2 | 3$\to$4 | 4$\to$1 | 4$\to$2 | 4$\to$3 | Avg.
---|---|---|---|---|---|---|---|---|---|---|---|---|---
w/ Exo (w/o endo) | 78.55 | 82.71 | 78.17 | 81.58 | 79.93 | 79.07 | 78.50 | 75.40 | 84.88 | 78.56 | 80.38 | 87.88 | 80.47
w/ Exo + SCA | 84.45 | 84.90 | 82.20 | 84.00 | 81.95 | 81.00 | 81.50 | 75.81 | 86.95 | 81.80 | 81.60 | 87.35 | 82.79
w/ Exo + SCA + MGA | 85.00 | 86.75 | 84.05 | 88.25 | 82.60 | 82.20 | 83.25 | 79.75 | 85.80 | 83.90 | 81.85 | 88.35 | 84.31
w/ Exo + SCA + 2-order | 85.20 | 87.40 | 84.90 | 88.35 | 82.30 | 82.00 | 81.10 | 76.75 | 87.15 | 81.35 | 84.40 | 87.30 | 84.02
w/ Exo + iSCA | 85.70 | 87.20 | 88.20 | 89.10 | 81.30 | 84.10 | 86.75 | 79.15 | 88.05 | 89.80 | 85.50 | 87.50 | 86.03
w/ Exo + SFA | 82.94 | 83.62 | 80.38 | 84.25 | 83.45 | 80.55 | 78.80 | 75.38 | 86.20 | 79.05 | 81.90 | 87.50 | 82.00
w/ Exo + SFA + MGA | 85.95 | 87.60 | 85.75 | 87.05 | 82.75 | 82.90 | 79.60 | 79.05 | 87.35 | 83.50 | 79.95 | 88.90 | 84.20
w/ Exo + SFA + 2-order | 84.50 | 87.40 | 84.25 | 87.00 | 84.90 | 82.45 | 82.15 | 80.95 | 87.75 | 85.45 | 83.60 | 88.45 | 84.90
w/ Exo + iSFA | 85.70 | 87.90 | 85.85 | 87.90 | 83.70 | 82.90 | 86.45 | 82.05 | 86.90 | 87.10 | 84.75 | 90.15 | 85.95
SEA++ | 86.85 | 90.30 | 90.85 | 90.25 | 83.85 | 85.40 | 89.80 | 81.50 | 88.10 | 91.70 | 86.95 | 92.55 | 88.18
To comprehensively evaluate the effectiveness of our endo-feature alignment
and the associated improvements, we have conducted extensive ablation studies
consisting of nine different variants. These variants can be categorized into
three aspects. The first aspect compares our method with a baseline that only
performs exo-feature alignment without utilizing any sensor-level alignment.
In the second and third aspects, we consider variants that combine exo-feature
alignment with one of the endo-feature alignment techniques: SCA or SFA. Each
aspect further includes four variants based on different improvements. We
utilize the variants with SCA for illustration. The “w/ Exo + SCA” variant
represents the use of the vanilla SCA for sensor correlation alignment. By
introducing Multi-Graph Alignment (MGA), we have the “w/ Exo + SCA + MGA”
variant, while still using alignment with first-order statistics. By
introducing alignment with second-order statistics without MGA, we have the
variant “w/ Exo + SCA + 2-order”. Finally, by incorporating both improvements,
i.e., MGA and second-order statistic alignment, we obtain the “w/ Exo + iSCA”
variant. The same variations were applied for SFA, resulting in four
additional variants.
TABLE III and TABLE IV show the ablation study on C-MAPSS and Opportunity HAR
respectively. Focusing on the average RMSE results on C-MAPSS for
illustration, we consider the variant “w/ Exo” as the baseline. We can first
observe that introducing SCA and SFA leads to improvements of 10.8% and 12.6%,
respectively, which indicates the effectiveness of endo-feature alignment. As
the endo-feature alignment still has minor limitations, we further introduce
MGA and second-order statistic alignment to improve the sensor-level
alignment. Introducing MGA leads to performance improvements of 13.3% (SCA)
and 14.9% (SFA), while introducing second-order statistic alignment improves
it by 16.5% (SCA) and 16.2% (SFA). The improvements indicate that MGA and
second-order statistic alignment both show effectiveness for better sensor-
level alignment. Thus, it is useful to combine them to further improve the
performance of our method. Combining both parts, the variants “w/ iSCA” and
“w/ iSFA” improve the performance by 18.3% and 19.7% respectively compared to
the baseline. Finally, both the sensor features and sensor correlations can be
aligned simultaneously. Thus, by combining iSCA and iSFA for endo-feature
alignment, we achieve a remarkable improvement of 24.1% compared to the
baseline. Similar trends are also observed in the experiments conducted on
Opportunity HAR, as shown in TABLE IV. For example, the average accuracy of
SEA++ increases by 7.61% compared to the model with exo-feature alignment
only. The results in the cross-domain scenarios of Opportunity HAR can further
verify the effectiveness of our proposed modules.
These experiments highlight the significance of endo-feature alignment in
achieving effective alignment at the sensor level for MTS-UDA. Furthermore,
the additional modules for endo-feature alignment, namely MGA and high-order
statistics alignment, prove to be highly effective in further improving our
model’s performance.
### V-D Sensitivity Analysis
Our method includes hyperparameters that require sensitivity analysis to
evaluate their impact on the method’s performance. We focus on four
hyperparameters: $\lambda_{SCA}$, $\lambda_{SFA}$, number of heads, and patch
sizes. The first two hyperparameters control the balance between sensor
correlation alignment and sensor feature alignment. The number of heads
determines the number of graph branches used to model spatial dependencies
between sensors. The patch sizes determine the number of sequential graphs
constructed. Conducting sensitivity analysis on these hyperparameters helps
determine their sensitivity and find optimal values. Notably, the sensitivity
analysis presented here pertains specifically to SEA++. Additional details
regarding the analysis related to SEA can be found in our appendix.
#### V-D1 Analysis for $\lambda_{SCA}$ and $\lambda_{SFA}$
Figure 4: The sensitivity analysis for $\lambda_{SCA}$ on C-MAPSS. Figure 5:
The sensitivity analysis for $\lambda_{SFA}$ on C-MAPSS. Figure 6: The
sensitivity analysis for $\lambda_{SCA}$ and $\lambda_{SFA}$ on Opportunity
HAR.
We have conducted experiments with various values of $\lambda_{SCA}$ and
$\lambda_{SFA}$ varying from 0.001 to 10 with an interval of $10\times$. Fig.
4 and 5 present the analysis of $\lambda_{SCA}$ and $\lambda_{SFA}$ on C-MAPSS
respectively, and Fig. 6 shows the analysis for $\lambda_{SCA}$ and
$\lambda_{SFA}$ on Opportunity HAR. From the results, we can find similar
trends for the two hyperparameters, which include two points. Firstly, our
method exhibits insensitivity to changes in the hyperparameters in most cases.
For example, our method is insensitive to $\lambda_{SCA}$ in cross-domain
scenarios such as 1$\to$4, 2$\to$1, and 2$\to$4 of C-MAPSS, and 1$\to$3,
2$\to$3, and 4$\to$2 of Opportunity HAR. Similarly, regarding $\lambda_{SFA}$,
we can also find the insensitivity in the scenarios 1$\to$2, 2$\to$1, and
3$\to$4 of C-MAPSS, and 1$\to$4, 3$\to$2, and 4$\to$3 of Opportunity HAR.
These results indicate that our method can achieve stable performance without
requiring extensive hyperparameter tuning, making it applicable to real-world
scenarios. Second, in the cases where our method is sensitive to the
hyperparameters, our method tends to achieve good performance when their
values are within the range of [0.001,1]. Taking the results on C-MAPSS as
examples, we observe that our method achieves poor performance when
$\lambda_{SCA}$ is set to 10 in the case of 3$\to$2, and when $\lambda_{SCA}$
is set to 0.001 in the case of 4$\to$1\. Similar trends can also be observed
for $\lambda_{SFA}$, e.g., $\lambda_{SFA}$=10 in 2$\to$3\. Based on these
observations, we recommend setting the values of these hyperparameters within
the range of 0.01 to 1, where our method consistently achieves good
performance.
#### V-D2 Analysis for the number of heads
Figure 7: The sensitivity analysis for different heads on C-MAPSS. Figure 8:
The sensitivity analysis for different heads on Opportunity HAR.
For the construction of each graph with MSGC, we have multiple branches with
differently initialized weights to model various aspects of sensor
correlations. In this part, we aim to evaluate the effect of the number of
branches on the model performance. Fig. 7 and 8 show the results on C-MAPSS
and Opportunity HAR respectively. In our analysis, we analyze the effect using
a maximum of 4 branches, as we observe that further increasing the number of
branches does not yield significant improvements. The results consistently
show that SEA++ obtains performance improvements with increasing the number of
branches. Specifically, SEA++ with just one branch tends to exhibit poorer
performance in most cases, such as in the 2$\to$3 cross-domain scenario of
C-MAPSS and the 2$\to$4 scenario of Opportunity HAR. With increasing the
number of branches, the performance improves. It is worth noting that the
improvements become less pronounced when the number of branches is large
enough, particularly when transitioning from 3 to 4 branches. This suggests
that using 3 branches is generally sufficient for effectively capturing the
spatial dependencies between sensors. Overall, the analysis demonstrates the
importance of leveraging multiple branches in MSGC for modeling the complex
relationships among sensors. Increasing the number of branches allows our
method to capture a more comprehensive understanding of sensor correlations,
leading to improved performance for MTS-UDA.
#### V-D3 Analysis for patch size
Figure 9: The sensitivity analysis for patch sizes on C-MAPSS. Figure 10: The
sensitivity analysis for patch sizes on Opportunity HAR.
To align the evolving distributions within MTS data, we segment an MTS sample
into multiple patches with a size of $d$, which are then utilized to construct
sequential graphs and conduct multi-graph alignment. Evaluating the effect of
patch size $d$ is important, as the size affects the construction of
sequential graphs. Notably, we conducted the analysis with different patch
sizes for C-MAPSS and Opportunity HAR as their samples have different lengths.
For C-MAPSS, the time length of each sample is fixed as 60, so we use the
patch sizes within [4,5,6,7,8,9,10] for patch size analysis. For Opportunity
HAR, the time length of each sample is fixed as 128, so we use the patch sizes
within [2,4,8,16,32].
Based on the analysis of C-MAPSS in Fig. 9 and Opportunity in Fig. 10, it is
evident that relatively large patch sizes tend to result in better performance
for our method. This observation holds true for several cross-scenarios, such
as 3$\to$1, 3$\to$2, and 3$\to$4 of C-MAPSS, where the best performance is
achieved with patch sizes of 8, 9, or 10. Similar observations can be found
for Opportunity HAR, such as 1$\to$3, 2$\to$1, and 3$\to$1, where the best
performance is achieved with patch sizes of 16 or 32. Conversely, setting too
small patch sizes leads to poor performance in most scenarios, such as 1$\to$3
and 1$\to$4 of C-MAPSS, and 1$\to$2 and 1$\to$3 of Opportunity HAR. The reason
behind this trend lies in the construction of sequential graphs and the
capturing of spatial dependencies for transferring across domains. By
segmenting MTS data into patches, we construct sequential graphs whose spatial
dependencies are captured for transferring across domains. In this process,
too small patch sizes may not contain enough information to fully capture the
complex spatial dependencies between sensors for each patch, resulting in
suboptimal performance in transferring the dependency information. Therefore,
it is recommended to use relatively large patch sizes to obtain better
performance using SEA++ for MTS-UDA.
## VI Conclusion
In this paper, we formulate Multivariate Time-Series Unsupervised Domain
Adaptation (MTS-UDA). We analyze the problems underlying this task and propose
SEnsor Alignment (SEA) to address these issues. To reduce the domain
discrepancy at both the local and global sensor levels, we design endo-feature
alignment and exo-feature alignment. At the local sensor level, endo-feature
alignment aligns sensor features and sensor correlations across domains,
preventing misalignment at the sensor level. To enhance the endo-feature
alignment, we further design SEA++, incorporating high-order statistic
alignment and a multi-graph alignment technique. These enhancements are
specifically designed to facilitate comprehensive alignment across domains and
effectively capture the evolving data distributions within MTS data. At the
global sensor level, we enforce restrictions on global sensor features to
reduce domain discrepancy. Our extensive experiments demonstrate the
effectiveness of SEA and SEA++ for MTS-UDA.
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# Unveiling a hidden bar-like structure in NGC 1087: kinematic and photometric
evidence using MUSE/VLT, ALMA and JWST
Carlos López-Cobá Institute of Astronomy and Astrophysics, Academia Sinica,
No. 1, Section 4, Roosevelt Road, Taipei 10617, Taiwan Lihwai Lin Institute
of Astronomy and Astrophysics, Academia Sinica, No. 1, Section 4, Roosevelt
Road, Taipei 10617, Taiwan Sebastián F. Sánchez Instituto de Astronomía,
Universidad Nacional Autonoma de México, Circuito Exterior, Ciudad
Universitaria, Ciudad de México 04510, Mexico
###### Abstract
We report a faint non-axisymmetric structure in NGC 1087 through the use of
JWST Near Infrared Camera (NIRCam), with an associated kinematic counterpart
observed as an oval distortion in the stellar velocity map, H$\alpha$ and CO
$J=2\rightarrow 1$ velocity fields. This structure is not evident in the MUSE
optical continuum images but only revealed in the near-IR with the F200W and
F300M band filters at $2\mu$m and $3\mu$m respectively. Due to its elongation,
this structure resembles a stellar bar although with remarkable differences
with respect to conventional stellar bars. Most of the near-IR emission is
concentrated within $6\arcsec~{}\sim 500$ pc with a maximum extension up to
1.2 kpc. The spatial extension of the large-scale non-circular motions is
coincident with the bar, which undoubtedly confirms the presence of a non-
axisymmetric perturbation in the potential of NGC 1087. The oval distortion is
enhanced in CO due to its dynamically cold nature rather than in H$\alpha$. We
found that the kinematics in all phases including stellar, ionized and
molecular, can be described simultaneously by a model containing a bisymmetric
perturbation; however, we find that an inflow model of gas along the bar major
axis is also likely. Furthermore the molecular mass inflow rate associated can
explain the observed star formation rate in the bar. This reinforces the idea
that bars are mechanisms for transporting gas and triggering star formation.
This work contributes to our understanding of non-axisymmetry in galaxies
using the most sophisticated data so far.
## 1 Introduction
Stellar bars or “bars” are one of the most visual signs of non-axisymmetry in
galaxies (de Vaucouleurs et al., 1991). Like other morphological structures in
galaxies, bars are not well defined, although they are relatively easy to
identity in composite images because of their bar-shape structure (e.g.,
Masters et al., 2011; Cheung et al., 2013), and their prominent bar dust lanes
(Athanassoula, 1992). NGC 1087 is a clear example where the combination of
high spatial resolution data allowed us to reveal a hidden bar that is only
observed in the near-infrared (NIR). The James Webb Space Telescope (JWST) is
allowing us to reveal structures not seen before due to the lack of resolution
and sensitivity. Their infrared bands are prone to detect the emission from
old-stellar structures like bars $\sim 10$ Gyr (Sánchez-Blázquez et al.,
2011). In the optical, dust obscuration can prevent their detection, affecting
the global statistics of galaxies hosting bars in the nearby Universe (e.g.,
Sellwood & Wilkinson, 1993). Apart from their optical and NIR characteristics,
bars leave imprints of non-axisymmetry in the kinematics of gas and stars
(e.g., Wong et al., 2004; Fathi et al., 2005; López-Cobá et al., 2022); thus
studying the dynamical effects of bars at local scales is crucial for
understanding the role they play in galaxy evolution. A comprehensive study of
bars could be addressed if kinematics and wide-range photometric observations
were accessible for a considerably large sample of galaxies. Yet, spatial
resolution plays an important role in separating the different structural
components of galaxies. For example, the large integral field spectroscopic
surveys like CALIFA (Sánchez et al., 2012) or MaNGA (Bundy et al., 2015),
although providing a large statistical sample of galaxies, their nominal
resolution of $2\farcs 5\sim 1$ kpc inhibits a detailed study of individual
galaxies, while biasing the sample towards bar-lengths larger than the FWHM
resolution. In a similar manner NIR photometric and optical catalogs provide
resolution of few arcseconds which again limits the detection of small bars
(Sheth et al., 2010).
In this work, we make use of the most sophisticated data to unravel the
detection of a bar in NGC 1087. This paper is structured as follows: in
Section 2 we describe the data and data analysis; in Section 3 we address the
detection of a faint stellar bar in the infrared while their ionized and
molecular counterpart are addressed in Section 4; Section 5 describes the oval
distortion and in Section 6 we present the discussion and conclusions.
## 2 Data and Data analysis
NGC 1087 is an intermediate Sc galaxy located at 14 Mpc (e.g., Kourkchi &
Tully, 2017); at this distance the physical spatial resolution is 68
pc$\mathrm{~{}arcsec^{-1}}$.
This work is based on public data from the Multi-Unit Spectroscopic Explorer
(MUSE, Bacon et al., 2010); data from the JWST Near Infrared Camera (NIRCam);
and ALMA CO $J=2\rightarrow$1 observations, in particular we used the data
products from the PHANGS–ALMA survey (e.g., Leroy et al., 2021a, b), namely
moments 0 and 1. The MUSE-VLT data was obtained as part of the recent release
of the MUSE-PHANGS datacubes (e.g., Schinnerer, 2021; Emsellem et al., 2022).
MUSE is an integral field spectrograph which provides spatially resolved
spectra over a $1\arcmin\times 1\arcmin$ field of view (FoV), covering the
optical spectrum, with a spectral resolution at full width at half maximum
(FWHM) of $\sim 2.6$Å. The estimated spatial resolution of the $2\arcmin\times
3\arcmin$ MUSE mosaic covering NGC 1087 is not worse than $0\farcs 9$/FWHM
(e.g., Emsellem et al., 2022). We used fully calibrated data products from
JWST from the JWST Science Calibration Pipeline version 1.10.1 (e.g., Bushouse
et al., 2022). Specifically, we made use of the PHANGS-JWST first result
products (e.g., Rosolowsky, Erik, 2022), namely, the NIRCam F200W filter (FWHM
$\sim 0\farcs 05$ and $0\farcs 031/\mathrm{pixel}$ sampling), which has its
nominal wavelength at $1.99\mu$m, in addition to the F360M and F335M band
filters to trace the stellar continuum. Finally the spatial resolution of the
ALMA data is $1\farcs 60$/FWHM following Leroy et al. (2021a). The CO moment 0
map was transformed to surface density ($\mathrm{\Sigma_{mol}}$), assuming a
standard Milky Way
$\mathrm{\alpha_{\mathrm{CO}}^{1-0}=4.35\,M_{\odot}/pc^{2}}$ conversion factor
and $\mathrm{\Sigma_{mol}[M_{\odot}pc^{2}]=6.7\,I_{CO(2-1)}[K\,km/s]}\cos i$
(e.g., Eq. 11 from Leroy et al., 2021a), with $i$ being the disk inclination
angle estimated in $44.5^{\circ}$.
The MUSE data analysis was made with the pypipe3d tool (e.g., Lacerda et al.,
2022). In short, pypipe3d performs a decomposition of the observed stellar
spectra into multiple simple stellar populations (SSPs) each with different
age and metalicities. For concordance with Emsellem et al. (2022), we use the
same stellar libraries based on the extended MILES (E-MILES, Vazdekis et al.,
2016), FWHM $=2.5$Å; We convolved the spectral resolution of the SSPs to that
of the MUSE line-spread function (e.g., Bacon et al., 2017). To increase the
signal-to-noise (SN) of the stellar continuum, we performed a Voronoi binning-
segmentation on a 2D-slice centered around the V-band, ensuring bin-sizes of
the order of the MUSE-FWHM resolution and SN $\sim 50$. This binned map will
serve to compute different properties of the underlying stellar continuum,
while the ionized gas properties are analyzed in a spaxel-wise sense. The
result of pypipe3d is a set of maps comprising information about the stellar
populations (stellar velocity, stellar mass, among others products), and the
ionized gas (emission-line fluxes including: H$\alpha$, H$\beta$, [S
ii]$\lambda 6717,6731$, [N ii]$\lambda 6584$, [O iii]$\lambda 5007$, [O
i]$\lambda 6300$, emission-line velocities, equivalent widths etc); see
Lacerda et al. (2022) for a thorough description of the analysis and
dataproducts. From this analysis we estimate the stellar mass of this object
in $\mathrm{\log M_{\star}/M_{\odot}=9.9}$.
Figure 1: Central panel: False color image showing the distribution of the
ionized gas and NIR continuum in NGC 1087 with fluxes taken from MUSE-VLT and
NIRCam imaging respectively (red: [S ii], yellow: H$\alpha$, blue: [O iii],
brown: F360M, orange: F335M, white: F200W). Right panels: A zoom-in of the
$30\arcsec\times 20\arcsec$ innermost region is shown to highlight the bar-
structure. The upper panel shows a true color image from the MUSE cube (R:
i-band, G: r, B: g). The black cross in the middle represents the NIR nucleus.
The middle panel shows the JWST-F200W image together with a set of isophotes
describing the bar NIR light distribution; the isophotes show a constant
orientation in the sky at $\phi^{\prime}_{bar}=304^{\circ}\pm 3^{\circ}$. Two
isophotes located at $6\arcsec$ and $14\arcsec$ are highlighted with thicker
lines for reference. In each case the FWHM resolution is shown with a yellow
circle. Bottom inset: Orientation of the object in the sky.
## 3 FAINT stellar bar
The optical continuum image of NGC 1087 exhibits a bright, fuzzy and
featureless nucleus with several dust lanes as revealed by the $gri$ MUSE
image in Figure 1, because of that it is unclear whether this object shows a
stellar bar in the optical. Two bright spots are observed in the central
region where the near-IR (NIR) nucleus is found. Unlike the majority of
galaxies hosting stellar bars, NGC 1087 does not show a well defined bar,
nucleus neither a star forming ring (e.g., de Vaucouleurs et al., 1991).
Conversely, the high spatial resolution from the F200W imaging filter allows
to resolve a clear elongated and clumpy structure of presumably stellar
clusters aligned along a preferential position angle (P.A.), which differs
from the disk orientation111From now on primed variables make reference to
values measured on the sky plane, otherwise in the disk plane.
$\phi_{disk}^{\prime}=359^{\circ}$, as observed in Figure 1. At $2\mu$m this
structure resembles a faint stellar bar and it is obscured in the optical due
to dust absorption. F200W traces mostly stellar continuum, hence, it is
expected that most of the continuum emission along this structure comes from
old stars (e.g., Leitherer et al., 1999). Elliptical isophotes with variable
position angle and ellipticity ($\varepsilon^{\prime}$) show a preferential
alignment of the $2\mu$m emission as observed in the top-right panel from
Figure 1.
Figure 2: NGC 1087 as observed with the F200W and F300M NIRCam band filters.
First panel: zoom around a $30\arcsec\times 20\arcsec$ size window containing
the bar-like structure. second panel: Best non-parametric model for the
$2\mu$m light distribution using the photometric version of the $\mathtt{XS}$
code (in prep.). The coordinates of the photometric center was estimated in
$\alpha_{2000}=2^{h}46^{m}25.170^{s}$,
$\delta_{2000}=-00^{\circ}29^{\prime}55.728\arcsec$, while the position angle
of the disk describing the bar light distribution was estimated in
$\phi_{bar}^{\prime}=305^{\circ}$ with a $0.5$ ellipticity. Third panel:
residuals of the modeling (observed-model). Fourth panel: the black line
represents the light distribution profile of the model shown in the second
panel, while the blue dashed line represents the cumulative light distribution
of the model. Distances are measured following these projection angles. Figure
3: Diagnostic diagrams for the ionized gas in the bar structure. Top panel: [O
i]6300/H$\alpha$ vs. [O iii]/H$\beta$ diagram. The black dashed line
represents the star forming demarcation curve from Kewley et al. (2006). Shock
grid models from the updated MAPPINGS-V (Sutherland et al., 2018) computed by
Alarie & Morisset (2019) are shown for $\mathrm{n_{e}=10\,cm^{-3}}$,
$\mathrm{Z_{ISM}=0.008~{}(equivalent~{}to~{}12+\log O/H=8.55)}$, and a wide
range of magnetic fields (blue lines) and pre-shock velocities (red lines).
The colors of the points map the F200W light distribution shown in the inset
figure. Bottom panel: EW(H$\alpha$) in absolute value vs. [N ii]/H$\alpha$
line ratio. Old and evolved stars are expected to produce EW(H$\alpha$)
$<3$Å.. Figure 4: Oxygen abundance in the bar-region computed with the
Pilyugin & Grebel (2016) calibrator. As in Figure 1 the white ellipses
describe the NIR bar. The innermost ellipse shows an average metallicity of
$\mathrm{12+\log O/H=8.5}$. Figure 5: Top: Specific star formation rate
computed through the H$\alpha$ based SFR and the stellar mass density maps,
$\mathrm{sSFR=SFR/M_{\star}}$. Bottom: Molecular surface density around the
bar structure. The H$\alpha$ flux is overlaid with reddish colors for
comparison. The yellow circles represent the FWHM spatial resolution in each
case. The white ellipses delineate the NIR bar extension. Figure 6: First
column: Line-of-sight velocities of CO, H$\alpha$ and the stellar velocity in
NGC 1087. Second column: Zoom-in around the bar-structure, with $\pm 25$ km
s-1 spaced iso-velocity shown on-top; the black straight line shows the P.A.
of the faint bar at $\phi_{bar}^{\prime}=305^{\circ}$. Third column: residual
velocities after subtracting the circular rotation in each velocity map.
Fourth column: Bisymmetric model after fixing the oval orientation at
$\phi_{bar}^{\prime}$. Fifth column: inflow model assuming the gas streams
along $\phi_{bar}^{\prime}$; the individual radial profiles and expressions
are shown in Figure 7. The root mean square of the models is shown at the
bottom right of each panel. All maps share the same color-bar except for the
residual maps which cover $\pm 50$ km s-1 following the same colors scheme.
### 3.1 Bar light modeling
The method adopted here is similar to the non-parametric method from Reese et
al. (2007), which is the same algorithm used by $\mathtt{XS}$ (e.g., Lopez-
Coba et al., 2021) for extracting the different velocities in the kinematic
models, as we explain in the following sections. Our non-parametric model
minimizes the function $\chi^{2}=(I_{obs}-I_{k}(r)w_{k}(r))^{2}$, where
$I_{obs}$ is the observed intensity map, $I_{k}$ represents a set of
intensities that will be estimated at different annuli, and $w_{k}$ is a set
of weighting factors that will serve for performing a linear interpolation
between the estimated $I_{k}$ intensities. The implementation on the F200W and
F300M band filters is shown in Figure 2.
The NIR light from the bar-structure can be successfully modeled by an
elliptical light distribution with with a constant orientation in the sky. The
P.A. of such ellipse is $\phi_{bar}^{\prime}=305^{\circ}$222Sky angles
relative the disk major axis can be translated to angles measured in the
galaxy plane and vise versa through, $\tan\phi_{bar}=\tan\Delta\phi/\cos i$,
with $\Delta\phi=\phi_{bar}^{\prime}-\phi_{disk}^{\prime}$. and
$\varepsilon^{\prime}_{bar}=0.5$, as shown in Figure 2. From this analysis the
estimated semi-major axis length of the NIR bar is
$a_{bar}^{\prime}=14\arcsec$ or $18\arcsec\sim 1.2$ kpc333Distances measured
at different P.A. in the sky plane can be translated to distances in the
galaxy plane following:
$r_{disk}^{2}=r_{sky}^{2}(\cos^{2}\Delta\phi+\sin^{2}\Delta\phi/\cos^{2}i)$.
on the disk at the considered distance. However, this structure is very
diffuse, with most of the 2$\mu$m emission $\sim 70\%$, concentrated within
the inner $6\arcsec\sim 500$ pc. The latter region encloses the two bright
spots observed in the MUSE optical continuum images.
## 4 Ionized and molecular gas
The central panel of Figure 1 shows the ionized gas distribution traced by the
H$\alpha$, [S ii] and [O iii] emission-line fluxes. In general, galaxies
hosting stellar bars do not frequently present cold or warm gas along the bar.
This is often explained as a lost of angular momentum after the gas encounters
the offset ridges, resulting in an infall of gas towards a central ring (e.g.,
Athanassoula, 1992). NGC 1087 shows plenty of ionized gas throughout the disk
and within the bar region. Furthermore, as we will see later, multiple
enhanced CO bloops are spatially coincident with the F200W emission. To
investigate the dominant ionizing source along the bar-like structure we made
use of line-ratios sensitive to the ionization, in particular the [O
i]6300/H$\alpha$ and [O iii]/H$\beta$. The [O i]6300/H$\alpha$ ratio has the
advantage of being a good indicator of the presence of shocks. The top panel
from Figure 3 shows these line-ratios color-coded with the $2\mu$m emission.
The observed ionized gas is compatible with being produced by star-formation
(SF) according to the Kewley et al. (2006) demarcation line. However, dust
lanes in bars are often associated with shocks as a result of the complex gas
dynamics (Athanassoula, 1992). The bar-region in NGC 1087 shows several dust
lanes, thus we adopt shock models to investigate whether shock ionization can
reproduce the observed line ratios. For this purpose we used the
photoionization grids from MAPPINGS V (e.g., Sutherland et al., 2018) computed
by Alarie & Morisset (2019); we adopt the Gutkin et al. (2016) metallicities.
The latter parameter was set from the oxygen gas-phase abundance, adopting the
Pilyugin & Grebel (2016) calibrator based on strong emission-lines (see Figure
4). For the bar-region we find $\mathrm{12+\log O/H\sim 8.5}$ corresponding to
$\mathrm{Z_{ISM}=0.008}$ in the Gutkin et al. (2016) models. For this
metallicity, the predicted emission-line intensities from shock models can not
explain the observed [O i]/H$\alpha$ line ratios as observed in the top panel
of Figure 3. Thus, if shocks are happening in the bar-region, then their
optical emission is not expected to be the dominant ionizing source. Although
there is still a possibility that a combination of SF plus shocks could
contribute to the observed line-ratios (e.g., Davies et al., 2016).
Additionally, Figure 3 shows the equivalent width of H$\alpha$
(EW($\rm{H}\alpha$)) for the ionized gas in the bar-region. Hot, old, and low-
mass evolved stars are characterized for producing
EW($\rm{H}\alpha$)$~{}\lesssim 3$Å (e.g., Stasińska et al., 2008; Cid
Fernandes et al., 2010; Lacerda et al., 2018). The large values of
EW($\rm{H}\alpha$) $\gtrsim 100$ Å found in the bar, can not be explained by
the ionizing continuum emitted by this population of stars. Therefore, if most
of the $2\mu$m emission observed along the bar is due to old stars, then their
ionizing continuum is not sufficient to explain the observed line-ratios.
The star formation rate (SFR) was estimated from the H$\alpha$ luminosity
(Kennicutt, 1998), after correcting the H$\alpha$ flux from dust extinction
adopting the Cardelli et al. (1989) extinction law with $\mathrm{R_{V}=3.1}$,
and assuming an intrinsic flux ratio of H$\alpha$/H$\beta$= 2.86 and ionized
gas temperature of T $\sim 10^{4}$ K corresponding to case B recombination
(e.g., Osterbrock, 1989). The integrated SFR within the NIR bar (i.e., Figure
2) is $\mathrm{SFR(H\alpha)_{bar}}=0.08~{}\mathrm{M_{\odot}/yr}$, while the
total $\mathrm{SFR(H\alpha)_{total}}=0.4~{}\mathrm{M_{\odot}/yr}$. The
specific star formation rate (sSFR) was obtained from dividing the SFR surface
density (=SFR/pixel area) by the stellar mass surface density as shown in
Figure 5. We note a clear enhancement in SF along the bar main axis;
furthermore, the molecular surface density shows an enhancement in the same
region.
Overall, our analysis suggests that the ionized gas in the bar structure is
mostly associated to SF processes. Following we investigate whether the CO and
SF concentration is induced by the bar potential.
## 5 Inner Oval distortion
If a perturbation in the gravitational potential, such as that induced by a
bar potential, is causing the $2\mu$m light distribution to elongate along a
preferred direction; then the particles in such structure are expected to
follow quasi-elliptical orbits (e.g., Athanassoula, 1992). Hence kinematics of
gas and stars should reflect this perturbation (Pence & Blackman, 1984).
Figure 6 shows the CO, ionized and stellar kinematics around the bar region.
The gas, being collisional, is more sensitive to non-axisymmetric
perturbations. This is clearly reflected in the CO moment 1 map where a strong
distortion of the semi-minor axis is observed. The former misalignment is a
signature of an oval distortion produced by a bar (e.g., Pence et al., 1988;
López-Cobá et al., 2022), here observed at high spatial resolution. The
distortion is also observed in the H$\alpha$ velocity map, however, since the
ionized gas traced by H$\alpha$ is hotter ($\mathrm{T\sim 10^{4}}$ K), and
therefore it presents a larger intrinsic velocity dispersion than the one of
the molecular gas traced with CO ($\mathrm{T\sim 10}$ K), it is less
pronounced in the ionized gas and is probably affected by local SF. The
stellar kinematics although affected by the pixel-coadding during the SSP
analysis, it still reveal a clear distortion in the iso-velocities near the
bar as noted in Figure 6.
So far, it is clear that the stars, the molecular gas traced with CO and the
ionized gas traced with H$\alpha$ respond in a similar way to the presence of
this bar-like structure detected in NGC 1087.
### 5.1 Kinematic interpretation of the oval distortion
The distortion in the velocity field such as the ones observed before could be
caused by an elongated potential. However, only a few kinematic models in the
literature attempt to describe the flow caused by an oval distortion (e.g.,
Spekkens & Sellwood, 2007; Maciejewski et al., 2012). A bisymmetric distortion
induced by a second order perturbation to the potential, such as that induced
by an elongated potential, has been shown to successfully reproduce the
velocity field in bars (e.g., Spekkens & Sellwood, 2007). Since the F200W
image evidences the presence of a faint bar-like structure oriented at
$305^{\circ}$ in the sky, we performed bisymmetric models over the stellar,
H$\alpha$ and CO velocity maps, with a fixed orientation of the oval
distortion.
The bisymmetric model from Spekkens & Sellwood (2007) is described by the
following expression:
$V_{\mathrm{LoS}}=\sin i\Big{(}V_{t}(r)\cos\theta-V_{2,t}(r)\cos
2\theta_{\mathrm{bar}}\cos\theta\\\ -V_{2,r}(r)\sin
2\theta_{\mathrm{bar}}\sin\theta\Big{)}+V_{\mathrm{sys}}$ (1)
where $\theta_{bar}=\theta-\phi_{bar}$; with $\theta$ being the azimuthal
angle measured on the disk plane from the line of nodes and $\phi_{bar}$ the
position angle of the bar-like structure on the disk plane. $V_{t}$ is the
tangential or circular rotation and $V_{2r}$ and $V_{2t}$ represent the radial
and tangential velocities that result from a bisymmetric distortion to the
gravitational potential.
Before this analysis we performed a circular rotation model to obtain the disk
projection angles and the rotational curves. In this case and in the
subsequent, we used the $\mathtt{XS}$ code for generating the kinematic models
(e.g., Lopez-Coba et al., 2021). This code derives an interpolated model over
a set of concentric rings evenly spaced $r_{k}$, minimizing the function
$\chi^{2}=(V_{obs}-V_{k}(r)w_{k}(r))^{2}/\sigma^{2}$, where $V_{obs}$ is the
observed velocity map; $V_{k}$ are the set of velocities inferred at $r_{k}$;
$w_{k}$ is a set of weights that depend on the specific kinematic model
adopted and will serve to create a 2D model; $\sigma$ is the error velocity
map. We refer to Lopez-Coba et al. (2021) for a thorough description of this
analysis. It is important to mention that the modeling does not assume any
parametric function on the velocity profiles.
A pure circular rotation, without non-circular motions, is described by the
first term on the right from Equation 1. The disk orientation of NGC 1087 was
estimated from modeling the H$\alpha$ and stellar velocity maps with circular
rotation only, obtaining $\phi_{disk}^{\prime}=358.9^{\circ}$,
$i=44.5^{\circ}$ and $V_{sys}=1526$ km s-1. The residual velocity maps from
this model i.e., $V_{obs}-V_{circular}$, are shown in the third column from
Figure 6. The circular rotation models leave large-scale residual velocities
of the order of $30$ km s-1 with a mirror symmetry about the nucleus; this
corresponds to de-projected amplitudes of $\sim 50$ km s-1 for the non-
circular velocities on the disk plane. These residual patterns have been
observed in larger scales in galaxies with long bars in H$\alpha$ (e.g., Lang
et al., 2020; López-Cobá et al., 2022) and molecular gas (e.g., Pence et al.,
1988; Mazzalay et al., 2014).
The bisymmetric model with a fixed oval orientation at $\phi_{bar}^{\prime}$
is shown in the fourth column in Figure 6. This model reproduces
simultaneously the twisted iso-velocities in the three velocity maps. The root
mean square (rms) of the models decrease compared with the circular rotation
one, therefore, in terms of the residuals, the LoS-velocities observed around
the bar region can be reproduced by a kinematic model that considers a bar-
like perturbation as the main source of non-circular motions. However, at such
high spatial resolution $\sim 70$ pc, there are still residual velocities that
the bisymmetric model can not account for, this is reflected in the 15 km s-1
rms in the models, which is similar or larger than the level of turbulence of
the ISM (e.g., Moiseev et al., 2015).
An alternative interpretation to the observed non-circular motions in the
H$\alpha$ and CO velocity maps is the presence of gas inflow induced by the
NIR-bar (Mundell & Shone, 1999); in fact, hydro-dynamical models simulating
bars predict inflow of gas along the offset ridges, or bar dust lanes, (e.g.,
Athanassoula, 1992). As observed in Figure 1, the MUSE image of NGC 1087 shows
several dust lanes in the central region making difficult the identification
of those associated with the bar. Assuming the spiral arms are trailing (see
bottom-right panel from Figure 1), NGC 1087 rotates counterclockwise in the
sky. Thus, positive (negative) residuals in the near (far) side represent
inflow; hence, a radial flow ($V_{rad}$) along the bar major axis is inflowing
if $V_{rad}<0$. We implemented a non-axisymmetric model, with a flow streaming
along $\phi_{bar}^{\prime}$. The non-axisymmetric inflow model is described by
the following expression:
$V_{LoS}=(V_{t}(r)\cos\theta+V_{rad}(r)\cos\theta_{bar}\sin\phi_{bar})\sin
i\\\ +V_{sys}$ (2)
where $V_{rad}$ is the radial velocity of the flow, and $\phi_{bar}$ is the
misalignment between the disk and the bar position angles on the disk plane.
This model assumes the gas is flowing along $\phi_{bar}$. This expression is
similar to Hirota et al. (2009) and Wu et al. (2021) assuming that the gas
flows parallel to the bar major axis.
In order to consider pixels likely affected by this motion, we consider the
elliptical region that better describes the bar-like light distribution and
defined in Figure 2. The 2D representation of this model is shown in Figure 6,
while the kinematic radial profiles of all models are shown in Figure 7. As
observed, the radial velocity $V_{rad}$ results negative in the model, with
maximum inflow velocities of the order of 50 km s-1.
Figure 7: Radial distribution of the different velocity components for the
considered kinematic models. $V_{t}$ is the disk pure circular rotation, or
rotational curve; $V_{2r}$ and $V_{2t}$ are the bisymmetric velocities;
$V_{rad}$ the radial flow from the non-axisymmetric flow model. In all non-
circular models the position angle of the oval distortion was fixed to
$\phi_{bar}^{\prime}=305^{\circ}$ (namely, $297^{\circ}$ in the galaxy plane).
Lines in red colors represent results of models computed on the H$\alpha$
velocity map, blue lines on the CO moment 1 and black lines on the stellar
velocity map. Shaded regions represent $1\sigma$ errors. The non-circular
velocities were estimated up to $18\arcsec$, which covers the de-projected
length of the bar. A $2\arcsec$ sampling step was adopted, i.e., larger than
the FWHM spatial resolution of the data.
The molecular mass flow rate ($\dot{M}_{mol}$) associated can be computed
following the expression (e..g, Di Teodoro & Peek, 2021):
$\dot{M}_{mol}(r)=2\pi r\Sigma_{mol}(r)V_{inflow}(r)$ (3)
with $\mathrm{\Sigma_{mol}}$ being the de-projected molecular mass surface
density, $V_{inflow}$ the CO inflow velocity and $r$ the galactocentric
distance. Figure 8 shows the spatially resolved $\dot{M}_{mol}$ and its
average radial profile. We find an average
$\dot{M}_{mol}\sim-20\mathrm{M_{\odot}/yr}$. For comparison, spiral arms
induce radial flows and radial velocities of the order of
$1\mathrm{M_{\odot}/yr}$ and $10$ km s-1 respectively, (e.g., Di Teodoro &
Peek, 2021).
Figure 8: Molecular mass inflow induced by the bar-like structure. Top panel:
2D distribution of $\dot{M}_{mol}$. Bottom panel: Average radial distribution
of $\dot{M}_{mol}$ in black and radial velocity profile of the inflow velocity
in red colors. Shadow black regions represent the standard deviation of
$\dot{M}_{mol}$ at each radial bin. A $\mathrm{SN>3}$ was applied to the
moment 0 map to exclude spurious CO detection.
## 6 Discussion and Conclusions
The combination of spatially resolved spectra provided by MUSE-VLT, with the
high spatial resolution from JWST and the ALMA CO $J=2\rightarrow 1$ data
allowed us to reveal a central kinematic oval distortion, as well as a small
scale elongated structure in the NIR continuum in NGC 1087. This elongated
structure is not evident in optical images, but only revealed with the NIRCam
F200W and F360M filters at $2\mu$m and $3\mu$m respectively, thanks to its
exquisite resolution. The optical counterpart of this bar is likely to be
affected by dust absorption and by the intense radiation from young stars. Its
preferential elongation suggests the presence of a non-axisymmetric
perturbation in the gravitational potential, in particular a bar-like type. At
first order, the 2$\mu$m light distribution of this structure can be described
with an exponential disk profile oriented at
$\phi_{bar}^{\prime}=305^{\circ}$, and the ellipticity of the elongated
structure is $\varepsilon^{\prime}=0.5$, (see Figure 2), with a half-major
axis length of $a_{bar}^{\prime}\sim 14\arcsec$ or $\sim$ 1.2 kpc in the
galaxy plane. Overall the shape of this structure resembles to a faint stellar
bar, although with remarkable differences with respect to conventional stellar
bars, in terms of bar-length, the lack of a SF nuclear ring and not clear
associated dust lanes along the bar axis. The same argument however applies
for an elongated bulge. The main reason this is not considered is due to the
large reservoirs of molecular gas observed, since bulges tend to be quiescent
structures with null or low SFRs (Shapiro et al., 2010). Furthermore, the [O
i]$/$H$\alpha$ and [O iii]$/$H$\beta$ line-ratios and large EWH$\alpha$ from
the bar (see Figure 1) indicate that the ionized gas is product from a recent
or ongoing SF event given its spatial coincidence with the CO emission.
The ionized and molecular gas kinematics and the stellar velocity, reveal
simultaneously the presence of an oval distortion in the velocity maps in
concordance with the extension of the faint bar. This evidences undoubtedly
the imprints of a non-axisymmetric perturbation in the potential, here
captured in different phases and enhanced in dynamically cold gas.
Kinematic models for a second order potential perturbation successfully
described the LoS velocities in the three velocity maps. We showed that this
is achieved just by fixing the orientation of the stream flow to the bar major
axis in the models, and the radial profiles of the noncircular velocities are
consistent with the residuals from circular rotation, with possible
differences in amplitudes due to assymmetric drift and other random motions
not considered in the model as we showed in Figure 7. Apart from the
elliptical motions considered in the bisymmetric models, radial inflows are
expected in bars, and these are observed to occur along the bar-dust lanes (Wu
et al., 2021; Sormani et al., 2023). Our implementation of an inflow model is
therefore physically motivated. An inflow of cold gas caused by a lose of
angular momentum could trigger the SF observed along the bar. Although a
fraction of ionized gas could arise by the shock with the dust lanes, the
contribution of shocks is expected to be minor. Moreover, the sign of the
radial flow suggests the gas is inflowing to the center at a maximum speed of
$50$ km s-1. This kinematic model yields the lowest rms despite being the
model with the lowest number of free variables. Statistically speaking, the
inflow scenario is favored over the bisymmetric model, although given the
complex behavior of gas in bars, the contribution of gas oval orbits can not
be ruled out. In fact both type of motions are expected to happen
simultaneously in bars (Athanassoula, 1992; Regan et al., 1999). To date,
there is no a 2D-kinematic model that includes such dynamics of bars. The gas
inflow model is in concordance with the observed enhancement in H$\alpha$ flux
along the bar. The inferred inflow velocities translate in molecular mass
inflow rates of $\dot{M}_{mol}=20~{}\mathrm{M_{\odot}/yr}$. This is larger
than the measured SFR(H$\alpha$) along the bar, thus we argue that a major
inflow of gas might be feeding the SF in the bar. In fact SF prevents the so-
called continuity problem (e.g., Simon et al., 2003; Maciejewski et al., 2012)
when often considering radial flow models. This large inflow rate is partially
due to the large inflow velocities; although this amplitude of non-circular
motions is observed in bars, the global inflow rate expected by hydrodynamical
models is lower (Athanassoula, 1992; Mundell & Shone, 1999).
To summarize, in this work we have taken the advantage of multiple archival
data to reveal a hidden non-axisymmetric structure in the optical, thanks to
the high angular resolution of the data. We were able to confirm the presence
of an oval distortion and successfully model it with a bar-like flow model. We
also show that the SFR in the bar could be explained by an inflow of gas along
the bar major-axis. Our results contribute to understand the overall picture
of non-axisymmetry in galaxies with the most sophisticated data so far.
Additionally, it highlights the importance of using the IR instead of optical
bands to detect stellar bars, which could increase the fraction of barred
galaxies detected in the local Universe.
## Acknowledgment
We thank the anonymous referee for their comments and suggestions which help
to improve the quality of this manuscript. CLC acknowledges support from
Academia Sinica Institute of Astronomy and Astrophysics. LL acknowledges the
Ministry of Science & Technology of Taiwan under the grant NSTC
112-2112-M-001-062 -.
Based on data obtained from the ESO Science Archive Facility with DOI(s):
https://doi.org/10.18727/archive/47 (catalog
https://doi.org/10.18727/archive/47). The specific PHANGS–JWST observations
analyzed can be can be found in MAST: https://doi.org/10.17909/9bdf-jn24
(catalog https://doi.org/10.17909/9bdf-jn24). This paper makes use of the
following ALMA data: ADS/JAO.ALMA#2018.1.01651.S.
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# Draft
Hitting time for Markov decision process
Ruihcao Jiang, Javad Tavakoli, Yiqiang Zhao
###### Abstract
We define the hitting time for a Markov decision process (MDP). We do not use
the hitting time of the Markov process induced by the MDP because the induced
chain may not have a stationary distribution. Even it has a stationary
distribution, the stationary distribution may not coincide with the
(normalized) occupancy measure of the MDP. We observe a relationship between
the MDP and the PageRank. Using this observation, we construct an MP whose
stationary distribution coincides with the normalized occupancy measure of the
MDP and we define the hitting time of the MDP as the hitting time of the
associated MP.
## 1 Introduction
††footnotetext: The first version of this paper pointed out some issues in the
old version of [3]. The authors of [3] have then addressed these issues
according to our suggestions in a new version. We therefore updated our paper,
in which we removed contents related to these issues.
In [3], Gromov-Wasserstein distance was used to compare different MDPs. In
their approach, a distance function is needed on the space of state-action
pairs of the underlying MDP. They used Euclidean distance. However, it is not
entirely clear how to define the distance between actions. Moreover, it is
desirable that the distance reflects some useful information about the MDP.
For this reason, we propose to use the first-hitting time quasi-distance in
places of the distance function. The rationale is as follows.
* •
The occupancy measure to the MDP is whats the stationary distribution to an
Markov process (MP);
* •
The stationary distribution of an MP is simply the inverse of the mean of the
first-return time;
* •
The first-return time is simply the diagonal element for the first-hitting
time matrix.
## 2 Hitting time of MDP
To define the first-hitting time of MDP, we first look at the occupancy
measure of MDP.
###### Definition 1 (Occupancy measure).
Let $(S,A,R,P,\gamma)$ be an MDP with the initial distribution $\rho_{0}(s)$
on $S$ and the stationary policy $\pi$, the occupancy measure $\rho:S\times
A\to\mathbb{R}$ is defined as follows.
$\rho_{\pi}(s,a)\coloneqq\sum_{t=0}^{\infty}\gamma^{t}\mathbb{P}\left(s_{t}=s,a_{t}=a|s_{0}\sim\rho_{0},\pi\right).$
(1)
We give a new interpretation of the occupancy measure. First, Tulcea’s theorem
(Proposition C.10 in [4]) says that a policy $\pi$ induces a Markov Process
(MP) on $S\times A$ with the transition probability
$P_{\pi}(s^{\prime},a^{\prime}|s,a)\coloneqq
P(s^{\prime}|s,a)\pi(a^{\prime}|s^{\prime})$ (2)
and the initial distribution $\rho_{0}(s,a)\coloneqq\rho_{0}(s)\pi(a|s)$. This
describes a process as follows: Given the state-action pair $(s,a)$, transit
to the next state $s^{\prime}$ according to the transition probability of the
original MDP; then immediately choose an action $a^{\prime}$ according to the
policy $\pi$.
Note that $\rho_{\pi}$ is not the stationary distribution (if it exists at
all) of $(S\times A,P_{\pi})$. First of all, $\rho_{\pi}$ is not a probability
measure as it sums to $\frac{1}{1-\gamma}$. Even after a normalization,
$\left(1-\gamma\right)\rho_{\pi}$ may differ from the stationary distribution.
Consider a scenario where a car moves straight on a straight road and if it
turns left or right, it will go off the road. Then none of the state-action
pairs related to steering will be occupied and hence $\rho_{\pi}$ is zero on
those pairs. However, for any stationary distribution, each component must be
strictly positive. We will see later that it does not matter whether $(S\times
A,P_{\pi})$ is ergodic or not. Treating $\rho_{\pi}$ and $\rho_{0}$ as
$1\times|S||A|$ vectors and $P_{\pi}$ as an $|S||A|\times|S||A|$ matrix, the
following equation is well-known.
$\rho_{\pi}=\left(I-\gamma P_{\pi}\right)^{-1}\rho_{0}.$ (3)
From Equation (3), we make the following key observation.
###### Observation (PageRank).
$(1-\gamma)\rho_{\pi}$ coincides with the personalized PageRank vector on
$S\times A$ with transition matrix $P_{\pi}$ and with probability $1-\gamma$
to restart with the initial distribution $\rho_{0}$.
Second, we look at the first-hitting time of an MP.
###### Definition 2 (First-hitting time of MP).
Let $\\{X_{t}\\}$ be an MP. The first-hitting time $T$ is the random variable
defined as follows.
$T_{ij}=\begin{cases}\inf_{t\in\mathbb{N}}\\{t\ |\ X_{0}=j,X_{1}\neq
i,\cdots,X_{t-1}\neq i,X_{t}=i\\}&\text{if}\ i\neq j\\\
0,&\text{otherwise.}\end{cases}$ (4)
If $i$ is never reached from $j$, $T_{ij}=+\infty$, where we have adopted the
convention that $\inf\emptyset=+\infty$.
By abuse of the notation, we call the expectation $\mathbb{E}T_{ij}$ of the
random variable $T_{ij}$ also the first-hitting time and denote it by
$T_{ij}$.
$T_{ij}\coloneqq\mathbb{E}T_{ij}=\begin{cases}\sum_{t=1}^{\infty}t\mathbb{P}(T_{ij}=t)&\text{if}\
i\neq j\\\ 0&\text{otherwise}.\end{cases}$ (5)
If the MP has transition matrix $P_{\pi}$, a one-step analysis shows that
$T_{ij}$ satisfies the following recursive relation.
$T_{ij}=\begin{cases}1+\sum_{k}T_{ik}P_{\pi}(k|j)&\text{if}\ i\neq j\\\
0&\text{otherwise}.\end{cases}$ (6)
By our Observation, we reduce the task of defining a first-hitting time for an
MDP $(S,A,R,P,\gamma)$ to that of defining a first-hitting time for a PageRank
$(S\times A,P_{\pi},1-\gamma)$. A PageRank can be described equivalently by a
new process $(S\times A,\tilde{P}_{\pi})$ with the transition matrix
$\tilde{P}_{\pi}(x^{\prime}|x)\coloneqq(1-\gamma)\rho_{0}(x^{\prime})+\gamma
P_{\pi}(x^{\prime}|x).$ (7)
$(S\times A,\tilde{P}_{\pi})$ is obviously Markovian. The initial condition
for this new process can be arbitrary, in particular, not necessarily
$\rho_{0}$ since it turns out that $(S\times A,\tilde{P}_{\pi})$ is ergodic
and its stationary distribution is given by $(1-\gamma)\rho_{\pi}$ (See
Proposition 1 and Corollary 1 of [1]). The ergodicity of $(S\times
A,\tilde{P}_{\pi})$ implies that the first-hitting time is finite on
supp$(\rho_{\pi})$. In this case, Equation (6) reads
$T_{ij}=\begin{cases}1+(1-\gamma)\sum_{k}T_{ik}\rho_{0}(k)+\gamma\sum_{k}T_{ik}P_{\pi}(k|j)&\text{if}\
i\neq j\\\ 0&\text{otherwise}.\end{cases}$ (8)
There is also a related quantity $L_{ij}$ satisfying the following recursive
relation.
$L_{ij}=\begin{cases}1+\gamma\sum_{k}L_{ik}P_{\pi}(k|j)&\text{if}\ i\neq j\\\
0&\text{otherwise}.\end{cases}$ (9)
The probabilistic interpretation of $L_{ij}$ is the expectation of the
discounted path length from $j$ to $i$ for the first time. $L_{ij}$ and
$T_{ij}$ are related by the following formula (Theorem 1(b) of [1]).
$T_{ij}=\frac{L_{ij}}{1-(1-\gamma)\sum_{k}L_{ik}\rho_{0}(k)}.$ (10)
###### Remark.
$L_{ij}$ is called the (1-$\gamma$)-discounted hitting time in [6] although
$L_{ij}$ is not the first-hitting time of $(S\times A,P_{\pi})$, whose first-
hitting time is given by Equation (6), nor of $(S\times A,\tilde{P}_{\pi})$,
whose first-hitting time is given by Equation (8). However, $L_{ij}$ is easier
to estimate empirically.
We have shown that
* •
The occupancy measure of an MDP $(S,A,R,P,\gamma)$ with the initial
distribution $\rho_{0}$ coincides with the PageRank vector of the personalized
PageRank $(S\times A,P_{\pi},1-\gamma)$ with the initial distribution
$\rho_{0}$.
* •
The PageRank vector of the personalized PageRank $(S\times
A,P_{\pi},1-\gamma)$ with the initial distribution $\rho_{0}$ coincides with
the stationary distribution of the ergodic MP $(S\times A,\tilde{P}_{\pi})$
with arbitrary initial distribution.
Based on the above facts, we propose the following definition.
###### Definition 3 (First-hitting time of MDP).
Let $(S,A,R,P,\gamma)$ be an MDP with a stationary policy $\pi$. Let $P_{\pi}$
be the induced transition matrix in Equation (2) and $\tilde{P}_{\pi}$ be the
modified transition matrix in Equation (7). The first-hitting time of the MDP
is defined as the first-hitting time of $(S\times A,\tilde{P}_{\pi})$ with its
expectation given by Equation (8).
Since we define the first-hitting time of MDP as the first-hitting time of
some MP, we immediately have the following.
###### Proposition 1.
The first-hitting time of an MDP gives rise to a quasi-distance
$T_{\pi}(i,j)\coloneqq T_{ij}$, i.e. a distance function without satisfying
the symmetry condition.
## 3 Hitting time quasidistance induces distance
It might be a concern that we use a quasi-distance, not a distance. In this
section, we show that applying $\mathcal{GW}$ to a quasi-metric measure spaces
still yield a distance (our Theorem 2).
We have constructed a triple $(S\times A,\rho_{\pi},T_{\pi})$ from the MDP
$(S,A,R,P,\gamma)$. Now we can apply the $\mathcal{GW}$ construction [5] to
define a quantity between two MDPs
###### Definition 4.
Let $(S_{1},A_{1},R_{1},P_{1},\gamma_{1})$ and
$(S_{2},A_{2},R_{2},P_{2},\gamma_{2})$ be MDPs. Denote by
$(\mathcal{X},\rho_{\mathcal{X}},T_{\mathcal{X}})$ and
$(\mathcal{Y},\rho_{\mathcal{Y}},T_{\mathcal{Y}})$ their induced triples,
respectively. Then
$\begin{split}&\mathcal{GW}((S_{1},A_{1},R_{1},P_{1},\gamma_{1}),(S_{2},A_{2},R_{2},P_{2},\gamma_{2}))\\\
\coloneqq&\mathcal{GW}((\mathcal{X},(1-\gamma_{1})\rho_{\mathcal{X}},T_{\mathcal{X}}),(\mathcal{Y},(1-\gamma_{2})\rho_{\mathcal{Y}},T_{\mathcal{Y}}))\\\
=&\min_{\mu}\frac{1}{2}\left[\sum_{\mathcal{X}\times\mathcal{Y}}\sum_{\mathcal{X}\times\mathcal{Y}}|T_{\mathcal{X}}(x,x^{\prime})-T_{\mathcal{Y}}(y,y^{\prime})|^{2}\mu(x,y)\mu(x^{\prime}.y^{\prime})\right]^{\frac{1}{2}},\end{split}$
(11)
where
$\mu\in\mathcal{M}((1-\gamma_{1})\rho_{\mathcal{X}},(1-\gamma_{2})\rho_{\mathcal{Y}})$
is the set of couplings, i.e.
$\mathcal{M}((1-\gamma_{1})\rho_{\mathcal{X}},(1-\gamma_{2})\rho_{\mathcal{Y}})=\left\\{\mu\left|\sum_{y\in\mathcal{Y}}\mu=(1-\gamma_{1})\rho_{\mathcal{X}}\
\text{and}\
\sum_{x\in\mathcal{X}}\mu=(1-\gamma_{2})\rho_{\mathcal{Y}}\right\\}\right.$
(12)
###### Remark.
It is necessary to normalize the occupancy measures. Otherwise, if
$\gamma_{1}\neq\gamma_{2}$, then no coupling exists since
$\sum_{x\in\mathcal{X}}\sum_{y\in\mathcal{Y}}\mu=\sum_{x\in\mathcal{X}}\rho_{\mathcal{X}}=\frac{1}{1-\gamma_{1}}\neq\frac{1}{1-\gamma_{2}}=\sum_{y\in\mathcal{Y}}\rho_{\mathcal{Y}}=\sum_{x\in\mathcal{X}}\sum_{y\in\mathcal{Y}}\mu.$
Different MDPs may give rise to the same triple. To eliminate this redundancy,
we introduce an equivalent relation on $\mathcal{MDP}$, the set of all MDPs.
###### Definition 5 (Equivalent relation on $\mathcal{MDP}$).
$\textup{MDP}_{1}$ and $\textup{MDP}_{2}$ are said to be equivalent, denoted
by $\textup{MDP}_{1}\sim\textup{MDP}_{2}$ if they give rise to the same
triple, i.e. there exists a bijection $\varphi:S_{1}\times A_{1}\to
S_{2}\times A_{2}$ s.t.
$T_{1}(x,x^{\prime})=T_{2}(\varphi(x),\varphi(x^{\prime}))$ and
$(1-\gamma_{1})\rho_{1}(x)=(1-\gamma_{2})\rho_{2}(\varphi(x))$ for all
$x,x^{\prime}\in S_{1}\times A_{1}$.
We have not yet called $\mathcal{GW}(\textup{MDP}_{1},\textup{MDP}_{2})$ a
distance. We know that $\mathcal{GW}$ is a distance function on the space of
metric-measure spaces after quotient out redundancy [5]. However, the first-
hitting time is only a quasi-distance, hence $(S\times
A,(1-\gamma)\rho_{\pi},T_{\pi})$ is not necessarily a metric-measure space. We
prove that $\mathcal{GW}$ is indeed a distance function on
$\mathcal{MDP}/\sim$.
###### Theorem 1.
$\mathcal{GW}$ is a distance on $\mathcal{MDP}/\sim$.
###### Proof.
Theorem 16 in [2] says that if $T_{\mathcal{X}}$ and $T_{\mathcal{Y}}$ are
measurable, then $\mathcal{GW}$ is a pseudo-distance. Hence,to show that
$\mathcal{GW}$ is a distance, we need to show that
$\mathcal{GW}(\textup{MDP}_{1},\textup{MDP}_{2})=0$ iff
$\textup{MDP}_{1}\sim\textup{MDP}_{2}$. The “if" part is trivial. We only need
to show
$\mathcal{GW}(\textup{MDP}_{1},\textup{MDP}_{2})=0\implies\textup{MDP}_{1}\sim\textup{MDP}_{2}$.
Let $\mu$ be a coupling s.t. Equation (11) evaluates to $0$. Then
$T_{\mathcal{X}}(x,x^{\prime})=T_{\mathcal{Y}}(y,y^{\prime})$ for all pairs of
$(x,y)$ and $(x^{\prime},y^{\prime})$ s.t. $\mu(x,y)>0$,
$\mu(x^{\prime},y^{\prime})>0$. Let $x=x^{\prime}$. Then
$T_{\mathcal{X}}(x,x)=T_{\mathcal{Y}}(y,y^{\prime})=0$. Since
$T_{\mathcal{Y}}$ is a quasi-distance, it implies that $y=y^{\prime}$. Hence
$\mu$ does not split mass i.e. $\mu$ induces a bijection $\varphi$ s.t.
$T_{\mathcal{X}}(x,x^{\prime})=T_{\mathcal{Y}}(\varphi(x),\varphi(x^{\prime}))$.
It remains to check measures. By the boundary condition Equation (12),
$\begin{split}(1-\gamma_{2})\rho_{\mathcal{Y}}(\varphi(x))&=\sum_{x\in\mathcal{X}}\mu(x,\varphi(x))\\\
&=\sum_{\varphi(x)\in\mathcal{Y}}\mu(x,\varphi(x))\\\
&=(1-\gamma_{1})\rho_{\mathcal{X}}(x)\end{split}$
as desired. ∎
###### Remark.
Intuitively, the definition of $\mathcal{GW}$ involves a minimization over all
couplings. If $T_{\pi}(x,x^{\prime})\neq T_{\pi}(x^{\prime},x)$, the
minimization will choose the smaller one.
###### Remark.
Theorem 16 in [2] only says that their $\mathcal{GW}$ is a pseudo-distance.
They explicitly constructs an example in their Remark 15 where their
$\mathcal{GW}$ fails to be a distance. This is because they assume nothing
about their network weight function $\omega$, except measurability. In
particular, $\omega(x,x)$ in their Remark 15 does not evaluate to zero.
However, for the first-hitting time, by definition $T(x,x)=0$.
## 4 Conclusion
We defined the hitting time for an MDP via a relationship between the MDP and
the PageRank. The hitting time is a quasi-distance. By applying $\mathcal{GW}$
construction, the hitting time quasi-distance induces a distance on the set of
discrete MDPs.
## References
* [1] K. Avrachenkov, A. Piunovskiy and Y. Zhang “Hitting Times in Markov Chains with Restart and their Application to Network Centrality” In _Methodology and Computing in Applied Probability_ 20, 2018, pp. 1173–1188 URL: https://doi.org/10.1007/s11009-017-9600-5
* [2] S. Chowdhury and F. Mémoli “The Gromov–Wasserstein distance between networks and stable network invariants” In _Information and Inference: A Journal of the IMA_ 8, 2019, pp. 757–787 URL: https://doi.org/10.1093/imaiai/iaz026
* [3] Arnaud Fickinger, Samuel Cohen, Stuart Russell and Brandon Amos “Cross-Domain Imitation Learning via Optimal Transport” In _International Conference on Learning Representations_ , 2022 URL: https://openreview.net/forum?id=xP3cPq2hQC
* [4] O. Hernández-Lerma and J. Lasserre “Discrete-Time Markov Control Processes”, Applications of Mathematics Springer, 1996 URL: https://doi.org/10.1007/978-1-4612-0729-0
* [5] F. Mémoli “Gromov–Wasserstein distances and the metric approach to object matching” In _Foundations of computational mathematics_ 11, 2011, pp. 417–487 URL: https://doi.org/10.1007/s10208-011-9093-5
* [6] P. Sarkar and G. Gordon “Random Walks with Random Projections” In _Workshop on Analyzing Networks and Learning with Graphs NIPS_ , 2009 URL: http://snap.stanford.edu/nipsgraphs2009/papers/sarkar-paper.pdf
|
# Robust Environment Perception for Automated Driving: A Unified Learning
Pipeline for Visual-Infrared Object Detection
Mohsen Vadidar1, Ali Kariminezhad1, Christian Mayr1, Laurent Kloeker2 and Lutz
Eckstein2 1The authors are with the Elektronische Fahrwerksysteme GmbH, 85080
Gaimersheim, Germany{mohsen.vadidar, ali.kariminezhad, christian5.mayr}@efs-
auto.de2The authors are with the research area Vehicle Intelligence &
Automated Driving, Institute for Automotive Engineering, RWTH Aachen
University, 52074 Aachen, Germany {laurent.kloeker<EMAIL_ADDRESS>aachen.de
###### Abstract
The RGB complementary metal-oxide-semiconductor (CMOS) sensor works within the
visible light spectrum. Therefore it is very sensitive to environmental light
conditions. On the contrary, a long-wave infrared (LWIR) sensor operating in
8-14 µm spectral band, functions independent of visible light.
In this paper, we exploit both visual and thermal perception units for robust
object detection purposes. After delicate synchronization and (cross-)
labeling of the FLIR [1] dataset, this multi-modal perception data passes
through a convolutional neural network (CNN) to detect three critical objects
on the road, namely pedestrians, bicycles, and cars. After evaluation of RGB
and infrared (thermal and infrared are often used interchangeably) sensors
separately, various network structures are compared to fuse the data at the
feature level effectively. Our RGB-thermal (RGBT) fusion network, which takes
advantage of a novel entropy-block attention module (EBAM), outperforms the
state-of-the-art network [2] by 10% with 82.9% mAP.
## I INTRODUCTION
A statistical projection of traffic fatalities in the United States for the
first half of 2021 shows that an estimated 20,160 people died in motor vehicle
traffic crashes. This represents an increase of about 18.4 percent as compared
to 17,020 fatalities that were reported in the first half of 2020 [3]. Looking
at the fatal accidents of 2019 based on the time, one can see that there are
1,000 more fatal accidents during the night-time compared to the day-time [4].
Given less average traffic during the night-time, the importance of visibility
in dark is inevitable.
The number of publications on RGB-IR sensor fusion for multi-spectral object
detection in the automotive sector has increased within the past two years.
However, the lack of data in this research area is still noticeable. There are
two main sources of data, namely FLIR thermal dataset [1] and KAIST multi-
spectral pedestrian detection benchmark [5], which provide a dataset
containing IR and RGB pair images. FLIR mainly provides three classes car,
pedestrian, and bicycle, whereas KAIST only contains pedestrians.
(a) RGB Frame
(b) IR Frame
Figure 1: A sample of IR and RGB frames in heavy rain from the same scene: one
can notice that the pedestrians are not detected on the RGB frame and the cold
vehicle coming out of a parking lot is not recognized on the IR frame.
Realizing the complementary data in this multispectral setup motivated us to
fuse the information.
Nevertheless, the FLIR dataset comes only with IR labels. That introduces the
first challenge to researchers. Previously published papers [2], [6] and [7]
have made various objections to the dataset. For instance, the usage of
different camera resolutions at multiple instances ranging from 0.3 to 3.1 MP,
misalignment of resolution/aspect ratio, a vast difference between the field
of views and resolutions between the sensors, and no possibility to align the
RGB and IR frames without having a correlation matrix. These issues currently
made fusing the data on this dataset an unsolved problem.
Within this work, we address the FLIR dataset challenges and propose a
solution to properly utilize it. Further, we introduce a CNN architecture to
fuse RGB-IR data. The proposed network outperforms the state-of-the-art
networks for multi-spectral object detection purposes [2]. In section II, the
latest advancements and publications on RGB-IR sensor fusion networks will be
presented. Later in section III, a cross-labeling algorithm and pre-processing
pipeline will be introduced to provide suitable labels for RGB frames. In
section IV, we start with an RGB network and compare its detection performance
with a monospectral IR network. The best monospectral detector will be
considered as our baseline. Afterward, we proceed with the simplest form of
fusion and develop our proposed RGBT network step by step. Finally, sections V
and VI are dedicated to results, analysis and conclusion of this work.
## II Related Works
In July 2020 Ravi Yadav et al. attempted fusing color and thermal images to
detect objects for self-driving applications [6]. They have used VGG16 [8] and
Faster-RCNN [9] as their encoder and detector, respectively. They achieved a
miss rate of 29% on the KAIST benchmark. However, they could not perform well
on the FLIR benchmark due to its complications. Later that month, Chaitanya
Devaguptapu et al. [10] have used the GAN framework to use the FLIR dataset
and produce RGB images from the IR frames. The idea was to borrow knowledge
from the data-rich pre-trained RGB domain and use it on IR images. Heng Zhang
et. al. introduced cyclic fuse and refine blocks for object detection [7] in
September 2020. The FSDD (Feature Fusion Single Shot Multibox Detector)
network [11] is used as their object detector and two VGG16 backbones are
implemented to work independently from each other for each sensor. No
information is given regarding the number of parameters or complexity of the
network. However, looking at the designed cyclic module, the number of its
occurrences, and considering the used VGG16, which is one of the most
expensive feature extractors with 138 million parameters, one can realize that
this network has more than 276 million parameters. In this work, a so-called
"well-aligned" version [7] of FLIR is released. However, as mentioned in the
paper, the misalignment problem made exploiting FLIR dataset for multispectral
object detection impractical. Therefore, the pre-processing part, could not
tackle all the issues from the dataset. We will have a look at a failed
attempt in the next section. Nevertheless, Cyclic Fuse-and-Refine Network
(CFR_3) could reduce the miss rate on the KAIST dataset and improve the mAP on
the FLIR benchmark to 72.39%. In November 2020 A. Sai Charan et al. attempted
to fuse IR and RGB data by giving weight to each sensor using attention
mechanism [12]. The attention module is placed in every residual block [13]
after the RELU activation function and first batch normalization. They have
tried both the late and mid-fusion approaches. Late fusion in this case is
done by pooling features of the thermal and RGB camera separately from their
respective region proposals and then concatenating them together. Comparing
the performance of the two mid and late fusions, mid-fusion has shown a better
result with 62.12% mAP. In January 2021 Heng Zhang et. al. proposed guided
attentive feature fusion (GAFF) [2]. This paper claims to be the first work
that is considering multi-spectral feature fusion as a sub-task in network
optimization. While bringing accuracy gains, GAFF has a lower computational
cost compared to common addition or concatenation methods. They reduced the
miss rate on the KAIST dataset by 2% and achieved an mAP of 72.9% on the FLIR
dataset. To the best of our knowledge, this is the highest mAP achieved using
the FLIR dataset. Thus, this work will be taken as the state-of-the-art
network on this benchmark. ZHANG et. al. in a comprehensive review studied
different methods and levels of fusion between visible and IR cameras for the
task of tracking [14]. Since the feature extraction process is the same in
tracking and object detection, this study can provide a good insight into
feature-level fusion using deep learning-based methods.
## III Dataset and Data Pre-processing
This section is an important part of our work, without which the rest of our
research would not be possible.
### III-A Dataset
Table I summarizes open-source datasets including IR frames. Most of the
datasets are concerned with pedestrian class only. Those which cover a wider
range of objects like the Dense dataset [15] confront other downsides such as
non-availability of RGB-IR data pairs, annotation issues, strong misalignment
between IR and RGB fields of view (FOV), and lack of extrinsic calibration
information (e.g. correlation matrix).
TABLE I: Available Datasets including IR Frames (NG stands for not given) Name | Year | Size (Frames) | Downsides
---|---|---|---
FLIR | 2020 | 13813 | IR label only
Dense | 2020 | 121500 | Narrow IR FOV
ZUT-FIR | 2020 | NG | IR data available only
SCUT-CV | 2019 | 211011 | IR data available only
| | | Pedestrian class only
CAMEL | 2018 | NG | Not on-board frames
Multispec. | 2017 | 7512 | Different Ref. Sys.
CVC-14 | 2016 | 3695 | Gray RGB
| | | Pedestrian class only
KAIST | 2015 | 95000 | Crowd label as human
| | | Pedestrian class only
CVC-09 | 2014 | 5990 | Pedestrian class only
### III-B Data Pre-processing
To include the Vulnerable Road Users (VRU) as well as motorized users in our
study, we have decided to work with FLIR dataset [1]. The non-availability of
a dataset for fusing the IR and RGB frames, led us to put a great deal of
effort into developing a reliable cross-labeling algorithm, additional manual
labeling of the missing objects, and checking the data one by one, to make
sure the entire dataset is reliable and usable.
Sensor data fusion between an IR and an RGB camera was a challenge to
researchers since there is no open-source dataset providing synchronized and
labeled RGB-IR image pairs. Given that the ground truth in FLIR dataset is
provided for IR frames only, we introduce a method to cross-label the RGB
frames using IR annotations semi-automatically.
Figure 2: IR-RGB overlay: Union of the filed of views in green and the
transferred labels from IR to RGB frame in red box
Algorithm 1 demonstrates our approach for pre-processing the data. We first
convert the RGB frames to gray-scale images and extract the edges. IR and RGB
frames are matrices of size $640\times 512$ and $1800\times 1600$,
respectively. Thus, we introduce a scaling-parameter to resize the IR frame
gradually in a loop, until the IR extracted edges fall on RGB edges. The
cross-correlation value between the two matrices is a good metric to
understand the accuracy of the alignment. The highest value is obtained, when
the edges of two frames are completely aligned. Since the two frames are
different in size, the cross-correlation between them takes place in a
sliding-window manner. As figure 2 illustrates, vertical and horizontal
offsets exist between the RGB and IR frames. Figure 3 represents a comparison
between our result and the "well-aligned" version [7]. Ultimately, we utilize
6924, 1982, and 3960 pair images for training, validation, and test,
respectively.
Result: X_Offset, Y_Offset, Scaling_Factor
1 load both frames
2 Smooth both frames by a Gaussian filter
3 Detect the edges using Canny edge detector
4 Use binary threshold to set the edge pixels to one and all the other pixels
to zero
5
6Scaling_Factor = 2 # A constant to resize the IR frame
7 while _Width_IR < Width_RGB_ do
8 Width_IR_new := Width_IR $\times$ Scaling_Factor
9 Resize IR binary frame with the Width_IR_new and the corresponding
Height_IR_new
10 Cross correlate (CC) the IR and RGB binary frames in a sliding window
manner and save the pixel coordinate of the maximum value from the correlation
matrix
11
12 if _CC_Value > Best_CC_Value_ then
13 Update Best_CC_Value, Pixel_coordinates, Best_Scaling_Factor
14
15 end if
16 Scaling_Factor += 0.001
17
18 end while
19Best_X_Offset, Best_Y_Offset := Pixel_coordinates
20 return Best_X_Offset, Best_Y_Offset, Best_Scaling_Factor
Algorithm 1 Cross Labeling Algorithm
(a) "well-aligned" version
(b) our version
Figure 3: Comparison of two cross-labeling methods: the left image represents
the previous attempt [7] to process the FLIR dataset [1] and transfer the
labels from IR to RGB coordinate system. The right picture illustrates the
result of our proposed cross-labeling algorithm.
## IV Network Structures
One of the most accurate networks on MSCOCO benchmark [16], namely Scaled-
Yolov4 [17] is selected as a starting point for this work. The performance of
each sensor will be evaluated separately and the best monospectral network
will be chosen as our baseline. The CSP (Cross-Stage-Partial-Connections) [18]
version of the network is used for both sensors. That is possible since the IR
sensor is delivering image-like data and the encoder is very good at picking
up the features and adapting itself to the IR sensors’ data texture (i.e.
domain adaptation).
To take the most of each monospectral backbone, we start with our vanilla
fusion. The designed network fuses the feature maps across scales as shown in
figures 4 and 5. The smallest scale is designed for large objects and the
biggest one detects the smaller objects. After concatenation, channels are
halved using $1\times 1$ convolution layers to prepare the feature maps for
the detector head.
The most crucial part of fusing visible and IR sensors is finding a way to
weight each backbone, depending on how informative the extracted features are.
As each sensor receives a different wavelength, we want to ensure that
complementary data from both spectrums increase the chance of detecting an
object as well as improving the robustness of the network. The goal is to
detect the objects, even in lack of strong emissions within visible-light
frequency. To do that, we integrate two Convolutional Block Attention Modules
(CBAM) [19], such that more weight is given to more informative features
depending on the input data (see figure 5).
Figure 4: RGBT Network Structure: Two identical encoders are used for feature
extraction from both RGB and IR frames. The feature-level fusion block
receives features in three scales from each encoder and fuses them. Then, the
data is passed to the detector, which is the same as CSP version of
Scaled_Yolov4. At last, three tensor outputs refer to three scales. Elements
of the fusion block are demonstrated in figure 5.
In what follows, we will discuss our proposed entropy-block attention module
(EBAM) and how it is a better substitute for the conventional CBAM [19]. In
this regard, we will proceed with channel attention, which can further be
generalized to spatial attention. Suppose, average and max pooling for feature
map (channel) $k$ are represented by $a_{k}$ and $m_{k}$, as:
$\displaystyle a_{k}$
$\displaystyle=\frac{1}{MN}\sum_{i}^{M}\sum_{j}^{N}X^{<k>}_{ij},$ (1)
$\displaystyle m_{k}$ $\displaystyle=\max_{i,j}\\{X^{<k>}_{ij}\\},$ (2)
respectively, where $\mathbf{X}^{<k>}$ is the matrix of $k$th feature map and
$X^{<k>}_{ij}$ is the $k$th feature map grid value for the grid located in
$i$th row and $j$th column. Further, we define the vector of average and max
poolings as
$\displaystyle\mathbf{a}$
$\displaystyle=[a_{1},\cdots,a_{k},\cdots,a_{K}]^{T},$ (3)
$\displaystyle\mathbf{m}$
$\displaystyle=[m_{1},\cdots,m_{k},\cdots,m_{K}]^{T}.$ (4)
According to the CBAM, the vectors $\mathbf{a}$ and $\mathbf{m}$ pass through
a shared multi-layer perceptron, for which the weights are optimized in the
backpropagation. Here, we assume a single layer for convenience in further
mathematical presentation. This can later be generalized to multi-layer
networks. Thus, the channel attention portion of the CBAM with a single-layer
perceptron delivers the following weight vector:
$\displaystyle\mathbf{c}_{\textrm{CBAM}}=\mathbf{W}(\mathbf{m}+\mathbf{a})+\mathbf{b},$
(5)
where the matrix of shared weights and the bias are represented by
$\mathbf{W}\in R^{K\times K}$ and $\mathbf{b}\in R^{K}$, respectively.
Therefore, the attention weight corresponding to the $k$th feature map can
readily be formulated as
$\displaystyle
c^{<k>}_{\textrm{CBAM}}=\mathbf{w}_{k}(\mathbf{m}+\mathbf{a})+b_{k},$ (6)
where $\mathbf{w}_{k}$ is the $k$th row of matrix $\mathbf{W}$. Interestingly,
one could observe that the average and max poolings are equally weighted due
to the shared layers among them. Consequently, the contribution of the average
and max poolings are not captured and only the sum of the values plays a role.
For instance, assume a particular feature map consists of multiple very high
values, while the majority of the grids contain low values. This way, the
average pooling downgrades the importance of that particular feature map.
However, due to the shared weights, this phenomenon can not be remedied in the
backpropagation by weight optimization. This motivates us to develop a more
robust and flexible attention mechanism as a replacement for CBAM.
Uncertainty of a data-driven feature is a valuable metric delivering a notion
for assessing how much extra information can still be mined from that feature.
$\displaystyle H_{k}=-\sum_{i=1}^{M}\sum_{j=1}^{N}P(X^{<k>}_{ij})\log
P(X^{<k>}_{ij}),$ (7)
where $P(X^{<k>}_{ij})$ are the probabilities delivered by the softmax
operation as,
$\displaystyle
P(X^{<k>}_{ij})=\frac{e^{X^{<k>}_{ij}}}{\sum_{i=1}^{M}\sum_{j=1}^{N}e^{X^{<k>}_{ij}}}.$
(8)
Notice that, equal probabilities maximizes the entropy.
$\displaystyle P(X^{<k>}_{ij})=\frac{1}{MN},\quad\forall i,j.$ (9)
It is important to discern that, the max entropy is reached if the max pooling
equals the average pooling of the feature map $k$, i.e., $a_{k}=m_{k}$, which
holds if:
$\displaystyle
X^{<k>}_{ij}=\frac{1}{\sum_{i}^{M}\sum_{j}^{N}X^{<k>}_{ij}},\forall i,j.$ (10)
This reflects the fact that in an extreme case the sum of max and average,
i.e., $m_{k}+a_{k}$, corresponds with the entropy upper-bound, i.e.,
$H^{\textrm{max}}_{k}$. This extreme case, i.e., $a_{k}=m_{k}$ can not be
differentiated by CBAM, since it considers only the sum. However, entropy
captures the interaction between max and average as follows:
$\displaystyle\lim_{m_{k}-a_{k}\rightarrow 0}H_{k}$
$\displaystyle=H^{\textrm{max}}_{k},$ (11)
$\displaystyle\lim_{m_{k}-a_{k}\rightarrow\infty}H_{k}$ $\displaystyle=0.$
(12)
That means, as the max pool gets closer to the average pool, the entropy
maximizes and as it gets higher value than the average pool, the entropy
approaches zero. It is important to mention that, the entropy of a random
process does not reduce except by providing information. Furthermore, notice
that more data does not necessarily mean more information. This observation
motivated us to design a network, which focuses on informative features in the
fusion process. This focus is represented by strongly weighting the highly-
informative feature maps. We call this new attention mechanism entropy-block
attention module (EBAM), for which the $k$th channel attention weight is given
by:
$\displaystyle c^{<k>}_{\textrm{EBAM}}=\mathbf{v}_{k}\mathbf{h}+b_{k},$ (13)
where $\mathbf{h}=[H_{1},\cdots,H_{k},\cdots,H_{K}]^{T}$. Note that
$\mathbf{v}_{k}$ is the channel attention weight vector designated for the
$k$th feature map optimized in the backpropagation. In what follows we
generalize the EBAM to capture both channel and spatial attention.
Let’s define the intermediate feature map $F\in R^{C\times H\times W}$ as a
tensor consisting of $\mathbf{X}^{<k>},\ \forall k$. That means,
$F=[\mathbf{X}^{<1>},\cdots,\mathbf{X}^{<k>},\cdots,\mathbf{X}^{<K>}]$. This
feature map tensor is the input to our attention module, from which attention
weights are derived. Hence, the generalized self-attention can be written as:
$W(F)=\alpha(F)\>\otimes\>F,$ (14)
where $\otimes$ denotes element-wise multiplication. $C,H$ and $W$ refer to
channel, height and width, respectively and $\alpha(F):R^{C\times H\times
W}\rightarrow R^{C\times H\times W}$ is an attention function of the feature
map $F$.
Figure 5: Upper block: Vanilla fusion is referred to a simplistic approach to
fuse the data by just concatenating the extracted features in each scale and
selecting the most valuable information by a convolution layer. Middle block:
CBAM is added as an attention module. Lower block: EBAM is designed to select
informative features more efficiently.
EBAM sequentially infers a 1D channel attention map $A_{c}\in R^{C\times
1\times 1}$ and a 2D spatial attention $A_{s}\in R^{1\times W\times H}$ as
illustrated in figure 6. The overall attention process can be summarized as:
$\displaystyle F^{\prime}=A_{c}(F)\otimes F,$ (15) $\displaystyle
F^{\prime\prime}=A_{s}(F^{\prime})\otimes F^{\prime},$
Figure 6: EBAM: detailed illustration of entropy-block attention module, which
contains channel and spatial domains. MLP refers to multilayer perceptron.
In the channel attention by giving more weight to more uncertain feature maps
(assuming each feature map is represented by a random variable), the network
is signaled to extract more information by prioritizing the gradient direction
in backpropagation. Therefore, the network is forced to gain more information
(i.e. kernel activation) from different kernels. As a matter of fact, the
proposed attention module operating only in the channel domain, showed a
relatively better performance than a CBAM including both spatial and channel
attention. One can calculate the explained channel attention as follows:
$\displaystyle A_{c}(F)$ $\displaystyle=\sigma(MLP(H(F))),$ (16)
$\displaystyle=\sigma(W_{0}(F^{C}_{H})),$
where $\sigma$ denotes the sigmoid function, $H$ is the entropy matrix, $F$
stands for feature map and $W_{0}$ represents the MLP weights.
The more the entropy of a random process, the more uncertain we are about the
outcome. In the case of the attention module in the spatial domain, each grid
pattern can contribute to detecting an object. As the training process goes
on, the grids, which refer to an object will have lower entropy (i.e. lower
uncertainty). Hence, if we use the entropy values directly, more weight will
be given to grids with high entropy, which would be the background information
in this case. Thus, more weights are intended to be allocated for the grids
with less entropy values through the feature maps.
Therefore, the entropy-based spatial attention can be denoted as follows:
$\displaystyle A_{s}(F^{\prime})$
$\displaystyle=\sigma(Conv.(1-\dfrac{H_{ij}}{\max_{\\{i,j\\}}H_{ij}})),$ (17)
$\displaystyle=\sigma(W_{1}(F^{S}_{H})),$
The networks are trained to the point, where an overfitting effect appeared.
The hyperparameters are kept unchanged as the original Scaled-Yolov4 [17]
implementation.
## V Results and Analysis
In this chapter, the results of our pre-processing, baseline, and fusion
networks are presented. The processed data and the code can be found in the
following GitHub link: https://github.com/SamVadidar/RGBT.
### V-A Data Pre-processing
Exploiting our proposed pre-processing step (i.e. algorithm 1), not only the
accuracy of the bounding boxes is improved, but more data (i.e. 13,848 images)
could be recovered from the original dataset. That means 34.6% more data with
respect to what Heng Zhang’s version [7] could use in training. Thus, a
dataset is produced, where the labels can be referred to both frames, which
are $640\times 512$ pixels. After running the pre-processing algorithm on the
original dataset, one receives 6924, 1982, and 3960 paired images for train,
validation, and test sets, respectively. Since the intention is to compare a
CMOS stand-alone sensor with a CMOS-IR sensor pair, the dataset is split into
night and day subsets. Therefore, one can test and evaluate a trained network
based on these two scenarios.
### V-B Studied Networks
To reduce the computational costs, all the network trainings and their
comparisons will be done using $320\times 320$ image size and only the final
network will be fine-tuned with the original images ($640\times 512$ pixels).
Generally speaking, car class being the largest object among the three, shows
the best mAP. On the contrary, the person class being the smallest in size is
the most challenging object to detect.
TABLE II: Summary of studied networks: Fusion+H_C refers to a fusion network with entropy-block attention modules in the channel domain only. RGBT contains the full network structure as discussed in chapter IV. CFR [7] and GAFF [2] illustrate the two best available mAPs (NG stands for not given) on the FLIR dataset [1]. RGBT* is trained and tested using previously published aligned version [7] of the FLIR dataset. Network |<EMAIL_ADDRESS>| Param.
---|---|---
| Person | Bicycle | Car | Overall |
RGB Baseline | 39.6% | 50.4% | 79.4% | 56.6% | 52.5 M
IR Baseline | 49.6% | 54.9% | 84.4% | 63.0% | 52.5 M
Vanilla Fusion | 56.9% | 56.7% | 82.0% | 65.2% | 81.8 M
Fusion+CBAM | 57.6% | 60.5% | 83.6% | 67.2% | 82.7 M
Fusion+H_C | 62.6% | 65.9% | 86.0% | 71.5% | 82.7 M
RGBT | 63.7% | 67.1% | 86.4% | 72.4% | 82.7 M
CFR(640) | 74.4% | 57.7% | 84.9% | 72.3% | $>$276 M
GAFF(640) | NG | NG | NG | 72.9% | NG
RGBT*(640) | 85.3% | 64.0% | 88.6% | 79.3% | 82.7 M
RGBT(640) | 80.1% | 76.7% | 91.8% | 82.9% | 82.7 M
A very interesting finding is that the infrared camera outperforms the CMOS
sensor even as a stand-alone sensor. Its performance in table II shows a 6.4%
overall improvement in the mAP metric using the same training pipeline, same
network, and an equal number of epochs. One can observe the improvement in all
classes, especially for the pedestrian class with a 10% jump in mAP. High
contrast in the frames between the objects and surroundings, very high sensor
sensitivity of less than 50 mK as well as an operating temperature of -40°C to
+85°C, delivers a data-rich infrared frame for feature extraction. Later in
this chapter, some qualitative and quantitative analyses will be illustrated
to support this statement.
The first attempt for fusing the two backbones (i.e. Vanilla Fusion) has shown
a 2.2% improvement in the overall mAP compared to the IR network (See table
II). The boldest improvement among the classes belongs to the pedestrian
class, which has experienced a drastic improvement of 7.3% with respect to the
IR network and 17.3% to the RGB network considering the mAP metric. Overall
8.6% mAP rise compared to the RGB network by adding no additional module,
motivated us to use the CBAM attention module [19]. By the way the CBAM is
implemented, each beneficial sensor data is enhanced and downsides are
compensated by focusing on the right and reliable features automatically.
Therefore, an additional performance gain of 2.2% compared to the vanilla
network is achieved and the bicycle class mAP is improved by 3.8%. After
designing the entropy-block, despite spatial attention absence, the Fusion+H_C
network could outperform the Fusion+CBAM network. One can see 4.3% mAP
improvement overall, and 5.0% for pedestrian class.
The proposed RGBT Network contains two complete (i.e. channel and spatial)
attention modules. The spatial attention added 0.9% on top of the channel-only
attention network and pushed the overall mAP up to 72.4%. The network shows a
noticeable overall mAP improvement to RGB and IR baselines with 15.8% and
9.4%, respectively. This shows that a fusion network can have an immense
impact on improving the mAP. Since the RGBT network performs better than both
IR and RGB object detectors, one can observe the importance of an efficient
feature-fusion by selecting the best features from each encoder through the
entropy-block attention module. This work could outperform the results of the
CFR network [7], by only utilizing half of the pixel resolution. We also have
trained and tested our proposed network with the same aligned version of the
FLIR dataset [7], which CFR and GAFF used. This helps to differentiate the
contribution of our preprocessing method from the proposed network. As the
table II illustrates, 3.6% mAP improvement is solely achieved by better
preprocessing the data. The test-set of the "well-aligned" dataset has almost
46% less number of pedestrian instances. Fewer test data and relatively easier
scenarios might be the underlying reason behind the higher mAP (i.e. 85.3%)
for this class. Considering the number of parameters and what the CFR paper
[7] reports, two VGG-16 networks [8] are used as separate feature extractors
of each sensor. Hence, knowing that each VGG-16 has around 138 million
parameters, one can find out that the number of the parameters in the state-
of-the-art network is greater than 276 million. Looking at the table II, it is
demonstrated that the RGBT network could outperform the CFR network by using
at least 70% less number of parameters. Therefore, not only the mAP is
improved in this work, but the complexity of the network is reduced
drastically. This will cause a considerably faster inference speed compared to
the CFR network. The GAFF, which showed slightly (i.e. 0.6%) better mAP than
CFR network, does not report the per class mAPs and the total number of
parameters of the complete network. Due to limited resources and long training
time, only our proposed network is trained with the original image size. Table
II shows that using the full resolution, one can reach 82.9% mAP with 10%
performance gain compared to the state-of-the-art network.
The results using the original images are compared to other networks in figure
7. 10.5% raise in the mAP due to a higher input resolution, which indicates
the importance of the pixel per object value. Working with FLIR dataset [1],
one can observe that it contains very small and challenging objects,
especially in the pedestrian class. Therefore, when the number of input
pixels, which are fed to the network is doubled (i.e. $640\times 640$), the
overall outcome elevates significantly.
Figure 7: Qualitative comparison of three scenarios: Columns from left to
right: RGB, IR, Fusion with CBAM, and Fusion with EBAM. Note that the fusion
results are represented using TGB (i.e. thermal, green, blue) instead of RGB
colormap.
Looking at figure 7, it is to be noted that the detected objects on the
sidewalk in the first two scenarios (i.e. first two rows) were not labeled.
That means the network is using the extracted features from other instances of
the pedestrian class to classify and localize the objects. Since in the
training process, detected objects, which do not have a corresponding label
are punished by the loss function, the performance of the network in this
scene is admirable. The third scenario (i.e. third-row) illustrates a group of
heavily occluded pedestrians sitting around a table. The night scenarios are
the most challenging scenarios for the RGB cameras and in this case, the
target object was neither labeled nor provided a sufficient amount of pixel
information. Despite these challenges, the RGBT network having only $320\times
320$ resolution could detect the pedestrians as well as the bicycles in front
of them.
## VI Conclusion and Outlook
In general, cameras are one of the most informative sensors on the vehicle.
However, they are highly sensitive to light and weather conditions. Such
disturbances affect their performance severely. In order to introduce
robustness to the system, the functionality of an infrared camera as a
complementary sensor is investigated. Two groups of researchers [2], [6] and
[7] have considered data pre-processing and preparation of FLIR dataset [1] as
one of the major challenges and referred to it as an unsolved problem. High-
quality data being the core of machine learning approaches motivated us to
address this issue with our cross-labeling algorithm. Hence, 34.6% more data
is retrieved with respect to the previous approach [7]. Considering the trend
in recent publications, the rally was to gain a single additional mAP
improvement, whereas thanks to the pre-processing pipeline and the entropy-
block attention module, the proposed RGBT network could outperform the state-
of-the-art network [2] by 10% mAP. Undoubtedly, the availability of higher
quantity and quality labeled data is essential for further investigation in
this domain. In our next step, we are planning to combine our RGB-Radar
network with the proposed RGBT network to extract depth information from the
detected objects. We also have seen that Radar can improve the mAP
considerably. At last, the network will be optimized by TensorRT for our
target hardware and run on our demonstrator vehicle to perform real-world
tests.
## ACKNOWLEDGMENT
The authors would like to thank EFS MLOps team, specially Mr. Sebastian Balz
for his unceasing support, without which the training and evaluation of the
networks would not be possible.
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|
Measuring Inconsistency over Sequences of Business Rule CasesThis research is part of the research project ”Handling Inconsistencies in Business Process Modeling“, which is funded by the German Research Association (reference number: DE1983/9-1).
Carl Corea1 Matthias Thimm2 Patrick Delfmann1
C. Corea et al.
Institure for Information Systems Research, University of Koblenz-Landau Institute for Web-Science and Technologies, University of Koblenz-Landau
In this report, we investigate (element-based) inconsistency measures for multisets of business rule bases. Currently, related works allow to assess individual rule bases, however, as companies might encounter thousands of such instances daily, studying not only individual rule bases separately, but rather also their interrelations becomes necessary, especially in regard to determining suitable re-modelling strategies. We therefore present an approach to induce multiset-measures from arbitrary (traditional) inconsistency measures, propose new rationality postulates for a multiset use-case, and investigate the complexity of various aspects regarding multi-rule base inconsistency measurement.
§ INTRODUCTION
In the context of Business Process Management, business rules are used as a central artifact to govern the execution of company activities [6]. To this aim, business rules are modelled to capture (legal) regulations as a declarative business logic. Then, given a new process instance (denoted as a case), instance-dependent facts are evaluated against the set of business rules for reasoning at run-time. For example, consider the following set of business rules in Figure <ref> (we will formalize syntax and semantics later) with the intuitive meaning that we have two rules stating that 1) platinum customers are credit worthy, and 2) customers with a mental condition are not credit worthy. Then, given a new customer case, in the example a new loan application, the facts set is evaluated against the rule set and the resulting rule base $\rb_1$ can be used to reason about the customer case.
Exemplary business rule base instance $\rb_1$.
The observant reader might have noticed, that the shown example yields an inconsistency, i.e., the contradictory conclusions $\mathit{creditWorthy, \neg creditWorthy}$. In fact, this is a current problem for companies, which can result from modelling errors in the business rules, or unexpected (case-dependent) facts. This problem has widely been acknowledged and has been addressed by a series of recent works, cf. e.g. [1, 5, 3].
While existing results allow to handle inconsistencies in a single business rule base instance as shown above, in practice, companies often face thousands of such instances daily. For example, the retailer Zalando reported that 37 million cases were executed in the first quarter of 2020 alone[<https://zln.do/2SFRnfC>]. As we will show in this work, considering not only single rule bases individually, but rather the entirety of all cases and their interrelations, can yield valuable insights, especially in regard to inconsistency resolution. For example, consider the following rule set, and assume there were four customer cases (with respective case-dependent facts), yielding the set of business rule cases $\ms_1$ shown in Figure <ref>:
Exemplary rule base instances, constructed over a seq. of case-dependent facts.
When auditing such an overview of rule base instances, two main questions are of interest from a business rules management perspective:
* How inconsistent in general was the entirety of process executions?
* Which specific rules were responsible for these inconsistencies from a global perspective?
Towards question 1, recent results studying inconsistency in sets of knowledge bases can easily be adapted to quantify the overall degree of inconsistency (cf. Section 2). While this is a beneficial step for companies, question 2 can however be seen as of much higher importance in the scope of improving business rules. Pin-pointing the culprits of inconsistency is an essential challenge for determining suitable resolution and re-modelling strategies. Here, new methods are needed that support companies in assessing which individual rules are highly problematic from a global perspective.
For instance, in Figure <ref>, the rule $a\rightarrow b$ is part of all inconsistencies and can therefore be seen as highly problematic. In this work, we therefore introduce novel means for an element-based assessment of inconsistency over a set of rule base instances by extending results from the field of inconsistency measurement [14]. Here, our contribution is as follows:
We present a novel approach for inducing element-based quantitative measures for multisets of business rule bases, allowing to pin-point problematic business rules from a global perspective (Section 3). Here, we also propose postulates that should be satisfied by respective measures for this use-case and analyze the proposed means w.r.t. these postulates. We implement our approach and perform run-time experiments with real-life data-sets, and also examine the complexity of central aspects regarding inconsistency measurement in multisets of business rule bases (Section 4). We present preliminaries in Section 2 and conclude in Section 5. Proofs for technical results are provided in a supplementary document[<https://bit.ly/2V2sIDw>].
§ PRELIMINARIES
Business Rule Bases. In this work, we consider a basic (monotonic) logic programming language to formalise business rule bases. A (business) rule base is then constructed over a finite set $\atoms$ of atoms, with $\lang$ being the corresponding set of literals, with a rule base $\rb$ being a set of rules $r$ of the form
\begin{align}\label{eq:rule}
r\,:\quad l_{1},\ldots,l_{m} \rightarrow l_{0}.
\end{align}
with every $l_i\in\lang$. Let $\allrbs$ denote all such rule bases. Also, we denote $head(r)=l_{0}$ and $body(r)=\{l_{1},\ldots,l_{m} \}$. If $body(r)=\emptyset$, $r$ is called a fact.
For a rule base $\rb$, we denote $\F(\rb)\subseteq\rb$ as the facts in $\rb$ and $\R(\rb)\subseteq\rb$ as the rules in $\rb$.
We recall the business rule base $\rb_1$. Then we have
\begin{align*}
\F(\rb_1) & = \{mentalCondition, platinumCustomer\}\\
\R(\rb_1) & = \{platinumCustomer \rightarrow creditWorthy,\\
& \qquad mentalCondition \rightarrow \neg creditWorthy\}.
\end{align*}
A set of literals $M$ is called closed w.r.t. $\rb$ if it holds that for every rule of the form <ref>: if $l_{1},\ldots,l_{m}\in M$ then $l_{0}\in M$. The minimal model of a rule base $\rb$ is the smallest closed set of literals (w.r.t. set inclusion). A set $M$ of literals is called consistent if it does not contain both $a$ and $\neg a$ for an atom $a$. We say a rule base $\rb$ is consistent if its minimal model is consistent. If $\rb$ is not consistent, we say $\rb$ is inconsistent, denoted as $\rb\models\perp$.
As discussed in the introduction, $\rb_1$ is not consistent. To assess inconsistency, the field of inconsistency measurement [7] has evolved, which studies quantitative measures to assess the severity of inconsistency. An inconsistency measure [7, 14] is a function $\inc: \allrbs \rightarrow \posRealInf$, where the semantics of the value are defined such that a higher value reflects a higher degree, or severity, of inconsistency. A basic inconsistency measure is the $\incmi$ inconsistency measure, which counts the number of minimal inconsistent subsets $\MI$ of a rule base $\rb$, defined via
\begin{align*}
\MI(\rb) = \{M \subseteq \rb \mid M \models \perp, \forall M' \subset M : M' \not\models \perp \}.
\end{align*}
We recall $\rb_1$. Then we have
\begin{align*}
\MI(\rb_1) &= \{M_1\}\\
M_{1} &= \{platinumCustomer,\\
& \qquad platinumCustomer \rightarrow creditWorthy,\\
& \qquad mentalCondition,\\
& \qquad mentalCondition \rightarrow \neg creditWorthy\},
\end{align*}
consequently, $\incmi(\rb_1)=1$.
As the concept of a ”severity“ of inconsistency is not easily characterisable, numerous inconsistency measures have been proposed, see [14] for an overview. To guide the development of inconsistency measures, various rationality postulates have been proposed, cf. [13] for an overview. For example, a widely agreed upon property is that of consistency, which states that an inconsistency measure should return a value of 0 w.r.t. a rule base $\rb$ iff $\rb$ is consistent.
As mentioned, various other postulates exist and we will revisit some of them later when introducing culpability measures for multisets of rule bases.
Measuring Overall Inconsistency in Multisets of Business Rule Bases.
In this work, we are not only interested in measuring inconsistency in single business rule bases, but rather in a series of corresponding business rule base instances. As motivated in the introduction, companies currently apply a set of business rules in order to assess a stream of (case-dependent) fact sets. Therefore, given a stream of fact sets $f = \F_1,...,\F_n$, we consider multisets of business rule bases which are constructed by matching the individual fact sets in $f$ to a shared rule set $\R$. To clarify, a multiset of rule bases is an n-tuple $\ms=(\{\F_1\cup\R\},...,\{\F_n\cup\R\})=(\rb_1, ..., \rb_n)$. Let $\allmss$ denote all such multisets.
An initial question for companies is to gain an overview of the overall inconsistency w.r.t. all business rule base instances in $\ms$. For that, we define an inconsistency measure for a multiset of rule bases as follows.
An inconsistency measure for a multiset of rule bases is a function $m: \allmss \rightarrow \posRealInf$.
In other words, an inconsistency measure for a multiset of rule bases is a function that assigns a non-negative numerical value to an n-tuple of rule bases. Similar to classical inconsistency measures, the intuition is that a higher value reflects a higher degree of inconsistency of the multiset of rule bases. For simplicity, we refer to such measures as multi-rb measures where appropriate.
For the intended use-case of gaining insights about the severity of inconsistency regarding the entirety of process instances, existing inconsistency measures can be adapted to induce multi-rb measures via a summation.
Given an inconsistency measure $\inc$ and a multiset of rule-bases $\ms$, the $\Sigma$-induced multi-rb measure $m_\inc^\Sigma$ is defined as $m_\inc^\Sigma: \allmss \rightarrow \posRealInf$ with $m_\inc^\Sigma(\ms)=\sum_{B \in \ms} \inc(B)$.
We recall the introduced $\MI$-inconsistency measure $\incmi$. Correspondingly, given a multiset of rule bases $\ms$, $\incmi$ can be used to $\Sigma$-induce the multi-rb measure $m_{\incmi}^{\Sigma}(\ms) = \sum_{B \in \ms} \incmi(B)$. Considering again the exemplary multiset $\ms_1$ of business rules from Figure <ref>, with cases $b_1-b_4$, we thus have $m_{\incmi}^{\Sigma}(\ms_1) = \incmi(b_1) + ... + \incmi(b_4) = 1+1+1+2 = 5$.
Note that the approach in [12], who—roughly speaking—measures inconsistency in a multiset of knowledge bases by performing a multiset union on all sets and then measuring inconsistency on this union, is not applicable for our use-case, as we are not interested in the disagreement between the individual instances, but rather want to gain an overview of inconsistencies in all instances.
While the above discussion showed how existing means can be used to measure the overall degree of inconsistency for a multiset of rule bases, in the following, we develop techniques for an element-based assessment of inconsistency over a multiset of rule bases.
§ CULPABILITY MEASURES FOR MULTISETS OF BUSINESS RULE BASES
In the field of inconsistency measurement, a culpability measure $\culp$ [8] is a function that assigns a non-negative numerical value to elements of a rule base. This quantitative assessment is also referred to as an inconsistency value. Again, the intuition is that a higher inconsistency value reflects a higher blame, that the specific element carries in the context of the overall inconsistency. In this section, we investigate culpability measures that can assess the blame that a rule carries in the context of the overall multiset inconsistency.
§.§ Baseline measures and basic properties
Given a multiset of business rule bases $\ms$, let $\R(\ms)$ denote the shared rule set of the respective business rule bases in $\ms$. Furthermore, let $\allrs_\allmss$ denote the set of all possible rules that can appear in these shared rule sets. Then, a culpability measure for a multiset of rule bases is defined as follows.
A culpability measure for a multiset of rule bases is a function $\culp^m:\allmss \times \allrs_\allmss \rightarrow \posRealInf$.
Similar to $\Sigma$-induced inconsistency measures, existing culpability measures can be exploited to entail $\Sigma$-induced culpability measures.
Given a culpability measure $\culp$, a multiset of rule-bases $\ms$ and a rule $r\in\R(\ms)$, a $\Sigma$-induced multi-rb culpability measure $m_{\culp}^{\Sigma}$ is defined as $m_{\culp}^{\Sigma}:\allmss \times \allrs_\allmss \rightarrow \posRealInf$ with $m_{\culp}^{\Sigma}(\ms,r)=\sum_ {B\in\ms}\culp(B,r)$.
Two baseline measures proposed in [9] are the $\culp_D$ and $\culp_{\#}$ measures.
Let a rule base $\rb$ and a rule $r\in\rb$, then
* $\culp_D(\rb,r) =
\begin{cases}
1 & \text{if } \exists M \in \MI(\rb):r\in M \\
0 & \, \text{otherwise}
\end{cases}$
* $\culp_{\#}(\rb,r) = |\{M \in \MI(\rb) \mid r \in M\}|$
Using $\Sigma$-induction, we can use these baseline culpability measures to entail the multi-rb culpability measures $m_{\culp_D}^{\Sigma}$ and $m_{\culp_{\#}}^{\Sigma}$.
We recall the multiset of rule bases $\ms_1$ from Figure <ref>. For the shown rule $a\rightarrow b$, we have that
\begin{align*}
m_{\culp_D}^{\Sigma}(\ms_1, a\rightarrow b) & = 4\\
m_{\culp_{\#}}^{\Sigma}(\ms_1, a\rightarrow b) & = 5
\end{align*}
Regarding multi-rb culpability measures, we propose the following rationality postulates based on an application of postulates for traditional culpability measures [8]. For that, we consider a multiset of business rules $\ms$ and a rule $r\in\R(\ms)$. Also, we define a rule $r\in\R(\ms)$ as a free formula if $r \notin M, \forall M \in \bigcup_{b\in\ms}\MI(b)$. We denote the set of all free formulas of $\R(\ms)$ as $\Free(\ms)$. We then propose the following postulates.
Rule Symmetry () $\culp^m(\ms,r)=\culp^m((\rb_{1},...\rb_n),r)$, for any permutation of the order of $\rb_1$ to $\rb_n$.
Rule Minimality () if $r\in \Free(\ms)$, then $\culp^m(\ms,r)=0$.
The first postulate states that the order of rule bases in the multiset should not affect the inconsistency value of an individual rule. The second postulate states that the inconsistency value of a rule is zero if this rule is a free formula w.r.t. the multiset of business rule bases.
$m_{\culp_D}^{\Sigma}$ and $m_{\culp_{\#}}^{\Sigma}$ satisfy and .
The second postulate was adapted from a postulate for traditional culpability measures, namely
Minimality ()Let a rule $r\in\rb$, if $r\not\in M, \forall M \in \MI(\rb)$, then the inconsistency value of r is zero.
As this is a commonly satisfied postulate, this allows for a generalization of the previous proposition.
Any $\Sigma$-induced multi-rb culpability measure satisfies . Given a culpability measure $\culp$ satisfying , any multi-rb culpability measure $\Sigma$-induced via $\culp$ satisfies .
In the following, given a multiset of business rules $\ms$ and a multi-rb culpability measure $\culp^m$, we consider all rules of $\R(\ms)$ as a vector $(r_1,...r_n)$, and denote $V^{\culp^m}(\ms)$ as the vector of corresponding multi-rb culpability values of all rules in $\R(\ms)$ w.r.t. $\culp^m$, i.e., $V^{\culp^m}(\ms) = (\culp^m(\ms,r_1),...,\culp^m(\ms,r_n))$. Next, let $\hat{V}^{\culp^m}(\ms)=\mathit{max}_{r\in\R(\ms)}(\culp^m(\ms,r))$ denote the largest multi-rb culpability value w.r.t. $\culp^m$ for all rules. Last, we denote adding a rule $r$ to the shared rule set $\R(\ms)$ of a multiset $\ms$ as $\ms\cup \{r\}$ by a slight missuse of notation, i.e., given $\ms = (\rb_1,...,\rb_n), \ms\cup \{r\}= (\rb_1 \cup \{r\},...,\rb_n\cup \{r\})$.
This allows to adapt some further desirable properties.
Multiset Consistency () $\hat{V}^{\culp^m}(\ms)=0$ iff $\nexists \rb\in\ms:\rb\models\perp$.
Multiset Monotony () Let a multiset of business rule bases $\ms$ and a rule r, $\hat{V}^{\culp^m}(\ms \cup \{r\}) \geq \hat{V}^{\culp^m}(\ms)$
Multiset Free formula independence () If a rule r is a free formula of $(\ms\cup\{r\})$, then $\hat{V}^{\culp^m}(\ms \cup r) = \hat{V}^{\culp^m}(\ms)$
The first property states that the largest multi-rb culpability value for a rule can only be zero if all business rule bases of the multiset are consistent. The second property demands that adding a rule to the shared rule set can only increase the culpability values. Similar to this property, the third postulate demands that adding a free formula to the shared rule set does not alter the culpability values.
$m_{\culp_D}^{\Sigma}$ and $m_{\culp_{\#}}^{\Sigma}$ satisfy , and .
Next to the introduced baseline culpability measures $\culp_D$ and $\culp_{\#}$, various other culpability measures have been proposed (cf. [10]), which could also be used to $\Sigma$-induce multi-rb culpability measures. While an analysis of such measures w.r.t. the introduced postulates could be interesting, we refrain from such a specific analysis and rather show a more generalized approach in the following section, namely how arbitrary inconsistency measures can be used to induce multi-rb culpability measures using Shapley inconsistency values.
§.§ (Adjusted) Shapley Inconsistency Values for Multi-RB measures
Next to designing specific culpability measures, an important approach in element-based analysis is to decompose the assessment of inconsistency measures (in order to derive corresponding culpability measures) by means of Shapley inconsistency values [8]. Given an inconsistency measure $\inc$ and a rule base $\rb$, the intuition is that the overall blame mass $\inc(\rb)$ is distributed amongst all elements in $\rb$, by applying results from game theory. The advantage of this approach is that arbitrary inconsistency measures can be applied to derive a corresponding element-based assessment. The amount of blame that an individual element is assigned relative to $\inc(\rb)$ is also referred to as the payoff.
Let $\inc$ be an inconsistency measure, $\rb$ be a rule base and $\alpha \in \rb$. Then, the Shapley inconsistency value of $\alpha$ w.r.t. $\inc$, denoted $S_\alpha^{\inc}$ is defined via
\begin{align*}
S_\alpha^{\inc}(\rb) = \sum_{B\subseteq\rb}\frac{(b-1)!(n-b)!}{n!}(\inc(\rb)-\inc(\rb\setminus \alpha))
\end{align*}
where $b$ is the cardinality of $B$, and $n$ is the cardinality of $\rb$.
Consider the rule base $\rb_2 = \{a,a\rightarrow b,a\rightarrow \neg b\}$. Then, for the Shapley inconsistency values w.r.t. $\incmi$, for all elements $e$ in $\rb$ we have that $S^{\incmi}_{e}(\rb_2) = \frac{1}{12}+\frac{1}{4} = \frac{1}{3}$.
As shown in Example <ref>, all elements were assigned an equal payoff. This makes sense w.r.t. $\incmi$ if all elements in the rule base are considered with equal importance. However, in our setting, knowledge contained in rule bases is distinguished into facts and rules. Here, facts have a different veracity than rules, as they are usually provided by a given (non-negotiable) case input and have to be kept "as-is" [6]. In the scope of inconsistency resolution, we are therefore only interested in identifying blamable rules, as these should be considered for re-modelling. Consequently, an element-based assessment for rule bases should only assign a payoff to blamable rules, and not facts. Correspondingly, this has to be considered when applying Shapley's game theoretical approach to distribute a blame mass over all elements. As recently discussed in [3], the Shapley inconsistency value can accordingly be adjusted as follows. For that, let $\Free(B)$ denote the free formula in a rule base $B$, i.e., all $r\in B: r\not\in M, \forall M\in\MI(B)$.
Let $\inc$ be a rule-based inconsistency measure, $\rb$ be a rule base and $\alpha \in \rb$. Then, the adjusted Shapley inconsistency value of $\alpha$ w.r.t. $\inc$, denoted $S*_\alpha^{\inc}$ is defined via
\begin{align*}
\begin{cases}
0 & \text{if } \alpha \in \F(\rb) \\
\sum\limits_{B\subseteq\rb}( \mathit{CoalitionPayoff}_{\alpha,\rb}^{\inc}(B) + \mathit{AdditionalPayoff}_{\alpha,\rb}^{\inc}(B) ) & \text{otherwise}
\end{cases}\\
\intertext{with}
&\mathit{CoalitionPayoff}_{\alpha,\rb}^{\inc}(B) = \frac{(b-1)!(n-b)!}{n!}(\inc(B)-\inc(B\setminus \alpha))\\
\intertext{being the payoff for an element for any coalition $B\subseteq\rb$, and}
= \left\{\begin{array}{ll}
0& \text{if } r \in \Free(B)\\
\frac{\sum_{f\in\F(B)}\mathit{CoalitionPayoff}_{f,\rb}^{\inc}(B)}{|r'\in \R(B) \text{ s.t. } r' \notin \Free(B)|} & \text{otherwise}
\end{array}\right.
\intertext{being the additional payoff that blamable rules receive, by shifting the blame mass from (given) facts to blamable rules.}
\end{align*}
Consider the rule bases $\rb_{2} = \{a, a \rightarrow b, a \rightarrow \neg b\}$ and $\rb_{3} = \{a, a \rightarrow b, \neg b\}$. Then for the adjusted Shapley inconsistency values w.r.t. $\incmi$, we have that $S*_a^{\incmi}(\rb_2) = 0, S*_{a\rightarrow b}^{\incmi}(\rb_2) = \frac{1}{3} (+\frac{1}{3}/2) = \frac{1}{2}$, and $S*_{a\rightarrow \neg b}^{\incmi}(\rb_2) = \frac{1}{3} (+\frac{1}{3}/2) = \frac{1}{2}$. Also, we have that $S*_a^{\incmi}(\rb_3) = 0, S*_{\neg b}^{\incmi}(\rb_3) = 0$, and $S*_{a\rightarrow \neg b}^{\incmi}(\rb_3) = \frac{1}{3} (+\frac{2}{3}) = 1$ . For the first rule base $\rb_2$ (containing two rules), the blame is evenly distributed amongst both rules. For the second rule base $\rb_3$, the single rule receives the entire blame. This assessment makes sense in a business rule setting, as the given fact input is evaluated against a set of humanly modelled rules, and any inconsistencies arise due to modelling errors in the set of business rules. The adjusted Shapley inconsistency values can thus be used for pin-pointing problematic rules for re-modelling purposes.
This element-based measure can consequently also be used to identify problematic rules over a multiset of cases, by inducing the corresponding multi-rb measure $m^\Sigma_{S*^I}$ (cf. Example <ref>). As mentioned, an advantage of using the adjusted Shapley measure is that arbitrary inconsistency measures can be used to derive an element-based assessment over a set of cases, based on company needs. Furthermore, we can identify the following properties for an adjusted Shapley value $m^\Sigma_{S*^I}$. For this,
we assume that any inconsistency measure $\inc$ used to derive adjusted Shapley inconsistency values satisfies the basic properties of consistency', monotony' and free formula independence' as defined in [8][Let a rule base $\rb$ and an inconsistency measure $\inc$. Consistency' states that $\inc(\rb)=0$ iff $\rb$ is consistent. Monotony' states that if $\rb\subseteq \rb'$ then $\inc(\rb)\leq \inc(\rb')$. Free formula independence' states that If $\alpha\in\Free(\rb)$ then $\inc(\rb)=\inc(\rb\setminus\{\alpha\})$.].
The adjusted multi-rb shapley inconsistency value satisfies , and .
Also, regarding the relation of inconsisteny measures and the corresponding $\Sigma-$induced Shapley inconsistency values for multi-rb analysis, we propose the following postulates.
Distribution ()$\sum_{\alpha\in\R(\ms)}m^\Sigma_{S*^I}(\ms,\alpha) = m^\Sigma_I(\ms)$
Upper Bound ()
$\hat{V}^{S*^I}(\ms) \leq m^\Sigma_I(\ms)$
The first postulate states that the sum adjusted multi-rb Shapley inconsistency values over all rules is equal to the overall blame mass of the original multi-rb inconsistency measure $\inc$ (used as a parameter to derive the corresponding Shapley values). Also, the second property states that the adjusted multi-rb Shapley inconsistency values for an individual element cannot be greater than the overall assessment of the original multi-rb inconsistency measure $\inc$.
As we are only interested in identifying problematic rules (e.g. for re-modelling), it would also be plausible to adapt the property of fact minimality as proposed in [3] for a multi-rb use-case.
Fact Minimality ()$m^\Sigma_{S^I}(\ms,\alpha) = 0 \forall \alpha\notin\R(\ms)$.
This property states that (non-negotiable) facts should not be assigned any blame value in an element-based multi-rb assessment.
The adjusted multi-rb shapley inconsistency value satisfies , and .
We conclude with an example illustrating the introduced multi-rb measures.
We recall the set of business rule bases $\ms_1$ from Figure <ref> and its shared rule set $\R(\ms_1)=\{r_1,...,r_6\}$. A multi-rb assessment w.r.t. the introduced measures is then as follows.
\begin{align*}
r_1: a &\rightarrow b & m_{\culp_D}^{\Sigma}(\ms_1, r_1) &= 4 & m_{\culp_{\#}}^{\Sigma}(\ms_1, r_1)&= 5 & m_{S*^{\incmi}}^{\Sigma}(\ms_1, r_1)&= 2\\
r_2: c &\rightarrow \neg b & m_{\culp_D}^{\Sigma}(\ms_1, r_2) &= 3 & m_{\culp_{\#}}^{\Sigma}(\ms_1, r_2)&= 3 & m_{S*^{\incmi}}^{\Sigma}(\ms_1, r_2)&= 1.5\\
r_3: b &\rightarrow x & m_{\culp_D}^{\Sigma}(\ms_1, r_3) &= 2 & m_{\culp_{\#}}^{\Sigma}(\ms_1, r_3)&= 2 & m_{S*^{\incmi}}^{\Sigma}(\ms_1, r_3)&= 0.5\\
r_4: x &\rightarrow z & m_{\culp_D}^{\Sigma}(\ms_1, r_4) &= 2 & m_{\culp_{\#}}^{\Sigma}(\ms_1, r_4)&= 2 & m_{S*^{\incmi}}^{\Sigma}(\ms_1, r_4)&= 0.5\\
r_5: y &\rightarrow \neg z & m_{\culp_D}^{\Sigma}(\ms_1, r_5) &= 2 & m_{\culp_{\#}}^{\Sigma}(\ms_1, r_5)&= 2 & m_{S*^{\incmi}}^{\Sigma}(\ms_1, r_5)&= 0.5\\
\cline{1-8}
& & & & && m_{\incmi}^{\Sigma}(\ms_1)&= 5
%& & \footnotesize{\emph{Sum:}} & \incmiQ(\rb_{13})&= 2 & \incpQ(\rb_{13})&= 6
\end{align*}
As can be seen, rule $r_1$ is classified as most problematic my all measures. This makes sense, as this rule is a cause of inconsistency over all cases for $\ms_1$. Hence, this rule should be prioritized in the scope of re-modelling and improving the set of business rules. Here, our proposed approach of multi-case inconsistency measurement can support modellers in identifying highly problematic rules from a global perspective, by recommending an order in which rules should be attended to, e.g. $<r1,r2,r3,r4,r5>$ for the shown example. A further important aspect for an application in practice is that the proposed measures can be combined to obtain multivariate metrics for a more fine-grained analysis. For example, as the $m_{\culp_D}^{\Sigma}(r)$ is equivalent to the number of distinct cases in which a rule $r$ is part of an inconsistency, this measure can be used to normalize and explain other measures. For example, a normalization via $m_{\culp_D}^{\Sigma}$ can be used to explain if a high $m_{\culp_{\#}}^{\Sigma}$-value originates from a few highly inconsistent cases (which might be outliers) or if the corresponding rule contributes a smaller amount towards inconsistency but in a vast majority of cases (in which case it might be sensible to re-consider this rule).
In this section, we have shown how arbitrary culpability measures can be transformed into multi-rb measures. Also, we have shown that the proposed measures satisfy desirable properties. Our results are summarized in Table <ref>[Proofs can be found in the supplementary document (<https://bit.ly/2V2sIDw>)].
$m^\Sigma_{\culp_D}$ n/a n/a
$m^\Sigma_{\culp_{\#}}$ n/a n/a
$m^\Sigma_{S^{I}}$ $^{c,i}$ $^{c,i}$ $^{m}$ $^{i}$
c: If $\inc$ satisfies consistency'
m: If $\inc$ satisfies monotony'
i: If $\inc$ satisfies free formula independence'
Compliance with rationality postulates of the investigated measures.
§ TOOL SUPPORT AND EVALUATION
We implemented our approach to assess element-based inconsistency over sequences of business rule cases[<https://gitlab.uni-koblenz.de/fg-bks/multi-rb-inconsistency-measurement/>]. Our implementation takes as input a shared business rule base and a sequence of fact sets, and can then computes the most problematic rules w.r.t. the multiset of rule bases. The $m^\Sigma_{\culp_D}$ and $m^\Sigma_{\culp_{\#}}$ measures can be used out-of-the-box, however, arbitrary culpability measures can be added based on company needs. To evaluate our tool, we then performed run-time experiments and investigated the computational complexity regarding various aspects of measuring inconsistency over sequences of business rule cases.
§.§ Run-Time Experiments
In the following, we present the results of run-time experiments with real-life and synthetic data-sets.
§.§.§ Evaluation with real-life data sets.
To evaluate the feasibility of applying our approach in practice, we conducted run-time experiments with real-life data sets of the Business Process Intelligence (BPI) challenge[<https://data.4tu.nl/search?q=bpi+challenge>]. This yearly scientific challenge from the field of process management provides real-life process logs for evaluating approaches in an industrial setting. In a nutshell, we mined a rule set from each event log and then measured inconsistency over all cases of the respective log. Here, we analyzed the data-sets from the last four years, i.e., BPI'17 (log of a loan application process with 31,509 cases), BPI'18 (log of a fund distribution process with 43,809 cases), BPI'19 (log of an application process with 251,734 cases), and BPI'20 (log of a travel expense claim process with 10,500 cases). From these event logs, declarative constraints of the general business rule form in (<ref>) can be mined using the results from [5]. In this way, we were able to mine a rule set from each of the provided data sets. The resulting number of rules for the respective rule sets is provided in Table <ref>. Also, the mined rule sets can be found online[<https://bit.ly/365Vs4C>]. We refer the reader to [5] for further details on the mining technique. Then, for each data set, we analyzed all cases as follows:
For each data set, a shared rule base $R$ was mined as described above. Then, for all cases $C_1,...,C_n$, the individual case-dependent fact inputs $F_1,...,F_n$ were extracted from the log. We then constructed a multiset of rule bases $B_1,...,B_n$, where every $B_i=(R,F_i)$. We then applied our implementation to analyze inconsistencies over $B_1,...,B_n$ and measured the run-time. The results of our experiments are shown in Table <ref>. The experiments were run on a machine with 3 GHz Intel Core i7 processor, 16 GB RAM (DDR3) under macOS.
Dataset # of Rules # of Cases Runtime # of inconsistent Cases
BPI'17 50 31.509 5657s 31.509 (100%)
BPI'18 84 43.809 3967s 0 (0%)
BPI'19 51 251.734 1610s 434 (0.17%)
BPI'20 330 10.500 1329s 323 (3.07%)
Runtimes for analyzing all cases for the considered BPI data sets
As can be seen, an analysis of all cases was feasible for all data sets.
A central assumption of our approach is that a global perspective over all cases should be considered as opposed to viewing cases individually. Interestingly, this was also confirmed by our experiments with the above real-life data sets:
For every individual rule base instance, we computed the $\culp_{\#}$ values for all rules and then ranked all rules by this value (rank 1 meaning that this rule is the most problematic element, and so on). If $n$ rules had the same $\culp_{\#}$ value, they were assigned the sum of the occupied ranks divided by $n$ (e.g. if two rules had the highest $\culp_{\#}$ value, they were awarded the rank (1+2)/2 = 1.5, and the next rule had the rank 3). Figure <ref> shows the distribution of all assigned ranks for the rules for the BPI'17 data set over all cases. For readability, rules that did not participate in any inconsistencies are omitted.
[col sep=comma]boxPlotData/bpi17.txt
boxplot/draw direction = y,
x axis line style = opacity=0,
axis x line* = bottom,
axis y line = left,
enlarge y limits,
xtick = 1, 2, 3, 4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,
xticklabel style = align=center, font=, rotate=60,
xticklabels = r1,r2,r3,,$\cdots$,,,,,,,,,,,,,,,,,,,,,r26,
xtick style = draw=none,
ylabel = Rank,
ytick = 1,10,20,30,
in 1,...,26
+[boxplot, draw=black] table[y index=] ;
Rank distribution for the individual rules of the BPI'17 rule set over all cases.
While there were some rules that had the same rank in all cases (e.g. r2), there were many rules where the respective local rankings had a large variability (e.g. r3). This shows that the global perspective as proposed in this work should be strongly considered in the scope of auditing.
In general, we see the above experiments as positive in regard to applying our approach in practice. As the analyzed data-sets were unrelated, no further comparison of run-times can be made. Therefore, we further assess our approach with synthetic data sets.
§.§.§ Evaluation with synthetic data sets.
We created a generator for synthetic rule base instances. Our generator can produce set of business rule base instances, based on a shared rule set and a sequence of fact sets relative to this rule set. As parameters, our generator takes the desired rule base size and a desired number of cases. Then, the generator constructs a multiset of rule bases as follows:
The set of business rule instances is constructed over a (potentially infinite) alphabet $\mathfrak{A}=<a,b,...>$. Then, for a desired number of rules $n_r$, a rule set $R=\{r_1,...,r_{n_r}\}$ is generated, where every $r_i$ is of the form $\mathfrak{A}_i \rightarrow \neg\mathfrak{A}_{i+1}$. For example, for a parameter of $2$ desired rules, the resulting rule set is $R=\{a\rightarrow\neg b, b\rightarrow \neg c\}$. To then generate a desired number of random cases $n_c$, a multiset of fact sets $F=\{F_1,...,F_{n_c}\}$ is initialized. Then, each of the fact sets is populated by adding atoms of the rule base, based on a user-defined probability. For example, for the exemplary rule set $R$ of size $2$, a random fact set relative to $R$ could be any element of $\{\emptyset,a,b,c,ab,ac,bc,abc\}$. In this way, the generator can create a set of random rule base instances $B=\{B_1,...,B_{n_c}\}$, where every $B_i=(R,F_i)$.
An advantage of our generator is that the structure of the contained rules is similar in all rule bases, which thus allows for better comparability. We consequently used our generator to analyze multiple sets of business rule cases with different parameters and measured the run-times. As parameter settings, based on the observed sizes from the real-life data-sets, we selected as parameters the rule base size from 10,20,...,100, and the number of cases from 10.000,20.000,...,100.000 and then tested every possible combination. Thus, a total of 100 (10x10) different configurations were tested (see above for hardware). The results of our experiments are shown in Figure <ref>. The smallest setting (10 rules and 10.000 cases) took around 90s, where the largest configuration (100 rules and 100.000 cases) took around 50 minutes. As can be seen, the run-time scales proportionally with the size of the rule base and the number of cases. Thus, we could not identify any of these two factors to be a dominant limiting factor to the run-times in our experiments.
scaled y ticks=false
[xlabel=Rule base size, ylabel=Number of cases, zlabel=Run-time in seconds, xlabel style=sloped like x axis, ylabel style=sloped,yticklabels=1,2,50T,100T,5,6,7,8,9,10, view/az=35, zmin=0, grid=major]
3[surf, shader=faceted]
Run-times for the analysis of 10x10 synthetic sets of rule base instances
To summarize, both the evaluation with real-life data sets and with synthetic data sets yielded feasible run-times. To extend this empirical analysis, we continue with an investigation of computational complexity in regard to our proposed approach.
§.§ Complexity Analysis[Proofs can be found in the supplementary document (<https://bit.ly/2V2sIDw>)]
We assume familiarity with basic concepts of computational complexity and basic complexity classes such as $\textsf{P}$ and $\textsf{NP}$, see [11] for an introduction. We first observe that the satisfiability problem for our formalism of business rules bases is tractable (note that similar observations have been made before on similar formalisms, see e. g., [4]).
Let $\rb$ be a rule base. The problem of deciding whether $\rb$ is consistent can be solved in polynomial time.
Then, the complexity of deciding whether a certain rule is contributing to the overall inconsistency is as follows.
Let $\ms$ be a multiset of rule bases with $\ms=(\rb_1, ..., \rb_n)=(\{\F_1\cup\R\},...,\{\F_n\cup\R\})$ and let $r\in \rb$. The problem of deciding whether there is a $i\in\{1,\ldots,n\}$ and $M\in\MI(\rb_i)$ s. t. $r\in M$ is -complete.
The following two results deal with the computational complexity of computing the baseline measure $\culp_{\#}$.
Let $\rb$ be a rule base and $M\subseteq \rb$. The problem of deciding whether $M\in\MI(\rb)$ can be solved in polynomial time.
For our final result note that $\#\textsf{P}$ is the complexity class of counting problems where the problem of deciding whether a particular element has to be counted is in $\textsf{P}$, cf. [15].
Let $\ms$ be a multiset of rule bases with $\ms=(\rb_1, ..., \rb_n)=(\{\F_1\cup\R\},...,\{\F_n\cup\R\})$ and let $r\in \R$. The problem of determining $|\{M \in \MI(\rb_i) \mid i\in \{1,\ldots,n\} r \in M\}|$ is $\#\textsf{P}$-complete.
§ CONCLUSION
In this work, we have shown how arbitrary culpability measures (for single rule bases) can be automatically transformed into multi-rb measures while maintaining desirable properties. This is highly needed in practice, as companies are often faced with thousands of rule bases daily, and thus need means to assess inconsistency from a global perspective. Here, our proposed measures can be used to gain fine-grained insights into inconsistencies in sequences of business rule bases. For the analyzed (real-life) data-sets, the proposed multi-cases analysis could be performed in a feasible run-time. Intuitively, the number of cases or the size of the rule base affect the run-time of our approach. Here, we plan to develop more efficient algorithms in future work. As a main takeaway, our results indicate that the interrelations of individual cases need to be considered for business rules management, which should be addressed more in future works.
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§ APPENDIX A: PROOFS OF TECHNICAL RESULTS
$m_{\culp_D}^{\Sigma}$ and $m_{\culp_{\#}}^{\Sigma}$ satisfy and .
We consider $m_{\culp_D}^{\Sigma}$ and $m_{\culp_{\#}}^{\Sigma}$ in turn. For this, let $\ms$ be a multiset of rule bases, $B$ any rule base in $\ms$, and $r$ any rule in a rule base $B$.
* We start with the measure $m_{\culp_D}^{\Sigma}$. To show , as $m_{\culp_D}^{\Sigma}(\ms,r)=\sum_ {B\in\ms}\culp_D(B,r)$, we have that $m_{\culp_D}^{\Sigma}(\ms,r)=m_{\culp_D}^{\Sigma}((\rb_{1},...\rb_n),r)$, for any permutation of the order of $\rb_1$ to $\rb_n$ due to commutativity via $\sum_{B_i\in\ms}\culp_D(B_i,r) = \culp_D(B_1,r)+ ... + \culp_D(B_n,r)$, with $n=|\{B\in\ms\}|$. For , recall that a rule $r$ is defined as a free formula in $\ms$ if $r \notin M, \forall M \in \bigcup_{b\in\ms}\MI(b)$. Consequently, if $r\in\Free(\ms)$, then $\sum_ {B\in\ms}\culp_D(B,r) = 0$ per definition.
* The proofs for $m_{\culp_{\#}}^{\Sigma}$ are analogous, i.e., $m_{\culp_{\#}}^{\Sigma}(\ms,r)=m_{\culp_{\#}}^{\Sigma}((\rb_{1},...\rb_n),r)$, for any permutation of the order of $\rb_1$ to $\rb_n$ due to commutativity, and $\sum_ {B\in\ms}\culp_{\#}(B,r) = 0$ if $r\in\Free(\ms)$, as $\culp_{\#}(B,r) = |\{M \in \MI(B) \mid r \in M\}|$ for any $\rb\in\ms$.
Any $\Sigma$-induced multi-rb culpability measure satisfies . Given a culpability measure $\culp$ satisfying , any multi-rb culpability measure $\Sigma$-induced via $\culp$ satisfies .
Let $\ms$ be a multiset of rule bases, $B$ any rule base in $\ms$, and $r$ any rule in a rule base $B$. To show , as $m_{\culp^m}^{\Sigma}(\ms,r)=\sum_ {B\in\ms}\culp^m(B,r)$, $\culp^m(\ms,r)=\culp^m((\rb_{1},...\rb_n),r)$ for any permutation of the order of $\rb_1$ to $\rb_n$ due to commutativity via $\sum_{B_i\in\ms}\culp^m(B_i,r) = \culp^m(B_1,r)+ ... + \culp^m(B_n,r)$, with $n=|\{B\in\ms\}|$. To show , given a culpability measure $\culp$, if we have for a rule $r\in B$: $r\not\in M, \forall M \in \MI(B)$ and $\culp(B,r)=0$ , then $\culp^m = \sum_ {B\in\ms}\culp(B,r) = 0$ per assumption.
$m_{\culp_D}^{\Sigma}$ and $m_{\culp_{\#}}^{\Sigma}$ satisfy , and .
We consider $m_{\culp_D}^{\Sigma}$ and $m_{\culp_{\#}}^{\Sigma}$ in turn. For this, let $\ms$ be a multiset of rule bases, $B$ any rule base in $\ms$, and $r$ any rule in a rule base $B$.
* We start with the measure $m_{\culp_D}^{\Sigma}$. To show , observe that $\culp_D(X,r)$ for a consistent rule base $X$ is $0$ for any rule $r$ per definition, as $\MI(X)=\emptyset$. Thus, if $\nexists B\in\ms:B\models\perp$, then $m_{\culp_D}^{\Sigma}(\ms)=0$. In turn, $\hat{V}^{m_{\culp_D}^{\Sigma}}(\ms)=\mathit{max}_{r\in\R(\ms)}(\culp_D(\ms,r))=0$. For the other direction, assume that $\hat{V}^{m_{\culp_D}^{\Sigma}}(\ms)>0$ for a case where $\nexists \rb\in\ms:\rb\models\perp$. This would mean, that there must exist a rule $r$ in a consistent rule base $X$ of $\ms$, s.t. $\culp_D(X,r)\neq 0$. This contradicts $\culp_D$ by definition. To show , observe that if a rule $r'$ is added, for any rule base $B_i$, we have that $\culp_D(B_i,r')= 0$ or $1$. Thus, $m_{\culp_D}^{\Sigma}(\ms\cup \{r'\})\geq m_{\culp_D}^{\Sigma}(\ms)$. Hence, $\hat{V}^{m_{\culp_D}^{\Sigma}}(\ms \cup \{r'\})=\mathit{max}_{r\in\R(\ms)}(\culp_D(\ms\cup \{r'\},r)) \geq \hat{V}^{m_{\culp_D}^{\Sigma}}(\ms)$. To show , recall that a rule $r'$ is defined as a free formula in $\ms$ if $r' \notin M, \forall M \in \bigcup_{b\in\ms}\MI(b)$. Consequently, if $r'\in \Free(\ms\cup \{r'\})$, then $|\MI(B_i\cup \{r'\})|=|\MI(B_i)|$ for all $B\in\ms$. In turn, for any other rule $r$ in a rule base $B_i\in\ms$, $\culp_D(B_i,r)=\culp_D(B_i\cup\{r'\},r)$. In result, if $r'\in \Free(\ms\cup\{r'\})$, then $\hat{V}^{m_{\culp_D}^{\Sigma}}(\ms \cup \{r'\}) = \hat{V}^{m_{\culp_D}^{\Sigma}}(\ms)$.
* The proofs for $m_{\culp_{\#}}^{\Sigma}$ are analogous, i.e., if $\nexists B\in\ms:B\models\perp$ then $m_{\culp_{\#}}^{\Sigma}(\ms)=0$ as $|\MI(B_i)|=0$ for any $B_i\in\ms$, $|\MI(B\cup \{r\})|\geq |\MI(B)|$ for any rule $r$, resp. $|\MI(B\cup \{r\})| = |\MI(B)|$ if $r$ is a free formula in $(B\cup \{r\})$.
The adjusted multi-rb shapley inconsistency value satisfies , , (, and ).
Let $\ms$ be a multiset of rule bases, $B$ any rule base in $\ms$, and $r$ any rule in a rule base $B$. Also, recall that we assume any inconsistency measure $\inc$ that is used to derived an adjusted multi-rb shapley inconsistency value $m^\Sigma_{S*^{I}}$ satisfies consistency', monotony' and free formula independence'. We then consider the individual properties in turn. To show , observe that for a consistent rule base $B$, we have that $\inc(B)=0$ per assumption of consistency'. It follows that for a consistent rule base $B$, $S*_\alpha^{\inc}(B)=0$ for any $\alpha\in B$, thus $\hat{V}^{m^\Sigma_{S*^{I}}}(\ms)=\mathit{max}_{r\in\R(\ms)}(m^\Sigma_{S*^{I}}(B,r))=0$ if $\nexists \rb\in\ms:\rb\models\perp$. For the only if direction, assume we would have a rule $\alpha$ s.t. $S*_\alpha^{\inc}(B)\neq0$ for a consistent rule base $B$. This would mean $\inc(B)-\inc(B\setminus\alpha)>0$ for a consistent rule base $B$, which contradicts the assumption of consistency'. To show , observe that $\inc(B\cup \{r\}) \geq \inc(B)$ for any rule $r$ if $\inc$ satisfies monotony'. Therefore, $S*_\alpha^{\inc}(B\cup \{r\})\geq S*_\alpha^{\inc}(B)$ for any $\alpha\in B$. It follows that $m^\Sigma_{S*^I}(\ms \cup \{r\},\alpha)\geq m^\Sigma_{S*^I}(\ms,\alpha)$ for any rule $\alpha$, and thus $\hat{V}^{m^\Sigma_{S*^{I}}}(\ms\cup\{r\})\geq \hat{V}^{m^\Sigma_{S*^{I}}}(\ms)$. The proof for is analogous, i.e., $\inc(\MI(B\cup \{r\})) = \inc(\MI(B))$ for any rule $r\in\Free(B\cup\{r\})$ if $\inc$ satisfies free formula independence'. Next, follows from Proposition <ref>. Last, to show , it suffices to show that $S*^{\inc}$ satisfies minimality, which has been shown in [3], i.e., for any fact $f$, $S*_f^{\inc}=0$ per definition, and for any free rule $\alpha$, $\inc(B)-\inc(B\setminus\alpha)$ in the last part of the summand of the coalition payoff will always equate to $0$ due to the consistency' and free formula independence' assumption of the underlying measure $\inc$, thus, for any $r_i\in\R(\rb)$, if $r_i\not\in M, \forall M \in \MI(\rb)$, then $S*_{r_i}^{\inc}=0$.
The adjusted multi-rb shapley inconsistency value satisfies , and .
Let $\ms$ be a multiset of rule bases, $B$ any rule base in $\ms$, and $r$ any rule in a rule base $B$. We then consider the individual properties in turn. The proof for follows the proof in [3]: We recall the adjusted Shapley inconsistency value
\begin{align*}
\scriptsize{
\begin{cases}
0 & \text{if } \alpha \in \F(\rb) \\
\sum\limits_{B\subseteq\rb} \mathit{CoalitionPayoff}_{\alpha,\rb}^{\inc}(B) +
\sum\limits_{B\subseteq\rb}\mathit{AdditionalPayoff}_{\alpha,\rb}^{\inc}(B) & \text{otherwise}
\end{cases}
\end{align*}
In the following, we abbreviate CoalitionPayoff as CP and AdditionalPayoff as AP for readability. Then, for a set of rule bases $\ms$, we consider the sum of all adjusted Shapley values (for all elements in $\ms$ over all rule bases $\rb\in\ms$).
\begin{align*}
&\sum_{\alpha \in \rb} \sum_ {B\in\ms} S*_\alpha^{\inc}(\rb)\\
&=\sum_{\alpha \in \rb} \sum_ {B\in\ms}
\begin{cases}
0 & \text{if } \alpha \in \F(\rb) \\
\sum\limits_{B\subseteq\rb} \mathit{CP}_{\alpha,\rb}^{\inc}(B) +
\sum\limits_{B\subseteq\rb}\mathit{AP}_\alpha^{\inc}(B) & \text{otherwise}
\end{cases}\\
&=\sum_{\alpha \in \rb}\sum_ {B\in\ms}\sum\limits_{B\subseteq\rb} \mathit{CP}_{\alpha,\rb}^{\inc}(B) - \sum_{f \in \F(\rb)}\sum_ {B\in\ms}\sum\limits_{B\subseteq\rb} \mathit{CP}_{f,\rb}^{\inc}(B)+\sum_{\alpha \in \R(\rb)}\sum_ {B\in\ms}\sum\limits_{B\subseteq\rb}\mathit{AP}_{\alpha,\rb}^{\inc}(B)\\
\intertext{Following [8], the first summand can be rewritten.}
&=\sum_ {B\in\ms}\inc(\rb) - \sum_{f \in \F(\rb)}\sum_ {B\in\ms}\sum\limits_{B\subseteq\rb} \mathit{CP}_{f,\rb}^{\inc}(B)+\sum_{\alpha \in \R(\rb)}\sum_ {B\in\ms}\sum\limits_{B\subseteq\rb}\mathit{AP}_{\alpha,\rb}^{\inc}(B)\\
\intertext{Then}
&=\sum_ {B\in\ms}\inc(\rb) - \sum_{f \in \F(\rb)}\sum_ {B\in\ms}\sum\limits_{B\subseteq\rb} \mathit{CP}_{f,\rb}^{\inc}(B)+\sum_{\alpha \in \R(\rb)}\sum_ {B\in\ms}\sum\limits_{B\subseteq\rb} \left\{\begin{array}{ll}
0& \text{if } r \in \Free(B)\\
\frac{\sum_{f\in\F(B)}\mathit{CP}_{f,\rb}^{\inc}(B)}{|r\in \R(B) \text{ s.t. } r \notin \Free(B)|} & \text{otherwise}
\end{array}\right. \\
&=\sum_ {B\in\ms}\inc(\rb) - \sum_{f \in \F(\rb)}\sum_ {B\in\ms}\sum\limits_{B\subseteq\rb} \mathit{CP}_{f,\rb}^{\inc}(B)+\sum_{r \in \Free(\R(\rb))}\sum_ {B\in\ms}\sum\limits_{B\subseteq\rb} 0 + \sum_{r \not\in \Free(\R(\rb))}\sum_ {B\in\ms}\sum\limits_{B\subseteq\rb} \frac{\sum_{f\in\F(B)}\mathit{CP}_{f,\rb}^{\inc}(B)}{|r\in \R(B) \text{ s.t. } r \notin \Free(B)|}\\
&=\sum_ {B\in\ms}\inc(\rb) - \sum_{f \in \F(\rb)}\sum_ {B\in\ms}\sum\limits_{B\subseteq\rb} \mathit{CP}_{f,\rb}^{\inc}(B) + \sum_{r \not\in \Free(\R(\rb))}\sum_ {B\in\ms}\sum\limits_{B\subseteq\rb} \frac{\sum_{f\in\F(B)}\mathit{CP}_{f,\rb}^{\inc}(B)}{|r\in \R(B) \text{ s.t. } r \notin \Free(B)|}\\
&=\sum_ {B\in\ms}\inc(\rb) - \sum_{f \in \F(\rb)}\sum_ {B\in\ms}\sum\limits_{B\subseteq\rb} \mathit{CP}_{f,\rb}^{\inc}(B) + \sum\limits_{B\subseteq\rb}\sum_ {B\in\ms} \sum_{f\in\F(B)}\mathit{CP}_{f,\rb}^{\inc}(B)\\
&=\sum_ {B\in\ms}\inc(\rb) - \sum_{f \in \F(\rb)}\sum_ {B\in\ms}\sum\limits_{B\subseteq\rb} \mathit{CP}_{f,\rb}^{\inc}(B) + \sum_{f\in\F(B)}\sum_ {B\in\ms}\sum\limits_{B\subseteq\rb}\mathit{CP}_{f,\rb}^{\inc}(B)\\
&=\sum_ {B\in\ms}\inc(\rb)
\end{align*}
To show , observe that due to $\sum_{\alpha \in \rb} \sum_ {B\in\ms} S*_\alpha^{\inc}(\rb)=\sum_ {B\in\ms}\inc(\rb)$ via distribution, we have that $\hat{V}^{m_{S*^I}^{\Sigma}}(\ms)=\mathit{max}_{r\in\R(\ms)}(m_{S*^I}(\ms,r)) \leq m^\Sigma_I(\ms)$. Last, to show , observe that $S*^I_f(B)=0$ for any fact $f\in B$ per definition, thus
$m^\Sigma_{S^I}(\ms,\alpha) = 0 \forall \alpha\notin\R(\ms)$.
Let $\rb$ be a rule base. The problem of deciding whether $\rb$ is consistent can be solved in polynomial time.
The minimal model $M$ of $\rb$ can be determined as follows:
* $M=\F(\rb)$
* Let $r\in \R(\rb)$ be s. t. $body(r)\subseteq M$
* If there is no such rule, then return $M$
* Otherwise, $M:=M\cup \{head(r)\}$ and continue with 2.
I can be seen that $M$ is both closed and minimal and therefore the minimal model of $\rb$. Both, the algorithm above and checking whether $M$ is inconsistent are polynomial, therefore deciding whether $\rb$ is consistent is polynomial.
Let $\ms$ be a multiset of rule bases with $\ms=(\rb_1, ..., \rb_n)=(\{\F_1\cup\R\},...,\{\F_n\cup\R\})$ and let $r\in \rb$. The problem of deciding whether there is a $i\in\{1,\ldots,n\}$ and $M\in\MI(\rb_i)$ s. t. $r\in M$ is -complete.
For -membership consider the following non-deterministic algorithm:
* Guess $i\in\{1,\ldots,n\}$
* Guess a set $M\subseteq \rb_i$ with $r\in M$
* If $M$ is consistent, return False
* For each $x\in M$, if $M\setminus\{r\}$ is inconsistent return False
* Return True
Observe that the above algorithm runs in polynomial non-deterministic time (due to consistency checks being polynomial, cf. Proposition <ref>) and returns True iff $r$ is contained in a minimal inconsistent subset of at least one of $\rb_1, ..., \rb_n$.
In order to show -hardness, we reduce the problem 3Sat to the above problem. For that, let $I=\{c_1,\ldots,c_n\}$ be a set of clauses $c_i=\{l_{i,1},l_{i,2},l_{i,2}\}$ where each $l_{i,j}$ is a literal of the form $a$ or $\neg a$ (with an atom a). 3Sat then asks whether there is an assignment $i:A\rightarrow \{\textsc{True},\textsc{False}\}$ that satisfies all clauses of $I$, where $A$ is the set of all atoms appearing in $I$. We introduce new atoms $k_1,\ldots,k_n$ for each of the clauses and a new atom $s$ (indicating satisfiability) and define $\rb_I$ through
\begin{align*}
\F(\rb_I) & = \{ a, \neg a\mid a\in A\} \cup\{\neg s\}\\
\R(\rb_I) & = \{l_{i,j} \rightarrow k_i \mid j=1,2,3, i=1,\ldots, n\}\cup \{r=k_1,\ldots, k_n\rightarrow s\}
\end{align*}
We now claim that $I$ is satisfiable iff $r$ is in a minimal inconsistent subset of $\rb_I$ (which is a special case of our problem with $\ms=(\rb_I)$). So assume $I$ is satisfiable and let $i$ be a satisfying assignment. Observe that $M$ defined via
\begin{align*}
M & = \{a \mid i(a)=\textsc{True} \} \cup \{\neg a \mid i(a)=\textsc{False}\}\cup\{\neg s\}\cup \R(\rb_I)
\end{align*}
is inconsistent: as $i$ is a satisfying assignment, each $k_i$ ($i=1,\ldots,n$) can be derived in $M$; then $s$ can also be derived, producing a conflict with $\neg s$. On the other hand, note that $M\subseteq \{r\}$ is consistent. It follows that there is a minimal inconsistent set $M'\subseteq M$ with $r\in M'$.
Now assume that there is a minimal inconsistent set $M\subseteq \rb_I$ with $r\in M$. First observe that there is no atom $a$ s. t., $a,\neg a\in M$ (otherwise $M\setminus\{r\}$ would still be inconsistent). Let $i:A\rightarrow \{\textsc{True},\textsc{False}\}$ be any assignment with $i(a)=\textsc{True}$ if $a\in M$ and $i(a)=\textsc{False}$ if $\neg a\in M$. It follows that each clause $c_1,\ldots,c_n$ is satisfied by $i$ (as each $k_i$ could be derived in $M$) and so $i$ is a satisfying assignment for $I$.
Let $\rb$ be a rule base and $M\subseteq \rb$. The problem of deciding whether $M\in\MI(\rb)$ can be solved in polynomial time.
Deciding whether $M$ is inconsistent and $M\setminus \{x\}$ for each $x\in \rb$ is consistent can each be solved in polynomial time due to Proposition <ref>. It follows that deciding $M\in\MI(\rb)$ can be solved in polynomial time.
Let $\ms$ be a multiset of rule bases with $\ms=(\rb_1, ..., \rb_n)=(\{\F_1\cup\R\},...,\{\F_n\cup\R\})$ and let $r\in \R$. The problem of determining $|\{M \in \MI(\rb_i) \mid i\in \{1,\ldots,n\} r \in M\}|$ is $\#\textsf{P}$-complete.
Membership follows from Proposition <ref> as deciding for a given $M$ whether $M\in \{M \in \MI(\rb_i) \mid i\in \{1,\ldots,n\}, r \in M\}$ is in $\textsf{P}$.
The proof of $\#\textsf{P}$-hardness is analogous to the proof of Proposition 5 in [2]. Observe that in the reduction the notion of “issue” coincides with notion of a minimal inconsistent subset containing the rule $\pi\leftarrow \alpha_1,\ldots,\alpha_n,\delta_1,\ldots,\delta_m$ if we add facts $a,\neg a$ for each atom $a$ occurring the in input instance.
|
# Robust Anthropomorphic Robotic Manipulation through Biomimetic Distributed
Compliance
Kai Junge 1∗ Josie Hughes1 1CREATE Lab, EPFL, Lausanne, Switzerland
$\ast$ Corresponding author. Email<EMAIL_ADDRESS>
###### Abstract
The impressive capabilities of humans to robustly perform manipulation relies
on compliant interactions, enabled through the structure and materials
spatially distributed in our hands. We propose by mimicking this distributed
compliance in an anthropomorphic robotic hand, the open-loop manipulation
robustness increases and observe the emergence of human-like behaviours. To
achieve this, we introduce the ADAPT Hand equipped with tunable compliance
throughout the skin, fingers, and the wrist. Through extensive automated pick-
and-place tests, we show the grasping robustness closely mirrors an estimated
geometric theoretical limit, while ‘stress-testing’ the robot hand to perform
800+ grasps. Finally, 24 items with largely varying geometries are grasped in
a constrained environment with a success rate of 93%. We demonstrate the hand-
object self-organization behavior underlines this extreme robustness, where
the hand automatically exhibits different grasp types depending on object
geometries. Furthermore, the robot grasp type mimics a natural human grasp
with a direct similarity of 68%.
## 1 Introduction
The human ability to robustly pick, place, and manipulate objects in uncertain
environments is common place, and we perform it with ease [1]. Whilst our
planning and sensory-motor capabilities play a key role, these compliant and
robust interactions are also heavily influenced by our hand morphology,
kinematics, and dynamics [2, 3]. From softness in the skin creating stable
contact, to compliant muscle synergies[4] forming diverse ranges of grasp
types[5], this physical intelligence can enable emergent and self-organized
behaviours which result in a robust response to uncertainties in the
environment[6]. This serves as inspiration for advancing the baseline open-
loop capabilities of anthropomorphic robotic manipulators. Such physical
intelligence will enable robotic hands with fluidic, human like dynamic
behaviours and is essential for fully leveraging the potential of rapidly
advancing control and learning strategies [7, 8].
A number of different approaches have been explored for advancing open-loop
robustness and versatility of anthropomorphic robotic hands. Incorporating
compliance underpins all these methodologies, with many taking a embodied
intelligence[9], or physical intelligence perspective[10]. One approach is to
develop fully compliant hands such as the RBO Hand I/II/III [11, 12, 13],
BCL-26 Hand[14], and the Shorthose Hand[15]. Typically pneumatically operated,
they have shown robustness to object geometry in grasping tasks through their
continuously compliant structure. In particular, the RBO Hand III has shown
significant robustness for dexterous in-hand manipulation of a rubix cube with
purely open-loop actions [16]. However, fully soft hands can experience some
limitations in terms of the possible force application, repeatability, and
agility. An alternative approach is tendon actuated soft-rigid hands, which
are inspired by and mimic the human musculoskeletal structure and actuation
mechanisms[17]. These can leverage rolling contact condyloid and synovial
joints held together by compliant ligaments [18, 19, 20] providing natural,
robust behaviour with resistance to impact and joint friction[21, 22, 23, 24].
Compliance embedded within the actuation mechanism is also important for
generating open-loop robustness. Although not in an anthropomorphic form
factor, a compliant differential and underactuated tendon routing can enable
multi-finger grippers which can robustly grasp objects with varying geometry
with a single actuator[25, 26, 27]. The theoretical framework for adaptive
synergies builds upon this to similarly exploit compliance to achieve diverse
behaviour for an anthropomorphic hand[28, 29]. The requirements for compliance
extends beyond the hand, with the wrist motion contributing to coordinating
the behaviour of passive, i.e. non-actuated hands [30]. This combination of
wrist motion and bio-inspired passivity in the structure has been demonstrated
to enable tasks such as dynamic and varied piano playing [31] and grasping
[32].
In all these varied approaches it is evident that compliance advances open-
loop robustness. However, in comparison to human hands, the incorporation of
compliance remains limited to distinct spatial regions. Furthermore, we lack
metrics or approaches for the systematic assessment of the inclusion of
compliance [33]. While there are benchmarks for hands, these are often a high
level task based metric which is heavily affected by the control strategy[34,
35, 36], object focused corresponding to a sparse fail/success metric[37], or
focus on low level static capabilities such as grasp taxonomies[5], range of
motion [38], and hardware features[39].
Figure 1: An anthropomorphic robot hand designed with biomimetic distribution
of compliance leads to a emergence of robust and self-organizing behavior. A)
The ADAPT Hand is compliant in its skin, finger, and wrist, which operate in
the 0.1cm, 1cm, and 10cm displacement ranges. B) Through identical open-loop
waypoints (four waypoints describing the full hand-arm motion for this grasp),
the hand can grasp objects (a large apple vs three small items) successfully.
The distributed compliance allows for the hand to be robust upon unknown
environmental interactions, and a self-organization to take place between the
hand and objects, resulting in different grasps.
We propose that matching the magnitude and distribution of compliance to that
of humans is essential for matching their robustness and fluidity in
interactions. Specifically, that by incorporating distributed and varying
compliance in the skin, fingers, and wrist of an anthropomorphic robot hand,
human-like behaviour can be generated with minimal open-loop control.
A similar focus has been explored for locomotion[40, 41], where it has been
seen that the distribution of compliance across the body and environmental
interactions leads to emergent and stable self-organizing gait patterns[42].
By introducing such strategically located compliance into robotic hands, we
can explore the resulting hand-environment self-organization to provide
robustness through unconscious selection of robust grasp configurations (see
Fig.1B).
This fundamentally requires a robotic hand where the compliance can be
spatially distributed. We introduce our ADAPT (Adaptive Dexterous
Anthropomorphic Programmable sTiffness) Hand, shown in Fig.1, which features a
soft skin covering the contact surface, series elastic actuated fingers/thumb,
and an impedance controlled wrist, all of which are tuned to match their
biological counterpart. Each compliant element acts at a different interaction
length scale, providing contact stability, pose adaptation, and motion
conditioning.Starting from the skin and the fingers, we demonstrate that
matching the compliance to that of human hands leads to stable and robust
manipulation tasks in comparison to a non-matched rigid configuration. This
leverages the tunability of the ADAPT hand, enabling us to compare and
contrast bio-inspired distributed compliance and a rigid hand. Extending to
the full hand, and using a robotic setup to autonomously perform pick-and-
place tasks, we quantify the open-loop robustness for grasping approaches with
respect to an estimated theoretical limit based on object and hand geometry.
Using the same setup, we also evaluate the robustness of the ADAPT Hand design
through an extensive autonomous experiment with over 500 grasps across 9+
hours of uninterrupted pick-and-place operation. The hand showed a success
rate of 97%. Finally, by introducing a compliant wrist we show how the
distributed compliance leads to self-organization of emergent grasp types,
which matches that of a human (direct similarity of 68%), with a success rate
of 93% to handle 24 items spanning a few to 100s of millimeters.
## 2 Results
### 2.1 ADAPT Hand
Figure 2: A) The ADAPT hand. B) Finger design with two independent actuation
and a series spring on the MCP flexor. C) The force-displacement measurement
curve for the skin, finger, and wrist for the robot and a human. For the skin
and finger, the rigid configuration of the ADAPT Hand is also shown for
comparison.
The ADAPT Hand is a bio-inspired anthropomorphic compliant robotic hand
platform, with joint kinematics that reflect an human adult’s hand (detailed
in Section 4.1) and the compliance reflecting a relaxed human adult’s hand.
The hand is designed to have independently tunable compliance in the skin,
finger, and wrist used to investigate the impact of spatially distributed
compliance on the robustness and performance of manipulation tasks. The hand,
shown in Fig.2A, has 12 actuators corresponding to 20 degrees of freedom, and
leverages tendon driven operation. The defining feature of the robot hand
design is the finger (shown in Fig.2B), where two actuators control the
flexion-extension action. Two antagonistic tendons actuate the
metacarpophalangeal(MCP) joint, while a single tendon is responsible for the
flexion motion of the proximal inter-phalangeal(PIP) and distal inter-
phalangeal(DIP) joints. The two actuators controls the finger motion
independently (see Fig.2B right). The flexor tendon of the MCP joint has a
series spring attached, which provides the finger compliance. The abduction-
adduction motion of the fingers are coupled (controlled by one actuator) and
series elastic. The thumb is designed in a similar way, but has two
antagonistic joints at the carpometacarpal(CMC) joint (base of the thumb).
Despite being underactuated, the actuated range of motion can achieve all 33
grasp taxonomies[5] and all 10 positions on the Kapandji score[38] (see
Fig.S1).
Fig.2C shows a force-displacement curve for each type of compliance for the
robot and the human. The variability of different curves for the human is
resultant of multiple measurement trials. The skin compliance is tuned through
the selection of materials to cover the underlying rigid structure. The chosen
skin (EcoFlex20) matches that of a human, especially in comparison to a fully
rigid skin (10 to 40 times stiffer). The finger stiffness is tuned by swapping
the aforementioned series spring in the MCP joint. Compared to the human-
matched finger, the average stiffness of a rigid finger (which is realised by
removing the series spring) is a factor 30 larger. The plateauing force
profile (on the human and tuned finger) from a low stiffness joint means the
fingertip exerts near-constant forces under large displacements - suggesting
the open-loop force application will be robust to large disturbances. The
wrist compliance is achieved by an impedance controlled robot arm. The force
profile of the compliance in the vertical direction is shown, where the
impedance control parameter is tuned to match a human wrist.
Throughout this work, the actuation signals are manually programmed open-loop
signals. That is, an operator will record key waypoints for both the
finger/thumb joints and the wrist 6dof pose (detailed in Section 4.2). During
execution, the recorded waypoints are linearly interpolated and directly
replayed. For every task in the work, the motion is programmed to best mimic
how a human would perform them.
### 2.2 Skin: Contact stability
Figure 3: A) Results from the finger sliding experiment. Schematic (top-left),
box plot and raw values for the shear force of rigid and soft skins measured
at the midpoint of the interaction (top-right), example time series of the
shear force of rigid and soft skins for the two sliding motions (bottom). B)
Results from the knob turning experiment. Schematic (top) and turn angles for
the soft and rigid skins as the diameter $d$ and resisting torque $\tau$ is
varied (bottom). C) Results from the finger gaiting experiment. Schematic
(top) and completed gaits for soft and rigid skins as the held block width $w$
is varied (bottom).
The skin plays a crucial role in the robot-environment interaction, being the
direct interface between the hand and the environment. The cutaneous
compliance in human skin offers increased local deformation, assisting with
contact stability and shear force generation. To measure the corresponding
contact stability offered by matching the skin stiffness we contrast the
matched skin to a rigid one made from PLA. The PLA skin is coated in a thin
layer of EcoFlex20 to ensure the surface friction properties are consistent
and we isolate the effect of compliance.
The main impact of the skin compliance is the increased shear forces generated
for a given motion. This shear force was analyzed for two sliding motions on a
plate (slide front and back motions: refer to Fig.S2) for different relative
displacements between the finger and the sliding surface. On the top right of
Fig.3A the shear force measured via a loadcell at the midpoint of the
interaction motion is plotted. Across a range of motions and surface
environments, the shear force is constantly higher with the soft skin in
comparison to the rigid case. The bottom of Fig.3A shows two representative
shear force profiles for the two sliding motions. The temporal force profile
is similar for the soft and rigid, but the generated force has typically twice
the force magnitude.
#### 2.2.1 Skin compliance on task performance
This demonstrated capacity of the soft skin to generate higher shear force
under the same action contributes to contact level stability. To measure the
resulting performance, we use the task of turning a cylindrical knob with the
middle finger and thumb with a predefined finger motion. The turn angle
$\theta$ is used as the performance metric, while the environment is varied by
independently varying the diameter $d$ and resistance torque $\tau$ of the
knob (Fig.3B top). Resultant turn angles for each environmental variation are
shown in the bottom of Fig.3B where across all tasks the soft skin is
outperforming. In particular, variations in $\tau$ impliy that the soft skin
is more robust, shown by a a far greater performance drop for the rigid skin
of 37$\deg$ compared to 18$\deg$ for the soft skin.
The effect of the softer skin also extends to tasks explicitly requiring
contact stability, such as finger gating when holding an object, as the width
$w$ of the block is varied from low to high. For a given, pre-determined
finger gating pattern, the performance is measured by counting the number of
completed gaits until failure. The results given in Figure Fig.3C show that
the soft skin leads to greater contact stability, and thus a higher
performance. The rigid one shows increasing performance as $w$ increases, as a
higher width results in higher holding forces. For the soft skin, there is in
fact a reduction in performance with the largest width object, however, this
is still almost twice the performance of the rigid setting.
In both these experiments, the soft skin creates a larger contact area with
the environment which results in a more stable contact, as measured through
the higher stabilizing shear forces. This leads to consistently higher task
performances and in some cases increased robustness to environmental
uncertainty.
### 2.3 Finger: Pose adaptation
Figure 4: A) Key frames from two finger sliding motions(left). By overdriving
the MCP joint, pseudo force control is possible with good repeatability
(right). B) Trajectories of relative changes in the three joint angles for the
sliding motions executed by a human, soft robot finger, and a rigid robot
finger. C) Schematic for the finger sliding experiment (top). Maximum forces
recorded as the sliding plate is displaced in the $z$ and $\theta$ directions
for the two motions/soft and rigid fingers (bottom). D) Schematic for the knob
turning experiment (top). Tendon waypoints required for the motion(mid) and
turn angles as the environment is varied (bottom) for the soft and rigid
fingers E) Schematic for the finger gaiting experiment (top). Completed gaits
(mid) and average holding forces (bottom) for soft and rigid fingers as the
held block width $w$ is varied. F) Success rates for the three cubes rotated
in-hand (top). Pictorial sequence of the in-hand manipulation sequence
(bottom).
The flexion-extension motion of the finger originates from the series
elastically actuated MCP joint. In this section we show how the series elastic
MCP joint can enable centimeter scale pose adaptation of the finger, resulting
in robust behavior.
When external contact forces are applied the entire finger will adaptive
comply to the environment since the MCP joint is located at the stem of the
finger. By combining this series elastic actuated joint with the PIP/DIP
actuation, the fingertip can move to follow a surface with consistent contact.
For example, Fig.4A shows two sliding motions (slide front and slide back),
achieved by flexing or extending the PIP/DIP joints respectively, which is
achieved through three waypoint transitions (detailed in Fig.S2). The
compliant MCP mechanism also offers pseudo force-control, by overdriving the
joint. For instance, if the finger makes contact when the MCP joint is at
$q\degree$, instead the position demand can be set to
$q+\Delta_{\mathrm{MCP}}\degree$. By doing so, the finger maintains a stable
contact with the environment at a similar force magnitude even when the
specific shape of the contact surface is unknown. The right side of Fig.4A
shows the normal and shear force profile of the slide back motion under
$\Delta_{\mathrm{MCP}}=0,\ 7.5,\ 15$mm for two repeats. The force magnitude
increases almost linearly with the additional $\Delta_{\mathrm{MCP}}$. This
plot also shows the repeatability of the soft finger, as two repeats of the
same action are near identical (an average RMSE of $3.6\pm 4.3$g across all
experiments).
#### 2.3.1 Human motion comparison
Through the compliant MCP joint, the ADAPT Hand finger motion shows a
kinematic resemblance to that a human finger at the joint displacement level.
For the slide back and front motions, a similar motion was performed by a
human whilst visually recording its pose.
Fig.4B shows the joint evolution of the finger during the two sliding motions
for a human, the robot with a soft MCP, and the robot with a rigid MCP. In
both motions, the soft finger is most similar to the human motion. For the
slide back motion, the difference is most notable in the relative motion of
the MCP and PIP joints. The two joint angles are near consistent on the rigid
finger, implying that all the motion is through the DIP joint (the image above
the plot shows this in effect). For the slide front motion, the soft finger
shows even higher similarity with the human while the rigid finger barely
moves. The slide front motion inherently relies on the MCP joint to flex as
the PIP/DIP joints are extending, which the rigid finger cannot achieve
without extra waypoints.
Although the results shown through this comparison are only indicative of the
similarities between the human and robot finger motions, it motivates the
necessity of a compliant MCP joint in the finger structure to provide a
natural motion of the PIP/DIP joints while maintaining contact.
#### 2.3.2 Measuring robustness through finger compliance
In Fig.4A and B we demonstrate human-like sliding motions with minimal
planning (only three waypoints) by leveraging the compliant MCP joint. The
robustness (behavior invariance to changes in the environment) is measured by
comparing task performances between the soft and rigid fingers while varying
the environment.
Firstly, we consider the consistency of the forces generated in the two
sliding motions. While displacing the finger and the surface in the $z$ and
$\theta$ directions (see Fig.4C top), the variability in applied forces for
the rigid and soft fingers is measured.
The scatter plots in Fig.4C compare the normal and shear forces ($F_{V}$,
$F_{H}$) for the two sling motions between the soft and rigid fingers for
combinations of $\Delta_{z}$ and $\Delta_{\theta}$. The MCP overdrive
($\Delta_{\mathrm{MCP}}$) on the soft finger is chosen to match the maximum
force applied to the rigid finger when $\Delta_{z},\ \Delta_{\theta}$=0 as
shown by the opaque scatter point. The error bars show the standard deviation
of maximum force recorded, where the soft finger is on average 2.4 times
lower. The lower variability in interaction forces implies the soft finger is
less influenced (i.e.: more robust) by position variability.
We can also return to the knob-turning task to evaluate the robustness when
actuating the fingers. Here the turn angle of the knob is measured while the
size of the knob and robot displacements are varied (see Fig.4D top). For the
soft finger, this motion can be is achieved with only three position waypoints
reusing the slide back and front motions for the thumb and middle finger
respectively. If the same waypoints are executed on the rigid finger, the
robot is damaged from high forces. Instead, for the rigid finger, precise
motion planning (15 waypoints) is necessary to describe the path specifically
around the knob. The waypoints needed for this task are shown in the
timeseries graph in Fig.4D, which highlights how the compliance in the finger
simplifies control.
The performance of the rigid and soft fingers is summarized in the scatter
plot (Fig.4D bottom) for three environmental settings: default settings (when
the motion was programmed), reduced knob diameter, and displacement of the
knob location by 1cm. Overall the soft finger is higher performing. As the
rigid finger requires waypoints that follow the contour of the knob, the
interaction forces can vary largely compared to the soft finger, leading to
inconsistent contact force application and thus lower turn angle. For example,
with a smaller knob the turn angle increases in the soft finger (since the
same turn distance covers a larger angle for a smaller $d$) by 10$\degree$,
the turn angle for the rigid finger reduces by 8$\degree$. Moreover, when the
knob is displaced, the rigid finger is damaged from the high reaction forces.
In the finger gaiting task, the number of completed gaits until failure and
the average holding force $F_{\mathrm{Hold}}$ is recorded while the width of
the block $w$ is varied (Fig.4E top). Similar to knob-turning, a separate
trajectory for the rigid finger was programmed which carefully tracks the
contact with the block. Unlike the knob turning, the number of waypoints
remain identical, but a high precision is still required for the MCP joint
angle to follow the width of the block. The raw measurements for the completed
gaits and $F_{\mathrm{Hold}}$ for the three $w$ settings are shown in Fig.4E
as two performance metrics. For both metrics, soft finger has a higher
consistency as $w$ changes in comparison to the rigid finger. This is a direct
extension of the result in Fig.4C, as the interaction forces are consistent
under uncertainty in displacement, resulting in a more robust behavior.
#### 2.3.3 In-hand cube re-orientation
Throughout Fig.4A-E, there are clear benefits to the motions, but they are
simple. While using the same waypoint planning methodology, we can demonstrate
a more complex cube re-orientation task (see Video S1).
Fig.4F captures key frames of the robot using its fingers, thumb, and palm to
continuously re-orient a cube with a total of 12 waypoints (number of
waypoints marked at every frame). While the entire motion is complex, it can
be formed by combining the simpler motions shown in Fig.4C-E. The same
programmed motion is repeated 20 times for cubes with three different sizes,
with 100% success rate of the small and medium cubes and 90% for the large
cube.
### 2.4 Systematic pick-and-place robustness assessment of the ADAPT Hand
Figure 5: A) Robotic setup to conduct large quantities of automatic pick-and-
place experiments while controlling the displacement of the object. B)
Measured limits on object displacement (orange axis), estimated geometric
limits based on object size and hand closure motion. C) Success and failed
grasps throughout the two experiments: robustness assessment and uninterrupted
pick-and-place totalling 845 grasps.
By introducing compliance into the robot skin and finger we have shown low
level measurements of improvements in the robustness and stability for
interactions that require individual or multiple fingers. Extending to include
all five digits, we now explore how this compliance influences a fundamental
manipulation task, pick-and-place. Specifically we want to explore the
robustness/repeatability of the hardware design through many 100s of picks,
and provide quantifiable measurements for the robustness of the open-loop
pick-and-place performance. To do this, we evaluate the robots picking success
as an object is iteratively displaced further away from the central point of
the palm in the x and y direction. We compare this to an estimated theoretical
limit defined purely geometrically (see Fig.S3).
To systematically perform this large scale evaluation of robustness, a robotic
system surrounding the hand is developed to automatically perform pick-and-
place tasks. This comprises of a secondary arm with a movable plate and an
overhead camera (see Fig.5A). The process begins with Robot 1 introducing an
arbitrary offset $\Delta_{M}$ as defined manually. To do this, the system
first identifies existing displacement of the object from the center of the
place: $\Delta_{O}$. By moving the plate by $\Delta_{M}-\Delta_{O}$, the
object is now displaced by $\Delta_{M}$ from the plate center. Robot 2 with
the ADAPT Hand can then execute a open-loop pick-and-place motion. This
process is outlined in Fig.5A and can be performed continuously for many hours
with minimal intervention.
#### 2.4.1 Measuring open-loop robustness
Using this setup, we experimentally measure the ADAPT Hand’s robustness to
displacement as compared to the estimated theoretical upper limit. First, five
objects with different geometries were chosen, and used to program a separate
grasp for each object mimicking how a human would grasp. Then, five extra
unseen objects but with similar geometries were added to the object set. The
tested and unseen objects are labeled in Fig.5B. For each objects,
$\Delta_{M}$ was varied in the horizontal and vertical directions
independently until two consecutive failed grasps were recorded.
Fig.5B shows ten pre-grasp poses of the hand overlayed with results of
measured and approximate theoretical limits of possible object displacements.
The blue shaded area indicate an estimated geometric theoretical limit. This
is defined as when the center of mass of each object could be placed within
the area spanned by the motion of the fingers without colliding with the
fingertips at the start of the grasp (see Fig.S3). The measured limits on the
vertical and horizontal directions are indicated by the orange error bars.
Surprisingly, the hand is close to or exceeds the theoretical geometric limits
of open-loop pick-and-place capabilities, despite the grasps are programmed
with approximate motions and half the objects have never been evaluated. When
the objects are displaced at the millimeter scale, the contact stability from
the skin compliance can maintain a similar grasp configuration (as the
programmed one). At larger centimeter scale displacements, the effect of the
finger pose adaptation is greater. The same objects under the same motion can
be held by different fingers, while still being able to hold the object stably
(examples of such grasps are shown in Fig.S4). Disregarding the measurements
from the Empty coke can (since it is a clear outlier which skews results), the
ratio of the measured over the theoretical limit on the vertical and
horizontal axis are 1.2$\pm$0.2 and 0.8$\pm$0.2. On average, the vertical
robustness exceeds the theoretical limit because even when objects collide
with fingertips, the grasp is successful (an assumption use to derive the
theoretical limit). The horizontal robustness is lower than the theoretical
limit, since at large deviations the forces from the finger are too focused on
one side forcing the object out of the hand. For the coke can, the measured
limits far exceeds the theoretical because of the round shape (grasp is still
successful even placed at the fingertips), and the extremely lightweight
nature (can grasp just at the ends).
#### 2.4.2 Extended period of operation
Using the continuous test capabilities of the robotic system in Fig.5A the
robustness and repeatability of the ADAPT Hand robot design can be evaluated
for a large, uninterrupted trial. The robot system completed 500 grasps with
$\Delta_{M}$=0 with a total success rate of 97%. During this operation, the
robot’s hardware and software system was fully untouched. In fact, the
majority of the failures (15/16) are from grasping the tape. This is
reasonable in hindsight given the zero position of the tape was already close
to the edge of the robustness limits (the point of intersection of the orange
error bars in Fig.5B lies near its edge).
Fig.5C shows a time series of all experiment performed using the pick-and-
place system. Divided by the blue dotted line, the first set of experiments
are for assessing the robustness described in Fig.5B. The second set
represents the uninterrupted robot operation. In the combined two experiments,
the ADAPT Hand was actively used over 16 hours and 845 grasps without any
modifications to the hardware/software. The only visible damage to the
hardware was the ware/dirt on the silicone fingertips shown in Fig.S5. Video
S2 shows a sped up version of this extended trial, while details to the raw
video can be found in section S1.
### 2.5 Self-organizing grasps
Figure 6: A) Key frames overlayed with wrist motion (blue arrow), finger
motion (red arrow), and new contact regions (pink shade) for the human and
robot grasping an object off the table. The number of waypoints for the wrist
or hand illustrated underneath the frames. B) Objects used for this experiment
with failed trials indicated by red circles above the image. C) Four grasps
identified on the human and robot. D) Object geometry plotted with the color
indicating the grasp type observed. E) Direct comparison between robot and
human grasps for every object.
In the previous section, the focus was on the skin and finger compliance which
offers robustness and stability to local interactions. A compliant wrist
enables the scale of these interactions to be expanded, for example grasping
while sliding and object off a table (Fig.6A). Through open-loop trajectories
of the hand and wrist which mimics human grasping of grasping objects from a
table, we explore how robust this is to different objects, and also how
different grasp types emerge.
When a human accomplishes this task, three behaviors within the trajectory can
be identified that are invariant to the object being grasped (top row of
Fig.6A shows an example of grasping a lemon with its key frames and the
finger/wrist movements). Firstly, the approaching motion is always the hand
moving downwards until making contact, either with the table or the object.
Secondly, after the initial contact is made the fingers continue to hold its
contact during the motion, either with the table or the object. Thirdly, the
grasping motion shown by the Flex fingers, Wrist up, and Retract wrist frames
is a single continuous motion where the fingers and thumb curls to form a
grasp as the wrist moves away from the table.
By using a wrist with human-matched compliance in addition to the skin and
fingers, the observed human behaviors can be robustly replicated on the robot
(as shown in the bottom two rows of Fig.6A). When approaching the table, the
compliance allows for a safe and controlled contact of the fingers and/or palm
as the wrist presses into the table (first two frames). The grasping motion
(the final three frames of Fig.6A) shows a coordinated flexing movement of
fingers and thumb as the wrist moves away while constantly maintaining contact
with the environment, matching with the second and third observations from the
human (last three frames). This motion relies on the distributed compliance,
in particular the compliance of the wrist conditions(provides the necessary
conditions) the interaction of the fingers with the environment. During the
grasping sequence, only the wrist is actively actuated (see the sequence of
waypoints executed in Fig.6A bottom). The fingers are only actuated before the
final wrist motion to from a grasp, where the movement is constrained by the
table. The wrist movement gradually removes the constraint of the table
enabling the fingers self-organize and grasp the object, while the compliance
of the wrist maintains contact with the surface.
Using the same open-loop motion, 24 objects of varying geometry were chosen to
be grasped from the table. Fig.6B shows each object, spanning from flat and
thin objects such as a pencil, to a large and bulky objects to an apple. Each
object was grasped three times with varying placement locations in the
vicinity of the robot hand motion path, with an overall success rate of 93%
(70/75 tests) failing to grasp only three objects (phone, bolt, and lemon).
Video S3 shows this grasping experiment for all items from two view angles.
#### 2.5.1 Emergence of discrete grasp types
The robustness of the robot to grasp a wide variety of objects through
identical commands can be explained through self-organization. Consider the
two robot motions given in Fig.6A. In the top row, the lemon is a large object
where the robot makes multiple contacts throughout the motion (see pink
highlights). On the bottom row, the robot only makes contact with a pencil
just when the grasp happens. The resultant grasp of the two objects are also
different: the lemon is held with a power grasp while the pencil is held only
with the fingertips, which is appropriate if a human were to grasp the two
objects.
For all the objects, the grasp motions were analyzed for the robot and the
human, and categorized to five distinct grasp types shown in Fig.6C. The grasp
types are categorized based on the hand posture and which parts of the hand
interact with the object which relate with the grasping taxonomies (see
section 4.3.3). Video S3 shows the four grasp types identified for the robot.
The scatter plot in Fig.6C shows the object geometry (length, width, height)
clustered by the observed grasp types of the robot. The clustering shows the
robot hand self-organizes to a discrete set of grasp types based on the
object’s geometry. For flat or small objects the grasp is using its fingertips
(blue cluster). As the height increases, the robot starts to use its thumb
(green cluster), slowly transitioning into a form of power grasp (yellow, red
clusters).
Like the example in Fig.6A, the robot grasp types observed match closely with
a human. It is know that a human will vary the grasping strategy based on the
object geometry (and other factors too)[43], which is indeed observed when
asked to grasp the objects used in this experiment.
The matrix in Fig.6C directly compares grasps observed for the robot and
human. Categorically the human exhibits one extra grasp type, the finger
surface grasp where all fingers a kept straight, which is impossible for the
robot as the programmed motion curls the PIP/DIP finger joints. Despite this
mismatch of four versus five grasps, the strong correspondence between the two
clear by 68% (17/24 objects) lying on the leading diagonal, and all other
objects on directly adjacent cells. Assuming the human is choosing the
optimally stable grasp through prior knowledge, the robot having a biomimicry
of the kinematics and distributed stiffness is able to self-organize to choose
the same.
## 3 Discussion
In this work, we present an anthropomorphic robot hand, the ADAPT Hand,
designed with a biomimetic distribution of compliance across different lengths
scales in the skin, finger, and wrist. Starting from low level interactions of
the skin and finger levels, we show the presence of a compliant skin and MCP
joint on the finger leads to higher performance and robust to environmental
uncertainty across three of tasks. Through the ADAPT Hand’s tunable
compliance, a direct comparison between a compliant and rigid robotic hardware
is made. Expanding the task to include all five digits, the robustness within
the hand is measured to lie close to an estimated theoretical limit for a
pick-and-place task. Using the same measurement setup, the robustness and
repeatability of the hardware platform itself is shown through a damage-less
execution of over 800 grasps and 15 hours of operation. Finally, a compliant
wrist motion is introduced to grasp 24 objects through an interaction-rich
motion across a tabletop surface. This motion which fully combines the
distribution of compliance across the robot body, where a wrist motion which
conditions the passive pose adaptation of the fingers while contact stability
from the skin enhances the grasp. The resultant discrete grasp types are self-
organized based on the object geometry, similar to that of a human. Not only,
a comparison between human and robot grasp shows 68% of grasps are directly
matching. Overall, we demonstrate an physical intelligence approach towards
anthropomorphic robot design which considers the interaction and motion at all
the lengths scales of robotic manipulation. The extreme robustness given
simplest form of actuation control input culminating in the self-organizing
grasps.
Although the proposed concept realized by the ADAPT Hand show remarkable
performance, there are clear limitations to be addressed in the future. At the
heart of this work is the distributed compliance, which can negatively affect
certain tasks. For instance the soft skin and finger limits the ability to
exert high forces in one direction. This means strongly pinching an object or
button pressing tasks are challenging. When using the full hand for
manipulation, the passive adaptation is simply a result of force equilibrium
working in a desired way. This means, when the external forces are too large
(e.g.: an object that is too heavy such as the phone in Fig.6B), the execution
fails. The same is true for balancing forces between multiple fingers and a
held object. Since the fingers will passively adapt, tasks such as controlled
object re-orientation is difficult.
Sensor feedback and control is one obvious next step to tackle these
limitations. This can mean sensing of the joint torques/tendon forces as well
as tactile sensing to span the skin surface. However, the control method using
the sensor feedback is not straightforward for this robot. This is because the
manipulation capabilities of this hand is driven through the emergent
behaviours which arise from the interaction between the compliant robot and
the environment. Although the actuation signals are position commands, each
waypoint is programmed with a higher level intention such as continue to apply
force downwards or move roughly in this direction instead of reaching a
particular position or force. The sensory-motor control should hence not
follow some explicit trajectory, but should rather complement the emergent of
behavior. This could be achieved through combining high level intention in the
form of the open-loop waypoints and a controller to regulate contact forces,
similar to a shared-control scheme used in[44] used to stabilize a noisy EMG
signal. Alternatively a bio-inspired sensory-motor control, such as central
pattern generators(CPGs) used to transition between walking and swimming gaits
for a salamander robot[45], can lead to emergent efficient behaviors. While
the concept of CPGs do not directly apply to manipulation, a sensory-motor
coordination method that drives emergent behaviours[9] is necessary to
maintain and extend the robust interactions.
The ADAPT Hand as a hardware platform itself also has room for improvement.
One direction is to increase the level of biomimicry of the human hand. The
current hand although is bio-inspired, clear differences are present
especially in the joint placement and the skin/flesh distribution. For
example, there is no material covering the MCP joints. Consequentially,
multiple grasp formations do not translate well from a human to the robot.
Another limitation is the lack of a wrist just below the hand. While the 7dof
robot arm can orient the hand in any angle, in practice the arm must move a
large displacement to resemble small wrist motions seen in a human. Likewise
to the hand itself, increasing the biomimicry in the arm kinematics can enable
more complex human motions to be realised by the robot. Not only could this
simplify manual programming or teleoperation by a human, but one could imagine
mapping motions learnt from the abundance of videos of humans to be applied to
this biomimetic robotic hand-arm structure.
## 4 Methods
Figure 7: A) Actuated, underactuated, and passive joints for the ADAPT Hand.
B) Finger design and tendon routing. C) Series elastic actuation design for
the the MCP joint. D) Soft and rigid skin designs. E) Abduction/adduction
linkage mechanism. F) Waypoint recording and replay method used to program the
robot motion.
### 4.1 ADAPT Hand Hardware
The ADAPT Hand is a custom designed and fabricated anthropomorphic robot hand,
with its four fingers and thumb having dimensions similar to an adult human.
The entire hand is fabricated from commercially available 3D printed
Polylactic acid(PLA) and Thermoplastic Polyurethane(TPU) with shore hardness
98A/65D, and uses tendons (cables) for its actuation. Fig.7A shows the ADAPT
Hand without the skin, showing the joint kinematics and actuation scheme. A
total of 12 servo motors (Dynamixel XM430-W210-R) located beneath the hand
actuates the 20 joints (four joints per finger/thumb). Having less actuators
than joints means certain joints are underactuated (shown by subscripts a, b,
c in Fig.7A) or passive (P in Fig.7A). Details on the CAD model is presented
in section S1.
#### 4.1.1 Finger design
The finger design shown in Fig.7B is a key feature of the ADAPT Hand, where
the same design is used across the five digits. One actuator controls the pin-
jointed MCP joint with antagonistic tendons (the thumb has an extra
antagonistic joint, but rotate 90 degrees to make the 2DoF CMC joint). The
PIP/DIP flexure joints which are coupled by a single flexor tendon is actuated
by another motor. The pin joint in the MCP joint allows the routing of the
PIP/DIP flexor to pass through the center of rotation, fully decoupling the
two axis of actuation. The extension forces for the PIP/DIP come from the
combined effect of the TPU flexure joint and the elastic thread, placed on the
backside of the finger (shown by a black line in Fig.7B).
The compliance at the finger level is generated by a series elastic MCP joint
which is achieved by routing the MCP flexor tendon around a pulley connected
to an extension spring (shown in Fig.7C). Replacing or removing the series
spring can change the finger compliance (as in Fig.2B).
#### 4.1.2 MCP abduction-adduction motion
A notable feature of the mechanical design of the ADAPT Hand is the series-
elastic linkage mechanism driving the abduction-adduction axis of the MCP
joints for the index, ring, and little fingers. The mechanism and its open and
closed states are shown in Fig.7E. The linkage mechanism connects to the MCP
pin joint with a series elastic TPU material, making the MCP joint compliant
in 2 axes. Having an actuated spread axis of the MCP joints increases the
workspace (thus the capability of the hand) such as holding large objects.
#### 4.1.3 Dexterity of the ADAPT Hand
Combining all the mechanisms of the fingers, the ADAPT Hand can be actuated to
produce dexterous motions. Starting from the zero position (all fingers
straight), the hand can be actuated to produce all 33 grasp taxonomies[5].
Likewise, the hand is able to complete all 10 postures on the Kapandji test.
The results for both tests are shown in Fig.S1.
#### 4.1.4 Skin design
A modular skin fully covers one side of the ADAPT Hand. Fig.7D shows the index
and middle finger equipped with a rigid and soft skin respectively. The skins
are identical in their geometry and is approximately 3mm offset from the
bones. The soft skin is fabricated from cast EcoFlex20 and the rigid skin is
3D printed PLA. A thin ($\approx 0.5$mm) layer of EcoFlex20 is glued on the
surface of the rigid skin to maintain the surface friction property.
### 4.2 ADAPT Hand motion programming
In all experiments, the ADAPT Hand (including the robotic arm) operates
through manually programmed open-loop motions. For both the hand and arm, a
series of manually determined key waypoints are recorded to then be played
back (see Fig.7F). The source code details are presented in section S1, with
the software system integration and hardware interfacing which allows the
programming and replay of waypoints on the hand and arm described in Fig.S6.
#### 4.2.1 Programming the hand
The ADAPT Hand is controlled by directly commanding the tendon displacements
for each actuator (which is proportional to the motor angle). To simplify the
procedure to manually record waypoints, the hand (fingers) is operated using a
custom built signal mixer box with 12 linear potentiometers which map to the
position of each actuator. As illustrated in the top left of Fig.7F, two
waypoints can be simply defined by varying the linear potentiometer positions.
Being an interactive device, the mixer box allows for quickly programming
motions while having full control over the tenon positions.
Once a set of one or more waypoints (slider position) are recorded, the robot
can smoothly move between the waypoints as in the top right of Fig.7F.
#### 4.2.2 Programming the arm
The Franka Research 3 robot arm is controlled using a gravity compensated
impedance control introduced in [46], where the end effector 6 dof pose and
corresponding stiffness can be commanded. To program the arm, the end effector
stiffness is set to zero which allows the arm to be manually moved around.
Likewise to the hand, after recording few key poses (see bottom left of
Fig.7F), the 6 dof waypoints are interpolated and replayed.
For motions which involve both the hand and arm waypoints are replayed
sequentially, meaning the hand and arm are not actively actuated at the same
time.
### 4.3 Experimental setup and procedure
#### 4.3.1 Measuring human compliance
The compliance (force displacement characteristics) for a human is measured by
recording the reaction force on a loadcell and its Cartesian position as it is
moved by a 6-axis robot arm (UR5), while the human is instructed to relax. The
measurement setup and movement directions for the skin, finger and wrist is
shown in Fig.S7. The forces are recorded for multiple repeats to capture the
variation in the muscle activity.
#### 4.3.2 Measuring low-level interactions
Three tasks: finger sliding, knob turning, and finger gaiting, were conducted
to characterise the low level interactions (such as contact forces and
kinematics) of the ADAPT Hand skin/finger with the environment. The
experimental setup are shown in Fig.8A,B,C for the three tasks respectively.
Figure 8: Experimental setup and independent (in blue) and dependent (in
orange) variables for the finger sliding experiment (A), knob turning
experiment (B), and the finger gaiting experiment (C).
In the finger sliding task, a single finger interacts with a wooden plate
through two sliding motions generated by combining a flexing motion of the MCP
joint and a flexing or extending motion of the PIP/DIP joints (see Fig.S2).
Fig.8A shows the experimental setup where the ADAPT Hand (rigid held by a UR5
arm) interacts with the wooden plate mounted above two load cells measuring
the vertical ($F_{V}$) and horizontal ($F_{H}$) forces. The finger joint angle
data used in Fig.4B were extracted by recording the April tag markers
throughout the motion. For the two motions, five independent variables were
combinatorialy tested with two repeats which are: pose offset $\Delta\theta$
($\pm 10\deg$), $\Delta z$ ($\pm 10$mm), soft and rigid configurations for the
skin and finger, and overdrive of the MCP tendon $\Delta_{\mathrm{MCP}}$ ($\pm
7.5$mm) (only for the soft finger).
In the knob turning task, the middle finger and thumb was used to turn a knob
shown in Fig.8B. The knob turn angle $\theta$ is used to assess the
performance measured by a position encoder (AMS AS5048B). The environment was
varied in three ways: the $x$ and $y$ position offset($\pm 10$mm each), the
diameter of the knob $d$($45\pm 15$mm), and the reaction torque of the knob
$\tau$ (low: $3.3\pm 0.6$Nmm, High: $18.8\pm 3.9$Nm). The reaction force is
modulated by varying the vertical forces applied on the knob which rests on a
plastic surface. When the finger is in the soft configuration, the motion is
near-identical to the sliding motions introduced in the finger sliding
experiment. In the rigid finger configuration, a secondary motion is
programmed to replicate the same motion to ensure the robot doesn’t damage
itself.
In the finger gaiting task, a plastic block is held between the thumb and four
fingers shown in Fig.8C. Starting from all four fingers contacting the block,
a finger gaiting pattern is executed (shown by 1, 2, 3, 4) in repeat until
eventually the block is dropped. Only the width $w$ of the block is varied
during this experiment ($\pm 5$mm), while the number of completed gaits and
the holding force $F_{\mathrm{grip}}$ is measured by an inbuilt load cell.
#### 4.3.3 Grasp type categorization
The grasp types shown in Fig.6C for both the robot and human are categorized
based on which part of the hand is used to hold/interact with the object and
its posture. Each grasp is also related with the grasp taxonomies in[5]. The
Finger surface grasp (only present for the human) is achieved by keeping the
four fingers straight, and using that as a surface to push the object against
by the thumb tip (corresponds with #22:Parallel extension taxonomy). The
Fingertip grasp uses only the tips of the fingers and thumb to hold the object
(corresponds with #6-8:Prismatic 2-4 finger taxonomy). The Tip and thumb grasp
uses the fingertips and the middle and/or proximal phalanges (corresponds with
#10:Power disk taxonomy). The Power (small) and Power (large) grasps are both
power grasps where one or more phalanges of the fingers/thumb and the palm is
used, distinguished based on the diameter of the grasp (corresponds to
#2:Small diameter and #1:Large diameter taxonomies).
Although only 24 objects are used for the experiment, 25 grasps are recorded
because the paper tape generated two distinct grasp types.
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## Author contributions
K.J and J.H conceived the idea and designed the research. K.J developed the
robotic systems, performed the experiment, and analyzed the data. K.J and J.H
wrote the manuscript.
## Competing interests
The authors declare no competing interests in this work.
## Supplementary Material
## S1 Links to external resources
The uncut video of the extended pick and place operation (section 2.4.2) can
be found here. All the code used to operate the ADAPT Hand can be found here.
The CAD model of the ADAPT Hand can be found here. All data collected from the
experiments alongside any other questions should be forwarded to the
corresponding author.
Figure S1: The ADAPT Hand showing all 33 grasp taxonomies and the 10 Kapandji
score posture. Figure S2: Visual representation of the tendon actuation space
for the flexion-extension motion of a single finger. The axis show the two
independent motion control for the finger. In the sliding experiment, the
slide back and slide front motions are developed by moving between the four
waypoints in a clockwise or counterclockwise fashion. $\Delta_{\mathrm{MCP}}$
shows the how much the MCP joint is overdriven. Figure S3: The flowchart used
to estimate the geometrical limits for open-loop grasping an object. First,
the directions of the finger/thumb motion for the grasp is noted. Then, an
approximate area spanned by the finger/thumb in this grasping motion is
obtained. Assuming the center of mass of the object must lie within the
fingertips, an area where the object would collide with the fingertips is
found. Finally, subtracting the two areas gives the estimated geometrical
limits of where the center of mass of the object can be placed. Figure S4:
Four examples of the ADAPT Hand grasping objects when an artificial offset is
given. A: Even with a large offset, the compliance and the motion of the hand
self-centers the coke can to be grasped. B, C, D: Examples where the resultant
grasp differs from that of $\Delta_{M}=0$. Figure S5: Photograph to show the
ware and tare of the silicone fingertips before and after the extended pick
and place experiment. Figure S6: A high level software and system integration
diagram used to control the ADAPT Hand. The heart of the control is the
teaching and playback system described in section 4.2. When teaching
(recording) different motions, the hand is actuated using a signal mixer box
where the sliders correspond to demand tendon positions. No control input is
given to the arm to be moved freely by hand. In the playback mode, the
recorded waypoints (a sequence of csv files for the hand or the arm) will be
executed sequentially either to the hand or arm. The full system is developed
using ROS2 to interface between microcontrollers (micro-ros), custom
processes, ADAPT Hand (ROS2 control driver), and the Franka impedance
controller[46]. Figure S7: The measurement of human compliance (force-
displacement characteristics) using a UR5 robot arm mounted with a 10kg
loadcell. The robot arm is moved in different axis to while the human is
instructed to relax.
|
# E Pluribus Unum Ex Machina:
Learning from Many Collider Events at Once
Benjamin Nachman<EMAIL_ADDRESS>Physics Division, Lawrence Berkeley
National Laboratory, Berkeley, CA 94720, USA Berkeley Institute for Data
Science, University of California, Berkeley, CA 94720, USA Jesse Thaler
<EMAIL_ADDRESS>Center for Theoretical Physics, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA The NSF AI Institute for Artificial
Intelligence and Fundamental Interactions
###### Abstract
There have been a number of recent proposals to enhance the performance of
machine learning strategies for collider physics by combining many distinct
events into a single ensemble feature. To evaluate the efficacy of these
proposals, we study the connection between single-event classifiers and multi-
event classifiers under the assumption that collider events are independent
and identically distributed (IID). We show how one can build optimal multi-
event classifiers from single-event classifiers, and we also show how to
construct multi-event classifiers such that they produce optimal single-event
classifiers. This is illustrated for a Gaussian example as well as for
classification tasks relevant for searches and measurements at the Large
Hadron Collider. We extend our discussion to regression tasks by showing how
they can be phrased in terms of parametrized classifiers. Empirically, we find
that training a single-event (per-instance) classifier is more effective than
training a multi-event (per-ensemble) classifier, as least for the cases we
studied, and we relate this fact to properties of the loss function gradient
in the two cases. While we did not identify a clear benefit from using multi-
event classifiers in the collider context, we speculate on the potential value
of these methods in cases involving only approximate independence, as relevant
for jet substructure studies.
††preprint: MIT-CTP 5271
###### Contents
1. I Introduction
2. II The Statistics of Per-Ensemble Learning
1. II.1 Review of Per-Instance Learning
2. II.2 Per-Ensemble Binary Classification
3. II.3 Comparing the Loss Gradients
4. II.4 Per-Ensemble Regression
1. II.4.1 Maximum Likelihood
2. II.4.2 Classifier Loss
3. II.4.3 Direct Regression
5. II.5 Beyond Regression
3. III Empirical Studies
1. III.1 Classifiers: Multi-Event from Single-Event
1. III.1.1 Two Gaussian Example
2. III.1.2 Dijet Resonance Search
3. III.1.3 Top Quark Mass Measurement
2. III.2 Classifiers: Single-Event from Multi-Event
3. III.3 Comparison of Regression Strategies
1. III.3.1 Gaussian Mean Example
2. III.3.2 Top Quark Mass Measurement
4. III.4 Beyond Regression Example
4. IV Conclusions
5. A Deriving Maximum Likelihood Classifier Loss
## I Introduction
Modern machine learning techniques are being widely applied to enhance or
replace existing analysis techniques across collider physics [1, 2, 3, 4, 5,
6]. These approaches hold great promise for new particle searches, for
Standard Model measurements, and for high-energy nuclear physics
investigations. A subset of these proposals have advocated for a multi-event
strategy whereby a machine-learned function acts on multiple collision events
at the same time [7, 8, 9, 10, 11, 12, 13, 14]. This multi-event (per-
ensemble) strategy contrasts with more typical single-event (per-instance)
machine learning methods that process one event at a time, although both
strategies make use of many events during the training process.
Intuitively, an ensemble approach might seem like a more promising learning
strategy because there is more information contained in $N>1$ collision events
than in one single event. There is, however, an important distinction between
the amount of information contained in a data set and the amount of
information needed to encode a machine-learned function. For this reason,
there need not be a gain from using multi-event strategies over single-event
strategies in the context of machine learning.
In this paper, we show that when directly compared on the same task, there is
indeed no informational benefit from training a function that processes
multiple events simultaneously compared to training a function that processes
only a single event at a time. This fact can be easily understood from the
statistical structure of collision data. To test for a practical benefit, we
perform empirical comparisons of per-ensemble and per-instance methods on
benchmark tasks relevant for the Large Hadron Collider (LHC), finding that
single-event (per-instance) methods are more effective for the cases we
studied.
To an excellent approximation, collider events are statistically independent
and identically distributed (IID). In simulation, this is exactly true up to
deficiencies in random number generators. In data, there are some small time-
dependent effects from changing conditions and there are also some
correlations between events introduced by detector effects with timescales
longer than a typical bunch crossing. These event-to-event correlations,
however, are truly negligible when considering the set of events typically
used for physics analysis that are selected by triggers. The probability for
two events next to each other in time to be saved by the triggers is
effectively zero, since triggers save only a tiny fraction of events. The IID
nature of collision events therefore ensures that the information content is
the same for ensembles of events and for single events drawn from an ensemble.
In equations, the probability to observe $N$ events $x_{i}$ is
$p(\\{x_{1},\ldots,x_{N}\\}|\theta)=\prod_{i=1}^{N}p(x_{i}|\theta),$ (1)
where $\theta$ represents possible parameters of the generative model, such as
the physics process being studied or the values of coupling constants. The
optimal classifier to distinguish whether events have been generated via
$\theta_{A}$ or via $\theta_{B}$ depends only on the per-ensemble likelihood
ratio [15]:
$\frac{p(\\{x_{1},\ldots,x_{N}\\}|\theta_{A})}{p(\\{x_{1},\ldots,x_{N}\\}|\theta_{B})}=\prod_{i=1}^{N}\frac{p(x_{i}|\theta_{A})}{p(x_{i}|\theta_{B})},$
(2)
which by the IID assumption only depends on knowing the per-instance
likelihood ratio $p(x_{i}|\theta_{A})/p(x_{i}|\theta_{B})$. This equality
explains the informational equivalence of per-ensemble and per-event learning.
Given the simplicity of Eq. (2), why are we writing a whole paper on this
topic (apart from the opportunity to invoke a gratuitously Latinate paper
title that incorporates an aspiration for national unity)? The studies in
Refs. [7, 8, 9, 10, 11, 12, 13, 14] find that per-ensemble learning is
effective for their respective tasks, in some cases arguing why per-instance
learning is deficient. It is certainly true that a set of events
$\\{x_{1},\ldots,x_{N}\\}$ contains more information than a single event
$x_{i}$ drawn from this set. What we will show in this paper is that if one
carefully combines the per-instance information, one can recover the per-
ensemble benefit, with the potential for a substantially reduced training
cost. We emphasize that our analysis does not contradict the studies in Refs.
[7, 8, 9, 10, 11, 12, 13, 14]; rather this work suggests the possibility of
achieving the same or better results by replacing per-ensemble learning with
per-instance learning. There may be specialized contexts where per-ensemble
learning is superior, particularly if the training procedure itself can be
made simpler, such as in the linear regression approach of Ref. [12]. This
paper also gives us a chance to mention some facts about loss functions that
are well known in the statistics literature but might not be as well
appreciated in collider physics. Moving away from the IID case, we speculate
on the relevance of our analysis for jet substructure tasks where there is a
notion of approximate independence of emissions.
The remainder of this paper is organized as follows. In Sec. II, we provide
the formal statistical basis for building multi-event classifiers from single-
event classifiers, and vice versa, under the IID assumption. We also explain
how regression tasks can be translated into the language of per-instance
parametrized classification. In Sec. III, we present empirical studies that
corroborate these analytic results. Our conclusions are given in Sec. IV.
## II The Statistics of Per-Ensemble Learning
### II.1 Review of Per-Instance Learning
Suppose that a collider event is represented by features in
$\mathbb{E}=\mathbb{R}^{M}$ and we are trying to train a binary classifier to
learn a target in $[0,1]$. Let $c:\mathbb{E}\rightarrow[0,1]$ be a function
that processes a single event, with the goal of distinguishing events being
generated by $\theta_{A}$ ($c\to 1$) versus those generated by $\theta_{B}$
($c\to 0$). Such a function can be obtained by minimizing an appropriate loss
functional, such as the binary cross entropy:
$\displaystyle L_{\rm BCE}[c]=-\int dx\,\Big{(}$ $\displaystyle
p(x|\theta_{A})\log c(x)$
$\displaystyle~{}+p(x|\theta_{B})\log(1-c(x))\Big{)},$ (3)
where $p(x|\theta)$ is the probability density of $x\in\mathbb{E}$ given class
$\theta$. Here and throughout this discussion, we consider the infinite
statistics limit such that we can replace sums over events by integrals. We
have also dropped the prior factors $p(\theta_{i})$, assuming that one has
equal numbers of examples from the two hypotheses during training. While this
is often true in practice, it is not strictly necessary for our main
conclusions, though it does simplify the notation. It is well-known [16, 17]
(also in high-energy physics [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30]) that an optimally trained $c$ will have the following property:
$\displaystyle\frac{c(x)}{1-c(x)}=\frac{p(x|\theta_{A})}{p(x|\theta_{B})},$
(4)
such that one learns the per-instance likelihood ratio. By the Neyman–Pearson
lemma [15], this defines the optimal single-event classifier.
Loss Name | $A(f)$ | $B(f)$ | $\operatorname*{argmin}_{f}L[f]$ | Integrand of $-\min_{f}L[f]$ | Related Divergence/Distance
---|---|---|---|---|---
Binary Cross Entropy | $\log f$ | $\log(1-f)$ | $\frac{p_{A}}{p_{A}+p_{B}}$ | $p_{A}\log\frac{p_{A}}{p_{A}+p_{B}}+(A\leftrightarrow B)$ | $2\big{(}\text{Jensen-Shannon}-\log 2\big{)}$
Mean Squared Error | $-(1-f)^{2}$ | $-f^{2}$ | $\frac{p_{A}}{p_{A}+p_{B}}$ | $-\frac{p_{A}p_{B}}{p_{A}+p_{B}}$ | $\frac{1}{2}\big{(}\text{Triangular}-1\big{)}$
Square Root | $\frac{-1}{\sqrt{f}}$ | $-\sqrt{f}$ | $\frac{p_{A}}{p_{B}}$ | $-2\sqrt{p_{A}p_{B}}$ | $2\big{(}\text{Hellinger}^{2}-1\big{)}$
Maximum Likelihood Cl. | $\log f$ | $1-f$ | $\frac{p_{A}}{p_{B}}$ | $p_{A}\log\frac{p_{A}}{p_{B}}$ | Kullback–Leibler
Table 1: Examples of loss functionals in the form of Eq. (5), with the
associated location and value of the loss minimum, using the shorthand
$p_{i}\equiv p(x|\theta_{i})$. We have used the symbol $f$ in all cases to
denote the classifier, but some choices require explicit constraints on $f$ to
be either non-negative or in the range $[0,1]$. In the last column, we
indicate the relation of the loss minimum to statistical divergences and
distances, up to an overall scaling and offset. See Ref. [31] for additional
relations.
There are many loss functionals that satisfy this property. Consider a more
general loss functional that depends on a learnable function
$f:\mathbb{E}\rightarrow\mathbb{R}$ (which unlike $c$ may or may not map to
$[0,1]$) as well as fixed rescaling functions $A:\mathbb{R}\to\mathbb{R}$ and
$B:\mathbb{R}\to\mathbb{R}$:
$\displaystyle L[f]=-\int dx\,$
$\displaystyle\Big{(}p(x|\theta_{A})\,A(f(x))+p(x|\theta_{B})\
B(f(x))\Big{)}.$ (5)
Taking the functional derivative with respect to $f(x)$, the extremum of
$L[f]$ satisfies the property:
$\displaystyle-\frac{B^{\prime}(f(x))}{A^{\prime}(f(x))}=\frac{p(x|\theta_{A})}{p(x|\theta_{B})}.$
(6)
As long as $-B^{\prime}(f)/A^{\prime}(f)$ is a monotonic rescaling of $f$ and
the overall loss functional is convex, then the function $f(x)$ learned by
minimizing Eq. (5) defines an optimal classifier. In many cases, the minimum
value of $L[f]$ itself is interesting in the context of statistical
divergences and distances [31], and a few examples are shown in Table 1.
To simplify the following discussion, we will focus on the “maximum likelihood
classifier” (MLC) loss:
$\displaystyle L_{\text{MLC}}[f]=-\int dx\,\Big{(}$ $\displaystyle
p(x|\theta_{A})\log f(x)$ $\displaystyle~{}+p(x|\theta_{B})\,(1-f(x))\Big{)}.$
(7)
This is of the general form in Eq. (5) with $A(f)=\log f$ and $B(f)=1-f$. To
our knowledge, the MLC was first introduced in the collider physics context in
Refs. [32, 33], although with an exponential parametrization of $f(x)$. We
call Eq. (7) the MLC loss to distinguish it from the related maximum
likelihood loss that is often used to fit generative models [34, 35, 36].
Using Eq. (6), the minimum of this loss functional yields directly the
likelihood ratio:
$\operatorname*{argmin}_{f}L_{\text{MLC}}[f]=\frac{p(x|\theta_{A})}{p(x|\theta_{B})},$
(8)
which will be useful to simplify later analyses.111 A variation of Eq. (8)
holds for $A(f)=\log C(f)$ and $B(f)=1-C(f)$, where $C(f)$ is any
monotonically increasing function with range that covers $(0,\infty)$. In this
case, $C(\operatorname*{argmin}_{f}L[f])=p(x|\theta_{A})/p(x|\theta_{B})$.
This can be useful in practice if $C(f)$ is everywhere positive, since $f$ can
take on negative values and still yield a valid likelihood ratio. See Fig. 10
for an empirical study of $C(f)=\exp f$. The MLC loss functional value at the
minimum is
$-\min_{f}L_{\text{MLC}}[f]=\int
dx\,p(x|\theta_{A})\log\frac{p(x|\theta_{A})}{p(x|\theta_{B})},$ (9)
which is the Kullback–Leibler (KL) divergence, also known as the relative
entropy from $p(x|\theta_{B})$ to $p(x|\theta_{A})$. See App. A for an
intuitive derivation of Eq. (7).
### II.2 Per-Ensemble Binary Classification
To move from single-event classification to multi-event classification, we
want to learn a classification function $f_{N}$ that can process $N$ events
simultaneously. Here, we are using $f_{N}:\mathbb{E}^{N}\rightarrow\mathbb{R}$
instead of $c_{N}:\mathbb{E}^{N}\rightarrow[0,1]$ to avoid algebraic
manipulations like Eq. (4). We will use the vector notation
$\vec{x}=\\{x_{1},\dots,x_{N}\\}$ (10)
to represent an element of $\mathbb{E}^{N}$. Our goal is to distinguish
whether $\vec{x}$ is drawn from $p(\vec{x}|\theta_{A})$ ($f_{N}\to\infty$) or
from $p(\vec{x}|\theta_{B})$ ($f_{N}\to 0$). Note that we are trying to
classify a pure event ensemble as coming from either $\theta_{A}$ or
$\theta_{B}$, which is a different question than trying to determine the
proportion of events drawn from each class in a mixed event ensemble. For
$N=1$, $f_{1}$ is the same as $f$ discussed in Eq. (5).
If $f_{N}$ is trained optimally, then the classification performance of
$f_{N}$ evaluated on $N>1$ events will be better than the performance of
$f_{1}$ evaluated on a single event, as relevant to the discussions in Refs.
[7, 8, 9, 10, 11, 12, 13, 14]. The key point of this paper is that one can
construct a classifier $f_{1\to N}$ that is built only from $f_{1}$, acts on
$N$ events, and has the same asymptotic performance as $f_{N}$.
Using the MLC loss in Eq. (7), but now applied to $N$ events, we have
$\displaystyle L_{\text{MLC}}[f_{N}]=-\int d^{N}x\,\Big{(}$ $\displaystyle
p(\vec{x}|\theta_{A})\,\log f_{N}(\vec{x})$
$\displaystyle~{}+p(\vec{x}|\theta_{B})\,(1-f_{N}(\vec{x}))\Big{)},$ (11)
whose minimum is the per-ensemble likelihood ratio:
$\operatorname*{argmin}_{f_{N}}L_{\text{MLC}}[f_{N}]=\frac{p(\vec{x}|\theta_{A})}{p(\vec{x}|\theta_{B})}.$
(12)
By the Neyman–Pearson lemma, this yields the optimal per-ensemble classifier.
On the other hand, once we have trained a single-event classifier $f_{1}$
using Eq. (7), we can build a multi-event classifier $f_{1\to N}$ without any
additional training:
$f_{1\to
N}(\vec{x})\equiv\prod_{i=1}^{N}f_{1}(x_{i})\quad\rightarrow\quad\frac{p(\vec{x}|\theta_{A})}{p(\vec{x}|\theta_{B})},$
(13)
where in the last step we have combined the solution found in Eq. (8) with the
IID condition in Eq. (2). Whereas minimizing Eq. (11) requires sampling over
$\mathbb{E}^{N}$, constructing $f_{1\to N}$ only requires sampling over
$\mathbb{E}$, which is a considerable reduction in computational burden for
large $N$. The technical details of carrying out this procedure are explained
in Sec. III.1.
Going in the converse direction, we can learn a single-event classifier
$f_{N\to 1}$ starting from a constrained multi-event classifier
$\tilde{f}_{N}$. Using weight sharing, we can minimize Eq. (11) subject to the
constraint that $\tilde{f}_{N}$ takes the functional form:
$\tilde{f}_{N}(\\{x_{1},\dots,x_{N}\\})=\prod_{i=1}^{N}f_{N\to 1}(x_{i}),$
(14)
where $f_{N\to 1}(x)$ is a learnable function. Under the IID assumption,
$\tilde{f}_{N}$ can still learn the per-ensemble likelihood ratio, but the
learned $f_{N\to 1}(x)$ will now be the per-instance likelihood ratio, at
least asymptotically.222In the case that the two samples are composed of
mixtures of two categories, then the learned $f_{N\to 1}(x)$ will be the ratio
of the mixed sample likelihoods, which is monotonically related to the optimal
pure sample classifier, as discussed in Ref. [37]. An examination of this
converse construction is presented in Sec. III.2.
### II.3 Comparing the Loss Gradients
We have shown that the per-ensemble classifier $f_{N}$ and the composite per-
event classifier $f_{1\to N}$ have the same asymptotic information content,
but one might wonder if there is nevertheless a practical performance gain to
be had using per-ensemble learning.
Under the IID assumption, the optimal $f_{N}$ takes the form of
$\tilde{f}_{N}$ in Eq. (14), and in our empirical studies, we found no benefit
to letting $f_{N}$ have more functional freedom. Therefore, to get a sense of
the efficacy of per-ensemble versus per-instance training, we can compare the
effective loss functions for $f_{N\to 1}$ and $f_{1}$. Since the inputs and
outputs of these functions are the same (i.e. $\mathbb{E}\to\mathbb{R}$), we
can do an apples-to-apples comparison of their behavior under gradient
descent. The following analysis assumes that the neural network training
occurs in the vicinity of the global minimum of the loss function.
For the per-ensemble case, plugging Eq. (14) into Eq. (11) and using the IID
relation in Eq. (1), we find the effective loss functional:
$\displaystyle L_{\rm MLC}[f_{N\to 1}]+1$ $\displaystyle=-N\int
dx\,p(x|\theta_{A})\,\log f_{N\to 1}(x)$ $\displaystyle\quad+\left(\int
dx\,p(x|\theta_{B})f_{N\to 1}(x)\right)^{N}.$ (15)
This is to be contrasted with the per-instance loss functional from Eq. (7),
repeated for convenience with the $f_{1}$ notation and typeset to be parallel
to the above:
$\displaystyle L_{\text{MLC}}[f_{1}]+1$ $\displaystyle=-\int
dx\,p(x|\theta_{A})\,\log f_{1}(x)$ $\displaystyle\quad+\int
dx\,p(x|\theta_{B})\,f_{1}(x).$ (16)
To understand the loss gradients, we can Taylor expand the learned functions
about the optimal solution:
$\displaystyle f_{N\to 1}(x)$
$\displaystyle=\frac{p(x|\theta_{A})}{p(x|\theta_{B})}+\epsilon(x),$ (17)
$\displaystyle f_{1}(x)$
$\displaystyle=\frac{p(x|\theta_{A})}{p(x|\theta_{B})}+\epsilon(x).$ (18)
Plugging these into their respective loss functionals and looking at the
leading-order variations, we have:
$\displaystyle\frac{\delta L_{\rm MLC}[f_{N\to 1}]}{N}$ $\displaystyle=\int
dx\,\frac{\big{(}p(x|\theta_{B})\,\epsilon(x)\big{)}^{2}}{2\,p(x|\theta_{A})}$
$\displaystyle\quad+\frac{N-1}{2}\left(\int
dx\,p(x|\theta_{B})\,\epsilon(x)\right)^{2},$ (19) $\displaystyle\delta L_{\rm
MLC}[f_{1}]$ $\displaystyle=\int
dx\,\frac{\big{(}p(x|\theta_{B})\,\epsilon(x)\big{)}^{2}}{2\,p(x|\theta_{A})}.$
(20)
These expressions are quadratic in $\epsilon(x)$, which means that we are
expanding around the correct minimum.
The expression for $\delta L_{\rm MLC}[f_{1}]$ involves a single integral over
$x$, so under gradient descent, the value of $\epsilon(x)$ can be
independently adjusted at each point in phase space to find the minimum. By
contrast, $\delta L_{\rm MLC}[f_{N\to 1}]$ has an additional piece involving
an integral squared, so even if at a given point in phase space $x_{0}$ we
have achieved $\epsilon(x_{0})=0$, gradient descent will tend to push
$\epsilon(x_{0})$ away from the correct value until $\epsilon(x)=0$
everywhere. This correlated structure explains the slower convergence of
$L_{\rm MLC}[f_{N\to 1}]$ compared to $L_{\rm MLC}[f_{1}]$ in our empirical
studies. While we focused on the MLC loss to simplify the algebra, the
appearance of these (typically counterproductive) correlations in the loss
gradient appears to be a generic feature of per-ensemble learning.
### II.4 Per-Ensemble Regression
While the discussion above focused on binary classification, the same basic
idea applies to regression problems as well. The goal of regression is to
infer parameters $\theta$ from the data $\vec{x}$. There are a variety of
approaches that can be used for this task, and each can be connected to
parametrized per-instance classification.
#### II.4.1 Maximum Likelihood
Maximum likelihood is the most common strategy for inference in collider
physics. Symbolically, we are trying to find
$\theta_{\rm ML}=\operatorname*{argmax}_{\theta}p(\vec{x}|\theta).$ (21)
One way to determine $\theta_{\rm ML}$ is with a two-step approach. First, one
can train a parametrized classifier $f(x,\theta)$ [26, 38] using, e.g., the
per-instance MLC loss:
$\displaystyle L_{\rm MLC}[f]=-\int dx\,\Big{(}$ $\displaystyle
p(x|\theta)\,p(\theta)\log f(x,\theta)$
$\displaystyle~{}+p(x|\theta_{0})\,p(\theta)\,(1-f(x,\theta))\Big{)}.$ (22)
The top line corresponds to a synthetic dataset where every event is generated
from $p(x|\theta)$ with different $\theta$ values drawn from the probability
density $p(\theta)$. The bottom line corresponds to a synthetic dataset where
every event is generated using the same $p(x|\theta_{0})$ for fixed
$\theta_{0}$ and then augmented with a value $\theta$ that follows from
$p(\theta)$ independently of $x$. Minimizing Eq. (22) with respect to
$f(x,\theta)$, the asymptotic solution is the likelihood ratio:
$f(x,\theta)=\frac{p(x|\theta)}{p(x|\theta_{0})},$ (23)
where the factors of $p(\theta)$ have canceled out. Second, one can estimate
$\theta_{\rm ML}$ by using the IID properties of the event ensemble to relate
likelihoods to the classifier output $f(x,\theta)$:
$\displaystyle\theta_{\rm ML}$
$\displaystyle=\operatorname*{argmin}_{\theta}\left\\{-\sum_{i=1}^{N}\log
p(x_{i}|\theta)\right\\}$
$\displaystyle=\operatorname*{argmin}_{\theta}\left\\{-\sum_{i=1}^{N}\log\frac{p(x_{i}|\theta)}{p(x_{i}|\theta_{0})}\right\\}$
$\displaystyle\approx\operatorname*{argmin}_{\theta}\left\\{-\sum_{i=1}^{N}\log
f(x_{i},\theta)\right\\}.$ (24)
Thus, even though maximum likelihood regression uses information from the full
event ensemble, only a parametrized per-instance classifier is required for
this procedure.
#### II.4.2 Classifier Loss
Two recent proposals for parameter estimation are explicitly built on
classifiers for regression [18, 19]. For any classifier, one can perform the
following optimization:333Note that Ref. [18] used the (non-differentiable)
area under the curve instead of the classifier loss, as it is not sensitive to
differences in the prior $p(\theta)$ between the two data sets.
$\displaystyle\theta_{\rm
CL}=\operatorname*{argmax}_{\theta^{\prime}}\left\\{\begin{matrix}\text{Loss
of a classifier trained}\cr\text{to distinguish $\theta^{\prime}$ from
$\theta_{\rm data}$}\end{matrix}\right\\}.$ (25)
Here, we are imagining that the $\theta^{\prime}$ samples come from synthetic
data sets. The appearance of a maximum instead of minimum in Eq. (25) is
because, as highlighted in Table 1, it is negative loss functions that
correspond to statistical divergences and distances.
In general, the $\theta_{\rm CL}$ that minimizes the classifier loss will be
different from the $\theta_{\rm ML}$ that maximizes the likelihood. For the
special case of the MLC loss, though, they are the same in the asymptotic
limit if we set $\theta_{A}=\theta_{\rm data}$ and
$\theta_{B}=\theta^{\prime}$. To see this, recall from Eq. (9) that after
training, the value of the MLC loss is related to the KL divergence:
$\displaystyle\operatorname*{argmax}_{\theta^{\prime}}\\{\min_{f}L_{\text{MLC}}[f]\\}$
$\displaystyle\hskip
5.69054pt=\operatorname*{argmax}_{\theta^{\prime}}\left\\{-\int
dx\,p(x|\theta_{\rm data})\log\frac{p(x|\theta_{\rm
data})}{p(x|\theta^{\prime})}\right\\}$ $\displaystyle\hskip
5.69054pt\approx\operatorname*{argmax}_{\theta^{\prime}}\left\\{\sum_{i=1}^{N}\log\frac{p(x_{i}|\theta^{\prime})}{p(x_{i}|\theta_{\rm
data})}\right\\}$ $\displaystyle\hskip
5.69054pt=\operatorname*{argmin}_{\theta^{\prime}}\left\\{-\sum_{i=1}^{N}\log
p(x_{i}|\theta^{\prime})\right\\}$ $\displaystyle\hskip 5.69054pt=\theta_{\rm
ML}\,,$ (26)
where the sum is over data events.
#### II.4.3 Direct Regression
In terms of information content, a regression model trained in the usual way
can be built from a parametrized classification model. Suppose that
$\theta\in\mathbb{R}^{Q}$ and $g_{N}:\mathbb{E}^{N}\rightarrow\mathbb{R}^{Q}$
is a regression model trained with the mean squared error loss:
$L_{\rm MSE}[g_{N}]=-\int
d^{n}x\,p(\vec{x},\theta)\Big{(}g_{N}(\vec{x})-\theta\Big{)}^{2}$ (27)
It is well known that the optimally trained $g_{N}$ will be related to the
expectation value of $\theta$:
$g_{N}(\vec{x})=\mathbb{E}[\theta|\vec{x}]=\int
d\theta\,\theta\,p(\theta|\vec{x}).$ (28)
Other loss functions approximate other statistics, as discussed in Ref. [39].
For example, the mean absolute error loss approximates the median of $\theta$.
Ultimately, all direct regression methods are functionals of
$p(\theta|\vec{x})$.
We can relate $p(\theta|\vec{x})$ to a parametrized classifier
$f_{N}(\vec{x},\theta)$ trained to distinguish $\theta$ from a baseline
$\theta_{0}$:
$\displaystyle
p(\theta|\vec{x})=\frac{p(\vec{x}|\theta)\,p(\theta)}{p(\vec{x})}$
$\displaystyle=\frac{p(\vec{x}|\theta)\,p(\theta)}{\int
d\theta^{\prime}\,p(\vec{x}|\theta^{\prime})\,p(\theta^{\prime})}$
$\displaystyle=\frac{\frac{p(\vec{x}|\theta)}{p(\vec{x}|\theta_{0})}\,p(\theta)}{\int
d\theta^{\prime}\,\frac{p(\vec{x}|\theta^{\prime})}{p(\vec{x}|\theta_{0})}\,p(\theta^{\prime})}$
$\displaystyle=\frac{f_{N}(\vec{x},\theta)\,p(\theta)}{\int
d\theta^{\prime}\,f_{N}(\vec{x},\theta^{\prime})\,p(\theta^{\prime})},$ (29)
where $p(\theta)$ is the probability density of $\theta$ used during the
training of $g_{N}$. Following the same logic as Sec. II.2, the per-ensemble
classifier $f_{N}(\vec{x},\theta)$ can be related to a per-instance classifier
$f_{1}(x,\theta)$. Therefore, even though $g_{N}$ acts on $N$ events, it has
the same information content as a parametrized classifier that acts on single
events.
Performing regression via Eqs. (28) and (29) is straightforward but tedious.
In practice, one would train a parametrized per-instance classifier
$f_{1}(x,\theta)$ as in Eq. (23), multiply it to construct
$f_{N}(\vec{x},\theta)=\prod_{i=1}^{N}f_{1}(x_{i},\theta)$, and then sample
over values of $\theta$ to approximate the integrals. We show examples of the
above regression strategies in Sec. III.3
### II.5 Beyond Regression
In addition to classification and regression, a standard machine learning task
is density estimation. While some classical machine learning methods like
$k$-nearest neighbors [40, 41] do require multi-instance information at
prediction time, many of the standard deep learning solutions to implicit or
explicit generative modeling are built on per-instance functions. Such methods
include generative adversarial networks [42],444 In the context of adversarial
training, it may be beneficial to use per-ensemble information in the
discriminator to mitigate mode collapse, as utilized in Ref. [14]. This is
also the philosophy behind mini-batch discrimination [43]. variational
autoencoders [44], and normalizing flows [45].
One reason for computing explicit densities is to estimate the distance to a
reference density. A common set of tools for this task are the $f$-divergences
mentioned earlier. As discussed in Ref. [31] and highlighted in Table 1, there
is a direct mapping between the loss value of a per-instance classification
task and a corresponding $f$-divergence between the underlying probability
densities.
A related quantity is the mutual information between two random variables $X$
and $Y$:
$\displaystyle I(X,Y)=\int dx\,dy\,p(x,y)\log\frac{p(x,y)}{p(x)\,p(y)}.$ (30)
For example, $Y$ could be binary (a class label) and then $I(X,Y)$ would
encode how much information (in units of nats) is available in $X$ for doing
classification. This can be helpful in the context of ranking input features,
and was studied in the context of quark/gluon jet classification in Ref. [46].
Naively, Eq. (30) might seem like it requires estimating the densities $p(x)$,
$p(y)$, and $p(x,y)$, which in turn may require ensemble information (see e.g.
Ref. [47] for a study in the context of HEP). On the other hand, Eq. (30)
takes the same form as the KL divergence in Eq. (9). Therefore, this quantity
can be estimated using a similar strategy as in earlier sections, by training
a classifier to distinguish data following $p(x,y)$ from data following
$p(x)\,p(y)$ using the MLC loss. The value of the loss at the minimum will be
an estimate of the mutual information. A simple example of this will be
studied in Sec. III.4.
## III Empirical Studies
We now present empirical studies comparing per-instance and per-ensemble data
analysis strategies to highlight the points made in Sec. II. Our analyses are
based on three case studies: a simple two Gaussian example, searching for
dijet resonances, and measuring the top quark mass.
### III.1 Classifiers: Multi-Event from Single-Event
As argued in Sec. II.2, under the IID assumption we can build multi-event
classifiers from single-event classifiers. We now demonstrate how to construct
$f_{1\rightarrow N}$ defined in Eq. (13), comparing its performance to
$f_{N}$.
#### III.1.1 Two Gaussian Example
(a) (b)
Figure 1: Classification in the two Gaussian example. (a) A histogram of the
Gaussian random variable $X$, for the “signal” ($x_{0}=0.1$) and background
($x_{0}=-0.1$). (b) ROC curves for various binary classifiers. From the
single-event classifier $f_{1}$, we can construct a multi-event classifier
$f_{1\to 10}$ that matches the performance of a classifier trained on 10
events simultaneously ($f_{10}$).
Our first case study involves one-dimensional Gaussian random variables. As
shown in Fig. 1a, we consider two Gaussian distributions
$X\sim\mathcal{N}(\pm\epsilon,1)$, with slightly different means
($x_{0}=\pm\epsilon$) but the same variance ($\sigma=1$). Here, the “signal”
has positive mean while the “background” has negative mean, and we take
$\epsilon=0.1$ for concreteness.
Both the per-instance ($f_{1}$) and per-ensemble ($f_{N}$) classifiers are
parametrized by neural networks and implemented using Keras [48] with the
Tensorflow backend [49] and optimized with Adam [50]. We use the binary cross
entropy loss function so Eq. (4) is needed to convert the classifier output to
a likelihood ratio. Each classifier consists of two hidden layers with 128
nodes per layer. Rectified Linear Unit (ReLU) activation functions are used
for the intermediate layers while sigmoid activation is used for the last
layer. The only difference between the per-instance and per-ensemble networks
is that the input layer has one input for $f_{1}$ but $N$ inputs for $f_{N}$.
We train each network with 50,000 events to minimize the binary cross entropy
loss function, and we test the performance with an additional 50,000 events.
For each network, we train for up to 1000 epochs with a batch size of 10%,
which means that the number of batches per epoch is the same, as is the number
of events considered per batch. The training is stopped if the validation loss
does not decrease for 20 consecutive epochs (early stopping). For the ensemble
network, we take $N=10$. We did not do any detailed hyperparameter
optimization for these studies.
In Fig. 1b, we show the performance of the resulting classifiers $f_{1}$ and
$f_{10}$. We checked that the $f_{1}$ classifier parametrized by a neural
network has essentially the same performance as an analytic function derived
by taking the ratio of Gaussian probability densities, which means that the
neural network $f_{1}$ is nearly optimal. As expected, the per-instance
classifier $f_{1}$ has a worse receiver operating characteristic (ROC) curve
than the per-ensemble classifier $f_{10}$. This is not a relevant comparison,
however, because the two are solving different classification tasks (i.e.
classifying individual events as coming from signal or background versus
classifying an ensemble of $N=10$ events as all coming from signal or
background). With Eq. (13), we can use $f_{1}$ to build a $10$-instance
classifier $f_{1\rightarrow 10}$, whose ROC curve is nearly identical to
$f_{10}$, if not even slightly better. Thus, as expected from Eq. (2), all of
the information in the 10-instance classifier is contained in the per-instance
classifier.
#### III.1.2 Dijet Resonance Search
(a)
(b)
Figure 2: Classification in the dijet resonance search example. (a,b)
Histograms of the four jet features for the signal ($W^{\prime}\to XY$) and
background (QCD dijet) processes. (c) ROC curves for various binary
classifiers. The multi-event classifier $f_{1\to 3}$ (built from $f_{1}$)
outperforms three classifiers trained on triplets of events:
$f_{3}^{\text{list}}$ with randomly ordered inputs, $f_{3}^{\text{sort}}$ with
sorted inputs, and $f_{3}^{\text{set}}$ based on the deep sets/PFN strategy in
Eq. (31) with built-in permutation invariance.
(c)
We now consider an example from collider physics, motivated by a search for
new beyond-the-Standard-Model (BSM) particles in a dijet final state. The
simulations used for this study were produced for the LHC Olympics 2020
community challenge [51]. The background process involves generic quantum
chromodynamics (QCD) dijet events with a requirement of at least one such jet
with transverse momentum $p_{T}>1.3$ TeV. The signal process involves the
production of a hypothetical new resonance $W^{\prime}$ with mass
$m_{W^{\prime}}=3.5$ TeV, which decays via $W^{\prime}\rightarrow XY$ to two
hypothetical particles $X$ and $Y$ of masses 500 GeV and 100 GeV,
respectively. Each of the $X$ and $Y$ particles decays promptly into pairs of
quarks. Due to the mass hierarchy between the $W^{\prime}$ boson and its decay
products, the final state is characterized by two large-radius jets with two-
prong substructure. The background and signal are generated using Pythia 8.219
[52, 53]. A detector simulation is performed with Delphes 3.4.1 [54, 55, 56]
using the default CMS detector card. Particle flow objects are used as inputs
to jet clustering, implemented with FastJet 3.2.1 [57, 58] and the
anti-$k_{t}$ algorithm [59] using $R=1.0$ for the radius parameter. Events are
required to have a reconstructed dijet mass within the range
$m_{JJ}<[3.3,3.7]\,\text{GeV}$.
Four features are used to train our classifiers: the invariant mass of the
lighter jet, the mass difference of the leading two jets, and the
$N$-subjettiess ratios $\tau_{21}$ [60, 61] of the leading two jets. The
observable $\tau_{21}$ quantifies the degree to which a jet is characterized
by two subjets or one subjet, with smaller values indicating two-prong
substructure. The mass features are recorded in units of TeV so that they are
numerically $\mathcal{O}(1)$. Histograms of the four features for signal and
background are shown in Figs. 2a and 2b. The signal jet masses are localized
at the $X$ and $Y$ masses and the $\tau_{21}$ observables are shifted towards
lower values, indicating that the jets have two-prong substructure.
We train a per-instance classifier ($f_{1}$) and a per-ensemble classifier
($f_{3}$) using the same tools as for the Gaussian example above, again using
binary cross entropy for the loss function. Because signal and background are
so well separated in this example, we restrict our attention to $N=3$ to avoid
saturating the performance. Note that this is an artificially constructed
classification problem, since in a more realistic context one would be trying
to estimate the signal fraction in an event ensemble, not classify triplets of
events as all coming from signal or background.
For $f_{1}$, the neural network architecture is the same as Ref. [18] with
four hidden layers, each with 64 nodes and ReLU activation, and an output
layer with sigmoid activation. For $f_{3}$, the neural network involves
$4\times 3=12$ inputs, and the penultimate hidden layer is adjusted to have
128 nodes, yielding a marginal performance gain. In both cases, about 100,000
events are used for testing and training, with roughly balanced classes. All
of the networks are trained for up to 1000 epochs with the same early stopping
condition as in the Gaussian case and with a batch size of 10%. Following Eq.
(13), we construct a tri-event classifier $f_{1\rightarrow 3}$ from $f_{1}$.
The ROC curves for $f_{3}$ and $f_{1\to 3}$ are shown in Fig. 2c, with $f_{1}$
also shown for completeness. Interestingly, the $f_{1\rightarrow 3}$
classifier trained on single events significantly outperforms $f_{3}$ trained
on multiple events. There are a variety of reasons for this, but one important
deficiency of the $f_{3}$ classifier is that it does not respect the
permutation symmetry of its inputs. Because events are IID distributed, there
is no natural ordering of the events, but the fully connected architecture we
are using imposes an artificial ordering. Inspired by Ref. [12], we can break
the permutation symmetry of the inputs by imposing a particular order on the
events. Specifically, we train a network $f_{3}^{\text{sort}}$ where the
triplet of events is sorted by their leading jet mass. Using
$f_{3}^{\text{sort}}$ yields a small gain in performance seen in Fig. 2, but
not enough to close the gap with $f_{1\rightarrow 3}$.
(a) (b)
Figure 3: Classification in the top quark mass example. (a) A histogram of
$m_{b_{1}\mu\nu}$ for top quark masses of 172.5 GeV and 175 GeV. The “wgt.”
curve is explained later in Sec. III.3.2, where we test the performance of a
likelihood reweighting. (b) The difference in efficiency for the 172.5 GeV top
quark mass sample (true positive) and the 175 GeV top quark mass sample (false
positive) as a function of the true positive rate for various binary
classifiers. Once again, a multi-event classifier ($f_{1\to 20}$) built from
the single-event classifier ($f_{1}$) has the best performance. For the
classifiers trained to process 20 events simultaneously, the deep sets/PFN
approach ($f_{20}^{\text{set}}$) does better than sorting the inputs
($f_{20}^{\text{sort}}$).
A more powerful way to account for the permutation symmetry among events is to
explicitly build a permutation-invariant neural network architecture. For this
purpose, we use the deep sets approach [62]. In the particle physics context,
deep sets were first used to construct particle flow networks (PFNs) [63],
where the inputs involve sets of particles. Here, we are interested in sets of
events, though we will still use the PFN code from the
https://energyflow.network/ package. Following Refs. [62, 63], we decompose
our set-based classifier as:
$\displaystyle
f^{\text{set}}_{N}(\vec{x})=F\left(\sum_{i=1}^{N}\Phi(x_{i})\right),$ (31)
where $F:\mathbb{R}^{L}\rightarrow[0,1]$ and
$\Phi:\mathbb{E}\rightarrow\mathbb{R}^{L}$ are neural networks that are
simultaneously optimized. The network $\Phi$ embeds single events $x_{i}$ into
a $L$-dimensional latent space. The sum operator in Eq. (31) guarantees that
$f^{\text{set}}_{N}$ is invariant under permutations $x_{\sigma(i)}$ for
$\sigma\in S_{N}$, the permutation group acting on $N$ elements. We use the
default parameters from the PFN code, with $L=128$, $\Phi$ having two hidden
layers with 100 nodes each, and $F$ having three hidden nodes with 100 nodes
each. The same learning strategy (up to 1000 epochs, early stopping, 10% batch
size) as the other networks is used for the PFN.
The performance of $f^{\text{set}}_{3}$ is shown in Fig. 2, which gets much
closer to matching the performance of $f_{1\rightarrow 3}$. Part of this
improvement is due to enforcing the permutation symmetry, though there is also
a potential gain from the fact the PFN we used for $f^{\text{set}}_{3}$ has
more trainable weights than the fully connected network for
$f^{\text{sort}}_{3}$. All of the $f_{3}$ variants were considerably more
difficult to train than $f_{1\to 3}$, likely for the reason discussed in Sec.
II.3. Thus, we have empirical evidence for the superiority of single-event
training for multi-event classification.
#### III.1.3 Top Quark Mass Measurement
Our third and final example is motivated by the top quark mass measurement, as
recently studied in Refs. [18, 12]. Extracting the top quark mass is really a
regression problem, which we investigate in Sec. III.3. Here, we consider a
related classification task to distinguish two event samples generated with
different top quark masses (172.5 GeV and 175 GeV). This is a realistic
hypothesis testing task that requires full event ensemble information, though
only per-instance training as we will see.
We use the same dataset as Ref. [18]. Top quark pair production is generated
using Pythia 8.230 [52, 53] and detector effects are modeled with Delphes
3.4.1 [54, 55, 56] using the default CMS run card. After the production and
decay steps $t\bar{t}\to bW^{+}\bar{b}W^{-}$, one of the $W$ bosons is forced
to decay to $\mu^{+}\nu$ while the other $W$ boson decays hadronically. Each
event is recorded as a variable-length set of objects, consisting of jets,
muons, and neutrinos. At simulation-level, the neutrino is replaced with the
missing transverse momentum. Generator-level and simulation-level jets are
clustered with the anti-$k_{t}$ algorithm using $R=0.4$ and the simulation-
level jet is labeled as $b$-tagged if the highest energy parton inside the
nearest generator-level jet ($\Delta R<0.5$) is a $b$ quark. Jets are required
to have $p_{T}>20$ GeV and they can only be $b$-tagged if $|\eta|<2.5$.
Furthermore, jets overlapping with the muon are removed.
Events are only saved if they have at least two $b$-tagged jets and at least
two additional non $b$-tagged jets. The $b$-jet closest to the muon in
rapidity-azimuth is labeled $b_{1}$. Of the remaining $b$-tagged jets, the
highest $p_{T}$ one is labeled $b_{2}$. The two highest $p_{T}$ non-$b$-tagged
jets are labeled $j_{1}$ and $j_{2}$, and typically come from the $W$ boson.
(Imposing the $W$ mass constraint on $j_{1}$ and $j_{2}$ would yield lower
efficiency, though without significantly impacting the results.) The four-
momentum of the detector-level neutrino ($\nu$) is determined by solving the
quadratic equation for the $W$ boson mass; if there is no solution, the mass
is set to zero, while if there are two real solutions, the one with the
smaller $|p_{z}|$ is selected. Four observables are formed for performing the
top quark mass extraction, given by the following invariant masses:
$m_{b_{1}\mu\nu}$, $m_{b_{2}\mu\nu}$, $m_{b_{1}j_{1}j_{2}}$, and
$m_{b_{2}j_{1}j_{2}}$. A histogram of $m_{b_{1}\mu\nu}$ is shown for
illustration in Fig. 3a.
We use the same neural network architectures and training procedure as in the
BSM example above, with 1.5 million events per fixed-mass sample. The only
difference is that the batch size is set to 0.1% in order to keep the number
of examples to be $\mathcal{O}(1000)$. For the per-ensemble classifier, we
take $N=20$, though of course for a realistic hypothesis testing situation,
$N$ would be as large as the number of top quark events recorded in data. To
capture the permutation invariance of the inputs, we construct
$f_{20}^{\text{set}}$ using the deep sets approach in Eq. (31). We also build
a classifier $f_{1\to 20}$ from the per-instance classifier $f_{1}$ using Eq.
(13).
In Fig. 3b, we see that $f_{1\to 20}$ and $f_{20}^{\text{set}}$ have
comparable performance, though $f_{1\to 20}$ is noticeably better. Some of
this improvement may be due to differences in the network architecture, but we
suspect that most of the gain is due to the more efficient training in the
per-instance case. We checked that very poor performance is obtained for a
classifier $f_{20}$ lacking permutation invariance, with a ROC curve that was
not that much better than $f_{1}$ alone. Explicitly breaking the invariance by
sorting the inputs based on $m_{b_{1}\mu\nu}$ does help a little, as indicated
by the $f_{20}^{\text{sort}}$ curve in Fig. 3b, but does not reach the set-
based approach.
Figure 4: Computational performance of single-event versus multi-event
training. Shown is the efficiency for the 175 GeV sample (false positive) for
a fixed 50% efficiency for the 172.5 GeV sample (true positive), plotted as a
function of training epoch. Single-event training ($f_{1\to 20}$) outperforms
multi-event training ($f_{20}^{\text{set}}$), where both methods go through
the full data set per epoch.
Given the similar performance of $f_{1\rightarrow 20}$ and
$f_{20}^{\text{set}}$, it is interesting to examine which learning strategy is
more computationally efficient. In Fig. 4, we compare the performance as a
function of the training epoch, using the difference of the true and false
positive rates at a fixed 50% signal efficiency. In each epoch, both
$f_{1\rightarrow 20}$ and $f_{20}^{\text{set}}$ see the full ensemble of
events, so this is an apples-to-apples comparison as far as data usage is
concerned. In particular, we plot this information per epoch instead of per
compute time to avoid differences due to the structure of the neural networks.
(There is not an easy way to control for possible differences in the training
time due to the differences in the network structures, since the underlying
tasks are different.) The $f_{1\rightarrow 20}$ classifier trains much faster,
in agreement with the analysis in Sec. II.3, even though the ultimate
asymptotic performance is similar for both classifiers. Once again, we see
better empirical behavior from $f_{1\to 20}$ trained on one event at a time
version $f_{20}^{\text{set}}$ trained on multiple events
simultaneously.555Away from the asymptotic limit, one could try to improve the
empirical per-ensemble performance through data augmentation. Data
augmentation is a generic strategy to help neural networks learn symmetries,
and the IID structure can be reinforced by showing the network new ensembles
built from sampling instances from the existing ensembles.
### III.2 Classifiers: Single-Event from Multi-Event
In general, one cannot take a multi-event classifier $f_{N}$ and extract a
single-event classifier $f_{1}$. It is, however, possible to construct a
special $\tilde{f}_{N}$ network such that one can interpret a subnetwork as a
per-event classifier, as discussed in Sec. II.2. When using the MLC loss
function, we can use the functional form in Eq. (14), where $\tilde{f}_{N}$ is
a product of $f_{N\to 1}$ terms. Training $\tilde{f}_{N}$, where the only
trainable weights are contained in $f_{N\to 1}$, we can learn a single-event
classifier $f_{N\to 1}$ from multi-event samples.
For the binary cross entropy loss used in our case studies, where Eq. (4) is
needed to convert the classifier to a likelihood ratio, we have to introduce a
slightly different structure than Eq. (14). Let $f_{N}^{\text{set}}$ be a
permutation-invariant classifier, as defined in Eq. (31) using the deep
sets/PFN strategy. Taking the latent space dimension to be $L=1$, the $\Phi$
network can be interpreted as a single-event classifier. Because the $\Phi$
network outputs are pooled via summation, we can build an optimal multi-event
classifier if $\Phi$ learns the _logarithm_ of the likelihood ratio; cf. Eq.
(2). With this insight, we can fix the $F$ function to achieve the same
asymptotic performance as a trainable $F$ by setting:
$\displaystyle
F(\vec{x})=\frac{\exp\big{(}\sum_{i=1}^{N}\Phi(x_{i})\big{)}}{1+\exp\big{(}\sum_{i=1}^{N}\Phi(x_{i})\big{)}}\,.$
(32)
Using Eq. (4), one can check that this $F$ is monotonically related to the
ensemble likelihood ratio. Similarly, $\Phi$ will be monotonically related to
the optimal $f_{1}$, which we call $f_{N\to 1}$ for the remainder of this
discussion.
Figure 5: Revisiting the ROC curves for the two Gaussian example from Fig. 1b.
The multi-event classifier $\tilde{f}_{10}$ with the restricted functional
form in Eq. (32) has the same performance as $f_{10}$ with no restrictions.
Using $\tilde{f}_{10}$, we can construct a single-event classifier
$\tilde{f}_{10\to 1}$ with the same performance as $f_{1}$ trained directly.
This construction is demonstrated in Fig. 5 for the Gaussian example. We see
that the deep sets architecture with the fixed form of Eq. (32)
($\tilde{f}^{\rm set}_{10}$) has the same or better performance as the
10-instance fully-connected classifier with more network capacity ($f_{10}$).
Similarly, the $\Phi$ function used as a single-event classifier
($f_{10\rightarrow 1}$) has nearly the same performance as an independently
trained single-event classifier ($f_{1}$).
Figure 6: Revisiting the ROC curves for the dijet resonance search example in
Fig. 2c. The set-based multi-event classifiers $\tilde{f}_{3}^{\rm set}$ and
$f_{3}^{\rm set}$ have similar performance, but we can use the former to
construct a single-event classifier $f_{3\to 1}$. This construction is not as
effective as performing single-event training directly ($f_{1}$).
The same conclusion holds for the BSM classification task, shown in Fig. 6.
The only difference between the set-based architectures
$\tilde{f}_{3}^{\text{set}}$ and $f_{3}^{\text{set}}$ is that the former uses
the fixed functional form in Eq. (32). The fact that they achieve nearly the
same performance is ensured by the IID relation in Eq. (2). The per-instance
$f_{3\rightarrow 1}$ network extracted from $\tilde{f}_{3}^{\text{set}}$ is
not quite as powerful as the $f_{1}$ network trained independently on single
events, as expected from the gradient issue discussed in Sec. II.3. While we
found no benefit to extracting a single-event classifier from a multi-event
classifier, it is satisfying to see these IID-derived theoretical predictions
borne out in these empirical examples.
### III.3 Comparison of Regression Strategies
We now consider the regression methods introduced in Sec. II.4. For
classification, the mapping between per-instance and per-ensemble information
is relatively straightforward. For regression, though, per-ensemble regression
is structurally dissimilar from per-instance regression because of the need to
integrate over priors on the regression parameters. Nevertheless, we can
perform per-ensemble regression by first mapping the problem to per-instance
parametrized classification.
We compare three different regression strategies for our empirical studies.
The first method is a maximum-likelihood analysis, using the form in Eq. (24)
based on the single-event parametrized classifier in Eq. (23). The second
method is per-instance direct regression, using the construction in Eqs. (28)
and (29) based on the same classifier as above. The third method is per-
ensemble direct regression, based on minimizing the mean squared error loss in
Eq. (27).
#### III.3.1 Gaussian Mean Example
Our first regression study is based on the same one-dimensional Gaussian
distributions as Sec. III.1.1. The prior distribution for the Gaussian means
is taken to be uniform with $\mu\in[-0.5,0.5]$, while the variance is fixed at
$\sigma=1$. A training dataset is created from 100 examples each from 10,000
values of the Gaussian mean, for a total of one million training data points.
For the reference sample $p(x|\theta_{0})$ needed to build the single-event
parametrized classifier $f(x,\mu)$ in Eq. (23), we create a second dataset
with one million examples drawn from a standard normal distribution (i.e.
$\mu=0$). To implement the $p(\theta)$ term in the second line of Eq. (22),
each example $x_{i}$ from the reference dataset is assigned a random mean
value picked from the variable-mean dataset.
We train a parametrized neural network to distinguish the variable-mean
datasets from the reference dataset. This network takes as input two features:
one component of $\vec{x}$ and the random mean value $\mu$. The architecture
consists of three hidden layers with $(64,128,64)$ nodes per layer and ReLU
activation. The output layer has a single node and sigmoid activation. Binary
cross entropy is used to train the classifier and Eq. (4) is used to convert
it to the likelihood ratio form $f(x,\mu)$. The model is trained for 1000
epochs with early stopping and a batch size of 10% of the training statistics.
The same learned function $f(x,\mu)$ is used for both the maximum likelihood
analysis and per-instance direct regression. For the maximum-likelihood
analysis, the optimization in Eq. (24) is performed over a fixed grid with 20
evenly spaced values in $\mu\in[-0.5,0.5]$. For per-instance direct
regression, the function $f_{N}(\vec{x},\mu)$ in Eq. (29) is constructed by
taking a product of $f(x,\mu)$ outputs over all 100 examples in a given
ensemble data point $\vec{x}$. The integrals in Eqs. (28) and (29) are
approximated by evaluating $f_{N}(\vec{x},\mu)$ at 20 evenly spaced $\mu$
values between $-0.5$ and $0.5$ and then adding their values; this is possible
because the prior is uniform.
The per-ensemble direct regression approach uses a neural network $g_{N}$ that
takes as input 100 values (i.e. all of $\vec{x}$) and predicts a single mean
value. This network has the same architecture as $f(x,\mu)$, except it
directly takes as input $\vec{x}$ and has linear (instead of a sigmoid)
activation for the output layer, since the predicted mean can be both positive
or negative. It is trained to minimize the mean squared error loss in Eq.
(27).
Figure 7: Comparison of regression methods with the Gaussian example, with
the predicted value of the mean plotted against the true value of the mean.
The regression involves analyzing 100 instances drawn from the same Gaussian
distribution. Bands are the standard deviation of the predictions over 10,000
generated samples. The per-instance direct regression uses single-event
training, yet achieves comparable performance to per-ensemble direct
regression that processes 100 events simultaneously.
In Fig. 7, we see that all three approaches give nearly the same results in
terms of bias and variance. Strictly speaking, maximum likelihood and direct
regression are different tasks so their behavior could be different. For per-
instance and per-ensemble direct regression, they are constructed to yield the
same asymptotic behavior, but there will be differences due to, e.g., the
finite approximations to the integrals. Note that maximum likelihood and per-
instance direct regression only use neural networks that process per-instance
inputs; information about the rest of the events is used only through the
training procedure. Thus, we have empirical evidence that per-ensemble
regression can be accomplished via per-instance training.
#### III.3.2 Top Quark Mass Measurement
As a physics example of regression, we consider extracting the top quark mass.
Here, the top quark mass is the regression target and the setup is similar to
the Gaussian example above. We use the same event generation as Sec. III.1.3,
but now with top quark mass parameters sampled uniformly at random in
$m_{t}\in[170,180]~{}\text{GeV}$. As with the Gaussian example, a variable-
mass dataset is created. In this case, we have 100 events for each of 100,000
sampled top quark mass values. The reference sample uses a top quark mass of
172.5 GeV. Due to event selection effects, the actual number of events for
each top quark mass value varies from set-to-set, with a mean of about 40
events. Because this event selection has a slight top quark mass dependence,
this yields an effective non-uniform prior on $m_{t}$, which we account for
when assigning dummy mass values to the reference sample.
The parametrized classifier now takes five inputs: the four mass features from
Sec. III.1.3 ($m_{b_{1}\mu\nu}$, $m_{b_{2}\mu\nu}$, $m_{b_{1}j_{1}j_{2}}$, and
$m_{b_{2}j_{1}j_{2}}$) plus the top quark mass used for event generation. The
neural network has three hidden layers with 50 nodes per layer and ReLU
activation, and a single node output layer with sigmoid activation. We train
100 models and take the median as the classifier output, using Eq. (4) to
convert it to the likelihood ratio $f(x,m_{t})$. Each model is trained for
1000 epochs with early stopping with a patience of 20 epochs and a batch size
of 0.1%. To test the fidelity of the training, we extract the estimated
likelihood ratio of $m_{t}=175~{}\text{GeV}$ over $m_{t}=172.5~{}\text{GeV}$
and use it to reweight the $172.5~{}\text{GeV}$ sample. From Fig. 3a, we see
that we achieve good reweighting performance despite the relatively limited
training data.
(a) (b)
Figure 8: Regression in the top quark mass example. (a) An estimate of the log
likelihood for samples generated with 172.5 and 175 GeV top quark masses. The
vertical axis has been shifted such that the minimum value is at zero. Note
that the axis represents the average log likelihood which is a factor of
$N_{\text{events}}$ different from the total log likelihood. (b) Correlation
between the per-instance predicted mass and the per-ensemble predicted mass in
the context of direct regression. The per-ensemble mass values are put in bins
of 0.1 GeV width, and the bands represent the standard deviation of the per-
instance mass values in each bin.
The maximum likelihood analysis is performed by scanning the learned log
likelihood estimate over a fixed grid with 100 uniformly spaced steps in
$m_{t}\in[170,180]~{}\text{GeV}$. In Fig. 8a, we show this scan where the
target data comes from the high statistics 172.5 GeV and 175 GeV samples from
Sec. III.1.3. As desired, the minimum of the parabolic shapes are near the
input top quark masses.
For the per-instance direct regression, we follow the same strategy as in the
Gaussian case to convert $f(x,m_{t})$ into an estimate of
$\mathbb{E}[m_{t}|\vec{x}]$. The integrals in Eqs. (28) and (29) are
approximated by sampling 50 random top quark masses per set of 100 following
the probability density from the training dataset. Because 40 events are
insufficient to make a precision measurement of the top quark mass, we find a
noticeable bias between the estimated and true top mass values, which is
exacerbated by edge effects at the ends of the training range. For this
reason, we do not show a direct analog to Fig. 7, though this bias could be
overcome with much larger training datasets with many more than 100 examples
per mass value.
For the per-ensemble direct regression, we use the deep sets approach in Eq.
(31) to handle the permutation-invariance of the inputs. This approach is also
well suited to handle the large variation in the number of events in each set
due to the event selection effect. We again use PFNs for our practical
implementation. We use the default PFN hyperparameters from the
https://energyflow.network/ package, except we use linear activation in the
output layer and the mean squared error loss function. We found that it was
important for the model accuracy to standardize both the inputs and outputs of
the network. Note that this is a different per-ensemble direct regression
setup than used in Ref. [12], which found excellent performance using linear
regression on sorted inputs.
In Fig. 8b, we compare the output of per-ensemble direct regression to the
output of per-instance direct regression. We find a very strong correlation
between these two very different approaches to computing the same quantity
$\mathbb{E}[m_{t}|\vec{x}]$. The band in Fig. 8b is the standard deviation
over data sets with a true mass in the same one of the 100 bins that are
evenly spaced between 170 and 180 GeV. A key advantage of the per-instance
approach is that it does not need to be retrained if more events are acquired.
By contrast, the per-ensemble approach is only valid for event samples that
have the same sizes as were used during training.
### III.4 Beyond Regression Example
As remarked in Sec. II.5, the ideas discussed above apply to learning tasks
beyond just standard classification and regression. As one simple example to
illustrate this, we consider the Gaussian classification task from Sec.
III.1.1 and compute the mutual information between the Gaussian feature and
the label. This quantifies how much information is available in the feature
for classification and can be directly compared with other features and other
classification tasks.
Figure 9: Mutual information between a Gaussian feature and a label, where
the “ signal” ($x_{0}=\epsilon$) and “ background” ($x_{0}=-\epsilon$) have
opposite means. The estimate using the MLC loss approach shows good agreement
with the exact analytic expression.
For this illustration, $10^{5}$ events are generated each from two Gaussian
distributions with means $\pm|\epsilon|$ for fixed $\epsilon$. The mutual
information is estimated using a per-instance classifier as described in Sec.
II.5 and also computed analytically via Eq. (30). For the per-instance
classifier, we use a neural network that processes two inputs (label and
feature), has two hidden layers with ReLU activation, and has a single node
sigmoid output. The classification task is to distinguish the nominal dataset
from one where the labels are assigned uniformly at random to the features.
The value of the MLC loss yields an estimate of the mutual information.
The mutual information results are presented in Fig. 9, as a function of
$\epsilon$. As expected, the neural network strategy yields an excellent
approximation to the analytic calculation. Note that this strategy does
require any binning and naturally extends to high-dimensional data, since the
core component is a neural network classifier. We leave an investigation of
this approach in the particle physics context to future work.
## IV Conclusions
We have demonstrated a connection between classifiers trained on single events
and those that process multiple events at the same time. One can take a
generic single-event classifier and build an $N$-event classifier using simple
arithmetic operations. Such classifiers tend to out-perform generic $N$-event
classifiers, since we can enforce the IID assumptions into the learning task.
This performance gap can be mostly recovered by deploying a classifier that
respects the permutation invariance of the set of $N$ events. We used the deep
sets/PFN architecture [62, 63] for this purpose, but other set-based
architectures such as graph neural networks [64, 65] would also be
appropriate.
An amusing feature of the deep sets approach is that we can use it to reverse-
engineer a single-event classifier from a multi-event classifier by
restricting the latent space to be one-dimensional and fixing a static output
function. Even after enforcing these additional structures, though, we found
both theoretically and empirically that the loss function gradients are better
behaved for single-event classifiers than multi-event classifiers. Going
beyond classification, we explained how various regression tasks can be
phrased in terms of per-instance parametrized classification, yielding similar
performance to per-ensemble direct regression. We also mentioned how to
compute distances and divergences between probability densities without
requiring explicit density estimation. These results hold for any data sample
satisfying the IID property.
Ultimately, we did not find any formal or practical advantage for training a
multi-event classifier instead of a single-event classifier, as least for the
cases we studied. With a carefully selected multi-event architecture, one can
achieve similar performance to a scaled-up per-event classifier, but the
latter will typically train faster. For direct regression, the per-ensemble
strategy might be conceptually simpler than the per-instance method, though
the per-instance methods allow for a simpler treatment of variably-sized data
sets. Note that there may be situations where a simplifying assumption (e.g.
the linear regression model in Ref. [12]) could yield better per-ensemble
behavior than indicated by our case studies. At minimum, we hope this paper
has demystified aspects of per-ensemble learning and highlighted some
interesting features of the MLC loss function.
Going beyond the IID assumption, the duality between per-instance classifiers
and per-ensemble classifiers could have applications to problems with
approximate independence. For example, flavor tagging algorithms have
traditionally exploited the approximate independence of individual track
features within a jet [66, 67]. Similarly, emissions in the Lund jet plane
[68, 69] are approximately independent, with exact independence in the
strongly ordered limit of QCD. In both contexts, the instances are particles
(or particle-like features) and the ensemble is the jet. A potentially
powerful training procedure for these situations might be to first train a
per-particle classifier, then build a per-jet classifier using the
constructions described in this paper, and finally let the network train
further to learn interdependencies between the particles.
## Code and Data
The code for this paper can be found at
https://github.com/bnachman/EnsembleLearning. The physics datasets are hosted
on Zenodo at Ref. [70] for the top quark dataset and Ref. [71] for the BSM
dataset.
###### Acknowledgements.
We thank Anders Andreassen, Patrick Komiske, and Eric Metodiev for discussions
about the MLC loss. We thank Rikab Gambhir and Ian Convy for discussions about
mutual information. We thank Adi Suresh for discussions about the regression
task with the classifier loss. We thank Katherine Fraiser, Yue Lai, Duff
Neill, Bryan Ostdiek, Mateusz Ploskon, Felix Ringer, and Matthew Schwartz for
useful comments on our manuscript. BN is supported by the U.S. Department of
Energy (DOE), Office of Science under contract DE-AC02-05CH11231. JT is
supported by the National Science Foundation under Cooperative Agreement
PHY-2019786 (The NSF AI Institute for Artificial Intelligence and Fundamental
Interactions, http://iaifi.org/), and by the U.S. DOE Office of High Energy
Physics under grant number DE-SC0012567. BN would also like to thank NVIDIA
for providing Volta GPUs for neural network training.
## Appendix A Deriving Maximum Likelihood Classifier Loss
Beyond just the practical value of learning the likelihood ratio, the MLC loss
in Eq. (7) has a nice interpretation in terms of learning probability
distributions.
Consider trying to learn a function $f(x)$ that is a normalized probability
distribution, up to a Jacobian factor $j(x)$:
$\int dx\,j(x)f(x)=1.$ (33)
We are given samples from a probability distribution $q(x)$, and we want to
learn $f(x)$ such that
$f(x)\to\frac{q(x)}{j(x)}.$ (34)
In other words, we want to learn a function $f(x)$ that reproduces the sampled
distribution $q(x)$ after including the Jacobian factor. This problem was
studied in Ref. [34], albeit in a context where $f(x)$ had a restricted
functional form such that Eq. (33) was automatically enforced.
Figure 10: A demonstration of the MLC loss for learning the likelihood ratio
directly, using the Gaussian example from Fig. 1a. The linear (lin) and
exponential (exp) parametrizations perform similarly. Shown for comparison is
likelihood ratio computed using the binary cross entropy (BCE) loss that
requires the manipulation in Eq. (4).
One strategy to accomplish this is to minimize the cross entropy of $f(x)$
with respect to $q(x)$, since the smallest cross entropy is obtained when
$f(x)$ has the same information content as $q(x)$. The associated loss
functional is:
$L[f]=-\int dx\,q(x)\log f(x)-\lambda\left(1-\int dx\,j(x)f(x)\right),$ (35)
where the first term is the cross entropy and $\lambda$ is a Lagrange
multiplier to enforce the normalization condition in Eq. (33). Taking the
functional derivative of Eq. (35) with respect to $f(x)$ and setting it equal
to zero, we find the extremum condition:
$-\frac{q(x)}{f(x)}+\lambda\,j(x)=0.$ (36)
Multiplying both sides of this equation by $f(x)$ and integrating over $x$ to
set the Lagrange multiplier, we find that Eq. (36) is solved for
$\lambda=1,\qquad f(x)=\frac{q(x)}{j(x)},$ (37)
so $f(x)$ learns the $q(x)/j(x)$ ratio as desired.
In the special case that $j(x)$ is itself a normalized probability
distribution, we can substitute for the Lagrange multiplier and rewrite Eq.
(35) in the following form:
$L[f]=-\int dx\,\Big{(}q(x)\log f(x)+j(x)(1-f(x))\Big{)}.$ (38)
Identifying $q(x)=p(x|\theta_{A})$ and $j(x)=p(x|\theta_{B})$, this is
precisely the MLC loss in Eq. (7). Therefore, we have an intuitive
understanding of the MLC loss as trying to maximize the (log) likelihood of
$f(x)$ with respect to $p(x|\theta_{A})$, subject to the constraint that
$f(x)\,p(x|\theta_{B})$ is a proper probability distribution.
In Fig. 10, we plot the learned likelihood ratio between the two Gaussian
samples from Fig. 1a, comparing the performance of MLC against binary cross
entropy and the exact analytic expression. In all cases, a network is trained
with 100 epochs and early stopping with a patience of 10 epochs. We also
compare the MLC loss against the $C(f)=\exp f$ variant discussed in footnote
1. We see that both the linear (i.e. $C(f)=f$) and exponential
parametrizations perform similarly in the region with ample data. That said,
the exponential parametrization has a more robust extrapolation towards the
edges, yielding similar behavior to binary cross entropy. Note that the
exponential parametrization of the MLC loss was used in Ref. [32].
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|
Vol.0 (200x) No.0, 000–000
11institutetext: Department of Astronomy, Beijing Normal University, Beijing
100875, China;
22institutetext: Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100101, China;
33institutetext: School of Astronomy and Space Science, University of Chinese
Academy of Sciences, Beijing 100049, China;
44institutetext: School of Astronomy and Space Science, Nanjing University,
Nanjing 210023, Jiangsu, China
# Detecting and Monitoring Tidal Dissipation of Hot Jupiters in the Era of
SiTian
Fan Yang 112233 Wei<EMAIL_ADDRESS>2233 Xing<EMAIL_ADDRESS>11 Hui<EMAIL_ADDRESS>44 Ji-Lin Zhou 44 Su-Su Shan 2233 Jie Zheng
22 Wei-Kai Zong 11 Ming Yang 44 Yu Bai 22 Song Wang 22 Jia-Chen Zheng 11 Yu-Ru
Xu 11 Yu-Feng Li 11 You-Jun Lu 2233 Ji-Feng Liu 2233
###### Abstract
Transit Timing Variation (TTV) of hot Jupiters provides direct observational
evidence of planet tidal dissipation. Detecting tidal dissipation through TTV
needs high precision transit timings and long timing baselines. In this work,
we predict and discuss the potential scientific contribution of SiTian Survey
in detecting and analyzing exoplanet TTV. We develop a tidal dissipation
detection pipeline for SiTian Survey that aims at time-domain astronomy with
72 1-meter optical telescopes. The pipeline includes the modules of light
curve deblending, transit timing obtaining, and TTV modeling. SiTian is
capable to detect more than 25,000 exoplanets among which we expect $\sim$50
sources showing evidence of tidal dissipation. We present detection and
analysis of tidal dissipating targets, based on simulated SiTian light curves
of XO-3b and WASP-161b. The transit light curve modeling gives consistent
results within 1$\sigma$ to input values of simulated light curves. Also, the
parameter uncertainties predicted by Monte-Carlo Markov Chain are consistent
with the distribution obtained from simulating and modeling the light curve
1000 times. The timing precision of SiTian observations is $\sim$ 0.5 minutes
with one transit visit. We show that differences between TTV origins, e.g.,
tidal dissipation, apsidal precession, multiple planets, would be significant,
considering the timing precision and baseline. The detection rate of tidal
dissipating hot Jupiters would answer a crucial question of whether the planet
migrates at an early formation stage or random stages due to perturbations,
e.g., planet scattering, secular interaction. SiTian identified targets would
be constructive given that the sample would extend tenfold.
###### keywords:
planets and satellites: gaseous planets, planets and satellites: physical
evolution, planets and satellites: dynamical evolution and stability, planets
and satellites: detection
## 1 Introduction
Tidal migration is one of mechanisms that may explain why hot Jupiters occur
at such close orbits, though direct observational evidence of orbit decay is
only achieved for a single planet, namely WASP-12b, in the past decades (Patra
et al., 2017; Dawson & Johnson, 2018; Turner et al., 2021). The first data
release of Transiting Exoplanet Survey Satellite (TESS; Ricker et al., 2015)
in 2018, leads to the identification of four new candidates through TTV
detection (Dong et al., 2021; Davoudi et al., 2021; Shan et al., 2021; Yang &
Chary, 2021, submitted; Yang & Wei, 2021). TTV monitoring depends on the
timing precision and baseline length which are key technical specifications
for time-domain facilities, e.g., Zwicky Transient Facility (ZTF; Bellm et
al., 2019), Large Synoptic Survey Telescope (LSST; Ivezić et al., 2019),
Tsinghua University-Ma Huateng Telescopes for Survey (TMTS; Zhang et al.,
2020; Lin et al., 2021), Antarctic Survey Telescopes times 3 (AST3; Ma et al.,
2018; Zhang et al., 2019), and SiTian (Liu et al., 2021; Zhu et al., 2021).
The latter is aiming to tune down the false-positive probability of time-
domain signals, by implementing a global network of three-color photometric
monitoring of one-quarter of the sky at a cadence of 30min, down to a
detection limit of $V\sim 21$ mag (Liu et al., 2021).
It may suggest a TTV signal if any constant period ephemeris can not fit well
the observed timings. TTV could originate from multiple physical processes,
e.g., tidal dissipation, apsidal precession, R$\o$mer effect, and mass loss
(Ragozzine & Wolf, 2009; Valsecchi et al., 2015; Patra et al., 2017; Ou et
al., 2021). Another TTV generation process is the interaction between planets
in a multi-planet system which should cause oscillation in transit timing
residual in an observable timescale (Holman et al., 2010; Weiss et al., 2013).
However, it has been reported that a planet companion should not be close or
massive enough to induce observable TTVs for hot Jupiters (Huang et al.,
2016). Distinguishing among the above mentioned various physical origins
requires long-term high precision monitoring of transit timings continuously.
Combining observations from wide field transit surveys, e.g., Kepler, TESS
(Borucki et al., 2010; Ricker et al., 2015) and follow-up observations using
focused telescopes e.g., CHEOPS (Benz et al., 2021), has been proved to be
feasible in TTV detecting and related researches (Borsato et al., 2021).
Sariya et al. (2021) report TTVs of HAT-P-12b in the baseline of ten years,
using light curves from small ground-based telescopes. The transit timings of
WASP-32b are monitored and no significant TTV is found with the available data
(Sun et al., 2015). WASP-43b is reported as a candidate showing period decay
(Jiang et al., 2016) which arises wide attention and scientific discussions
(Davoudi et al., 2021; Garai et al., 2021). WASP-4b presents a significant TTV
and is furtherly explained by the R$\o$mer effect (Bouma et al., 2019, 2020).
HAT-P-25 is reported with no significant TTV which sets a limit on the
possible planet-planet interactions (Wang et al., 2018). The success in
observation in turn motivates the development of planet formation and
evolution theory (Dawson & Johnson, 2018; Wang et al., 2019; Liu & Ji, 2020).
The sample for tidal dissipation studies would be significantly enlarged in
the next decade, especially through the joint survey of ground-based
telescopes, e.g., SiTian and the next generation exoplanet space telescopes,
e.g., Habitable ExoPlanet Survey (HEPS; Yu et al., 2019, 2020), Earth 2.0
Transit Planet Survey, and shall be studied in details by ARIEL (Tinetti et
al., 2018), HABItable Terrestrial planetary ATmospheric Surveyor (HABITATS;
Wang et al., 2020).
In this work, we discuss the potential contribution of SiTian in detecting and
investigating tidal decaying hot Jupiters. We present general-purpose tools
for tidal dissipation detection, using the simulated light curves of SiTian.
The detection pipeline contains four major steps, i.e., light curve
generation, contamination light deblending, transit timing detection, and
transit timing modeling. The latter three modules are described given that
light curve generation would be integrated into SiTian science processing
pipeline (Liu et al., 2021). The paper is organized as follows. In Section 2,
we introduce the SiTian exoplanet observation strategy and the generation of
simulated SiTian light curves. In Section 3, we present the tidal dissipation
identification pipeline and apply it to the simulated light curves of
WASP-161b and XO-3b. In Section 4, we discuss the potential contribution of
SiTian in planet tidal dissipation. In Section 5, a summary is presented.
## 2 SiTian Exoplanet Observational Strategy and Simulated Transit Light
Curves Generation
SiTian is an integrated network of 1-meter telescopes, aiming at time-domain
astronomy (Liu et al., 2021). It shall produce a huge amount of the light
curves, that would contribute to time-domain researches of targets with
physical size ranging from galaxy clusters to planets (Liu et al., 2019, 2020;
Yang et al., 2022, 2020b, 2020a; Lennon et al., 2021; Wang et al., 2021; Yang
et al., 2021; Ngeow et al., 2021). Monitoring the same area of the sky
simultaneously in three bands ($u$, $g$, $i$), SiTian will deliver high
precision timing measurements, enabling detections of TTVs. The commissioning
of SiTian is expected to be before or around 2030, and the first three proto-
type equipment may start its operation in the middle of 2022.
### 2.1 SiTian Technical Specification for Exoplanet Research
SiTian will eventually accomplish with a worldwide network of 72 telescopes,
monitoring a sky area of $\sim$ 30,000 deg2 (Liu et al., 2021). The main
monitoring sky coverage is $\sim$ 10,000 deg2 which can be observed by
telescopes located in China. The typical exposure time is 1 minute, resulting
in a 5 $\sigma$ brightness limit of 21.1 mag in the $g$-band (Liu et al.,
2021). The photometric precision is expected to reach 1$\%$ for point sources
with $V=16$ mag (Liu et al., 2021), assuming the detector has a QE of 70$\%$,
read noise of 7 e-, an optical throughput of 70$\%$, filter transmission
fraction of 70$\%$, seeing of 2.5 arcsec, and night-sky brightness of 21.1 mag
arcsec-2. The applied CMOS detector GSENSE4040 has a readout noise of 3.7 e-
(Liu et al., 2021). A classic good observing site has an average night-sky
brightness around or better than 22.0 mag arcsec-2 and a median seeing size
better than 1 arcsec, for example, the Lenghu site(Deng et al., 2021), which
is the most promising major Chinese site for SiTian. Using these parameters
for calculation, the photometric precision is predicted as shown in Table 1.
The photometric precision is $\sim$ 700 parts per million (ppm) at $g$-band
and 1075 ppm at $i$-band for 12 mag targets (as shown in Figure 1). The
photometric uncertainties of sources brighter than 12 mag are dominated by
photon noise. In addition, the SiTian mission would occupy at least three
4-meter class spectroscopic telescopes for following-up observation.
Figure 1: The simulated and model-predicted light curves. The blue points are
the simulated SiTian light curve and the red points are binned data for
clarity. The black line shows the best-fit model light curves, while the
yellow region presents 1 $\sigma$ significance region. The fitting residuals
are shown below the light curves. The top panel refers to WASP-161b and the
bottom panel refers to XO-3b. Table 1: The expected limiting magnitude and
photometric precision of a 1 minute single exposure for one SiTian telescope,
based on the technical specifications from Liu et al. (2021).
| $u$ | $g$ | $i$
---|---|---|---
Limiting magnitudea | 20.2 | 21.1 | 20.4
Photometric Precision at Certain Magnitude
Magnitude | $\sigma(u)$ | $\sigma(g)$ | $\sigma(i)$
16 Mag | 0.85$\%$ | 0.48$\%$ | 0.76$\%$
12 Magb | 1258 ppm | 708 ppm | 1075 ppm
Note. (a) 5 $\sigma$ limit magnitude. (b) The photometric errors of brighter
sources or longer exposure time are dominated by Poisson error.
The data will be reduced by the SiTian collaboration in two modes, i.e.,
online and offline (Liu et al., 2021). The quick data reduction, light curve
retrieval, and classification will be performed in the online mode immediately
after observation with a typical delay of 1 minute. In addition, alerts for
transient events and other interesting targets will be triggered. Full data
products, e.g. time-domain images, light curves, and target catalogs would be
available and released to the public after reducing by the offline pipeline.
The capability of an observation to detect transit signal in a planetary
system can be represented by the signal to noise ratio (S/R) as:
${\rm
S/R}=\sqrt{\frac{t_{obs}}{P}}\frac{R_{p}^{2}}{R_{\ast}^{2}}/\sigma_{total},\\\
$ (1)
where $t_{obs}$ is the total integration time for the source, $P$ the orbit
period, $R_{p}$ and $R_{\ast}$ the radius of the planet and host star,
respectively. $\sigma_{total}$ is the total uncertainty in the duration of
transit event, taking into account Poisson photon noise, uncorrected stellar
variability, and equipment noise (Borucki et al., 2010). For example, a star
with $g$=12 observed by SiTian would have a $\sigma_{total}$ of 36 ppm when
the t${}_{obs}\sim$6.5 hr, a typical transit duration of hot Jupiters (Akeson
et al., 2013; Yang et al., 2022, 2021). A typical stellar variability level
for a low variability star like Sun is $\sim$ 10 ppm on the timescale of a
planet transit (Jenkins, 2002). This stellar variability is significantly
smaller than Poisson photon noise. The $\sigma_{total}$ is $\sim 239$ ppm for
a 16 mag star in the same stacked time.
The average observation time tobs can be estimated by multiplying the survey
time and the ratio of exposure time to scanning cadence. The total field of
view (FOV) of SiTian is 600 deg2. The cadence for the main sky coverage
(10,000 deg2) is 30 minutes. Assuming 8-hour observation time per night and
300 observing nights per year, the average annual tobs is 3.3 days. For
Neptune-sized planet (3$R_{\oplus}$) orbiting $g$=12 mag stars with orbital
periods of 30 days, the average S/N is 8.2 in one year observation, superior
to the empirical exoplanet detection threshold S/R = 7.1 (Borucki et al.,
2010; Fressin et al., 2013; Christiansen et al., 2015, and references
therein). For a $g=16$ Sun-like star, the S/R for an orbiting Jupiter-sized
planet is $\sim$ 16 in one-year observation. The S/R is the same for a Neptune
orbiting an M-type star. A more detailed discussion of planet detecting rate
shall be presented in the following work, describing the pipeline of exoplanet
detecting (Yang et al. in preparation). Here, we present a simple and general
calculation.
The number of planets that can be detected by a survey depends on three
factors, i.e., the planet occurrence rate, star counts, and solid angle. The
solid angle equals to $R_{p}$/$a$, where $R_{p}$ is planet radius and $a$ is
orbital semi-major axis (Borucki et al., 2010). The planet occurrence rate is
estimated to be $\sim$ 0.15 for SiTian’s capability, based on the knowledge
and lessons gained from the Kepler mission (Howard et al., 2012; Fressin et
al., 2013; Christiansen et al., 2015).
Taking the advantage of the high spatial resolution of $\sim$ 1 arcsec, and
considering the number density of bright stars, SiTian is planned to monitor
the galactic plane for exoplanet research. There are about 6.5 million stars
brighter than 12 mag in the galactic plane with galactic longitude $b$ between
0∘ and 20∘ (Robin et al., 2003). This leads to an expected planet detection of
25,000 if applying an average solid angle of 2.5$\%$ (Akeson et al., 2013).
The expected detected planet would thereby be 250 per year, assuming 1$\%$
SiTian time is allocated for bright sources. We note that the blending of
nearby sources is correctable (cf. details in Section 2.2) for bright target
transit observation (Yang et al., 2022, 2021, 2021). Thereby, monitoring
crowded stars in the Galactic plane is an efficient strategy for SiTian to
detect transit planets.
The expected number of stars with $g\sim$15-17 mag and with absolute $b$
between 20 and 40 ∘ is $\sim$ 1 million(Robin et al., 2003). The occurrence of
Jupiter-sized planets around Sun-like stars is about 0.06 (Fressin et al.,
2013), which implies the detection of 1500 Jupiter size exoplanets orbiting
Sun-like stars by SiTian. In addition, about 1400 Neptune size exoplanets
orbiting M-type star are expected to be detected, considering the small size
of M-type stars and their large abundance of $\sim$ 70$\%$ (Chabrier, 2003).
The detection of these 2900 exoplanets around faint stars are byproducts of
SiTian’s main scientific objectives. We emphasize that with only 1$\%$
observing time on bright stars, SiTian may discover additional 2500 exoplanets
in ten-year observation and up to 25000 planets with 10$\%$ time.
### 2.2 SiTian Data for Transit Timing Analysis and Simulated Transit Light
Curves
The proposed SiTian Mission would significantly extend the sample size of
exoplanets. The expected multiple timing measurements of these newly
discovered targets from SiTian may already allow a transit timing analysis.
Meanwhile, SiTian’s main survey or follow-up discretionary program of the
known planets with reported timing observation and TTV evidence will provide
opportunities for a full TTV study of these targets. The planet identification
package will detect exoplanet candidates and provide their preliminary orbital
and planetary parameters (Yang et al., in preparation). For the research of
TTV, it is necessary to build a specialized package to obtain more precise
transit parameters, especially the transit timings. The potential targets for
the pipeline include already-known hot Jupiters and new transiting hot
Jupiters detected from the SiTian survey. Below we describe our simulation of
SiTian’s capability on the study of TTV and tidal dissipation.
The deblending of the light curve is crucial for the transit planet survey
programs with poor spatial resolution and is particularly important for
studies relying on high precision transit depth measurements (Yang et al.,
2022, 2021). We note that the high spatial resolution and sampling are one of
the major advantages of SiTian.
We model and remove the blending light from the unsolved sources in the
vicinity of the target for the TESS image which has a pixel size of 21 arcsec
(Yang et al., 2022). Firstly, the correlation between the fraction of blended
light and the distance of the contaminating star to the target is built. The
step-by-step description of blending-distance correlation is available in our
previous work (Yang et al., 2022). With this relation, one can calculate the
contamination fraction of every source and their sum is the total blending
fraction. The calculation needs external information from the Gaia catalog
(Gaia Collaboration et al., 2018) on the flux and position of individual stars
inside the aperture of the target star.
The deblending method that we have developed delivers TESS brightness highly
consistent the Gaia brightness (Yang et al., 2022). The derived transit depths
are consistent within 1$\sigma$ with those given by the TESS Pre-Search Data
Conditioning (PDC; Smith et al., 2012) modules for the comparison samples
(Yang et al., 2022, 2021). The deblending package has been provided as a
general-purpose software 111https://github.com/sailoryf/TESS_Deblending/, and
will be used for the deblending and correction of the SiTian light curves.
For this purpose, we first simulate the “observed” SiTian $g$-band light
curves of two hot Jupiters with the host star of 12 mag. The sampling interval
is set as 1 minute with the input observational uncertainty of 708 ppm (as
shown in Table 1). The input parameters are set as the same as those of
WASP-161b and XO-3b, which shows TTV evidence (Yang & Chary, 2021, submitted;
Yang & Wei, 2021). For each source, we simulate 1000 transit light curves. The
light curves are all set with a total observation time of 8 hours. The
simulation light curves are de-trended by a polynomial fit to the several-hour
out-of-transit baseline data, which is shown to be a valid approach in (Yang
et al., 2022, 2021, 2021).
WASP-161b is a hot Jupiter orbiting an F6-type star every 5.41 days (Barkaoui
et al., 2019) and is reported to bear significant TTV related to tidal
dissipation (Yang & Chary, 2021, submitted). The giant planet has a mass
($M_{p}$) of 2.49$\pm$0.21$M_{J}$ and a radius ($R_{p}$) of $1.14\pm
0.06R_{J}$. The host star has a mass of 1.39$\pm$0.14$M_{\odot}$, a radius of
$1.71\pm 0.08R_{\odot}$. Applying a combined-fit to both the radial velocity
curve and the TESS 2-minute cadence light curve, Yang & Chary (2021,
submitted) has an orbital eccentricity of $e$ = 0.34$\pm$0.03, an inclination
of 89.58$\pm$0.28, and a semi-major axis in stellar radii ($a/R_{\ast}$) of
6.57$\pm$0.45 (Yang & Chary, 2021, submitted). In addition, a significant TTV
is reported by combining the TESS observation and the archival timing
measurements.
XO-3b is an another hot Jupiter with significant TTVs (Shan et al., 2021; Yang
& Wei, 2021). The planet-star system has an orbit period ($P_{orb}$) of 3.19
days, an $a$ of 4.95$\pm$0.18, an inclination of 84.20$\pm$0.54, and an $e$ of
0.27587${}^{+0.00071}_{-0.00067}$ (Winn et al., 2008; Bonomo et al., 2017;
Stassun et al., 2017). The planet has an $M_{p}$ of 11.70$\pm$0.42$M_{J}$ and
an $R_{p}$ of 1.217$\pm$0.073$R_{J}$. The host star has an $M_{\ast}$ of
1.213$\pm$0.066$M_{\odot}$ and an $R_{\ast}$ of 1.377$\pm$0.083$R_{\odot}$.
When simulating the observed light curve, and performing light curve model
fitting, we used the classic planet transit model assuming a Keplerian orbit
from Mandel & Agol (2002). The parameters include $R_{p}/R_{\ast}$,
$a/R_{\ast}$, transit mid-point $T_{C}$, inclination, the argument of
periapsis, the time of periapse passage, the longitude of the ascending node,
and quadratic limb darkening coefficients (a1 and a2). The limb darkening
coefficients are inserted from limb darkening models, depending on the stellar
type of the host star and observational band (Claret, 2000; Claret & Bloemen,
2011; Yang et al., 2021, 2021). We set the transit midpoint as the time zero
reference for comparison purposes. Examples of synthetic light curves for
WASP-161b and XO-3b are as shown in Figure 1. The blending light from sources
nearby is negligible given that no comparable sources within 10′′ are detected
and SiTian data has a pixel resolution of 1′′.
Figure 2: The posterior distribution obtained from the MCMC fitting to the
simulated light curves of WASP-161b (top), XO-3b (bottom). The dashed lines in
the diagonal histograms give the 1 $\sigma$ regions. The corner plot uses the
routine from Foreman-Mackey (2016).
## 3 The pipeline for Tidal Dissipation Candidates Identification
Our pipeline to search for the tidal dissipation candidates is firstly
developed for the TESS data, including four main modules: the light curve
generation and detrending, the light curve deblending from contamination by
additional stars inside the aperture, transit timing determinations, and
multi-epoch timing analysis. The light curve generation and detrending will be
performed by the SiTian data reduction pipeline (Liu et al., 2021) which
should be more specializing for SiTian data. We have described light curve
deblending in Section 2.2. Below, we describe timing obtaining, and timing
modeling modules in detail. Our pipeline is based on the usage of functions
from NumPy (Harris et al., 2020), Astropy (Astropy Collaboration et al.,
2013), EXOFAST (Eastman et al., 2013), and PYMC (Patil et al., 2010). NumPy
and Astropy provide basic statistics and astronomical calculations. EXOFAST
utilize the transit model from Mandel & Agol (2002). PYMC furnishes the Markov
Chain Monte-Carlo (MCMC) technique.
### 3.1 Modeling Transit Light Curve and Obtaining Transit Timing
The Markov Chain Monte-Carlo technique (Patil et al., 2010; Foreman-Mackey et
al., 2013), or alternatively, the multimodal nested sampling algorithms
(MULTINEST; Feroz et al., 2009) are widely used for the fitting to transit
light curves. These two methods have been compared in previous work and the
results obtained are consistent (Yang et al., 2021). In our pipeline, we
currently use the MCMC method and may add the MULTINEST method as an option in
future versions.
The fitting procedure applies the Mandel & Agol (2002) model assuming
Keplerian orbit, to generate theoretical transit light curves. All parameters
involved in light curve generation are set as free parameters. Uniform priors
ranging through the whole reasonable space are used for all these parameters,
except for the limb darkening coefficients, which use Gaussian priors with a
$\sigma$ of 0.05. The central priors of the limb darkening coefficients are
dependent on stellar spectral types and can be inferred via interpolation of
the classic limb darkening model (Claret, 2000; Claret & Bloemen, 2011; Yang
et al., 2021). The spectral types of the planet host stars can be found from
Gaia catalogs. The induced uncertainties in our work can be up to a few
percent in radius ratios and are not significant for the timing study (Yang et
al., 2021).
The MCMC algorithm omits some first steps as burn-in and applies a number of
iterations for probability statistics. More iteration should more likely give
an effective MCMC fitting. However, the numbers of the omitted steps and the
applicable steps linearly relate to the consuming time of MCMC fitting. In
practice, we show that 30,000 as burn-in numbers and 50,000 as applicable
steps are high enough for the TESS light curve fitting (Yang et al., 2021,
2021).
An investigation has been performed for the simulated SiTian light curves, to
determine the threshold of the chain steps for securing stable MCMC results.
For each of WASP-161b and XO-3b, 100 light curves are generated using their
reported parameters as input, respectively. We then fit the light curves and
monitor the stability of the MCMC results for the 100 run. The monitoring
shows that the MCMC fitting turns stable when 10,000 steps are used as burn-in
and 20,000 steps are used as analysis. Considering possible extra fluctuation
due to the diversity of planet parameters and light curve quality in real
applications, we apply 30,000 as burn-in and 50,000 for a statistic in the
pipeline. We note that a tenfold-length chain will be used for high priority
sources which already show tidal dissipation evidence, for example, WASP-161
b, and XO-3 b. The probability statistics of their parameters are shown in
Figure 2.
The MCMC fitting obtains a $T_{C}$ of 0.00027$\pm$0.00031 days (0.39$\pm$0.44
minutes) for WASP-161b and a $T_{C}$ of 0.00010$\pm$0.00022 days
(0.14$\pm$0.32 minutes) for XO-3b, consistent with the input $T_{C}$ of 0. We
apply further simulations to investigate whether the MCMC fitting is robust.
We generate another 1000 light curves for each planet and fit them
individually. The resulting distributions are well consistent within 1$\sigma$
to the MCMC derived distributions in fitting one particular light curve.
Taking $T_{C}$ as example, the distributions are 0.00000$\pm$0.00038 days
(0$\pm$0.54 minutes) for WASP-161b, and 0.00001$\pm$0.00027 days
(0.01$\pm$0.39 minutes) for XO-3b, as shown in Figure 3. Therefore, we
conclude that the MCMC fitting gives precise timing measurement and the timing
precision obtained for the simulated SiTian light curve is $\sim$ 0.5 minutes
for a single transit observation.
Figure 3: The distribution of the best-fit $T_{C}$ values of the 1000
simulated light curves. Upper panel for WASP-161 b and lower for XO-3 b. The
vertical dash lines show the input value of 0.
### 3.2 Transit Timing Modeling and Tidal Dissipation Identification
For hot Jupiters, observed TTV signal can be accounted for by various
mechanisms, e.g., tidal dissipation, apsidal precession, R$\o$mer effect,
multi-planet interaction, and mass loss. Exploring these physical processes is
among the main scopes of SiTian for the exoplanet research. The apsidal
precession scenario can be identified or rejected by studying the shape of the
light curve (Jordán & Bakos, 2008; Antoniciello et al., 2021; Yang & Wei,
2021). For the mass-loss case, the TTV signal is dependent on the mass rate
and amplitude, which can be well constrained by orbital parameters. The
R$\o$mer effect can be tackled by long-term radial velocity monitoring of the
host star, or high-resolution imaging search for stellar-mass companion(Siverd
et al., 2018; Yang et al., 2021).
In principle, a TTV signal induced by tidal dissipation should persist a
constant period derivative, which can be easily spotted out from transit
timing data using our developed pipeline (as shown in Figure 4).
Figure 4: The archival timings (black points) and the predicted new SiTian
timing observations (red diamonds) of WASP-161b (top) and XO-3b (bottom). The
black horizontal line gives the best fitted constant period model. The red
dashed curve presents tidal dissipation models as described by Yang & Chary
(2021, submitted); Yang & Wei (2021) with the yellow region showing 1 $\sigma$
region encompassing 68$\%$ archival timings. The simulated SiTian data are to
be taken in 2022, 2027, and 2032, which can effectively distinguish between
the linear and the quadratic models.
As predicted by the tidal dissipation model, timing variation follows a
quadratic function of the transit epochs (Patra et al., 2017; Yang & Chary,
2021, submitted):
$t_{\mathrm{tra}}(N)=t_{0}+NP+\frac{1}{2}\frac{dP}{dN}N^{2},\\\ $ (2)
where $t_{\mathrm{tra}}(N)$ is the timing of Nth transit, $t_{0}$ is the zero
point.
It was found in previous works that timings of WASP-161b and XO-3b
significantly favor a quadratic function model than linear models (Yang &
Chary, 2021, submitted; Yang & Wei, 2021). We present the predicted SiTian
timing measurements of WASP-161b and XO-3b in 2022, 2027, and 2031 (as shown
in Figure 4). Transit midpoints are generated according to Equation 2. We
recalculate BICs to estimate the evidence among different models. The
$\Delta$BIC of WASP-161b and XO-3b is 8.8, 383 obtained from archival timings,
favoring a quadratic model compared to a linear model. The $\Delta$BIC of
WASP-161b and XO-3b between linear and quadratic models would be larger than
5.5$\times$105 and 1.8$\times$104, favoring quadratic models.
Adding the future SiTian measurements is shown to be highly capable of
distinguishing the difference between the constant period model and the period
decaying model. In our tidal dissipation detection module, a target with the
BIC difference between the best fit quadratic model and the best fit linear
model larger than 5 will be flagged as a candidate planet with tidal
dissipation. These candidates shall be further studied and verified by follow-
up observations including more transit epochs measuring and radial velocity
monitoring.
We present timings and following-up strategy of KELT-19Ab as an example.
KELT-19Ab is reported to hold a maximum stellar acceleration of 4 m s-1 yr-1
caused by the binarity of its host star (Siverd et al., 2018). Applying the
relation between the stellar acceleration and transit period derivative from
Bouma et al. (2020), the transit period would present a derivative of 5.32 ms
yr-1. Shan et al. (2021) report a possible but not significant period
derivative of 112$\pm$94 ms yr-1.
Timings obtained from SiTian would potentially distinguish these different
scenarios. SiTian timing predictions are obtained following the same process
as described above with planet parameters taken from (Siverd et al., 2018).
The timing would present a difference at $\sim$ 1.5 minutes in 2022 between a
linear and a quadratic model obtained from archival timings available (as
shown in Figure 5). The difference would be significant when one has timings
in 2027 and 2031. The $\Delta$BIC obtained from archival timings between
linear and quadratic models is 2.1, slightly favoring a quadratic model. It
would derive a $\Delta$BIC of 1963 until 2031 if the observational timings are
as predicted by the quadratic model (seen in Figure 5). Moreover, the stellar
acceleration would be detectable with SiTian observation in 2031. It would
reveal a 1-minute difference between the constant period and the stellar
acceleration models. The period derivative due to acceleration is taken from
Shan et al. (2021).
Figure 5: Timings of KELT-19Ab with symbols the same to Figure 4. The green
line presents the period derivative caused by stellar acceleration calculated
by Shan et al. (2021).
Moreover, long-term transit observation can also help to discriminate between
possible TTV origins. Planets in a multi-planet system are usually not massive
and close enough to each other to cause TTVs with an amplitude larger than 10
minutes (Holman et al., 2010; Huang et al., 2016). Therefore, it may rule out
the possibility of the multi-planet origin, if either a monotonic TTV or a
large TTV amplitude is observed, as shown in Figure 4. For another important
arguing possible origin, apsidal precession, the transit duration variation
(TDV) should be more significant than TTV (Pál & Kocsis, 2008; Ragozzine &
Wolf, 2009). The TDV can be obtained by modeling the transit light curves as
well. No evidence of TDV in the archival data is found for WASP-161b and XO-3b
(Yang & Chary, 2021, submitted; Yang & Wei, 2021). We expect that the SiTian
project may provide additional observation on TDV.
## 4 discussion
The observed TTV signal and its interpretation are highly dependent on the
data quality of timing observation. Any unexpected factors in deriving the
transit timing parameters may change the period derivative (detailed example
in Yang & Chary, 2021, submitted; Shan et al., 2021), and thus leads to
different explanations. Therefore, long-baseline transit monitoring with high
precision is crucial for enhancing understanding of the tidal dissipation
process.
We have introduced the capacity of SiTian in modeling the tidal dissipation of
WASP-161b and XO-3b (as shown in Figure 4). For them, we have scheduled multi-
epoch transit observations from 2022 with the SiTian prototype telescopes. In
addition, continuous observation of at least one transit per five years is
proposed. This observation would improve the accuracy and robustness of the
period derivative, which is important for modeling the dissipation process for
these two benchmark planets.
The migration is believed to be the reason that giant gas planets can exist at
such en-close orbit (Dawson & Johnson, 2018). The migration is probably due to
one of the two scenarios: tidal migration that loses angular momentum by e.g.,
planet scattering, stellar companion scattering, and secular interaction (Wu &
Lithwick, 2011; Naoz et al., 2011); or migration due to disk friction in the
early stage of planet formation (Lin et al., 1996). A high eccentricity system
showing period decaying favors the former model (Yang & Chary, 2021,
submitted; Yang & Wei, 2021). Also, the tidal dissipation occurrence rate
should be higher for the former model. It is likely that both migration
mechanisms may contribute separately to different sources, or even act in the
same source while in different evolutionary stages.
Interestingly, the system parameters of the five reported tidal dissipation
exoplanets show obvious diversity (Patra et al., 2017; Dong et al., 2021;
Davoudi et al., 2021; Yang & Wei, 2021; Yang & Chary, 2021, submitted).
Extending the available sample crucial for further statistic investigations.
Using the methods introduced by Dong et al. (2018); Zhu & Dong (2021), we are
performing statistic analysis to find possible relationships between the
eccentricity, orbital radius or semi-major axis, obliquity, metallicity, and
planet-multiplicity with no conclusive results yet due to the limited sample
size.
A large number of candidate systems with evidence of tidal dissipation may be
discovered or identified by SiTian, which will be of great importance for
understanding tidal migration. The detection number depends on the real
occurrence rate of planets with tidal dissipation, which is one of the most
important questions to answer by future observations, e.g. SiTian. The current
fraction of such systems is 5 out of more than 4000. We note that planets
other than hot Jupiters bearing tidal dissipation should be rarer. We expect
to discover $\sim$ 50 new tidal dissipation star-planet systems taking into
account at least one order of magnitude larger sample size. Moreover, other
targets showing TTV signals but caused by other mechanisms such as planet-
planet interactions, apsidal precession, will be identified by our pipeline as
byproducts.
For targets with high scientific interests, following-up observation would be
promptly scheduled with the SiTian 4-meter class spectroscopic following-up
telescopes (Liu et al., 2021). The following-up observation would give us
information of e.g., host star properties, binarity, and planet multiplicity.
In addition, future advanced facilities will provide important opportunities
for deep investigations. For example, the HABItable Terrestrial planetary
ATmospheric Surveyor (HABITATS), which is a proposed mission for a 4-6 meter
space telescope aiming at modeling the atmospheric features of exoplanets down
to Earth-like planets(Wang et al., 2020). The high-precision transmission
spectra from HABITATS will help us understand the atmosphere properties of
tidal dissipation targets, including the composition, thermal structures,
inflation, and escape. This information shall yield a comprehensive
understanding of the undergoing physical processes in the tidal dissipating
sources.
## 5 Summary
We describe a pipeline for tidal dissipation detection as software preparation
for the ongoing SiTian survey which is expected to have the first light of the
three prototype telescopes in the middle of 2022. The SiTian survey will have
72 telescopes in total with full installation till 2030 (Liu et al., 2021). We
have estimated SiTian’s capability for detecting exoplanets, based on its
technical parameters. SiTian is expected to discovery 5,000 to 25,000 new
exoplanets. Assuming a similar tidal dissipation occurrence, there would be
$\sim$ 50 hot Jupiters showing tidal dissipation evidence.
The pipeline for tidal dissipation detection has modules of light curve
deblending, transit light curve modeling, and timing modeling for tidal
dissipation detection. The light curve deblending is based on our developed
algorithm for TESS light curve deblending (Yang et al., 2022). For each
target, a relationship between the contamination fraction and the distance to
the target will be built, based on the brightness and position measurements
for individual stars from the GAIA catalog (Gaia Collaboration et al., 2018),
and then corrected to obtain “uncontaminated” light curves for SiTian.
We set the reported period decaying candidates of WASP-161b and XO-3b as
examples to describe our pipeline and the contribution of SiTian data. We
simulate SiTian light curves for WASP-161b and XO-3b with a photometric
precision of $\sim$ 700 ppm in 1-minute cadence, estimated from SiTian’s
technical parameters. The MCMC technique is applied when modeling transit
light curves. We show that our transit timing measurement has a precision of
0.5 minutes for a single transit observation.
We conclude that the inclusion of SiTian data shall provide key evidence to
discriminate between various TTV origin models. For WASP-161b and XO-3b, a
tidal dissipation origin seems to be the most likely explanation. The proposed
future SiTian measurements shall significantly improve the model confidence
and robustness.
We thank SiTian Collaboration for the support in preparing this work. The work
made use of the NASA Exoplanet Archive
222https://exoplanetarchive.ipac.caltech.edu/index.html (Akeson et al., 2013).
We would like to thank Jin-Ping Zhu for the careful reading and constructive
suggestions. We thank Bo Zhang for the helpful discussion. Fan Yang and Su-Su
Shan are supported by funding from the Cultivation Project for LAMOST
Scientific Payoff and Research Achievement of CAMS-CAS. Xing Wei acknowledges
National Natural Science Foundation of China (NSFC; No.11872246, 12041301),
and the Beijing Natural Science Foundation (No. 1202015). Wei Wang is
supported by the National Natural Science Foundation of China (NSFC) grants
No. 11988101, 42075123, the National Key RD Program of China No.
2019YFA0405102, and the science research grants from the China Manned Space
Project with NO. CMS-CSST-2021-B12.
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# Early Planet Formation in Embedded Disks (eDisk) VI:
Kinematic Structures around the Very Low Mass Protostar IRAS 16253-2429
Yusuke Aso Korea Astronomy and Space Science Institute, 776 Daedeok-daero,
Yuseong-gu, Daejeon 34055, Republic of Korea Woojin Kwon Department of Earth
Science Education, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul
08826, Republic of Korea SNU Astronomy Research Center, Seoul National
University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, Republic of Korea Nagayoshi
Ohashi Academia Sinica Institute of Astronomy and Astrophysics, 11F of
Astronomy-Mathematics Building, AS/NTU, No.1, Sec. 4, Roosevelt Rd, Taipei
10617, Taiwan Jes K. Jørgensen Niels Bohr Institute, University of
Copenhagen, Øster Voldgade 5–7, DK 1350 Copenhagen K., Denmark John J. Tobin
National Radio Astronomy Observatory, 520 Edgemont Rd., Charlottesville, VA
22903 USA Yuri Aikawa Department of Astronomy, Graduate School of Science,
The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Itziar
de Gregorio-Monsalvo European Southern Observatory, Alonso de Cordova 3107,
Casilla 19, Vitacura, Santiago, Chile Ilseung Han Division of Astronomy and
Space Science, University of Science and Technology, 217 Gajeong-ro, Yuseong-
gu, Daejeon 34113, Republic of Korea Korea Astronomy and Space Science
Institute, 776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea
Miyu Kido Department of Physics and Astronomy, Graduate School of Science and
Engineering, Kagoshima University, 1-21-35 Korimoto, Kagoshima, Kagoshima
890-0065, Japan Patrick M. Koch Academia Sinica Institute of Astronomy and
Astrophysics, 11F of Astronomy-Mathematics Building, AS/NTU, No.1, Sec. 4,
Roosevelt Rd, Taipei 10617, Taiwan Shih-Ping Lai Institute of Astronomy,
National Tsing Hua University, No. 101, Section 2, Kuang-Fu Road, Hsinchu
30013, Taiwan Center for Informatics and Computation in Astronomy, National
Tsing Hua University, No. 101, Section 2, Kuang-Fu Road, Hsinchu 30013, Taiwan
Department of Physics, National Tsing Hua University, No. 101, Section 2,
Kuang-Fu Road, Hsinchu 30013, Taiwan Academia Sinica Institute of Astronomy
and Astrophysics, 11F of Astronomy-Mathematics Building, AS/NTU, No.1, Sec. 4,
Roosevelt Rd, Taipei 10617, Taiwan Chang Won Lee Division of Astronomy and
Space Science, University of Science and Technology, 217 Gajeong-ro, Yuseong-
gu, Daejeon 34113, Republic of Korea Korea Astronomy and Space Science
Institute, 776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea
Jeong-Eun Lee Department of Physics and Astronomy, Seoul National University,
1 Gwanak-ro, Gwanak-gu, Seoul 08826, Republic of Korea Zhi-Yun Li University
of Virginia, 530 McCormick Rd., Charlottesville, Virginia 22904, USA Zhe-Yu
Daniel Lin University of Virginia, 530 McCormick Rd., Charlottesville,
Virginia 22904, USA Leslie W. Looney Department of Astronomy, University of
Illinois, 1002 West Green St, Urbana, IL 61801, USA Suchitra Narayanan
Institute for Astronomy, University of Hawai‘i at Mānoa, 2680 Woodlawn Dr.,
Honolulu, HI 96822, USA Nguyen Thi Phuong Korea Astronomy and Space Science
Institute, 776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea
Department of Astrophysics, Vietnam National Space Center, Vietnam Academy of
Science and Technology, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam Jinshi
Sai (Insa Choi) Academia Sinica Institute of Astronomy and Astrophysics, 11F
of Astronomy-Mathematics Building, AS/NTU, No.1, Sec. 4, Roosevelt Rd, Taipei
10617, Taiwan Kazuya Saigo Department of Physics and Astronomy, Graduate
School of Science and Engineering, Kagoshima University, 1-21-35 Korimoto,
Kagoshima, Kagoshima 890-0065, Japan Alejandro Santamaría-Miranda European
Southern Observatory, Alonso de Cordova 3107, Casilla 19, Vitacura, Santiago,
Chile Rajeeb Sharma Niels Bohr Institute, University of Copenhagen, Øster
Voldgade 5–7, DK 1350 Copenhagen K., Denmark Shigehisa Takakuwa Department of
Physics and Astronomy, Graduate School of Science and Engineering, Kagoshima
University, 1-21-35 Korimoto, Kagoshima, Kagoshima 890-0065, Japan Academia
Sinica Institute of Astronomy and Astrophysics, 11F of Astronomy-Mathematics
Building, AS/NTU, No.1, Sec. 4, Roosevelt Rd, Taipei 10617, Taiwan Travis J.
Thieme Institute of Astronomy, National Tsing Hua University, No. 101, Section
2, Kuang-Fu Road, Hsinchu 30013, Taiwan Center for Informatics and
Computation in Astronomy, National Tsing Hua University, No. 101, Section 2,
Kuang-Fu Road, Hsinchu 30013, Taiwan Department of Physics, National Tsing
Hua University, No. 101, Section 2, Kuang-Fu Road, Hsinchu 30013, Taiwan
Kengo Tomida Astronomical Institute, Graduate School of Science, Tohoku
University, Sendai 980-8578, Japan Jonathan P. Williams Institute for
Astronomy, University of Hawai‘i at Mānoa, 2680 Woodlawn Dr., Honolulu, HI
96822, USA Hsi-Wei Yen Academia Sinica Institute of Astronomy and
Astrophysics, 11F of Astronomy-Mathematics Building, AS/NTU, No.1, Sec. 4,
Roosevelt Rd, Taipei 10617, Taiwan
###### Abstract
Precise estimates of protostellar masses are crucial to characterize the
formation of stars of low masses down to brown-dwarfs (BDs;
$M_{*}<0.08~{}M_{\sun}$). The most accurate estimation of protostellar mass
uses the Keplerian rotation in the circumstellar disk around the protostar. To
apply the Keplerian rotation method to a protostar at the low-mass end, we
have observed the Class 0 protostar IRAS 16253-2429 using the Atacama Large
Millimeter/submillimeter Array (ALMA) in the 1.3 mm continuum at an angular
resolution of $0\farcs 07$ (10 au), and in the 12CO, C18O, 13CO ($J=2-1$), and
SO ($J_{N}=6_{5}-5_{4}$) molecular lines, as part of the ALMA Large Program
Early Planet Formation in Embedded Disks (eDisk). The continuum emission
traces a non-axisymmetric, disk-like structure perpendicular to the associated
12CO outflow. The position-velocity (PV) diagrams in the C18O and 13CO lines
can be interpreted as infalling and rotating motions. In contrast, the PV
diagram along the major axis of the disk-like structure in the 12CO line
allows us to identify Keplerian rotation. The central stellar mass and the
disk radius are estimated to be $\sim$0.12-0.17 $M_{\sun}$ and $\sim$13-19 au,
respectively. The SO line suggests the existence of an accretion shock at a
ring ($r\sim 28$ au) surrounding the disk and a streamer from the eastern side
of the envelope. IRAS 16253-2429 is not a proto-BD but has a central stellar
mass close to the BD mass regime, and our results provide a typical picture of
such very low-mass protostars.
Circumstellar disks (235) — Protostars (1302) — Low mass stars (2050)
††facilities: ALMA††software: astropy (Astropy Collaboration et al., 2013,
2018), CASA (McMullin et al., 2007), eDisk script (Tobin et al., 2023),
ptemcee (Foreman-Mackey et al., 2013; Vousden et al., 2016), SLAM (Aso & Sai,
2023)
## 1 Introduction
The low-mass end of star formation is connected to brown dwarf (BD) formation.
Investigating such low-mass regimes is crucial to comprehensively
understanding star formation. BDs are characterized by their masses that are
not enough to fuse hydrogen ($M_{*}<0.08~{}M_{\sun}$) and are as numerous as
hydrogen-burning stars (Chabrier, 2002). The formation process of BDs can be
similar to or different from those of low-mass stars (Padoan & Nordlund,
2004). Theoretical studies have suggested various mechanisms for BD formation,
such as turbulent fragmentation of a molecular cloud (e.g., Bate, 2012), disk
fragmentation (e.g., Stamatellos & Whitworth, 2009), ejection from multiple
young stellar systems (e.g., Basu et al., 2012), photo-erosion of a prestellar
core by OB stars (e.g., Whitworth & Zinnecker, 2004), and eroding outflows
(e.g., Machida et al., 2009). A point of view for testing these scenarios is
whether or not physical quantities in star formation and BD formation follow
the same scaling laws. Kim et al. (2019) reported that their sample of 15
proto-BD candidates have different scaling laws, than the laws among $\sim 60$
low-mass protostars, between the outflow force versus the luminosity and
between the outflow force and the envelope mass. Protostars around the BD
threshold i.e., proto-BDs or very low-mass protostars, are then important to
determine down to which mass such a scaling law of protostars holds. Recent
observational studies have aimed to identify and characterize proto-BDs, as
well as pre-BDs (de Gregorio-Monsalvo et al., 2016; Huélamo et al., 2017;
Santamaría-Miranda et al., 2021). To study star formation in this very low
mass regime, it is, therefore, necessary to establish a method for precisely
estimating the central mass of each protostar down to the mass regime around
$M_{*}\sim 0.1~{}M_{\sun}$.
The most direct method for estimating a central protostellar mass is to
identify the Keplerian rotation in a circumstellar disk, as demonstrated in
previous observations toward young stellar objects in the typical low mass
regime ($M_{*}\gtrsim 0.2~{}M_{\sun}$; Yen et al., 2017). In contrast,
previous studies of two representative proto-BD candidates, IC348-SMM2E and
L328-IRS, estimated masses in indirect methods, which do not verify that the
rotational velocity has the radial profile of the Keplerian rotation but
merely assume that the observed gas motion is the Keplerian rotation or use
other kinematics. The central mass of IC348-SMM2E is estimated to be $\sim
0.002-0.024~{}M_{\sun}$ from its luminosity, mass accretion rate, and
efficiencies of mass accretion (Palau et al., 2014). This range includes a
dynamical mass of $\sim 16~{}M_{\rm Jup}$ estimated on the assumption that the
velocity gradient in the C18O $J=2-1$ line is due to the Keplerian rotation,
based on Submillimeter Array (SMA) observations (Palau et al., 2014). The
central mass of L328-IRS is estimated to be $\sim 0.012-0.023~{}M_{\sun}$ from
its mass accretion rate converted from its outflow force (Lee et al., 2018).
The outflow force is estimated using ALMA observations in the 12CO $J=2-1$
line and converted to the mass accretion rate, assuming an entrainment
efficiency of 0.25, a ratio of mass loss and accretion rates of 0.1, and a
wind velocity of $15~{}{\rm km~{}s^{-1}}$. Meanwhile, Lee et al. (2018)
explained the C18O and 13CO $J=2-1$ emission in L328-IRS by the Keplerian
rotation and suggested a central mass, $\sim 0.27$-$0.30~{}M_{\sun}$,
significantly larger than the BD mass regime. In addition to the disregard of
Keplerian-disk identification, numerical simulations predict that the disk
mass can be comparable to the central stellar mass during the early phase with
$M_{*}\lesssim 0.1~{}M_{\sun}$ (Machida et al., 2010, 2014). Such a massive
disk may cause the rotational velocity to deviate from the Keplerian rotation
determined only by $M_{*}$, preventing us from directly estimating $M_{*}$ in
observations. It is thus crucial to verify whether the direct method with the
Keplerian disk identification can be applied for protostars down to the
$M_{*}\sim 0.1$-$M_{\sun}$ regime as is for the typical low mass protostars.
### 1.1 Target IRAS 16253-2429
The Class 0 protostar IRAS 16253-2429 (hereafter I16253) in the Ophiuchus
star-forming region is a good target for the verification in that mass regime.
I16253 has a bolometric luminosity of $L_{\rm bol}=0.16~{}L_{\sun}$ and a
bolometric temperature of $T_{\rm bol}=42$ K (Ohashi et al., 2023). Its
internal luminosity is estimated to be $L_{\rm int}\sim 0.08~{}L_{\sun}$ from
the infrared luminosity of $L_{\rm IR}=0.046~{}L_{\sun}$ measured in
$1.25-70~{}\micron$ and an empirical ratio of $L_{\rm int}/L_{\rm IR}\sim 1.7$
(Dunham et al., 2008). This protostar is thus classified as a very low
luminosity object (VeLLO), which suggests that this protostar may eventually
evolve into a very-low mass star or a BD. We observed this protostar as a part
of the ALMA large program “Early Planet Formation in Embedded Disks” (eDisk)
(see the overview paper by Ohashi et al., 2023). The main goal of the program
is to reveal signs of planet formation in disks in the course of protostellar
evolution. In addition to this main goal, I16253 was observed to investigate
the protostellar evolution and the disk formation down to the BD mass regime.
The central stellar mass of this target in previous works has been estimated
to be a wide range of $\sim 0.02$ to $\sim 0.12~{}M_{\sun}$, depending on
methods. Tobin et al. (2012), who observed I16253 in the N2H+
$J,F_{1},F=1,2,2-0,1,1$ line using the Combined Array for Research in
Millimeter-wave Astronomy (CARMA) at a resolution of $\sim 9\arcsec$,
estimated the stellar mass to be $0.1~{}M_{\sun}$ based on comparisons of a
position-velocity (PV) diagram across the associated outflow on a 6000-au
scale between the observations and a model of the rotating, collapsing
envelope (UCM model; Ulrich, 1976; Cassen & Moosman, 1981). Yen et al. (2017)
estimated the stellar mass to be $0.02^{+0.02}_{-0.01}~{}M_{\sun}$ based on
C18O $J=2-1$ observations of the protostar using ALMA at an angular resolution
of $\sim 1\arcsec$; they reproduced the observed PV diagram along the major
and minor axes of the C18O envelope with a model envelope having a rotation
conserving its specific angular momentum and a radial free-fall motion. Hsieh
et al. (2019) used a similar model of the envelope and estimated the stellar
mass to be $\sim 0.028~{}M_{\sun}$ by reproducing PV diagrams of 12CO and C18O
$J=2-1$ obtained with ALMA at an angular resolution of $0\farcs 1-0\farcs 4$.
Additionally, Hsieh et al. (2019) also suggested the stellar mass of
$0.12~{}M_{\sun}$ with the assumption that the 12CO emission arises from a
Keplerian disk; they found offsets of the 12CO emission, $\pm 7.8$ au away
from the center, at two velocity channels of $V_{\rm LSR}-V_{\rm sys}=\pm
3.4~{}{\rm km~{}s^{-1}}$ by 2D Gaussian fitting. This mass derived on the
assumption of Keplerian rotation can be different from the one derived on the
assumption of free-fall motion, if the infall velocity is slower than the
free-fall velocity, as reported in other protostars (Ohashi et al., 2014; Aso
et al., 2015, 2017; Sai et al., 2022). These previous works show controversy
about the central stellar mass of I16253 around the BD mass regime (i.e.,
$M_{*}<0.08~{}M_{\sun}$).
Previous observations using the Walraven Photometer estimated the distance of
the Ophiuchus star-forming region to be $125\pm 25$ pc (de Geus et al., 1989),
while the same observation estimated the distance at a location of
$(l,b)=(353.161\arcdeg,15.936\arcdeg)$ closest to I16253
$(353.2\arcdeg,16.5\arcdeg)$ to be 140 pc. This distance is consistent with an
estimate using Hipparcos ($120-160$ pc; Knude & Hog, 1998) and a new estimate
by Gaia DR2, $139^{+9}_{-10}$ pc, at a location of
$(353.2\arcdeg,16.6\arcdeg)$ closest to I16253 (Zucker et al., 2020). In
addition to the I16253 distance, the average distance of the whole Ophiuchus
region was updated by Gaia DR2 to $144\pm 9$ pc (Zucker et al., 2019). We
adopt the Gaia DR2 distance of 139 pc as the distance of I16253 in this paper.
This work aims to estimate the central mass of the Class 0 protostar I16253
more accurately and precisely than the previous works and reveal the kinematic
structures around the protostar. Our observations have $>2.5$ times better
sensitivity in the CO isotopologue lines than that of previous observations in
the same lines at similar angular and velocity resolutions to those of our
observations. We aim to reveal kinematic structures, such as a rotating disk,
an infalling envelope, and an outflow, using the molecular line emission
detected at the higher signal to noise ratio. The rest of the paper has the
following structure. Section 2 describes key points of the settings and data
processing of our I16253 observations. Section 3 shows the observed results in
the 1.3-mm dust continuum and the 12CO, 13CO, C18O, and SO lines. We analyze
the asymmetry in the continuum image and radial profiles of the rotational
velocity in Section 4. The rotational velocity is found to follow the
Keplerian rotation, which allows us to directly estimate the central mass of
I16253. We discuss kinetic structures around I16253, showing a schematic
picture (Figure 18) of those structures, in Section 5. A summary of our
results and interpretation is in Section 6.
## 2 Observations
The details of the observing strategy, spectral setups, and data reduction
process are presented in Ohashi et al. (2023)111The scripts used for
reduction, including the self-calibration, can be found at
https://github.com/jjtobin/edisk (Tobin et al., 2023).. Here we briefly
describe the key points specific to I16253 and summarize the observational
parameters in Table 1. We observed I16253 using ALMA in Cycle 8 with the
antenna configuration of C-8 on 2021 October 5, 26, 27, and 28
(2019.1.00261.L). The total observing time with C-8 is $\sim 310$ min (5.2
hr), while the on-source observing time is $\sim 60$ min. The number of
antennas with C-8 was 46, and the projected baseline length ranges from 52 to
10540 m. Any Gaussian component with FWHM$\gtrsim 2.8\arcsec$ ($\sim 400$ au)
was resolved out by $\gtrsim 63\%$ with this shortest baseline length (Wilner
& Welch, 1994). The phase center is $(\alpha_{\rm ICRS},\delta_{\rm
ICRS})=(16^{\rm h}28^{\rm m}$21$\fs$6, $-24^{\circ}36\arcmin 23\farcs 4)$.
To recover the more extended structure, we also observed I16253 using ALMA in
Cycle 8 with the antenna configuration of C-5 on 2022 June 14 and 15
(2019.A.00034.S). The total observing time with C-5 is $\sim 170$ min (2.8
hr), while the on-source observing time is $\sim 30$ min. The number of
antennas with C-5 was 43, and the projected baseline length ranges from 14 to
1290 m. The resolving out scale is $10\arcsec$ ($\sim 1400$ au) with this
shortest baseline length. The phase center is the same as with C-8.
The ALMA observations were set up to cover spectral windows including CO
isotopologues ($J=2-1$), SO ($J_{N}=6_{5}-5_{4}$), and other molecular lines
at Band 6. The spectral window for the 12CO ($J=2-1$) line has 3840 channels
covering a 940 MHz bandwidth at an original frequency resolution of 224 kHz.
Spectral windows for the 13CO, C18O ($J=2-1$), SO ($J_{N}=6_{5}-5_{4}$), and
H2CO ($3_{21}-2_{20}$) lines have 960 channels covering a 59 MHz bandwidth at
an original frequency resolution of 61 kHz. Spectral windows for the other
lines have 3840 channels covering a 1.9 GHz bandwidth at an original frequency
resolution of 488 kHz. Channels were binned to produce the velocity
resolutions of 0.32, 0.17, 0.17, $0.17~{}{\rm km~{}s^{-1}}$, respectively, to
make maps in the 13CO, C18O, SO, and H2CO lines. The velocity resolution for
other lines is $1.34~{}{\rm km~{}s^{-1}}$ (Appendix A). Continuum maps were
made using line-free channels of the spectral windows including two wide
spectral windows with a $\sim 1.8$ GHz bandwidth centering at $\sim 220$ to
$\sim 230$ GHz. The absolute flux density accuracy of ALMA is $\sim 10\%$ in
this frequency band.
All the imaging procedure was carried out with Common Astronomical Software
Applications (CASA) (McMullin et al., 2007) version 6.2.1. The visibilities
were Fourier transformed and cleaned with Briggs weighting and a robust
parameter of $-2.0$, 0.0, 0.5, or 2.0, using the CASA task tclean at a pixel
size of $0\farcs 02$. Continuum images adopt robust$=-2.0$, 0.0, and 2.0 to
show the most compact, intermediate, and most extended components,
respectively. The line images adopt robust$=0.5$ and 2.0 to show compact
components and extended components, respectively. The line images do not have
signal-to-noise (S/N) ratios high enough for robust$=-2$. The continuum images
with robust$=-2$ and 2 are produced with tapering parameters of 3 and 1
M$\lambda$ ($\sim 0\farcs 06$ and $\sim 0\farcs 18$), respectively. The
tapering parameter of 3 M$\lambda$ was selected to focus more on extended
components with robust$=2$, while the tapering parameter of 1 M$\lambda$ was
selected to increase the S/N ratio of the image with robust$=-2$. All the line
images are produced with a tapering parameter of 2 M$\lambda$ ($\sim 0\farcs
09$) to increase the S/N ratio. The resultant angular resolution is $0\farcs
04-0\farcs 29$ for the continuum emission and $0\farcs 17-0\farcs 38$ for the
line emission. We also performed self-calibration for the continuum data using
tasks in CASA (tclean, gaincal, and applycal). While all the maps are primary-
beam corrected in this paper with the primary beam size of $26\arcsec$, the
root-mean-square noise levels of the line maps were measured in emission-free
channels/areas before the primary-beam correction, which indicates the
sensitivity achieved toward the phase center.
Table 1: Summary of the parameters of our ALMA observations toward the Class 0
protostar I16253.
Date | 2021 October 5, 26, 27, & 28 | 2022 June 14 & 15
---|---|---
Projected baseline range | 52–10540 m | 14–1290 m
Maximum Recoverable Scale | ${0\farcs 62}$ | ${2\farcs 9}$
Bandpass/flux calibrator | J1517-2422 | J1517-2422
Check source | J1650-2943 | J1650-2943
Phase calibrator | J1633-2557 | J1700-2610
| Continuum | 12CO $J=2-1$ | 13CO $J=2-1$ | C18O $J=2-1$ | SO $J_{N}=6_{5}-5_{4}$
Frequency (GHz) | 225 | 230.5380000 | 220.3986842 | 219.5603541 | 219.9494420
Freq./vel. width | $\sim 2$ GHz | $0.32~{}{\rm km~{}s^{-1}}$ | $0.17~{}{\rm km~{}s^{-1}}$ | $0.17~{}{\rm km~{}s^{-1}}$ | $0.17~{}{\rm km~{}s^{-1}}$
Beam (P.A.) | $0\farcs 29\times 0\farcs 24\ (86\arcdeg)^{a}$ | $0\farcs 35\times 0\farcs 25\ (76\arcdeg)^{a}$ | $0\farcs 37\times 0\farcs 26\ (77\arcdeg)^{a}$ | $0\farcs 38\times 0\farcs 27\ (78\arcdeg)^{a}$ | $0\farcs 37\times 0\farcs 26\ (78\arcdeg)^{a}$
noise rms | $23~{}{\rm\mu Jy~{}beam^{-1}}^{a}$ | $1.4~{}{\rm mJy~{}beam^{-1}}^{a}$ | $2.5~{}{\rm mJy~{}beam^{-1}}^{a}$ | $1.9~{}{\rm mJy~{}beam^{-1}}^{a}$ | $2.3~{}{\rm mJy~{}beam^{-1}}^{a}$
Beam (P.A.) | | $0\farcs 073\times 0\farcs 048\ (77\arcdeg)^{c}$
---
$0\farcs 043\times 0\farcs 034\ (73\arcdeg)^{d}$
$0\farcs 17\times 0\farcs 13\ (88\arcdeg)^{b}$ | $0\farcs 17\times 0\farcs 13\ (87\arcdeg)^{b}$ | $0\farcs 18\times 0\farcs 14\ (87\arcdeg)^{b}$ | $0\farcs 18\times 0\farcs 14\ (87\arcdeg)^{b}$
noise rms | | $22~{}{\rm\mu Jy~{}beam^{-1}}^{c}$
---
$100~{}{\rm\mu Jy~{}beam^{-1}}^{d}$
$1.8~{}{\rm mJy~{}beam^{-1}}^{b}$ | $3.0~{}{\rm mJy~{}beam^{-1}}^{b}$ | $2.1~{}{\rm mJy~{}beam^{-1}}^{b}$ | $2.5~{}{\rm mJy~{}beam^{-1}}^{b}$
Note. — The beam and noise rms values are shown for different robust
parameters with superscripts of (a) Robust$=2$, (b) 0.5, (c) 0, and (d) $-2$.
## 3 Results
### 3.1 1.3 mm continuum
Figure 1 shows continuum images with different robust and tapering parameters.
Figure 1(a) shows that the continuum emission with the robust parameter of 0
traces a disk-like structure on a $\sim 40$ au scale. This emission appears
more extended to the southeast than to the northwest from the emission peak.
This asymmetry is investigated in detail in Section 4.1. Two dimensional (2D)
Gaussian fitting to this robust$=0$ continuum image provides the central
position of $(\alpha_{\rm ICRS},\ \delta_{\rm ICRS})=(16^{\rm h}28^{\rm
m}21\fs 615,\ -24\arcdeg 36\arcmin 24\farcs 33$) with an uncertainty of (0.6
mas, 0.3 mas). The fitting, including the error bar calculation, used the CASA
task imfit. The fitting result also provides the deconvolved size of $107\pm
2$ mas $\times$ $40\pm 2$ mas with a major axis in the direction of P.A. of
$113\arcdeg\pm 1\arcdeg$. We adopt this direction as the major axis of the
protostellar system, whici is perpendicular to the outflow direction,
P.A.$=200\arcdeg-205\arcdeg$ (Yen et al., 2017). The aspect ratio corresponds
to an inclination angle of ${\rm arccos}(40/107)\sim 68\arcdeg$, assuming a
geometrically thin disk.
The peak intensity and flux density derived from the Gaussian fitting are
$5.10\pm 0.02~{}{\rm mJy~{}beam^{-1}}$ and $11.5\pm 0.2$ mJy, respectively.
The flux densities integrated over $0\farcs 4\times 0\farcs 4$ (the compact
emission) and $10\arcsec\times 10\arcsec$ (the maximum recoverable scale of
our ALMA observations) regions are $12.4\pm 0.3$ and $25\pm 2$ mJy,
respectively. The flux density $F_{\nu}$ can be converted to a lower limit of
gas mass, assuming that the emission is optically thin: $M_{\rm
gas}=d^{2}F_{\nu}/\kappa_{\nu}B_{\nu}(T)$, where $d$, $\kappa_{\nu}$, and
$B_{\nu}$ are the target distance, dust mass opacity, and the Planck function
with an average temperature $T$, respectively. For this conversion, two
different temperatures are adopted: $T=20$ and 27 K. $T=20$ K is derived from
the relation between 850-$\micron$ continuum fluxes and model gas masses for
44 young stellar objects (Andrews & Williams, 2005). $T=27$ K is calculated
from the empirical relation, $T=43~{}{\rm K}~{}(L_{\rm
bol}/1~{}L_{\sun})^{0.25}$, from Tobin et al. (2020). With these temperatures,
$d=139$ pc, $\kappa_{\nu}=2.3~{}{\rm cm}^{2}~{}{\rm g}^{-1}$ (Beckwith et al.,
1990), and a gas-to-dust mass ratio of 100, $F_{\nu}\simeq 12$ mJy corresponds
to a gas mass of $M_{\rm gas}\simeq(1.4-2.1)\times 10^{-3}~{}M_{\sun}$.
Figure 1(b) adopts the robust parameter of $-2$ and the 3-M$\lambda$ taper to
focus on the central region with a reasonable S/N ratio. The plotted spatial
range is half of Figure 1(a). This image shows an extension or a secondary
peak to the southeast, separated from the central peak by $\sim 0\farcs 1$
roughly along the major axis. This is consistent with the extension seen in
the image with robust$=0$ (Figure 1a).
Figure 1(c), adopting the robust parameter of 2 and the 1-M$\lambda$ taper,
shows the extended emission over $\sim 600$ au from the continuum emission
peak to the northwest, as well as the compact ($\sim 100$ au) emission. This
large-scale structure is also reported in Yen et al. (2017).
Figure 1: Images of the 1.3 mm continuum emission produced with (a) the robust
parameter of 0.0 and no taper, (b) the robust parameter of $-2.0$ and a 3
M$\lambda$ ($\sim 0\farcs 06$) taper, (c) the robust parameter of 2.0 and a 1
M$\lambda$ ($\sim 0\farcs 18$) taper. The brightness temperature $T_{b}$ is
calculated from the Rayleigh-Jeans approximation. The contour levels are
$3,6,12,24,48,96,192\sigma$, where $1\sigma$ is (a) 0.022, (b) 0.100, (c)
$0.023~{}{\rm mJy~{}beam^{-1}}$ (Table 1). The filled ellipses at the bottom-
left corners denote the synthesized beams: (a) $0\farcs 073\times 0\farcs 048$
($77\arcdeg$), (b) $0\farcs 043\times 0\farcs 034$ ($73\arcdeg$), and (c)
$0\farcs 29\times 0\farcs 24$ ($86\arcdeg$). The diagonal lines denote the
major and minor axes of the emission in the image in panel (a):
P.A.$=113\arcdeg$ and $23\arcdeg$. The square in panel (a) shows the plotted
range of panel (b).
### 3.2 12CO $J=2-1$
Figure 2(a) shows the integrated intensity (moment 0) and the mean velocity
(moment 1) maps in the 12CO emission produced with the robust parameter of 2.
Positive and negative intensities are integrated for the moment 0 map, while
intensities above the $3\sigma$ level are integrated for the moment 1 map. The
noise level of each moment 0 map is calculated by the noise propagation from
the noise level of the data cube (Table 1) used to make the moment 0 map:
$\sigma_{\rm mom0}=\sigma_{\rm cube}dv\sqrt{N_{\rm ch}}$, where $\sigma_{\rm
mom0}$, $\sigma_{\rm cube}$, $dv$, and $N_{\rm ch}$ are the noise level of the
moment 0 map, the noise level of the cube data, the velocity width per channel
(Table 1), and the number of integrated channels, respectively. The moment 0
and 1 maps of the other lines in this paper are also made in the same method.
The 12CO emission traces a clear bipolar outflow whose axis is perpendicular
to the major axis of the disk-like structure. An additional eastern component
results from a large-scale structure around the systemic velocity, which is
velocity resolved out by the interferometric observation. While the emission
in the northern outflow lobe is overall blueshifted, redshifted emission can
also be found on the northern side near the outflow axis. Similarly, the
southern lobe is mostly redshifted but partly blueshifted. This is expected if
a given outflow lobe crosses the plane of the sky that contains the central
source, with the outflow axis close to the plane of sky (Cabrit, 1989). A part
of the lobe in the front of the plane of the sky appears blue-shifted, while
the other part of the lobe behind the plane of the sky appears red-shifted.
Using a radially-expanding parabolic outflow model, Yen et al. (2017)
estimated the outflow shape and inclination by fitting a moment 0 map and a PV
diagram along the outflow axis of the 12CO emission observed at an angular
resolution of $\sim 1\arcsec$. Their best model provides an inclination angle
of $60\arcdeg-65\arcdeg$ ($0\arcdeg$ means pole-on) and an opening angle of
$60\arcdeg-70\arcdeg$, calculated from the parabolic shape at the distance
from the protostar along the outflow axis of $z\sim 5\arcsec-10\arcsec$. This
inclination is consistent with that of the disk-like structure detected in the
dust continuum emission (Section 4.1). Thus, their model is consistent with
our observations which reveal blue- and red-shifted outflow components to the
northeast and southwest, respectively.
Figure 2: Moment 0 (integrated intensity) and 1 (mean velocity) maps in the
12CO $J=2-1$ line. The contour maps show the moment 0 map, while the color
images show the moment 1 map. (a) Moment maps with the robust parameter of
2.0, integrated from $V_{\rm LSR}=-6.0$ to $14.0~{}{\rm km~{}s^{-1}}$. (b)
Zoom-in view of the moment maps with the robust parameter of 0.5, integrated
from $V_{\rm LSR}=-6.0$ to $14.0~{}{\rm km~{}s^{-1}}$. The contour levels are
in (a) $12\sigma$ steps from $12\sigma$ with $1\sigma=3.2~{}{\rm
mJy~{}beam^{-1}}~{}{\rm km~{}s^{-1}}$ and (b) $6\sigma$ steps from $6\sigma$
with $1\sigma=3.5~{}{\rm mJy~{}beam^{-1}}~{}{\rm km~{}s^{-1}}$. The filled
ellipse at the bottom-left corner shows the synthesized beam: (a) $0\farcs
35\times 0\farcs 25$ ($76\arcdeg$) and (b) $0\farcs 17\times 0\farcs 13$
($88\arcdeg$). The diagonal lines are the major and minor axes of the
continuum emission, P.A.$=113\arcdeg$ and $23\arcdeg$. The green contour in
panel (b) shows the 5$\sigma$ level of the continuum emission (Figure 1a).
Figure 2(b) shows a zoom-in view of the moment 0 and 1 maps with a different
robust parameter of 0.5. The green contour shows the 5$\sigma$ level of the
continuum emission (Figure 1a) for comparison. The moment 0 map shows local
peaks within $\sim 0\farcs 4$ from the protostellar position, which is
reported in Hsieh et al. (2019) as a quadruple peak. In addition to the
velocity gradient along the outflow axis on a $1\arcsec$ scale, this figure
also shows a velocity gradient perpendicular to the outflow axis near the
midplane on a $0\farcs 3$ scale, i.e., transition from a blueshifted part at
P.A.$=113\arcdeg$ to a redshifted part at P.A.$=-67\arcdeg$. This velocity
gradient is expected for rotating gas and analyzied in more detail in Section
4.2.
Figure 4 shows a position-velocity (PV) diagram of the 12CO emission along the
outflow axis from the robust parameter 2 images and a width of $0\farcs 6$
($\sim 2\times$ beam size). The width allows us to increase the S/N ratio but
still focus on the velocity structure on the outflow axis. Because of this
width, the noise level of this PV diagram is better than the one in Table 1.
This figure also shows that both northern and southern lobes have both blue-
and red-shifted parts. The outflow velocity is $|V-V_{\rm sys}|\sim
1.5-2.5~{}{\rm km~{}s^{-1}}$ at any position except for the central
$1\arcsec-2\arcsec$ region. The central component appears not to be a part of
the outflow because of its $\sim 3$ times larger line widths. The outer
($>2\arcsec$) emission shows several ($\sim 5-9$) local peaks or extensions
within $10\arcsec$ ($\sim 1400$ au) in each lobe, based on visual inspection
(short lines in Figure 4). This may be suggestive of episodic mass ejection
(Lee, 2020). The interval angular scale, $10\arcsec/7=1.4\arcsec$, is
significantly larger than the beam size of $0\farcs 35$. With the inclination
angle of $65\arcdeg$ (Yen et al., 2017), the deprojected length $l$ and the
deprojected velocity $v$ of the outflow can be calculated as
$l=l^{\prime}/\cos i$ and $v=v^{\prime}/\sin i$, respectively, where
$l^{\prime}$ and $v^{\prime}$ are the projected length and velocity. The
interval $t$ of the mass ejection is a time required for material to move from
a peak to the next peak. This can be calculated with the number of peaks $n$
within the length: $t=l^{\prime}/n/v^{\prime}$. Then, the values measured
above provide a typical interval of $1400~{}{\rm au}/\cos 65\arcdeg/(5\ {\rm
to}\ 9)/(1.5\ {\rm to}\ 2.5~{}{\rm km~{}s^{-1}}/\sin 65\arcdeg)=600\ {\rm to}\
2000$ year. The episodic mass ejection is thought to be caused by episodic
accretion bursts. The interval of accretion bursts is investigated in
statistical studies using FUor objects. Park et al. (2021) reported accretion-
burst intervals of $500-2300$ yr in a statistical study of FUor outbursts
based on the fraction of outbursting sample in a set of NEOWISE observations.
The fraction of variable sources is similar between VeLLOs like I16253 and
more luminous objects in their sample. The uncertainty comes from the number
of outbursts ranging from 2 to 9 and the total number of sample protostars
ranging from 735 to 1059. This accretion-burst interval is consistent with the
mass-ejection interval of I16253, implying that accretion bursts likely
occurred in I16253.
Figure 3: Figure 4: Position-velocity diagram in the 12CO $J=2-1$ line along
the outflow axis (P.A.$=23\arcdeg$) with a width of $0\farcs 6$. The positive
offset corresponds to the northern side. The contour levels are in $6\sigma$
steps from $6\sigma$, where $1\sigma$ is $0.85~{}{\rm mJy~{}beam^{-1}}$. The
dashed lines denote the systemic velocity ($V_{\rm LSR}=4~{}{\rm
km~{}s^{-1}}$) and the protostellar position. The orange short lines denote
the position of local peaks or extensions visually identified.
This PV diagram shows a strong emission component at offsets of $<-10\arcsec$.
This component is not confined around the outflow axis but extended over the
redshifted lobe. This may imply that there could be more material on the
southern side than on the northern side, although our observation did not
target such a large spatial scale.
### 3.3 C18O and 13CO $J=2-1$
Figure 5 shows the moment 0 and 1 maps in the C18O and 13CO lines in
robust$=0.5$ and 2. The two lines show an overall structure extended along the
major axis (P.A.$=113\arcdeg$) and a velocity gradient primarily along the
same direction. The integrated intensity maps with robust$=0.5$ show double
peaks with a separation of $\sim 0\farcs 2-0\farcs 3$, and the integrated
intensity is stronger at the eastern, blueshifted peaks than at the western,
redshifted peaks, by $3\sigma$ in both lines. The integrated intensity maps
with robust$=2$ show more clearly that the eastern emission is stronger than
the western emission. The detected emission size is larger in the C18O line
than in the 13CO line with both robust parameters. This is because the 13CO
emission tends to be optically thicker and more extended than the C18O
emission, and thus the 13CO emission is resolved out more severely around the
systemic velocity.
Figure 5: Moment 0 and 1 maps in the C18O $J=2-1$ line emission and the 13CO
$J=2-1$ line emission. The contour maps show the moment 0 map, while the color
images show the moment 1 map. The contour levels are in $3\sigma$ steps in the
robust$=0.5$ images and $6\sigma$ steps in the robust$=2$ images starting from
$3\sigma$. The C18O emission is integrated from 0.8 to $7.1~{}{\rm
km~{}s^{-1}}$. These moment 0 maps have $1\sigma$ of 2.2 (robust$=0.5$) and
$2.0~{}{\rm mJy~{}beam^{-1}}~{}{\rm km~{}s^{-1}}$ (robust$=2$). The 13CO
emission is integrated from 0.7 to $7.3~{}{\rm km~{}s^{-1}}$. These moment 0
maps have $1\sigma$ of 2.8 (robust$=0.5$) and $2.7~{}{\rm
mJy~{}beam^{-1}}~{}{\rm km~{}s^{-1}}$ (robust$=2$). The beam sizes are (a)
$0\farcs 18\times 0\farcs 14$ ($87\arcdeg$), (b) $0\farcs 38\times 0\farcs 27$
($78\arcdeg$) (c) $0\farcs 17\times 0\farcs 13$ ($87\arcdeg$), and (d)
$0\farcs 37\times 0\farcs 26$ ($77\arcdeg$). The diagonal lines are the major
and minor axes of the continuum emission, P.A.$=113\arcdeg$ and $23\arcdeg$.
### 3.4 SO $J_{N}=6_{5}-5_{4}$
Figure 6(a) shows the integrated intensity and mean velocity maps in the SO
line produced with robust$=0.5$ to focus on the central compact emission. The
SO emission shows a double-peaked structure with a separation of $\sim 0\farcs
2-0\farcs 3$ and the velocity gradient along the major axis, like the C18O and
13CO lines. On the other hand, the mean velocity of the SO emission is larger
outside the peaks than inside the peaks, unlike the C18O and 13CO lines.
Figure 6(b) shows the maps produced with robust$=2$ to focus on extended
emission. The central double peaks are not spatially resolved in this map. The
velocity structure in the central $1\arcsec$ region is overall the same as
that in Figure 6(a): eastern blueshifted emission and western redshifted
emission. In addition, this figure shows an extended structure to the east
from the central protostar. This structure is extended ($\sim 3\arcsec$)
beyond the C18O and 13CO emission ($\sim 1\arcsec$) in Figure 5. An inner part
($\lesssim 1\arcsec$ from the protostellar position) of this extended
structure shows velocities blueshifted from the systemic velocity, $V_{\rm
LSR}=4~{}{\rm km~{}s^{-1}}$, which is consistent with the C18O and 13CO
velocities. In contrast, the outer part ($\gtrsim 1\arcsec$) shows slightly
redshifted velocities ($V_{\rm LSR}-V_{\rm sys}\sim 0.3~{}{\rm km~{}s^{-1}}$).
This redshifted velocity on the eastern side is different from the velocity
gradient seen in the C18O and 13CO lines and also different from the velocity
in the eastern cavity wall of the 12CO outflow (Figure 2a). This SO component
is discussed in more detail in Section 5.2.
Figure 6: Moment 0 and 1 maps in the SO $J_{N}=6_{5}-5_{4}$ line emission. The
contour maps show the moment 0 map, while the color images show the moment 1
map. (a) The robust parameter is 0.5. The emission is integrated from 1.7 to
$6.3~{}{\rm km~{}s^{-1}}$. The contour levels are in $3\sigma$ steps from
$3\sigma$ with $1\sigma=2.3~{}{\rm mJy~{}beam^{-1}}~{}{\rm km~{}s^{-1}}$. The
beam size is $0\farcs 18\times 0\farcs 14$ ($87\arcdeg$). (b) The robust
parameter is 2. The emission is integrated from 1.7 to $6.3~{}{\rm
km~{}s^{-1}}$. The contour levels are in $3\sigma$ steps from $3\sigma$ until
$15\sigma$ and then in $10\sigma$ steps with $1\sigma=2.1~{}{\rm
mJy~{}beam^{-1}}~{}{\rm km~{}s^{-1}}$. The beam size is $0\farcs 37\times
0\farcs 26$ ($78\arcdeg$). The diagonal lines are the major and minor axes of
the continuum emission, P.A.$=113\arcdeg$ and $23\arcdeg$.
## 4 Analysis
### 4.1 Non-axisymmetry of the continuum image
The 1.3 mm continuum emission with robust$=0$ (Figure 1a) is more extended in
the southeastern side ($\sim 30$ au) than in the northwestern side ($\sim 20$
au). We verify that the extended components are real by comparing images made
with various robust parameters. To extract the extended component, an
axisymmetric model of the dust continuum emission is constructed in this
subsection. The model is made from a radial profile of intensities in the unit
of Jy pixel-1, before beam convolution, set with free parameters of
$I(r=0),I(dr),I(2dr),...,I(11dr)$. The grid separation $dr$ is half of the
beam minor-axis. The largest radius $11dr=0\farcs 26$ covers the entire
continuum emission with robust$=0$. Then, the circular 2D intensity
distribution is projected by the inclination factor of $\cos i$ in the
direction of the minor axis and rotated by the position angle $pa$; these two
angles are also free parameters. This elliptical 2D intensity in the unit of
Jy pixel-1 is convolved with the observational beam to make the model image in
the unit of ${\rm Jy~{}beam^{-1}}$. Hence, this model does not assume any
shape, such as a Gaussian function but only the axisymmetry. This modeling
method follows the one in Aso et al. (2021). The central position is not a
free parameter but fixed at the observed peak position, $(\alpha_{\rm ICRS},\
\delta_{\rm ICRS})=(16^{\rm h}28^{\rm m}21\fs 6153,\ -24\arcdeg 36\arcmin
23\farcs 325)$, which coincides with the Gaussian center derived in Section
3.1. The best-fit parameters are obtained by minimizing $\chi^{2}$ between the
observed and model images, divided by the number of pixels in the beam,
through the Markov chain Monte Carlo (MCMC) methods with the public Python
package ptemcee. The numbers of free parameters, steps, and walkers per
parameter are 14, 8000, and 16, respectively. The first half (4000) steps were
removed for the burn-in. The derived best-fit angles are $i=64.1\arcdeg\pm
0.5\arcdeg$ and $pa=114.3\arcdeg\pm 0.6\arcdeg$, and Figure 8 shows the
posterior distribution of these two angles produced by the MCMC fitting. The
uncertainties of these angles are derived as the 16 and 84 percentiles.
Figure 7: Figure 8: Corner plot of the MCMC fitting to the continuum image.
Two angles, $i$ and $pa$, are plotted among the 14 free parameters. The dashed
lines show the 16, 50, and 84 percentiles.
Figure 9(a) shows the comparison between the best-fit axisymmetric model and
the observed image. The overall structure in the observed image is reproduced,
but the model emission is weaker than the observed emission in the southeast,
while it is stronger in the northwest, as expected. The contour map in Figure
9(b) shows the residual image after subtracting the model from the observed
image with robust$=0$, while the color map shows the observed continuum image
with robust$=-2$ (same as Figure 1c). The strong residual is located in the
southeast. This residual overlaps with the extension in the robust$=-2$ image,
supporting that this excess from the symmetric component is not due to the
specific robust parameter. The direction of the residual and extension
coincides with the direction of the SO extended structure and the stronger
peak on the eastern side in the C18O and 13CO moment 0 maps.
Figure 9: (a) Axisymmetric model (red) fitted to the observed 1.3 mm continuum
emission (black) with the robust parameter of 0.0. The contour levels are the
same as Figure 1(b): 3,6,12,24,48,96,192$\sigma$. (b) Residual (contour map)
between the model and observation in panel (a), overlaid on the robust$=-2$
continuum image (color map; Figure 1c). The contour levels are the same:
$3,6,12,24\sigma$.
### 4.2 Keplerian rotation in the 12CO emission
Previous observational studies have not yet identified any part of gas motion
in I16253 having a radial profile of the Keplerian rotation but assumed that
the observed gas follows Keplerian rotation. The Keplerian disk is crucial to
estimate the central stellar mass in the protostellar phase and verify whether
I16253 is a proto-BD or a very low-mass protostar. To tackle this problem, we
analyze the position-velocity (PV) diagrams along the major axis. Figure 10(a)
shows the PV diagrams in the 12CO (blue contours), C18O (red contours), and
13CO (color) lines. The C18O and 13CO lines show a typical shape of an
infalling plus rotating motion: a distorted diamond shape with emission in all
the four quadrants, as reported in Hsieh et al. (2019). In contrast, the 12CO
emission is concentrated on the first (upper right) and third (lower left)
quadrants (the western redshifted emission and the eastern blueshifted
emission), showing a clear velocity gradient, in $\pm(0\farcs 1-0\farcs 2)$ at
$|V-V_{\rm sys}|\gtrsim 2~{}{\rm km~{}s^{-1}}$. Figure 10(b) shows the PV
diagrams in the three lines along the minor axis. The C18O and 13CO emission
can be seen in all four quadrants in the minor-axis PV diagram with a diamond
shape, which is typical for the infall motion (e.g., Ruíz-Rodríguez et al.,
2022). The 12CO emission is mainly concentrated around the center, with
additional components in the second and fourth quadrants (the northern
blueshifted emission and the southern redshifted emission). While the
additional components show a velocity gradient due to the outflow motion, the
main component shows no velocity gradient along the minor axis. These PV
diagrams suggest that the 12CO line traces a purely rotating, i.e., Keplerian
disk, while the C18O and 13CO lines trace the infalling rotating envelope
previously identified. This difference among individual molecular lines can be
understood by the different strengths of emission and the different missing
fluxes in the interferometric observations. The 12CO emission is strong enough
to be detected even in the expected compact disk ($r\sim 0\farcs 1-0\farcs 2$
in the PV diagram as well as in the continuum image). The 13CO and C18O
emission is not detected sufficiently in this compact region at the high
velocities where the 12CO emission is detected. The nondetection in the 13CO
and C18O lines suggests the 12CO emission is not optically thick at the high
velocities. In contrast to the high velocities, the 12CO emission is more
extended and strongly resolved out at the low velocities where the 13CO and
C18O emission traces the envelope.
Figure 10: Position-velocity diagrams in the 12CO (blue; robust$=0.5$), C18O
(red; robust$=2.0$), and 13CO (color; robust$=2.0$) $J=2-1$ lines along the
(a) major and (b) minor axes. The positive offset corresponds to the western
and northern sides in panels (a) and (b), respectively. The contour levels are
in $6\sigma$ steps from $6\sigma$, where $1\sigma$ is $1.4~{}{\rm
mJy~{}beam^{-1}}$ for 12CO, $2.1~{}{\rm mJy~{}beam^{-1}}$ for C18O, and
$3.0~{}{\rm mJy~{}beam^{-1}}$ for 13CO. The beam sizes in these diagrams are
$\sim 0\farcs 15$ for 12CO, $\sim 0\farcs 33$ for C18O, and $\sim 0\farcs 32$
for 13CO (Table 1).
In order to verify whether the 12CO line traces the Keplerian rotation, we
first find the emission ridge (e.g., Yen et al., 2013; Aso et al., 2015; Sai
et al., 2020) and edge (e.g., Seifried et al., 2016; Alves et al., 2017) in
the major-axis PV diagram and fit them with power-law functions. The ridge is
a center of the emission along the positional axis at each velocity, while the
edge is an outer boundary of the emission along the positional axis at each
velocity. The analysis and fitting process was performed through the Python
open package pvanalysis in Spectral Line Analysis/Modeling (SLAM; Aso & Sai,
2023)222https://github.com/jinshisai/SLAM, and the detail is described in the
overview paper of the eDisk project (Ohashi et al., 2023). We thus mention
here the settings specific to the case of I16253. The ridge point is defined
as the intensity-weighted mean position $x_{m}$ at each velocity $v$:
$x_{m}(v)=\int I(x,v)xdx/\int I(x,v)dx$. $x_{m}$ is calculated using the
intensities above the $5\sigma$ level along the positional axis at each
velocity in the range of $|V-V_{\rm sys}|>2~{}{\rm km~{}s^{-1}}$. This
velocity range is selected because the emission is located in the first and
third quadrants in this velocity range. Similarly, using the 1D intensity
profile, the edge point is defined, at each velocity in the same velocity
range, as the outer position where the intensity is at the $5\sigma$ level. In
addition, the tool uses velocities higher than or equal to the velocity at
which the derived edge/ridge radius is largest so that the relation between
radius and velocity can be consistent with spin-up rotation. The ridge and
edge tend to under- and overestimate, respectively, the radius at a given
velocity (Maret et al., 2020, the ridge and edge here correspond to their
“centroid” and “first emission contour”, respectively), and thus we adopt the
former and the latter as the lower and upper limits of radius in this paper.
Figure 11 shows the estimated ridge and edge points overlaid on the PV diagram
in the linear and logarithmic scales. The logarithmic diagram is made by
avaraging the emission in the first and third quadrants of the linear diagram.
The edge and ridge radii are separately fitted with a power-law relation
between either edge or ridge radius $R$ and the velocity $V$:
$\displaystyle V=V_{b}\left(\frac{R}{R_{b}}\right)^{-p}+V_{\rm sys},$ (1)
$\displaystyle p=p_{\rm in}\ {\rm if}\ R<R_{b}\ {\rm else}\ p_{\rm in}+dp.$
(2)
This fitting uses the MCMC method with the tool of emcee, where the numbers of
walker per free parameter, burn-in steps, adopted steps are 16, 2000, and
1000, respectively. The error bar of each free parameter is defined as the 16
and 84 percentiles. First, we adopt a single power-law function, where the
free parameters are the power-law index $p$ and the radius $R_{b}$ at a fixed
middle velocity $V_{b}$. $dp=0$ and $V_{\rm sys}=4.0~{}{\rm km~{}s^{-1}}$ are
fixed. The fixed parameter $V_{b}$ is an arbitrary reference velocity where
$R_{b}$ is derived from the fitting. The central stellar mass $M_{*}$ is
calculated from the radius with the fixed inclination angle $i=65\arcdeg$ as
$M_{*}=R(V)(V/\sin i)^{2}/G$, where $R(V)$ is calculated from Equation (1) as
a function of $V$. The calculated $M_{*}$ can thus depend on $V$ if $p\neq
0.5$. The uncertainty of $M_{*}$ is calculated through the error propagation.
The inclination angle is adopted from an outflow model ($60\arcdeg-65\arcdeg$;
Yen et al., 2017) and our continuum analysis ($64.3\arcdeg$ in Section 4.1).
Even with $i=60\arcdeg$, the calculated stellar mass is only 10% higher. The
best-fit parameters are $p=0.5$ and $M_{*}=0.09~{}M_{\sun}$ with the ridge
points, while they are $p=0.67$ and $M_{*}=0.3-0.4~{}M_{\sun}$ with the edge
points. The range of edge $M_{*}$, $0.3-0.4~{}M_{\sun}$ is because $p_{\rm
in}\neq 0.5$. This range is wider than the statistical uncertainty. The
fitting results are summarized in Table 2 including statistical uncertainties.
The uncertainty of the distance adds a relative uncertainty of $\sim 7\%$. The
best-fit functions are plotted in Figure 11. Figure 11(a) shows that the best-
fit functions trace the ridge and edge of the observed PV diagram. Figure
11(b) clearly shows that the ridge and edge indices are close to each other,
as well as to that of Keplerian rotation $(p=0.5)$. The ridge and edge points
were also fitted with a double power-law function. The best-fit parameters are
summarized in Table 2. The double power-law fitting yields similar power-law
indices $p=0.5-0.6$ for the most part of the fitted velocity range, except for
only the lowest velocity channel. These results indicate that the 12CO line
traces the Keplerian disk around I16253.
Figure 11: Edge and ridge points, estimated for each velocity channel,
overlaid on the 12CO major-axis PV diagram (robust$=0.5$) in the (a) linear
and (b) logarithmic scales. The positive radius corresponds to the western
side. The velocity is the relative velocity to the systemic velocity of
$V_{\rm LSR}=4~{}{\rm km~{}s^{-1}}$. The white curves/lines are the best-fit
power-law functions (Section 4.2). The contour levels are in the $3\sigma$
steps from $3\sigma$, where $1\sigma$ is $1.4~{}{\rm mJy~{}beam^{-1}}$. Table
2: Power-law fitting to the edge and ridge points in the major-axis 12CO PV
diagram.
| Single power
---|---
| $R_{b}$ (au) | $V_{b}$ (${\rm km~{}s^{-1}}$) | $p_{\rm in}$ | $dp$ | $M_{*}(M_{\sun})$
Edge | $24.7\pm 0.7$ | 3.18 (fixed) | $0.67\pm 0.04$ | 0 (fixed) | $0.34\pm 0.01-0.43\pm 0.02$
Ridge | $6.2\pm 0.7$ | 3.20 (fixed) | $0.5\pm 0.1$ | 0 (fixed) | $0.09\pm 0.02$
| Double power
| $R_{b}$ (au) | $V_{b}$ (${\rm km~{}s^{-1}}$) | $p_{\rm in}$ | $dp$ | $M_{*}(M_{\sun})$
Edge | $45\pm 10$ | $2.2\pm 0.5$ | $0.63\pm 0.07$ | $2\pm 4$ | $0.3\pm 0.2-0.4\pm 0.2$
Ridge | $13\pm 2$ | $2.3\pm 0.2$ | $0.5\pm 0.1$ | $4\pm 4$ | $0.09\pm 0.04$
Note. — The uncertainties of $R_{b}$ and $M_{b}$ in this table are before the
uncertainty of the distance, $\sim 7\%$, is incorpolated.
By identifying the Keplerian rotation, we have estimated the central stellar
mass to be $M_{*}=0.09-0.34~{}M_{\sun}$ directly (kinematically) for the first
time. Even with the lower limit, I16253 already has sufficient mass to evolve
into a low-mass star, beyond the brown-dwarf mass regime
($M_{*}>0.08~{}M_{\sun}$). Maret et al. (2020) demonstrated that the central
stellar masses derived from the ridge and edge points under- and overestimate
the actual stellar mass by $\sim 30\%$ and $\sim 100\%$ in the case where the
beam size is a few tens percent larger than the observed disk size. The reason
why the ridge underestimates the stellar mass is that a part of the gas inside
the Keplerian radius has the same line-of-sight velocity as the gas at the
Keplerian radius, and this inner emission shifts the ridge inward. This effect
is also discussed quantitatively by Aso et al. (2015). The reason why the edge
overestimates the stellar mass is that the observational beam causes emission
outside the outermost, i.e., Keplerian radius, and this outer emission shifts
the edge outward. The condition in Maret et al. (2020) is applicable to our
case (the disk radius of I16253 is discussed in Section 5.1). Then, the
central stellar mass is likely within $M_{*}=0.09\times 1.3$ to
$0.34/2~{}M_{\sun}=0.12$ to $0.17~{}M_{\sun}$. This mass is consistent with a
value derived from the 12CO $J=2-1$ emission at two velocity channels by
assuming Keplerian rotation ($\sim 0.12~{}M_{\sun}$; Hsieh et al., 2019).
### 4.3 Linear velocity gradient in the SO emission
Figure 13 shows the PV diagram in the SO emission along the major axis. The SO
emission is detected within $|V-V_{\rm sys}|\sim\pm 2~{}{\rm km~{}s^{-1}}$,
which is similar to the velocity range where the 13CO and C18O emission is
present (see Figure 14 for the comparison of the SO and C18O emission). The
shape of the SO PV diagram is represented by a linear velocity gradient from
the eastern blueshifted emission to the western redshifted emission in the
velocity range of $|V-V_{\rm sys}|<1.3~{}{\rm km~{}s^{-1}}$, unlike the CO
isotopologues. The SO emission outside this velocity range appears to overlap
a component traced by the CO isotopologue emissions (see Figure 14 for the
comparison), although it is difficult to discuss it given the limited number
of the channels at the common velocity range. The linear velocity gradient can
also be seen in the moment 1 map (Figure 6a), where the high velocities are
located at the outermost parts along the major axis. To evaluate the linear
velocity gradient quantitatively, the ridge points are found at each velocity,
as plotted in Figure 13, and fitted with a linear function of radius $r$
passing $(r,v)=(0,0)$, using emcee with the same condition as that in Section
4.2. The best-fit gradient is estimated to be $g=0.056\pm 0.005~{}{\rm
km~{}s^{-1}}~{}{\rm au}^{-1}$, after the correction for the inclination angle
of $i=65\arcdeg$. This value is slightly smaller than a previous result of
$0.082\pm 0.004~{}{\rm km~{}s^{-1}}~{}{\rm au}^{-1}$ ($d=139$ pc) by Yen et
al. (2017). This difference is mainly due to the different angular
resolutions, $\sim 0\farcs 2$ in the present observations and $\sim 1\arcsec$
in Yen et al. (2017). Their result may thus include SO emission at larger
radii and slower velocities. The SO emission is located within the velocities
of $|V-V_{\rm sys}|<2~{}{\rm km~{}s^{-1}}$, which are lower than those showing
the Keplerian rotation in the 12CO PV diagram (Figure 11), implying that the
SO emission traces a different part from the 12CO emission. The SO velocity
structure and the velocity gradient are discussed in more detail in Section
5.1 along with the infalling and rotating motions traced in the C18O and 13CO
emission.
Figure 12: Figure 13: Ridge points, found for each velocity, overlaid on the
SO PV diagram (robust$=2$) along the major axis in the linear scale. The
positive radius corresponds to the western side. The velocity is relative to
the systemic velocity of $V_{\rm LSR}=4~{}{\rm km~{}s^{-1}}$. The white line
is the best linear function (Section 4.3). The contour levels are in the
$3\sigma$ steps from $3\sigma$, where $1\sigma$ is $2.3~{}{\rm
mJy~{}beam^{-1}}$.
## 5 Discussion
### 5.1 Disk, Ring, and Envelope
We suggest a picture of the I16253 system based on our results and the
previous results. Hsieh et al. (2019) estimated the specific angular momentum
of the envelope to be $j\sim 45~{}{\rm km~{}s^{-1}}{\rm au}$ ($d=139~{}pc$),
by reproducing the PV diagrams along the major and minor axes in the C18O and
12CO lines with an infalling and rotating envelope model. When the specific
angular momentum and the central stellar mass are given, the velocity field of
a protostellar envelope can be predicted by the UCM envelope model (Ulrich,
1976; Cassen & Moosman, 1981), where the velocity field consists of ballistic,
parabolic flows from an outer boundary in the rigid-body rotation. Figure 14
compares the maximum line-of-sight velocity of the UCM envelope model, as well
as that of the Keplerian disk model, at each position and the observed PV
diagrams in the C18O, SO, and 12CO lines along the major and minor axes. The
model parameters are the specific angular momentum of $j\sim 45~{}{\rm
km~{}s^{-1}}{\rm au}$ and the central stellar mass of $M_{*}\sim
0.14~{}M_{\sun}$ (middle of $0.12-0.17~{}M_{\sun}$ in Section 4.2). With the
specific angular momentum and the central stellar mass, the centrifugal radius
is calculated to be $R_{c}=j^{2}/GM_{*}\sim 16$ au ($0\farcs 12$);
$M_{*}=0.12-0.17~{}M_{\sun}$ corresponds to $R_{c}=13-19$ au. $R_{c}=16$ au is
used as the disk radius to draw the model curves in Figure 14. The C18O PV
diagrams are consistent with the model maximum velocity both in the major and
minor axes in the velocity range lower than the disk velocities, $|V-V_{\rm
sys}|<2~{}{\rm km~{}s^{-1}}$. The obtained $R_{c}$ is close to the radius of
the continuum emission (the red $6\sigma$ contour in Figure 9a), except for
the southeastern extension, supporting that this is the disk radius. This is
also consistent with a relation suggested by Aso & Machida (2020) that the
Gaussian deconvolved radius of the 1.3 mm continuum emission ($\sim 7.4$ au
for I16253; Section 3.1) is $\sim 0.5$ as large as the disk radius for an
evolutionary phase from $M_{*}\sim 0.1$ to $0.4~{}M_{\sun}$; this relation is
based on their synthetic observations of a magnetohydrodynamics simulation of
protostellar evolution.
Figure 14: Comparison between the predicted maximum line-of-sight velocities
(white dashed and dotted curves) and the observed PV diagrams (C18O in red, SO
in green, and 12CO in blue). (a) PV diagram along the major axis. The dashed
curves represent the UCM envelope model with a central stellar mass of
$M_{*}=0.14~{}M_{\sun}$ (Section 4.2) and the specific angular momentum
$j=45~{}{\rm km~{}s^{-1}}~{}{\rm au}$ (Hsieh et al., 2019). The envelope
maximum velocity is calculated only up to $|V-V_{\rm sys}|=2~{}{\rm
km~{}s^{-1}}$. The dotted curves show the maximum line-of-sight velocities of
the Keplerian disk with a disk radius of 16 au. (b) PV diagram along the minor
axis. The dashed curves represent the same UCM envelope as for the major axis.
The SO emission shows a different shape from the envelope and from the disk in
the major-axis PV diagram, whereas the detected velocity range is similar to
that of the C18O (and 13CO) emission as shown in Figure 14. The linear
velocity gradient in the major-axis PV diagram, along with the double peak
structure in the moment 0 map at a higher angular resolution (Figure 6a),
suggests that the SO emission traces a ring close to edge-on due to accretion
shock between the infalling envelope and the disk, as reported in other
protostellar systems (e.g., Yen et al., 2014; Ohashi et al., 2014; Sakai et
al., 2016). The linear velocity gradient in the SO PV diagram $0.056~{}{\rm
km~{}s^{-1}}~{}(r/1~{}{\rm au})$ intersects with the envelope rotation
$45~{}{\rm km~{}s^{-1}}~{}(r/1~{}{\rm au})^{-1}$ at $r=28$ au. This radius is
close to the outermost radius of the SO ridge points (Figure 13). Recent
theoretical studies predict that the accretion shock around the disk occurs at
the radius of $\sim 1.5R_{c}$ (Shariff et al., 2022). The inferred radius of
the SO ring in I16253 is approximately consistent with this prediction. In
conclusion, we suggest that I16253 has the Keplerian disk with $r\sim 16\pm 3$
au traced in the 12CO line, which is surrounded by the shock at $r\sim 28$ au
traced in the SO line due to the mass accretion from the envelope traced in
the C18O and 13CO lines.
### 5.2 Streamer in the SO emission
In addition to the shocked ring, the SO emission also shows an extended
structure to the east as shown in Figure 6(b). Its slightly redshifted
velocity in the outer ($\gtrsim 1\arcsec$) region cannot be explained by the
rotation, the infall motion on the midplane nor the outflow seen in our
observations toward I16253, implying a different motion. A possible
explanation is a streamer as reported in other protostellar systems on
$\lesssim 1000$ au scales (e.g., Yen et al., 2019; Garufi et al., 2022; Thieme
et al., 2022) (see also Kido et al., 2023) as well as on larger scales (e.g.,
Pineda et al., 2020). Hence, we constructed a streamer model by extracting
ballistic, parabolic flows from the UCM envelope model used in Section 5.1.
The free parameters are the two directional angles $(\theta_{0},\phi_{0})$ to
specify the initial polar and azimuthal angle of the streamer trajectory,
respectively. The polar angle $\theta_{0}$ is $0\arcdeg$ at the disk northern
pole and $90\arcdeg$ at the midplane. The azimuthal angle $\phi_{0}$ is
$0\arcdeg$ in the direction of the observer on the midplane and $90\arcdeg$ on
the right seen from the observer. The inclination and position angles of the
midplane are fixed at $65\arcdeg$ and $113\arcdeg$, respectively. By visual
inspection, we found that
$(\theta_{0},\phi_{0})=(60\arcdeg,240\arcdeg-270\arcdeg)$ can reasonably
reproduce the observed SO distribution and velocity. This suggests that the SO
gas is gravitationally bound if it came from these $(\theta_{0},\phi_{0})$
angles, since the UCM envelope model describes how material infalls toward the
central gravitational source. Figure 16 shows the distribution of the model
streamer projected on the plane-of-sky and the line-of-sight velocity of the
streamer model and the rotating ring at a radius of 28 au. The ring rotates in
the same velocity as the UCM envelope model. The model ring appears to
reproduce the size and line-of-sight velocity of the observed compact
component (Figures 6a), while the model streamer appears to reproduce the
extended structure to the east and its line-of-sight velocity (Figure 6b).
Meanwhile, the observed extended structure appears less confined than the
model. This is probably because our model did not consider any beam blurring
effect or radiative transfer effect. More sophisticated modeling including
such effects will help to verify whether the anisotropy in the SO line is
caused by the streamer.
Figure 15: Figure 16: Model map of the line-of-sight velocity of a streamer
in the UCM envelope ($M_{*}=0.14~{}M_{\sun}$ and $j=45~{}{\rm
km~{}s^{-1}}~{}{\rm au}$ are the same as for Figure 14) and a Keplerian ring
at 28 au. The streamer comes from $30\arcdeg$ above the midplane (i.e.,
$\theta_{0}=60\arcdeg$) and $0\arcdeg-20\arcdeg$ away from the left to the
front seen from the observer around the rotational axis (i.e.,
$\phi_{0}=240\arcdeg-270\arcdeg$).
Ballistic streamers are simulated in a theoretical study of the cloudlet
capture in a protostellar system (Hanawa et al., 2022). The simulation shows
that the streamers can be recognized as enhanced molecular line emission. Such
an enhancement is seen in the eastern peak ($3\sigma$ higher than the western
peak) of the C18O and 13CO moment 0 maps in I16253 (Figure 5), as well as in
the SO extended emission. The eastern extension in the continuum emission
(Figures 1b and 9b) could also be explained by the enhancement due to the
streamer since it is located on the eastern side.
Tobin et al. (2010) show the envelope of I16253 in the $8~{}\micron$
extinction observations at a $\sim 2\arcsec$ resolution using InfraRed Array
Camera (IRAC) on the Spitzer telescope. The extinction is mainly concentrated
in the $\sim 40\arcsec$-long outflow cavity wall, and the northeastern wall
shows higher extinction than the other walls. This may suggest the existence
of inhomogeneities in the ambient material that could produce the anisotropic
streamer observed in our SO result. On the other hand, if the anisotropy in
the SO line is due to an inhomogeneous density structure, similar anisotropy
could also be seen in the C18O and 13CO lines, unlike our results. This may
suggest that the anisotropy in the SO line could be caused by other factors
than density, such as temperature, shock, or chemistry. Figure 18 summarizes
the structures identified around I16253 in our ALMA observations, along with
the large-scale envelope, denoting the radius or length of each structure.
Figure 17: Figure 18: Schematic picture of the Keplerian disk, ring,
envelope, streamer, outflow identified in the CO isotopologue and SO lines,
and the eastern excess in the 1.3 mm continuum emission in the Class 0
protostar IRAS16253-2429. The gray background represents the large-scale
envelope identified in 8 $\micron$ extinction (Tobin et al., 2010). The radii
or lengths for each of the structures are indicated as well in units of au.
### 5.3 Mass Accretion and Other Quantities in I16253
The central stellar mass of I16253 has been directly estimated to be
$0.12-0.17~{}M_{\sun}$ by identifying the Keplerian rotation in its disk in
this paper, which ranged from $\sim 0.02$ to $\sim 0.12~{}M_{\sun}$ in
previous works (Section 1). In contrast, some previous works estimated the
central stellar mass of a protostar or a proto-BD from the mass accretion rate
from the disk to the central object $\dot{M}_{\rm acc}$. We discuss the
uncertainties for the method using $\dot{M}_{\rm acc}$ along with the
accretion rate derived from the updated stellar mass.
The central stellar mass of I16253 that we have estimated requires a mass
accretion rate of $\dot{M}_{\rm acc}=L_{\rm bol}R_{*}/GM_{*}=(0.9-1.3)\times
10^{-7}~{}M_{\sun}~{}{\rm yr}^{-1}$, where the bolometric luminosity is
$L_{\rm bol}=0.16~{}L_{\sun}$, the stellar radius is assumed to be
$R_{*}=3~{}R_{\sun}$, $G$ is the gravitational constant, and the stellar mass
is $M_{*}=0.12-0.17~{}M_{\sun}$. The stellar radius is predicted in numerical
simulations to be within $\sim 30\%$ of $3~{}R_{\sun}$ (e.g., Masunaga &
Inutsuka, 2000; Vorobyov & Basu, 2015). In comparison, Hsieh et al. (2016)
estimated the mass accretion rate of I16253 from its outflow force. The
outflow force, $F_{\rm out}\sim 7\times 10^{-7}~{}M_{\sun}~{}{\rm
km~{}s^{-1}}~{}{\rm yr}^{-1}$, was measured through IRAM 30-m and APEX
observations in 12CO lines at an angular resolution of $\sim 11\arcsec$. This
force was converted to a mass accretion rate of $\dot{M}_{\rm acc}\sim 5\times
10^{-7}~{}M_{\sun}~{}{\rm yr}^{-1}$ (after the distance correction from 125 to
139 pc) as $\dot{M}_{\rm acc}=F_{\rm out}/\epsilon V_{W}f_{\rm ent}$, where
the ratio between mass loss and accretion rates
$\epsilon=\dot{M}_{W}/\dot{M}_{\rm acc}$ is assumed to be 0.1, the wind (jet)
velocity $V_{W}$ is assumed to be 150 ${\rm km~{}s^{-1}}$, and the entrainment
efficiency $f_{\rm ent}$ is assumed to be 0.1. $F_{\rm out}$ is estimated from
observations in the 12CO $J=2-1$ line using IRAM 30-m and 12CO $J=7-6$ and
$J=6-5$ lines using APEX, with the optical depth correction and assumptions
that the kinetic temperature is 40 K and the H2 number density is
$10^{5}~{}{\rm cm}^{-3}$. This mass accretion rate is $\sim 5$ times higher
than the one derived from the updated stellar mass, whereas being similar to
that of two proto-BD candidates, IC348-SMM2D and L328-IRS, $\sim 2-9\times
10^{-7}~{}M_{\sun}~{}{\rm yr}^{-1}$.
This difference can be explained by large uncertainties in the conversion from
the outflow force to the mass accretion rate. The mass loss and accretion
ratio $\epsilon=\dot{M}_{\rm W}/\dot{M}_{\rm acc}$, the jet velocity $V_{W}$,
and the efficiency $f_{\rm ent}$ can vary from $\sim 0.01$ to $\sim 0.5$
(Ellerbroek et al., 2013; Podio et al., 2021), $\sim 30$ to $\sim 160~{}{\rm
km~{}s^{-1}}$ (jets in SiO $5-4$; Podio et al., 2021), and 0.1 to 0.25 (André
et al., 1999), respectively, in the protostellar phase. A theoretical model
called X-wind suggests that $\epsilon$ and $V_{W}$ are anti-correlated and the
factor of $\epsilon V_{W}$ varies only around $\sim 50$ to $\sim 70~{}{\rm
km~{}s^{-1}}$ (Najita & Shu, 1994) in the T Tauri phase. However, these
velocities are $\gtrsim 4$ times higher than the typical velocity for
protostars, $\sim 15~{}{\rm km~{}s^{-1}}$, as adopted in Hsieh et al. (2016),
and the anti-correlation is not observationally confirmed in the protostellar
phase. Those large uncertainties imply that $\dot{M}_{\rm acc}$ calculated by
Hsieh et al. (2016) could be an order of magnitude lower. The abovementioned
extreme values provide a range of $4\times 10^{-8}$ to $2\times
10^{-5}~{}M_{\sun}~{}{\rm yr}^{-1}$. For this reason, if the central stellar
mass is estimated from the Keplerian rotation, the mass accretion rate could
also be estimated better than from the outflow observations.
The updated mass accretion rate of I16253, $\sim 1\times
10^{-7}~{}M_{\sun}~{}{\rm yr}^{-1}$, cannot provide $M_{*}=0.14~{}M_{\sun}$
within the lifetime of Class 0 $\lesssim$0.2 Myr. In other words, the updated
central stellar mass is large against the luminosity of I16253, as the
required time can be estimated to be $M_{*}/\dot{M}_{\rm
acc}=GM_{*}^{2}/R_{*}L_{\rm bol}\sim 1$ Myr. This suggests that I16253 likely
experienced an accretion burst (or stronger accretion) in the past. The
presence of an accretion burst is supported by the episodic mass ejection as
seen in the 12CO PV diagram along the outflow axis (Section 3.2). The Class 0
age is also long enough to experience bursts with a typical interval of 2400
years in the Class 0 phase (Hsieh et al., 2019). Once an accretion burst
occurs, the luminosity of a protostar increases resulting in higher
temperatures in the surrounding envelope at a given radius. This causes CO
molecules to sublimate from the icy grain mantles in an extended region of the
envelope. After the protostar has returned to its quiescent state and the
luminosity decreased again, the molecules remain in the gas-phase for a period
of time before freezing out again. This freeze-out time-scale depends on the
density at a given radius (Rodgers & Charnley, 2003) but is typically of order
$10^{4}$ years where CO sublimation due to an accretion burst occurs around
Solar-type protostars (e.g., Jørgensen et al., 2015). From this point of view,
the size of the C18O emission observed in I16253 (Figure 5a), $\gtrsim 150$
au, also supports that an accretion burst happened because this size is much
larger than expected from the current luminosity of this protostar: The C18O
emission radius is predicted to be $\sim 30$ au with the current luminosity of
I16253, $0.16~{}L_{\sun}$ (Jørgensen et al., 2015) (see also Lee, 2007).
Based on the measured $M_{*}$ and the above discussion about the mass
accretion and the luminosity, we conclude that I16253 is not a proto-BD even
though its luminosity and mass accretion rate are similar to those of the two
proto-BD candidates. The core (or envelope) mass of I16253 is estimated to be
$\sim 1~{}M_{\sun}$ from 1.1 mm observations with Bolocam on the CSO telescope
at an angular resolution of $31\arcsec$ (Young et al., 2006) and $8~{}\micron$
observations with IRAC on the Spitzer telescope at an angular resolution of
$2\arcsec$ (Tobin et al., 2010). If this envelope mass accretes onto I16253
with a typical star formation efficiency ($30\%\pm 10\%$ calculated from the
dense core mass function and initial mass function; Alves et al., 2007), a
mass of 0.2 to 0.4 $M_{\sun}$ will be added to the central stellar mass. This
also supports an idea that I16253 will ultimately obtain mass high enough to
fuse hydrogen (i.e., $M_{*}>0.08~{}M_{\sun}$).
The Keplerian disk around I16253 has been kinematically identified for the
first time. Without any clear identification of the Keplerian disk, previous
works have attempted to estimate the central stellar mass using the infalling
motion or the outflow force (through the mass accretion rate) in I16253 and
suggested this protostar as a proto-BD candidate. Similarly, the central
stellar mass of proto-BD candidates was estimated in those methods in some
previous works (Section 1), without identifying a Keplerian disk. Our study
demonstrates that a proto-BD and a very low mass protostar must be identified
through the dynamical central stellar mass derived from the Keplerian rotation
of its disk, rather than its infall motion or outflow force.
The accurate mass estimation enables us to compare physical quantities in the
I16253 system with scaling relations among young stellar objects. For example,
its disk radius, $16-19$ au, appears consistent with or slightly lower than
the scaling relation between the disk radius and the central stellar mass
found with the protostellar sample of Yen et al. (2017). Its mass accretion
rate, $(0.9-1.3)\times 10^{-7}~{}M_{\sun}~{}{\rm yr}^{-1}$, and disk mass,
$\gtrsim 2\times 10^{-3}~{}M_{\sun}$, are consistent with the scaling relation
found with the Class I sample of Fiorellino et al. (2022). More observations
around the BD mass threshold with accurate mass estimations will be necessary
to determine whether these scaling relations hold down to the BD mass regime
and bridge star formation and BD formation.
## 6 Conclusions
As a part of the ALMA large project eDisk, we have observed the Class 0
protostar IRAS16253-2429, which has been suggested to be a proto-brown dwarf
candidate in previous works, in the 1.3 mm continuum, 12CO $J=2-1$, C18O
$J=2-1$, 13CO $J=2-1$, SO $J_{N}=6_{5}-5_{4}$, and other molecular lines at an
angular resolution of $0\farcs 07$ ($\sim 10$ au). Our results provide a
typical picture of protostars with a very low stellar mass close to the brown
dwarf threshold (Figure 18). The main results are summarized below.
1. 1.
The continuum emission shows structures from a $\sim 600$-au scale down to a
$\sim 15$-au scale. The main component shows a disk-like structure with a
radius of $\sim 20$ au. The emission is extended to the southeast along the
major axis. These extensions can be interpreted as an enhancement due to a
streamer from the east and the near-side (southwestern side) wall of the disk-
like structure.
2. 2.
The 12CO emission overall traces a clear bipolar outflow up to a $\sim
3000$-au scale. The outflow suggests episodic mass ejections. Furthermore, the
12CO emission on the midplane shows a velocity gradient along the disk’s major
axis, implying rotation of the disk.
3. 3.
We analyzed the 12CO major-axis position-velocity diagram by the edge and
ridge methods and identified a Keplerian disk in IRAS16253-2429 for the first
time. From this identification of the Keplerian rotation in both methods, the
central stellar mass is estimated to be $0.12-0.17~{}M_{\sun}$. This mass
leads us to conclude that IRAS16253-2429 is unlikely a proto-brown dwarf but a
very low mass protostar.
4. 4.
The C18O and 13CO emissions trace the infalling and rotating envelope as
reported in previous observational works. The major- and minor-axis position-
velocity diagrams in these lines are consistent with the UCM envelope model
with the central stellar mass of $0.14~{}M_{\sun}$ and the disk (centrifugal)
radius of $r\sim 16$ au. Their moment 0 maps exhibit stronger emission
intensities on the eastern peak, which could result from the streamer
mentioned below.
5. 5.
The SO emission shows a different velocity structure from the CO
isotopologues. Its double peaks in the moment 0 map and linear velocity
gradient in the major-axis position-velocity diagram suggest that this
emission traces a ring ($r\sim 28$ au) due to the accretion shock between the
disk and the envelope. This emission also shows a streamer from the eastern
side, which can be explained with a ballistic, parabolic flow at $30\arcdeg$
above the midplane extracted from the UCM envelope model.
## Acknowledgements
This paper makes use of the following ALMA data: ADS/JAO.ALMA#2019.1.00261.L
and ADS/JAO.ALMA#2019.A.00034.S. ALMA is a partnership of ESO (representing
its member states), NSF (USA) and NINS (Japan), together with NRC (Canada),
MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the
Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and
NAOJ. The National Radio Astronomy Observatory is a facility of the National
Science Foundation operated under cooperative agreement by Associated
Universities, Inc. W.K. was supported by the National Research Foundation of
Korea (NRF) grant funded by the Korea government (MSIT)
(NRF-2021R1F1A1061794). N.O. acknowledges support from National Science and
Technology Council (NSTC) in Taiwan through the grants NSTC 109-2112-M-001-051
and 110-2112-M-001-031. J.K.J. acknowledge support from the Independent
Research Fund Denmark (grant No. 0135-00123B). J.J.T. acknowledges support
from NASA XRP 80NSSC22K1159. N.O. acknowledges support from National Science
and Technology Council (NSTC) in Taiwan through the grants NSTC
109-2112-M-001-051 and 110-2112-M-001-031. Y.A. acknowledges support by NAOJ
ALMA Scientific Research Grant code 2019-13B, Grant-in-Aid for Scientific
Research (S) 18H05222, and Grant-in-Aid for Transformative Research Areas (A)
20H05844 and 20H05847. IdG acknowledges support from grant
PID2020-114461GB-I00, funded by MCIN/AEI/10.13039/501100011033. PMK
acknowledges support from NSTC 108-2112- M-001-012, NSTC 109-2112-M-001-022
and NSTC 110-2112-M-001-057. SPL and TJT acknowledge grants from the National
Science and Technology Council of Taiwan 106-2119-M-007-021-MY3 and
109-2112-M-007-010-MY3. J.K.J. acknowledges support from the Independent
Research Fund Denmark (grant No. 0135-00123B). C.W.L. is supported by the
Basic Science Research Program through the National Research Foundation of
Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-
2019R1A2C1010851), and by the Korea Astronomy and Space Science Institute
grant funded by the Korea government (MSIT; Project No. 2022-1-840-05). JEL
was supported by the National Research Foundation of Korea (NRF) grant funded
by the Korean government (MSIT) (grant number 2021R1A2C1011718). ZYL is
supported in part by NASA 80NSSC20K0533 and NSF AST-1910106. ZYDL acknowledges
support from NASA 80NSSC18K1095, the Jefferson Scholars Foundation, the NRAO
ALMA Student Observing Support (SOS) SOSPA8-003, the Achievements Rewards for
College Scientists (ARCS) Foundation Washington Chapter, the Virginia Space
Grant Consortium (VSGC), and UVA research computing (RIVANNA). LWL
acknowledges support from NSF AST-2108794. S.N. acknowledges support from the
National Science Foundation through the Graduate Research Fellowship Program
under Grant No. 2236415. Any opinions, findings, and conclusions or
recommendations expressed in this material are those of the authors and do not
necessarily reflect the views of the National Science Foundation. R.S.
acknowledge support from the Independent Research Fund Denmark (grant No.
0135-00123B). S.T. is supported by JSPS KAKENHI grant Nos. 21H00048 and
21H04495, and by NAOJ ALMA Scientific Research grant No. 2022-20A. JPW
acknowledges support from NSF AST-2107841. H.-W.Y. acknowledges support from
the National Science and Technology Council (NSTC) in Taiwan through the grant
NSTC 110-2628-M-001-003-MY3 and from the Academia Sinica Career Development
Award (AS-CDA-111-M03).
## Appendix A Other molecular lines
Figure 19 shows the moment 0 and 1 maps of the lines observed in the eDisk
project toward I16253, except for the 12CO, C18O, 13CO, and SO lines. These
lines were observed from the wide spectral windows otherwise intended for
continuum measurements. The three H2CO lines have different upper state
energies, depending on the rest frequencies (10.5 K; 218.22 GHz, 57.6 K;
218.47 GHz, 57.6 K; 218.76 GHz). The three cyclic C3H2 lines also have
different upper state energies (28.2 K; 217.82 GHz, 25.0 K; 217.94 GHz, 24.9
K; 218.16 GHz). The CH3OH, DCN, and SiO lines have upper state energies of
35.0, 10.4 and 20.8 K, respectively. In comparison, the 12CO, C18O, 13CO, and
SO lines have upper state energies of 5.5, 5.3, 5.3, and 24.4 K, respectively.
All the panels in Figure 19 are made from three channels centered at the
systemic velocity ($V_{\rm LSR}=2.66-5.34~{}{\rm km~{}s^{-1}}$) and show the
same spatial and velocity ranges.
Figure 19: Moment 0 and 1 maps in 5 other molecular lines (9 transitions)
observed in the ALMA observations of the eDisk project. These images are made
with the robust parameter of 2. The emission is integrated from 2.66 to
$5.34~{}{\rm km~{}s^{-1}}$ (3 channels) for all lines. The contour levels are
in $3\sigma$ steps from $3\sigma$, where $1\sigma$ is $1.5~{}{\rm
mJy~{}beam^{-1}}~{}{\rm km~{}s^{-1}}$. The ellipse at the lower left corner in
each panel denotes the synthesized beam, $\sim 0\farcs 38\times 0\farcs 27\
(78\arcdeg)$. The diagonal lines are the major and minor axes of the continuum
emission, P.A.$=113\arcdeg$ and $23\arcdeg$.
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|
# Gravitational Portals with Non-Minimal Couplings
Simon Clérya<EMAIL_ADDRESS>Yann Mambrinia,b
<EMAIL_ADDRESS>Keith A. Olivec<EMAIL_ADDRESS>Andrey
Shkerinc<EMAIL_ADDRESS>Sarunas Vernerc<EMAIL_ADDRESS>a Université Paris-
Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France b CERN, Theoretical Physics
Department, Geneva, Switzerland c William I. Fine Theoretical Physics
Institute, School of Physics and Astronomy, University of Minnesota,
Minneapolis, MN 55455, USA
###### Abstract
We consider the effects of non-minimal couplings to curvature of the form
$\xi_{S}S^{2}R$, for three types of scalars: the Higgs boson, the inflaton,
and a scalar dark matter candidate. We compute the abundance of dark matter
produced by these non-minimal couplings to gravity and compare to similar
results with minimal couplings. We also compute the contribution to the
radiation bath during reheating. The main effect is a potential augmentation
of the maximum temperature during reheating. A model independent limit of
$\mathcal{O}(10^{12})$ GeV is obtained. For couplings
$\xi_{S}\gtrsim\mathcal{O}(1)$, these dominate over minimal gravitational
interactions.
††preprint: UMN–TH–4116/22††preprint: FTPI–MINN–22/07††preprint: CERN-
TH-2022-025
## I Introduction
Promoting a field theory Lagrangian from a Lorentz-invariant one to a
generally-covariant one necessarily leads to an interaction between the fields
of the theory and the gravitational field. In the case of a scalar field, $S$,
the natural generalization of this minimal interaction scenario is to
introduce a non-minimal coupling term of the form
$\propto\xi_{S}S^{2}R\;.$ (1)
Here $R$ is the Ricci scalar and $\xi_{S}$ is a non-minimal coupling constant.
This non-minimal coupling to gravity proved to be useful in many applications
to cosmology. Examples include Higgs inflation Bezrukov:2007ep ;
Lebedev:2021xey , where $S$ is associated with the Higgs field degree of
freedom $h$ — the only scalar degree of freedom in the Standard Model,
preheating Ema:2016dny , where $S$ is associated with the inflaton field
$\phi$, and non-perturbative production of dark matter nonminprod , where $S$
represents the scalar dark matter particle $X$.
In the general case, when the fields $\phi$, $h$, and $X$ are all different,
the question arises as to what extent they must interact with each other in
order to successfully reheat the Universe and generate the right amount of
dark matter. Recent studies have shown that interactions via gravity alone, to
which the fields are coupled minimally, is enough for these purposes. Indeed,
the perturbative gravitational production of dark matter through graviton
exchange can play a dominant role during reheating with processes involving
the inflaton MO ; CMOV ; Barman:2021ugy as well as thermal bath particles
CMOV ; Haque:2021mab . Further, the minimal gravitational coupling can lead to
the completion of the reheating process for certain types of the inflationary
potential, $V(\phi)\sim\phi^{k}$ with $k>2$ CMOV ; Haque:2022kez . Thus,
gravity is strong enough to mediate perturbative channels of reheating and
dark matter production.
The purpose of this work is to study how the inclusion of the non-minimal
coupling terms of the form (1) affect the gravitational production of dark
matter and radiation during reheating. Note that the presence of these terms
is unavoidable: if there were no such couplings at tree level, they would
still be generated by quantum corrections Callan:1970ze . We study particle
production in the processes $hh\rightarrow XX$, $\phi\phi\rightarrow hh$, and
$\phi\phi\rightarrow XX$ which are induced by the non-minimal couplings. Here
$\phi$ represents the inflaton background oscillating around its minimum after
the end of inflation gravprod . Since the scalar fields couple directly to the
curvature scalar $R$, the oscillating background causes the effective masses
of the fields to change non-adiabatically and leads to particle production.
This regime of particle creation has been considered in several different
contexts, including gravitational production of scalar gravscalar ; ema ,
fermion gravferm , and vector dark matter gravvector .
Our main interest is to compare the (dark) matter production channels induced
by the non-minimal couplings with the production via the s-channel graviton
exchange that sets minimal possible production rates. We will see for which
values of the couplings the rates are enhanced, and what are the consequences
on the dark matter density or the temperature attained during reheating.
Throughout the work we adopt the Starobinsky inflationary potential staro ,
although our results are largely independent of the particular form of the
potential. As for the potentials for the fields $h$ and $X$, we take them to
be renormalizable polynomials. We also assume no direct interaction between
$\phi$, $h$, and $X$.
Working in the perturbative regime implies that the non-minimal couplings must
satisfy $|\xi_{S}|\ll M_{P}^{2}/\langle S\rangle^{2}$, where $\langle
S\rangle$ is the vacuum expectation value of $S=\phi,h,X$. The value of
$\xi_{h}$ is constrained from collider experiments as $|\xi_{h}|\lesssim
10^{15}$ higgscons1 .111Note that in the case of Higgs inflation, $\xi_{h}$ is
fixed from CMB measurements Bezrukov:2007ep . Furthermore, the lower bound on
$\xi_{h}$ comes from the fact that the Standard Model electroweak vacuum may
not be absolutely stable HiggsStab . To prevent the vacuum decay due to
quantum fluctuations during inflation fluc , the effective mass of the Higgs
field induced by the non-minimal coupling must be large enough; this gives
$\xi_{h}\gtrsim 10^{-1}$ higgstree ; higgsloop (see also Markkanen:2018pdo
).222This estimate assumes no new physics interfering the RG running of the
Higgs self-coupling constant until inflationary energy scales.
The paper is organized as follows: The framework for our computation is
presented in Section II. We discuss non-minimal gravitational couplings of the
inflaton, the Higgs boson, and a dark matter scalar in detail. We calculate
the dark matter production rates either from scattering in the thermal bath or
from oscillations in the inflaton condensate. We compare similar processes
obtained from the minimal gravitational particle production. We choose the
Starobinsky model of inflation and discuss the reheating epoch when the
inflaton begins oscillating. In Section III we discuss the resulting abundance
of dark matter produced from the thermal bath and directly from scattering of
the inflaton condensate. We also compute the effects of the non-minimal
couplings on the maximum temperature attained during reheating. We then
compare different processes in Section IV, before summarizing our results in
Section V.
## II The framework
### II.1 Scalar-gravity Lagrangian
The theory we consider comprises 3 scalar fields non-minimally coupled to
gravity: the inflaton $\phi$, the Higgs field333We consider the Higgs boson as
a surrogate for any additional scalars with Standard Model couplings. $H$, for
which we adopt the Unitary gauge, $H=(0,h)^{T}/\sqrt{2}$, and the dark matter
candidate $X$. The relevant part of the action takes the form444The metric
signature is chosen as $(+,-,-,-)$.
$\mathcal{S}\;=\;\int
d^{4}x\sqrt{-\tilde{g}}\left[-\frac{M_{P}^{2}}{2}\Omega^{2}\tilde{R}+\mathcal{L}_{\phi}+\mathcal{L}_{h}+\mathcal{L}_{X}\right]$
(2)
with the conformal factor $\Omega^{2}$ given by
$\Omega^{2}\;\equiv\;1+\frac{\xi_{\phi}\phi^{2}}{M_{P}^{2}}+\frac{\xi_{h}h^{2}}{M_{P}^{2}}+\frac{\xi_{X}X^{2}}{M_{P}^{2}}\,.$
(3)
Here $M_{P}=2.4\times 10^{18}\,\rm{GeV}$ is the reduced Planck mass, and the
tilde used in Eq. (2) indicates that the theory is considered in the Jordan
frame. For the scalar field Lagrangians we have
$\mathcal{L}_{S}\;=\frac{1}{2}\tilde{g}^{\mu\nu}\partial_{\mu}S\partial_{\nu}S-V_{S}\,,~{}~{}~{}S=\phi,h,X\,.$
(4)
Next, we specify the scalar field potentials. For a model of inflation, we
choose the well-motivated Starobinsky model for which staro
$V_{\phi}\;=\;\frac{3}{4}m_{\phi}^{2}M_{P}^{2}\left(1-e^{-\sqrt{\frac{2}{3}}\frac{\phi}{M_{P}}}\right)^{2}\,.$
(5)
In what follows, we work in the perturbative regime with $\phi\ll M_{P}$,
hence the potential is approximated as
$V_{\phi}\;\simeq\;\frac{1}{2}m_{\phi}^{2}\phi^{2}\,.$ (6)
The inflaton mass, $m_{\phi}$, is fixed by the amplitude of scalar
perturbations inferred from CMB measurements Planck ; for the potential (5)
this gives $m_{\phi}=3\times 10^{13}$ GeV building .
The potential for the Higgs field is taken as follows
$V_{h}\;=\;\frac{1}{2}m_{h}^{2}h^{2}+\frac{1}{4}\lambda_{h}h^{4}\,.$ (7)
Here $m_{h}$ and $\lambda_{h}$ are the Higgs mass and quartic coupling,
correspondingly. Note that both parameters undergo the renormalization group
(RG) running. In what follows we take a weak scale mass, which is a good
approximation at the time of reheating and our results are insensitive to
$\lambda_{h}$. Finally, the dark matter potential is simply given by
$V_{X}\;=\;\frac{1}{2}m_{X}^{2}X^{2}\,.$ (8)
To study the reheating in the theory (2), it is convenient to remove the non-
minimal couplings by performing the redefinition of the metric field. Leaving
the details to Appendix A, we write the action (2) in the Einstein frame,
$\begin{split}\mathcal{S}\;=\;\int
d^{4}x\sqrt{-g}&\left[-\frac{M_{P}^{2}}{2}R+\frac{1}{2}K^{ij}g^{\mu\nu}\partial_{\mu}S_{i}\partial_{\nu}S_{j}\right.\\\
&\qquad\qquad\qquad\left.-\frac{V_{\phi}+V_{h}+V_{X}}{\Omega^{4}}\right]\,.\end{split}$
(9)
Here the indices $i,j$ enumerate the fields $\phi,h,X$, and the kinetic
function is given by
$K^{ij}\;=\;6\frac{\partial\log\Omega}{\partial
S_{i}}\frac{\partial\log\Omega}{\partial
S_{j}}+\frac{\delta^{ij}}{\Omega^{2}}\,.$ (10)
Note that the scalar field kinetic term is not canonical. In general, it is
impossible to make a field redefinition that would bring it to the canonical
form, unless all three non-minimal couplings vanish.555 Such a redefinition
exists if the three-dimensional manifold spanned by the fields $\phi$, $h$ and
$X$ is flat. One can show that it is not the case if at least one of the
couplings is non-zero. For the theory (9) to be well-defined, the kinetic
function (10) must be positive-definite. Computing the eigenvalues, one
arrives at the condition
$\Omega^{2}>0\,,$ (11)
which is satisfied automatically for positive values of the couplings. Note
that the negative couplings are also allowed for certain scalar field
magnitudes.
In what follows, we will be interested in the small-field limit
$\frac{|\xi_{\phi}|\phi^{2}}{M_{P}^{2}}\;,~{}~{}\frac{|\xi_{h}|h^{2}}{M_{P}^{2}}\;,~{}~{}\frac{|\xi_{X}|X^{2}}{M_{P}^{2}}\ll
1\,.$ (12)
We can expand the kinetic and potential terms in the action (9) in powers of
$M_{P}^{-2}$. We obtain a canonical kinetic term for the scalar fields and
deduce the leading-order interactions induced by the non-minimal couplings.
The latter can be brought to the form
$\mathcal{L}_{\rm{non-min.}}\;=\;-\sigma_{hX}^{\xi}h^{2}X^{2}-\sigma_{\phi
X}^{\xi}\phi^{2}X^{2}-\sigma_{\phi h}^{\xi}\phi^{2}h^{2}\,,$ (13)
where the $\sigma_{ij}^{\xi}$ are functions of the couplings $\xi_{i}$,
$\xi_{j}$, the masses $m_{i}$, $m_{j}$, and the Mandelstam variables; see
Appendix A for details.
The small-field approximation (12) implies the bound $\sqrt{|\xi_{S}|}\lesssim
M_{P}/\langle S\rangle$ with $S=\phi,h,X$. Since the inflaton value at the end
of inflation is $\phi_{\rm end}\sim M_{P}$ and afterwards
$\langle\phi^{2}\rangle\sim a^{-3}$, where $a$ is the cosmological scale
factor, then $|\xi_{\phi}|\lesssim(a/a_{\rm end})^{3}$. In particular, at the
onset of inflaton oscillations
$|\xi_{\phi}|\lesssim 1\,.$ (14)
Note that since our calculations involve the effective couplings $\sigma_{\phi
X}^{\xi}$ ($\sigma_{\phi h}^{\xi}$), which depend both on $\xi_{\phi}$ and
$\xi_{X}$ ($\xi_{h}$), the relatively small value of $|\xi_{\phi}|$ can, in
principle, be compensated by a large value of the other couplings.
In Fig. 1, we show the scattering processes obtained from the Lagrangian (13).
These contribute to reheating (when $h$ is in the final state) and dark matter
production (when $X$ is in the final state).
Figure 1: Feynman diagram for the 4-point interactions between the inflaton
$\phi$, the dark matter scalar candidate $X$, and the Higgs boson $h$, given
by the Lagrangian (13).
Finally, in evaluating the cosmological parameters, it is important to stay
within the validity of the low-energy theory. The cutoff of the theory can be
estimated as (see, e.g., higgscons2 )
$\Lambda\sim\frac{M_{P}}{\text{max}_{i}\>|\xi_{i}|}\,.$ (15)
In particular, the temperature of reheating must not exceed $\Lambda$.
### II.2 Graviton exchange
Let us first consider the case of vanishing $\xi_{\phi,\,h,\,X}$, i.e., the
case of the minimal coupling of the scalar fields to gravity MO ; CMOV ; ema ;
Garny:2015sjg ; Tang:2017hvq ; Chianese:2020yjo ; Redi:2020ffc . It was argued
in MO ; CMOV that the interaction between the dark and visible sectors
induced by gravity leads to unavoidable contributions to reheating and dark
matter production, in the thermal bath or via the scattering of the inflaton
condensate, through the graviton exchange processes shown in Fig. 2. It is
therefore important to compare the minimal gravitational particle production
to similar processes obtained from the Lagrangian in Eq. (13) with non-minimal
couplings.
Figure 2: Feynman diagram for the (dark) matter production through the
gravitational scattering of the inflaton or the Higgs boson from the thermal
bath.
To study the universal gravitational interactions in minimally coupled
gravity, we expand the space-time metric around flat space using
$g_{\mu\nu}\simeq\eta_{\mu\nu}+2h_{\mu\nu}/M_{P}$, where $h_{\mu\nu}$ is the
canonically-normalized perturbation. The gravitational interactions are
characterized by the following Lagrangian,
${\cal L}_{\rm
min.}=-\frac{1}{M_{P}}h_{\mu\nu}\left(T^{\mu\nu}_{h}+T^{\mu\nu}_{\phi}+T^{\mu\nu}_{X}\right)\,,$
(16)
where the stress-energy tensor is given by
$T^{\mu\nu}_{S}\;=\;\partial^{\mu}S\partial^{\nu}S-g^{\mu\nu}\left[\frac{1}{2}\partial^{\alpha}S\partial_{\alpha}S-V_{S}\right]\,.$
(17)
Note that in this work, we consider only the Higgs field in the visible
sector. Generalization to the complete spectrum of the Standard Model is
straightforward, and we leave it for future work.
For models with minimally coupled gravity, the processes
$\phi/h(p_{1})+\phi/h(p_{2})\rightarrow{h}/X(p_{3})+{h}/X(p_{4})$ can be
parametrized by
$\mathcal{M}^{00}\propto
M_{\mu\nu}^{0}\Pi^{\mu\nu\rho\sigma}M_{\rho\sigma}^{0}\;,$ (18)
where the graviton propagator for the canonically-normalized field
$h_{\mu\nu}$ with exchange momentum $k=p_{1}+p_{2}$ is given by
$\Pi^{\mu\nu\rho\sigma}(k)=\frac{\eta^{\mu\rho}\eta^{\nu\sigma}+\eta^{\mu\sigma}\eta^{\nu\rho}-\eta^{\mu\nu}\eta^{\rho\sigma}}{2k^{2}}\,,$
(19)
and the partial amplitude, $M_{\mu\nu}^{0}$, is given by
$\displaystyle
M_{\mu\nu}^{0}=\frac{1}{2}\left[p_{1\mu}p_{2\nu}+p_{1\nu}p_{2\mu}-\eta_{\mu\nu}p_{1}\cdot
p_{2}-\eta_{\mu\nu}V_{S}^{\prime\prime}\right]\,,$ (20)
with analogous expression for the final state in terms of outgoing momenta
$p_{3,4}$ and the final state potential. In Fig. 2 we show the s-channel
graviton exchange scattering obtained from the Lagrangian (16) for the
production of dark matter from either the Higgs field or the inflaton
condensate as well as the reheating process (the production of Higgs bosons
from the inflaton condensate).
### II.3 Production rates
In this work, we consider three processes:
1. A.
The production of dark matter from the scattering of thermal Higgs bosons
(assuming reheating is produced by inflaton decay). In this case, the dark
matter is populated via a freeze-in mechanism throughout the reheating period.
2. B.
The production of dark matter from direct excitations of the inflaton
condensate. This process, which can be viewed as gravitational inflaton
scattering, is independent of the presence of a thermal bath.
3. C.
The creation of a radiative bath at the start of reheating arising from the
Higgs boson production through gravitational inflaton scattering. Since such a
process is unavoidable in minimally coupled gravity, it is interesting to know
when such a process becomes dominant in models with non-minimal couplings
$\xi_{i}$.
The thermal dark matter production rate $R(T)$ for the process $hh\rightarrow
XX$ can be calculated from666We include the symmetry factors associated with
identical initial and final states in the definition of
$|\overline{{\cal{M}}}|^{2}$, and a factor of 2 is explicitly included in the
definition of the rate to account for the production of 2 identical particles.
gravitino
$R(T)=\frac{2\times N_{h}}{1024\pi^{6}}\int
f_{1}f_{2}E_{1}\mathop{}\\!\mathrm{d}E_{1}E_{2}\mathop{}\\!\mathrm{d}E_{2}\mathop{}\\!\mathrm{d}\cos\theta_{12}\int|\overline{{\cal
M}}|^{2}\mathop{}\\!\mathrm{d}\Omega_{13}\,,$ (21)
where $E_{i}$ is the energy of particle $i=1,2$, $\theta_{13}$ and
$\theta_{12}$ are the angles formed by momenta ${\bf{p}}_{1,3}$ and
${\bf{p}}_{1,2}$, respectively. $N_{h}=4$ is the number of internal degrees of
freedom for 1 complex Higgs doublet, $|\overline{{\cal M}}|^{2}$ is the matrix
amplitude squared with all symmetry factors included. This accounts for the
explicit factor of 2 in the numerator of Eq. (21). The thermal distribution
function of the incoming Higgs particles is given by the Bose-Einstein
distribution
$f_{i}=\frac{1}{e^{E_{i}/T}-1}\,.$ (22)
The rate for minimal gravitational interactions from Eq. (16) was derived in
CMOV ; Bernal:2018qlk . The rate we use here differs in two respects. As noted
earlier, we only include Higgs scalars in the initial state whereas in CMOV ;
Bernal:2018qlk , all Standard Model particle initial states were included.
Secondly, we keep terms depending on the dark matter mass which had not
previously been taken into account. This allows us to consider dark matter
masses approaching the inflaton mass and/or the reheating temperature.
For minimal (non-minimal) gravitational interactions, we find that the thermal
dark matter production rate can be expressed as
$\displaystyle R_{X}^{T,\,(\xi)}(T)$ $\displaystyle=$
$\displaystyle\beta_{1}^{(\xi)}\frac{T^{8}}{M_{P}^{4}}+\beta_{2}^{(\xi)}\frac{m_{X}^{2}T^{6}}{M_{P}^{4}}+\beta_{3}^{(\xi)}\frac{m_{X}^{4}T^{4}}{M_{P}^{4}}\,,$
(23)
where the coefficients $\beta_{1,\,2,\,3}^{(\xi)}$ are given in Appendix B by
Eqs. (84-86) (Eqs. (80-82)). The ratio of the non-minimal to minimal rate is
shown in Fig. 3. However, we note that when $\xi_{i}\sim\mathcal{O}(1)$ both
rates are comparable and interference effects become significant. The full
coefficients $\beta_{1,\,2,\,3}$ including interference are given by Eqs.
(87-89) from Appendix B. We leave the comparison of the effects on dark matter
production from the two rates for the next section.
Figure 3: Contours of the ratio of the dark matter production rates from the
thermal bath based on non-minimal gravitational interactions to those based on
minimal interactions. The ratio is displayed in the $(\xi_{h},\xi_{X})$ plane.
Note that as discussed in the Introduction, negative values of $\xi_{h}$ may
require new physics (such as supersymmetry) to stabilize the Higgs vacuum.
The rate for dark matter produced from inflaton oscillations of the inflaton
condensate for a potential of the form $V=\lambda\phi^{k}$ were considered in
detail in GKMO2 ; CMOV . The time-dependent inflaton can be written as
$\phi(t)=\phi_{0}(t)\mathcal{P}(t)$, where $\phi_{0}(t)$ is the time-dependent
amplitude that includes the effects of redshift and $\mathcal{P}(t)$ describes
the periodicity of the oscillation. The dark matter production rate is
calculated by writing the potential in terms of the Fourier modes of the
oscillations Ichikawa:2008ne ; Kainulainen:2016vzv ; GKMO2 ; CMOV
$V(\phi)=V(\phi_{0})\sum_{n=-\infty}^{\infty}{\cal P}_{n}^{k}e^{-in\omega
t}=\rho_{\phi}\sum_{n=-\infty}^{\infty}{\cal P}_{n}^{k}e^{-in\omega t}\,.$
(24)
For $k=2$ (the only case considered here), the frequency of oscillation is
simply, $\omega=m_{\phi}$.
The rate generated by non-minimal couplings can be readily calculated using
the Lagrangian (13), which leads to
$R_{X}^{\phi,\,\xi}\;=\;\frac{2\times\sigma_{\phi
X}^{\xi~{}2}}{\pi}\frac{\rho_{\phi}^{2}}{m_{\phi}^{4}}\Sigma_{0}^{k}\,,$ (25)
where
$\Sigma_{0}^{k}=\sum_{n=1}^{\infty}|{\cal
P}^{k}_{n}|^{2}\sqrt{1-\frac{4m_{X}^{2}}{E_{n}^{2}}}\,,$ (26)
and $E_{n}=n\omega$ is the energy of the $n$-th inflaton oscillation mode. For
$k=2$, only the second Fourier mode in the sum contributes, with
$\sum|\mathcal{P}^{2}_{n}|^{2}=\frac{1}{16}$. Thus, the rate becomes
$R_{X}^{\phi,\,\xi}\;=\;\frac{2\times\sigma_{\phi
X}^{\xi~{}2}}{16\pi}\frac{\rho_{\phi}^{2}}{m_{\phi}^{4}}\sqrt{1-\frac{m_{X}^{2}}{m_{\phi}^{2}}}\,,$
(27)
where $\rho_{\phi}$ is the energy density of the inflaton and the interaction
term $\sigma_{\phi X}^{\xi}$ is given in Appendix A by Eq. (75).
It was shown in MO that the dark matter production rate through the exchange
of a graviton, computed from the partial amplitude (18), is
$R^{\phi}_{X}=\frac{2\times\rho_{\phi}^{2}}{256\pi
M_{P}^{4}}\left(1+\frac{m_{X}^{2}}{2m^{2}_{\phi}}\right)^{2}\sqrt{1-\frac{m_{X}^{2}}{m_{\phi}^{2}}}\,,$
(28)
which can be written in the same form as (27) by defining an effective
coupling $\sigma_{\phi X}$
$\sigma_{\phi
X}\;=\;-\frac{m_{\phi}^{2}}{4M_{P}^{2}}\left(1+\frac{m_{X}^{2}}{2m_{\phi}^{2}}\right)\,.$
(29)
A comparison of the non-minimal to minimal rates for the production of dark
matter from inflaton scattering is shown in Fig. 4.
Figure 4: Contours of the ratio of the dark matter production rates from
oscillations in the inflaton condensate based on non-minimal gravitational
interactions to those based on minimal interactions. The ratio is displayed in
the $(\xi_{\phi},\xi_{X})$ plane.
For the production of Higgs bosons through inflaton condensate scattering, we
follow a similar procedure, and from the Lagrangian (13) we find
$R_{h}^{\phi,\,\xi}\;\simeq\;N_{h}\frac{2\times\sigma_{\phi
h}^{\xi~{}2}}{16\pi}\frac{\rho_{\phi}^{2}}{m_{\phi}^{4}}\,,$ (30)
where we assumed that $m_{h}\ll m_{\phi}$, $N_{h}=4$ is the number of internal
degrees of freedom for 1 complex Higgs doublet, and $\sigma_{\phi h}^{\xi}$ is
given in Appendix A by Eq. (76).
On the other hand, it was argued in CMOV that the scattering
$\phi\phi\rightarrow hh$ through the graviton exchange can also be
parameterized by an effective coupling
$\mathcal{L}_{h}=-\sigma_{\phi h}\phi^{2}h^{2}\,,$ (31)
with
$\sigma_{\phi h}\;=\;-\frac{m_{\phi}^{2}}{4M_{P}^{2}}\,,$ (32)
and the rate $R_{h}^{\phi}$ is given by the analogous expression to (30) with
$\sigma_{\phi h}^{\xi}$ replaced by $\sigma_{\phi h}$.
The full four-point coupling of course is given by the sum $\sigma_{\phi
h/X}^{\xi}+\sigma_{\phi h/X}$. However, except for values where the two are
similar, which occurs when $12\xi^{2}+5\xi\simeq\frac{1}{2}$ (assuming
$m_{X}\ll m_{\phi}$ and taking the $\xi_{i}$ to be equal to $\xi$), either the
minimal or the non-minimal contribution dominates. Thus, for the most part, we
will consider separately the minimal and non-minimal contributions. Note that
for two values of $\xi$ ($\xi\sim-1/2$ and 1/12) destructive interference
could occur causing the entire rate to vanish (at the tree level).
## III Particle Production with a Non-Minimal Coupling
Given the rates $R_{i}^{j}$ calculated in the previous section, we compute the
evolution for the gravitational (minimal and non-minimal) contribution to the
reheating processes and the dark matter density for the three reactions
outlined above.
### III.1 $h~{}h\rightarrow X~{}X$
The gravitational scattering of thermal Higgs bosons leads to the production
of massive scalar dark matter particles $X$. The dark matter number density
$n_{X}$ can be calculated from the classical Boltzmann equation
$\frac{dn_{X}}{dt}+3Hn_{X}=R^{T}_{X}\,,$ (33)
where $H=\frac{\dot{a}}{a}$ is the Hubble parameter and the right-hand side of
the equation represents the dark matter production rate. It is more practical
to rewrite the above equation in terms of the scale factor $a$ rather than the
parameters $t$ or $T$.
We proceed by introducing the comoving number density $Y_{X}=na^{3}$ and
rewriting the Boltzmann equation as
$\frac{dY_{X}}{da}=\frac{a^{2}R^{T}_{X}(a)}{H(a)}\,.$ (34)
Since the production rate (23) is a function of the temperature of the thermal
bath, it is necessary to determine the relation between $T$ and $a$ in order
to solve the Boltzmann equation as a function of the scale factor $a$. For the
Starobinsky potential in Eq. (5), at the end of inflation, the inflaton starts
oscillating about a quadratic minimum, and we find the following energy
conservation equations777For the inflaton scattering with
$V(\phi)\sim\phi^{k}$, where $k>2$, see Bernal:2019mhf ; GKMO1 ; GKMO2 ; GKMOV
; Haque:2021mab ; Haque:2022kez ; Ahmed:2021fvt .
$\displaystyle\frac{d\rho_{\phi}}{dt}+3H\rho_{\phi}=-\Gamma_{\phi}\rho_{\phi}\,,$
(35)
$\displaystyle\frac{d\rho_{R}}{dt}+4H\rho_{R}=\Gamma_{\phi}\rho_{\phi}\,,$
(36)
where $\rho_{\phi}$ and $\rho_{R}$ are the energy density of the inflaton and
radiation, respectively, $\Gamma_{\phi}$ is the inflaton decay rate, and for a
quadratic minimum, we are able to set the equation of state parameter
$w_{\phi}=\frac{P_{\phi}}{\rho_{\phi}}\simeq 0$. We will assume that reheating
occurs due to an effective inflaton coupling to the Standard Model fermions,
given by the interaction Lagrangian
$\mathcal{L}_{\phi-SM}^{y}\;=\;-y\phi\bar{f}f\,,$ (37)
where $y$ is a Yukawa-like coupling, $f$ is a Standard Model fermion, and the
inflaton decay rate is
$\Gamma_{\phi}\;=\;\frac{y^{2}}{8\pi}m_{\phi}\,.$ (38)
If we solve the Friedmann equations (35, 36), we find GKMO1 ; GKMO2 ; CMOV
$\rho_{\phi}(a)=\rho_{\rm end}\left(\frac{a_{\rm end}}{a}\right)^{3}$ (39)
and
$\rho_{R}(a)=\rho_{\rm RH}\left(\frac{a_{\rm
RH}}{a}\right)^{\frac{3}{2}}\frac{1-\left(\frac{a_{\rm
end}}{a}\right)^{\frac{5}{2}}}{1-\left(\frac{a_{\rm end}}{a_{\rm
RH}}\right)^{\frac{5}{2}}}\,,$ (40)
where $a_{\rm{end}}$ is the scale factor at the end of inflation, $\rho_{\rm
end}\equiv\rho_{\phi}(a_{\rm{end}})$ is the inflaton energy density at the end
of inflation when there is no radiation present, $a_{\rm{RH}}$ is the scale
factor at reheating, and $\rho_{\rm
RH}\equiv\rho_{R}(a_{\rm{RH}})=\rho_{\phi}(a_{\rm{RH}})$ is the energy density
at reheating. We note that these equations are strictly valid for $a_{\rm
end}\ll a\ll a_{\rm{RH}}$ and the end of inflation occurs when $\ddot{a}=0$
which corresponds to $\rho_{\rm end}=\frac{3}{2}V(\phi_{\rm{end}})$. For the
Starobinsky potential, $\rho_{\rm end}\simeq 0.175m_{\phi}^{2}M_{P}^{2}$ egno5
.
The radiation energy density can be parameterized as
$\rho_{R}=\frac{g_{T}\pi^{2}}{30}T^{4}\equiv\alpha T^{4}\,,$ (41)
where $g_{T}$ is the number of relativistic degrees of freedom at the
temperature $T$. The maximum temperature is attained when the radiation energy
density reaches its peak at $\rho_{R}(a_{\rm{max}})=\alpha T_{\rm{max}}^{4}$.
It was shown in GKMO1 that the ratio of $a_{\rm max}$ to $a_{\rm end}$ is
given by
$\frac{a_{\rm max}}{a_{\rm end}}=\left(\frac{8}{3}\right)^{\frac{2}{5}}\simeq
1.48\,.$ (42)
Using Eq. (40) we can then express the production rate from gravitational
scattering of thermal particles (23) as a function of the scale factor $a$
$R_{X}^{T,\,(\xi)}(a)\simeq\beta_{1}^{(\xi)}\frac{\rho_{\rm
RH}^{2}}{\alpha^{2}M_{P}^{4}}\left(\frac{a_{\rm
RH}}{a}\right)^{3}\left[\frac{1-\left(\frac{a_{\rm
end}}{a}\right)^{\frac{5}{2}}}{1-\left(\frac{a_{\rm end}}{a_{\rm
RH}}\right)^{\frac{5}{2}}}\right]^{2}\,,$ (43)
where we assumed that $m_{X}\ll m_{\phi},\,T$, and thus neglected the terms
$\beta_{2,\,3}^{(\xi)}$. If we use
$H\simeq\frac{\sqrt{\rho_{\phi}(a)}}{\sqrt{3}M_{P}}$, which is valid for $a\ll
a_{\rm RH}$, we can rewrite Eq. (34) as
$\frac{dY_{X}^{\xi}}{da}=\frac{\sqrt{3}M_{P}}{\sqrt{\rho_{\rm
RH}}}a^{2}\left(\frac{a}{a_{\rm
RH}}\right)^{\frac{3}{2}}R_{X}^{T,\,(\xi)}(a)\,.$ (44)
We find that the solution to this equation is
$\displaystyle~{}~{}n^{T,\,\xi}_{X}(a_{\rm
RH})=\frac{2\beta_{1}^{\xi}}{\sqrt{3}\alpha^{2}M_{P}^{3}}\frac{\rho_{\rm
RH}^{3/2}}{(1-(a_{\rm end}/a_{\rm RH})^{\frac{5}{2}})^{2}}\times$
$\displaystyle\left(1+3\left(\frac{a_{\rm end}}{a_{\rm
RH}}\right)^{\frac{5}{2}}-\frac{25}{7}\left(\frac{a_{\rm end}}{a_{\rm
RH}}\right)^{\frac{3}{2}}-\frac{3}{7}\left(\frac{a_{\rm end}}{a_{\rm
RH}}\right)^{5}\right)\,,$ (45)
where we integrated Eq. (44) in the interval $a_{\rm end}<a<a_{\rm RH}$.
The relic abundance is given by book
$\Omega_{X}h^{2}=1.6\times 10^{8}\frac{g_{0}}{g_{\rm RH}}\frac{n(T_{\rm
RH})}{T_{\rm RH}^{3}}\frac{m_{X}}{1~{}{\rm GeV}}\,,$ (46)
and if we combine it with Eq. (III.1), we obtain
$\displaystyle\Omega_{X}^{T,\,(\xi)}h^{2}\;=\;\frac{2}{3}\Omega_{k}^{(\xi)}$
$\displaystyle\left[1+3\left(\frac{\rho_{\rm{RH}}}{\rho_{\rm{end}}}\right)^{\frac{5}{6}}-\frac{25}{7}\left(\frac{\rho_{\rm{RH}}}{\rho_{\rm{end}}}\right)^{\frac{1}{2}}\right.$
(47)
$\displaystyle-\left.\frac{3}{7}\left(\frac{\rho_{\rm{RH}}}{\rho_{\rm{end}}}\right)^{\frac{5}{3}}\right]\,,$
with
$\Omega_{k}^{(\xi)}=1.6\times 10^{8}\frac{g_{0}}{g_{\rm
RH}}\frac{\beta_{1}^{(\xi)}\sqrt{3}}{\sqrt{\alpha}}\frac{m_{X}}{1~{}\rm{GeV}}\frac{T_{\rm
RH}^{3}}{M_{P}^{3}}\left[1-\left(\frac{\rho_{\rm RH}}{\rho_{\rm
end}}\right)^{\frac{5}{6}}\right]^{-2},$ (48)
where $g_{0}=43/11$ and we use the Standard Model value $g_{\rm RH}=427/4$.
We observe that $\Omega_{X}^{T,\,\xi}\propto\beta_{1}^{\xi}\,T_{\rm{RH}}^{3}$.
Therefore large values of the couplings $\xi_{h}$ and $\xi_{X}$ would require
a decrease in the reheating temperature. In Section IV we compare the
scattering rates and the dark matter abundances with the minimally coupled
case.
### III.2 $\phi~{}\phi\rightarrow X~{}X$
Another mode of dark matter production is through the scattering of the
inflaton itself. Whereas the graviton exchange channel was treated with care
in MO ; CMOV , in the case of non-minimal coupling it suffices to replace
$R_{X}^{T,\,\xi}$ in Eq. (44) with the production rate (27),
$\frac{dY_{X}^{\xi}}{da}=\frac{\sqrt{3}M_{P}}{\sqrt{\rho_{\rm
RH}}}a^{2}\left(\frac{a}{a_{\rm
RH}}\right)^{\frac{3}{2}}R_{X}^{\phi,\,\xi}(a)\,,$ (49)
and to integrate between $a_{\rm end}$ and $a_{\rm RH}$, which leads to
$n_{X}^{\phi,\,\xi}(a_{\rm RH})=\frac{\sigma_{\phi
X}^{\xi~{}2}\rho_{\rm{RH}}^{3/2}M_{P}}{4\sqrt{3}\pi
m_{\phi}^{4}}\left[\left(\frac{a_{\rm{RH}}}{a_{\rm{end}}}\right)^{\frac{3}{2}}-1\right]\sqrt{1-\frac{m_{X}^{2}}{m_{\phi}^{2}}}\,.$
(50)
For $a_{\rm RH}\gg a_{\rm end}$, using Eq. (39) we can express
$n_{X}^{\phi,\,\xi}$ as a function of $\rho_{\rm end}$:
$n_{X}^{\phi,\,\xi}(a_{\rm RH})\simeq\frac{\sigma_{\phi X}^{\xi~{}2}\rho_{\rm
RH}\sqrt{\rho_{\rm end}}M_{P}}{4\sqrt{3}\pi
m_{\phi}^{4}}\sqrt{1-\frac{m_{X}^{2}}{m_{\phi}^{2}}}\,,$ (51)
and we find
$\displaystyle\frac{\Omega_{X}^{\phi,\,\xi}h^{2}}{0.12}\simeq\frac{1.3\times
10^{7}\sigma_{\phi
X}^{\xi~{}2}\rho_{\rm{RH}}^{1/4}M_{P}^{2}}{m_{\phi}^{3}}\frac{m_{X}}{1\,\rm{GeV}}\sqrt{1-\frac{m_{X}^{2}}{m_{\phi}^{2}}}\,,$
(52)
where we assumed the Starobinsky value for $\rho_{\rm end}$. The analogous
expression for models with minimally coupled gravity is found by replacing
$\sigma_{\phi X}^{\xi}\rightarrow\sigma_{\phi X}$.
Up to this point we have assumed that the radiation is produced via the direct
inflaton decay to a fermion pair. In the next subsection we discuss an
unavoidable radiation production channel when the inflaton condensate
scattering produces Higgs bosons in models with minimal and non-minimal
coupling to gravity.
### III.3 $\phi~{}\phi\rightarrow h~{}h$
Gravitational processes that produce dark matter can also populate the thermal
bath in the same way. Even if this Planck-suppressed production mechanism does
not dominate throughout the entire reheating process, it was shown in CMOV
that for $T_{\rm RH}\lesssim 10^{9}$ GeV it is graviton exchange that
dominates the production of the thermal bath at the very beginning of the
reheating, when $\rho_{\phi}\sim\rho_{\rm end}$. In fact, it was shown that
the maximal temperature reached, $T_{\rm max}$, (which can be considered as an
absolute lower bound on $T_{\rm max}$) is $T_{\rm max}\sim 10^{12}$ GeV. It is
therefore natural to determine the value of the couplings ($\xi_{\phi}$,
$\xi_{h}$), for which non-minimal gravitational processes generate the thermal
bath at early times, and the maximal temperature which can be attained by
these processes.
Following the discussion in the previous subsection, to compute the radiation
energy density produced by gravitational couplings we implement the rate
$R_{h}^{\phi,\,\xi}$ (30) into the Friedmann equation (36)
$\frac{d\rho_{R}}{dt}+4H\rho_{R}\;\simeq\;N_{h}\frac{\sigma_{\phi
h}^{\xi~{}2}}{8\pi}\frac{\rho_{\phi}^{2}}{m_{\phi}^{3}}\,,$ (53)
where we took into account that each scattering corresponds to an energy
transfer of $2m_{\phi}$.888Or equivalently that each Higgs quanta carries an
energy $m_{\phi}$. The solution to this equation is
$\rho_{R}\;=\;N_{h}\frac{\sqrt{3}\sigma_{\phi
h}^{\xi~{}2}}{4\pi}\frac{\rho_{\rm
end}^{3/2}M_{P}}{m_{\phi}^{3}}\left[\left(\frac{a_{\rm{end}}}{a}\right)^{4}-\left(\frac{a_{\rm{end}}}{a}\right)^{\frac{9}{2}}\right]\,.$
(54)
Note that the dependence on the scale factor $a$ is very different from that
found in Eq. (40) due to inflaton decay. Indeed, the Higgs bosons produced by
gravitational scattering (minimal as well as non-minimal) are redshifted to a
greater extent because of the high dependence of the rate on their energy due
to the form of the energy-momentum tensor $T_{\mu\nu}^{0}$. Since
$\rho_{R}\propto a^{-4}$ in Eq. (54) (at large $a$) and $\rho_{\phi}\propto
a^{-3}$ in Eq. (39), reheating through this process does not occur (i.e.,
$\rho_{R}$ never comes to dominate the total energy at late times) and
inflaton decay is necessary.999This conclusion is avoided if the inflaton
potential about minimum is approximated by $\phi^{k}$ with a higher power of
$k>4$ CMOV ; Haque:2022kez .
However, as in the case of the reheating from the inflaton decay, the energy
density in Eq. (54) exhibits a maximum when $a=a_{\rm max}=(81/64)a_{\rm
end}$. The maximum radiation density is then,
$\rho_{\rm{max}}^{\xi}\;\simeq\;N_{h}\frac{\sigma_{\phi
h}^{\xi~{}2}}{12\sqrt{3}\pi}\frac{\rho_{\rm
end}^{3/2}M_{P}}{m_{\phi}^{3}}\left(\frac{8}{9}\right)^{8}\,,$ (55)
and from this expression we find that the maximum temperature produced by
gravitational interactions is given by
$\displaystyle T_{\rm{max}}^{\xi}$ $\displaystyle\simeq$ $\displaystyle
6.5\times 10^{11}\left(\frac{|\sigma_{\phi
h}^{\xi}|}{10^{-11}}\right)^{\frac{1}{2}}{\rm GeV}$ $\displaystyle\simeq$
$\displaystyle 1.8\times
10^{12}\sqrt{|\xi|}\left(|5+12\xi|\right)^{\frac{1}{2}}\left(\frac{m_{\phi}}{3\times
10^{13}\,{\rm GeV}}\right){\rm GeV}\,,$
where we took $\xi_{\phi}=\xi_{h}=\xi$ in the last equality. The analogous
expression for models with minimally coupled gravity is found by replacing
$\sigma_{\phi h}^{\xi}\rightarrow\sigma_{\phi h}$.
To compare the maximum temperature obtained by non-minimal interactions with
respect to minimal gravitational interactions, we can rewrite Eq. (III.3) (now
including minimal interactions in $T_{\rm max}^{\xi}$) as
$T_{\rm{max}}^{\xi}\simeq 1.3\times 10^{12}\left(\frac{|\sigma_{\phi
h}^{\xi}+\sigma_{\phi h}|}{\sigma_{\phi h}}\right)^{\frac{1}{2}}{\rm GeV}\,.$
(57)
The value of $\xi$ for which the maximum temperature generated by the non-
minimal coupling surpasses the one from graviton exchange is shown in Fig. 5
and is determined using
$\sqrt{\frac{|\sigma_{\phi h}^{\xi}|}{|\sigma_{\phi
h}|}}=\sqrt{2|\xi|}\left(|5+12\xi|\right)^{\frac{1}{2}}>1$ (58)
which is satisfied when $\xi>1/12$ or $\xi<-1/2$, as discussed earlier.
Figure 5: The maximum temperature during reheating generated separately by
minimal and non-minimal gravitational scattering of Higgs bosons in the
thermal bath.
As noted above and discussed in CMOV , minimal (and non-minimal) gravitational
interactions for a quadratic inflaton potential do not lead to the completion
of the reheating process, thus requiring additional inflaton interactions for
decay. Although radiation density produced in scattering falls off faster than
that from decay, at early time, the radiation density may in fact dominate and
determine $T_{\rm max}$. To determine when the $\phi~{}\phi\rightarrow h~{}h$
process leads to the maximum temperature, we rewrite Eq. (40) as:
$\rho^{y}_{R}\;=\;\frac{\sqrt{3}y^{2}m_{\phi}M_{P}^{3}}{20\pi}\left(\frac{\rho_{\rm
end}}{M_{P}^{4}}\right)^{\frac{1}{2}}\left[\left(\frac{a_{\rm
end}}{a}\right)^{\frac{3}{2}}-\left(\frac{a_{\rm
end}}{a}\right)^{4}\right]\,.$ (59)
Using Eq. (42), we find that the maximum radiation density produced by the
inflaton decay is given by
$\rho_{\rm
max}^{y}\;=\;\frac{\sqrt{3}y^{2}m_{\phi}M_{P}^{3}}{32\pi}\left(\frac{\rho_{\rm
end}}{M_{P}^{4}}\right)^{\frac{1}{2}}\left(\frac{3}{8}\right)^{\frac{3}{5}}\,.$
(60)
The maximum temperature is therefore determined by (non-minimal) gravitational
interactions when
$y^{2}\lesssim N_{h}\frac{8\rho_{\rm end}\sigma_{\phi
h}^{\xi~{}2}}{9m_{\phi}^{4}}\left(\frac{8}{9}\right)^{8}\left(\frac{8}{3}\right)^{\frac{3}{5}}$
(61)
or
$y\lesssim 1.6~{}\sigma_{\phi h}^{\xi}\sqrt{\frac{\rho_{\rm
end}}{m_{\phi}^{4}}}\simeq 5.4\times 10^{4}~{}\sigma_{\phi
h}^{\xi}\left(\frac{3\times 10^{13}\,\rm{GeV}}{m_{\phi}}\right)\,.$ (62)
This leads to the following reheating temperature:
$\displaystyle T_{\rm{RH}}\;$ $\displaystyle\lesssim$
$\displaystyle\;3.1\times 10^{19}\sigma_{\phi
h}^{\xi}\left(\frac{m_{\phi}}{3\times 10^{13}\,\rm{GeV}}\right)^{-1/2}{\rm
GeV}$ (63) $\displaystyle\lesssim$ $\displaystyle 2.4\times
10^{9}\left(\frac{m_{\phi}}{3\times
10^{13}}\right)^{\frac{3}{2}}\xi(5+12\xi)~{}{\rm GeV}$
where $T_{\rm RH}$ is given by GKMO2
$\rho_{\phi}(a_{\rm RH})=\alpha T_{\rm
RH}^{4}=\frac{12}{25}\Gamma_{\phi}^{2}M_{P}^{2}=\frac{3y^{4}m_{\phi}^{2}M_{P}^{2}}{400\pi^{2}}\,,$
(64)
when the reheating temperature is determined by inflaton decay.
The primary effect of the gravitational scattering processes on reheating is
the augmentation of $T_{\rm max}$ for sufficiently small inflaton decay
coupling, $y$. This can be seen in Fig. 6 where we show the evolution of the
energy density of radiation from scattering and decay as well as the energy
density of the inflaton as a function of $a/a_{\rm end}$ for $\sigma_{\phi
h}^{\xi}=0$ and $\sigma_{\phi h}^{\xi}/\sigma_{\phi h}=100$, respectively.
Figure 6: Evolution of the inflaton density (blue) and the total radiation
density (red), with radiation density produced from inflaton decays (dashed
orange) and $\phi~{}\phi\rightarrow h~{}h$ scattering processes
$\rho_{R}^{\sigma,\,\xi}$ (dotted green) and $\rho_{R}^{\sigma}$ (dash-dotted
purple) with $\sigma_{\phi h}^{\xi}/\sigma_{\phi h}=100$ (or
$\xi_{\phi}=\xi_{h}=\xi\simeq-2.3~{}{or}~{}1.8$), as a function of $a/a_{\rm
end}$ for a Yukawa-like coupling $y=10^{-8}$ and $\rho_{\rm end}\simeq
0.175\,m_{\phi}^{2}M_{P}^{2}$ $\simeq 9\times 10^{62}\,\rm{GeV}^{4}$. The
black dashed lines corresponds to the ratios $a_{\rm{int}}/a_{\rm{end}}\simeq
150$ and $6500$, which agrees with Eq. (66). The numerical solutions are
obtained from Eqs. (35), (36), and (53).
As we saw in Eq. (58), minimal gravitational interactions dominate over non-
minimal interactions when $\sigma_{\phi h}^{\xi}<\sigma_{\phi h}$ or when
$12\xi_{\phi}\xi_{h}+3\xi_{h}+2\xi_{\phi}<\frac{1}{2}\,,$ (65)
when we neglect contributions proportional to the Higgs mass. In this case,
the maximum temperature is determined by gravitational interactions when
$y\lesssim 2.1\times 10^{-6}$ from Eq. (62) using $\sigma_{\phi h}$ from Eq.
(32). The evolution of the energy densities in this case is shown in Fig. 6
with $y=10^{-8}$. However as the energy density of radiation after the maximum
falls faster than $\rho_{\phi}$, reheating in the Universe is determined by
the inflaton decay. For a sufficiently small coupling $y$, the energy density
from the decay dominates the radiation density at $a>a_{\rm int}$, where
$\frac{a_{\rm int}}{a_{\rm end}}\simeq\left(\frac{5\sigma_{\phi
h}^{2}N_{h}\rho_{\rm end}}{y^{2}m_{\phi}^{4}}\right)^{2/5}\simeq
1.6\left(\frac{\sigma_{\phi h}M_{P}}{ym_{\phi}}\right)^{4/5}\,.$ (66)
For $\sigma_{\phi h}=3.8\times 10^{-11}$, $m_{\phi}=3\times 10^{13}$ GeV, and
$y=10^{-8}$ we have $a_{\rm int}\approx 160a_{\rm end}$, as seen in the
figure.
When Eq. (65) is not satisfied, non-minimal interactions may dominate as shown
in the bottom panel of Fig. 6, for $\sigma_{\phi h}^{\xi}=100\sigma_{\phi h}$
and $y=10^{-8}$. The cross-over can be determined from Eq. (66) with the
replacement $\sigma_{\phi h}\rightarrow\sigma_{\phi h}^{\xi}$. In this
example, $a_{\rm int}\approx 6500a_{\rm end}$.
## IV Results
We now turn to some general results that may be obtained from the framework
described above. Concerning the gravitational production of dark matter from
the thermal bath, the difficulty of populating the Universe via the exchange
of a graviton was already known Bernal:2018qlk ; CMOV . Summing the minimal
and non-minimal contributions in Eq. (47), we find for $\rho_{\rm
RH}\ll\rho_{\rm end}$
$\displaystyle\frac{\Omega_{X}^{T}}{0.12}$ $\displaystyle\simeq$
$\displaystyle\left[1+30f(\xi_{h},\xi_{X})\right]\left(\frac{T_{\rm
RH}}{10^{14}~{}{\rm GeV}}\right)^{3}\left(\frac{m_{X}}{4.0\times 10^{9}~{}{\rm
GeV}}\right)$ (67) $\displaystyle=$
$\displaystyle\left[1+120\xi^{2}(1+6\xi+12\xi^{2})\right]$
$\displaystyle\times\left(\frac{T_{\rm RH}}{10^{14}~{}{\rm
GeV}}\right)^{3}\left(\frac{m_{X}}{4.0\times 10^{9}~{}{\rm GeV}}\right)$
with
$f(\xi_{h},\xi_{X})=\xi_{h}^{2}+2\xi_{h}\xi_{X}+\xi_{X}^{2}+12\xi_{h}\xi_{X}\left(\xi_{h}+\xi_{X}+4\xi_{h}\xi_{X}\right)$
where we assumed $\xi_{h}=\xi_{X}=\xi$ in the last equality, for simplicity.
It is clear that, if we set $\xi=0$, i.e. if we consider only graviton
exchange, the reheating temperature necessary to obtain a reasonable density
respecting the data Planck is dangerously close to the mass of the inflaton,
even for extremely large dark matter masses. This problem had already been
raised in Bernal:2018qlk and resolved in MO ; CMOV by considering the dark
matter produced from the (minimal) gravitational inflaton scattering.
On the other hand, from Eq. (67) we see that there is another solution to this
tension if one allows for non-minimal gravitational couplings. Indeed, it is
easy to see that for values of $\xi_{i}\gtrsim 0.1$
($f(\xi_{h},\xi_{X})\gtrsim{\frac{1}{30}}$), non-minimal gravitational
production dominates over graviton exchange. In this case, it becomes easier
to obtain the correct dark matter density for more reasonable values of
$T_{\rm RH}$ and/or $m_{X}$. For example, for a common value
$\xi=\xi_{h}=\xi_{X}=1$, a temperature of $T_{\rm RH}\sim 1.2\times 10^{13}$
GeV, thus slightly below the inflaton mass, is sufficient to produce an EeV
dark matter candidate, whereas for $\xi=1000$, $T_{\rm RH}\sim 10^{11}$ GeV
will saturate the relic density for a 2.6 TeV dark matter mass. We show this
result in Fig. 7 where we plot the reheating temperature needed to satisfy the
relic density constraint as function of $m_{X}$ for different value of $\xi$.
For each value of $\xi$, the relic density exceeds $\Omega_{X}h^{2}=0.12$
above the corresponding curve. As one can see, the line for $\xi=0$ is in the
upper corner of the figure at high values of $T_{\rm RH}$ and $m_{X}$ and
these drop significantly at higher values of $\xi$.
Figure 7: Region of parameter space respecting the relic density constraint
$\Omega_{X}h^{2}=0.12$ in the plane ($m_{X}$,$T_{\rm RH}$) for different
values of $\xi=\xi_{h}=\xi_{X}$ and $\rho_{\rm end}\simeq
0.175\,m_{\phi}^{2}M_{P}^{2}$ in the case of gravitational production from the
thermal bath $h~{}h\rightarrow X~{}X$. Both minimal and non-minimal
contributions are taken into account.
As was shown in MO ; CMOV , another possibility to avoid the necessity of high
reheating temperatures and/or dark matter masses is the production of matter
from the oscillations within the inflaton condensate when the energy stored in
the condensate is much larger than the reheating temperature. A simple
comparison between Eqs. (47) and (52) shows that the production of dark matter
via inflaton scattering when $\xi_{i}\neq 0$ generally dominates over the
production of dark matter from the thermal bath:
$\displaystyle\frac{\Omega_{X}^{\phi,\,\xi}}{\Omega_{X}^{T,\,\xi}}$
$\displaystyle\simeq$ $\displaystyle 34\frac{(\sigma_{\phi
X}^{\xi})^{2}}{\beta_{1}^{\xi}}\frac{M_{P}^{5}}{T_{\rm RH}^{2}m_{\phi}^{3}}$
(68) $\displaystyle\simeq$ $\displaystyle 185\frac{M_{P}m_{\phi}}{T_{\rm
RH}^{2}}\frac{(5+12\xi)^{2}}{1+6\xi+12\xi^{2}}\gg 1\,,\ $
where we took $\xi=\xi_{\phi}=\xi_{h}=\xi_{X}$ and $m_{X}\ll m_{\phi}$ in the
last equality. We are therefore able to state that the relic density of dark
matter generated by the non-minimal gravitational scattering of the inflaton
is always much more abundant than that produced by the thermal bath.
Dark matter production from inflaton scattering via minimal graviton exchange
also dominates over minimal gravitational thermal production CMOV . This state
of affairs is anything but surprising. Indeed, the energy available in the
inflaton condensate at the onset of oscillations is much greater than that
available in the thermal bath during the reheating process. As the scattering
cross-sections are themselves highly dependent on the energies through the
energy-momentum tensor, it is quite normal that inflaton scattering is the
dominant process for both minimal and non-minimal gravitational couplings.
Since inflaton scattering dominates in both the minimal and non-minimal
gravitational interactions we can compare the two. We obtain
$\frac{\Omega_{X}^{\phi,\,\xi}}{\Omega_{X}^{\phi}}=\frac{\sigma_{\phi
X}^{\xi~{}2}}{\sigma_{\phi X}^{2}}\simeq 4\xi^{2}(5+12\xi)^{2}\,,$ (69)
and we see again that non-minimal interactions dominate when $\xi>1/12$ or
$<-1/2$.
We show in Fig. 8 the region of the parameter space in the ($m_{X}$, $T_{\rm
RH}$) plane allowed by the relic density constraint, adding all of the minimal
and non minimal gravitational contributions, from inflaton scattering and as
well as Higgs scattering from the thermal bath taking
$\xi_{\phi}=\xi_{h}=\xi_{X}=\xi$. As expected, for $\xi=0$ we recover the
result found in CMOV . As one can see, the difficulty in the gravitational
production from the thermal bath is indeed alleviated as a reheating
temperature $T_{\rm RH}\simeq 10^{11}$ GeV allows for the production of a PeV
scale dark matter candidate. If in addition we introduce the non-minimal
couplings $\xi$, the necessary reheating temperature to fit the Planck data
may be as low as the electroweak scale for a GeV candidate if $\xi\gtrsim
1000$.
Figure 8: Region of parameter space respecting the relic density constraint
$\Omega_{X}h^{2}=0.12$ in the plane ($m_{X}$,$T_{\rm RH}$) for different
values of $\xi_{\phi}=\xi_{h}=\xi_{X}=\xi$ and $\rho_{\rm end}\simeq
0.175\,m_{\phi}^{2}M_{P}^{2}$ in the case of production from gravitational
inflaton scattering $\phi~{}\phi\rightarrow X~{}X$. Both minimal and non-
minimal contributions are taken into account.
Finally, we note that given the dark matter mass and reheating temperature (if
that sector of beyond the Standard Model physics were known), the contours in
Fig. 8 allow us to place an upper bound on the non-minimal couplings, $\xi$.
We can rewrite Eq. (52) as
$\displaystyle\frac{\Omega_{X}h^{2}}{0.12}$ $\displaystyle=$ $\displaystyle
4.1\times 10^{-7}(12\xi^{2}+5\xi+\frac{1}{2})^{2}\left(\frac{T_{\rm
RH}}{10^{10}{\rm GeV}}\right)$ (70)
$\displaystyle\times\left(\frac{m_{X}}{1{\rm
GeV}}\right)\left(\frac{m_{\phi}}{3\times 10^{13}{\rm GeV}}\right)\,,$
when $m_{X}\ll m_{\phi}$ and $\xi=\xi_{\phi}=\xi_{X}$. Then, for example, if
$m_{X}=1$ TeV, and $T_{\rm RH}=10^{9}$ GeV, we obtain an upper limit of
$|\xi|\lesssim 4$.
## V Conclusions
In this paper, we have generalized the minimal gravitational interactions in
the early Universe, i.e., the s-channel exchange of a graviton, to include
non-minimal couplings of all scalars to the Ricci curvature $R$. We consider a
scalar sector $S_{i}$ consisting of the inflaton condensate $\phi$, the Higgs
field $H$ and a dark matter candidate $X$, and we have analyzed the impact of
couplings of the type $\xi_{i}S_{i}^{2}R$ on the reheating process and dark
matter production. The latter can be generated by the thermal Higgs scattering
or excitations of the inflaton, both through minimal and non-minimal
gravitational couplings. Whereas the Higgs scattering through the exchange of
a graviton necessitates a very large reheating temperature and/or dark matter
mass in order to fulfill Planck CMB constraints ($T_{\rm RH}\simeq 10^{14}$
GeV with $m_{X}\simeq 10^{9}$ GeV), for $\xi\gtrsim 0.1$, the non-minimal
coupling dominates the process and alleviates the tension. For $\xi\simeq
1000$, a dark matter mass of $\sim 1$ PeV with $T_{\rm RH}\simeq 10^{10}$ GeV
will satisfy the constraint, see Fig. 7. However, thermal production is not
the sole source of dark matter production through gravity. When we include the
contribution (necessarily present) of the inflaton scattering, we showed that
the energy stored in the condensate at the end of inflation compensates
largely the reduced gravitational Planck coupling. These processes yield the
correct relic abundance through minimal graviton exchange for a dark matter
mass of $\sim 10^{8}$ GeV with $T_{\rm RH}\simeq 10^{10}$ GeV, and the
constraint is satisfied for a dark matter mass of $\sim 100$ GeV and $T_{\rm
RH}\gtrsim 10^{4}$ GeV if one adds non-minimal couplings of the order
$\xi\simeq 100$ as we show in Fig. 8. Gravitational inflaton scattering also
affects the reheating process, producing a maximum temperature $\simeq
10^{12}$ GeV with minimal couplings, reaching as large as $T_{\rm
max}^{\xi}\simeq 5|\xi|T_{\rm max}\simeq 10^{14}$ GeV for $\xi=100$ as one can
see in Fig. 5. This result can be re-expressed as an upper limit to $|\xi|$
given values of $m_{X}$ and $T_{\rm RH}$.
We can not over-emphasize that all of our results are unavoidable, in the
sense that they are purely gravitational, and do not rely on physics beyond
the Strandard Mode. The relic density of dark matter, and maximum temperature
of the thermal bath computed here should be considered as lower bounds, that
should be implemented in any extension of the Standard Model, whatever is its
nature.
Note added : During the completion of the manuscript, some overlapping results
were presented in Aoki:2022dzd .
Acknowledgements. The authors want to thank Emilian Dudas for useful
discussions. This work was made possible by with the support of the Institut
Pascal at Université Paris-Saclay during the Paris-Saclay Astroparticle
Symposium 2021, with the support of the P2IO Laboratory of Excellence (program
“Investisse ments d’avenir” ANR-11-IDEX-0003-01 Paris-Saclay and
ANR-10-LABX-0038), the P2I axis of the Graduate School Physics of Université
Paris-Saclay, as well as IJCLab, CEA, IPhT, APPEC, the IN2P3 master projet
UCMN and EuCAPT ANR-11-IDEX-0003-01 Paris-Saclay and ANR-10-LABX-0038). This
project has received support from the European Union’s Horizon 2020 research
and innovation programme under the Marie Sk $-$ ${\rm l}$ odowska-Curie grant
agreement No 860881-HIDDeN. The work of K.A.O. and A.S. was supported in part
by DOE grant DE-SC0011842 at the University of Minnesota.
## Appendix
### A PARTICLE PRODUCTION WITH A NON-MINIMAL COUPLING
The full Jordan frame action we consider is given by Eq. (2). The conformal
transformation to the Einstein frame is given by
$g_{\mu\nu}\;=\;\Omega^{2}\tilde{g}_{\mu\nu}\,,$ (71)
where $g_{\mu\nu}$ is the Einstein frame spacetime metric and the conformal
factor is expressed by Eq. (3). It can readily be shown that the scalar
curvature transforms as (see, e.g., Fujii:2003pa )
$\tilde{R}=\Omega^{2}\left[R+6g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\ln\Omega-6g^{\mu\nu}\left(\nabla_{\mu}\ln\Omega\right)\left(\nabla_{\nu}\ln\Omega\right)\right]\,.$
(72)
After eliminating the total divergence term, we find the Einstein frame action
(9).
To find the effective interaction terms we assume the small field limit (12)
and expand the conformal factors in the Einstein frame action. We find the
following effective interaction Lagrangian:
$\displaystyle\mathcal{L}_{\rm{eff}}\;$ $\displaystyle=$
$\displaystyle\;-\frac{1}{2}\left(\frac{\xi_{\phi}\phi^{2}}{M_{P}^{2}}+\frac{\xi_{X}X^{2}}{M_{P}^{2}}\right)\partial^{\mu}h\partial_{\mu}h-\frac{1}{2}\left(\frac{\xi_{h}h^{2}}{M_{P}^{2}}+\frac{\xi_{X}X^{2}}{M_{P}^{2}}\right)\partial^{\mu}\phi\partial_{\mu}\phi-\frac{1}{2}\left(\frac{\xi_{\phi}\phi^{2}}{M_{P}^{2}}+\frac{\xi_{h}h^{2}}{M_{P}^{2}}\right)\partial^{\mu}X\partial_{\mu}X$
(73) $\displaystyle+$
$\displaystyle\frac{6\xi_{h}\xi_{X}hX}{M_{P}^{2}}\partial^{\mu}h\partial_{\mu}X+\frac{6\xi_{h}\xi_{\phi}h\phi}{M_{P}^{2}}\partial^{\mu}h\partial_{\mu}\phi+\frac{6\xi_{\phi}\xi_{X}\phi
X}{M_{P}^{2}}\partial^{\mu}\phi\partial_{\mu}X+m_{X}^{2}X^{2}\left(\frac{\xi_{\phi}\phi^{2}}{M_{P}^{2}}+\frac{\xi_{h}h^{2}}{M_{P}^{2}}\right)$
$\displaystyle+$ $\displaystyle
m_{\phi}^{2}\phi^{2}M_{P}^{2}\left(\frac{\xi_{X}X^{2}}{M_{P}^{2}}+\frac{\xi_{h}h^{2}}{M_{P}^{2}}\right)+m_{h}^{2}h^{2}\left(\frac{\xi_{\phi}\phi^{2}}{M_{P}^{2}}+\frac{\xi_{X}X^{2}}{M_{P}^{2}}\right)\,,$
and we can rewrite the above Lagrangian in terms of the effective couplings as
Eq. (13), with
$\displaystyle\sigma_{hX}^{\xi}\;$ $\displaystyle=$
$\displaystyle\;\frac{1}{4M_{P}^{2}}\left[\xi_{h}(2m_{X}^{2}+s)+\xi_{X}(2m_{h}^{2}+s)\right.$
(74) $\displaystyle+$
$\displaystyle\left.\left(12\xi_{X}\xi_{h}(m_{h}^{2}+m_{X}^{2}-t)\right)\right]\,,$
$\sigma_{\phi
X}^{\xi}\;=\;\frac{1}{2M_{P}^{2}}\left[\xi_{\phi}m_{X}^{2}+12\xi_{\phi}\xi_{X}m_{\phi}^{2}+3\xi_{X}m_{\phi}^{2}+2\xi_{\phi}m_{\phi}^{2}\right]\,,$
(75) $\sigma_{\phi
h}^{\xi}\;=\;\frac{1}{2M_{P}^{2}}\left[\xi_{\phi}m_{h}^{2}+12\xi_{\phi}\xi_{h}m_{\phi}^{2}+3\xi_{h}m_{\phi}^{2}+2\xi_{\phi}m_{\phi}^{2}\right]\,,$
(76)
where $s,t$ are the Mandelstam variables. The latter couplings assume an
inflaton condensate in the initial state rather than a thermal Higgs in the
initial state accounting for the lack of symmetry in the three couplings.
### B THERMAL PRODUCTION
In this appendix we calculate the thermal dark matter production rate
$R_{X}^{T,\,\xi}(T)$ arising from the effective four-point interaction
$\sigma_{hX}h^{2}X^{2}$, where $\sigma_{hX}$ is given by Eq. (74). We also
calculate the production rate $R_{X}^{T}(T)$ for the thermal scattering
processes mediated by gravity alone, ${\rm{SM}}~{}{\rm{SM}}\rightarrow X~{}X$,
that are unavoidable in models with a minimal coupling to gravity
($\xi_{\phi,h,X}=0$) Bernal:2018qlk ; CMOV , and compare the two results.
The production rate $R_{X}^{T,\,\xi}(T)$ can be computed from Eq. (21). The
matrix element squared is given by
$|\overline{{\cal M}}^{hX,\,\xi}|^{2}\;=\;4\sigma_{hX}^{\xi~{}2}\,,$ (77)
where in the limit where the Higgs boson mass is neglected, the Mandelstam
variables $s$ and $t$ are given by
$\displaystyle t\,=$
$\displaystyle\,\dfrac{s}{2}\left(\sqrt{1-\frac{4m_{X}^{2}}{s}}\cos\theta_{13}-1\right)+m_{X}^{2}\,,$
(78) $\displaystyle s\,=$
$\displaystyle\,2E_{1}E_{2}\left(1-\cos\theta_{12}\right)\,.$ (79)
We find the following coefficients for Eq. (23)
$\displaystyle\beta_{1}^{\xi}$
$\displaystyle\;=\;\frac{\pi^{3}}{2700}\left[\xi_{h}^{2}+2\xi_{h}\xi_{X}+\xi_{X}^{2}+12\xi_{h}\xi_{X}\left(\xi_{h}+\xi_{X}+4\xi_{h}\xi_{X}\right)\right]\,,$
(80) $\displaystyle\beta_{2}^{\xi}$
$\displaystyle\;=\;\frac{\zeta(3)^{2}\xi_{h}}{2\pi^{5}}\left[\xi_{h}+\xi_{X}+6\xi_{h}\xi_{X}-12\xi_{h}\xi_{X}^{2}\right]\,,$
(81) $\displaystyle\beta_{3}^{\xi}$
$\displaystyle\;=\;\frac{\xi_{h}^{2}}{576\pi}\,.$ (82)
Similarly, using Eqs. (18)-(20), we find the matrix element squared for
minimally coupled gravity:
$|\overline{{\cal
M}}^{hX}|^{2}\;=\;\frac{1}{4M_{P}^{4}}\frac{\left(t(s+t)-2m_{X}^{2}t+m_{X}^{4}\right)^{2}}{s^{2}},$
(83)
where we have neglected the Higgs field mass. We find the coefficients:
$\displaystyle\beta_{1}$ $\displaystyle\;=\;\frac{\pi^{3}}{81000}\,,$ (84)
$\displaystyle\beta_{2}$
$\displaystyle\;=\;-\frac{\zeta(3)^{2}}{30\pi^{5}}\,,$ (85)
$\displaystyle\beta_{3}$ $\displaystyle\;=\;\frac{1}{4320\pi}\,.$ (86)
Note that when both contributions are kept, and we neglect $m_{h}\ll m_{X}$,
the full coefficients (including interference) are given by
$\displaystyle\beta_{1}^{\xi}$ $\displaystyle=$
$\displaystyle\frac{\pi^{3}}{81000}\left[30\xi_{h}^{2}\left(12\xi_{X}(4\xi_{X}+1)+1\right)\right.$
(87)
$\displaystyle\left.+10\xi_{h}(6\xi_{X}+1)^{2}+10\xi_{X}(3\xi_{X}+1)+1\right]\,,$
$\displaystyle\beta_{2}^{\xi}$ $\displaystyle=$
$\displaystyle-\frac{\zeta(3)^{2}}{60\pi^{5}}\left[2+10\xi_{X}\right.$ (88)
$\displaystyle\left.+5\xi_{h}\left(1+6\xi_{X}+6\xi_{h}\left(6\xi_{X}(2\xi_{X}-1)-1\right)\right)\right]\,,$
$\displaystyle\beta_{3}^{\xi}$ $\displaystyle=$
$\displaystyle\frac{1}{8640\pi}\left[2+5\xi_{h}\left(32\xi_{h}-2\right)\right]\,.$
(89)
which reduces to Eqs. (84-86) when all $\xi_{i}=0$.
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|
f_{\alpha}\cdot f_{\beta}\bigr{)}\cdot_{g}f_{I}\Bigr{)}$
$\displaystyle=\Bigl{(}\sum_{\alpha<\beta}g(\nabla_{e_{i}}^{L.C.,g}\tau_{g_{0},g}e_{\alpha},\tau_{g_{0},g}e_{\beta})\cdot
e_{\alpha}\cdot e_{\beta}\Bigr{)}\cdot e_{I}.$
We already know that
$g(\nabla_{e_{i}}^{L.C.,g}e_{\alpha},e_{\beta})={}^{g}\Gamma_{i\alpha}^{\beta}$
depends smoothly on $g$. Thus, the assignment
$g\mapsto{p\mathcal{G}_{g,g_{0}}}_{\ast}\nabla^{\mathfrak{S}_{g}}-\nabla^{\mathfrak{S}_{g_{0}}}$
is a smooth map
$\mathrm{Riem}(M)\rightarrow\Gamma\bigl{(}M;\mathrm{Hom}(\mathfrak{S}_{g_{0}},T^{\vee}M\otimes\mathfrak{S}_{g_{0}})\bigr{)}$
and hence it is also a smooth map
$\mathrm{Riem}(M)\rightarrow\Psi\mathrm{DO}^{0}(\mathfrak{S}_{g},T^{\vee}M\otimes\mathfrak{S}_{g}).$
This implies that $g\mapsto{\not{\mathfrak{D}}}_{g}$ is smooth as it is a
composition of smooth maps. ∎
The next result is classical and follows immediately from the theory developed
in [LawsonMichelsonSpin]*Chapter III.5.
###### Proposition 4.1.33.
Let $M$ be a closed manifold and let $P\in\Psi\mathrm{Dir}(M)$ be a pseudo
Dirac operator with underlying Riemannian metric $g\in\mathrm{Riem}(M)$. Then
$P$ is a self-adjoint, unbounded Fredholm operator on
$L^{2}(\mathfrak{S}_{g})$ with domain $H^{1}(\mathfrak{S}_{g})$.
###### Proof.
By definition, $P$ is symmetric, so $P-{\not{\mathfrak{D}}}_{g}$ is a
symmetric pseudo differential operator of order zero on $\mathfrak{S}_{g}$.
Thus the difference extends to a bounded, symmetric, and hence self-adjoint,
operator on $L^{2}(\mathfrak{S}_{g})$. Since the sum of a self-adjoint
operator with a self-adjoint bounded operator is again self-adjoint
[higson2000analytic]*Exercise 1.9.20,
$P={\not{\mathfrak{D}}}_{g}+(P-{\not{\mathfrak{D}}}_{g})$ is self-adjoint.
Gårdings inequality, see [LawsonMichelsonSpin]*Thm III.5.2, implies that the
minimal domain of $P$ agrees with $H^{1}(\mathfrak{S}_{g})$. It is classical
that elliptic pseudo differential operators on vector bundles over closed
manifolds are unbounded Fredholm operators, see [LawsonMichelsonSpin]*Thm
III.5.2. ∎
##### Relations to K-theory
Our interests in pseudo Dirac operators arise from their connection to
$KO$-theory, which we will outline in this section. All presented results have
already appeared in [ebert2017indexdiff] in a different language and we do not
claim originality for the results presented here; rather we translate his
results into our language, fill some gaps, and give more details. Needless to
say, the presentation here closely follows [ebert2017indexdiff]. Throughout
this section, we assume that $M^{d}$ is a _closed_ spin manifold of positive
dimension.
We start with recalling the connection between $KO$-theory and Clifford-linear
Fredholm operators. Let $H$ be a separable real Hilbert space with
$\mathord{\mathbb{Z}}_{2}$-grading $\iota$ and graded $Cl_{d,0}$-right action
$\mathbf{r}$, which we assume to be a $\ast$-homomorphism. For brevity, we
simply call it a $Cl_{d,0}$-Hilbert space. Call
$\omega_{d,0}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\iota\mathbf{r}(e_{1}\cdots e_{d})$ the _Chirality element_. Recall that
$Cl_{d,0}$ has, up to isomorphism, exactly one irreducible representation if
$d\not\equiv 0\mod 4$, and exactly two irreducible representations if
$d\equiv-1\mod 4$. In the latter case, the irreducible representations are
distinguished as different eigenspaces of the Chirality element.
The triple $(H,\iota,\mathbf{r})$ is called _ample_ if $H$ contains all
irreducible representations with infinite multiplicity.
###### Definition 4.1.34.
Let $(H,\iota,\mathbf{r})$ be an ample $Cl_{d,0}$-Hilbert space. We call an
operator $F$ on $H$ an $Cl_{d,0}$_-Fredholm operator_ if it is Clifford-
linear, odd, and self-adjoint. If $d\equiv-1\mod 4$, we further require that
$\omega_{p,0}F\iota$ is neither essentially positive nor essentially negative,
that means, there exist infinite dimensional, closed subspaces that are
orthogonal to each other and on those $F$ restricts to a positive or negative
operator, respectively.
Let $\mathrm{Fred}^{d,0}(H)$ be the space of all $Cl_{d,0}$-Fredholm operators
on $H$ equipped with the norm topology.
By a classical result of Atiyah and Singer, these spaces represent real
$K$-theory.
###### Theorem 4.1.35 (Theorem [atiyah1969indexskew]).
If $H$ is ample, then $\mathrm{Fred}^{d,0}(H)$ is a classifying space for
$KO^{-d}$. That means there is a natural bijection
$\mathrm{ind}\colon[X,\mathrm{Fred}^{d,0}(H)]\rightarrow KO^{-d}(X)$
for every compact $CW$-complex $X$.
A detailed construction of this map was carried out in the master’s thesis of
Jonathan Glöckle [gloeckle2019master]*Theorem 2.17. For more details on this
space like Bott-periodicity or Morita equivalences, we refer the reader to
[ebert2017indexdiff]. To identify the space of invertible pseudo Dirac
operators as a classifying space for $KO$-theory, we first need to introduce
auxiliary spaces.
###### Definition 4.1.36.
Let $\Psi\mathrm{Dir}^{\times}(M)$ be the subspace of $\Psi\mathrm{Dir}(M)$
consisting of all invertible pseudo Dirac operators and denote $[-1,1]$ by
$I$. Provided $\Psi\mathrm{Dir}^{\times}(M)_{g_{0}}$ is non-empty for a fixed
metric $g_{0}$, we define, for a fixed choice of base point
$B\in\Psi\mathrm{Dir}^{\times}(M)$, the spaces
${\mathcal{X}}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\\{P\colon I\rightarrow\Psi\mathrm{Dir}(M)\,:\,P\text{ smooth,
}P(-1)=B,\ P(1)\in\Psi\mathrm{Dir}^{\times}(M)\\}$
and
${\mathcal{X}}_{g_{0}}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\\{P\colon I\rightarrow\Psi\mathrm{Dir}(M)_{g_{0}}\,:\,P\text{ smooth,
}P(-1)=B,\ P(1)\in\Psi\mathrm{Dir}^{\times}(M)_{g_{0}}\\}.$
###### Lemma 4.1.37.
If $\Psi\mathrm{Dir}^{\times}(M)_{g_{0}}$ is non-empty, then the evaluation
map
$\mathrm{ev}_{1}\colon{\mathcal{X}}\rightarrow\Psi\mathrm{Dir}^{\times}(M)$
is a weak homotopy equivalence that also restricts to a weak homotopy
equivalence between ${\mathcal{X}}_{g_{0}}$ and
$\Psi\mathrm{Dir}^{\times}(M)_{g_{0}}$.
###### Proof.
We only prove the first statement; for the second statement, one simply needs
to add the subscript ${}_{g_{0}}$ to $\Psi\mathrm{Dir}^{\times}(M)$ and the
space $\mathcal{Y}$ defined below.
Let $\mathcal{Y}$ be the set of all _continuous_ paths $P\colon
I\rightarrow\Psi\mathrm{Dir}(M)$ having the same boundary conditions as
elements in ${\mathcal{X}}$. Smoothing theory yields that the inclusion of
${\mathcal{X}}$ into $\mathcal{Y}$ is a weak homotopy equivalence.
The map
$\mathrm{ev}_{1}\colon\mathcal{Y}\rightarrow\Psi\mathrm{Dir}^{\times}(M)$ is a
fibration. Indeed, given a commutative square
$\textstyle{A\times
0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{\mathcal{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ev}_{1}}$$\textstyle{A\times
I\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{\Psi\mathrm{Dir}^{\times}(M)}$
we define the lift $\mathcal{H}\colon A\times I\rightarrow\mathcal{Y}$ by
$\mathcal{H}(a,s)(t)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\frac{(1-t)}{2}\cdot F(a)+\frac{(1+t)}{2}\cdot h(a,s).$
This is a continuous map because the affine structure on $\Psi\mathrm{Dir}(M)$
is (jointly) continuous with respect to the Fréchet structure. It is also a
lift for $\mathrm{ev}_{1}(\mathcal{H})=h$, showing that $\mathrm{ev}_{1}$ is a
fibration.
The homotopy fibre of $\mathrm{ev}_{1}$ is therefore the preimage of $B$,
which is the set of all closed curves in $\Psi\mathrm{Dir}(M)$ that start (and
end) at $B$. This subspace is clearly convex and therefore contractible,
implying that $\mathcal{Y}$ and $\Psi\mathrm{Dir}^{\times}(M)$ are weak
homotopy equivalent. ∎
The auxiliary path spaces ${\mathcal{X}}$ and ${\mathcal{X}}_{g_{0}}$ help to
identify $\Psi\mathrm{Dir}^{\times}(M)$ as classifying space for $KO$-theory.
We remark that, for a closed manifold $M^{d}$ of positive dimension $d$, the
$Cl_{d,0}$-Hilbert space $H=L^{2}(M,\mathfrak{S}_{g_{0}})$ is always ample.
Indeed, the right action of the Chirality element decomposes
$\mathfrak{S}_{g_{0}}$ into non-zero subbundles $\mathfrak{S}_{g_{0}}^{\pm}$
on which the Chirality element acts as $\pm\mathrm{id}$, hence
$L^{2}(M,\mathfrak{S}_{g_{0}})\cong L^{2}(M,\mathfrak{S}_{g_{0}}^{+})\oplus
L^{2}(M,\mathfrak{S}_{g_{0}}^{-})$
is a decomposition into infinite dimensional eigenspaces of the eigenvalues
$\pm 1$ of the Chirality element.
The following theorem is a collection of results proved by Johannes Ebert in
[ebert2017indexdiff]*Section 4.1:
###### Theorem 4.1.38.
Assume that $\Psi\mathrm{Dir}^{\times}(M)_{g_{0}}$ is non-empty. Then the
diagram of continuous maps
$\textstyle{{\mathcal{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega\mathrm{Fred}^{d,0}\bigl{(}L^{2}(M,\mathfrak{S}_{g_{0}})\bigr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{X}}_{g_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\Omega\mathrm{Fred}^{d,0}\bigl{(}L^{2}(M,\mathfrak{S}_{g_{0}})\bigr{)}}$
commutes. The horizontal maps are given by
$P_{t}\mapsto\frac{\mathcal{G}_{g_{t},g_{0}}P_{t}\mathcal{G}_{g_{t},g_{0}}^{-1}}{(1+(\mathcal{G}_{g_{t},g_{0}}P_{t}\mathcal{G}_{g_{t},g_{0}}^{-1})^{2})^{1/2}}$
and the lower map is a weak homotopy equivalence. Here
$\Omega\mathrm{Fred}^{d,0}(L^{2}(\mathfrak{S}_{g_{0}}))$ denotes the space of
paths into $\mathrm{Fred}^{d,0}(L^{2}(\mathfrak{S}_{g_{0}}))$ whose start and
end-points are invertible operators.
###### Proof.
For fixed $t$, these maps certainly take values in the space of self-adjoint,
odd, Clifford-linear Fredholm operators. From the discussion in
[ebert2017indexdiff]*Section 4.1 it is not clear why the horizontal map takes
values in the subspace with the additional condition that $\omega F\iota$ is
neither essentially positive nor essentially negative if $d\equiv-1\mod 4$.
In order to show this, first recall from [LawsonMichelsonSpin]*Theorem III.5.8
that the spectrum of every self-adjoint elliptic pseudo differential operator
$P$ of positive order on a bundle over a closed manifold consists of countably
many eigenvalues only, that the eigenspaces are finite dimensional, and that
the eigenvalues go to infinity. The operator $P/\sqrt{1+P^{2}}$ is the unique
operator that acts as $\lambda/\sqrt{1+\lambda^{2}}\cdot\mathrm{id}$ on the
$\lambda$-eigenspace of $P$.
The Chirality element satisfies
$\omega_{d,0}^{2}=(-1)^{\frac{d(d+1)}{2}}\quad\text{ and
}\quad\omega_{d,0}^{\ast}=(-1)^{\frac{d(d+1)}{2}}\omega_{d,0}.$
Thus, it is a self-adjoint, odd involution if $d\equiv-1\mod 4$. Since $P$,
and therefore $P/\sqrt{1+P^{2}}$ too, commute with $\omega_{d,0}$, we can
simultaneously diagonalise the two operators.
The grading $\mathrm{e/o}$ anti-commutes with $P$, so the grading sends the
$\lambda$-eigenspace of $P$ to the $-\lambda$-eigenspaces of $P$. The
corresponding statement holds true for the $\pm 1$-eigenspaces of
$\omega_{d,0}$ for the same reason.
Abbreviate $P/\sqrt{1+P^{2}}$ to $F$. Since the grading $\mathrm{e/o}$
commutes with $\omega_{d,0}F$, the operator $\omega_{d,0}F\mathrm{e/o}$ is
positive definite on
$H^{+}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\bigoplus_{\lambda>0}\mathrm{Eig}(P;\lambda)\cap\mathrm{Eig}(\omega_{d,0};1)\oplus\mathrm{Eig}(P;-\lambda)\cap\mathrm{Eig}(\omega_{d,0};-1)$
and negative definite on
$H^{-}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\bigoplus_{\lambda>0}\mathrm{Eig}(P;\lambda)\cap\mathrm{Eig}(\omega_{d,0};-1)\oplus\mathrm{Eig}(P;-\lambda)\cap\mathrm{Eig}(\omega_{d,0};1).$
By the discussion above, the two subspaces are infinite dimensional. Thus,
$\omega_{d,0}F\mathrm{e/o}$ cannot be essentially positive or essentially
negative.
The upper horizontal map is continuous because conjugation with invertible
functions is a continuous operation as
$\Psi\mathrm{DO}^{\bullet}(\mathfrak{S}_{g_{0}})$ is a Fréchet algebra, and
bounded functional calculus is continuous by [ebert2017indexdiff]*Proposition
3.7. The weak equivalence statement is [ebert2017indexdiff]*Proposition 4.3. ∎
The space $\Psi\mathrm{Dir}^{\times}(M)_{g_{0}}$ is unsuitable for our
purposes. Because we want to compare different metrics, we need the bigger
space $\Psi\mathrm{Dir}^{\times}(M)$. Luckily, the canonical inclusion is a
weak homotopy equivalence. This is the main result of
[ebert2017indexdiff]*Section 4.2 formulated in our language.
###### Theorem 4.1.39.
The inclusion
$\Psi\mathrm{Dir}^{\times}(M)_{g_{0}}\hookrightarrow\Psi\mathrm{Dir}^{\times}(M)$
is a weak homotopy equivalence.
The identification between $\Psi\mathrm{Dir}^{\times}(M)$ and
$\Omega\mathrm{Fred}^{d,0}(L^{2}(\mathfrak{S}_{g_{0}}))$ is the space-level
version of the index difference of Hitchin [hitchin1974harmonic].
###### Definition 4.1.40.
Fix base points $g_{0}\in\mathrm{Riem}(M)$ and
$B\in\Psi\mathrm{Dir}^{\times}(M)$. For any choice of section
$s\colon\Psi\mathrm{Dir}^{\times}(M)\rightarrow\mathcal{X}$ of
$\mathrm{ev}_{1}$, we call the composition
$\textstyle{\mathrm{inddif}(\text{--},B)\colon\Psi\mathrm{Dir}^{\times}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\mathrm{ev}_{1}}$$\textstyle{\Omega\mathrm{Fred}^{d,0}(L^{2}(\mathfrak{S}_{g_{0}}))}$
the _(operator) index difference_.
If $g_{0}$ is a psc-metric then we call the pre-composition with the Dirac
operator
$\textstyle{\alpha(\text{--},g_{0})\colon\mathrm{Riem}^{+}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\not{\mathfrak{D}}}_{(\text{--})}}$$\textstyle{\Psi\mathrm{Dir}^{\times}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{inddif}(\text{--},{\not{\mathfrak{D}}}_{g_{0}})}$$\textstyle{\Omega\mathrm{Fred}^{d,0}(L^{2}(\mathfrak{S}_{g_{0}}))}$
the _index difference_.
We summarise our achievement of this subsection in the following theorem.
###### Theorem 4.1.41.
Let $M^{d}$ be a closed spin manifold of positive dimension. If the space of
all invertible pseudo Dirac operators $\Psi\mathrm{Dir}^{\times}(M)$ is non-
empty, then it is a classifying space for $KO^{-(d+1)}(\mathrm{pt})$. The
natural transformation is given by the _Hitchin index difference_
$\textstyle{[X;\Psi\mathrm{Dir}^{\times}(M)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{inddif}(\text{--},B)_{\ast}}$$\textstyle{[X;\Omega\mathrm{Fred}^{d,0}(L^{2}(\mathfrak{S}_{g_{0}}))]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{adjunction}}$$\scriptstyle{\cong}$$\textstyle{KO^{-(d+1)}(X)}$$\textstyle{KO^{-d}(\Sigma
X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{[\Sigma
X;\mathrm{Fred}^{d,0}(L^{2}(\mathfrak{S}_{g_{0}}))]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ind}}$$\scriptstyle{\cong}$
Our primary goal is to show that the index difference factors through the
concordance set. To this end, we will construct another model for real
$K$-theory that is built out of $\Psi\mathrm{Dir}^{\times}(M)$ in the same way
as $\widetilde{\mathcal{R}^{+}_{\bullet}}(M)$ is built out of
$\mathrm{Riem}^{+}(M)$. We will pursue this construction in Section 4.4 by
“blockifying” the objects of this section. In the next section, we lay the
necessary analytical groundwork.
### 4.2 Block Operators
In this section, we discuss the foundations of _block pseudo differential
operators._ These are operators on $M\times\mathord{\mathbb{R}}^{n}$ that can
be formally thought of as pseudo differentials operators with the property
that if we think of $\mathord{\mathbb{R}}^{n}$ as a cube of infinite length,
then they decompose into a product operator near each face. We will discuss
the required combinatorial language to deal with those operators, present a
construction of a partition of unity that respects the decomposition of a
block pseudo differential operator near infinity, and discuss the application
of these partitions to the support of these block operators. Finally, we
define the operator suspension of a family of pseudo Dirac operator and show
that it is a block operator.
We assume that the reader is familiar with the concept of pseudo differential
operators and their extension of Atiyah and Singer, otherwise he or she may
consult Appendix B and the references therein.
#### Block Bundles and Basics of Block Operators
The seemingly harmless space $\mathord{\mathbb{R}}^{n}$ can be decomposed into
products $\mathord{\mathbb{R}}^{p}\times\mathord{\mathbb{R}}^{n-p}$ in various
ways. It will be important to keep track of these decompositions, which
requires notation that seems quite heavy (and maybe unnecessary) at the
beginning. For
${\mathrm{dom}\,}{\bm{\varepsilon}}\subseteq\mathbf{n}=\\{1,\dots,n\\}$ and
${\bm{\varepsilon}}\colon{\mathrm{dom}\,}{\bm{\varepsilon}}\rightarrow\mathord{\mathbb{Z}}_{2}$
we define, for a given $M$ the sets
$\displaystyle
U_{R}({\bm{\varepsilon}})\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=M\times\\{x\in\mathord{\mathbb{R}}^{n}\,:\,{\bm{\varepsilon}}(i)x_{i}>R\text{
for }i\in{\mathrm{dom}\,}{\bm{\varepsilon}}\\},$
$\displaystyle\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\mathrm{Map}(\mathbf{n}\setminus{\mathrm{dom}\,}{\bm{\varepsilon}},\mathord{\mathbb{R}})\cong\\{x\in\mathord{\mathbb{R}}^{n}\,:\,{\bm{\varepsilon}}(i)x_{i}=R\\}$
$\displaystyle\mathord{\mathbb{R}}^{{\bm{\varepsilon}}}_{R}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\\{x\in\mathord{\mathbb{R}}^{|{\mathrm{dom}\,}{\bm{\varepsilon}}|}\,:\,{\bm{\varepsilon}}(i_{j})x_{j}>R,\,i_{j}\in{\mathrm{dom}\,}{\bm{\varepsilon}},\,i_{j}<i_{j+1}\\}.$
We would like to emphasise that the identification between
$\mathrm{Map}(\mathbf{n}\setminus{\mathrm{dom}\,}{\bm{\varepsilon}},\mathord{\mathbb{R}})$
and $\\{x\in\mathord{\mathbb{R}}^{n}\,:\,{\bm{\varepsilon}}(i)x_{i}=R\\}$ is
only canonical once $R$ is fixed.
For such an ${\bm{\varepsilon}}$ let
$c({\bm{\varepsilon}})^{-1}\colon\mathbf{n}\rightarrow\mathbf{n}\setminus{\mathrm{dom}\,}{\bm{\varepsilon}}\sqcup{\mathrm{dom}\,}{\bm{\varepsilon}}$
be the unique map that partitions $\mathbf{n}$ into
$\mathbf{n}\setminus{\mathrm{dom}\,}{\bm{\varepsilon}}$ and
${\mathrm{dom}\,}{\bm{\varepsilon}}$ in an order preserving manner. We write
${\bm{\eta}}\geq{\bm{\varepsilon}}$ if
${\mathrm{dom}\,}{\bm{\eta}}\supseteq{\mathrm{dom}\,}{\bm{\varepsilon}}$ and
${\bm{\eta}}|_{{\mathrm{dom}\,}{\bm{\varepsilon}}}={\bm{\varepsilon}}$. For
all ${\bm{\eta}}\geq{\bm{\varepsilon}}$ let
$c({\bm{\eta}},{\bm{\varepsilon}})^{-1}$ be the unique map that partitions
${\mathrm{dom}\,}{\bm{\eta}}^{c}$ into
${\mathrm{dom}\,}{\bm{\eta}}\setminus{\mathrm{dom}\,}{\bm{\varepsilon}}$ and
${\mathrm{dom}\,}{\bm{\varepsilon}}^{c}$ in an order preserving manner and let
$\mathrm{per}({\bm{\eta}},{\bm{\varepsilon}})^{-1}$ be the unique map that
partitions ${\mathrm{dom}\,}{\bm{\eta}}$ into
${\mathrm{dom}\,}{\bm{\eta}}-{\bm{\varepsilon}}$ and
${\mathrm{dom}\,}{\bm{\varepsilon}}$ in an order preserving manner. Here,
${\bm{\eta}}-{\bm{\varepsilon}}$ denotes ${\bm{\eta}}$ restricted to
${\mathrm{dom}\,}{\bm{\eta}}\setminus{\mathrm{dom}\,}{\bm{\varepsilon}}$.
For each ${\bm{\eta}}\geq{\bm{\varepsilon}}$ we have a commutative diagram
$\textstyle{\mathbf{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c({\bm{\eta}})^{-1}}$$\textstyle{{\mathrm{dom}\,}{\bm{\eta}}^{c}\sqcup{\mathrm{dom}\,}{\bm{\eta}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}\sqcup\mathrm{per}({\bm{\eta}},{\bm{\varepsilon}})^{-1}}$$\textstyle{{\mathrm{dom}\,}{\bm{\eta}}^{c}\sqcup{\mathrm{dom}\,}{\bm{\eta}}-{\bm{\varepsilon}}\sqcup{\mathrm{dom}\,}{\bm{\varepsilon}}}$$\textstyle{\mathbf{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c({\bm{\varepsilon}})^{-1}}$$\textstyle{{\mathrm{dom}\,}{\bm{\varepsilon}}^{c}\sqcup{\mathrm{dom}\,}{\bm{\varepsilon}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{c({\bm{\eta}},{\bm{\varepsilon}})^{-1}\sqcup\mathrm{id}}$
These maps induce linear maps on Euclidean spaces by pulling back their
inverse. For example,
$\displaystyle c({\bm{\varepsilon}})\colon\mathord{\mathbb{R}}^{n}$
$\displaystyle\rightarrow\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})\times\mathord{\mathbb{R}}^{{\bm{\varepsilon}}}$
$\displaystyle(x_{1},\dots,x_{n})$
$\displaystyle\mapsto(x_{c({\bm{\varepsilon}})^{-1}(1)},\dots,x_{c({\bm{\varepsilon}})^{-1}(|{\mathrm{dom}\,}{\bm{\varepsilon}}^{c}|)};x_{c({\bm{\varepsilon}})^{-1}(|{\mathrm{dom}\,}{\bm{\varepsilon}}^{c}|+1)},\dots,x_{c({\bm{\varepsilon}})^{-1}(n)}).$
We make the following notational convention: For a block metric $g$ on
$M\times\mathord{\mathbb{R}}^{n}$, we denote by $g({\bm{\varepsilon}})$ the
restriction to $M\times\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})$ if we
interpret $\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})$ as
$\\{x\in\mathord{\mathbb{R}}^{n}\,:\,{\bm{\varepsilon}}(i)x_{i}=R\\}$ and by
$\partial^{{\bm{\varepsilon}}}g$ the restriction to
$M\times\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})$ if we interpret
$\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})$ as
$\mathord{\mathbb{R}}^{|{\mathrm{dom}\,}({\bm{\varepsilon}})^{c}|}$. The
identification between these two is the pull-back with
${\delta^{R{\bm{\varepsilon}}}}$.
The pull-back of $c({\bm{\varepsilon}})$ coordinates yields an isometry
$c({\bm{\varepsilon}})\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\mathrm{id}_{M}\times
c({\bm{\varepsilon}})\colon\left(U_{R}({\bm{\varepsilon}}),g|_{U({\bm{\varepsilon}})}\right)\rightarrow\bigl{(}M\times\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})\times\mathord{\mathbb{R}}_{R}^{\bm{\varepsilon}},\underbrace{g({\bm{\varepsilon}})\oplus\langle\cdot,\cdot\rangle}_{=c({\bm{\varepsilon}})_{\ast}g|_{U_{R}({\bm{\varepsilon}})}}\bigr{)}.$
This fits into the following diagram of isometries:
$\textstyle{U_{R}({\bm{\eta}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c({\bm{\eta}})}$$\textstyle{(M\times\mathord{\mathbb{R}}^{n})({\bm{\eta}})\times\mathord{\mathbb{R}}_{R}^{{\bm{\eta}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}\times\mathrm{per}({\bm{\eta}},{\bm{\varepsilon}})}$$\textstyle{(M\times\mathord{\mathbb{R}}^{n})({\bm{\eta}})\times\mathord{\mathbb{R}}_{R}^{{\bm{\eta}}-{\bm{\varepsilon}}}\times\mathord{\mathbb{R}}_{R}^{\bm{\varepsilon}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c({\bm{\eta}},{\bm{\varepsilon}})}$$\textstyle{U_{R}({\bm{\varepsilon}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c({\bm{\varepsilon}})}$$\textstyle{(M\times\mathord{\mathbb{R}}^{n})({\bm{\varepsilon}})\times\mathord{\mathbb{R}}_{R}^{\bm{\varepsilon}}.}$
Since all of these maps are induced by permutations on Euclidean spaces, they
are Pin-structure preserving isometries because each Euclidean space has, up
to equivalence, only one $\mathrm{Pin}^{-}$-structure.
Functoriality of the spinor bundle construction, see Lemma 4.1.17, yields a
splitting on the associated spinor bundles:
$\Phi_{\bm{\varepsilon}}=\mathfrak{S}(c({\bm{\varepsilon}}))\colon\mathfrak{S}_{U_{R}({\bm{\varepsilon}})}\rightarrow\mathfrak{S}_{g({\bm{\varepsilon}})}\boxtimes
Cl_{{\bm{\varepsilon}},0}=\mathfrak{S}_{g({\bm{\varepsilon}})\oplus\langle\cdot,\cdot\rangle},$
where $Cl_{{\bm{\varepsilon}},0}$ denotes $Cl_{|{\bm{\varepsilon}}|,0}$. This
gives rise to morphisms between spinor bundles:
$\textstyle{\mathfrak{S}|_{M\times
U_{R}({\bm{\eta}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{{\bm{\eta}}}}$$\textstyle{\mathfrak{S}_{g({\bm{\eta}})}\boxtimes
Cl_{{\bm{\eta}},0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}\times\mathfrak{S}(\mathrm{per}({\bm{\varepsilon}},{\bm{\eta}}))}$$\textstyle{\mathfrak{S}_{g({\bm{\eta}})}\boxtimes
Cl_{{\bm{\eta}}-{\bm{\varepsilon}},0}\boxtimes
Cl_{{\bm{\varepsilon}},0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{C}_{{\bm{\eta}},{\bm{\varepsilon}}}}$$\textstyle{\mathfrak{S}|_{M\times
U_{R}({\bm{\varepsilon}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{{\bm{\varepsilon}}}}$$\textstyle{\mathfrak{S}_{g({\bm{\varepsilon}})}\boxtimes
Cl_{{\bm{\varepsilon}},0}.}$
###### Definition 4.2.1.
Let $g$ be a block metric on $M\times\mathord{\mathbb{R}}^{n}$ and let
$A\in\\{\mathrm{id},\mathrm{e/o}\\}$. A pseudo differential operator
$P\in\overline{\Psi\mathrm{DO}}^{m}(\mathfrak{S}_{g})$ is of _block form_ (_of
type_ $A$) if there are $R>r>0$ such that:
* (1)
The block metric $g$ decomposes outside of $M\times rI^{n}$.
* (2)
If $\sigma\in\Gamma_{c}(\mathfrak{S}_{g})$ is supported within
$M\times(RI^{n})^{\circ}$, then $P\sigma$ is supported within
$M\times(RI^{n})^{\circ}$.
* (3)
The operator $P$ restricts555This means that if a section is supported within
$U_{r}({\bm{\varepsilon}})$, then $P\sigma$ is also supported within this set.
to $M\times U_{r}({\bm{\varepsilon}})$ and there is a
$P({\bm{\varepsilon}})\in\overline{\Psi\mathrm{DO}}^{m}(\mathfrak{S}_{g({\bm{\varepsilon}})})$
such that
$(\Phi_{\bm{\varepsilon}})_{\ast}P|_{M\times
U_{r}({\bm{\varepsilon}})}=P({\bm{\varepsilon}})\boxtimes\mathrm{id}+A\boxtimes
D({\bm{\varepsilon}}),$
where
$P({\bm{\varepsilon}})\in\overline{\Psi\mathrm{DO}}^{m}(\mathfrak{S}_{g({\bm{\varepsilon}})})$
and $D({\bm{\varepsilon}})$ is a differential operator of order $m$ on the
trivial bundle
$Cl_{{\bm{\varepsilon}},0}\rightarrow\mathord{\mathbb{R}}^{\bm{\varepsilon}}_{r}$
with constant coefficients.
We call $M\times RI^{n}$ _the core_ of $P$.
The constants $R$ and $r$ are not unique. We will see in Lemma 4.2.6 below
that property (2) and (3) hold for all larger constants $R^{\prime}>R$ and
$r^{\prime}>r$ as well.
###### Lemma 4.2.2.
Let $P\in\overline{\Psi\mathrm{DO}}^{m}(\mathfrak{S}_{g})$ be a block operator
of type $A$. Then the operators $P({\bm{\varepsilon}})$ and
$D({\bm{\varepsilon}})$ are uniquely determined by $P$.
###### Proof.
We prove the statement by induction over
$|{\mathrm{dom}\,}{\bm{\varepsilon}}|$. On
${\mathrm{dom}\,}{\bm{\varepsilon}}=\emptyset$, there is nothing to prove. For
the induction step, it suffices to prove the following statement: Every
operator of the form $P=Q\boxtimes\mathrm{id}+A\boxtimes
D\in\overline{\Psi\mathrm{DO}}^{m}(\mathfrak{S}_{g_{M}}\boxtimes Cl_{n,0})$ on
a _not necessarily closed_ spin manifolds $M$ with product metrics
$g=g_{M}\oplus\langle\cdot,\cdot\rangle$ uniquely deterimes $Q$ and $D$. We
will prove this statement by expressing $Q$ and $D$ through $P$.
Let $x^{1},\dots,x^{n}$ be the standard coordinates of
$\mathord{\mathbb{R}}^{n}$ and $\partial_{j}$ the derivation into the $j$-th
direction. The operator $D$ can be written as
$D=\sum_{|{\bm{\alpha}}|\leq m}a_{\bm{\alpha}}\partial^{\bm{\alpha}},$
for some constants $a_{\bm{\alpha}}\in\mathrm{End}(Cl_{n,0})$. For a point
$p\in\mathord{\mathbb{R}}^{n}$, choose smooth functions $f_{1},\dots,f_{n}$
with compact support satisfying $f_{j}(x)=(x_{j}-p_{j})$ in some neighbourhood
$V$ of $p$. Choose further $u\in\Gamma_{c}(M,\mathfrak{S}_{g_{M}})$ and
$v\in\Gamma_{c}(V,Cl_{n,0})$. Finally, for each multiindex ${\bm{\alpha}}$
with $|{\bm{\alpha}}|\leq m$, define the multiplication operator
$\mu_{\bm{\alpha}}$ by
$\displaystyle\mu_{\bm{\alpha}}\colon\Gamma_{c}(M\times\mathord{\mathbb{R}}^{n},\mathfrak{S}_{g_{M}}\boxtimes
Cl_{n,0})$
$\displaystyle\rightarrow\Gamma_{c}(M\times\mathord{\mathbb{R}}^{n},\mathfrak{S}_{g_{M}}\boxtimes
Cl_{n,0})$ $\displaystyle\sigma$
$\displaystyle\mapsto\prod_{j=1}^{n}f_{j}^{\alpha_{j}}\cdot\sigma.$
The Leibniz rule implies
$[P,\mu_{\bm{\alpha}}](u\otimes v)(m,p)=A(u)(m)\otimes a_{\bm{\alpha}}v(p),$
so $D$ is uniquely determined by $P$.
It remains to determine $Q$ from $P$ and $D$. Since $P-A\boxtimes D$ is
$\mathcal{C}^{\infty}(\mathord{\mathbb{R}}^{n})$-linear on
$\Gamma_{c}(M\times\mathord{\mathbb{R}}^{n},\mathfrak{S}_{g_{M}}\boxtimes
Cl_{n,0})$ we can define a linear map
$\tilde{Q}\colon\Gamma_{c}(M,\mathfrak{S}_{g_{M}})\rightarrow\Gamma(M,\mathfrak{S}_{g_{M}})$
as follows: Fix a point $p\in\mathord{\mathbb{R}}^{n}$ and a function
$f\in\mathcal{C}^{\infty}(\mathord{\mathbb{R}}^{n},\mathord{\mathbb{R}})$ that
is identically $1$ near $x$ and compactly supported. Set then
$\tilde{Q}(\sigma)(m)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=(P-A\boxtimes D)(\sigma\otimes f\cdot 1)(m,p).$
Plugging in the decomposition gives
$\displaystyle\tilde{Q}(\sigma)(m)$ $\displaystyle=(P-A\boxtimes
D)(\sigma)(m,p)$ $\displaystyle=Q\boxtimes\mathrm{id}(\sigma\otimes f\cdot
1)(m,p)$ $\displaystyle=Q(\sigma)(m)\otimes f(p)=Q(\sigma)(m)\otimes 1,$
which can be identified with $Q(\sigma)$ under the inclusion
$\mathfrak{S}_{g_{M}}\subseteq\mathfrak{S}_{g_{M}}\boxtimes Cl_{n,0}$. ∎
###### Corollary 4.2.3.
If $P$ is a block operator on $\mathfrak{S}_{g}$, then $P({\bm{\varepsilon}})$
is a block operator on $\mathfrak{S}_{g({\bm{\varepsilon}})}$. Equivalently,
if
${\delta^{2\rho{\bm{\varepsilon}}}}\colon\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})\xrightarrow{\cong}\\{x\in\mathord{\mathbb{R}}^{n}\,:\,{\bm{\varepsilon}}(i)x_{i}=2\rho\\}$
denotes the isometric inclusion that plugs $2{\bm{\varepsilon}}(j)\rho$ into
the $j$-th coordinate for all $j\in{\mathrm{dom}\,}{\bm{\varepsilon}}$, then
${\delta^{2\rho{\bm{\varepsilon}}}}^{\ast}P({\bm{\varepsilon}})=({\delta^{2\rho{\bm{\varepsilon}}}})^{-1}_{\ast}P({\bm{\varepsilon}})$
is a block operator on $\mathfrak{S}_{\partial^{{\bm{\varepsilon}}}g}$ for all
$\rho>R$, where $R$ denotes the size of the core of $P$. These block operators
satisfy conditions (1)-(3) for the same constants as $P$.
###### Proof.
Since $g$ decomposes outside of $rI^{n}$, it also decomposes outside of $\rho
I^{n}$. If we identify $\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})$ with the
affine subspace
$\\{x\in\mathord{\mathbb{R}}^{n}\,:\,{\bm{\varepsilon}}(j)x_{j}=\rho\\}$, then
the restricted metric
$g({\bm{\varepsilon}})=g\mathord{\upharpoonleft}_{\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})}$
is again a block metric that decomposes outside of
$rI^{n}\cap\mathord{\mathbb{R}}^{n}{\bm{\varepsilon}})\cong
rI^{n-|{\mathrm{dom}\,}{\bm{\varepsilon}}|}$.
Set
$m\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=n-|{\mathrm{dom}\,}{\bm{\varepsilon}}|$. We verify axiom $(2)$ for
${{\delta^{2r{\bm{\varepsilon}}}}}^{\ast}P({\bm{\varepsilon}})$ by contra-
position. Assume there is a section
$\sigma\in\Gamma_{c}(\mathfrak{S}_{\partial^{\varepsilon}_{{\bm{\varepsilon}}}g({\bm{\varepsilon}})})$
with support in $M\times(RI^{m})^{\circ}$ such that
${{\delta^{2r{\bm{\varepsilon}}}}}^{\ast}P({\bm{\varepsilon}})(\sigma)$ is not
supported within $M\times(RI^{m})^{\circ}$. Pick a smooth function
$f\colon\mathord{\mathbb{R}}\rightarrow[0,1]$ that is supported within $(r,R)$
and define on $R_{r}^{\bm{\varepsilon}}$ the real valued function
$f^{\bm{\varepsilon}}(x)=\prod_{j=1}^{|{\mathrm{dom}\,}{\bm{\varepsilon}}|}f(x_{j}).$
Then the section $\sigma\otimes f^{\bm{\varepsilon}}$ is supported within the
interior of $M\times RI^{m}\times RI^{|{\mathrm{dom}\,}{\bm{\varepsilon}}|}$
but, by assumption, the support of
$\bigl{(}P({\bm{\varepsilon}})\boxtimes\mathrm{id}+A\boxtimes
D\bigr{)}(\sigma\otimes f^{\bm{\varepsilon}})$ is not supported within
$M\times RI^{m}\times RI^{|{\mathrm{dom}\,}{\bm{\varepsilon}}|}$. Thus
$\tau\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt={\Phi_{\bm{\varepsilon}}}^{-1}_{\ast}(\sigma\otimes
f^{\bm{\varepsilon}})$ is a section of $\mathfrak{S}_{g}$ that is supported
within the interior of $M\times RI^{n}$. But the support of $P(\tau)$ does not
lie in $M\times RI^{n}$, contradicting axiom $(2)$ of $P$.
We first give an informal proof for (3). We know that
$\Phi_{{\delta^{{\bm{\varepsilon}}}}({\bm{\eta}})\,\ast}P|_{U_{r}({\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}))}=P\bigl{(}{\delta^{{\bm{\varepsilon}}}}({\bm{\eta}})))\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes
D({\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}))\bigr{)}.$
If we permute the (fixed) coordinates parametrised by
${\mathrm{dom}\,}\,{\bm{\varepsilon}}$ “to the right”, using
$\mathrm{per}({\delta^{{\bm{\varepsilon}}}}({\bm{\eta}})),{\bm{\varepsilon}})$
we can identify
$\mathord{\mathbb{R}}^{{\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}))}_{r}$ with
$\mathord{\mathbb{R}}^{{\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}))-{\bm{\varepsilon}}}_{r}\times\mathord{\mathbb{R}}^{{\bm{\varepsilon}}}_{r}$.
The uniqueness statement of Lemma 4.2.2 implies (with abuse of notation)
$P({\bm{\varepsilon}})=P\bigl{(}{\delta^{{\bm{\varepsilon}}}}({\bm{\eta}})\bigr{)}\otimes\mathrm{id}+\mathrm{per}({\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}),{\bm{\varepsilon}})_{\ast}D\bigl{(}{\delta^{{\bm{\varepsilon}}}}({\bm{\eta}})\bigr{)}.$
Now the formal proof of (3): The inclusion ${\delta^{2r{\bm{\varepsilon}}}}$
maps $U^{m}_{r}({\bm{\eta}})$ into
$U^{n}_{r}({\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}))$. Furthermore, the
following diagram commutes:
$\textstyle{M\times
U^{m}_{r}({\bm{\eta}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c^{m}({\bm{\eta}})}$$\scriptstyle{{\delta^{2r{\bm{\varepsilon}}}}}$$\textstyle{M\times\mathord{\mathbb{R}}^{m}({\bm{\eta}})\times\mathord{\mathbb{R}}^{{\bm{\eta}}}_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}\times{\delta^{{\bm{\varepsilon}}}}}$$\textstyle{M\times
U^{n}_{r}({\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c({\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}))}$$\textstyle{M\times\mathord{\mathbb{R}}^{n}({\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}))\times\mathord{\mathbb{R}}_{r}^{{\delta^{{\bm{\varepsilon}}}}({\bm{\eta}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{per}({\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}),{\bm{\varepsilon}})}$$\textstyle{M\times\mathord{\mathbb{R}}^{n}({\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}))\times\mathord{\mathbb{R}}^{{\delta^{{\bm{\varepsilon}}}}({\bm{\eta}})-{\bm{\varepsilon}}}_{r}\times\mathord{\mathbb{R}}_{r}^{{\bm{\varepsilon}}}}$
This implies
$(\Phi_{\bm{\eta}})_{\ast}({{\delta^{2r{\bm{\varepsilon}}}}}^{\ast}P({\bm{\varepsilon}})))=(\mathrm{id}\times{\delta^{{\bm{\varepsilon}}}})^{\ast}\bigl{(}P({\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}))\otimes\mathrm{id}+A\otimes\mathrm{per}({\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}),{\bm{\varepsilon}})_{\ast}D({\delta^{{\bm{\varepsilon}}}}({\bm{\eta}}))\bigr{)},$
so ${{\delta^{2r{\bm{\varepsilon}}}}}^{\ast}P({\bm{\varepsilon}})$ is indeed
of block form. ∎
The converse of this lemma is also true.
###### Lemma 4.2.4.
Let $m>0$ and $P\in\overline{\Psi\mathrm{DO}}^{m}(\mathfrak{S}_{g})$ be of
block form of type $A$. For each differential operator $D$ of order $m$ with
constant coefficients on
$\mathcal{C}_{c}^{\infty}(\mathord{\mathbb{R}},Cl_{1,0})$, the operator
$Q\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=P\boxtimes\mathrm{id}+A\boxtimes
D\in\overline{\Psi\mathrm{DO}}^{m}(\mathfrak{S}_{g}\boxtimes
Cl_{1,0})=\overline{\Psi\mathrm{DO}}^{m}(\mathfrak{S}_{g\oplus\mathrm{d}x_{n+1}^{2}})$
is also block operator of type $A$.
###### Proof.
If $m>0$, then $Q$ is an element of
$\overline{\Psi\mathrm{DO}}^{m}(\mathfrak{S}_{g\oplus\mathrm{d}x_{n+1}^{2}})$
by Theorem B.0.28.
Axiom (1) is trivially satisfied. Axiom (2) also holds because $A\boxtimes D$
acts as a differential operator in the additional dimension. Axiom (3)
requires some definitions. Let
${\bm{\varepsilon}}\colon\mathbf{n+1}\rightarrow\mathord{\mathbb{Z}}_{2}$ be
given.
If $n+1\in{\mathrm{dom}\,}{\bm{\varepsilon}}$, then
$U_{r}({\bm{\varepsilon}})=U_{r}(p_{n+1}({\bm{\varepsilon}}))\times\\{{\bm{\varepsilon}}(n+1)x_{n+1}>r\\}$
and
$\Phi_{{\bm{\varepsilon}}}=\Phi_{p_{n+1}({\bm{\varepsilon}})}\times\mathrm{id}$.
In this case, we have
$P({\bm{\varepsilon}})=Q(p_{n+1}({\bm{\varepsilon}}))\boxtimes\mathrm{id}_{\mathord{\mathbb{R}}}\text{
and
}D^{Q}({\bm{\varepsilon}})=D^{P}(p_{n+1}({\bm{\varepsilon}}))\boxtimes\mathrm{id}.$
If $n+1\notin{\mathrm{dom}\,}{\bm{\varepsilon}}$, then
$U_{r}({\bm{\varepsilon}})=U_{r}^{n}({\bm{\varepsilon}})\times\mathord{\mathbb{R}}$,
where $U_{r}^{n}({\bm{\varepsilon}})\subseteq\mathord{\mathbb{R}}^{n}$, and
$\Phi_{\bm{\varepsilon}}$ is given by the decomposition
$\textstyle{\mathfrak{S}|_{M\times
U_{r}({\bm{\varepsilon}})}=\mathfrak{S}\mathord{\upharpoonleft}_{M\times
U_{r}^{n}({\bm{\varepsilon}})}\boxtimes
Cl_{1,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{\bm{\varepsilon}}\boxtimes\mathrm{id}}$$\textstyle{\mathfrak{S}_{g({\bm{\varepsilon}})}\boxtimes
Cl_{{\bm{\varepsilon}},0}\boxtimes
Cl_{1,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{swap}}$$\scriptstyle{\mathrm{id}\boxtimes}$$\textstyle{\mathfrak{S}_{g({\bm{\varepsilon}})}\boxtimes
Cl_{1,0}\boxtimes Cl_{{\bm{\varepsilon}},0}.}$
In this case, we have
$Q({\bm{\varepsilon}})=P({\bm{\varepsilon}})\boxtimes\mathrm{id}+A\boxtimes
D\quad\text{ and }\quad D^{Q}({\bm{\varepsilon}})=D^{P}({\bm{\varepsilon}}).$
inline]Figure out if $\mathrm{Cycl}(1:1+|{\mathrm{dom}\,}{\bm{\varepsilon}}|$
or $\mathrm{Cycl}(1:1+|{\mathrm{dom}\,}{\bm{\varepsilon}}|^{-1}$. Define the
correct map in an ad-hoc manner as the conceptual notation will be used later.
With these definitions the equation
${\Phi_{\bm{\varepsilon}}}_{\ast}Q|_{M\times
U_{r}({\bm{\varepsilon}})}=Q({\bm{\varepsilon}})\boxtimes\mathrm{id}+A\boxtimes
D^{Q}({\bm{\varepsilon}})$
is easily verified so that $Q$ is a block operator. ∎
Pseudo differential operators are not local by nature. Block operators on the
other hand are local in certain directions on special classes of open sets. In
order to use local-to-global principles for block operators, we need partition
of unities that are adapted to the specific block form of a block operator.
###### Definition 4.2.5.
Let $\chi\colon\mathord{\mathbb{R}}\rightarrow[0,1]$ be a compactly supported,
smooth, and even function that is identically $1$ near the origin. We denote
the extensions by zero of
$\chi^{-1}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=(1-\chi)|_{\mathord{\mathbb{R}}_{\leq
0}},\quad\chi^{0}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\chi,\quad\text{ and
}\quad\chi^{1}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=(1-\chi)|_{\mathord{\mathbb{R}}_{\geq 0}}$
with the same symbols. More generally, we define on $\mathord{\mathbb{R}}^{n}$
or $M\times\mathord{\mathbb{R}}^{n}$ the partition of unity
$\biggl{\\{}\chi^{\bm{\alpha}}(m,x)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\prod_{j=1}^{n}\chi^{{\bm{\alpha}}(j)}(x_{j})\biggr{\\}}.$
parameterised ${\alpha\colon\mathbf{n}\rightarrow\mathord{\mathbb{Z}}_{3}}$.
As an application of this partition, we prove that the constants in the
definition of a block operator are actually lower bounds.
###### Lemma 4.2.6.
Let $P\in\overline{\Psi\mathrm{DO}}^{m}(\mathfrak{S}_{g})$ be a block operator
and let $R>r$ be constants such that the conditions (1)-(3) of $P$ hold. Then
these conditions also hold for all $R^{\prime}>R$ and $r^{\prime}>r$ with
$R^{\prime}>r^{\prime}$,
###### Proof.
Condition (1) is clearly satisfied for all $r^{\prime}>r$.
We continue with the proof of condition (3). Let
$\sigma\in\Gamma_{c}(\mathfrak{S}_{g})$ be a section with
$\mathrm{supp}\,\,\sigma\subseteq U_{r^{\prime}}({\bm{\varepsilon}})$. Since
the support is compact, we find a slightly bigger
$r^{\prime\prime}>r^{\prime}$ such that $\mathrm{supp}\,\,\sigma\subseteq
U_{r^{\prime\prime}}({\bm{\varepsilon}})$. Let
$\chi\colon\mathord{\mathbb{R}}\rightarrow[0,1]$ be a smooth function that is
identically $1$ near $\mathord{\mathbb{R}}_{\geq r^{\prime\prime}}$ and
identically zero near $\mathord{\mathbb{R}}_{\leq r^{\prime}}$.
For
${\bm{\varepsilon}}\colon{\mathrm{dom}\,}{\bm{\varepsilon}}\subseteq\mathbf{n}\rightarrow\mathord{\mathbb{Z}}_{2}$
define
$\chi^{\bm{\varepsilon}}(x)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\prod_{j\in{\mathrm{dom}\,}{\bm{\varepsilon}}}\chi({\bm{\varepsilon}}(j)x_{j}).$
This function is supported within $U_{r^{\prime}}({\bm{\varepsilon}})$. If we
do not notationally distinguish $\chi^{\bm{\varepsilon}}$ and
$(\Phi_{\bm{\varepsilon}})_{\ast}\chi^{\bm{\varepsilon}}=\chi^{\bm{\varepsilon}}\circ\Phi_{\bm{\varepsilon}}^{-1}$,
we have
$\displaystyle(\Phi_{\bm{\varepsilon}})_{\ast}(P\chi^{\bm{\varepsilon}}\sigma)$
$\displaystyle=P({\bm{\varepsilon}})\boxtimes\mathrm{id}(\chi^{\bm{\varepsilon}}\sigma)+A\boxtimes
D(\chi^{\bm{\varepsilon}}\sigma)$
$\displaystyle=\chi^{\bm{\varepsilon}}\cdot(P({\bm{\varepsilon}})\boxtimes\mathrm{id})(\sigma)+D(\chi^{\bm{\varepsilon}})\cdot(A\boxtimes\mathrm{id})(\sigma)+\chi^{\bm{\varepsilon}}\cdot(A\boxtimes
D)(\sigma),$
which is supported within $U_{r^{\prime}}({\bm{\varepsilon}})$, too. Thus, $P$
also restricts to $U_{r^{\prime}}({\bm{\varepsilon}})$ and necessarily
decomposes there as in (3).
Condition (2) uses the construction of Definition 4.2.5. We will prove
condition (2) inductively over $n$, the dimension of the Euclidean factor of
$M\times\mathord{\mathbb{R}}^{n}$. For $n=0$ there is nothing to prove, and we
can continuue with $n\geq 1$. Let
$\chi\colon\mathord{\mathbb{R}}\rightarrow[0,1]$ be a smooth and even function
that is identically $1$ near $[-r,r]$ and that vanishes near
$\mathord{\mathbb{R}}\setminus(-R,R)$. Let
$\\{\chi^{\bm{\alpha}}\\}_{{\bm{\alpha}}\in\mathord{\mathbb{Z}}_{3}^{n}}$ be
the corresponding partition of unity on $M\times\mathord{\mathbb{R}}^{n}$ from
Definition 4.2.5. We clearly have
$P(\sigma)=P(\chi^{\mathbf{0}}\sigma)+\sum_{{\bm{\alpha}}\in\mathord{\mathbb{Z}}_{3}^{n}\setminus\mathbf{0}}P(\chi^{\bm{\alpha}}\sigma).$
The summand $\chi^{\mathbf{0}}\sigma$ is supported within
$M\times(RI^{n})^{\circ}$, so, by condition (2) of $P$, also
$P(\chi^{\mathbf{0}}\sigma)$ is supported within $M\times(RI^{n})^{\circ}$.
Recall the notation
$\hat{{\bm{\alpha}}}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt={\bm{\alpha}}|_{\mathrm{supp}\,{\bm{\alpha}}}\colon\mathrm{supp}\,\,{\bm{\alpha}}\rightarrow\mathord{\mathbb{Z}}_{2}$.
The section $\chi^{\bm{\alpha}}\sigma$ is supported within
$U_{r}(\hat{{\bm{\alpha}}})$. If ${\bm{\alpha}}\neq\mathbf{0}$, then
$\hat{{\bm{\alpha}}}$ is not the empty map so that
$P(\chi^{\bm{\alpha}}\sigma)$ is also supported within
$U_{r}(\hat{{\bm{\alpha}}})$ by the proof of condition (3) above.
Since $\mathrm{supp}\,\sigma\subseteq M\times(R^{\prime}I^{n})^{\circ}$ we can
find a smooth and even function
$\psi\colon\mathord{\mathbb{R}}\rightarrow[0,1]$ that vanishes near
$\mathord{\mathbb{R}}\setminus(-R^{\prime},R^{\prime})$ and such that
$\psi^{\mathbf{0}}(x)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\prod_{j=1}^{n}\psi(x_{j})$ is identically $1$ on
$\mathrm{supp}\,\,\sigma$. The block form of $P$ implies
$\displaystyle P(\chi^{\mathbf{{\bm{\alpha}}}}\sigma)$
$\displaystyle=P(\chi^{\mathbf{{\bm{\alpha}}}}\psi^{\mathbf{0}}\sigma)$
$\displaystyle\begin{split}&=\prod_{j\in\mathrm{supp}\,{\bm{\alpha}}}\psi\circ\mathord{\mathrm{pr}}_{j}\cdot
P\left(\Bigl{(}\prod_{j\in\mathrm{Null}({\bm{\alpha}})}\psi\circ\mathord{\mathrm{pr}}_{j}\Bigr{)}\cdot\chi^{\bm{\alpha}}\sigma\right)\\\
&\qquad\quad+D(\hat{{\bm{\alpha}}})\left(\prod_{j\in\mathrm{supp}\,{\bm{\alpha}}}\psi\circ\mathord{\mathrm{pr}}_{j}\right)\cdot
P\left(\Bigl{(}\prod_{j\in\mathrm{Null}\,{\bm{\alpha}}}\psi\circ\mathord{\mathrm{pr}}_{j}\Bigr{)}\cdot\chi^{\bm{\alpha}}\sigma\right).\end{split}$
By Corollary 4.2.3, $P({\bm{\alpha}}|_{\mathrm{supp}\,{\bm{\alpha}}})$ is a
block operator on
$\mathfrak{S}_{g({\bm{\alpha}}|_{\mathrm{supp}\,{\bm{\alpha}}})}$ that satisfy
conditions (1) - (3) for the same constants as $P$ and $A\boxtimes
D({\bm{\alpha}}|_{\mathrm{supp}\,{\bm{\alpha}}})$ is a differential operator
and hence only support-decreasing. This and the induction assumption, implies
that the section
$P\left(\Bigl{(}\prod_{j\in\mathrm{Null}({\bm{\alpha}})}\psi\circ\mathord{\mathrm{pr}}_{j}\Bigr{)}\cdot\chi^{\bm{\alpha}}\sigma\right)$
must be supported within
$\\{|x_{j}|<R^{\prime}\,:\,j\in\mathrm{Null}\,{\bm{\alpha}}\\}\cap
U_{r}({\bm{\alpha}}|_{\mathrm{supp}\,{\bm{\alpha}}})$. It follows that
$\prod_{j\in\mathrm{supp}\,{\bm{\alpha}}}\psi\circ\mathord{\mathrm{pr}}_{j}\cdot
P\left(\prod_{j\in\mathrm{Null}({\bm{\alpha}})}\psi\circ\mathord{\mathrm{pr}}_{j}\chi^{\bm{\alpha}}\sigma\right)$
and
$D(\hat{{\bm{\alpha}}})\left(\prod_{j\in\mathrm{supp}\,{\bm{\alpha}}}\psi\circ\mathord{\mathrm{pr}}_{j}\right)\cdot
P\left(\prod_{j\in\mathrm{Null}\,{\bm{\alpha}}}\psi\circ\mathord{\mathrm{pr}}_{j}\chi^{\bm{\alpha}}\sigma\right)$
are supported within $M\times(R^{\prime}I^{n})^{\circ}\cap
U_{r}({\bm{\alpha}}|_{\mathrm{supp}\,\,{\bm{\alpha}}})$, which finishes the
proof of (2). ∎
###### Corollary 4.2.7.
Let $P\in\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{g})$ be a block operator
of type $A$ whose core is contained in $M\times RI^{n}$. For
${\bm{\alpha}}\colon\mathbf{n}\rightarrow\mathord{\mathbb{Z}}_{3}$ and
$\rho^{\prime}>\rho>R$ define
$V_{\rho<\rho^{\prime}}({\bm{\alpha}})\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=M\times\\{x\in\mathord{\mathbb{R}}^{n}\,:\,x_{j}\in(-\rho^{\prime},\rho^{\prime})\text{
if }j\in\mathrm{Null}({\bm{\alpha}}),\alpha_{j}x_{j}>\rho\text{
otherwise}\\}.$
If $\sigma\in\Gamma_{c}(\mathfrak{S}_{g})$ is supported within
$V_{\rho<\rho^{\prime}}({\bm{\alpha}})$, then $P\sigma$ is also supported
there.
###### Proof.
If ${\bm{\alpha}}=\mathbf{0}$, then
$V_{\rho<\rho^{\prime}}(0)=M\times\rho^{\prime}I^{n}$ and the claim follows
from Lemma 4.2.6.
Assume now that ${\bm{\alpha}}\neq\mathbf{0}$, so that
$V_{\rho<\rho^{\prime}}({\bm{\alpha}})\subseteq
U_{\rho}(\hat{{\bm{\alpha}}})$. For each $u\in\Gamma_{c}(\mathfrak{S}_{g})$
supported with in $V_{\rho<\rho^{\prime}}({\bm{\alpha}})$ we deduce with abuse
of notation
$\displaystyle
Pu=(P(\hat{{\bm{\alpha}}})\boxtimes\mathrm{id})(\chi^{\bm{\alpha}}u)+\mathrm{e/o}\boxtimes
D(\hat{{\bm{\alpha}}})(\chi^{\bm{\alpha}})(\chi^{\bm{\alpha}}u)$
The second summand is a differential operator and hence support-decreasing.
We know from Corollary 4.2.3 that $P(\hat{{\bm{\alpha}}})$ is again a block
operator whose core is contained in
$M\times\rho^{\prime}I^{|\mathrm{Null}({\bm{\alpha}})|}$. Since
$V_{\rho<\rho^{\prime}}({\bm{\alpha}})\cong\rho^{\prime}I^{|\mathrm{Null}({\bm{\alpha}})|}\times\mathord{\mathbb{R}}_{\rho}^{\hat{{\bm{\alpha}}}}$,
it follows from the previous Lemma that
$\mathrm{supp}\,\,P({\bm{\alpha}})(u(\text{--},y))\subseteq\rho^{\prime}I^{|\mathrm{Null}({\bm{\alpha}})|}$
and hence
$\mathrm{supp}\,\,P({\bm{\alpha}})(u)\subseteq\rho^{\prime}I^{|\mathrm{Null}({\bm{\alpha}})|}\times\mathord{\mathbb{R}}_{\rho}^{\hat{{\bm{\alpha}}}}=V_{\rho<\rho^{\prime}}({\bm{\alpha}}).$
∎
The technique used in the previous proofs can be used to prove that block
operators are local in the following limited sense.
###### Corollary 4.2.8.
Let $P$ be a block operator whose core is contained in $M\times\rho I^{n}$. If
$\rho^{\prime}>\rho$ and $V\subseteq U_{\rho^{\prime}}({\bm{\varepsilon}})$ is
an open set that corresponds under $U_{\rho^{\prime}}({\bm{\varepsilon}})\cong
M\times\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})\times\mathord{\mathbb{R}}^{{\bm{\varepsilon}}}_{\rho^{\prime}}$
to $M\times\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})\times
V^{\bm{\varepsilon}}$, then $(Pu)|_{V}$ only depends on (the germ of)
$u|_{\bar{V}}$.
###### Proof.
It suffices to show that if $u$ vanishes identically near $\bar{V}$, then $Pu$
vanishes identically on $V$. Let $\phi\colon
M\times\mathord{\mathbb{R}}^{n}\rightarrow\mathord{\mathbb{R}}_{\geq 0}$ be a
smooth function such that $\phi^{-1}(\mathord{\mathbb{R}}_{>0})=V$ and such
that $\phi$ only depends on variables indexed by
${\mathrm{dom}\,}({\bm{\varepsilon}})$. Let
$\chi\colon\mathord{\mathbb{R}}\rightarrow[0,1]$ be a smooth even function
such that $\chi$ is identically $1$ on $[-\rho,\rho]$ and whose support lies
in $(-\rho^{\prime},\rho^{\prime})$. Denote by $\chi^{\bm{\alpha}}$ the
partition of unity from Definition 4.2.5.
Since $\mathrm{supp}\,\phi\subseteq U_{\rho^{\prime}}({\bm{\varepsilon}})$, we
conclude that $\mathrm{supp}\,\,\phi$ and
$V_{\rho<\rho^{\prime}}(\hat{{\bm{\alpha}}})$ have non-empty intersection only
if $\hat{{\bm{\alpha}}}\geq{\bm{\varepsilon}}$. We know from Corollary 4.2.7
that $\mathrm{supp}\,(P\chi^{\bm{\alpha}}u)\subseteq
V_{\rho<\rho^{\prime}}({\bm{\alpha}})$, so we conclude $\phi\cdot
P\chi^{\bm{\alpha}}u=0$ if $\hat{{\bm{\alpha}}}\geq{\bm{\varepsilon}}$ is
false. On the other hand, if $\hat{{\bm{\alpha}}}\geq{\bm{\varepsilon}}$, then
$V_{\rho<\rho^{\prime}}({\bm{\alpha}})\subseteq U_{\rho}({\bm{\varepsilon}})$
and the block decomposition of $P$ implies
$\displaystyle\phi\cdot P\chi^{\bm{\alpha}}u$
$\displaystyle=P\phi\chi^{\bm{\alpha}}u+[\phi,P]\chi^{\bm{\alpha}}u=P\chi^{\bm{\alpha}}\underbrace{\phi
u}_{=0}-i^{-1}\underbrace{\mathrm{symb}_{1}(P)(\text{--},\mathrm{d}\phi)\chi^{\bm{\alpha}}u}_{=0\text{
on }V}$ $\displaystyle=0\qquad\text{ on }V$
because $u|_{V}=0$. Since
$\\{\chi^{\bm{\alpha}}\\}_{{\bm{\alpha}}\in\mathord{\mathbb{Z}}_{3}^{n}}$ is a
partition of unity, the previous calculations imply $\phi Pu|_{V}=0$ from
which we deduce $Pu|_{V}=0$. ∎
###### Definition 4.2.9.
A block operator $P\in\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{g})$ is
called _block operator of Dirac-type_ , if $A=\mathrm{e/o}$ and
$D({\bm{\varepsilon}})={\not{\mathit{D}}}_{\mathord{\mathbb{R}}^{\bm{\varepsilon}}}$
is the Dirac operator of the Euclidean metric on
$\mathord{\mathbb{R}}^{\bm{\varepsilon}}$ for all
${\bm{\varepsilon}}\colon{\mathrm{dom}\,}{\bm{\varepsilon}}\rightarrow\mathord{\mathbb{Z}}_{2}$,
and if the principal symbol satisfies
$\mathrm{symb}_{1}(P)=\mathrm{symb}_{1}({\not{\mathfrak{D}}}_{g})$.
The motivating examples for this definition are the Dirac operator of a block
metric and the adjoint operator of a block map of pseudo Dirac operators.
###### Example 4.2.10.
We apply the discussion of Example 4.1.18 to show that
${\not{\mathfrak{D}}}_{g}$ is a block operator (of Dirac type), if $g$ is a
block metric on $M\times\mathord{\mathbb{R}}^{n}$. Assume that the block
metric decomposes outside of $M\times rI^{n}$. Then, since differential
operators are only support decreasing, we can choose any $R>r$ so that
condition (1) and (2) are trivially satisfied. It remains to show that
${\not{\mathfrak{D}}}_{g}$ decomposes appropriately on
$U_{r}({\bm{\varepsilon}})$. Since the defining sequence of the Dirac operator
(4.1.10) is natural with respect to Pin-structure preserving, isometric
diffeomorphisms, we have
$\displaystyle(\Phi_{{\bm{\varepsilon}}})_{\ast}{\not{\mathfrak{D}}}_{g}|_{M\times
U_{r}({\bm{\varepsilon}})}$
$\displaystyle={\not{\mathfrak{D}}}_{c({\bm{\varepsilon}})_{\ast}g}={\not{\mathfrak{D}}}_{g({\bm{\varepsilon}})+\langle\cdot,\cdot\rangle_{\mathord{\mathbb{R}}^{|{\mathrm{dom}\,}{\bm{\varepsilon}}|}}}={\not{\mathfrak{D}}}_{g({\bm{\varepsilon}})}\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}_{\mathord{\mathbb{R}}^{|{\mathrm{dom}\,}{\bm{\varepsilon}}|}}.$
The decomposition of the spinor bundle in Example 4.1.18 has a weaker version.
If $U\subseteq\mathord{\mathbb{R}}^{n}$ is open and $g\colon
U\rightarrow\mathrm{Riem}(M)$ is a smooth map, then $TU$ is still an
orthogonal complement of $TM$ in $T(M\times U)$ with respect to
$\mathrm{susp}(g)$. This induces a splitting of the spinor bundle
$\mathfrak{S}_{\mathrm{susp}(g)}\cong\mathfrak{S}_{g}\otimes Cl_{n,0}$.
###### Definition 4.2.11.
Let $P\colon U\rightarrow\Psi\mathrm{DO}^{m}(\mathfrak{S})$ be a smooth map,
where the target carries the (subspace topology of the) amplitude topology. It
comes with a smooth map $g\colon U\rightarrow\mathrm{Riem}(M)$. Define on
$\mathfrak{S}_{\mathrm{susp}(g)}$ the operators that are given under the
isomorphism $\mathfrak{S}_{\mathrm{susp}(g)}\cong\mathfrak{S}_{g}\otimes
Cl_{n,0}$ by
$\displaystyle P^{\mathrm{ex}}(\sigma\otimes
e_{I})(m,t)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\left(P_{t}\sigma(\cdot,t)\right)\otimes e_{I},$
$\displaystyle\mathrm{susp}(P)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=P^{\mathrm{ex}}+\sum_{k=1}^{n}\partial_{t_{k}}\cdot\nabla_{\partial_{t_{k}}}^{\mathfrak{S}_{\mathrm{susp}\,g}}.$
We call $\mathrm{susp}(P)$ the _operator suspension of_ $P$.
By the parametrised version of the Atiyah-Singer exterior tensor theorem
B.0.29 we have
$\mathrm{susp}(P)\in\overline{\Psi\mathrm{DO}}^{m}(\mathfrak{S}_{\mathrm{susp}(g)})$
if $m>0$.
We are mostly interested in the case where $U=\mathord{\mathbb{R}}^{n}$ and
$P$ is a smooth block map. Note that this implies that the underlying map of
Riemannian metrics is also a smooth block map. However, in some proofs, we
need the operator suspension on more general open subsets.
###### Lemma 4.2.12.
Let $m>0$ and let
$P\colon\mathord{\mathbb{R}}^{n}\rightarrow\Psi\mathrm{DO}^{m}(\mathfrak{S})$
be a smooth block map, where the target carries the (subspace topology of the)
amplitude topology and underlying block map $g$.
Then $P^{\mathrm{ex}}$ and $\mathrm{susp}(P)$ are a block operators of type
$\mathrm{id}$ and $\mathrm{e/o}$, respectively. If $P$ takes values in
$\Psi\mathrm{Dir}(M)$, then $\mathrm{susp}(P)$ is a block operator of Dirac
type.
###### Proof.
We will only prove the statement for $\mathrm{susp}(P)$ in the case that $P$
takes values in $\Psi\mathrm{Dir}(M)$, for the other cases are similar. Since
$P$ and $g$ are block maps, there is an $r>0$ such that $P$ and $g$ are
independent of $x_{i}$ if $|x_{i}|>r$. Then $\mathrm{susp}(g)$ decomposes
outside of $M\times rI^{n}$. Since
$\mathrm{susp}(P)-\mathrm{e/o}\otimes{\not{\mathit{D}}}_{\mathord{\mathbb{R}}^{n}}$
is $\mathcal{C}^{\infty}(\mathord{\mathbb{R}}^{n})$-linear, the operator
restricts to open subsets of the form $M\times U$ with
$U\subseteq\mathord{\mathbb{R}}^{n}$ open, and thus $P$ does it, too.
It remains to show that $P$ decomposes appropriately on
$U_{r}({\bm{\varepsilon}})$. On this subset, a permutation of coordinates
$c({\bm{\varepsilon}})\colon M\times U_{r}({\bm{\varepsilon}})\rightarrow
M\times\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})\times\mathord{\mathbb{R}}^{\bm{\varepsilon}}_{r}$
is an isometry if the domain carries the metric
$g|_{U_{r}({\bm{\varepsilon}})}$ and the right hand side has
$\mathrm{susp}(g)({\bm{\varepsilon}})\oplus\langle\cdot,\cdot\rangle$. Thus
the spinor bundle decomposes into
$\mathfrak{S}_{\mathrm{susp}(g)({\bm{\varepsilon}})}\boxtimes
Cl_{{\bm{\varepsilon}},0}\cong\mathfrak{S}_{g}\otimes
Cl_{n-|{\bm{\varepsilon}}|,0}\boxtimes Cl_{{\bm{\varepsilon}},0}$
over
$(M\times\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}}))\times\mathord{\mathbb{R}}^{\bm{\varepsilon}}_{r}$
via $\Phi_{\bm{\varepsilon}}=\mathfrak{S}(c({\bm{\varepsilon}}))$. Since
$P|_{U_{r}({\bm{\varepsilon}})}$ is independent of the coordinates indexed by
${\mathrm{dom}\,}{\bm{\varepsilon}}$, we derive the equation
$\displaystyle(\Phi_{\bm{\varepsilon}})_{\ast}(\mathrm{susp}(P)|_{M\times
U_{r}({\bm{\varepsilon}})})$
$\displaystyle=(\Phi_{\bm{\varepsilon}})_{\ast}\bigl{(}P\otimes\mathrm{id}\bigr{)}+(\Phi_{\bm{\varepsilon}})_{\ast}\bigl{(}\mathrm{e/o}_{\mathfrak{S}_{g}}\otimes{\not{\mathit{D}}}_{\mathord{\mathbb{R}}^{n}}\bigr{)}$
$\displaystyle\begin{split}&=P_{c({\bm{\varepsilon}})^{-1}(\text{--})}\otimes\mathrm{id}+(\mathrm{e/o}_{\mathfrak{S}_{g}}\otimes{\not{\mathit{D}}}_{\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})})\boxtimes\mathrm{id}_{Cl_{{\bm{\varepsilon}},0}}\\\
&\qquad+\mathrm{e/o}_{\mathfrak{S}_{g}}\otimes\mathrm{e/o}_{Cl_{n-|{\bm{\varepsilon}}|,0}}\boxtimes{\not{\mathit{D}}}_{\mathord{\mathbb{R}}^{\bm{\varepsilon}}_{r}}\end{split}$
$\displaystyle\begin{split}&=\Bigl{(}(P|_{M\times\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})})_{c({\bm{\varepsilon}})^{-1}(\text{--})}\otimes\mathrm{id}_{Cl_{n-|{\bm{\varepsilon}}|,0}}+\mathrm{e/o}_{\mathfrak{S}_{g}}\otimes{\not{\mathit{D}}}_{\mathord{\mathbb{R}}^{n}({\bm{\varepsilon}})}\Bigr{)}\boxtimes\mathrm{id}\\\
&\qquad+\mathrm{e/o}_{\mathfrak{S}_{\mathrm{susp}(g)({\bm{\varepsilon}})}}\boxtimes{\not{\mathit{D}}}_{\mathord{\mathbb{R}}^{\bm{\varepsilon}}},\end{split}$
which shows that $\mathrm{susp}(P)$ decomposes as required. ∎
So far, we have only studied block operators on the level of vector bundles
over manifolds. Before we are going to construct the block version of
$\Psi\mathrm{Dir}^{\times}(M)$, we will study its (functional) analytic
properties in the next section.
### 4.3 Analytical Properties of Block Operators
The purpose of this section is to derive the needed analytical properties of
block operators. We will show that each block operator realises to a bounded
operator on positive Sobolev spaces and that the realisations of block
operators of Dirac type satisfy analogous versions of the fundamental elliptic
estimate. Moreover, we will show that if they are self-adjoint and invertible
near infinity, then they are also Fredholm operators. Many of these properties
rely on the structure of a block metric near infinity because we are working
on non-compact manifolds.
In the following, we only consider pseudo differential operators of order
$m\in\\{0,1\\}$.
###### Lemma 4.3.1.
Let $g$ be a block metric on $M\times\mathord{\mathbb{R}}^{n}$ and let
$P\in\overline{\Psi\mathrm{DO}}^{m}(\mathfrak{S}_{g})$ be a block operator.
Then $P$ extends to a bounded linear operator
$P\colon H^{s}(\mathfrak{S}_{g})\rightarrow H^{s-m}(\mathfrak{S}_{g})$
for all $s\in\mathord{\mathbb{Z}}$.
###### Proof.
We prove this lemma by induction over $n$. For $n=0$, this is classical
because $M$ is closed, see Theorem B.0.18. For $n>0$, we pick appropriate
$R>r>0$ such that $P$ restricts to $M\times(RI^{n})^{\circ}$ and decomposes on
$\\{\varepsilon x_{i}>r\\}$ for all $i\in\\{1,\dots,n\\}$ and all
$\varepsilon\in\mathord{\mathbb{Z}}_{2}$. Let
$\chi\colon\mathord{\mathbb{R}}\rightarrow[0,1]$ be a smooth, symmetric
function that is identically $1$ on $(-r,r)$ and vanishes near
$\mathord{\mathbb{R}}\setminus(-R,R)$. Let
$\\{\chi^{\bm{\alpha}}\\}_{{\bm{\alpha}}\in\mathord{\mathbb{Z}}_{3}^{n}}$ be
the partition of unity obtained from Definition 4.2.5. By the triangle
inequality, it suffices to show that $P\cdot\chi^{\bm{\alpha}}$ extends to a
bounded operator.
If ${\bm{\alpha}}=\mathbf{0}$, then $P\chi^{\mathbf{0}}$ restricts to an
operator on the relatively compact set $M\times(RI^{n})^{\circ}$. Thus, by
Theorem B.0.18, $P\chi^{\mathbf{0}}$ extends to a bounded operator
$H^{s}(\mathfrak{S}_{g})\rightarrow H^{s-1}(\mathfrak{S}_{g})$.
If ${\bm{\alpha}}\neq\mathbf{0}$, then there is a tupel
$(i,\varepsilon)\in\\{1,\dots,n\\}\times\mathord{\mathbb{Z}}_{2}$ such that
$\mathrm{supp}\,\chi^{\bm{\alpha}}$ is contained in $U_{r}((i,\varepsilon))$.
The induction hypothesis and [palais1965seminar]*Corollary XIV.1 applied to
the discrete Hilbert chains
$H^{s}(M\times\mathord{\mathbb{R}}^{n-1};\mathfrak{S}_{g(i,\varepsilon)})$ and
$H^{s}(\mathord{\mathbb{R}};Cl_{1,0})$ imply
$P(i,\varepsilon)\boxtimes\mathrm{id}+A\boxtimes D(i,\varepsilon)\in
Op^{1}(H^{s}(\mathfrak{S}_{g(i,\varepsilon)}\boxtimes Cl_{1,0})),$
which means that this operator induces a bounded linear map
$H^{s}(\mathfrak{S}_{g(i,\varepsilon)}\boxtimes Cl_{1,0})\rightarrow
H^{s-1}(\mathfrak{S}_{g(i,\varepsilon)}\boxtimes Cl_{1,0})$
for all $s\in\mathord{\mathbb{Z}}$. Since $P$ decomposes into
$P(i,\varepsilon)\boxtimes\mathrm{id}+A\boxtimes D(i,\varepsilon)$ on
$\mathfrak{S}_{g}|_{U_{r}((i,\varepsilon))}\cong\mathfrak{S}_{g(i,\varepsilon)}\boxtimes
Cl_{1,0}|_{\varepsilon\mathord{\mathbb{R}}_{>r}}$, it follows
$P\chi^{\bm{\alpha}}\in Op^{1}(\mathfrak{S}_{g})$. ∎
Recall that a sequence of operators $P_{n}$ converges to $P$ in the Atiyah-
Singer topology if the $P_{i}$ converge uniformly on compact subsets to $P$.
Thus, it is not clear that the fundamental ellitpic estimate carries over to
elements in the Atiyah-Singer closure. However, this is the case, if the
operator $P$ increases the support in a controlled manner.
###### Lemma 4.3.2.
Let $M$ be a not necessarily closed manifold and $E,F\rightarrow M$ be vector
bundles over $M$. Let $P\in\overline{\Psi\mathrm{DO}}^{1}(E,F)$ be an elliptic
pseudo differential operator whose principal symbol is the principal symbol of
an actual pseudo differential operator (i.e. smooth) and that has the
following property: For all compact neighbourhoods $K\subseteq M$ there is a
relatively compact open subset $W$ such that if $\mathrm{supp}\,u\subseteq K$
then $\mathrm{supp}\,Pu\subseteq W$.
Then, for all compact $K\subseteq M$, there is a compact $L\subseteq M$ such
that $P$ induces a bounded map $H_{K}^{s}(E)\rightarrow H_{L}^{s-1}(F)$, where
$H_{K}^{s}(E)$ denotes the subspace of all section with support in $K$. The
operator $P$ furthermore satisfies the elliptic estimate:
$||u||_{s}\leq C\left(||u||_{s-1}+||Pu||_{s-1}\right).$
###### Proof.
The subspace $\Gamma_{c}(E)$ is dense in $H^{s}(E)$ for all
$s\in\mathord{\mathbb{Z}}$ and the multiplication with a smooth compactly
supported function induces a continuous endomorphism on all Sobolev spaces.
Thus, if $\chi$ is a smooth compactly supported function that is identically
$1$ on $K$, then we can approximate each element of $H_{K}^{s}(E)$ by an
element of $\chi\cdot\Gamma_{c}(E)$. If $L$ is the compact subset from the
assumption assigned to $\mathrm{supp}\,\,\chi$, then a simple approximation
argument shows that $P\colon H_{K}^{s}(E)\rightarrow H^{s-1}_{L}(F)$ is well
defined and continuous, because $H^{s-1}_{L}(F)$ is a closed subspace.
To prove the elliptic estimate we argue as follows. Pick smooth functions
$\rho_{0}$ and $\rho_{1}$ such that $\rho_{0}$ is identically $1$ on $K$ and
supported within $L^{\circ}$ and $\rho_{1}$ is identically $1$ on
$\mathrm{supp}\,\rho_{0}$ and supported within $L^{\circ}$. Pick further a
pseudo-inverse $R$ for $P$, i.e., an actual pseudo differential operator of
order $-1$ whose principal symbol is the inverse of $\mathrm{symb}_{1}(P)$.
Then $\rho_{0}-\rho_{0}RP\rho_{1}\in\overline{\Psi\mathrm{DO}}^{0}_{L}(E)$ and
its principal symbol vanishes. Since $\Psi\mathrm{DO}^{-1}(E)$ lies dense in
$\ker\mathrm{symb}_{0}$, we can approximate it by actual pseudo differential
operators of order $-1$, see Lemma B.0.24, so there is a
$Q\in\Psi\mathrm{DO}^{-1}_{L}(E)$ such that
$||\rho_{0}-\rho_{0}RP\rho_{1}-Q||_{s,s}<1/2$.
With that knowledge we can derive the required elliptic estimate from the
elliptic estimate of actual pseudo differential operators. For any $u\in
H_{K}^{s}(E)$ we have $\rho_{j}u=u$ and thus
$\displaystyle||u||_{s}$
$\displaystyle=||(\rho_{0}-\rho_{0}RP\rho_{1})u+\rho_{0}RP\rho_{1}u||_{s}$
$\displaystyle\leq||(\rho_{0}-\rho_{0}RP\rho_{1})u||_{s}+||\rho_{0}RPu||_{s}$
$\displaystyle\leq||(\rho-\rho_{0}RP\rho_{1})u-Qu||_{s}+||Qu||_{s}+||\rho_{0}R||_{s-1,s}||Pu||_{s-1}$
$\displaystyle\leq
1/2||u||_{s}+||Q||_{s-1,s}||u||_{s-1}+||\rho_{0}R||_{s-1,s}||Pu||_{s-1},$
where we used the fact that $\rho_{0}R$ is an actual pseudo differential
operators of order $-1$ that maps sections to sections that are supported
within $\mathrm{supp}\,\rho_{0}$, to deduce that $\rho_{0}R$ induces bounded
maps between Sobolev spaces. The chain of inequalities can be rewritten as
$\displaystyle||u||_{s}$ $\displaystyle\leq
2\left(||Q||_{s-1,s}||u||_{s-1}+||\rho_{0}R||_{s-1,s}||Pu||_{s-1}\right)$
$\displaystyle\leq
2\max\\{||Q||_{s-1,s},||\rho_{0}R||_{s-1,s}\\}\left(||u||_{s-1}+||Pu||_{s-1}\right),$
which gives the second claim. ∎
Since the constants in the fundamental elliptic estimate may depend on the
compact subsets, the result does not immediately carry over to block
operators. To carry the result over, we use the following partition of unity
which is due to Bunke [bunke2009index], who introduced it in special local
coordinates for manifolds with corners.
###### Lemma 4.3.3.
Let $P\in\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{g})$ be a block operator
of Dirac type whose core is contained in $M\times(-\rho,\rho)^{n}$ and that
decomposes on all $U_{\rho-1/2}({\bm{\varepsilon}})$. For
${\bm{\alpha}}\in\mathord{\mathbb{Z}}_{3}^{n}$, set
$V_{\rho}({\bm{\alpha}})=\bigl{\\{}(m,x)\in
M\times\mathord{\mathbb{R}}^{n}\,:\,\alpha_{j}x_{j}>\rho-1/2\text{ if
}\alpha_{j}\neq 0,\,x_{j}\in(-\rho,\rho)\text{ otherwise}\bigr{\\}}.$
There is a continuous partition of unity $(\phi_{0},\dots,\phi_{2n})$ that
satisfies the following three properties:
1. (i)
Each $\phi_{l}$ is almost everywhere differentiable, has bounded derivative
that is zero at infinity, which means
$\lim_{s\to\infty}\mathrm{sup}\\{|\mathrm{d}\phi_{l}|(x)\,|\,x\in
M\times\mathord{\mathbb{R}}^{n}\setminus(-s,s)^{n}\\}=0$.
2. (ii)
If $u$ is supported within $V_{\rho}({\bm{\alpha}})$ with ${\bm{\alpha}}\neq
0$, then $[P,\phi_{l}]u=\mathbf{c}(\mathrm{grad}(\phi_{l}))(u)$.
3. (iii)
$\phi_{0}$ is compactly supported and for each $l>0$ there is a unique tupel
$(i,\varepsilon)\in\\{1,\dots,n\\}\times\mathord{\mathbb{Z}}_{2}$ such that
$\phi_{l}$ is supported within $\\{\varepsilon x_{i}>\rho\\}$.
###### Proof.
Fix $0<a<(2\sqrt{n})^{-1}$. Let $\chi\colon\mathord{\mathbb{R}}_{\geq
0}\rightarrow[0,1]$ be a smooth function whose zero set is
$\mathord{\mathbb{R}}_{\leq a}$ and that is identically $1$ near
$\mathord{\mathbb{R}}_{\geq 2a}$. We denote its mirror symmetric extension to
$\mathord{\mathbb{R}}$ again with $\chi$. For $i\in\\{1,\dots,n\\}$, we define
$\widetilde{\chi}_{i}\colon S^{n-1}\rightarrow[0,1]$ via
$\widetilde{\chi}_{i}(x)=\chi(\mathord{\mathrm{pr}}_{i}(x))$ and set
$\chi_{i}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\frac{\widetilde{\chi_{i}}}{\sum_{j=1}^{n}\widetilde{\chi}_{j}}.$
The denominator is always non-zero because it vanishes at $x$ if and only if
all summands vanish at $x$, which implies $||x||^{2}<n\cdot a<1$. The
$n$-tuple of functions $(\chi_{1},\ldots,\chi_{n})$ has the following three
evident properties:
1. 1.
$\chi_{i}(x)=0$ if $|x_{i}|<a$,
2. 2.
$\sigma^{\ast}\chi_{i}=\chi_{\sigma^{-1}(i)}$ for each permutation
$\sigma\in\Sigma_{n}$,
3. 3.
$\sum_{j}\chi_{j}=1$.
For each
${\bm{\alpha}}=(\alpha_{1},\ldots,\alpha_{n})\in\mathord{\mathbb{Z}}_{3}^{n}$,
we define the set
$\displaystyle A_{\bm{\alpha}}$
$\displaystyle\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\\{(m,x)\in
M\times\mathord{\mathbb{R}}^{n}\,:\,x_{j}\in[-\rho,\rho]\text{ if
}\alpha_{j}=0,\ \alpha_{j}x_{j}\geq\rho\\}$ $\displaystyle\cong
M\times[-\rho,\rho]^{|\mathrm{Null}({\bm{\alpha}})|}\times\prod_{j\in\mathrm{supp}\,{\bm{\alpha}}}\alpha_{j}\mathord{\mathbb{R}}_{\geq\rho}.$
The complement of $M\times(-\rho,\rho)^{n}$ is the union of all
$A_{\bm{\alpha}}$ with ${\bm{\alpha}}\neq 0$. For
$(i,\varepsilon)\in\\{1,\ldots,n\\}\times\mathord{\mathbb{Z}}_{2}$, we define
$\hat{\phi}^{\varepsilon}_{i}|_{A_{\bm{\alpha}}}(m,x)=\begin{cases}\chi_{i}\left(\frac{\hat{x}}{||\hat{x}||}\right),&\text{if
}\alpha_{i}\neq 0,\\\ 0,&\text{if }\alpha_{i}=0,\end{cases}$
where
$\hat{x}=(|\alpha_{1}|x_{1}-\alpha_{1}\rho,\dots,|\alpha_{n}|x_{n}-\alpha_{n}\rho)$.
The assignments agree on the overlaps $A_{\bm{\alpha}}\cap A_{\bm{\beta}}$
yielding well defined functions on the complement of
$M\times(-\rho,\rho)^{n}$. Each $\hat{\phi}^{\varepsilon}_{i}$ is continuous
because it is continuous on all $A_{\bm{\alpha}}$. For $0<\delta<1$, we pick a
smooth function $\xi\colon\mathord{\mathbb{R}}_{\geq 0}\rightarrow[0,1]$ whose
zero set is $[0,\rho+\delta]$ and which is identically $1$ near
$\mathord{\mathbb{R}}_{\geq\rho+1}$. Its mirror-symmetric extension to
$\mathord{\mathbb{R}}$ is again denoted by $\xi$. Set
$\xi_{i}(x):=\xi(\mathord{\mathrm{pr}}_{i}(x))$ and extend
$\hat{\phi_{i}^{\varepsilon}}$ to $M\times\mathord{\mathbb{R}}^{n}$ via
$\phi_{i}^{\varepsilon}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\xi_{i}\hat{\phi}_{i}^{\varepsilon}$.
By construction, all $\phi_{i}^{\varepsilon}$ sum up to $1$ on the complement
of $M\times(-\rho+1,\rho+1)^{n}$ and the sum is bounded from above by 1
everywhere. Each $\phi^{\varepsilon}_{i}$ is supported within $\\{\varepsilon
x_{i}\geq\rho+\delta\\}$. Thus, by setting
$\phi_{0}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=1-\sum_{(i,\varepsilon)}\phi_{i}^{\varepsilon}$, we have constructed a
continuous partition of unity.
The function $\phi_{0}$ is supported within $M\times(-\rho-1,\rho+1)^{n}$,
which proves property (iii).
The points at which the partition of unity might not be differentiable lie in
$\bigcup_{{\bm{\alpha}}\neq{\bm{\beta}}}A_{\bm{\alpha}}\cap A_{{\bm{\beta}}}$,
which is a zero-set. Outside this set, the differential is given by
$\mathrm{d}\phi_{i}^{\varepsilon}=\hat{\phi}_{i}^{\varepsilon}(x)\xi^{\prime}(x_{i})\mathrm{d}x_{i}+\xi(x_{i})\mathrm{d}\chi_{i}\cdot\left(\frac{\hat{x}\cdot\hat{x}^{T}-||\hat{x}||^{2}\mathrm{id}}{||\hat{x}||^{3}}\right)\mathrm{diag}(|\alpha_{1}|,\dots,|\alpha_{n}|).$
Since $\mathrm{d}\chi_{i}$ is bounded, we conclude
$|\mathrm{d}\phi_{i}^{\varepsilon}|\in
L^{\infty}(M\times\mathord{\mathbb{R}}^{n})\text{ and
}\lim_{s\to\infty}\mathrm{sup}\\{|\mathrm{d}\phi_{i}^{\varepsilon}|(x)\,|\,x\in
M\times\mathord{\mathbb{R}}^{n}\setminus(-s,s)^{n}\\}=0,$
showing property (i).
By construction, the restriction of each $\phi^{\varepsilon}_{i}$ to
$V_{\rho}({\bm{\alpha}})$ with ${\bm{\alpha}}\neq 0$ depends only on the
coordinates $x_{j}$ indexed by $j\in\mathrm{supp}\,{\bm{\alpha}}$. If
$u\in\Gamma_{c}(\mathfrak{S}_{g})$ is supported within
$V_{\rho}({\bm{\alpha}})$ with ${\bm{\alpha}}\neq\mathbf{0}$, then the block
form of $P$ implies
$\displaystyle[P,\phi_{i}^{\varepsilon}]u$
$\displaystyle=\left(P({\bm{\alpha}})+\sum_{j\in\mathrm{supp}\,{\bm{\alpha}}}e_{j}\cdot\partial_{j}\right)\phi_{i}^{\varepsilon}u-\phi_{i}^{\varepsilon}\cdot\left(P({\bm{\alpha}})+\sum_{j\in\mathrm{supp}\,{\bm{\alpha}}}e_{j}\cdot\partial_{j}\right)u$
$\displaystyle=\phi^{\varepsilon}_{i}(P({\bm{\alpha}})-P({\bm{\alpha}}))u+\sum_{j\in\mathrm{supp}\,{\bm{\alpha}}}[e_{j}\cdot\partial_{x_{j}},\phi_{i}^{\varepsilon}]u$
$\displaystyle=\mathrm{grad}(\phi_{i}^{\varepsilon})\cdot
u=\mathbf{c}(\mathrm{grad}(\phi_{i}^{\varepsilon}))(u).$
This shows property (ii). ∎
We use this partition of unity to derive the fundamental elliptic estimates
for block operators of Dirac type. The idea of the following two propositions
is borrowed from [ebert2017indexdiff]*Prop 3.3.
###### Proposition 4.3.4.
Let $P\in\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{g})$ be an elliptic
block operator of Dirac type. Then $P$ satisfies the fundamental elliptic
inequality, which means that there is a $C>0$ such that
$||u||_{1}\leq C\left(||u||_{0}+||Pu||_{0}\right)$
for all $u\in H^{1}(\mathfrak{S}_{g})$ .
###### Proof.
The proof will be carried out by induction over $n$, the dimension of the
Euclidean part of our block manifolds $M\times\mathord{\mathbb{R}}^{n}$. For
$n=0$ our base manifold is compact and the theorem follows from Lemma 4.3.2.
Assume the statement for $n-1$ and pick an elliptic block operator
$P\in\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{g})$ over a block metric
$g\in\widetilde{\mathcal{R}_{n}}(M)$. Pick $R>r>0$ such that $g$ decomposes
outside $M\times(-r,r)^{n}$ and $M\times(-R,R)^{n}$ is a core for $P$. Since
$P$ is of block form the classical elliptic estimate gives us
$||u||_{1}\leq C_{0,R}\left(||u||_{0}+||Pu||_{0}\right)$
for some $C_{0,R}$ and all sections $u$ with support in $M\times(-R,R)^{n}$.
If $u$ is supported within $U_{r}(i,\varepsilon)=\\{\varepsilon x_{i}>r\\}$,
then we can identify $\mathfrak{S}_{g}|_{U_{r}(i,\varepsilon)}$ with
$\mathfrak{S}_{g(i,\varepsilon)}$ and $P$ with
$P(i,\varepsilon)\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}$.
Since $g$ is a product metric here, we can apply Fubini’s theorem to get:
$\displaystyle||u||_{1}^{2}$
$\displaystyle=\int_{\varepsilon\mathord{\mathbb{R}}_{>r}}||u(t)||_{1,g(i,\varepsilon)}^{2}\mathrm{d}t+\int_{\varepsilon\mathord{\mathbb{R}}_{>r}}||\partial_{t}u(t)||_{0,g(i,\varepsilon)}^{2}\mathrm{d}t$
$\displaystyle\leq\int
C^{2}\left(||u(t)||^{2}_{0,g(i,\varepsilon)}+||P(i,\varepsilon)u||^{2}_{0,g(i,\varepsilon)}\right)+||\partial_{t}u(t)||_{0,g(i,\varepsilon)}^{2}\mathrm{d}t$
$\displaystyle\leq
C^{2}\int||u(t)||^{2}_{0,g(i,\varepsilon)}+||P(i,\varepsilon)u||^{2}_{0,g(i,\varepsilon)}+||\partial_{t}u(t)||_{0,g(i,\varepsilon)}^{2}\mathrm{d}t$
$\displaystyle=C^{2}\int||u(t)||^{2}_{0,g(i,\varepsilon)}+\langle
P(i,\varepsilon)^{2}u,u\rangle_{0,g(i,\varepsilon)}-\langle\partial^{2}_{t}u(t),u(t)\rangle_{0,g(i,\varepsilon)}\mathrm{d}t$
$\displaystyle=C^{2}\int||u(t)||^{2}_{0,g(i,\varepsilon)}+\langle
P(i,\varepsilon)^{2}u-\partial^{2}_{t}u(t),u\rangle_{0,g(i,\varepsilon)}\mathrm{d}t$
$\displaystyle=C^{2}\int||u(t)||_{0,g(i,\varepsilon)}^{2}+\langle(P(i,\varepsilon)\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}})^{2}u(t),u(t)\rangle\mathrm{d}t$
$\displaystyle=C^{2}\int||u(t)||^{2}_{0,g(i,\varepsilon)}+||(Pu)(t)||_{0,g(i,\varepsilon)}^{2}\mathrm{d}t$
$\displaystyle=C^{2}(||u||_{0}^{2}+||Pu||_{0}^{2}),$
from which we immediately get the fundamental elliptic inequality for this
case.
Finally, we need to patch these inequalities together. We define the set
$L\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\\{0\\}\cup\\{1,\dots,n\\}\times\mathord{\mathbb{Z}}_{2}$ and let
$(\phi_{l})_{l\in L}$ be the partition of unity from Lemma 4.3.3 adapted to
$P$. We use this partition of unity to patch the inequalities together:
$\displaystyle||u||_{1}$
$\displaystyle=||\sum_{L}\phi_{l}u||_{1}\leq\sum_{L}||\phi_{l}u||_{1}$
$\displaystyle\leq
C_{0}(||\phi_{0}u||_{0}+||P\phi_{0}u||_{0})+\sum_{L\setminus\\{0\\}}C_{l}(||\phi_{l}u||_{0}+||P\phi_{l}u||_{0})$
$\displaystyle\leq(2n+1)\max\\{C_{l}\\}||u||_{0}+\sum_{L}C_{l}(||\phi_{l}Pu||_{0}+||[P,\phi_{l}\cdot]u||_{0})$
$\displaystyle\leq(2n+1)\max\\{C_{l},||[P,\phi_{l}\cdot]||_{0,0}\\}\left(||u||_{0}+||Pu||_{0}\right).$
This concludes the induction step. ∎
###### Definition 4.3.5.
A block operator $P\in\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{g})$ of
Dirac type is _invertible_ if its induced operator
$H^{1}(\mathfrak{S}_{g})\rightarrow H^{0}(\mathfrak{S}_{g})$ is an
isomorphism.
###### Remark 4.3.6.
By Proposition 4.3.4 the defining condition is equivalent to require that $P$
is an invertible unbounded operator $H^{0}(\mathfrak{S}_{g_{P}})$. Recall that
this means that the extension
$P\colon\overline{\Gamma_{c}(\mathfrak{S}_{g_{P}})}^{||\text{--}||_{P}}\rightarrow
H^{0}(\mathfrak{S}_{g})$ is invertible, where the domain is the completion of
$\Gamma_{c}(\mathfrak{S}_{g_{P}})$ with respect to the graph norm
$||\text{--}||_{P}=||\text{--}||+||P(\text{--})||$.
If the block operator $P$ has invertible faces, then the previous proposition
can be improved.
###### Proposition 4.3.7.
Let $P$ be a block operator of Dirac type whose faces $P(i,\varepsilon)$ are
invertible. Then $P$ satisfies the elliptic estimate
$||u||_{1}\leq
C\left(||Pu||_{0}+||\phi_{0}u||_{0}+\sum_{l=1}^{2n}||[P,\phi_{l}\cdot]u||_{0}\right),$
where $\\{\phi_{l}\\}$ is the partition of unity from Lemma 4.3.3.
Furthermore, $P$ has closed image and finite dimensional kernel.
If $P$ is self-adjoint, then $P$ is an (unbounded) Fredholm operator with
domain $H^{1}(M\times\mathord{\mathbb{R}}^{n};\mathfrak{S}_{g})$.
The proof of this proposition requires the main Fredholm Lemma.
###### Lemma 4.3.8 ([salamon2018functional]*Lemma 4.3.9).
Let $X,Y$ be Banach spaces and $Z$ be a normed vector space. Let $A\colon
X\rightarrow Y$ be a bounded linear map and let $K\colon X\rightarrow Z$ be a
compact operator. If there is a constant $C$ such that for all $x\in X$ the
inequality
$||x||_{X}\leq C(||Ax||_{Y}+||Kx||_{Z})$
holds, then $A$ has closed image and finite dimensional kernel.
###### Proof of Proposition 4.3.7.
Pick $R>r$ such that $M\times(-R,R)^{n}$ is a core of $P$ and that the block
operator $P$ decomposes on $U_{r}(i,\varepsilon)$ under the identification
$U_{r}(i,\varepsilon)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\\{(m,x)\in M\times\mathord{\mathbb{R}}^{n}\,:\,\varepsilon
x_{i}>r\\}\cong
M\times\mathord{\mathbb{R}}^{n}(i,\varepsilon)\times\\{\varepsilon x_{i}>r\\}$
into the operator
$P(i,\varepsilon)\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}$.
For each $u\in\Gamma_{c}(M\times\mathord{\mathbb{R}}^{n},\mathfrak{S}_{g})$
with support in $U_{r}(i,\varepsilon)$, we have
$\displaystyle||Pu||_{0}^{2}$
$\displaystyle=\int_{M\times\mathord{\mathbb{R}}^{n}}|Pu|^{2}=\int_{M\times\mathord{\mathbb{R}}^{n}}|P(i,\varepsilon)u+\partial_{i}\cdot\partial_{i}u|^{2}$
$\displaystyle=\int_{M\times\mathord{\mathbb{R}}^{n}}\langle
u,(P(i,\varepsilon)+\partial_{i}\cdot\partial_{i})^{2}u\rangle=\int_{M\times\mathord{\mathbb{R}}^{n}}\langle
u,P(i,\varepsilon)^{2}u-\partial_{i}^{2}u\rangle$
$\displaystyle=\int_{M\times\mathord{\mathbb{R}}^{n}}|P(i,\varepsilon)u|^{2}+|\partial_{i}u|^{2}$
$\displaystyle=\int_{\varepsilon\cdot\mathord{\mathbb{R}}_{\geq
r}}||(P(i,\varepsilon)u)(x)||^{2}_{L^{2}(M\times\mathord{\mathbb{R}}^{n}(i,\varepsilon),\mathfrak{S}_{g})}+\int_{M\times\mathord{\mathbb{R}}^{n}}|\partial_{i}u|^{2}$
$\displaystyle\geq\int_{\varepsilon\cdot\mathord{\mathbb{R}}_{\geq
r}}c^{-2}_{(i,\varepsilon)}||u(x)||^{2}_{H^{1}(M\times\mathord{\mathbb{R}}^{n}(i,\varepsilon),\mathfrak{S}_{g})}+\int_{M\times\mathord{\mathbb{R}}^{n}}|\nabla_{\partial_{i}}u|^{2}$
$\displaystyle\geq\mathrm{min}\\{c^{-2}_{(i,\varepsilon)},1\\}||u||^{2}_{H^{1}(M\times\mathord{\mathbb{R}}^{n},\mathfrak{S}_{g})}.$
In this chain of equations, we used in the second line that $P$ is symmetric
and that $P(i,\varepsilon)$ anti-commutes with $e_{i}$. In the fourth line, we
applied Fubini’s theorem, while in the fifth line, we used the invertibility
of $P(i,\varepsilon)$ and that $\nabla_{\partial_{i}}=\partial_{i}$. In the
last line we estimated against the minimum and identified the expression as
the Sobolev-1-norm.
On the other hand, if $u$ is supported within $M\times(-R,R)^{n}$, then $Pu$
is again supported within this relatively compact subset and the classical
elliptic estimate implies
$||u||_{1}\leq C(||u||_{0}+||Pu||_{0}).$
We now use the special adapted partition of unity from Lemma 4.3.3 with
$\rho=R$ and the notation from there to patch the inequalities together. Using
that $\phi_{l}$ is contained in some $U_{r}(i,\varepsilon)$ for $l\neq 0$, the
previous estimates yield
$\displaystyle||u||_{1}$
$\displaystyle=||\sum_{L}\phi_{l}u||_{1}\leq\sum_{L}||\phi_{l}u||_{1}$
$\displaystyle\leq
C_{0}(||\phi_{0}u||_{0}+||P\phi_{0}u||_{0})+\sum_{L\setminus\\{0\\}}C_{l}||P\phi_{l}u||_{0}$
$\displaystyle\leq
C_{0}||\phi_{0}u||_{0}+\sum_{L}C_{l}(||\phi_{l}Pu||_{0}+||[P,\phi_{l}\cdot]u||_{0})$
$\displaystyle\leq
C_{0}||\phi_{0}u||_{0}+\max\\{C_{l}\cdot||\phi_{l}||_{0,0},1\\}\cdot\left(\sum_{L}||Pu||_{0}+||[P,\phi_{l}\cdot]u||_{0}\right)$
$\displaystyle\leq
C\left(||Pu||_{0}+||\phi_{0}u||_{0}+\sum_{L}||[P,\phi_{l}\cdot]u||_{0}\right),$
where $C_{l}$ is the inverse of $\min\\{c_{(i,\varepsilon)^{-1}}\\}$ for the
pair $(i,\varepsilon)$ corresponding to $l$.
It turns out that $\phi_{0}$ and all $[P,\phi_{l}\cdot]$ are compact
operators. Indeed, $\phi_{0}$ is compactly supported, so $\phi_{0}\cdot$ is a
compact operator by the classical Rellich-Lemma.
To see that
$T_{l}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=[P,\phi_{l}\cdot]$ is compact, note first that it is an element of
$\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{g})$. Symbol calculus shows that
its principal symbol vanishes. Pick a partition of unity as in Definition
4.2.5 such that $\chi^{\mathbf{0}}$ is identically $1$ on $M\times[-R,R]^{n}$.
We decompose it into
$T_{l}=\sum_{{\bm{\alpha}}\in\mathord{\mathbb{Z}}_{3}^{n}}T_{l}\chi^{\bm{\alpha}}$.
By Lemma 4.2.6 the summand $T_{l}\chi^{\mathbf{0}}$ is supported within
$\mathrm{supp}\,\chi^{\mathbf{0}}$ in the sense that
$\mathrm{supp}\,T_{l}\chi^{\mathbf{0}}u\subseteq\mathrm{supp}\,\chi^{\mathbf{0}}$
and $T_{l}\chi^{\bm{\alpha}}u=0$ if
$\mathrm{supp}\,\chi^{\bm{\alpha}}\cap\mathrm{supp}\,u=\emptyset$. Since the
principal symbols of $T_{l}\chi^{\mathbf{0}}$ is zero, the Atiyah-Singer exact
sequence, see Lemma B.0.24, implies that $T_{l}\chi^{\mathbf{0}}$ is compact.
If ${\bm{\alpha}}\neq 0$, then
$\hat{{\bm{\alpha}}}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt={\bm{\alpha}}|_{\mathrm{supp}\,{\bm{\alpha}}}$ is not the empty map and
$\chi^{\bm{\alpha}}u$ is supported within $U_{R}(\hat{{\bm{\alpha}}})$. Thus,
property (ii) of the partition of unity $\\{\phi_{l}\\}$ yields
$T_{l}\chi^{\bm{\alpha}}u=\mathrm{grad}\phi_{l}\cdot\chi^{\bm{\alpha}}u.$
By Lemma 4.3.3 (i) and the generalised Rellich lemma for complete manifolds,
see [bunke2009index]*p.50, we conclude that $T_{l}\chi_{\bm{\alpha}}$ is also
compact.
A finite sum of compact operators is still compact, so $T_{l}$ is a compact
operator. Lemma 4.3.8 now yields that $P$ has finite kernel and closed image.
If $P$ is a self-adjoint (unbounded) operator on
$L^{2}(M\times\mathord{\mathbb{R}}^{n},\mathfrak{S}_{g})$, then the maximal
domain agrees with the minimal domain. The elliptic estimate implies that the
graph norm of $P$ is equivalent to $||\text{--}||_{1}$, so the (minimal)
domain of $P$ agrees with
$H^{1}(M\times\mathord{\mathbb{R}}^{n},\mathfrak{S}_{g})$. The kernel and the
cokernel of a self-adjoint operator are isomorphic, so the cokernel is also
finite dimensional and $P$ is a Fredholm operator. ∎
Usually, it is not easy to show that a (pseudo) differential operator is self-
adjoint. On complete manifolds, there is, however, a general criterion, the
boundness of the propagation speed.
###### Proposition 4.3.9 ([higson2000analytic]*Prop. 10.2.11).
Let $N$ be a complete Riemannian manifold and let $D$ be a symmetric
differential operator on $M$. If the _propagation speed_
$c_{D}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\sup\\{||\mathrm{symb}_{1}(D)(x,\xi)||\,:\,x\in M,\xi\in
T^{\vee}_{x}M,||\xi||=1\\}$
is finite, then $D$ is (essentially) self-adjoint, i.e. its closure is a self-
adjoint operator.
###### Corollary 4.3.10.
Let $g\in\widetilde{\mathcal{R}_{n}}(M)$ be a block metric on
$M\times\mathord{\mathbb{R}}^{n}$. Then ${\not{\mathfrak{D}}}_{g}$ is a self-
adjoint operator.
More generally, if $P\in\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{g})$ is a
symmetric, block operator of Dirac type, such that
$P-{\not{\mathfrak{D}}}_{g}$ extends to a bounded operator on
$H^{0}(\mathfrak{S}_{g})=L^{2}(\mathfrak{S}_{g})$, then $P$ is a self-adjoint
operator.
###### Proof.
A block metric is complete by Proposition 3.1.5 and the propagation speed
obviously satisfies $c_{{\not{\mathfrak{D}}}_{g}}=1$. By the preceding
proposition, ${\not{\mathfrak{D}}}_{g}$ is therefore self-adjoint.
The second statement follows from Exercise 1.9.21 of [higson2000analytic]. ∎
A common strategy to verify that a self-adjoint operator is invertible is to
verify that it squares to a uniformly positive operator. We recall this
concept.
###### Definition 4.3.11.
An unbounded operator $T$ on a Hilbert space $H$ is called _non-negative_ , if
$\langle Tx,x\rangle\geq 0$ for all $x\in{\mathrm{dom}\,}T$. It is called
_uniformly positive_ , if there is a constant $c>0$ such that $\langle
Tx,x\rangle\geq c||x||^{2}$.
An unbounded operator is _bounded from below_ if there is a positive constant
$c>0$ such that $||Tx||\geq c||x||$ for all $x\in{\mathrm{dom}\,}T$.
Clearly, a closed unbounded operator $T$ is bounded from below if and only if
$T^{\ast}T$ is uniformly positive. The next theorem is an immediate
consequence from unbounded functional calculus, see
[werner2006funktionalanalysis].
###### Lemma 4.3.12.
Let $T$ be a self-adjoint unbounded operator on a separable Hilbert space.
1. (i)
If $T$ is uniformly positive, then $T$ is invertible.
2. (ii)
If $T$ is closed and invertible, then $T^{\ast}T$ is uniformly positive.
Being uniformly positive is an open condition for block operators of Dirac
type in the space $\mathrm{Hom}(H^{1},H^{0})$, where
$H^{j}=H^{j}(\mathfrak{S}_{g})$.
###### Lemma 4.3.13.
Let $P,Q\in\mathrm{Hom}(H^{1},H^{0})$ be bounded operators. Let $P$ be bounded
from below and satisfy a fundamental elliptic estimate. If $Q$ is sufficiently
close to $P$ in the operator norm of $\mathrm{Hom}(H^{1},H^{0})$, then $Q$ is
also bounded from below.
###### Proof.
Given $c\in(0,1)$ such that $||Px||_{0}>c||x||_{0}$ and a constant $C=C(P)$
making the fundamental elliptic estimate valid for $P$. We claim that all
operators $Q$ satisfying the inequality
$||P-Q||_{1,0}<\frac{c}{(c+1)C}$
are bounded from below.
Indeed, the fundamental elliptic estimate applied to $P$ and the assumed
inequality yield
$\displaystyle||Qu||_{0}$
$\displaystyle\geq\bigl{|}||Pu||_{0}-||(Q-P)u||_{0}\bigr{|}$
$\displaystyle\geq||Pu||_{0}-||P-Q||_{1,0}C(||u||_{0}+||Pu||_{0})$
$\displaystyle=(1-||P-Q||_{1,0}\cdot C)||Pu||_{0}-||P-Q||_{1,0}C||u||_{0}.$
Using uniform positivity and the assumed inequality a second time, we continue
the estimation with
$\displaystyle||Qu||_{0}$ $\displaystyle\geq
c(1-C)||P-Q||_{1,0}||u||_{0}-C||P-Q||_{1,0}||u||_{0}$
$\displaystyle=\underbrace{\bigl{(}c-(c+1)C||P-Q||_{1,0}\bigr{)}}_{>0}||u||_{0}$
This proves the claim. ∎
##### Gluing Techniques
In the forthcoming sections we will carry out several cut and paste arguments
for block operators. Although we will start and end with a block operator on a
complete manifold, during the procedures we will consider operators on non-
complete manifolds. Being self-adjoint or invertible are global properties of
an operator that are usually lost when we restrict the operator to an open
subset, so we cannot expect them to hold through the entire cut and paste
procedure. On the other hand, being symmetric and bounded from below are
properties that are preserved under such restrictions and the purpose of this
section is to show that they are also preserved under certain gluing
constructions.
The first gluing technique for block operator roughly says that being
symmetric is a sheaf-like property, provided the operators are suitably local.
###### Lemma 4.3.14.
Let $E\rightarrow M$ be a vector bundle over a not necessarily compact
manifold $M$. Let $\\{U_{\alpha}\\}_{\alpha\in A}$ be a locally finite open
cover and let $\\{P_{\alpha}\\}_{\alpha\in A}$ be a family of operators
$P_{\alpha}\colon\Gamma_{c}(U_{\alpha},E|_{U_{\alpha}})\rightarrow\Gamma_{c}(U_{\alpha},E|_{U_{\alpha}})\quad\text{
and }\quad P_{\alpha}\in\overline{\Psi\mathrm{DO}}^{1}(E|_{U_{\alpha}})$
such that $P_{\alpha}|_{U_{\alpha}\cap U_{\beta}}=P_{\beta}|_{U_{\alpha}\cap
U_{\beta}}\colon\Gamma_{c}(U_{\alpha\beta},E_{U_{\alpha\beta}})\rightarrow\Gamma_{c}(U_{\alpha\beta},E_{U_{\alpha\beta}})$
for all $\alpha,\beta\in A$. Assume further that there is a partition of unity
$\\{\chi_{\alpha}\\}_{\alpha\in A}$ that is subordinated to the cover
$\\{U_{\alpha}\\}_{\alpha\in A}$ such that the operators acts as derivation on
the partition of unity, that is
$[P_{\alpha},\chi_{\beta}\cdot]u=i^{-1}\mathrm{symb}_{1}(P)(\text{--},\mathrm{d}\chi_{\beta})\cdot
u.$
Then each $P_{\alpha}$ is symmetric if and only if the operator
$P\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\sum_{\alpha\in A}P_{\alpha}\circ\chi_{\alpha}$
is symmetric.
###### Proof.
Assume that $P$ is symmetric, then also $P_{a}$ is symmetric for all $a\in A$
because for all $u\in\Gamma_{c}(U_{a},E|_{U_{a}})$ we have
$Pu=\sum_{\alpha\in A}P_{\alpha}\chi_{\alpha}u=\sum_{\\{\alpha:U_{\alpha}\cap
U_{a}\neq\emptyset\\}}P_{a}\chi_{\alpha}u=P_{a}u.$
For the converse, let $u,v\in\Gamma_{c}(M,E)$ be two sections. Set
$u_{\alpha}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\chi_{\alpha}u$ and
$v_{\beta}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\chi_{\beta}v$. By bilinearity, it suffices to show that
$(P_{\alpha}u_{\alpha},v_{\beta})=(u_{\alpha},P_{\beta}v_{\beta})$ for all
$\alpha,\beta\in A$.
To verify this, we rely on the following two facts that are immediate
consequences from the derivation property:
* (I)
$\mathrm{supp}\,\,P_{\beta}u_{\alpha}\subseteq\mathrm{supp}\,\,\chi_{\alpha}$
for all $\alpha,\beta\in A$.
* (II)
If $u|_{\\{\chi_{\alpha}\neq 0\\}}=0$, then
$(P_{\beta}u)|_{\\{\chi_{\alpha}\neq 0\\}}=0$.
Indeed, property (I) follows from
$\displaystyle P_{\beta}u_{\alpha}$
$\displaystyle=P_{\beta}\chi_{\alpha}u=\chi_{\alpha}P_{\beta}u+[P_{\beta},\chi_{\alpha}]u=\chi_{\alpha}u+i^{-1}\mathrm{symb}_{1}(P_{\beta})(\text{--},\mathrm{d}\chi_{\alpha})u$
and that
$\mathrm{symb}_{1}(P_{\beta})(\text{--},\mathrm{d}\chi_{\alpha})=\mathrm{symb}_{1}(P_{\beta})(\text{--},0)=0$
on $\\{\chi_{\alpha}=0\\}$. Property (II) follows from
$\displaystyle(\chi_{\alpha}P_{\beta}u)|_{\chi_{\alpha}\neq
0}=P_{\beta}\underbrace{\chi_{\alpha}u}_{=0}-i^{-1}\mathrm{symb}_{1}(P_{\beta})(\text{--},\mathrm{d}\chi_{\alpha})\underbrace{u|_{\chi_{\alpha}\neq
0}}_{=0}.$
Now, for each $\alpha\in A$, we pick a smooth function $\psi_{\alpha}$ that is
identically $1$ near $\mathrm{supp}\,\,\chi_{\alpha}$ and supported within
$U_{\alpha}$. Symmetry of $P$ now follows from the calculation
$\displaystyle(P_{\alpha}u_{\alpha},v_{\beta})$
$\displaystyle\overset{(\mathrm{I})}{=}(\psi_{\alpha}P_{\alpha}u_{\alpha},v_{\beta})=(P_{\alpha}u_{\alpha},\psi_{\alpha}v_{\beta})$
$\displaystyle=(u_{\alpha},P_{\alpha}\psi_{\alpha}v_{\beta})$
$\displaystyle=(u_{\alpha},P_{\beta}\psi_{\alpha}v_{\beta})\qquad\text{because
}P_{\alpha}|_{U_{\alpha\beta}}=P_{\beta}|_{U_{\alpha\beta}}$
$\displaystyle=(u_{\alpha},(P_{\beta}\psi_{\alpha}v_{\beta})|_{\mathrm{supp}\,\,\chi_{\alpha}})$
$\displaystyle=(u_{\alpha},P_{\beta}v_{\beta})$
in which we use (II) and linearity of $P_{\beta}$ to deduce that
$(P_{\beta}\psi_{\alpha}v_{\beta})|_{\mathrm{supp}\,\,\chi_{\alpha}}$ only
depends on (the germ of)
$\psi_{\alpha}v_{\beta}|_{\mathrm{supp}\,\,\chi_{\alpha}}=v_{\beta}|_{\mathrm{supp}\,\,\chi_{\alpha}}$.
∎
The second gluing technique yields that two invertible block operators with
matching faces can be glued together to an invertible operator, provided the
“gluing strip” is sufficiently large. We carry out the gluing construction in
a more general setup as we allow our underlying Riemannian manifolds to be
non-complete. This forces us, to work with the following weaker notion that is
equivalent to invertibility if the operator in question is self-adjoint.
###### Definition 4.3.15.
Let $P$ be a densely defined operator on a Hilbert space $H$. We call $P$
_(uniformly) bounded from below_ , if there is a $c>0$ such that $||Pv||\geq
c||v||$ for all $v\in\mathrm{dom}(P)$. More generally, an operator
$P\in\overline{\Psi\mathrm{DO}}^{1}(E)$ is bounded from below, if
$P\colon\Gamma_{c}(M,E)\rightarrow\Gamma(M,E)\subset L^{2}_{loc}(M,E)$ is
bounded from below (meaning we also allow $||P\sigma||_{0}=\infty$).
We work in the following setup.
###### Setup 4.3.16.
Let $H\subseteq M$ be a separating hypersurface that is closed as a subset but
not necessarily compact. Set $M\setminus H=M^{-}\sqcup M^{+}$ and let
$\Phi\colon H\times(-a,a)\hookrightarrow M$ be an open embedding that
restricts to the inclusion on $H\times\\{0\\}=H$ and satisfies
$\Phi(H\times(-a,0))\subseteq M^{-}$ and $\Phi(H\times(0,a))\subseteq M^{+}$.
We further denote $M_{\pm
a}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=M^{\pm}\setminus\Phi(H\times(-a,a))$ and, for all
$t,t_{1},t_{2}\in(-a,a)$, we set
$\displaystyle
M_{<t}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=M_{-a}\,\cup\,\Phi(H\times(-a,t)),\qquad
M_{>t}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\Phi(H\times(t,a))\,\cup\,M_{a},\quad\text{and}$ $\displaystyle
M_{(t_{1},t_{2})}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\Phi(H\times(t_{1},t_{2})).$
###### Definition 4.3.17.
Let $\mathcal{A}\subseteq\mathcal{C}^{\infty}(M,\mathord{\mathbb{R}})$ be a
sub-algebra of smooth functions on $M$. An operator
$P\in\overline{\Psi\mathrm{DO}}^{1}(E)$ is an $\mathcal{A}$_-derivation_ , if
$[P,f\cdot]=i^{-1}\mathrm{symb}_{1}(P)(\text{--},\mathrm{d}f)$
for all $f\in\mathcal{A}$.
We are interested in the algebra
$\mathcal{A}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=(\mathord{\mathrm{pr}}_{2}\circ\Phi^{-1})^{\ast}\mathcal{C}_{c}^{\infty}((-a,a),\mathord{\mathbb{R}}))$
considered as a sub-algebra of $\mathcal{C}^{\infty}(M,\mathord{\mathbb{R}})$
(via extension by zero).
###### Lemma 4.3.18.
Let $E\rightarrow M$ be a vector bundle over a manifold as in setup 4.3.16.
Assume that $P\in\overline{\Psi\mathrm{DO}}^{1}(E)$ restricts to $M_{>b}$ and
$M_{<-b}$ for some $0<b<a$ and that $P$ is an $\mathcal{A}$-derivation. If
$\sigma\in\Gamma_{c}(M,E)$ is supported within $M_{(t_{1},t_{2})}$, then
$P\sigma$ is also supported within $M_{(t_{1},t_{2})}$ for all
$-a<t_{1}<t_{2}<a$. In particular, $(P\sigma)|_{M_{(t_{1},t_{2})}}$ only
depends on the germ of $\sigma|_{\overline{M}_{(t_{1},t_{2})}}$.
Analogous statements hold for $M_{>t}$ and $M_{<t}$ for all $t\in(-b,b)$.
###### Proof.
We only prove the statement for $M_{(t_{1},t_{2})}$. The other proofs are
quite similar. Assume $\sigma$ is supported within this set. Since the support
is compact, there are $t_{1}<t_{1}^{\prime}<t_{2}^{\prime}<t_{2}$ such that
$\sigma$ is also supported within $M_{(t_{1}^{\prime}t_{2}^{\prime})}$. Let
$\chi\colon(t_{1},t_{2})\rightarrow[0,1]$ be identically $1$ on
$(t_{1}^{\prime},t_{2}^{\prime})$ and compactly supported. To ease the
notation, we denote $\chi\circ\mathord{\mathrm{pr}}_{2}\circ\Phi^{-1}$ (and
its extension by zero) again with $\chi$. We have
$\displaystyle P\sigma$ $\displaystyle=P\chi\sigma=\chi
P\sigma+[P,\chi\cdot]\sigma$ $\displaystyle=\chi
P\sigma+i^{-1}\mathrm{symb}_{1}(P)(\text{--},\chi^{\prime})\sigma$
$\displaystyle=\chi P\sigma,$
because $\chi\equiv 1$ on $\mathrm{supp}\,\sigma$. Thus, $P\sigma$ is
contained in $\mathrm{supp}\,\chi\subseteq M_{(t_{1},t_{2})}$.
To prove that $P\sigma|_{M_{(t_{1},t_{2})}}$ only depends on the germ of
$\sigma|_{\overline{M_{(t_{1},t_{2})}}}$, it suffices to show that if $\sigma$
is identically zero near $\overline{M_{(t_{1},t_{2})}}$, then $P\sigma$
vanishes identically on $M_{(t_{1},t_{2})}$. But if $\sigma$ vanishes near
$\overline{M_{(t_{1},t_{2})}}$, then, by compactness of
$\mathrm{supp}\,\sigma$, there are $t_{1}^{\prime}<t_{1}$ and
$t_{2}^{\prime}>t_{2}$ such that $\mathrm{supp}\,(\sigma)\cap
M_{(t_{1}^{\prime},t_{2}^{\prime})}=\emptyset$. Thus $\mathrm{supp}\,(\sigma)$
is supported within $M_{<t_{1}^{\prime}}$ and $M_{>t_{2}^{\prime}}$, so
$P\sigma$ must be supported there, too. ∎
###### Definition 4.3.19.
Let $R<b<a$ and let $P_{1}$, $P_{2}$ be operators on $M_{<a}$ and $M_{>-a}$,
respectively. Assume that $P_{1}$ and $P_{2}$ restrict to $M_{<-b}$ and
$M_{>b}$, respectively, and that they are $\mathcal{A}$-derivations. By the
previous Lemma, the two operators restrict to $M_{(-R,R)}$ and we assume that
the restrictions agree. This allows us to patch the two operators together:
Let $\chi\colon\mathord{\mathbb{R}}\rightarrow[0,1]$ is a smooth function that
is identically $1$ near $\mathord{\mathbb{R}}_{\leq-R}$ and identically $0$
near $\mathord{\mathbb{R}}_{\geq R}$. We extend he function via $0$ and $1$ to
$M$ and denote the result again with $\chi$. We define
$P_{1}\cup
P_{2}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=P_{1}(1-\chi)+P_{2}\chi.$
Since $P_{1}$ and $P_{2}$ agree on $M_{(-R,R)}$ and act as
$\mathcal{A}$-derivations, the result does not depend on the choice of $\chi$.
An application of Lemma 4.3.14 shows that $\cup$ preserves symmetry.
###### Corollary 4.3.20.
The operator $P_{1}\cup P_{2}$ is symmetric if and only if the operators
$P_{1}$ and $P_{2}$ are symmetric.
###### Proof.
The only if direction is easy because restrictions of symmetric operators are
symmetric and $P_{1}$ agrees with $P_{2}$ on $M_{(-R,R)}$.
For the other direction, we will prove that $M_{<R}$ and $M_{>-R}$ together
with $\chi$ and $1-\chi$ from Definition 4.3.19 satisfy the condition of Lemma
4.3.14 for $P_{1}$ and $P_{2}$. To this end, fix a function
$\eta\colon(-a,a)\rightarrow[0,1]$ with compact support and
$\eta|_{(-b,b)}\equiv 1$. Let $\eta^{-1}$, $\eta^{0}$, and $\eta^{1}$ be the
functions obtained from the construction in Definition 4.2.5 but considered as
function on $M$. They form a partition of unity.
For all compactly supported sections $\sigma\in\Gamma_{c}(M_{<R},E)$, the
section $\eta^{-1}\sigma$ is supported within $M_{<-b}$, so that
$P_{1}\eta^{-1}\sigma$ is also supported there by assumption. This implies
$\displaystyle P_{1}\chi\sigma$
$\displaystyle=P_{1}\chi\eta^{-1}\sigma+P_{1}\chi\eta^{0}\sigma=P_{1}\eta^{-1}+P_{1}\chi\eta^{0}\sigma$
$\displaystyle=\chi P_{1}\eta^{-1}\sigma+\chi
P_{1}\eta^{0}\sigma+[P_{1},\chi]\eta^{0}\sigma$ $\displaystyle=\chi
P_{1}(\eta^{-1}+\eta^{0})\sigma+i^{-1}\mathrm{symb}_{1}(P_{1})(\text{--},\chi^{\prime})\cdot\eta^{0}\sigma$
$\displaystyle=\chi
P_{1}\sigma+i^{-1}\mathrm{symb}_{1}(P_{1})(\text{--},\chi^{\prime})\sigma.$
Note that the last equation follows from $\chi^{\prime}\equiv 0$ on
$\mathrm{supp}\,\eta^{-1}$. Of course, this implies that $P_{1}$ acts on
$1-\chi$ as a derivation.
The corresponding statement for $P_{2}$ and $(1-\chi)$ can be derived in an
analogous manner. ∎
###### Lemma 4.3.21.
Let $P_{1}$ and $P_{2}$ be operators on $M_{<a}$ and $M_{>-a}$ as in setup
4.3.19. Assume that
$||\mathrm{symb}_{1}(P_{1})(\text{--},\mathrm{d}t)||_{\infty,M_{(-R,R)}}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\\{||\mathrm{symb}_{1}(P_{1})(x,\mathrm{d}t)||_{\mathrm{op}}\,:\,x\in
M_{(-R,R)}\\}\leq C.$
If there is a section $u\in\Gamma_{c}(M,E)$ such that $||(P_{1}\cup
P_{2})u||_{0}<\varepsilon||u||_{0}$, then there is a $v\in\Gamma_{c}(M,E)$
that is supported within $M_{<R}$ that satisfies
$||P_{1}v||_{0}^{2}<4\left(\varepsilon^{2}+\frac{C^{2}}{R^{2}}\right)||v||_{0}^{2}$
or is supported within $M_{>-R}$ and satisfies an analogous inequality with
$P_{1}$ replaced by $P_{2}$.
###### Proof.
Abbreviate $P_{1}\cup P_{2}$ to $Q$. Let $u\in\Gamma_{c}(M,E)$ with
$||Qu||_{0}^{2}<\varepsilon^{2}||u||_{0}^{2}$. Assume without loss of
generality, that $||u||_{0,M_{\leq 0}}^{2}\geq 1/2||u||_{0}^{2}$. Let
$\eta\colon\mathord{\mathbb{R}}\rightarrow[0,1]$ be a smooth function that
satisfies $\eta\equiv 1$ on $\mathord{\mathbb{R}}_{\leq 0}$, $\eta\equiv 0$ on
$\mathord{\mathbb{R}}_{\geq 1}$, and $|\eta^{\prime}|\leq 1.2$. Set
$\eta_{R}(t)=\eta(R^{-1}t)$ and extend this function as described above to
$M$. We denote the result again with $\eta_{R}$.
The norm splits into
$\displaystyle||P_{1}\eta_{R}u||_{0}^{2}=||P_{1}\eta_{R}u||_{0,M_{\leq-R}}^{2}+||P_{1}u||_{0,M_{(-R,R)}}^{2}$
and, by Lemma 4.3.18, the summands only depend on the germs of the
restrictions to these domains. For example,
$||P_{1}\eta_{R}u||_{0,M_{\leq-R}}^{2}=||P_{1}\eta_{R}u|_{M_{\leq-R}}||_{0,M_{\leq-R}}^{2}=||P_{1}u||_{0,M_{\leq-R}}^{2}.$
Since also $Q$ satisfies the assumptions of Lemma 4.3.18, we have
$(P_{1}\eta_{R}u)|_{M\leq-R}=P_{1}(\eta_{R}u|_{M\leq-R})=(P_{1}u)|_{M\leq-R}=(Qu)|_{M\leq-R}$
and because $P_{1}$ and $P_{2}$ agree on $M_{(-R,R)}$ we have
$\displaystyle(Q\eta_{R}u)|_{M_{(-R,R)}}$
$\displaystyle=Q\eta_{R}(u|_{M_{(-R,R)}})=P_{1}\eta_{R}(u|_{M_{(-R,R)}})$
$\displaystyle=\eta_{R}P_{1}(u|_{M_{(-R,R)}})+[P,\eta_{R}\cdot](u|_{M_{(-R,R)}})$
$\displaystyle=(\eta_{R}Qu)|_{M_{(-R,R)}}+i^{-1}\mathrm{symb}_{1}(P_{1})(\text{--},\eta_{R}^{\prime}\mathrm{d}t)(u)|_{M_{(-R,R)}}.$
This implies
$\displaystyle||P_{1}\eta_{R}u||_{0}^{2}$
$\displaystyle=||Qu||^{2}_{0,M_{\leq-R}}+||\eta_{R}Qu+\mathrm{symb}_{1}(P_{1})(\text{--},\eta_{R}^{\prime}\mathrm{d}t)(u)||_{0,M_{(-R,R)}}^{2}$
$\displaystyle\leq||Qu||^{2}_{0,M_{\leq-R}}+2||\eta_{R}Qu||^{2}_{0,M_{(-R,R)}}+2||\mathrm{symb}_{1}(P_{1})(\text{--},\eta_{R}^{\prime}\mathrm{d}t)(u)||_{0,M_{(-R,R)}}^{2}$
$\displaystyle\leq
2||Qu||^{2}_{0,M_{<R}}+2||\mathrm{symb}_{1}(P_{1})(\text{--},\eta_{R}^{\prime}\mathrm{d}t)(u)||_{0,M_{(-R,R)}}^{2}$
$\displaystyle<2\varepsilon^{2}||u||^{2}_{0}+2\frac{C^{2}}{R^{2}}||u||_{0}^{2},$
where we used in the last inequality that
$||Qu||_{0}^{2}<\varepsilon^{2}||u||^{2}_{0}$ and that
$\displaystyle\quad\
||\mathrm{symb}_{1}(P_{1})(\text{--},\eta_{R}^{\prime}\mathrm{d}t)(u)||_{0;M_{(-R,R)}}$
$\displaystyle\leq||\mathrm{symb}_{1}(P_{1})(\text{--},\eta_{R}^{\prime}\mathrm{d}t)||_{0,0;M_{(-R,R)}}||u||_{0,M_{(-R,R)}}$
$\displaystyle\leq||\mathrm{symb}_{1}(P_{1})(\text{--},\eta_{R}^{\prime}\mathrm{d}t)||_{\infty;M_{(-R,R)}}||u||_{0}$
$\displaystyle\leq\frac{|\eta^{\prime}|_{\infty}}{R}||\mathrm{symb}_{1}(P_{1})(\text{--},\mathrm{d}t)||_{\infty;M_{(-R,R)}}||u||_{0}$
$\displaystyle\leq\frac{C}{R}||u||_{0}$
From $1/2||u||_{0}^{2}\leq||u||_{0,M_{\leq 0}}^{2}$ it follows that
$||u||_{0}^{2}\leq 2||\eta_{R}u||_{0}^{2}$. Putting all inequalities together,
we get
$||P_{1}\eta_{R}u||_{0}^{2}<4\left(\varepsilon^{2}+C^{2}/R^{2}\right)||\eta_{R}u||_{0}^{2},$
so the claim follows for
$v\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\eta_{R}u$. ∎
The following implication is an indispensable tool in the following sections
because it allows us to glue invertible block operators of Dirac type
together. Note that, in this set up, we are allowed to choose the length of
the “gluing-strip” to be arbitrarily large without increasing the propagation
speed of the operator.
###### Corollary 4.3.22.
Let $P_{1}$ and $P_{2}$ be as in the previous lemma. If they are bounded from
below with lower bounds $c_{1}$ and $c_{2}$ respectively and if
$R>\max\\{C^{2}/c_{1}^{2},C^{2}/c_{2}^{2}\\}$, then $P_{1}\cup P_{2}$ is
bounded from below with lower bound
$c_{P_{1}\cup
P_{2}}^{2}=\frac{\min\left\\{{c_{1}^{2}-C^{2}/R^{2}},c_{2}^{2}-C^{2}/R^{2}\right\\}}{4}.$
###### Proof.
By assumption on $R$, the constant $c_{P_{1}\cup P_{2}}$ is positive. Assume
that there is a section $u\in\Gamma_{c}(M,E)$ such that $||(P_{1}\cup
P_{2})(u)||_{0}<c_{P_{1}\cup P_{2}}||u||_{0}$, then the previous Lemma implies
the existence of $v\in\Gamma_{c}(M,E)$ that is, without loss of generality,
supported within $M_{<R}$ and satisfies
$\displaystyle||P_{1}v||_{0}^{2}$ $\displaystyle<4\left(c_{P_{1}\cup
P_{2}}^{2}+C^{2}/R^{2}\right)||v||_{0}^{2}<c_{1}^{2}||v||_{0}^{2},$
which contradicts that $P_{1}$ is bounded from below by $c_{1}$. ∎
### 4.4 Foundations of the Operator Concordance Set
With the theory developed in the previous section we are finally in the
position to define the operator concordance set
$\widetilde{\Psi\mathrm{Dir}}^{\times}_{\bullet}(M)$ as the cubical subset of
all block Dirac operators.
But, first we give a combinatorial model for the classifying space for real
$K$-theory, by considering the cubical set of block maps into
$\Psi\mathrm{Dir}(M)$ and $\Psi\mathrm{Dir}^{\times}(M)$ instead of the spaces
themselves.
###### Definition 4.4.1.
Let $\Psi\mathrm{Dir}_{\bullet}(M)$ be the cubical set whose $n$-cubes
consists of smooth block maps
$P\colon\mathord{\mathbb{R}}^{n}\rightarrow\Psi\mathrm{Dir}(M)$. The
connecting maps are given by
$\partial^{\varepsilon}_{i}P=\lim_{R\to\infty}P\circ{\delta^{R\varepsilon}_{i}}\qquad\text{
and }\qquad\sigma_{i}P=P\circ p_{i}.$
Let $\Psi\mathrm{Dir}^{\times}_{\bullet}(M)$ be the cubical subset consisting
of all smooth block maps with values in $\Psi\mathrm{Dir}^{\times}(M)$.
Recall that $\Psi\mathrm{Dir}(M)$ is an affine bundle over $\mathrm{Riem}(M)$
with fibre $\Psi\mathrm{Dir}(M)_{g}$, the set of all self-adjoint, odd,
Clifford-linear pseudo differential operators of order $1$ whose principal
symbol agree with the principal symbol of the Dirac operator
${\not{\mathfrak{D}}}_{g}$. The map
$\Psi\mathrm{Dir}(M)\rightarrow\mathrm{Riem}(M)$ assigns to an operator $P$
its underlying metric $g$. This map induces a map of cubical sets
$\Psi\mathrm{Dir}_{\bullet}(M)\rightarrow\mathcal{R}^{+}_{\bullet}(M)$.
Now, we are going to construct the combinatorial reference space.
###### Definition 4.4.2.
Let $g$ be a block metric on $M\times\mathord{\mathbb{R}}^{n}$. A block
operator $P\in\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{g})$ of Dirac type
is called a _block Dirac operator_ if it is symmetric and
$P-{\not{\mathfrak{D}}}_{g}$ extends to a bounded endomorphism on
$H^{0}(\mathfrak{S}_{g})=L^{2}(\mathfrak{S}_{g})$.
An immediate consequence of Corollary 4.3.10 is that a block Dirac operator is
automatically essentially self-adjoint.
For the definition of the operator concordance set it will be important to
trace permutation of coordinates. A permutation
$\sigma\in\mathrm{Aut}(\\{1,\dots,n\\})=S_{n}$ induces a linear map
$\sigma\colon(x_{1},\dots,x_{n})\mapsto(x_{\sigma^{-1}(1)},\dots,x_{\sigma^{-1}(n)}).$
If $g$ is a block metric on $M\times\mathord{\mathbb{R}}^{n}$, then
$\sigma^{\ast}g$ is also a block metric on $M\times\mathord{\mathbb{R}}^{n}$.
Since $\mathord{\mathbb{R}}^{n}$ carries up to equivalence only one
$\mathrm{Pin}^{-}$-structure, permutations are Pin structure preserving.
Functoriality of the spinor bundle construction yields a map, which we denote
with
$\mathfrak{S}(\sigma)\colon\mathfrak{S}_{\sigma^{\ast}g}\rightarrow\mathfrak{S}_{g}\quad\text{
or equivalently
}\quad\mathfrak{S}(\sigma)\colon\mathfrak{S}_{g}\rightarrow\mathfrak{S}_{\sigma_{\ast}g}.$
We are particularly interested in cyclic permutations.
###### Definition 4.4.3.
For positive integers $a<b$ let $\mathrm{cycl}(a,b)$ be the cyclic permutation
given by $a\mapsto a+1$ and $b\mapsto a$ and let
$\mathrm{cycl}(b,a)=\mathrm{cycl}(a,b)^{-1}$. We denote the induced maps on
spinor bundles with $\mathrm{Cycl}(a,b)$ and $\mathrm{Cycl}(b,a)$,
respectively.
###### Definition 4.4.4.
Let $\widetilde{\Psi\mathrm{Dir}}_{\bullet}(M)$ be the cubical set whose set
of $n$-cubes is given by
$\widetilde{\Psi\mathrm{Dir}}_{n}(M)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\\{P\in\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{g})\,:\,g\in\widetilde{\mathcal{R}_{n}}(M),\,P\text{
block Dirac operator}\\}$
and whose connecting maps are given by
$\displaystyle\partial^{\varepsilon}_{i}P$
$\displaystyle\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\lim_{R\to\infty}{{\delta^{R\varepsilon}_{i}}}^{\ast}P(i,\varepsilon)\in\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{\partial^{\varepsilon}_{i}g}),$
$\displaystyle\sigma_{i}P$
$\displaystyle\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\mathrm{Cycl}(i,n+1)_{\ast}(P\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}})\in\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{\sigma_{i}g})$
###### Remark 4.4.5.
A comment to the (subtle) notation. Recall that if a block operator $P$ with
underlying block metric $g$ decomposes on $M\times
U_{r}(i,\varepsilon)=\\{(m,x)\,:\,\varepsilon x_{i}>r\\}$ into
$P(i,\varepsilon)\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}$
under the isometry $U_{r}(i,\varepsilon)\cong M\times\\{\varepsilon
x_{i}=r\\}\times\varepsilon\mathord{\mathbb{R}}_{>r}$ given by permutation,
then the operator $P(i,\varepsilon)$ restricts to a block operator on
$\mathfrak{S}_{g(i,\varepsilon)}\rightarrow M\times\\{\varepsilon x_{i}=R\\}$
for all sufficiently large $R$ and is independent of $R$. The map
${\delta^{R\varepsilon}_{i}}\colon(M\times\mathord{\mathbb{R}}^{n-1},\partial^{\varepsilon}_{i}g)\rightarrow\bigl{(}M\times\\{\varepsilon
x_{i}=R\\},g(i,\varepsilon)\bigr{)}$
is a Pin-structure preserving isometry and thus provides a bundle isometry
$\mathfrak{S}({\delta^{R\varepsilon}_{i}})$ between the corresponding spinor
bundles that induces an isomorphism between the corresponding sections with
compact support. We use this isomorphism to pull $P(i,\varepsilon)$, an
operator on $\Gamma_{c}(\mathfrak{S}_{g(i,\varepsilon)})$, back to
${{\delta^{R\varepsilon}_{i}}}^{\ast}P(i,\varepsilon)$, an operator on
$\Gamma_{c}(\mathfrak{S}_{\partial^{\varepsilon}_{i}g})$.
For the degeneracy map, we use that the cyclic permutation
$\mathrm{cycl}(i,n+1)\colon\left(M\times\\{\varepsilon
x_{i}>r\\},\sigma_{i}g\right)\rightarrow\left(M\times\mathord{\mathbb{R}}^{n}\times\varepsilon\mathord{\mathbb{R}}_{>r},g\oplus\mathrm{d}x_{n+1}^{2}\right)$
is an isometry and the natural isomorphism $\mathfrak{S}_{g}\boxtimes
Cl_{1,0}\cong\mathfrak{S}_{g\oplus\mathrm{d}x_{n+1}^{2}}$.
###### Lemma 4.4.6.
The connecting maps of $\widetilde{\Psi\mathrm{Dir}}_{\bullet}(M)$ are well-
defined and satisfy the cubical identities.
The proof requires the following simple functional analytic result.
###### Lemma 4.4.7.
Let $H_{1}$, $H_{2}$ be Hilbert spaces and $A\colon H_{1}\rightarrow H_{1}$ be
an unbounded operator. Then $A\otimes\mathrm{id}$ is a bounded operator on
$H_{1}\otimes H_{2}$ if and only if $A$ is bounded on $H_{1}$. In this case,
the operator norms satisfy
$||A\otimes\mathrm{id}||_{\mathrm{op}}=||A||_{\mathrm{op}}$.
###### Proof.
Since the algebraic tensor product $H_{1}\odot H_{2}$ lies dense in
$H_{1}\otimes H_{2}$, so does
${\mathrm{dom}\,}(A\otimes\mathrm{id})\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt={\mathrm{dom}\,}A\odot H_{2}$. If $A$ is bounded, then we can
continuously extend it to an operator on $H_{1}\otimes H_{2}$ and the operator
norm of this extension necessarily agrees with $||A||_{\mathrm{op}}$.
Conversely, assume that $A\otimes\mathrm{id}$ is bounded. Let $v\in H_{2}$ be
a unit-length vector. The map $\iota_{v}\colon H_{1}\rightarrow H_{1}\otimes
H_{2}$ given by $x\mapsto x\otimes v$ is linear and bounded. The map $x\otimes
y\mapsto\langle y,v\rangle\cdot x$ extends from $H_{1}\odot H_{2}$ to a
bounded linear map $p_{v}\colon H_{1}\otimes H_{2}\rightarrow H_{1}$. From
$A=p_{v}\circ A\otimes\mathrm{id}\circ\iota_{v}$ we deduce that $A$ is a
bounded operator. ∎
###### Proof of Lemma 4.4.6.
By Corollary 4.2.3, $\partial^{\varepsilon}_{i}P$ is a again a block operator.
Symbol calculus shows that the principal symbol of
$\partial^{\varepsilon}_{i}P$ agrees with the principal symbol of
${\not{\mathfrak{D}}}_{\partial^{\varepsilon}_{i}g}$, so
$\partial^{\varepsilon}_{i}P$ is a block operator of Dirac type. In a similar
fashion, but using Lemma 4.2.4 instead, we see that $\sigma_{i}P$ is a block
operator of Dirac type.
We need to show that $\partial^{\varepsilon}_{i}P$ and $\sigma_{i}P$ are
symmetric operators if $P$ is symmetric. We start with
$\sigma_{i}P=\mathrm{Cycl}(i,n+1)_{\ast}(P\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}})$.
Since $\mathrm{Cycl}(i,n+1)$ induces an isometry between the corresponding
Hilbert spaces of square integrable sections, the push-forward
$\mathrm{Cycl}(i,n+1)_{\ast}$ maps symmetric operators to symmetric operators.
Thus, it suffices to prove that
$P\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}$ is a symmetric
operator on
$L^{2}(M\times\mathord{\mathbb{R}}^{n+1};\mathfrak{S}_{g\oplus\mathrm{d}x_{n+1}^{2}})\cong
L^{2}(M\times\mathord{\mathbb{R}}^{n};\mathfrak{S}_{g})\otimes
L^{2}(\mathord{\mathbb{R}},Cl_{1,0}),$
which is true because the tensor product of two symmetric operators is again
symmetric and all four operators in question $P$, $\mathrm{id}$,
$\mathrm{e/o}$ and ${\not{\mathit{D}}}$ are symmetric.
To argue that $\partial^{\varepsilon}_{i}P$ is symmetric, we proceed as
follows: On
$U(i,\varepsilon)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=U_{R}(i,\varepsilon)$, for some sufficiently large $R>0$, we can
identify the symmetric operator $P|_{U(i,\varepsilon)}$ with
$P(i,\varepsilon)\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}_{\mathord{\mathbb{R}}}$
via an isometric bundle morphism. The second summand is symmetric, so
$P(i,\varepsilon)\boxtimes\mathrm{id}$ must be a symmetric operator on
$\mathfrak{S}_{g(i,\varepsilon)\oplus\mathrm{d}x_{n}^{2}}\rightarrow(M\times\mathord{\mathbb{R}}^{n})(i,\varepsilon)\times\varepsilon\mathord{\mathbb{R}}_{>R}$.
This implies that $P(i,\varepsilon)$ is symmetric on
$\mathfrak{S}_{g(i,\varepsilon)}$ and consequently that
$\partial^{\varepsilon}_{i}P={\delta^{R\varepsilon}_{i}}^{\ast}P(i,\varepsilon)$
is symmetric because
${\delta^{R\varepsilon}_{i}}\colon(M\times\mathord{\mathbb{R}}^{n-1},\partial^{\varepsilon}_{i}g)\rightarrow(M\times\mathord{\mathbb{R}}^{n}(i,\varepsilon),g(i,\varepsilon))$
is an isometry.
From
$\displaystyle\quad\ \sigma_{i}P-{\not{\mathfrak{D}}}_{\sigma_{i}g}$
$\displaystyle=\mathrm{Cycl}(i,n+1)_{\ast}\left(P\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}\right)-\mathrm{Cycl}(i,n+1)_{\ast}\left({\not{\mathfrak{D}}}_{g}\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}\right)$
$\displaystyle=\mathrm{Cycl}(i,n+1)_{\ast}((P-{\not{\mathfrak{D}}}_{g})\boxtimes\mathrm{id})$
and Lemma 4.4.7 it follows that the difference
$\sigma_{i}P-{\not{\mathfrak{D}}}_{\sigma_{i}g}$ extends to a bounded operator
on $H^{0}(\mathfrak{S}_{\sigma_{i}g})$ with operator norm
$||P-{\not{\mathfrak{D}}}_{g}||_{0,0}$.
To see that
$\partial^{\varepsilon}_{i}P-{\not{\mathfrak{D}}}_{\partial^{\varepsilon}_{i}g}$
extends to a bounded operator on
$H^{0}(\mathfrak{S}_{\partial^{\varepsilon}_{i}g})$ we argue as follows. For
$R>0$ sufficiently large the operator $P-{\not{\mathfrak{D}}}_{g}$ restricts
to $\Gamma_{c}(U_{R}(i,\varepsilon),\mathfrak{S}_{g})$. Since
$P-{\not{\mathfrak{D}}}_{g}$ extends to a bounded operator on
$H^{0}(\mathfrak{S}_{g})$, the restriction extends to a bounded operator on
$H^{0}(U_{R}(i,\varepsilon),\mathfrak{S}_{g})$, which is isometric equivalent
to $H^{0}(M\times\\{\varepsilon
x_{i}=R\\},\mathfrak{S}_{g(i,\varepsilon)})\otimes
L^{2}(\varepsilon\mathord{\mathbb{R}}_{>R},Cl_{1,0})$. Under this
identification, the restriction of $P-{\not{\mathfrak{D}}}_{g}$ agrees with
$\left(P(i,\varepsilon)-{\not{\mathfrak{D}}}_{g(i,\varepsilon)}\right)\boxtimes\mathrm{id}$.
By Lemma 4.4.7, this implies that
$P(i,\varepsilon)-{\not{\mathfrak{D}}}_{g(i,\varepsilon)}$ is bounded. Thus,
$\partial^{\varepsilon}_{i}P-{\not{\mathfrak{D}}}_{\partial^{\varepsilon}_{i}g}={\delta^{R\varepsilon\,\ast}_{i}}\left(P(i,\varepsilon)-{\not{\mathfrak{D}}}_{g(i,\varepsilon)}\right)$
is a bounded operator on $H^{0}(\mathfrak{S}_{\partial^{\varepsilon}_{i}g})$.
Lastly, we need to show, that the connecting maps satisfy the cubical
identities. For $i<j$ and $R>0$ sufficiently large, we have
$\displaystyle\partial^{\varepsilon}_{i}\partial^{\omega}_{j}P$
$\displaystyle=\partial^{\varepsilon}_{i}\left({\delta^{\omega
R}_{j}}^{\ast}P(j,\omega)\right)$ $\displaystyle={\delta^{\varepsilon
R}_{i}}^{\ast}\left({\delta^{\omega
R}_{j}}^{\ast}\left(P(j,\omega)\right)(i,\varepsilon)\right)$
$\displaystyle=\left({\delta^{\omega R}_{j}}{\delta^{\varepsilon
R}_{i}}\right)^{\ast}P(i,j,\varepsilon,\omega)$
$\displaystyle=\left({\delta^{\varepsilon R}_{i}}{\delta^{\omega
R}_{j-1}}\right)^{\ast}P(i,j,\varepsilon,\omega)$
$\displaystyle={\delta^{\omega R}_{j-1}}^{\ast}\left({\delta^{\varepsilon
R}_{i}}^{\ast}\left(P(i,\varepsilon)\right)(j-1,\omega)\right)$
$\displaystyle=\partial^{\omega}_{j-1}\partial^{\varepsilon}_{i}P.$
For $i<j$ and a sufficiently large $R>0$, we calculate
$\displaystyle\partial^{\varepsilon}_{i}\sigma_{j}(P)$
$\displaystyle={\delta^{R\varepsilon}_{i}}^{\ast}(\sigma_{j}(P)(i,\varepsilon))$
$\displaystyle={\delta^{R\varepsilon}_{i}}^{\ast}\mathrm{Cycl}(j,n+1)_{\ast}\left(P\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}\right)(\mathrm{cycl}(j,n+1)^{-1}(i),\varepsilon)$
$\displaystyle={\delta^{R\varepsilon}_{i}}^{\ast}\mathrm{Cycl}(j,n+1)_{\ast}(P(i,\varepsilon)\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}})\quad\quad\quad\text{
for }i<j$
$\displaystyle=\left({\delta^{R\varepsilon}_{i}}^{-1}\right)_{\ast}\mathrm{Cycl}(j,n+1)_{\ast}(P(i,\varepsilon)\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}})$
$\displaystyle=\mathrm{Cycl}(j-1,n)_{\ast}\left({\delta^{R\varepsilon}_{i}}^{-1}\right)_{\ast}\left(P(i,\varepsilon)\boxtimes\mathrm{id}+\mathrm{e/o}_{\mathfrak{S}_{g(i,\varepsilon)}}\boxtimes{\not{\mathit{D}}}\right)$
$\displaystyle=\mathrm{Cycl}(j-1,n)_{\ast}(\partial^{\varepsilon}_{i}P\boxtimes\mathrm{id}+\mathrm{e/o}_{\mathfrak{S}_{\partial^{\varepsilon}_{i}g}}\boxtimes{\not{\mathit{D}}})$
$\displaystyle=\sigma_{n-1}\partial^{\varepsilon}_{i}P,$
and analogously
$\partial^{\varepsilon}_{i}\sigma_{j}P=\sigma_{j}\partial^{\varepsilon}_{i-1}P$
for $i>j$. Finally, with a slight abuse of notation in the last lines, we
calculate
$\displaystyle\partial^{\varepsilon}_{i}\sigma_{i}P$
$\displaystyle={\delta^{R\varepsilon}_{i}}^{\ast}\left((\sigma_{i}P)(i,\varepsilon)\right)$
$\displaystyle={\delta^{R\varepsilon}_{i}}^{\ast}\left(\mathrm{Cycl}(i,n+1)_{\ast}(P\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}})(i,\varepsilon)\right)$
$\displaystyle={\delta^{R\varepsilon}_{i}}^{-1}_{\ast}\mathrm{Cycl}(i,n+1)_{\ast}\left((P\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}})(\mathrm{cycl}(i,n+1)^{-1}(i),\varepsilon)\right)$
$\displaystyle={\delta^{R\varepsilon}_{i}}^{-1}_{\ast}\mathrm{Cycl}(i,n+1)_{\ast}\left((P\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}})(n+1,\varepsilon)\right)$
$\displaystyle={\delta^{R\varepsilon}_{n+1}}^{-1}_{\ast}\left((P\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}})(n+1,\varepsilon)\right)=P.$
For $i\leq j$, we use the identity
$\mathrm{cycl}(i,n+2)\circ\mathrm{cycl}(j,n+1)=\mathrm{cycl}(j+1,n+2)\circ\mathrm{cycl}(i,n+1)\circ\tau_{n+1,n+2}$
in which $\tau_{n+1,n+2}$ denotes the transposition that interchanges $n+1$
and $n+2$, to conclude
$\displaystyle\sigma_{i}\sigma_{j}(P)$
$\displaystyle=\sigma_{i}(\mathrm{Cycl}(j,n+1)_{\ast}(P\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}))$
$\displaystyle\begin{split}&=\mathrm{Cycl}(i,n+2)_{\ast}\left(\mathrm{Cycl}(j,n+1)_{\ast}(P\boxtimes\mathrm{id}+\mathrm{e/o}_{\mathfrak{S}_{g}}\boxtimes{\not{\mathit{D}}})\boxtimes\mathrm{id}\right.\\\
&\quad\left.+\mathrm{e/o}_{\mathfrak{S}_{\sigma_{j}(g)}}\boxtimes{\not{\mathit{D}}}\right)\end{split}$
$\displaystyle\begin{split}&=\mathrm{Cycl}(i,n+2)_{\ast}\circ\mathrm{Cycl}(j,n+1)_{\ast}\biggl{(}P\boxtimes\mathrm{id}\boxtimes\mathrm{id}\\\
&\quad+\left.\mathrm{e/o}_{\mathfrak{S}_{g}}\boxtimes{\not{\mathit{D}}}\boxtimes\mathrm{id}+\mathrm{e/o}_{\mathfrak{S}_{\sigma_{n+1}(g)}}\boxtimes{\not{\mathit{D}}}\right)\end{split}$
$\displaystyle\begin{split}&=\mathrm{Cycl}(i,n+2)_{\ast}\circ\mathrm{Cycl}(j,n+1)_{\ast}\biggl{(}P\boxtimes\mathrm{id}\boxtimes\mathrm{id}\\\
&\quad\left.+\mathrm{e/o}_{\mathfrak{S}_{g}}\boxtimes{\not{\mathit{D}}}\boxtimes\mathrm{id}+\mathrm{e/o}_{\mathfrak{S}_{g}}\boxtimes\mathrm{e/o}\boxtimes{\not{\mathit{D}}}\right)\end{split}$
$\displaystyle\begin{split}&=\mathrm{Cycl}(j+1,n+2)_{\ast}\circ\mathrm{Cycl}(i,n+1)_{\ast}\circ{\tau_{n+1,n+2}}_{\ast}\biggl{(}P\boxtimes\mathrm{id}\boxtimes\mathrm{id}\\\
&\quad\left.+\mathrm{e/o}_{\mathfrak{S}_{g}}\boxtimes{\not{\mathit{D}}}\boxtimes\mathrm{id}+\mathrm{e/o}_{\mathfrak{S}_{g}}\boxtimes\mathrm{e/o}\boxtimes{\not{\mathit{D}}}\right)\end{split}$
$\displaystyle\begin{split}&=\mathrm{Cycl}(j+1,n+2)_{\ast}\circ\mathrm{Cycl}(i,n+1)_{\ast}\biggl{(}P\boxtimes\mathrm{id}\boxtimes\mathrm{id}\\\
&\quad\left.+\mathrm{e/o}_{\mathfrak{S}_{g}}\boxtimes{\not{\mathit{D}}}\boxtimes\mathrm{id}+\mathrm{e/o}_{\mathfrak{S}_{g}}\boxtimes\mathrm{e/o}\boxtimes{\not{\mathit{D}}}\right)\end{split}$
$\displaystyle=...=\sigma_{j+1}\sigma_{i}(P).$
In the firth line, we can drop ${\tau_{n+1,n+2}}_{\ast}$ in the second factor
to go to the sixth line because
${\not{\mathit{D}}}\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}=\partial_{x_{n+1}}\cdot\frac{\partial}{\partial{x_{n+1}}}+\partial_{x_{n+2}}\cdot\frac{\partial}{\partial{x_{n+2}}}$
is symmetric in the $(n+1)$-th and $(n+2)$-th coordinate. ∎
We come now to one of the most essential definitions of this thesis.
###### Definition 4.4.8.
Let $\widetilde{\Psi\mathrm{Dir}}^{\times}_{\bullet}(M)$ be the sequence of
subsets of $\widetilde{\Psi\mathrm{Dir}}_{\bullet}(M)$, where
$\widetilde{\Psi\mathrm{Dir}}^{\times}_{n}(M)$ consists of all block Dirac
operator $P\in\widetilde{\Psi\mathrm{Dir}}_{n}(M)$ that extend to an
invertible operator $P\colon H^{1}(\mathfrak{S}_{g_{P}})\rightarrow
H^{0}(\mathfrak{S}_{g_{P}})$.
The proof that the face maps of $\widetilde{\Psi\mathrm{Dir}}_{\bullet}(M)$
restricts to $\widetilde{\Psi\mathrm{Dir}}^{\times}_{\bullet}(M)$ requires the
following lemma that roughly says that positivity descends to hypersurfaces.
###### Lemma 4.4.9.
Let $\mathfrak{S}_{g}\rightarrow M$ be a spinor bundle over a Riemannian
manifold $(M,g)$. For $Q\in\overline{\Psi\mathrm{DO}}^{1}(E)$ symmetric and
odd, define
$P\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=Q\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}$ on
$\mathfrak{S}\boxtimes Cl_{1,0}\rightarrow M\times(-a,a)$ for all
$0<a\leq\infty$.
If $||Pu||_{0}\geq c||u||_{0}$ for all
$u\in\Gamma_{c}(M\times(-a,a),\mathfrak{S}_{g}\boxtimes Cl_{1,0})$, then
$||Qv||_{0}^{2}\geq\max\\{0,c-4/a^{2}\\}||v||_{0}^{2}$
for all $v\in\Gamma_{c}(M,\mathfrak{S}_{g})$.
###### Proof.
Let $v\in\Gamma_{c}(M,\mathfrak{S}_{g})$ and
$f\in\Gamma_{c}((-a,a),Cl_{1,0})$. The operator $P$ is again symmetric by the
proof of Lemma 4.4.6 and odd. Using Fubini’s theorem and that $P$ is bounded
from below, we get
$\displaystyle||Pv\otimes f||_{0}^{2}$ $\displaystyle=\langle P^{2}(v\otimes
f),v\otimes f\rangle$ $\displaystyle=\langle Q^{2}(v)\otimes
f-v\otimes\Delta_{\mathord{\mathbb{R}}}(f),v\otimes f\rangle$
$\displaystyle=||Q(v)\otimes f||_{0}^{2}+||v\otimes f^{\prime}||_{0}^{2}$
$\displaystyle=||Q(v)||_{0}^{2}||f||_{0}^{2}+||v||_{0}^{2}||f^{\prime}||_{0}^{2}$
$\displaystyle\geq c||v||_{0}^{2}||f||_{0}^{2}=c||v\otimes f||_{0}^{2}.$
The function
$\varphi\colon(-1,1)\rightarrow\mathord{\mathbb{R}}\quad\text{ given by
}\quad\varphi(t)=\begin{cases}1+t,&\text{if }t\leq 0,\\\ 1-t,&\text{if }t\geq
0,\end{cases}$
satisfies $||\varphi||_{0}^{2}=2/3$ while $||\varphi^{\prime}||_{0}^{2}=2$.
Pick a compactly supported function $f$ that is sufficiently
$||\text{--}||_{1}$-close to $\varphi$ such that
$||f^{\prime}||_{0}^{2}/||f||_{0}^{2}\leq 4$.
By the transformation formula, the assignment $f\mapsto
a^{-1/2}f(a^{-1}\cdot)$ is an isometry $L^{2}(-1,1)\rightarrow L^{2}(-a,a)$.
Thus,
$||f_{a}^{\prime}||_{0}^{2}/||f_{a}||_{0}^{2}=a^{-2}||f^{\prime}||_{0}^{2}/||f||_{0}^{2}\leq
4/a^{2}$.
In conclusion, if we plug into $f_{a}$ in the previous inequality, we obtain
$\displaystyle||Qv||_{0}^{2}$
$\displaystyle\geq\left(c-||f_{a}^{\prime}||_{0}^{2}/||f_{a}||_{0}^{2}\right)||v||_{0}^{2}$
$\displaystyle\geq\left(c-4/a^{2}\right)||v||_{0}^{2},$
which is positive if $c>4/a^{2}$. ∎
###### Lemma 4.4.10.
The connecting maps of $\widetilde{\Psi\mathrm{Dir}}_{\bullet}(M)$ send
invertible elements to invertible elements. Thus,
$\widetilde{\Psi\mathrm{Dir}}^{\times}_{\bullet}(M)$ is a cubical subset.
###### Proof.
Fix a sufficiently large $R>0$ such that $g=g_{P}$ decomposes into
$g(i,\varepsilon)\oplus\mathrm{d}x_{i}^{2}$ on $U_{R}{(i,\varepsilon)}$ and,
by abusing notation, $P$ decomposes into
$P(i,\varepsilon)\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}_{\mathord{\mathbb{R}}}$.
Since $P$ is bounded from below, its restriction $P|_{U_{R}(i,\varepsilon)}$
is also bounded from below. Lemma 4.4.9 implies that $P(i,\varepsilon)$, and
hence $\partial^{\varepsilon}_{i}P$ too, is bounded from below. Since
$\partial^{\varepsilon}_{i}P$ is self-adjoint, it must be invertible.
To see that $\sigma_{i}(P)$ is invertible, it suffices to check that
$P\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}$ is invertible.
Since $P\boxtimes\mathrm{id}$ and $\mathrm{e/o}\boxtimes{\not{\mathit{D}}}$
anti-commute we have
$\left(P\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}\right)^{2}=P^{2}\boxtimes\mathrm{id}-\mathrm{id}\boxtimes\Delta.$
Both summands are non-negative operators. Since $P$ is invertible, $P^{2}$ is
positive and therefore the sum is positive and therefore invertible. ∎
We would like to know whether $\widetilde{\Psi\mathrm{Dir}}_{\bullet}(M)$ and
$\widetilde{\Psi\mathrm{Dir}}^{\times}_{\bullet}(M)$ are Kan sets. For
$\widetilde{\Psi\mathrm{Dir}}_{\bullet}(M)$ even more is true.
###### Proposition 4.4.11.
The cubical set $\widetilde{\Psi\mathrm{Dir}}_{\bullet}(M)$ is combinatorially
contractible.
The proposition follows immediately from the following elementary lemma.
###### Lemma 4.4.12.
Let $P_{(j,\omega)}$ and $P_{(k,\eta)}$ be two block Dirac operators on
$M\times\mathord{\mathbb{R}}^{n-1}$. Assume that $j\leq k$, that
$\partial^{\omega}_{j}P_{(k,\eta)}=\partial^{\eta}_{k-1}P_{(j,\omega)}$ and
that the two operators decompose outside of $M\times\rho I^{n}$. Then the
operators $\sigma_{j}P_{(j,\omega)}$ and $\sigma_{k}P_{(k,\eta)}$ agree on the
set $\\{\omega x_{j}>\rho,\eta x_{k}>\rho\\}$.
###### Proof.
Abbreviate $\\{\omega x_{j}>\rho,\eta x_{k}>\rho\\}$ to $U$. If $j=k$ there is
nothing to prove for $U$ is either empty or $(j,\omega)=(k,\eta)$. In the
other cases, it follows from Lemma 3.1.10 that the underlying metrics agree on
$U$. The proof of Lemma 4.2.4 shows that $\sigma_{j}P_{(j,\omega)}$ restricts
to $U$. Furthermore, for $j<k$, the calculation
$\displaystyle\sigma_{j}P_{(j,\omega)}|_{U}$ |
# Remarks on existence and uniqueness of the solution for stochastic partial
differential equations
Benny Avelin Uppsala University, Department of Mathematics, Sweden
<EMAIL_ADDRESS>and Lauri Viitasaari Aalto University School of
Business, Department of Information and Service Management, Finland
<EMAIL_ADDRESS>
###### Abstract.
In this article we consider existence and uniqueness of the solutions to a
large class of stochastic partial differential of form
$\partial_{t}u=L_{x}u+b(t,u)+\sigma(t,u)\dot{W}$, driven by a Gaussian noise
$\dot{W}$, white in time and spatial correlations given by a generic
covariance $\gamma$. We provide natural conditions under which classical
Picard iteration procedure provides a unique solution. We illustrate the
applicability of our general result by providing several interesting
particular choices for the operator $L_{x}$ under which our existence and
uniqueness result hold. In particular, we show that Dalang condition given in
[5] is sufficient in the case of many parabolic and hypoelliptic operators
$L_{x}$.
Mathematics Subject Classifications (2010): 60H15, 60G15, 35C15, 35K58, 35S10.
Keywords: Stochastic partial differential equations, existence and uniqueness,
mild solution, semilinear parabolic equations, hypoelliptic equations.
## 1\. Introduction
In this article we consider the stochastic partial differential equation
(SPDE) of form
$\frac{\partial u}{\partial
t}(t,x)=(L_{x}u)(t,x)+b(t,u(t,x))+\sigma(t,u(t,x))\dot{W}(t,x),\hskip
14.22636ptt\geq 0,x\in\mathbb{R}^{d}$ (1.1)
with initial condition $u(0,x)=u_{0}(x)$. Here $b$ and $\sigma$ are assumed to
be Lipschitz continuous functions and bounded on compacts in the spatial
variable $x$, and uniformly in $t$. The stochastic force $\dot{W}$ is assumed
to be Gaussian with correlation structure that is white in the time variable
$t$ and given by a generic covariance $\gamma$ in the spatial variable $x$.
SPDEs have been a subject of active research in the literature for recent
years, and the basic theory is already rather well-established. Especially,
initiated by the seminal paper by Dalang [5], stochastic heat equations, in
which case $L_{x}=\Delta$ is the Laplace operator, with Gaussian noise that is
white in time have received a lot of attention. In this case one obtains the
existence and uniqueness of the solution provided that the following so-called
Dalang’s condition
$\int_{\mathbb{R}^{d}}\frac{1}{\beta+2|\xi|^{2}}\hat{\gamma}(d\xi)<\infty$
(1.2)
holds for some $\beta_{0}>0$ (in which case it holds for all
$\beta>\beta_{0}$). Here $\hat{\gamma}$ denotes the non-negative measure
arising as the Fourier transform of the spatial covariance $\gamma$ of the
noise $\dot{W}$. One also observes that here the term $|\xi|^{2}$ corresponds
to the Fourier multiplier arising from the Laplace operator $\Delta$. This
leads to a natural extension to the case where $L_{x}$ is the
$L^{2}$-generator of a Levy process. In this case 1.2 is replaced by
$\int_{\mathbb{R}^{d}}\frac{1}{\beta+2\text{Re}\Psi(\xi)}\hat{\gamma}(d\xi)<\infty,$
(1.3)
where $\Psi(\xi)$ arises from the characteristic exponent of the associated
Levy process. Here $\text{Re}\Psi(\xi)$ is again a non-negative function, and
if 1.3 holds for some $\beta>0$, then it holds for all $\beta>0$. As a
particular interesting example, this covers the case of the stochastic
fractional heat equation where $L_{x}$ is given by the fractional Laplace
operator $-(-\Delta)^{\alpha}$, $\alpha\in(0,1]$. Indeed, then
$\Psi(\xi)=|\xi|^{2\alpha}$ and one recovers the classical heat equation and
condition 1.2 by plugging in $\alpha=1$. This generalisation is studied, among
others, in [10] where the existence and uniqueness result was given under
suitable assumptions on the coefficients $b,\sigma$.
A standard technique to prove existence and uniqueness of the solution to 1.1
is based on the so-called fundamental solution $G_{t}(x)$ (or the Green
kernel) associated to the equation $\partial_{t}u=L_{x}u$. Then one obtains a
candidate for the solution through convolutions with the kernel $G_{t}(x)$. In
particular, in the settings mentioned above one obtains that $G_{t}$ is the
density of the underlying Levy process and forms a semigroup. After that,
using the positivity of $G_{t}$ as well, one obtains the solution through
Picard iteration.
In this article we provide a general existence and uniqueness result for the
solution to 1.1. Our condition, given by Equation 2.6 below, is similar to
1.3. However, in our results we do not require non-negativity of the
fundamental solutions $G_{t}$ as we only consider upper bounds for $|G_{t}|$.
Furthermore, we do not require the associated function, given by $\Psi(\xi)$
in 1.3, to be non-negative. This fact allows to consider a larger class of
operators $L_{x}$ in 1.1, in the case where we do not have a Gaussian upper
bound on the fundamental solution. Our main contribution is that, by a careful
analysis on the essential requirements in the classical arguments, we get a
remarkably strong generalization. As such, we are able to cover a host of
equations which was previously not known in the literature, including the
important Kolmogorov equations. Finally, we stress that similar considerations
can be applied to equations of form 1.1, where $\partial_{t}$ is replaced with
more general operator $L_{t}$, see Section 4.
The rest of the paper is organised as follows. In Section 2 we present our
assumptions and provide the existence and uniqueness result, Theorem 2.2. In
Section 2.1, we illustrate the applicability of our result by providing a
detailed discussion on several interesting examples. The proof of Theorem 2.2
is postponed to Section 3 where we also recall some basic facts on stochastic
calculus. We end the paper with some concluding remarks.
## 2\. General existence and uniqueness result
We consider the stochastic partial differential equation
$\frac{\partial u}{\partial
t}(t,x)=(L_{x}u)(t,x)+b(t,u(t,x))+\sigma(t,u(t,x))\dot{W}(t,x),\hskip
14.22636ptt\geq 0,x\in\mathbb{R}^{d}$ (2.1)
where $L_{x}$ is a suitable differential operator (acting on the variable
$x$), and coefficients $b$ and $\sigma$ are assumed to be spatially Lipschitz
continuous functions, uniformly in $s$ over compacts. That is, for any $T>0$
we have for all $t\in[0,T]$ that
$|b(t,x)-b(t,y)|\leq L_{b,T}|x-y|$
and
$|\sigma(t,x)-\sigma(t,y)|\leq L_{\sigma,T}|x-y|.$
Note that then the functions $b$ and $\sigma$ are spatially locally bounded,
uniformly in $s$ over compacts. That is,
$\sup_{s\in[0,T],y\in K}\max(|\sigma(s,y)|,b(s,y)|)<\infty$
for all compact sets $K\subset\mathbb{R}^{d}$ and all finite $T>0$. For the
(centered) Gaussian noise $\dot{W}$ we assume that the covariance is given by
$\mathbb{E}\left[\dot{W}(t,x)\dot{W}(s,y)\right]=\delta_{0}(t-s)\gamma(x-y),$
where $\delta_{0}$ denotes the Dirac delta and $\gamma$ are non-negative and
non-negative definite measures. That is, the noise can be described by a
centered Gaussian family with covariance
$\mathbb{E}[W(\phi)W(\psi)]=\int_{0}^{\infty}\int_{\mathbb{R}^{d}}\mathcal{F}\phi(\cdot,s)(\xi)\overline{\mathcal{F}(\psi)(\cdot,s)}(\xi)\hat{\gamma}(d\xi)ds,$
(2.2)
where $\hat{\gamma}$ is the spectral measure of $\gamma$, $\mathcal{F}$
denotes the Fourier transform, and $\phi,\psi$ are suitable functions. We
provide rigorous treatment on the construction of the Gaussian noise $\dot{W}$
in Section 3.1 that contains a brief introduction to Gaussian analysis and
stochastic integration required for our analysis.
We denote by $G_{t}(x)$ the fundamental solution (or the Green kernel)
associated to the operator $M=\partial_{t}-L_{x}$, in the sense that for any
"nice enough" intial data $u_{0}$ we have that
$\displaystyle u(x,t)=\int G_{t}(x-y)u_{0}(y)dy$
satisfies $Mu=0$ and $u\to u_{0}$ as $t\to 0$.
Consider now our original equation 2.1. For the initial condition
$u(0,x)=u_{0}(x)$, we assume that $u_{0}(x)$ is deterministic and satisfies,
for every $T>0$,
$\sup_{t\in(0,T],x\in\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}|G_{t}(x-y)||u_{0}(y)|dy<\infty.$
(2.3)
We prove that under certain conditions, equation 2.1 admits a unique mild
solution in the following sense.
###### Definition 2.1.
We say that a random field $u(t,x)$, adapted to the filtration generated by
$\dot{W}$, is a mild solution to 2.1 if for all
$(t,x)\in\mathbb{R}_{+}\times\mathbb{R}^{d}$, the process $(s,y)\rightarrow
G_{t-s}(x-y)\sigma(u(s,y))\textbf{1}_{[0,t]}(s)$ is integrable and we have
$\displaystyle u(t,x)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}G_{t}(x-y)u_{0}(y)dy+\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)b(s,u(s,y))dyds.$
(2.4) $\displaystyle+$
$\displaystyle\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)\sigma(s,u(s,y))W(ds,dy).$
Here the stochastic integral exists in the sense of Dalang-Walsh, see Section
3.1.
The following existence and uniqueness result is the main result of this
paper. The proof follows standard arguments and is presented in Section 3.
###### Theorem 2.2.
Let $G_{s}$ be the fundamental solution to 2.1 and assume that there exists an
integrable function $g_{s}(x)=\mathcal{F}^{-1}(e^{-s\hat{f}})$,
$\hat{f}\geq-C$ such that $|G_{s}(x)|\leq g_{s}(x)$ and there exists
$\beta_{0}>2C$ such that,
$\int_{0}^{\infty}e^{-\beta_{0}s}\|g_{s}\|_{L^{1}(\mathbb{R}^{d})}ds<\infty$
(2.5)
and
$\int_{\mathbb{R}^{d}}\frac{1}{\beta_{0}+2\hat{f}(\xi)}\hat{\gamma}(d\xi)<\infty$
(2.6)
Then 2.1 admits a unique mild solution.
###### Remark 1.
Observe that the condition 2.5 is independent of the chosen covariance
$\gamma$, and is rather mild. Indeed, in many examples $g_{s}$ is given by a
density of some generating stochastic process at time $s$, and hence we have
$\|g_{s}\|_{L^{1}(\mathbb{R}^{d})}=1$ for all $s>0$. Moreover, by carefully
examining our proof one observes that 2.5 can be replaced with a weaker
condition $\int_{0}^{T}\|g_{s}\|_{L^{1}(\mathbb{R}^{d})}ds<\infty$ for all
$T>0$ finite. Finally, we note that 2.5 can be omitted in the case $b\equiv
0$.
###### Remark 2.
The proof of Theorem 2.2 actually gives more, however for purposes of
exposition we chose to present the simplified form 2.6. For a discussion about
further cases (including for instance the wave equation) see, Section 4.
###### Remark 3.
Note that if 2.6 is satisfied for some $\beta_{0}>0$, then it is automatically
valid for all $\beta>\beta_{0}$ as well.
### 2.1. Examples
In this section we outline some examples of quite general linear equations for
which Theorem 2.2 can be applied.
#### 2.1.1. Linear operators with Gaussian upper bounds
Many parabolic and hypoelliptic evolution operators satisfy a Gaussian upper
bound for their fundamental solution, namely an estimate of the following type
$\displaystyle G_{s}(x)\leq g(t)e^{-\frac{C|x|_{\mathbb{G}}^{2}}{t}}$ (2.7)
where $\mathbb{G}$ is a homogeneous Lie group. The reason for writing it as
above is that if we consider $|x|_{\mathbb{G}}=|x|$, being the standard
Euclidean norm, we can cover a big class of parabolic operators. But for
hypoelliptic equations, we get a nontrivial norm induced by the corresponding
Lie-group.
Let us now consider some interesting examples satisfying 2.7 with the
Euclidean norm.
Let $L$ in $\mathbb{R}^{n}$ be given as
$\displaystyle
L=\sum_{i,j=1}^{n}a_{ij}(x,t)\partial_{ij}+\sum_{j=1}^{n}b_{i}(x,t)\partial_{i}+c(x,t)$
then if the matrix $a_{ij}(x,t)$ is uniformly elliptic, the functions $a,b,c$
are bounded and uniformly Hölder continuous with exponent $\alpha$, then there
exists positive constants $c_{1},c_{2}$ such that
$\displaystyle|G_{t}(x)|\leq c_{1}t^{-n/2}e^{-c_{2}\frac{|x|^{2}}{t}}$ (2.8)
where $G_{t}(x)$ is the fundamental solution to $\partial_{t}-L$, see [8, 9].
As such Theorem 2.2 is applicable, and since the Fourier transform of a
Gaussian is a Gaussian we get that if
$\displaystyle\int_{\mathbb{R}^{d}}\frac{1}{\beta_{0}+2C(c_{1})|\xi|^{2}}\hat{\gamma}(d\xi)<\infty$
(2.9)
then there exists a mild solution to 2.1. That is, the condition reduces to
the standard one for the heat equation, [5]. For instance, if $\gamma$ is
given by the Riesz kernel, i.e. $\hat{\gamma}=|\xi|^{\lambda-d}d\xi$, then the
above is verified as long as $\lambda<2$. In the case when $\gamma$ is in
$W^{k,1}(\mathbb{R}^{n})$ then
$\displaystyle|\hat{\gamma}|\leq\frac{C}{(1+|\xi|)^{k}},$
from this we see that 2.9 is verified if $k>n-2$. Finally we remark that if
$\gamma=\delta_{0}$ (white noise), then 2.9 only holds in dimension 1.
In fact the above can be extended as follows, if we instead consider the
divergence form operator
$\displaystyle L=\sum_{i,j=1}^{n}\partial_{i}(a_{ij}(x,t)\partial_{j}\cdot),$
with $a_{ij}$ still an elliptic matrix but now only bounded and measurable,
then $\partial_{t}-L$, Aronson [1] proved that 2.8 holds also in this case.
The case for non-divergence form equations with rough coefficients was treated
in [7].
Let us now consider the hypoelliptic setting and consider some relevant
examples. Let us begin with some notation. We consider the hypoelliptic
evolution operator
$\displaystyle M=\sum_{i=1}^{m}X_{i}^{2}+X_{0}-\partial_{t},$
where $X_{i}$ are smooth vector fields in $\mathbb{R}^{n}$ for $i=0,\ldots,m$,
and usually $m<n$. Such an equation induces an interesting geometry. Namely,
let us denote
$\displaystyle Y=X_{0}-\partial_{t},\quad\text{and}\quad\lambda\cdot
X=\lambda_{1}X_{1}+\ldots\lambda_{m}X_{m}$
then a curve in $\gamma:[0,R]\to\mathbb{R}^{n}\times[0,T]$ is $M$-admissible
if it is absolutely continuous and
$\displaystyle\gamma^{\prime}(s)=\lambda(s)\cdot
X(\gamma(s))+Y(\gamma(s)),\quad\text{a.e. in [0,R]}.$
That is, the curve has a direction at each point in the tangent-bundle
described by the vector-fields $X,Y$. If we assume that the vector fields are
such that
* •
every two points $(x,t),(\xi,\tau)\in\mathbb{R}^{n}\times[0,T]$ can be
connected with an $M$ admissible curve,
* •
there exists a homogeneous Lie group
$\mathbb{G}=(\mathbb{R}^{n}\times[0,T],\circ,\delta_{\lambda})$ for which
$X,Y$ are left translation invariant. Furthermore, that $X$ is
$\delta_{\lambda}$-homogeneous of degree 1 and $Y$ is
$\delta_{\lambda}$-homogeneous of degree 2.
Then the operator $M$ is hypoelliptic (distributional solutions are smooth),
see [3, 11]. This allows us to write $\mathbb{R}^{n}=V_{1}\oplus\ldots\oplus
V_{l}$, i.e. the vector field $X$ stratifies $\mathbb{R}^{n}$ into a direct
sum of subspaces, such that if we write $x=x^{(1)}+\ldots+x^{(l)}$, where
$x^{(k)}\in V_{k}$, then the dilations becomes simple multiplication in the
sense that
$\displaystyle\delta_{\lambda}(x,t)=(\lambda
x^{(1)}+\ldots+\lambda^{l}x^{(l)},\lambda^{2}t).$
Furthermore if we introduce the $\delta_{\lambda}$ homogeneous norm as
$\displaystyle|x|_{\mathbb{G}}=\max\left\\{|x_{i}^{(k)}|^{\frac{1}{k}},k=1,\ldots,l,i=1,\ldots,m_{k}\right\\}$
Then, for such operators $M$ we have that there exists (see [11]) a positive
constant $C$ such that
$\displaystyle|G_{t}(x)|\leq\frac{C}{t^{\frac{Q-2}{2}}}e^{-\frac{|x|_{\mathbb{G}}^{2}}{Ct}}.$
From this we see that we can apply Theorem 2.2 again in this context, however
now $\gamma$ needs to be adapted to the geometry of $\mathbb{G}$. Furthermore,
for certain hypoelliptic operators we even have positivity of the fundamental
solution, [2].
A prototypical example of a hypoelliptic operator is the following (well known
Kolmogorov operator)
$\displaystyle
K=\sum_{j=1}^{n}\partial_{x_{j}}^{2}+\sum_{j=1}^{n}x_{j}\partial_{y_{j}}-\partial_{t}$
where we are working in $\mathbb{R}^{2n}\times[0,T]$ and we denote the first
$n$ coordinates as $x_{i}$ and the other $n$ as $y_{i}$. The operator $K$ is
hypoelliptic in the above sense [3]. Furthermore, the fundamental solution for
$n=1$ is given by
$\displaystyle G_{t}(x,y)=\frac{\sqrt{3}}{2\pi
t^{2}}\exp\left(-\frac{x^{2}}{t}-\frac{3xy}{t^{2}}-\frac{3y^{2}}{t^{3}}\right)$
Considering the above "heat type" kernel in the finite interval $[0,T]$, it is
clear that we can bound it from above by
$g_{t}(x)=\frac{C}{t^{2}}\exp\left(-\frac{1}{C}\frac{x^{2}+y^{2}}{t}\right)$
for a positive constant $C$. As such the same conclusion as in Section 2.1.1
holds.
Operators of this type often appear in the context of SDE’s where the noise is
only in some directions, like the kinetic equations, for which the density
satisfies the kinetic Fokker-Planck equation. The kinetic Fokker-Planck is
hypoelliptic, see [13] and the references therein.
#### 2.1.2. Fractional heat equation
Consider the fractional evolution equation defined as
$\displaystyle\partial_{t}u=-(-\Delta)^{s}u+Cu,\quad 0<s\leq 1,$
where $C=0$ corresponds to the fractional heat equation. The fourier transform
of said equation is
$\displaystyle\partial_{t}\hat{u}=-|\xi|^{2s}\hat{u}+C\hat{u}.$
As such, the fundamental solution is given by
$\hat{G}_{t}(\xi)=e^{-(|\xi|^{2s}-C)t}$, and we can thus apply Theorem 2.2
with $b\equiv 0$ and $\hat{f}\geq-C$, together with Remark 1.
#### 2.1.3. Mixture operators
We formally consider an equation of the following type
$\displaystyle\partial_{t}u=L_{1}u+L_{2}u.$
If $G_{1},G_{2}$ are the fundamental solutions of
$\displaystyle\partial_{t}u=L_{1}u,\quad\partial_{t}u=L_{2}u,$
respectively, then $G=G_{1}\ast G_{2}$ is a fundamental solution of the above,
(under some integrability assumptions). Indeed, we have
$\displaystyle\partial_{t}(G_{1}\ast G_{2})$
$\displaystyle=(\partial_{t}G_{1}\ast
G_{2})+(G_{1}\ast\partial_{t}G_{2})=(L_{1}G_{1})\ast
G_{2}+G_{1}\ast(L_{2}G_{2})$ $\displaystyle=(L_{1}+L_{2})(G_{1}\ast G_{2}).$
Thus for instance we can combine Sections 2.1.1 and 2.1.2 and apply Theorem
2.2 to get the existence of a mild solution to 2.1.
## 3\. Proof of Theorem 2.2
### 3.1. Preliminaries on stochastic calculus
In this section we introduce stochastic analysis with respect to the noise
$\dot{W}$.
Denote by $C_{c}^{\infty}\left([0,\infty)\times\mathbb{R}^{d}\right)$ the
class of $C^{\infty}$ functions on $[0,\infty)\times\mathbb{R}^{d}$ with
compact support. We consider a Gaussian family of centered random variables
$\left(W(\varphi),\varphi\in
C^{\infty}_{c}\left([0,\infty)\times\mathbb{R}^{d}\right)\right)$
on some complete probability space $\left(\Omega,\mathcal{F},P\right)$ such
that
$\displaystyle\mathbb{E}[W(\varphi)W(\psi)]$ (3.1) $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\varphi(s,y)\psi(s,y^{\prime})\gamma(y-y^{\prime})dydy^{\prime}ds:=\langle\varphi,\psi\rangle_{\mathfrak{H}}.$
By taking the Fourier transform, this can equivalently be written as 2.2. Note
that here $\gamma$ is not a function, and hence 3.1 should be understood as
$\mathbb{E}[W(\varphi)W(\psi)]=\int_{0}^{\infty}\int_{\mathbb{R}^{d}}\varphi(s,y)\left[\psi(s,\cdot)\ast\eta\right](y)dyds,$
(3.2)
where $\ast$ denotes the convolution. For the simplicity of our presentation,
we use notation $\gamma(y-y^{\prime})dydy^{\prime}$ in Sections 3.2 and 3.3
from which change of variable transformations are easier to follow.
We denote by $\mathfrak{H}$ the Hilbert space defined as the closure of
$C_{c}^{\infty}\left([0,\infty)\times\mathbb{R}^{d}\right)$ with respect to
the inner product (3.1). As a result, we obtain an isonormal process
$(W(\varphi),\varphi\in\mathfrak{H})$, which consists of a Gaussian family of
centered random variable such that, for every $\varphi,\psi\in\mathfrak{H}$,
$\mathbb{E}[W(\varphi)W(\Psi)]=\langle\varphi,\psi\rangle_{\mathfrak{H}}.$
The filtration $\mathbb{F}=(\mathcal{F}_{t})_{t\geq 0}$ associated to the
random noise $W$ is generated by random variables $W(\varphi)$ for which
$\varphi\in\mathfrak{H}$ has support contained in $[0,t]\times\mathbb{R}^{d}$.
Let us now define the stochastic integral with respect to $\dot{W}$. For every
random field $\\{X(s,y),s\geq 0,y\in\mathbb{R}^{d}\\}$ such that
$\mathbb{E}\|X\|_{\mathfrak{H}}^{2}=\mathbb{E}\int_{0}^{\infty}\int_{\mathbb{R}^{d}}X(s,y)X(s,y^{\prime})\gamma(y-y^{\prime})dydy^{\prime}<\infty,$
we can define the stochastic integral
$\int_{0}^{\infty}\int_{\mathbb{R}^{d}}X(s,y)W(ds,dy)$
in the sense of Dalang-Walsh (see, e.g. [5, 14]). It follows that we have the
Isometry
$\mathbb{E}\left[\int_{0}^{\infty}\int_{\mathbb{R}^{d}}X_{n}(s,y)W(ds,dy)\right]^{2}=\mathbb{E}\|X_{n}\|_{\mathfrak{H}}.$
(3.3)
Moreover, we have the following version of the Burkholder-Davis-Gundy
inequality: for any $t\geq 0$ and $p\geq 2$,
$\displaystyle\left|\left|\int_{0}^{\infty}\int_{\mathbb{R}^{d}}X(s,y)W(ds,dy)\right|\right|_{p}^{2}$
$\displaystyle\leq
c_{p}\int_{0}^{t}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\|X(s,y)X(s,y^{\prime})\|_{\frac{p}{2}}\gamma(y-y^{\prime})dydy^{\prime}ds.$
(3.4)
### 3.2. Auxiliary results
Set
$I(s)=\|g_{s}\|_{L^{1}(\mathbb{R}^{d})}+\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}g_{s}(y)g_{s}(y^{\prime})\gamma(y-y^{\prime})dydy^{\prime}.$
(3.5)
By taking the Fourier transform, we have
$I(s)=\|g_{s}\|_{L^{1}(\mathbb{R}^{d})}+\int_{\mathbb{R}^{d}}|\hat{g}_{s}(\xi)|^{2}\hat{\gamma}(d\xi)=\|g_{s}\|_{L^{1}(\mathbb{R}^{d})}+\int_{\mathbb{R}^{d}}e^{-2s\hat{f}(\xi)}\widehat{\gamma}(d\xi).$
###### Lemma 3.1.
Suppose that 2.5-2.6 hold for some $\beta_{0}$ and set
$\Upsilon(\beta)=\int_{0}^{\infty}e^{-\beta s}I(s)ds.$
Then $\Upsilon:(\beta_{0},\infty)\rightarrow(0,\infty)$ is well-defined and
non-increasing in $\beta$. Moreover, $\lim_{\beta\to\infty}\Upsilon(\beta)=0$.
###### Proof.
By Tonelli’s theorem, we have
$\int_{0}^{\infty}e^{-\beta
s}\int_{\mathbb{R}^{d}}e^{-2s\hat{f}(\xi)}\widehat{\gamma}(d\xi)ds=\int_{\mathbb{R}^{d}}\int_{0}^{\infty}e^{-\beta
s-2s\hat{f}(\xi)}ds\hat{\gamma}(d\xi)=\int_{\mathbb{R}^{d}}\frac{1}{\beta+2\hat{f}(\xi)}\hat{\gamma}(d\xi)$
for every $\beta$ such that $\beta+2\hat{f}(\xi)>0$. Since $I(s)$ is non-
negative, it is clear that $\Upsilon(\beta)$ is non-increasing which concludes
the proof. ∎
###### Remark 4.
We remark that for our purposes, it would actually suffice to consider
$\Upsilon_{T}(\beta)=\int_{0}^{T}e^{-\beta s}I(s)ds$
for each fixed $T<\infty$. Hence we could replace the condition 2.5 with
$\int_{0}^{T}\|g_{s}\|_{L^{1}(\mathbb{R}^{d})}ds<\infty$, cf. Remark 1.
The following Proposition is the main technical ingredient. The result follows
directly from [5, Lemma 15] adapted to our context.
###### Proposition 3.2.
Let $I(s)$ be given by 3.5 and suppose that 2.5-2.6 hold. Let $\iota>0$,
$\beta>\beta_{0}$, and $T\in(0,\infty)$ be fixed, and let $h_{n}$ be a
sequence of non-negative functions such that $\sup_{t\in[0,T]}h_{0}(t)<\infty$
and, for $n\geq 1$, we have
$h_{n}(t)\leq\iota\int_{0}^{t}h_{n-1}(s)e^{-\beta(t-s)}I(t-s)ds.$ (3.6)
Then the series
$H(\iota,p,t):=\sum_{n\geq 0}h_{n}(t)$ (3.7)
converges uniformly in $t\in[0,T]$.
### 3.3. Proof of Theorem 2.2
###### Proof of Theorem 2.2.
Let $T>0$ be fixed and finite. We consider the standard Picard iterations by
setting $u_{0}(t,x)=\int_{\mathbb{R}^{d}}G_{t}(x-y)u_{0}(y)dy$ and, for $n\geq
1$ and a given $\beta>\beta_{0}$,
$\begin{split}e^{-\beta t}u_{n+1}(t,x)&=e^{-\beta
t}u_{0}(t,x)+\int_{0}^{t}e^{-\beta
t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)b(s,u_{n}(s,y))dyds\\\
&+\int_{0}^{t}e^{-\beta
t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)\sigma(s,u_{n}(s,y))W(ds,dy),\hskip
14.22636ptt\geq 0,x\in\mathbb{R}^{d}.\end{split}$
We first prove that, for each $n\geq 1$, $u_{n}(t,x)$ is well-defined and
satisfies
$\sup_{t\in(0,T]}\sup_{x\in\mathbb{R}^{d}}\mathbb{E}\left[e^{-p\beta
t}\left|u_{n}(t,x)\right|^{p}\right]<\infty.$ (3.8)
We first note that if 3.8 holds for some $n\geq 0$, it follows from the
Lipschitz continuity of $\sigma$ that
$\begin{split}\mathbb{E}\left[e^{-p\beta
s}|\sigma(s,u_{n}(s,y))|^{p}\right]&\leq C\left[\mathbb{E}\left[e^{-p\beta
s}|\sigma(s,u_{0}(s,y))|^{p}\right]\right.\\\
&+\left.\mathbb{E}\left[e^{-p\beta
s}|u_{n}(s,y))-u_{0}(s,y)|^{p}\right]\right].\end{split}$
By the boundedness of $\sigma$ on compacts and 2.3, we get
$\sup_{s\in[0,T],y\in\mathbb{R}^{d}}\mathbb{E}\left[e^{-p\beta
s}|\sigma(s,u_{0}(s,y))|^{p}\right]=\sup_{s\in[0,T],y\in\mathbb{R}^{d}}e^{-p\beta
s}|\sigma(s,u_{0}(s,y))|^{p}<\infty$
and consequently,
$\mathbb{E}\left[e^{-p\beta s}|\sigma(s,u_{n}(s,y))|^{p}\right]\leq
C\left(1+\sup_{s\in[0,T],y\in\mathbb{R}^{d}}\mathbb{E}\left[e^{-p\beta
s}|u_{n}(s,y)|^{p}\right]\right)<\infty.$
This in turn gives us, by Hölder’s inequality, that
$\sup_{s\in[0,T]}\sup_{y,y^{\prime}\in\mathbb{R}^{d}}e^{-\beta
s}\left|\left|\sigma(s,u_{n}(s,y))\sigma(s,u_{n}(s,y^{\prime}))\right|\right|_{\frac{p}{2}}<\infty.$
Exactly the same way, we obtain
$\sup_{s\in[0,T]}\sup_{y\in\mathbb{R}^{d}}\mathbb{E}\left[e^{-p\beta
s}|b(s,u_{n}(s,y))|^{p}\right]<\infty.$
In view of 3.3 and 3.4 together with the boundedness of $u_{0}(t,x)$ and
$|G_{t}(y)|\leq g_{t}(y)$, applying the Minkowski integral inequality leads to
$\displaystyle\mathbb{E}\left[e^{-p\beta
t}\left|u_{n+1}(t,x)\right|^{p}\right]$ $\displaystyle\leq$ $\displaystyle
C\left(\left[e^{-\beta t}u_{0}(t,x)\right]^{p}+\Big{\|}\int_{0}^{t}e^{-\beta
t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)b(s,u_{n}(s,y))dyds\Big{\|}_{p}^{p}\right.$
$\displaystyle+$ $\displaystyle\left.\Big{\|}\int_{0}^{t}e^{-\beta
t}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)G_{t-s}(x-y^{\prime})\right.$
$\displaystyle\phantom{kukkuu}\left.\times\sigma(s,u_{n}(s,y))\sigma(s,u_{n}(s,y^{\prime}))\gamma(y-y^{\prime})dy^{\prime}dyds\Big{\|}_{\frac{p}{2}}^{\frac{p}{2}}\right)$
$\displaystyle\leq$ $\displaystyle
C\left[1+\left(\int_{0}^{t}e^{-\beta(t-s)}\int_{\mathbb{R}^{d}}g_{t-s}(x-y)e^{-\beta
s}\|b(s,u_{n}(s,y))\|_{p}dyds\right)^{p}\right.$ $\displaystyle+$
$\displaystyle\left.\left(\int_{0}^{t}e^{-\beta(t-s)}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}g_{t-s}(x-y)g_{t-s}(x-y^{\prime})\right.\right.$
$\displaystyle\phantom{kukkuu}\left.\left.\times e^{-\beta
s}\left|\left|\sigma(s,u_{n}(s,y))\sigma(s,u_{n}(s,y^{\prime}))\right|\right|_{\frac{p}{2}}\gamma(y-y^{\prime})dy^{\prime}dyds\right)^{\frac{p}{2}}\right]$
$\displaystyle\leq$ $\displaystyle
C\left[1+\left(\int_{0}^{t}e^{-\beta(t-s)}I(t-s)ds\right)^{p}+\left(\int_{0}^{t}e^{-\beta(t-s)}I(t-s)ds\right)^{\frac{p}{2}}\right]$
$\displaystyle\leq$ $\displaystyle
C\left(1+\Upsilon^{p}(\beta)+\Upsilon^{\frac{p}{2}}(\beta)\right).$
which is finite for $\beta>\beta_{0}$. Since $u_{0}(t,x)$ is uniformly bounded
on $[0,T]\times\mathbb{R}^{d}$, 3.8 for all $n\geq 0$ thus follows from
induction and hence $u_{n+1}$ is well-defined. The same arguments applied to
$H_{n}(t):=\sup_{x\in\mathbb{R}^{d}}\left[e^{-\beta
pt}\mathbb{E}\left[\left|u_{n+1}(t,x)-u_{n}(t,x)\right|^{p}\right]\right]^{\frac{1}{p}}$
gives us
$\displaystyle H_{n}(t)$ $\displaystyle\leq
C\left[\int_{0}^{t}e^{-\beta(t-s)}\int_{\mathbb{R}^{d}}g_{t-s}(x-y)H_{n-1}(s)dyds\right.$
$\displaystyle+\left(\int_{0}^{t}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}e^{-2\beta
t}g_{t-s}(x-y)g_{t-s}(x-y^{\prime})\left|\left|\sigma(s,u_{n}(s,y))-\sigma(s,u_{n-1}(s,y))\right|\right|_{p}\right.$
$\displaystyle\phantom{kukkuu}\times\left.\left.\left|\left|\sigma(s,u_{n}(s,y^{\prime}))-\sigma(s,u_{n-1}(s,y^{\prime}))\right|\right|_{p}\gamma(y-y^{\prime})dy^{\prime}dyds\right)^{\frac{1}{2}}\right]$
$\displaystyle\leq
C\left[\int_{0}^{t}e^{-\beta(t-s)}I(t-s)H_{n-1}(s)ds+\left(\int_{0}^{t}e^{-2\beta(t-s)}I(t-s)H^{2}_{n-1}(s)ds\right)^{\frac{1}{2}}\right].$
Here Hölder’s inequality gives
$\left(\int_{0}^{t}e^{-2\beta(t-s)}I(t-s)H^{2}_{n-1}(s)ds\right)^{\frac{1}{2}}\leq
C\int_{0}^{t}e^{-\beta(t-s)}I(t-s)H_{n-1}(s)ds$
leading to
$H_{n}(t)\leq C\int_{0}^{t}e^{-\beta(t-s)}I(t-s)H_{n-1}(s)ds.$
By noting that $b(s,u_{0}(s,y))$ and $\sigma(s,u_{0}(s,y))$ are uniformly
bounded over $[0,T]\times\mathbb{R}^{d}$, we obtain from the above
computations that $\sup_{s\in[0,T]}H_{0}(s)<\infty$. Consequently, it follows
from Proposition 3.2 that $\sum_{n\geq 1}H_{n}(t)$ converges uniformly on
$[0,T]$, and hence the sequence $u_{n}$ converges in $L^{p}(\Omega)$ to some
process $u\in L^{p}(\Omega)$, uniformly on $[0,T]\times\mathbb{R}^{d}$. From
the uniform convergence one can deduce further that $u$ satisfies 2.4, and
thus we have obtained the existence of the mild solution. Proving the
uniqueness in a similar fashion and noting that $T>0$ was arbitrary concludes
the proof. ∎
## 4\. Concluding remarks
In this article we have considered general SPDEs 2.1 and showed that one can
obtain existence and uniqueness of the mild solution by classical arguments,
initiated by Dalang [5], for a very large class of operators $L_{x}$. Our
condition 2.6 is similar to the Dalang condition 1.2 for the stochastic heat
equation and actually, our examples in Section 2.1 reveals that 1.2 is indeed
sufficient for a very large class of operators $L_{x}$. By carefully examining
the above proofs, we observe that even more is true. Indeed, condition 2.6
could be replaced with a condition $|G_{s}(x)|\leq g_{s}(x)$ and, for all
$T>0$,
$\int_{0}^{T}\int_{\mathbb{R}^{d}}e^{-\beta
s}[\hat{g}_{s}(\xi)]^{2}\hat{\gamma}(d\xi)ds<\infty.$
This formulation allows to consider even more general operators $L_{t}$ in the
time variable instead of considering merely $L_{t}=\partial_{t}$. In
particular, with this formulation one can easily recover the existence and
uniqueness result of [5] for the stochastic wave equation (in one or two
dimensions) with $L_{t}=\partial_{tt}$. As another example, we can obtain the
existence and uniqueness result in the case of the fractional power of the
full heat operator $(\partial_{t}-\Delta)^{s}$, studied for instance in [12].
## References
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* [3] A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni. Stratified Lie groups and potential theory for their sub-Laplacians. Springer Science & Business Media, 2007.
* [4] L. Chen and R. Dalang. Moments, intermittency and growth indices for nonlinear stochastic fractional heat equation. Stoch. Partial Differ. Equ. Anal. Comput., 3(3): 360–397, 2015.
* [5] R. Dalang. Extending the Martingale Measure Stochastic Integral With Applications to Spatially Homogeneous S.P.D.E.’s. Electron. J. Probab. Volume 4, paper no. 6, 29 pp., 1999.
* [6] L. Debbi and M. Dozzi. On the solutions of nonlinear stochastic fractional partial differential equations in one spatial dimension. Stoch. Proc. Appl., 115: 1761–1781, 2005.
* [7] L. Escauriaza. Bounds for the fundamental solutions of elliptic and parabolic equations: In memory of eugene fabes. Communications in Partial Differential Equations, 25.5-6 (2000): 821–845.
* [8] A. Friedman. Partial differential equations of parabolic type. Courier Dover Publications, 2008.
* [9] A. Friedman. Stochastic differential equations and applications. Vol. 1. Probability and Mathematical Statistics, Vol. 28. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975.
* [10] M. Foondun and D. Khoshnevisan. On the stochastic heat equation with spatially-colored random forcing. Transactions of the American Mathematical Society, 365: 409–458, 2013.
* [11] A.E. Kogoj and E. Lanconelli. An invariant Harnack inequality for a class of hypoelliptic ultraparabolic equations. Mediterranean Journal of Mathematics 1.1 (2004): 51–80.
* [12] K. Nyström and O. Sande. Extension properties and boundary estimates for a fractional heat operator. Nonlinear Analysis 140 (2016): 29–37.
* [13] C. Villani, Hypocoercivity. No. 949-951. American Mathematical Soc., 2009.
* [14] J. B. Walsh. An Introduction to Stochastic Partial Differential Equations. In: École d’été de probabilités de Saint-Flour, XIV—1984, 265-439. Lecture Notes in Math. 1180, Springer, Berlin, 1986.
|
# Resolution Complete In-Place Object Retrieval
given Known Object Models
Daniel Nakhimovich, Yinglong Miao, and Kostas E. Bekris The authors are with
the Dept. of Computer Science, Rutgers, New Brunswick, NJ. Email: {d.nak,
<EMAIL_ADDRESS>The work is partially supported by NSF awards
1934924 and 2021628. The opinions expressed here are those of the authors and
do not reflect the positions of the sponsor.
###### Abstract
This work proposes a robot task planning framework for retrieving a target
object in a confined workspace among multiple stacked objects that obstruct
the target. The robot can use prehensile picking and in-workspace placing
actions. The method assumes access to 3D models for the visible objects in the
scene. The key contribution is in achieving desirable properties, i.e., to
provide (a) safety, by avoiding collisions with sensed obstacles, objects, and
occluded regions, and (b) resolution completeness ($\tt RC$) - or
probabilistic completeness ($\tt PC$) depending on implementation - which
indicates a solution will be eventually found (if it exists) as the resolution
of algorithmic parameters increases. A heuristic variant of the basic $\tt RC$
algorithm is also proposed to solve the task more efficiently while retaining
the desirable properties. Simulation results compare using random picking and
placing operations against the basic $\tt RC$ algorithm that reasons about
object dependency as well as its heuristic variant. The success rate is higher
for the $\tt RC$ approaches given the same amount of time. The heuristic
variant is able to solve the problem even more efficiently than the basic
approach. The integration of the $\tt RC$ algorithm with perception, where an
RGB-D sensor detects the objects as they are being moved, enables real robot
demonstrations of safely retrieving target objects from a cluttered shelf.
## I INTRODUCTION
Robotic manipulation has the potential of being integrated into daily lives of
people, such as in household service areas [1, 2]. A useful skill for such
household settings involves the retrieval of a target object from a confined
and cluttered workspace, such as a fridge or a shelf, which may also require
the rearrangement of other objects in the process. In this context, it is
important to consider how to safely retrieve objects while minimizing the time
spent or the amount of pick and place operations, so as to assist humans
efficiently.
Figure 1: (Top) Setup for the real demonstration using an RGB-D sensor,
robotiq gripper, and Yaskawa Motoman robot to retrieve the target bottle.
(Bottom Left) The camera view in which objects are occluded. (Bottom Right)
The corresponding voxel map.
One of the challenging aspects of these problems that requires explicit
reasoning relates to heavy occlusions in the scene, as the sensor is often
mounted on the robot and has limited visibility. These visibility constraints
complicate the task planning process, as rearranging one object can limit
placements for others and can introduce new occlusions. Moreover, real-world
scenes in household setups are often unstructured and involve objects with
complex spatial relationships, such as objects stacked on each other.
Many previous efforts on object retrieval have focused on cases where blocking
objects are extracted from the workspace [3, 4], which simplifies the
challenge as it does not require identifying temporary placement locations for
the objects within the confined space. In-place rearrangement has been
considered in some prior efforts [5]. While this prior method is efficient, it
is not complete as it limits the reasoning on the largest object in the scene
to analyze object traversability [4]. Alternatives use machine learning to
guide the decision making [6, 7, 8], which is an exciting direction but does
not easily allow for performance guarantees, such as resolution completeness.
Setups where object stacking arise have received less attention and most
solutions that do consider stacking are dependent on machine learning for
reasoning [9, 10, 11]. Some works have proposed testbeds [12] that can help
evaluate solutions in this domain.
This work focuses on object retrieval in clutter where occlusions arise and
objects may be initially stacked under the assumption of known object models
(e.g. Figure 2). It aims at a theoretical understanding to show the algorithm
has safety and $\tt RC$ guarantees. A heuristic variant improves the practical
efficiency. Key features of the proposed $\tt RC$ framework are the following:
* $\bullet$
it employs an adaptive dependency graph data structure inspired by solutions
in object rearrangement with performance guarantees [13] that express a larger
variety of object relationships than previously considered (namely occlusion
dependencies);
* $\bullet$
it computes the occlusion volume of detected objects as a heuristic to inform
the planning process;
* $\bullet$
it reasons about the collision-free placement of objects in the confined
workspace efficiently by utilizing a voxelized representation of the space;
* $\bullet$
it achieves $\tt RC$ (or $\tt PC$) depending on the implementation of the
underlying sampling subroutines;
* $\bullet$
it provides an early termination criterion when a solution cannot be found for
the given resolution.
Simulation results, using a model of a Yaskawa Motoman manipulator for
rearranging objects on a tabletop as shown in Fig. 1, evaluate the proposed
$\tt RC$ framework against a baseline using random picking and placing
operations. Both variants of the $\tt RC$ reason about object dependencies.
One doesn’t use heuristics and one is heuristically guided while retaining
$\tt RC$. Both of the $\tt RC$ approaches outperform the baseline. The
heuristically guided solution is able to solve the problem more efficiently
than the basic $\tt RC$ solution. The integration of the proposed approach
with perception, where an RGB-D sensor detects the objects as they are being
moved, provides real robot demonstrations of safe object retrieval from a
cluttered shelf.
## II RELATED WORK
Some works on object retrieval rely on geometric analysis of object occlusion
[3, 4]. They provide theoretical insights but frequently do not limit actions
to in-place rearrangement of blocking objects. Specifically, one method
constructs a dependency graph taking into account objects that jointly occlude
a region and objects that block others [3]. The occlusion volume is used to
estimate belief regarding the target object position and helps to construct an
optimal A* algorithm. An alternative constructs a Traversability graph
(T-graph) [4], where the edges encode if the largest object in the scene can
be moved between two poses. It then constructs an algorithm to extract the
target object, but is limited as the traversability edges are too
constraining. The POMDP formulation is popular for the task [14, 15], which
allows the application of general POMDP solvers. The POMDP formulation was
also adopted by the work that formalizes object retrieval in unstructured
scenes as ”mechanical search” [16].
Alternatives rely on learning-based methods to solve such challenges, such as
reinforcement learning [6] or target belief prediction [7, 9, 17]. They report
good performance but do not provide theoretical guarantees given the black-box
nature of the solutions. In particular, a reinforcement learning solution [6]
uses the rendered top-down projections of the scene to predict the target
poses. A recent follow-up effort [7] on previous work [17] estimates the 1D
position belief of the target object on the shelf via machine learning. It
then constructs a policy based on the distribution change after applying
pushing and suction actions. It incorporates stacking and unstacking actions,
where object stacking is represented by a tree structure. Other works such as
[18] utilize learning for planning grasps to greedily empty bins of complex
and novel objects.
A related work [19] proposes a complete framework to safely reconstruct all
objects in the scene amidst object occlusions. Nevertheless, object retrieval
may not require reconstructing all objects and requires a search procedure
that is more task-driven for efficiency. There are also previous works [20,
19] that construct a voxelization of the environment to model object
occlusions, similar to the current work. This representation is used to
compute an object’s occlusion volume, which provides heuristic guidance.
Object spatial relationships are often represented by scene graphs [10, 11],
or implicitly in machine learning solutions [16, 21, 22].
What stands out in this work is that it proposes a general template for a $\tt
RC$ or $\tt PC$ approach to task retrieval in occluded environments that only
relies on basic motion and perception primitives. This modular nature allows
for quick sim-to-real transfer and passive performance improvement as the
primitives are improved over time. Furthermore, an efficient implementation is
demonstrated utilizing a voxelized representation of the environment for quick
collision filtering of object placements as well as providing an effective
heuristic to rank object manipulations. Thus, the framework enables effective
in-workspace manipulation.
## III PROBLEM STATEMENT
Consider an environment with a set
$\mathbb{S}=\\{o_{1},...,o_{n}\\}\subset\mathbb{O}$ of $n$ objects for which
there are available 3D models. The objects are stably resting on a support
surface. Objects are allowed to be initially stacked and occlude each other
from the camera view.
The robot has one fixed RGB-D sensor at its disposal. Discovered objects are
those recognized given the observation history. An objects is assumed to be
recognized once an image segmentation process recognizes it as an individual
object in the observed image. Similarly, a perception method for detecting the
target object once observed is assumed. The region of the workspace occluded
by object $o_{i}$ at pose $s_{i}$ is denoted as $O_{i}(s_{i})$. Similarly the
space uniquely occluded by object $o_{i}$ at pose $s_{i}$, called the direct
occlusion space, is denoted as $\tilde{O}_{i}(s_{i})$. The proposed algorithms
gradually removes occlusions and recognizes objects. A motion planner is used
to plan pick-and-place actions.
While objects can start out stacked, they are not re-stacked and are only
placed on the ground surface during actions. While objects can start out
stacked, no reasoning about stability and ability to re-stack objects is
considered. Thus, once an action to pick up a stacked object is taken, that
object will only be placed on the ground surface. For further assumptions and
required properties of the motion planner see section V.
The objective for the object retrieval task is to determine a sequence of pick
and place actions in order to discover and subsequently retrieve the target
object; the target need not be directly visible or pickable from the robot’s
sensors. The corresponding solution should provide desirable guarantees: (a)
safety, by avoiding collisions with sensed obstacles and objects as well as
occluded regions, and (b) resolution completeness (RC) - or alternatively
probabilistic completeness, depending on the implementation of the underlying
motion planner, grasping process and object placement sampling. The
optimization objective is to minimize the number of performed actions until
the target object is retrieved.
## IV METHOD
The proposed pipeline is detailed in Algorithm 1. First a voxelized
representation of the scene and a dependency graph are computed (lines 3,4,5),
which are detailed in subsection IV-A. The dependency graph contains a belief
state of the current scene based on visibility and reachability constraints.
All the sinks of this directed graph represent likely pickable objects. The
ranks are described later (see subsection IV-C). If the target object is
pickable then it is retrieved and the pipeline is terminated line 6.
Otherwise, a placement is planned (see subsection IV-B) one at a time for each
object in a shuffled order biased by the ranks (TryMoveOne line 7). The first
successful plan is executed.
If no placement is found for any of the pickable objects, then a fallback
procedure is called to try and pick one object and move it temporarily out of
view of the camera; the first successful plan is executed, the scene is re-
sensed, and a new placement is sampled for the object or the object is simply
put back at the same spot (MoveOrPlaceback line 10). At this stage, the
pipeline could restart if a new object was discovered (lines 12-15). Otherwise
the same set of pickable objects are tested for new placements in the scene
(line 16). If these two operations of “moving an object to look behind
it”(MoveOrPlaceback) followed by retrying to “move one of the pickable objects
to a new spot”(TryMoveOne) fail for all objects, then the pipeline can return
and report failure for the current resolution (line 22). If the sequence of
operations succeeds, then the pipeline can restart (line $19\rightarrow 2$).
Algorithm 1 $\tt RC\\_Pipeline$($\tt target$)
1:$\tt failure\leftarrow false$
2:while $\tt failure=false$ do
3: $\tt space\leftarrow UpdateVoxelsFromImage()$
4: $\tt dg\leftarrow{\tt DepGraph(space)}$
5: $\tt sinks,ranks\leftarrow RankSinks(target,dg)$
6: if $\tt target\in sinks$ then $\tt break$
7: if $\tt{\tt TryMoveOne(sinks,ranks)}=false$ then
8: $\tt failure\leftarrow true$
9: for $\tt sink\in sinks$ do
10: if $\tt{\tt MoveOrPlaceback(sink)}=false$ then
11: $\tt continue$
12: $\tt space\leftarrow UpdateVoxelsFromImage()$
13: if $\tt DidDiscoverObject(space)$ then
14: $\tt failure\leftarrow false$
15: $\tt break$
16: if $\tt{\tt TryMoveOne(sinks,\emptyset)}=false$ then
17: $\tt continue$
18: $\tt failure\leftarrow false$
19: $\tt break$
20:if $\tt not~{}failure$ then
21: $\tt Retrieve(target)$
22:$\tt return~{}failure$
\begin{overpic}[width=145.69568pt,trim=0.0pt 140.525pt 40.15pt
22.08249pt,clip]{figures/sim_exp/step1_ws.png}
\put(3.0,10.0){\huge$\displaystyle 1)$} \end{overpic}
\begin{overpic}[width=145.69568pt,trim=0.0pt 140.525pt 40.15pt
22.08249pt,clip]{figures/sim_exp/step2_ws.png}
\put(3.0,10.0){\huge$\displaystyle 2)$} \end{overpic}
\begin{overpic}[width=145.69568pt,trim=0.0pt 140.525pt 40.15pt
22.08249pt,clip]{figures/sim_exp/step3_ws.png}
\put(3.0,10.0){\huge$\displaystyle 3)$} \end{overpic}
\begin{overpic}[width=145.69568pt,trim=0.0pt 140.525pt 40.15pt
22.08249pt,clip]{figures/sim_exp/step4_ws.png}
\put(3.0,10.0){\huge$\displaystyle 4)$} \end{overpic}
\begin{overpic}[width=145.69568pt,trim=0.0pt 140.525pt 40.15pt
22.08249pt,clip]{figures/sim_exp/step5_ws.png}
\put(3.0,10.0){\huge$\displaystyle 5)$} \end{overpic}
\begin{overpic}[width=145.69568pt,trim=0.0pt 140.525pt 40.15pt
22.08249pt,clip]{figures/sim_exp/step6_ws.png}
\put(3.0,10.0){\huge$\displaystyle 6)$} \end{overpic}
\begin{overpic}[width=106.23477pt]{figures/sim_exp/step1_dg_c.png}
\put(-1.0,5.0){\huge a)} \end{overpic}
\begin{overpic}[width=106.23477pt]{figures/sim_exp/step2_dg.png}
\put(-1.0,5.0){\huge b)} \end{overpic}
\begin{overpic}[width=106.23477pt]{figures/sim_exp/step3_dg.png}
\put(-1.0,5.0){\huge c)} \end{overpic}
\begin{overpic}[width=106.23477pt]{figures/sim_exp/step4_dg.png}
\put(-1.0,5.0){\huge d)} \end{overpic}
Figure 2: Images (1)-(6) show a simulated experiment from initial
configuration to final one action at a time with corresponding camera views in
the top left. The corresponding generated dependency graphs transitioning
between the images (1)-(2), (2)-(3), (3)-(4), and (4)-(5) are shown in images
(a)-(d). The colors of nodes corresponds with the objects in the scene and the
red object (labeled ‘T’) is the target object for the trial. The last graph
between images 5-6 is not shown since it is trivial having no dependencies
between any objects.
### IV-A Voxel Map and Dependency Graph
A 3D occlusion voxel grid within the workspace is constructed from RGB-D
images. First the point cloud (in world frame) is generated using the RGB-D
image and the inverse of the camera projection. These points are down-sampled
into a voxelization of the scene. Given segmentation image information, each
object is associated with a portion of the voxel grid. Object geometry is used
to label voxels as occupied and remaining associated voxels as occluded. The
occluded regions of objects may intersect when jointly occluded.
The dependency graph is a directed graph where each node represents a visible
(or target) object and a labeled edge $(o_{i},o_{j},r)$ represents a relation
$r$ from object $o_{i}$ to $o_{j}$ that necessitates $o_{j}$ to be picked and
placed before $o_{i}$ could be picked. Valid relations in this work include
“below”, “grasp blocked by”, and – for the prediction of the target object –
“hidden by”. See Figure 2 for a sequence of such dependency graphs generated
during an example experiment.
Object x is defined to be “below” object y ($x\xrightarrow{below}y$) if object
x touches object y and the z-coordinate of the center of mass of x is less
than that of y. Note that this isn’t guaranteed to capture all intuitive cases
of one object being below another for non-convex objects. This relation is
computed using object models and poses given by the perception system.
Object x has its “grasp blocked by” y ($x\xrightarrow{blocked}y$) if there are
no collision free grasp poses for object x and the arm is in collision with y
for one or more valid grasp poses. (Grasp poses are sampled and tested by
inverse kinematics (IK) for discovered objects) Grasp poses are sampled using
inverse kinematics (IK) to discovered objects. (or, if the grasping pose
collides with objects, an edge to each collided object is added) If there
exists a collision free grasp for an object, no blocking edges are added;
otherwise, an edge to each object that has a collision with the arm is added.
Note that although this relation guarantees that the source object blocks the
target, it doesn’t capture all such reachability dependencies. This however is
not an issue for completeness as Algorithm 1 will eventually try to grasp all
objects in the case of motion planning failure.
The target object t is possibly “hidden by” x ($t\xrightarrow{hidden}x$) if
the target isn’t sensed in the scene and object x is touching the table. This
relation is used to keep track of the belief state of where the target is.
Each edge is assigned a probability based on the volume of the occluded space
behind object x (see subsection IV-C).
### IV-B Placement Sampling
A valid placement is one that doesn’t collide with another object or any
undiscovered area of the workspace. Instead of randomly sampling x,y
coordinates for an object and checking for collision at that point, we create
a grid matching the horizontal extents of the workspace and add a collision
mask which is the shadow of the object occupancy and occlusion voxel grid
looking from a birds eye view. This mask is then convolved with the shadow of
the object that is to be placed. The occupied pixel indices indicate
collision-free placements and are converted to world coordinates. Object
orientations can be enumerated by rotating the object shadow.
### IV-C Target Object Prediction
Intelligent object location prediction is achieved by applying a heuristic
which ranks the pickable objects determined by the dependency graph (line 4 in
Algorithm 1). This is done by augmenting the dependency graph edges with a
weight $p\in(0,1]$ estimating the probability for the relation
$o_{i}\overset{r}{\longrightarrow}o_{j}$ to be true. To rank a pickable
object, the sum of products of edge weights of all simple paths between the
target and the object is computed. When sampling from the list of pickable
objects, this rank is used as a probability weight.
For the “below” relation $p=1$ since object segmentation is assumed to be
reliable. For the “grasp blocked by” relation p is equal to the fraction of
total sampled grasps for which that object is in collision with the arm. Note,
the weights of grasp blocking edges coming out of any object need not add to
one (hence don’t truly represent probabilities) since the arm could be
colliding with multiple objects for any single grasp. For the “hidden by”
relation, the goal is to encourage knowledge gain of the environment. This is
done by normalizing the volume of the direct occlusion region of each stack of
objects and assigning the inverse as the probability estimate that the target
is hidden behind each stack. This heuristic biases the pipeline towards
discovering large volumes of occluded workspace.
From an algorithmic point of view, there is technically no reason to normalize
the output of the heuristic, however, representing the heuristics as
probabilities is insightful since - without prior knowledge - the probability
the hidden object to be in a larger volume is larger than the probability of
it being in a smaller volume. Furthermore, modeling the dependency graph edges
with probabilities as opposed to non-normalized weights is conducive for
exploring future work which might seek to combine the probabilities based on
the proposed volumetric-heuristics with priors based on the semantics of the
objects involved or from an additional human instruction (see section VII
work).
## V RESOLUTION COMPLETENESS
Given a (formally) complete motion planner (finds solution in finite time if
exists), a continuous space grasp sampler, and continuous placement sampler,
the proposed pipeline would be $\tt PC$ Given a $\tt RC$ motion planner, a
discrete space grasp sampler, and discrete placement sampler, the proposed
pipeline would be $\tt RC$.
To show the $\tt PC$ or $\tt RC$ of the algorithm proposed in this work, a
simpler version of the algorithm is analysed first. Without loss of
generality, the actual proposed algorithm will be likewise proven complete.
Consider a much simpler algorithm that, at every iteration, tries to pick an
object at random and, if is not the target object, subsequently place it
randomly in the explored region of the workspace; call this RAND-ACT.
###### Lemma V.1
RAND-ACT is $\tt PC$ or $\tt RC$ (depending on the planning and sampling
subroutines).
###### Proof:
At every iteration, RAND-ACT attempts to perform a random action.
Consequently, RAND-ACT executes a random walk on the space of all actions.
Since pick and place actions are reversible, if a sequence of such valid
actions exists, this algorithm will eventually perform it (or an augmented
version of the sequence - i.e. placing an object back to where it was picked
from) in the limit (or finite time for RC subroutines). ∎
###### Corollary V.2
Algorithm 1 is $\tt PC$ or $\tt RC$ (depending on the planning and sampling
subroutines)).
###### Proof:
Indeed Algorithm 1 is really a fancy implementation of RAND-ACT. At every
iteration, the dependency graph is used to identify (and heuristically rank)
the currently pickable objects. One of these is chosen randomly for a pick and
place action via the $\tt TryMoveOne$ subroutine. The $\tt MoveOrPlaceback$
subroutine acts as a fallback in case there are no new discovered placements
found; it delays placement sampling till after the environment is re-sensed
with the picked object moved out of the way. Notice further, that even if the
pickable objects are sampled weighted according to their ranking rather than
uniformly, the action space is still explored entirely because each action has
a positive probability of being sampled. Thus, w.l.o.g. Algorithm 1 is $\tt
PC$ or $\tt RC$ as well. ∎
Failure Detection: The implementation in this work uses the $\tt RC$ approach.
In addition to $\tt RC$, Algorithm 1 takes a step closer towards achieving
general completeness by actually detecting certain unsolvable cases within the
completeness constraints of the motion and sampling subroutines. The
detectable unsolvable instances for which the algorithm will return failure in
finite time are as follows.
* $\bullet$
No object can be grasped. This could happen if two objects each block the
grasp of the other.
* $\bullet$
No objects can be placed anywhere (except its current spot). This could happen
in a highly cluttered scene where the only valid placement for each object is
to just put it back where it was.
Thus, Algorithm 1 has stronger guarantees than $\tt RC$ but is not formally
complete since it may run forever by juggling two objects between two
placements.
A Caveat: A fundamental assumption in the argument presented is that actions
are reversible. This is not always true in practice depending on the
implementation of the sampling subroutines. And indeed, the implementation of
the placement sampling process proposed in subsection IV-B implies
irreversible actions for scenes with stacked objects because it does not
consider the possibility of re-stacking objects. Thus, the proposed pipeline
(as implemented) is only resolution complete on the any sub-task where the
objects are no longer stacked. Implementation of stacking actions is planned
for future work.
## VI EXPERIMENTS
Figure 3: Execution on the real robot : (a) Initial scene where the red bottle
is hidden. (b) The robot moves the yellow bottle, which occludes the most
space. (c) The robot moves the second yellow bottle, revealing the red bottle.
(d and e) The robot moves the green and the blue bottles to reach the red
bottle. (f) Target is now reachable.
Simulated experiments and the real demonstrations are performed with the
Yaskawa Motoman sda10f, with a robotiq 85 gripper attached on the right arm.
The simulated trials are randomly generated by picking random objects,
dimensions, and collision-free placements within the specified workspace. 20
scenes with each of 6, 8, 10, 12, and 14 objects were used, all of which
contain objects occluded from the camera. Each of the 100 trials is a unique
scene. The target object in each scene was selected to be the hidden object
with the most objects above it, if any.
All tested algorithms were given 20 minutes to run before being terminated. A
trial run is considered successful only if the target object was retrieved
within the time limit. Discovering but failing to pick up the object was still
considered a failure.
Comparisons of the success rate, number of actions for solved trials, total
run-time for solved trials, and number of timed out trials are shown in Figure
4 for 3 algorithms. The algorithms compared are the baseline random action
approach (blue in figure), the proposed resolution complete pipeline without
the object ranking heuristic (orange) and with it (green). For the resolution
complete approaches timing-out is not the only failure mode since they could
detect certain infeasible problems (section V: Failure Detection)
The success rate of the resolution complete approaches is higher than that of
the random baseline. Although its not directly apparent from the plotted
results in Figure 4, the resolution complete approaches (as expected) always
found a solution when the random baseline found a solution; however in one
such trial the resolution complete approach without heuristic exceeded the 20
min time-limit. It is also clear that the heuristic approach has better
success than the non heuristic approach even though they are both complete.
Looking at the data for timed-out experiments, it becomes clear that the
increased success of the heuristic approach is due to timing out less
frequently. This also coincides with the data showing that the heuristic
approach overwhelmingly finds solutions faster and with fewer object
manipulations. In fact, while the non-heuristic RC approach started timing out
linearly with increase in the number of objects, the heuristic approach had
virtually no issue until the scenes got very cluttered with 14 objects.
It is clear that for all methods, success rate starts dropping off
significantly at around 14 objects. This marks the difficulty level for the
given industrial Motoman robot and the workspace. A more compact robot with a
streamlined end-effector (such as the “bluction” tool [7]) could scale to more
cluttered scenes.
Figure 4: On top are shown the graphs of the: success rate (left) and number
of timed out trials (right). On the bottom are the number of actions and the
total runtime for the subset of trials in which all algorithms were
successful.
### VI-A Integration with Perception & Real Robot Demonstration
The pipeline is directly transferable to scenarios on the real robot to
retrieve a target red bottle from a cluttered shelf. Due to time constraints,
a simple implementation of a perception system is used which only segments and
detects colored cylinders without stacking. Despite the simplifications, a
scene with significant object occlusion is still demonstrated with a
successful retrieval. The proposed pipeline (with heuristic) is run online and
communicates with the robot controller and the RGBD camera for execution and
sensing. The camera extrinsic matrix is estimated by a classical robot-camera
calibration procedure using ArUco markers [23]. For object recognition, the
perception component is implemented via plane fitting, DBSCAN segmentation
[24], and cylinder fitting using Open3D [25]. The plane fitting algorithm
extracts the boundaries of the workspace, which is used to construct the
collision geometries in MoveIt [26]. The inliers of each segmented cylinder
are used to produce the segmentation mask for the RGBD image, which is used to
label occlusion correspondence for each object. To ensure safety, additional
cubic collision geometries are added to the planning scene to avoid collisions
between the robot and the camera. 111Videos can be found at
https://sites.google.com/scarletmail.rutgers.edu/occluded-obj-retrieval.
Extensive experiments of the proposed pipeline were not performed on the real
robot but the demonstration presented was performed a few times and the
pipeline was observed to have qualitatively similar performance as in
simulated experiments; however, calibration and perception issues were
observed to lead to pipeline failure.
## VII DISCUSSION
It’s worth mentioning that the physical execution accounts for over 60% of
time used for the trials. This shows that there could be room for performance
improvement by performing scene perception asynchronously, since a lot can
still be sensed while the robot is moving. Further performance improvement can
be found by parallelizing the planning of picks and placements for multiple
objects as well.
While this work applies heuristics for selecting objects based on the
occlusion volume, additional information regarding effective placements can
also improve practical performance. In order to solve a larger variety of
problems it would be useful to adapt the placement primitive to allow placing
objects on top of others when there is limited space on the workspace surface.
Another direction is to integrate the task planner with human instructions.
For instance, it would be helpful to use human language to identify the target
as well as influence the search at some regions over others. Additional
heuristics can also be obtained from semantic reasoning of the scene when
objects of the same category tend to be placed closer [27]. Since current
experiments only include simple geometries, such as cylinders and rectangular
prisms, future work can investigate more complex objects where state-of-the-
art perception algorithms are necessary. This would also be necessary for
realistic human-robot integration.
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|
# Disorder induced transition from type-I to type-II superconductivity in the
Dirac semimetal PdTe2
M. V. Salis<EMAIL_ADDRESS>J. P. Lorenz Y. K. Huang A. de Visser
<EMAIL_ADDRESS>Van der Waals - Zeeman Institute, University of Amsterdam,
Science Park 904, 1098 XH Amsterdam, The Netherlands
###### Abstract
We report a doping study directed to intentionally induce disorder in PdTe2 by
the isoelectronic substitution of Pt. Two single-crystalline batches
Pd1-xPtxTe2 have been prepared with nominal doping concentrations $x=0.05$ and
$x=0.10$. Sample characterization by energy dispersive x-ray spectroscopy
(EDX) revealed Pt did not dissolve homogeneously in the crystals. For the
nominal value $x=0.10$ small single crystals cut from the batch appeared to
have $x=0.09$, as well as the non stoichiometric composition
Pd0.97Pt<0.004Te2.03. Magnetic and heat capacity measurements demonstrate a
transition from type-I to type-II superconducting behavior upon increasing
disorder. From transport measurements we calculate a residual resistivity
$\rho_{0}=1.4~{}\mu\Omega$cm suffices to turn PdTe2 into a superconductor of
the second kind.
## I introduction
Recently, interest in transition metal dichalcogenides has increased
significantly due to their extraordinary electronic properties. Notably, the
opportunity to realize novel quantum states arising from the topologically
non-trivial band structure, as found by density functional theory Soluyanov
_et al._ (2015); Huang _et al._ (2016); Yan _et al._ (2017); Bahramy _et
al._ (2018), attracts much attention. The formation of both type-I and type-II
bulk Dirac cones has been predicted Bahramy _et al._ (2018). Of special
interest in this family is the semimetal PdTe2 since it undergoes a
superconducting transition at $T_{c}\sim 1.7$ K Guggenheim _et al._ (1961).
Furthermore, PdTe2 is classified as a type-II Dirac semimetal, as uncovered by
angle-resolved photoemission spectroscopy and ab initio electronic structure
calculations Bahramy _et al._ (2018); Yan _et al._ (2015); Fei _et al._
(2017); Noh _et al._ (2017); Clark _et al._ (2018). A type-II Dirac
semimetal is characterized by a Dirac cone with a tilt parameter $k>1$ leading
to broken Lorentz invariance Soluyanov _et al._ (2015). It is predicted that
for Dirac semimetals with $k\approx 1$, meaning close to the topological
transition at $k=1$, superconductivity is generally of the second type (type-
II) Rosenstein _et al._ (2018). For $k>1$, superconductivity becomes of the
first kind (type-I). Interestingly, PdTe2 Leng _et al._ (2017); Salis _et
al._ (2021) is a type-I superconductor and based on its $T_{c}$ Shapiro et al.
Shapiro _et al._ (2018) estimated $k\approx 2$. In view of the effect
topology has on superconductivity in these systems, it is of interest to
investigate whether the superconductivity type can be altered by, for
instance, doping.
Superconductivity in PdTe2 has been explored in great detail. Type-I
superconductivity was uncovered with help of magnetic and transport
measurements on single crystals Leng _et al._ (2017). The intermediate state,
a hallmark of type-I behavior, was observed through the dc magnetization
curves and the differential paramagnetic effect in the ac susceptibility data.
Here, a bulk critical field $B_{c}(0)=13.6$ mT was determined in conjunction
with a surface critical field $B_{c}^{S}(0)=34.9$ mT. Moreover, the
temperature dependence of the surface superconductivity did not follow the
Saint-James $-$ de Gennes model Saint-James and de Gennes (1963). Peculiarly,
from resistance measurements a critical field $B_{c}^{R}(0)=0.32$ T was
deduced. Weak-coupling conventional superconductivity in PdTe2 was
demonstrated via measurements of the heat capacity Amit and Singh (2018);
Salis _et al._ (2021), penetration depth Teknowijoyo _et al._ (2018); Salis
_et al._ (2018), scanning tunneling microscopy and spectroscopy (STM/STS)
Clark _et al._ (2018); Das _et al._ (2018); Sirohi _et al._ (2019), and
side junction tunneling spectroscopy Voerman _et al._ (2019).
Superconductivity is partly attributed to a van Hove singularity situated at
$\sim 30$ meV above the Fermi level Kim _et al._ (2018); van Heumen _et al._
(2019).
On the other hand, a mixed type-I and type-II superconducting state was
concluded from STM/STS Das _et al._ (2018); Sirohi _et al._ (2019) and point
contact spectroscopy (PCS) Le _et al._ (2019) measurements. In a magnetic
field a range of critical fields was observed at the surface, which was
explained by spatially separated type-I and type-II regions. However, later
muon spin rotation measurements Leng _et al._ (2019a) and scanning squid
magnetometry Garcia-Campos _et al._ (2021) provide solid evidence for bulk
type-I superconductivity probed on the microscopic and macroscopic scale,
respectively. Finally, evidence for bulk type-I superconductivity was attained
through heat capacity measurements by demonstrating the presence of latent
heat Salis _et al._ (2021). Measurements under hydrostatic pressure show that
superconductivity is still present at 5.5 GPa Furue _et al._ (2021) and
remains of the first kind at least till 2.5 GPa Leng _et al._ (2019b).
Substitution or doping studies using PdTe2 are scarce. Kudo et al. Kudo _et
al._ (2016) examined Pd substitution in AuTe2 by preparing a series of
Au1-xPdxTe2 samples. Bulk superconductivity emerges at $x\approx 0.55$ with
$T_{c}\approx 4.0$ K as evidenced by heat capacity measurements. At lower
$x$-values the Te-Te dimer connections stabilize a monoclinic crystal
structure in which superconductivity is absent Kudo _et al._ (2016). The
strong-coupled nature of superconductivity near $x\approx 0.55$ is attributed
to a large density of states (DOS) at the Fermi level. Further increasing the
Pd content results in weak coupling superconductivity with lower transition
temperatures, as expected from approaching the stoichiometric end compound
PdTe2. Ryu investigated Cu doping in PdTe2 by preparing a series of CuxPdTe2
samples Ryu (2015). Optimal doping was found near $x=0.05$ with bulk
superconductivity at $T_{c}\approx 2.6$ K Ryu (2015); Hooda and Yadav (2018).
The increase of $T_{c}$ is attributed to an increase in the DOS at the Fermi
level due to the hybridization of Te-p and Cu-d orbitals along the $c$-axis,
effectively reducing the 2D-nature of this layered material. This is in-line
with the Cu atoms being intercalated in the Van der Waals gaps. STM/STS
measurements provide evidence that Cu0.05PdTe2 is a homogeneous type-II
superconductor Vasdev _et al._ (2019). This change, compared to the STM/STS
data on PdTe2 Das _et al._ (2018); Sirohi _et al._ (2019) that revealed a
mixed type-I/II behavior, is explained by Cu intercalation inducing disorder.
This effectively reduces the electron mean free path $l_{e}$ and the coherence
length $\xi$, thus increasing the Ginzburg Landau (GL) parameter
$\kappa=\frac{\lambda}{\xi}$ to larger than the $1/\sqrt{2}$ threshold for
type-I behavior.
Here we report the results of a doping study, directed to intentionally induce
disorder in PdTe2 by substituting Pd by iso-electronic Pt. We have prepared
Pd1-xPtxTe2 crystals with nominal doping concentrations $x=0.05$ and $x=0.10$.
Sample characterization by energy dispersive x-ray spectroscopy (EDX) revealed
that Pt did not dissolve homogeneously in the crystals. Notably, small
crystals cut from the nominal $x=0.10$ batch appeared to have $x=0.09$, or the
non stoichiometric composition Pd0.97Pt<0.004Te2.03. Transport, magnetic and
heat capacity measurements demonstrate a transition from type-I to type-II
superconducting behavior upon increasing disorder.
## II Experimental
PdTe2 crystallizes in the trigonal CdI2 structure (space group $P\bar{3}m1$).
Two single-crystalline batches Pd1-xPtxTe2 were prepared with $x=0.05$ and
$x=0.10$ using a modified Bridgman technique Lyons _et al._ (1976). The same
technique was previously used to prepare PdTe2 single crystals Leng _et al._
(2017). Small flat crystals were cut from the prepared batches by a scalpel.
The crystals have an area of $2\times 3$ mm2 and a thickness of about 0.3 mm.
Scanning electron microscopy with energy dispersive x-ray spectroscopy
(SEM/EDX) was carried out with help of a Hitachi table top microscope TM3000.
For details of the SEM/EDX results we refer to the Supplemental Material file
Sup . SEM micrographs taken on cut crystals and other sample pieces revealed
the final composition can deviate from the nominal one and that Pt did not
dissolve in the same amount in all pieces. In fact for the cut crystal with a
nominal Pt content of 5 at.% no Pt was detected. This crystal has a
stoichiometric composition with a Pd:Te ratio of 1:2 (the error in these
numbers is 1%). Transport, ac susceptibility and heat capacity measurements
were carried out on this sample, which we labeled #ptnom5. For the experiments
on the 10 at.%Pt concentration we used two crystals. One sample had a
composition close to the nominal $x=0.10$ composition Pd0.91Pt0.09Te2. This
sample, labeled #ptnom10res, was used for transport experiments only. EDX on
the second sample showed a small Te excess and a very small Pt content ($<$
0.4%). Its composition is Pd0.97Pt<0.004Te2.03. This sample was used for
transport, ac susceptibility and heat capacity measurements and it is labeled
#ptnom10. We remark that the EDX determined compositions above each yield the
average over a large part of the sample surface and are thus representative
for the specific sample. The experimental results on the doped samples are
compared with previous resistance, ac susceptibility and heat capacity data
taken on a crystal with the stoichiometric 1:2 composition to within 0.5% as
determined by EDX Leng _et al._ (2017); Salis _et al._ (2021). In the
following this sample is labelled #pdte2.
Resistance measurements were performed using the standard four point method in
a Quantum Design Physical Property Measurement System (PPMS) down to 2.0 K.
Data at lower temperatures were collected in a 3-He refrigerator (Heliox,
Oxford Instruments) down to 0.3 K using a low frequency (16 Hz) ac-resistance
bridge (Linear Research LR700). The ac susceptibility was measured in the
Heliox with a custom-made coil set. Data were also taken with the LR700
bridge, operated at a driving field of 0.026 mT. The heat capacity was
measured using the dual slope thermal relaxation calorimetry technique Stewart
(1983), using a home-built set-up Salis _et al._ (2021), where each data
point is the average of four dual slope measurements. The increase in
temperature $\Delta T$ in the measurement of the heat capacity is always in
between 1$\%$ and 1.6$\%$ of the bath temperature of the particular
measurement. In the ac susceptibility and specific heat experiment the dc
magnetic field was applied in the $ab$-plane. The demagnetization factor of
the crystals is $N\simeq 0.1$, which implies the intermediate state is formed
between $(1-N)H_{c}\simeq 0.9H_{c}$ and $H_{c}$ in the case of type-I
superconductivity. The resistance and ac susceptibility measurements in field
have been carried out by applying the field above $T_{c}$ and subsequently
cooling in field, while the specific heat data in field were taken after zero
field cooling and then applying the field.
## III Results
Figure 1: Temperature dependence of the resistivity of crystals #ptnom5 (red
circles), #ptnom10 (blue circles) and #ptnom10res (green circles). The data
for #pdte2 (black circles) are taken from Ref. Leng _et al._ (2017).
The resistivity of samples #ptnom5, #ptnom10 and #ptnom10res in the
temperature range 2-300 K is shown in figure 1, where we have also traced the
data for crystal #pdte2 reported in Ref. Leng _et al._ (2017). The curves for
#ptnom5 and #pdte2 are very similar with a residual resistivity value,
$\rho_{0}$, taken at 2 K, of 0.75 and 0.76 $\mu\Omega$cm, respectively. This
is in agreement with both samples having the same stoichiometric 1:2
composition. The residual resistance ratio, RRR= $\rho(300$K$)/\rho_{0}$,
amounts to 40 and 30, respectively. For the non-stoichiometric sample #ptnom10
$\rho_{0}$ has increased to 3.6 $\mu\Omega$cm and RRR = 12. The
$\rho_{0}$-value of the substituted sample #ptnom10res is considerably higher
as expected, and equals 16.3 $\mu\Omega$cm. Its RRR is 3.
Figure 2: Resistance as a function of temperature around the superconducting
transition for crystal #ptnom5 (panel (b)) and #ptnom10res (panel (c)) in zero
field (black curves) and small applied fields, $\mu_{0}H_{a}$, as indicated.
The data in panel (a) for #pdte2 are taken from Ref. Leng _et al._ (2017).
Figure 3: Ac susceptibility of crystals #ptnom5 (panel (b)) and #ptnom10
(panel (c)) measured in zero field (black curves) and small applied dc fields
as indicated. The field is applied in the $ab$-plane. The data of #pdte2 are
taken from Ref. Leng _et al._ (2017). Note the ac driving field applied to
take the data in panels (b) and (c) is a factor 10 smaller than in panel (a).
The resistance as a function of temperature around the superconducting
transition in zero field and applied magnetic fields of crystals #ptnom5 and
#ptnom10res is depicted in figure 2. Again, the data for #pdte2, shown in
panel (a), are taken from Ref. Leng _et al._ (2017). The critical temperature
in zero field, $T_{c}(0)$, here defined by the onset of the transition, is
1.87 K and 1.56 K for the stoichiometric samples #ptnom5 and #pdte2,
respectively. Surprisingly, the higher $T_{c}$ and RRR for #ptnom5 indicate it
has a somewhat higher purity than sample #pdte2. For the substituted sample
the superconducting transition shows several steps and $T_{c}$ is lower. It
ranges from 1.44 to 1.12 K. In a magnetic field superconductivity is rapidly
suppressed. The data in panels (b) and (c) of figure 2 show these crystals
also have superconducting resistance paths in fields above the critical field
$B_{c}(0)$ determined by ac susceptibility and heat capacity (see below and
figure 5). The $B_{c}^{R}(0)$ values that can be deduced are however not as
large as the value $B_{c}^{R}(0)\approx$ 0.3 T for $H$ $\|~{}c$ reported for
PdTe2 (see figure S6 in the Supplemental Material file of Ref. Leng _et al._
(2017)).
In figure 3 we show the in-phase component of the ac susceptibility,
$\chi_{ac}^{\prime}$, in arbitrary units measured on crystals #ptnom5 and
#ptnom10 in the temperature range 0.3-2.0 K. Again the data are compared with
those of #pdte2 (data in S.I. units taken from Ref. Leng _et al._ (2017)).
The onset $T_{c}$ values are 1.64 K and 1.85 K for #pdte2 and #ptnom5 and
compare well to the values determined above from the resistivity. The onset
$T_{c}$ value for #ptnom10 is 1.91 K, but the transition is rather broad (the
width is 0.3 K) with a slow decrease below $T_{c}$. The resistance of this
sample was only measured in the PPMS down to 2.0 K. The RRR-value of 12 tells
us the disorder is enhanced, which is also reflected in the broad transition.
The $\chi_{ac}^{\prime}(T)$ data measured in applied magnetic fields for
#pdte2 and #ptnom5 show pronounced peaks below $T_{c}$ that are due to the
differential paramagnetic effect (DPE). The DPE is due to the positive $dM/dH$
($M$ is the magnetization) in the intermediate state Hein and Falge (1961).
The intermediate phase is due to the sample shape and is present when the
demagnetization factor, $N$, is finite. Observation of red a DPE that largely
exceeds the Meissner signal can therefore be used as solid proof for type-I
superconductivity. Most importantly, the DPE is absent for crystal #ptnom10,
which provides the first piece of evidence it is a type-II superconductor.
Figure 4: Electronic specific heat, $C_{el}$, of crystals #ptnom5 (panel (b))
and #ptnom10 (panel (c)) measured in zero field (black curves) and small
applied dc fields as indicated. The field is directed in the $ab$-plane. The
data of #pdte2 are taken from Ref. Salis _et al._ (2021).
In figure 4 we show the electronic specific heat, $C_{el}$, of crystals
#ptnom5 and #ptnom10 in the temperature range 0.3-2.0 K. The
$C_{el}(T)$-curves are obtained by subtracting the phononic contribution from
the measured $C$ in the standard way, i.e. by using the relation $C=\gamma
T+\beta T^{3}$, where $\gamma$ is the Sommerfeld coefficient and $\beta$ the
phononic coefficient. The data are compared with $C_{el}$ of PdTe2 reported in
Ref. Salis _et al._ (2021) (panel (a) of figure 4). This PdTe2 crystal was
cut from the same batch as the samples studied in Ref. Leng _et al._ (2017)
and we also label it #pdte2. The onset $T_{c}$ values of crystals #pdte2 and
#ptnom5 are 1.62 K and 1.75 K and compare well to the values determined above.
The onset $T_{c}=1.60$ K for #ptnom10 is however lower than the value 1.91 K
determined by $\chi_{ac}^{\prime}(T)$.
The $\gamma$-values of the three crystals in panel (a), (b) and (c) of figure
4 amount to 4.4, 4.5 and 4.7 mJ/molK2 and the $\beta$-values are 0.7, 1.1 and
1.0 mJ/molK4, respectively. These $\gamma$-values are very similar, which
indicates the density of states near the Fermi level is not affected much by
doping. The $\beta$-values do show some variation, which is not correlated
with the amount of disorder, and likely related to an experimental uncertainty
because of the small temperature interval in which $\beta$ is obtained. To
examine the strength of the electron-phonon coupling, the step size $\Delta
C|_{T_{c}}$ is analysed using the BCS relation $\Delta C|_{T_{c}}/\gamma
T_{c}=1.43$, where $T_{c}$ is the superconducting transition temperature, here
taken as the onset of superconductivity. For crystal #pdte2 a ratio $\Delta
C|_{T_{c}}/\gamma T_{c}=1.42$ is found Salis _et al._ (2021), which is close
to the textbook value of 1.43 for a weakly coupled BCS superconductor. For
crystal #ptnom5 a ratio of 1.41 is found, which presents a minute change from
the textbook value. However, for crystal #ptnom10 we determine a ratio of
1.48, suggesting that superconductivity is slightly more than weakly coupled.
Next we discuss the electronic specific heat measured in applied magnetic
fields (figure 4). Distinguishing between type-I and type-II superconductivity
via heat capacity can be achieved by observing the presence or absence of
latent heat. The extra energy necessary to facilitate a first order phase
transition is reflected in the heat capacity as an increased value of $C$ at
the transition. A type-I superconductor has a second order phase transition in
zero field, but a first order one in field. While for a type-II superconductor
the transition remains second order in an applied field. The excess $C_{el}$
above the standard BCS heat capacity in panel (a) provided solid thermodynamic
evidence PdTe2 is a type-I superconductor Salis _et al._ (2021).
Surprisingly, for crystal #ptnom5 (panel b) the excess specific heat becomes
more pronounced as illustrated by the sharp peaks below $T_{c}(B)$. Thus the
contribution of the latent heat to $C_{el}$ is much larger, which indicates
the transition has a stronger first order character than observed for crystal
#pdte2. On the other hand, for crystal #ptnom10 the data in panel (c) show
latent heat is absent, which provides the second piece of evidence of type-II
superconductivity, in-line with the $\chi_{ac}^{\prime}$-data.
Finally, we trace the temperature variation of the critical field, $B_{c}(T)$,
extracted from the ac susceptibility (figure 3) and specific heat data (figure
4). The $B-T$ phase diagram is reported in figure 5. For crystals #pdte2 and
#ptnom5 we identify $T_{c}(B)$ by the onset in $C_{el}$ and the onset of the
DPE in $\chi_{ac}^{\prime}(T)$. $B_{c}(T)$ follows the standard BCS quadratic
temperature variation $B_{c}(T)=B_{c}(0)[1-(T/T_{c})^{2}]$, with
$B_{c}(0)=14.2$ mT and $T_{c}$ = 1.63 K for #ptpde2 Leng _et al._ (2017);
Salis _et al._ (2021), and $B_{c}(0)=15.9$ mT and $T_{c}$ = 1.77 K for
#ptnom5. For crystal #ptnom10 the transition in $\chi_{ac}^{\prime}(T)$ is
rather broad. Here we identify $T_{c}$ by the onset temperature in $C_{el}$,
which corresponds to the temperature where the magnetic transition is complete
in $\chi_{ac}^{\prime}(T)$. The $B-T$ phase-line provides further evidence for
type-II superconductivity. It compares well to the Werthamer-Helfand-Hohenberg
(WHH) model curve Werthamer _et al._ (1966) for an orbital-limited weak-
coupling spin-singlet superconductor with an upper critical field
$B_{c2}(0)=21.8$ mT.
Figure 5: Critical field $B_{c}(T)$ of crystal #pdte2 and #ptnom5 and upper
critical field $B_{c2}(T)$ of crystal #ptnom10 extracted from the specific
heat (closed squares) and ac susceptibility (closed circles) data.
## IV Discussion
From the sample preparation side our goal was to prepare Pd1-xPtxTe2 crystals
with $x=0.05$ and $x=0.10$. The SEM/EDX micrographs showed that Pt did not
dissolve as expected in these crystals and that the single-crystalline batches
are inhomogeneous. Crystals cut from the nominal $x=0.05$ batch appeared to be
undoped and have the 1:2 stoichiometry. From the nominal $x=0.10$ batch we
managed to obtain a crystal with $x=0.09$, and a non-stoichiometric crystal
Pd0.97Pt<0.004Te2.03. Specific heat and ac-susceptibility measurements on this
last crystal #ptnom10 demonstrated we could make a doping-induced transition
to type-II superconductivity.
To observe type-II superconductivity the disorder should be large enough such
that the threshold $\kappa=1/\sqrt{2}$ can be overcome. The effect of
controlled non-magnetic disorder on the normal and superconducting properties
of PdTe2 was recently studied by electron irradiation by Timmons et al.
Timmons _et al._ (2020). The residual resistivity was found to increase from
a pristine crystal value of 0.6 $\mu\Omega$cm to 2.4 $\mu\Omega$cm for a
irradiation dose of 2.4 C/cm2, while at the same time $T_{c}$ decreased from
1.76 K to 1.65 K as identified by reaching the zero resistance state $R=0$.
Assuming a linear relation between $\rho_{0}$ and $T_{c}$, $T_{c}$ decreases
at a rate of 0.046 K/$\mu\Omega$cm. With this rate we estimate for crystal
#ptnom10res ($\Delta\rho_{0}=15.5~{}\mu\Omega$cm) $T_{c}=0.9$ K, which
compares favorably to the measured $T_{c}=1.1$ K ($R=0$), given the crude
approximation. In this electron irradiation work no discussion was made
whether disorder is strong enough to induce type-II behavior.
For crystal #ptnom10 the coherence length $\xi$ can be calculated from the
relation $B_{c2}(0)=\Phi_{0}/2\pi\xi^{2}$, where $\Phi_{0}$ is the flux
quantum. From figure 5 we determine $B_{c2}(0)=21.8$ mT and obtain $\xi=123$
nm. The coherence length can be related to the electron mean free path,
$l_{e}$, via Pippard’s relation $1/\xi=1/\xi_{0}+1/l_{e}$, where $\xi_{0}$ is
the intrinsic coherence length given by the BCS value Tinkham (1996). With
$\xi_{0}=1.8~{}\mu$m Salis _et al._ (2018) and $\xi=~{}123$ nm we obtain
$\l_{e}=132$ nm. As expected, this value is reduced compared to $\l_{e}=531$
nm calculated from the residual resistivity value
$\rho_{0}=0.76~{}\mu\Omega$cm Salis _et al._ (2018) of nominally pure PdTe2.
Reversely, using the experimental value $\rho_{0}=3.6~{}\mu\Omega$cm (figure
1) we calculate $\l_{e}=112$ nm for crystal #ptnom10, which is close to the
value $\l_{e}=132$ nm derived from Pippard’s relation.
Next we calculate $\kappa=\lambda/\xi$ of crystal #ptnom10. In their
controlled disorder study Timmons et al. Timmons _et al._ (2020) measured the
penetration depth and found that upon increasing the disorder $\lambda$ stays
nearly constant Timmons _et al._ (2020) at a value of 220 nm. This is in-line
with the minute change in the $\gamma$-value reported above. With $\xi=123$ nm
we calculate $\kappa=1.8$, which is in agreement with superconductivity being
of the second kind. For crystals #pdte2 and #ptnom5 we calculate
$\kappa\simeq$ 0.5-0.6 Leng _et al._ (2017). Here $\xi\simeq$ 440-370 nm is
estimated from the GL relation $\xi=\Phi_{0}/(2\sqrt{2}\pi B_{c}\lambda_{L})$
Tinkham (1996), where $\lambda_{L}\propto(m_{e}/n_{s})^{1/2}$ is the London
penetration depth with $m_{e}$ the effective electron mass and $n_{s}$ the
superfluid density.
Another way to provide an estimate of $\kappa$ of crystal #ptnom10 is from the
GL relation $\kappa=B_{c2}/\sqrt{2}B_{c}$. The thermodynamic critical field,
$B_{c}$, can be determined from the specific heat by the relation $\Delta
C|_{T_{c}}=4B_{c}(0)^{2}/\mu_{0}T_{c}$ Poole _et al._ (2007), where $C$ is in
units of J/m3. From $\Delta C|_{T_{c}}$ in figure 4 (panel (c)) we calculate
$B_{c}(0)=11.1$ mT. We remark this value is close to the calculated value
$B_{c}(0)=12.6$ mT reported for PdTe2 Leng _et al._ (2017). Using
$B_{c}(0)=11.1$ mT and $B_{c2}=21.8$ mT in the expression above, we calculate
$\kappa=1.4$, which is similar to the value of 1.8 directly estimated from the
ratio $\lambda/\xi$.
We remark that for Type-I superconductivity $B_{c}(0)$ can also be obtained
from the latent heat with help of the Clausius-Clapeyron relation. We
calculate $B_{c}(0)$ = 11.2 mT and 11.1 mT for #pdte2 and #ptnom5,
respectively, in good agreement with the values obtained from $\Delta
C|_{T_{c}}$ in zero field Sup .
Our results are of relevance for the observation of a mixed type-I and type-II
superconducting state in PdTe2 probed by surface sensitive techniques Das _et
al._ (2018); Sirohi _et al._ (2019); Le _et al._ (2019). Our doping study
shows that nominal pure PdTe2 crystals can already be close to the type-I/II
border. Using the value $\lambda=230$ nm Timmons _et al._ (2020), we
calculate $\xi=310$ nm at the threshold value $\kappa=1/\sqrt{2}$. This
implies $l_{e}$ should be smaller than 375 nm for type-II superconductivity,
or $\rho_{0}>1.4~{}\mu\Omega$cm. From the resistivity graph reported in Ref.
Das _et al._ (2018) we deduce $\rho_{0}\simeq 1~{}\mu\Omega$cm, which indeed
is not far from the type-I/II border. Thus it is plausible inhomogeneities
give rise to the mixed phase observation reported in Refs. Das _et al._
(2018); Sirohi _et al._ (2019); Le _et al._ (2019).
An unsolved aspect of superconductivity in PdTe2 is the observation of surface
superconductivity detected in the screening signal $\chi_{ac}^{\prime}(T)$
measured in small applied dc fields Leng _et al._ (2017, 2019b). The
extracted surface critical field $B_{c}^{S}(0)=34.9$ mT exceeds the value
predicted by the Saint-James $-$ de Gennes model Saint-James and de Gennes
(1963) $B_{c3}=2.39\times\kappa B_{c}=16.3$ mT. Recently, the GL model at the
superconducting-insulator boundary was revisited Samoilenka and Babaev (2021)
and it was shown that $T_{c}$ and the third critical field $B_{c3}$ can be
enhanced to exceed the Saint-James $-$ de Gennes value, which is worthy to
explore further. On the other hand, it is tempting to attribute the surface
superconductivity in PdTe2 to superconductivity of the topological surface
state detected by ARPES Bahramy _et al._ (2018); Yan _et al._ (2015); Fei
_et al._ (2017); Noh _et al._ (2017); Clark _et al._ (2018). We remark that
the $\chi_{ac}^{\prime}(T)$ data for the doped crystals, reported in figure 3
panel (b) and (c), also show superconducting screening signals above the
$B_{c}(0)$ and $B_{c2}(0)$-values reported in figure 5. Likewise, the
resistance traces in figure 2 reveal $B_{c}^{R}(0)$ is similarly enhanced.
These screening signals of enhanced superconductivity are however not as
pronounced as reported for PdTe2 in Ref. Leng _et al._ (2017). Nonetheless,
the robustness of superconducting screening signals above $B_{c}(0)$ or
$B_{c2}(0)$ upon doping, as well as under high pressure Leng _et al._
(2019b), calls for further experiments.
## V Conclusion
The Dirac semimetal PdTe2 is a type-I superconductor with $T_{c}=1.7$ K. We
have carried out a doping study directed to intentionally increase the
disorder and induce type-II superconductivity. Two single-crystalline batches
Pd1-xPtxTe2 have been prepared with nominal doping concentrations $x=0.05$ and
$x=0.10$. Sample characterization by energy dispersive x-ray spectroscopy
(EDX) on small crystals cut from the batches revealed that Pt did not dissolve
homogeneously in the crystals. In fact the nominal $x=0.05$ crystal appeared
to be undoped and have the stoichiometric 1:2 composition. From the nominal
$x=0.10$ batch we obtained a small single crystal with $x=0.09$, as well as a
crystal with the non stoichiometric composition Pd0.97Pt<0.004Te2.03. The
presence of the differential paramagnetic effect in the ac susceptibility and
latent heat in the heat capacity demonstrate the nominal $x=0.05$ crystal is a
type-I superconductor, just like PdTe2. The absence of these effects for
Pd0.97Pt<0.004Te2.03 revealed it is a type-II superconductor with an upper
critical field $B_{c2}=21.8$ mT. The analysis of $B_{c2}$ and resistance data
using Pippard’s model convincingly show PdTe2 can be turned into a
superconductor of the second kind when the residual resistivity
$\rho_{0}>1.4~{}\mu\Omega$cm.
Acknowledgement: This work is part of the Projectruimte programme with project
number 680-91-109, which is financed by the Dutch Research Council (NWO).
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Supplemental Material: Disorder induced transition from type-I to type-II
superconductivity in the Dirac semimetal PdTe2
Content
1\. SEM/EDX mapping experimental and results
Fig.S1 SEM/EDX mapping of crystal #ptnom5
Fig.S2 SEM/EDX mapping of crystal #ptnom10
Fig.S3 SEM/EDX mapping of crystal #ptnom10res
2\. Latent heat and Clausius-Clapeyron relation
1\. SEM/EDX mapping: experimental
Scanning electron microscopy with energy dispersive x-ray spectroscopy
(SEM/EDX) was carried out with help of a Hitachi table top microscope TM3000.
The acceleration voltage in all measurements is 15 kV. Of each single-
crystalline boule prepared with a certain nominal Pt content small thin
crystals were isolated with typical size 2 $\times$ 3 mm2. On each of these
crystals we have investigated the composition by EDX in several areas of
typically 200 $\times$ 200 $\mu$m2. Overall the composition in the sleected
and measured crystals was found to be homogenous. The table below gives the
labels of the doped crystals, the nominal Pt content and the EDX determined
composition.
Crystal | Nominal Pt cotent | EDX compostion
---|---|---
#ptnom5 | 5 at. $\%$ | PdTe2
#ptnom10 | 10 at. $\%$ | Pd0.97Pt<0.004Te2.03
#ptnom10res | 10 at.$\%$ | Pd0.91Pt0.09Te2
Typical SEM/EDX results for crystals #ptnom5, #ptnom10 and #ptnom10res are
given in Fig. S1, Fig. S2 and Fig. S3, respectively. Shown are:
Top panel: The SEM spectrum in cps/eV (counts per second per electron-volt)
with Pd, Pt and Te peaks labelled.
Middle panel: Table with quantitative results of the composition analysis.
Lower left panel: SEM picture of the crystal with scanned area for the
composition analysis indicated.
Lower right panel: Pd, Pt and Te element distribution in the scanned area.
Figure S1 SEM/EDX mapping of crystal #ptnom5
Figure S2 SEM/EDX mapping of crystal #ptnom10
Figure S3 SEM/EDX mapping of crystal #ptnom10res
2\. Latent heat and Clausius-Clapeyron relation
As mentioned in the Discussion section in the manuscript an estimate for the
thermodynamic critical field $B_{c}(0)$ can be obtained by evaluating the
relation $\Delta C=4B_{c}(0)^{2}/\mu_{0}T_{c}$, where $\Delta C$ is the step
at $T_{c}$ in the specific heat at $B=0$. Alternatively, for a type-I
superconductor in a magnetic field $B_{c}(0)$ can be calculated from the
latent heat, $L$. Using the Clausius-Clapeyron equation the entropy change is
given by $\Delta S=L(T)/T=-2\mu_{0}B_{c}(T)\frac{dB_{c}(T)}{dT}$, with
$B_{c}(T)=B_{c}(0)(1-(T/T_{c})^{2})$. $\Delta S$ was calculated by graphically
integrating the $C/T$ data in field and subtracting the data taken at $B=0$.
This procedure is carried out for all applied field values, and from $\Delta
S(B)$ we obtain $B_{c}(0)$ as a fit parameter. For crystals #pdte2 and
#ptnom5, that exhibit type-I superconductivity, we obtain $B_{c}(0)=11.2$ mT
and 11.1 mT, respectively. We remark these values are close to the $B_{c}(0)$
values evaluated at $B=0$. On the other hand, the $B_{c}(0)$-values derived
from $\Delta C$ and $\Delta S$ are smaller than the value derived directly
from the experiment. In the table below we compare the $B_{c}(0)$ values
obtained in different ways.
Crystal | $B_{c}(0)$ from $\Delta C$ | $B_{c}(0)$ from $\Delta S$ | $B_{c}(0)$ experiment
---|---|---|---
#pdte2 | 10.9 mT | 11.2 mT | 14.2 mT
#ptnom5 | 11.8 mT | 11.1 mT | 15.9 mT
#ptnom10 | 11.1 mT | - | -
|
$\mathcal{S}:\mathbb{T}_{p}\rightarrow\mathbb{Z}[p^{-1}][\mathrm{X}^{\ast}(\hat{\mathrm{T}})]^{\mathrm{W}_{\mathbf{G}}}$
where $\mathrm{W}_{\mathbf{G}}$ is the absolute Weyl group of $\mathbf{G}$. We
fix a datum $\mathbf{T}\subset\mathbf{B}\subset\mathbf{G}$ of the diagonal
torus contained in the upper triangular Borel subgroup as in the global case.
This determines a positive Weyl chamber $\mathrm{P}^{+}$ in
$\mathrm{X}_{\ast}(\mathrm{T})\cong\mathrm{X}^{\ast}(\hat{\mathrm{T}})$. An
element $\nu\in\mathrm{X}^{\ast}(\hat{\mathrm{T}})$ determines an irreducible
representation $V_{\nu}$ of the dual group
$\hat{\mathrm{G}}=\mathrm{GSpin}(5)$ whose character is denoted by
$\chi_{\nu}\in\mathrm{R}(\hat{\mathrm{G}})=\mathbb{Z}[\mathrm{X}^{\ast}(\hat{\mathrm{T}})]^{\mathrm{W}_{\mathbf{G}}}$.
It is well-known that $\mathbb{T}_{p}$ is isomorphic to
$\mathbb{Z}[\mathrm{T}^{\pm}_{p,0},\mathrm{T}_{p,1},\mathrm{T}_{p,2}]$ where
$\begin{split}&\mathrm{T}_{p,0}=c_{\nu_{0}}=\mathrm{char}(\mathbf{G}(\mathbb{Z}_{p})\begin{pmatrix}p&&&\\\
&p&&\\\ &&p&\\\ &&&p\\\ \end{pmatrix}\mathbf{G}(\mathbb{Z}_{p}))\text{ for
}\nu_{0}=(1,1,1,1)\\\
&\mathrm{T}_{p,2}=c_{\nu_{2}}=\mathrm{char}(\mathbf{G}(\mathbb{Z}_{p})\begin{pmatrix}p&&&\\\
&p&&\\\ &&1&\\\ &&&1\\\ \end{pmatrix}\mathbf{G}(\mathbb{Z}_{p}))\text{ for
}\nu_{2}=(1,1,0,0)\\\
&\mathrm{T}_{p,1}=c_{\nu_{1}}=\mathrm{char}(\mathbf{G}(\mathbb{Z}_{p})\begin{pmatrix}p^{2}&&&\\\
&p&&\\\ &&p&\\\ &&&1\\\ \end{pmatrix}\mathbf{G}(\mathbb{Z}_{p}))\text{ for
}\nu_{1}=(2,1,1,0)\\\ \end{split}$
and where $\mathrm{char}(\cdot)$ is the characteristic function.
The following spherical Hecke operator will appear naturally in later
computations
$\mathrm{T}_{p^{2},2}=c_{2\nu_{2}}=\mathrm{char}(\mathbf{G}(\mathbb{Z}_{p})\begin{pmatrix}p^{2}&&&\\\
&p^{2}&&\\\ &&1&\\\ &&&1\\\ \end{pmatrix}\mathbf{G}(\mathbb{Z}_{p}))\text{ for
}2\nu_{2}=(2,2,0,0).$
We will also need the following two Hecke operators in
$\mathbb{Z}[\mathrm{K}_{\\{2\\}}\backslash\mathbf{G}(\mathbb{Q}_{p})/\mathrm{K}_{\\{0\\}}]$
$\displaystyle\mathrm{T}_{02}=\mathrm{char}(\mathrm{K}_{\\{2\\}}\begin{pmatrix}p&&&\\\
&p&&\\\ &&1&\\\ &&&1\\\ \end{pmatrix}\mathrm{K}_{\\{0\\}})$
$\displaystyle\mathrm{T}_{20}=\mathrm{char}(\mathrm{K}_{\\{0\\}}\begin{pmatrix}p^{-1}&&&\\\
&p^{-1}&&\\\ &&1&\\\ &&&1\\\ \end{pmatrix}\mathrm{K}_{\\{2\\}}).$
In the computations below, we will follow mostly the notations and conventions
in [Gross] and use results from there freely. We have the following useful
formula [Gross, 3.12]
$p^{\langle\nu,\rho\rangle}\chi_{\nu}=\mathcal{S}(c_{\nu})+\sum_{\xi<\nu}d_{\nu}(\xi)\mathcal{S}(c_{\xi}).$
The integers $d_{\nu}(\xi)$ are given by $\mathrm{P}_{\xi,\nu}(p)$ for the
Kazhdan-Lusztig polynomial $\mathrm{P}_{\xi,\nu}$ evaluated at $p$ and $2\rho$
is the sum of the positive roots. Since the weight $\nu_{2}$ is minuscule, we
immediately have
$p^{\frac{3}{2}}\chi_{\nu_{2}}=\mathcal{S}(c_{\nu_{2}})=\mathcal{S}(\mathrm{T}_{p,2}).$
On the other hand for the non-minuscule weight $\nu_{1}$, we have
$p^{2}\chi_{\nu_{1}}=\mathcal{S}(c_{\nu_{1}})+\mathcal{S}(c_{0})=\mathcal{S}(\mathrm{T}_{p,1})+1.$
The following identity is the main result of this subsection. It can perhaps
be proved by a massive double coset computation whereas our proof uses
crucially the Satake isomorphism to simplify the computation. This identity
seems not to exist already in the literature and therefore we give full
details which we think are of independent interest.
###### Proposition 6.9.
The following identity holds in $\mathbb{T}_{p}$
$\mathrm{T}_{p^{2},2}=\mathrm{T}^{2}_{p,2}-(p+1)\mathrm{T}_{p,1}-(p+1)(p^{2}+1).$
###### Proof.
The representation $V_{\nu_{1}}$ is the $5$-dimensional orthogonal
representation and $V_{\nu_{2}}$ is the standard representation. Therefore
$\wedge^{2}V_{\nu_{2}}=V_{\nu_{1}}\oplus\mathbb{C}$. We also have
$\wedge^{2}V_{\nu_{1}}=V_{2\nu_{2}}$ is the adjoint representation of
dimension $10$. Thus we have
$\displaystyle V^{\otimes 2}_{\nu_{2}}$
$\displaystyle=V_{2\nu_{2}}+\wedge^{2}V_{\nu_{2}}$
$\displaystyle=V_{2\nu_{2}}+V_{\nu_{1}}+\mathbb{C}$
Taking characters, we have
$\chi^{2}_{\nu_{2}}=\chi_{2\nu_{2}}+\chi_{\nu_{1}}+1$ and hence
$p^{3}\chi^{2}_{\nu_{2}}=p^{3}\chi_{2\nu_{2}}+p^{3}\chi_{\nu_{1}}+p^{3}$.
Therefore
$\displaystyle p^{3}\chi_{2\nu_{2}}$
$\displaystyle=p^{3}\chi^{2}_{\nu_{2}}-p^{3}\chi_{\nu_{1}}-p^{3}$
$\displaystyle=\mathcal{S}(\mathrm{T}^{2}_{p,2})-p\mathcal{S}(\mathrm{T}_{p,1})-(p+p^{3}).$
On the other hand, we also have
$\displaystyle p^{3}\chi_{2\nu_{2}}$
$\displaystyle=\mathcal{S}(c_{2\nu_{2}})+d_{2\nu_{2}}(\nu_{1})S(c_{\nu_{1}})+d_{2\nu_{2}}(0)$
$\displaystyle=\mathcal{S}(\mathrm{T}_{p^{2},2})+d_{2\nu_{2}}(\nu_{1})S(\mathrm{T}_{p,1})+d_{2\nu_{2}}(0).$
* •
$d_{2\nu_{2}}(\nu_{1})=1$; This follows from the fact that
$\mathrm{P}_{\nu_{1},2\nu_{2}}$ has degree strictly less than $\langle
2\nu_{2}-\nu_{1},\rho\rangle=1$ and the constant coefficient is $1$.
* •
$d_{2\nu_{2}}(0)=1+p^{2}$; This follows from the fact that $2\nu_{2}$ is the
highest weight of the adjoint representation, which implies that
$\mathrm{P}_{0,2\nu_{2}}(p)=\sum_{i=1}p^{m_{i}-1}$ where $m_{i}$ are the
exponents of the Weyl group of type $\mathrm{C}_{2}$ which are $1$ and $3$,
see [Gross, 4.6].
Equating the two expressions of $p^{3}\chi_{2\nu_{2}}$ gives the desired
result. ∎
### 6.2. Computing the level raising matrix
We now calculate the entries of the level raising matrix
$\mathcal{T}_{\mathrm{lr}}$ term by term. The main result is the following
proposition.
###### Proposition 6.10.
The level raising matrix $\mathcal{T}_{\mathrm{lr}}$ is given by
$\mathcal{T}_{\mathrm{lr}}=-2\begin{pmatrix}&\mathrm{T}^{-1}_{p,0}\mathrm{T}_{p,1}+(p+1)(p^{2}+1)&(p+1)\mathrm{T}_{02}\\\
&(p+1)\mathrm{T}_{20}&\mathrm{T}^{-1}_{p,0}\mathrm{T}_{p,1}+(p+1)(p^{2}+1)\\\
\end{pmatrix}.$
###### Proof.
To begin the proof, note that we can identify both
$\bigoplus\limits_{\sigma\in\mathrm{Z}_{\\{1\\}}(\overline{\mathrm{B}})}\mathrm{H}^{3}_{\\{\sigma\\}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))$
and
$\bigoplus\limits_{\sigma\in\mathrm{Z}_{\\{1\\}}(\overline{\mathrm{B}})}\mathrm{R}^{3}\Phi_{\sigma}(\mathcal{O}_{\lambda}(2))$
with $\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{Pa}}(\overline{\mathrm{B}})]$
ignoring the Galois actions. The level raising diagram (7.2) reduces to
(6.11)
${\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]}$${\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]}$${\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))}$${\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{Pa}}(\overline{\mathrm{B}})]}$${\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))}$${\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]}$${\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]}$$\scriptstyle{\mathrm{inc}^{\\{0\\}}_{!}}$$\scriptstyle{\mathrm{inc}^{\\{2\\}}_{!}}$$\scriptstyle{\beta}$$\scriptstyle{\alpha}$$\scriptstyle{\mathrm{inc}^{\ast}_{\\{0\\}}}$$\scriptstyle{\mathrm{inc}^{\ast}_{\\{2\\}}}$
By Proposition 5.8, the map $\alpha$ can be realized as a Gysin map and
$\beta$ can be realized as a restriction map. The situation is similar to that
of [LTXZZ, Proposition 5.8.8]. We proceed to analyze each entry.
#### 6.2.1. $\mathcal{T}_{\mathrm{lr},\\{00\\}}$
Since $\mathrm{Z}_{\\{i\\}}(\overline{\mathrm{B}})$ corresponds to the set of
vertex lattices of type $i$ for $i\in\\{0,1,2\\}$. Therefore to understand the
map
$\mathcal{T}_{\mathrm{lr},\\{00\\}}=\mathrm{inc}^{\ast}_{\\{0\\}}\circ\alpha\circ\mathrm{N}^{-1}_{\Sigma}\circ\beta\circ\mathrm{inc}^{\\{0\\}}_{!}$,
we need to understand the possible relative position of vertex lattices
$\Lambda_{0}$, $\Lambda_{\mathrm{Pa}}$ and $\Lambda^{\prime}_{0}$ of type $0$,
$1$ and $0$. It not hard to see that the only possible situation is that
$\Lambda_{\mathrm{Pa}}\subset^{1}\Lambda_{0}$ and
$\Lambda_{\mathrm{Pa}}\subset^{1}\Lambda^{\prime}_{0}$. There are two possible
cases.
Case $(1)$: $\Lambda_{0}=\Lambda^{\prime}_{0}$. Then we have
$p\Lambda^{\vee}_{0}\subset^{1}p\Lambda^{\vee}_{\mathrm{Pa}}\subset^{2}\Lambda_{\mathrm{Pa}}\subset^{1}\Lambda_{0}$.
Giving such a $\Lambda_{\mathrm{Pa}}$ is equivalent to giving a complete flag
in $\Lambda_{0}/p\Lambda_{0}$. The number of such flags is given by
$[\mathbf{G}(\mathbb{Z}_{p}):\mathrm{Kl}]=(p+1)(p^{2}+1)$ where $\mathrm{Kl}$
is the Klingen parahoric.
Case $(2)$: $\Lambda_{0}\neq\Lambda^{\prime}_{0}$. Then we have
$\Lambda_{0}\cap\Lambda^{\prime}_{0}=\Lambda_{\mathrm{Pa}}$ and the relative
position between $\Lambda_{0}$ and $\Lambda^{\prime}_{0}$ is given by
$\mathrm{K}_{\\{0\\}}\begin{pmatrix}p&&&\\\ &1&&\\\ &&1&\\\ &&&p^{-1}\\\
\end{pmatrix}\mathrm{K}_{\\{0\\}}.$
This operator is the same as $\mathrm{T}^{-1}_{p,0}\mathrm{T}_{p,1}$ by
identifying $\mathrm{K}_{\\{0\\}}$ with the hyperspecial group
$\mathbf{G}(\mathbb{Z}_{p})$. Therefore the entry
$\mathcal{T}_{\mathrm{lr},\\{00\\}}$ is given by
$\mathcal{T}_{\mathrm{lr},\\{00\\}}=-2(\mathrm{T}^{-1}_{p,0}\mathrm{T}_{p,1}+(p+1)(p^{2}+1)).$
Here the coefficient $-2$ comes from the self-intersection number of the class
$\eta_{\sigma}$ as in Lemma 5.7.
#### 6.2.2. $\mathcal{T}_{\mathrm{lr},\\{22\\}}$
To understand the map
$\mathcal{T}_{\mathrm{lr},\\{22\\}}=\mathrm{inc}^{\ast}_{\\{2\\}}\circ\alpha\circ\mathrm{N}^{-1}_{\Sigma}\circ\beta\circ\mathrm{inc}^{\\{2\\}}_{!}$,
we need to understand the possible relative position of vertex lattices
$\Lambda_{2}$, $\Lambda_{\mathrm{Pa}}$ and $\Lambda^{\prime}_{2}$ of type $2$,
$1$ and $2$. It not hard to see that the only possible way is that
$\Lambda_{2}\subset^{1}\Lambda_{\mathrm{Pa}}$ and
$\Lambda^{\prime}_{2}\subset^{1}\Lambda_{\mathrm{Pa}}$. The rest of the
analysis is completely the same as in the case of
$\mathcal{T}_{\mathrm{lr},\\{00\\}}$. Once we identify $\mathrm{K}_{\\{2\\}}$
with the hyperspecial group $\mathbf{G}(\mathbb{Z}_{p})$, the entry
$\mathcal{T}_{\mathrm{lr},\\{22\\}}$ is given by
$\mathcal{T}_{\mathrm{lr},\\{22\\}}=-2(\mathrm{T}^{-1}_{p,0}\mathrm{T}_{p,1}+(p+1)(p^{2}+1)).$
Here the coefficient $-2$ comes from the self-intersection number of the class
$\eta_{\sigma}$ as in Lemma 5.7.
#### 6.2.3. $\mathcal{T}_{\mathrm{lr},\\{02\\}}$
To understand the map
$\mathcal{T}_{\mathrm{lr},\\{02\\}}=\mathrm{inc}^{\ast}_{\\{2\\}}\circ\alpha\circ\mathrm{N}^{-1}_{\Sigma}\circ\beta\circ\mathrm{inc}^{\\{0\\}}_{!}$,
we need to understand the possible relative position of vertex lattices
$\Lambda_{0}$, $\Lambda_{\mathrm{Pa}}$ and $\Lambda^{\prime}_{2}$ of type $0$,
$1$ and $2$. We find that the only possible situation is that
$\Lambda_{2}\subset^{1}\Lambda_{\mathrm{Pa}}\subset^{1}\Lambda_{0}=\Lambda^{\vee}_{0}\subset^{1}\Lambda^{\vee}_{\mathrm{Pa}}\subset^{1}\Lambda^{\vee}_{2}.$
It follows each such $\Lambda_{\mathrm{Pa}}$ corresponds to a line in the two
dimensional space $\Lambda_{0}/\Lambda_{2}$.
Therefore the entry $\mathcal{T}_{\mathrm{lr},\\{02\\}}$ is given by
$\mathcal{T}_{\mathrm{lr},\\{02\\}}=-2(p+1)\mathrm{char}(\mathrm{K}_{\\{0\\}}\begin{pmatrix}p&&&\\\
&p&&\\\ &&1&\\\
&&&1\end{pmatrix}\mathrm{K}_{\\{2\\}})=-2(p+1)\mathrm{T}_{02}.\ $
#### 6.2.4. $\mathcal{T}_{\mathrm{lr},\\{20\\}}$
To understand the map
$\mathcal{T}_{\mathrm{lr},\\{20\\}}=\mathrm{inc}^{\ast}_{\\{0\\}}\circ\alpha\circ\mathrm{N}^{-1}_{\Sigma}\circ\beta\circ\mathrm{inc}^{\\{2\\}}_{!}$,
we need to understand the possible relative positions of the vertex lattices
$\Lambda_{0}$, $\Lambda_{\mathrm{Pa}}$ and $\Lambda^{\prime}_{2}$ of type $0$,
$1$ and $2$. We find that the only possible way is that
$\Lambda_{2}\subset^{1}\Lambda_{\mathrm{Pa}}\subset^{1}\Lambda_{0}=\Lambda^{\vee}_{0}\subset^{1}\Lambda^{\vee}_{\mathrm{Pa}}\subset^{1}\Lambda^{\vee}_{2}.$
Each $\Lambda_{\mathrm{Pa}}$ corresponds to a line in the two dimensional
space $\Lambda_{0}/\Lambda_{2}$.
Therefore the entry $\mathcal{T}_{\mathrm{lr},\\{20\\}}$ is given by
$\mathcal{T}_{\mathrm{lr},\\{20\\}}=-2(p+1)\mathrm{char}(\mathrm{K}_{\\{0\\}}\begin{pmatrix}p^{-1}&&&\\\
&p^{-1}&&\\\ &&1&\\\
&&&1\end{pmatrix}\mathrm{K}_{\\{2\\}})=-2(p+1)\mathrm{T}_{02}.\ $
This finishes the proof of Proposition 6.10. ∎
We will compute the determinant of the level raising matrix and the
supersingular matrix that we will define in the next section. For this reason,
we will next compute the composite $\mathrm{T}_{20}\circ\mathrm{T}_{02}$ and
$\mathrm{T}_{02}\circ\mathrm{T}_{20}$ in terms of elements in the spherical
Hecke algebra.
###### Lemma 6.12.
The composite $\mathrm{T}_{20}\circ\mathrm{T}_{02}$ and
$\mathrm{T}_{02}\circ\mathrm{T}_{20}$ are both given by
$\mathrm{T}^{-1}_{p,0}\mathrm{T}_{p^{2},2}+(p+1)\mathrm{T}^{-1}_{p,0}\mathrm{T}_{p,1}+(p^{2}+1)(p+1).$
As a result, if $\mathrm{T}_{p,0}$ acts trivially, then
$\mathrm{T}_{20}\circ\mathrm{T}_{02}=\mathrm{T}_{02}\circ\mathrm{T}_{20}=\mathrm{T}^{2}_{p,2}$
###### Proof.
The proof of this lemma is similar to that of the above proposition.
#### 6.2.5. $\mathrm{T}_{20}\circ\mathrm{T}_{02}$
Let $\Lambda_{0}$ be a vertex lattice of type $0$. Then
$\mathrm{T}_{20}\circ\mathrm{T}_{02}(\Lambda_{0})$ classifies a pair of vertex
lattices $(\Lambda_{2},\Lambda^{\prime}_{0})$ of type $2$ and $0$. We need to
understand all the possible relative position of the vertex lattices
$\Lambda_{0}$, $\Lambda_{2}$ and $\Lambda^{\prime}_{0}$. The only possible
situation is given below
$\displaystyle
p\Lambda^{\vee}_{0}\subset^{2}\Lambda_{2}\subset^{2}\Lambda_{0}=\Lambda^{\vee}_{0}\subset^{2}\Lambda^{\vee}_{2}$
$\displaystyle
p\Lambda^{\prime\vee}_{0}\subset^{2}\Lambda_{2}\subset^{2}\Lambda^{\prime}_{0}=\Lambda^{\prime\vee}_{0}\subset^{2}\Lambda^{\vee}_{2}.$
Note that $\Lambda_{2}$ determines an isotropic subspace of the symplectic
space
$\Lambda_{0}\cap\Lambda^{\prime}_{0}/p\Lambda^{\vee}_{0}+p\Lambda^{\prime\vee}_{0}$
of possible dimensions in the set $\\{0,2,4\\}$.
When
$\dim_{\mathbb{F}}\Lambda_{0}\cap\Lambda^{\prime}_{0}/p\Lambda^{\vee}_{0}+p\Lambda^{\prime\vee}_{0}=0$,
then
$\displaystyle\Lambda_{0}\cap\Lambda^{\prime}_{0}\subset^{2}\Lambda_{0}$
$\displaystyle\Lambda_{0}\cap\Lambda^{\prime}_{0}\subset^{2}\Lambda^{\prime}_{0}.$
Hence this case contributes to the $\mathrm{T}_{20}\circ\mathrm{T}_{02}$ by
the double coset operator
$\mathrm{K}_{\\{0\\}}\begin{pmatrix}p&&&\\\ &p&&\\\ &&p^{-1}&\\\
&&&p^{-1}\end{pmatrix}\mathrm{K}_{\\{0\\}}.$
Once we identify $\mathrm{K}_{\\{0\\}}$ with $\mathbf{G}(\mathbb{Z}_{p})$, we
see this is the same as $\mathrm{T}^{-1}_{0,p}\mathrm{T}_{p^{2},2}$.
When
$\dim_{\mathbb{F}}\Lambda_{0}\cap\Lambda^{\prime}_{0}/p\Lambda^{\vee}_{0}+p\Lambda^{\prime\vee}_{0}=2$,
there are $p+1$ choices of $\Lambda_{2}$. And we have
$\displaystyle\Lambda_{0}\cap\Lambda^{\prime}_{0}\subset^{1}\Lambda_{0}$
$\displaystyle\Lambda_{0}\cap\Lambda^{\prime}_{1}\subset^{1}\Lambda^{\prime}_{0}.$
Hence this case contributes to the $\mathrm{T}_{20}\circ\mathrm{T}_{02}$ by
the double coset operator
$(p+1)\mathrm{K}_{\\{0\\}}\begin{pmatrix}p&&&\\\ &1&&\\\ &&1&\\\
&&&p^{-1}\end{pmatrix}\mathrm{K}_{\\{0\\}}.$
Once we identify $\mathrm{K}_{\\{0\\}}$ with $\mathbf{G}(\mathbb{Z}_{p})$, we
see this is the same as $(p+1)\mathrm{T}^{-1}_{0,p}\mathrm{T}_{p,1}$.
When
$\dim_{\mathbb{F}}\Lambda_{0}\cap\Lambda^{\prime}_{0}/p\Lambda^{\vee}_{0}+p\Lambda^{\prime\vee}_{0}=4$,
there are $[\mathbf{G}(\mathbb{Z}_{p}):\mathrm{Sie})]=(p+1)(p^{2}+1)$ choices
of $\Lambda_{2}$. And we have
$\displaystyle\Lambda_{0}\cap\Lambda^{\prime}_{0}\subset^{0}\Lambda_{0}$
$\displaystyle\Lambda_{0}\cap\Lambda^{\prime}_{0}\subset^{0}\Lambda^{\prime}_{0}.$
Hence this case contributes to the $\mathrm{T}_{20}\circ\mathrm{T}_{02}$ by
the constant $(p^{2}+1)(p+1)$. All in all, we obtain the following formula
(6.13)
$\mathrm{T}_{20}\circ\mathrm{T}_{02}=\mathrm{T}^{-1}_{p,0}\mathrm{T}_{p^{2},2}+(p+1)\mathrm{T}^{-1}_{p,0}\mathrm{T}_{p,1}+(p^{2}+1)(p+1).$
#### 6.2.6. $\mathrm{T}_{02}\circ\mathrm{T}_{20}$
The computation is almost the same to the above. Let $\Lambda_{2}$ be a vertex
lattice of type $2$. Then $\mathrm{T}_{02}\circ\mathrm{T}_{20}(\Lambda_{2})$
classifies pair of vertex lattices $(\Lambda_{0},\Lambda^{\prime}_{2})$ of
type $0$ and $2$. We need to understand the relative position of vertex
lattices $\Lambda_{0}$, $\Lambda_{2}$ and $\Lambda^{\prime}_{2}$. The only
possible situation is given below
$\displaystyle
p\Lambda^{\vee}_{0}\subset^{2}\Lambda_{2}\subset^{2}\Lambda_{0}=\Lambda^{\vee}_{0}\subset^{2}\Lambda^{\vee}_{2};$
$\displaystyle
p\Lambda^{\vee}_{0}\subset^{2}\Lambda^{\prime}_{2}\subset^{2}\Lambda_{0}=\Lambda^{\vee}_{0}\subset^{2}\Lambda^{\prime\vee}_{2}.$
Note that $\Lambda_{0}$ is determined by an isotropic subspace of the
symplectic space
$\Lambda_{2}\cap\Lambda^{\prime}_{2}/\Lambda^{\vee}_{2}+\Lambda^{\prime\vee}_{2}$
of possible dimensions $\\{0,2,4\\}$.
When
$\dim_{\mathbb{F}}\Lambda_{2}\cap\Lambda^{\prime}_{2}/\Lambda^{\vee}_{2}+\Lambda^{\prime\vee}_{2}=0$,
then in this case we have
$\displaystyle\Lambda_{2}\subset^{2}\Lambda_{2}+\Lambda^{\prime}_{2}$
$\displaystyle\Lambda^{\prime}_{2}\subset^{2}\Lambda_{2}+\Lambda^{\prime}_{2}.$
Hence this case contributes to the $\mathrm{T}_{02}\circ\mathrm{T}_{20}$ by
the double coset operator
$\mathrm{K}_{\\{2\\}}\begin{pmatrix}p^{-1}&&&\\\ &p^{-1}&&\\\ &&p&\\\
&&&p\end{pmatrix}\mathrm{K}_{\\{2\\}}.$
Once we identify $\mathrm{K}_{\\{2\\}}$ with $\mathbf{G}(\mathbb{Z}_{p})$, we
see this is the same as $\mathrm{T}^{-1}_{0,p}\mathrm{T}_{p^{2},2}$.
When
$\dim_{\mathbb{F}}\Lambda^{\vee}_{2}\cap\Lambda^{\prime\vee}_{2}/\Lambda_{2}+\Lambda^{\prime}_{2}=2$,
there are $p+1$ choices of $\Lambda_{0}$. And in this case we have
$\displaystyle\Lambda_{2}\subset^{1}\Lambda_{2}+\Lambda^{\prime}_{2}$
$\displaystyle\Lambda^{\prime}_{2}\subset^{1}\Lambda_{2}+\Lambda^{\prime}_{2}.$
Hence this case contributes to the $\mathrm{T}_{20}\circ\mathrm{T}_{02}$ by
the Hecke operator
$(p+1)\mathrm{K}_{\\{2\\}}\begin{pmatrix}p^{-1}&&&\\\ &1&&\\\ &&1&\\\
&&&p\end{pmatrix}\mathrm{K}_{\\{2\\}}.$
Once we identify $\mathrm{K}_{\\{2\\}}$ with the hyperspecial subgroup
$\mathbf{G}(\mathbb{Z}_{p})$, we see this is the same as
$(p+1)\mathrm{T}^{-1}_{0,p}\mathrm{T}_{p,1}$.
When
$\dim_{\mathbb{F}}\Lambda^{\vee}_{2}\cap\Lambda^{\prime\vee}_{2}/\Lambda_{2}+\Lambda^{\prime}_{2}=4$,
there are $[\mathbf{G}(\mathbb{Z}_{p}):\mathrm{Sie})]=(p+1)(p^{2}+1)$ choices
of $\Lambda_{0}$. And in this case we have
$\displaystyle\Lambda_{2}\subset^{0}\Lambda_{2}+\Lambda^{\prime}_{2}$
$\displaystyle\Lambda^{\prime}_{2}\subset^{0}\Lambda_{2}+\Lambda^{\prime}_{2}.$
Hence this case contributes to the $\mathrm{T}_{20}\circ\mathrm{T}_{02}$ by
the constant $(p^{2}+1)(p+1)$. All in all, we obtain the following formula
(6.14)
$\mathrm{T}_{20}\circ\mathrm{T}_{02}=\mathrm{T}^{-1}_{0,p}\mathrm{T}_{p^{2},2}+(p+1)\mathrm{T}^{-1}_{0,p}\mathrm{T}_{p,1}+(p^{2}+1)(p+1).$
The last claim in the lemma follows immediately from Proposition 6.9. This
finishes the proof of the lemma. ∎
The final result of this section is the computation of the determinant of the
level raising matrix modulo $\mathfrak{m}$.
###### Proposition 6.15.
Let $(\pi,\Sigma,\mathfrak{m})$ be the datum considered as in the beginning of
the section. Suppose the maximal ideal $\mathfrak{m}$ appears in the support
of $\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]$.
Let $[\alpha_{p},\beta_{p},p^{3}\beta^{-1},p^{3}\alpha^{-1}_{p}]$ be the Hecke
parameter of $\pi$ at $p$. Then the determinant of the level raising matrix
modulo $\mathfrak{m}$ is given by
$\mathrm{det}\mathcal{T}_{\mathrm{lr}/\mathfrak{m}}=4p^{-2}\prod_{\mathrm{u}=\pm
1}(\alpha_{p}+p^{3}\alpha^{-1}_{p}-\mathrm{u}p(p+1))(\beta_{p}+p^{3}\beta^{-1}_{p}-\mathrm{u}p(p+1)).$
###### Proof.
Since $\pi$ has trivial central character, $\mathrm{T}_{p,0}$ acts trivially
on
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]/\mathfrak{m}$.
We therefore have
$\mathrm{T}_{20}\circ\mathrm{T}_{02}=\mathrm{T}_{02}\circ\mathrm{T}_{20}=\mathrm{T}_{p^{2},2}+(p+1)\mathrm{T}_{p,1}+(p^{2}+1)(p+1)=\mathrm{T}^{2}_{p,2}$
by Lemma 6.13. Then
$\displaystyle\mathrm{det}\mathcal{T}_{\mathrm{lr}/\mathfrak{m}}$
$\displaystyle=4((\mathrm{T}_{p,1}+p(p+1))^{2}-(p+1)^{2}\mathrm{T}_{20}\mathrm{T}_{02})$
$\displaystyle=4(\mathrm{T}_{p,1}+p(p+1)-(p+1)\mathrm{T}_{p,2})(\mathrm{T}_{p,1}+p(p+1)+(p+1)\mathrm{T}_{p,2})$
$\displaystyle=4p^{-2}$
$\displaystyle[(\alpha_{p}+p^{3}\alpha^{-1}_{p})(\beta_{p}+p^{3}\beta^{-1}_{p})-p(p+1)(\alpha_{p}+\beta_{p}+p^{3}\alpha^{-1}_{p}+p^{3}\beta^{-1}_{p}+p^{2}(p+1)^{2})]$
$\displaystyle[(\alpha_{p}+p^{3}\alpha^{-1}_{p})(\beta_{p}+p^{3}\beta^{-1}_{p})+p(p+1)(\alpha_{p}+\beta_{p}+p^{3}\alpha^{-1}_{p}+p^{3}\beta^{-1}_{p}+p^{2}(p+1)^{2})]$
$\displaystyle=4p^{-2}\prod_{\mathrm{u}=\pm
1}(\alpha_{p}+p^{3}\alpha^{-1}_{p}-\mathrm{u}p(p+1))(\beta_{p}+p^{3}\beta^{-1}_{p}-\mathrm{u}p(p+1)).$
This finishes the proof. ∎
## 7\. Tate cycles on the quaternionic unitary Shimura variety
In this section, our goal is to establish the following priniciple: the
localized cohomology group
$\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}$
is generated by cycles coming from the supersingular locus. On the dual side,
similar result also holds for
$\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}$,
however we need to show this cohomology group contains no
$\mathcal{O}_{\lambda}$-torsion first. We will establish the full result
simultaneously with our main theorem on arithmetic level raising in Corollary
9.8. Let $(\pi,\Sigma,\mathfrak{m})$ be the datum considered as in the
beginning of the last section.
###### Construction 7.1.
Consider the supersingular locus
$\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa}}(\mathrm{B})$, we have the
following restriction morphism
$\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))\rightarrow\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))$
and we denote the composite of this restriction morphism with the natural map
$\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))\rightarrow\mathrm{H}^{2}(\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))$
by
$\mathrm{inc}^{\ast}_{\\{\mathrm{ss}\\}}:\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))\rightarrow\mathrm{H}^{2}(\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1)).$
We also have the natural Gysin map
$\mathrm{inc}^{\\{\mathrm{ss}\\}}_{!}:\mathrm{H}^{2}(\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))\rightarrow\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2)).$
We can fit the two maps
$(\mathrm{inc}^{\ast}_{\\{\mathrm{ss}\\}},\mathrm{inc}^{\\{\mathrm{ss}\\}}_{!})$
in the above construction in the diagram below.
(7.2)
${\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]}$${\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]}$${\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))}$${\mathrm{H}^{2}(\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))}$${\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))}$${\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]}$${\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]}$$\scriptstyle{\mathrm{inc}^{\\{0\\}}_{!}}$$\scriptstyle{\mathrm{inc}^{\\{2\\}}_{!}}$$\scriptstyle{\mathrm{inc}^{\ast}_{\\{\mathrm{ss}\\}}}$$\scriptstyle{\mathrm{inc}^{\\{\mathrm{ss}\\}}_{!}}$$\scriptstyle{\mathrm{inc}^{\ast}_{\\{0\\}}}$$\scriptstyle{\mathrm{inc}^{\ast}_{\\{2\\}}}$
The above diagram sets up the intersection matrix for the supersingular locus
which we will elaborate below.
###### Construction 7.3.
We obtain naturally the following four maps from the above diagram
$\displaystyle\mathcal{T}_{\mathrm{ss},\\{00\\}}=\mathrm{inc}^{\ast}_{\\{0\\}}\circ\mathrm{inc}^{\\{\mathrm{ss}\\}}_{!}\circ\mathrm{inc}^{\ast}_{\\{\mathrm{ss}\\}}\circ\mathrm{inc}^{\\{0\\}}_{!}$
$\displaystyle\mathcal{T}_{\mathrm{ss},\\{02\\}}=\mathrm{inc}^{\ast}_{\\{2\\}}\circ\mathrm{inc}^{\\{\mathrm{ss}\\}}_{!}\circ\mathrm{inc}^{\ast}_{\\{\mathrm{ss}\\}}\circ\circ\mathrm{inc}^{\\{0\\}}_{!}$
$\displaystyle\mathcal{T}_{\mathrm{ss},\\{20\\}}=\mathrm{inc}^{\ast}_{\\{0\\}}\circ\mathrm{inc}^{\\{\mathrm{ss}\\}}_{!}\circ\mathrm{inc}^{\ast}_{\\{\mathrm{ss}\\}}\circ\mathrm{inc}^{\\{2\\}}_{!}$
$\displaystyle\mathcal{T}_{\mathrm{ss},\\{22\\}}=\mathrm{inc}^{\ast}_{\\{2\\}}\circ\mathrm{inc}^{\\{\mathrm{ss}\\}}_{!}\circ\mathrm{inc}^{\ast}_{\\{\mathrm{ss}\\}}\circ\circ\mathrm{inc}^{\\{2\\}}_{!}.$
The resulting matrix
$\mathcal{T}_{\mathrm{ss}}=\begin{pmatrix}&\mathcal{T}_{\mathrm{ss},\\{00\\}}&\mathcal{T}_{\mathrm{ss},\\{02\\}}\\\
&\mathcal{T}_{\mathrm{ss},\\{20\\}}&\mathcal{T}_{\mathrm{ss},\\{22\\}}\\\
\end{pmatrix}$
will be referred to as the supersingular matrix. We can localize the above
diagram at the maximal ideal $\mathfrak{m}$ and write the resulting matrix as
$\mathcal{T}_{\mathrm{ss},\mathfrak{m}}$, however all the matrix entries will
still be denoted by symbols without referring to $\mathfrak{m}$. If we need to
consider this matrix modulo $\mathfrak{m}$, then we will denote it by
$\mathcal{T}_{\mathrm{ss}/\mathfrak{m}}$. The entries of this matrix will be
denoted without referring to $\mathfrak{m}$.
### 7.1. Computing the supersingular matrix
We compute the entries of the supersingular matrix using the same procedure as
we did for computing the level raising matrix.
###### Proposition 7.4.
The supersingular matrix $\mathcal{T}_{\mathrm{ss}}$ is given by
$\mathcal{T}_{\mathrm{ss}}=\begin{pmatrix}&4p^{2}(p+1)^{2}+\mathrm{T}_{20}\circ\mathrm{T}_{02}&-4p(p+1)\mathrm{T}_{02}\\\
&-4p(p+1)\mathrm{T}_{20}&4p^{2}(p+1)^{2}+\mathrm{T}_{02}\circ\mathrm{T}_{20}\\\
\end{pmatrix}.$
###### Proof.
We will prove the proposition in an entry by entry manner.
#### 7.1.1. $\mathcal{T}_{\mathrm{ss},\\{00\\}}$
Recall
$\mathcal{T}_{\mathrm{ss},\\{00\\}}=\mathrm{inc}^{\ast}_{\\{0\\}}\circ\mathrm{inc}^{\\{\mathrm{ss}\\}}_{!}\circ\mathrm{inc}^{\ast}_{\\{\mathrm{ss}\\}}\circ\mathrm{inc}^{\\{0\\}}_{!}$.
If the maps
$\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))\xrightarrow{\mathrm{inc}^{\ast}_{\\{\mathrm{ss}\\}}}\mathrm{H}^{2}(\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))\xrightarrow{\mathrm{inc}^{\\{\mathrm{ss}\\}}_{!}}\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))$
factor through
$\mathrm{H}^{2}(\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa},\\{0\\}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))$.
Then the contribution is $4p^{2}(p+1)^{2}$. This follows from the fact that
for each $\Lambda_{0}\in\mathcal{L}_{\\{0\\}}$ the degree of the variety
$\mathrm{DL}(\Lambda_{0})$ has degree $(p+1)$ and the normal bundle has degree
$-2p$ by Lemma 2.10. If the maps
$\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))\xrightarrow{\mathrm{inc}^{\ast}_{\\{\mathrm{ss}\\}}}\mathrm{H}^{2}(\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))\xrightarrow{\mathrm{inc}^{\\{\mathrm{ss}\\}}_{!}}\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))$
factor through
$\mathrm{H}^{2}(\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa},\\{2\\}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))$,
then as we have seen before we need to understand the relative position of the
vertex lattices $\Lambda_{0}$, $\Lambda_{2}$ and $\Lambda^{\prime}_{0}$ of
type $0$, $2$ and $0$. The only possible situation is given by
$\displaystyle
p\Lambda^{\vee}_{0}\subset^{2}\Lambda_{2}\subset^{2}\Lambda_{0}=\Lambda^{\vee}_{0}\subset^{2}\Lambda^{\vee}_{2}$
$\displaystyle
p\Lambda^{\prime\vee}_{0}\subset^{2}\Lambda_{2}\subset^{2}\Lambda^{\prime}_{0}=\Lambda^{\prime\vee}_{0}\subset^{2}\Lambda^{\vee}_{2}.$
As we have seen, this is parametrized by
$\mathrm{T}_{20}\circ\mathrm{T}_{02}$. Therefore this entry is given by
$\mathcal{T}_{\mathrm{ss},\\{00\\}}=(p+1)^{2}+\mathrm{T}_{20}\circ\mathrm{T}_{02}.$
#### 7.1.2. $\mathcal{T}_{\mathrm{ss},\\{02\\}}$
If the maps
$\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))\xrightarrow{\mathrm{inc}^{\ast}_{\\{\mathrm{ss}\\}}}\mathrm{H}^{2}(\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))\xrightarrow{\mathrm{inc}^{\\{\mathrm{ss}\\}}_{!}}\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))$
factor through
$\mathrm{H}^{2}(\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa},\\{0\\}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))$.
This case contributes by $-2p(p+1)\mathrm{T}_{02}$. If the maps factor through
$\mathrm{H}^{2}(\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa},\\{2\\}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))$.
This case contributes by $-2p(p+1)\mathrm{T}_{02}$. Therefore the total
contribution gives the entry
$\mathcal{T}_{\mathrm{ss},\\{02\\}}=-4p(p+1)\mathrm{T}_{02}.$
#### 7.1.3. $\mathcal{T}_{\mathrm{ss},\\{20\\}}$
The computation is the same as in the case for
$\mathcal{T}_{\mathrm{ss},\\{02\\}}$. Thus this entry is given by
$\mathcal{T}_{\mathrm{ss},\\{20\\}}=-4p(p+1)\mathrm{T}_{20}.$
#### 7.1.4. $\mathcal{T}_{\mathrm{ss},\\{22\\}}$
The computation is exactly the same as in the case for
$\mathcal{T}_{\mathrm{ss},\\{00\\}}$. Thus this entry is given by
$\mathcal{T}_{\mathrm{ss},\\{22\\}}=4p^{2}(p+1)^{2}+\mathrm{T}_{02}\circ\mathrm{T}_{20}.$
This finishes the proof of Proposition 7.4. ∎
###### Proposition 7.5.
Let $(\pi,\Sigma,\mathfrak{m})$ be the datum considered as in the beginning of
the section. Suppose the maximal ideal $\mathfrak{m}$ appears in the support
of $\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]$.
Suppose that $[\alpha_{p},\beta_{p},p^{3}\beta^{-1},p^{3}\alpha^{-1}_{p}]$ is
the Hecke parameter of $\pi$ at $p$. Then determinant of the supersingular
matrix modulo $\mathfrak{m}$ is given by
$\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{ss}/\mathfrak{m}}=\prod_{\mathrm{u}=\pm
1}(\alpha_{p}+\beta_{p}+p^{3}\alpha^{-1}_{p}+p^{3}\beta^{-1}_{p}-2\mathrm{u}p(p+1))^{2}.$
###### Proof.
Since $\pi$ has trivial central character, then $\mathrm{T}_{p,0}$ acts
trivially on
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]/\mathfrak{m}$.
We therefore have
$\mathrm{T}_{20}\circ\mathrm{T}_{02}=\mathrm{T}_{02}\circ\mathrm{T}_{20}=\mathrm{T}_{p^{2},2}+(p+1)\mathrm{T}_{p,1}+(p^{2}+1)(p+1)=\mathrm{T}^{2}_{p,2}$
by Lemma 6.13. Then we have
$\displaystyle\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{ss}/\mathfrak{m}}$
$\displaystyle=(\mathrm{T}^{2}_{p,2}+4p^{2}(p+1))^{2}-16p^{2}(p+1)^{2}\mathrm{T}^{2}_{p,2}$
$\displaystyle=(\mathrm{T}^{2}_{p,2}+4p^{2}(p+1)-4p(p+1)\mathrm{T}_{p,2})$
$\displaystyle\phantom{abc}(\mathrm{T}^{2}_{p,2}+4p^{2}(p+1)+4p(p+1)\mathrm{T}_{p,2})$
$\displaystyle=(\alpha_{p}+\beta_{p}+p^{3}\beta^{-1}_{p}+p^{3}\alpha^{-1}_{p}-2p(p+1))^{2}$
$\displaystyle\phantom{abc}(\alpha_{p}+\beta_{p}+p^{3}\beta^{-1}_{p}+p^{3}\alpha^{-1}_{p}+2p(p+1))^{2}$
$\displaystyle=\prod_{\mathrm{u}=\pm
1}(\alpha_{p}+\beta_{p}+p^{3}\beta^{-1}_{p}+p^{3}\alpha^{-1}_{p}-2\mathrm{u}p(p+1))^{2}.$
∎
### 7.2. Tate classes in the quaternionic unitary Shimura variety
We fix the datum $(\pi,\Sigma,\mathfrak{m})$ as in the beginning of this
section. Let $p$ be a level raising special prime for $\pi$ of length $m$. We
make the following assumption regarding to the vanishing of the nearby cycle
cohomology of the quaternionic unitary Shimura variety.
###### Assumption 7.6.
Let
$\mathrm{H}^{d}_{(\mathrm{c})}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}))_{\mathfrak{m}}$
be the usual or compact support nearby cycle cohomology of the special fiber
$\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B})$. We assume the maximal ideal
$\mathfrak{m}$ satisfies the following properties
1. (1)
The natural morphism
$\mathrm{H}^{d}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}))_{\mathfrak{m}}\xrightarrow{\sim}\mathrm{H}^{d}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}))_{\mathfrak{m}}$
is an isomorphism for all $d$.
2. (2)
The cohomology
$\mathrm{H}^{d}_{(\mathrm{c})}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}))_{\mathfrak{m}}=0$
for $d\neq 3$, and that the middle degree cohomology
$\mathrm{H}^{3}_{(\mathrm{c})}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}))_{\mathfrak{m}}$
is a finite free $\mathcal{O}_{\lambda}$ module.
###### Remark 7.7.
The first assumption is usually known in literature as that of $\mathfrak{m}$
being a non-Eisenstein ideal. It is satisfied if $\mathfrak{m}$ is associated
to a cuspidal automorphic representation of general type. The second
assumption is more serious. This is reasonable assumption in light of the
recent work of Caraiani-Scholze [CS17, CS19] and recent advances by Koshikawa
[Kos19, Kos21]. Although they deal with certain unitary Shimura varieties, the
methods are general enough to treat at least our cases: if one assumes that
$\pi$ has an unramified type $\mathrm{I}$ component which is generic modulo
$l$, then combining the proof of Koshikawa [Kos21] with a semi-perversity
result of the pushforward along the Hodge-Tate period map [San22], one can
prove that $\mathfrak{m}$ satisfies $(2)$.
If we make the assumption that $\mathrm{X}_{\mathrm{Pa}}(\mathrm{B})$ has good
reduction at $l$ and that $\pi$ is ordinary at $l$, then the method of [MT02]
could be used to show that the Assumption 7.6 is fulfilled.
We also conjecture that the product of Shimura curves for $\mathrm{B}$
embedded in $\mathrm{Sh}(\mathrm{B},\mathrm{K}_{\mathrm{Pa}})$ has an affine
complement. Then the above assumption can be proved if one assumes that the
residual Galois representation has large image. This is the analogue of the
fact that the Igusa divisor on the classical Siegel threefold has an affine
complement.
###### Definition 7.8.
If $\mathfrak{m}$ satisfies either of the two conditions below, we call
$\mathfrak{m}$ a generic maximal ideal.
1. (1)
We say $\mathfrak{m}$ is generic level raising at $p$ if it is level raising
special at $p$ and Assumption 7.6 is satisfied.
2. (2)
We say $\mathfrak{m}$ is generic non-level raising at $p$ if the Hecke
parameters for $\pi$ satisfies
$\displaystyle\alpha_{p}+p^{3}\alpha^{-1}_{p}\not\equiv\pm(p+p^{2})\mod\lambda$
$\displaystyle\beta_{p}+p^{3}\beta^{-1}_{p}\not\equiv\pm(p+p^{2})\mod\lambda$
and Assumption 7.6 is satisfied.
###### Theorem 7.9.
Let $\mathfrak{m}$ be a generic maximal ideal as in the above definition, we
have the following statements.
1. (1)
The map from Construction 6.4
$(\mathrm{inc}^{\\{0\\}}_{!,\mathfrak{m}}+\mathrm{inc}^{\\{2\\}}_{!,\mathfrak{m}}):\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\rightarrow\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}$
is an isomorphism.
2. (2)
The map from Construction 6.4
$(\mathrm{inc}^{\ast}_{\\{0\\},\mathfrak{m}},\mathrm{inc}^{\ast}_{\\{2\\},\mathfrak{m}}):\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}\rightarrow\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$
is surjective and whose kernel is the torsion part of
$\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}$.
###### Proof.
Suppose for the moment that $\mathfrak{m}$ is generic level raising at $p$.
Then the determinant of the supersingular matrix
$\mathrm{det}\mathcal{T}_{\mathrm{ss}/\mathfrak{m}}$ is given by
$\prod_{\mathrm{u}=\pm
1}(\alpha_{p}+\beta_{p}+p^{3}\beta^{-1}_{p}+p^{3}\alpha^{-1}_{p}-2\mathrm{u}p(p+1))^{2}.$
This is non-zero by our definition of the level raising condition. This
immediately implies that we have an injection
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\hookrightarrow\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}$
by the Nakayama’s lemma. Note that
$\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}$
is a free $\mathcal{O}_{\lambda}$-module as it injects into
$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda})(1))_{\mathfrak{m}}$
which is torsion free by the genericness of $\mathfrak{m}$. Therefore to
finish the proof in this case, we need to show that
$2\mathrm{rank}_{\mathcal{O}_{\lambda}}\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\geq\mathrm{rank}_{\mathcal{O}_{\lambda}}\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}.$
For this purpose, let $\vec{\pi}$ be an automorphic representation of
$\mathbf{G}(\mathbb{A}_{\mathbb{Q}})$ of general type. It suffices to show
that
$2\dim_{\overline{\mathbb{Q}}_{l}}\overline{\mathbb{Q}}_{l}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})][\iota_{l}\vec{\pi}^{\infty
pq}]\geq\dim_{\overline{\mathbb{Q}}_{l}}\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\overline{\mathbb{Q}}_{l}(1))[\iota_{l}\vec{\pi}^{\infty
pq}].$
Consider the co-specialization exact sequence
$\displaystyle 0$
$\displaystyle\rightarrow\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\overline{\mathbb{Q}}_{l}(1))[\iota_{l}\vec{\pi}^{\infty
pq}]\rightarrow\bigoplus_{\sigma\in\mathrm{Z}_{\\{1\\}}(\overline{\mathrm{B}})}\mathrm{H}^{3}_{\\{\sigma\\}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\overline{\mathbb{Q}}_{l})(1))[\iota_{l}\vec{\pi}^{\infty
p}]$
$\displaystyle\rightarrow\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\overline{\mathbb{Q}}_{l})(1))[\iota_{l}\vec{\pi}^{\infty
p}]\rightarrow\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\overline{\mathbb{Q}}_{l}(1))[\iota_{l}\vec{\pi}^{\infty
p}]\rightarrow 0$
We can canonically identify the space
$\bigoplus\limits_{\sigma\in\mathrm{Z}_{\\{1\\}}(\overline{\mathrm{B}})}\mathrm{H}^{3}_{\\{\sigma\\}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\overline{\mathbb{Q}}_{l})[\iota_{l}\vec{\pi}^{\infty
p}]$
with the space
$\overline{\mathbb{Q}}_{l}[\mathrm{Z}_{\mathrm{Pa}}(\overline{\mathrm{B}})][\iota_{l}\vec{\pi}^{\infty
p}]$. Note that to complete $\vec{\pi}^{p\infty}$ to a representation
$\vec{\pi}$ of $\mathbf{G}(\mathbb{A}_{\mathbb{Q}})$ occuring in
$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\overline{\mathbb{Q}}_{l})(1))$,
$\vec{\pi}_{p}$ must be of type $\mathrm{II}\mathrm{a}$ as a representation of
$\mathbf{G}(\mathbb{Q}_{p})$ by the proof of Theorem 4.5. On the other hand,
those $\vec{\pi}_{p}$ completing $\vec{\pi}^{\infty p}$ in
$\overline{\mathbb{Q}}_{l}[\mathrm{Z}_{\mathrm{Pa}}(\overline{\mathrm{B}})][\iota_{l}\pi^{\infty
p}]$ can be either the unramified principal series or of type
$\mathrm{II}\mathrm{a}$. In the proof of Theorem 4.5, we have seen the the
Jacquet–Langlands correspondence preserves the automorphic multiplicity. Thus
by the uniqueness of local newforms [BS, Theorem 5.6.1] or [Sch05, Theorem
2.3.1], those $\vec{\pi}_{p}$ completing $\vec{\pi}^{\infty p}$ in
$\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\overline{\mathbb{Q}}_{l}(1))[\iota_{l}\vec{\pi}^{\infty
p}]$ can only be unramified unless the map
$\bigoplus_{\sigma\in\mathrm{Z}_{\\{1\\}}(\overline{\mathrm{B}})}\mathrm{H}^{3}_{\\{\sigma\\}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\overline{\mathbb{Q}}_{l})(1))[\iota_{l}\vec{\pi}^{\infty
p}]\rightarrow\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\overline{\mathbb{Q}}_{l})(1))[\iota_{l}\vec{\pi}^{\infty
p}]\ $
is zero. Howerver this is impossible as otherwise, it would imply the
isomorphism
$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\overline{\mathbb{Q}}_{l})(1))[\iota_{l}\vec{\pi}^{\infty
p}]\rightarrow\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\overline{\mathbb{Q}}_{l}(1))[\iota_{l}\vec{\pi}^{\infty
p}]$
which is absurd by considering the inertia action. By the oldforms principle
[BS, Theorem 5.6.1] or [Sch05, Theorem 2.3.1], we have the desired
$2\dim_{\overline{\mathbb{Q}}_{l}}\overline{\mathbb{Q}}_{l}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})][\iota_{l}\vec{\pi}^{\infty
p}]\geq\dim_{\overline{\mathbb{Q}}_{l}}\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\overline{\mathbb{Q}}_{l}(1))[\iota_{l}\vec{\pi}^{\infty
p}].$
Next we show the same result under the assumption that $\mathfrak{m}$ is
generic non-level raising. Suppose that $\mathfrak{m}$ is not level raising
special. Then the determinant of the level raising matrix
$\mathrm{det}\mathcal{T}_{\mathrm{lr}/\mathfrak{m}}$ is given by
$\mathrm{det}\mathcal{T}_{\mathrm{lr}/\mathfrak{m}}=4p^{-2}\prod_{\mathrm{u}=\pm
1}(\alpha_{p}+p^{3}\alpha^{-1}_{p}-\mathrm{u}p(p+1))(\beta_{p}+p^{3}\beta^{-1}_{p}-\mathrm{u}p(p+1)).$
This is non-zero by our assumption. This immediately implies that we have an
injection
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\hookrightarrow\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}$
by the Nakayama’s lemma. The rest of the argument is the same as in the
previous case. Finally, the second part of the proposition is dual to the
first part. In particular, the map
$(\mathrm{inc}^{\ast}_{\\{0\\},\mathfrak{m}},\mathrm{inc}^{\ast}_{\\{2\\},\mathfrak{m}}):\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}\rightarrow\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$
is surjective. Using the same argument as in the first part, we see that
$(\mathrm{inc}^{\ast}_{\\{0\\},\mathfrak{m}},\mathrm{inc}^{\ast}_{\\{2\\},\mathfrak{m}})$
is an isomorphism up to the torsion of
$\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}$.
The second part now follows ∎
We finish this section with several remarks about this theorem. First of all,
this result can be understood as the statement that the cohomology of the
special fiber
$\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}$
is generated by Tate cycles coming from the supersingular locus. This result
along with [LTXZZ, Proposition 6.3.1] can be seen as an analogue of the main
result of [XZ] in the bad reduction case for low degree cohomology groups
localized at generic maximal ideals. It seems that these cohomology groups
always correspond to oldforms. It would be of great interest to formulate some
predictions in a similar manner of [XZ]. Secondly, we have established the
following analogue of the Ihara’s lemma in the setting of definite
quaternionic unitary groups.
###### Corollary 7.10.
There is an injection
$\mathbf{Ih}_{\mathfrak{m}}:\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\xhookrightarrow{\phantom{aaaa}}\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{1\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}.$
###### Proof.
We have shown in the above proof that the composite of the two maps
$\displaystyle\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\xrightarrow{(\mathrm{inc}^{\\{0\\}}_{!,\mathfrak{m}}+\mathrm{inc}^{\\{2\\}}_{!,\mathfrak{m}})}\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}$
$\displaystyle\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}\xhookrightarrow{\phantom{aa}\alpha\phantom{aa}}\bigoplus_{\sigma\in
Z_{\\{1\\}}(\overline{\mathrm{B}})}\mathrm{H}^{3}_{\\{\sigma\\}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda})(1))=\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{1\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$
gives rise to an injection which we call
$\mathbf{Ih}_{\mathfrak{m}}:\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\xhookrightarrow{\phantom{aaaa}}\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{1\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}.$
∎
Note that there is no natural degeneracy map from neither
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$
nor
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$
to
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{1\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$.
Instead, these maps should be the analogues of the level raising operators in
the sense of [BS]. Finally suppose that $\mathfrak{m}$ is generic level
raising at $p$, we will show that
$\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}$
is torsion free following the method of [LTXZZ] using Galois deformation
argument.
## 8\. Galois deformation rings and cohomology of Shimura varieties
In this section, we recall the main results in [Wang22a] where we use the
method of [LTXZZa] to show that the cohomology of the quaternionic unitary
Shimura varieties are free over certain universal deformation rings. These
results are necessary as we lack of torsion vanishing results on the
cohomology of the special fibers of these quaternionic unitary Shimura
varieties.
### 8.1. Galois representation for $\mathrm{GSp}_{4}$
First we recall some results on associating Galois representations to cuspidal
automorphic representations of $\mathrm{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ and
the local-global compatibility of Langlands correspondence. Let $\pi$ be a
cuspidal automorphic representation of
$\mathrm{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ with trivial central character. We
assume that $\pi$ has weight $(k_{1},k_{2})$ and trivial central character. In
this article, we will only consider discrete automorphic representation of
$\mathrm{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ that is of general type. This
means there is a cuspidal automorphic representation $\Pi$ of
$\mathrm{GL}_{4}(\mathbb{A}_{\mathbb{Q}})$ of symplectic type such that for
each place $v$ of $\mathbb{Q}$, the $L$-parameter obtained from
$\mathrm{rec}_{\mathrm{GT}}(\pi_{v})$ by composing with the embedding
$\mathrm{GSp}_{4}\hookrightarrow\mathrm{GL}_{4}$ is precisely
$\mathrm{rec}_{\mathrm{GL}_{4}}(\Pi_{v})$ which is the Langlands parameter
attached to $\Pi_{v}$. Here $\Pi$ is of symplectic type if the partial
$L$-function $L^{\mathrm{S}}(s,\Pi,\wedge^{2})$ has a pole at $s=1$ for any
finite set $S$ of places of $\mathbb{Q}$. In this case, we say $\Pi$ is the
transfer of $\pi$.
We now recall some general results of Mok and Sorensen on the existence of
Galois representations attached to $\pi$. This theorem is very general and we
temporarily drop the assumption that $\pi$ is of general type.
###### Theorem 8.1.
Suppose $\pi$ is a cuspidal automorphic representation of
$\mathrm{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ of weight $(k_{1},k_{2})$ where
$k_{1}\geq k_{2}\geq 3$ and $k_{1}\equiv k_{2}\mod 2$. Suppose that $\pi$ has
trivial central character. Let $l\geq 5$ be a fixed prime.Then there is a
continuous semisimple representation
$\rho_{\pi,\iota_{l}}:\mathrm{G}_{\mathbb{Q}}\rightarrow\mathrm{GSp}_{4}(\overline{\mathbb{Q}}_{l})$
satisfying the following properties.
1. (1)
$\mathrm{c}\circ\rho_{\pi,\iota_{l}}=\epsilon^{-3}_{l}$;
2. (2)
For each finite place $v$, we have
$\mathrm{WD}(\rho_{\pi,\iota_{l}}|_{\mathrm{G}_{\mathbb{Q}_{v}}})^{\mathrm{F}-\mathrm{ss}}\cong\mathrm{rec}_{\mathrm{GT}}(\pi_{v}\otimes|\mathrm{c}|^{-3/2})^{\mathrm{ss}};$
3. (3)
The local representation $\rho_{\pi,\iota_{l}}|_{\mathrm{G}_{\mathbb{Q}_{l}}}$
is de Rham with Hodge–Tate weights
$(2-\frac{k_{1}+k_{2}}{2},-\frac{k_{1}-k_{2}}{2},1+\frac{k_{1}-k_{2}}{2},-1+\frac{k_{1}+k_{2}}{2}).$
4. (4)
If $\pi$ is unramified at $l$, then
$\rho_{\pi,\iota_{l}}|_{\mathrm{G}_{\mathbb{Q}_{v}}}$ is moreover crystalline
at $l$.
5. (5)
If $\rho_{\pi,\iota_{l}}$ is irreducible, then for all finite place $v$,
$\rho_{\pi,\iota_{l}}|_{\mathrm{G}_{\mathbb{Q}_{v}}}$ is pure.
###### Remark 8.2.
Several remarks about this theorem are in order. The construction of the
Galois representation in this case is given in [Lau97, Lau05, Wei05, Tay93]
using the Langlands–Kottwitz method. The statements about local-global
compatibilities are mostly proved in [Sor10] and completed in [Mok05]. In fact
the authors use the strong transfer of cuspidal automorphic representation
from $\mathrm{GSp}_{4}$ to $\mathrm{GL}_{4}$ to construct the desired Galois
representation in the totally real field case. The Harish-Chandra parameter
$(\mu_{1},\mu_{2})$ of $\pi$ used more often has the following relation with
the weight of $\pi$ used in this theorem: $k_{1}=\mu_{1}+1$ and
$k_{2}=\mu_{2}+2$. In this article, we will be interested in the case $\pi$ is
of general type and has trivial central character and weights $(3,3)$. In our
normalization, this is the case when these representation appear in the
cohomology of quaternionic unitary Shimura varieties with trivial
coeffficient.
###### Definition 8.3.
Let $\pi$ be a cuspidal automorphic representation of
$\mathrm{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ as in the previous remark. We say
a number field $E\subset\mathbb{C}$ is a strong coefficient field of $\pi$ if
for every prime $\lambda$ of $E$ there exits a continuous homomorphism
$\rho_{\pi,\lambda}:\mathrm{G}_{\mathbb{Q}}\rightarrow\mathrm{GSp}_{4}(E_{\lambda})$
up to conjugation, such that for every isomorphism
$\iota_{l}:\mathbb{C}\xrightarrow{\sim}\overline{\mathbb{Q}}_{l}$ inducing the
prime $\lambda$, the representations
$\rho_{\pi,\lambda}\otimes_{E_{\lambda}}\overline{\mathbb{Q}}_{l}$ and
$\rho_{\pi,\iota_{l}}$ are conjugate.
Let $p$ be a place at which $\pi$ is an unramified type $\mathrm{I}$
representation. Then the Frobenius eigenvalues of
$\rho_{\pi,\lambda}(\mathrm{Frob}_{p})$ agree with the Hecke parameter
$[\alpha_{p},\beta_{p},\gamma_{p},\delta_{p}]$ of $\pi_{p}$ by Theorem 8.1
$(2)$. We will always make the following assumption on $\rho_{\pi,\lambda}$.
###### Assumption 8.4.
The Galois representation
$\rho_{\pi,\lambda}:\mathrm{G}_{\mathbb{Q}}\rightarrow\mathrm{GSp}_{4}(E_{\lambda})$
is residually absolutely irreducible.
The above assumption allows us to define the residual Galois representation
(8.5)
$\overline{\rho}_{\pi,\lambda}:\mathrm{G}_{\mathbb{Q}}\rightarrow\mathrm{GSp}_{4}(k)$
which is unique up to conjugation. Note it also follows that $\pi$ is
necessarily of general type if assumption 8.4 is effective.
### 8.2. Galois deformation rings
We summarize the main results in [Wang22a]. Suppose we are given a Galois
representation
$\overline{\rho}:\mathrm{G}_{\mathbb{Q}}\rightarrow\mathrm{GSp}_{4}(k)$. We
will write the restriction
$\overline{\rho}|_{\mathrm{G}_{\mathbb{Q}_{v}}}:\mathrm{G}_{\mathbb{Q}_{v}}\rightarrow\mathrm{GSp}(k)$
by $\overline{\rho}_{v}$.
Let $\Sigma_{\mathrm{min}}$ and $\Sigma_{\mathrm{lr}}$ be two sets of non-
archimedean places of $\mathbb{Q}$ away from $l$ and such that
* •
$q$ is contained in $\Sigma_{\mathrm{lr}}$;
* •
$\Sigma_{\mathrm{min}}$, $\Sigma_{\mathrm{lr}}$ and $\\{p\\}$ are mutually
disjoint;
* •
for every $v\in\Sigma_{\mathrm{lr}}\cup\\{p\\}$, the prime $v$ satisfies
$l\nmid(v^{2}-1)$.
###### Definition 8.6.
For a Galois representation
$\overline{\rho}:\mathrm{G}_{\mathbb{Q}}\rightarrow\mathrm{GSp}(k)$ and
$\psi=\epsilon^{-3}_{l}$, we say the pair $\overline{\rho}$ is rigid for
$(\Sigma_{\mathrm{min}},\Sigma_{\mathrm{lr}})$ if the followings are
satisfied.
* •
For every $v\in\Sigma_{\mathrm{min}}$, every lifting of $\overline{\rho}_{v}$
is minimally ramified in the sense of [Wang22a, Definition 3.4];
* •
For every $v\in\Sigma_{\mathrm{lr}}$, the generalized eigenvalues of
$\overline{\rho}_{v}(\phi_{v})$ contain the pair $\\{v^{-1},v^{-2}\\}$ exactly
once;
* •
At $v=l$, $\overline{\rho}_{v}$ is regular Fontaine-Laffaille crystalline as
in [Wang22a, Definition 3.10];
* •
For $v\not\in\Sigma_{\mathrm{min}}\cup\Sigma_{\mathrm{lr}}\cup\\{l\\}$, the
representation $\overline{\rho}_{v}$ is unramified.
Suppose that $\overline{\rho}$ is rigid for
$(\Sigma_{\mathrm{min}},\Sigma_{\mathrm{lr}})$. We consider a global
deformation problem in the sense of [Wang22a, Definition 2.2] of the form
$\
\mathcal{S}^{\ast}=(\overline{\rho},\psi,\Sigma_{\mathrm{min}}\cup\Sigma_{\mathrm{lr}}\cup\\{p\\}\cup\\{l\\},\\{\mathcal{D}_{v}\\}_{v\in\Sigma_{\mathrm{min}}\cup\Sigma_{\mathrm{lr}}\cup\\{p\\}\cup\\{l\\}})$
where $\ast=\\{\mathrm{ram},\mathrm{unr},\mathrm{mix}\\}$.
* •
For $v\in\Sigma_{\mathrm{min}}$, $\mathcal{D}_{v}$ is the local deformation
problem $\mathcal{D}^{\mathrm{min}}_{v}$ classifying all the minimal ramified
liftings defined in [Wang22a, Defintion 3.4];
* •
For $v\in\Sigma_{\mathrm{lr}}\cup\\{q\\}$, $\mathcal{D}_{v}$ is the local
deformation problem $\mathcal{D}^{\mathrm{ram}}_{v}$ defined as in [Wang22a,
Definition 3.6 (2)] classifying certain ramified liftings;
* •
For $v=l$, $\mathcal{D}_{v}$ is the local deformation problem
$\mathcal{D}^{\mathrm{FL}}_{v}$ defined as in [Wang22a, Definition 3.1.1]
classifying regular Fontaine–Laffaille crystalline liftings;
* •
For $v=p$, depending on $\ast\in\\{\mathrm{ram},\mathrm{unr},\mathrm{mix}\\}$,
we have
* –
$\mathcal{D}_{v}$ is the local deformation problem
$\mathcal{D}^{\mathrm{unr}}_{v}$ which is given by [Wang22a, Definition 3.6
(2)] when $\ast=\mathrm{unr}$;
* –
$\mathcal{D}_{v}$ is the local deformation problem
$\mathcal{D}^{\mathrm{ram}}_{v}$ which is given by [Wang22a, Definition 3.6
(3)] when $\ast=\mathrm{ram}$;
* –
$\mathcal{D}_{v}$ is the local deformation problem
$\mathcal{D}^{\mathrm{mix}}_{v}$ defined in [Wang22a, Definition 3.6 (1)] when
$\ast=\mathrm{mix}$;
* •
For each global deformation problem $\mathcal{S}^{\ast}$ with
$\ast=\\{\mathrm{ram},\mathrm{unr},\mathrm{mix}\\}$, we have its corresponding
universal deformation ring denoted by
$\mathrm{R}^{\ast}=\mathrm{R}^{\mathrm{univ}}_{\mathcal{S}^{\ast}}$ for
$\ast=\\{\mathrm{ram},\mathrm{unr},\mathrm{mix}\\}$.
###### Construction 8.7.
When $\mathrm{B}$ is indefinite, we choose the level
$\mathrm{K}=\mathrm{K}_{\mathrm{Pa}}$ for the Shimura variety
$\mathrm{Sh}(\mathrm{B},\mathrm{K}_{\mathrm{Pa}})$ in the following way.
* •
For $v\not\in\Sigma_{\mathrm{lr}}\cup\Sigma_{\mathrm{min}}$ or $v=l$, then
$\mathrm{K}_{v}$ is hyperspecial;
* •
For $v\in\Sigma_{\mathrm{lr}}-\\{p,q\\}$, then $\mathrm{K}_{v}$ is the
paramodular subgroup of $\mathrm{GSp}(\mathbb{Z}_{v})$;
* •
For $v\in\Sigma_{\mathrm{min}}$, then $\mathrm{K}_{v}$ is contained in the
pro-$v$ Iwahori subgroup $\mathrm{Iw}_{1}(v)$;
* •
For $v=p,q$, then $\mathrm{K}_{v}$ is the paramodular subgroup
$\mathbf{G}(\mathrm{B})(\mathbb{Z}_{v})$ of $\mathrm{GU}_{2}(\mathrm{D})$ and
$\mathrm{GU}_{2}(\mathrm{D}^{\prime})$ respectively;
Let
$\Sigma_{\mathrm{Pa}}=\Sigma_{\mathrm{min}}\cup\Sigma_{\mathrm{lr}}\cup\\{p,q\\}$.
We denote by $\mathbf{T}^{\mathrm{ram}}$ the image of
$\mathbb{T}^{\Sigma_{\mathrm{Pa}}}$ in
$\operatorname{End}_{\mathcal{O}_{\lambda}}(\mathrm{H}^{3}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}))).$
###### Construction 8.8.
We choose the level $\overline{\mathrm{K}}=\mathrm{K}_{\mathrm{H}}$ for the
Shimura variety $\mathrm{Sh}(\overline{\mathrm{B}},\mathrm{K}_{\mathrm{H}})$
when $\overline{\mathrm{B}}$ is definite in such the following way.
* •
For $v\not\in\Sigma_{\mathrm{lr}}\cup\Sigma_{\mathrm{min}}$ or $v=l$ or $v=p$,
then $\overline{\mathrm{K}}_{v}$ is hyperspecial;
* •
For $v\in\Sigma_{\mathrm{lr}}$, then $\overline{\mathrm{K}}_{v}$ is the
paramodular subgroup of $\mathrm{GSp}(\mathbb{Z}_{v})$;
* •
For $v\in\Sigma_{\mathrm{min}}$, then $\overline{\mathrm{K}}_{v}$ is contained
in the pro-$v$ Iwahori subgroup $\mathrm{Iw}_{1}(v)$;
* •
For $v=q$, then $\overline{\mathrm{K}}_{v}$ is the paramodular subgroup
$\mathbf{G}(\overline{\mathrm{B}})(\mathbb{Z}_{v})$ of
$\mathrm{GU}_{2}(\mathrm{D}^{\prime})$.
Let
$\Sigma_{\mathrm{H}}=\Sigma_{\mathrm{min}}\cup\Sigma_{\mathrm{lr}}\cup\\{q\\}$.
We denote by $\mathbf{T}^{\mathrm{unr}}$ the image of
$\mathbb{T}^{\Sigma_{\mathrm{H}}}$ in
$\operatorname{End}_{\mathcal{O}_{\lambda}}(\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})])$.
Let $\overline{\rho}$ be a Galois representation
$\overline{\rho}:\mathrm{G}_{\mathbb{Q}}\rightarrow\mathrm{GSp}_{4}(k)$
and let $\mathfrak{m}$ be the maximal ideal corresponding to $\overline{\rho}$
in $\mathbb{T}^{\Sigma}$ for
$\Sigma\in\\{\Sigma_{\mathrm{Pa}},\Sigma_{\mathrm{H}}\\}$, in the sense that
the characteristic polynomial
$\mathrm{det}(\mathrm{X}-\overline{\rho}(\mathrm{Frob}_{v}))$ is congruent to
the Hecke polynomial $\mathrm{Q}_{v}(\mathrm{X})$ given by
$\mathrm{X}^{4}-\mathrm{T}_{v,2}\mathrm{X}^{3}+(v\mathrm{T}_{v,1}+(v^{3}+v)\mathrm{T}_{v,0})\mathrm{X}^{2}-v^{3}\mathrm{T}_{v,0}\mathrm{T}_{v,2}\mathrm{X}+v^{6}\mathrm{T}^{2}_{v,0}$
modulo $\mathfrak{m}$ for each $v$ not in $\Sigma$. We will also assume that
the image of $\overline{\rho}$ contains $\mathrm{GSp}_{4}(\mathbb{F}_{l})$ and
that
$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}))_{\mathfrak{m}}=\mathrm{H}^{3}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}))_{\mathfrak{m}}.$
The following theorem is the main result of [Wang22a] proved using the Taylor-
Wiles method.
###### Theorem 8.9.
Let $\overline{\rho}$ and $\mathfrak{m}$ be given as above. Suppose the
following assumptions hold.
* (D1)
the image of $\overline{\rho}(\mathrm{G}_{\mathbb{Q}})$ contains
$\mathrm{GSp}_{4}(k)$;
* (D2)
$\overline{\rho}$ is rigid for $(\Sigma_{\mathrm{min}},\Sigma_{\mathrm{lr}})$;
* (D3)
For every finite set $\Sigma^{\prime}$ of nonarchimedean places containing
$\Sigma$ and every open compact subgroup $\mathrm{K}^{\prime}$ of $\mathrm{K}$
satisfying $\mathrm{K}^{\prime}_{v}=\mathrm{K}_{v}$ for all
$v\not\in\Sigma^{\prime}$, we have
$\mathrm{H}^{d}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(k))_{\mathfrak{m}^{\prime}}=0$
for $d\neq 3$ where
$\mathfrak{m}^{\prime}=\mathfrak{m}\cap\mathbb{T}^{\Sigma^{\prime}}$.
Then the following holds true.
1. (1)
If $\mathbf{T}^{\mathrm{unr}}_{\mathfrak{m}}$ is non-zero, then we have an
isomorphim of complete intersection rings:
$\mathrm{R}^{\mathrm{unr}}=\mathbf{T}^{\mathrm{unr}}_{\mathfrak{m}}$
and
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$
is a finite free $\mathbf{T}^{\mathrm{unr}}_{\mathfrak{m}}$-module.
2. (2)
If $\mathbf{T}^{\mathrm{ram}}_{\mathfrak{m}}$ is non-zero, then we have an
isomorphim of complete intersection rings:
$\mathrm{R}^{\mathrm{ram}}=\mathbf{T}^{\mathrm{ram}}_{\mathfrak{m}}$
and
$\mathrm{H}^{3}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}))_{\mathfrak{m}}$
is a finite free $\mathbf{T}^{\mathrm{ram}}_{\mathfrak{m}}$-module.
###### Remark 8.10.
Let $\pi$ be a cuspidal automorphic representation of general type. Assume
that $\overline{\rho}_{\pi,\lambda}(\mathrm{G}_{\mathbb{Q}})$ contains
$\mathrm{GSp}_{4}(\mathbb{F}_{l})$ for sufficiently large $l$, then we have
shown that $\overline{\rho}_{\pi,\lambda}$ is indeed rigid, see [Wang22a,
Theorem 5.5].
## 9\. Arithmetic level raising and applications
Now we are ready to prove our main result on arithmetic level raising on the
quaternionic Shimura varieties studied in the previous sections. First we
recall the setting we are in. Let $\pi$ be a cuspidal automorphic
representation of $\mathrm{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ with weight
$(3,3)$ and trivial central character. Suppose that $\pi$ is of general type
and $E$ be the coefficient field of $\pi$. Let $\Sigma_{\pi}$ be the minimal
set of finite places outside of which $\pi$ is unramified. We fix an
isomorphism $\iota_{l}:\mathbb{C}\xrightarrow{\sim}\overline{\mathbb{Q}}_{l}$
which induces a place $\lambda$ in $E$ over $l$. We fix the following datum.
* •
A finite set $\Sigma_{\mathrm{min}}$ of places of $\mathbb{Q}$ contained in
$\Sigma_{\pi}$;
* •
A finite set $\Sigma_{\mathrm{lr}}$ of places of $\mathbb{Q}$ which is
disjoint from $\Sigma_{\mathrm{min}}$ and contained in $\Sigma_{\mathrm{lr}}$;
* •
A finite set of places $\Sigma$ containing
$\Sigma_{\mathrm{min}}\cup\Sigma_{\mathrm{lr}}$ which is away from $l$.
* •
We have, by Construction 3.3 $(2)$, a homomorphism
$\phi_{\pi}:\mathbb{T}^{\Sigma}\rightarrow\mathcal{O}_{E}.$
and its $\lambda$-adic avatar
$\phi_{\pi,\lambda}:\mathbb{T}^{\Sigma}\rightarrow\mathcal{O}_{\lambda}$ for
the valuation ring $\mathcal{O}_{\lambda}\subset E_{\lambda}$.
* •
We can associate to it a Galois representation
$\rho_{\pi,\lambda}:\mathrm{G}_{\mathbb{Q}}\rightarrow\mathrm{GSp}_{4}(E_{\lambda})$
which we assume is residually absolutely irreducible as in Assumption 8.4.
* •
Let $p$ be a prime which is level raising special of length $m$ for $\pi$ in
the sense of Definition 3.2.
* •
Let $q$ be a prime such that $l\nmid q^{2}-1$. Suppose that $\pi$ is of type
$\mathrm{II}\mathrm{a}$ and $q$ is contained in $\Sigma_{\mathrm{lr}}$. In
this case $\pi_{q}$ admits Jacquet-Langlands transfer to a representation of
the group $\mathrm{GU}_{2}(\mathrm{D}^{\prime})$ where $\mathrm{D}^{\prime}$
which is the quaternion division algebra over $\mathbb{Q}_{q}$.
* •
We choose the level $\mathrm{K}_{\mathrm{Pa}}$ and $\Sigma_{\mathrm{Pa}}$ as
in Construction 8.7 for the Shimura varieity
$\mathrm{Sh}(\mathrm{B},\mathrm{K}_{\mathrm{Pa}})$ whose integral model is
$\mathrm{X}_{\mathrm{Pa}}(\mathrm{B})$ and whose special fiber is
$\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B})$; and choose the level
$\mathrm{K}_{\mathrm{H}}$ and $\Sigma_{\mathrm{H}}$ as in Construction 8.8 for
the Shimura set $\mathrm{Sh}(\overline{\mathrm{B}},\mathrm{K}_{\mathrm{H}})$
which is also denoted by $\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})$.
* •
We introduced two ideals
(9.1)
$\displaystyle\mathfrak{m}=\mathbb{T}^{\Sigma\cup\\{p\\}}\cap\ker(\mathbb{T}^{\Sigma}\xrightarrow{\phi_{\pi,\lambda}}\mathcal{O}_{\lambda}\rightarrow\mathcal{O}_{\lambda}/\lambda)$
$\displaystyle\mathfrak{n}=\mathbb{T}^{\Sigma\cup\\{p\\}}\cap\ker(\mathbb{T}^{\Sigma}\xrightarrow{\phi_{\pi,\lambda}}\mathcal{O}_{\lambda}\rightarrow\mathcal{O}_{\lambda}/\lambda^{m})$
as in Construction 3.3 $(3)$ and we assume that $\mathfrak{m}$ is generic
level raising at $p$ in the sense of 7.8.
###### Construction 9.2.
We define the following map
$\nabla:\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]\rightarrow\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]$
by sending
$(x,y)\rightarrow(\mathrm{T}^{-1}_{p,0}\mathrm{T}_{p,1}+(p+1)(p^{2}+1))x-(p+1)\mathrm{T}_{20}y$
for
$(x,y)\in\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]$.
We also write
$\nabla_{\mathfrak{m}}:\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\rightarrow\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$
for the localization of $\nabla$ at $\mathfrak{m}$
Note that the composition of the level raising matrix
$\mathcal{T}_{\mathrm{lr},\mathfrak{m}}:\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\rightarrow\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$
with the map $\nabla_{\mathfrak{m}}$ is given by sending
$(x,y)\rightarrow\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr},\mathfrak{m}}\cdot
x$ for
$(x,y)\in\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]$
where we have
$\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr},\mathfrak{m}}=(\mathrm{T}^{-1}_{p,0}\mathrm{T}_{p,1}+(p+1)(p^{2}+1))^{2}-(p+1)^{2}\mathrm{T}_{20}\circ\mathrm{T}_{02}$
by Proposition 6.10.
###### Proposition 9.3.
Let $p$ be a level raising special prime for $\pi$ and $\mathfrak{m}$ be the
maximal ideal as above. We have a surjective morphism
$\mathrm{F}_{-1}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}})\twoheadrightarrow\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\mathrm{B})]_{\mathfrak{m}}/\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr},\mathfrak{m}}$
whose kernel can be identified with the torsion part of
$\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}$.
###### Proof.
Note that, by Proposition 5.4, we have an isomorphism
$\mathrm{F}_{-1}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda})(1))_{\mathfrak{m}})\xrightarrow{\sim}\mathrm{Coker}(\alpha\circ\mathrm{N}^{-1}_{\Sigma}\circ\beta)_{\mathfrak{m}}.$
On the other hand, by Proposition 7.9 we have an isomorphism
$(\mathrm{inc}^{\\{0\\}}_{!,\mathfrak{m}}+\mathrm{inc}^{\\{2\\}}_{!,\mathfrak{m}}):\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\rightarrow\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}$
and a surjection
$(\mathrm{inc}^{\ast}_{\\{0\\},\mathfrak{m}},\mathrm{inc}^{\ast}_{\\{2\\},\mathfrak{m}}):\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}\rightarrow\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$
whose kernel is given by the torsion part of
$\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}$.
It follows that we have a surjection
$\mathrm{F}_{-1}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda})(1))_{\mathfrak{m}})\rightarrow\frac{\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}}{\mathcal{T}_{\mathrm{lr},\mathfrak{m}}(\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}})}.$
The kernel of this map is given by
$\frac{\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))^{\mathrm{tor}}_{\mathfrak{m}}}{\mathrm{Im}(\alpha\circ\mathrm{N}^{-1}_{\Sigma}\circ\beta)_{\mathfrak{m}}\cap\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))^{\mathrm{tor}}_{\mathfrak{m}}}.$
Composing this with the map
$\nabla_{\mathfrak{m}}:\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\rightarrow\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$,
we therefore obtain a map
$\mathrm{F}_{-1}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda})(1))_{\mathfrak{m}})\rightarrow\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}/\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr},\mathfrak{m}}.$
which is surjective by Nakayama’s lemma.
It remains to show that
$\mathrm{Im}(\alpha\circ\mathrm{N}^{-1}_{\Sigma}\circ\beta)_{\mathfrak{m}}\cap\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))^{\mathrm{tor}}_{\mathfrak{m}}$
is zero. By [Nek07, Proposition 4.2.2(1)], the group
$\mathrm{F}_{-1}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(E_{\lambda})(1))_{\mathfrak{m}})$
vanishes. It follows that the $\mathcal{O}_{\lambda}$-rank of
$\mathrm{Im}(\alpha\circ\mathrm{N}^{-1}_{\Sigma}\circ\beta)_{\mathfrak{m}}$ is
the same as the $\mathcal{O}_{\lambda}$-rank of
$\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}$.
Next, using the excision exact sequences, we have isomorphisms
(9.4)
$\displaystyle\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),E_{\lambda}(2))\xrightarrow{\sim}\mathrm{H}^{4}(\overline{\mathrm{X}}^{\mathrm{sm}}_{\mathrm{Pa}}(\mathrm{B}),E_{\lambda}(2))$
$\displaystyle\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}^{\mathrm{sm}}_{\mathrm{Pa}}(\mathrm{B}),E_{\lambda}(1))\xrightarrow{\sim}\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),E_{\lambda}(1))$
where
$\overline{\mathrm{X}}^{\mathrm{sm}}_{\mathrm{Pa}}(\mathrm{B})=\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B})-\overline{\mathrm{X}}^{\mathrm{sing}}_{\mathrm{Pa}}(\mathrm{B})$,
using the fact that the singular locus
$\overline{\mathrm{X}}^{\mathrm{sing}}_{\mathrm{Pa}}(\mathrm{B})$ is zero
dimensional. Therefore Poincare duality for
$\overline{\mathrm{X}}^{\mathrm{sm}}_{\mathrm{Pa}}(\mathrm{B})$ implies that
we have
$\dim_{E_{\lambda}}\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),E_{\lambda}(2))_{\mathfrak{m}}=\dim_{E_{\lambda}}\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),E_{\lambda}(1))_{\mathfrak{m}}.$
Since the source
$\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}$
of $\mathrm{Im}(\alpha\circ\mathrm{N}^{-1}_{\Sigma}\circ\beta)_{\mathfrak{m}}$
is free over $\mathcal{O}_{\lambda}$, the intersection
$\mathrm{Im}(\alpha\circ\mathrm{N}^{-1}_{\Sigma}\circ\beta)_{\mathfrak{m}}\cap\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))^{\mathrm{tor}}_{\mathfrak{m}}$
is trivial. The proposition is proved. ∎
Now we are ready to state and prove our main result on arithmetic level
raising for the quaternionic Shimura variety studied in this article. We first
recall a list of running assumptions below.
* •
We assume that
$\mathrm{H}^{i}_{(\mathrm{c})}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}))_{\mathfrak{m}}=0$
for $i\neq 3$, and that
$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}))_{\mathfrak{m}}\xrightarrow{\sim}\mathrm{H}^{3}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}))_{\mathfrak{m}}$
is a finite free $\mathcal{O}_{\lambda}$ module. This is Assumption 7.6.
* •
We assume that for $\pi$, the associated Galois representation
$\rho_{\pi,\lambda}:\mathrm{G}_{\mathbb{Q}}\rightarrow\mathrm{GSp}_{4}(E_{\lambda})$
is residually absolutely irreducible. This is Assumption 8.4.
###### Theorem 9.5 (Arithmetic level raising for $\mathrm{GSp}_{4}$).
Suppose that $\pi$ is a cuspidal automorphic representation of
$\mathrm{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ of general type. We assume that
the assumptions in 7.6, 8.4 recalled above are effective. Let $p$ be a level
raising special prime for $\pi$ of depth $m$. We assume further that
1. (1)
$l\geq 3$ and $l\nmid p^{2}-1$;
2. (2)
$\overline{\rho}_{\pi,\lambda}$ is rigid for
$(\Sigma_{\mathrm{min}},\Sigma_{\mathrm{lr}})$ as in Definition 8.6;
3. (3)
The image of $\overline{\rho}_{\pi,\lambda}(\mathrm{G}_{\mathbb{Q}})$ contains
$\mathrm{GSp}_{4}(\mathbb{F}_{l})$;
4. (4)
$\mathfrak{m}$ is in the support of
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]$ and is
generic level raising.
Then following holds.
1. (1)
There is an isomorphism
$\mathrm{H}^{1}_{\mathrm{sing}}(\mathbb{Q}_{p^{2}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})\xrightarrow{\sim}\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}/\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr}}.$
2. (2)
The above isomorphism induces an isomorphism
$\mathrm{H}^{1}_{\mathrm{sing}}(\mathbb{Q}_{p^{2}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}/\lambda^{m}(1)))_{\mathfrak{m}})\xrightarrow{\sim}\mathcal{O}_{\lambda}/\lambda^{m}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}.$
### 9.1. Proof of Theorem 9.5
Consider the universal deformation problem
$\mathcal{S}^{\ast}=(\overline{\rho}_{\pi,\lambda},\psi,\Sigma_{\mathrm{min}}\cup\Sigma_{\mathrm{lr}}\cup\\{p\\}\cup\\{l\\},\\{\mathcal{D}_{v}\\}_{v\in\Sigma_{\mathrm{min}}\cup\Sigma_{\mathrm{lr}}\cup\\{p\\}\cup\\{l\\}})$
where $\ast=\\{\mathrm{ram},\mathrm{unr},\mathrm{mix}\\}$. Recall that
$\mathrm{Spf}\phantom{.}\mathrm{R}^{\mathrm{ram}}$ and
$\mathrm{Spf}\phantom{.}\mathrm{R}^{\mathrm{unr}}$ are subspaces of
$\mathrm{Spf}\phantom{.}\mathrm{R}^{\mathrm{mix}}$. Thus we have surjections
$\displaystyle\mathrm{R}^{\mathrm{mix}}\twoheadrightarrow\mathrm{R}^{\mathrm{ram}}$
$\displaystyle\mathrm{R}^{\mathrm{mix}}\twoheadrightarrow\mathrm{R}^{\mathrm{unr}}.$
we define
$\mathrm{R}^{\mathrm{cong}}=\mathrm{R}^{\mathrm{unr}}\otimes_{\mathrm{R}^{\mathrm{mix}}}\mathrm{R}^{\mathrm{ram}}$.
We have a universal lifting
$\rho^{\mathrm{mix}}:\mathrm{G}_{\mathbb{Q}}\rightarrow\mathrm{GSp}_{4}(\mathrm{R}^{\mathrm{mix}}).$
It also induces two representations
$\displaystyle\rho^{\mathrm{ram}}:\mathrm{G}_{\mathbb{Q}}\rightarrow\mathrm{GSp}_{4}(\mathrm{R}^{\mathrm{ram}})$
$\displaystyle\rho^{\mathrm{unr}}:\mathrm{G}_{\mathbb{Q}}\rightarrow\mathrm{GSp}_{4}(\mathrm{R}^{\mathrm{unr}})$
by the above surjections.
Denote by $\mathrm{P}_{\mathbb{Q}_{p}}$ be the maximal closed subgroup of the
inertia subgroup $\mathrm{I}_{\mathbb{Q}_{p}}\subset G_{\mathbb{Q}_{p}}$ of
pro-order coprime to $l$. Then
$G_{\mathbb{Q}_{p}}/\mathrm{P}_{\mathbb{Q}_{p}}\cong
t^{\mathbb{Z}_{l}}\rtimes\phi^{\widehat{\mathbb{Z}}}_{p}$ is a $p$-tame group.
By definition, $\rho^{\mathrm{mix}}$ is trivial on
$\mathrm{P}_{\mathbb{Q}_{p}}$. Let $\overline{\mathrm{v}}_{0}$ and
$\overline{\mathrm{v}}_{1}$ be eigenvectors in
$k^{4}=(\mathcal{O}_{\lambda}/\lambda)^{4}$ for
$\rho^{\mathrm{mix}}(\phi^{2}_{p})$ with eigenvalues $p^{-2}$ and $p^{-4}$,
respectively. By Hensel’s lemma, they lift to $\mathrm{v}_{0}$ and
$\mathrm{v}_{1}$ in $(\mathrm{R}^{\mathrm{mix}})^{\oplus 4}$ for
$\rho^{\mathrm{mix}}(\phi^{2}_{p})$ with eigenvalues $\mathrm{s}_{0}$ and
$\mathrm{s}_{1}$ lifting $p^{-2}$ and $p^{-4}$. Let
$\mathrm{x}\in\mathrm{R}^{\mathrm{mix}}$ be the unique element such that
$\rho^{\mathrm{mix}}(t)\mathrm{v}_{1}=\mathrm{x}\mathrm{v}_{0}+\mathrm{v}_{1}$.
It follows from the relation
$\rho^{\mathrm{mix}}(\phi^{2}_{p})\rho^{\mathrm{mix}}(t)\rho^{\mathrm{mix}}(\phi^{-2}_{p})=\rho^{\mathrm{mix}}(t)^{p^{2}}$
that $\mathrm{x}(\mathrm{s}-p^{-2})=0$. By the definitions of
$\mathrm{R}^{\ast}$ for $\ast=\\{\mathrm{ram},\mathrm{unr},\mathrm{mix}\\}$,
we have
$\displaystyle\mathrm{R}^{\mathrm{unr}}=\mathrm{R}^{\mathrm{mix}}/(\mathrm{x})$
$\displaystyle\mathrm{R}^{\mathrm{ram}}=\mathrm{R}^{\mathrm{mix}}/(\mathrm{s}_{0}-p^{-2})$
$\displaystyle\mathrm{R}^{\mathrm{cong}}=\mathrm{R}^{\mathrm{mix}}/(\mathrm{x},\mathrm{s}_{0}-p^{-2}).$
By Theorem 8.9,
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$
is a free over $\mathbf{T}^{\mathrm{unr}}_{\mathfrak{m}}$ and we have an
isomorphism $\mathrm{R}^{\mathrm{unr}}\cong\mathbf{T}_{\mathfrak{m}}$. Thus
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$
is a free $\mathrm{R}^{\mathrm{unr}}$-module of rank $d_{\mathrm{unr}}$. Note
that the characteristic polynomial of $\rho^{\mathrm{mix}}(\phi^{2}_{p})$ can
written as
$(\mathrm{X}-\mathrm{s}_{0})(\mathrm{X}-p^{-6}\mathrm{s}^{-1}_{0})\mathrm{Q}(\mathrm{X})$
where $\mathrm{Q}(\mathrm{X})$ is a polynomial in
$\mathrm{R}^{\mathrm{unr}}[\mathrm{X}]$ whose reduction in
$\mathcal{O}_{\lambda}/\lambda[\mathrm{X}]=k[\mathrm{X}]$ does not contain
$p^{-2}$ and $p^{-4}$ as its roots. By Proposition 6.15 and the equations
$\displaystyle(\alpha_{p}+p^{3}\alpha^{-1}_{p}-p(p+1))(\alpha_{p}+p^{3}\alpha^{-1}_{p}+p(p+1))=\alpha^{2}_{p}+p^{6}\alpha^{-2}_{p}-p^{2}(p^{2}+1),$
$\displaystyle(\beta_{p}+p^{3}\beta^{-1}_{p}-p(p+1))(\beta_{p}+p^{3}\beta^{-1}_{p}+p(p+1))=\beta^{2}_{p}+p^{6}\beta^{-2}_{p}-p^{2}(p^{2}+1),$
we have
$(\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr},\mathfrak{m}})\cdot\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}=(\mathrm{s}_{0}-p^{-2})\cdot\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}.$
Thus we have
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}/\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr},\mathfrak{m}}\cong\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\otimes_{\mathrm{R}^{\mathrm{unr}}}\mathrm{R}^{\mathrm{cong}}$
and it follows that
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}/\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr},\mathfrak{m}}$
is a free $\mathrm{R}^{\mathrm{cong}}$-module of rank $d_{\mathrm{unr}}$.
Since we have a surjection
$\mathrm{F}_{-1}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{c}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})\twoheadrightarrow\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\mathrm{B})]_{\mathfrak{m}}/\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr},\mathfrak{m}}$
by Proposition 9.3, we have $\mathbf{T}^{\mathrm{ram}}_{\mathfrak{m}}$ is non-
zero. Moreover
$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}}$
is a free $\mathrm{R}^{\mathrm{ram}}$-module by Theorem 8.9 $(2)$, say of rank
$d_{\mathrm{ram}}$. Consider the $\mathrm{R}^{\mathrm{ram}}$-module
$\mathrm{H}:=\operatorname{Hom}_{\mathrm{G}_{\mathbb{Q}}}((\mathrm{R}^{\mathrm{ram}})^{\oplus
4},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})$
where $\mathrm{G}_{\mathbb{Q}}$ acts on $(\mathrm{R}^{\mathrm{ram}})^{\oplus
4}$ via $\rho^{\mathrm{ram}}$ which is free of rank $d_{\mathrm{ram}}$. It
follows this that we have an isomorphism
$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}))_{\mathfrak{m}}\xrightarrow{\sim}\mathrm{H}\otimes_{\mathrm{R}^{\mathrm{ram}}}(\mathrm{R}^{\mathrm{ram}})^{\oplus
4}.$
It follows from that
$\mathrm{H}^{1}_{\mathrm{sing}}(\mathbb{Q}_{p^{2}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})\xrightarrow{\sim}\mathrm{H}\otimes_{\mathrm{R}^{\mathrm{ram}}}\mathrm{H}^{1}_{\mathrm{sing}}(\mathbb{Q}_{p^{2}},(\mathrm{R}^{\mathrm{ram}})^{\oplus
4}(1))$
We still denote by $\mathrm{v}_{0}$ and $\mathrm{v}_{1}$ the projections of
$\mathrm{v}_{0}$ and $\mathrm{v}_{1}$ in $(\mathrm{R}^{\mathrm{ram}})^{\oplus
4}$ of $\mathrm{G}_{\mathbb{Q}}$-modules. Then a straightforward computation
shows that
$\mathrm{H}^{1}_{\mathrm{sing}}(\mathbb{Q}_{p^{2}},(\mathrm{R}^{\mathrm{ram}})^{\oplus
4}(1))\xrightarrow{\sim}\mathrm{R}^{\mathrm{ram}}\mathrm{v}_{0}/\mathrm{x}\mathrm{v}_{0}\cong\mathrm{R}^{\mathrm{cong}}.$
Hence we have
$\mathrm{H}^{1}_{\mathrm{sing}}(\mathbb{Q}_{p^{2}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})$
is a free $\mathrm{R}^{\mathrm{cong}}$-module of rank $d_{\mathrm{ram}}$.
###### Lemma 9.6.
We have an equality
$d_{\mathrm{unr}}=d_{\mathrm{ram}}.$
###### Proof.
Let
$\eta^{\mathrm{unr}}\in\mathrm{Spec}\phantom{.}\mathrm{R}^{\mathrm{unr}}[1/l](\overline{\mathbb{Q}}_{l})$
be in the support of
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$
which gives rise to an automorphic representation $\pi^{\mathrm{unr}}$ of
$\mathrm{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ such that
$\overline{\rho}_{\pi,\lambda}\otimes_{k}\overline{\mathbb{F}}_{l}$ and
$\rho_{\pi^{\mathrm{unr}},\iota_{l}}$ are residually isomorphic. Then we have
$d_{\mathrm{unr}}=\dim_{\overline{\mathbb{Q}}_{l}}\overline{\mathbb{Q}}_{l}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})][\iota_{l}\phi_{\pi^{\mathrm{unr}}}].$
Similarly, let
$\eta^{\mathrm{ram}}\in\mathrm{Spec}\phantom{.}\mathrm{R}^{\mathrm{ram}}[1/l](\overline{\mathbb{Q}}_{l})$
be in the support of
$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}}$
which gives rise to an automorphic representation $\pi^{\mathrm{ram}}$ of
$\mathrm{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ such that
$\rho_{\pi^{\mathrm{ram}},\iota_{l}}$ is residually isomorphic to
$\overline{\rho}_{\pi,\lambda}\otimes_{k}\overline{\mathbb{F}}_{l}$. Then we
have
$4d_{\mathrm{ram}}=\dim_{\overline{\mathbb{Q}}_{l}}\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\overline{\mathbb{Q}}_{l}))[\iota_{l}\phi_{\pi^{\mathrm{ram}}}].$
Note that $\pi^{\mathrm{unr}}$ and $\pi^{\mathrm{ram}}$ are necessarily of
general type. By considering the transfer of $\pi^{\mathrm{unr}}$ and
$\pi^{\mathrm{ram}}$ to $\mathrm{GL}_{4}(\mathbb{A}_{\mathbb{Q}})$, we have
$d_{\mathrm{ram}}=d_{\mathrm{unr}}$ by [LTXZZ, Lemma 6.4.2]. ∎
_Proof of Theorem 9.5._ Now we can prove our main theorem on arithmetic level
raising. We have proved that we have a surjection
$\mathrm{F}_{-1}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{c}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})\twoheadrightarrow\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\mathrm{B})]_{\mathfrak{m}}/\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr},\mathfrak{m}}$
where the target is free of rank $d_{\mathrm{unr}}$ over
$\mathrm{R}^{\mathrm{cong}}$. On the other hand, we have an injection
$\mathrm{F}_{-1}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{c}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})\hookrightarrow\mathrm{H}^{1}_{\mathrm{sing}}(\mathbb{Q}_{p^{2}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1))_{\mathfrak{m}})$
and the target is free of rank $d_{\mathrm{ram}}$ over
$\mathrm{R}^{\mathrm{cong}}$. Since $d_{\mathrm{unr}}=d_{\mathrm{ram}}$, we
have the desired isomorphism
(9.7)
$\mathrm{H}^{1}_{\mathrm{sing}}(\mathbb{Q}_{p^{2}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})\xrightarrow{\sim}\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}/\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr}}.$
This finishes the proof of the first part of the theorem.
For the second part, by the freeness of
$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}}$
as a $\mathbf{T}^{\mathrm{ram}}_{\mathfrak{m}}$-module. It follows that the
natural map
$\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})/\mathfrak{n}\xrightarrow{\sim}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}}/\mathfrak{n})$
is an isomorphism. We also have the short exact sequence as in Proposition 6.1
$\displaystyle 0$
$\displaystyle\rightarrow\mathrm{F}_{-1}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})\rightarrow\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda})(1))_{\mathfrak{m}})$
$\displaystyle\rightarrow\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}}/\mathrm{F}_{-1}\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}}\rightarrow
0$
which is split by the $\mathrm{G}_{\mathbb{F}_{p^{2}}}$-action. And it follows
that
$\mathrm{F}_{-1}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})/\mathfrak{n}\xrightarrow{\sim}\mathrm{F}_{-1}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}}/\mathfrak{n})$
is an isomorphism. Since we have shown above that there is an isomorphism
$\mathrm{F}_{-1}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{c}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})\xrightarrow{\sim}\mathrm{H}^{1}_{\mathrm{sing}}(\mathbb{Q}_{p^{2}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}),$
it follows that
$\mathrm{H}^{1}_{\mathrm{sing}}({\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})/\mathfrak{n}\xrightarrow{\sim}\mathrm{H}^{1}_{\mathrm{sing}}({\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}}/\mathfrak{n}).$
On the other hand, we have
$\mathbf{T}^{\mathrm{ram}}/\mathfrak{n}\cong\mathcal{O}_{\lambda}/\lambda^{m}$
and thus
$\mathrm{H}^{1}_{\mathrm{sing}}({\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}}/\mathfrak{n})\cong\mathrm{H}^{1}_{\mathrm{sing}}({\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}/\lambda^{m}(1)))_{\mathfrak{m}}$
by the freeness of
$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}}$
as $\mathbf{T}^{\mathrm{ram}}_{\mathfrak{m}}$-module. Finally the second claim
follows from the observation that
$\displaystyle\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}/\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr}}$
$\displaystyle\cong\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\otimes_{\mathrm{R}^{\mathrm{unr}}}\mathrm{R}^{\mathrm{cong}}$
$\displaystyle\cong\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\otimes_{\mathrm{R}^{\mathrm{unr}}}\mathbf{T}^{\mathrm{ram}}_{\mathfrak{m}}/\mathfrak{n}$
$\displaystyle\cong\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\otimes\mathcal{O}_{\lambda}/\lambda^{m}$
and the isomorphism 9.7. $\square$
###### Corollary 9.8.
We maintain the assumptions in Theorem 9.5.
1. (1)
$\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda})_{\mathfrak{m}}$
is a torsion free $\mathcal{O}_{\lambda}$-module. In particular, the map from
Construction 6.4
$(\mathrm{inc}^{\ast}_{\\{0\\},\mathfrak{m}},\mathrm{inc}^{\ast}_{\\{2\\},\mathfrak{m}}):\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}\rightarrow\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$
is an isomorphism.
2. (2)
$\mathrm{H}^{3}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda})_{\mathfrak{m}}$
and
$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda})_{\mathfrak{m}}$
are both torsion free $\mathcal{O}_{\lambda}$-modules.
###### Proof.
Recall we have proved that the kernel of the natural map
$\mathrm{F}_{-1}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{n}_{c}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}})\twoheadrightarrow\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\mathrm{B})]_{\mathfrak{m}}/\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr},\mathfrak{m}}$
can be identified with the torsion part of
$\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}$.
We have shown in the above proof that this surjection is an isomorphism and
the result follows.
For the second part, we have an injection
$\mathrm{H}^{3}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}\hookrightarrow\mathrm{H}^{3}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(2)))_{\mathfrak{m}}$
and
$\mathrm{H}^{3}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(2)))_{\mathfrak{m}}$
is a torsion free $\mathcal{O}_{\lambda}$-module by our assumption. Thus
$\mathrm{H}^{3}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}$
is torsion free.
Next, we show the cohomology group
$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}$
is torsion free. We consider the long exact sequence
$\cdots\rightarrow\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}\rightarrow\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}\rightarrow\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}/\lambda(1))_{\mathfrak{m}}\rightarrow\cdots$
induced by the short exact sequence
$0\rightarrow\mathcal{O}_{\lambda}\rightarrow\mathcal{O}_{\lambda}\rightarrow\mathcal{O}_{\lambda}/\lambda\rightarrow
0.$
We need to show the injection
$\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}/\lambda\hookrightarrow\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}/\lambda(1))_{\mathfrak{m}}$
is in fact surjective. For this, note that we have established that
$\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}$
is torsion free and thus
$\dim_{k}\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}/\lambda=\dim_{E_{\lambda}}\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),E_{\lambda}(1))_{\mathfrak{m}}$.
But we have
$\displaystyle\dim_{k}\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}/\lambda(1))_{\mathfrak{m}}=\dim_{k}\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}/\lambda(2))_{\mathfrak{m}};$
$\displaystyle\dim_{E_{\lambda}}\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),E_{\lambda}(1))_{\mathfrak{m}}=\dim_{E_{\lambda}}\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),E_{\lambda}(2))_{\mathfrak{m}}.$
by the Poincare duality and (9.4). By the considering the same long exact
sequence and the reasoning as above but applied to
$\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}$,
we have
$\dim_{k}\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}/\lambda(2))_{\mathfrak{m}}=\dim_{E_{\lambda}}\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),E_{\lambda}(2))_{\mathfrak{m}}$
and hence
$\dim_{k}\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}/\lambda(1))_{\mathfrak{m}}=\dim_{E_{\lambda}}\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),E_{\lambda}(1))_{\mathfrak{m}}.$
It follows that
$\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}/\lambda\hookrightarrow\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}/\lambda(1))_{\mathfrak{m}}$
is in fact surjective and we are done. ∎
### 9.2. Level raising for $\mathrm{GSp}_{4}$
Let $\pi$ be a cuspidal automorphic representation of
$\mathrm{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ of general type with weight
$(3,3)$ and trivial central character. Let $p$ be a level raising special
prime for $\pi$ of length $m$. From the arithmetic level raising theorem
proved in the last subsection, we can deduce a level raising theorem for $\pi$
which can be applied to deduce level raising theorems in [Wang22b]. This is
the content of this subsection.
Suppose the component $\pi_{q}$ of $\pi$ at the prime $q$ is of type
$\mathrm{I}\mathrm{I}\mathrm{a}$. Then $\pi$ admits a global Jacquet–Langlands
transfer to an automorphic representation of
$\mathbf{G}(\overline{\mathrm{B}})(\mathbb{A}_{\mathbb{Q}})$. In particular
the maximal ideal $\mathfrak{m}$ associated to $\pi$ in the Construction 3.3
$(3)$ lies in the support of
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]$ as a
$\mathbb{T}^{\Sigma\cup\\{p\\}}$-module.
###### Theorem 9.9.
Let $\pi$ be an automorphic representation of
$\mathrm{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ as above. Suppose that $\pi$
satisfy all the assumptions in Theorem 9.5. Let $p$ be a level raising special
prime for $\pi$ of length $m$.
Then there exists an automorphic representation $\pi^{\prime}$ of
$\mathrm{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ of general type with weight
$(3,3)$ and trivial central character such that
1. (1)
the component $\pi^{\prime}_{p}$ at $p$ is of type
$\mathrm{I}\mathrm{I}\mathrm{a}$;
2. (2)
we have an isomorphism of the residual Galois representation
$\overline{\rho}_{\pi,\lambda}\cong\overline{\rho}_{\pi^{\prime},\lambda}.$
In this case we will say $\pi^{\prime}$ is a level raising of $\pi$.
###### Proof.
As remark in the above, the localization
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$
is non-zero. By Theorem 9.5, we have an isomorphism
$\mathrm{H}^{1}_{\mathrm{sing}}(\mathbb{Q}_{p^{2}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})\xrightarrow{\sim}\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}/\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr}}.$
In particular
$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}}$
is non-zero. Since
$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}}$
is a free $\mathrm{R}^{\mathrm{ram}}$-module, we can find a point
$\eta^{\mathrm{ram}}\in\mathrm{Spec}\phantom{.}\mathrm{R}^{\mathrm{ram}}[1/l]$
in the support of
$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}}.$
Then we can find an automorphic representation $\pi^{\prime}$ of
$\mathrm{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ which satisfy the isomorphism
$\overline{\rho}_{\pi,\lambda}\cong\overline{\rho}_{\pi^{\prime},\lambda}.$
In particular $\pi^{\prime}$ is of general type as
$\overline{\rho}_{\pi^{\prime},\lambda}$ is absolutely irreducible. By the
Picard-Lefschetz formula applied to the Shimura variety
${\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B})$ , we see that the monodromy of the
associated Weil-Deligne representation of $\rho_{\pi^{\prime},\lambda}$ is
necessarily is of the form
$\begin{pmatrix}0&&&\\\ &0&1&\\\ &&0&\\\ &&&0\\\ \end{pmatrix}$
up to conjugation. Since $\pi^{\prime}$ is of general type and hence tempered
at $p$, it follows that $\pi^{\prime}$ is of type
$\mathrm{I}\mathrm{I}\mathrm{a}$ by the local Langlands correspondence for
non-supercuspidal representations, see for example [Sch05, Table 2]. The
theorem is proved. ∎
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# An Attention Module for Convolutional Neural Networks
Zhu Baozhou1, Peter Hofstee12, Jinho Lee3, Zaid Al-Ars1
###### Abstract
Attention mechanism has been regarded as an advanced technique to capture
long-range feature interactions and to boost the representation capability for
convolutional neural networks. However, we found two ignored problems in
current attentional activations-based models: the approximation problem and
the insufficient capacity problem of the attention maps. To solve the two
problems together, we initially propose an attention module for convolutional
neural networks by developing an AW-convolution, where the shape of attention
maps matches that of the weights rather than the activations. Our proposed
attention module is a complementary method to previous attention-based
schemes, such as those that apply the attention mechanism to explore the
relationship between channel-wise and spatial features. Experiments on several
datasets for image classification and object detection tasks show the
effectiveness of our proposed attention module. In particular, our proposed
attention module achieves $1.00\%$ Top-1 accuracy improvement on ImageNet
classification over a ResNet101 baseline and $0.63$ COCO-style Average
Precision improvement on the COCO object detection on top of a Faster R-CNN
baseline with the backbone of ResNet101-FPN. When integrating with the
previous attentional activations-based models, our proposed attention module
can further increase their Top-1 accuracy on ImageNet classification by up to
$0.57\%$ and COCO-style Average Precision on the COCO object detection by up
to $0.45$. Code and pre-trained models will be publicly available.
## Introduction
Convolutional neural networks have demonstrated to be the gold-standard to
solving various problems in the field of computer vision, including image
classification (He et al. 2016a), object detection (Liu et al. 2016; Ren et
al. 2015), and segmentation (Chen et al. 2017). To improve their performance,
many researchers explored various aspects of CNN design and implementation (Gu
et al. 2018).
To enrich the representation power of CNNs, we can build substantially deeper
convolutional neural networks. For example, VGGNet (Simonyan and Zisserman
2015) stacks very small $3\times 3$ convolutional layers, while ResNet (He et
al. 2016a) stacks residual blocks with skip connections. GoogLeNet (Szegedy et
al. 2015) uses multi-scales of processing to capture spatial correlation. Wide
ResNet (Zagoruyko and Komodakis 2016) shows that the width increase of
residual networks can enlarge the representation capability and reuse the
features better. Xception (Chollet 2017) and ResNeXt (Xie et al. 2017) expose
new dimensions to increase cardinalities. Besides, recent literature
(Jaderberg et al. 2015; Gregor et al. 2015; Xu et al. 2015) have investigated
the attention mechanism since it can improve not only the representation power
but also the representation of interests. Convolutional neural networks can
extract informative features by blending cross-channel and spatial information
(Hu et al. 2018). Attention modules (Woo et al. 2018; Linsley et al. 2019) can
learn ”where” and ”what” to attend in channel and space axes, respectively, by
focusing on important features and suppressing unnecessary ones of the
activations. Dynamic Filter Networks (Jia et al. 2016; Li et al. 2019b)
generate the filters conditioned on the input and show the flexibility power
of such filters because of their adaptive nature, which has become popular in
prediction (Klein, Wolf, and Afek 2015) and Natural Language Processing (Wu et
al. 2019). Both Dynamic Filter Networks and attention-based models are
adaptive based on the inputs, but there are significant differences between
them. Attention-based models (Hu et al. 2018; Woo et al. 2018) produce
attention maps using the attention mechanism to operate on the activations of
convolution. On the contrary, Dynamic Filer Networks (Su et al. 2019; Liu et
al. 2019) generate input information-specific kernels, such as position-
specific kernels (Su et al. 2019), semantic label map-specific kernels (Liu et
al. 2019), and few-shot learning setting-specific kernels (Zhao et al. 2018),
which work as the weights of convolution. Our proposed attention module
leverages the attention mechanism to compute the attention maps for attending
the activations of convolution, so it is clear to categorized the models
applied with our proposed attention module as attention-based models instead
of Dynamic Filter Networks.
Based on our analysis, the approximation problem and the insufficient capacity
problem of the attention maps are ignored in current attentional activations-
based models. Motivated by solving the two problems together, we develop an
attention module and inspect the complementary relationship between our
proposed attention module and previously published attention-based models,
such as the attention augmented models and the attentional activations-based
models. Our contributions are summarized as follows.
* •
We point out and analyze two ignored problems of the current attentional
activations-based models: the approximation problem and the insufficient
capacity problem of the attention maps. To address the two problems together,
we originally propose an attention module by developing an AW-convolution,
where the shape of the attention maps matches that of the weights instead of
the activations.
* •
Our proposed attention module is a complementary method to previous attention
mechanism-based modules, such as Attention Augmented (AA) convolution (Bello
et al. 2019), the SE (Hu, Shen, and Sun 2018) and CBAM (Woo et al. 2018)
modules in the attentional activations-based models. Integrating with our
proposed attention module, the accuracy of AA-Net, SE-Net, and CBAM-Net will
be improved further.
* •
We use image classification and object detection tasks to demonstrate the
effectiveness of our proposed attention module. With negligible computational
complexity increase, our proposed attention module can boost the image
classification and object detection task performance, and it can achieve
better accuracy when integrating with other attention-based models.
## Related work
In this section, we discuss the recent developments of network engineering and
attention mechanism.
### Network engineering
”Network engineering” has been one of the most active research areas since it
targets building powerful convolutional neural networks on image
classification, which are the backbones of various computer vision tasks and
ensure their remarkable performance (Kornblith, Shlens, and Le 2019).
Increasing the depth of convolutional neural networks has been regarded as an
intuitive way to boost performance, which is the philosophy of VGGNet
(Simonyan and Zisserman 2015) and ResNet (He et al. 2016a). In addition, since
the skip connection from ResNet shows a strong ability to assist the gradient
flow, WideResNet (Zagoruyko and Komodakis 2016), PyramidNet (Han, Kim, and Kim
2017), Inception-ResNet (Szegedy et al. 2017), and ResNeXt (He et al. 2016b;
Xie et al. 2017) are ResNet-based versions proposed to explore further the
influence of the width, the increase of the width, the multi-scale and the
cardinality of convolution, respectively. In terms of efficiency, DenseNet
(Huang et al. 2017) reuses the feature maps by concatenating the feature maps
from different layers. In particular, MobileNet (Howard et al. 2017, 2019) and
ShuffleNet (Ma et al. 2018) series present the advantage of depthwise
convolution and the shuffle operation between various group convolutions,
respectively. Another design approach uses automated neural architecture
search, which achieves state-of-the-art performance regarding both accuracy
and efficiency across a range of computer vision tasks (Tan et al. 2019).
### Attention mechanism
The attention mechanism plays an important role in the human vision perceptron
since it can allocate the available resources to selectively focus on
processing the salient part instead of the whole scene (Rensink 2000; Corbetta
and Shulman 2002). Multiple attention mechanisms are used to address a known
weakness in convolution (Chen et al. 2018; Szegedy et al. 2015; Ke, Maire, and
Yu 2017; Chen et al. 2019; Hu, Shen, and Sun 2018; Linsley et al. 2019), by
capturing long-range information interactions (Bello et al. 2018; Vinyals,
Fortunato, and Jaitly 2015). The Inception family of architectures (Szegedy et
al. 2015, 2017), Multigrid Neural Architectures (Ke, Maire, and Yu 2017), and
Octave Convolution (Chen et al. 2019) aggregate the scale-space information,
while Squeeze-and-Excitation Networks (Hu, Shen, and Sun 2018) and Gather-
Excite (Hu et al. 2018) adaptively recalibrate channel-wise response by
modeling interdependency between channels. GALA (Linsley et al. 2019), CBAM
(Woo et al. 2018), and BAM (Park et al. 2018) refine the feature maps
separately in the channel and spatial dimensions. Attention Modules (Wang et
al. 2017) and self-attention (Vaswani et al. 2017; Bello et al. 2019) can be
used to exploit global context information. Precisely, non-local networks
(Wang et al. 2018) deploy self-attention as a generalized global operator to
capture the relationship between all pairwise convolutional feature maps
interactions. Except for applying the attention mechanism to computer vision
tasks (Li et al. 2019a), it has been a widespread adoption to modeling
sequences in Natural Language Processing (Yang et al. 2019).
## Proposed attention module
In this section, we analyze the two ignored problems in current attentional
activations-based models: the approximation problem and the insufficient
capacity problem of the attention maps. To address the two problems together,
we develop an attention module that mainly refers to the AW-convolution, where
the shape of attention maps matches that of the weights rather than the
activations. Besides, we refine the branch of calculating the attention maps
to achieve a better trade-off between efficiency and accuracy. Last but not
least, we integrate our proposed attention module with other attention-based
models to enlarge their representational capability.
### Motivation
First, we define basic notations in a traditional convolutional layer. In a
traditional convolutional layer, the input activations, weights, and output
activations are denoted as $I$, $K$, and $O$, respectively. For the input
activations $I\in R^{N\times C_{1}\times H\times W}$, $N$, $C_{1}$, $H$, and
$W$ refer to the batch size, the number of input channels, the height, and
width of the input feature maps, respectively. For the weights $K\in
R^{C_{2}\times C_{1}\times h\times w}$, $C_{2}$, $h$ and $w$ refer to the
number of output channels, the height and width of the weights, respectively.
For the output activations $O\in R^{N\times C_{2}\times H\times W}$, it is
computed as the convolution between the input activations $I$ and the weights
$K$. In particular, every individual value of the output activations
${O_{[l,p,m,n]}}$ is calculated as follows.
$\displaystyle{O_{[l,p,m,n]}}=\text{Convolution}(I,K)$ (1)
$\displaystyle=\sum\limits_{o=1}^{{C_{1}}}{\sum\limits_{j=1}^{h-1}{\sum\limits_{k=1}^{w-1}{{I_{[l,o,m^{\prime}+j,n^{\prime}+k]}}\times{K_{[p,o,j,k]}}}}}$
where $l=0,...,N-1$, $m=0,...,H-1$, $n=0,...,W-1$, $o=0,...,C_{1}-1$,
$p=0,...,C_{2}-1$, $m^{\prime}=m-\frac{h-1}{2}$, $n^{\prime}=n-\frac{w-1}{2}$.
To apply the attention mechanism on the input activations $I$, previous
attentional activations-based models produce the channel attention maps
$A_{c}\in R^{N\times C_{1}\times 1\times 1}$ and spatial attention maps
$A_{s}\in R^{N\times 1\times H\times W}$ separately. For example, applying the
channel attention maps $A_{c}$ on the input activations $I$ is presented as
$O=\text{Convolution}((I\odot{A_{c}}),K)$, where $\odot$ refers to the
Hadamard product and broadcasting during element-wise multiplication is
omitted.
#### Approximation problem of the attention maps
Instead of directly computing the three-dimensional attention map (N is
omitted, otherwise the attention maps are of four dimensions.), all the
current attentional activations-based models produce the attention maps
separately into the channel attention maps $A_{c}$ and spatial attention maps
$A_{s}$, which leads to the approximation problem of attention maps. However,
to thoroughly attend the input activations $I$, we need to compute the
attention maps $A_{f}\in R^{N\times C_{1}\times H\times W}$ and apply it as
$O=\text{Convolution}((I\odot{A_{f}}),K)$, which requires too much
computational and parameter overhead.
Inspired by convolution, we adopt local connection and attention maps sharing
to reduce the size of the attention maps. We compute the attention maps
$A_{a}\in R^{N\times C_{1}\times h\times w}$ as follows, where $\otimes$ is a
special element-wise multiplication since it only works associated with
convolution.
$\displaystyle{O_{[l,p,m,n]}}=\text{Convolution}(I\otimes A_{a},K)$ (2)
$\displaystyle=\sum\limits_{o=1}^{{C_{1}}}{\sum\limits_{j=1}^{h-1}{\sum\limits_{k=1}^{w-1}{({{I_{[l,o,m^{\prime}+j,n^{\prime}+k]}}}\times{{A_{a}}_{[l,o,j,k]}})\times{K_{[p,o,j,k]}}}}}$
##### Insufficient capacity problem of the attention maps
To compute different channels of the output activations of the convolution,
the input activations are constrained to be recalibrated by the same attention
maps, which indicates the insufficient capacity of the attention maps. As each
channel of the feature maps is considered as a feature detector, different
channels of the output activations of the convolution expect the input
activations to be adapted by different attention maps.
Take two channels of output activations of a convolutional layer as an
example, the two channels are responsible for recognizing rectangle shape and
triangle shape, respectively. Thus, it is reasonable for the two channels to
expect that there are different attention maps for attending the input
activations of the convolution (i.e., the attention maps to compute the
channel of recognizing the rectangle shape should be different from the
attention maps to compute the channel of recognizing the triangle shape). To
meet this expectation, we need to compute the attention maps $A_{ic}\in
R^{N\times C_{2}\times C_{1}\times 1\times 1}$ and apply it on the input
activations as follows.
$\displaystyle{O_{[l,p,m,n]}}=\text{Convolution}(I\odot{{A_{ic}}_{[l,p,:,:,:]}},K)$
(3)
$\displaystyle=\sum\limits_{o=1}^{{C_{1}}}{\sum\limits_{j=1}^{h-1}{\sum\limits_{k=1}^{w-1}{({{I_{[l,o,m^{\prime}+j,n^{\prime}+k]}}}\times{{A_{ic}}_{[l,p,o,0,0]}})\times{K_{[p,o,j,k]}}}}}$
To solve the approximation problem and the insufficient capacity problem of
the attention maps together (i.e., combining the solution of Equation.2 and
the solution of Equation. 3), we introduce our proposed attention module by
developing the AW-convolution. Specifically, we propose to compute the
attention maps $A\in R^{N\times C_{2}\times C_{1}\times h\times w}$ and apply
it as follows where the attention maps ${A_{[l,:,:,:,:]}}$ has the same shape
as that of the weights instead of the input activations. In this paper,
”Attentional weights” refers to the element-wise multiplication result between
the attention maps and the weights. Similarly, ”Attentional activations”
refers to the element-wise multiplication result between the attention maps
and the activations in previous attentional activations-based models. Thus,
$I\otimes A$ and ${A_{[l,:,:,:,:]}}\odot K$ represent the attentional
activations and attentional weights, respectively. To reduces half the number
of element-wise multiplications, we calculate attentional weights instead of
attentional activations as follows.
$\displaystyle{O_{[l,p,m,n]}}=\text{Convolution}(I\otimes A,K)$ (4)
$\displaystyle=\sum\limits_{o=1}^{{C_{1}}}{\sum\limits_{j=1}^{h-1}{\sum\limits_{k=1}^{w-1}{{{I_{[l,o,m^{\prime}+j,n^{\prime}+k]}}}\times({A_{[l,p,o,j,k]}}\times{K_{[p,o,j,k]}}})}}$
$\displaystyle=\text{Convolution}(I,{A_{[l,:,:,:,:]}}\odot K)$
$\displaystyle=\text{AW-Convolution}(I,A\odot K)$
(a) The AW-convolution architecture.
(b) The architecture of calculating attention maps $A$.
Figure 1: The architecture of our proposed attention module.
### AW-convolution in proposed attention module
The AW-convolution in our proposed attention module is presented in Figure
1(a). In this figure, the attention maps $A$ has five dimensions, which is
computed from the input activations $I$ as $A=F_{1}(I)$. $F_{1}$ is a function
to calculate the attention maps $A$ given the input activations $I$. Then, the
attentional weights $AK\in R^{N\times C_{2}\times C_{1}\times h\times w}$ is
calculated as $AK=F_{2}(A,K)=K+A\odot K$. $F_{2}$ is a function to calculate
the attentional weights $AK$ given the weights $K$ and the attention maps $A$.
Finally, the output activations $O$ is calculated from the input activations
$I$ and the attentional weights $AK$ as follows.
$\displaystyle{O_{[l,p,m,n]}}={F_{3}}(I,AK)$ (5) $\displaystyle=\text{AW-
Convolution}(I,AK)$
$\displaystyle=\sum\limits_{i=1}^{{C_{1}}}{\sum\limits_{j=1}^{h-1}{\sum\limits_{k=1}^{w-1}{{I_{[l,o,m^{\prime}+j,n^{\prime}+k]}}\times
A{K_{[l,p,o,j,k]}}}}}$
$\displaystyle=\text{Convolution}(I,A{K_{[l,:,:,:,:]}})$
where $F_{3}$ is a function to calculate the output activations $O$ given the
input activations $I$ and the attentional weights $AK$. Compared with the
traditional convolution, the attentional weights $AK$ of the AW-convolution in
our proposed attention module has five dimensions rather than four dimensions,
which are different from each other for every individual sample of the input
activations batch to convolute.
It is also worth explaining the definition of the function $F_{2}$.
$AK=K+A\odot K$ instead of $AK=A\odot K$ is used to describe the function
$F_{2}$ since it can be regarded as a residual design as follows.
$\displaystyle O={F_{3}}(I,AK)$ (6) $\displaystyle=\text{AW-
Convolution}(I,{F_{2}}(A,K))$ $\displaystyle=\text{Convolution}(I,K)+\text{AW-
Convolution}(I,A\odot K)$
### Calculating the attention maps $A$
As shown in Figure 1(b), the architecture to compute the attention maps $A$
(i.e., the definition of the function $F_{1}$) is presented, which can be
expressed as follows. Avgpool2d aggregates feature responses from the whole
spatial extent and embeds them into $A_{0}$, and Pointconv1 and Pointconv2
followed by Relu redistribute the pooled information to capture the dynamic
and no-linear dependencies between channels and spatial spaces.
$\displaystyle A={F_{1}}(I)=\text{Expand}{{}_{{C_{1}}}}({A_{2}})$ (7)
$\displaystyle=\text{Expand}{{}_{{C_{1}}}}(\text{Pointconv2}({A_{1}}))$
$\displaystyle=\text{Expand}{{}_{{C_{1}}}}(\text{Pointconv2}(\text{Pointconv1}({A_{0}})))$
$\displaystyle=\text{Expand}{{}_{{C_{1}}}}(\text{Pointconv2}(\text{Pointconv1}(\text{Avgpool2d}(I))))$
where Pointconv1 and Pointconv2 are pointwise convolutions. We add Batch
Normalization and Relu layers after Pointconv1, while adding Batch
Normalization and Sigmoid layers after Pointconv2, and they are omitted here
to provide a clear expression.
In Figure 1(b), Expand function along $C_{1}$ dimension, denoted as
$\text{Expand}_{C_{1}}$, is used as an example, and Expand function can be
also executed along $N$, $C_{2}$, $h$, and $w$ dimensions in a similar way.
$\text{Expand}{{}_{C_{1}}}$ function is used to expand the tensor $A_{2}\in
R^{N\times(C_{2}C_{1}/r_{C_{1}})\times h\times w}$ into the attention maps
$A\in R^{N\times C_{2}\times C_{1}\times h\times w}$ with the reduction ratio
$r_{C_{1}}$, including necessary squeeze, reshape, and expand operations.
$\text{Expand}_{C_{1}}$ can be expressed as follows.
$\displaystyle A=\text{Expand}{{}_{{C_{1}}}}({A_{2}})$ (8)
$\displaystyle={A_{2}}.\text{reshape}(N,{C_{2}},{C_{1}}/{r_{C_{1}}},h,w)$
$\displaystyle.\text{unsqueeze(dim=3)}\text{}$
$\displaystyle\text{.expand}(N,{C_{2}},{C_{1}}/{r_{C_{1}}},{r_{C_{1}}},h,w)\text{}$
$\displaystyle\text{.reshape}(N,{C_{2}},{C_{1}},h,w)$
Calculating the five-dimension attention maps $A$ is not an easy computational
task without careful design. Thus, we analyze the additional computational
complexity of an AW-convolution compared with a traditional convolution as a
reference to refine this design. Considering the trade-off between
computational complexity and accuracy, all the experiments in the remainder of
this paper use the same settings for the architecture of calculating the
attention maps $A$ in our proposed attention module, including
$r_{C1}={C_{1}}$, $r_{C_{2}}$ = $r_{hw}$ = 1, $r$ = 16, used in all the
stages, and $AK=K+A\odot K$ as the definition for the function $F_{2}$. The
details of refining the architecture of calculating the attention maps $A$ are
in Section C of Supplementary Material.
### Integrating with other attention-based modules
(a) Integrating with SE-ResNet/CBAM-ResNet.
(b) Integrating with AA-Wide-ResNet.
Figure 2: The schema of bottlenecks and blocks when integrating with our
proposed attention module.
In this section, we show how to integrate our proposed attention module with
the previous attention-based convolutional neural networks to demonstrate the
complementary relationship between our proposed attention module and other
attention-based modules. Since applying our proposed attention module is using
the AW-convolution to replace the traditional convolution, we can easily
integrate our proposed attention module with any convolutional neural networks
consisting of traditional convolution, including all the recently developed
attention-based models (Hu, Shen, and Sun 2018; Woo et al. 2018; Park et al.
2018; Linsley et al. 2019; Bello et al. 2019).
We choose the recent attentional activations-based models, i.e., SE-Net and
CBAM-Net, as examples to show how to integrate our proposed attention module
with other attention-based models. Here we use the popular ResNet (He et al.
2016a) as the backbone to apply the attention mechanism. As shown in Figure
2(a), the left side is the structure of a primary bottleneck in ResNet. The
middle one is the structure of a bottleneck with SE/CBAM modules in SE-
ResNet/CBAM-ResNet. Integrating the central bottleneck with our proposed
attention module is completed by replacing its $3\times 3$ convolution with a
$3\times 3$ AW-convolution, and its final structure in AW-SE-ResNet/AW-CBAM-
ResNet is shown on the right side. In summary, our proposed attention module
is a general module to be integrated seamlessly with any CNNs architectures,
including previous attention-based CNNs.
In Figure 2(b), we integrate our proposed attention module with Attention
Augmented (AA) convolutional networks and describe the architecture of their
possible AW-AA-blocks. Experiments of AW-AA-Wide-ResNet (Zagoruyko and
Komodakis 2016; Bello et al. 2019) on CIFAR-100 image classification
(Krizhevsky and Hinton 2009), as shown in Table 2, suggest that integrating
our proposed attention module with attention-based models should be explored
carefully since different integration architectures achieve different
accuracy. In some cases, an improper integration architecture leads to a small
accuracy drop. With a careful design for integration, our proposed attention
module is complementary to AA convolution and improves the accuracy of AA-Net
further.
In particular, we augment Wide-ResNet-28-10 by replacing the first convolution
of all the residual blocks with Attention Augmented convolution, which is the
AA-Wide-ResNet baseline. Here we set $N_{h}=8$ heads, $k=2v=0.2$, and a
minimum of $20$ dimensions per head for the keys as in (Bello et al. 2019). In
this figure, we can develop four possible architectures to integrate our
proposed attention module with AA-Wide-ResNet. On the left side of this
figure, we can build AA-Wide-ResNet-0 by replacing the second $3\times 3$
convolution with our AW-convolution in an AA-block. On the right side, there
are three traditional convolutions in an AA convolution, including conv1,
conv2, conv3. The conv1 is parallel to the multi-head attention, while the
conv2 and conv3 are used to calculate QKV (i.e., queries, keys, and values)
and to output attention maps, respectively. On the medial side, we can replace
one traditional convolution from conv1, conv2, or conv3 with our AW-conv1, AW-
conv2, or AW-conv3 to construct AW-AA-Wide-ResNet-1, AW-AA-Wide-ResNet-2, or
AW-AA-Wide-ResNet-3, respectively. The Top-1 accuracy of AW-AA-Wide-ResNet-0
is $1.87\%$ higher than the AA-Wide-ResNet baseline, while AW-AA-Wide-ResNet-3
shows worse performance by $0.10\%$ drop.
Model | Top-1 Error | Top-5 Error | GFLOPs | Parameters (M)
---|---|---|---|---
ResNet50 Baseline (He et al. 2016a) | $24.56\%$(+$0.00\%$) | $7.50\%$ | $3.86$ | $25.56$
AW-ResNet50 | $23.38\%$(+$1.18\%$) | $6.79\%$ | $3.87$ | $25.72$
SE-ResNet50 (Hu, Shen, and Sun 2018) | $23.14\%$(+$1.42\%$) | $6.70\%$ | $3.87$ | $28.09$
AW-SE-ResNet50 | $22.72\%$(+$1.84\%$) | $6.47\%$ | $3.88$ | $28.25$
AW-CBAM-ResNet50 (MaxPool) | $22.82\%$(+$1.74\%$) | $6.41\%$ | $3.89$ | $28.25$
AW-CBAM-ResNet50 (Spatial) | $23.20\%$(+$1.36\%$) | $6.58\%$ | $3.90$ | $28.25$
ResNet101 Baseline (He et al. 2016a) | $23.38\%$(+$0.00\%$) | $6.88\%$ | $7.57$ | $44.55$
AW-ResNet101 | $22.38\%$(+$1.00\%$) | $6.21\%$ | $7.58$ | $44.95$
SE-ResNet101 (Hu, Shen, and Sun 2018) | $22.35\%$(+$1.03\%$) | $6.19\%$ | $7.58$ | $49.33$
AW-SE-ResNet101 | $21.78\%$(+$1.60\%$) | $5.74\%$ | $7.59$ | $49.73$
AW-CBAM-ResNet101 (MaxPool) | $21.64\%$(+$1.74\%$) | $5.76\%$ | $7.60$ | $49.73$
AW-CBAM-ResNet101 (Spatial) | $22.32\%$(+$1.06\%$) | $6.18\%$ | $7.61$ | $49.73$
MobileNet Baseline (Howard et al. 2017) | $31.39\%$(+$0.00\%$) | $11.51\%$ | $0.569$ | $4.23$
SE-MobileNet (Hu, Shen, and Sun 2018) | $29.97\%$(+$1.42\%$) | $10.63\%$ | $0.581$ | $5.07$
AW-SE-MobileNet | $29.41\%$(+$1.98\%$) | $10.59\%$ | $0.623$ | $5.52$
CBAM-MobileNet (Woo et al. 2018) | $29.01\%$(+$2.38\%$) | $9.99\%$ | $0.611$ | $5.07$
AW-CBAM-MobileNet (Spatial) | $28.82\%$(+$2.57\%$) | $9.98\%$ | $0.652$ | $5.52$
Table 1: Comparisons of attention-based models on ImageNet classfication.
## Experimental results
In this section, we use extensive experiments to demonstrate the effectiveness
of our proposed attention module. We use ResNet (He et al. 2016a), MobileNet
(Howard et al. 2017), SSD300 (Liu et al. 2016), and Faster R-CNN (Ren et al.
2015) as the baseline models, and various variants of these models are
developed, including using our proposed attention module for these baseline
models and integrating our proposed attention module with their attentional
activations-based models. The datasets to train these models include CIFAR-100
classification (Krizhevsky and Hinton 2009), ImageNet classification (Deng et
al. 2009), VOC object detection datasets (Everingham et al. 2015) (The
experimental results are included in Section E of Supplementary Material.),
and COCO object detection datasets (Lin et al. 2014). To have a better
interpretation of our proposed attention module, its feature visualizations
using Grad-CAM (Selvaraju et al. 2017) can be found in Section F of
Supplementary Material. All the data augmentation and training settings can be
found in Section A of Supplementary Material.
### ImageNet image classification
To investigate the performance of our proposed attention module on high-
resolution images, we train ResNet50 and ResNet101 (He et al. 2016a) and their
attention-based variants on the ImageNet classification dataset (Deng et al.
2009). According to the results shown in Table 1, our proposed attention
module is complementary to other attentional activations-based models. AW-
ResNet50 achieves a $1.18\%$ Top-1 error reduction compared with the ResNet50
baseline. Integrating with our proposed attention module, SE-ResNet50 (Hu,
Shen, and Sun 2018) can improve further by $0.42\%$ Top-1 accuracy. The Top-1
accuracy of our AW-SE-ResNet101 is $1.60\%$ and $0.57\%$ higher than that of
ResNet101 and SE-ResNet101, respectively. To integrate with CBAM-ResNet (Woo
et al. 2018) more carefully, we define CBAM-ResNet (MaxPool) and CBAM-ResNet
(Spatial) separately. In CBAM-ResNet (MaxPool), we do not deploy the spatial
attention maps, while we do not use max-pooled features in CBAM-ResNet
(Spatial). The Top-1 accuracy of AW-CBAM-ResNet50 (MaxPool) and AW-CBAM-
ResNet50 (Spatial) are better than AW-ResNet50 by $0.56\%$ and $0.18\%$,
respectively, but worse than AW-SE-ResNet50. The number of additional
parameters for our proposed attention module is $0.16$ M, which is much
smaller than $2.83$ M (i.e., one-sixteenth) of SE and CBAM modules. Moreover,
it takes only $0.01$ GFLOPs to apply our proposed attention module on the
ResNet50 model on ImageNet classification, which is comparable with $0.01$
GFLOPs and $0.04$ to adopt the SE and CBAM modules and is negligible in terms
of FLOPs to implement the baseline model. The computational complexity
analysis introduced by the attention mechanism can be found in Section B of
Supplementary Material.
Model | Top-1 Error | Top-5 Error | GFLOPs | Parameters (M)
---|---|---|---|---
AA-Wide-ResNet Baseline (Bello et al. 2019) | $28.01\%$($+0.00\%$) | $7.92\%$ | $3.89$ | $8.43$
AW-AA-Wide-ResNet-0 | $26.14\%$($+1.87\%$) | $7.43\%$ | $3.90$ | $8.50$
AW-AA-Wide-ResNet-1 | $27.17\%$($+0.84\%$) | $7.49\%$ | $3.89$ | $8.48$
AW-AA-Wide-ResNet-2 | $27.09\%$($+0.92\%$) | $8.00\%$ | $3.89$ | $8.45$
AW-AA-Wide-ResNet-3 | $28.11\%$($-0.10\%$) | $8.24\%$ | $3.89$ | $8.44$
ResNet50 Baseline (He et al. 2016a) | $22.33\%$($+0.00\%$) | $5.83\%$ | $1.22$ | $23.71$
AW-ResNet50 | $19.87\%$($+2.46\%$) | $4.76\%$ | $1.23$ | $23.87$
SE-ResNet50 (Hu, Shen, and Sun 2018) | $20.43\%$($+1.90\%$) | $5.01\%$ | $1.23$ | $26.24$
AW-SE-ResNet50 | $19.00\%$($+3.33\%$) | $4.51\%$ | $1.24$ | $26.40$
CBAM-ResNet50 (Woo et al. 2018) | $19.46\%$($+2.87\%$) | $4.56\%$ | $1.24$ | $26.24$
AW-CBAM-ResNet50 | $18.94\%$($+3.39\%$) | $4.76\%$ | $1.25$ | $26.40$
Table 2: Comparisons of attention-based models on CIFAR-100 classfication. Backbone | Detector |<EMAIL_ADDRESS>0.95] |<EMAIL_ADDRESS>|<EMAIL_ADDRESS>
---|---|---|---|---
ResNet101-FPN (Lin et al. 2017) | Faster R-CNN | $37.13$($+0.00\%$) | $58.28$ | $40.29$
ResNet101-AW-FPN | Faster R-CNN | $37.76$($+0.63\%$) | $59.17$ | $40.91$
ResNet101-SE-FPN (Hu, Shen, and Sun 2018) | Faster R-CNN | $38.11$($+0.98\%$) | $59.41$ | $41.33$
ResNet101-AW-SE-FPN | Faster R-CNN | $38.45$($+1.32\%$) | $59.70$ | $41.86$
ResNet101-CBAM-FPN (Woo et al. 2018) | Faster R-CNN | $37.74$($+0.61\%$) | $58.84$ | $40.77$
ResNet101-AW-CBAM-FPN | Faster R-CNN | $38.19$($+1.06\%$) | $59.52$ | $41.43$
Table 3: Comparisons of attention-based Faster R-CNN on COCO.
#### Resource-constrained architecture
Driven by demand for mobile applications, many depthwise convolution-based
models are developed to take care of the trade-off between accuracy and
efficiency. To inspect the generalization of our proposed attention module in
this resource-constrained scenario, we conduct the ImageNet classification
(Deng et al. 2009) with the MobileNet architecture (Howard et al. 2017). Since
our proposed attention module is efficient in terms of both the storage and
computational complexity, integrating it into the light-weight architecture is
worth exploring. We apply our proposed attention module to pointwise
convolution instead of depthwise convolution in every two depthwise separable
convolutions. When integrating with the CBAM models (Woo et al. 2018), we
remove the max-pooled features and keep spatial attention maps. As shown in
Table 1, AW-SE-MobileNet and AW-CBAM-MobileNet achieve $0.56\%$ and $0.19\%$
Top-1 accuracy improvements compared with SE-MobileNet (Hu, Shen, and Sun
2018) and CBAM-MobileNet, respectively. It is an impressive result that the
Top-1 accuracy of AW-CBAM-MobileNet is $2.57\%$ better than that of the
MobileNet baseline. For the MobileNet model, our proposed attention module
increases the computation by $0.041$ GFLOPs, while SE and CBAM modules need
$0.012$ and $0.041$ GFLOPs, respectively. Also, the required parameters for
our proposed attention module are $0.45$ M, which is much less than $0.84$ M
for SE and CBAM modules.
### CIFAR-100 image classification
According to the results shown in Table 2, we conclude that our proposed
attention module can boost the CIFAR-100 (Krizhevsky and Hinton 2009) accuracy
of both the ResNet50 baseline model (He et al. 2016a) and their attentional
activations-based models with negligible additional computational complexity.
AW-ResNet50 achieves a $2.46\%$ Top-1 error reduction compared with the
ResNet50 baseline. Integrating with our proposed attention module, SE-ResNet50
(Hu, Shen, and Sun 2018) and CBAM-ResNet50 (Woo et al. 2018) can increase
Top-1 accuracy by $1.43\%$ and $1.52\%$, respectively. In terms of the
computational complexity, our proposed attention module requires $0.01$
GFLOPs, which is acceptable compared with $0.01$ GFLOPs for the SE module,
$0.02$ GFLOPs for the CBAM module, and $1.22$ GFLOPs for the baseline model.
Besides, we only introduce $0.16$ M parameters for our proposed attention
module, which is less than $2.53$ M parameters for the SE and CBAM modules. We
train ResNet with various depths on CIRAR-100 image classification to show
that our proposed attention module works for CNNs with different depths, which
are in Section D of Supplementary Material.
### Object Detection on COCO
To show the generalization of our proposed attention module, we apply it to
object detection tasks. We evaluate our proposed attention module further on
the COCO dataset (Lin et al. 2014), which contains $118K$ images (i.e.,
train2017) for training and $5K$ images (i.e., val2017) for validating. We use
Faster R-CNN (Ren et al. 2015) as the detection method with the ResNet101-FPN
backbone (Lin et al. 2017). Here we intend to evaluate the benefits of
applying our proposed attention module on the ResNet101-FPN backbone (Lin et
al. 2017), where all the lateral and output convolutions of the FPN adopt our
AW-convolution. The SE and CBAM modules are placed right before the lateral
and output convolutions. As shown in Table 3, applying our proposed attention
module on ResNet101-FPN boosts<EMAIL_ADDRESS>0.95] by $0.63$ for the Faster R-CNN
baseline. Integrating with attentional activations-based models, Faster R-CNNs
with the backbones of ResNet101-AW-SE-FPN and ResNet101-AW-CBAM-FPN outperform
Faster R-CNNs with the backbones of ResNet101-SE-FPN and ResNet101-CBAM-FPN by
$0.34$ and $0.45$ on COCO’s standard metric AP.
## Conclusion
In this paper, We analyze the two ignored problems in attentional activations-
based models: the approximation problem and the insufficient capacity problem
of the attention maps. To address the two problems together, we propose an
attention module by developing the AW-convolution, where the shape of the
attention maps matches that of the weights rather than the activations, and
integrate it with attention-based models as a complementary method to enlarge
their attentional capability. We have implemented extensive experiments to
demonstrate the effectiveness of our proposed attention module, both on image
classification and object detection tasks. Our proposed attention module alone
shows noticeable accuracy improvement compared with baseline models. More
importantly, integrating our proposed module with previous attention-based
models, such as AA-Net (Bello et al. 2019), SE-Net (Hu, Shen, and Sun 2018),
and CBAM-Net (Woo et al. 2018), will further boost their performance.
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|
Vortex Creep Heating in Neutron Stars
Motoko Fujiwaraa,b*** E-mail address<EMAIL_ADDRESS>Koichi
Hamaguchia,c††† E-mail address<EMAIL_ADDRESS>Natsumi
Nagataa‡‡‡ E-mail address<EMAIL_ADDRESS>
and Maura E. Ramirez-Quezadaa,d§§§ E-mail address: me.quezada@hep-
th.phys.s.u-tokyo.ac.jp,
aDepartment of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113–0033, Japan
b Physik-Department, Technische Universität, München, James-Franck-Straße,
85748 Garching, Germany
cKavli IPMU (WPI), University of Tokyo, Kashiwa, Chiba 277–8583, Japan
dDual CP Institute of High Energy Physics, C.P. 28045, Colima, México
Recent observations of old warm neutron stars suggest the presence of a
heating source in these stars, requiring a paradigm beyond the standard
neutron-star cooling theory. In this work, we study the scenario where this
heating is caused by the friction associated with the creep motion of neutron
superfluid vortex lines in the crust. As it turns out, the heating luminosity
in this scenario is proportional to the time derivative of the angular
velocity of the pulsar rotation, and the proportional constant $J$ has an
approximately universal value for all neutron stars. This $J$ parameter can be
determined from the temperature observation of old neutron stars because the
heating luminosity is balanced with the photon emission at late times. We
study the latest data of neutron star temperature observation and find that
these data indeed give similar values of $J$, in favor of the assumption that
the frictional motion of vortex lines heats these neutron stars. These values
turn out to be consistent with the theoretical calculations of the vortex-
nuclear interaction.
## 1 Introduction
A neutron star (NS) is incredibly dense and exists under extreme conditions of
pressure and temperature that cannot be found in other places in the universe.
While the internal structure of NSs remains elusive, indirect evidence
suggests the existence of a neutron superfluid in the inner crust. The first
indication of superfluidity in NSs came from the observation of non-zero
pairing energy associated with attractive forces, leading to the formation of
an energy gap and hence superfluidity [1]. This scenario was predicted to
occur in stars with neutron cores [2]. After the discovery of pulsars, the
energy gap in NS matter has been further studied; see, e.g., Refs. [3, 4, 5]
for a recent review on superfluidity in NSs.
In a rotating NS, the irrotational property of a superfluid requires the
formation of vortex lines, whose distribution determines the angular velocity
of the superfluid component. In the inner crust region, these vortex lines are
fixed at certain positions by the interactions with nuclei and cannot move
freely. This preserves the rotational speed of the superfluid component and
prevents it from following the slowdown of the pulsar rotation, giving rise to
the deviation in the rotational speed between the superfluid and other
components. This deviation increases until the vortex lines are forced to move
by the Magnus force, which increases as the difference in the rotational speed
increases. This vortex-line dynamics leads to some observational consequences.
A well-known example is the pulsar glitch phenomenon, namely, sudden changes
in the rotational frequency of NSs,111See Refs. [6, 7, 8, 9] for a recent
review on pulsar glitches. which could be attributed to an avalanche of
unpinning of superfluid vortex lines [10, 11]. Another phenomenon is the
heating effect caused by the friction associated with the creep motion of
vortex lines [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], which is the
subject of the present paper.
Our motivation to revisit this heating mechanism is provided by the recent
observations of old warm NSs [23, 24, 25, 26, 27, 28, 29], whose observed
temperature is considerably higher than that predicted in the standard NS
cooling scenario [30, 31, 32, 33, 34, 35]. This work aims to study whether
these observations can be explained by the vortex-creep heating effect. With
this objective in mind, we focus on the following characteristic property of
the vortex-creep heating. As we see below in detail, the heating luminosity in
this heating mechanism is proportional to the time derivative of the angular
velocity of the pulsar rotation, and the proportional constant is determined
only by the NS structure and the vortex-nuclear interactions. As a result, the
value of this proportional constant, denoted by $J$ in this paper, is almost
universal over NSs. In addition, we can obtain the value of $J$ for an old NS
by observing its temperature and its pulsar motion since, at late times, the
heating luminosity balances with the luminosity of photon emission, which is
determined by the surface temperature of the NS. As it turns out, the present
data of old warm NSs indeed show similar values of $J$ for these stars, in
agreement with the prediction of the vortex-creep heating scenario. We also
find that these values are compatible with the $J$ parameter evaluated from
the calculations of the nuclear pinning force available in the literature.
The remainder of this paper is structured as follows. In Section 2, we provide
a brief overview of the thermal evolution of NSs, focusing on the isothermal
phase. In Section 3, we describe the vortex creep heating mechanism for NSs
and explain how we can relate a universal parameter with the late-time
temperature prediction. In Section 4, we summarize the numerical evaluation of
the pinning force, from which we calculate the parameter $J$. In Section 5, we
study the recent observations of old and warm NSs to assess the current status
of the vortex creep heating hypothesis. Finally, we summarize our findings and
conclude our discussion in Section 6.
## 2 Thermal evolution of neutron star
This section reviews the surface temperature prediction of NSs based on their
thermal evolution and sources of heating and cooling. The temperature
distribution within NSs is characterized by a local core temperature $T$,
which exhibits a temperature gradient only during the early stages of the NS,
typically within the first 10–100 years of its existence [36, 37]. The high
thermal conductivity of the highly degenerate electron gas causes the core to
become isothermal over time, reaching thermal equilibrium. Hence, the red-
shifted temperature of NSs,
$T^{\infty}(\bar{r},t)=T(\bar{r},t)e^{\phi(\bar{r})}$, where $\phi(\bar{r})$
specifies the gravitational redshift, reaches a constant value
$T^{\infty}(\bar{r},t)\simeq T^{\infty}(t)$ and only the outermost layers
exhibit an appreciable temperature gradient. The relativistic equation
describing the thermal evolution of NSs after thermal relaxation is given by
[38, 39]
$\displaystyle
C(T^{\infty})\frac{dT^{\infty}}{dt}=-L_{\nu}^{\infty}(T^{\infty})-L_{\gamma}^{\infty}(T^{\infty})+L_{\rm
H}^{\infty}.$ (2.1)
Here, $C$ represents the total heat capacity of the NS and is temperature-
dependent. The right-hand side of the equation expresses the red-shifted
luminosity for three different processes, namely, neutrino cooling
$L_{\nu}^{\infty}$, photon cooling $L_{\gamma}^{\infty}$, and heating source
$L_{\rm H}^{\infty}$.
At temperatures below a few $\times~{}10^{9}~{}\mathrm{K}$, the NS becomes
transparent to neutrinos, allowing them to escape without interacting with the
stellar matter and carrying away energy. Therefore, the cooling process during
the early stage of the star’s life is dominated by neutrino emission. At later
times, typically for $t\gtrsim 10^{5}~{}\mathrm{yrs}$,222The dominance of
photon emission might be delayed for massive NSs, in which the rapid neutrino
emission via the direct Urca process could occur; we do not consider this
possibility in what follows. photon emission dominates neutrino emission. The
thermal photon emission follows a blackbody spectrum. The photon luminosity
$L_{\gamma}$ can be described by the Stefan-Boltzmann law and related to
surface temperature $T_{\rm s}$ as
$\displaystyle L_{\gamma}=4\pi R_{\rm NS}^{2}\sigma_{\rm SB}T_{\rm s}^{4},$
(2.2)
in the local reference frame of the NS, where $\sigma_{\rm SB}$ is the
Stephan-Boltzmann constant and $R_{\rm NS}$ the NS radius. To relate the
internal temperature $T$ of a NS to its surface temperature $T_{\rm s}$, we
use the heat envelope model proposed by Potekhin et al. in 1997 [40].
According to this model, the observed thermal emission from old isolated NSs
can be explained by the heat trapped in a thin envelope surrounding the star’s
crust.
If we have an internal heating source, the heating luminosity will balance
with the photon luminosity at a sufficiently late time,
$\displaystyle L^{\infty}_{\rm H}$ $\displaystyle\simeq L^{\infty}_{\gamma},$
(2.3)
which determines the surface temperature. It is pointed out that NSs have some
internal heating mechanisms, such as vortex creep heating, rotochemical
heating, and magnetic field decay. These heating effects may become visible at
late times and can operate even for isolated NSs. These internal mechanisms
are comprehensively compared with the observed surface temperature in Refs.
[20, 22], which is recently revisited by Ref. [41].
We note in passing that the balance equation (2.3) generically holds with good
accuracy for old NSs that are older than $10^{5}~{}\mathrm{yrs}$. This can be
seen by estimating the typical timescale $\tau_{\rm eq}$ for the relaxation
into the equilibrium state:
$\displaystyle\tau_{\rm eq}$ $\displaystyle\simeq\frac{CT^{\infty}}{4\pi
R_{\rm NS}^{2}\sigma_{\rm SB}T_{\rm s}^{4}}$ (2.4) $\displaystyle\sim 3\times
10^{4}~{}\mathrm{yrs}\left(\frac{C}{10^{35}~{}\mathrm{erg/K}}\right)\left(\frac{R_{\rm
NS}}{11.43~{}{\rm km}}\right)^{-2}\left(\frac{T_{\rm
s}}{10^{5}~{}\mathrm{K}}\right)^{-4}\left(\frac{T^{\infty}}{10^{6}~{}\mathrm{K}}\right)~{},$
(2.5)
where we have used $T^{\infty}\sim 10^{6}~{}\mathrm{K}$ corresponding to the
surface temperature, at the equilibrium phase, of $T_{\rm s}\sim
10^{5}~{}\mathrm{K}$.333To derive the typical scale of $\tau_{\rm eq}$, we
fitted the relation between the NS internal temperature $T$ and the NS surface
temperature $T_{\rm s}$ for $t_{\rm age}\lesssim 10^{6}~{}\mathrm{yrs}$ [40]
by assuming $T\propto T_{\rm s}^{2}$. It is found that this timescale is
shorter than the age of old NSs, assuring the equilibrium condition (2.3).
## 3 Review of vortex creep heating
In this section, we will review vortex creep heating, where the presence of a
superfluid in the inner crust of a NS plays a key role. In this region, vortex
lines are thought to exist as a consequence of the NS rotation. In Sec. 3.1,
we will derive the equation of motion for this rotational motion by
introducing two different angular velocities for the inner crust superfluid
and the other part of the star. In Sec. 3.2, we will consider the dynamics of
a vortex line and evaluate its radial velocity. Finally, in Sec. 3.3, we will
assess the effect of vortex creep on the late-time temperature prediction of
NSs.
### 3.1 Equations of motion for crust and superfluid
To describe a rotating NS, let us divide it into two components depending on
how external torque exerts on it, the crust component and the superfluid
component [42]. The crust component comprises a rigid crust, a lattice of
nuclei, and charged particles tightly coupled to electromagnetic field lines
[42]. This component is directly affected by external torque provided by
pulsar magnetic radiation. The superfluid component refers to ${}^{1}S_{0}$
superfluid of neutrons.444The neutron triplet (${}^{3}P_{2}$) superfluid in
the core region is classified into the crust component in the two-component
model [42]. This is because the neutron superfluid in this region is expected
to coexist with the proton superconductor and be tightly coupled to the crust
component [43, 44]. This superfluid phase is believed to appear in the inner
crust region based on pairing gap evaluations [3, 4, 5]. This component is
just indirectly affected by the external torque through the interaction with
the crust component.
Figure 1: The structure of a NS and vortex lines. Left: The grey, blue, and
brown regions represent the outer crust, inner crust, and core regions,
respectively. The red lines represent vortex lines. Right: A single vortex
line in the inner crust. The vortex line in the neutron superfluid is attached
to the outer crust and has two boundaries.
On the left side of Fig. 1, we show a schematic diagram of a NS. The grey
region is the outer crust, composed of ions and electrons. The blue layer
represents the inner crust region and has an approximately $\sim
1~{}\mathrm{km}$ thickness. In this region, nuclei, electrons, and neutrons
exist, and neutrons are expected to be in the form of the neutron singlet
superfluid. The brown region is the NS core, whose internal structure remains
uncertain and a subject of ongoing research and debate. In the two-component
model, the neutron superfluid in the blue layer is classified into the
superfluid component, while all the other parts are classified into the crust
component.
Based on this two-component model, we derive the equations of motion and
describe the rotation. For later convenience, we divide the system into a thin
disc and introduce the cylindrical coordinate $(r,\varphi,z)$. Since we treat
the crust component as a rigid body, its angular velocity $\Omega_{\rm c}(t)$
is independent of $r$. On the other hand, the superfluid angular velocity
varies with $r$, and we denote it by $\Omega_{\rm s}(t,r)$. The equation of
motion for the crust component is
$\displaystyle I_{\rm c}\dot{\Omega}_{\rm c}(t)=N_{\rm ext}(t)+N_{\rm
int}(t),$ (3.6)
where $I_{\rm c}$ represents the moment of inertia of the crust component, and
the dot represents the derivative with respect to time $t$. On the right-hand
side, we divide the torque into two parts: The first term $N_{\rm ext}(t)$
represents the external torque acting on the system. The second term $N_{\rm
int}(t)$ corresponds to the effects of internal torques in the system
$\displaystyle N_{\rm int}(t)$ $\displaystyle=-\int
dI_{p}(r)\frac{\partial\Omega_{\rm s}}{\partial t}(t,r),$ (3.7)
where $dI_{p}$ represents the differential inertial momenta. The integral is
taken over the region where the crust component interacts with the superfluid
component, called the pinning region. These two components connect through the
pinning force between the vortex line and the nuclei-like object in the inner
crust, as we will discuss in Sec. 3.2.
The equations of the superfluid component are obtained by introducing two
fundamental measures of rotation in the fluid: vorticity and circulation. The
vorticity vector $\bm{\omega}$ characterizes the microscopic features of
rotation and is a locally determined value defined as the curl of the fluid
velocity $\bm{v}$,
$\displaystyle\bm{\omega}\equiv\nabla\times\bm{v}.$ (3.8)
In contrast, the circulation $\Gamma$ measures the macroscopic rotation and is
defined over a finite region. It is given by the line integral of the fluid
velocity $\bm{v}$ around a closed path $C$, which can be expressed as a
surface integral over any surface $S$ with the boundary $C$,
$\displaystyle\Gamma\equiv\oint_{C}\bm{v}\cdot
d\bm{\ell}=\iint_{S}\bm{\omega}\cdot d\bm{S},$ (3.9)
where $d\bm{\ell}$ and $d\bm{S}$ denote the line and surface elements,
respectively. We used Stokes’ theorem to obtain the last expression.
Superfluid motion obeys the potential flow condition,
$\displaystyle\nabla\times\bm{v}{{}_{\rm s}}=\bm{0},$ (3.10)
where $\bm{v}{{}_{\rm s}}$ denotes the superfluid velocity. This condition,
implying the absence of the vorticity in the superfluid, holds because the
superfluid velocity is proportional to the gradient of the phase of the
condensate wave-function of the superfluid. Nevertheless, we still have a
nonzero circulation if there exists a singular object known as a vortex line.
In Fig. 1, we show a schematic picture of a single vortex line in the NS inner
crust. The vortex line is a string-like configuration with a thickness of the
order of femtometers (red curve). The circulation for each vortex line is
quantized in units of
$\displaystyle\kappa$ $\displaystyle\equiv\frac{h}{2m_{n}},$ (3.11)
where $h$ is the Planck constant and $m_{n}$ is the neutron mass. This
quantization follows from the condition that the wave-function of the
condensate is single-valued, and thus the change in its phase must be $2\pi k$
with $k$ as an integer. Since a vortex line with $k=1$ is energetically
favored and stable, the system with larger angular velocity contains a larger
number of vortex lines [45]. The number of vortex lines will be saturated if
the total circulation reaches that of the rigid rotation as a whole system,
which is expected in NSs. The vortex lines in the inner crust have boundaries
corresponding to the normal matter in the outer crust. Under this
circumstance, the circulation for the contour $C$ in Fig. 1 is uniquely
determined and, because of the potential flow condition (3.10), can be
regarded as topological as it remains unchanged under deformations of $C$
(unless it passes through another vortex line).
We may express the superfluid velocity on average in the same form as the
normal fluid,555This relation is confirmed by observing the shape of the free
surface for rotating superfluid in liquid He system [46, 47, 48].
$\displaystyle\Braket{\bm{v}_{\rm s}}=\bm{\Omega}_{\rm s}\times\bm{r}~{},$
(3.12)
where $\bm{r}$ denotes the position vector from the center of the NS. The
total circulation of a superfluid system is equal to the sum of the
circulation of each vortex line. This means that the number of vortex lines is
directly related to the superfluid angular velocity $\bm{\Omega}_{\rm s}$. By
substituting Eq. (3.12) in Eq. (3.9), we obtain
$\displaystyle\Gamma_{\rm superfluid}$
$\displaystyle=\int_{C}d\bm{\ell}\cdot\bigl{(}\bm{\Omega}_{\rm
s}(t,r)\times\bm{r}\bigr{)}=\int_{0}^{r}dr^{\prime}~{}2\pi r^{\prime}\kappa
n(t,r^{\prime}),$ (3.13)
where we choose the integral path $C$ around the edge of the disc with radius
$r$, and $n(t,r)$ is the number density of the vortex lines per unit area.
Noting that only the radial motion of vortex lines changes $n$ due to the
axial symmetry around the rotation axis, we obtain the following conservation
law,
$\displaystyle\frac{\partial n}{\partial t}+\nabla\cdot(nv_{r}\bm{e}_{r})=0,$
(3.14)
where $v_{r}$ is the vortex velocity in the radial direction $\bm{e}_{r}$,
which we call the creep rate. Combining Eqs. (3.13) and (3.14), we obtain the
equation of motion for the superfluid component:
$\displaystyle\frac{\partial\Omega_{\rm s}}{\partial t}$
$\displaystyle=-\left(2\Omega_{\rm s}+r\frac{\partial\Omega_{\rm s}}{\partial
r}\right)\frac{v_{r}}{r}.$ (3.15)
The NS rotation is described by Eqs. (3.6) and (3.15) coupled through a
nonzero $v_{r}$. In other words, by switching off the radial motion of the
vortex line, we have $\partial\Omega_{\rm s}/\partial t=0$. In this case,
$N_{\rm int}$ turns out to be zero, and the two equations of motion are
decoupled.
### 3.2 Dynamics of a vortex line
In the inner crust, a vortex line feels two forces, the pinning force and the
Magnus force. The pinning force arises from the interaction between a vortex
line and a nuclear-like object within the inner crust, resulting in the
pinning of the vortex line to the lattice of nuclei where the energy is
minimized [10]. As long as the pinning force is dominant, the vortex lines
remain attached to the crust and move at the same velocity as the crust
component:
$\displaystyle\bm{v}_{\rm VL}(t,r)=\bm{\Omega}_{\rm c}(t)\times\bm{r},$ (3.16)
where $\bm{v}_{\rm VL}(t,r)$ denotes the velocity of a vortex line.
One way to quantify the pinning force is to compare the energies associated
with different configurations of a vortex line.
Figure 2: The nuclear pinning and interstitial pinning for the vortex line
pinning configurations. Red cylinders and black spheres represent vortex lines
and nuclei, respectively.
In Fig. 2, we show two possibilities for the pinning configurations, the
nuclear pinning and the interstitial pinning. The difference in the energies
between these two configurations defines the pinning energy,
$\displaystyle E_{\rm pin}\equiv E_{\rm NP}-E_{\rm IP},$ (3.17)
where $E_{\rm NP}$ ($E_{\rm IP}$) denotes the energy of the nuclear
(interstitial) pinning configuration. A nuclear pinning configuration occurs
when the pinning energy is negative, and the vortex line is directly attached
to the nuclei lattice. Conversely, when the pinning force is repulsive, the
vortex line is pinned in the interstitial regions. For a nuclear pinning
configuration, a rather crude estimate of the pinning force per unit length is
then given by
$\displaystyle f_{\rm pin}|_{\rm NP}$ $\displaystyle\simeq\frac{|E_{\rm
pin}|}{\Delta r\Delta L},$ (3.18)
where $\Delta r$ is the distance between the nuclear and interstitial pinning
positions and $\Delta L$ is the distance between the successive pinning sites
along a vortex line. These two quantities are expected to be of the order of
the Wigner-Seitz radius $R_{\rm WS}$, the radius of an imaginary sphere whose
volume is equal to the average volume per nucleus in each region.
Due to this pinning effect, the relative velocity between the superfluid and
the vortex lines is developed,
$\displaystyle\delta\bm{v}\equiv\bm{v}_{\rm s}-\bm{v}_{\rm
VL}=\delta\bm{\Omega}\times\bm{r},$ (3.19)
where we use Eqs. (3.12) and (3.16) to obtain the last expression and
introduce the relative angular velocity,
$\displaystyle\delta\bm{\Omega}$ $\displaystyle\equiv\bm{\Omega}_{\rm
s}-\bm{\Omega}_{\rm c}.$ (3.20)
This velocity difference induces the Magnus force per unit length of a vortex
line,
$\displaystyle\bm{f}_{\rm Mag}$
$\displaystyle=\rho\,(\delta\bm{v})\times\bm{\kappa},$ (3.21)
where $\rho$ is the superfluid density and $\bm{\kappa}$ is the vorticity
vector, which is parallel to the vortex line (hence, parallel to the
rotational axis of the NS) and has the absolute value
$|\bm{\kappa}|\equiv\kappa$.
Figure 3: The Magnus force acting on a vortex line. We define the direction of
$\bm{\kappa}$ as the direction of the right-hand screw.
We see that the Magnus force always acts in the outward direction, as
illustrated in Fig. 3, since the crust component rotates slower than the
superfluid component due to the deceleration by the external torque.
Vortex lines overcome the trapping through thermal fluctuations or quantum
tunneling [49, 50], and start to creep outward. If this creep rate is rapid
enough, the superfluid rotation can smoothly follow the change in the crust
rotation, and the system can reach a steady state. To see if this is the case
for the NSs of our interest, let us briefly review the evaluation of the
vortex creep rate $v_{r}$ following Ref. [50]. In this analysis, the pinning
force is modeled by a periodic potential with a period equal to the span of
the nuclei lattice $(\sim R_{\rm WS})$ and a height equal to the pinning
energy $(\sim|E_{\rm pin}|)$. The Magnus force is considered as a bias that
tilts the periodic potential. Let us introduce the transition rate for a
vortex line to move from a local minimum into the next local minimum as ${\cal
R}_{\rm VC}$. The zero-point frequency in the vicinity of local minima of the
potential $\omega_{0}$ controls ${\cal R}_{\rm VC}$ and is obtained through
the quantization of the vortex system [49, 50]. If the vortex tension is
negligible compared to the pinning force, we obtain666In Ref. [50], the case
where the vortex tension dominates the pinning force is also studied, and the
conclusion of the steady state turns out to remain unchanged.
$\displaystyle\omega_{0}\simeq\frac{\pi\kappa\Lambda}{4R_{\rm WS}^{2}}\simeq
1.2\times 10^{20}\,{\rm s}^{-1}\left(\frac{R_{\rm WS}}{50~{}{\rm
fm}}\right)^{-2}\left(\frac{\Lambda}{2}\right),$ (3.22)
where $\Lambda$ characterizes the vortex tension,
$T_{v}=\rho\kappa^{2}\Lambda/(4\pi)$, and is evaluated as
$2\lesssim\Lambda\lesssim 10$ in the inner crust [51]. The transition rate is
then given by [49, 50]
$\displaystyle{\cal R}_{\rm
VC}\simeq\frac{\omega_{0}}{2\pi}\,e^{-\frac{|E_{\rm pin}|}{k_{\rm B}T_{\rm
eff}}},$ (3.23)
where $T_{\rm eff}$ is defined by
$\displaystyle k_{\rm B}T_{\rm
eff}\equiv\frac{\hbar\omega_{0}}{2}\coth\left(\frac{T_{\rm
q}}{T}\right)\sim\begin{cases}k_{\rm B}T&(T\gg T_{\rm q})\\\
\frac{\hbar\omega_{0}}{2}&(T\ll T_{\rm q})\end{cases},$ (3.24)
with $T$ being the temperature of the inner crust and
$\displaystyle T_{\rm q}\equiv\frac{\hbar\omega_{0}}{2k_{\rm B}}\simeq
3.8\times
10^{8}~{}\mathrm{K}~{}\left(\frac{\omega_{0}}{10^{20}~{}\mathrm{s}^{-1}}\right).$
(3.25)
As can be seen from these expressions, the unpinning occurs predominantly
through thermal activation (quantum tunneling) for $T\gg T_{\rm q}$ ($T\ll
T_{\rm q}$). In particular, for NSs as old as those considered in the
following analysis, their internal temperature is $\lesssim 10^{8}$ K.
Therefore, the vortex creep motion in these NSs is triggered by quantum
tunneling. By using the transition rate in Eq. (3.23), we evaluate the creep
rate as $v_{r}\simeq{\cal R}_{\rm VC}\cdot R_{\rm WS}$.
With the vortex creep rate obtained above, we can estimate the typical
distance at which a vortex line travels through the creep motion during the
lifetime of the NS:
$\displaystyle v_{r}\times t_{\rm age}$ $\displaystyle\simeq R_{\rm
WS}\cdot\frac{\omega_{0}}{2\pi}\cdot e^{-\frac{2|E_{\rm
pin}|}{\hbar\omega_{0}}}\times t_{\rm age}$ $\displaystyle\simeq 160~{}{\rm
km}\times\left(\frac{\omega_{0}}{10^{20}~{}\rm
s^{-1}}\right)\left(\frac{R_{\rm WS}}{50~{}\rm fm}\right)\left(\frac{t_{\rm
age}}{10^{5}~{}\rm yr}\right),$ (3.26)
where we set $|E_{\rm pin}|=1$ MeV as a representative value. We find that
this distance is considerably larger than the crust thickness ($\simeq 1$ km),
implying that the vortex creep motion would reach a steady state in old NSs.
Once the system enters the steady phase, the crust and superfluid components
decelerate at the same rate, i.e.,
$\displaystyle\frac{\partial\Omega_{\rm s}(t,r)}{\partial t}=\dot{\Omega}_{\rm
c}\equiv\dot{\Omega}_{\infty}~{}.$ (3.27)
Note that $\Omega_{\rm c}$ and $|\dot{\Omega}_{\infty}|$ are identified as the
current observed values of the angular velocity and deceleration rate of NSs,
respectively. As a consequence, the relative angular velocity between the
crust and superfluid components becomes independent of time, and this is found
to be fairly close to the critical angular velocity determined by the
condition $f_{\rm pin}=f_{\rm Mag}$ [50]:
$\displaystyle\delta\Omega_{\infty}\simeq\delta\Omega_{\rm cr}~{},$ (3.28)
with
$\displaystyle\delta\Omega_{\rm cr}$
$\displaystyle\equiv|\delta\bm{\Omega}|_{f_{\rm pin}=f_{\rm Mag}}=\frac{f_{\rm
pin}}{\rho\kappa r}.$ (3.29)
### 3.3 Prediction of surface temperature
As vortices move outward, the rotational energy of the superfluid is
dissipated through their frictional interaction with the normal components of
the inner crust, which heats the NS. This heating luminosity is computed as
[12, 15]
$\displaystyle L_{\mathrm{H}}$ $\displaystyle=N_{\rm ext}\Omega_{\rm
c}(t)-\frac{d}{dt}\left[\frac{1}{2}I_{\rm c}\Omega_{\rm
c}^{2}(t)+\frac{1}{2}\int dI_{\rm p}\Omega_{\rm s}^{2}(t,r)\right]$
$\displaystyle=\int dI_{\rm p}\left[\Omega_{\rm c}(t)-\Omega_{\rm
s}(t,r)\right]\frac{\partial\Omega_{\rm s}(t,r)}{\partial t}$
$\displaystyle=\int dI_{\rm p}\,\delta\Omega\left|\frac{\partial\Omega_{\rm
s}(t,r)}{\partial t}\right|~{},$ (3.30)
where we use Eqs. (3.6) and (3.7), and the inertial momenta $I_{\rm p}$ is
integrated over the region where the pinning process efficiently occurs. This
expression is further simplified in the steady state with the condition
(3.27),
$\displaystyle L_{\mathrm{H}}$ $\displaystyle=J|\dot{\Omega}_{\infty}|~{},$
(3.31)
where we define,
$\displaystyle J\equiv\int dI_{\rm p}\delta\Omega_{\infty}~{},$ (3.32)
and $\delta\Omega_{\infty}$ denotes the steady-state value of the relative
angular velocity. As we see, the heating luminosity is proportional to the
current deceleration rate of the pulsar. Note that the value of $J$ can be
estimated from Eqs. (3.27), (3.29), and (3.32) by specifying the value of
$f_{\rm pin}$ and the region of pinning. We will evaluate the proportional
coefficient $J$ in Sec. 4.2.
In particular, if this heating luminosity balances with the photon luminosity,
which we expect to occur for old NSs as we discussed in Sec. 2, the surface
temperature $T_{\rm s}$ can be estimated using Eq. (2.3) as
$\displaystyle T_{\rm s}^{\rm eq}$
$\displaystyle\equiv\left(\frac{J|\dot{\Omega}_{\infty}|}{4\pi R_{\rm
NS}^{2}\sigma_{\rm SB}}\right)^{\frac{1}{4}}$ (3.33) $\displaystyle\simeq
1.0\times
10^{5}~{}\mathrm{K}\left(\frac{J}{10^{43}~{}\mathrm{erg~{}s}}\right)^{\frac{1}{4}}\left(\frac{|\dot{\Omega}_{\infty}|}{10^{-14}~{}\mathrm{s}^{-2}}\right)^{\frac{1}{4}}\left(\frac{R_{\rm
NS}}{11.43~{}\mathrm{km}}\right)^{-\frac{1}{2}}.$ (3.34)
In the steady vortex creep scenario, therefore, the surface temperature is
predicted as a function of $J$, $|\dot{\Omega}_{\infty}|$, and $R_{\rm NS}$,
free from the uncertainty of the initial condition and the subsequent
temperature evolution. We will examine this prediction against observation in
Sec. 5.
## 4 Theoretical approaches for the vortex pinning
To compute the energy dissipation due to vortex creep, we evaluate the
parameter $J$ in Eq. (3.32). We first review the calculation of the pinning
force $f_{\rm pin}$ available in the literature in Sec. 4.1; the values of
$f_{\rm pin}$ which our analysis is based on are summarized in Appendix A.
Then, in Sec. 4.2, we estimate possible ranges of $J$ using the results in
Sec. 4.1, which will be compared with observation in the subsequent section.
### 4.1 Evaluation of pinning force
To evaluate the pinning force, we need to analyze a nucleon many-body system
at high densities, which generically suffers from technical difficulties due
to less-known properties of nuclear interactions. A traditional method of
treating nuclear interactions is to model a form of the interaction and fit it
to experimental data, such as nucleon-nucleon scattering [52, 53, 54]. For
example, the Argonne interaction [54], which we consider in the following
analysis, is a two-body nucleon potential fitted to nucleon scattering data
and deuteron properties. This sort of bare interaction does not include in-
medium effects. The many-body calculation based on bare interaction is
necessary to obtain the properties of nucleon systems. At the same time, it is
still challenging to perform it in general due to, e.g., the strong repulsive
core. An alternative method is to use an effective interaction incorporating
in-medium effect phenomenologically. Skyrme-type interactions [55, 56] are
well-known examples, which consist of contact (zero-range) interactions with
momentum-dependent coefficients. The parameters of the interactions are
determined by fitting to the experimental data of binding energies and radii
of several nuclei. There are many sets of fitting parameters used in the
literature [57, 58, 59], such as SLy4 [60] and SkM* [61]. There are also
finite-range interactions, such as the Gogny interaction [62].
There are several approaches to analyzing the nuclear matter, and the
following two are often used for the calculation of the pinning energy:
* •
Quantum approach
A standard method to analyze a quantum multi-body system is to calculate the
energy levels of a single particle in the mean field of a self-consistent
potential by solving the corresponding Schrödinger equation. A neutron pairing
interaction is then considered to determine the pairing field as in the BCS
theory. These solutions are obtained via an iterative process such that they
satisfy the self-consistent conditions. This method is called the Hartree-
Fock-Bogoliubov (HFB) method and adopted in Refs. [63, 64, 65, 66].
* •
Semi-classical approach
The quantum approach based on the HFB method often requires a high
computational cost. To evade this, a semi-classical approach based on the
Thomas-Fermi approximation is also frequently used, where nuclear matter is
regarded as a many-body system of nucleons subject to the Pauli exclusion
principle and moving independently from each other in a mean-field potential.
The energy of the system for a given chemical potential is obtained by the
variational principle.777Strictly speaking, we minimize the modified
Hamiltonian defined by $H^{\prime}=H-\mu N$, where $\mu$ and $N$ denote the
chemical potential and the number of particles, respectively. We also note
that for NSs, the temperature $T$ can be regarded as zero; thus, the free and
internal energy are equivalent. This approach is used in Refs [67, 68, 69, 70,
71, 72].
We can then estimate the pinning force from the pinning energy obtained above.
There are two approaches for this calculation:
* •
Microscopic calculation
We may estimate the pinning force through Eq. (3.18) for the nuclear pinning
configuration. We refer to this estimation as the microscopic approach since,
as illustrated in the most right window in Fig. 4, it focuses on the
microscopic scale of $\mathcal{O}(R_{\rm WS})$. This approach considers the
interaction between a vortex and the single nucleus in the Wigner-Seitz cell,
and thus the interaction of the vortex with other distant nuclei is neglected.
We obtain the pinning force per unit length by just multiplying the pinning
force per nucleus with the number of nuclei in the unit length along the
vortex line ($\simeq 1/R_{\mathrm{WS}}$).
* •
Mesoscopic calculation
Vortex lines are much longer than the lattice spacing; therefore, each vortex
line pins onto a large number of nuclei in reality. Such a vortex line does
not align to the crystal axis over its total length in general. The mesoscopic
approach considers this realistic configuration by taking the average of the
force exerted on a vortex over the possible directions of the vortex line with
respect to the crystal lattice. This calculation focuses on the length-scale
$L\sim(10^{2}$–$10^{3})\times R_{\rm WS}$, for which the vortex line can be
regarded as a straight line, as illustrated in the middle window in Fig. 4—we
call this scale the mesoscopic scale. The derived pinning force thus tends to
be smaller than those obtained with the microscopic calculation.
Figure 4: The landscape of a vortex-line configuration at different length-scales. | Semi-classical | Quantum
---|---|---
Microscopic | Mean field: | Pairing: | Hartree-Fock: | Pairing:
Woods-Saxon | Argonne | SLy4 | contact force
$f_{\mathrm{pin}}=(1-7)\times 10^{-3}$ [71] | $f_{\mathrm{pin}}=(3-4)\times 10^{-4}$ [65]
Mesoscopic | Mean field: | Pairing: | Hartree-Fock: | Pairing:
Woods-Saxon | Argonne | SLy4 | contact force
$f_{\mathrm{pin}}=8\times 10^{-7}-4\times 10^{-4}$ [72] | $f_{\mathrm{pin}}=5\times 10^{-7}-8\times 10^{-5}$ [66]
Table 1: Calculations of the vortex pinning force considered in this work.
The first row in each column lists the potentials for the mean-field and
pairing interactions used in the evaluations; the second row shows the range
of the pinning force in the inner crust in units of
$\mathrm{MeV}\cdot\mathrm{fm}^{-2}$.
All in all, we have $2\times 2=4$ combinations for the prescription of the
pinning force calculation. In the following discussion, we consider a
representative calculation with a specific choice of nuclear interactions for
each combination, as summarized in Table 1.
For the microscopic semi-classical approach, we consider the calculation in
Ref. [71] with a Woods-Saxon potential for the mean-field potential and the
Argonne interaction for the neutron-neutron pairing interaction. It is found
that the nuclear pinning configuration (see Fig. 2) occurs only in high-
density regions. The pinning force per nuclear pinning site is estimated by
$E_{\mathrm{pin}}/R_{\mathrm{WS}}$ for this configuration in Ref. [71]. To
convert this into the pinning force per unit length $f_{\mathrm{pin}}$, we
multiply this by a factor of $1/(2R_{\mathrm{WS}})$, as cubic cells with the
side length $2R_{\mathrm{WS}}$ are used in the calculation of Ref. [71]. As a
result, we obtain the values of $f_{\mathrm{pin}}=\text{($1$--$7$)}\times
10^{-3}~{}\mathrm{MeV}\cdot\mathrm{fm}^{-2}$, depending on the position in the
inner crust. We list quantities relevant to this calculation in Table 3 in
Appendix A.888To get some ideas about the dependence of this calculation on
the choice of potentials, we note that Ref. [71] also considers the case where
the Gogny interaction is used instead of the Argonne interaction. In this
case, the nuclear pinning occurs in higher density regions, and the pinning
forces tend to be larger by a factor of a few, compared with the calculation
with the Argonne interaction; see Table 4 in Appendix A for this calculation.
Notice that the values of the pinning force shown here should be regarded as a
ballpark estimate. The lower limit ($f_{\mathrm{pin}}=1\times
10^{-3}~{}\mathrm{MeV}\cdot\mathrm{fm}^{-2}$) could have been overestimated
because of the discretization of the positions at which the pinning energy is
estimated; as seen in Table 3, there is a density region of $\rho\sim
10^{13}~{}\mathrm{g}\cdot\mathrm{cm}^{-3}$ around which
$E_{\mathrm{pin}}\simeq 0$, leading to a very small pinning force if we use
Eq. (3.18). The upper limit could also be underestimated since the pinning
force obtained by this equation corresponds to the average taken over the
distance between the nuclear center and the interstitial position.
For the microscopic quantum approach, we consider the results given in Ref.
[65], where the SLy4 Skyrme interaction is used for the Hartree-Fock
calculation and a density-dependent contact interaction is used for the
neutron-neutron pairing interaction. The parameters of the contact interaction
are discussed in Ref. [73]. Table 5 summarises the relevant quantities for
this calculation. In contrast to the semi-classical analysis [71], nuclear
pinning occurs in lower-density regions in this case. However, it also occurs
in the highest-density regions (see Table 5 in Appendix A).999The qualitative
feature described here does depend on the choice of the nuclear interactions.
As shown in Ref. [65], if we use the SkM* interaction instead of SLy4, the
nuclear pinning never occurs at high densities. The values of the pinning
force estimated from the pinning energy and the Wigner-Seitz radius as in the
semi-classical calculation are $f_{\mathrm{pin}}=\text{($3$--$4$)}\times
10^{-4}~{}\mathrm{MeV}\cdot\mathrm{fm}^{-2}$. See Ref. [65] for detailed
discussions regarding the difference between the semi-classical and quantum
results.
The semi-classical mesoscopic calculation is given in Ref. [72], where the
nuclear potentials are taken to be the same as in the semi-classical
microscopic calculation in Table 1. The resultant values of the pinning force
are found to be $f_{\mathrm{pin}}=8\times 10^{-7}$ – $4\times
10^{-4}~{}\mathrm{MeV}\cdot\mathrm{fm}^{-2}$, which are summarized in Table 6.
We see that these values are considerably smaller than those for the semi-
classical microscopic calculation in Ref. [71] due to the averaging over the
vortex-line directions.
For the quantum mesoscopic calculation, we consider the result given in Ref.
[66], where the SLy4 interaction and a contact interaction are used for the
Hartree-Fock calculation and pairing interactions, respectively, as in the
quantum microscopic calculation in Table 1. As summarized in Table 7, the
pinning force is found to be in the range $f_{\mathrm{pin}}=5\times
10^{-7}-8\times 10^{-5}~{}\mathrm{MeV}\cdot\mathrm{fm}^{-2}$.101010If we use
the SkM* potential instead of SLy4, we obtain slightly smaller values of
$f_{\mathrm{pin}}$, as also shown in Table 7. We again find that these values
are much smaller than those obtained with the quantum microscopic approach.
Before concluding this section, we note that we can also calculate the pinning
force with a three-dimensional dynamical simulation of a vortex [74, 75, 76,
77].111111See also Ref. [78] for a kinetic approach. Such a calculation tends
to be costly; thus, a certain degree of simplification is usually required for
the moment. Besides, the current evaluation is limited to a few benchmark
values of density. The estimated values of the pinning force are consistent
with the above estimates.
### 4.2 Theoretical evaluation of $J$
For the evaluation of $J$ in Eq. (3.32), it is convenient to change the
coordinate from cylindrical coordinates $(r,\varphi,z)$ to spherical
coordinates $(R,\theta,\phi)$:
$\displaystyle J$ $\displaystyle=\int_{\rm pin}dr\,dz\,d\varphi\,\rho
r^{3}\cdot\frac{f_{\mathrm{pin}}}{\rho\kappa r}\simeq\int_{R_{\rm in}}^{R_{\rm
out}}dR\,d\theta\,d\phi\,R^{3}\sin^{2}\theta\cdot\frac{f_{\rm
pin}}{\kappa}\,,$ (4.35)
where we approximate $\delta\Omega_{\infty}$ in Eq. (3.32) by
$\delta\Omega_{\mathrm{cr}}$ in Eq. (3.29), which holds with good accuracy in
the situation of our interest [50] as we mentioned in Sec. 3.2. We perform the
integral over the range $[R_{\rm in},R_{\rm out}]$ where the pinning force is
evaluated. We use the Akmal-Pandharipande-Ravenhall (APR) [79] equation of
state to determine the NS core size and the equation of state tabulated in
Crust_EOS_Cat_HZD-NV.dat in NSCool [80] based on Refs. [81, 82] to determine
the density distribution in the crust.
For the evaluation method of pinning force, we focus on the mesoscopic
approach shown in Table 1, since for the microscopic calculation, the pinning
force is obtained only in a limited region in the crust, as can be seen in
Table 3–5. Considering that the evaluation of the pinning force suffers from
large uncertainty depending on calculation methods, we make the following
crude approximation in the calculation of the above integral—we neglect the
density dependence of $f_{\mathrm{pin}}$ and fix it to a value in the range
shown in Table 1. We thus obtain a range of $J$ accordingly, which we regard
as the uncertainty of this pinning force estimation. As a result, we obtain
$J=3.9\times 10^{40}-1.9\times 10^{43}~{}\mathrm{erg}\cdot\mathrm{s}$ for the
semi-classic mesoscopic calculation and $J=1.7\times 10^{40}-2.7\times
10^{42}~{}\mathrm{erg}\cdot\mathrm{s}$ for the quantum mesoscopic calculation.
## 5 Vortex creep heating vs. observation
No. | Type | Name | $\log_{10}t_{\rm sd}$ | $\log_{10}t_{\rm kin}$ | $\log_{10}|\dot{\Omega}|$ | $\log_{10}T_{\rm s}$ | $\log_{10}J_{\rm obs}$
---|---|---|---|---|---|---|---
| | | [yr] | [yr] | [s-2] | [K] | [erg s]
1. | [Y] | PSR B1706-44 | $4.2$ | — | $-10.3$ | $5.68$–$6.34$ [83] | $41.9$–$44.6$
2. | [O] | PSR J1740+1000 | $5.1$ | — | $-11.2$ | $5.89$ [84] | $43.8$
3. | [Y] | PSR B2334+61 | $4.6$ | — | $-11.3$ | $5.76$ [85] | $43.3$
4. | [O] | PSR B0656+14 | $5.0$ | — | $-11.6$ | $5.87$ [86] | $44.1$
5. | [O] | PSR J0633+1746 | $5.5$ | — | $-11.9$ | $5.71$ [87] | $43.7$
6. | [Y] | PSR J0538+2817 | $5.8$ | $4.60^{+0.18}_{-0.30}$[88] | $-11.9$ | $6.02$ [88] | $45.0$
7. | [O] | PSR B1055-52 | $5.7$ | — | $-12.0$ | $5.81$ [89] | $45.1$
8. | [X] | RX J1605.3+3249 | $4.5$ | $5.66^{+0.04}_{-0.07}$ [90] | $-12.1$ | $5.86$ [91] | $44.5$
9. | [O] | PSR J2043+2740 | $6.1$ | — | $-12.1$ | $<5.95$ [92] | $<44.8$
10. | [O] | PSR J1741-2054 | $5.6$ | — | $-12.2$ | $5.85$ [93] | $44.6$
11. | [O] | PSR J0357+3205 | $5.7$ | — | $-12.4$ | $5.62$ [94] | $43.8$
12. | [O] | PSR B0950+08 | $7.2$ | — | $-13.6$ | $4.78$–$5.08$ [29] | $41.7$–$42.9$
13. | [X] | RX J0420.0-5022 | $6.3$ | — | $-13.8$ | $5.74$ [95] | $45.8$
14. | [M] | PSR J0437-4715 | $9.20$ | — | $-14.0$ | $5.54$ [25] | $45.1$
15. | [X] | RX J1308.6+2127 | $6.2$ | $5.74^{+0.16}_{-0.26}$ [96] | $-14.2$ | $6.08$ [97] | $47.5$
16. | [X] | RX J0720.4-3125 | $6.3$ | $5.93^{+0.07}_{-0.26}$ [98] | $-14.2$ | $6.02$ [99] | $47.3$
17. | [M] | PSR J2124-3358 | $9.58$ | — | $-14.3$ | $4.70$–$5.32$ [26] | $42.0$–$44.5$
18. | [X] | RX J1856.5-3754 | $6.6$ | $5.66^{+0.04}_{-0.05}$[98] | $-14.4$ | $5.65$ [100] | $46.0$
19. | [X] | RX J2143.0+0654 | $6.6$ | — | $-14.5$ | $5.67$–$6.06$ [101, 102] | $46.1$–$47.8$
20. | [X] | RX J0806.4-4123 | $6.5$ | — | $-14.6$ | $6.01$ [103] | $47.6$
21. | [O] | PSR J0108-1431 | $8.3$ | — | $-15.1$ | $4.43$–$4.74$ [28] | $41.8$–$43.1$
22. | [O] | PSR J2144-3933 | $8.4$ | — | $-16.4$ | $<4.62$ [104] | $<43.8$
Table 2: The data of the NSs considered in this paper. We classify them into
four types—ordinary pulsars younger than $10^{5}~{}\mathrm{yrs}$ [Y], ordinary
pulsars older than $10^{5}~{}\mathrm{yrs}$ [O], XDINSs [X], and millisecond
pulsars [M]. The values of $t_{\rm sd}$ and $\dot{\Omega}$ without references
are computed from the data given in the ATNF pulsar catalogue [105, 106]. The
value of $J_{\rm obs}$ is evaluated as in Eq. (5.36) with $R_{\rm
NS}=11.43~{}\mathrm{km}$.
We now compare the prediction of the vortex-creep heating mechanism with
observation. For this purpose, it is useful to calculate the following
quantity for each NS:121212We neglect the gravitational redshift factor since
its effect is within the $\mathcal{O}(1)$ uncertainty of $J$ discussed below.
$\displaystyle J_{\rm obs}$ $\displaystyle\equiv\frac{4\pi R_{\rm
NS}^{2}\sigma_{\rm SB}T_{\rm s}^{4}}{|\dot{\Omega}|}~{}.$ (5.36)
As evident from Eqs.(2.2), (2.3), and (3.31), this corresponds to the $J$
parameter inferred from the observation of each NS. This inference assumes the
steady creeping of vortices (discussed in Sec.3.2) and the balance between
vortex-creep heating luminosity and photon cooling luminosity, which we expect
to hold if the NS is older than $\sim 10^{5}$ years. Since NSs are comparable
in size and mass, we expect that $J_{\rm obs}$ is also roughly equal (up to a
factor of $\mathcal{O}(1)$) for every NS. We test this expectation by using
the data of $J_{\rm obs}$ for old NSs. We also compare the values of
$J_{\mathrm{obs}}$ with the theoretical computations given in Sec. 4.2.
In Table 2, we list the values of $J_{\mathrm{obs}}$ for the NSs we consider
in this paper. We select isolated NSs older than $10^{4}$ yrs. In evaluating
$J_{\mathrm{obs}}$, we have just assumed $R_{\rm NS}=11.43~{}\mathrm{km}$ for
all NSs, as the radius is poorly known for most NSs; we keep in mind that this
may introduce an $\mathcal{O}(1)$ error in the determination of
$J_{\mathrm{obs}}$. We also show the age, surface temperature, and
$\dot{\Omega}=2\pi\dot{P}/P^{2}$ of the NSs, ($P$ and $\dot{P}$ are the period
and its time derivative, respectively). Regarding the NS age, we use the
kinetic age $t_{\rm kin}$ if available. Otherwise, we use the spin-down age
$t_{\rm sd}=P/(2\dot{P})$. We calculate $t_{\rm sd}$ and $\dot{\Omega}$ from
$P$ and $\dot{P}$ given in the Australia Telescope National Facility (ATNF)
pulsar catalogue [105, 106]. Notice that the surface temperatures of some of
the old NSs in this table are much higher than the predicted temperature in
the standard NS cooling scenario [30, 31, 32, 33, 34, 35].
It is important to note that not all NSs listed in Table 2 are useful for
testing the vortex-creep heating. As we have discussed in Sec. 2, photon
emission becomes the dominant cooling source for NSs older than $\sim 10^{5}$
years. For younger NSs, this may not be the case, so
$L^{\infty}_{\gamma}\lesssim L^{\infty}_{\rm H}$ instead of (2.3), for which
the values of $J_{\mathrm{obs}}$ in Table 2 may be underestimated. To
distinguish such young NSs from others, we indicate them by the type [Y] in
the table. Another class of NSs that are inappropriate for our test is the
X-ray Dim Isolated NSs (XDINSs). These NSs are considered to be descendants of
magnetars [103, 95] that experienced the decay of magnetic fields before. This
may make these NSs hotter than ordinary NSs of the same age [107, 95, 108,
109], resulting in an overestimate of $J_{\mathrm{obs}}$. We denote these NSs
by the type [X]. The rest of the NSs, which we use for the test of the vortex-
creep heating, are classified into old ordinary pulsars [O] and millisecond
pulsars [M].
The uncertainty in the determination of $J_{\mathrm{obs}}$ stems mainly from
that in the surface temperature, which is significant due to its quartic
dependence on $T_{\rm s}$. Generically speaking, it is very difficult to
identify all of the sources of uncertainties in the measurement of the NS
surface temperature, and it is often the case that the error shown in the
literature is only a part of them, such as those from the spectrum fitting,
the determination of the distance and/or radius of the NS, and so on. At
present, it is fair to say that the NS temperature measurement typically
suffers from $\mathcal{O}(1)$ uncertainty, as can be seen in, e.g., Ref.
[110]. Motivated by this, we include a factor of two uncertainty in $T_{\rm
s}$ for the stars in Table 2 for which only the central value is presented.
For the other stars, we describe our prescription for the error estimation in
Appendix B. We have checked that the errors thus obtained are similar to or
more conservative than those adopted in Ref. [110].
Figure 5: The values of the $J$ parameter obtained from the observation. The
grey triangles, blue circles, green inverse triangles, and orange stars
correspond to the young ordinary pulsars ([Y]), the old ordinary pulsars
([O]), the XDINSs ([X]), and the millisecond pulsars ([M]), respectively. The
points with an arrow indicate upper limits. The red shaded region shows
observationally favored range, $J\simeq
10^{42.9\text{--}43.8}~{}\mathrm{erg}\cdot\mathrm{s}$. For comparison, we also
show the values of $J$ estimated with the mesoscopic calculations by black
bars.
In Fig. 5, we show the range of $J_{\mathrm{obs}}$ estimated as described
above for each NS listed in Table 2. The grey triangles, green inverse
triangles, blue circles, and orange stars represent the young ordinary pulsars
([Y]), the XDINSs ([X]), the old ordinary pulsars ([O]), and the millisecond
pulsars ([M]), respectively. The points with an arrow indicate that we only
have an upper limit on $J_{\rm obs}$ for those NSs. Recall that we are
concerned only with the NSs represented by the blue [O] and orange [M] points.
It is found that the estimated values of $J_{\mathrm{obs}}$ for these NSs are
in the same ballpark, $J\sim 10^{43}~{}\mathrm{erg}\cdot\mathrm{s}$, even
though their $|\dot{\Omega}|$’s distribute over orders of magnitude. This is
in good agreement with the prediction of the vortex-creep heating mechanism.
On the other hand, $J_{\mathrm{obs}}$ for the green points (XDINSs [X]) tend
to be larger than this, as expected.
We also show the theoretical estimations given in Sec. 4.2 in the upper panel
of Fig. 5. We see that the semi-classical mesoscopic calculation is consistent
with the observation, given that this theoretical estimation suffers from a
NS-dependent uncertainty of $\mathcal{O}(1)$ coming from the integration in
Eq. (4.35), in addition to that from the estimation of $f_{\mathrm{pin}}$. The
quantum mesoscopic calculation can explain some of the points with a small
$J_{\mathrm{obs}}$, but they are not large enough to explain, e.g., that of
J0437-4715. However, we note that this theoretical estimation is still allowed
by the observations since it just results in a lower heating luminosity than
the observed one. If this is the case, the vortex-creep heating may operate
but there exists another heating mechanism that dominates the vortex-creep
heating, such as the rotochemical heating [111, 112, 113, 114, 115, 116, 117,
118, 119].
It is premature to establish the existence of the vortex-creep heating, as
well as to conclude if an extra heating mechanism is required to be present.
To that end, we need to accumulate more data on the surface temperature of old
NSs with high accuracy, which we anticipate to be provided by future optical,
UV, and X-ray observations.131313See, for instance, Ref. [120]. Nevertheless,
obtaining a current compilation of the value of $J$ suggested by the
observation is intriguing. Considering intrinsic $\mathcal{O}(1)$ uncertainty
in $J$, we determine its rough range by requiring that it covers the range
suggested by B0950+08, which favors the smallest value, and satisfies the
upper limit set by J2144-3944 based on non-observation of thermal flux. This
yields
$\displaystyle J\simeq 10^{42.9-43.8}~{}\mathrm{erg}\cdot\mathrm{s}~{},$
(5.37)
which we show as the red band in Fig. 5.
(a) Ordinary pulsars (b) Millisecond pulsars
Figure 6: The evolution of NS surface temperature with (without) the vortex
creep heating effect in the red band (black dashed line). The band corresponds
to the range of $J$ in Eq. (5.37). The dots with error bars show the observed
temperatures, presented in Table 2, with the same colors as in Fig. 5.
Finally, in Fig. 6, we show the evolution of NS surface temperature with
(without) the vortex creep heating effect in the red band (black dashed
line).141414In these plots, we use the APR equation of state [79] for a NS
mass of $1.4M_{\odot}$ to calculate the NS structure. For Cooper pairing gap
models, we use the SFB model [121] for the neutron singlet pairing, the model
“b” in Ref. [33] for the neutron triplet pairing, and the CCDK model [122] for
the proton singlet pairing. The temperature evolution at late times scarcely
depends on the choice of these models. The band corresponds to the range of
$J$ in Eq. (5.37). The dots with error bars show the observed temperatures in
Table 2, with the same colors as in Fig. 5. In Fig. 6(a), we take
$P\dot{P}=10^{-15}$ s and the initial period $P_{0}=10~{}\mathrm{ms}$ to
calculate $|\dot{\Omega}(t)|$, where we assume that the external torque is
dominated by magnetic dipole radiation.151515In this case, we have
$\dot{\Omega}\propto-\Omega^{3}$, i.e., $P\dot{P}=\text{constant}$, and by
solving this we obtain
$\displaystyle|\dot{\Omega}|(t)=\frac{\pi}{\sqrt{2P\dot{P}}}\left[t+t_{\mathrm{sd},0}\right]^{-3/2},$
with $t_{\mathrm{sd},0}\equiv P_{0}^{2}/(2P\dot{P})$. For the choice of
parameters in Fig. 6(a) and Fig. 6(b), $t_{\mathrm{sd},0}\simeq 2\times
10^{3}$ and $5\times 10^{7}$ years, respectively. The value of $P\dot{P}$ is
related to the surface magnetic flux density $B_{s}$. In the ATNF pulsar
catalogue [105, 106], $B_{s}=3.2\times 10^{19}(P\dot{P})^{1/2}$ G is used for
this relation, with which we have $B_{s}\simeq 1.0\times 10^{12}$ G and
$5.8\times 10^{8}$ G in Fig. 6(a) and Fig. 6(b), respectively. These values
are typical for ordinary pulsars. Nevertheless, we note that
$|\dot{\Omega}(t)|$ obtained with these parameters do not exactly agree with
the observed values of $|\dot{\Omega}|$ in Table 2, so the data points shown
in this figure should be regarded as just an eye guide. In Fig. 6(b), we set
$P\dot{P}=3.3\times 10^{-22}$ s, which is the observed value for PSR
J2124-3358, and $P_{0}=1~{}\mathrm{ms}$. As we see in these plots, the
predicted temperature with the vortex creep heating starts to deviate from
that in the standard cooling scenario at $t\sim 10^{5}$ years and remains high
enough at later times to be compatible with the observed data.
## 6 Conclusion and discussion
We have revisited the vortex-creep heating mechanism in light of recent
observations of old warm NSs. As we have seen, this heating mechanism gives a
characteristic prediction that the heating luminosity is proportional to
$|\dot{\Omega}|$, with the proportional constant $J$ having an almost
universal value over NSs since the NS structure and the vortex-nuclear
interactions determine it. We have found that this prediction agrees with the
observational data of old NSs, with the favored range of $J$ in the same
ballpark as the theoretical calculations.
Notice that the scenario where vortex creep heating dominates all NSs can
readily be overturned if we discover a NS having $J$ much smaller than those
presented in Fig. 5. On the other hand, if we find a NS with a larger $J$, we
can disfavor our scenario only after excluding the existence of other heating
sources specific to this NS, such as accretion from its environment.
It is possible that other heating mechanisms also work in old NSs. Indeed, we
have already considered potential heating caused by the decay of magnetic
fields in XDINSs (see Sec. 5), and we have not used these NSs in our test of
the vortex-creep heating mechanism for this reason. Another heating mechanism
that may operate without relying on exotic phenomena is provided by the out-
of-equilibrium beta processes, which is dubbed rotochemical heating [111, 112,
113, 114, 115, 116, 117, 118, 119]. It is known that this rotochemical heating
mechanism can increase the surface temperature of old NSs up to $\sim 10^{6}$
K. Thus, its heating luminosity can be comparable to or even dominate the
vortex-creep heating. It would be worthwhile to study the vortex creep heating
in the presence of rotochemical heating and compare its prediction with the
temperature observations of old warm NSs. The NS heating caused by the
accretion of dark matter particles is also widely discussed in the literature
[123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137,
138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152,
153, 154, 155]. In this case, the surface temperature at late times is
predicted to be $\text{a few}\times 10^{3}$ K, and thus this heating effect
could be concealed by the vortex heating mechanism, making it improbable to
observe the dark matter signature through the temperature observation of old
NSs. A detailed study of this issue will be given in the forthcoming paper
[156].
## Acknowledgments
MF thanks Kazuyuki Sekizawa and Tomoya Naito for the fruitful discussion about
the current situation and the possible future directions in the pinning force
evaluation. This work is supported in part by the Collaborative Research
Center SFB1258 and by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy - EXC-2094 -
390783311 [MF], JSPS Core-to-Core Program (No. JPJSCCA 20200002 [MF]), the
Grant-in-Aid for Innovative Areas (No.19H05810 [KH and MRQ], No.19H05802 [KH],
No.18H05542 [MF and NN]), Scientific Research B (No.20H01897 [KH and NN]),
Young Scientists (No.21K13916 [NN]).
## Appendix A Pinning force
In this appendix, we show the density dependence of the quantities relevant to
the pinning force calculations discussed in Sec. 4.1. Tables 3 and 4 are for
the microscopic semi-classical approach in Ref. [71], where the Argonne and
Gogny interactions are used for the nuclear pairing interaction, respectively.
The element corresponding to the cell nuclear composition and Wigner-Seitz
radius $R_{\rm WS}$ are derived for each baryon density $\rho$ in Ref. [81].
In Ref. [71], the pinning force is evaluated only for the nuclear pinning
configuration, and thus we show the values of $E_{\mathrm{pin}}$ and
$f_{\mathrm{pin}}=|E_{\mathrm{pin}}|/(2R_{\mathrm{WS}}^{2})$ only for this
case. The labels of NP and IP in the last column indicate the nuclear and
interstitial pinnings, respectively.
zone | Element | $\rho$ | $R_{\rm WS}$ | $\xi$ | $E_{\rm pin}$ | $f_{\rm pin}$ | config
---|---|---|---|---|---|---|---
| | [g cm-3] | [fm] | [fm] | [MeV] | [MeV fm-2] |
1 | ${}^{320}_{40}\mathrm{Zr}$ | $1.5\times 10^{12}$ | $44.0$ | $7.02$ | — | — | IP
2 | ${}^{1100}_{50}\mathrm{Sn}$ | $9.6\times 10^{12}$ | $35.5$ | $4.34$ | — | — | IP
3 | ${}^{1800}_{50}\mathrm{Sn}$ | $3.4\times 10^{13}$ | $27.0$ | $8.54$ | $-5.2$ | $0.0036$ | NP
4 | ${}^{1500}_{40}\mathrm{Zr}$ | $7.8\times 10^{13}$ | $19.4$ | $11.71$ | $-5.1$ | $0.0068$ | NP
5 | ${}^{982}_{32}\mathrm{Ge}$ | $1.3\times 10^{14}$ | $13.8$ | $8.62$ | $-0.4$ | $0.0011$ | NP
Table 3: Quantities relevant to the pinning force calculation obtained with the microscopic semi-classical approach in Ref. [71], where the Argonne potential is used for the nuclear pairing interaction. $\rho$, $R_{\mathrm{WS}}$, and $\xi$ are the mass density, Wigner-Seitz radius, and coherence length, respectively. zone | Element | $\rho$ | $R_{\rm WS}$ | $\xi$ | $E_{\rm pin}$ | $f_{\rm pin}$ | config
---|---|---|---|---|---|---|---
| | [g cm-3] | [fm] | [fm] | [MeV] | [MeV fm-2] |
1 | ${}^{320}_{40}\mathrm{Zr}$ | $1.5\times 10^{12}$ | $44.0$ | $7.76$ | — | — | IP
2 | ${}^{1100}_{50}\mathrm{Sn}$ | $9.6\times 10^{12}$ | $35.5$ | $4.07$ | — | — | IP
3 | ${}^{1800}_{50}\mathrm{Sn}$ | $3.4\times 10^{13}$ | $27.0$ | $3.93$ | — | — | IP
4 | ${}^{1500}_{40}\mathrm{Zr}$ | $7.8\times 10^{13}$ | $19.4$ | $7.78$ | $-7.5$ | $0.010$ | NP
5 | ${}^{982}_{32}\mathrm{Ge}$ | $1.3\times 10^{14}$ | $13.8$ | $8.62$ | $-5.9$ | $0.015$ | NP
Table 4: Quantities relevant to the pinning force calculation obtained with the microscopic semi-classical approach in Ref. [71], where the Gogny potential is used for the nuclear pairing interaction. zone | Element | $\rho$ | $n$ | $R_{\rm WS}$ | $\xi$ | $E_{\rm pin}$ | $f_{\rm pin}$ | config
---|---|---|---|---|---|---|---|---
| | [g cm-3] | [fm-3] | [fm] | [fm] | [MeV] | [MeV fm-2] |
1a | ${}^{320}_{40}\mathrm{Zr}$ | $1.7\times 10^{12}$ | $0.001$ | $43.3$ | $4.43$ | $-1.08$ | $0.00029$ | NP
1b | ${}^{320}_{40}\mathrm{Zr}$ | $3.4\times 10^{12}$ | $0.002$ | $40.0$ | $4.21$ | $-1.20$ | $0.00038$ | NP
1c | ${}^{320}_{40}\mathrm{Zr}$ | $6.7\times 10^{12}$ | $0.004$ | $36.9$ | $3.93$ | — | — | IP
2a | ${}^{1100}_{50}\mathrm{Sn}$ | $1.3\times 10^{13}$ | $0.008$ | $33.0$ | $4.04$ | — | — | IP
2b | ${}^{1100}_{50}\mathrm{Sn}$ | $1.8\times 10^{13}$ | $0.011$ | $31.0$ | $4.12$ | — | — | IP
2c | ${}^{1100}_{50}\mathrm{Sn}$ | $2.8\times 10^{13}$ | $0.017$ | $28.0$ | $4.70$ | — | — | IP
3a | ${}^{1800}_{50}\mathrm{Sn}$ | $4.3\times 10^{13}$ | $0.026$ | $24.5$ | $6.05$ | — | — | IP
3b | ${}^{1800}_{50}\mathrm{Sn}$ | $6.2\times 10^{13}$ | $0.037$ | $21.4$ | $8.75$ | $-0.41$ | $0.00045$ | NP
Table 5: Relevant quantities for the pinning force calculation obtained with
the microscopic quantum approach in Ref. [65].
The results for the microscopic quantum approach in Ref. [65] are summarized
in Table 5, where the SLy4 Skyrme interaction is used for the mean-field
interaction. The Wigner-Seitz radius $R_{\rm WS}$ shown in this table is
interpolated from the plot in Ref. [81]. For the evaluation of
$f_{\mathrm{pin}}$, we again use the formula
$f_{\mathrm{pin}}=|E_{\mathrm{pin}}|/(2R_{\mathrm{WS}}^{2})$ and show the
values only for the nuclear pinning, just for easy comparison with the semi-
classical calculations.
zone | | | $f_{\rm pin}$ [MeV fm-2] | |
---|---|---|---|---|---
$L=100R_{\rm WS}$ | $L=500R_{\rm WS}$ | $L=1000R_{\rm WS}$ | $L=2500R_{\rm WS}$ | $L=5000R_{\rm WS}$
1 | $7.63\times 10^{-6}$ | $2.29\times 10^{-6}$ | $1.49\times 10^{-6}$ | $9.30\times 10^{-7}$ | $7.68\times 10^{-7}$
2 | $2.12\times 10^{-5}$ | $6.34\times 10^{-6}$ | $4.06\times 10^{-6}$ | $2.62\times 10^{-6}$ | $2.12\times 10^{-6}$
3 | $1.43\times 10^{-4}$ | $4.49\times 10^{-5}$ | $2.74\times 10^{-5}$ | $1.54\times 10^{-5}$ | $1.14\times 10^{-5}$
4 | $3.84\times 10^{-4}$ | $1.10\times 10^{-4}$ | $6.89\times 10^{-5}$ | $4.23\times 10^{-5}$ | $3.32\times 10^{-5}$
5 | $7.85\times 10^{-5}$ | $2.71\times 10^{-5}$ | $1.89\times 10^{-5}$ | $1.36\times 10^{-5}$ | $1.12\times 10^{-5}$
Table 6: The pinning force obtained in the semi-classical mesoscopic approach
for different values of $L$ over which the forces exerted on a vortex are
integrated [72]. The zones correspond to those in Table 3.
In Table 6, we show the pinning force obtained in the semi-classical
mesoscopic approach for different values of $L$ over which the forces exerted
on a vortex are integrated [72]. The zone numbers in the first column
correspond to those in Table 3. We show the calculation in which the reduction
of the pairing gap due to the polarization effects in the nuclear matter is
not included, corresponding to the choice of the reduction factor $\beta=1$
introduced in Ref. [72]. Because of the averaging procedure, we find that a
larger $L$ results in a smaller value of $f_{\mathrm{pin}}$.
model | zone | $E_{\rm pin}$ | config | | $f_{\rm pin}$ [MeV fm-2] |
---|---|---|---|---|---|---
| | [MeV] | | $L=1000R_{\rm WS}$ | $L=2500R_{\rm WS}$ | $L=5000R_{\rm WS}$
SLy4 | 1a | $-0.72$ | NP | $1.39\times 10^{-6}$ | $7.38\times 10^{-7}$ | $5.40\times 10^{-7}$
1b | $-0.91$ | NP | $1.97\times 10^{-6}$ | $1.08\times 10^{-6}$ | $8.00\times 10^{-7}$
1c | $-0.89$ | NP | $2.20\times 10^{-6}$ | $1.19\times 10^{-6}$ | $8.74\times 10^{-7}$
2a | $2.73$ | IP | $5.61\times 10^{-6}$ | $3.72\times 10^{-6}$ | $2.95\times 10^{-6}$
2b | $3.01$ | IP | $7.52\times 10^{-6}$ | $5.03\times 10^{-6}$ | $4.00\times 10^{-6}$
2c | $10.00$ | IP | $1.47\times 10^{-5}$ | $1.01\times 10^{-5}$ | $8.08\times 10^{-6}$
3a | $11.78$ | IP | $3.25\times 10^{-5}$ | $2.31\times 10^{-5}$ | $1.88\times 10^{-5}$
3b | $9.85$ | IP | $8.47\times 10^{-5}$ | $6.41\times 10^{-5}$ | $5.31\times 10^{-5}$
SkM* | 1a | $-0.72$ | NP | $3.61\times 10^{-7}$ | $1.60\times 10^{-7}$ | $9.50\times 10^{-8}$
1b | $-0.91$ | NP | $2.20\times 10^{-7}$ | $8.72\times 10^{-8}$ | $4.08\times 10^{-8}$
1c | $-0.89$ | NP | $2.83\times 10^{-6}$ | $1.82\times 10^{-6}$ | $1.44\times 10^{-6}$
2a | $2.73$ | IP | $5.68\times 10^{-6}$ | $3.73\times 10^{-6}$ | $2.93\times 10^{-6}$
2b | $3.01$ | IP | $7.58\times 10^{-6}$ | $5.01\times 10^{-6}$ | $4.00\times 10^{-6}$
2c | $10.00$ | IP | $1.25\times 10^{-5}$ | $8.50\times 10^{-6}$ | $6.76\times 10^{-6}$
3a | $11.78$ | IP | $2.54\times 10^{-5}$ | $1.80\times 10^{-5}$ | $1.45\times 10^{-5}$
3b | $9.85$ | IP | $8.00\times 10^{-5}$ | $5.93\times 10^{-5}$ | $4.79\times 10^{-5}$
Table 7: The pinning force obtained in the quantum mesoscopic calculation
where the Skytme interactions, SLy4 and SkM*, are used for the mean-field
potential [66]. The zones correspond to those in Table 5.
Table 7 shows the pinning force obtained in the quantum mesoscopic calculation
where the SLy4 and SkM* Skytme interactions are used for the Hartree-Fock
calculation [66]. The zones correspond to those in Table 5. We again show the
results obtained without including the polarization effect, i.e., $\beta=1$ as
in the previous case.
Figure 7: The values of $f_{\rm pin}$ given in the tables in this appendix
against the density $\rho$. The filled and opened markers correspond to the
nuclear and interstitial pinnings.
Finally, we plot the values of the pinning force for each density region in
Fig. 7. The filled and opened markers correspond to the nuclear and
interstitial pinnings. As we see, the values of $f_{\rm pin}$ are distributed
in the range $10^{-8}$–$10^{-2}$ MeV $\cdot$ fm-2, depending on the evaluation
scheme and the selected nuclear potential.
## Appendix B Selection criteria of NS data
We explain how we choose the range of uncertainty of $T_{\rm s}$ for each NS
shown in Table 2.
* •
No. 1, PSR B1706-4: In Ref. [83], the X-ray data of PSR B1706-44 obtained in
XMM-Newton is fitted by the BB (blackbody), BB+PL (power law), and
atmosphere+PL models, and only BB+PL and atmosphere+PL models result in
acceptable $\chi^{2}$ values. The atmosphere model includes the light-element
NS atmosphere (e.g. dominated by Hydrogen) and shows large Wien excesses in
the high-energy region. Therefore, atmosphere+PL models tend to favor lower
temperature and larger radius for the emitting area. We selected the minimum
and maximum among the BB+PL and atmosphere+PL models,
$T^{\infty}=(0.48-2.2)\times 10^{6}~{}\mathrm{K}$, to include the uncertainty
coming from the choice of fitting models.
* •
No. 9, PSR J2043+2740: Ref. [157] studied the XMM-Newton data of PSR
J2043+2740. Using the BB + PL model, the upper bound is derived as
$T^{\infty}_{s}<6.27\times 10^{5}~{}\mathrm{K}$, where $R_{\rm
NS}=10~{}\mathrm{km}$ is assumed for the emission radius. On the other hand,
Ref. [158] also fitted the X-ray data of the XMM-Newton and obtained an even
higher BB temperature, $T^{\infty}_{s}\simeq 9\times 10^{5}~{}\mathrm{K}$ with
the radiation radius $R^{\infty}\simeq
2~{}\mathrm{km}~{}\text{\cite[cite]{[\@@bibref{Number}{Zavlin:2004wt}{}{}]}}$.
Although the fitted radius is smaller than the expected NS radius, it is too
large to be interpreted as the magnetic cap radius. Thus, we can not exclude
the possibility that this BB temperature corresponds to the emission from the
NS surface. To evaluate $T_{\rm s}$ conservatively, we chose the highest value
of the BB temperature as an upper bound.
* •
No. 12, PSR B0950+08: In Ref. [29], the optical-UV flux of PSR B0950+08
obtained in the Hubble Space Telescope (HST) far-UV (FUV) detector is
analyzed. The best-fit temperature is obtained as $T_{\rm s}=(6-12)\times
10^{4}~{}\mathrm{K}$, and we decided to use the proposed value. Note that the
conservative upper bound is also derived as $T_{\rm s}<1.7\times
10^{5}~{}\mathrm{K}$ by varying the parameter, such as the ratio of NS radius
and distance.
* •
No. 17, PSR J2124-3358: Ref. [26] analyzed the optical data from the
J2124-3358. The BB+PL model gives the following possible range $T_{\rm
s}\in[0.5,2.1]\times 10^{5}~{}\mathrm{K}$ with the uncertainty from the
distance. We decided to select the original range, which is almost the same
uncertainty added by hand in the way we described in Sec. 5.
* •
No. 19, RX J2143.0+0654: In Ref. [101], the X-ray data in XMM-Newton is fitted
using the BB absorption model, and the authors obtain the BB temperature
$k_{\rm B}T^{\infty}=104.0\pm 0.4~{}\mathrm{eV}$ with the BB radius as
$R^{\infty}=(3.10\pm 0.04)~{}\mathrm{km}$, where distance is fixed to be
$d=500~{}\mathrm{pc}$. This value is smaller than the typical NS radius, which
implies that this BB temperature is not from the surface but from the small
areas around the magnetic caps. In Ref. [102], the authors fitted the data
from Large Binocular Telescope (optical) by combining with the X-ray data in
XMM-Newton. Using the BB absorption model, $k_{\rm B}T^{\infty}=105.1\pm
0.9~{}\mathrm{eV}$ is obtained, which is consistent with the result in Ref.
[101]. They also perform fitting using the two-component BB model and the
hotter (cooler) component is obtained as $k_{\rm B}T=104~{}\mathrm{eV}$
($k_{\rm B}T=40~{}\mathrm{eV}$). It is impossible to eliminate uncertainty
from the model selection to fit the data from this situation, and thus we
choose all the possible ranges of the temperature, $k_{\rm B}T_{\rm
s}=40$–$106~{}\mathrm{eV}$.
* •
No. 20, PSR J0108-1431: The X-ray data from the direction of J1080-1431
observed in XMM-Newton is fitted by the BB+PL model [159] with $k_{\rm
B}T=110^{+30}_{-10}~{}\mathrm{eV}$ and $R_{\rm
NS}=43^{+16}_{-9}~{}\mathrm{m}$. This small emission radius implies that this
BB component is not the cool surface temperature but the hot magnetic pole
component. We can interpret this result as the surface temperature is much
cooler than the magnetic pole, and thus, the hot component dominates the
observed flux. The latest analysis [28] analyses both the XMM-Newton and
optical data (HST, VLT). In particular, HST F140LP detected thermal emission,
and they put the conservative upper bound on the surface temperature as
$T_{\rm s}<5.9\times 10^{4}~{}\mathrm{K}$. To derive this conservative bound,
they included uncertainty from the parallax distance [160]. Furthermore, they
obtain the value of $T_{\rm s}$ by assuming the FUV flux is dominated by a
thermal component, $T_{\rm s}=27000-55000~{}\mathrm{K}$. We selected this
range to represent the uncertainty.
* •
No. 22, PSR J2144-3933: The upper bound is obtained for the surface
temperature of J2144-3933 using XMM-Newton data (combining with the optical
data of Very Large Telescope (VLT)) [161] as $T_{\rm s}<2.3\times
10^{5}~{}\mathrm{K}$. The latest analysis [104] used deep optical and FUV
observation data by HST and derived the upper bound on the surface temperature
of J2144-3933. The conservative upper bound on the surface temperature is
derived based on the non-detection,
$\displaystyle T_{\rm s}<4.2\times 10^{4}~{}\mathrm{K},$ (B.38)
where a range of NS radius $R_{\rm NS}=[11,13]~{}\mathrm{km}$ and parallax
distance $d=172^{+20}_{-15}~{}\mathrm{pc}$ is considered to estimate the
uncertainty [160]. In this analysis, NS mass is fixed as $M_{\rm
NS}=1.4~{}M_{\odot}$.
Let us also comment on the rejected observational data from our list.
* $-$
PSR B1929+10: The BB+PL fit is performed for the X-ray data [162]. However,
the magnetic pole component is reported to dominate the temperature because
the fitted radiation radius is much smaller than the NS radius. We conclude
this data is not appropriate to test the vortex creep heating.
* $-$
XMMU J1732-344: We also omit XMMU J1732-344 from our list because the observed
value of $|\dot{\Omega}|$ is not determined. Once its pulsation data is fixed,
it is worth studying whether the vortex creep heating can explain this data;
its thermal emission is expected to exceed the value expected from minimal
cooling [163, 164] with its kinetic time information [163].
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1.2 0.95
# We both think you did wrong - How agreement shapes and is shaped by indirect
reciprocity - DRAFT
Marcus Krellner1,⋆ and The Anh Han1
1 School Computing, Engineering and Digital Technologies, Teesside University
⋆ Corresponding authors<EMAIL_ADDRESS>
## Abstract
Humans judge each other’s actions, which at least partly functions to detect
and deter cheating and to enable helpfulness in an indirect reciprocity
fashion. However, most forms of judging do not only concern the action itself,
but also the moral status of the receiving individual (to deter cheating it
must be morally acceptable to withhold help from cheaters).
This is a problem, when not everybody agrees who is good and who is bad.
Although it has been widely acknowledged that disagreement may exist and that
it can be detrimental for indirect reciprocity, the details of this crucial
feature of moral judgments have never been studied in depth.
We show, that even when everybody assesses individually (aka privately), some
moral judgement systems (aka norms) can lead to high levels of agreement. We
give a detailed account of the mechanisms which cause it and we show how to
predict agreement analytically without requiring agent-based simulations, and
for any observation rate. Finally, we show that agreement may increase or
decrease reputations and therefore how much helpfulness (aka cooperation)
occurs.
Keywords:cooperation, evolutionary game theory, indirect reciprocity, donation
game, private assessments, analytical predictions
## 1 Introduction
From simplest replicators, evolution has worked its wonders and today we
marvel at the success, diversity and complexity of life. Many wonders of life
were enabled by cooperation. Cells work together to form multicellular
organisms, former bacteria serve as the powerplants of cells in exchange for
eternal protection from an outside world, and individual organism work
together in communities of thousands and millions to become the most
widespread forms of life there are, ants/wasps and, in similar but different
way, humans.
Indirect Reciprocity (IR) is one of the few other mechanisms to facilitate
cooperation among self-interested agents (Rand and Nowak,, 2013, Nowak and
Sigmund,, 2005, perc2017statistical). You help me, without receiving any
direct returns (that is, in that moment you behave altruistically). But you
will be rewarded by others in the future. Because others have observed your
behavior and will like you for it. I.e. they judge your action and form an
opinion about you. And because they like you, they decide to help you.
Indirect reciprocity allows cooperation (aka continued help amongst helpers)
in large groups of unrelated individuals, even if they do not interact enough
to establish direct reciprocity (Schmid et al.,, 2022). Hence, indirect
reciprocity has been the focus of many influential studies of evolutionary
game theory (Nowak and Sigmund, 1998a , Ohtsuki and Iwasa, (2004), see also
Sigmund, (2016)) and is even considered as an important foundation of human
morality (Nowak and Sigmund,, 2005).
Traditionally, studies simplified the mechanism of IR, assuming that judgments
are unanimous (also called public assessments). More realistically, everybody
would make their own judgements and therefore have their own opinions (aka
private assessments). When the public assessment assumption is removed,
dynamics and outcomes of IR models change (Brandt and Sigmund,, 2004, Okada,
2020b, ), often to the worse for the evolution of cooperation (Uchida and
Sasaki,, 2013, Hilbe et al.,, 2018). The main reason is the emergence and
spreading of disagreements about someone’s reputation, which had not been
possible with public assessments. Disagreements can significantly disturb IR
strategies that are stable under the public assessment regime, because they
hinder a universal principle: if you withhold help towards somebody bad, this
should not be judged as bad 111at least if done by somebody good, which can be
seen as in general judging it neutral or even good (Okada, 2020b, , Krellner
and Han,, 2022). (Panchanathan and Boyd,, 2003, Ohtsuki and Iwasa,, 2006). It
is problematic to apply this rule, if potential helpers and observers may
disagree. You withhold help to me, since you do not like me. But some
observers like me, so they will disapprove of your action. So, if there was
disagreement about my reputation, your reputation can get tainted.
In addition, a disagreement can cause further disagreements. If two observers
disagree about me, they may also judge you differently for withholding help.
That is, they now disagree about another person’s reputation. Even a single
disagreement can cascade through the population, even if no further errors
occur, so that all opinions become bad or essentially random (Hilbe et al.,,
2018).
Private assessment and disagreements are not only detrimental for IR
strategies, but also lead to new challenges for IR research. Whereas for
public assessment, there had long been exhaustive investigations of hundreds
of strategies (Ohtsuki and Iwasa,, 2004)[santos] using analytical models,
something similar for private assessment was only achieved in recent years
(Okada, 2020b, , Perret et al.,, 2021). And these models still require
restricting assumptions. Namely that, in an infinite population, only a finite
number of players observe. That is, the probability to observe $\psi$ and
hence also the fraction of opinions changed by a single interaction are
required to be negligible ($\psi\rightarrow 0$).
Analytical models and most models of IR in general (Okada, 2020a, ) consider
opinions as binary. That is, an individual is judged as either good (1) or bad
(0). To understand the state of opinions in a population, it is important to
know the average reputation $r$: the probability of a random opinion being
good, as well as the average probability of an agreement $a$: the probability
that two randomly selected players have the same opinion about another random
player.
The models mentioned above only considered $r$ (hence we shall call them
R-models for short). They assume that, when the observation probability is
very small ($\psi\to 0$), the probability of agreement $a$ only depends on $r$
and is given by $\widehat{a}=r^{2}+(1-r)^{2}$ (i.e. probability that two
random opinions are either both good or both bad) (Okada et al.,, 2018).
Figure 1:
a) Two dimensions of global opinion state. Graph shows agreement-reputation
space, with the solid lines $a=1$ and $a=\widehat{a}$ indicating outer borders
of possible agreement values in grey. (Subsequently, values above
$a=\widehat{a}+0.05$ will be considered as significant additional agreement,
since values very close to $\widehat{a}$ could conceivably be the result of
noise or finite properties of the simulations; or be just too insignificant to
make a difference). Case 1 has minimal agreement for the specific global
average reputation $r$. Case 3 has maximal agreement for the same $r$ and case
2 an intermediate one for the same $r$ still.
b) Model of simplified image matrices. We simplify the image matrices for the
three cases by removing all information about whose opinions it is and what
specific player it is about. We only consider whether the opinion is about a
player of group 1 or group 2 . The result can be imagined as two containers
(left and right, separated by the double line) holding good opinions (dark
blue areas). We may call the containers or groups ”lucky” on the left and
”unlucky” on the right. In case 1, both containers are filled to the same
height, representing that the reputations of lucky and unlucky group, $r_{L}$
and $r_{U}$, are equal to the global average. In case 2, some good opinions
from the unlucky container were ’poured’ into the lucky container, keeping $r$
unchanged since we do not add or remove any good opinions. In case 3, all
remaining of these good opinions were poured as well, filling the lucky
container exactly to the brim. It is apparent that in case 3, all players of
the lucky group have only good opinions about them, therefore agreement about
a player of this group is always $100\%$. The same applies to the unlucky
group, which has only bad opinions about them.
c) Size of the groups and precise state of agreement d. The complete
transition from case 1 to case 3 can only be achieved under a specific
condition. The size of the groups (the width of the containers) must be $r$
and $1-r$, respectively. With this, the areas $A_{L}$ and $A_{U}$ are of equal
size. That means, for minimal agreement as in case 1, there are exactly as
many good opinions about the unlucky group as there are bad opinions about the
lucky one. Therefore, they can be entirely exchanged. The central feature of
the analytical model introduced in this paper, is the parameter $d$, which
describes the percentage of good opinions from area $A_{U}$ exchanged with bad
opinions from area $A_{L}$. This parameter is independent of $r$ and can
obtain any value in the interval $[0,1]$ (note that both areas disappear for
the extreme cases $r=0$ and $r=1$, which is why $d$ is arbitrary in these
cases). For details see equations 2 and 4.
In theory, $a$ can be much higher. The population could for example consist of
two groups, one with perfect reputation, the other with an abysmal one, see
case 3 in Figure 1. The average reputation is still $r=0.6$. But all opinions
about a player of the unlucky group are bad. Which means that the opinions of
any two players about such a player will always be the same. This is similarly
true for the lucky group in which all opinions are good. In case 3, average
agreement is therefore 100%. There are of course also intermediate states,
such as case 2. The left side of the figure shows the entire agreement-
reputation space and how agreement can range anywhere between $\widehat{a}$
and 1. Note that $0.5\leq\widehat{a}\leq 1$ (since $0\leq r\leq 1$).
There are obvious mechanisms that cause disagreement and push state of the
population towards $\widehat{a}$, as discussed above. However, there could be
other mechanisms that increase agreement. For example, if you donated your
help, observers may not care about their current opinion about the recipient
and like you either way (which is the case for image scoring (Nowak and
Sigmund, 1998b, ) or standing (Sudgen,, 1986, Leimar and Hammerstein,, 2001)).
Hence, aside from a few who miss-perceive your action due to noise, all
observers will have the same opinion about you.
The existence of additional agreement, i.e. states where agreement is
significantly above $\widehat{a}$, was already implicitly shown by the work of
Fujimoto and Ohtsuki, (2022), who investigated the exact distributions of
reputations for four strategies. They found that even infinite populations
sometimes have two or even up to an infinite number of distinct reputation
states, that a player could be in. It was not stated by the authors but any
split of players into groups with different reputation states entails that the
opinions of this population have additional agreement (see a proof sketch in
Supplementary Information (SI)). Nevertheless, their seminal work included an
analytical model to predict distributions of strategies exactly, which will
likely be foundational for future analytical models of IR. Yet, two problems
remain. The number of strategies it can be applied to is small and the model
is limited to the condition of full observation (i.e., $\psi=1$).
In conclusion, in the literature so far, there have been only two types of
analytical models of private assessment, and they are all limited to either
$\psi=1$ or $\psi\to 0$. Both cases are extreme ends of possible natural
conditions. Full observation seems impossible due to physical constraints.
Even if all interactions between people were public, there are still
limitations to the attention observers can pay. Nominal observation rates, on
the other hand, make IR less effective in the short-term, and may cause it to
be replaced by direct reciprocity (Schmid et al., 2021a, ). As Fujimoto and
Ohtsuki, (2022) state themselves, a model for intermediate observation rates
is necessary, especially to study IR under private assessment in an exhaustive
fashion, like the defining work of Ohtsuki and Iwasa, (2004).
Overall, our investigation fills two major gaps in the literature. We provide
the first the exhaustive investigation of agreement in reputation dynamics
under private assessment. We test whether additional agreement is common for
high or intermediate observation rates, but also if the assumption of the
R-models always holds for $\psi\to 0$. We test this with simulations of a vast
number of strategies and conditions. Second, we define the first A-R-model,
that moves beyond the one-dimensional approach of the R-models, to capture the
two-dimensional space of agreement and reputation. It is capable of predicting
long-term behavior of said simulations. Both contributions advance the
understanding of indirect reciprocity under private assessment and lay
important groundwork for exhaustive evolutionary investigations.
## 2 Methods
### 2.1 Simulation
We examine if the average agreement $a$ is greater than assumed in the
R-model, $\widehat{a}=r^{2}+(1-r)^{2}$. For that, we run simulations with
different strategies and conditions. To limit complexity, following the
example of Fujimoto and Ohtsuki, (2022), we consider homogeneous populations,
i.e. within a simulation all agents follow the same strategy.
We have chosen a framework to model many different strategies. The first
characteristic of a strategy is its assessment rules, given by vector
$\alpha$. It defines how an observer judges the action, i.e. cooperation $C$
or defection $D$, depending on its opinion about the recipient, i.e. good $G$
or bad $B$. An assessment can be one of three forms: the action is approved of
(+1), disapproved of (-1) or ignored (0). If the player approves of an action,
their opinion of the donor will be good afterwards (which is the same as to
say that it has increased within the limit of the two opinion values, 0 and 1,
aka bad and good). If the action is disapproved of, the opinion will be bad.
And if the action is ignored, the opinion does not change.
The second characteristic of the strategy is given by its action rules
$\beta$. They define what players will do, when they meet a recipient they
like or dislike. These actions are cooperate (1) or defect (0). In the
following, if we refer to $\alpha$ and $\beta$ as a whole, in the order given
below. But we will sometimes refer to specific values or pairs of values, such
$\alpha_{CG}$: cooperation towards good. With this framework we can model 324
different strategies. For example the commonly studied strategy “staying”
(Okada et al.,, 2017, Sasaki et al.,, 2017) is described as follows:
$\begin{matrix}staying:&\begin{matrix}\alpha=\\{\alpha_{CG},\alpha_{DG},\alpha_{CB},\alpha_{DB}\\}=(1,-1,0,0)\\\
\beta=\\{\beta_{G},\beta_{B}\\}=(1,0)\end{matrix}\par\end{matrix}$ (1)
Of all the possible strategies of our framework, we study 171 strategies with
unique behavior. Each of the possible strategies has a mirror image, which
behaves equivalently in regard to cooperation. If we were to label opinion
states as blue and red (instead of good and bad), an mirror image would
cooperate in the exact same way to the same partners, but would think of them
as blue instead of red or vice versa. We obtain such a mirror image of a
strategy by exchanging the symbols +1 and -1 in the assessment rules and
flipping them for good and bad recipients
($\alpha_{m}=(x_{1},x_{2},x_{3},x_{4})$ to
$\alpha=(-x_{3},-x_{4},-x_{1},-x_{2})$) as well as flipping values of the
action rules ($\beta=(xy)$ to $\beta_{m}=(yx)$). Note, that some strategies
are their own mirror strategy. This focus on unique strategies conveniently
leaves us with only three possible action rules: (1,1): unconditional
cooperators or AllC, (1,0): conditional cooperators or Coco222sometimes,
somewhat unfortunately, called discriminators and (0,0): unconditional
defectors or AllD (see SI for examples of mirror images).
For our simulations, we consider a well-mixed population of size $N=100$.
Every agent can have either a good (1) or a bad (0) opinion about each other
agent. Hence, the state of the population can be described by the $N\times N$
image matrix $M(t)$ of the population at time $t$ (Uchida,, 2010). Initially
all entries are filled by a fair coin toss. Other studies have reported, that
other initial conditions almost never change the outcomes (Hilbe et al.,,
2018).
Time is discrete. In total, each time step consists of three parts. First, a
donor $do$ and a recipient $re$ are drawn at random from the population. The
donor then decides whether to cooperate. Note, these two steps are usually
referred to as the donation game. To apply IR, donors base their decision on
their action rule and their current opinion about the recipient. In the third
part, opinions about the donor are updated due to observations.
These observations and updating can be broken down further. First, for each
player (except the donor and recipient) it is decided if they will observe the
interaction. For each player, who observes, it is determined if it observes
accurately or if the observation is altered by a perception error, i.e. with a
probability $\epsilon$ they will perceive the opposite of what the donor is
actually doing. Next, individually perceived action and individual opinion of
the observer about the recipient are combined for the private assessment of
the donor. The opinion is updated if assessment is 1 or -1, but left as is if
assessment is 0. Finally, each observer (whether they made an assessment or
not) may change their opinion to a random value if they commit a cognitive
error with probability $\mu$.
We study the behavior of the simulation in the long term and extract precise
values for average reputation $r$ and average agreement $a$. In order to
ensure accurate and reliable results, the simulations are run until we are
reach an objective level of confidence about the measured average values (for
details see SI.)
Note, in our version of the donation game, the donor and recipient do not
update their opinion. I.e. players ignore the information of interactions they
are themselves involved. This means we study pure indirect reciprocity,
without any direct reciprocity (Schmid et al., 2021a, ). Because of that, the
diagonal of the image matrix is never updated nor used. We therefore exclude
it from all computations of averages. For a detailed motivation of this design
see SI.
### 2.2 Predictive A-R-Model
How could we define a new model which can represent both $r$ and $a$, but is
otherwise as simple as possible? It has to be able to represent a continuous
transition of $a$, from the baseline of the former prediction $\widehat{a}$ to
its maximum 1 (see transition from 1 to 3 in figure 1). The transition must be
possible for any $0\leq r\leq 1$ while keeping $r$ constant. As mentioned, we
could first nominally divide the population into two groups, lucky and
unlucky. To increase agreement a minimal step, we could take away a random
good opinion about an unlucky player and exchange with a bad opinion about a
lucky player. I.e. we decrease the average opinion of unlucky players and
increase the average opinion about lucky players. We could continue to do so
until lucky players have the average reputation $r_{L}=1$ and unlucky players
have the average reputation $r_{U}=0$, hence both agreement about lucky
players $a_{L}$ and agreement about unlucky players $a_{U}$ is 1.
We can do this only if the number of good opinions about unlucky players
$n_{GU}$ is equal to the number of bad opinions about the lucky players
$n_{BL}$. This is the case, if the size of the lucky group (or the percentage
of lucky players) is equal to $r$, hence the size of the unlucky group is
$1-r$. With this assumption, we may define a single value $d$, which is both
the percentage of good opinions about unlucky players stolen and the
percentage of bad opinions about lucky players replaced (see figure 1c).
$r_{L}=r+d(1-r)\quad\&\quad r_{U}=r-dr,\quad 0\leq d\leq 1.$ (2)
With this model we define the two dimensional space with just two variables.
We assume a specific distribution of reputations, namely that there are
exactly two groups (in general there could be up to $N$ groups in a population
of $N$ players). The goal of our model is not to represent the exact
distribution. Just to represent $a$ as well as $r$, for which the two-group
scenario provides a minimal model. It allows us to to model all agreement
states in all reputation states with a single parameter $d$.
Both $r$ and $d$ can range between 0 and 1 (whereas the minimum of $a$ would
depend on current $r$). Note that $d$ is undefined for $r=0$ and $r=1$ (see
equation 3), but it does not need to be. In the case of $r=0$, $d$ vanishes
from $r_{U}$ and the size of the lucky group is zero, hence $n_{GU}$ is zero
for any $d$. The opposite is true for $r=1$ respectively. Besides these
extreme cases, $d$ is given by
$d=\frac{2^{1/2}(r(1-r)(-2r^{2}+2r+a-1))^{1/2}}{2r(1-r)}=\left(1-\frac{1-a}{2r(1-r)}\right)^{1/2},\
0<r<1.$ (3)
This close form of $d$ is derived by solving the equation (4) below for $d$,
by replacing $r_{L}$ and $r_{U}$ with the equations in (2). The global average
agreement $a$ is given by the sum of agreements about members in each group
discounted by the group size, i.e. $a=ra_{L}+(1-r)a_{U}$. The agreement for a
group is derived by its average reputation, that is
$a_{L}=r_{L}^{2}+(1-r_{L})^{2}$ and $a_{U}=r_{U}^{2}+(1-r_{U})^{2}$. Thus,
$a=r(r_{L}^{2}+(1-r_{L})^{2})+(1-r)(r_{U}^{2}+(1-r_{U})^{2}).$ (4)
We will now use $r$, $r_{L}$ and $r_{U}$ to predict the probabilities of
events of our simulation, assuming an infinite population. To model a time
step of the simulation, we need to distinguish between onetime events and
repeated events. There are three events that happen only once per time step:
picking a donor, picking a recipient and the decision of the donor. For these
we need to know three kinds of probabilities for later computation. First, a
pair of probabilities $q$ about the donor’s group affiliation: $q_{L}=r$
(lucky group) and $q_{U}=1-r$ (unlucky group). Second, four probabilities $p$
representing the combinations of donor’s choice and recipient’s group. For
example, $p_{CL}$, the probability that the donor cooperates and the recipient
is in the lucky group (where $\beta$ is the set of the donor’s action rules
and $1-r_{L}$ is the chance the donor has a bad opinion of the recipient from
the lucky group), is given as follows
$p_{CL}=r\Big{(}r_{L}\beta_{G}+(1-r_{L})\beta_{B}\Big{)}$ (5)
or in general
$\begin{matrix}p_{jk}=q_{k}\Big{(}r_{k}c_{jG}+(1-r_{k})c_{jB}\Big{)},\\\
\text{where }j\in\\{C,D\\},\ k\in\\{L,U\\}\text{ and
}c_{Cx}=\beta_{x},c_{Dx}=1-\beta_{x}.\end{matrix}$ (6)
Third, a 2-by-4 matrix $G$ showing how likely the opinion of the donor is good
after assessment. It depends on the previous reputation of the donor’s group
$r_{i}$ and the four assessment rules $\alpha$ (see equation 1). For example,
since staying judges cooperation with bad recipient as neutral,
$\alpha_{CB}=0$, the probability to be considered good is equal to the
previous reputation, which is $r_{L}$ if the donor belongs to the lucky group
$G_{L,CB}=r_{L},$ (7)
or in general
$\begin{matrix}G_{i,mn}=\begin{matrix}1,&\text{if}&\alpha_{mn}=1\\\
r_{i},&\text{if}&\alpha_{mn}=0\\\ 0,&\text{if}&\alpha_{mn}=-1\end{matrix},\\\
\\\ \text{where }i\in\\{L,U\\},m\in\\{C,D\\},n\in\\{G,B\\}.\end{matrix}$ (8)
These are the probabilities that depend onetime events. There are also
probabilities of repeated events, namely observations, since there can be many
observers. Each observation includes the action the observer perceives and
what opinion it had about the recipient. Hence there can be four kinds of
observations, and we compute their probability $O$ for each of the four
combinations of onetime events $p$ described in equation 6. For example, in
the event that the donor cooperated with a lucky recipient, we can compute the
probability $O_{CL,CG}$, that the observer has a good opinion about this lucky
recipient and that the observer perceives the cooperation without a perception
error $\epsilon$.
$O_{CL,CG}=r_{L}(1-\epsilon),$ (9)
or in general
$\begin{matrix}O_{jk,mn}=r_{i}^{*}e,\\\ \\\
\begin{matrix}r_{x}^{*}=r_{x}&\text{if}&n=G\\\
r_{x}^{*}=1-r_{x}&\text{if}&n=B\\\ e=\epsilon&\text{if}&m\neq j\\\
e=1-\epsilon&\text{if}&m=j\\\ \end{matrix}\\\ \\\ \text{where
}j,m\in\\{C,D\\},k\in\\{L,U\\},n\in\\{G,B\\}.\end{matrix}$ (10)
We then take into account observation rates $\psi$ and cognitive errors $\mu$,
which are also repeated events. Each observation might not take place, leaving
opinions unchanged. When it takes place, the resulting opinion may be altered
by a cognitive error and its value would be reversed. We therefore adjust $G$
to $G^{*}$.
$G^{*}_{i,mn}=(1-\psi)r_{i}+\psi(G_{i,mn}(1-\mu)+(1-G_{i,mn})\mu).$ (11)
With these probabilities we can now compute the probability of the eight
possible interactions, that is, the combination of all three onetime events,
as follows
$\Pi_{ijk}=q_{i}p_{jk},$ (12)
and the expected reputation of the donor for each of these combinations
$\Gamma_{ijk}=\sum_{m,n}G^{*}_{i,mn}O_{jk,mn}.$ (13)
Finally, we can now compute the expected change of reputation
$\Delta_{r}=\Big{(}\sum_{i,j,k}\Gamma_{ijk}\Pi_{ijk}\Big{)}-r$ (14)
and the expected change of agreement
$\Delta_{a}=\Big{(}\sum_{i,j,k}((\Gamma_{ijk})^{2}+(1-\Gamma_{ijk})^{2})\Pi_{ijk}\Big{)}-a.$
(15)
Note, as it was when computing agreement in the simulation, the instances of
interaction have to be computed separately. Each of the eight possible
interactions predicts a specific agreement about a donor’s image. We compute
each instance and only then compute the expected value of $\Delta_{a}$. Note
also, that we do not assume that the donor can be attributed to the lucky or
unlucky group after assessment. These groups are simplifications of the real
image matrix that we use to describe its state with only two parameters.
Information about the exact expected reputations of the donor and hence the
agreement about it for the eight possible interactions is lost. Our model does
not give exact predictions.
To validate our analytical model, we create a numerical algorithm to find the
stable equilibria points in the $r\times a$ space (see SI for a detailed
description). We use it to find the equilibria for all cases which we
simulated and compare the results. For additional comparison we also use an
analytical approach that models agreement always as $\widehat{a}$, aka a
R-model (Perret et al.,, 2021), and compare the fit of both predictive models.
## 3 Results
(a) Simulations. Agreement and reputation averaged across runs for homogeneous
populations with 100 players. Not shown are cases where range of averaged
reputations between runs exceeded 0.1 (2.9% of cases for Cocos). For Cocos,
$25.2\%$ have significant additional agreement (above the broken line).
(b) Predictions. Unique stable states of the model for infinitely large
homogeneous populations. Not shown are cases where multiple equilibria were
found which ranged more than 0.01 in total distance in A-R space (for Cocos,
27 cases spread over only 3 strategies. See SI for some examples.)
Figure 2: Comparing results of simulations and analytical predictions. Values
for 171 unique strategies in 15 conditions (namely,
$\epsilon\in\\{0,0.001,0.01,0.05,0.1,0.2,0.4\\}$ and
$\mu\in\\{0.001,0.01,0.05\\}$). Reputation $r$ (x-axis) can range between $0$
and $1$. Agreement $a$ can range between $a=\widehat{a}$ and $a=1$ (solid
black lines). Values above $a_{th}=\widehat{a}+0.05$ (broken black line) are
considered as significant additional agreement. Figure 3: Results for Cocos
with other observation rates $\psi$. On the left are the simulation results,
and on the right, the predictions of the A-R-model. Values for all 81 unique
conditional cooperators, with conditions and excluding criteria the same as
those given in figure 2.
As described above, we studied 171 strategies with unique behavior. Each
homogeneous population of 100 players is first tested for the observation rate
$\psi=1$ in 15 conditions. The conditions span an exhaustive range of
perception errors $\epsilon\in\\{0,0.001,0.01,0.05,0.1,0.2,0.4\\}$ and a
reasonable range of cognitive error $\mu\in\\{0.001,0.01,0.05\\}$. Note, this
full observation is the natural opposite to limiting assumption of the R-model
where $\psi\to 0$.
Figure 2(a) shows the results of our simulations as reddish dots. They are
depicted separately for the three kinds as action rules: Coco, AllC and AllD.
For AllC and AllD, agreement is always virtually minimal. But for Coco
strategies, many points lay far above. Sometimes agreement is almost maximal
($a\to 1$), while $\widehat{a}$ predicted it to be the smallest possible value
($a=0.5$ at $r=0.5$). For Cocos, 25.2% of the measured agreements were at
least 0.05 higher than expected (indicated by the broken black line).
Note that we removed cases in which results differed significantly across
runs. Averaging them could give a false impression of average agreement. For
example, two runs with $r_{1}=0.3$ and $r_{2}=0.7$ and each minimal
$a_{i}\to\widehat{a}=0.58$ would average to $r=0.5$ and $a=0.58$. For this
averaged reputation, the minimal agreement would be $\widehat{a}=0.5$, so the
average measurement would be $0.08$ higher than that. In other words, although
the actual agreement in each run was minimal, the averaged agreement would
indicate additional agreement above the threshold. We exclude cases in which
the range of $r$ in a single condition is above $0.1$ ($2.8\%$ of cases in
condition $\psi=1$, see SI for examples of excluded cases and alternative
figure with cases included). It is clear that our findings are not caused by
such artifacts and that for $\psi=1$ results differed substantially from the
prediction $\widehat{a}$ with more agreement than expected for many
conditional cooperators.
Next, we show that these results can be replicated for $\psi=1$ by our A-R-
model, see blue dots in figure 2(b). Excluded are again cases for with
multiple equilibria (given the much higher precision of the analytical
approach we excluded all cases in which the total distance in the agreement-
reputation space, similar to equation 16, exceeded 0.01, see SI for examples
of excluded cases). The general patterns of the simulation could be
replicated. AllCs and AllDs have virtually no additional agreement whereas
Cocos do. The qualitative fit between simulation and model is striking.
Before we quantitatively describe the fit, we expand our investigation to
other observation rates. We show only results for Cocos, since these are the
most interesting cases, in figure 3. We studied three intermediate observation
rates $\psi\in\\{.75,.5,.25\\}$ as well as a special case, where every round
would have exactly one observer. A single observer is the smallest amount of
meaningful observation possible. No observation would simply not change the
image matrix and have no impact on reputation or agreement. It is the closest
one can get to the limiting assumption of the R-Model $\psi\to 0$.333Even
taking a very small probabilistic observation rate such as $\psi=0.0001$ would
have actually increased the number of observers. All rounds in which no
observation takes place are simply ignored. The rest may have one or more
observers. In addition, simulating many rounds just to be ignored would
increase the computational demand unnecessarily for no benefit.
Both simulation and prediction show a steady decline in agreement, until $a$
is virtually minimal when there is only a single observer. This is a
qualitative fit of what the R-model assumes. Adjusting the observation rate in
the A-R-model replicates it. For all observation rates, qualitative fit is
still high. And, similarly to full observation, intermediate observation rates
still show significant additional agreement.
We quantified the fit of simulation and predictions in two ways. Our A-R-model
makes predictions about agreement $a_{p}$ as well as reputation $r_{p}$. We
therefore compute the absolute distance to simulated agreement $a_{s}$ and
reputation $r_{s}$ by
$\Delta=\sqrt{(a_{p}-a_{s})^{2}+(r_{p}-r_{s})^{2}}.$ (16)
We excluded values not shown in figures 2 and 3 (if either simulation or
prediction had to be excluded, we excluded the entire pair). Results for all
observation rates are shown in the left of table 1. For both maximal and
intermediate observation range, the fits are excellent. The median of absolute
distance between the simulations and the model is 0.003. For 92.7-95.7% of
predictions, the deviation is smaller than 0.01. For the minimal observation
rate, fits are slightly worse. The median 0.005 and only 72.3% of predictions
are less than 0.01 of target.
We also compared the predictions of our A-R-model with predictions of the
R-model. Since the R-model can only predict reputation $r_{o}$, we compared
the two models in that regard. The difference in deviation is shown in the
right of table 1. Keep in mind that the R-model was designed only for $\psi\to
0$, which is most closely met in the single observer condition. The new A-R-
model is not often better as the R-model in this case. And at least for one
case, it is much worse, as indicated by the range. However, the A-R-model is
superior for all high and intermediate observation rates $\psi$. Its
predictions are better in $4.6\%$ to $11.5\%$ of all cases (AllD, AllC and
Cocos), and the accuracy is increased by as much as $0.5$, which is half of
the maximum range of $r$. Higher accuracy is especially prevalent for Cocos,
to which all evolutionary successful strategies belong, such as image scoring
(Nowak and Sigmund, 1998a, ) and the leading-eight (Ohtsuki and Iwasa,, 2006).
Here, for $\psi=1$, the A-R-model is better in $24\%$ of cases. Considering an
independently changing average agreement seems to have increased the
prediction of average reputation substantially.
Table 1: Deviations between Model Predictions and Simulation | A-R-model | | | Difference A-R-model minus R-model | |
---|---|---|---|---|---|---
| absolute deviation | | | deviation in reputation | |
| median | $<.01$ | $<.05$ | $<-.01$ | |
(A-R better) | $>.01$ | | | | |
(A-R worse) | range | | | | |
$\psi=1$ | 0.003 | 92.7% | 99% | 11.5% | 0% | -0.498:0.006
$\psi=0.75$ | 0.003 | 95.7% | 99% | 9.7% | 0.1% | -0.497:0.051
$\psi=0.5$ | 0.003 | 95.4% | 98.9% | 7.8% | 0.2% | -0.495:0.053
$\psi=0.25$ | 0.003 | 94.7% | 98.4% | 4.6% | 0.1% | -0.491:0.014
single observer | 0.005 | 72.3% | 86% | 1.2% | 0.4% | -0.034:0.334
## 4 Discussion
We now summarize our findings and their immediate implications, before
highlighting remaining issues and limitations. We further compare our A-R-
model with the recent model of Fujimoto and Ohtsuki, (2022) to highlight the
differences and common ground. We then discuss some important potential
extensions of the A-R-model, which would enable it to implement recently
proposed enhancements to IR strategies, such as generous assessment (Schmid et
al., 2021a, ) and pleasing (Krellner and Han,, 2021). We will close with
possible real world implications of the discovered additional agreement.
In the first part of our report, we showed that additional agreement can
emerge in indirect reciprocity under private assessment. For high or
intermediate observation rates, opinions about a person were shared much more
often than mere chance. Therefore, the assumption of the minimal agreement
$\widehat{a}$, on which most previous models (R-models) relied, cannot be
generalized to these circumstances. We could on the other hand confirm that
the assumption is reasonable for very low observation rates, such as a single
observer for each interaction. But we showed, that results about evolutionary
success with solitary observers (Okada et al.,, 2018, Okada, 2020b, , Perret
et al.,, 2021) cannot yet be generalized to other observation rates.
The second part of our report makes an important step towards that goal.
Previous R-models can only model average reputation and therefore must rely on
the assumption of minimal agreement. Our A-R-model is able to represent
average levels of agreement and reputation independently. It predicts both
with astounding accuracy and does so for any intermediate or high observation
rates. It outperforms the predictions of R-models in these circumstances by a
large margin.
Making precise measurements or predictions of reputation in particular is the
basis for studies on the evolutionary stability of any reputation-based IR
strategies (Okada, 2020a, ). An individual’s reputation directly corresponds
to how much cooperation they receive, hence it determines their payoffs and
even the payoffs of others (from which the may receive cooperation, hence
cause costs to them).
Parallel to this work, other researchers have discovered another way to
predict opinion dynamics more accurately than by average reputation alone.
Fujimoto and Ohtsuki, (2022) were able to predict precise distributions of
reputations, e.g. 80% of players who would have a reputation of about 0.8, 15%
of 0.2, and 5% of 0.25. This was another hugely important step for analytical
models of IR under private assessment and is, of course, closely related to
the current paper.
Our model assumes a simplified distribution of reputations. We treat the
populations as if there were at most two groups, and as if their size was
given by the average reputation. Fujimoto and Ohtsuki, (2022) show that this
is often not the case, that there can be more groups or other compositions.
Their analytical approach is exact and uses no simplification. However, as the
fit of our model shows, this level of detail may not be necessary.
And the level of detail in their model seems to come at a cost as their
analysis is limited to only four strategies. It seems reasonable however that
their framework could model all strategies, that care about action as well as
reputation of the recipient, but apply only binary assessment rules (i.e.
assess each action as either good or bad, but not as neutral). This would
cover 64 strategies (mirror images included) compared to the 324 of our
study.444Indeed, in a preprint, Fujimoto2023 modelled 16 strategies, focusing
on a single action rule, instead of four. If they had looked at all actions
rules and not exclude mirror images, they could have modelled 64 strategies.
However, since they were only concerned with levels of cooperation, the
limitation to Cocos ($\beta=(1,0)$) was perfectly justified. AllD and AllC
with different assessment rules may held different reputations, yet the all
cooperate the same.
Their model also covers only a single observation rate $\psi=1$. The authors
themselves state that being able to model arbitrary observation rates would be
a very important extension of their approach. Our A-R-model is already able to
do that. And, it seems at least possible, that their approach is not capable
of dealing with intermediate observation rates, i.e. $\psi<1$. Because their
model currently relies on the fact that the new reputation of the donor is
entirely independent of their current reputation. If some players do not
observe, their new opinions depend (entirely) on their current opinions, since
their opinions are just kept as they are. Their model could no longer be based
on one dimension (the current reputation of recipients) but would have to
incorporate entirely new dimension (the current reputation of the donor). This
increase in complexity might be a serious problem.
These problems for arbitrary observation rates also concern the modelling of
more complex, so-called 3rd-order strategies (Santos et al.,, 2021), which
also use the current reputation of the donor in their assessment. In contrast,
in our A-R-model, the current reputation of the donor is already incorporated.
It will be straightforward to extend our model to consider all 2048 3rd-order
strategies (mirror images included) (Ohtsuki,, 2004).
We envisage the following additional extensions of our model. First, we can
include strategies that do not use deterministic rules but probabilistic ones
(Schmid et al., 2021a, , Schmid et al., 2021b, ). Instead of assessing
defection always as bad, observers may only do so 80% of the time (in other
words they are sometimes generous in the judgment). The same could be applied
to their action rules. They may want to cooperate even with bad individuals
about 10% of the time. Second, we can include pleasing (Krellner and Han,,
2020, 2021). Instead of granting or refusing help only based on the donor’s
own opinion (which can be easily disturbed by perception error), the donor
pools some of the opinions of others and act like the majority would decide.
Some of these extensions are as easy as replacing -1, 0 or 1 in the action and
assessment rules of this paper with -0.8 and 0.1.
Figure 4: Detailed comparison between models for staying norm (see equation
1). Parameters are $\psi=1$, $\epsilon_{p}=0.1$ and $\mu=0.01$. Bottom graph
shows the two-dimensional space of the A-R-model. Dark blue areas indicate
likely changes in either agreement or reputation or both, whereas light yellow
areas indicate little to no change. Arrows indicate the direction of the
change. The precise predictions of the model is indicated by a blue x.
Simulations are shown in red, the averages of single runs are shown as red
circles and their total average as a red +. Top graph shows one-dimensional
R-model in comparison. Change in reputation and its direction is depicted on
the y-axis in addition to the arrows. The stable point (green x) is where the
x-axis is crossed with a downward slope. The white line in the A-R-model
indicates a shift from reputation decrease (below) to reputation increase
(above).
This paper is the first to study agreement in such detail. Agreement is the
key feature that distinguishes previous research on IR under public assessment
from the more recent research on private assessment. Public assessment fixes
agreement at its maximum (all agree all the time), but private assessment does
not fix it at the opposite state i.e. its minimum $\widehat{a}$. Public and
private are not opposite sides of the same coin. Rather private assessment
allows agreement to vary, opening a new dimension of complexity to the
dynamics of reputation-based IR.
Our A-R-model allows one to study agreement and reputation in even more detail
than was reported within the results of this paper. We focused on the stable
states of the population. But one could study the direction and likelihood of
change in each state, as seen in figure 4 for the example of the staying
strategy (see also SI for other important strategies, such as image scoring
(Nowak and Sigmund, 1998a, ) and the leading-eight (Ohtsuki and Iwasa,,
2006)). One important insight this grants is about the stability of other
regions. For staying, there exist somewhat stable areas with reputations lower
than $r=0.3$ and minimal agreement. The existence of additional relatively
stable regions has large implications for the short-term behavior of the
population.
Seeing the direction of change in the entire space also allows us to make
educated guesses about another dynamic. The white line in the A-R-model of
figure 4 indicates a shift from reputation decrease (below) to reputation
increase (above). This highlights how the trend in reputation depends on the
state of agreement. It seems reasonable, that increasing agreement could
increase reputation as well. Increase in agreement could be done by a form of
opinion synchronization, for example by some players gossiping about their
observation or opinions to make other opinions fall into line with their own.
Hilbe et al., (2018) suggested that any IR strategy would profit of, or indeed
rely on, some sort of opinion synchronization to maintain stable cooperation
under private assessment. With the A-R-model, we can visualize which
strategies can actually profit from such mechanisms. Some are not affected,
e.g. image scoring, some actually show the opposite pattern, such as ”GKGB”
(Okada, 2020b, ), which seems only stable under private assessment, but not
under public one (see SI for examples). In the future, we can seek a way to
incorporate a probability for opinion synchronization in the A-R-model, to
predict new stable points under various synchronization regimes.
Related to that is a last alteration of the model. In the current
investigation, we considered private assessment in accordance with most
literature (Okada, 2020a, ). Every individual or player in the population
observes independently. But we can imagine a situation, where only a few
players observe, who then share their assessment with many others. Consider
for example the extreme case of a single observer that shares with the entire
population. This would correspond to public assessment (Ohtsuki and Iwasa,,
2004), where no disagreement is possible ($a=1$). In a more realistic
scenario, multiple observers are at least possible, and they may judge the
interaction differently due to individual errors or different previously held
opinions. The information of their judgments may also fail to reach some
players. In such scenarios, disagreement can exist. However, all players who
got the news from the same observer will most likely have the same opinion
(even if information transfer is noisy). It is a form of built-in opinion
synchronization. We can therefore expect agreement to be higher than in the
scenario of the current paper. The amount of agreement reported here might
only be the lower end for private assessment scenarios.
Lastly, we look at the reasons additional agreement emerges. As discussed
above, there can be no additional agreement if every player has the same
reputation. Additional agreement requires at least two groups exist (but up to
$N$-many), with different average reputations. Such groups can only emerge, if
players with the same strategy face different fates. For example, a donor of
strategy $\alpha=(1,-1,1,-1)$ and $\beta=(1,0)$ (image scoring) can get lucky
by meeting a recipient they believe to be good. Hence the donor cooperates and
earns a high reputation (everybody who observes, and does not make an error,
believes the donor to be good). But another donor with the same strategy might
get unlucky, because they believe the recipient to be bad. Hence they defect
and most observers will now think this donor is bad.
In general, groups can emerge if different onetime events cause different
expected reputations. Such onetime event are which recipient is met or which
decision is made. In the image scoring example, if errors happen with
probability $0.1$, we expect the reputation of a donor to be either $0.1$ if
they defected or $0.9$ if they cooperated. Different decisions as donor seem
indeed to be the most important factor for additional agreement. That is why
only conditional cooperators (Coco) show additional agreement, but
unconditional cooperators (AllC) or defectors (AllD) show none. Because some
Cocos cooperate and some defect, they can form groups with different
reputations and therefore have additional agreement.
Is additional agreement a good thing? Our results for strategies such as
staying show that it can increase reputations. The reputations studied here
correspond to how often a player finds another player of the same strategy
worth of cooperation (and Cocos also act on it). More agreement can increase
reputations, hence increase cooperation rates within the strategy, hence
increase the stability of that strategy. This could keep defectors at bay and
increase cooperation in general. Evolutionary investigations need to confirm
these assumptions, but in that regard, agreement might be a good thing.
However, we also showed another consequence of agreement. It is connected with
differences in reputations. Such differences inevitably lead to short-term
inequality between players. Over the long run, each player of the same
strategy will alternate between being lucky and being unlucky, so each will
earn the same pay-offs. But some players may earn much less than others if
only a few consecutive interactions are considered. This can be a problem, if
the game (or life) would require a player to earn a minimal amount to sustain
themselves. It may also be a problem if such inequality itself has bad
consequences for individuals or society.
Research on IR seems to continue to converge (Okada, 2020a, , Krellner and
Han,, 2022). However, there seem to be still a lot of potential to deepen our
understanding of the processes, as demonstrated in Fujimoto and Ohtsuki,
(2022) and the current paper. The understanding of agreement is central for
any analytical model of IR under private assessment. It is central in
understanding what benefits or problems that opinion synchronization or
different observation rates may bring. And it is as central to all research on
reputation-based IR under private assessment, since agreement can change
independently of reputation and can significantly alter the latter as well.
Our A-R-models provides the means to study the evolutionary dynamics of
indirect reciprocity under private assessment for yet the largest strategy
space and widest range of conditions.
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* Sudgen, (1986) Sudgen, R. (1986). The Economics of Rights, Cooperation and Welfare. Basic Blackwell.
* Uchida, (2010) Uchida, S. (2010). Effect of private information on indirect reciprocity. Physical Review E, 82(3):036111.
* Uchida and Sasaki, (2013) Uchida, S. and Sasaki, T. (2013). Effect of assessment error and private information on stern-judging in indirect reciprocity. Chaos, Solitons & Fractals, 56:175–180.
|
# LLMs Still Can’t Avoid Instanceof: An Investigation Into GPT-3.5, GPT-4 and
Bard’s Capacity to Handle Object-Oriented Programming Assignments
Bruno Pereira Cipriano<EMAIL_ADDRESS>0000-0002-2017-7511 Lusófona
University, COPELABSCampo Grande, 376LisbonPortugal1700-097 and Pedro Alves
0000-0003-4054-0792<EMAIL_ADDRESS>Lusófona University,
COPELABSCampo Grande, 376LisbonPortugal1700-097
(2024)
###### Abstract.
Large Language Models (LLMs) have emerged as promising tools to assist
students while solving programming assignments. However, object-oriented
programming (OOP), with its inherent complexity involving the identification
of entities, relationships, and responsibilities, is not yet mastered by these
tools. Contrary to introductory programming exercises, there exists a research
gap with regard to the behavior of LLMs in OOP contexts. In this study, we
experimented with three prominent LLMs - GPT-3.5, GPT-4, and Bard - to solve
real-world OOP exercises used in educational settings, subsequently validating
their solutions using an Automatic Assessment Tool (AAT). The findings
revealed that while the models frequently achieved mostly working solutions to
the exercises, they often overlooked the best practices of OOP. GPT-4 stood
out as the most proficient, followed by GPT-3.5, with Bard trailing last. We
advocate for a renewed emphasis on code quality when employing these models
and explore the potential of pairing LLMs with AATs in pedagogical settings.
In conclusion, while GPT-4 showcases promise, the deployment of these models
in OOP education still mandates supervision.
programming assignments, teaching, object-oriented programming, object-
oriented design, OOP best practices, large language models, gpt-3, gpt-4, bard
††journalyear: 2024††copyright: rightsretained††conference: 46th International
Conference on Software Engineering: : Software Engineering Education and
Training; April 14–20, 2024; Lisbon, Portugal††booktitle: 46th International
Conference on Software Engineering: : Software Engineering Education and
Training (ICSE-SEET ’24), April 14–20, 2024, Lisbon, Portugal††doi:
10.1145/3639474.3640052††isbn: 979-8-4007-0498-7/24/04††ccs: Applied computing
Computer-assisted instruction
## 1\. Introduction
In the last few years, researchers introduced us to Large Language Models
(LLMs), which are tools that can predict the next word in a sequence, after
being trained on large amounts of data, and taking into consideration a large
number of parameters. Popular implementations of LLMs are ChatGPT, a chatbot
developed by OpenAI on top of their Generative Pre-trained Model (GPT), and
GitHub Copilot111https://github.com/features/copilot, a developer support tool
that can generate code from annotations and is also based on GPT.
As CS educators, we are interested in determining how these tools can be used
to support students in their programming learning process, both inside and
outside of the classroom. We are specifically interested in the possibilities
of using these tools for teaching and learning object-oriented design and
programming.
Previous research has shown that LLMs have the capacity to generate computer
program code from natural language descriptions (Xu et al., 2022).
Furthermore, GPT-based models, such as Codex, have been shown to be able to
solve most types of exercises that are used in introductory programming
classes (Finnie-Ansley et al., 2022). Additionally, researchers have reported
that GPT-3.5 can handle object-oriented programming assignments, managing to
reach decent to good scores, with the caveat that the generated code is not
up-to-par with the industry’s best practices of that paradigm (Cipriano and
Alves, 2023a).
However, both the availability and the capabilities of these tools are
increasing rapidly. Updated models, such as GPT-4222Launched on March 14,
2023, promise even better performance than their older versions (OpenAI, 2023;
Bubeck et al., 2023), while new tools, such as Bard333Launched in the US & UK
on Mar 21, 2023; in EU & Brazil on Jul 13, 2023., a chatbot developed by
Google on top of their Language Model for Dialogue Applications (LaMDA)
(Sundar, 2023; Thoppilan et al., 2022), are becoming available to the general
public. These new models justify performing further research on these topics,
to confirm and/or update the findings that resulted from the previous works.
Thus, we decided to explore these newer tools to learn their capabilities and
limitations, as well as identify opportunities to integrate them into our
courses, similarly to what is being done by other instructors (Lau and Guo,
2023; Denny et al., 2023; Daun and Brings, 2023; Leinonen et al., 2023;
Liffiton et al., 2023).
As such, we performed an evaluation of some of the most recently available
LLMs, GPT-4, and Bard, following previous research focused on object-oriented
design, programming and best-practices using GPT-3.5 (Cipriano and Alves,
2023a). This evaluation was based on assignments that are automatically graded
by an Automatic Assessment Tool (AAT).
This paper makes the following contributions:
* •
Presents and compares the performance (as measured by an AAT) of GPT-3.5,
GPT-4, and Bard using 6 real-world assignments that have been used to grade
students in an Object-Oriented Programming university course;
* •
Identifies and categorizes the errors found in the code generated by the
tested LLMs;
* •
Provides full logs of the interactions used to get each model to solve one of
the assignments, in an annotated prompt/reply format;
* •
Presents a set of recommendations for CS Educators wishing to adapt their
classes to the availability of these technologies.
## 2\. Related work
In the last couple of years, a relevant body of research related with
measuring LLMs’ ability to handle introductory programming exercises has been
published (Chen et al., 2021; Finnie-Ansley et al., 2022, 2023; OpenAI, 2023;
Reeves et al., 2023; Wermelinger, 2023; Destefanis et al., 2023). Those
introductory exercises are usually solved with a single function that receives
a set of parameters and transforms them into an expected output. For example,
in (Chen et al., 2021), researchers managed to solve 70% of 164 Python
programming exercises using code generated by Codex, a GPT-based language
model trained on code publicly available in GitHub. In (Finnie-Ansley et al.,
2022), Codex was evaluated using 23 introductory programming assignments, and
managed to make the 17th position when ranked alongside 71 students. The
authors of (Reeves et al., 2023) focused on Codex’s capacity to handle Python-
based Parsons problems (Parsons and Haden, 2006) and revealed that the model
solved 50% of the cases when indentation errors were considered, and 80% when
those errors were ignored. Finally, a study (Destefanis et al., 2023) based on
single-function Java exercises from CodingBat444https://codingbat.com,
reported that GPT-3.5 solved 90.6% of the functions, while Bard solved 53.1%.
Some papers have also delved into GPT’s capacity to handle OOP assignments,
which typically require the implementation of multiple classes that interact
with each other and have mutable states. In one of these studies (Savelka et
al., 2023b), researchers assessed the performance of two GPT-3.5 models
(“text-davinci-002” and “text-davinci-003”) on various introductory and
intermediate Python programming exercises. Some of the intermediate exercises
involved OOP, and the models’ solutions were evaluated using an AAT. The first
model achieved a success rate of 52.9% in the tests, while the second model
scored 70.6%. However, due to the unavailability of the exercise statements,
we could not ascertain whether the type of OOP exercises was equivalent to
those presented in this research paper. The same authors have updated their
research by performing the evaluations using GPT-4, which reached an improved
score of 82.4% (Savelka et al., 2023a). In another study (Ouh et al., 2023),
researchers attempted to utilize ChatGPT (versions 3.5 and 4) for solving OOP
exercises. Some of these exercises involved introductory-level object usage
without inheritance or polymorphism, and the chatbot managed to provide
partially correct solutions. Other exercises introduced inheritance, but a UML
class diagram with the solution was provided to the students, who only had to
implement the corresponding code. As it was not feasible to supply the UML
diagram to the model, the generated solution remained incomplete. Despite
encompassing OOP exercises, this study’s tasks are either very elementary or
highly guided. Another study (Cai et al., 2023), proposed the utilization of
LLMs as low-code tools. While directly using ChatGPT led to solutions with
weak object-oriented design, they were able to achieve good results by pre-
guiding the model on OOP best practices. Finally, the authors of (Cipriano and
Alves, 2023a), evaluated GPT-3.5 (“text-davinci-003”) using both functional as
well as code quality criteria and found that, although GPT-3.5 was able to
pass an average of 77.63% of unit tests, it only passed an average of 50% of
the code quality evaluations.
As far as we know, no research focused on Bard’s (or LaMDA’s) ability to
handle object-oriented assignments has been published.
## 3\. The assignments
This research was based on assignments used for student evaluation in a
Computer Engineering degree. The course occurs on the 2nd curricular year of
the degree and focus mostly on teaching Object-Oriented Design and Programming
using the Java programming language. In this course, students are expected to
learn how to implement object-oriented software solutions following the
paradigm’s best practices (Wegner, 1990), with a strong focus on issues like
code readability, modularity and extensibility. For example, students are
expected to understand the drawbacks of using type testing, such as those
permitted by ‘instanceof’, to decide program behaviors, since that technique
tends to make programs harder to modify.
The course’s assignments (e.g., tests and projects, among others) are
evaluated using an open-source AAT, called Drop Project (Cipriano et al.,
2022). It evaluates if the student’s code is respecting the assignment’s
requirements using teacher-defined unit tests. It is also capable of verifying
if the student’s code follows the expected code quality rules and guidelines.
In this course, some assignments are configured to warn the teacher about the
use of certain keywords that allow the program to work while disregarding some
of the aforementioned best practices. An example of this is the usage of the
‘instanceof’ keyword or the ‘getClass()’ function. Due to the limitations of
the plugin used by the AAT for the code quality validations (Checkstyle
(Ivanov et al., 2023)), some quality validations are implemented using unit
tests. A case in point is verifying whether a certain class has been declared
as abstract.
To evaluate the LLMs, a set of assignments that have been used in this course
as mini-tests focused on inheritance and polymorphism was used. Each
assignment has a different business domain, as well as different requirements.
All the assignments have the following goals: identify and implement entity
relationships (both composition and inheritance), implement some getters and
setters, implement a non-trivial ‘toString()’ function, and, implement some
functions that have to create, query and/or manipulate objects of several
classes. Note that the relationships aren’t directly provided to students, who
must infer which classes are the super and sub-classes. This approach sets our
study apart from others where assignments give more explicit directions (Ouh
et al., 2023).
⬇
We want to implement a software to help an IT company that provides
ITConsultant services with the management of their employees. In this
challenge, there exist two types of employees: those who pertain to the human
resources management area (HRWorker), and the IT (information technology)
experts, who pertain to ITConsultant.
The classes Employee, HRWorker and ITConsultant must be created (...) however
it should not be possible to instantiate an object using the super class’s
constructor.
A company is characterized by its name. A company can have multiple employees,
but each employee only the belongs to a single company.
A employee is identified by its id (int), name (String) and monthly salary
(int).
(...)
Add, to the appropriate classes, the following methods:
- A public int getId() method, which returns the employee’s ID.
- A public int getSalary() method, which returns the employee’s salary.
- A public int getValue() method, which returns the IT consultant’s hourly rate.
(...)
(Within the Company class) add a public String toString() method, which must
be implemented to return the following:
- "The company <name> does not have employees." - if the company does not have any employee;
- "The company <name> has <number_employees> employees:" - if the company has at least one employee. Besides that, the information for each employee should be display, considering the respective type.
Listing 1: Partial instructions for the ITCompany assignment
Moreover, each assignment has behaviors based on the object type to assess
whether students can devise solutions without resorting to explicit type
checks like those allowed by the ‘instanceof’ keyword and ‘getClass()’
function. Finally, in some cases, we ask for the student ID to be used as the
value of an object’s attribute, to guarantee unique solutions among students.
Listing 1 presents a partial example of an assignment’s text.
Six such assignments were used in this study. The selected assignments were
originally used to grade students in different school years (2018/19, 2021/22,
and 2022/23).
## 4\. Methodology
To interface with GPT-3.5, openAI’s Playground555available at
https://platform.openai.com/playground (free account) was used. The selected
model was “text-davinci-003”. The “temperature” parameter, which controls
randomness, was left at the default value of 0.7, and the “maximum length”
parameter was left at the default of 256. GPT-4 was studied using OpenAI’s
ChatGPT Plus666available at https://chat.openai.com/ (paid account). Finally,
to study Bard, we used the respective chat interface 777available at
https://bard.google.com/. Note that neither ChatGPT Plus nor Bard allows
changing the models’ parameters. The experiments were done in different time
periods, since we did them when the models became available to us: the
original experiment with GPT-3.5 was done in December 2022, the experiment
with GPT-4 was done in May 2023, and finally, the experiment with Bard was
done in July 2023. To evaluate the models’ output, version 0.9.7 of the Drop
Project AAT was used.
For each model and assignment, the following algorithm was performed:
1. (1)
paste the assignment’s text into the LLM’s text input area
2. (2)
submit it
3. (3)
examine the output to determine whether all the mandatory classes and
functions are present
1. (a)
if they are, move to step 4
2. (b)
otherwise, prompt for the missing code and repeat step 3
4. (4)
inspect the code for syntactic, logic, and output format errors
5. (5)
manually fix any syntactic / compilation errors
6. (6)
submit the LLM’s generated code to the AAT
7. (7)
analyze the AAT’s output
## 5\. It works but it’s not OO - how LLM(s) tried to solve the IT Company
assignment
Figure 1. Simplified UML class diagram for the “IT Company” assignment,
excluding non-essential attributes, methods, and constructors. Students only
receive textual instructions, not this diagram.
This section presents a high level overview of our attempts to use each model
to solve one of the 6 assignments: the “IT Company” assignment. In this
assignment, students are expected to design the object model of an IT Company
with the following concepts: ‘Company’, ‘Employee’, ‘ITConsultant’, and
‘HRWorker’. The concepts and their attributes are explained to the students by
text. However, the text does not explicitly indicate the relations that exist
between the classes: it is up to the student to infer that there exists a
composition relationship — between the ‘Company’ and the ‘Employee’ classes —,
and an inheritance relationship where the ‘Employee’ class is the super-class
and the ‘ITConsultant’ and ‘HRWorker’ classes are the sub-classes. Students
are also expected to understand that the ‘Employee’ class should be abstract,
since the text mentions that it must not be possible to instantiate that
class. Finally, students must implement several mandatory functions,
identifying the class of the hierarchy where each function belongs. For
example, there is an attribute that only makes sense for the ‘ITConsultant’
class: the hourly rate. As such, only that class should have that attribute
and the respective getter, which must be called ‘getValue()’. Also mandatory
is the creation of a function called ‘getEmployeeHighestHourlyRate()‘ in the
‘Company’ class. This function must return the ‘Employee’ with the highest
hourly rate. However, since the hourly rate concept only applies to the
‘ITConsultant’ class, the student must find the proper Object-Oriented design
to implement this challenge. As such, if a student uses ‘instanceof’ or
‘getClass()’, a penalty will be applied to the respective grade. Figure 1
presents an overview of this assignment’s required classes, as well as the
most relevant design expectations. Listing 1 presents part of this
assignment’s instructions. A full version of the assignment, including both
instructions as well as unit tests is available
online.888https://github.com/drop-project-edu/itCompanyAssignment Refer to
(Anonymous, 2023) for full logs of these interactions.
### 5.1. Main analysis
GPT-3.5 required 18 interactions to generate a solution that included all
mandatory elements. Issues found in that solution:
* •
Function ‘getEmployeeHighestHourlyRate()’ uses ‘instanceof’.
* •
Fails 1 test because the ‘Employee’ class is not abstract.
* •
Fails 2 tests because the package name was incorrectly included in
‘ITConsultant.toString()’ function’s return value.
* •
Fails the test for the ‘Main.myCompany()’ function due to using an uninformed
number in the solution.999This problem was expected, since we did not input
the student ID to the models.
* •
Failed to import the ‘ArrayList’ class.
Bard required 11 interactions to generate a solution that included all
mandatory elements. Issues found in that solution:
* •
Function ‘getEmployeeHighestHourlyRate()’ uses ‘instanceof’.
* •
Fails 1 test because the ‘Employee’ class is not abstract.
* •
Fails 2 tests due to incorrect usage of the downcast operator in the
‘getEmployeeHighestHourlyRate()’.
* •
Fails 2 tests due to not respecting the expected format of the
‘HRWorker.toString()’ and ‘Company.toString()’ functions.
* •
Fails the same test related to the ‘Main.myCompany()’ function that the other
models failed, with a difference in the implementation: instead of guessing a
number, Bard’s code attempts to obtain the student ID by using
‘System.getEnv()’, resulting in a ‘NullPointerException’.101010See footnote 9
* •
Failed to import the ‘ArrayList’ class.
* •
Declared 3 attributes of the ‘Employee’ class as private, and tried to access
them directly in the ‘ITConsultant’ sub-class.
GPT-4 required 4 interactions to generate a solution that included all
mandatory elements. Issues found in that solution:
* •
Function ‘getEmployeeHighestHourlyRate()’ uses ‘instanceof’.
* •
Fails 1 unit test because it did not fully respect the format of the
‘Company.toString()’ function (a new line was missing).
* •
Fails the test related to the ‘Main.myCompany()’ function due to using an
uninformed number in the code.111111See footnote 9
In summary… These experiments show that these 3 LLMs were able to partially
solve the ‘IT Company’ assignment, although with some errors, both in terms of
basic syntax (e.g. missing library importations), as well as more complex
object-oriented programming errors (e.g. not declaring a class as abstract, or
using explicit type testing to decide program behaviors). This is similar to
what happened in the other 5 assignments, as described in Section 6.
### 5.2. Attempts to improve the LLM(s)’ solutions
After obtaining each model’s solutions for the “IT Company” assignment and
verifying that all of them were breaking one of our code quality rules — the
restriction not to use ‘instanceof’ —, we attempted to guide the models toward
better solutions, similar to how we expect a student would do after seeing the
AAT’s feedback, and to get a better grade. Note that these extra prompts are
not part of our main methodology and are not represented in Table 1.
These attempts to improve the models’ solutions were done using an ad-hoc
methodology where we identified problems with each model’s solution, and
attempted to give it a direct indication of the problem, to see what it would
do. The first extra prompt was the same for all models: ‘Change the code to
not use instanceof’. Afterward, we re-prompted based on what each model gave
us. The prompts for GPT-3.5 and Bard ended up being similar because the
respective intermediate solutions had similar problems from the object-
oriented perspective: they both were assuming that all sub-classes of
‘Employee’ have the ‘getValue()’ function, which is not coherent with the rest
of the respective solutions.
Besides the initial extra prompt, GPT-3.5 required two more prompts to reach
an acceptable solution: as a reply to the third prompt, it suggested adding
the ‘getValue()’ function to both the super-class and the sub-class with an
implementation that would return 0. In this case, 0 is being used as a flag to
indicate that the object should be ignored, which is coherent with the model’s
implementation of ‘getEmployeeHighestHourlyRate()’. Although that solution is
more OO, it is somewhat risky: such a design would be problematic if a
function ‘getEmployeeLowestHourlyRate()’ is needed at some point in the
future.
Bard had a behavior similar to that of GPT-3.5, although it required a fourth
prompt to reach a solution acceptable from the OOP perspective. Bard’s fourth
solution was equivalent to GPT-3.5’s third solution, thus having the
equivalent problems. Note that, although the OO solution is acceptable, Bard
implemented the ‘getValue()’ function incorrectly: it is returning the value
of the salary instead of the value of the hourly rate. Since we were focused
on getting Bard to solve the OO issue, we ignored this implementation detail.
Finally, GPT-4 only needed the extra initial prompt to reach an acceptable OO
solution. The model suggested adding an abstract version of ‘getHourlyRate()’
to the ‘Employee’ class, as well as adding a concrete version of that function
(returning -1, which is much safer than returning 0) to the ‘HRWorker’ class.
It also gave us the code for those changes, as well as a version of the
‘getEmployeeHighestHourlyRate()’ function that only considers objects for
which ‘getValue()’ returns more than 0.
In summary… Our experiments show that it was possible and easy to guide GPT-4
towards a fully working solution that respects the object-oriented best
practice of avoiding explicit type testing. Also, while the other models also
managed to reach acceptable, although risky, OO solutions, they required more
effort (GPT-3.5: 3 prompts; Bard: 4 prompts) and generated some minor errors
(e.g. Bard’s incorrect return of the salary attribute in the ‘getValue()’
function). This extra effort is coherent with the findings from other
researchers (Wermelinger, 2023).
## 6\. Overall Findings
This section presents the 3 models’ performances across the 6 assignments. As
Table 1 shows, all models needed multiple prompts and no model passed all the
tests.
GPT-4 generally had the best performance, requiring fewer interactions than
the other models, generating fewer compilation errors and, in general, having
better results in the unit tests. Bard always had equal or worse unit test
results than GPT-3.5, except in the “Home Banking” assignment, where it
surpassed both GPT models.
In terms of the Code Quality validation, we observed that GPT-4 yields the
best results: while GPT-3.5 and Bard respected the code quality rules in 3 out
of the 6 assignments, GPT-4 did it in 4 out of 6 assignments. Listing 2 shows
an example of the type of code that these models tend to generate: it (mostly)
works, but tends to decide behaviors using explicit type testing, via
‘instanceof’ and/or ‘getClass()’ . In summary, similarly to what previous
research found for GPT-3.5 (Cipriano and Alves, 2023a), the two newer models
still can’t avoid using ‘instanceof’.
It should be noted that, in the “Condominium Mgt.” assignment, GPT-4 managed
to pass the code quality, as well as 13/14 tests. Considering that the single
test failure was expected121212See footnote 9, we can consider that GPT
successfully solved one of the OOP assignments.
Table 1. Evaluation of the 3 models’ solutions for each assignment. G-3.5 is GPT-3.5. The values are related to the first solution that contained all mandatory classes and functions. For example, GPT-3.5 needed 5 prompts to generate all classes and functions for the first assignment, that solution had 4 compilation errors, the code quality failed and it passed 8/13 unit tests. | Nr. of prompts | Nr. of Compilation errors | Code quality Ok? | Tests passed
---|---|---|---|---
Assignment | G-3.5 | GPT-4 | Bard | G-3.5 | GPT-4 | Bard | G-3.5 | GPT-4 | Bard | G-3.5 | GPT-4 | Bard
Realtor Agency | 5 | 2 | 2 | 4 | 0 | 8 | No | Yes | Yes | | 8/13
---
(61.54%)
| 12/13
---
(92.31%)
| 7/13
---
(53.85%)
IT Company | 18 | 4 | 11 | 1 | 0 | 4 | No | No | No | | 9/13
---
(69.23%)
| 11/13
---
(84.62%)
| 7/13
---
(53.85%)
Home Banking | 7 | 2 | 10 | 1 | 0 | 14 | Yes | Yes | Yes | | 7/13
---
(53.85%)
| 7/13
---
(53.85%)
| 9/13
---
(69.23%)
Condominium Mgt. | 5 | 2 | 5 | 12 | 0 | 3 | Yes | Yes | Yes | | 11/14
---
(78.57%)
| 13/14
---
(92.86%)
| 10/14
---
(71.43%)
Home Cinema A/V | 5 | 2 | 6 | 3 | 0 | 2 | Yes | Yes | No | | 8/10
---
(80%)
| 9/10
---
(90%)
| 8/10
---
(80%)
Railway Co. Vehicles | 11 | 6 | 8 | 4 | 3 | 10 | No | No | No | | 11/16
---
(68.75%)
| 12/16
---
(75%)
| 10/16
---
(62.5%)
Average | 8,5 | 3 | 7 | 4,17 | 0,5 | 6,83 | 50% | 66.67% | 50% | 68,66% | 81,44% | 65,14%
Listing 2: A function generated by Bard for the “Condominium Mgt.” assignment
that fails our Code Quality validations. Besides using explicit type-testing
(i.e. uses instanceof) Bard also failed to implement the proper business rules
since the formula for the ‘Garage’ should multiply the area by 3 instead of 4
(see line 3).
⬇
1public int calculateCondominiumPayment() {
2 int value = baseValue;
3 value += area * 4;
4 if (this instanceof Apartment) {
5 value += ((Apartment) this).getFloorNumber() * 3;
6 value += ((Apartment) this).getNrRooms() * 2;
7 }
8 else if (this instanceof Store) {
9 value += ((Store) this).getNrFronts() * 2;
10 value += ((Store) this).getNrDoors() * 2;
11 }
12 else if (this instanceof Garage) {
13 value += Math.abs(((Garage) this).getNrFloor()) * 2;
14 }
15 return value;
16}
### 6.1. Problem categorization
This section presents a brief categorization of the problems observed across
assignments and models. After each problem, we indicate which models displayed
it at least in one assignment.
Issues related with syntax
* •
Failed to include library imports. [GPT-3.5, Bard]
* •
Created minor compilation errors (e.g. missing “}”). [GPT-3.5, Bard]
* •
Failed to declare a required package. [GPT-3.5]
* •
Generated incoherent code (e.g. called a function with less arguments than it
receives). [Bard]
Issues related with OOP concepts
* •
Used instanceof or getClass() to decide program behaviours. [all]
* •
Failed to declare a class as abstract. [all]
* •
Failed to correctly identify and implement business rules. [all]
* •
Did not respect the ‘toString()’ function’s required format. [all]
* •
Failed to apply inheritance best practices. [all]
* •
Suggested solutions with code duplication. [all]
* •
Accessed a super-class’s private fields from its sub-classes. [GPT-3.5, Bard]
* •
Declared constructors with problems (e.g. added extra arguments). [GPT-3.5]
* •
Failed to properly apply the downcast operator. [Bard]
* •
Declared abstract methods in the super-class but failed to implement them in
the sub-classes. [GPT-4]
## 7\. Recommendations for CS Educators
Students are likely to misuse tools like GPT-3.5 and Bard, which offer decent-
to-good OOP code generation for free. Superior models like GPT-4 are currently
available at a cost. With the evident progress from GPT-3.5 to GPT-4, observed
both in other research (Savelka et al., 2023a) as well as in our study, we
believe that educators should adapt to this emerging landscape.
### 7.1. Give more weight to code quality evaluations
For educators focusing on object-oriented programming, given our observations,
as well as previous research (Keuning et al., 2023), we recommend putting more
weight on evaluating items such as code quality, design patterns and other
similar aspects. The course’s focus should change from just producing
“functional code” to producing “functional and high-quality code”. The
assessment work can be scaled using an AAT with the capacity to evaluate
functional and quality requirements.
### 7.2. Use LLMs in your classes
Consider embracing LLMs in your classes. One option is the adoption of in-
class exercises where students have to 1) interact with LLMs to generate code,
and then, 2) evaluate the respective solutions. This process would be
supervised by the teacher, who would help the students analyze and critique
the LLMs’ output. This has the advantage of showing students that these models
should not be trusted blindly, and that they should inspect and test the
generated code, possibly improving their critical thinking, similarly to other
mistake-finding exercises (Naumova, 2023). This exercise can be enhanced by
having students validate their findings through an AAT using teacher-defined
tests or by creating their own unit tests.
### 7.3. Adopt project-based learning
If you currently rely solely on using small assignments for assessments, we
recommend that you consider including also project-based evaluations. Projects
will require more complex code and interactions to be implemented, which will
possibly require more complex interactions with the LLMs to obtain a fully
working solution that respects the object-oriented best practices. The
projects can be incremental, in order to allow students to grow and apply
their knowledge incrementally. For example: consider having a first project
delivery where the students do not need to use inheritance, followed by
another delivery where inheritance is needed, followed by another delivery
where some of the requirements change, and so on. Although LLMs’ performance
over larger assignments has not yet been measured, there are known benefits to
project-based learning (Mills et al., 2003; Shin, 2018).
## 8\. On the importance of citing your sources
One of the interesting features of Bard is its ability to display sources for
parts of the generated content. In the case of the “IT Company” assignment,
Bard displayed a single source: a GitHub repository with a Java
course.131313https:github.com/camilaabrantes/CursoJava
We performed a brief analysis of the repository and found some OOP examples
that are incorrect. For example, the repository contains a class called
‘Product’ with a ‘getPrice()’ method, which is extended by the sub-class
‘ImportedProduct’ that redefines the price calculation formula to take into
account a customs fee. Instead of overriding the ‘getPrice()’,
‘ImportedProduct’ declares a new method ‘totalPrice()’. This is clearly wrong,
in light of OOP best practices.
Although we are unsure of how much of this repository actually contributed to
Bard’s reply, we suspect that it is the use of data sources that have not been
curated that results in the generation of code that does not follow the OOP
best practices.
Note that GPT-3.5 and GPT-4 do not disclose any information regarding their
sources. Since the only publicly available information are high-level
descriptions of the training datasets (e.g. a filtered version of Common Crawl
141414https://commoncrawl.org/the-data, English Wikipedia, among others)
(Brown et al., 2020), it was not possible to perform this analysis for those
models.
## 9\. Limitations
The 3 experiments were done using different interaction interfaces. This may
have some kind of impact on the model’s replies, namely in terms of the number
of prompts required to obtain a solution.
These models’ output is not deterministic: repeating the experiments for each
assignment might have led us to different results.
We did not employ any Prompt Engineering (PE) techniques. The models were
initially inputted with the original assignments’ text. The prompts used to
obtain missing elements or to guide the models’ toward better solutions
consisted of very literal and direct requests. It is possible that the models
would have given better if some PE techniques were applied.
## 10\. Conclusions and Future Work
In this work, we present the performance of 3 LLMs when solving several OOP
assignments.
From our observations, GPT-4 demonstrated better performance than GPT-3.5,
needing fewer interactions and producing code with fewer issues. Additionally,
GPT-4 consistently passed more unit tests than GPT-3.5. In comparison, Bard
lagged behind both GPT versions in number of issues, passed tests, and the
number of interactions needed to guide it toward a reasonable solution.
If the 3 models were students, we would say that GPT-3.5 is a student that has
learned the basics of programming but struggles with object-oriented design
and programming concepts; GPT-4 is a more experienced student who, with
minimal guidance, is able to achieve good solutions; and Bard is a student of
a level slightly below GPT-3.5, with less autonomy than the others and more
issues with basic programming elements. In the end, all these students
struggled somehow and made some mistakes.
But can we blame students for making mistakes when they had a bad teacher?
Perhaps the challenges encountered by LLMs in adhering to OOP best practices
are not due to inherent reasoning limitations often associated with this
technology but rather attributed to suboptimal choices in their information
resources. We have verified the poor quality of the source referenced by Bard,
and while GPT-3.5 and GPT-4 do not disclose their sources, it is plausible
that they too might have relied on inadequate references. The significance of
LLMs disclosing their sources, and the more general issue of explaining their
reasoning has been a topic of repeated discussion (Arrieta et al., 2020). We
not only endorse such explanations but also emphasize the necessity of
curating high quality sources for training these models.
As for future work, we plan to investigate how these models handle larger OOP
assignments (e.g. 10-15 classes) following our recommendation to switch to
project-based learning. We also plan on experimenting new ways of presenting
OOP exercises to the students, with the goal of creating barriers to naive
‘copy-and-prompt’ approaches. One of the possible approaches is the creation
of diagram-based and video-based assignments.
## 11\. Data Availability
Since this work is based on real assignments that we use for student
assessment in our course, publishing the full assignment and/or interactions
dataset would require us to fully recreate all the assignments, ensuring we
use unpublished materials when evaluating our students. As such, we have
decided to release just one of the assignments: the IT Company Assignment.
This was achieved by creating the respective GitHub repository
151515https://github.com/drop-project-edu/itCompanyAssignment, as well as
publishing the respective interaction logs, for each LLM, in Zenodo (Cipriano
and Alves, 2023b).
###### Acknowledgements.
This research was funded by the Fundação para a Ciência e a Tecnologia under
Grant No.: UIDB/04111/2020 (COPELABS).
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|
# Atomic transitions for adaptive optics
Rui Yang School of Physics and Astronomy, Yunnan University, South Section,
East Outer Ring Road, Chenggong District, Kunming, 650500, China Department
of Physics and Astronomy, University of British Columbia, 6224 Agriculture
Road, Vancouver BC, V6T 1Z1,Canada Joschua Hellemeier Department of Physics
and Astronomy, University of British Columbia, 6224 Agriculture Road,
Vancouver BC, V6T 1Z1,Canada Paul Hickson Department of Physics and
Astronomy, University of British Columbia, 6224 Agriculture Road, Vancouver
BC, V6T 1Z1,Canada National Astronomical Observatories, Chinese Academy of
Sciences, A20 Datun Road, Chaoyang District, Beijing, China Corresponding
author<EMAIL_ADDRESS>
###### Abstract
Adaptive optics systems using sodium laser guide stars are widely employed at
major astronomical observatories. It is natural to ask whether other atomic
species might offer advantages. In this paper we review all abundant atoms and
ions in the upper atmosphere, including Na, Fe, Mg+, Si+, Ca+, K and also the
non-metallic species N, N+, O, H, considering their potential for adaptive
optics. Return fluxes for all transitions that can be excited using either one
or two wavelengths were computed. We also considered multi-wavelength
emission, comparing the performance of different transitions for polychromatic
laser guide star (PLGS) adaptive optics. We find that of all the mesospheric
metals, Na is the most suitable for both monochromatic and polychromatic laser
guide stars, providing about six times more return flux than the best
transitions in Fe. For high-altitude observatories, excitation at 330 nm in Na
should give the highest PLGS performance. Atomic O, N and N+ have strong
transitions and very high abundances in the mesosphere. This makes them
potential candidates for the generation of intense laser guide stars by
amplified spontaneous emission, if a suitable excitation process can be
demonstrated. Direct excitation by CW lasers is impractical as all transitions
from the ground state are beyond the atmospheric cutoff. Nevertheless, it may
be possible using high-power pulsed lasers.
††journal: josab
## 1 Introduction
Adaptive optics (AO) systems are employed by many ground-based telescopes to
compensate atmospheric turbulence, thereby greatly improving the resolution of
astronomical images [1]. Many of these systems employ laser guide star (LGS)
beacons, which allow sensing of atmospheric turbulence in any desired
direction, increasing sky coverage [2]. Large telescopes typically employ
resonant LGS systems in which transitions in mesospheric atom are excited,
typically the $D_{2}$ transition of neutral atomic sodium. Although other
chemical species have been considered [3, 4], sodium is preferred due to its
relatively-high abundance and strong resonance lines.
Despite the impressive success of AO systems employing sodium LGS, there are
several reasons to reconsider other atoms and ions. The performance of such AO
systems is limited by the return flux of the LGS, due to photon shot noise.
Advances have been made both in increasing laser power, and improving the
efficiency of sodium LGS [5, 6, 7, 8, 9]. For example, simulations indicate
that repumping atoms from the $D_{2b}$ ground state, in combination with
circular polarization, can increase the return flux by a factor of $3-4$ [6].
Nevertheless, any technique that might further increase the return flux would
be valuable.
Several new techniques that might potentially achieve high-intensity LGS are
discussed in [10] . One possible approach is amplified spontaneous emission
(ASE) [11]. In this idea, laser pumping of an abundant atmospheric atomic or
molecular species could produce a population inversion that would amplify
radiation generated by spontaneous emission. A critical requirement for ASE is
a suitable transition for which the optical depth $\tau$, defined as the
integral of the absorption coefficient along the propagation path, is negative
and has an absolute value that exceeds unity. This is a consequence of the
equation of radiative transfer, from which it follows that the gain in
intensity is proportional to $\exp(-\tau)$.
Another motivation for considering alternatives to sodium is the problem of
atmospheric tilt indeterminacy [12]. Backscattered photons follow the same
path as the transmitted beam, so the apparent position of the LGS is
independent of wavefront tilt induced by the atmosphere. A number of
techniques have been proposed to break this degeneracy (see, for example, [13,
14] and references therein). A promising approach is that of polychromatic
laser guide stars (PLGS) [15, 16]. In this technique, a laser is used to
generate emission at two or more widely-separated wavelengths. Because of
differential refraction, the backscattered light propagates along different
paths through the atmosphere. Thus one need not rely on natural guide stars to
sense the atmospheric tilt. The differential positions of the PLGS at
different wavelengths is small and must be measured with high precision.
Creating sufficiently-intense polychromatic emission is a challenge.
There is also a related problem of focus indeterminacy, as the sodium centroid
altitude varies on short timescales [17]. This makes it difficult to separate
atmospheric focus variations from other focus errors. This problem primarily
affects very-large telescopes as the wavefront error due to focus variations
increases in proportion to the square of the telescope aperture diameter [18].
The present paper examines atoms and ions in the upper atmosphere that might
be useful for AO. We consider the abundance and distribution of all relevant
species and compute the relative return flux for transitions that can be
excited by ground-based lasers. Our aim is to identify potential transitions
for monochromatic and PLGS, and to assess the feasibility of ASE.
## 2 Mesospheric atoms and parameters of interest
Meteoric ablation is the primary source of metallic species in the mesosphere
and lower thermosphere (MLT) region of the atmosphere, at altitudes between 80
and 120 km. In this altitude range, the atmospheric density decreases from
$10^{-5}$ to $10^{-8}$ kg m-3 and collision rates are in the range
$10^{3}-10^{5}$ s-1. This is also the coldest region of the atmosphere. At
mid-latitude sites, the temperature at the mesopause ranges from $178-192$ K,
at an altitude of $85-102$ km, depending on the season [19].
Typical values of the column density $N$, altitude of the centroid of the
density distribution, and the RMS width (vertical extent) for each species are
listed in Table 1. For sodium, we use recent data by Gardner [20] to update
earlier values listed by Happer [3]. The centroid altitude and RMS width for
non-metallic species were computed from the MSIS-90 atmospheric model [21].
These were evaluated at midnight, January 1 for latitude $+30^{\circ}$ and
longitude $0^{\circ}$.
The major meteoric species are Na, Fe, Mg and Si. There are also two less-
abundant species, Ca and K. These metals exist as layers of atoms between
about 80 and 105 km and occur primarily as atomic ions at higher altitudes.
Below 85 km they form compounds: oxides, hydroxides, and carbonates, which
polymerize into nanometer-sized meteoric smoke particles. Above 85 km,
photolysis of N2, O2 and H2O by extreme ultraviolet (EUV) radiation leads to
high concentrations of nitrogen, oxygen and hydrogen atoms. These atoms attack
metallic compounds such as hydroxides and oxides, reducing them and thereby
returning the metals to their atomic phase.
Table 1: Parameters of mesospheric atoms and ions Species | $N$ | Altitude | RMS width | Sources
---|---|---|---|---
| m-2 | km | km |
Na | $4.0$$\times 10^{13}$ | $91.5$ | $4.6$ | [20]
Fe | $10.2$$\times 10^{13}$ | $88.3$ | $4.5$ | [20]
Mg | $1$$\times 10^{13}$ | $89.4$ | $4.2$ | [23, 22]
Mg+ | $8$$\times 10^{13}$ | $94.6$ | $7.0$ | [23, 22]
Si+ | $4$$\times 10^{13}$ | $111.0$ | $10.1$ | [24]
Ca | $3.4$$\times 10^{11}$ | $90.5$ | $3.5$ | [20]
Ca+ | $7.2$$\times 10^{11}$ | $95.0$ | $3.6$ | [20]
K | $4.5$$\times 10^{11}$ | $91.0$ | $4.7$ | [20]
N | $1$$\times 10^{18}$ | $223.0$ | $60.2$ | [25, 21]
N+ | $1$$\times 10^{18}$ | - | - | [25]
O | $6.5$$\times 10^{21}$ | $110.3$ | $28.7$ | [26, 21]
H | $2.3$$\times 10^{18}$ | $95.6$ | $38.9$ | [27, 21]
In a typical AO application, a laser is used to excite atoms from the ground
state to an excited state. The atom then returns to the ground state via
spontaneous and stimulated emission. Only spontaneous emission contributes to
backscattered radiation since stimulated emission propagates in the same
direction as the laser radiation. The LGS return flux can be computed from the
equation of radiative transfer. For an optically-thin medium (optical depth
$\tau\ll 1$) the specific intensity $I_{\nu}$ (W m-2 Hz-1 sr-1) of the
backscattered radiation is given by [10]
$I_{\nu}(\nu)=\frac{h\nu}{4\pi}NA_{21}x\varphi(\nu)=\frac{2h\nu}{\lambda^{2}}N\sigma_{21}(\nu)x.$
(1)
Here $\lambda$ is the wavelength, $\nu=c/\lambda$ is the frequency, $c$ is the
speed of light in vacuum and $N$ is the column density of atoms of this
species. $A_{21}$ is the Einstein A coefficient for the transition and the
function $\varphi(\nu)$ is the mean line profile, normalized so that its
integral over frequency $\nu$ is unity. The dimensionless parameter $x$ is the
occupation fraction of the upper state, i.e. the equilibrium fraction of atoms
found in this state under continuous excitation by a laser having specific
energy density $\rho_{\nu}$, and $\sigma_{21}(\nu)$ is the cross section for
stimulated emission. The second equality follows from the relation between
stimulated and spontaneous emission, $B_{21}$ = $\lambda^{3}A_{21}/8\pi h$,
and the definition
$\sigma_{21}(\nu)\equiv\frac{h\nu}{c}B_{21}\varphi_{\nu}(\nu)=\frac{\lambda^{2}}{8\pi}A_{21}\varphi(\nu).$
(2)
Here the subscript 1 denotes the lower (ground) state, 2 denotes the upper
state, and $B_{21}$ is the Einstein B coefficient for stimulated emission.
For a CW laser having beam area $\mathcal{A}$ in the MLT, the return flux
$\Phi$, in units of photons s-1 m-2 is related to the intensity by
$\Phi=\frac{\mathcal{A}}{z^{2}}\int_{0}^{\infty}\frac{I_{\nu}(\nu)}{h\nu}d\nu,$
(3)
where $z$ is the line-of-sight distance to the LGS.
In the mesosphere, Doppler broadening is the dominant line-broadening process.
As the temperature varies within the MLT, the line profile is more precisely
written as
$\varphi(\nu)=\frac{1}{N}\int_{0}^{\infty}n(z)\varphi(\nu,z)dz,$ (4)
where $n(z)$ is the number density of the atomic species under consideration
at distance $z$ along the line of sight. In practice, the function
$\varphi(\nu)$ is well-represented by a Gaussian
$\varphi(\nu)=\frac{2}{w}\sqrt{\frac{\ln 2}{\pi}}e^{-4\ln
2(\nu-\nu_{0})^{2}/w^{2}},$ (5)
where $w$ is the line width at half maximum intensity. It depends on the
characteristic temperature $T$ in the MLT and the mass $m$ of the atomic
species,
$w=\frac{1}{\lambda_{0}}\left(\frac{8k_{B}T\ln 2}{m}\right)^{1/2}$ (6)
where $k_{B}$ is Boltzmann’s constant. Adopting $T=180$ K results in a width
for the sodium D line of 1.02 GHz.
The absorption cross section $\sigma_{12}(\nu)$ is given by
$\sigma_{12}(\nu)=\frac{g_{2}}{g_{1}}\sigma_{21}(\nu),$ (7)
where $g_{1}$ and $g_{2}$ are the statistical weights of the lower and upper
state, respectively. Tabulated values of cross sections
$\sigma_{21}\equiv\sigma_{21}(\nu_{0})$ are computed at the line center
$\nu_{0}$.
Substituting Eqs. (1) and (2) into Eq. (3), we obtain
$\displaystyle\Phi$ $\displaystyle=\frac{\mathcal{A}}{4\pi
z^{2}}NA_{21}\int_{0}^{\infty}x(\nu)\varphi(\nu)d\nu$
$\displaystyle\simeq\frac{\mathcal{A}}{4\pi
z^{2}}NA_{21}x(\nu_{0})\varphi(\nu_{0})\Delta\nu,$ (8)
where $\Delta\nu$ is the laser line width. The second equality follows because
the line width for a CW laser, and therefore the frequency range over which
$x$ is nonzero, is typically much smaller than the Doppler width. From this we
see that the atomic quantity that best characterizes the LGS return flux, is
$\varepsilon=NA_{21}x(\nu_{0}).$ (9)
We shall call $\varepsilon$ the _return flux coefficient_ for the transition.
It has units of photons s-1 m-2.
Also of interest is the dimensionless product $N\sigma_{21}$, which determines
the maximum possible optical depth. Specifically, $\tau$ is bounded by
$-N\sigma_{21}\leq\tau\leq N\sigma_{12}$ [10]. Therefore, ASE is not possible
if $N\sigma_{21}<1$.
Several other physical processes also play a role in the response of atoms to
laser excitation. Atomic collisions effectively reset atoms that have been
excited, either returning them to the ground state, or causing a transition to
a different excited state. The latter process makes the atom unavailable for
excitation by the laser. At the same time, collisions impart a velocity to the
atom, generally pushing it out of the velocity interval that can be excited by
the laser (the “velocity class”). In equilibrium this is balanced by
collisions that move atoms into the velocity class. Those atoms are in the
ground state as the collision energies, typically $\sim 0.02$ eV, are not
sufficient to excite the atom from the ground state. Therefore any individual
atom is effectively “reset” to the ground state on a time scale equal to the
mean time between collisions.
Collision rates vary by species and altitude. Rates for the species of
interest are calculated in Appendix A, for a range of altitudes.
In order to be useful for AO, a transition should have a high spontaneous
transition rate to the lower state (Einstein $A$ coefficient). Also, as most
atoms will be in the ground state, there should be a path to the upper state
from the ground state, with a high transition probability (Einstein $B$
coefficient). As the Einstein coefficients are related, it is sufficient to
consider transitions that have a large $A$ coefficient.
A second consideration is that the upper state should not have many
transitions to metastable states, which have low $A$ coefficients. In such a
situation, atoms can become trapped in those states, reducing the number of
atoms available for the desired transition. Ultimately, atoms trapped in
metastable states will be reset at the collision rate.
The performance of PLGS will critically depend on the wavelengths and return
flux of the available transitions. For amplitude $\theta$ of a Zernike mode
(except for piston), the ratio of the difference in amplitude $\Delta\theta$
at different wavelengths, is related to the ratio of refracting indexes by
[15]
$p\equiv\frac{\theta}{\Delta\theta}=\frac{n_{\lambda_{\text{obs}}}-1}{n_{\lambda_{1}}-n_{\lambda_{2}}},$
(10)
where $n_{\lambda_{1}}$ and $n_{\lambda_{2}}$ are the refractive indexes at
wavelengths of the measurement and $n_{\lambda_{obs}}$ is the refractive index
of the wavelength at which the amplitude of the Zernike mode is determined.
The dimensionless ratio $p$ is called the _penalty factor_. Smaller values of
$p$ correspond to greater PLGS performance. We take $\lambda_{\text{obs}}=500$
nm as the reference wavelength.
The impact of return flux and penalty factor can be captured in a single merit
function. The signal-to-noise ratio for the measurement of the atmospheric
wavefront tilt is proportional to the square root of the return flux and
inversely proportional to the penalty factor [15]. Therefore,
$\varepsilon^{1/2}/p$ is a measure of relative PLGS performance. To make this
dimensionless, we divide by the same quantity evaluated for a reference
system. We thus define the PLGS merit function
$q\equiv\frac{p_{0}}{p}\left(\frac{\varepsilon}{\varepsilon_{0}}\right)^{1/2},$
(11)
where $\varepsilon_{0}$ and $p_{0}$ are reference values. For these we use the
330.298/2208.370 nm transitions in Na I, which is the highest-performance
system employing single-photon excitation. In all cases, the return flux
coefficient used is that of the weaker line.
When discussing atomic transitions we use spectroscopic notation and refer to
Fe as Fe I, Fe+ as Fe II, etc. We refer to transitions having wavelengths
greater than 300 nm as “visible” transitions. Shorter-wavelengths are greatly
attenuated by the atmosphere so we refer to those as “vacuum-ultraviolet”
transitions. We refer to transitions having an Einstein $A$ coefficient
greater than $10^{6}$ s-1 as “strong transitions”, and “weak transitions”
otherwise. This distinction is somewhat arbitrary but serves to generally
separate permitted lines from forbidden lines. An excitation scheme that uses
two different transitions to reach an excited state will be called a “two-
step” process. One that uses two photons to excite a single transition, using
a virtual intermediate state, will be called a “two-photon” process.
Transition wavelengths described in the text and listed in the tables in the
following sections are in air for wavelengths greater than 185 nm and in
vacuum otherwise.
## 3 Analysis and Results
The data set employed for this analysis is the atomic database of the National
Institute of Standards and Technology (NIST) [28]. Complete data for all
transitions were downloaded and a computer program was written to apply
various criteria to identify transitions of interest for AO. We examined all
excited states that could be reached from the ground state by at most two
successive excitations using wavelengths greater than 300 nm. For species in
which no such transitions are possible, two-photon excitation was considered.
For every such state, all possible downward transitions were examined, and the
excitation fraction $x$ of the excited state was computed according to the
methods presented in Appendix B. A specific energy density of
$\rho_{\nu}=10^{-12}$ J m-1 Hz-1 was assumed for the laser excitation. This is
at the high end of range of energy densities that can be achieved with current
AO lasers. A complete listing of all such transitions that have $A\geq 10^{7}$
s-1 is available from the authors upon request. From this and the data of
Table 1, the return flux coefficient was computed. For PLGS, the penalty
factor $p$ and merit function $q$ were computed for all downward transitions.
Our results are shown in Tables 2 – 9. Table 2 lists all transitions, for all
species, that have a return flux coefficient $\varepsilon>10^{18}$ photons s-1
m-2. In this, and subsequent tables, $\lambda_{\text{ex}}$ refers to the
excitation wavelength(s) employed to reach the excited state, and $\lambda$ is
the wavelength of the downward transition. Unless otherwise specified, values
of $p$ are computed with respect to the smallest wavelength.
We now discuss individual atomic species in more detail.
### 3.1 Sodium
Sodium is most-often employed for the generation of LGS. Neutral sodium is
used as it is more abundant than ionized forms. Averaged over all seasons and
latitudes, the observed ratio of Na+ to Na is between 0.2 and 0.25 [29]. An
energy level diagram of neutral sodium, indicating the lifetimes and
wavelengths for selected transitions is shown in Fig. 1. For simplicity, fine-
structure splitting of the levels is not shown.
At visible wavelengths, Na I has four strong transitions from the ground
state, at 330.237, 330.298, 588.995 and 589.592 nm. The latter two form the
sodium D doublet. A fifth transition, at 388.390 nm, has a very-small rate
($A=6.95\times 10^{-4}$ s-1).
The transition between the 3S1/2 ground state and the 3P3/2 excited state, the
D2 line, has been of interest for adaptive optics since 1985 [2]. Laser-
induced excitation followed by spontaneous emission produces the backscattered
photons of the LGS. The LGS return flux is limited by the low abundance and
small absorption cross section, $N\sigma_{12}\simeq 3$%. In practice, the D2b
line, which involves only transitions between the $F=2$ lower and $F=1,2$ and
3 upper sub-levels, is used for AO. A CW laser can excite 3/4 of the available
atoms to the upper state, corresponding to the ratio $15/20$ of the number of
magnetic substates in the upper levels to the total. Current AO lasers are
sufficiently powerful that the excitation fraction closely approaches this
limit. However, only a fraction of the total number of sodium atoms are
available for excitation because of limited spectral overlap as the laser
linewidth is typically much smaller than the sodium linewidth. This limitation
might be overcome with further development of efficient broad-band modeless
pulsed lasers [30].
Figure 1: Low-lying energy levels of the sodium atom and transitions of
interest. Table 2. Transitions for a monochromatic LGS having $\lambda\geq
300$ nm, $\lambda_{\text{ex}}\geq 150$ nm and $\log(\varepsilon)>18.3$.
---
Species | $\lambda_{\text{ex}}$ | $\lambda$ | $\log(\varepsilon)$ | $\log(N\sigma_{21})$
| (nm) | (nm) | |
Na I | $588.995$ | $588.995$ | $21.20$ | $-1.50$
| $589.592$ | $589.592$ | $21.07$ | $-1.50$
| $330.237$ | $2205.647$ | $20.00$ | $-0.75$
| $330.237$ | $588.995$ | $19.83$ | $-1.50$
| $330.298$ | $2208.370$ | $19.82$ | $-0.75$
| $330.237$ | $1140.377$ | $19.82$ | $-1.19$
| $330.298$ | $588.995$ | $19.65$ | $-1.50$
| $330.298$ | $1140.377$ | $19.65$ | $-1.19$
| $330.237$ | $330.237$ | $19.61$ | $-3.61$
| $330.237$ | $589.592$ | $19.52$ | $-1.50$
| $330.237$ | $1138.144$ | $19.52$ | $-1.49$
| $330.298$ | $330.298$ | $19.44$ | $-3.61$
| $330.298$ | $589.592$ | $19.37$ | $-1.50$
| $330.298$ | $1138.144$ | $19.35$ | $-1.49$
| $285.281$ | $5426.880$ | $19.20$ | $-0.25$
| $285.281$ | $588.995$ | $19.18$ | $-1.50$
| $285.301$ | $5434.170$ | $18.98$ | $-0.25$
| $285.301$ | $588.995$ | $18.95$ | $-1.50$
| $285.281$ | $1140.377$ | $18.93$ | $-1.19$
| $285.281$ | $1074.645$ | $18.91$ | $-2.65$
| $285.281$ | $589.592$ | $18.87$ | $-1.50$
| $285.281$ | $616.075$ | $18.78$ | $-2.54$
| $268.034$ | $588.995$ | $18.71$ | $-1.50$
| $285.301$ | $1140.377$ | $18.71$ | $-1.19$
| $285.301$ | $1074.930$ | $18.68$ | $-2.65$
| $285.301$ | $589.592$ | $18.67$ | $-1.50$
| $285.281$ | $3414.261$ | $18.64$ | $-0.45$
| $285.281$ | $1138.144$ | $18.63$ | $-1.49$
| $268.034$ | $10\,807.960$ | $18.61$ | $0.14$
| $285.301$ | $616.075$ | $18.56$ | $-2.54$
| $285.281$ | $2205.647$ | $18.51$ | $-0.75$
| $285.281$ | $615.423$ | $18.49$ | $-2.84$
| $268.043$ | $588.995$ | $18.45$ | $-1.50$
| $285.301$ | $3414.261$ | $18.42$ | $-0.45$
| $268.034$ | $1140.377$ | $18.41$ | $-1.19$
| $285.301$ | $1138.144$ | $18.41$ | $-1.49$
| $268.034$ | $589.592$ | $18.40$ | $-1.50$
| $268.043$ | $10\,823.050$ | $18.36$ | $0.14$
| $259.387$ | $588.995$ | $18.34$ | $-1.50$
| $285.281$ | $3407.755$ | $18.34$ | $-0.75$
| $330.237$ | $9090.560$ | $18.33$ | $-0.57$
| $330.237$ | $819.482$ | $18.33$ | $-1.15$
| $268.034$ | $2440.182$ | $18.32$ | $-2.08$
| $268.034$ | $864.993$ | $18.31$ | $-3.44$
Fe I | $371.993$ | $371.993$ | $20.40$ | $-2.08$
| $385.991$ | $385.991$ | $18.72$ | $-2.26$
| $344.061$ | $344.061$ | $18.49$ | $-2.16$
| $367.991$ | $373.713$ | $18.18$ | $-2.14$
| $296.690$ | $373.486$ | $18.17$ | $-1.33$
| $216.677$ | $349.710$ | $18.13$ | $-2.42$
| $302.064$ | $302.064$ | $18.13$ | $-1.68$
| $261.871$ | $381.584$ | $18.08$ | $-1.21$
| $302.064$ | $382.042$ | $18.07$ | $-1.43$
| $344.061$ | $349.057$ | $18.05$ | $-2.59$
| $213.202$ | $340.746$ | $18.04$ | $-1.62$
| $437.593$ | $646.271$ | $18.04$ | $-3.59$
| $319.166$ | $516.749$ | $18.03$ | $-2.43$
Ca I | $422.673$ | $422.673$ | $19.72$ | $-3.34$
K I | $766.490$ | $766.490$ | $19.05$ | $-3.21$
| $769.896$ | $769.896$ | $18.92$ | $-3.21$
Mg I | $202.582$ | $1710.866$ | $18.15$ | $-1.55$
| $202.582$ | $1182.819$ | $18.15$ | $-1.63$
Quantitative results for single-photon excitation in Na I are shown in Table
2. In Table 3 we list all transitions in Na I that are available with two-step
excitation and have a return flux coefficient $\varepsilon>10^{20}$ photons
s-1 m-2. There are many, but even the strongest is two times weaker that the
sodium D2 line.
Of particular interest for PLGS are transitions from the 4P3/2 state [16].
This state may be populated directly from the ground state by absorption of a
330 nm photon. De-excitation occurs through two pathways. Approximately 29%
decay directly to the ground state, emitting a 330 nm photon and 70% decay to
the 4S1/2 state, emitting a 2206 nm photon. That state can decay to the ground
state via two pathways. About 67% decay via the 3P3/2 state, emitting 1140 and
589 nm photons. The remaining decay via the 3P1/2 state emitting 1138 and 590
nm photons. About 1.5% of the atoms in the 4P3/2 state return to the ground
state via the 3D5/2 state, emitting 9093, 819 and 589 nm photons.
Table 3. Na I transitions for a monochromatic LGS employing two-step
excitation having $\lambda\geq 300$ nm and $\log(\varepsilon)>20$.
---
$\lambda$ | $\lambda_{\text{ex}}$ | $\log(\varepsilon)$ | $\log(N\sigma_{21})$
(nm) | (nm) | |
$819.482$ | 588.995+819.482 | $21.00$ | $-1.15$
$818.326$ | 589.592+818.326 | $20.89$ | $-1.23$
$818.326$ | 588.995+819.479 | $20.68$ | $-1.23$
$589.592$ | 588.995+819.479 | $20.68$ | $-1.50$
$1140.377$ | 589.592+1138.144 | $20.31$ | $-1.19$
$588.995$ | 589.592+1138.144 | $20.31$ | $-1.50$
$1140.377$ | 588.995+1140.377 | $20.22$ | $-1.19$
$568.820$ | 588.995+568.820 | $20.19$ | $-2.26$
$819.479$ | 589.592+818.326 | $20.19$ | $-1.93$
$588.995$ | 589.592+818.326 | $20.19$ | $-1.50$
$588.995$ | 330.237+2337.915 | $20.18$ | $-1.50$
$568.263$ | 589.592+568.263 | $20.10$ | $-2.34$
$568.820$ | 330.237+2337.915 | $20.08$ | $-2.26$
$1138.144$ | 589.592+1138.144 | $20.01$ | $-1.49$
It is also possible to use a 330 nm photon to excite directly to the 4P1/2
state, rather than the 4P3/2 state, with a comparable transition rate. That
state decays directly to the ground state with a 29% probability, to the 4S1/2
state with a 70% probability, and to the 3D3/2 state with a 1.6% probability,
producing a 9140 nm photon. The 3D3/2 state decays to ground state via the
3P1/2 state, producing 818 and 590 nm photons, or via the 3P3/2 state,
producing 819 and 589 nm photons.
We searched for all possible PLGS line combinations employing single-photon
excitation in Na I, with the condition that they have a return flux greater
than $10^{17}$ photons s-1 m-2 (four orders of magnitude weaker than the Na D2
lines) and a penalty factor less than 50. The results are shown in Table 4.
The only possibilities are the two already mentioned.
Table 4. Na I transitions for a PLGS. All transitions having $q>0.4$ and
$\lambda_{\text{ex}}\geq 300$ nm are listed.
---
$\lambda_{\text{ex}}$ | $q$ | $p$ | $\lambda_{1}$ | $\log(\varepsilon_{1})$ | $\lambda_{2}$ | $\log(\varepsilon_{2})$
(nm) | | | (nm) | | (nm) |
$330.237$ | $1.00$ | $18.59$ | $330.237$ | $19.61$ | $2205.647$ | $20.00$
| $0.94$ | $19.74$ | $330.237$ | $19.61$ | $1140.377$ | $19.82$
| $0.84$ | $19.74$ | $330.237$ | $19.61$ | $1138.144$ | $19.52$
| $0.72$ | $25.86$ | $330.237$ | $19.61$ | $588.995$ | $19.83$
| $0.65$ | $25.84$ | $330.237$ | $19.61$ | $589.592$ | $19.52$
$330.298$ | $0.82$ | $18.60$ | $330.298$ | $19.44$ | $2208.370$ | $19.82$
| $0.77$ | $19.75$ | $330.298$ | $19.44$ | $1140.377$ | $19.65$
| $0.69$ | $19.75$ | $330.298$ | $19.44$ | $1138.144$ | $19.35$
| $0.59$ | $25.87$ | $330.298$ | $19.44$ | $588.995$ | $19.65$
| $0.54$ | $25.85$ | $330.298$ | $19.44$ | $589.592$ | $19.37$
Several transitions involving two-step excitation have been discussed in the
literature. The processes of 3S${}_{1/2}\to$ 3P${}_{3/2}\to$ 4D5/2 ($589+569$
nm) [15], 3S${}_{1/2}\to$ 3P${}_{3/2}\to$ 4S1/2 ($589+1140$ nm) [31] and
3S${}_{1/2}\to$ 3P${}_{3/2}\to$ 3D5/2 ($589+819.7$ nm) [32]. Excitation via
the $3P_{1/2}$ state is also possible but with somewhat lower transition
rates.
Atoms excited to the 3P levels can be further excited to the 4D, 4S or 3D
levels by absorption of visible or infrared light, provided that this is done
within a few ns. Sodium can also be excited from the ground state to the 4D5/2
level by absorption of two 578 nm photons. This is a non-linear process
involving a virtual intermediate state, requiring picosecond pulses and a
laser irradiance on the order of $10^{10}$ W m-2 [31].
All possible combinations employing two-step excitation in Na I are listed in
Table 5. The selection criteria is $q>0.4$. The strongest of these is
$589+569$ nm excitation producing lines at 569 and 2334 nm. These transitions
would provide a SNR that is 0.79 times the single-photon excitation
transitions shown in Table 4. That is roughly equivalent to a decrease to 65%
in LGS return flux.
Table 5. Na I transitions for a PLGS employing two-step excitation. All
transitions that have $q>0.4$ and $\lambda_{\text{ex}}\geq 300$ nm are listed.
The last line for each excitation scheme shows the return flux coefficient for
the first excitation wavelength.
---
$\lambda_{\text{ex}}$ | $q$ | $p$ | $\lambda_{1}$ | $\log(\varepsilon_{1})$ | $\lambda_{2}$ | $\log(\varepsilon_{2})$
(nm) | | | (nm) | | (nm) |
588.995 | $0.79$ | $18.55$ | $330.237$ | $19.41$ | $2337.915$ | $19.95$
\+ 568.820 | $0.79$ | $18.59$ | $330.237$ | $19.41$ | $2205.647$ | $19.79$
| $0.75$ | $19.74$ | $330.237$ | $19.41$ | $1140.377$ | $19.62$
| $0.67$ | $19.74$ | $330.237$ | $19.41$ | $1138.144$ | $19.32$
| $0.55$ | $26.69$ | $330.237$ | $19.41$ | $568.820$ | $20.19$
| $0.51$ | $25.84$ | $330.237$ | $19.41$ | $589.592$ | $19.32$
| $0.45$ | $60.81$ | $568.820$ | $20.19$ | $2337.915$ | $19.95$
| | | $588.995$ | $20.78$ | |
589.592 | $0.71$ | $18.56$ | $330.298$ | $19.32$ | $2334.843$ | $19.86$
\+ 568.263 | $0.71$ | $18.60$ | $330.298$ | $19.32$ | $2208.370$ | $19.70$
| $0.67$ | $19.75$ | $330.298$ | $19.32$ | $1140.377$ | $19.60$
| $0.66$ | $19.75$ | $330.298$ | $19.32$ | $1138.144$ | $19.30$
| $0.59$ | $18.56$ | $330.298$ | $19.32$ | $2337.896$ | $19.16$
| $0.51$ | $25.87$ | $330.298$ | $19.32$ | $588.995$ | $19.82$
| $0.49$ | $18.60$ | $330.298$ | $19.32$ | $2205.647$ | $19.00$
| $0.49$ | $26.71$ | $330.298$ | $19.32$ | $568.819$ | $19.40$
| $0.49$ | $26.73$ | $330.298$ | $19.32$ | $568.263$ | $20.10$
| $0.41$ | $60.69$ | $568.263$ | $20.10$ | $2334.843$ | $19.86$
| | | $589.592$ | $20.65$ | |
588.995 | $0.57$ | $18.56$ | $330.298$ | $19.13$ | $2334.843$ | $19.67$
\+ 568.819 | $0.57$ | $18.60$ | $330.298$ | $19.13$ | $2208.370$ | $19.51$
| $0.54$ | $19.75$ | $330.298$ | $19.13$ | $1140.377$ | $19.41$
| $0.53$ | $19.75$ | $330.298$ | $19.13$ | $1138.144$ | $19.11$
| $0.48$ | $18.56$ | $330.298$ | $19.13$ | $2337.896$ | $18.97$
| $0.41$ | $25.85$ | $330.298$ | $19.13$ | $589.592$ | $19.98$
| $0.40$ | $18.60$ | $330.298$ | $19.13$ | $2205.647$ | $18.81$
| $0.40$ | $26.71$ | $330.298$ | $19.13$ | $568.819$ | $19.21$
| $0.40$ | $26.73$ | $330.298$ | $19.13$ | $568.263$ | $19.91$
| | | $588.995$ | $20.96$ | |
330.237 | $0.51$ | $18.56$ | $330.298$ | $19.03$ | $2334.843$ | $19.57$
\+ 2337.896 | $0.51$ | $18.60$ | $330.298$ | $19.03$ | $2208.370$ | $19.41$
| $0.51$ | $18.60$ | $330.298$ | $19.03$ | $2205.647$ | $19.64$
| $0.48$ | $19.75$ | $330.298$ | $19.03$ | $1140.377$ | $19.66$
| $0.48$ | $19.75$ | $330.298$ | $19.03$ | $1138.144$ | $19.36$
| $0.43$ | $18.56$ | $330.298$ | $19.03$ | $2337.896$ | $18.87$
| | | $330.237$ | $19.25$ | |
588.995 | $0.50$ | $126.58$ | $589.592$ | $20.68$ | $818.326$ | $20.68$
\+ 819.479 | $0.23$ | $126.21$ | $589.592$ | $20.68$ | $819.479$ | $19.99$
| | | $588.995$ | $20.91$ | |
589.592 | $0.50$ | $83.39$ | $588.995$ | $20.31$ | $1140.377$ | $20.31$
\+ 1138.144 | $0.35$ | $83.50$ | $588.995$ | $20.31$ | $1138.144$ | $20.01$
| | | $589.592$ | $20.87$ | |
588.995 | $0.49$ | $18.27$ | $330.237$ | $18.99$ | $5426.880$ | $19.01$
\+ 498.281 | $0.49$ | $18.28$ | $330.237$ | $18.99$ | $5013.920$ | $19.30$
| $0.49$ | $18.59$ | $330.237$ | $18.99$ | $2205.647$ | $19.37$
| $0.47$ | $19.09$ | $330.237$ | $18.99$ | $1477.974$ | $19.49$
| $0.46$ | $19.74$ | $330.237$ | $18.99$ | $1140.377$ | $19.30$
| $0.46$ | $19.74$ | $330.237$ | $18.99$ | $1138.144$ | $18.99$
| | | $588.995$ | $20.78$ | |
589.592 | $0.47$ | $18.37$ | $330.237$ | $18.94$ | $3414.261$ | $19.49$
\+ 615.423 | $0.47$ | $18.37$ | $330.237$ | $18.94$ | $3407.755$ | $19.19$
| $0.46$ | $18.59$ | $330.237$ | $18.94$ | $2208.370$ | $19.03$
| $0.46$ | $18.59$ | $330.237$ | $18.94$ | $2205.647$ | $19.33$
| $0.44$ | $19.74$ | $330.237$ | $18.94$ | $1140.377$ | $19.33$
| $0.44$ | $19.74$ | $330.237$ | $18.94$ | $1138.144$ | $19.03$
| | | $589.592$ | $20.86$ | |
588.995 | $0.44$ | $18.37$ | $330.237$ | $18.88$ | $3414.261$ | $19.42$
\+ 616.075 | $0.44$ | $18.37$ | $330.237$ | $18.88$ | $3407.755$ | $19.12$
| $0.43$ | $18.59$ | $330.237$ | $18.88$ | $2208.370$ | $18.96$
| $0.43$ | $18.59$ | $330.237$ | $18.88$ | $2205.647$ | $19.26$
| $0.41$ | $19.74$ | $330.237$ | $18.88$ | $1140.377$ | $19.26$
| $0.40$ | $19.74$ | $330.237$ | $18.88$ | $1138.144$ | $18.96$
| | | $588.995$ | $21.03$ | |
589.592 | $0.44$ | $18.28$ | $330.298$ | $18.88$ | $5434.170$ | $18.92$
\+ 497.854 | $0.44$ | $18.29$ | $330.298$ | $18.88$ | $5007.670$ | $19.21$
| $0.43$ | $18.60$ | $330.298$ | $18.88$ | $2208.370$ | $19.26$
| $0.42$ | $19.10$ | $330.298$ | $18.88$ | $1476.749$ | $19.40$
| $0.40$ | $19.75$ | $330.298$ | $18.88$ | $1140.377$ | $19.28$
| $0.40$ | $19.75$ | $330.298$ | $18.88$ | $1138.144$ | $18.98$
| | | $589.592$ | $20.66$ | |
330.237 | $0.40$ | $60.81$ | $568.820$ | $20.08$ | $2337.915$ | $19.85$
\+ 2337.915 | | | $330.237$ | $19.26$ | |
### 3.2 Iron
The average iron abundances near the density peak at 90 km is two orders of
magnitude higher than that of most other mesospheric metals, and twice that of
sodium. Although its cross section is small, the key product $N\sigma_{21}$ is
still larger than that of all other metallic elements except sodium. The
abundance ratio Fe${}^{+}/$Fe is approximately 0.2 above 90 km [33].
Atomic iron has the largest number of spectral lines of all mesospheric
species. The iron-group elements have complex transitions with thousands of
lines from the vacuum ultraviolet to the infrared. An energy-level diagram for
Fe I is shown in Fig. 2. The ground state is split into three fine-structure
levels. Only the lowest-energy a5D4 state is populated in thermal equilibrium
in the mesosphere. There are five strong visible-wavelength transitions from
this ground state, at 344.061, 367.991, 371.993, 382.444 and 385.991 nm. Most
of these can decay to the other two ground states. The z5D3 state, reached by
the 372 nm transition from the ground state, has a moderately-strong
($A=1.15\times 10^{6}$ s-1) downward transition to the metastable a5F4 state.
Figure 2: Selected states and transitions of Fe I.
The strongest transition in Fe I using single-photon excitation has a return
flux coefficient of $2.51\times 10^{20}$ photons s-1 m-2 (Table 2). This is
six times weaker than the sodium D2 line. Two-step excitation (Table 6)
provides more possibilities, but these are all weaker by at least an order of
magnitude.
Fe I has combinations of transitions which have promising wavelengths and
penalty factors for PLGS. However, for one-step and two-step excitation, $q$
does not exceed $q>0.18$ for any combination of wavelengths (Tables 7 and 8).
Fe II has no transitions from the ground state for wavelengths greater than
260 nm. This rules out excitation by ground-based lasers unless two-photon
excitation is used. That would require pulsed lasers with very high
irradiance.
Table 6. Fe I transitions for a monochromatic LGS employing two-step
excitation. All transitions that have $\log(\varepsilon)>19$ and
$\lambda_{\text{ex}}\geq 300$ nm are listed.
---
$\lambda$ | $\lambda_{\text{ex}}$ | $\log(\varepsilon)$ | $\log(N\sigma_{21})$
(nm) | (nm) | |
355.492 | 437.593+355.492 | $19.32$ | $-1.21$
314.399 | 385.991+314.399 | $19.31$ | $-1.73$
381.764 | 371.993+381.764 | $19.29$ | $-2.37$
380.198 | 371.993+380.198 | $19.08$ | $-2.70$
Table 7. Fe I transitions for a PLGS employing one-step excitation. All
transitions that have $q>0.1$ and $\lambda_{\text{ex}}\geq 300$ nm are listed.
---
$\lambda_{\text{ex}}$ | $q$ | $p$ | $\lambda_{1}$ | $\log(\varepsilon_{1})$ | $\lambda_{2}$ | $\log(\varepsilon_{2})$
(nm) | | | (nm) | | (nm) |
319.323 | 0.10 | 26.39 | 319.323 | $17.90$ | 517.160 | $17.90$
### 3.3 Magnesium
Magnesium is one of the more abundant meteoritic constituents, 9.6% by mass,
so that meteoric ablation should inject large quantities of this metal into
the MLT region [22]. Compared to other meteoric metals, magnesium has the
largest ionization fraction, with a ratio Mg+/Mg in the range 4 to 12 [22].
Mg I has six strong transitions from the ground state, all in the vacuum
ultraviolet. The strongest of these goes to the 31P1 state, with a wavelength
of 285.213 nm and a rate ($A_{21}=4.91\times 10^{8}$ s-1) that is an order of
magnitude greater than that of the sodium D lines. The only other downward
transitions from this upper state are forbidden lines that have negligible
rates. In vacuum, this would be a strong candidate for LGS, however, the
wavelength is beyond the atmospheric cutoff.
Table 8. Fe I transitions for a PLGS employing two-step excitation. All
transitions that have a de-excitation $q>0.1$ and $\lambda_{\text{ex}}\geq
300$ nm are listed. In brackets in the last line for each excitation scheme
the return flux coefficient for the first excitation wavelength is shown.
---
$\lambda_{\text{ex}}$ | $q$ | $p$ | $\lambda_{1}$ | $\log(\varepsilon_{1})$ | $\lambda_{2}$ | $\log(\varepsilon_{2})$
(nm) | | | (nm) | | (nm) |
437.593 | $0.12$ | $25.41$ | $361.207$ | $18.17$ | $973.857$ | $18.01$
\+ 361.207 | $0.11$ | $27.33$ | $371.993$ | $18.85$ | $973.857$ | $18.01$
| | | $437.593$ | $16.06$ | |
385.991 | $0.12$ | $40.78$ | $371.993$ | $18.49$ | $561.564$ | $18.49$
\+ 532.418 | $0.10$ | $44.57$ | $371.993$ | $18.49$ | $532.418$ | $18.38$
| | | $385.991$ | $18.18$ | |
Figure 3: Selected states and transitions of Mg II.
Mg II has two transitions from the ground state that have rates greater than
$10^{8}$ s-1. These go to the 32P1/2 and 32P3/2 states, with wavelengths of
280.271 and 279.553 nm respectively. Both of these upper states return
directly to the ground state, with no other significant transitions. Given the
high abundance of Mg+ (twice that of Na), these would be very good candidates
for LGS, but they are beyond the atmospheric cutoff.
Mg II does have strong emission lines in the visible region [34] (Fig.3).
However, populating these levels from the ground state requires photons having
wavelengths of 124 nm or less. This probably rules out Mg+ for ground-based
adaptive optics.
### 3.4 Silicon
Silicon is one of the most abundant elements in cosmic dust, comprising around
11% by mass, and meteoric ablation injects a significant amount of Si into the
atmosphere [24]. However, neutral Si is oxidized to SiO very rapidly by
reaction with O2 at the altitude of maximum ablation, around 90 km. Si+, SiO,
and Si(OH)4 are predicted to be the major silicon species above 80 km. Below
97 km the dominant sink is Si(OH)4. The column density of Si+ varies
seasonally by an order of magnitude and peaks at $4.0\times 10^{9}$ cm-2
during the summer [24].
Figure 4: Selected states and transitions of Si II.
Singly-ionized Silicon has several strong visible-light transitions, as shown
in Fig. 4 [35]. However, excitation from the ground state requires UV photons.
The ground state is a doublet (32P1/2, 32P3/2), separated by 0.0356 eV. This
is comparable to the mean particle energy of $3kT/2\simeq 0.023$ eV, so both
states will be populated. Transitions can occur from the lower ground state to
the 4P1/2 state, followed by a 637 nm visible-light transition, but this
requires a 123-nm photon. The lowest-energy transitions from either ground
state to the 4P levels requires a 123-nm photon. There is a transition from
the ground state to the 32P3/2 state, but this is quite weak
($A_{21}=2.54\times 10^{6}$ s-1). This likely rules out SI II as a viable
candidate for AO, and indeed no transitions met our thresholds for inclusion.
### 3.5 Calcium
Lidar measurements of calcium showed a peak density of $2\times 10^{7}$ atoms
m-3, about 200 times lower than the typical sodium atom density and 400 times
lower than that of iron [36]. The annual average Ca+/Ca column density ratio
is 2.4, the second largest ratio after magnesium. In comparison, the ratios of
Na+/Na and Fe+/Fe are only about 0.2 [37]. Ca+ is the dominant form above 90
km altitude, peaking near 105 km.
Ca I has a strong transition from the ground state to the 31P1 state with a
rate of $2.18\times 10^{8}$ s-1 and a wavelength of 422.673 nm. The upper
state returns directly to the ground state, with no other transitions. The
only problem here is the low abundance. As can be seen in Table 2, the return
flux coefficient is about 30 times smaller than that of the sodium D2 line.
Table 9. Ca I transitions for a monochromatic LGS employing two-step
excitation having wavelength $\lambda\geq 300$ nm and return-flux coefficient
$\log(\varepsilon)>19$.
---
$\lambda$ | $\lambda_{\text{ex}}$ | $\log(\varepsilon)$ | $\log(N\sigma_{21})$
(nm) | (nm) | |
$585.745$ | 422.673+585.745 | 19.07 | -3.43
$430.774$ | 657.278+430.774 | 19.01 | -3.35
Fig. 5 shows the relevant energy levels and transitions of Ca+ [38].
Excitation from the ground state to the 4P1/2 and 4P3/2 levels is possible
using 397 nm and 393 nm photons, respectively. These states decay quickly to
the ground state and also, with 7% probability, to the 3D3/2 and 3D5/2 states,
emitting 866 nm, 850 nm and 854 nm photons. These states are metastable, so
most of the atoms excited from the ground state using a UV laser will become
trapped in these states. No lines of Ca II met the criteria for inclusion in
Table 2.
Figure 5: Selected states and transitions of Ca II.
### 3.6 Potassium
Potassium has relatively low abundance in the upper atmosphere, having a
column density that is about two orders of magnitude less than that of Na. For
neutral potassium atoms, many transitions covering the visible light range
have high transition probabilities. The relevant energy levels and transitions
are shown as Fig. 6. Its persistent lines are 769.896, 766.490, 404.414 and
404.721 nm. Other strong lines having slightly smaller transition
probabilities occur at higher energy levels. Direct transitions to and from
the ground state are possible at 766.490 and 769.896 nm. These have cross
sections that are comparable to that of the sodium D2 line. However, the
relatively low column density of potassium results in return flux coefficients
that are two orders of magnitude smaller.
Figure 6: Selected states and transitions of K I. Figure 7: Energy level
diagram of N I (left) and N II (right).
Potassium can also be excited directly from the ground state to the 5P1/2 and
5P3/2 levels. These have a path back to the ground state via the 3D states and
4P states, producing strong lines at 3139.265, 3160.163, 1169.024 and 1176.967
nm, in addition to the 766.490 and 769.896 nm doublet. This would be an
attractive system for PLGS if it was not for the relatively low abundance.
### 3.7 Nitrogen
In the mesosphere, nitrogen molecules are ionized and dissociated by energetic
electrons and protons, producing N${}_{2}^{+}$ by direct ionization and N+ by
dissociative ionization. Secondary electrons associated with ionization
produce N in both the ground and 2D states. At 80 km atomic nitrogen has a
peak density of $2\times 10^{8}$ cm-3, falling to $10^{6}$ cm-3 at 70 km [25].
The estimated column density of atomic nitrogen is $\sim 1\times 10^{14}$
cm-2.
The most relevant energy levels and transitions of atomic NI and singly-
ionized NII are shown in Fig.7. It can be seen that NII has many more strong
emission lines in the visible range than does NI. Furthermore, nitrogen-
discharge experiments [39] show that the optical emission spectrum in the
visible and UV regions, is quite complex with multiple-peaks and many blended
lines.
Direct transitions from the ground state require vacuum-ultraviolet photons,
even using two-photon excitation. This likely rules out nitrogen for ground-
based AO systems.
### 3.8 Oxygen
Atomic oxygen (O) is a fundamental component in chemical aeronomy of Earth’s
MLT region extending from approximately 50 km to over 100 km altitude.
Primarily, O is generated through photolysis of molecular oxygen by UV
radiation. The peak atomic oxygen concentrations, $\sim 6.0-6.5\times 10^{17}$
m-3, are found at an altitude of approximately 95 km [26]. Assuming an atomic
oxygen column width of 10 km in the mesosphere, the estimated column density
of atomic oxygen is $\sim 6.0-6.5\times 10^{21}$m-2.
Energy level diagrams for O and O+ are shown in Fig. 8. The triplet at $\sim
130$ nm has been observed in the cometary spectra from above the Earth’s
atmosphere [40]. These lines arise when O atoms in the ground state are
excited to the 5S1 state by solar photons. In the visible spectrum region,
several astrophysically-important forbidden transitions exist. For O, these
are the red doublet at 630.030 nm and 636.377 nm, arising from transitions
between the 2D2 and the 2P2 and the 2P1 levels respectively, and the green
line at 557.734 nm from the 2D2 and 2S0 levels. These are magnetic dipole and
electric quadrupole transitions respectively. These forbidden oxygen emission
lines likely arise from atoms produced directly in the excited 2S0 or 2D2
states by photo-dissociation of parent molecules such as H2O, CO and CO2. In
O+, the doublet at 372.602 and 372.882 nm arises from transitions from the
2D3/2 and 2D5/2 levels to the 2S3/2 ground state. As with nitrogen,
$N\sigma_{21}$ is high for permitted transitions in oxygen, but low for the
forbidden transitions.
In O, strong visible lines arise from transitions between the 3P and 3S
levels, and also the 5S1 level. Visible lines can also occur from states above
the 5S1 energy level. Each line is a doublet or triplet due to hyperfine
structure. However, all these lines require vacuum-ultraviolet photons for
excitation from the ground state.
In O+, strong visible lines result from transitions from the 3P and 3D levels
to the 3P levels, and also the 3S1/2 and 3P3/2 levels. In this case there are
no direct transitions from the ground state to the upper states involved, and
transitions to the lower states require photons of wavelength 54 nm or less.
Figure 8: Strong resonance transitions in O I (left) and O II (right).
### 3.9 Hydrogen
Atomic hydrogen (H) is also a fundamental component in the photochemistry and
energy balance of the mesopause region between approximately 80 and 100 km
altitude. H is generated primarily by photolysis of water vapor and
participates in a highly-exothermic reaction with ozone. During the day,
atomic hydrogen concentrations peak in excess of $2.25\times 10^{14}$ m-3 at
higher latitudes near 85 km [27]. The estimated column density of atomic
hydrogen is $\sim 2\times 10^{18}$ m-2.
Hydrogen and hydrogen-like ions have familiar Lyman, Balmer, and Paschen
spectral series. Hydrogen emission spectral lines in the visible range include
Balmer lines at 410.173, 434.047, 486.134, and 656.283 nm wavelengths,
corresponding to transitions to the $n=2$ level. All energy levels for a given
principal quantum number $n$ are degenerate. Transitions from the ground state
require vacuum-ultraviolet photons having wavelengths of 122 nm or less.
### 3.10 Carbon
Meteoric ablation injects a significant amount of carbon into the mesosphere.
However, atomic carbon is a very short-lived species. It is stabilized in
various multi-atomic structures having different molecular configurations. The
carbon monoxide (CO) mixing ratio increases steeply with altitude in the
mesosphere and in the lower part of the thermosphere is a result of the
photolysis of carbon dioxide (CO2) [41]. Additionally, as the chemical
lifetime of CO is much longer than the characteristic dynamical times, this
gas is an excellent tracer of dynamics due to its strong 4.7 $\mu$m
transition.
The most common oxidation state of carbon in inorganic compounds is $+4$,
while $+2$ is found in carbon monoxide. The four outer electrons of carbon are
valence electrons, so its ionization energies are much higher than those of
most other elements. Short wavelength UV photons are needed to excite the
atoms from the ground state, making excitation using a ground-based laser
unfeasible. Also, there is little information about the detailed distribution
and abundance of carbon and their ions in mesosphere, so carbon and related
species are not considered further here.
## 4 Discussion
It can be seen from Table 2, that the sodium $D_{2}$ line provides the highest
return flux of all possible transitions, including two-step excitation
schemes. This is due to its relatively-large natural abundance and large cross
section compared to other metal atoms. Mg+ and Si+ have stronger lines, but
lack an effective scheme for excitation from the ground state. Although the
column density of Fe is twice that of sodium, its lower cross-section results
in a return flux that is one to two orders of magnitude smaller.
In contrast, the Mg+ lines at 280.271 and 279.553 nm have transition rates
that are an order of magnitude greater than that of the sodium D lines, and
the column density of Mg+ is twice that of sodium. These would be very strong
lines if they were not beyond the atmospheric cutoff. Similarly, Si+ has
strong transitions, and a column density comparable to that of Na, but vacuum-
ultraviolet excitation is required.
Potassium, like Ca+, has a column density that is smaller than that of sodium
by two orders of magnitude, so its return flux is small.
As with monochromatic LGS, Na yields the highest performance for PLGS from
both one-step and two-step excitation. According to our results, the next-
ranked element, Fe, provides at most 10% of the SNR of the best-suited
transitions in Na. If atmospheric absorption can be compensated by
sufficiently-high laser power the one-step excitation at 330 nm yields a
higher SNR than the two-step excitation at $569+589$ nm. At an altitude of
4200 m, the atmospheric transmission at 330 nm is about 60% of the
transmission at 600 nm. The most-promising combinations of transitions for
PLGS include de-excitations at 330 nm, however atmospheric absorption will
decrease the return flux at 330 nm. For two-step excitation, an alternative
approach could be to combine the return flux from the D1 and D2 line at 589
nm. For an excitation at $589+569$ nm, the photons at 589 nm can only be used
if it was ensured that the photons were emitted from fully-coincident laser
spots in the mesosphere. Even though the two-step excitation adds wavelengths
at about 3410 and 2338 nm, the PLGS performance does not exceed that of the
one-step excitation, since the penalty factor does not decrease significantly
for infrared wavelengths.
Our analysis used statistical weights provided by NIST, which give the number
of hyperfine structure states. Magnetic sub-states arising from Zeeman
splitting in an external magnetic field are not considered. This is a
limitation to our approach. Excitation with optimized laser polarization,
which can provide a greater return flux, can be assessed by employing a more-
detailed treatment of single transitions using Bloch-equations. Nevertheless,
we believe that the simple approach that we employ is useful for a first
assessment of potential transitions.
For a species in which essentially all atoms are in the ground state, the
optical depth at the line center is equal to the product of column density,
$\tau_{0}=N\sigma_{12}(\nu_{0})=g_{2}N\sigma_{21}(\nu_{0})/g_{1}$. From the
values listed in Table C1, we see that all transitions of mesospheric metals
have optical depth less than 0.2, so the atmosphere is optically thin
($\tau<1$). In contrast, the abundant species of N I and O I have $\tau_{0}\gg
1$ for most permitted transitions, which means that the atmosphere is opaque
at those wavelengths.
The optical depth for stimulated emission is closely related to the optical
depth for absorption. In a fully-excited medium, $\tau\sim-\sigma_{21}N$.
Significant amplification by stimulated emission can only occur in a medium
for which $-\tau\gg 1$. To achieve an optical depth of order unity,
$N_{2}A_{21}\geq 10^{23}$ m-2s-1 is required. For a typical transition rate,
$A_{21}\sim 10^{7}$ s-1, the column density of the medium must therefore
exceed $10^{16}$ m-2 to achieve ASE. This would appear to rule out all
metallic species in the mesosphere.
An interesting new development is the use of high-powered lasers to create a
plasma in the atmosphere. Rairoux et al. [42] have demonstrated ASE in
backscattered fluorescence of nitrogen from filaments generated by intense
ultra-fast Ti:sapphire laser pulses propagating over a distance of up to
several km. As it propagates along the filament, the backscattered radiation
is amplified by a population inversion resulting from direct ionization. The
laser excitation leads to plasma formation and the consequence is a white-
light laser pulse. Backscattered fluorescence from N2 molecules and ions shows
the exponential increase with increasing filament length expected for ASE. It
has the potential to generate very bright LGS, but a terawatt femtosecond
Ti:sapphire laser system is required to produce the $10^{17}$ W/m2 intensity
that is necessary to generate non-linear effects and plasma. Nevertheless, the
potential of a white-light N2 ASE for adaptive optics might well be realized
as laser technology continues to improve.
Other factors, in addition to LGS return flux, also affect AO performance.
With large-aperture telescopes, LGS appear elongated for subapertures of the
pupil that are offset from the axis of the laser launch telescope. This
increases noise and reduces the wavefront-sensing accuracy. In this regard, an
atomic species that has a high centroid altitude and small vertical extent
would be favored. In practice, the altitude is limited by the rapidly
decreasing atmospheric density and there is not a large difference between the
atomic species listed in Table 1. If an LGS can be created by a high-power
pulsed laser, an important consideration will be to limit the extent of the
emitting region along the line of sight.
## 5 Summary and conclusions
We have reviewed transitions of atoms and ions in the upper atmosphere of
interest for AO. Besides Na, which is used extensively by current systems, Fe,
Mg, Si, Ca, K all have potential uses. Iron has the highest abundance and the
largest number of transitions. However, its return flux is less than that of
sodium due to lower cross sections.
Our results confirm the general presumption of the LGS community that Fe is
not as well-suited as Na for PLGS. However, we find that one-step excitation
at 330 nm may be more suitable than two-step excitation at $569+589$ nm.
Detailed simulations, using Bloch equations and including the effects of
Zeeman splitting, would be a useful next step to evaluate the potential of
these excitation schemes.
Mg+ and Si+ have high abundance and several very-strong visible-light
transitions. However they require vacuum ultraviolet photons for excitation
that appear to be implausible even with two-photon excitation.
Amplified spontaneous emission using metallic species appears to be unfeasible
because of the small optical depths. A high abundance is required for ASE, so
nitrogen and oxygen are likely the only atoms that could be used.
## Appendix A Atomic collision rates in the mesosphere and lower thermosphere
We wish to estimate the collision rates for various metallic atoms and ions in
the mesosphere and lower thermosphere (MLT) region. The atmospheric
composition in the MLT is essentially the same as in the troposphere,
consisting almost entirely of three species, N2, O2 and Ar. Their fractions,
by volume, are approximately 78%, 21% and 1% respectively. The species of
interest to us are Na, Fe, Ca, Si, Mg, K, N, O, H and their ions.
The collision rate $R$, for a given atom, is given in terms of the collision
cross-section Q and the number densities $n$ for the dominant atomic and
molecular species. For atom $i$, interacting with field atoms or molecules
$j$,
$R_{i}=\sum_{j}n_{j}Q_{ij}v_{ij},$ (A1)
where $v_{ij}$ is the relative velocity. For a Maxwell-Boltzmann distribution
at temperature $T$, the RMS relative velocity of an atom or molecule of mass
$m_{i}$ and one of mass $m_{j}$ is,
$v_{ij}=\sqrt{v_{i}^{2}+v_{j}^{2}}=\sqrt{\frac{8kT}{\pi\mu_{ij}}},$ (A2)
where $\mu_{ij}=m_{i}m_{j}/(m_{i}+m_{j})$ is the reduced mass. The low-energy
collision cross section between two atoms is well-approximated by the hard-
sphere model,
$Q_{ij}=\pi(r_{i}+r_{j})^{2},$ (A3)
where $r_{i}$ and $r_{j}$ are the atomic radii. For a diatomic molecule $j$,
the situation is more complicated. Modelling the molecule by two spheres of
radius $r_{j}$ in contact, the appropriate geometrical cross section is the
projected area $A$ of two spheres of radius $r=r_{i}+r_{j}$, having centres
separated by a distance $d$. The separation depends on the angle $\theta$
between the axis of the molecule and the velocity vector,
$d=2r_{j}\sin\theta$. To find the mean cross-section we average over this
angle, assuming an isotropic velocity distribution. From plane geometry,
$A=2\pi
r^{2}-2r^{2}\arccos\left(\frac{d}{2r}\right)+\frac{d}{2}\sqrt{4r^{2}-d^{2}}.$
(A4)
The frequency with which a particular angle $\theta$ appears is proportional
to $\sin\theta$, so
$\displaystyle Q_{ij}$
$\displaystyle=\frac{4}{\pi}\int_{0}^{\pi/2}A\sin^{2}\theta d\theta,$
$\displaystyle=2\pi r^{2}+\frac{r^{4}+2r^{2}r_{j}^{2}-3r_{j}^{4}}{\pi
r_{j}^{2}}\text{arcsinh}\left(\frac{r_{j}}{\sqrt{r^{2}-r_{j}^{2}}}\right)$
$\displaystyle\quad-\frac{r^{3}-3rr_{j}^{2}}{\pi
r_{j}}+\frac{32r_{j}r}{3\pi\sqrt{2\pi}}{{}_{3}F_{2}}\left[2,\frac{1}{2},\frac{1}{2};\frac{5}{2},\frac{3}{2};\left(\frac{r_{j}}{r}\right)^{2}\right],$
(A5)
where ${}_{3}F_{2}$ is a generalized hypergeometric function.
Atomic radii are taken from [43]. For the atmospheric density and temperature
the MSIS-E-90 atmospheric model was used [21]. These were evaluated, for a
latitude of $30^{\circ}$ ond longitude of $0^{\circ}$, at midnight on the
first day of each month, and the average over a full year was computed. The
resulting mean cross sections for the atoms and ions of interest, colliding
with N2, O2 and Ar, are listed in Table A1. The computed collision rates are
listed in Tables A2 and A3. For comparison, Holzlöhner et al. [6] estimated a
Na – O2 collision rate of 1/(35 $\mu$s) at an altitude of 92 km, which is
within a factor of two of our estimate.
Table A1. Mean collision cross sections.
---
Atom | mass | radius | cross section
| (amu) | (nm) | (nm2)
| | | N2 | O2 | Ar
Na | 22.990 | 0.191 | 0.565 | 0.550 | 0.277
Fe | 55.845 | 0.127 | 0.315 | 0.303 | 0.171
Mg | 24.305 | 0.162 | 0.443 | 0.429 | 0.226
Si | 28.085 | 0.109 | 0.257 | 0.247 | 0.145
Ca | 40.078 | 0.197 | 0.593 | 0.577 | 0.288
K | 39.098 | 0.237 | 0.790 | 0.771 | 0.370
N | 14.007 | 0.070 | 0.152 | 0.144 | 0.097
O | 15.999 | 0.066 | 0.143 | 0.135 | 0.093
Ar | 39.948 | 0.106 | 0.248 | 0.238 | 0.141
H | 1.008 | 0.070 | 0.152 | 0.144 | 0.097
Table A2. Collision rates, in units of $10^{2}$ s-1, for metallic species.
---
altitude | Ca | Fe | K | Mg | Na | Si
(km) | | | | | |
$75$ | 2449 | 1219 | 3283 | 2062 | 2675 | 1150
$80$ | 1108 | 551.7 | 1486 | 933.3 | 1211 | 520.7
$85$ | 484.1 | 241.0 | 649.1 | 407.7 | 528.8 | 227.5
$90$ | 203.7 | 101.4 | 273.1 | 171.5 | 222.5 | 95.70
$95$ | 82.66 | 41.15 | 110.8 | 69.60 | 90.28 | 38.84
$100$ | 32.65 | 16.26 | 43.77 | 27.49 | 35.65 | 15.34
$105$ | 13.08 | 6.515 | 17.54 | 11.01 | 14.28 | 6.147
$110$ | 5.662 | 2.821 | 7.590 | 4.766 | 6.181 | 2.661
$115$ | 2.740 | 1.365 | 3.673 | 2.306 | 2.991 | 1.288
$120$ | 1.489 | 0.742 | 1.996 | 1.253 | 1.625 | 0.700
Table A3. Collision rates, in units of $10^{2}$ s-1, for non-metallic species.
---
altitude | Ar | H | N | O
(km) | | | |
$75$ | 1023 | 2592 | 833.3 | 748.7
$80$ | 463.0 | 1173 | 377.1 | 338.8
$85$ | 202.3 | 512.3 | 164.7 | 148.0
$90$ | 85.10 | 215.5 | 69.30 | 62.27
$95$ | 34.54 | 87.46 | 28.13 | 25.27
$100$ | 13.64 | 34.54 | 11.11 | 9.982
$105$ | 5.468 | 13.83 | 4.451 | 4.000
$110$ | 2.367 | 5.984 | 1.926 | 1.731
$115$ | 1.146 | 2.895 | 0.932 | 0.838
$120$ | 0.623 | 1.573 | 0.507 | 0.455
## Appendix B Transition rates in a multi-level atom
In a multi-level atom, with states $j$, $j=0,1,\cdots,n-1$, the fraction of
atoms $x_{j}$ for which the electron is in state $j$ is determined by $n$ rate
equations
$\dot{x}_{j}=\Gamma_{jk}x_{k}.$ (B1)
Here $\Gamma_{jk}$ specifies the net transition rate between states $j$ and
$k$. These equations are not linearly independent because the occupations
fractions are related by the constraint
$\sum x_{k}=1.$ (B2)
This condition can be used to eliminate one of the variables, $x_{0}$ say,
which results in a set of $n-1$ linearly-independent inhomogeneous equations
$\dot{x}_{j}=M_{jk}x_{k}+b_{j},\quad j=1,2,\cdots,n-1.$ (B3)
The quantities $M_{jk}$ and $b_{j}$ depend on transition rates and the
radiation energy densities at the transition wavelengths. If the radiation
energy densities can be regarded as independent of time, these quantities are
constant and the equations are readily solved using standard techniques. The
general solution is
$\boldsymbol{x}=\sum
c_{k}\boldsymbol{a}_{k}e^{\lambda_{k}t}-M^{-1}\boldsymbol{b},$ (B4)
where $\lambda_{k}$ and $c_{k}$ are the eigenvalues and eigenvectors of $M$.
The steady-state solution is found by equating the left hand side of Eqn (B3)
to zero,
$\boldsymbol{x}=-M^{-1}\boldsymbol{b},$ (B5)
Funding. Yunnan Provincial Department of Education; Natural Sciences and
Engineering Research Council of Canada (RGPIN-2019-04369); Chinese Academy of
Sciences, CAS President’s International Fellowship Initiative (2017VMA0013).
Acknowledgement. We thank Profs. D. Budker and R. Holzlöhner for comments on
an earlier version of the manuscript. PH thanks the National Astronomical
Observatories, Chinese Academy of Sciences for hospitality during a sabbatical
visit.
Disclosures. The authors declare no conflict of interests.
Data availability. Data underlying the results presented in this paper are
available in Refs. [21] and [28]. Selected data for strong transitions in the
atomic species discussed here are available from the authors upon request.
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|
# Knowledge Prompts: Injecting World Knowledge into
Language Models through Soft Prompts
Cicero Nogueira dos Santos, Zhe Dong, Daniel Cer, John Nham,
Siamak Shakeri, Jianmo Ni, Yun-hsuan Sung
Google Research
$\\{$cicerons, zhedong, cer, jnham, siamaks, jianmon<EMAIL_ADDRESS>
###### Abstract
Soft prompts have been recently proposed as a tool for adapting large frozen
language models (LMs) to new tasks. In this work, we repurpose soft prompts to
the task of injecting world knowledge into LMs. We introduce a method to train
soft prompts via self-supervised learning on data from knowledge bases. The
resulting soft knowledge prompts (KPs) are task independent and work as an
external memory of the LMs. We perform qualitative and quantitative
experiments and demonstrate that: (1) KPs can effectively model the structure
of the training data; (2) KPs can be used to improve the performance of LMs in
different knowledge intensive tasks.
## 1 Introduction
Very large neural language models (LMs) are known to perform well on knowledge
intensive natural language understanding (NLU) tasks, because they memorize a
significant amount of world knowledge from the training data. The larger the
LM, the more facts it can memorize at the training time, and the better the
results at the inference time Roberts et al. (2020). Despite their success,
these models also present some important drawbacks such as: the parametric
memory of these models have a fixed size and cannot grow (or shrink) over time
without fully retraining the model; there is no control in terms of which part
of the memory stores data about what; facts that do not co-occur frequently in
the training data are not well represented in the model; very large models are
needed to memorize enough data in order to perform well on knowledge intensive
tasks such as generative question answering; and at last, but not the least,
the memorized knowledge gets obsolete overtime, and requires re-training the
model for refreshness.
In this work, we employ soft prompts to overcome some of these issues of LMs.
_Soft prompts_ Lester et al. (2021); Li and Liang (2021); Hambardzumyan et al.
(2021) have been recently proposed as a tool for adapting large frozen LMs to
new tasks. Nevertheless, we repurpose soft prompts to the task of injecting
world knowledge into LMs. The goal is to train an external memory that is
composed of a large set of soft prompts that encode world knowledge. We
introduce a method to train knowledge driven soft prompts via self-supervised
learning on data from knowledge bases. The resulting soft prompts, which we
call _knowledge prompts_ (KPs), function as an auxiliary memory of the LMs
that is activated when solving knowledge intensive tasks. Different from
regular applications of soft prompts that concatenate a fixed small set of
embeddings to every input, our approach learns a very large set of KPs, which
are sparsely activated depending on the input.
We focus on entity-centric KPs, which means that each prompt primarily encodes
information about one entity from a knowledge base. We use Wikidata Vrandečić
and Krötzsch (2014) triples as our training data and train KPs for the top
1.1M entities, based on the number of triples. We present a qualitative
analysis of KPs using t-SNE plots and k-nearest neighbors approaches. In terms
of quantitative analysis, we show experimental results for three knowledge
intensive tasks: question answering, fact checking and relation
classification. For all datasets, the use of KPs improves the performance of
the T5 baseline. Our experimental results demonstrate that KPs are an
effective way to expand the memory of frozen LMs.
Figure 1: Training of Knowledge Prompts: given a serialized KB triple where
one of the entities has been masked out, the frozen LM has to predict the
masked entity given the input and the knowledge prompt of the non-masked
entity, which is _Michelle Obama_ in the example. The cross-entropy loss is
computed and the error is back-propagated through the frozen LM in order to
update the KP.
The main contributions of this work are the following:
* •
we propose a self-supervised approach to train knowledge driven soft prompts
that can be used to inject world knowledge into LMs.
* •
we demonstrate that knowledge prompts can effectively model the structure of
the training data and can also improve the performance of LMs on knowledge
intensive tasks.
* •
this work sheds light on the usability of soft prompts for storing data rather
than storing instructions on how to solve specific tasks.
## 2 Methods
### 2.1 Soft Prompts
Different approaches have been recently proposed to train soft prompts Lester
et al. (2021); Li and Liang (2021); Hambardzumyan et al. (2021). One of the
most popular methods, and probably the simplest one, consists of the following
steps (Lester et al., 2021):
* (1)
for a task in the dataset, prepend a fixed number of embeddings (soft prompts)
to the word embeddings of every input;
* (2)
during finetuning, update the soft prompt while keeping all the other
parameters of the LM frozen.
Despite its simplicity, this approach has demonstrated to be very effective
when used with large language models.
### 2.2 Soft Knowledge Prompts
We are interested in training _soft knowledge prompts_ (KPs) to encode world
knowledge, which could work as an external memory for LMs. In this work, we
focus on the training of _entity-centric KPs_ , each of which stores the
knowledge related to a specific entity from a knowledge base (KB). In other
words, the KP of an entity encodes information from the KB triples that
mention the entity either as a subject or an object. We adopt KB triples from
Wikidata Vrandečić and Krötzsch (2014), as a simple and trustworthy source of
world knowledge.
#### 2.2.1 KP Training
We train KPs with a masked language modeling (MLM) objective (Devlin et al.,
2019; Taylor, 1953), where the goal is to generate the object entity of a KB
triple given the subject entity and relation, and vice versa. As an example,
the input / target pair "Germany capital <MASK>" / "Berlin" will be used to
update the KP for Germany, while the pair "<MASK> capital Berlin" / "Germany"
will be used to update the KP for Berlin.
The KPs are randomly initialized, and are updated only when the corresponding
entities appear (not masked) in the input. This makes the training of KPs
sparse and parallelizable.
Given an input triple with the object entity being masked, a training
iteration has the following steps:
* (1)
retrieve the KP of the subject entity, which is a simple lookup operation;
* (2)
concatenate the KP to the sequence of word embeddings of the input text;
* (3)
predict the object entity name and compute the cross-entropy loss;
* (4)
back-propagate the error through the frozen LM to the KP, and update the KP
using stochastic gradient descent;
Figure 1 illustrates our proposed method for training knowledge prompts.
Notice that this method is general, and can be used with any textual input as
long as the entities of interest are already identified/linked.
#### 2.2.2 Using KPs in Downstream Tasks
Using KPs during the finetuning of the LM is straightforward. Given the input
sequence, e.g. a question, the relevant KPs are retrieved and concatenated to
the word embeddings of the input, to generate or predict the answer. At the
finetuning time, we freeze the KPs and only update the parameters of the LM,
as the KPs are used as pre-trained external knowledge. One can also use task
specific soft prompts instead of finetuning the parameters of the LM as in
Lester et al. (2021), however in this work we focused on finetuning the LM as
we use relatively small LMs.
Retrieving KPs that are relevant to the input sequence is crucial for good
performance in the downstream task. KPs are useful only if they contain the
knowledge that is helpful to solve the input at hand. In this work we employ
entity linking as a way to retrieve relevant KPs during training/inference for
downstream tasks. Given an input, we first perform entity linking to identify
the entities mentioned in the input. Then, we simply do a lookup operation to
retrieve the KPs of the identified entities.
#### 2.2.3 Why Knowledge Prompts?
Some advantages of the proposed approach include:
* •
It allows a better control of what information is stored, by choosing what
examples are used to train the KPs.
* •
KPs are trained independently, therefore the training can be massively
parallelized.
* •
As the LM is kept frozen during the training of KPs, we do not mess up with
the language generation/understanding capabilities of the LM.
* •
KPs can increase the capacity of small LMs in a dynamic way. We can add/remove
KPs at any time. Moreover, if information about a single entity changes, we
can update that entity’s KP without changing other KPs. This addresses the
freshness issue of the LMs.
## 3 Related Work
Most works on soft prompting focus on the problem of learning a set of
embeddings that can _reprogram_ a frozen LM to solve a specific task Lester et
al. (2021); Li and Liang (2021); Hambardzumyan et al. (2021). Some key aspects
that distinct KPs from these approaches include: (1) in KPs, the goal is to
create an external memory that stores world knowledge. (2) while traditional
soft prompt approaches learn a small set of embeddings our approach learns a
very large set of KPs, which are sparsely accessed by the LM. (3) different
from regular soft prompts, KPs are not task-specific.
Our work is related to the recent body of work on memory augmented LMs. The
most related approaches are the Entities as Experts (EaE) model proposed by
Févry et al. (2020) and the Fact Injected Language Model (FILM) proposed by
Verga et al. (2021). In both papers the authors present methods to train
entity embeddings during the pretraining of the LM. During inference time, the
model uses a separate classification head to identify entities, which are then
used to retrieve entity embeddings that are merged to the current entity
representation. de Jong et al. (2022) proposed the TOME model, which employs
embeddings in a similar way to EaE but with a much larger granularity. Instead
of modeling entities, their method embeds entity mentions, which results in a
very large external memory. In their experiments, they use up to 150M entries
in the external memory while EaE, FILM and our method use about 1M entries
only. Compared to these three models, our proposed approach trains an external
LM memory in a quite different way. Our method uses frozen pretrained LMs
while those methods train the external memory and the LMs together. Our method
concatenates the relevant memory entry to the input as if they were additional
word embeddings, while the other three approaches merge the relevant memory
entries to the contextual embedding of the identified entities. Additionally,
while EaE, FILM and TOME adapt the memory for each new task, our approach uses
the exact same external memory for different tasks.
This work is also related to multiple recent papers on integrating knowledge
bases into LMs Zhang et al. (2019); Peters et al. (2019); Poerner et al.
(2020); Sun et al. (2020); Wang et al. (2021); Agarwal et al. (2021). On of
the key differences between KPs and all these methods is the use of soft
prompts to integrate knowledge bases to frozen LMs.
## 4 Experimental Setup
### 4.1 KP Training Data
We adopt Wikidata triples Vrandečić and Krötzsch (2014) as our source of data
to train KPs. We start with the set of 45M triples that was previously
preprocessed by Agarwal et al. (2021). Next, we filter out triples whose
subject entity appears less than 12 times as subject entity in the dataset.
This results in a set of 23M triples containing 1.1M distinct subject
entities, which form our entity vocabulary and, respectively, the number of
KPs in our experiments.
### 4.2 KP Training Setup
We adopt the T5.1.1 model family Raffel et al. (2020) and perform experiments
with three model sizes: small, base and large, which contain 60M, 220M and
770M parameters, respectively. We use the T5.1.1 checkpoints that Lester et
al. (2021) adapted from the original T5.1.1 checkpoints by running an
additional 100K training steps using the “LM” objective discussed in Raffel et
al. (2020). Just like reported by Lester et al. (2021), we also noticed that
these adapted checkpoints make the training of soft prompts easier. Although
we use an encoder-decoder LM, our approach is not limited to this type of
architecture and can be used with encoder only models like BERT Devlin et al.
(2019) or decoder only models like GPT2 Radford et al. (2019).
The input length for training KPs is normally short because our examples are
masked serialized triples (concatenation of Subject/Object entity and a
relation). Therefore, we set the input length to 64, which allows us to use
very large batch sizes: between 4K and 8K, depending on the model size. Note
that the objective when training KPs is to memorize the training data. Hence,
we let KP training run for up to 200 epochs.
In the beginning of the training, KPs are initialized by randomly sampling
from the word embeddings matrix. This allows KPs to start from a region that
is known by the LM, which makes the training smoother and less sensitive to
hyperparameters. After training, KPs are kept frozen during LM finetuning for
downstream tasks. Therefore, for each model size, the exact same set of KPs is
used in our experiments with the different downstream tasks and datasets.
### 4.3 Entity Linking
In all experiments where KPs are used, we first preprocess the input text
using Google Cloud Natural Language API 111https://cloud.google.com/natural-
language/docs/analyzing-entities to perform entity linking.
### 4.4 Downstream Tasks
We perform experiments with three different knowledge intensive tasks: (1)
question answering (QA), (2) fact checking and (3) relation classification. In
terms of datasets, for question Answering experiments we use Entity Questions
Sciavolino et al. (2021) and TriviaQA Joshi et al. (2017) datasets. For fact
checking we use the FEVER dataset Thorne et al. (2018). For relation
classification, we use the TACRED dataset Zhang et al. (2017).
In the question answering experiments we follow the closed-book QA (CBQA)
setup of Roberts et al. (2020). In this setup, the model has no access to
external text, which means that there is no retrieval step and the model has
to solve the task using the world knowledge it acquired from pretraining and
finetuning data only. During training, we try to use the default
hyperparameters as much as possible except for the learning rate, which is
finetuned on the development sets. Following previous works we use exact
matching (EM) as the evaluation metric in CBQA.
## 5 Results and Discussion
Figure 2: t-SNE visualization of 100K KPs trained with T5-Base model on
Wikidata. KPs form well separated clusters whose member entities are very
similar in terms of their properties and the relations they belong to.
.
### 5.1 Qualitative Results
We perform a qualitative assessment of KPs through different experiments
including t-SNE visualizations, analysis of entity similarity in KP space and
evaluation of KPs for QA when golden entity linking is provided.
#### 5.1.1 t-SNE visualization of KPs
One of the main goals in our qualitative assessment of KPs is to check whether
the learned KPs can model the structure of the training data. An approach that
can give us some clue about the data structure learning aspect are t-SNE
visualizations van der Maaten and Hinton (2008). In Fig. 2, we show a t-SNE
visualization of 100K randomly selected KPs that were trained using T5-BASE
model on Wikidata triples. We can see in Fig. 2 that KPs form multiple well
separated clusters. Zooming in into these clusters we can notice that they are
very coherent. There are clusters about companies, celestial objects, movies,
locations, etc. This is a good indication that, although trained
independently, KPs encapsulate a notion of similarity between entities that
aligns well with the KB structure.
#### 5.1.2 k-Nearest Neighbors in KP Space
We investigate further the quality of the entity similarity captured by KPs
using cosine similarity in the KP space to retrieve the k-nearest neighbors of
different entities. In Table 1 we show the top four neighbours of four
different entities. We present results for T5 models of three different sizes.
The top two entities (_Barack Obama_ and _Roger Waters_) are cherry picked
popular entities to make it easier for the reader to grasp the quality of the
results. The bottom two entities (_Fairmont station_ and _Iacobeni , Sibiu_)
were randomly selected. The KP space learned by the three model sizes can
produce high quality nearest neighbors for the four different entities. For
instance, in the case of the search entity _Fairmont station_ , which is a
streetcar stop in Salt Lake City - Utah - USA, all the retrieve entities are
also streetcar stops in Salt Lake City. Similar results can be seen for the
other entities, where the retrieved neighbors share multiple properties (e.g.
same occupation and nationality) with the search entity.
Search Entity | Top 4 neighbors in KP space
---|---
| Small | Base | Large
Barack Obama | Donald Trump | Bill Clinton | Richard Nixon
George W. Bush | Donald Trump | Donald Trump
Michelle Obama | Ronald Reagan | Michelle Obama
| Jimmy Carter | Jimmy Carter | Harrison Ford
Roger Waters | George Harrison | David Gilmour | Tom Waits
Pink Floyd | Syd Barrett | Freddie Mercury
Kris Kristofferson | Eddie Vedder | David Gilmour
| James Hetfield | Brian Wilson | Cliff Edwards
Fairmont station | 500 East station | 700 East station | 700 East station
300 East station | 300 East station | Historic Gardner station
Sugarmont station | 500 East station | 300 East station
| 700 East station | Fairpark station | Decker Lake station
Iacobeni , Sibiu | Pălatca | Mica , Cluj | Sic , Cluj
Racovita , Timis | Gârbău , Cluj | Gârbău , Cluj
Movila , Ialomita | Aiton , Cluj | Mihăileni , Sibiu
| Mosna , Sibiu | Râsca , Cluj | Mica , Cluj
Table 1: Nearest neighbors in the knowledge prompt space for different search entities and model sizes. The top two entities were cherry picked while the two bottom ones were randomly selected. Model | Simple Questions
---|---
| Zero-shot Learning | Finetuning
| no KPs | KPs | no KPs | KPs
T5-SMALL | 0.1 | 4.3 | 32.9 | 54.1
T5-BASE | 0.0 | 8.8 | 35.1 | 58.9
T5-LARGE | 0.0 | 8.8 | 36.6 | 58.3
Table 2: Comparing LMs performances on Simple Questions Bordes et al. (2015)
with and without KPs as a knowledge source, measured by exact match (EM) score
(%). The KPs are retrieved using golden entity linking information. Most
performant results for each setup is marked in bold.
#### 5.1.3 KPs as a Knowledge Source for LMs
In order to assess in a controlled manner whether KPs can be used as a
knowledge source for LMs, we perform an experiment on closed-book QA using the
Simple Questions dataset Bordes et al. (2015). This dataset is a good fit for
our purpose because it contains golden information about the entity involved
in the question (golden entity linking). We use the Simple Questions version
that was processed by Diefenbach et al. (2017) to align the original Freebase
entities to Wikidata entities. We further preprocessed the dataset to contain
only questions involving one of the 1.1M entities for which we trained KPs.
In Table 2, we present two sets of experiments for models of different sizes.
In the first experiment we check whether the use of KPs can improve the
performance of the models for zero-shot learning. In this scenario, we can see
that T5 models without KPs performs very poorly and achieve exact match (EM)
score of 0 percent. The use of KPs boosts the performance of all model sizes,
with the base and large models achieving EM of 8.8. In the finetuning
scenario, the use of KPs also brings a significant boost in performance. In
particular, for T5-Base the improvement is of $\sim$24 points in EM. We know
the improvement is actually larger than this because for some questions there
are multiple good answers (e.g. songwriter and singer are valid occupations
for _John Lennon_), but the Simple Questions dataset list a single answer
only.
These experimental results indicate that KPs are an effective approach to
store information about entities in a way that can be readily used by the LM
without any adaptation of the KPs for the downstream QA task.
### 5.2 Quantitative Results
Model | Entity Questions | TriviaQA
---|---|---
| Dev | Test | Dev | Test
| no KPs | KPs | no KP | KPs | no KPs | KPs | no KP | KPs
T5-SMALL | 23.3 | 30.6 | 23.6 | 30.8 | 17.4 | 19.4 | 21.4 | 23.8
T5-BASE | 24.5 | 33.3 | 25.4 | 33.1 | 22.7 | 24.9 | 27.1 | 28.1
T5-LARGE | 26.8 | 33.8 | 26.7 | 33.9 | 28.3 | 28.9 | 32.0 | 32.5
Table 3: Comparing LMs performances on EntityQuestions (Sciavolino et al.,
2021) and TriviaQA (Joshi et al., 2017) with and without KPs as a knowledge
source, measured by exact match (EM) score (%).
#### 5.2.1 Closed-book Question Answering
Table 3 presents experimental results for two closed-book QA (CBQA) datasets
and different T5 model sizes. KPs provide a significant performance
improvement on the Entity Questions dataset, which is an entity-centric QA
task. For instance, the improvement for T5-Base is of 7.7 points in EM.
Interestingly, T5-SMALL + KPs outperforms T5-LARGE model by a large margin,
30.8 vs 26.7, respectively. Although to a smaller extent, KPs also bring
performance improvements for all model sizes on TriviaQA dataset.
In Table 4 we compare the performance of T5-Base/Large + KPs with other
recently published results on CBQA for TriviaQA. To the best of our knowledge,
there is no previous work that reports CBQA results for the Entity Questions
dataset. T5 + KPs model does not perform as well as the other CBQA approaches
on TriviaQA. We believe this is mainly due to the following factors: (1) in
EaE, TOME and T5-11B+SSM, the full LM is heavily pretrained on Wikipedia text
using entity focused objective functions, which is known to make the model
very effective to QA Guu et al. (2020). In our method, we use an entity
focused objective to train KPs only while the LM is frozen. Note in Table 3,
no KPs column, that our initial baseline is poor. (2) Models like EaE and TOME
update their external memory component during finetuning for a new task. We
keep KPs frozen when finetuning for a new task. (3) Our model generates the
answer token by token, which is more prone to errors compared to the entity
retrieval approaches used in EaE and TOME. (4) We conjecture that the
additional pretraining steps performed by Lester et al. (2021) might hurt the
performance for CBQA.
Model | TriviaQA
---|---
| Dev | Test
T5-BASE + KPs | 24.9 | 28.1
T5-LARGE + KPs | 28.9 | 32.5
BERT-base Févry et al. (2020) | 28.9 |
EaE Févry et al. (2020) | 43.2 | 53.4
T5-11B+SSM Roberts et al. (2020) | 51.0 | 60.5
TOME-2 de Jong et al. (2022) | 54.6 | 65.8
Table 4: Comparison with other closed-book QA approaches for TriviaQA.
#### 5.2.2 Fact Checking
We present experimental results for the fact checking task in Table 5. The use
of KPs brings significant improvements for the three model sizes on the FEVER
dataset. Compared to recent works that use LMs with external memory, our
T5-Base + KPs model has similar performance to EaE (Févry et al., 2020), and
T5-LARGE + KPs achieves results competitive to TOME-2 (de Jong et al., 2022)
model. TOME-2 achieves better results than EaE and T5 + KPs because of the
granularity of its memory. While TOME-2 has an external memory with 150M
entries that store fine-grained information about entities, both EaE and our
model have a memory with about 1M entries only. Our KP training method allows
to increase the granularity of KPs in a straightforward manner. For instance,
one can use multiple KPs per entity, where each KP is trained using a subset
of the triples that mention the entity. We leave the investigation on multiple
KPs per entity as a future work.
Model | FEVER
---|---
| no KPs | KPs
T5-SMALL | 60.2 | 61.4
T5-BASE | 61.3 | 63.4
T5-LARGE | 63.0 | 65.2
EaE (Févry et al., 2020) | 63.6
TOME-2 (de Jong et al., 2022) | 68.1
Table 5: Comparing LMs performances on fact checking dataset, FEVER (test
split) (Thorne et al., 2018), with and without KPs as a knowledge source,
measured by accuracy (%). Two baselines, Entities-as-Experts (EaE, Févry et
al. (2020)) and MentionMemory (TOME-2, de Jong et al. (2022)), are included.
#### 5.2.3 Relation Classification
Table 6 presents experimental results for the relation classification task
using the original TACRED dataset. Following previous papers, we use F1 as the
metric and report results for the test set. Similar to the other two tasks, in
relation classification KPs also provide performance improvements for all
three model sizes. Interestingly, T5-Base + KPs outperform T5-LARGE by almost
one point in F1.
Model | TACRED
---|---
| no KPs | KPs
T5-SMALL | 64.3 | 66.1
T5-BASE | 68.3 | 70.0
T5-LARGE | 69.1 | 69.8
EaE Févry et al. (2020) | 70.2
KnowBERT Peters et al. (2019) | 71.5
Table 6: Experimental results on relation classification.
Compared to previous papers that use knowledge augmented approaches, T5-Base +
KPs achieves performance similar to EaE and is competitive with KnowBERT. It
is important to note that KnowBERT uses entity types as input while both EaE
and our method do not use that additional information.
### 5.3 Ablation Experiments
#### 5.3.1 KP$\rightarrow$Encoder vs. KP$\rightarrow$Decoder
The use of Encoder-Decoder LM gives us the flexibility to introduce KPs at
either encoder or decoder. In all experiments presented so far in the paper,
KPs are concatenated to the word embeddings of the input sequence and given as
input to the encoder (KP$\rightarrow$Encoder). However, one can instead
concatenate KPs to the output of the encoder, which makes KPs accessible only
by the decoder via cross-attention (KP$\rightarrow$Decoder). In Table 7 we
present a comparison of the results of KP$\rightarrow$Encoder and
KP$\rightarrow$Decoder for the QA datasets. In both cases, KPs were trained
using T5-Base model and we report results for the dev set. We believe
KP$\rightarrow$Encoder achieves better performance because it allows
interaction (self-attention) between input and KPs in the encoder, which gives
the model more opportunity to select and extract the correct information from
KPs. On the other hand, the advantage of KP$\rightarrow$Decoder is that its
training is faster because it is a simpler optimization problem as the error
does not have to be back-propagated through the frozen encoder and KPs are
used via cross-attention in the decoder only. KP$\rightarrow$Decoder requires
3x less training iterations to converge compared to KP$\rightarrow$Enc.
Dataset | KP$\rightarrow$Enc | KP$\rightarrow$Dec
---|---|---
Simple Questions | 58.9 | 57.7
Entity Questions | 33.3 | 32.8
TriviaQA | 24.9 | 24.2
Table 7: Comparison of models that inject KPs into encoder
(KP$\rightarrow$Enc) vs models that inject KPs into decoder
(KP$\rightarrow$Dec).
#### 5.3.2 Entity Linking vs. Searching in KP Space
Retrieving relevant KPs given an input is a fundamental task that has direct
impact on the usefulness of KPs. Beyond entity linking, another approach to
retrieve KPs is to transform the input into a single dense vector, then search
for the most similar vectors in the KP space. We have experimented with this
strategy by training an external encoder that creates a vector representation
of the input. The external encoder has the same architecture and size of the
respective T5 model. We use the TEKGEN dataset Agarwal et al. (2021), which
contains Wikipedia sentences mapped to Wikidata triples, as a source of noisy-
labeled data to train the encoder via contrastive loss. KPs are kept frozen
during the training of the input encoder.
Table 8 presents a comparison between T5 + KP models that use either entity
linking or search in the KP space. The results were computed on TriviaQA. We
can see in the results that entity linking performs better for both model
sizes. We conjecture that searching in KP space does not work well because KPs
are not optimized to be used in search. KPs are trained to memorize knowledge
in a way that can later be extracted by the LM.
Model | Ent. Linking | KP Search
---|---|---
T5-SMALL+KPs | 19.4 | 17.9
T5-BASE+KPs | 24.9 | 23.2
Table 8: Evaluation of the LM performances on TriviaQA with KPs retrieved with
Entity Linking and Searching in KP space.
## 6 Conclusions
We present a new method for training soft prompts that can be used to extend
the world knowledge of LMs. We demonstrate the generality and usefulness of
the resulting KPs by employing the same set of KP to improve the LM
performance in three different tasks: question answering, fact checking, and
relation classification. Although in this work we focused on the use of KPs
for injecting knowledge into LMs, we believe that entity-centric KPs can be
seen as a general purpose knowledge base embedding approach. We leave the
investigation of KPs as a general KB embebedding approach for future work.
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|
§ INTRODUCTION
Verbal autopsy (VA) is a well established approach to ascertain
cause-of-death when medical certification and full autopsies are not
feasible or practical [Garenne, 2014, Taylor et al., 1983].
After a death is identified, a specially-trained fieldworker interviews
the caregivers (usually family members) of the decedent. A typical VA
interview includes a set of structured questions with categorical or
quantitative responses and a narrative section that records the `story'
of the death from the respondent's point of view
[World Health Organization and others, 2012]. Currently, there are multiple commonly used
questionnaires with overlapping, but not identical questions.
The process of inferring a cause from VA data consists of two
components. First, there must be some external information about the
relationship between causes and symptoms. This is similar to the role of
training data in supervised learning tasks. In the context of VA,
training dataset of cases with cause of death assigned by in person
autopsy is rare, and collecting such data is extremely time and
resource-intensive, and requires strong assumptions about the
generalizability of deaths in the training set to the population of
interest. A more common practice is to obtain training data using
clinically trained, experienced physicians read a fraction of the
interviews and determine causes. To address the fact that physicians
frequently do not agree on causes, VA interviews are often read by two
physicians, and sometimes three, and the final causes are determined
through a consensus mechanism <cit.>. Another
means of obtaining this information is directly through expert opinion.
A common practice is to ask groups of physicians to rate the likelihood
of a symptoms occurring given a particular cause of death, which can be
converted into a set of probabilities of observing a symptom given a
particular cause. Such expert knowledge can then be used for modeling.
The second component is the algorithmic or statistical method that
actually assigns the cause of death by combining the expert or learned
cause-symptom relationship with the symptoms observed in the VA
interviews from the target population.
In all, therefore, there are three key pieces required to analyze VA
data: (i) VA survey data itself, (ii) inputs that give information about
the association between symptoms and causes, and (iii) a statistical or
algorithmic method for assigning a likely cause. The current state of VA
literature usually does not distinguish between these three, in part
because existing software for algorithmic and statistical methods
require a specific set of inputs and survey format. This restriction
prevents robust comparison between methods and contexts. A health agency
in one region may, for example, want to analyze VA data using the same
VA algorithm as a neighboring region to ensure estimates are comparable.
Unless the agencies used the same survey format, however, this is not
possible with existing tools. The openVA package [Li et al., 2021]
addresses this issue through an open-source toolkit that (i) performs
data processing and conversion between existing survey formats, (ii)
implements multiple currently available algorithmic and statistical
methods, and (iii) provides visualization and tools for interpreting
§.§ The openVA package
The openVA comprises a suite of R [R Core Team, 2020]
packages for the analysis of verbal autopsy data. The goal of this
package is to provide researchers with an open-sourced tool that
supports flexible data input format and allows easier data manipulation
and model fitting. The openVA family consists of four core
required packages that are on CRAN, InterVA4, InterVA5,
InSilicoVA, and Tariff, and an optional package
nbc4va. Each of these packages implements one coding algorithm.
Three survey formats are supported currently: the WHO 2012 and WHO 2016
instrument after standard dichotomization procedures into binary
variables [World Health Organization and others, 2012, 26], the Institute for
Health Metrics and Evaluation (IHME) questionnaire in the format of the
Population Health Medical Research Consortium (PHMRC) dataset
[Murray et al., 2011], and customized binary data with training
The main focus of this paper is to provide a general introduction to the
implementation details of the included algorithms both in terms of the
underlying methodology, and through a series of case studies. For each
of the algorithms discussed, there is a standalone R package
available on CRAN, and three of them, InterVA-4, InterVA-5 and Tariff,
are also available in compiled software program distributed by the
original authors. The openVA package has four major contributions:
* It provides a standardized and user-friendly interface for analysts to fit and evaluate each method on different types of data input using standard syntax. Previously, most of the methods are designed to be used specifically with their own input formats and are usually incompatible with others. The openVA package closes this gap and allows easier and fair model comparison of multiple algorithms on the same data. This significantly facilitates further research on VA algorithms.
* It provides a series of functionalities to summarize and visualize results from multiple algorithms, which is helpful for analysts not familiar with data manipulation and visualization in R.
* It does not directly implement any algorithms for coding VA data[A special case is when we extend the InterVA-4 algorithm to the scenario where symptoms and causes are not in the pre-defined set provided by the original InterVA software, the extension is included in the openVA package instead of the InterVA4 and InterVA-5 packages for a faithful replication of the InterVA methods themselves.], so that it is possible for a research group to maintain their own algorithm implementations callable from the openVA package, while also make it available to the general users as a standalone piece of software. For example, the nbc4va was developed and maintained independently by researchers at the Center for Global Health Research in Toronto, but is designed so that it can be seamlessly integrated into the openVA package.
* It is fully open-sourced, and can be run on multiple platforms. The open-sourced nature of openVA significantly expands its potential for methodological research and its suitability for integration within a larger data analysis pipeline. Compared to the alternative implementations, the InterVA-4 and InterVA-5 software are distributed as compiled software that can only be run on Windows system. They provide the source codes as an additional code script, which are difficult to modify and re-compile. Tariff, as implemented through the SmartVA-Analyze application [Serina et al., 2015], is also primarily distributed as a compiled application that can only be run on Windows system [Institute for Health Metrics and Evaluation, 2021]. Their source codes was recently made available under the open source MIT License on GitHub [Institute for Health Metrics and Evaluation, 2021].
The rest of this paper is organized as follows: In Section
<ref> we briefly introduce the main component packages and
the underlying algorithms. We demonstrate model fitting with the
openVA package for different input data formats in Section
<ref> to <ref>. We first show in Section
<ref> how to prepare the input data. We then demonstrate
fitting different VA models in Section <ref>, and
functionalities to summarize results in Section <ref>. We
then discuss how additional information can be incorporated into
modeling VA data in Section <ref>. Section <ref>
briefly surveys additional packages and software developments built
around the openVA package. We end in Section <ref>
with a discussion of remaining issues and limitations of the existing
automated VA algorithms and proposes new functionalities to be included
in openVA package.
§ STRUCTURE OF THE OPENVA SUITE
The openVA suite of packages currently consists of four standalone
packages that are available on the CRAN and one optional package hosted
on Github. In this section, we first provide a brief introduction to
these five packages, and we discuss the mathematical details behind each
algorithm in the next subsection.
* InterVA4 [Li et al., 2014, Li et al., 2019] is an R
package that implements the InterVA-4 model
[Byass et al., 2012]. It provides replication of both
InterVA software version 4.02 and the later release of version 4.03
update [Byass, 2015]. The standard input of InterVA4 is in
the form of a pre-defined set of indicators, based on the 2012 WHO VA
instrument (see
The default InterVA-4 algorithm cannot be applied to other data input
format because its internal built-in prior information is specific to
a fixed set of indicators and causes. The same restriction is also
maintained in InterVA4 package. However, the mathematical
formulation of InterVA-4 model is completely generalizable to other
binary input format. The generalized algorithm is described in Section
<ref>, and also implemented in the openVA
* InterVA5 [Thomas et al., 2021] is an R package that
implements the InterVA-5 model developed by Peter Byass
The InterVA-5 model updates the previous version in several ways.
First, the input data must adhere to the format of the 2016 WHO VA
instrument [26]. Second, changes have been made
to the data processing steps, which are described in Section
<ref>. It is also worth noting that the model outputs
have expanded by the inclusion of the most likely Circumstances Of
Mortality CATegory, or COMCAT, among the results – the categories
include: culture, emergency, health systems, inevitable, knowledge,
resources, or an indeterminant combination of multiple factors
<cit.>. Despite these
changes, the mathematical formulation of InterVA-5 is identical to
that of InterVA-4.
* InSilicoVA [Li et al., 2021] is an R package that
implements the InSilicoVA algorithm, a Bayesian hierarchical framework
for cause-of-death assignment and cause-specific mortality fraction
estimation proposed in McCormick et al., 2016. It is originally designed to
work with the WHO VA instrument, i.e., the same input data as in
InterVA software, but is also generalizable to other data input
format. It is a fully probabilistic algorithm and could incorporate
multiple sources of information, such as known sub-population in the
dataset, and physician coding when such data is available. The MCMC
sampler is implemented in Java for improved speed.
* Tariff [Li et al., 2018] is an R package that
implements the Tariff algorithm [James et al., 2011]. It most closely
reflects the description of Tariff 2.0 method
[Serina et al., 2015]. The Tariff algorithm is developed by the
Institute for Health Metrics and Evaluation (IHME) and officially
implemented in the SmartVA-Analyze software [Institute for Health Metrics and Evaluation, 2021].
However, as the developers of this R package are not
affiliated with the authors of the original algorithm, there are some
discrepancies in implementation. The source code of the two versions
of Tariff was not publicly available at the time when the Tariff
package was created, so the package was developed based solely on the
descriptions in the published work. Despite the difference in
implementation, Tariff is able to achieve comparable results as
the published work as demonstrated in McCormick et al., 2016. More detailed
descriptions of the Tariff implementations are also discussed in
the supplement of McCormick et al., 2016. The later released
Python source codes of SmartVA-Analyze has been
incorporated in the web application extension of the openVA
package, which we briefly discuss in Section <ref>.
* nbc4va [Miasnikof et al., 2015, Wen et al., 2018] is an R
package that implements the Naive Bayes Classifier for VA encoding. It
calculates the conditional probabilities of symptoms given causes of
death from training dataset instead of using physician provided
values. nbc4va is developed and maintained by researchers at the
Center for Global Health Research in Toronto, but is designed so that
it can be seamlessly integrated into openVA. Currently the
nbc4va package is hosted on Github and is an optional package
that users can choose to load separately.
The openVA package is hosted on CRAN and can be installed with the
following commands. For the analysis in this paper, we also install the
nbc4va package separately from Github. The versions of the
supporting packages can be checked in R using the
openVA_status() function.
R> library(openVA)
R> library(devtools)
R> install_github("rrwen/nbc4va")
R> library(nbc4va)
R> openVA_status()
§.§ Overview of VA cause-of-death assignment methods
The general modeling framework for model VA data consist of first
converting the surveys into a series of binary responses to questions
about each death, and then the two main goals of a typical VA analysis
are to estimate the population cause-specific mortality fractions
(CSMF), and the probability distribution or rankings of cause-of-death
(COD) for each individual death. In this section, we formally compare
the modeling approaches utilized by each algorithm. We adopt the
following notations. Consider \(N\) deaths, each with \(S\) binary
indicators of symptoms. Let \(s_{ij}\) denote the indicator for the
presence of \(j\)-th symptom in the \(i\)-th death, which can take
values from 0, 1, or NA (for missing data). We consider a pre-defined
set of causes is of size \(C\). For the \(i\)-th death, denote the COD
by \(y_i \in \{1, ..., C\}\) and the probability of dying from cause
\(k\) is denoted by \(P_{ik}\). For the population, the CSMF of cause
\(k\) is denoted as \(\pi_k\), with \(\sum_{k=1}^C \pi_k = 1\).
* InterVA4 [Byass et al., 2012] and InterVA5 [Byass, 2018] algorithm calculate the probability of each COD given the observed symptoms using the Bayes rule, so that
\[
P_{ik} = \frac{\pi_{k}^{(0)} \prod_{j=1}^S P(s_{ij}=1|y_{i}=k) \mathbf{1}_{s_{ij} = 1}}
{\sum_{k' = 1}^C \pi_{k'}^{(0)} \prod_{j=1}^S P(s_{ij}=1|y_{i}=k') \mathbf{1}_{s_{ij} = 1}}
\]
where both the prior distribution of each causes, $\pi_{k}^{(0)}$, and the conditional probability of each symptom given each cause, $P(s_{ij}=1|y_{i}=k)$, are fixed values provided in the algorithm. The conditional probabilities, $P(s_{ij}=1|y_{i}=k)$, used in InterVA algorithms are represented as rankings with letter grades instead of numerical value [Byass et al., 2012]. For example, $P(s_{ij}=1|y_{i}=k) = A+$ is translated into $P(s_{ij}=1|y_{i}=k) = 0.8$, etc. The standard InterVA software only supports the fixed set of symptoms and causes where such prior information is provided. For a different data input format, this formulation can be easily generalized if training data is available. We include in the openVA package an extension of the algorithm that calculates $\hat P(s_{ij}=1|y_{i}=k)$ from the empirical distribution in the training data and then maps to letter grades with different truncation rules. Details of the extended InterVA algorithm can be found in McCormick et al., 2016.
After the individual COD distributions are calculated, InterVA-4 utilizes a series of pre-defined rules to decide up to top three most likely COD assignments and truncates the probabilities for the rest of the CODs to 0 and adds an 'undetermined' category so that the probabilities sum up to 1 (See the user guide of Byass, 2015). Then the population-level CSMFs are calculated as the aggregation of individual COD distribution, such that
\[
\pi_k = \sum_{i=1}^N P^*_{ik}
\]
where $P^*_{ik}$ denotes the individual COD distribution after introducing the undetermined category.
* Naive Bayes Classifier [Miasnikof et al., 2015] is very similar to the InterVA algorithm with two major differences. First, instead of considering only symptoms that present, NBC algorithm also considers symptoms that are absent. Second, the conditional probabilities of symptoms given causes are calculated from training data instead of given by physicians, which is similar to our extension of InterVA discussed above. Similar to InterVA, the NBC method can be written as
\[
P_{ik} = \frac{\pi_{k}^{(0)} \prod_{j=1}^S (P(s_{ij}=1|y_{i}=k) \mathbf{1}_{s_{ij} = 1} + P(s_{ij} \neq 1|y_{i}=k) \mathbf{1}_{s_{ij} \neq 1})}
{\sum_{k' = 1}^C \pi_{k'}^{(0)} \prod_{j=1}^S (P(s_{ij}=1|y_{i}=k') \mathbf{1}_{s_{ij} = 1}+ P(s_{ij} \neq 1|y_{i}=k') \mathbf{1}_{s_{ij} \neq 1})}
\]
and the CSMFs are calculated by $\pi_k = \sum_{i=1}^N P_{ik}$.
* InSilicoVA algorithm [McCormick et al., 2016] assumes a generative model that characterizes both CSMF at the population level, and the COD distributions at the individual level. In short, the core generative process assumes
\begin{eqnarray} \nonumber
s_{ij} | y_i = k &\propto& \mbox{Bernoulli}(P(s_{ij} | y_i = k)) \\\nonumber
y_i | \pi_1, ..., \pi_C &\propto& \mbox{Multinomial}(\pi_1, ..., \pi_C) \\\nonumber
\pi_k &=& \exp \theta_k / \sum_{k=1}^C \exp \theta_k \\\nonumber
\theta_k &\propto& \mbox{Normal}(\mu, \sigma^2)
\end{eqnarray}
and hyperpriors are also placed on $P(s_{ij} | y_i = k)$, $\mu$, and $\sigma^2$. The priors for $P(s_{ij} | y_i = k)$ are set by the rankings used in InterVA-4 if the data is prepared into InterVA format, or learned from training data if otherwise. Parameter estimation is performed using Markov Chain Monte Carlo (MCMC), so that a sample of posterior distribution of $\pi_k$ can be obtained after the sampler converges. Additional heuristics of data processing steps introduced since the original publication of the method is discussed in Section <ref> of the Appendix.
* Tariff algorithm [James et al., 2011] differs from all other three methods in that it does not calculate an explicit probability distribution of COD for each death. Instead, for each death $i$, a Tariff score is calculated for each COD $k$ so that
\[
Score_{ik} = \sum_{j = 1}^{S} \mbox{Tariff}_{kj}\mathbf{1}_{s_{ij}=1}
\]
where the symptom-specific Tariff score $\mbox{Tariff}_{kj}$ is defined as
\[
\mbox{Tariff}_{kj} = \frac{n_{kj} - median(n_{1j}, n_{2j}, ..., n_{Cj})} {IQR(n_{1j}, n_{2j}, ..., n_{Cj})}
\]
where $n_{kj}$ is the count of how many deaths from cause $k$ contain symptom $j$ in the training data. The scores calculated are then turned into rankings by comparing to a reference distribution of scores calculated from re-sampling the training dataset to have a uniform COD distribution. It is worth noting that the Tariff algorithm produces the COD distribution for each death in terms of their rankings instead of the probability distributions. And thus the CSMF for each cause $k$ is calculated by the fraction of deaths with cause $k$ being the highest ranked cause, i.e.,
\[
\pi_k = \frac{\sum_{i=1}^N\mathbf{1}_{y_i = k}}{N}
\]
In addition to the different model specifications underlying each
algorithm, two major distinctions are most significant in understanding
the four methods for practitioners to interpret the model results.
First, the missing indicators are assumed to be equivalent to `absence'
in InterVA, NBC, and Tariff, but strictly as `missing' in InSilicoVA.
Second, the CSMFs are calculated in three different ways. Tariff
calculates CSMF as the proportion of the most likely COD assignments in
the dataset. InterVA-4 calculates CSMF as the aggregated distribution of
up to the top three most likely CODs. And InSilicoVA estimates CSMF
directly from the parametric model using MCMC. Some further feature
comparisons are summarized in Table <ref>.
Feature InterVA Tariff NBC InSilicoVA
Exact replication in the current openVA package Yes No Yes Yes
Implementable without training dataset Yes No No Yes
Can produce instantaneous results for single death Yes Yes Yes No
Accounts for absence of symptoms No No Yes Yes
Accounts for missing symptoms No No No Yes
Provides individual COD distribution Yes No Yes Yes
Direct estimation of CSMF and its uncertainty No No No Yes
Comparison the features of the four VA methods implemented in the openVA package.
§ DATA PREPARATION
In the openVA package, we consider two main forms of standardized
questionnaire: the WHO instrument and the IHME questionnaire. In this
section, we focus on describing these two data formats and tools to
clean and convert data into standardized forms for VA algorithms.
Pre-processing the raw data collected from the Open Data Toolkit is
usually performed with additional packages and software outside of the
analysis pipeline in R. We briefly mention software for data
preprocessing in Section <ref>.
§.§ Standardizing format: WHO
For users familiar with InterVA software and the associated data
processing steps, the standard input format from the WHO 2012 and 2016
instrument is usually well understood. For the 2012 instrument, the data
expected by the InterVA-4 software is organized into a data frame where
each row represents one death, and contains \(246\) fields, starting
from the first item being the ID of the death. The \(245\) items
following the ID each represent one binary variable of
symptom/indicator, where `presence' is coded by `Y', and `absence' is
coded by an empty cell. The details of this format, as well as the
translation from WHO 2012 instrument, could be found at
To accommodate updates for the WHO 2016 instrument
[26], InterVA-5 software accepts a data frame
with \(354\) columns that include \(353\) columns of symptom/indicators
followed by an additional column for the record ID. It should be noted
that the R package InterVA5 retains the format with the
record ID residing in the first column. Another important update with
InterVA-5 is that it acknowledge the difference between both “Yes” and
“No” (or “Y/y” and “N/n”, which is different from the coding
scheme in InterVA-4) are processed as relevant responses, while all
other responses are treated as missing values and ignored. With respect
to the list of causes of death, InterVA-5 utilizes the WHO 2016 COD
categories, which is nearly identical to the WHO 2012 COD categories
(used by InterVA-4) except that haemorrhagic fever and dengue fever are
two separate categories in the 2016 COD categories.
The same input format is inherited by the openVA package, except
for one modification. We further distinguish `missing' and `absent' in
the input data frame explicitly. This is conceptually easy to
understand: knowing that a symptom does not exist provides some
information to the possible cause assignment, while a missing symptom
does not. Missing data could arise from different stages of the data
collection process. Although in theory, most of the VA algorithms could
benefit from distinguishing `missing' from `absent', InSilicoVA is the
only algorithm that has implemented with missing data. We highly
recommend users to pre-process all the raw data so that a `missing'
value in the data spreadsheet is coded by a `.' (following the
stata practice), and an `absent' is coded by a empty cell, as
in the standard InterVA-4 software. For WHO 2016 data, both `.' and `-'
(following the default coding scheme of InterVA-5 software) are
interpreted as missing values. For methods other than InSilicoVA, the
`missing' and `absent' will be considered the same internally and thus
will not introduce compatibility problem.
§.§ Standardizing format: PHMRC
For studies that need to compare with existing work using PHMRC gold
standard dataset [Murray et al., 2011], a practical challenge is
that although the data is publicly accessible, the pre-processing steps
described from the relevant publications are not clear enough nor easy
to implement. The openVA package provides functions to clean up
the PHMRC gold standard data in the best way we could replicate from
First, we allow users to download part or all of the PHMRC gold standard
data directly from its on-line repository:
R> PHMRC_first1000 <- read.csv(getPHMRC_url("adult"), nrows = 1000)
The data can then be cleaned up into a set of dichotomized symptoms, as
described in Section 2 of the supplement material of McCormick et al., 2016.
This dichotomization process was originally described in
Murray et al., 2011. However, following the procedures
described in the original paper, we are not able to reproduce the
thresholds provided with the on-line supplement materials. Therefore we
have included both the thresholds used in Murray et al., 2011
and the thresholds calculated as in their description based on the data
that user inputs. See the following codes for the summary of the two
different transformation results.
R> convert.default <- ConvertData.phmrc(PHMRC_first1000, phmrc.type = "adult",
+ cutoff = "default", cause = "va34")
The first column is site, assign IDs to each row by default
1000 deaths in input. Format: adult
168 binary symptoms generated
Number of Yes 21023
Number of No 124644
Number of Not known 22333
R> convert.adapt <- ConvertData.phmrc(PHMRC_first1000, phmrc.type = "adult",
+ cutoff = "adapt", cause = "va34")
The first column is site, assign IDs to each row by default
1000 deaths in input. Format: adult
168 binary symptoms generated
Number of Yes 21711
Number of No 123956
Number of Not known 22333
Notice that the original PHMRC data are useful for comparing and
validating new methods, as well as for using as training data, but the
cleaning functions require only the columns to be exactly as the PHMRC
gold standard dataset on-line, so they could also be used for new data
that are pre-processed into the same
format[However, since calculating the thresholds adaptively requires the knowledge of underlying cause-of-deaths, the latter approach could not be applied to unlabeled datasets.].
§.§ Other formats
In addition to the standard formats discussed previously, researchers
might also be interested in using customized dichotomous symptoms data
in their analysis. The openVA package also supports customized
input as long as they are dichotomous. In such case, neither the
built-in conditional probability matrix of InterVA nor the PHMRC gold
standard dataset could be used to learn the relationship between
training and testing data, thus some training data with known causes of
death is necessary for all three algorithm. The ConvertData()
function can be used to convert data in a different coding scheme into
the format used by the openVA package. This is demonstrated in the
following small example.
R> toyid <- c("d1", "d2")
R> toycause <- c("A", "B")
R> toyData <- matrix(c("Yes", "No", "Don't know",
+ "Yes", "Refused to answer", "No"),
+ byrow = TRUE, nrow = 2, ncol = 3)
R> toyData <- cbind(toyid, toycause, toyData)
R> colnames(toyData) <- c("ID", "Cause", "S1", "S2", "S3")
R> toyData
ID Cause S1 S2 S3
[1,] "d1" "A" "Yes" "No" "Don't know"
[2,] "d2" "B" "Yes" "Refused to answer" "No"
R> toyDataNew <- ConvertData(toyData, yesLabel = "Yes", noLabel = "No",
+ missLabel = c("Don't know", "Refused to answer"))
R> toyDataNew
ID Cause S1 S2 S3
1 d1 A Y .
2 d2 B Y .
§.§ Convert data between standardized formats
The openVA package supports customized set of symptoms to be used
as input for all methods, thus it is usually less important to convert
data from one format to another, since the conversion inevitably creates
loss of information. We recognize the exact mapping of symptoms between
different format can be useful for some applications. For example, a
full mapping of PHMRC dataset into the InterVA format enables the use of
physician provided conditional probabilities included in the InterVA
software. This remains as an important feature to be added to the
package in the future.
§ FITTING VA CAUSE-OF-DEATH ASSIGNMENT MODELS
In this section, we demonstrate model fitting process in the
openVA package using two types of datasets: (1) a random sample of
\(1,000\) deaths from ALPHA network without gold standard labels
collected with the WHO 2012 instrument, and (2) the adult VA records in
the PHMRC gold standard data, with the \(1,554\) records from Andhra
Pradesh, India used as testing set and the rest as training dataset. In
the first case without gold standard training data, only InterVA and
InSilicoVA can be fitted. All four methods can be fitted on the second
§.§ Modeling data collected with WHO 2012 questionnaire
The randomly sampled VA records from ALPHA network sites are already
included in the openVA package and can be loaded directly.
R> data(RandomVA1)
R> dim(RandomVA1)
[1] 1000 246
R> head(RandomVA1[, 1:10])
ID elder midage adult child under5 infant neonate male female
1 d1 Y Y
2 d2 Y Y
3 d3 Y Y
4 d4 Y Y
5 d5 Y Y
6 d6 Y Y
The codeVA() function provides a standardized syntax to fit
different VA models. Internally, the codeVA() function organizes
the input data into standardized formats, checks for incompatibility of
the data and specified model, and calls the specified model fitting
functions. It returns a classed object of the specified model class. In
this example, we use the version 4.03 of InterVA software, which is the
latest release of the original software compatible with the WHO 2012
instrument. Any additional model-specific parameters can be passed
through the arguments of codeVA(). Here we specify the HIV and
malaria prevalence levels required by the InterVA model to be `high'.
R> fit_inter_who <- codeVA(data = RandomVA1, data.type = "WHO2012",
+ model = "InterVA", version = "4.03",
+ HIV = "h", Malaria = "h")
R> summary(fit_inter_who)
InterVA-4 fitted on 1000 deaths
CSMF calculated using reported causes by InterVA-4 only
The remaining probabilities are assigned to 'Undetermined'
Top 5 CSMFs:
cause likelihood
Undetermined 0.156
HIV/AIDS related death 0.122
Stroke 0.072
Reproductive neoplasms MF 0.058
Pulmonary tuberculosis 0.055
We can implement InSilicoVA method with the similar syntax. We use the
default parameters and run the MCMC for \(10,000\) iterations. Setting
the auto.length argument to FALSE specifies that the algorithm
does not automatically increase the length of the chain when convergence
failed. In addition to this simple interface, the algorithm
implementation in the InSilicoVA packages allows many more types
of model customization and extensions that cater to specific practical
modeling needs. We will briefly discuss some unique features in Section
R> fit_ins_who <- codeVA(RandomVA1, data.type = "WHO2012", model = "InSilicoVA",
+ Nsim = 10000, auto.length = FALSE)
R> summary(fit_ins_who)
InSilicoVA Call:
1000 death processed
10000 iterations performed, with first 5000 iterations discarded
250 iterations saved after thinning
Fitted with re-estimated conditional probability level table
Data consistency check performed as in InterVA4
Top 10 CSMFs:
Mean Std.Error Lower Median Upper
Other and unspecified infect dis 0.266 0.0168 0.235 0.265 0.301
HIV/AIDS related death 0.102 0.0091 0.085 0.102 0.119
Renal failure 0.101 0.0108 0.084 0.101 0.123
Other and unspecified neoplasms 0.062 0.0089 0.046 0.061 0.080
Other and unspecified cardiac dis 0.058 0.0076 0.044 0.058 0.075
Digestive neoplasms 0.050 0.0077 0.033 0.050 0.065
Acute resp infect incl pneumonia 0.048 0.0073 0.034 0.049 0.063
Pulmonary tuberculosis 0.039 0.0068 0.025 0.039 0.054
Stroke 0.038 0.0061 0.027 0.038 0.052
Other and unspecified NCD 0.034 0.0089 0.018 0.034 0.052
§.§ Modeling the PHMRC data
For the PHMRC gold standard dataset, we first load the complete dataset
from its on-line repository, and then organize them into training and
testing splits.
R> PHMRC_all <- read.csv(getPHMRC_url("adult"))
R> is.test <- which(PHMRC_all$site == "AP")
R> test <- PHMRC_all[is.test, ]
R> train <- PHMRC_all[-is.test, ]
R> dim(test)
\end{CodeInput}
\begin{CodeOutput}
[1] 1554 946
\end{CodeOutput}
\begin{CodeInput}
R> dim(train)
\end{CodeInput}
\begin{CodeOutput}
[1] 6287 946
\end{CodeOutput}
\end{CodeChunk}
In order to fit the models on the PHMRC data, we specify
\code{data.type = "PHMRC"}, and the column of the causes of death label
in the training data. The rest of the syntax is similar to the previous
example. When the input consist of both training and testing data, the
InterVA and InSilicoVA algorithm requires a small modification that map
the conditional probabilities of symptoms given causes in the training
dataset to a letter grade system assumed by the model. In such case, the
\code{version} argument for the InterVA algorithm is no longer needed.
There are several ways to do this conversion, specified by the
\code{convert.type} argument. The \code{convert.type = "quantile"}
performs the mapping so that the percentile of each rank stays the same
as the original \(P_{s|c}\) matrix in InterVA software. Alternatively we
can also use the original fixed values of translation, and assign letter
grades closest to each entry in \(\hat{P}_{s|c}\). This conversion is
specified by \code{convert.type = "fixed"}, and is more closely aligned
to the original InterVA and InSilicoVA setting. Finally, we can also
directly use the values in the \(\hat{P}_{s|c}\) without converting them
to ranks and re-estimating the values associated with each rank. This
can be specified by \code{convert.type = "empirical"}. In this
demonstration, we assume the fixed value conversion.
\begin{CodeChunk}
\begin{CodeInput}
R> fit_inter <- codeVA(data = test, data.type = "PHMRC", model = "InterVA",
+ data.train = train, causes.train = "gs_text34",
+ phmrc.type = "adult", convert.type = "fixed")
\end{CodeInput}
\end{CodeChunk}
\begin{CodeChunk}
\begin{CodeInput}
R> fit_ins <- codeVA(data = test, data.type = "PHMRC", model = "InSilicoVA",
+ data.train = train, causes.train = "gs_text34",
+ phmrc.type = "adult", convert.type = "fixed",
+ Nsim=10000, auto.length = FALSE)
\end{CodeInput}
\end{CodeChunk}
The NBC and Tariff method, on the other hand, does not need to perform
such conversion, and only need to change the \code{model} argument.
\begin{CodeChunk}
\begin{CodeInput}
R> fit_nbc <- codeVA(data = test, data.type = "PHMRC", model = "NBC",
+ data.train = train, causes.train = "gs_text34",
+ phmrc.type = "adult")
\end{CodeInput}
\end{CodeChunk}
\begin{CodeChunk}
\begin{CodeInput}
R> fit_tariff <- codeVA(data = test, data.type = "PHMRC", model = "Tariff",
+ data.train = train, causes.train = "gs_text34",
+ phmrc.type = "adult")
\end{CodeInput}
\end{CodeChunk}
Notice that we do not need to transform the PHMRC data manually as
described in the previous section. Data transformations are performed
automatically within the \code{codeVA()} function. The arguments in
\code{ConvertData.phmrc()} can also be passed into \code{codeVA()}.
\section{Summarizing results}
\label{sec:results}
In this section we demonstrate how to summarize results, extract output,
and visualize and compare fitted results. All the fitted object returned
by \code{codeVA()} are \code{S3} objects, where readable summary of
model results can be obtained with the \code{summary()} function as
shown in the previous section. In addition, several other metrics are
commonly used to evaluate and compare VA algorithms at either the
population or individual levels. In the rest of this section, we show
how to easily calculate and visualize some of these metrics with the
\pkg{openVA} package.
\subsection{CSMF Accuracy}
We can extract the CSMFs directly using the \code{getCSMF()} function.
The function returns a vector of the point estimates of the CSMFs, or a
matrix of posterior summaries of the CSMF for the InSilicoVA algorithm.
\begin{CodeChunk}
\begin{CodeInput}
R> csmf_inter <- getCSMF(fit_inter)
R> csmf_ins <- getCSMF(fit_ins)
R> csmf_nbc <- getCSMF(fit_nbc)
R> csmf_tariff <- getCSMF(fit_tariff)
\end{CodeInput}
\end{CodeChunk}
One commonly used metric to evaluate the CSMF estimates is the so-called
CSMF accuracy, defined as \[
CSMF_{acc} = 1 - \frac{\sum_i^C CSMF_i - CSMF_i^{(true)}}{2(1 - \min CSMF^{(true)})}
\] We use the empirical distribution in the test data to calculate the
true CSMF distribution and evaluate the CSMF accuracy using the
\code{getCSMF_accuracy()} function. As discussed previously, the default
CSMF calculation is slightly different for different methods. For
example, the InterVA algorithm creates the additional category of
\texttt{Undetermined} by default, which is not in the true CSMF
categories and needs to be specified. The creation of the undetermined
category can also be suppressed by \code{interVA.rule = FALSE} in the
\code{getCSMF()} function call. For the InSilicoVA algorithm, we will
use the posterior mean to calculate the point estimates of the CSMF
\begin{CodeChunk}
\begin{CodeInput}
R> csmf_true <- table(c(test$gs_text34, unique(PHMRC_all$gs_text34))) - 1
R> csmf_true <- csmf_true / sum(csmf_true)
R> c(getCSMF_accuracy(csmf_inter, csmf_true, undet = "Undetermined"),
+ getCSMF_accuracy(csmf_ins[, "Mean"], csmf_true),
+ getCSMF_accuracy(csmf_nbc, csmf_true),
+ getCSMF_accuracy(csmf_tariff, csmf_true))
\end{CodeInput}
\begin{CodeOutput}
[1] 0.58 0.74 0.77 0.68
\end{CodeOutput}
\end{CodeChunk}
\subsection{Individual COD summary}
At each individual level, we can extract the most likely cause-of-death
assignment from the fitted object using the \code{getTopCOD()} function.
\begin{CodeChunk}
\begin{CodeInput}
R> cod_inter <- getTopCOD(fit_inter)
R> cod_ins <- getTopCOD(fit_ins)
R> cod_nbc <- getTopCOD(fit_nbc)
R> cod_tariff <- getTopCOD(fit_tariff)
\end{CodeInput}
\end{CodeChunk}
With the most likely COD assignment, other types of metrics based on
individual COD assignment accuracy can be similarly constructed by
users. The summary methods can also be called for each death ID. For
example, using Tariff method, we can extract the fitted rankings of
causes for the death with ID 6288 by
\begin{CodeChunk}
\begin{CodeInput}
R> summary(fit_inter, id = "6288")
\end{CodeInput}
\begin{CodeOutput}
InterVA-4 fitted top 5 causes for death ID: 6288
Cause Likelihood
Stroke 0.509
Pneumonia 0.318
COPD 0.081
Other Infectious Diseases 0.064
Renal Failure 0.013
\end{CodeOutput}
\end{CodeChunk}
Notice that the direct call of summary for InSilcoVA does not provide
uncertainty estimates for individual COD assignments. This is because
the calculation of individual posterior probabilities of COD
distribution is relatively time-consuming and memory-intense. To obtain
individual-level uncertainty measurement, we can either run the MCMC
chain with the additional argument \code{indiv.CI = 0.95} when calling
\code{codeVA()}, or update the fitted object directly with the saved
posterior draws.
\begin{CodeChunk}
\begin{CodeInput}
R> fit_ins <- updateIndiv(fit_ins, CI = 0.95)
R> summary(fit_ins, id = "6288")
\end{CodeInput}
\begin{CodeOutput}
InSilicoVA fitted top causes for death ID: 6288
Credible intervals shown: 95%
Mean Lower Median Upper
Stroke 0.5043 0.3485 0.5083 0.6361
Pneumonia 0.4116 0.2615 0.4083 0.5834
Other Infectious Diseases 0.0660 0.0411 0.0642 0.0966
Epilepsy 0.0099 0.0064 0.0097 0.0142
COPD 0.0053 0.0031 0.0052 0.0079
Malaria 0.0007 0.0005 0.0007 0.0011
Diabetes 0.0005 0.0003 0.0005 0.0009
Acute Myocardial Infarction 0.0004 0.0003 0.0004 0.0006
Falls 0.0004 0.0001 0.0004 0.0013
Renal Failure 0.0004 0.0002 0.0003 0.0005
\end{CodeOutput}
\end{CodeChunk}
For \(N\) deaths, \(C\) causes, the posterior mean of individual COD
distributions returned by the InSilicoVA model, along with median and
with credible intervals can be represented by a
\((N \times C \times 4)\)-dimensional array. The function
\code{getIndivProb()} extracts this summary in the form of a list of
\(4\) matrices of dimension \(N\) by \(C\), which can then be saved to
other formats to facilitate further analysis. For other methods, the
point estimates of individual COD distribution are returned as the \(N\)
by \(C\) matrix.
\begin{CodeChunk}
\begin{CodeInput}
R> fit_prob <- getIndivProb(fit_inter)
R> dim(fit_prob)
\end{CodeInput}
\begin{CodeOutput}
[1] 1554 34
\end{CodeOutput}
\end{CodeChunk}
\subsection{Visualization}
The previous sections discuss how results could be extracted and
examined in \proglang{R}. In this subsection, we show some visualization
tools provided in the \pkg{openVA} package for presenting these results.
The fitted CSMFs for the top causes can be easily visualized by the
\code{plotVA()} function. The default graph components are specific to
each algorithm and individual package implementations, with options for
further customization.
\begin{CodeChunk}
\begin{CodeInput}
R> plotVA(fit_inter, title = "InterVA")
R> plotVA(fit_ins, title = "InSilicoVA", bw = TRUE)
R> plotVA(fit_nbc, title = "NBC")
R> plotVA(fit_tariff, title = "Tariff")
\end{CodeInput}
\begin{figure}[ht]
{\centering \includegraphics[width=0.49\linewidth,]{openVA-JSS_files/figure-latex/vis-1-1} \includegraphics[width=0.49\linewidth,]{openVA-JSS_files/figure-latex/vis-1-2} \includegraphics[width=0.49\linewidth,]{openVA-JSS_files/figure-latex/vis-1-3} \includegraphics[width=0.49\linewidth,]{openVA-JSS_files/figure-latex/vis-1-4}
\caption[Plot of top 10 CSMFs estimated by InterVA, InSilicoVA, NBC, and Tariff]{Plot of top 10 CSMFs estimated by InterVA, InSilicoVA, NBC, and Tariff.}\label{fig:vis-1}
\end{figure}
\end{CodeChunk}
The CSMFs can also be aggregated for easier visualization of groups of
causes. For InterVA-4 cause list, we included an example grouping built
into the package, so the aggregated CSMFs can be compared directly. In
practice, the grouping of causes of deaths often needs to be determined
according to context and the research question of interest. Changing the
grouping can be easily achieved by modifying the \code{grouping}
argument in \code{stackplotVA()} function. For example, to facilitate
the new category of \texttt{Undetermined} returned by InterVA, we first
modify the grouping matrix to include it as a new cause:
\begin{CodeChunk}
\begin{CodeInput}
R> data(SampleCategory)
R> grouping <- SampleCategory
R> grouping[,1] <- as.character(grouping[,1])
R> grouping <- rbind(grouping, c("Undetermined", "Undetermined"))
R> compare <- list(InterVA4 = fit_inter_who,
+ InSilicoVA = fit_ins_who)
R> stackplotVA(compare, xlab = "", angle = 0, grouping = grouping)
\end{CodeInput}
\begin{figure}[H]
{\centering \includegraphics[width=.7\linewidth,]{openVA-JSS_files/figure-latex/vis-2-1}
\caption[Comparing aggregated CSMF for InterVA-4 and InSilicoVA, adding undetermined category]{Comparing aggregated CSMF for InterVA-4 and InSilicoVA, adding undetermined category.}\label{fig:vis-2}
\end{figure}
\end{CodeChunk}
The ordering of the stacked bars can also be changed to reflect the
structures within the aggregated causes.
\begin{CodeChunk}
\begin{CodeInput}
R> order.group <- c("TB/AIDS", "Communicable", "NCD", "External", "Maternal",
+ "causes specific to infancy", "Undetermined")
R> stackplotVA(compare, xlab = "", angle = 0, grouping = grouping,
+ order.group = order.group)
\end{CodeInput}
\begin{figure}[H]
{\centering \includegraphics[width=.7\linewidth,]{openVA-JSS_files/figure-latex/vis-3-1}
\caption[Comparing aggregated CSMF for InterVA-4 and InSilicoVA, with the causes reordered]{Comparing aggregated CSMF for InterVA-4 and InSilicoVA, with the causes reordered.}\label{fig:vis-3}
\end{figure}
\end{CodeChunk}
\section{Incorporating additional information}
\label{sec:insilico}
Among the VA methods discussed in this paper, the InSilicoVA algorithm
[McCormick \emph{et~al.}, 2016] allows more flexible modifications to the Bayesian
hierarchical model structure when additional information is available.
In this section, we illustrate two features unique to the InSilicoVA
method: jointly estimating CSMFs from multiple population, and
incorporating partial and potentially noisy physician codings in the
\subsection{Sub-population specific CSMFs}
In practice researchers may want to estimate and compare CSMFs for
different regions, time periods, or demographic groups in the
population. Running separate models on subsets of data can be
inefficient and does not allow each parameter estimation to borrow
information across different groups. The generative framework adopted by
InSilicoVA allows the specification of sub-population in analyzing VA
data. Consider an input dataset with \(G\) different sub-population, we
can estimate different CSMFs \(\pi^{(g)}\) for \(g = 1, ..., G\) for
each sub-population, while assuming the same conditional probability
matrix, \(P_{s|c}\) and other hyperpriors. As an example, we show how to
estimate different CSMFs for sub-populations specified by sex and age
groups, using a randomly sampled ALPHA dataset with additional columns
specifying the sub-population each death belongs to.
\begin{CodeChunk}
\begin{CodeInput}
R> data(RandomVA2)
R> head(RandomVA2[, 244:248])
\end{CodeInput}
\begin{CodeOutput}
stradm smobph scosts sex age
1 . . . Men 60+
2 . . . Women 60-
3 . . . Women 60-
4 . . . Women 60+
5 . . . Women 60-
6 . . . Women 60-
\end{CodeOutput}
\end{CodeChunk}
Then we can fit the model with one or multiple additional columns
specifying sub-population membership for each observation.
\begin{CodeChunk}
\begin{CodeInput}
R> fit_sub <- codeVA(RandomVA2, model = "InSilicoVA",
+ subpop = list("sex", "age"), indiv.CI = 0.95,
+ Nsim = 10000, auto.length = FALSE)
\end{CodeInput}
\end{CodeChunk}
Functions discussed in the previous sections works in the same way for
the fitted object with multiple sub-populations. Additional
visualization tools are also available. By specify
\code{type = "compare"}, we can plot the CSMFs for two sub-populations
on the same plot.
\begin{CodeChunk}
\begin{CodeInput}
R> plotVA(fit_sub, type = "compare", title = "Comparing CSMFs", top = 3)
\end{CodeInput}
\begin{figure}[H]
{\centering \includegraphics[width=0.9\linewidth,]{openVA-JSS_files/figure-latex/ins3-1}
\caption[Comparing CSMF for different sub-population]{Comparing CSMF for different sub-population.}\label{fig:ins3}
\end{figure}
\end{CodeChunk}
By default, the comparison plots will select all the CODs that are
included in the top \(K\) for each of the sub-populations. We can also
plot only subsets of them by specifying with causes are of interest.
\begin{CodeChunk}
\begin{CodeInput}
R> plotVA(fit_sub, type = "compare", title = "Comparing CSMFs",
+ causelist = c("HIV/AIDS related death",
+ "Pulmonary tuberculosis",
+ "Other and unspecified infect dis",
+ "Other and unspecified NCD"))
\end{CodeInput}
\begin{figure}[H]
{\centering \includegraphics[width=0.8\linewidth,]{openVA-JSS_files/figure-latex/ins4-1}
\caption[ Comparing the fraction of deaths due to selected CODs for different sub-populations]{ Comparing the fraction of deaths due to selected CODs for different sub-populations.}\label{fig:ins4}
\end{figure}
\end{CodeChunk}
If a single sub-population is selected by the \code{which.sub} argument,
the top CSMFs of the chosen sub-population will be plotted in the same
way as before.
\begin{CodeChunk}
\begin{CodeInput}
R> plotVA(fit_sub, which.sub = "Women 60-", title = "Women 60-")
\end{CodeInput}
\begin{figure}[H]
{\centering \includegraphics[width=0.8\linewidth,]{openVA-JSS_files/figure-latex/ins5-1}
\caption[Top 10 CSMF for a specified sub-population]{Top 10 CSMF for a specified sub-population.}\label{fig:ins5}
\end{figure}
\end{CodeChunk}
The \code{stackplot()} function can also be used to compare different
sub-populations in aggregated cause groups.
\begin{CodeChunk}
\begin{CodeInput}
R> stackplot(fit_sub)
R> stackplot(fit_sub, type = "dodge")
\end{CodeInput}
\begin{figure}[H]
{\centering \includegraphics[width=0.49\linewidth,]{openVA-JSS_files/figure-latex/ins6-1} \includegraphics[width=0.49\linewidth,]{openVA-JSS_files/figure-latex/ins6-2}
\caption[Comparing aggregated CSMF for four different sub-population]{Comparing aggregated CSMF for four different sub-population.}\label{fig:ins6}
\end{figure}
\end{CodeChunk}
\subsection{Physician coding}
When physician coded cause of death is available for all or a subset of
the death, we can incorporate such information in the InSilicoVA model.
The physician coded causes can be either the same as the CODs used for
the final algorithm, or a higher level aggregation of them. When there
are more than one physician codes for each death and the physician
identity is known, we can first de-bias the multiple codes provided from
different physicians using the process described in McCormick \emph{et~al.}, 2016.
For the purpose of implementation, we only need to specify which columns
are physician IDs, and which are their coded causes respectively.
\begin{CodeChunk}
\begin{CodeInput}
R> data(SampleCategory)
R> data(RandomPhysician)
R> head(RandomPhysician[, 245:250])
\end{CodeInput}
\begin{CodeOutput}
smobph scosts code1 rev1 code2 rev2
1 . . NCD doc9 NCD doc6
2 . . NCD doc4 NCD doc3
3 . . NCD doc1 NCD doc5
4 . . TB/AIDS doc4 TB/AIDS doc7
5 . . TB/AIDS doc5 TB/AIDS doc9
6 . . Communicable doc9 Communicable <NA>
\end{CodeOutput}
\end{CodeChunk}
\begin{CodeChunk}
\begin{CodeInput}
R> doctors <- paste0("doc", c(1:15))
R> causelist <- c("Communicable", "TB/AIDS", "Maternal",
+ "NCD", "External", "Unknown")
R> phydebias <- physician_debias(RandomPhysician,
+ phy.id = c("rev1", "rev2"), phy.code = c("code1", "code2"),
+ phylist = doctors, causelist = causelist, tol = 0.0001, max.itr = 100)
\end{CodeInput}
\end{CodeChunk}
The de-biased step essentially creates a prior probability distribution
for each death over the broader categories of causes. Then to run
InSilicoVA with the de-biased physician coding, we can simply pass the
fitted object from the previous step to the model. Additional arguments
are needed to specify the external cause category, since they are
handled by separated heuristics, and the unknown category, which is
equivalent to an uniform probability distribution over all other
categories, i.e., the same as the case where no physician coding exist.
\begin{CodeChunk}
\begin{CodeInput}
R> fit_ins_phy <- codeVA(RandomVA1, model = "InSilicoVA",
+ phy.debias = phydebias, phy.cat = SampleCategory,
+ phy.external = "External", phy.unknown = "Unknown",
+ Nsim = 10000, auto.length = FALSE)
\end{CodeInput}
\end{CodeChunk}
This can be compared with the previous results without including
physicians codes.
\begin{CodeChunk}
\begin{CodeInput}
R> plotVA(fit_ins_who, title = "Without physician coding")
R> plotVA(fit_ins_phy, title = "With physician coding")
\end{CodeInput}
\begin{figure}[!h]
{\centering \includegraphics[width=0.7\linewidth,]{openVA-JSS_files/figure-latex/phy3-1} \includegraphics[width=0.7\linewidth,]{openVA-JSS_files/figure-latex/phy3-2}
\caption[Comparing fitted CSMF with and without physicians]{Comparing fitted CSMF with and without physicians}\label{fig:phy3}
\end{figure}
\end{CodeChunk}
\section{Other related software packages}
\label{sec:other}
Since the release of the \pkg{openVA} package on CRAN, there have been
many new developments in both methodology and software implementations
that build on the \pkg{openVA} suite of packages and further extended
its functionalities. Here we briefly survey some of the related methods,
software packages, and ongoing work in the VA community that are related
to \pkg{openVA}.
First, the ability to easily fit and compare existing methods using the
\pkg{openVA} package facilitated the development of several new VA
methods in the last several years. Most of the development focuses on
building statistical models to relax the conditional independence
assumption of symptoms given a cause of death
\citep[e.g.,][]{li2017mix, tsuyoshi2017, moran2021bayesian}. These
methods tend to be computationally more demanding compared to the
algorithms currently included in the \pkg{openVA} package, but usually
provide improved inference. It is future work to include some of these
latest developments to the \pkg{openVA} package for routine use. Most of
these methods have publicly available implementations, such as the
\pkg{farva} package [Moran, 2020] on Github. In another direction of
research, a class of transfer learning methods focuses on building
models to correct for bias in existing methods when predicting out of
domain. These methods take the predicted cause of death assignments and
distributions obtained with the \pkg{openVA} package and learn an
ensemble of the predictions calibrated to the target population
[Datta \emph{et~al.}, 2020, Fiksel \emph{et~al.}, 2021]. The \pkg{calibratedVA} package
[Fiksel and Datta, 2018] is available to implement these models.
Outside of research community that develops new VA algorithms,
\pkg{openVA} has also been used extensively by governments and health
care organizations, particularly in locations that lack a strong vital
registration system and use VA data to identify the leading causes of
death. To facilitate the work of these users, \pkg{openVA} has been
wrapped into a web application, the openVA App [Thomas, 2021], using
the \pkg{shiny} package. The open source openVA App is available on
GitHub and provides an intuitive interface to \pkg{openVA} that does not
require one to learn \proglang{R}, but provides visual and tabular
output produced by the different VA algorithms. It also runs the
official Tariff algorithm by calling the \proglang{Python} source code
of SmartVA-Analyze [Institute for Health Metrics and Evaluation}, 2021] and processing the output to be
consistent with the other algorithms. The \pkg{openVA} \proglang{R}
package has also been a key component in a larger data analysis pipeline
that pulls VA data from an Open Data Kit (ODK) Collect server and
deposits the assigned causes of death to another server running the
District Health Information Software 2 (DHIS2), which is a common
information management system used in low and middle-income countries.
This open source software is implemented as a \proglang{Python} package,
\pkg{openva-pipeline}, and is available on GitHub and the Python Package
Index [Thomas \emph{et~al.}, 2021]. Finally, the \proglang{R} package
\pkg{CrossVA} [Thomas \emph{et~al.}, 2020] and the \proglang{Python} package
\pkg{pyCrossVA} [Choi and Thomas, 2021] provide additional toolkit to convert
raw VA data from its original format from the ODK server to the
standardized formats discussed before. Both packages are open source and
available on GitHub. The \pkg{pyCrossVA} package is also available on
the Python Package Index.
\section{Conclusion}
\label{sec:conclusion}
In this paper, we introduce the \pkg{openVA} package. This is the first
open-sourced software that implements and compares the major VA methods.
The \pkg{openVA} package allows researchers to easily access the most
recent tools that are previously difficult to work with or unavailable.
It also enables the compatibility of multiple data input formats and
significantly reduces the tedious work to pre-process different data
formats specific to each algorithm. The software framework of the
\pkg{openVA} package allows the integration of new methods in the
future. The \pkg{openVA} package makes all the steps involved in
analyzing VA data, i.e., data processing, model tunning and fitting,
result summarization, and evaluation metrics, transparent and
reproducible. This contributes significantly to the public health
community using VA.
Finally, we note that several features that will be helpful for future
development. First, many users of the \pkg{openVA} package may not be
familiar with the command line tools or does not have access to
\proglang{R} on their local machines. A well designed graphical user
interface can be very useful in such settings. The work on the shiny web
application, openVA app, is a first step towards making the package more
accessible. The authors intend to extend it to a better front end hosted
on secure centralized servers. Second, although in this paper, we aim to
provide users with all the available methods for assigning causes of
death, without comparing the accuracy and robustness between them, much
future work is needed to systematically assess, compare, and combine
these methods in a better analysis framework. Finally, the development
of VA algorithms is still an active area of research and it would be
possible to extend the \pkg{openVA} suite to incorporate better VA
algorithms and new types of data such as free-text narratives.
\section*{Acknowledgement}
This work was supported by grants K01HD078452, R01HD086227, and
R21HD095451 from the Eunice Kennedy Shriver National Institute of Child
Health and Human Development (NICHD).
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\appendix
\section{New changes to the InSilicoVA algorithm}
\label{app:insilico}
Since the original publication of McCormick \emph{et~al.}, 2016, several major
changes have been introduced in the InSilicoVA algorithm implementation
to improve cause-of-death assignments. Two major changes are summarized
\subsection{Removal of physically impossible causes}\label{sec:impossible}
The originally proposed InSilicoVA assumes all causes of death are
possible for each observation. The impact form such assumption is mild
when data are abundant, but could be problematic when either sample size
is small or the proportion of missing data is high. In both cases,
physically impossible causes might get assigned with non-ignorable
posterior mass. Since the version 1.1.5 of \pkg{InSilicoVA}, the
algorithm automatically checks and removes impossible causes before
fitting the model. The \(k\)-th cause is defined as physically
impossible for the \(i\)-th death if \(P(s_{ij}=1|y_{i}=k) = 0\) for any
\(j\) representing either sex or age group that presents. We then
consider a cause to be physically impossible to the underlying
population if it is impossible for all the observations of the input
data. For example, with the new implementation, CSMF for
pregnancy-related causes will not be estimated if the input data consist
of only male deaths.
\subsection{Structured symptom dependence}\label{sec:dependence}
Another change in InSilicoVA algorithm is the implementation of the data
consistency check. Inherited from the InterVA-4 algorithm, InSilicoVA
performs a data consistency check before fitting the algorithm as
described in Algorithm \ref{alg:check}. This consistency check consists
of two parts. First, it removes indicators whose presence contradicts
the existence of a lower-level indicator. For example, if the question
\textit{pregnant at the time of death} is answered with an `Yes', then
the algorithm makes sure that the item \textit{child} is set to `No'.
The second part of the algorithm checks the existence of each
higher-level indicator when lower-level indicators that implied by it
exists. Again, if the question \textit{pregnant at the time of death} is
answered with an `Yes', then the algorithm makes sure that the item
\textit{female} is set to `Yes' as well.
\begin{algorithm}
\caption{Data Check}
\label{alg:check}
\begin{algorithmic}[1]
\Statex
\Require
\Statex $\mathbf{s}_j$ = vector of $j=1 \mbox{ to } 245$ indicators for one death \vspace{3pt}
\Statex $\mbox{notask}(s)$: higher-level indicator(s) whose presence ensures indicator $s$ does not exist \vspace{3pt}
\Statex $\mbox{anc}(s)$: higher-level indicator(s) that must exist if indicator $s$ exists \vspace{3pt}
\Statex
\Ensure $\mathbf{s}^*$ = modified vector of $j=1 \mbox{ to } 245$ indicators for one death \vspace{3pt}
\Statex
\For{each indicator $j = 1 \mbox{ to } 245$}
\State Set $ \mathbf{s}^*_j \leftarrow \mathbf{s}_j$ \Comment{Initialize}
\EndFor
\For{each indicator $j = 1 \mbox{ to } 245$} \label{line:start}
\If{there exists $\mathbf{s}^*_{\ell}$ in $\mbox{notask}(\mathbf{s}^*_j)$ and $\mathbf{s}^*_{\ell}$ is `yes'}\label{line:firstif}
\State $\mathbf{s}^*_j \leftarrow$ `no' \Comment{This indicator is nonsensical, set to `No'} \label{line:changetomissing}
\EndIf
\If{there exists $\mathbf{s}^*_{\ell}$ in $\mbox{anc}(\mathbf{s}_j)$ and $\mathbf{s}^*_j$ is `yes'}\label{line:secondif}
\State $\mathbf{s}^*_\ell \leftarrow$ `yes' \Comment{Set more general version of this indicator to `yes'}
\EndIf
\EndFor \label{line:end}
\State Repeat loop in lines \ref{line:start} to line \ref{line:end} again \Comment{Processes 2-level hierarchies}
\Statex
\end{algorithmic}
\end{algorithm}
This data check procedure ensures the collected data are internal
consistent, yet it might introduce additional complications under the
common assumption that symptoms are conditionally independent given any
cause of death. This can be illustrated with a simple example below. In
calculating the conditional probability of observing two symptoms, e.g.,
`infant', and `recent abortion', we can decompose
\begin{eqnarray}\nonumber
\mbox{Pr}(\mbox{infant} = Y \;\&\; \mbox{recent abortion} = N \;|\; \mbox{some cause})
\mbox{Pr}(\mbox{infant} = Y \;|\; \mbox{some cause})\\\nonumber
\mbox{Pr}(\mbox{recent abortion} = N \;|\; \mbox{some cause}).
\end{eqnarray} However, such independence does not hold since the two
symptoms are mutually exclusive, i.e., the presence of symptom `infant'
implies no recent abortion. In fact, the structure of the WHO instrument
assures that the lower-level question will not be asked at all if the
interview is about an infant death. Thus in practice, the above
probability calculation yields an underestimate of the target value and
it should be calculated simply with \[
\mbox{Pr}(\mbox{infant} = Y \;\&\; \mbox{recent abortion} = N \;|\; \mbox{some cause})
\mbox{Pr}(\mbox{infant} = Y \;|\; \mbox{some cause})
\] To reflect the different joint probability decompositions when the
known symptom hierarchy is available, it turns out we can easily modify
the data check algorithm by imputing missing instead of absence to the
lower level symptoms in line \ref{line:changetomissing}. We have found
that such a change usually yield more reasonable CSMF estimates,
especially for child and neonate deaths, in practice. Also it should be
noticed that this problem stems from the conditional independence
assumption that has been adopted in all other methods besides
InSilicoVA. The computational trick here only eliminates the bias from
mutually exclusive symptoms known from the hierarchical structure of VA
questionnaires. More systematic ways to deal with symptom correlations
are needed to further improve the estimation.
With the update of the WHO VA questionnaire to the 2016 format, three
major changes were made to the data check procedure implemented by
InterVA-5. First, similar to InSilicoVA, InterVA-5 also imputes missing
instead of absence to the lower level symptoms in line
\ref{line:changetomissing}. Second, the hierarchical structure of the
2016 VA questionnaire makes use of both ``Yes'' and ``No'' (or ``Y'' and
``N'') as substantive values that trigger the asking of lower-level
questions. For example, if the indicator/question ``Did the baby ever
cry?'' has the response ``No'', then the following indicator/question,
``Did the baby cry immediately after birth, even if only a little bit?''
should not be asked -- as implemented with the notask(\(s\)) function in
Algorithm \ref{alg:check}. Thus, additional steps are needed in the
InterVA-5 data check which test the value of each indicator with the
appropriate substantive response that implies subsequent action (i.e.,
asking or not asking a different indicator/question) in line
\ref{line:firstif} and \ref{line:secondif}. The last major change is
that an additional check is made for indicators/questions that only
apply to neonates. If a record for a non-neonate contains a value for an
indicator/question that only applies to neonates, then that value is
cleared in the working copy of the data file. The same check rules are
used by InSilicoVA for WHO 2016 data as well. The implementation is
slightly different between InSilicoVA and InterVA-5. InterVA-5 updates
\(s_j^*\) to be missing in line \ref{line:changetomissing} only when it
is recorded to be the corresponding substantive value, since the
non-substantive value is considered the same as missing values in the
calculation. InSilicoVA, on the other hand, make the update regardless
of the value of \(s_j^*\), since the non-substantive value and missing
are treated differently in the algorithm.
\end{document} |
# Servicifying zk-SNARKs Execution for
Verifiable Off-chain Computations ††thanks:
Alvaro Alonso Domenech, Jonathan Heiss, Stefan Tai Information System
Engineering
TU Berlin
Berlin, Germany
<EMAIL_ADDRESS>
###### Abstract
Zk-SNARKs help scale blockchains with Verifiable Off-chain Computations (VOC).
zk-SNARK DSL toolkits are key when designing arithmetic circuits but fall
short of automating the subsequent proof-generation step in an automated
manner. We emphasize the need for portability, interoperability, and
manageability in VOC-based solutions and introduce a Proving Service that is
designed to provide a scalable and reusable solution for generating zk-SNARK
proofs leveraging clouds.
###### Index Terms:
Blockchain, Zero-knowledge Proofs, zk-SNARKs, ZoKrates, Proving, Service
Architecture, Cloud Computing
## I Introduction
The succinctness property and the short verification times of zk-SNARK s come
at the cost of large computational complexity, static execution models, and
memory overheads during the proof generation process [1]. These limitations
represent a problem in VOC applications which require handling large and
varying workloads, like rollups. This type of application would benefit from
scalable, interoperable, and manageable system environments like clouds.
However, DSL toolkits like ZoKrates [2] or Circom [3] concentrate on circuit
development and put little emphasis on how to integrate these circuits into
production systems. Bridging the gap between advanced cryptography and modern
systems engineering also helps to create standardized benchmarks for proving
systems and tools.
## II Proving Service
We introduce a service-oriented approach for VOC that facilitates the use of
the cryptographic procedures of zk-SNARK s within cloud system architectures.
Our system allows for executing arithmetic circuits as encapsulated
application logic in containers that are deployable to different machines
realizing scalability, provide interoperability with other services, and are
better manageable. For that, we give an overview of the service-oriented
system architecture and describe the internals of the proving service.
### II-A Service Oriented Verifiable Off-chain Computation
The problems above can be fulfilled through a service-oriented approach as
depicted in Figure 1. Starting from a higher-level system’s perspective, we
treat the proving service as a black box which, upon a request, returns the
proof together with the computation’s output. Following the VOC model [2], we
distinguish between the blockchain infrastructure hosting the verifier
contract and a cloud-based off-chain infrastructure that runs outside the
consensus protocol and hosts the consumer and the proving service.
Figure 1: Proving Service Model
The consumer service is responsible for managing the proving service’s inputs
and outputs and interacting with the relying verifier contract. It receives
data from external sources and translates them into a proof request. Upon a
request through the prover client, the proving service executes the VOC and
returns the ZKP attesting to the computational integrity of the VOC. The
consumer service then submits the ZKP to the verifier contract through its
verifier client. On submission, the verifier contract verifies the proof
computed by the proving service.
Figure 2: Proving Service Architecture Figure 3: Proving Time and Memory
Consumption
### II-B Service Architecture
The Proving Service depicted in Figure 2 is an application service that runs
stand-alone on the prover’s off-chain infrastructure. It serves proof requests
by exposing the proving-related operations through an Application Program
Interface (API) as minimal functionality to the consumer service.
The procedure can be summarized in four simple steps: First, the proving
service receives the Proof Request containing the public ($x$) and private
($x^{\prime}$) proof arguments and the identifier of the addressed Executable
Constraint System ($ecs$)($id$). Second, using the $id$ the API fetches the
corresponding $ecs$ and private key ($pk$) from the Proof Registry which is
the persistent storage component containing these large, recurrently requested
files needed for proving. The proof registry manages different reusable pairs
of $ecs$ and $pk$, each addressable through a unique $id$.
Third, the Proving Instance is executed in two stages: the witness
computations and the proof generation. Fourth, the ZKP is returned to the
consumer service through the API upon successful execution.
## III ZoKrates-API
The previous service-oriented architecture serves as a technology-agnostic
blueprint for building proving services for DSL circuits. For evaluation, we
technically instantiate the proving service for ZoKrates [2] and present the
ZoKrates-API111https://github.com/ZK-Plus/zokrates-api as a ready-to-use open-
source software. We servicify ZoKrates by wrapping an API around the ZoKrates
interpreter, the central component of the ZoKrates software that previously
has only been addressable through a Command Line Interface and a Javascript
library. The ZoKrates-API exposes the methods of the ZoKrates interpreter
through HTTP endpoints. We containerized the ZoKrates-API using Docker, making
the services easily deployable among a wide range of machines and allowing us
to further leverage cloud-native tools like Kubernetes for horizontal
scalability, manageability, and observability. Furthermore, the ZoKrates-API
supports multi-threading so a single instance can compute multiple proofs in
parallel.
## IV Evaluation
To test the presented implementation, we deployed the containerized ZoKrates-
API on a Kubernetes cluster. As a workload for our experiments, we generated a
large number of EdDSA signatures[4] which amounts for more than $2^{29}$ of
circuit constraints similar to [5].
We conducted three experiments (see Figure 3) to measure the average proving
time per signature in [sec] and the memory consumption in [gb] using various
cluster configurations. Figure 3A) shows a 33% improvement in proving time for
a single machine when choosing an appropriate machine size. An argument in
favour of vertical escalability. For a single machine as well, Figure 3B)
shows that the proving time can be brought down drastically when few parallel
threads are enabled, though the gains plateaued rapidly due to the increasing
resources needed. Running the same experiments in parallel VMs instead of
threads, Figure 3C) demonstrate a better approach to scaling proving as the
computational burden is distributed over several machines, proportional
increasing the processing time.
## V conclusion
As the experiments demonstrate, significant performance improvements can be
gained from horizontal (more nodes) and vertical (larger nodes) scalability of
an arbitrary zk-SNARK proof. We facilitate this process by leveraging modern
cloud-native architectures such as Docker and Kubernetes.
## References
* [1] E. Ben-Sasson, A. Chiesa, E. Tromer, and M. Virza, “Scalable zero knowledge via cycles of elliptic curves,” _Algorithmica_ , vol. 79, pp. 1102–1160, 2017.
* [2] J. Eberhardt and S. Tai, “Zokrates - scalable privacy-preserving off-chain computations,” in _IEEE International Conference on Blockchain_. IEEE, 2018.
* [3] M. Bellés-Muñoz, M. Isabel, J. L. Muñoz-Tapia, A. Rubio, and J. Baylina, “Circom: A circuit description language for building zero-knowledge applications,” _IEEE Transactions on Dependable and Secure Computing_ , 2022.
* [4] J. Heiss, A. Busse, and S. Tai, “Trustworthy pre-processing of sensor data in data on-chaining workflows for blockchain-based iot applications,” in _Service-Oriented Computing: 19th International Conference, ICSOC 2021, Virtual Event, November 22–25, 2021, Proceedings 19_. Springer, 2021, pp. 133–149.
* [5] A. Chiesa, R. Lehmkuhl, P. Mishra, and Y. Zhang, “Eos: Efficient private delegation of zksnark provers,” in _USENIX Security Symposium. USENIX Association_ , 2023.
|
11institutetext: Tecnológico de Monterrey, School of Engineering and Sciences,
Mexico. 22institutetext: Universidad de las Americas Puebla, Department of
Chemical, Food and Environmental Engineering, Puebla, 72810, Mexico.
33institutetext: Universitat Politècnica de Catalunya. EEBE, Eduard Maristany
16, 08019 Barcelona. Catalonia, Spain.
# Comparing Machine Learning based Segmentation Models on Jet Fire Radiation
Zones††thanks: Currently under review for the Mexican Conference on AI (MICAI
2021)
Carmina Pérez-Guerrero 11 Adriana Palacios 22 Gilberto Ochoa-Ruiz 11 Christian
Mata 33 Miguel Gonzalez-Mendoza 11 Luis Eduardo Falcón-Morales 11
###### Abstract
Risk assessment is relevant in any workplace, however there is a degree of
unpredictability when dealing with flammable or hazardous materials so that
detection of fire accidents by itself may not be enough. An example of this is
the impingement of jet fires, where the heat fluxes of the flame could reach
nearby equipment and dramatically increase the probability of a domino effect
with catastrophic results. Because of this, the characterization of such fire
accidents is important from a risk management point of view. One such
characterization would be the segmentation of different radiation zones within
the flame, so this paper presents an exploratory research regarding several
traditional computer vision and Deep Learning segmentation approaches to solve
this specific problem. A data set of propane jet fires is used to train and
evaluate the different approaches and given the difference in the distribution
of the zones and background of the images, different loss functions, that seek
to alleviate data imbalance, are also explored. Additionally, different
metrics are correlated to a manual ranking performed by experts to make an
evaluation that closely resembles the expert’s criteria. The Hausdorff
Distance and Adjusted Random Index were the metrics with the highest
correlation and the best results were obtained from the UNet architecture with
a Weighted Cross-Entropy Loss. These results can be used in future research to
extract more geometric information from the segmentation masks or could even
be implemented on other types of fire accidents.
###### Keywords:
semantic segmentation deep learning computer vision jet fires.
## 1 Introduction
In the industrial environment there are certain activities such as the storage
of fuels or the transportation of hazardous material, that can be involved in
severe accidents that affect the industrial plant or activity border, as well
as external factors like human health, environmental damage, property damage,
and others. An overall knowledge of the features and characteristics of major
accidents is required to prevent and manage them or, in the worst case
scenario, take action to reduce and control their severity and aftermath.
There are some accidents that are relatively well known and researched,
however, there are other accidents that have not been broadly explored.
Sometimes the detection of fire, in an industrial setting, is not enough to
make the correct decisions when managing certain fire accidents. An example of
this, is the impingement of jet flames on nearby pipes, depending on the heat
fluxes of the flame, the pipe wall could heat up quickly and reach dangerous
temperatures that lead to severe domino effect sequences. These heat fluxes
can be defined as three main areas of interest within the flame [6], and the
definition of their geometric characteristics and localization becomes
valuable information in risk management.
Semantic segmentation could be an useful approach for this kind of flame
characterization, and the same procedure could even be employed in other fire
related accidents. To explore this proposal, different segmentation
methodologies are evaluated on a set of images from real jet fires of propane
to accurately segment the radiation zones within the flames. This different
zones are illustrated in Figure 1 and are defined across this paper as the
Central Zone, the Middle Zone and the Outer Zone.
The rest of this paper is organized as follows. Section 2 describes the
previous work present in the literature related to the problem and different
semantic segmentation methods. Section 3 describes the approaches used to
perform the exploration research. Section 4 explains the data set used for the
experiments and the pre-processing methods applied. Section 5 describes the
different evaluation metrics and loss functions explored during the
experiments. Section 6 contains the training protocol used for the experiments
with Deep Learning architectures. Section 7 explains the testing procedure
applied to the segmentation methods. Section 8 presents the results of the
exploration research. This section also offers a discussion of the future work
that can stem from the knowledge obtained. Finally, Section 9 summarizes the
major findings of the presented work.
(a)
(b)
Figure 1: Image (a) is an infrared visualization of an horizontal propane jet
flame. Image (b) is the corresponding ground truth segmentation, with the
segment names indicated. Modified from [6].
## 2 State of the Art
There has been previous research work regarding the use of Computer Vision and
image processing for the detection and monitoring of flares in the context of
industrial security. For example, Rodrigues and Yan [13] used imaging sensors
combined with digital image processing to determine the size, shape and region
of the flare, successfully characterizing the dynamism of the fire.
Another example is the work done in Janssen and Sepasian [7], where, by
separating the flare form the background using temperature thresholding, and
adding false colors to represent different temperature regions, a system was
created, capable of tracking the flare size for automated event signaling.
### 2.1 Deep Learning Architectures
Deep learning algorithms, such as Convolutional Neural Networks, have shown
outstanding performance in many complex tasks, such as image recognition,
object detection, and semantic and instance segmentation [9]. Advancements on
these methods have been increasing rapidly, with constant research being
published regarding better and more robust algorithms. Some important
architectures present in the literature are summarized in Table 1 with their
advantages and disadvantages described.
Table 1: Summary of the selected architectures. Source | Architecture | Pros | Cons
---|---|---|---
[4] | DeepLabv3 | | Recovers detailed structures
---
lost due to spatial invariance.
Efficient approximations via
probabilistic inference.
Wider receptive fields.
| Low accuracy on small
---
scaled objects.
Has trouble capturing
delicate boundaries of objects.
[2] | SegNet | | Efficient inference in terms of memory
---
and computational time.
Small number of trainable parameters.
Improved boundary delineation.
| Input image must be fixed.
---
Performance drops with a
large number of classes and
is sensitive to class imbalance.
[14] | UNet | | Can handle inputs of arbitrary sizes.
---
Smaller model weight size.
Precise localization of regions.
| Size of network comparable
---
to the size of features.
Significant amount of time
to train.
High GPU memory footprint
in larger images.
[10] | | Attention
---
UNet
| Avoids the use of multiple similar
---
feature maps.
Focuses on the most informative features
without additional supervision.
Enhances the results of the UNet architecture.
| Adds more weight parameters
---
The time for training increases,
especially for long sequences
In general, these architectures were selected due to their efficiency during
segmentation and their capacity to accurately portray the shape of such
dynamic figures obtained from the flames.
### 2.2 Traditional Computer Vision Methods
Different traditional segmentation methods are included in the analysis as a
baseline for the results given by the Deep Learning architectures, these
methods are: Gaussian Mixture Model (GMM), K-means clustering, Thresholding,
and Chan-Vese segmentation.
* •
GMM is one of the main methods applied to fire and smoke image segmentation.
It offers a clear definition of their dynamic shape, but sometimes missed
pixels from the inner parts of the fire and smoke [1].
* •
K-means clustering provides shape-based image segmentation [18] and has been
previously used for fire segmentation [15] [1]. However, this method does not
guarantee continuous areas [18]
* •
Thresholding is the simplest method of image segmentation and has a fast
operation speed. It is most commonly used in region-based segmentation, but it
is sensitive to noise and gray scale unevenness [17].
* •
The Chan-Vese segmentation [3] belongs to the group of active contour models,
which present some advantages for infrared image segmentation, since the edges
obtained are smooth and are represented with closed curves. However, this kind
of segmentation is very sensitive to noise and depends heavily on the location
of the initial contour, so it needs to be manually placed near the image of
interest [19].
## 3 Proposed Approach
To explore the semantic segmentation of radiation zones within the flames as
characterization of fire incidents, a group of 4 traditional segmentation
methods and 4 deep learning architectures are explored. These segmentation
methods are trained and tested using a data set of jet fire images obtained
from videos of an experiment performed in an open field.
To properly evaluate and compare the different segmentation approaches,
several metrics with different evaluation methods are correlated to manual
rankings performed by two experts in the field, this is to make sure that the
evaluation is the most representative of a fire engineer’s perception of good
segmentation.
The best model found from this exploratory research will then be used in
future work, not included in this paper, to extract other geometric
characteristics from the resulting segmentation masks.
## 4 Data set
Investigators from Universitat Politècnica de Catalunya performed an
experiment to produce horizontal jet fires at subsonic and sonic gas exit
rates. The experiment was filmed using an infrared thermographic camera, more
specifically, an AGEMA 570 by Flir Systems. The video was saved in four frames
per second, resulting in a total of 201 images with a resolution of 640 x 480
pixels. After obtaining the infrared visualizations of the flames, a
segmentation was performed of the three radiation zones within the fire, the
results were validated by experts in the field and the result was a ground-
truth for each of the infrared images. The images were saved as Matlab files
that contain a temperature matrix corresponding to the temperature values
detected by the camera for the infrared images, and the a label matrix
corresponding to the segments for the ground-truth segmentations. These files
were then exported as PNG files to be used by the different segmentation
algorithms described in Section 2.1.
### 4.1 Image Processing
To enhance the characteristics of the jet fires represented in the infrared
images, and to reduce their variance, a process of image normalization was
employed, which also helps in the convergence of the Deep Learning methods.
The ground truth images were transformed into labeled images, where a label id
was used instead of the original RGB values. Given the small number of samples
of the data set, and to increase the variability of the input during the
training of the models, data augmentation techniques were applied. These
processing of the images can increase the performance of the models, can avoid
the over-fitting to the training samples, and can help the models to also
perform well for instances that may not be present in the original data set.
Horizontal flipping, random cropping, and random scaling were applied in
parallel of the training workflow, so for each iteration of training, a
different augmented image was inputted. The probability of horizontal flipping
was set to 50%, and random scaling had values that ranged between 0.7 and 2.0.
## 5 Metrics and Loss Functions
### 5.1 Metrics
An analysis was performed to select evaluation metrics that were more
representative of an expert’s evaluation of the segmentation. The metrics
analysed are separated into groups that describe their evaluation method. This
diversity is important because the consideration of particular properties
could prevent the discovery of other particular errors or lead to over or
underestimating them. To compare the metrics enlisted in Table 2, a group of
images were evaluated with the metrics and two manual rankings performed by
two experts in the field. A Pearson pairwise correlation is used to perform
this comparison at segmentation level.
Table 2: Summary of the metrics analysed in this paper. The ”Group” column describes the method group that the metric belongs to. Based on [16]. Metric | Group
---|---
Jaccard Index | Spatial Overlap Based
F-measure | Spatial Overlap Based
Adjusted Rand Index | Pair Counting Based
Mutual Information | Information Theoretic Based
Cohen’s Kappa | Probabilistic Based
Hausdorff Distance | Spatial Distance Based
Mean Absolute Error | Performance Based
Mean Square Error | Performance Based
Peak Signal to Noise Ratio | Performance Based
### 5.2 Loss Functions
The proportion of the different radiation zones within each flame is
different. The Outer Zone tends to be the largest segment and the central zone
is usually smaller than the Middle Zone, this differences could affect the
overall segmentation obtained during training, which is why the loss functions
employed in this research work were focused on dealing with this class
imbalance. The implemented loss functions are summarized in Table 3.
Table 3: Summary of the loss functions implemented in this paper. Source | Loss Function | Description
---|---|---
[12] | | Weighted Cross-Entropy
---
Loss
| Combines Log Softmax and Negative Log Likelihood.
---
Useful for multiple classes that are unbalanced.
[8] | Focal Loss | | Addresses foreground-background class imbalance.
---
Training is focused on a sparse set of hard examples.
[5] | | Generalized Wasserstein
---
Dice Loss (GWDL)
| Semantically-informed generalization of the Dice score.
---
Based on the Wasserstein distance on the probabilistic
label space.
## 6 Training
The PyTorch framework [12] was used for the implementation of the Deep
Learning architectures. To maintain the weights of the Convolutional Neural
Networks as small as possible, a weight decay strategy was used with L2
regularization. The learning rate had an initial value of 0.0001 and used an
ADAM optimizer during training. The class weights used by the Weighted Cross-
Entropy and Focal losses were computed according to the ENet custom class
weighing scheme [11] and are defined as the following: 1.59 for background,
10.61 for Outer zone, 17.13 for Middle zone, and 22.25 for Central zone. The
class distances used for the loss function of GWDL is defined as 1 between the
background and the radiation zones, and as 0.5 between the zones themselves.
The training was performed using an Nvidia DGX workstation that has 8 Nvidia
GPUs and allows a batch size of 4 as maximum, which is the batch size used for
the models. The data was split in 80% for training and 20% for testing and
validation, resulting in a total of 161 images for training, 20 images for
testing, and 20 images for validation. The models were trained for up to 5000
epochs with an Early Stopping strategy to avoid over fitting. The resulting
loss values for the best models can be visualized in Fig. 2.
(a) Loss values for the DeepLabv3 model. Early Stopping took place at epoch
701.
(b) Loss values for the SegNet model. Early Stopping took place at epoch 1274.
(c) Loss values for the UNet model. Early Stopping took place at epoch 1460.
(d) Loss values for the Attention UNet model. Early Stopping took place at
epoch 1560.
Figure 2: The Weighted Cross-Entropy loss values for the best models obtained
for each architecture explored.
## 7 Testing
Testing and Validation was performed on 20% of the data set, which represent a
total of 40 images. The results are compared using the metrics with the
highest correlation to the manual ranking of experts. The results for each
Deep Learning architecture are first compared across all 3 different loss
functions mentioned in Section 5.2. The best performing combination, for each
architecture, is then compared to the results obtained from the other 4
traditional segmentation models. The time each method takes to segment the
whole data set of 201 images is also taken into account. The goal of the
comparison is to find the best overall model for the segmentation of the
radiation zones within the flames.
## 8 Results and Discussion
### 8.1 Selected Metrics
The results of the correlation between the metrics, mentioned in Section 5,
and the manual rankings performed by experts can be observed in Fig. 3.
Figure 3: Heatmap representing the Pearson Correlation Coefficient for the
metrics in Section 5 and the manual rankings Rank1 and Rank2 done by experts
in the field of the problem.
The highest Pearson correlation value was given by the Hausdorff Distance,
with a value of -0.32 to the first manual ranking, this negative relationship
is because a smaller Hausdorff Distance is preferable and the manual ranking
assigned higher values as the segmentation improved. The second highest
correlations were given by the Adjusted Rand Index in both manual rankings,
with a value of 0.29 for the first one and 0.27 for the second one, this
positive relationship is because the values of the Adjusted Rand Index go from
0 to 1, with 1 being the best result, and showing a similar behaviour to the
manual ranking that assigns higher values to better segmentations.
### 8.2 Best Loss Function
Each combination of Deep Learning architecture and Loss Function was evaluated
using the Hausdorff Distance as main metric. The best results across all
architectures were obtained when using the Weighted Cross-Entropy Loss, this
can be observed in Fig 4. In general the Focal Loss also showed good results,
being close to the Weighted Cross-Entropy results and most of the times
surpassing the GWDL results, with the only exception being the Attention UNet
model, where the mean Hausdorff Distance with Focal Loss is larger than with
GWDL, but has a much closer distribution. The most dramatic difference
happened with the SegNet architecture, which showed very poor results with
GWDL.
(a) Hausdorff Distance for all the DeepLabv3 models.
(b) Hausdorff Distance for all the SegNet models.
(c) Hausdorff Distance for all the UNet models.
(d) Hausdorff Distance for all the Attention UNet models.
Figure 4: The Hausdorff Distance for all the model and loss function
combinations across the validation set. For each model the order of loss
functions is Weighted Cross-Entropy, Focal and GWDL from left to right.
### 8.3 Traditional and Deep Learning Segmentation
A comparison was done between the traditional Computer Vision methods
mentioned in Section 2.2 and the Deep Learning models mentioned in Section
2.1,using the Weighted Cross-Entropy loss. The Hausdorff Distance and the
Adjusted Rand Index are used to evaluate the models across the testing set and
the time each method takes to perform the segmentation on all the images from
the data set is also taken into account. These results are summarized in Table
4 and the distribution of the results can be visualized in Fig 5. Overall the
best performing models are observed to be the UNet and Attention UNet models.
Even if Attention UNet achieved a slightly better Adjusted Rand Index score,
UNet obtained a better Hausdorff distance and segmentation time, therefore we
can say that the best model is the one that uses the UNet architecture.
Table 4: Mean Hausdorff Distance and Adjusted Rand Index values for all the segmentation models across the testing set, as well as the time in seconds that each method takes to segment the whole data set. The best results are in bold. Method | Hausdorff Distance | Adjusted Rand Index | Time (s)
---|---|---|---
GMM | 1288.10 | 0.9156 | 2723.8
K-means | 1000.63 | 0.8855 | 3035.1
Thresholding | 1029.08 | 0.9152 | 30.7
Chan-Vese | 1031.90 | 0.8568 | 18177.5
DeepLabv3 | 784.86 | 0.9514 | 17.1
SegNet | 692.73 | 0.9381 | 16.4
UNet | 586.46 | 0.9504 | 15.7
Attention UNet | 601.05 | 0.9592 | 17.7
(a) Hausdorff distance for all segmentation models.
(b) Adjusted Rand Index for all segmentation models.
Figure 5: The Hausdorff Distance and Adjusted Rand Index values for all the
segmentation models across the testing set.
The difference in the segmentation of each method can also be visualized in
Fig 6. It can be observed that the shape of the Outer and Middle zones are
generally well represented in most of the segmentation models, however the
most important differences are found in the Central zone, where the Deep
Learning architectures defined more clearly it’s shape. The similar results of
the UNet and Attention UNet architectures are also observed in this sample
segmentation masks, with a really close resemblance to the ground truth.
(a) GMM.
(b) K-means.
(c) Thresholding.
(d) Chan-Vese.
(e) DeepLabv3.
(f) SegNet.
(g) UNet.
(h) Attention UNet.
Figure 6: Sample segmentations of all the models.
### 8.4 Discussion
Overall, the Deep Leaning algorithms greatly outperformed the traditional
Computer Vision methods and the best proposed model would be a UNet
architecture with a Weighted Cross-Entropy loss function. The segmentation
masks obtained from this model could be used in the future to further extract
other geometric characteristics of the flame, such as length, area and lift-
off distance. This additional information would improve greatly the decision
making process involved in the risk assessment and management of fire related
accidents that can take place in an industrial setting. Furthermore, the
metrics of Hausdorff Distance and Adjusted Rand Index can be used to evaluate
other segmentation approaches that may try to solve similar problems in the
future, having the certainty that the evaluation would be a close
representation of an expert’s opinion.
## 9 Conclusions
The semantic segmentation of radiation zones within the flames can be used to
characterize fire accidents, such as jet fires, and the information obtained
from the segmentation can prove to be critical when dealing with risk
management in industrial settings. The exploratory research presented in this
paper continued to show that Deep Learning architectures greatly outperform
other traditional Computer Vision approaches. It was also found that for this
specific problem, the best loss function to train a Deep Learning model is a
Weighted Cross-Entropy Loss, the best architecture to be used is UNet, and the
best evaluation metrics are both the Hausdorff Distance and the Adjusted Rand
Index. All this knowledge can be later used in future research focused on
extracting even more geometric information from the segmentation masks. This
could bring about a more complete characterization analysis of jet fires and
the methods applied to this types of fire accidents could then be used on
other fire scenarios.
#### Acknowledgements
This research is supported in part by the Mexican National Council of Science
and Technology (CONACYT).
This research is part of the project 7817-2019 funded by the Jalisco State
Council of Science and Technology (COECYTJAL).
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|
# Long time solutions for 1D cubic dispersive equations, Part II: the focusing
case
Mihaela Ifrim Department of Mathematics, University of Wisconsin, Madison
<EMAIL_ADDRESS>and Daniel Tataru Department of Mathematics, University of
California at Berkeley<EMAIL_ADDRESS>
###### Abstract.
This article is concerned with one dimensional dispersive flows with cubic
nonlinearities on the real line. In a very recent work, the authors have
introduced a broad conjecture for such flows, asserting that in the defocusing
case, small initial data yields global, scattering solutions. Then this
conjecture was proved in the case of a Schrödinger dispersion relation. In
terms of scattering, our global solutions were proved to satisfy both global
$L^{6}$ Strichartz estimates and bilinear $L^{2}$ bounds. Notably, no
localization assumption is made on the initial data.
In this article we consider the focusing scenario. There potentially one may
have small solitons, so one cannot hope to have global scattering solutions in
general. Instead, we look for long time solutions, and ask what is the time-
scale on which the solutions exist and satisfy good dispersive estimates. Our
main result, which also applies in the case of the Schrödinger dispersion
relation, asserts that for initial data of size $\epsilon$, the solutions
exist on the time-scale $\epsilon^{-8}$, and satisfy the desired $L^{6}$
Strichartz estimates and bilinear $L^{2}$ bounds on the time-scale
$\epsilon^{-6}$. To the best of our knowledge, this is the first result to
reach such a threshold.
###### Contents
1. 1 Introduction
2. 2 Density-flux and interaction Morawetz identities
3. 3 Strichartz and bilinear $L^{2}$ bounds
4. 4 Long time energy estimates
5. 5 Optimality remarks
## 1\. Introduction
The question of obtaining long time solutions for one dimensional dispersive
flows with quadratic/cubic nonlinearities has attracted a lot of attention in
recent years. One can distinguish two different but closely related types of
results that have emerged, as well as several successful approaches.
On one hand, _normal form methods_ have been developed in order to extend the
lifespan of solutions, beginning with [28] in the late ’80’s. Somewhat later,
around 2000, the _I-method_ , introduced in [5] brought forth the idea of
constructing better almost conserved quantities. These two ideas serve well in
the study of semilinear flows, where it was later understood that they are
connected [3].
Neither of these techniques can be directly applied to quasilinear problems.
Addressing this problem, it was discovered in the work of the authors and
collaborators [16], [18] that one can adapt the normal form method to
quasilinear problems by constructing energies which simultaneously capture
both the quasilinear and the normal form structures. This idea was called the
_modified energy method_ , and can also be seen in some way as a quasilinear
adaptation of the I-method. An alternate approach, also in the quasilinear
setting, is provided by the _flow method_ of [15], where a better normal form
transformation is constructed using a well chosen auxiliary flow.
On the other hand, the further goal of obtaining scattering, global in time
solutions for one dimensional dispersive flows with quadratic/cubic
nonlinearities has also been extensively studied in the last two decades for a
number of models, under the assumption that the initial data is both _small_
and _localized_ ; without being exhaustive, see for instance [13, 14, 25, 23,
17]. The nonlinearities in these models are primarily cubic, though the
analysis has also been extended via normal form and modified energy methods to
problems which also have nonresonant quadratic interactions; several such
examples are [1, 18, 10, 19, 24], see also further references therein, as well
as the authors’ expository paper [22].
If instead one considers initial data which is just _small_ , without any
localization assumption, then the problem becomes much more difficult, because
this allows for far stronger nonlinear interactions over long time-scales. One
also needs to distinguish between the focusing and the defocusing problems. In
a recent paper [21], the authors have introduced a broad global well-posedness
(GWP) conjecture, which applies to both semilinear and quasilinear problems:
###### Conjecture 1 (Non-localized data defocusing GWP conjecture).
One dimensional dispersive flows on the real line with cubic defocusing
nonlinearities and small initial data have global in time, scattering
solutions.
The main result of [21] asserts that this conjecture is true under suitable
assumptions, most notably that the dispersion relation is the Schrödinger
dispersion relation. That was the first global in time well-posedness result
of this type. Notably, scattering here is interpreted in a weak sense, to mean
that the solution satisfies global $L^{6}$ Strichartz estimates and bilinear
$L^{2}$ bounds. This is because of the strong nonlinear effects, which
preclude any kind of classical scattering. The precise result is stated later
in Theorem 3.
Our interest in this article is instead in the focusing case of the same
problem. Since 1D focusing dispersive problems typically admit small solitons,
a global result as stated in the above conjecture simply cannot hold. Even if
global solutions exist (as it is the case for instance for the cubic focusing
NLS problem) the presence of solitons will defeat any kind of global decay
estimates. For this reason, in the focusing case we will rethink the problem
as a question about the lifespan of solutions with small initial data.
Precisely, if the initial data has size $\epsilon$ when measured in a suitable
$H^{s}$ Sobolev norm, what can be said about the lifespan of the solutions as
a function of $\epsilon$ ?
Following the lead of our earlier paper, we begin by formulating the focusing
counterpart of the previous conjecture. Then we will prove that the conjecture
is true under suitable assumptions. Our main conjecture is as follows:
###### Conjecture 2 (Non-localized data focusing conjecture).
One dimensional dispersive flows on the real line with cubic nonlinearities
and small initial data of size $\epsilon$ have solutions which remain of
comparable size at least on an $\epsilon^{-8}$ time-scale.
The main result of this paper, see Theorem 1 below, asserts that this
conjecture is valid under the additional assumption that the dispersion
relation is of Schrödinger type. As part of this result, we also prove that
our long time solutions satisfy both $L^{6}$ Strichartz estimates and bilinear
$L^{2}$ bounds on suitable time-scales. This is akin to our earlier work on
the defocusing case, but with the difference that in the defocusing case such
estimates were proved globally in time.
For reference purposes, we note some intermediate lifespan thresholds which
can be reached with methods which were developed earlier:
* •
A cubic lifespan $\epsilon^{-2}$ can be reached using direct energy estimates,
using only the fact that the nonlinearity is cubic.
* •
A quintic lifespan $\epsilon^{-4}$ can be reached by more accurate energy
estimates using a quartic energy correction. This requires the cubic
nonlinearity to be conservative, as defined later in the introduction.
* •
In the case of a perturbative nonlinearity, a quintic lifespan can also be
obtained by directly using Strichartz estimates.
Heuristically, if it were possible to directly combine the last two ideas
above, that would lead to the $\epsilon^{-8}$ threshold in the present paper.
However, the price to achieve that would be very steep, as one would need to
assume both a perturbative nonlinearity and high regularity. By comparison,
our new result provides a much more robust approach, which is both of a
nonperturbative nature and far more efficient in terms of the regularity
requirements. We show here how dispersive and normal form tools can be
combined very efficiently in order to make substantial gains.
In the present paper we aim for a reasonably simple setting, where our model
problem is borderline semilinear, and where we prove results we expect to be
optimal. This should also serve as a baseline for further developments. In
particular, we believe that our methods can be also applied in non-
perturbative, quasilinear settings.
### 1.1. Cubic NLS problems in one space dimension
The fundamental model for one-dimensional dispersive flows with cubic
nonlinearity in one space dimension is the cubic nonlinear Schrödinger (NLS)
flow,
(1.1) $\left\\{\begin{aligned} &iu_{t}+u_{xx}=\pm 2u|u|^{2}\\\
&u(0)=u_{0},\end{aligned}\right.$
with $u:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{C}$. This comes in a
defocusing (+) and a focusing (-) version.
The above cubic NLS flows are globally well-posed in $L^{2}$ both in the
focusing and in the defocusing case, though the global behavior differs in the
two cases. Both of these model problems are completely integrable, and one may
study their global behavior using inverse scattering tools [9], [2].
In the defocusing case, the inverse scattering approach allows one to treat
the case of localized data, and show that global solutions scatter at
infinity, see for instance [9]. This can also be proved in a more robust way,
without using inverse scattering, under the assumption that the initial data
is small and localized, see [17] and references therein. Much less is known in
terms of scattering for nonlocalized $L^{2}$ data. However, if more regularity
is assumed for the data, then we have the following estimate due to Planchon-
Vega [26], see also the work of Colliander-Grillakis-Tzirakis [4]:
(1.2)
$\|u\|_{L^{6}_{t,x}}^{6}+\|\partial_{x}|u|^{2}\|_{L^{2}_{t,x}}^{2}\lesssim\|u_{0}\|_{L^{2}_{x}}^{3}\|u_{0}\|_{H^{1}_{x}}.$
This allows one to estimate the $L^{6}$ Strichartz norm of the solution, i.e.
to prove some type of scattering or dispersive decay. This estimate was
improved and extended to $L^{2}$ solutions as a corollary of the results in
our previous paper [21]. Precisely, we have
(1.3)
$\|u\|_{L^{6}_{t,x}}^{2}+\|\partial_{x}|u|^{2}\|_{L^{2}_{t}(\dot{H}^{-\frac{1}{2}}_{x}+cL^{2}_{x})}\lesssim\|u_{0}\|_{L^{2}_{x}}^{2},\qquad
c=\|u_{0}\|_{L^{2}_{x}}.$
On the other hand, the focusing problem admits small solitons, so the
solutions cannot in general scatter at infinity. If in addition the initial
data is localized, then one expects the solution to resolve into a
superposition of (finitely many) solitons, and a dispersive part; this is
called _the soliton resolution conjecture_ , and is known to hold in a
restrictive setting, via the method of inverse scattering, see e.g. [2].
Our interest here is in focusing problems, but without any integrability
assumptions, and even without assuming any conservation laws. The model we
consider is similar to the one in [21], namely
(1.4) $\left\\{\begin{aligned} &iu_{t}+u_{xx}=C(u,\bar{u},u)\\\
&u(0)=u_{0},\end{aligned}\right.$
where $u$ is a complex valued function,
$u:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{C}$. Here $C$ is a trilinear
translation invariant form, whose symbol $c(\xi_{1},\xi_{2},\xi_{3})$ can
always be assumed to be symmetric in $\xi_{1},\xi_{3}$; see [21] for an
expanded discussion of multilinear forms. The arguments $u,\bar{u}$ and $u$ of
$C$ are chosen so that our equation (1.4) has the phase rotation symmetry,
$u\to ue^{i\theta}$, as it is the case in many examples of interest. The
symbol $c(\xi_{1},\xi_{2},\xi_{3})$ will be required to satisfy the following
set of assumptions, which are similar to [21]:
1. (H1)
Bounded and regular:
(1.5) $|\partial_{\xi}^{\alpha}c(\xi_{1},\xi_{2},\xi_{3})|\leq
c_{\alpha},\qquad\xi_{1},\xi_{2},\xi_{3}\in\mathbb{R},\,\mbox{ for every
multi-index $\alpha$}.$
2. (H2)
Conservative:
(1.6) $\Im c(\xi,\xi,\eta)=0,\qquad\xi,\eta\in\mathbb{R},\mbox{ where }\Im
z=\mbox{imaginary part of }z\in\mathbb{C}.$
In addition to these two conditions, in [21] we have also employed a
defocusing assumption, namely
1. (H3)
Defocusing:
(1.7) $c(\xi,\xi,\xi)\geq c_{0}>0,\qquad\xi\in\mathbb{R}\mbox{ and
}c_{0}\in\mathbb{R^{+}}.$
Here one might think that we should require the opposite, namely
1. (H4)
Focusing:
(1.8) $-c(\xi,\xi,\xi)\geq c_{0}>0,\qquad\xi\in\mathbb{R}\mbox{ and
}c_{0}\in\mathbb{R^{+}}.$
But as it turns out, no such assumption is needed here, as the result of this
paper applies equally regardless of any sign condition; so we will simply drop
it.
Using the same assumptions (H1), (H2) as in [21] is convenient here because it
will allow us to reuse a good part of the analysis there, up to the point
where the defocusing assumption is needed.
Repeating a similar comment in [21], one should view both our choices of the
Schrödinger dispersion relation and the uniform bounds in (H1) not as
fundamental, but rather as a balance between the generality of the result on
one hand, and a streamlined exposition on the other hand. This choice places
our model in the semilinear class, but just barely so.
The simplest example of such a trilinear form $C$ is of course $C=\pm 1$,
which corresponds to the classical one-dimensional cubic NLS problem. But this
problem has too much structure, in particular it is completely integrable, and
also globally well-posed in $L^{2}$.
At the other end, both our use of the linear Schrödinger operator and the
boundedness condition (H1) are non-optimal, and we hope to relax both of these
restrictions in subsequent work.
### 1.2. The main result
Our main result asserts that long time well-posedness holds for our problem
for small $L^{2}$ data. In addition, our solutions not only satisfy uniform
$L^{2}$, but also space-time $L^{6}$ Strichartz estimates, as well as bilinear
$L^{2}$ bounds, on appropriate time-scales:
###### Theorem 1.
Consider the problem (1.4) where the cubic nonlinearity $C$ satisfies the
assumptions (H1) and (H2). Assume that the initial data $u_{0}$ is small,
$\|u_{0}\|_{L^{2}_{x}}\leq\epsilon\ll 1,$
Then the solution $u$ exists on on a time interval
$I_{\epsilon}=[0,c\epsilon^{-8}]$ and has the following properties for every
interval $I\subset I_{\epsilon}$ of size $|I|\leq\epsilon^{-6}$:
1. (i)
Uniform $L^{2}$ bound:
(1.9) $\|u\|_{L^{\infty}_{t}(I_{\epsilon};L^{2}_{x})}\lesssim\epsilon.$
2. (ii)
Strichartz bound:
(1.10) $\|u\|_{L^{6}_{t,x}(I\times\mathbb{R})}\lesssim\epsilon^{\frac{2}{3}}.$
3. (iii)
Bilinear Strichartz bound:
(1.11)
$\|\partial_{x}(u\bar{u}(\cdot+x_{0}))\|_{L^{2}_{t}(I;H_{x}^{-\frac{1}{2}})}\lesssim\epsilon^{2},\qquad
x_{0}\in\mathbb{R}.$
The local well-posedness in $L^{2}$ for the problem (1.4) was already proved
in [21], so the emphasis here and later in the proof is on the lifespan bound
and the long time estimates in the theorem.
We remark that the intermediate time-scale $\epsilon^{-6}$ does not have an
intrinsic meaning from a scaling perspective, but is instead connected to the
unit frequency-scale which is implicit in (H1), and which motivates the
frequency decomposition on the unit frequency-scale which is used in the proof
of both the present result and the earlier result in [21]. One could also use
a smaller frequency-scale for this decomposition, which in turn corresponds to
a smaller size for $|I|$. This is however not needed in the proof of the
$\epsilon^{-8}$ result, so, in order to avoid cluttering the theorem, we omit
the details. But the interested reader should see Remark 1.1 below.
A natural question to ask is whether this result is optimal. On one hand, the
bounds (1.10) and (1.11) are sharp for the cubic NLS, and likely for any
focusing flow (i.e. which satisfies (H4) at least in some region); this is
discussed in Section 5. But the $\epsilon^{-8}$ lifespan bound is not optimal
for any flow satisfying our hypotheses. Indeed, if for instance the $L^{2}$
norm is conserved (as is the case for the focusing NLS) then global well-
posedness follows. However, we conjecture that
###### Conjecture 3.
The result in Theorem 1 is sharp for generic focusing flows satisfying our
hypotheses.
It it also interesting to see how our theorem applies to the focusing cubic
NLS problem. There we can also consider large data simply by scaling. Global
well-posedness in $L^{2}$ is relatively straightforward there, but some of the
estimates we prove are new:
###### Theorem 2.
Consider the focusing 1-d cubic NLS problem (1.1)(-) with $L^{2}$ initial data
$u_{0}$. Then the global solution $u$ satisfies the following bounds in all
intervals $I$ with size $|I|>\|u_{0}\|_{L^{2}_{x}}^{-4}$:
1. (i)
Uniform $L^{2}$ bound:
(1.12) $\|u\|_{L^{\infty}_{t}L^{2}_{x}}\lesssim\|u_{0}\|_{L^{2}_{x}}.$
2. (ii)
Strichartz bound:
(1.13)
$\|u\|_{L^{6}_{t,x}(I\times\mathbb{R})}\lesssim\|u_{0}\|_{L^{2}_{x}}(I\|u_{0}\|_{L^{2}_{x}}^{4})^{\frac{1}{6}}.$
3. (iii)
Bilinear Strichartz bound:
(1.14)
$\|\partial_{x}|u|^{2}\|_{L^{2}_{t}(I;\dot{H}_{x}^{-\frac{1}{2}}+cL^{2}_{x})}\lesssim\|u_{0}\|_{L^{2}_{x}}^{2},\qquad
c^{2}=\|u_{0}\|_{L^{2}_{x}}^{2}(|I|\|u_{0}\|_{L^{2}_{x}}^{4}).$
Here the Strichartz estimates are fairly easy to obtain directly, but the
bilinear Strichartz bounds are new. Returning to our discussion after Theorem
1, the numerology in this application helps clarify the earlier comment about
the choice of the time-scales in Theorem 1:
###### Remark 1.1.
In the context of Theorem 1, a scaling argument shows that the bounds (1.13)
and (1.14) hold for all intervals $I$ so that
$\epsilon^{-4}\leq|I|\leq\epsilon^{-6}$.
One may gain further insights into our result for focusing problems by
comparing it with our earlier result in [21].
###### Theorem 3 ([21]).
Under the above assumptions (H1), (H2) and (H3) on the symbol of the cubic
form $C$, small initial data
$\|u_{0}\|_{L^{2}_{x}}\leq\epsilon\ll 1,$
yields a unique global solution $u$ for (1.4), which satisfies the following
bounds:
1. (i)
Uniform $L^{2}$ bound:
(1.15) $\|u\|_{L^{\infty}_{t}L^{2}_{x}}\lesssim\epsilon.$
2. (ii)
Strichartz bound:
(1.16) $\|u\|_{L^{6}_{t,x}}\lesssim\epsilon^{\frac{2}{3}}.$
3. (iii)
Bilinear Strichartz bound:
(1.17)
$\|\partial_{x}(u\bar{u}(\cdot+x_{0}))\|_{L^{2}_{t}H_{x}^{-\frac{1}{2}}}\lesssim\epsilon^{2},\qquad
x_{0}\in\mathbb{R}.$
One may observe here that the estimates are similar in the two cases, and the
only difference is the time-scale on which the estimates hold: in the
defocusing case this is global, while in the focusing case it is finite and
depends on the solution size. For this reason, the proofs of Theorem 3 and (1)
are closely related, and we will take advantage of this within the proof.
For the convenience of the reader, we also recall the main ideas in the proof
of the last theorem in [21], which are equally employed here:
_1\. Energy estimates via density flux identities._ This is a classical idea
in pde’s, and particularly in the study of conservation laws. The novelty in
[21] is that this analysis is carried out in a nonlocal setting, where both
the densities and the fluxes involve translation invariant multilinear forms.
The densities and the fluxes are not uniquely determined here, so careful
choices need to be made.
_2\. The use of energy corrections._ This is an idea originally developed in
the context of the so called I-method [5] or more precisely the second
generation I-method [8], whose aim was to construct more accurate almost
conserved quantities. In [21] this idea is instead implemented at the level of
density-flux identities.
_3\. Interaction Morawetz bounds._ These were originally developed in the
context of the three-dimensional NLS problems by Colliander-Keel-Stafillani-
Takaoka-Tao in [6], and have played a fundamental role in the study of many
nonlinear Schrödinger flows, see e,g. [7, 27], and also for one-dimensional
quintic flows in the work of Dodson [11, 12]. Our take on this is somewhat
closer to the one-dimensional approach of Planchon-Vega [26], though recast in
the setting and language of nonlocal multilinear forms.
_4\. Tao’s frequency envelope method._ This is used as a way to accurately
track the evolution of the energy distribution across frequencies. Unlike the
classical implementation relative to dyadic Littlewood-Paley decompositions,
in [21] we adapt and refine this notion for lattice decompositions instead.
This is also very convenient as a bootstrap tool, see e.g. Tao [29], [30] but
with the added twist of also bootstrapping bilinear Strichartz bounds, as in
the authors’ paper [20].
### 1.3. An outline of the paper
To a large extent, the proof of our main result mirrors the proof of the
global result in the defocusing case in [21]. The primary difference is in how
the $L^{6}$ Strichartz norms are handled, both globally and in a frequency
localized setting.
Section 2 reviews two of the main ideas in [21], namely the construction of
modified density-flux identities for the mass/momentum in a frequency
localized setting, as well as the interaction Morawetz identities associated
to those density-flux relations.
The proof of the main $L^{6}$ Strichartz bounds and the bilinear $L^{2}$
estimates is done in a frequency localized setting, using a bootstrap argument
based on a frequency decomposition on the unit scale, where the components are
measured using a maximal frequency envelope, another notion introduced in
[21]. This is described in Section 3, and leads to dispersive bounds on the
$\epsilon^{-6}$ time-scale. It is within this argument where the $L^{6}$ norms
are treated differently from the defocusing case.
In order to advance from the $\epsilon^{-6}$ to the $\epsilon^{-8}$ time-scale
it suffices to propagate the $L^{2}$ bound (i.e. the mass) on the larger time-
scale. However the mass is not a conserved quantity, so instead it is better
to propagate the bounds for the modified mass. This analysis is carried out in
Section 4.
Finally, in the last section of the paper we discuss the optimality of our
result, or rather the optimality of the $L^{6}$ and the bilinear $L^{2}$
estimates on the $\epsilon^{-6}$ time-scale. This is done by considering the
obvious enemies, namely the solitons, in the focusing NLS context.
### 1.4. Acknowledgements
The first author was supported by the Sloan Foundation, and by an NSF CAREER
grant DMS-1845037. The second author was supported by the NSF grant
DMS-2054975 as well as by a Simons Investigator grant from the Simons
Foundation.
## 2\. Density-flux and interaction Morawetz identities
A key role in the proof of the results in both [21] and in the present paper
is played by the approximate conservation laws for the mass and the momentum.
Rather than considering them directly, we instead consider several
improvements:
* •
the conservation laws are written in density-flux form, rather than as
integral identities, where both the densities and the fluxes are multilinear
forms.
* •
we improve the accuracy of these conservation laws by using well chosen
quartic corrections for both the densities and the fluxes, with $6$-linear
errors.
* •
we use these densities and fluxes not only globally in frequency, but also in
a frequency localized setting.
The aim of this section is to provide an overview of these density-flux
identities, following the set-up of [21]. We conclude the section with an
overview of the interaction Morawetz identities obtained in [21] from the
above density flux identities.
### 2.1. Resonances and multilinear forms
A key role in our analysis is played by four wave resonances. Given three
input frequencies $(\xi_{1},\xi_{2},\xi_{3})$ in the cubic nonlinearity $C$,
the output is at frequency
$\xi_{4}=\xi_{1}-\xi_{2}+\xi_{3}.$
The three wave interaction is resonant if
$\xi_{4}^{2}=\xi_{1}^{2}-\xi_{2}^{2}+\xi_{3}^{2}.$
To rewrite these relations in a symmetric fashion we use the notations
$\Delta^{4}\xi=\xi_{1}-\xi_{2}+\xi_{3}-\xi_{4},\qquad\Delta^{4}\xi^{2}=\xi_{1}^{2}-\xi_{2}^{2}+\xi_{3}^{2}-\xi_{4}^{2},$
The first expression is Galilean invariant but not the second, which is why we
also use the adjusted, Galilean invariant expression
${\tilde{\Delta}}^{4}\xi^{2}=\Delta^{4}\xi^{2}-2\xi_{avg}\Delta^{4}\xi,$
where $\xi_{avg}$ represents the average of the four frequencies.
With these notations, the resonant set is described as
$\mathcal{R}=\\{[\xi]=(\xi_{1},\xi_{2},\xi_{3},\xi_{4});\
\Delta^{4}\xi=0,\Delta^{4}\xi^{2}=0\\}$
which can be explicitly characterized as
$\mathcal{R}=\\{[\xi];(\xi_{1},\xi_{3})=(\xi_{2},\xi_{4})\\}$
Quadruples in the resonant set can be described by two parameters, namely
(2.1) $\displaystyle{\delta\xi^{\text{hi}}}=$ $\displaystyle\
\max\\{|\xi_{1}-\xi_{2}|+|\xi_{3}-\xi_{4}|,|\xi_{1}-\xi_{4}|+|\xi_{3}-\xi_{2}|\\},$
$\displaystyle{\delta\xi^{\text{med}}}=$ $\displaystyle\
\min\\{|\xi_{1}-\xi_{2}|+|\xi_{3}-\xi_{4}|,|\xi_{1}-\xi_{4}|+|\xi_{3}-\xi_{2}|\\},$
These distance parameters are carefully defined so that they can also be used
outside the resonant set to characterize frequency quadruples. This is very
useful in the density-flux relations later on.
### 2.2. The conservation of mass
The starting point of the analysis in [21] is to consider energy estimates for
our flow from a density-flux perspective. In the simplest case, we start with
the mass density
$M(u,\bar{u})=|u|^{2},$
whose linear flux is given by the momentum
$P(u,\bar{u})=2i\Im(u\partial_{x}\bar{u}).$
These can be viewed as translation invariant bilinear forms with symbols
$m(\xi,\eta)=1,\qquad p(\xi,\eta)=\xi+\eta.$
Integrating the densities we obtain the familiar mass and momentum,
$\mathbf{M}(u)=\int_{\mathbb{R}}M(u,\bar{u})\,dx,\quad\mathbf{P}(u)=\int_{\mathbb{R}}M(u,\bar{u})\,dx.$
At the nonlinear level, we have the density flux relation
$\partial_{t}M(u,\bar{u})=\partial_{x}P(u,\bar{u})+C^{4}_{m}(u,\bar{u},u,\bar{u}),$
where $C^{4}_{m}$ is a symmetric translation invariant real multilinear form
which depends on our cubic nonlinearity $C$. A key observation in [21] is
that, under the conservative assumption (H2) on the nonlinearity, the mass
density admits a quartic correction which is accurate to sixth order.
Precisely, the correction has the form
(2.2) $M^{\sharp}(u)=M(u)+B^{4}_{m}(u,\bar{u},u,\bar{u}),$
and the associated density flux relation has the form
(2.3)
$\partial_{t}M^{\sharp}(u)=\partial_{x}(P(u)+R^{4}_{m}(u,\bar{u},u,\bar{u}))+R^{6}_{m}(u,\bar{u},u,\bar{u},u,\bar{u}).$
with suitable translation invariant multilinear forms $R^{4}_{m}$ and
$R^{6}_{m}$. The corresponding integral corrected mass is
$\mathbf{M}^{\sharp}(u)=\int_{\mathbb{R}}M^{\sharp}(u)\,dx.$
The choice of the symbols $b^{4}_{m}$ and $r^{4}_{m}$ above depends on the
behavior of $c^{4}_{m}$ near the resonant set $\mathcal{R}$, precisely they
have to solve the division problem
(2.4) $c^{4}_{m}+i\Delta^{4}\xi^{2}\,b^{4}_{m}=i\Delta^{4}\xi\,r^{4}_{m}.$
This is possible due to our condition (H2), which implies that $c^{4}_{m}=0$
on $\mathcal{R}$. But the choice is not uniquely determined, so it is
important to make a good one, i.e. which insures good symbol bounds. To
achieve this, in [21] we decompose the phase space for frequency quadruples
into three overlapping regions which can be separated using cutoff functions
which are smooth on the unit scale:
1. i)
The full division region,
$\Omega_{1}=\\{{\delta\xi^{\text{med}}}\lesssim 1\\},$
which represents a full unit size neighbourhood of the resonant set
$\mathcal{R}$.
2. ii)
The region
$\Omega_{2}=\\{1+|\Delta^{4}\xi|\ll{\delta\xi^{\text{med}}}\\},$
where ${\tilde{\Delta}}^{4}\xi^{2}$ must be elliptic,
${\tilde{\Delta}}^{4}\xi^{2}\approx{\delta\xi^{\text{hi}}}{\delta\xi^{\text{med}}}$,
and thus we will favor division by the symbol ${\tilde{\Delta}}^{4}\xi^{2}$.
3. iii)
The region
$\Omega_{3}=\\{1\ll{\delta\xi^{\text{med}}}\lesssim|\Delta^{4}\xi|\\},$
we will instead divide by $\Delta^{4}\xi$; this is compensated by the
relatively small size of this region.
Since we will also need this in the present paper, we state the result in the
following
###### Proposition 2.1 ([21]).
Assume that the nonlinearity $C$ satisfies the conditions (H1), (H2). Then
there exist multilinear forms $B^{4}_{m}$ $R^{4}_{m}$ and $R^{6}_{m}$ so that
the relation (2.3) holds for solutions $u$ to (1.4), and so that the symbols
$c^{4}_{m}$ and $r^{4}_{m}$ satisfy the bounds
1. i)
Size
(2.5) $\displaystyle|\partial^{\alpha}r^{4}_{m}|\lesssim$ $\displaystyle\
\frac{1}{\langle{\delta\xi^{\text{med}}}\rangle},$
$\displaystyle|\partial^{\alpha}b^{4}_{m}|\lesssim$ $\displaystyle\
\frac{1}{\langle{\delta\xi^{\text{hi}}}\rangle\langle{\delta\xi^{\text{med}}}\rangle}.$
2. ii)
Support: $b^{4}$ is supported in $\Omega_{1}\cup\Omega_{2}$ and
$\tilde{r}^{4}$ is supported in $\Omega_{1}\cup\Omega_{3}$.
In addition, we have the fixed time bound
(2.6)
$\left|\int_{\mathbb{R}}B^{4}_{m}(u,u,u,u)\,dx\right|\lesssim\|u\|_{L^{2}_{x}}^{4}.$
We make several remarks concerning this result:
* •
No bound for the symbol $r^{6}_{m}$ is provided in the proposition. This is
because $r^{6}_{m}$ is obtained directly as the contribution of $C$ to the
time derivative of $b^{4}_{m}$.
* •
This proposition is a consequence of Lemma 4.1 and Lemma 7.1 in [21].
* •
The defocusing hypothesis (H3), which is used for the final result in [21],
plays no role here.
* •
The estimate (2.6) shows that the mass correction is perturbative for as long
as the solution $u$ remains small in $L^{2}$.
* •
A similar analysis applies for the momentum conservation law. But the
counterpart of the above Proposition for the momentum is less useful directly,
and instead it is used in [21] only in a frequency localized context.
### 2.3. Frequency localized density-flux identities
Instead of relying on the more standard Littlewood-Paley decomposition, the
analysis in [21] uses a frequency decomposition on the unit scale in
frequency. Given any integer $j$, we will use localized versions of the mass
in a unit size region around $j$. More generally, for an interval
$A\subset{\mathbb{Z}}$, we use a symbol $a_{0}$ which is frequency localized
in a unit neighbourhood of $A$. At the level of bilinear forms, we will use
the symbol
$a(\xi,\eta)=a_{0}(\xi)a_{0}(\eta).$
Corresponding to such $a$ we define quadratic localized mass, momentum and
energy densities by the symbols
$m_{a}(\xi,\eta)=a(\xi,\eta),\qquad
p_{a}(\xi,\eta)=(\xi+\eta)a(\xi,\eta),\qquad
e_{a}(\xi,\eta)=(\xi+\eta)^{2}a(\xi,\eta).$
The associated bilinear forms are denoted by $M_{a}$, $P_{a}$, respectively
$E_{a}$. If $A=\\{j\\}$ then we simply replace the subscript $a$ with $j$.
It is shown in [21], again under the assumptions (H1) and (H2), that one may
find quartic corrections $M^{\sharp}_{a}$ and $P^{\sharp}_{a}$ of the form
(2.7) $M^{\sharp}_{a}(u)=M_{a}(u)+B^{4}_{m,a}(u,\bar{u},u,\bar{u}),$ (2.8)
$P^{\sharp}_{a}(u)=P_{a}(u)+B^{4}_{p,a}(u,\bar{u},u,\bar{u}),$
for which we obtain density-flux identities akin to (2.3), namely
(2.9)
$\partial_{t}M^{\sharp}_{a}(u)=\partial_{x}(P_{a}(u)+R^{4}_{m,a}(u))+R^{6}_{m,a}(u),$
and
(2.10)
$\partial_{t}P^{\sharp}_{a}(u)=\partial_{x}(E_{a}(u)+R^{4}_{p,a}(u))+R^{6}_{p,a}(u).$
We will consider these relations together with their Galilean shifts obtaining
relations of the form
(2.11)
$(\partial_{t}-2\xi_{0}\partial_{x})M^{\sharp}_{a}(u)=\partial_{x}(P_{a,\xi_{0}}(u)+R^{4}_{m,a,\xi_{0}}(u))+R^{6}_{m,a,\xi_{0}}(u),$
respectively
(2.12)
$(\partial_{t}-2\xi_{0}\partial_{x})P^{\sharp}_{a,\xi_{0}}(u)=\partial_{x}(E_{a,\xi_{0}}(u)+R^{4}_{p,a,\xi_{0}}(u))+R^{6}_{p,a,\xi_{0}}(u).$
These correspond to the algebraic division relations
(2.13) $c^{4}_{m,a}+i\Delta^{4}(\xi-\xi_{0})^{2}b^{4}_{m,a}=i\Delta^{4}\xi
r^{4}_{m,a,\xi_{0}},$
respectively
(2.14)
$c^{4}_{p,a,\xi_{0}}+i\Delta^{4}(\xi-\xi_{0})^{2}b^{4}_{p,a,\xi_{0}}=i\Delta^{4}\xi
r^{4}_{p,a,\xi_{0}},$
where $c^{4}_{m,a}$ and $c^{4}_{p,a,\xi_{0}}$ are the density-flux sources
corresponding to uncorrected mass, respectively momentum.
The symbols above are connected in the obvious way. Precisely, we have
(2.15) $r^{4}_{m,a,\xi_{0}}=r^{4}_{m,a}-2\xi_{0}b^{4}_{m,a},\qquad$
and
(2.16) $P^{\sharp}_{a,\xi_{0}}=P^{\sharp}_{a}-2\xi_{0}M^{\sharp}_{a},\qquad
b^{4}_{p,a,\xi_{0}}=b^{4}_{p,a}-2\xi_{0}b^{4}_{m,a},$
and finally
(2.17)
$r^{4}_{p,a,\xi_{0}}=r^{4}_{p,a}-2\xi_{0}b^{4}_{p,a}-2\xi_{0}r^{4}_{m,a,\xi_{0}}.$
To use these density flux relations we need to have appropriate bounds for our
symbols:
###### Proposition 2.2.
Let $J\subset\mathbb{R}$ be an interval of length $r$, and
$d(\xi_{0},J)\lesssim r$. Assume that $a$ is supported in $J\times J$, with
bounded and uniformly smooth symbol. Then the relations (2.11) and (2.12) hold
with symbols $b^{4}_{m,a}$, $b^{4}_{p,a,\xi_{0}}$, $r^{4}_{m,a,\xi_{0}}$ and
$r^{4}_{p,a,\xi_{0}}$ which can be chosen to have the following properties:
1. i)
Support: they are all supported in the region where at least one of the
frequencies is in $J$.
2. ii)
Size:
(2.18)
$|b^{4}_{m,a}|\lesssim\frac{1}{\langle{\delta\xi^{\text{hi}}}\rangle\langle{\delta\xi^{\text{med}}}\rangle},\qquad|b^{4}_{p,a,\xi_{0}}|\lesssim\frac{r}{\langle{\delta\xi^{\text{hi}}}\rangle\langle{\delta\xi^{\text{med}}}\rangle},$
(2.19) $\displaystyle|r^{4}_{m,a,\xi_{0}}|\lesssim$ $\displaystyle\
\frac{1}{\langle{\delta\xi^{\text{med}}}\rangle}1_{\Omega_{1}\cup\Omega_{2}}+\frac{r}{\langle{\delta\xi^{\text{hi}}}\rangle\langle{\delta\xi^{\text{med}}}\rangle}1_{\Omega_{1}\cup\Omega_{3}},$
$\displaystyle|R^{4}_{p,a,\xi_{0}}|\lesssim$ $\displaystyle\
\frac{r}{\langle{\delta\xi^{\text{med}}}\rangle}1_{\Omega_{1}\cup\Omega_{2}}+\frac{r^{2}}{\langle{\delta\xi^{\text{hi}}}\rangle\langle{\delta\xi^{\text{med}}}\rangle}1_{\Omega_{1}\cup\Omega_{3}}.$
3. iii)
Regularity: similar bounds hold for all derivatives.
This is Proposition 4.3 in [21].
### 2.4. Interaction Morawetz identities
One way the desity flux relations above are used is to obtain more accurate
bounds for the mass propagation. However, another way to use them is via
interaction Morawetz identities, which yield bilinear estimates for the
interaction of different frequency portions of the solutions, or even the
self-interaction of unit frequency portions of solutions.
Given two frequency intervals $A$ and $B$ and corresponding mass/momentum
modified density associated to these intervals for two solutions $u,v$ to
(1.4), in [21] we define the associated _interaction Morawetz functional_ by
(2.20)
$\mathbf{I}_{AB}=\iint_{x>y}M^{\sharp}_{a}(u)(x)P^{\sharp}_{b,\xi_{0}}(v)(y)-P^{\sharp}_{a,\xi_{0}}(u)(x)M^{\sharp}_{b}(v)(y)\,dxdy,$
where we add several remarks:
* •
In applications, for the second solution $v$ we will simply choose a spatial
translation of the solution $u$.
* •
A velocity parameter $\xi_{0}$ is introduced on the right, but the functional
does not depend on $\xi_{0}$. This parameter plays a role, however, in
estimating the time derivative of $\mathbf{I}_{AB}$, and will be chosen to be
close to the sets $A$ and $B$.
* •
Such interaction functionals will be used in two settings:
1. (i)
the separated case $1\ll|A|\approx|B|\approx dist(A,B)$, and
2. (ii)
the self-interaction case $|A|+|B|+dist(A,B)\lesssim 1$.
The time derivative of the interaction Morawetz functional is computed using
the frequency localized mass density-flux (2.11) and the corresponding
momentum density-flux (2.12). This yields a localized _interaction Morawetz
identity_ ,
(2.21)
$\frac{d}{dt}\mathbf{I}_{AB}=\mathbf{J}^{4}_{AB}+\mathbf{J}^{6}_{AB}+\mathbf{J}^{8}_{AB}+\mathbf{K}^{8}_{AB},$
where the terms on the right are described as follows:
1. a)
the quartic contribution $\mathbf{J}^{4}_{a}$ is
$\mathbf{J}^{4}_{AB}=\int_{\mathbb{R}}M_{a}(u)(x)E_{b,\xi_{0}}(v)(x)+M_{b}(v)(x)E_{a,\xi_{0}}(u)(x)-2P_{a,\xi_{0}}(u)(x)P_{b,\xi_{0}}(v)(x)\,dx,$
and is used to capture bilinear $L^{2}$ bounds.
2. b)
The sixth order term $\mathbf{J}^{6}_{a}$ has the form
$\mathbf{J}^{6}_{AB}=\int_{\mathbb{R}}-(P_{a,\xi_{0}}B^{4}_{p,b,\xi_{0}}+P_{b,\xi_{0}}R^{4}_{m,a,\xi_{0}})+(M_{a}R^{4}_{p,b,\xi_{0}}+E_{b,\xi_{0}}B^{4}_{m,a,\xi_{0}})-\text{symmetric}\,dx,$
where in the symmetric part we interchange both the indices $a,b$ and the
functions $u,v$. In the defocusing case its symbol has a favorable sign on the
diagonal, and is used to capture the $L^{6}$ bound in the self-interaction
case. Here, it is estimated perturbatively.
3. c)
The eight-linear term
$\mathbf{J}^{8}_{AB}=\int_{\mathbb{R}}-R^{4}_{m,a,\xi_{0}}B^{4}_{p,b,\xi_{0}}+B^{4}_{m,a,\xi_{0}}R^{4}_{p,b,\xi_{0}}-\text{symmetric}\,dx.$
plays a perturbative role.
4. d)
The $8^{+}$-linear term $\mathbf{K}^{8}_{AB}$ has the form
$\mathbf{K}^{8}_{AB}=\iint_{x>y}M^{\sharp}_{a}(x)R^{6}_{p,a,\xi_{0}}+P^{\sharp}_{a,\xi_{0}}R^{6}_{m,a,\xi_{0}}-\text{symmetric}\,dxdy.$
This is a double integral, which also includes a $10$-linear term. It is also
estimated perturbatively.
## 3\. Strichartz and bilinear $L^{2}$ bounds
To obtain estimates for $L^{2}$ solutions $u$ to (1.4), a unit scale frequency
decomposition is needed,
$u=\sum_{k\in{\mathbb{Z}}}u_{k},\qquad u_{k}:=P_{k}u,$
where $P_{k}$ are multipliers with smooth symbols localized in a unit
neighbourhood of the integer frequency $k$. To measure the components $u_{k}$
we use a frequency envelope $\\{c_{k}\\}\in\ell^{2}$ in order to transfer
bounds from the initial data to the solutions.
These frequency envelopes are chosen to satisfy an adapted version of the
slowly varying property, originally introduced by Tao [29] in the context of
dyadic decompositions. Such a property is needed in order to account for the
nonlinear leakage of energy between nearby frequencies.
We recall the frequency envelope set-up in [21], associated to lattice
decompositions:
###### Definition 3.1.
A lattice frequency envelope $\\{c_{k}\\}$ is said to have the maximal
property if
(3.1) $Mc\leq Cc,$
where $Mc$ represents the maximal function of $c$. Here $C$ is a universal
constant.
Frequency envelopes that have this property will be called _admissible_. The
proof of the $L^{6}$ Strichartz and the bilinear $L^{2}$ bounds will be
phrased as a bootstrap argument relative to an admissible frequency envelope
for the initial data.
###### Theorem 4.
Let $u\in C[0,T;L^{2}]$ be a solution for the equation (1.4) with initial data
$u_{0}$ which has $L^{2}$ size at most $\epsilon$. Let $\\{c_{k}\\}$ be a
maximal frequency envelope for the initial data in $L^{2}$, also of size
$\epsilon$,
$\|u_{0k}\|_{L^{2}}\lesssim\epsilon c_{k}.$
Assume that
(3.2) $T\ll\epsilon^{-6}.$
Then the solution $u$ satisfies the following bounds in $[0,T]$:
1. (i)
Uniform frequency envelope bound:
(3.3) $\|u_{k}\|_{L^{\infty}_{t}L^{2}_{x}}\lesssim\epsilon c_{k},$
2. (ii)
Localized Strichartz bound:
(3.4) $\|u_{k}\|_{L_{t,x}^{6}}\lesssim(\epsilon c_{k})^{\frac{2}{3}},$
3. (iii)
Localized Interaction Morawetz:
(3.5)
$\|\partial_{x}|u_{k}|^{2}\|_{L^{2}_{t,x}}\lesssim\epsilon^{2}c_{k}^{2},$
4. (iv)
Transversal bilinear $L^{2}$ bound:
(3.6)
$\|\partial_{x}(u_{A}{\bar{u}}_{B}(\cdot+x_{0}))\|_{L^{2}_{t,x}}\lesssim\epsilon^{2}c_{A}c_{B}\,\langle
dist(A,B)\rangle^{\frac{1}{2}},$
for all $x_{0}\in\mathbb{R}$ whenever $|A|+|B|\lesssim\langle
dist(A,B)\rangle$.
This proposition mirrors a similar result in Section 7 of [21], with two key
differences. On one hand we drop the defocusing assumption (H3), and on the
other hand we limit the size of the time interval in (3.2).
The $L^{6}$ Strichartz estimates (1.10) and the bilinear $L^{2}$ bounds in
(1.11) follow from the estimates in the above proposition, by the same
arguments as those in Section 8 of [21].
To prove this theorem, we make a bootstrap assumption where we assume the same
bounds but with a worse constant $C$, as follows:
1. (i)
Uniform frequency envelope bound,
(3.7) $\|u_{k}\|_{L^{\infty}_{t}L^{2}_{x}}\lesssim C\epsilon c_{k},$
2. (ii)
Localized Strichartz bound,
(3.8) $\|u_{k}\|_{L^{6}_{t,x}}\lesssim C(\epsilon c_{k})^{\frac{2}{3}},$
3. (iii)
Localized Interaction Morawetz,
(3.9) $\|\partial_{x}|u_{k}|^{2}\|_{L^{2}_{t,x}}\lesssim
C\epsilon^{2}c_{k}^{2},$
4. (iv)
Transversal Interaction Morawetz,
(3.10)
$\|\partial_{x}(u_{k_{1}}\bar{u}_{k_{2}}(\cdot+x_{0}))\|_{L^{2}_{t,x}}\lesssim
C\epsilon^{2}c_{k_{1}}c_{k_{2}}\langle k_{1}-k_{2}\rangle^{\frac{1}{2}}$
uniformly for all $x_{0}\in\mathbb{R}$.
Then we seek to improve the constant in these bounds. The gain will come from
the fact that the $C$’s will always come paired either with extra $\epsilon$
factors, or with $T\epsilon^{6}$ factors.
To a large extent the proof largely repeats the proof of the corresponding
result in [21], so we review the steps and expand the portion where the
argument differs here.
_STEP 1:_ The proof of the energy bound (3.3). This is done by integrating the
density flux relation (2.9) for the localized mass $\mathbf{M}_{k}(u)$. The
argument in [21] applies unchanged.
_STEP 2:_ The proof of the $L^{6}$ Strichartz bound (3.3). In [21] this is
proved together with (3.5) by integrating the interaction Morawetz identity
applied to the pair $(u_{k},u_{k})$. However, in the focusing case the sign of
the $u_{k}^{6}$ contribution changes, and the same argument no longer applies.
Instead, here we will estimate the $L^{6}$ norm directly using an
interpolation argument. Precisely, for the function $v_{k}=|u_{k}|^{2}$, using
(3.7) and (3.9), we have the bounds
$\|v_{k}\|_{L^{\infty}_{t}L^{1}_{x}}\lesssim C^{2}\epsilon^{2}c_{k}^{2},$
respectively
$\|v_{k}\|_{L^{2}_{t}\dot{H}^{1}_{x}}\lesssim C^{2}\epsilon^{2}c_{k}^{2}.$
We interpolate between the two estimates in homogeneous Sobolev spaces, with
weights $5/9$ and $4/9$. We obtain
$\|v_{k}\|_{L^{\frac{9}{2}}_{t}\dot{W}^{\frac{4}{9},\frac{9}{7}}_{x}}\lesssim\|v_{k}\|_{L^{\infty}_{t}L^{1}_{x}}^{\frac{5}{9}}\|v_{k}\|_{L^{2}_{t}\dot{H}^{1}_{x}}^{\frac{4}{9}}\lesssim
C^{2}\epsilon^{2}c_{k}^{2}.$
By Sobolev embeddings $\dot{W}^{\frac{4}{9},\frac{9}{7}}$ embeds in $L^{3}$
so, using also Hölder’s inequality with respect to time, we obtain
$\|v_{k}\|_{L^{3}_{t}L_{x}^{3}}\lesssim
T^{\frac{1}{9}}\|v_{k}\|_{L^{\frac{9}{2}}L^{3}}\lesssim
T^{\frac{1}{9}}\|v_{k}\|_{L^{\frac{9}{2}}\dot{W}^{\frac{4}{9},\frac{9}{7}}}\lesssim
C^{2}T^{\frac{1}{9}}\epsilon^{2}c_{k}^{2}=C^{2}(T\epsilon^{6})^{\frac{1}{9}}\epsilon^{\frac{4}{3}}c_{k}^{2}.$
This implies the desired Strichartz bound (3.4) under the time constraint
(3.2).
_STEP 3:_ The proof of the bilinear $L^{2}$ bound (3.6). This is again exactly
as in [21], by applying the interaction Morawetz identity to the functions
$(u_{A},u_{B}(\cdot+x_{0}))$. We note that the $L^{6}$ bound is used as an
input in this proof, and the defocusing assumption (H2) is not needed. Here we
view (3.5) as a special case of (3.6), and no longer in conjunction with
(3.3).
## 4\. Long time energy estimates
The frequency envelope bounds in Theorem 4 provide us with uniform energy
bounds on the $\epsilon^{-6}$ time-scale, and so they do not suffice in order
to prove our main result in Theorem 1, which is on the $\epsilon^{-8}$ time-
scale. To fill in this gap, we will prove a direct energy estimate on the
$\epsilon^{-8}$ time-scale. Precisely, we will show the following:
###### Proposition 4.1.
Let $u$ be an $L^{2}$ solution for (1.4) in a time interval $[0,T]$. Assume
that the initial data for (1.4) satisfies
(4.1) $\|u_{0}\|_{L_{x}^{2}}\leq\epsilon\ll 1,$
and that $T\ll\epsilon^{-8}$. Then the solution $u$ satisfies
(4.2) $\|u\|_{L_{t}^{\infty}(0,T;L_{x}^{2})}\leq 4\epsilon.$
Once we have this proposition, a continuity argument based on the local well-
posedness for (1.4) in $L^{2}$ implies Theorem 1.
###### Proof.
It suffices to prove that the conclusion holds assuming that we have the
bootstrap assumption
(4.3) $\|u\|_{L^{\infty}(0,T;L^{2})}\leq 8\epsilon.$
Instead of tracking directly the mass $\mathbf{M}(u)=\|u\|_{L_{x}^{2}}^{2}$,
it is more efficient to work with the modified mass
$\mathbf{M}^{\sharp}(u)=\int_{\mathbb{R}}M^{\sharp}(u)\,dx.$
In view of the bound (2.6), we have
$\mathbf{M}(u)=\mathbf{M}^{\sharp}(u)+O(\epsilon^{4}).$
Since $\epsilon\ll 1$, we have
(4.4) $\mathbf{M}(u)(0)\leq 2\epsilon^{2},$
and it suffices to show that
(4.5) $\mathbf{M}(u)(t)\leq 4\epsilon^{2},\qquad t\in[0,T].$
In view of Proposition 2.3, the time evolution of $\mathbf{M}^{\sharp}$ is
given by
(4.6)
$\frac{d}{dt}\mathbf{M}^{\sharp}(u)=\int_{\mathbb{R}}R^{6}_{m}(u,{\bar{u}},u,{\bar{u}},u,{\bar{u}})\,dx.$
To bound its growth, we use the following
###### Lemma 4.2.
Assume that the bounds (3.3)-(3.6) for $u$ hold in a time interval $[0,T]$.
Then we have
(4.7)
$\|R^{6}_{m}\|_{L^{1}_{t,x}([0,T]\times\mathbb{R})}\lesssim\epsilon^{4}.$
This is Lemma 7.3 in [21]. We apply this lemma on time intervals of size
$\epsilon^{-6}$, where the bounds (3.3)-(3.6) hold in view of Theorem 4, and
then add up the results. Then for $T>\epsilon^{-6}$ we get
(4.8)
$\|R^{6}_{m}\|_{L^{1}_{t,x}([0,T]\times\mathbb{R})}\lesssim\epsilon^{4}(T\epsilon^{6}).$
Hence for $T\ll\epsilon^{-8}$ we arrive at
(4.9) $\|R^{6}_{m}\|_{L^{1}_{t,x}([0,T]\times\mathbb{R})}\ll\epsilon^{2}.$
This allows us to obtain (4.5) from (4.4), thereby concluding the proof of the
proposition.
∎
## 5\. Optimality remarks
As noted earlier, a key obstruction to global dispersive estimates in the
focusing case is given by the potential existence of small solitons. Here we
will restrict our attention to the simplest model, namely the focusing cubic
NLS, and test the optimality of our estimates on solitons for this model. It
is not so difficult to show that small solitons exist for our model whenever
the focusing assumption (H4) is satisfied in some frequency region.
All cubic NLS solitons are equivalent modulo scaling and Galilean
transformations. Our bounds are Galilean invariant, so we set the soliton
velocity to zero and we focus on scaling. Then the unit scale soliton has the
form
$u(x,t)=e^{it}Q(x),\qquad Q(x)=\operatorname{sech}(x).$
Rescaled to the frequency scale $\lambda$, this yields the solitons
$u_{\lambda}(x,t)=e^{it\lambda^{2}}Q_{\lambda}(x),\qquad
Q_{\lambda}(x)=\lambda Q(\lambda x),$
which has initial data size
$\|u_{0}\|_{L_{x}^{2}}^{2}=\|Q_{\lambda}\|_{L^{2}_{x}}^{2}=\int_{\mathbb{R}}\lambda^{2}Q^{2}(\lambda
x)\,dx\approx\lambda,$
so for Theorem 1 we will choose $\lambda=\epsilon^{2}$.
On the other hand,
$\|u_{\lambda}\|_{L^{6}_{t,x}(0,T,\mathbb{R})}^{6}=T\int_{\mathbb{R}}\lambda^{6}Q^{6}(\lambda
x)\,dx\approx T\lambda^{5}.$
Then it is easily seen that we have approximate equality in (1.10) and (1.13).
Similarly, we compute the bilinear Strichartz norm. Due to the bound from
below on $|I|$, we have $c\gtrsim\lambda^{-\frac{1}{2}}$ whereas $u$ is
concentrated at frequency $\lesssim\lambda$. Then
$\|\partial_{x}|u_{\lambda}|^{2}\|_{L^{2}_{t}(0,T;\dot{H}^{\frac{1}{2}}_{x}+cL^{2}_{x})}^{2}\approx
c^{-2}\|\partial_{x}|u_{\lambda}|^{2}\|_{L^{2}_{t,x}(0,T;L^{2})}^{2}=\lambda(T\lambda^{2})^{-1}T\int_{\mathbb{R}}\lambda^{6}\partial_{x}Q_{x}^{2}(\lambda
x)\,dx\approx\lambda^{2}.$
This corresponds to having equality in (1.11) and (1.14).
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|
# Salt-bearing disk candidates around high-mass young stellar objects
Adam Ginsburg Department of Astronomy, University of Florida, P.O. Box 112055,
Gainesville, FL, USA Brett A. McGuire Department of Chemistry, Massachusetts
Institute of Technology, Cambridge, MA 02139 National Radio Astronomy
Observatory, Charlottesville, VA 22903 Patricio Sanhueza National
Astronomical Observatory of Japan, National Institutes of Natural Sciences,
2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan Department of Astronomical
Science, SOKENDAI (The Graduate University for Advanced Studies), 2-21-1
Osawa, Mitaka, Tokyo 181-8588, Japan Fernando Olguin Institute of Astronomy
and Department of Physics, National Tsing Hua University, Hsinchu 30013,
Taiwan Luke T. Maud ESO Headquarters, Karl-Schwarzchild-Str 2 85748 Garching,
Germany Kei E. I. Tanaka Center for Astrophysics and Space Astronomy,
University of Colorado Boulder, Boulder, CO 80309, USA National Astronomical
Observatory of Japan, National Institutes of Natural Sciences, 2-21-1 Osawa,
Mitaka, Tokyo 181-8588, Japan Yichen Zhang Department of Astronomy,
University of Virginia, Charlottesville, VA 22904, USA RIKEN Cluster for
Pioneering Research, Wako, Saitama 351-0198, Japan Henrik Beuther Max Planck
Institute for Astronomy, Königstuhl 17, 69117, Heidelberg, Germany Nick
Indriolo AURA for the European Space Agency (ESA), Space Telescope Science
Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
###### Abstract
Molecular lines tracing the orbital motion of gas in a well-defined disk are
valuable tools for inferring both the properties of the disk and the star it
surrounds. Lines that arise only from a disk, and not also from the
surrounding molecular cloud core that birthed the star or from the outflow it
drives, are rare. Several such emission lines have recently been discovered in
one example case, those from NaCl and KCl salt molecules. We studied a sample
of 23 candidate high-mass young stellar objects (HMYSOs) in 17 high-mass star-
forming regions to determine how frequently emission from these species is
detected. We present 5 new detections of water, NaCl, KCl, PN, and SiS from
the innermost regions around the objects, bringing the total number of known
briny disk candidates to 9. Their kinematic structure is generally disk-like,
though we are unable to determine whether they arise from a disk or outflow in
the sources with new detections. We demonstrate that these species are
spatially coincident in a few resolved cases and show that they are generally
detected together, suggesting a common origin or excitation mechanism. We also
show that several disks around HMYSOs clearly do not exhibit emission in these
species. Salty disks are therefore neither particularly rare in high-mass
disks, nor are they ubiquitous.
††software: The source code underlying this work are available from github at
https://github.com/keflavich/saltmining/releases/tag/accepted-2022-11-04. This
work used CARTA (Comrie et al., 2021), Jupyter notebooks (Kluyver et al.,
2016). numpy (van der Walt et al., 2011; Harris et al., 2020), scipy (Virtanen
et al., 2020), astropy (Astropy Collaboration et al., 2013, 2018; The Astropy
Collaboration et al., 2022), spectral-cube (Ginsburg et al., 2019b), radio-
beam (Koch et al., 2021), and CASA-6
(https://casa.nrao.edu/casadocs/casa-5.6.0/introduction/casa6-installation-
and-usage). matplotlib (Hunter, 2007), astroquery (Ginsburg et al., 2019c),
and pyspeckit (Ginsburg et al., 2022).
doublespace
## 1 Introduction
Circumstellar accretion disks develop around forming new stars. While the
presence of disks around low-mass stars has been clear for decades, we have
only definitively demonstrated that accretion disks exist around high-mass
young stellar objects (HYMSOs) in the last decade (e.g., Beltrán & de Wit,
2016; Maud et al., 2017; Ilee et al., 2018; Motogi et al., 2019; Johnston et
al., 2020; Sanna et al., 2021; Moscadelli et al., 2021).
One limiting factor in the detection and subsequent characterization of HYMSOs
has been the lack of molecular emission lines that arise from the disk, but
are not confused with or absorbed by the surrounding molecular cloud. A select
few lines have recently been discovered that are uniquely produced in the
disks of some HMYSOs. Emission from salt molecules has been detected in the
surroundings of four stars in three star-forming regions: Orion Source I
(Ginsburg et al., 2019a), G17.64+0.16 (Maud et al., 2019), and a pair in IRAS
16547-4247 (Tanaka et al., 2020) (hereafter, SrcI, G17, and I16547,
respectively). Of these, SrcI and G17 are confirmed disks, while the I16547
pair remain candidates, and in all cases the salt emission comes from zones
within $\lesssim 100$ au of the central source. Each of these sources also
exhibits H2O emission from both the (candidate) disk and a slightly more
extended region, and thus we dub these objects ‘brinaries.’
Salts are detected in the atmospheres of AGB and post-AGB stars, giving some
clues as to the physical conditions needed to produce them. Salts have been
detected in CRL2688 (Highberger et al., 2003), IRAS+10216 (Cernicharo &
Guelin, 1987), IK Tauri, VY Canis Majoris (Milam et al., 2007; Decin et al.,
2016), VX Sgr (mentioned in passing in Danilovich et al., 2021), and
OH231.8+44.2 (Sánchez Contreras et al., 2022). Sánchez Contreras et al. (2022)
discovered a brinary disk surrounding the post-AGB mass transfer system
OH231.8+4.2, highlighting the similarity between birth and death among
moderately massive stars. However, there are also some non-detections in well-
surveyed sources, including the S-type AGB stars W Aql and $\pi^{1}$ Gru
(Homan et al., 2020; Danilovich et al., 2021).
The development of line lists in the infrared, and more complete lists in the
radio, has been driven in part by interest in salts as constituents of the
atmospheres of hot planets, in which these species are predicted to be
important components of upper cloud layers (Barton et al., 2014). There is
therefore some motivation to understand where salts occur in disks and in or
on dust grains.
Studies of (post)-AGB stars and models of planetary atmospheres provide some
clues about the physical conditions required to produce gas-phase salts. NaCl
is expected to change states (from solid to gas or vice-versa) around
$\sim$500–600 K at planetary atmospheric pressure (i.e., 1 bar; Woitke et al.,
2018). Decin et al. (2016) suggest that it comes off of grains at 100–300 K
based on the detection locations in supergiant stars. These studies provide a
first hint about where NaCl may come from if it precipitates out of cooling
gas, though it remains unclear if the same mechanisms apply in and around YSO
disks.
Motivated by the detection of salts in a few HMYSOs with ALMA in recent years,
we present a first search for salt-bearing disks in the ALMA archives. In
Section 2, we describe the ALMA observations we analyze. In Section 3, we
describe the analysis approach (3.1.1) and detections (3.3). We discuss the
chemical correlations observed in the sample and possible reasons for
(non)detections in Section 4, then conclude.
## 2 Data & Sample Selection
We utilize archival and new data from several projects. We select data sets
that have high angular resolution ($\lesssim 0.1$ ″) targeting high-mass star-
forming regions. Most of the sources in our sample come from the Digging into
the Interior of Hot Cores with ALMA (DIHCA; PI: Sanhueza) program, which is
surveying $\sim 30$ candidate disks. The main aims of the DIHCA survey are to
study the interior of massive hot cores to determine whether they form high-
mass stars collapsing monolithically or by fragmenting into binary (multiple)
systems and to search for accretion disks around high-mass stars. The DIHCA
targets were selected from the literature as regions with previous
interferometric (e.g., SMA) observations and having an expected flux of $>$0.1
Jy at 230 GHz. All clumps follow the empirical threshold for high-mass star
formation suggested by Kauffmann & Pillai (2010). We only examine a small
subset of the DIHCA sample here because not all data were available as of
March 2022. Because our sample is not uniformly selected, we can say little
about completeness; we can only search for general trends. Nevertheless, the
trends we find are interesting and suggest that observing a more uniform
sample in the future would be productive.
DIHCA observations of G335, G333.23, NGC6334, IRAS16562, G34.43mm1, G29.96,
and G351.77 were obtained during July 2019, and observations of G5.89, G11.92,
IRAS18089, and W33A were taken both in September 2017 and July 2019. The
observations were reduced using CASA (v5.4.0-70; McMullin et al., 2007). The
data were then phase self-calibrated in three steps with decreasing solution
intervals and the continuum subtracted following the procedure of Olguin et
al. (2021). Dirty cubes were produced with a Briggs weighting robust parameter
of 0.5 using the CASA tclean task. We use dirty image cubes for expediency, so
it is possible significant improved images of these objects could be obtained,
though we note that the targeted lines are generally faint and would not be
affected by cleaning with typical clean parameters.
Table 1: Observation Summary
Field | Source Name | $\theta_{\rm maj}$ | $\theta_{\rm maj}$ | $\theta_{\rm min}$ | PA | $\sigma$ | $f(>5\sigma)$ | $N_{\rm beams}$ | $\sigma_{\rm avg}$ | $v$ ref. line | Distance
---|---|---|---|---|---|---|---|---|---|---|---
| | $\mathrm{AU}$ | ′′ | ′′ | ∘ | $\mathrm{K}$ | | | $\mathrm{K}$ | | $\mathrm{kpc}$
Orion | SrcI | 20 | 0.050 | 0.039 | 72.7 | 19.3 | 0.05 | 43 | 2.9 | NaCl J=18-17 v=0 | 0.4
S255IR | SMA1 | 40 | 0.036 | 0.027 | 5.6 | 46.9 | 0.04 | 19 | 10.6 | no clear disk | 1.6k
G351.77 | mm12 | 50 | 0.027 | 0.022 | -89.3 | 6.0 | 0.08 | 24 | 1.2 | H2O | 2.2c
G351.77 | mm2 | 50 | 0.027 | 0.022 | -89.3 | 6.0 | 0.05 | 21 | 1.3 | H2O | 2.2c
G351.77 | mm1 | 50 | 0.027 | 0.022 | -89.3 | 6.0 | 0.14 | 41 | 0.9 | H2O | 2.2c
G17 | G17 | 50 | 0.038 | 0.022 | 44.4 | 10.0 | 0.01 | 75 | 1.2 | H2O | 2.2c
NGC6334I | mm1d | 50 | 0.074 | 0.041 | 62.8 | 19.6 | 0.10 | 20 | 4.3 | H2O | 1.3d
NGC6334I | mm1b | 50 | 0.074 | 0.041 | 62.8 | 19.6 | 0.23 | 27 | 3.7 | H2O | 1.3d
NGC6334I | mm2b | 50 | 0.074 | 0.041 | 62.8 | 19.6 | 0.07 | 7 | 7.0 | H2O | 1.3d
NGC6334IN | SMA1b/d | 50 | 0.074 | 0.042 | 63.4 | 17.5 | 0.16 | 278 | 1.0 | H2O | 1.3d
NGC6334IN | SMA6 | 50 | 0.074 | 0.042 | 63.4 | 17.5 | 0.06 | 22 | 3.7 | H2O | 1.3d
IRAS18162 | GGD27 | 90 | 0.099 | 0.066 | -89.0 | 6.9 | 0.01 | 36 | 1.1 | SO $6_{5}-5_{4}$ | 1.3a
IRAS18089 | I18089-1732 | 100 | 0.064 | 0.045 | 65.8 | 18.4 | 0.33 | 57 | 2.4 | CH3OH | 2.3f
G34.43 | mm1 | 110 | 0.105 | 0.069 | 59.5 | 9.2 | 0.35 | 53 | 1.3 | no clear disk | 1.6l
IRAS16562 | G345.4938+01.4677 | 120 | 0.106 | 0.053 | 81.6 | 10.5 | 0.04 | 25 | 2.1 | H30$\alpha$ | 2.3g
I16547 | A | 130 | 0.065 | 0.043 | 35.7 | 23.2 | 0.19 | 7 | 8.4 | H2O | 2.9c
I16547 | B | 130 | 0.065 | 0.043 | 35.7 | 23.2 | 0.18 | 5 | 9.5 | H2O | 2.9c
G5.89 | mm15 | 130 | 0.063 | 0.043 | 66.3 | 19.2 | 0.00 | 85 | 2.1 | H2O | 3.0b
G335 | ALMA1 | 140 | 0.066 | 0.041 | 47.3 | 20.4 | 0.29 | 118 | 1.9 | no clear disk | 3.3i
W33A | mm1-main | 170 | 0.103 | 0.067 | -86.4 | 6.5 | 0.17 | 51 | 0.9 | H2O | 2.6k
G333.23 | mm1 | 220 | 0.069 | 0.041 | 54.1 | 18.4 | 0.02 | 70 | 2.2 | SO $6_{5}-5_{4}$ | 5.3h
G333.23 | mm2 | 220 | 0.069 | 0.041 | 54.1 | 18.4 | 0.08 | 45 | 2.7 | SO $6_{5}-5_{4}$ | 5.3h
G11.92 | mm1 | 220 | 0.101 | 0.066 | -86.9 | 6.6 | 0.16 | 103 | 0.7 | SO $6_{5}-5_{4}$ | 3.3e
G29.96 | submm1 | 530 | 0.099 | 0.071 | 65.0 | 9.6 | 0.22 | 369 | 0.5 | no clear disk | 7.4j
Observation properties. The ‘Field’ name indicates the region of the ALMA
pointing. The ‘Source Name’ is the identifier of the disk candidate examined.
$\theta$ gives the beam parameters, with $\theta_{maj}$ [au] providing the
physical size using the adopted distance. $\sigma$ is the average noise level
of the field, which is averaged down by $N_{beams}^{1/2}$ to give $\sigma_{\rm
avg}$, the noise level in the stacked spectrum. $f(>5\sigma)$ is the fraction
of the stacked spectrum that is above five times $\sigma_{avg}$; it is used as
a diagnostic of the line crowding in the spectrum covering 219.2–220.8 GHz,
which is high for complex-molecule-rich regions. The $v$ reference line is the
line used to create a velocity map to produce stacked spectra. Section 3.1.4
provides additional details. Distances come from the following sources: a
Añez-López et al. (2020), b Sato et al. (2014); Fernández-López et al. (2021),
c Beuther et al. (2017), d Chibueze et al. (2014), e Sato et al. (2014), f Xu
et al. (2011), g Guzmán et al. (2020), h Whitaker et al. (2017), i Peretto et
al. (2013), j Kalcheva et al. (2018), k Reid et al. (2014), l Kurayama et al.
(2011), m Ilee et al. (2018),
We use the SrcI data from Ginsburg et al. (2018, 2019a). We use G17 data from
Maud et al. (2019). We use G351.77 data from both DIHCA and Beuther et al.
(2019). We use I16547 data from both DIHCA and Tanaka et al. (2020). For each
of these data sets, we refer the reader to the cited papers for the data
reduction description. We include summary statistics of these observations in
Table 1. We give additional details about the physical resolution, distance to
the targets, and the line used as a velocity guide (see §3.1) in Table 1.
Finally, we use proprietary data toward Sh 255-IR SMA1 (hereafter S255IR) from
project 2019.1.00492.S (PI Ginsburg). We use the archive-produced data
products, which were cleaned with the ALMA pipeline, and imaged them with CASA
6.4.3.4. The images were cleaned to a depth of 10 mJy using Briggs robust=0
weighting.
## 3 Results and Analysis
We search for lines of NaCl, KCl, H2O, SiS, and H30$\alpha$ in each of the
target pointings (see Table 2). Since each target was selected for having a
high luminosity or a strong HMYSO disk candidate beforehand, we used the
literature identification of existing sources as our starting point. We cut
out cubes centered on the brightest continuum source in the ALMA images
(except G5.89; see §B.4). We also searched fainter continuum sources,
selecting small sub-regions around each of the compact continuum peaks that
could plausibly contain disks. Because the source selection is based on a by-
eye examination of the data over a limited field of view (in most cases, only
the inner 5–10 arcseconds of the ALMA field of view was imaged), the sample
presented here has unknown completeness - the conclusions we draw will
therefore be only suggestive, not conclusive.
Most of our line detections come from stacked spectra of resolved disk-like
objects. We describe in Section 3.1 the line stacking approach used to obtain
higher signal-to-noise ratio spectra that represent average values over the
candidate disk. Section 3.3 describes the detections in individual sources and
shows some of the extracted images and spectra. Additional images and spectra
are displayed in the appendices.
The main result is the detection of the ‘brinary’ lines toward the 9 sources
(of which 3 are tentative detections) shown in Figures 1 and 2. These initial
figures show moment maps of the NaCl lines as described in Section 3.1.3.
Figure 1: Moment-0 (integrated intensity) images of the resolved sources in
NaCl lines. For SrcI (top-left), this is the integrated intensity of the NaCl
$J$=18-17 $v$=0 line. For the remainder, G17 (top right), G351 mm1 (bottom
left), and NGC 6334I mm1b (bottom right), these are the average of the
$J$=18-17 and $J$=17-16 transitions of both the $v$=0 and $v$=1 states. The
coordinates are given in RA/Dec offset from the central position specified
under the abscissa. The scalebars show physical sizes as labeled. The ellipses
in the corners show the full-width half-maximum beam ellipse.
Figure 2: Integrated intensity (moment-0) images of the NaCl stacked lines
toward G351mm2 (top left), G351mm12 (top middle), NGC 6334I mm2b - which is
only a tentative detection (top right), I16547A (bottom left) and B (bottom
center), and W33A (bottom right). These disk candidates are either unresolved
or marginally resolved.
### 3.1 Line stacking extraction
From source candidate identification from the continuum data, the analysis
forks down two different paths. We start by searching by eye for emission
associated with the H2O line, which is quite bright in SrcI, in §3.1.1, which
we then use as a kinematic reference. If water is not detected, we use a
different line as our kinematic reference as described in §3.1.2.
#### 3.1.1 Water-driven analysis
If the water line is detected, we use it to create a ‘velocity map’ following
this procedure:
1. 1.
Cut out a cube containing the region around the estimated central $v_{\rm
LSR}\pm 20\>\textrm{km~{}s}^{-1}$ in velocity and encompassing only the
candidate disk region in space.
2. 2.
Create a peak intensity map of the line in that region.
3. 3.
Create a threshold mask including only the (by-eye) estimated significant
emission in the peak intensity map. Use one iteration of binary erosion and
three to seven iterations of binary dilation to remove isolated bright noise
pixels and fill back in the mask.111Binary dilation refers to mathematical
morphology operations in which any pixel having value False and a neighbor
with the value True is set to True. Binary erosion is the inverse operation.
4. 4.
Create a volumetric threshold mask including only pixels above the estimated
noise level. Then, use one iteration of binary erosion followed by one to
three iterations of binary dilation to fill in the mask. As in the previous
step, this step is to eliminate isolated bright pixels, but because it is in
3D, a different threshold can be adopted.
5. 5.
Create a moment-1 (intensity-weighted velocity) map of the spatially and
volumetrically masked data cube.
The erosion and dilation steps are performed to exclude isolated bright pixels
and to maximally include all pixels associated with source emission.
We then use the velocity map to stack the spectra obtained across the disk
candidate. Each spectrum from each spatial pixel in the cube that has a
measured velocity is shifted such that the line peak is moved to 0 km s-1. The
spectra are then averaged to produce the stacked spectrum.
This stacking process assumes that all spectra through the candidate disk will
have similar peak intensity and width but different central velocities; this
assumption held well in the SrcI spectrum (Ginsburg et al., 2018). We assume
that the kinematics of our selected line are the same as the lines of
interest; this assumption is justified by position-velocity diagrams in
§3.3.2.
We demonstrate the advantage of this line stacking approach in Figure 3. The
signal-to-noise in targeted lines is substantially increased, and faint lines
appear that are otherwise missed. In the example figure, the most obvious case
is KCl $v$=4 $J$=31-30, which is not apparent in either the peak intensity or
average spectrum, but is strongly evident in the stacked spectrum. Those
sources that are best-resolved - in our sample, G351 and G17 - show the most
improvement.
Figure 3: Demonstration of the utility of stacking analysis toward G17 in one
spectral window. The black spectrum is stacked by shifting each individual
spectrum to match the location of the peak intensity of the H2O line; it is
the same in both panels. The top panel shows the peak intensity spectrum taken
over the same area as the stacked spectrum, and the bottom shows the mean
spectrum in blue. It is clear that the stacked spectrum has clearer features
and better signal-to-noise ratios than either the peak or average spectrum.
We label the resulting averaged spectra with known NaCl and KCl transitions
and selected other lines, including those of SiO, SiS, H2O, H30$\alpha$, and
several prominent other molecules (see Table 2). We then use these plots to
populate the detection table, Table 3. We regard the lines as firm detections
only if they:
* •
Are prominent, bright, and relatively isolated (e.g., H2O, as in Figure 4)
* •
Exhibit an appropriately broad linewidth (H30$\alpha$ is expected to have
$\sigma_{v}\gtrsim 5$ km s-1 because it comes from hot plasma ($T\sim 10^{4}$
K), while molecular species are expected to be narrower, coming from gas at
$T<1000$ K).
For species for which we label the vibrationally excited transitions, we
consider only the $v$=0, $v$=1, and $v$=2 levels, as higher levels are
expected to be weaker if present.
There are several cases where a compelling detection of one line of a species
(NaCl or KCl) is detected, but an adjacent state is not detected. For example,
the $J$=18-17 $v$=1 line is detected, but the $J$=16-17 $v$=1 line is not. In
these cases, we regard the detection as tentative and note it in Table 3 with
an asterisk. These were cases in which the line may still be present, but may
be hidden by either blending with neighboring species or absorption by a line
in the surrounding core.
We report detections qualitatively rather than quantitatively because the
detections are generally obvious and high signal-to-noise. When a detection is
ambiguous, it is not because of high noise but because of confusion with
neighboring lines (e.g., 41KCl $J$=31-30 is confused with NaCl $v=1$
$J=18-17$) or absorption by non-disk material (e.g., KCl $v$=4 $J$=31-30).
#### 3.1.2 Water nondetection-driven analysis
The path for water non-detections is less linear. If we are unable to clearly
identify emission in the H2O line, we search for other lines to use as the
basis for stacking. We first create a simple averaged spectrum over the
selected disk candidate’s emitting region by examining several different lines
in the cube. We then search for other plausible guiding lines, focusing on
those that exhibit gradients in the direction expected given known outflows
(i.e., we look at lines rotating perpendicular to outflows). We look at SiS,
SO, and, when truly desperate, $\textrm{CH}_{3}\textrm{OH}$ lines. If we are
able to identify a reasonably disk-like line from among these lines, we use it
to produce a velocity map as above (§3.1.1). If, after this search, we are
still unable to find a line that traces disk-like kinematics, we remove the
source from further consideration.
For both water- and non-water-driven stacking, we report the achieved noise
level in Table 1.
#### 3.1.3 Cube stacking
When NaCl is detected, it is generally seen in multiple transitions. The NaCl
$v$=0 and $v$=1 $J$=18-17 and $v$=1 and $v$=2 $J$=17-16 transitions are
present in the DIHCA observational setups and are not too badly contaminated
by neighboring lines. We therefore create ‘stacked’ NaCl cubes by cutting out
cubes centered on each of those transitions, smoothing them to a common beam,
regridding them to a common spectral resolution, and then averaging the cubes.
We use these stacked cubes for further analysis of the NaCl lines, producing
both moment-0 images and position-velocity diagrams (except for SrcI, for
which the signal-to-noise ratio in individual lines was high enough to produce
moment maps without stacking). The stacked cubes have higher signal-to-noise
than the individual cubes and de-emphasize contaminant lines that are adjacent
to the target lines, since the contaminants arise at different relative
velocities for each NaCl line (for example, while the 41KCl 31-30 line is 13
km s-1 to the red of the NaCl $v=1$ $J=18-17$ line, there is no line 13 km s-1
to the red of the three other NaCl lines included here). This stacking
approach is not strictly necessary for use or analysis of the NaCl lines, but
it aesthetically improves the resulting images and makes visual inspection and
comparison more straightforward.
We created these cubes whether or not we first noted an NaCl detection in the
spectrum. In cases where no detection was apparent in the stacked spetrum, we
nevertheless checked the stacked NaCl cubes to see if extended emission at the
expected velocity was apparent. No additional detections were obtained through
this approach.
#### 3.1.4 COMs
We identify the presence of complex organic molecules (COMs) in the spectrum
in a very broad-strokes manner. We do not identify specific species, though we
note that $\textrm{CH}_{3}\textrm{OH}$ and CH3CN are commonly detected, but
instead characterize the spectra by the richness of the ‘line forest.’ For
each observation, in the spectral range 219.2-220.8 GHz (which is COM-rich and
includes the CH3CN $J=12-11$ ladder), we measure the per-pixel noise
($\sigma_{\rm cube}$) by obtaining the standard deviation over the full field
of view over all pixels; this approach slightly overestimates the noise
because it includes signal in the noise estimate. For image cubes that were
not already continuum-subtracted, we estimate, and then subtract, the
continuum by performing pixel-by-pixel sigma clipping to 3-$\sigma$, then
taking the median across the spectral axis (i.e., as in Sánchez-Monge et al.,
2018). Then, for each extracted region around a candidate disk, we average the
spectra within that region, then determine what fraction of the spectrum
exceeds five times the expected noise level, where the expected noise level is
$\sigma_{\rm cube}n_{\rm beams}^{-1/2}$, where $n_{\rm beams}$ is the number
of beams included in the averaging area. Note that we only search for COMs in
emission; absorption by C- and O-bearing species is seen toward most sources,
but is not directly associated with the candidate disk, instead, it likely
comes from the surrounding envelope or molecular cloud.
Table 1 gives a summary of these statistics in addition to general properties
of the data.
### 3.2 Line Identification
We briefly discuss the key lines used for identification of NaCl, KCl, PN,
SiS, and SiO in this section. The summary of lines considered is in Table 2.
The NaCl $v$=0 and $v$=1 $J$=18-17 and $v$=1 and $v$=2 $J$=17-16 transitions
are present in the DIHCA observational setups. The $v=1$, $J=18-17$ line is
very close to the 41KCl 31-30 line, but the latter can be ruled out as a
contaminant because the 41KCl 29-28 is also included in the observations. In
Orion SrcI, the peak intensity ratio was NaCl $v=1$ $J=18-17$ $\approx
5\times$ 41KCl 29-28.
Table 2: Summary of spectroscopic lines used in this analysis
Line Name | Frequency | EU
---|---|---
| $\mathrm{GHz}$ | $\mathrm{K}$
SiS v=1 12-11 | 216.757603 | 1138.75
SiO v=0 5-4 | 217.104980 | 31.26
KCl v=4 29-28 | 217.228912 | 1733.21
41KCl 29-28 | 217.543178 | 156.71
SiS v=0 12-11 | 217.817644 | 67.95
NaCl v=2 17-16 | 217.979967 | 1128.38
H2CO $3_{0,3}-2_{0,2}$ | 218.222192 | 20.96
HC3N 24-23 | 218.324788 | 1084.99
CH3OH $4_{2,2}-3_{1,2}$ | 218.440063 | 1422.49
KCl v=3 29-28 | 218.579708 | 1345.06
C18O 2-1 | 219.560354 | 15.81
NaCl v=1 17-16 | 219.614936 | 614.51
HNCO $10_{10}-9_{9}$ | 219.798274 | 58.02
KCl v=2 29-28 | 219.936113 | 671.38
SO $6_{5}-5_{4}$ | 219.949440 | 34.98
K37Cl v=2 30-29 | 221.078543 | 948.25
41KCl v=1 31-30 | 231.088150 | 572.25
K37Cl 31-30 | 231.218839 | 177.68
H30$\alpha$ | 231.900928 | -
C30$\alpha$ | 232.016632 | -
KCl v=4 31-30 | 232.163002 | 1755.14
41KCl 31-30 | 232.499840 | 178.67
NaCl v=1 18-17 | 232.509950 | 625.67
Si33S 13-12 | 232.628545 | 78.16
H2O v2=1 $5_{5,0}-6_{4,3}$ | 232.686700 | 3461.91
K37Cl v=4 32-31 | 232.907553 | 1739.18
PN v=1 J=5-4 | 233.271800 | 1937.30
KCl v=3 31-30 | 233.605698 | 1367.12
NaCl v=0 18-17 | 234.251912 | 106.85
SiS v=1 13-12 | 234.812968 | 865.40
PN J=5-4 | 234.935663 | 33.83
KCl v=2 31-30 | 235.055578 | 687.16
K37Cl v=2 31-30 | 235.768235 | 970.53
SiS v=0 13-12 | 235.961363 | 79.28
Lines covered by one or more observations in this work. The frequency and
upper state energy levels are pulled from Splatalogue and refer either to
CDMS, SLAIM, or JPL values. For KCl v=3 and v=4 values, EU is drawn from the
modified version of the Barton et al. (2014) line list used in Ginsburg et al.
(2019a).
### 3.3 Salt detections
We detect salts in 9 sources (of which 3 are tentative) in 6 regions. Figure 1
gives an overview of those objects that are spatially resolved. In the
following sections, we describe the salt-bearing sources in more detail: G17
(§3.3.1), G351 mm1, mm2, and mm12 (§3.3.2), W33A (§3.3.4), NGC6334I mm1b and
mm2b (§3.3.3), and I16547 A and B (§3.3.5). The sixth region and disk, Orion
SrcI, is not discussed in detail here because the same analysis was already
done in Ginsburg et al. (2019a), but it is included in the discussion section
below.
Table 3: Summary of detections, tentative detections, and non-detections of
target species in the source sample
Source | disk | H2O | NaCl | KCl | SiO | RRL | COMs | SiS | SO | PN
---|---|---|---|---|---|---|---|---|---|---
Orion SrcI | yes | yes | yes | yes | yes | no | no | yes | yes | ?
G17 | yes | yes | yes | yes | yes | yes | no | yes | yes* | no
I16547A | yes-c | yes | yes* | no | yes | no | yes | yes | yes | yes
I16547B | yes-c | yes | yes* | no | yes | no | yes | yes | yes | yes
G351.77mm1 | yes-c | yes | yes | no | yes | no* | no | yes | no | yes
G351.77mm2 | unres | yes | yes* | no | no | yes | no | no | no | no*
G351.77mm12 | unres | yes | yes* | no | yes | no | no | yes | no | yes
W33A mm1-main | unres | yes* | yes* | no | yes | yes* | yes | yes* | yes | yes
NGC6334Imm1b | yes-c | yes | yes | yes | yes | no | yes* | yes | no* | yes*
G5.89 mm15 | cont | no | no | no | ? | no | no | no | no | no
IRAS18162 GGD27 | yes | no | no | no | ? | no | yes | no | yes | no
NGC6334IN SMA6 | no | yes* | no | no | no | no | yes | no | no | yes*
NGC6334IN SMA1b/d | no | no* | no | no | no | no | yes | no | no | no*
G11.92mm1 | yes | no | no | no | yes | no | yes | yes* | yes | yes*
IRAS18089 I18089-1732 | yes-c | no | no | no | yes | no | yes | yes* | yes | no*
IRAS16562 G345.4938+01.4677 | no | no | no | no | yes | yes | ext | no | ext | no
G333.23mm1 | no | no | no | no | no* | no | yes | no | yes | yes*
G333.23mm2 | yes* | no | no | no | no* | no | yes | no | yes | yes*
G335 ALMA1 | no | no | no | no | yes | no | yes | no | no* | no
G29.96 submm1 | no | no | no | no | yes | no | yes | no* | no* | no*
G34.43mm1 | no | no | no | no | no* | no* | yes | no | yes | no*
S255IR SMA1 | no* | no* | no | no | yes | yes | yes | no | yes | no
NGC6334Imm1d | yes* | yes* | no | no | no | no | no | no | no | yes*
NGC6334Imm2b | unres | yes | no* | no | yes | no | yes | yes | yes | yes*
We use ‘?’ to indicate “not observed”, ‘yes’ for a definitive detection, ‘no’
for definitive non-detections, ‘yes*’ for tentative detections, and ‘no*’ for
tentative nondetections, where this uncertainty can either be from line
confusion or low signal-to-noise, and ‘ext’ for those associated with the
envelope but not the disk. These all refer to detections in emission; COMs are
seen in absorption toward many sources, but we do not consider these. For SiO,
we do not attempt to distinguish between SiO from the outflow and from the
disk; in Orion SrcI, we know that SiO is present in both. For the ‘disk’
column, we either answer ‘yes’ for a clear disk detection from literature
kinematic characterization, ‘yes-c’ for a candidate disk toward which a
kinematic signature consisten with rotation has been observed, but for which
the kinematics have not yet been confirmed to be Keplerian, ‘continuum’ if a
disk-like (linear, $\lesssim 300$ au long) feature is seen in the continuum,
‘unresolved’ if the targeted source is too small for us to say, and ‘no’ if
neither of the above hold; ‘no’ does not indicate that no disk is present,
merely that we did not identify one. In many cases, we suspect a disk must be
present because there is an outflow, but we say ‘no’ if we can’t see it.
#### 3.3.1 G17
The G17 disk is the best resolved source in our sample after SrcI (Fig 1; Maud
et al., 2019). It is a confirmed Keplerian disk (Maud et al., 2019). Because
of its high signal-to-noise and well-resolved structure, we investigate it in
somewhat more detail than the other sources in the sample. Figure 4 shows the
stacked spectrum based on the water line as described in Section 3.1.1.
Figure 4: Stacked spectra toward G17 from the Maud et al. (2019) data set. The
stacking was based on the H2O line. Line IDs are shown. Different colors are
used for targeted species with multiple transitions in-band: orange for SiS,
blue for KCl, red for NaCl, magenta for H30$\alpha$, purple for PN, and green
for H2O. The remaining species, with only one transition marked, are shown in
black.
The salt detections toward G17 resemble that toward SrcI, with both NaCl and
KCl tracing the same rotational structure. Figure 5 shows position-velocity
diagrams perpendicular to the outflow axis measured by Maud et al. 2018 at
position angle $\theta\approx 135^{\circ}$ based on the large-scale CO
outflow. Maud et al. (2019) measured the disk position angle to be 25.9∘,
which traces the direction of maximum gradient in the H2O disk and through the
emission peak of the continuum structure. This angle is nearly perpendicular
to the large-scale outflow; we adopt this angle as the disk PA. Maud et al.
(2019) measured the disk inclination to be $i=40\pm 4^{\circ}$ from the axis
ratio of the continuum image, so we adopt that inclination when overplotting
Keplerian curves. As with SrcI, both the water and salt lines trace out orbits
consistent with Keplerian rotation around a high-mass star. Figure 6 shows
integrated intensity (moment-0) and intensity-weighted velocity (moment-1)
maps of the G17 disk.
Figure 5: Position-velocity diagrams extracted across the G17 disk. (top left)
KCl grayscale with NaCl contours (top right) NaCl grayscale with Keplerian
rotation curves drawn for a 15, 30, 40 $M_{\odot}$ star with a $i=40^{\circ}$
disk (Maud et al., 2019) in orange, green, and blue, respectively. (bottom
left) Water (H2O) in grayscale with thin NaCl contours. In all cases, the
velocity gradient matches that of the disk identified in Maud et al. (2019).
(bottom right) Position-velocity diagram of the H30$\alpha$ line toward G17,
with NaCl contours overlaid. The H30$\alpha$ emission is clearly confined to
within the NaCl disk. While there is a hint of a velocity gradient in the
H30$\alpha$ line, it is unclear whether this gradient traces rotation. Figure
6: Moment-0 (integrated intensity) and moment-1 (intensity-weighted velocity)
images of stacked NaCl (left) and H2O (right) for G17. The red and blue arrows
indicate the outflow direction from Maud et al. (2017). The dashed gray line
shows the direction the position-velocity diagram was extracted from (Figure
5).
G17 chemically resembles SrcI in several ways: the only lines seen directly
toward the source are H2O, NaCl, KCl, SiS, and SiO. There is little
‘contamination’ in the spectrum from COMs. As in SrcI, KCl lines are detected
at about half the peak brightness of NaCl lines with similar $E_{\rm U}$; no
$v$=0, $v$=1, or $v$=2 transitions of KCl are covered by the observations
(except KCl v=2 J=29-28, which is blended with SO $6_{5}-5_{4}$ and cannot be
clearly identified).
We measure enough lines to produce both rotational and vibrational diagrams
for KCl, but only a rotational diagram for NaCl (Figure 7). These plots
provide temperature and column density measurements of the target molecules if
the observed transitions are in local thermodynamic equilibrium (LTE). As in
SrcI, the rotational temperatures are much cooler ($T_{rot}\sim 35-60$ K) than
the vibrational temperatures ($T_{vib}\sim 900$ K), indicating that non-LTE
effects are important. We have yet to determine the underlying mechanism, but
several possibilities are discussed in Ginsburg et al. (2019a). We defer
further discussion of the excitation to a future work in which we will
integrate additional transitions.
Figure 7: Rotation diagrams from the fitted lines toward G17. (left) KCl
rotation diagram (right) NaCl rotation diagram
A key difference between SrcI and G17 is that G17 exhibits radio recombination
line (RRL) emission, while SrcI does not. Figure 5 shows the H30$\alpha$ RRL
in grayscale with NaCl contours on top, showing that the RRL comes from a
smaller region contained within the NaCl-bearing disk.
Both SrcI and G17 have central ‘holes’ in which there is no NaCl emission. In
SrcI, Ginsburg et al. (2019a) inferred the presence of this hole from the gas
kinematics, as the _apparent_ hole seen in Figure 1a is an observational
effect caused by high optical depth in the edge-on dust disk. In G17, which is
less inclined, the hole is directly observed (Figure 1b). Since the radiation
field is more energetic in G17 than in SrcI, it is possible that the salt hole
is related to the gas temperature.
#### 3.3.2 G351.77-0.54
Beuther et al. (2017) and Beuther et al. (2019) published high-resolution
observations of the G351.77-0.54 region, which we use here over the lower-
resolution DIHCA data. We focus on the two brightest mm sources, mm1 and mm2,
which both exhibit H2O and salt line emission. G351mm2 is a substantially
fainter source, but similarly exhibits brine lines. Spectra of the G351
sources and subsequent sources are presented in the Appendices.
We identify several lines in mm1, including NaCl $v$=1 and $v$=2 $J$=18-17 and
SiS $v$=0 and $v$=1 $J$=13-12 (these latter were clearly detected, and used
for disk kinematic study, in Beuther et al. (2019), but were listed as
‘unidentified’). Because of the slightly different spectral configuration
adopted in these observations as compared to the DIHCA observations, both the
$J$=12-11 and $J$=13-12 lines of SiS $v$=0 are covered and detected.
The morphology and range covered by NaCl and SiS are very similar (Appendix
A.1), though the gap in the center for SiS is more pronounced than for NaCl.
These molecules arise in similar, but not identical, regions.
The velocity gradients in NaCl, H2O, and SiS in mm1 appear to trace a bipolar
outflow. Beuther et al. (2019) discussed the SiS emission lines in detail,
comparing the velocity gradient observed in this line to that seen in SiO. The
SiO $J$=5-4 and CO $J$=6-5 lines both appear in an extended redshifted lobe to
the northwest of the source. The outflow is asymmetric and truncates at the
position of mm1, suggesting that mm1 is the source (Beuther et al., 2017).
Since the redshifted lobe occurs on the redshifted side of the observed SiS
velocity gradient, Beuther et al. (2019) interpreted the lines as part of an
outflow rather than a disk. While there is a blueshifted component opposite
the redshifted flow, it is detected in only two channels and is only seen in
the beam adjacent to the central source, so its direction cannot be determined
independent from the elongated red lobe. They also noted the presence of a
velocity gradient perpendicular to the outflow direction in CH3CN, suggesting
that CH3CN traces the disk in a disk-outflow system. The observed velocity
gradient in briny lines is perpendicular to the CH3CN disk, indicating either
that the emission comes from outflow or that the disk changes orientation with
scale.
We re-examine the kinematics of the now-identified lines here. Figure 8 shows
moment-0 and moment-1 maps of both NaCl (stacked) and H2O. The kinematics
resemble a disk, with kinematics consistent with orbital motion around a $\sim
20/\sin i$ $M_{\odot}$ central potential, However, the extended SiO / CO
$J$=6-5 outflow feature is parallel, rather than perpendicular, to the axis of
the velocity gradient. Assuming the SiO structure is tracing an outflow, the
velocity structure seen in Figure 8 is the base of the outflow. The lack of
any gradient perpendicular to the outflow direction suggests that the briny
lines are not tracing a disk wind in G351mm1, as they are in SrcI (Hirota et
al., 2017), since they would retain that rotation signature for at least some
distance above the disk. However, the lines are marginally resolved and
extended in the direction perpendicular to the outflow, suggesting that the
emission arises from an area at least comparable to be beam size ($\sim 50$
AU). The extended launching region is difficult to reconcile with the lack of
disklike kinematics. We are not able to definitively conclude on the nature of
the velocity gradient in mm1 and suggest that it should be studied further at
high resolution, particularly to trace the kinematics of the disk-outflow
system(s). Nevertheless, we note that the briny emission is limited to a
region $<100$ AU across.
Figure 8: Moment-0 (integrated intensity) and moment-1 (intensity-weighted
velocity) images of stacked NaCl (left) and H2O (right) for G351 mm1. The
position-velocity diagrams in Figure 14 are taken along position angle
$130^{\circ}$, parallel to the velocity gradient seen in the lower panels
here. The red arrow shows the direction of the outflow from Beuther et al.
(2019) that extends $\gtrsim 0.2$ pc to the northwest. A compact blueshifted
feature was also weakly detected bin SiO opposite the redshifted flow, but it
is unresolved. The blue side of the NaCl, H2O, and (in Appendix A.1) SiS and
PN corresponds with the SiO blueshifted lobe seen in Fig. 7 of Beuther et al.
(2019), but since its extent is limited to the $\lesssim 100$ AU scale shown
in this figure, we cannot confirm whether it is comprised of outflowing
material.
By contrast to mm1, mm2 has H2O emission (Appendix A.2), but only tentative
NaCl emission (the $v$=1 line is detected at $\sim{\mathrm{few}}-\sigma$, but
the $v$=2 line is below the noise). No SiS emission is detected. However,
H30$\alpha$ emission is fairly clearly detected. There is a velocity gradient
along the NW-SE axis in the H2O line (Appendix A.2). Curiously, this is
perpendicular to the direction of the gradient shown in Figure 9 of Beuther et
al. (2019), which shows the unidentified 231.986 GHz line, suggesting that
there may be perpendicular gradients here tracing outflow and disk. However,
no outflow is known toward this source, and we do not see clear signs of
outflow in any of the lines studied here, including SiO. While SiO is detected
in emission and absorption toward mm1, it is detected only in absorption
toward mm2.
Despite the presence of many KCl transitions in band, there are no detections
- but our limits on these lines are relatively weak, as all of the $v$=0 and
$v$=1 transitions tend to land in confused regions or come from doubly-rare
isotopologues.
A third source, G351mm12, also has salt detected. Appendix A.3 shows the
standard suite of figures for this detection. This source is barely resolved.
Its moment maps indicate a hint of velocity gradient, but the gradient is at
the limit of our sensitivity and may be spurious.
#### 3.3.3 NGC6334I
There are three sources within the NGC 6334I region that may exhibit brine
emission. Sources mm1b and mm2b are both notable for being faint continuum
sources adjacent to bright sources identified in lower-resolution data (the
‘b’ designation indicates that there are sources mm1a and mm2a that are
brighter).
mm1b exhibits clear signatures of H2O, multiple NaCl lines, SiS, and several
likely KCl detections (Appendix A.4). The H2O line shows hints of rotation
perpendicular to the outflow axis (Fig. 8 of Brogan et al. (2018) shows the
outflow, which is aligned to $\mathrm{PA}\approx 0^{\circ}$ to $-5^{\circ}$,
close to straight north-south), though the other lines do not as clearly
exhibit this signature. Brogan et al. (2018) note several other less prominent
outflows centered on this source, however, which hints that this object cannot
be interpreted as a single disk-outflow system. In the PV diagram of NaCl
(Fig. 9), we show curves at 10, 20, and 30 $M_{\odot}$ for an edge-on
Keplerian disk. Since the emission is confined to lower velocities than the 10
$M_{\odot}$ curve, it appears that this source is $<10$ $M_{\odot}$. However,
it may also be significantly inclined to the line of sight, in which case it
may be more massive. KCl is tentatively detected toward mm1b, with reasonably
strong peaks appearing in the KCl $v$=2 $J$=29-28 and K^37Cl $v$=0 $J$=31-30
lines. Some others that might be expected to be bright, e.g., KCl $v$=3
$J$=29-28 and $v$=3 $J$=31-30, are ambiguous or blended.
Figure 9: Position-velocity diagram of NaCl in the NGC 6334I mm1b disk.
Keplerian velocity curves for edge-on 10, 20, and 30 $M_{\odot}$ central
potentials are overplotted; the central potential here appears to have $M\sin
i<10$ $M_{\odot}$.
The other sources are less clear detections and are therefore considered only
candidate brinary sources. Similar to mm1b, mm2b has a reasonably clear H2O
detection and strong signs of SiS $v$=0 $J$=12-11 emission, but it has no
clear detection of any salt line (Appendix A.4). No clear outflow is present.
The SiS line profile is broad and somewhat different from that of water, so it
is not obvious that they trace the same kinematics.
mm1d has only has marginal SiS and H2O detections. We do not include it in the
figures or detection statistics. It is a strong candidate for follow-up
observations.
#### 3.3.4 W33A
We marginally detect H2O, H30$\alpha$, and NaCl toward the bright source at
the center of W33A (Appendix A.5). At the current resolution, the emitting
region is unresolved. The detection of any of these lines individually is
tentative because we have only one firm line detection for each of these
molecules and they are potentially blended with other emission lines. The PN
5-4, H2O, and NaCl v=1 18-17 lines are the most prominently detected. The NaCl
v=1 and v=2 J=17-16 lines are too weak to confirm the NaCl detection. While we
detect only one PN line, it is isolated enough that confusion is unlikely to
affect it, so it is a reasonably firm detection. This source is a prime
candidate for further followup.
#### 3.3.5 I16547
I16547A and B were reported to have salt, water, and SiS emission in Tanaka et
al. (2020). We confirm their detections both with their original data and with
coarser-resolution observations from the DIHCA program. Despite the clear
detection of those three briny species (see spectra in Appendix A.6 and A.7),
there is no sign of KCl in the data. The non-detection is in part driven by
confusion, in that many of the lower-$J$ transitions of KCl lay atop
transitions from other molecular species that are spatially extended and not
filtered out. Line stacking was not very helpful for these two targets because
they are only marginally resolved (Fig. 2). Nevertheless, velocity gradients
consistent with rotation are apparent in position-velocity diagrams extracted
along the direction of the maximum gradient (Appendices A.6 and A.7), which is
perpendicular to the outflow direction (Tanaka et al., 2020), suggesting that
these are both likely disks around high-mass YSOs.
### 3.4 Unsalted sources
The remainder of our targets do not have salt or water detections. We describe
them in slightly more detail in Appendix B. We note here that several of
these, i.e., G11.92, GGD27, and I18089, have clear extended disks that are
detected in other molecules (e.g., CH3CN) but not in brinary lines.
## 4 Discussion
### 4.1 Are briny lines disk-only tracers?
Many of our targets are only candidate disk sources, in that no resolved
Keplerian rotation curve has been observed. We therefore consider the
question: Could the briny emission be from outflows? Under the assumption that
outflows are driven by either a wind from a disk or from accretion from a disk
onto a star, the presence of an outflow still indicates the presence of a
disk, but that does not mean the lines we observe necessarily arise within
that disk.
In SrcI, it is clear that the water is partly outflowing, but it arose from a
disk wind and had a small observable scale height ($h<40$ au; see Fig. 10 of
Ginsburg et al., 2018). Even in that case, the water line was dominated by
disk kinematics, not outflow.
In most of the observed sources, a velocity gradient across the lines of
interest was observed. Such a velocity gradient can be produced by either
outflow ejection or disk rotation. In the best-resolved cases, G17 and G351,
the emission forms a complete ring, which is expected of a disk or disk wind.
The circular extent of the briny emission shows that it is not tracing a
collimated jet feature. However, in G351mm1, the direction of the gradient is
along the known outflow, suggesting that the briny lines do not trace a disk
or a disk wind; this source remains difficult to interpret.
While we cannot definitively determine the general origin of briny emission,
we observe here that it is restricted to radii $<300$ au in our full sample,
such that it is always consistent with arising in either the disk or the very
inner portion of the outflow.
### 4.2 Chemical similarities between the salt disks
We examine the general chemical properties of the brinary disks. Each has
several properties in common, but several differences. Table 3 lists the
detections and non-detections toward each examined source. We observe that
NaCl, H2O, and SiS lines are often detected toward the same sources, and
generally if one is absent, all are. In other words, they exhibit comparable
brightness when they are observed. This correlation hints that they come from
similar regions within disks or outflows, either because of excitation or
chemical (formation/destruction) conditions.
Figure 10: Cross-correlation plot made from Table 3. We cross-correlate purely
on the boolean value; those with a ‘yes’ are marked True, and those without a
‘yes’ are marked False. The rows are sorted in order of correlation with NaCl.
Figure 10 shows the cross-correlation of the boolean value (“yes” or “no”)
encoded in table 3; the tentative detections marked with ‘*’s are assigned to
their corresponding ‘yes’ or ‘no’ prefix. It shows that the presence of NaCl,
SiS, and H2O are strongly correlated. The presence of COMs in the spectra is
moderately anti-correlated with these ‘brinary’ species. RRLs are strongly
anti-correlated with COMs. The correlations among the other molecules (and
RRLs) are less pronounced.
We note that our data are extremely incomplete, as we explicitly select
against COM-bearing targets by selecting H2O-bearing disks. There are many
fainter disk candidates in the fields of view of our observations that we did
not include in our sample or in this plot. Most of these sources do exhibit
COM emission and do not exhibit RRL emission, though, so adding them to this
plot would generally strengthen the trends shown.
Because of the incompleteness of our sample, we caution against over-
interpretation. A repeat of this analysis with a consistently-selected sample
will be needed to draw firm conclusions. However, the correlations in the top-
left of the plot, between H2O, NaCl, and SiS appear firm - these species
coexist in this sample of disk candidates.
#### 4.2.1 The low detection rate of KCl is an observational effect
The KCl detections are not perfectly correlated with NaCl detections, but we
argue here that this is an observational selection effect. Most likely, KCl is
better-correlated with NaCl than is apparent in Figure 10.
In the targeted band, there are no $v$=1 or $v$=0 transitions of the most
common isotopologue (39K-35Cl) of KCl, such that all detections are of higher-
excitation, lower-brightness transitions. The targeted KCl lines are also
partly hidden by confusion with other lines. The KCl lines are strongly anti-
correlated with the presence of COMs, which is primarily (or entirely) because
of confusion: the KCl lines in this band are weaker than NaCl when they are
detected, so they are more difficult to distinguish from the molecular line
forest present in dense spectra. They land inconveniently near bright COM
lines in the 215-235 GHz range more often than the NaCl lines.
The comparison to SrcI illustrates some of these effects. In SrcI, the peak
brightness temperature of the NaCl and KCl lines with E${}_{U}<1000$ K were
similar, with $T_{\rm B,max}\sim 100-200$ K. The NaCl $v$=0 lines were no more
than twice as bright as the KCl $v$=0 lines. The SrcI data set was targeted on
outflow-tracing lines, including 12CO $J$=2-1 and SiO $v$=1 $J$=5-4, which
both have nearby $v$=0 and $v$=1 KCl transitions, while the DIHCA observations
and others presented here chose to target the CH3CN ladder at 220.4 GHz and
therefore did not cover these low-$v$ transitions.
#### 4.2.2 PN in the brine
The PN J=5-4 line appears to be detected in several of our sources. The PN v=1
J=5-4 transition, at 233.27182 GHz, is not detected. For the line-poor
brinaries, there is little confusion around this line, and its velocity lines
up perfectly with that of salts, so this identification is reasonably certain.
For the rest of the sample, the case is less clear; while there are some
tentative detections with clear lines at this velocity, the spectra are so
rich that we cannot definitively identify PN as the carrier species. The
presence of PN in the same regions as the highly-excited salt lines may
suggest that PN occupies a similar location within and binding energy to dust
grains. Rivilla et al. (2020) suggest that PN is present in the cavity walls
of an outflow toward AFGL 5142, but that it is released to the gas phase as
PH3, and the PN molecules are subsequently formed under the influence of the
star’s UV radiation. Given the lack of UV photons in these sources, as
indicated by the lack of correlation between PN and RRLs in Figure 10, our
data may indicate that PN is present in the grains and not formed in the gas
phase.
Figure 11: The integrated intensity (moment-0) maps of NaCl shown in Fig. 1
and 2, but now with each image scaled to the same physical resolution. The 100
AU scalebar, shown on the W33A mm1-main panel, applies to all panels. The
intensity scales are arbitrary, as the intent is only to show the disk
candidate structure.
#### 4.2.3 RRLs
Hydrogen recombination lines are likely to be produced in the ionized regions
surrounding accreting high-mass young stars once they have contracted onto the
main sequence. In our sample, few RRLs are detected. Their presence, or
absence, is only weakly anti-correlated with the presence of COMs and PN, but
has little correlation with other molecules. While we might expect RRL
emission to become detectable from accreting HMYSOs toward the end of their
accretion phase, as the ionization rate is able to overtake the accretion rate
of fresh neutral material, our data provide little evidence for this process.
### 4.3 Excitation
At least in the best-resolved cases, G17 and G351, vibrationally excited
states of NaCl are detected ($v$=1, and 2). This feature is in common with
SrcI, where states up to $v$=6 were convincingly detected. The $v$=2
detections in particular, with $E_{\rm U}\gtrsim 1000$ K, suggest that an
excitation pattern similar to that in SrcI, in which $T_{\rm vib}>T_{\rm
rot}$, is common. This general feature is also common in evolved stars that
exhibit salt emission, suggesting that these salt lines only appear in regions
with strong radiative backgrounds in the infrared. Figure 7 highlights the
high excitation, though we defer deeper analysis of the excitation properties
to a future work.
### 4.4 Spatial Resolution
Our observed sample has non-uniform spatial resolution (Table 1), which helps
explain several of the non-detections. We show a version of Figure 1 with all
images resized to the same physical scale shown in Figure 11, highlighting
that the detected NaCl disks are small. We did not detect NaCl toward any
sources observed with beam size $>300$ au, which included four of our targeted
fields. Five of the fifteen targeted fields included detections. We performed
a logistic regression of the detection of salt against resolution and found
that the likelihood of salt detection in our sample is $>50\%$ for resolution
$<120$ au, peaking at $\sim 80\%$ at infinite resolution. While our haphazard
sample selection prevents drawing strong conclusions, these results suggest
that salty regions are limited to small scales ($\lesssim 300$ au) and are
challenging to observe if they are not well-resolved. This observational
limitation highlights the need for more extensive extremely high-resolution
observations with ALMA’s longest-baseline configurations to obtain dynamical
mass measurements for high-mass YSOs.
It is possible that more sensitive observations with poor physical resolution
may detect brinary lines now that we know to look for them. However, the main
difficulty in detecting these lines is not raw sensitivity, but confusion with
other lines. It will be beneficial to search other parts of the spectrum for
more isolated brinary lines, which might be expected to be more common at
lower frequencies. It may be possible to determine the presence or absence of
brinary lines by stacking the various transitions across species to average
down the ‘contaminant’ noise provided by other lines; we leave investigation
of this possibility to future work.
## 5 Conclusions
* •
We have substantially increased the number of high-mass protostellar objects
with detection of salt, hot water, and SiS, increasing the number from three
to nine published detections across six regions.
* •
Salt, water, and SiS tend to coexist. PN may also coexist with these species.
When any of these molecules is detected in a HMYSO disk, all of them are
likely to be.
* •
Brinary sources are line-poor compared to typical hot cores. The lack of COMs
in the briny regions suggests that the chemistry of these regions is different
from hot cores, even when the central objects are surrounded by hot cores.
* •
These emission lines do not come from the same volume as ionized hydrogen.
While some of the HMYSO candidates targeted exhibit both RRL and brine
emission, the presence of an RRL is a poor predictor of whether NaCl and H2O
are detected. The resolved case of G17 shows that the ionized gas comes from a
smaller radius than the NaCl.
* •
With the nine brinaries presented here, we demonstrate that salt emission is
not rare. The primary reason it is not often detected is resolution: the
emission comes exclusively from small ($\lesssim 100$ au) size scales,
confirming that either chemistry or excitation restricts the millimeter lines
described here to disk-sized regions. Line confusion limits our ability to
detect these lines even when they are present, though confusion can be
alleviated with high spatial resolution.
* •
However, salt and water emission is also not ubiquitous in HMYSO disks. Some
of the most compelling Keplerian disks around HMYSOs in the literature, such
as GGD27 and G11.92mm1, show no sign of salt emission despite their
similarities to other disks and superior data quality.
There is clearly substantial future work to do with these data and expanded
samples of brinaries. Some of the more obvious questions about brinary lines
include:
* •
Are they correlated with source luminosity or stellar mass?
* •
What disks, or outflows, produce them?
* •
Do they occur around low-mass YSOs?
* •
Why are highly vibrationally excited lines ($v>3$) detected?
Many of these questions will require establishing less biased, more systematic
samples of YSO candidates. Others will require more detailed, multi-line
studies toward a limited sample.
Acknowledgements We thank the anonymous referee for a thorough and
constructive report that significantly improved the paper, particularly in
terms of structure and readability. AG acknowledges support from NSF AAG
2008101 and NSF CAREER 2142300. HB acknowledges support from the European
Research Council under the European Community’s Horizon 2020 frame-work
program (2014-2020) via the ERC Consolidator Grant ‘From Cloud to Star
Formation (CSF)’ (project number 648505). HB also acknowledges support from
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) –
Project-ID 138713538 – SFB 881 (“The Milky Way System”, subproject B01). P.S.
was partially supported by JSPS KAKENHI grant Nos. JP18H01259 and JP22H01271.
K.E.I.T. acknowledges support by JSPS KAKENHI grant Nos. JP19K14760,
JP19H05080, JP21H00058, and JP21H01145.
This paper makes use of the following ALMA data:
ADS/JAO.ALMA#2016.1.01036.S, ADS/JAO.ALMA#2017.1.00237.S,
ADS/JAO.ALMA#2016.1.00550.S, ADS/JAO.ALMA#2019.1.00492.S,
ADS/JAO.ALMA#2017.1.00098.S, ADS/JAO.ALMA#2018.1.01656.S, and
ADS/JAO.ALMA#2013.1.00260.S. ALMA is a partnership of ESO (representing its
member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST
and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the
Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and
NAOJ. In addition, publications from NA authors must include the standard NRAO
acknowledgement: The National Radio Astronomy Observatory is a facility of the
National Science Foundation operated under cooperative agreement by Associated
Universities, Inc.
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## Appendix A Additional figures for salted sources
For the salted sources, we present moment maps and spectra with lines
identified in the following Appendices.
### A.1 G351.77mm1
We show additional moment maps, spectra, and position-velocity diagrams from
G351.77mm1 in this section. This object is one of the most compelling, high
signal-to-noise objects in the sample, yet its kinematics are perplexing,
exhibiting disk-like rotation in the same direction as the larger-scale
outflow rather than rotating perpendicular to the outflow, as we would expect.
Figure 12 shows the stacked spectra. Figure 8 shows moment-0 and moment-1 maps
of NaCl and H2O. Figure 13 shows the same for SiS and PN. All four molecules
exhibit similar morphology and kinematics, including a prominent central hole,
which is strongly suggestive of a disk. Figure 14 shows moment-0 maps and
position-velocity diagrams, and Figure 15 more position-velocity diagrams,
illustrating that rotation is a plausible explanation for the observed
kinematics.
Figure 12: Stacked spectra from G351.77mm1 from the Beuther et al. (2019) data
set. The stacking was based on the H2O line. Line IDs are shown; no KCl
detections are clear. Different colors are used for targeted species with
multiple transitions in-band: orange for SiS, blue for KCl, red for NaCl,
magenta for H30$\alpha$, purple for PN, and green for H2O. The remaining
species, with only one transition marked, are shown in black. The PN line
identification should be taken with a grain of salt since it is the only
transition we observe from PN. Figure 13: Moment-0 and 1 images as in Figure
8, but for the SiS v=0 J=13-12 and PN 5-4 lines of G351 mm1.
Figure 14: Comparison of NaCl and SiS moment-0 and position-velocity maps of
G351.77mm1. (top left) NaCl stack moment 0 image, as seen in Figure 1. (top
right) SiS 13-12 moment 0 image, showing similar morphology. (bottom left)
NaCl stack cube position-velocity diagram extracted along the direction of
maximum gradient. (bottom right) SiS 13-12 position-velocity diagram extracted
along the direction of maximum gradient. Keplerian rotation curves assuming an
edge-on central source with the listed mass are shown in colored lines; these
curves are not fits to the data and do not account for inclination, they are
just provided to guide the eye. Furthermore, as discussed in Section 3.3.2,
the velocity gradient shown here may trace an outflow rather than a disk.
Figure 15: Two more position-velocity diagrams of G351 mm1, showing the 235
GHz PN 5-4 line (newly identified here) and the H2O line. The common structure
seen in these diagrams, and in Figure 14, justifies our assumption that these
have common kinematics. Keplerian rotation curves assuming an edge-on central
source with the listed mass are shown in colored lines; these curves are not
fits to the data and do not account for inclination, they are just provided to
guide the eye. Furthermore, the direction of maximum gradient, along which
these diagrams are extracted, points in the direction of the outflow.
### A.2 G351.77mm2
Figure 16 shows the stacked spectra of G351 mm2. Figure 17 shows the moment-0
and moment-1 maps of NaCl and H2O, which are only marginally resolved.
Figure 16: The G351 mm2 disk candidate stacked spectrum. Like mm1 (Fig. 12),
the stacking was done on the H2O line. However, other lines are at most weakly
detected; the NaCl v=1 J=18-17 line is evident, but no other clear detections
are present in this or other bands. In all spectra, but particularly in
spectral window 0, much of the spectrum is absorbed by material unassociated
with the disk; we cut off the absorption features to emphasize emission
features here. Figure 17: Moment-0 (integrated intensity) and moment-1
(intensity-weighted velocity) images of stacked NaCl (left) and H2O (right)
for G351 mm2.
### A.3 G351.77mm12
Figure 18 shows the stacked spectra of G351 mm12. This source is unresolved,
as shown in Fig 2. Figure 19 shows the moment-0 and moment-1 maps of NaCl and
H2O, which are only marginally resolved.
Figure 18: The G351 mm12 disk candidate stacked spectrum. See Figure 12 for
additional description. Figure 19: Moment-0 (integrated intensity) and
moment-1 (intensity-weighted velocity) images of stacked NaCl (left) and H2O
(right) for G351 mm12.
### A.4 NGC6334I
We show additional figures of NGC6334I, including spectra (Fig 20 and moment
maps (Fig 21. For mm2b, we show only the stacked spectrum (Fig 22), since the
source is unresolved
Figure 20: The NGC6334I mm1b disk stacked spectrum. See Figure 12 for
additional description. Figure 21: Moment-0 and 1 images as in Figure 8, but
for NGC 6334I mm1b. The outflow noted by Brogan et al. (2018) is shown at
PA=-5∘ with green arrows, as it appears to be primarily in the plane of the
sky. The solid gray line shows the orientation from which the position-
velocity diagram (Figure 9) is extracted.
Figure 22: The NGC6334I mm2b disk stacked spectrum. See Figure 12 for
additional description.
### A.5 W33A
The labeled, stacked spectrum from W33A is shown in Figure 23. The four-panel
moment map is not shown for this source because it is unresolved and shows no
structure; it is consistent with a point source.
Figure 23: The W33A disk stacked spectrum. See Figure 12 for additional
description.
### A.6 I16547A
The labeled, stacked spectrum from I16547A is shown in Figure 24. Moment-0 and
moment-1 images are shown in Figure 25. A position-velocity diagram, with
overlaid Keplerian curves for an edge-on orbit with masses labeled, is shown
in Figure 28. We show these to provide order-of-magnitude mass estimates, but
note that we have no constraint on the disk inclination and have not attempted
to model the extent of the disk emission. No outflow is observed toward A
(Tanaka et al., 2020).
Figure 24: The I16547A disk stacked spectrum. See Figure 12 for additional
description. Figure 25: Moment-0 and 1 images as in Figure 8, but for IRAS
16547A The solid gray line shows the orientation from which the position-
velocity diagram (Figure 28) is extracted, based on the angle determined in
Tanaka et al. (2020). The radio jet at PA=-16∘ is shown with arrows (Tanaka et
al., 2020).
### A.7 I16547B
The labeled, stacked spectrum from I16547B is shown in Figure 26. Moment-0 and
moment-1 images are shown in Figure 27. A position-velocity diagram, with
overlaid Keplerian curves for an edge-on orbit with masses labeled, is shown
in Figure 28. We show these to provide order-of-magnitude mass estimates, but
note that we have no constraint on the disk inclination and have not attempted
to model the extent of the disk emission. An SiO outflow is observed toward B,
perpendicular to the gradient in the PV diagram (Tanaka et al., 2020).
Figure 26: The I16547B disk stacked spectrum. See Figure 12 for additional
description. Figure 27: Moment-0 and 1 images as in Figure 8, but for IRAS
16547B The solid gray line shows the orientation from which the position-
velocity diagram (Figure 28) is extracted, based on the angle determined in
Tanaka et al. (2020). The red and blue arrows indicate the direction of the
SiO outflow noted in Tanaka et al. (2020).
Figure 28: Position-velocity diagrams of the I16547 disks in the stacked NaCl
lines. The overplotted curves show Keplerian rotation around central point
sources with masses indicated in the caption, assuming an edge-on inclination.
As noted in Tanaka et al. (2020), the disks appear to be counter-rotating
along similar position angles.
### A.8 Continuum
We show Figures 1 and 2 again, but this time with continuum contours overlaid,
in Figures 29 and 30.
Figure 29: Moment-0 (integrated intensity) images of the resolved sources in
NaCl lines as described in Figure 1. The red ellipse shows the continuum beam
corresponding to the contours, while the black ellipse shows the line image
beam. The contours are: SrcI $\sigma=0.21$ mJy/beam, contours at 25, 50, 75,
100, 125, 150 $\sigma$, G17 $\sigma=0.12$ mJy/beam, contours at 50, 100, 150,
200 $\sigma$, G351mm1 $\sigma=0.08$ mJy/beam, contours at 50, 75, 100, 125
$\sigma$, NGC6334Imm1b $\sigma=1$ mJy/beam, contours at 10, 15, 20, 25
$\sigma$.
Figure 30: As in Figure 2, but with continuum contours overlaid. The red
ellipse shows the continuum beam corresponding to the contours, while the
black ellipse shows the line image beam. The contours are: G351mm2
$\sigma=0.08$ mJy/beam, contours at 50, 75, 100, 125 $\sigma$, G351mm12
$\sigma=0.08$ mJy/beam, contours at 10, 20, 30 $\sigma$, NGC6334Imm2
$\sigma=1$ mJy/beam, contours at 2, 3, 4 $\sigma$, I16547A $\sigma=0.08$
mJy/beam, contours at 100, 150, 200, 250, 300 $\sigma$, I16547B $\sigma=0.08$
mJy/beam, contours at 75, 100, 125, 150 $\sigma$, W33A $\sigma=0.3$ mJy/beam,
contours at 20,40,60,80 $\sigma$.
## Appendix B Unsalted Sources
### B.1 I16562
IRAS16562-3959 is also known as G345.4938+01.4677. This source shows SO in
absorption against a continuum disk. It exhibits RRL emission that shows a
very tiny gradient; Guzmán et al. (2020) reported the detection of an ionized
disk in this system based on RRL emission. No other emission lines are
associated with the RRL possible disk. There is extensive CH3CN emission
around this source with kinematics more complicated than a simple Keplerian
disk.
### B.2 G333
We adopt naming from Stephens et al. (2015), though their resolution was only
$\sim 2.5\arcsec$.
G333.23mm1 is the bright central region. Morphologically, it looks like the
hub at the center of several converging filaments. There is no obvious sign of
a disk in any of the lines we examined. There is no hint of brinary lines.
G333.23mm2 shows some sign of a line gradient in $\textrm{CH}_{3}\textrm{OH}$,
hinting that a disk is present. However, no brinary lines are seen.
### B.3 G335
Olguin et al. (2022) performed an extensive study of this system using the
data presented here. G335-ALMA1 is the bright central source that shows signs
of rotation, but no clear disk. Instead, it appears dominated by inflowing
accretion filaments. The spectrum is extremely rich, and we adopt the same
$\textrm{CH}_{3}\textrm{OH}$ line as in Olguin et al. (2022) as our velocity
reference. No brinary lines are detected.
### B.4 G5.89
The G5.89 image is dominated by an extended HII region. The only disklike
source in the imaged field of view is mm15, so our cutout centered on that
source. It apparently exhibits _no_ line emission at all in the present data
set. We examined the other millimeter peaks in the region, but found no
obvious signatures of disks or brinary lines.
### B.5 GGD27
GGD27mm1 (otherwise referred to here as GGD27) is the driving source of
HH80/81 (Girart et al., 2017, 2018; Añez-López et al., 2020). It is also known
as IRAS 18162-2048. It has a clear, well-defined, massive ($M\sim 5M_{\odot}$)
disk orbiting a $\sim 20$ $M_{\odot}$ star (Añez-López et al., 2020). There is
no sign of NaCl, H2O, KCl, or SiS in the spectrum of the disk in our data or
previous observations (Girart et al., 2017). It is bright in SO $6_{5}-5_{4}$,
which we used to stack spectra to perform a deeper search.
This source is the archetype of non-salt-bearing disks. Its well-defined
Keplerian line profiles, high central mass, and line-poor spectrum demonstrate
that not all HMYSO disks exhibit salt emission. The system is very similar to
SrcI in terms of its central stellar mass, disk size (though GGD27’s is
2-4$\times$ larger in radius), and luminosity, suggesting that none of these
features are critical for releasing salt into the gas phase. As the driver of
the HH80/81 outflow, it is also clear that the simple presence of an outflow
does not determine whether brinary lines are produced. However, one notable
difference is that the HH80/81 jet is highly collimated (Rodríguez-Kamenetzky
et al., 2017; Qiu et al., 2019), while SrcI drives a broader disk wind (Hirota
et al., 2017; Tachibana et al., 2019), hinting that the driving mechanism of
the outflow may have a role in determining when salts are detectable. The
comparison of this source to SrcI and others in this sample will be useful for
future understanding of the origin of salts.
### B.6 IRAS18089
Sanhueza et al. (2021) observed this source at high resolution, though they
focused on the magnetic field. Previously, Beuther et al. (2004, 2005) showed
that this source was line-rich with SMA observations. We used a
$\textrm{CH}_{3}\textrm{OH}$ as the stacking line because the kinematics were
similar across all bright lines, including the SO $6_{5}-5_{4}$ line, but the
SO line was affected by strong absorption toward the inner disk. The line
kinematics were reasonably disk-like, but not consistent with a single
Keplerian disk; the velocity structure in this source requires more
sophisticated modeling. Nevertheless, there is no sign of brinary lines in the
averaged or stacked spectra.
### B.7 G34.43mm1
There is a structure toward the center of G34.43mm1 with a clear line
gradient, but it does not trace disk-like kinematics and is quite extended. On
the larger scale, G34.43mm1 drives a powerful outflow (Sanhueza et al., 2010).
While this is region is quite nearby (1.56 kpc Xu et al., 2011), it is not
among the salt-bearing candidates. Despite its relatively near distance, the
observed spatial resolution is only $\sim 400$ au, which is much larger than
the detected disks. The overall appearance of the inner region suggests that
streamers are feeding in to a central region. The chemically rich inflowing
material results in a very spectrally dense spectrum (e.g., Sanhueza et al.,
2012; Liu et al., 2020a), which likely prevents detection of brinary features
even if they are present. This is a good candidate for followup at higher
angular resolution.
### B.8 G29.96
Beuther et al. (2007) and Beltrán et al. (2011) studied this region at
$\sim$arcsecond resolution. The target region is centered on the brightest few
sources at the center. The distance to this object is unclear, with Beltrán et
al. (2011) reporting 3.5 kpc and Kalcheva et al. (2018) reporting 7.4 kpc.
There is no clear signature of any of the lines of interest, nor is there a
clear signature of rotation. There is a tentative detection of SiS v=0 12-11,
but neither of the SiS v=1 lines (13-12 or 12-11) appear, so this detection is
uncertain.
### B.9 S255IR
Sh 2-255 IR SMA1 (S255IR) was the recent ($<10$ years) site of a major
accretion outburst. As such it is a strong candidate for being actively heated
over its ‘normal’ level.
No disk is obvious in the data cube from rotational signatures. While several
authors (Zinchenko et al., 2015, 2020; Liu et al., 2020b) have noted bulk
rotation in the molecular gas around S255IR, these signatures are not evident
on the smaller ($\sim 70$ au) scales probed here.
There is, however, a hint of both H2O and H30$\alpha$ emission along the
innermost part of the outflow, within about one resolution element of the
central source. There is also a hint of SiS here. There is clearly SiO
emission with a velocity gradient along the outflow axis, but curiously it is
not very bright along the continuum jet. This source warrants followup at
higher resolution and sensitivity to try to identify the location of the
actual disk, though it is not yet clear which lines to use for this search.
### B.10 NGC6334IN
The NGC6334IN region contains several bright sources. The brightest in our
observations are SMA6 and SMA1 b/d (names from Sadaghiani et al., 2020).
Neither source shows clear signs of rotation in any line. SMA1b/d contains
several sources that may exhibit some hint of H2O emission, so we use that
line as the basis for stacking We do not detect any other lines. Notably, both
SO and SiO appear absent toward these sources. SMA6 also shows a hint of a
water line, but similarly shows no sign of any of the other targeted lines.
### B.11 G11.92mm1
Ilee et al. (2018) reported on the disk G11.92mm1, showing a much larger
($r\sim 800$ au) size than most of the sources in our sample. We use SO
$6_{5}-5_{4}$ as the guide line for stacking, since it shows clear disk
kinematics. No brinary lines are detected.
|
2010 Vol. 10 No. XX, 000–000
11institutetext: LPTA, Université Montpellier 2 - CNRS/IN2P3, 34095
Montpellier, France 22institutetext: Department of Astronomy, Nanjing
University, Nanjing 210093, P. R. China 33institutetext: European Southern
Observatory, Alonso de Córdova 3107, Santiago, Chile 44institutetext:
Department of Astronomy and Astrophysics, P. Universidad Catolica de Chile,
Casilla 306, Santiago, Chile
Received [year] [month] [day]; accepted [year] [month] [day]
# Low-ionization galaxies and evolution in a pilot survey up to z = 1
00footnotetext: Based on observations obtained in service mode at the European
Southern Observatory at Paranal
E. Giraud 11 Q.-S. Gu 22 J. Melnick 33 H. Quintana 44 F. Selman 33 I. Toledo
44 P. Zelaya 44
###### Abstract
We present galaxy spectroscopic data on a pencil beam of $10.75^{\prime}\times
7.5^{\prime}$ centered on the X-ray cluster RXJ0054.0-2823 at $z=0.29$. We
study the spectral evolution of galaxies from $z=1$ down to the cluster
redshift in a magnitude-limited sample at $\rm R\leq 23$, for which the
statistical properties of the sample are well understood. We divide emission-
line galaxies in star-forming galaxies, LINERs, and Seyferts by using
emission-line ratios of [OII], $\rm H\beta$, and [OIII], and derive stellar
fractions from population synthesis models. We focus our analysis on
absorption and low-ionization galaxies. For absorption-line galaxies we
recover the well known result that these galaxies have had no detectable
evolution since $z\sim 0.6-0.7$, but we also find that in the range $z=0.65-1$
at least 50% of the stars in bright absorption systems are younger than
2.5Gyr. Faint absorption-line galaxies in the cluster at $z=0.29$ also had
significant star formation during the previous 2-3Gyr, while their brighter
counterparts seem to be composed only of old stars. At $z\sim 0.8$, our
dynamically young cluster had a truncated red-sequence. This result seems to
be consistent with a scenario where the final assembly of E/S0 took place at
$z<1$. In the volume-limited range $0.35\leq z\leq 0.65$ we find that 23% of
the early-type galaxies have LINER-like spectra with $\rm H\beta$ in
absorption and a significant component of A stars. The vast majority of LINERs
in our sample have significant populations of young and intermediate-aged
stars and are thus not related to AGN, but to the population of ‘retired
galaxies’ recently identified by Cid-Fernandes et al. (2010) in the SDSS.
Early-type LINERs with various fractions of A stars, and E+A galaxies appear
to play an important role in the formation of the red sequence.
###### keywords:
cosmology: observations – galaxies: evolution - large scale structures -
evolution – RX J0054.0-2823
## 1 Introduction
In the course of an investigation of the diffuse intergalactic light in X-ray
emitting clusters at intermediate redshifts (Melnick et al., 1999), we
detected a puzzling S-shaped arc-like structure in the ROSAT cluster RX
J0054.0-2823 (Faure et al., 2007), which we tentatively identified as the
gravitationally lensed image of a background galaxy at a redshift between
z=0.5 and z=1.0. The cluster, however, is characterized by having three
dominant D or cD galaxies in the center, two of which are clearly interacting.
We designed an observing strategy that allowed us at the same time to observe
the arc, the diffuse Intra-Cluster Light (ICL), and a magnitude limited sample
of individual galaxies in the field taking advantage of the multi-object
spectroscopic mode of the FORS2 instrument on Paranal. By optimizing the mask
design (see below) we were able to obtain: (a) very deep observations of the
arc; (b) very deep long-slit observations of the ICL; and (c) redshifts and
flux distributions for 654 galaxies of which 550 are in the pencil beam and at
$0.275\leq z\leq 1.05$.
Our pencil beam sample covers a redshift range up to z = 1 (with some galaxies
up to z = 1.7). In standard cosmology with $H_{o}=75$ $\rm
km~{}s^{-1}~{}Mpc^{-1}$, $\Omega_{0,m}=0.30$, and $\Omega_{0,\Lambda}=0.70$,
this range provides a large leverage of about 3000 Mpc or 7 Gyr, which should
be sufficient to extract some of the most conspicuous characteristics on
galaxy evolution at $z\leq 1$. About half of all stars seem to be still
forming, mostly in disks, in this redshift range (Dickinson et al., 2003;
Hammer et al., 2005). Our spectroscopy provides a 50-60% complete sample of
the galaxies in a pencil beam of $\sim 10^{\prime}\times 10^{\prime}$,
centered on the cluster, uniformly down to R=23. Our sample compares in size
with the DEEP1 spectroscopic pilot survey (Weiner et al., 2005) but is smaller
than large surveys such as DEEP2 (e.q. Lin et al. 2008; Yan et al. 2009), VVDS
(e.q. Franzetti et al. 2007; Garilli et al. 2008), GOODS (e.q. Bell et al.
2005; Weiner et al. 2006). The advantage of a pilot survey is that it can be
handled rather easily by a single (or a few) researcher(s) to test new
methods, new ideas before applying these new methods to large samples.
The vast majority of our individual spectra reduced to zero redshift have S/N
ratios per $\rm 2.6\AA$ pixel larger than 3 at $\rm 4200\AA$. This resolution
is very well adapted to the detection of small equivalent width [OII]
emission, which is expected to be found in bulge dominated galaxies with small
disks, in some LINERs, in “mixed” mergers between E/S0 and star-forming
objects, and perhaps in some post-starbursts galaxies. The line of sight of
our field crosses three main structures: a dynamically young cluster at
$z=0.29$, an over-dense region with layers at $z=0.4-0.5$, and a mixed region
of field and possible layers from $z=0.6$ to $z=1$. According to morphology-
density relations (Dressler, 1980; Dressler et al., 1997; Melnick & Sargent,
1977; Smith et al., 2005; Postman et al., 2005; Cooper et al., 2006; Scoville
et al., 2007), we expect that over-dense regions will provide a rather large
number of red objects available to our study. Therefore red objects with or
without star formation, or with low photo-ionization is the subject which we
will focus on, having in mind the possible roles of E+A galaxies (Dressler &
Gunn, 1983; Norton et al., 2001; Blake et al., 2004; Goto, 2007; Yang et al.,
2008, and references therein) and of LINERs (Yan et al., 2006) in the
building-up of the red sequence.
We focus on galaxies with either low star-formation or low ionization which
appear at $z\leq 0.6$. We use line ratio diagnosis based upon [OII], $\rm
H\beta$, and [OIII], from Yan et al. (2006), to classify galaxies in LINERs,
star-forming galaxies, and Seyferts. This method, combined with visible
morphology, allow us to isolate a significant population of early-type LINERs,
and galaxies with diluted star-formation in later morphological types at
$z=0.35-0.6$.
Several studies suggest that the bulk of stars in early-type cluster galaxies
had a formation redshift of $z\geq 3$, while those in lower density
environments may have formed later, but still at $z\geq 1.5-2$ (for reviews
see Renzini, 2006, 2007). This may be in contradiction with the rise in the
number of massive red galaxies found by Faber et al. (2007) who concluded that
most early types galaxies reached their final form below $z=1$. Our data
include a clear red sequence at $z=0.29$ and a quite large number of
absorption systems up to $z\sim 1$ which we fit with population synthesis
models in order to search for age variations with $z$ and luminosity.
The paper is structured as follows. Section 2 presents details of the
observations and the data reduction procedures. Section 3 is on the resulting
redshift catalog. Section 4 presents an overview of variations in spectral
energy distribution with redshift for absorption and emission systems. Section
5 is dedicated to population variations with $z$ and luminosity in absorption
systems. Low-ionization galaxies are in 5.3. In Section 5.4 we suggest a
scenario in which early-type LINERs will become E/S0 galaxies once the A stars
die, and photo-ionization disappear. Summary and Conclusions are in Section 6.
## 2 Observations and data reduction
The observations (ESO program 078A-0456(A) were obtained with the FORS2
instrument (fors:2005, 2005) on the Cassegrain focus of the VLT UT1 telescope
in multi-object spectroscopy mode with the exchangeable mask unit (MXU). They
were acquired in service observing and were spread over two periods 78 and 80
to satisfy our observing conditions. FORS2 was equipped with two $\rm 2k\times
4k$ MIT CCDs with $15\rm\mu m$ pixels. These CCDs have high efficiency in the
red combined with very low fringe amplitudes. We used the grisms 300V and
600RI, both with the order sorting filter GG435. With this filter, the 300V
grism has a central wavelength at 5950 Å and covers a wavelength between
$4450-8700$ Å at a resolution of 112 Å$~{}{\rm mm}^{-1}$. The 600RI grism has
a central wavelength of 6780 Å and covers the 5120-8450 Å region at a
resolution of 55 Å ${\rm mm}^{-1}$. Combined with a detector used in binned
mode, the 300V grism has a pixel resolution of 3.36 Å pixel-1. The grisms were
used with a slit width of 1′′. In order to match the major and minor axis of
the ICL and the prominent arc-like feature rotation angles of $-343^{o}$,
$-85^{o}$, and $-55^{o}$ were applied. The slit lenghts used for the ICL
spectra are 56.5′′, 32.5′′, and 24.5′′, while those of typical galaxies vary
between 7′′ and 12′′. The ICL was located either on the master CCD or the
second one, resulting in a combined pencil beam field of $\rm
10.75^{\prime}\times 7.5^{\prime}$ (Figure 1).
A total of 30 hours of observing time including field acquisition, mask
positioning, and integration time were dedicated to our pencil beam redshift
survey of the J0054.0-2823 field. Each mask was filled with 39-49 slitlets in
addition to the ICL slits. In order to trace some of the apparent structures
connected to J0054.0-2823, and to reach beyond its Virial radius, we also
obtained MXU exposures of 8 FORS2 fields of $\rm 7^{\prime}\times 5^{\prime}$
adjacent to the pencil beam, so in total we obtained spectra of 730 individual
sources.
Figure 1: The central (pencil beam) field from R images obtained with the wide
field camera at the 2.2m telescope in La Silla
### 2.1 Mask preparation
Tables for preparing the masks and instrument setups were obtained with the
FORS Instrumental Mask Simulator111
http://www.eso.org/sci/observing/phase2/FORS/FIMS.html (FIMS, 2006). The
selection of the objects for the preparation of the slit masks of the pencil
beam field was done by using a photometric catalog in V and I which we had
derived from deep images obtained in a previous NTT run (Faure et al., 2007),
and pre-images in R from the VLT. The selection of the objects in the fields
adjacent to the pencil beam were obtained by using images taken with the WIde
Field Imager (WIFI; $34^{\prime}\times 33^{\prime}$) at the 2.2m telescope on
La Silla. Photometry in V and R from the WFI images are used throughout the
paper. The allocated time was divided in observing blocks (OBs) to be executed
in service mode. A typical OB of 1h execution time had a science integration
time of 2900s in two exposures of 1450s.
We estimated exposure times for E to Sb galaxies in the range z = 0.3 - 0.8.
Using the exposure time calculator of FORS, we obtained magnitude limits, the
major steps of which are given in Table LABEL:maglimit, which we used to
optimize the distribution of slitlets in the masks. After isolating bright
objects which did not require long exposure time, we prepared a grid with an
exposure time step $\rm 2\times 1450~{}s$ which we filled with galaxies having
V magnitudes such that the expected S/N ratio would be better than 2.8 (1
pixel along the dispersion). After receiving VLT pre-images in the red band,
we did a similar grid in R and adjusted the two grids. The masks were prepared
interactively with the FIMS tool and the R pre-images. We started to fill
masks with objects that require an exposure time $\rm\leq 2\times 1450~{}s$,
then moved to $\rm\leq 4,~{}6,~{}and~{}8\times 1450~{}s$. Because we prepared
sets of masks with slits in very different directions (those of the ICL long
and short axis in particular), objects that could not be targeted with a mask
in a given direction (i.e. such as any mask with running name ICL-s in Table
2) were targeted in a perpendicular one (i.e. masks with running name ICL-L),
an approach which made the mask filling quite efficient, in particular in
over-dense areas and field edges. Objects which were close to a predicted S/N
of 2.8 in an OB, were selected to be also observed in another OB as often as
possible. Some objects with expected good S/N in an OB, were re-observed in
another OB when there was no other target in the corresponding slit strip.
They provides a set of high S/N $(\sim 20)$ ratio spectra.
A total of 973 slitlets were selected, 621 in 14 different masks in the pencil
beam field, and 352 in 8 masks in the adjacent fields. Thirty five percent of
the sources of the pencil beam field were observed through different masks,
whereas the slitlets of the adjacent fields are all for different sources.
Table 1: Table used for preparing MXU plates of multiple Observing Blocks Number of OBs of 1h | Integration time | Magnitude limit in V | S/N for S0-Sb at $0.3\leq z\leq 0.8$
---|---|---|---
1 | 2900s | 24.4 - 24.8 | 2.8 - 5.2
2 | 5800s | 24.8 - 25.2 | 2.8 - 5.2
4 | 11600s | 25.2 - 25.6 | 2.8 - 5.2
The resulting list of masks and OBs, and the journal of observations are given
in Table 2. Spectra of the pencil beam field were obtained through masks with
running names Bright, ICL-L, ICL-s, and arc. ICL-L and ICL-s were obtained
with rotator angle $-343^{o}$ and $-85^{o}$ respectively, and arc with a
rotation of $-55^{o}$. Masks with names SE, E, NE, N, NW, W, SW1 & SW2 are on
adjacent fields. The observations were obtained during clear nights, with
seeing between $0.7^{\prime\prime}$ and $1.5^{\prime\prime}$ and dark sky.
Table 2: Journal of the MXU Observations Name | OB ID | Date | Exp. time (s) | # slitlets | Grism
---|---|---|---|---|---
Bright1 | 255728 | 20 Oct. 06 | $3\times 550$ | 34 | 600RI
Bright2 | 255726 | 23 Oct. 06 | $3\times 550$ | 38 | 600RI
SW1 | 255710 | 18 Oct. 06 | $3\times 550$ | 45 | 300V
SW2 | 255708 | 15 Oct. 06 | $3\times 550$ | 40 | 300V
W | 255712 | 19 Oct. 06 | $3\times 550$ | 49 | 300V
SE | 255706 | 3 Oct. 07 | $3\times 550$ | 42 | 300V
N | 255716 | 5 Oct. 07 | $3\times 550$ | 42 | 300V
NW | 255714 | 5 Oct. 07 | $3\times 550$ | 48 | 300V
NE | 255718 | 14 Oct. 07 | $3\times 550$ | 47 | 300V
E | 255704 | 15 Oct. 06 | $3\times 710$ | 39 | 600RI
ICL-s1 | 255750 | 12 Dec. 06 | $2\times 1450$ | 46 | 300V
ICL-s2 | 255748 | 15 Nov. 06 | $2\times 1450$ | 48 | 300V
ICL-L1 | 255761 | 12 Dec. 07 | $2\times 1450$ | 41 | 300V
ICL-L2 | 255763 | 9 Jan. 07 | $2\times 1450$ | 39 | 300V
arc2 | 255734 | 24 Nov. 06 | $2\times 1450$ | 48 | 300V
arc1 | 255736, 38 | 9 Jan. 07, 11 Sept. 07 | $4\times 1450$ | 43 | 300V
ICL-s3 | 255744, 46, 47 | 27 Oct. 06, 9 Nov. 06 | $6\times 1450$ | 47 | 300V
ICL-L3 | 255752, 59, 60 | 21 Sept. 07, 31 Oct. 07 | $6\times 1450$ | 49 | 300V
arc3 | 255730, 32, 33 | 17 Aug. 07 | $6\times 1450$ | 46 | 300V
ICL-s4 | 255739, 41, 42, 43 | 23 Oct. 06, 13 Nov. 06 | $8\times 1450$ | 46 | 300V
ICL-L4 | 255754, 56, 57, 58 | 15, 17 & 20 Nov. 06 | $8\times 1450$ | 49 | 300V
RI | 255720, 22, 23, 24, 25 | 13 Nov. 06, 12 & 14 Sept. 07, | | |
| | & 3 Oct. 07 | $10\times 1450$ | 47 | 600RI
### 2.2 Spectral extraction
The data were reduced by the ESO quality control group who provided us with
science products (i.e. sky subtracted, flat fielded and wavelength calibrated
spectra of our objects), together with calibration data: master bias (bias and
dark levels, read-out noise), master screen flats (high spatial frequency
flat, slit function), wavelength calibration spectra from He-Ar lamps, and a
set of spectrophotometric standards, which were routinely observed. The sky
subtracted and wavelength calibrated 2D spectra allowed a very efficient
extraction of about 60 % of the spectra. Nevertheless the pipeline lost a
significant fraction of objects, in particular when they were located on the
edges of the slitlets. To increase the efficiency of the spectral extraction
we performed a new reduction starting from frames that were dark subtracted,
flat-fielded and wavelength calibrated, but not sky subtracted, using a list
of commands taken from the LONG context of the MIDAS package. For each
slitlet, the position of the object spectrum was estimated by averaging 500
columns in the dispersion direction between the brightest sky lines and
measuring the maximum on the resulting profile. The sky background was
estimated on one side of the object, or on both, depending on each case.
Spatial distortion with respect to the columns was measured on the sky line at
5577 Å and used to build a 2D sky which was subtracted to the 2D spectrum.
Multiple exposures where then aligned and median averaged. The 1D spectra of
objects were extracted from 2D medians by using the optimal extraction method
in MIDAS.
### 2.3 Redshift identification
The identification of lines for determining the redshifts was done
independently by two methods and three of the authors. The 2D spectrum was
visually scanned to search for a break in the continuum, or an emission-line
candidate (e.g. [OII] $\lambda$3728.2 Å). A plot of the 1D spectrum was
displayed in the corresponding wavelength region to search for [OII], the Ca H
& K lines, and/or Balmer lines H$\epsilon$, H9 $\lambda$3835.4 Å, H8
$\lambda$3889.1 Å, H10 $\lambda$3797.9 Å, and H$\delta$. The redshift was then
confirmed by searching for the [OIII] doublet $\lambda$4958.9 & 5006.8 Å, and
H$\beta$ in emission if [OII] had been detected, or G and the Mgb band, if the
4000 Å break and (or) the H and K lines had been identified. The MgII
$\lambda$2799 Å line in absorption and, in some cases AlII $\lambda$ 3584 Å,
were searched to confirm a potential redshift $z\geq 0.65$, while in the cases
of low redshift candidates we searched for $\rm H\beta$, the NaD doublet
$\lambda$5890 & 5896 Å, and in a few cases H$\alpha$. The H$\gamma$ line, the
E (FeI+CaI $\lambda$5270 Å) absorption feature and, in some bright galaxies
the Fe $\lambda$4383 Å, Ca $\lambda$4455 Å, Fe $\lambda$4531 Å absorption
lines, were used to improve the redshift value. The resulting identification
ratio of galaxy redshifts is of the order of 90%. The 10% of so-called
unidentified include stars, objects with absorption lines which were not
understood, a few objects with low signal, and defects. Six QSO’s were also
found. An example of good spectrum of red galaxy, with its main absorption
lines identified, is shown in Figure 2.
Figure 2: Example of a spectrum of a red and bright galaxy with [OII] and the
main absorption lines identified
A second independent visual identification was performed using Starlink’s
Spectral Analysis Tool (SPLAT-VO), matching an SDSS reference table of
emission and absorption
lines222http://www.sdss.org/dr5/algorithms/linestable.html to the spectra.
After a first estimate of the redshift a cross-correlation was performed using
the FXCOR task on the RV package of IRAF333IRAF is distributed by the National
Optical Astronomy Observatory, which is operated by the Association of
Universities for Research in Astronomy, Inc., under cooperative agreement with
the National Science Foundation.. Due to the large span of redshifts, two sets
of templates were used. The first one consisting of 3 template spectra of
galaxies ($\lambda=3500-9000$Å with emission and absorption lines and a
dispersion of 3Å/pix) from the SDSS
survey444http://www.sdss.org/dr2/algorithms/spectemplates/index.html with
continuum subtraction using a spline3 order 5 fitting function. The second set
of templates were two average composite spectra of early type and intermediate
type galaxies ($\lambda=2000-7000$Å with only absorption lines and a
dispersion of 2Å/pix) from the K20
survey555http://www.arcetri.astro.it/$\sim$k20/spe_release_dec04/index.html
using a spline3 order 7 function for continuum subtraction. An interactive
selection of the wavelength range used in the cross correlation was done on
each spectrum avoiding contamination by sky lines. The spectra were re-binned
to the template dispersion (smaller for 300V spectra and larger for 600RI
spectra) , which gave the best results. Velocity errors were determined from
the quality of the cross-correlation, by using standard R value of Tonry &
Davies (Tonry:1979p5170, 1979). Here we used $R_{T}$ to differentiate it from
the R band magnitudes symbol. These values are provided in the IRAF task FXCOR
and explained in the reference quoted. In brief, $R_{T}$ is proportional to
the ratio of the fitted peak height and the antisymmetric noise as defined by
Tonry & Davies (1979). The redshifts, $R_{T}$ values, and velocity errors are
given in Table 5, which also includes the list of visually identified lines.
A third independent visual inspection was carried out when a discrepancy was
seen between the previous two sets of measurements, and also in the very few
cases were no redshift could be measured. For these spectra we first tried to
detect emission or absorption lines and then used Gaussian fits to establish
the line centroids and their errors and shifts. The redshift of each line was
measured independently and the galaxy redshift was obtained from the weighted
average of all lines. This third inspection resolved nearly all the few
remaining discrepancies so we have retained the cross-correlation values
whenever possible. We note that Xcorr failed in two instances: 1) for $z>0.8$
galaxies with low S/N and few weak absorption lines, and, 2) when no
absorption lines, but 1, 2 or 3 clear emission lines were present. In these
cases we used the visual line identifications and assigned a conservative
error of 300 km/s.
Spectra from more than one mask were obtained for 94 objects. Their final
velocities and velocity errors were calculated as error-weighted means from
multiple observations, although no significant disagreements were found. These
repeated observations serve as a check on the internal errors. Figure 3
presents the differences between the cross-correlation velocity measurements
for all galaxies with multiple observations. The representative full width
half maximum (FWHM) error is 200 km/s. In Figure 4 we have plotted the
relation between velocity errors and the Tonry $R_{T}$ value obtained in our
cross correlations. Most errors are $<300$ km/s even for $4>R_{T}>2$ and the
typical error is of order 80 km/s with the vast majority of the radial
velocities have errors below 200 km/s. We have only discarded a few values
with $R_{T}<1$ when there were no measurable emission lines.
Figure 3: Radial velocity differences for galaxies with multiple
observations. The objects with velocity discrepancies larger than $400$ km/s
are broad line QSO’s and one high-z galaxy. Figure 4: Relation between radial
velocity errors ($V_{err}$) and the Tonry $R_{T}$ parameter (Tonry:1979p5170,
1979) in redshifts obtained by cross-correlation. The points away from the
general trend (5 points with $R_{T}>10$ and 5 with $V_{err}>500~{}\rm
km~{}s^{-}1$) are 5 QSO’s and distant weak spectrum galaxies with emission
lines. Objects with $V_{err}>500~{}\rm km~{}s^{-}1$, marked in red, were not
used in combined spectra.
### 2.4 Flux Calibrations
The 1D spectra were divided by the response curve of the detector, which had
been determined from 4 spectrophotometric standard stars observed along the
runs, and reduced by the same method (bias, flat field, wavelength
calibration, and extraction) as the galaxy spectra. The thick absorption
telluric band of O2 centered at 7621 Å (unresolved line series) was not
removed from the observation response curve and was considered as a feature of
the global wavelength dependent efficiency.
The relative fluxes per wavelength of the corrected spectra can be compared
with stellar population models, in arbitrary unit, but are not calibrated in
flux. The spectra were re-binned to the z = 0 rest frame with relative flux
conservation. Because a significant fraction of spectra have a too low S/N
ratio for a meaningful comparison with population synthesis models, one may
either select the brightest objects or combine spectra of similar types. The
spectra taken at different locations of the MXU masks have different lengths
along the dispersion direction. In order to merge them the spectra were
normalized to have the same flux in the region 4050-4250 Å (see below).
### 2.5 Quality of the spectra
The final S/N ratio of the extracted spectra, corrected for the response
curve, and re-binned to zero redshift depends on a number of parameters:
seeing, night sky transparency and background, magnitude of the object and
integration time, wavelength of the S/N measurement, and redshift. To give an
idea of the final products we present in Table 3 a representative set of 28
spectra at various z, magnitudes, number of OBs and resulting S/N ratio
measured on zero redshift spectra in the wavelength range 4150-4250 Å which
corresponds well to the location where we will measure the main indexes of
this work. S/N ratios of spectra re-binned to zero redshift are for a pixel
element of $\rm 2.6~{}\AA$ throughout the paper. Table 3 gives also the names
of the OBs.
Table 3: Signal-to-noise ratio of representative spectra. The columns indicate respectively: the redshift $(z)$ of a selected object, its V and R Petrosian magnitudes, the number of observing blocks, N(OB), from which its spectrum is extracted, the S/N ratio measured in the wavelength range 4150-4250Å of the spectrum rebinned to zero redshift, the name of observing blocks from Table 2, and the grism used. Spectra from OB’s with running name “arc” have on the average higher S/N ratio than those with name “ICL” as illustrated by the two objects marked (*). $z$ | V | R | N(OB) | S/N | Name of OBs | Grism
---|---|---|---|---|---|---
0.2923 | 19.2 | 18.6 | 2 | 14 | arc1 | 300V
0.2932 | 20.3 | 19.3 | 2 | 18 | ICL-L1 & L2 | 300V
0.2928 | 21.6 | 20.5 | 3 | 17 | arc2 & ICL-L1 | 300V
0.2905 | 22.8 | 22.1 | 2 | 10 | ICL-L1 & L2 | 300V
0.2910 | 23.5 | 22.8 | 3 | 9 | arc1 & 2 | 300V
0.4486 | 21.3 | 20.0 | 1 | 6 | Bright2 | 600RI
0.4477 | 22.3 | 21.3 | 2 | 20 | arc1 | 300V
0.4148 | 23.0 | 22.3 | 2 | 7 | ICL-s1 & s2 | 300V
0.4538 | 23.1 | 22.0 | 5 | 14 | ICL-L4 & arc2 | 300V
0.5355 | 22.3 | 20.9 | 1 | 8 | arc2 | 300V
0.6309 | 22.7 | 21.4 | 4 | 11 | arc2 & 3 | 300V
0.6553 | 22.3 | 21.5 | 4 | 9 | ICL-s4 & arc2 | 300V
0.6282 | 23.5 | 22.5 | 5 | 9 | arc1 & 3 | 300V
0.6267 | 23.9 | 22.9 | 4 | 10 | arc3 | 300V
0.6864 | 22.6 | 21.9 | 1 | 4 | ICL-s2 | 300V
0.6886 | 23.0 | 22.0 | 7 | 13 | ICL-L3 & L4 | 300V
0.6861 | 23.0 | 22.1 | 4 | 8 | ICL-s4 | 300V
0.6864 | 23.5 | 22.3 | 4 | 10 | ICL-s3 & arc2 | 300V
0.6879 | 23.8 | 22.8 | 4 | 7 | ICL-s4 | 300V
0.8222 | 20.7 | 20.0 | 1 | 10 | ICL-s1 | 300V
0.8287 | 22.7 | 22.4 | 5 | 10 | ICL-s4 & arc2 | 300V
0.8249 | 23.2 | 22.6 | 3 | 8 | arc3 | 300V
0.8823 | 23.8 | 23.4 | 4 | 3.5 | ICL-s4 | 300V
0.9792 | 23.3 | 22.7 | 3 | 8 | arc3 (*) | 300V
0.9626 | 23.2 | 22.7 | 4 | 5 | ICL-L4 (*) | 300V
0.9637 | 23.4 | 23.2 | 3 | 6 | ICL-s3 | 300V
0.9809 | 23.8 | 23.7 | 5 | 6 | RI | 600RI
1.0220 | 24.1 | 23.3 | 4 | 3 | ICL-s4 | 300V
### 2.6 Spectral indexes
The 4000 Å break amplitude definition used in the present paper is the
‘narrow’ 4000 Å break defined by Balogh et al. (1999) as the flux ratio in the
range 4000-4100Å over 3850-3950Å (e.g. Kauffmann et al., 2003). The error in
D(4000) is calculated from the spectral noise in the two passbands. The
equivalent widths of [OII] and of $\rm H\delta$ were measured by using the
MIDAS context ALICE as follows: the continuum was obtained by linear
interpolation through two passbands each side of the line, a Gaussian was
fitted to the emission or absorption line, and an integration was done over
the resulting Gaussian profile above or below the continuum. The continuum and
line fits, and the integration were done interactively on a graphic window in
which the spectral region of the line was displayed. Table LABEL:integr lists
the wavelength ranges of the sidebands used to define the fluxes and continua.
Table 4: Wavelength bands used in the measurement of 4000 Å break amplitude, and in the determination of the continua of the [OII] and $\rm H\delta$ indexes (equivalent widths). Index | Blue band | Red band
---|---|---
D(4000) | 3850 - 3950 Å | 4000 - 4100 Å
EQW([OII]) | 3650 - 3700 Å | 3750 - 3780 Å
$\rm EQW(H\delta)$ | 4030 - 4070 Å | 4130 - 4180 Å
Uncertainties in equivalent widths were deduced from simple Monte Carlo: the
values of the equivalent widths are the average of 20 continuum determinations
and best Gaussian fits to the absorption or emission lines, and the errors in
equivalent widths are deduced from the Monte Carlo dispersion. The largest
index errors are for spectra in which $\rm H\delta$ is both in absorption and
in emission. In such cases the emission line was removed after fitting the
spectrum of an A star onto all Balmer lines to estimate the depth of $\rm
H\delta$ in absorption, and this step was added to the Monte Carlo. The errors
on indexes given in Tables of combined spectra throughout the paper are those
which were measured on combined spectra. They do not take into account the
astrophysical dispersions in the distributions of individual galaxies which
were used to build combined spectra. Those astrophysical dispersions are given
in relevant Tables concerning spectral variations.
Full observational measurement errors on indexes of individual spectra were
obtained by measuring $\rm D(4000)$ and $\rm EQW([OII])$ on spectra with
multiple observations. Thus $17\%$ of the spectra have typical errors of $4\%$
in D(4000) and $10\%$ in EQW([OII]); $54\%$ have typical errors of $8\%$ in
D(4000) and $20\%$ in EQW([OII]); and $14\%$ have poorer spectra with typical
errors of $16\%$ in D(4000) and $40\%$ in EQW([OII]).
### 2.7 Stellar Population Analysis
In order to study the stellar population quantitatively, we applied a modified
version of the spectral population synthesis code,
starlight666http://www.starlight.ufsc.br/ (Cid Fernandes et al., 2004; Gu et
al., 2006) to fit the observed and combined spectra. The code does a search
for the best-fitting linear combination of 45 simple stellar populations
(SSPs), 15 ages, and 3 metallicities ($0.2\,Z_{\odot}$, $1\,Z_{\odot}$,
$2.5\,Z_{\odot}$) provided by Bruzual & Charlot (2003) to match a given
observed spectrum $O_{\lambda}$. The model spectrum $M_{\lambda}$ is:
$M_{\lambda}(x,M_{\lambda_{0}},A_{V},v_{\star},\sigma_{\star})=M_{\lambda_{0}}\left[\sum_{j=1}^{N_{\star}}x_{j}b_{j,\lambda}r_{\lambda}\right]\otimes
G(v_{\star},\sigma_{\star})$ (1)
where
$b_{j,\lambda}=L_{\lambda}^{SSP}(t_{j},Z_{j})/L_{\lambda_{0}}^{SSP}(t_{j},Z_{j})$
is the spectrum of the $j^{\rm th}$ SSP normalized at $\lambda_{0}$,
$r_{\lambda}=10^{-0.4(A_{\lambda}-A_{\lambda_{0}})}$ is the reddening term,
$x$ is the population vector, $M_{\lambda_{0}}$ is the synthetic flux at the
normalization wavelength, and $G(v_{\star},\sigma_{\star})$ is the line-of-
sight stellar velocity distribution modeled as a Gaussian centered at velocity
$v_{\star}$ and broadened by $\sigma_{\star}$. The match between model and
observed spectra is calculated as
$\chi^{2}(x,M_{\lambda_{0}},A_{V},v_{\star},\sigma_{\star})=\sum_{\lambda=1}^{N_{\lambda}}\left[\left(O_{\lambda}-M_{\lambda}\right)w_{\lambda}\right]^{2}$,
where the weight spectrum $w_{\lambda}$ is defined as the inverse of the noise
in $O_{\lambda}$. The code yields a table with input and ouput parameters for
each component. Input parameters include individual stellar masses, ages,
metallicities, L/M, … and ouput parameters include luminosity fractions, mass
fractions, fit parameters of individual components …, and global parameters
such as velocity dispersion and extinction. For more details we refer to the
paper by Cid Fernandes et al. (2005). In the present work we use the standard
luminosity fraction in the rest frame of normalized spectra at $\rm
4050~{}\AA$, which we compare in different redshift bins.
Figure 5: Spectral fitting results with SSP models for the redshift $<z>=0.29$
bin. (a): Observed (thin black line), model (red line) and residuals for the
absorption spectrum. Points indicate bad pixels and emission-line windows that
were masked out during fitting. (b): Emission-line spectrum; (c): Total
spectrum.
Figure 5 shows an example of the fit for the averaged spectrum at $<z>=0.29$.
Panels (a), (b), and (c) correspond to absorption-line, emission-line, and all
spectra respectively. After fitting the spectra, we rebin the 45 SSPs into 5
components according to their age: I ($10^{6}\leq t<10^{8}$ yr), II
($10^{8}\leq t<5\times 10^{8}$ yr), III ($5\times 10^{8}\leq t<10^{9}$ yr), IV
($10^{9}\leq t<2.5\times 10^{9}$ yr), and V ($t\geq 2.5\times 10^{9}$ yr).
Components with the same age and different metallicities are combined
together.
## 3 The Catalog of Galaxies and Large Scale Structures in the Line of Sight
in the Pencil Beam
Table 5 presents positions, redshifts, Petrossian R-magnitudes ($m_{R}$), and
line identifications for the full sample of 654 galaxies observed in our
program. The radial velocities and the corresponding measurement errors are
also given. The full Catalogue from which Table 5 is extracted will be sent as
a public database to CDS. The rough data are presently in the public domain at
ESO.
Table 5: Properties of galaxies in the field of RX J0054.0-2823 obj | RA ($\alpha$) | DEC ($\delta)$ | z | $m_{R}$ | V | Verr | $R_{T}$ | Nobs | lines
---|---|---|---|---|---|---|---|---|---
| J2000 | J2000 | | | km/s | km/s | | |
23 | 13.598707 | -28.434965 | 0.79304 | 22.78 | 237912 | 161 | 4.7 | 1 | K–H
26 | 13.590379 | -28.416917 | 0.77636 | 22.26 | 232908 | 77 | 8.1 | 1 | [OII]–H10–H9–H
27 | 13.584442 | -28.394515 | 0.41463 | 22.29 | 124389 | 22 | 17.3 | 1 | [OII]–H9–H–H$\beta$–[OIII]
28 | 13.586628 | -28.438063 | 0.44877 | 22.02 | 134631 | 73 | 6.7 | 1 | K–H–G
30 | 13.580301 | -28.435437 | 0.29032 | 21.08 | 87096 | 68 | 11.3 | 1 | H9–K–H–H$\delta$–H$\alpha$
31 | 13.572009 | -28.380385 | 0.63267 | 21.32 | 189801 | 49 | 11.5 | 1 | [OII]–K–H
32 | 13.579012 | -28.439414 | 0.45335 | — | 136005 | 22 | 19.5 | 1 | [OII]–H$\gamma$–H$\beta$–[OIII]
33 | 13.574997 | -28.439377 | 0.44741 | 19.21 | 134223 | 80 | 11.4 | 1 | K–H–G–H$\beta$
34 | 13.573913 | -28.442855 | 0.63013 | 20.63 | 189039 | 87 | 7.9 | 1 | K–H–H$\delta$–G
35 | 13.571781 | -28.435423 | 0.44862 | 20.26 | 134586 | 73 | 9.8 | 1 | H9–K–H–G–H$\beta$
Figure 6 shows the R-magnitude histogram of the galaxies with measured
redshifts superimposed on the magnitude histogram of all galaxies in our
pencil-beam indicating that our observations sample uniformly at a rate of
50-60% the population of galaxies down to $\rm R=22.5$. The sampling seems
fairly representative in the magnitude bin $\rm R=22.5-23.0$, and sparse at
$\rm R>23$. The apparent increase in incompleteness toward brighter magnitudes
is due to a selection bias in the observations, which were designed to avoid
bright galaxies at redshifts $z\leq 0.25$.
Figure 6: R-magnitude histogram of galaxies with measured redshift in the
central beam.
Figure 7 presents the magnitude redshift relation and the cone diagrams for
the full sample. The points are color coded according to the presence or
absence of emission lines.
Figure 7: (a) Magnitude redshift relation for the full sample. The three
lines overploted over the measured points correspond to absolute R magnitudes
of -22.5, -20.5, and -18.5. The distances have been calculated using a
cosmology with $\Omega_{0,\Lambda}=0.70$, $\Omega_{0,m}=0.30$, $w=-1$, and
$\rm H_{0}=75km~{}s^{-1}~{}Mpc^{-1}$ (h = $\rm
H_{0}/75km~{}s^{-1}~{}Mpc^{-1}$). Red dots are galaxies with no emission lines
and blue dots are galaxies with emission lines. (b) Cone diagrams in Dec for
all the galaxies measured in the field of RX J0054.0-2823. The scales is in
Mpc calculated using the angular distance for the standard cosmology. The
detection threshold for emission-lines is $\rm EQW([OII])\sim 2-3$Å. (c) Same
as (b) but for RA.
A cursory inspection of Figure 7 reveals the presence of several conspicuous
structures - walls of objects spanning almost the entire field of view - over
the full range of redshifts covered by our observations. Ignoring objects with
$z<0.28$, we see structures centered at $z=0.29$ (our prime target); two
distinct structures at $z\sim 0.4$, which we will denote $z=0.415$ and
$z=0.447$; a rather complex structure at $z\sim 0.6$, with two main over-
densities at $z=0.58-0.63$, and $z=0.68$; a single rather sparsely populated
layer at $z=0.82$. In what follows, we will refer to these groups (including
the main cluster at $z=0.29$) as our pencil beam structures. Making bins
centered on the peaks of the redshift distribution maximizes the number of
objects in each bin and minimizes its redshift dispersion. So using the
apparent structures rather than a blind slicing appears well adapted to our
sample. If the structures are real, the objects of a given structure may have
a common history and this may also help to reduce the cosmic scatter.
The numbers of spectra observed in each structure are given in Table
LABEL:zdist. The redshift bins given in column (2) are chosen a posteriori to
fit the structures. The numbers of redshifts measured in each bin are given in
column (3), with the respective numbers of absorption and emission systems in
columns (4) & (5). We have determined the median distance to the nearest
object $\triangle\delta$ in each of the apparent structures in column (6), and
the median velocity dispersion, $\rm\sigma(V)\equiv c\sigma(z)/(1+z)$ in
column (7). We give a rough morphology of the structures in column (1). The
cluster at $z=0.293$ appears to have small projected separation and velocity
dispersion. Layers or filaments have a comoving velocity dispersion (dynamical
and cosmological) less than $\rm\sim 1500~{}km~{}s^{-1}$; clouds have
$\rm\sigma(V)>2000~{}km~{}s^{-1}$. The structures marked “filaments” are the
arc layers seen in Figure 7. Projected on the sky they seem to be filamentary,
but the median distances $\triangle\delta$ to the nearest object are
approximately 2/3 those expected for uniform distributions, so they are not
clearly different from 2D layers. The names of the bins that are used to
combine spectra are given in column (8). Large scale arc structures, as seen
in cone diagrams, are expected to be formed by infall of galaxies on
gravitational potentials: galaxies which are on the far side have a negative
infall velocity, while those on the nearby side have a positive infall
component, which when superimposed on the Hubble flow reduces the velocity
dispersion. This is presumably what we observe in the two filaments or layers
with low velocity dispersion at z = 0.4.
We combined the spectra in each structure using the median. This results in a
slightly lower total S/N (by $\sqrt{2}$), but allows to eliminate spurious
features.
Table 6: Apparent structures in the field of RX J0054.0-2823 Apparent Structure | z range | N | N(abs) | N(em) | $\triangle\delta~{}(kpc)$ | $c\sigma(z)/(1+z)$ | Composite name
---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
| 0.275 - 0.285 | 0 | 0 | 0 | | |
Cluster | 0.285 - 0.298 | 91 | 60 | 31 | 165 | 527 | SPEC029
| 0.298 - 0.320 | 5 | 1 | 4 | | |
| 0.320 - 0.330 | 12 | 7 | 5 | | |
| 0.330 - 0.390 | 28 | 7 | 21 | | |
| 0.390 - 0.430 | 35 | 6 | 29 | 490 | 3250 | SPEC0415
Filament (layer) | 0.432 - 0.440 | 29 | 5 | 24 | 500 | 350 | SPEC0415
| 0.440 - 0.444 | 1 | 1 | 0 | | |
Filament (layer) | 0.444 - 0.456 | 46 | 21 | 25 | 470 | 450 | SPEC0447
| 0.456 - 0.465 | 1 | 1 | 0 | | |
Cloud | 0.465 - 0.550 | 53 | 15 | 38 | 480 | 3370 |
Cloud | 0.550 - 0.620 | 56 | 15 | 41 | 520 | 3550 | SPEC063
Filament (layer) | 0.620 - 0.657 | 48 | 12 | 36 | 580 | 1450 | SPEC063
| 0.657 - 0.673 | 1 | 0 | 1 | | |
Filament (layer) | 0.673 - 0.696 | 43 | 12 | 31 | 650 | 1010 | SPEC068
| 0.710 - 0.790 | 22 | 4 | 18 | | |
Filament & cloud | 0.790 - 0.850 | 33 | 6 | 27 | 820 | 2050 | SPEC082
| 0.850 - 0.880 | 0 | 0 | 0 | | |
Cloud | 0.880 - 0.930 | 11 | 1 | 10 | | |
| 0.930 - 0.946 | 0 | 0 | 0 | | |
Cloud | 0.946 - 1.046 | 25 | 4 | 21 | 1080 | 4110 | SPEC099
### 3.1 Magnitudes
The R-band average magnitudes of galaxies in each redshift bin are given in
Table LABEL:magnitudes separately for absorption, “red” and “blue” emission-
line galaxies, together with the adopted distance moduli. The partition “red”
versus “blue” is defined by the median spectral slope in each redshift bin. In
a study on emission line galaxies (Giraud et al., 2010) we divided the sample
of emission-line galaxies in two halves: those with continuum slopes bluer
than the average and those with continuum slopes redder than the average in
each redshift bin. This was done interactively by displaying reduced 1D
spectra and using MIDAS. While a median partition is not necessarily a
physical partition, we showed that, in the present case, it divides ”young”
galaxies, for which the evolution is dominated by on-going star formation from
”old” galaxies where the evolution is dominated by changes in the older
stellar populations.
Table 7: Average R-band magnitudes of absorption systems (abs), and red and blue emission-line galaxies. The adopted distance moduli $(m-M)_{0}$ and the 4150-4250Å fluxes $f$ normalized to the blue galaxies at $z=0.9$ are also tabulated. $<z>$ | R(abs) | R(red) | R(blue) | $(m-M)_{0}$ | $f$(abs) | $f$(red) | $f$(blue)
---|---|---|---|---|---|---|---
0.29 | 19.80 | 20.12 | 20.97 | 40.18 | 0.74 | 0.72 | 0.48
$0.43$ | 20.24 | 20.59 | 20.95 | 40.86 | 1.08 | 0.92 | 0.75
$0.65$ | 21.50 | 21.60 | 21.94 | 41.51 | 1.42 | 1.28 | 0.86
$0.9$ | 22.45 | 22.13 | 22.35 | 41.98 | 2.08 | 1.70 | 1
We used the R-band photometry to calibrate individual spectra by convolving
each spectrum with a box filter 1290 Å wide, centered at $\lambda=6460$ Å.
Once the spectra were calibrated in the observer R-band, we measured the
average fluxes in the wavelength range 4150-4250Å of the galaxies, which we
normalized to the flux of blue emission galaxies at $<z>=0.9$ to compute the
luminosity index $f$. Thus $f$ (that is equal to 1 for blue galaxies at
$<z>=0.9$) is an indicator of AB(4200) that allows us to compare the
luminosities of red and blue galaxies at a given redshift and to investigate
luminosity variations with $z$. Thus Table LABEL:magnitudes clearly shows that
in each redshift bin, absorption-line and red emission-line galaxies are more
luminous than blue galaxies.
## 4 Composite spectra
Each galaxy spectrum was wavelength calibrated, corrected for instrument
response, re-binned to zero redshift, and normalized to have the same flux in
the wavelength range $\rm\triangle\lambda=4050-4250\AA$. Normalizing spectra
gives the same weight to all galaxies. As a consequence stellar fractions must
be understood as average stellar fractions per galaxy.
We have truncated the sample at $z=1.05$ and assembled the spectra in bins
centered on (pseudo) structures at 0.29, 0.41, 0.45, 0.63, 0.68, 0.82, and
0.99 to build high S/N composite spectra for each bin. In order to compensate
(or at least alleviate) for Malmquist bias we rejected objects fainter than
$\rm M_{R}=-18.8$ mostly at $z\leq 0.45$ (Figure 7a). A sample completely free
of Malmquist bias would require a cutoff at $\rm M_{R}\sim-20.5$. For clarity
of the figures, we often combined the mean spectra at $z=0.41$ & $0.45$ into a
single bin at $<z>=0.43$, the spectra at $z=0.63$ & $0.68$ into a bin at
$<z>=0.65$, and in some cases the spectra at $z=0.82$ & $0.99$ into a bin at
$<z>=0.9$. The spectra of galaxies in these four bins are presented in Figure
8 where we show the spectra of absorption systems (top) and emission line
galaxies (bottom) separately. The corresponding $4000\AA$ break amplitudes are
given in Table LABEL:D4000
Figure 8: Composite spectra of absorption systems (top); and emission line galaxies (bottom) normalized in the wavelength range $\rm\triangle\lambda=4050-4250$ Å. All individual galaxies are brighter than $\rm M_{R}=-18.8$ Table 8: 4000Å break amplitudes for absorption (abs) and emission (em) galaxies, and equivalent width of H$\delta$ for absorption galaxies with measurement errors. The S/N ratios of the combined spectra were measured in the interval 4050Å–4250Å. The magnitude cutoff is $\rm M_{R}=-18.8$ for all redshift bins. | Absorption systems | Emission systems
---|---|---
$<z>$ | D(4000) | $\rm EQW(H\delta)$ | S/N | D(4000) | S/N
0.29 | $1.67\pm 0.065$ | $-1.5\pm 0.2$ | 23 | $1.22\pm 0.02$ | 32
$0.43$ | $1.70\pm 0.06$ | $-1.5\pm 0.2$ | 22 | $1.22\pm 0.01$ | 52
$0.65$ | $1.60\pm 0.055$ | $-1.8\pm 0.2$ | 24 | $1.14\pm 0.01$ | 35
0.82 | $1.57\pm 0.06$ | $-2.4\pm 0.5$ | 18 | $1.07\pm 0.02$ | 28
0.99 | $1.43\pm 0.05$ | $-2.9\pm 0.3$ | 23 | $1.08\pm 0.02$ | 25
The most conspicuous spectral change with redshift is a decrease in flux
redward of the G-band from $<z>=0.29$ and $<z>=0.43$ to higher $z$ coupled to
an increase to the blue of [OII] from $<z>=0.65$ to $<z>=0.82$ and higher $z$
in emission-line galaxies. This systematic change of the continuum implies
that the galaxy population varies as a function of redshift: more star forming
galaxies at higher $z$ and more galaxies with old stars at lower $z$. This
spectral change, which is known, will not be studied further in this paper
except to quantify (in 5.3.1) the impact of LINER-like galaxies at $z=0.4--
0.9$. In the following section we concentrate on absorption systems and low-
ionization galaxies.
## 5 Absorption line systems
The spectral resolution of the 300V grism allows us to detect [OII] emission
down to $\rm EQW([OII])\sim 2-3~{}\AA$. We will call absorption-line galaxies
those for which any mechanism of ionization is low enough to preclude [OII]
detection at our detection level. Thus, our pure absorption-line sample
comprises mostly E, E+A, and S0 galaxies with no on-going star formation,
nuclear activity, or other mechanism of ionization.
### 5.1 Absorption systems as function of redshift
The normalized and combined spectra of absorption line systems presented in
Figure 8 (top) do not show any obvious change in their continuum and 4000Å
break amplitude up to $z\approx 0.6$ (Table LABEL:D4000). There is a moderate
decrease in the 4000Å break at $z\geq 0.65$ ranging from $5\%$ at $z\sim 0.65$
to $7\%$ at $z\sim 0.82$ and up to $15\%$ at $z\sim 1$, while the $\rm
H\delta$ absorption line becomes stronger at $z\geq 0.65$ (Table LABEL:D4000),
suggesting the presence of increasing numbers of A stars at higher redshifts.
The indexes suggest that these galaxies had the bulk of their star formation
at $z\geq 1$, while some of the systems at $z>0.8$ still had clearly
detectable star formation about 1 Gyr ago.
We have compared our spectral indexes at $z\sim 0.82$ with those measured by
Tran et al. (2007) in the rich cluster MS 1054-03 at z = 0.83 using the same
index definitions from Kauffmann et al. (2003). The average break amplitude
and $\rm H\delta$ index of absorption systems in MS 1054-03 are respectively
$\rm D(4000)(abs)=1.67\pm 0.00$ and $\rm EQW(H\delta)(abs)=-1.7\pm 0.0$ (Tran
et al., 2007, Table 4). Our absorption systems at $z\sim 0.82$ appear to have
younger stellar populations as indicated both by $\rm D(4000)$ and $\rm
EQW(H\delta)$ (Table LABEL:D4000). Therefore our absorption systems contain A
stars, but clearly less than composite field E+A galaxies at $<z>=0.6$ for
which $\rm D(4000)(abs)=1.36\pm 0.02$ and $\rm EQW(H\delta)(abs)=-4.6\pm 0.2$
(quoted in Tran et al. (2007, Table 4) from data in Tran et al. (2004)).
Consequently our average spectrum at $z\sim 0.82$ is intermediate between pure
E and pure E + A. In fact, our SSP models (Table LABEL:poptable) indicate that
absorption-line systems at $z\geq 0.65$ contain on average more than $50\%$ of
stars younger than 2.5Gyr per galaxy, while those at $z\geq 0.8$ had
significant star formation as recently as one Gyr ago (Table LABEL:poptable).
Table 9: Stellar population properties of normalized average absorption (abs) spectra in each redshift bin. The magnitude cutoff is at $\rm M_{R}=-18.8$, except for the 10 faintest absorption systems at $z=0.29$ where we used all the observed objects. The fractions indicated in all SSP Tables are standard luminosity fractions at $\rm 4050\AA$, as in Cid-Fernandes et al. (2010, and references therein) $<z>$ | log(Age): | $<8$ | $8-8.7$ | $8.7-9$ | $9-9.4$ | $>9.4$
---|---|---|---|---|---|---
$0.29$ | abs | 0.0% | 0.0% | 30.1% | 0.1% | 69.8%
| abs (10 brightest) | 0 | 0 | 17.4 | 0 | 82.4
| abs (10 faintest) | 0 | 0 | 12.0 | 66.4 | 21.7
$0.43$ | abs | 0.0 | 0.7 | 11.7 | 6.9 | 80.7
$0.65$ | abs | 0.0 | 0.0 | 18.2 | 38.5 | 43.3
$0.82$ | abs | 0.0 | 0.0 | 86.8 | 3.3 | 9.8
$0.99$ | abs | 0.0 | 0.0 | 42.3 | 0.0 | 57.7
Post-starburst E+A galaxies are thought to be in a transition phase between a
star-forming period and a passively evolving period. Being close to the phase
of shutdown or quenching of star formation, they probably play an important
role in the build-up of early-type systems (e.g. Wild et al, 2009; Yan et al.,
2009). Studies of intermediate redshift clusters at $0.3\leq z\leq 0.6$ have
found either a higher fraction of post-starburst galaxies in clusters than in
the field (Dressler et al., 1999; Tran et al., 2003, 2004), or a similar
fraction (Balogh et al., 1999). In fact, there is a strong variation in the
E+A fraction between the SSDS low redshift survey at $z\sim 0.07-0.09$, and
high $z$ surveys at $z\approx 0.5-1$ (VVDS, Wild et al, 2009), or $z\approx
0.7-0.9$ (DEEP2, Yan et al., 2009).
In order to search for E+A galaxies in our sample we built template spectra by
combining a pure E spectrum from our sample with various fractions of an A
stellar template. We then compared our models with absorption-line systems in
the range $\rm 1.2\leq D(4000)\leq 1.5$ assuming, by definition, that E+A
galaxies contain at least $\rm 25\%$ A stars. Using this (standard) definition
we searched our sample at $0.35\leq z\leq 1$ and found only 6 bona-fide E+A
galaxies. In fact all the objects were found at $0.68\leq z\leq 1$, which
makes our small number consistent with the VVDS and the DDEP2 surveys within a
factor of 2. The median of the normalized spectra of these 6 (as far as we can
judge from our images) elliptical galaxies is presented in Figure 9 (a). We
were surprised to find no E+As at $z\sim 0.4$, but we did find 4 objects with
early-type morphology and very small $\rm<EQW([OII])>\approx 3.5\AA$, which
probably would have been classified as E+As on lower resolution spectra. The
average spectrum of these 4 objects is shown in Figure 9 (b). Their $\rm R\sim
22$ magnitudes place them at the faint end of absorption line systems at the
corresponding redshifts.
Figure 9: (Top) Average spectrum of 6 E+A galaxies at $0.68\leq z\leq 1$. The
Balmer series $\rm H\delta$, $\rm H+H\epsilon$, H8, H9, H10, and H11 is very
promintent and $\rm<D(4000)>=1.40$. (Botton) Average spectrum of 4 galaxies in
the intermediate redshift range $0.4\leq z\leq 0.5$ with E/S0 morphological
type, showing a poststarburst E+A spectrum with still some star formation.
$\rm<EQW([OII])>\approx 3.5\AA$ and $\rm<D(4000)>=1.41$. Probably this
spectrum would have been classified as E+A at lower resolution.
### 5.2 Absorption-line galaxies as function of luminosity at $z=0.29$
Having tested bright absorption galaxies at various redshifts (with cut-off at
$\rm M_{R}=-18.8$), we now turn to faint absorption galaxies in the cluster at
$<z>=0.29$ by combining the spectra of the 10 faintest galaxies without
emission lines. Their average R-band magnitude is $\rm R=22$, which at a
distance modulus of 40.18 corresponds to $\rm M_{R}=-18.2$, and the faintest
object has $\rm M_{R}=-17.44$. Their mean indexes, $\rm D(4000)=1.55\pm 0.01$;
$\rm H\delta=-2.27\pm 0.04$, measured on the spectrum shown in Figure 10, are
consistent with a younger age than absorption-line galaxies with $\rm
M_{R}\leq-18.8$ (Table LABEL:D4000) in the same cluster. This is in agreement
with the well known evidence that the stellar populations in absorption
systems tend to be younger in low mass galaxies than in the more massive ones
(e.g. Renzini, 2006). The index values are in fact very close to those of our
absorption systems at z = 0.8 (Table LABEL:D4000) which, by selection effects,
are bright (Table LABEL:magnitudes and Figure 7).
The SSP models indicate that on average about 80% of the stars in the 10
faintest galaxies are younger than 2.5 Gyr (Table LABEL:poptable), i.e. were
born at $z<1$. In comparison, 80% of the stars contributing to the spectrum of
the brightest absorption galaxies in the cluster are older than 2.5 Gyr (Table
LABEL:poptable). To illustrate the spectral differences between bright and
faint systems at $z=0.29$, and the striking similarity between the spectra of
faint galaxies at $z=0.29$ and those of bright galaxies at $z=0.8$, we have
plotted in Figure 10 the average spectra of the 10 brightest and the 10
faintest absorption systems at $z=0.29$, and the average spectrum of
absorption galaxies at $z=0.82$. The effect of downsizing, (in the present
case the so-called ’archeological dowsizing’) where star formation shifts from
high mass galaxies at high redshifts, to low mass galaxies at low redshifts is
clearly exemplified in this figure.
Figure 10: Normalized spectra of the 10 brightest (in red) and the 10 faintest
(in blue) absorption-line galaxies in the cluster at $z=0.29$, and the full
sample of absorption-line systems at $z=0.82$ (in cyan).
At a redshift of $z\sim 0.8$ (i.e. $\sim 4$ Gy earlier), the red-sequence of
our unrelaxed (merging central system; elongated intra-cluster light and
galaxy distribution) cluster at z=$0.29$ was already in place, but was
truncated at brighter magnitudes because the faint absorption-line galaxies
were still copiously forming stars. This seems consistent with the observation
that some clusters at $z\simeq 1$ have red sequences truncated at faint limits
(Kodama et al., 2004; Koyama et al., 2007), and supports the picture of an
environmental dependence of red-sequence truncation presented by Tanaka et al.
(2005). This is also in agreement with scenarios where the final assembling of
the red-sequence can be observed well below $z=1$ (Faber et al., 2007).
As discussed above, the strict definition of E+A galaxy requires a mix of an
E-type spectrum with at least 25% A stars and no traces of star formation,
which in our sample implies no emission lines with equivalent widths larger
than 2-3Å. With this definition our $z=0.29$ cluster contains only one E+A
galaxy while the dense layers at $z\sim 0.4$, where the red-sequence is
already in place (layer in Figure 7), contains none. However, both in the
cluster and in the intermediate redshift layers we find plenty of galaxies
with early type morphologies, A stars, and very weak emission lines. In the
next section we present a closer look at these low-ionization emission line
galaxies.
### 5.3 Galaxy evolution and low-ionization emission-line galaxies (LINERs).
In an extensive work based on the SDSS survey, Yan et al. (2006) determined
the extent to which [OII] emission produced by mechanisms other than recent
star formation introduces biases in galaxy evolution studies based upon [OII]
only. They showed that the $\rm[OII]/H\beta$ ratio separates LINERs from star-
forming galaxies, while $\rm[OIII]/[OII]$ and $\rm[OIII]/H\beta$ separate
Seyferts from LINERs and star-forming galaxies. Using the classification
scheme of Yan et al. (2006) we divided our spectra in 3 main classes: LINERs,
with clearly detected [OII], but no ($3\sigma$) detection of [OIII] and $\rm
H\beta$ in emission after subtracting an E+A profile; Seyferts, with
$\rm[OIII]/H\beta\geq 3$; and star-forming galaxies, which are the objects
with clearly detected [OII] that are neither Seyferts nor LINERs after
subtracting an E+A profile. Typical spectra of low-ionization objects, star-
forming galaxies and Seyferts are shown in Figure 11.
Figure 11: Typical average spectra of low-ionization objects, star-forming
galaxies and Seyferts from the sample in the $<z>=0.415$ layer.
#### 5.3.1 The impact of LINERs in our previous results on red emission-line
galaxies
Because the spectral coverage in a rather large fraction of our objects at
$<z>=0.68$ and higher is truncated below $\rm 5000\AA$ in the rest frame, we
applied our classification scheme only to objects in the range $0.29\leq z\leq
0.65$. To extract $\rm H\beta$ in emission we built a series of E+A models,
combining an observed E spectrum with different fractions of an A stellar
template, ranging form $0.05\%$ to $80\%$ of the total luminosity. To
determine the best-fit model we minimized the continuum slope of the
difference between the spectrum and the E+A model. Thus, in the range
$0.35-0.55$ our sample contains 23% LINERs, 51% star-forming galaxies, 8%
Seyferts, and 14% uncertain types. The layer at $z=0.63$ has 18% LINERs, 50%
star-forming galaxies, 7% Seyferts, 13% uncertain types and 11% of truncated
spectra. Altogether, the fraction of LINERs among emission-line galaxies up to
$z=0.65$ in our pencil beam is $\rm\approx 22\%$. With an average
$\rm<D(4000)>=1.39\pm 0.18$. LINERs at $z\leq 0.65$ have a potentially
significant impact on the conclusions of Giraud et al. (2010) about the
evolution of red emission-line galaxies. To quantify this impact, we have
subtracted all LINERs from the sample of emission-line spectra in the $z=0.43$
bin, determined the new blue-to-red partition (as in Giraud et al. 2010;
section 5.1), and computed a new average spectrum for the red galaxies. This
(also cleaned of rare red Seyferts) is shown in Figure 12 where it is compared
with the mean red spectrum at $z=0.9$. We find that the differences in
continuum slope and D(4000) between $<z>=0.43$ and $<z>=0.9$ is reduced by a
factor of 2/3. The main difference between red galaxies with LINERs and those
without is the presence of young stellar population.
Figure 12: Average spectra of red emission-line galaxies after subtracting
early-type LINERs and galaxies with diluted star formation (and rare Seyferts)
from the sample in the $<z>=0.43$ bin and recalculating the median blue-to-red
partition, and of the red half of emission-line galaxies at $<z>=0.9$. The
spectra at $<z>=0.9$ were not classified because $\rm H\beta$ and [OIII] are
missing in most cases.
#### 5.3.2 Early-type LINERs
The fraction of nearby early-type galaxies hosting bona-fide (i.e. nuclear)
LINERs in the Palomar survey (Filippenko & Sargent, 1985; Ho et al. 1997a, )
was found to be $\sim 30\%$ (Ho et al. 1997b, ), but LINER-like emission line
ratios are also observed in extended regions (Phillips et al., 1986;
Goudfrooij et al., 1994; Zeilinger et al., 1996; Sarzi et al., 2006, and
references therein). A similar fraction of LINER-like ratios is found in the
SDSS at $0.05\leq z\leq 0.1$ in color-selected red galaxies (Yan et al.,
2006).
Because it is very difficult to disentangle early-type LINERs from spirals
with extended and diluted star formation by using only [OII] and $\rm H\beta$,
we make use of morphology to distinguish compact objects with low ellipticity
and profiles consistent with early type galaxies, from other morphologies:
apparent disks, high ellipticity, and irregular or distorted morphologies.
Images of early-type galaxies with low ionization spectra are shown in Figure
13.
Figure 13: Examples of early-type galaxies having low-ionization spectra, and
indicated redshifts.
Our visual early-type morphologies are the same as ZEST type T=1 (Scarlata et
al., 2007, Figure 4 (b), (c), (d)). In the $<z>=0.43$ bin we find that 92% of
the galaxies classified as star-forming objects have morphologies inconsistent
with early-types. At $<z>=0.43$ and in the $z=0.63$ layer, about half of the
LINERs have compact morphology while the other half are mainly bulge-dominated
disk galaxies, or “early disks” of ZEST type T=2.0 ((Bundy et al, 2009, Figure
4)). At $z=0.29$ all LINERs have disks. The spectra of galaxies with apparent
disks have an extended [OII] emission suggesting that they do have extended
star-formation. Average spectra of 11 early-type galaxies (E) and 10 later
types (hereafter S) resulting from our morphological classification are shown
in Figure 14.
Figure 14: Median spectra of 11 early-type galaxies and of 10 galaxies with
later type morphology (S) with low ionization at $z=0.4-0.5$.
The absence of $\rm H\beta$ in the S sample suggests that $\rm H\beta$ in
emission resulting from star formation is diluted in $\rm H\beta$ in
absorption from A and older stars. The closest spectral comparison in the
atlas of galaxy spectra (Kennicutt, 1992) is with an Sb galaxy. The rather
strong $\rm H\beta$ in absorption in early-types (E) combined with [OII]
suggests either a low fraction of young stars or a mechanism of photo-
ionization other than young stars as in (Fillipenko, 2003; Ho, 2004, and
references therein). In fact, the recent work by the SEAGAL collaboration (Cid
Fernandes et al., 2010, and references therein) has shown that the majority of
galaxies with LINER spectra in the SDSS can be explained as retired galaxies,
that is, galaxies that have stopped forming stars but still contain
appreciable amounts of gas that is being photoionized by intermediate-aged
post-AGB stars. In fact, the SEAGAL models with no young stars, but with
significant populations of 100Myr-1Gyr stars resemble remarkably well our
average LINER spectrum shown in Figure 11.
We calculated population synthesis models for our average spectra of LINERs
with early-type and late-type morphologies. The results, shown in Figure 15
and Table LABEL:earlylinerpop, indicate that both early-type and late-type
LINERs have significant populations of young and intermediate age stars, but
late-type (S) LINERs have much younger populations. In fact, the residuals of
the S-LINER fit show $\rm H\beta$ in emission stronger than [OIII], consistent
with the idea that they are red spirals with diluted star formation.
Table 10: Stellar population properties of an average of LINERs with early-type morphology and with morphology of later types Type | log Age | $\rm\chi^{2}$
---|---|---
| $<8$ | $8-8.7$ | $8.7-9$ | $9-9.4$ | $>9.4$ |
Early | 18.2 | 0 | 57.6 | 0 | 24.2 | 1.4
Later-type | 32.7 | 0 | 53.4 | 0 | 13.9 | 1.3
Figure 15: Spectral fitting with SSP models for an average spectrum of LINERs
with early-type morphology (left), and with morphology of later type (right).
Thus, our results are consistent with the interpretation that most early-type
LINERs at intermediate redshifts are in fact post-starburst galaxies, as
postulated by the SEAGAL collaboration for lower redshift objects. These
results indicate that LINERs and E+As depict the quenching phase in the
evolution of galaxies massive enough to retain significant amounts of gas
after the stellar-wind and supernova phases of the most massive stars.
### 5.4 The red limit of emission-line galaxies
At each $z$ we have selected galaxies with the reddest continuum (the reddest
quartile at each redshift bin) to construct the combined spectra of the red
envelope or red limit of emission line galaxies. Since we are working with
small numbers of galaxies, typically 5-10, it was necessary to combine the
samples at $z=0.82$ and $z=0.99$ to improve statistics. Nevertheless, because
our red emission-line galaxies are rather luminous, the combined spectra still
have high continuum S/N ratios (Table LABEL:redest). The common parts of the
red envelopes of spectra at $<z>=0.29$, $<z>=0.43$, $<z>=0.65$ are similar,
while the red limit at $<z>=0.9$ has noticeably stronger UV continuum. The
spectra in different bins are shown in Figure 7 of Giraud et al. (2010). Both
the continuum and the indexes of the red limit at $z\leq 0.65$ (i.e. $\rm
D(4000)\sim 1.35-1.45;EQW([OII])\sim 4-8$), are typical of nearby spirals with
prominent bulges and low star formation (Kennicutt, 1992; Kinney et al., 1996;
Balogh et al., 1999), or early-type LINERs. Up to $z\sim 0.7$ the populations
of red spirals and early-types can be well separated by their morphology. The
higher UV continuum and lower $\rm D(4000)$ of the limit spectrum at $<z>=0.9$
indicate that such red objects become rare at $z\geq 0.68$ in our sample.
Absorption systems have (by definition) already lost enough gas to suppress
any detected star formation by the time they first appear in our sample at
$z\simeq 1$. At $z=0.8-1$ emission-line galaxies in our sample are found to
have very strong star formation, which declines at lower $z$, the reddest
quartile being bluer than at lower $z$. Therefore the evolutionary paths of
bright absorption and emission systems might have been more separated at
$z\simeq 1$ than at lower redshift suggesting two different physical processes
of different time scales. In one we have early-type LINERs and E+A galaxies
that define the ”entrance gate” to the red sequence of passively evolving
galaxies. In the other we have red spirals with diluted star formation, that
are in a final phase of smooth star formation, possibly of a “main sequence”
(Noeske et al., 2007).
In Section 5.3 we found a large fraction of LINERs in layers at intermediate
$z$. More precisely, in the volume-limited range $0.35\leq z\leq 0.65$, we
find, gathering the counts of Section 5.3, that LINERs are $\rm 23\%$ of all
early-type galaxies with measured redshifts.
Table 11: Equivalent width of [OII], 4000 Å break amplitude, $\rm H\delta$ index, and the G-step of the red envelope of emission-line galaxies. The continuum S/N ratios are given in the last column. $<z>$ | EQW([OII]) | D(4000) | $\rm EQW(H\delta)$ | G step | $\rm S/N$
---|---|---|---|---|---
0.29 | $4.2\pm 0.3$ | $1.35\pm 0.07$ | $-1.8\pm 0.3$ | $1.224\pm 0.017$ | $\rm 19$
$0.43$ | $8.5\pm 0.2$ | $1.39\pm 0.06$ | $-2.5\pm 0.2$ | $1.268\pm 0.014$ | $\rm 24$
$0.65$ | $8.3\pm 0.2$ | $1.44\pm 0.06$ | $-3.0\pm 0.2$ | $1.273\pm 0.012$ | $\rm 29$
$0.9$ | $9.0\pm 0.3$ | $1.30\pm 0.07$ | $-3.9\pm 0.2$ | - | $\rm 18$
## 6 Summary and Conclusions
We have presented a catalogue of galaxy spectra in a pencil beam survey of
$\sim 10.75^{\prime}\times 7.5^{\prime}$, and used these data to make an
analysis of the spectral energy distribution of a magnitude limited sample up
to $z\sim 1$, concentrating on absorption and low ionization emission-line
systems. The redshift range has been divided in bins centered on the
structures that were detected in the (RA, Dec, $z$) pseudo-volume, and
corresponding to cosmic time slices of $\rm\sim 1Gyr$. Our sample is
reasonably complete for galaxies brighter than $\rm M_{R}=-18.8$ up to
$z\approx 0.5$; at $z\geq 0.75$ the cutoff is at -20.5.
From this analysis we reach the following conclusions:
1. 1.
We confirm in our pencil-beam sample the well known result Hamilton (1985)
that absorption-line galaxies do not show significant variations in their
continuum energy distributions up to $z=0.6$, and a moderate decrease of the
4000 Å break amplitude of 5% at $z\sim 0.65$, 7% at $z\sim 0.82$, and up to
15% at $z\sim 1$. Using stellar population synthesis models we find that
absorption-line galaxies at $z\geq 0.65$ show more than 50% of stars younger
than 2.5Gyrs, while those at $z\geq 0.8$ had star formation as recently as
1Gyr ago. This suggests that the red sequence is still in a buildup phase at
$z\leq 1$.
The faint absorption-line galaxies in our dynamically young cluster at
$z=0.29$ have indexes similar to those of bright absorption-line systems at
$z=0.8$, suggesting that faint galaxies without emission lines tend to be
younger than more massive galaxies with similar spectra. Our population
synthesis models indicate that about 50% of the stars contributing to the
luminosity of faint absorption-line galaxies at z = 0.29 were formed at $z<1$.
This is consistent with cases of truncated red sequences observed in some
high-$z$ clusters and suggests that clusters with truncated red-sequences may
be dynamically young.
2. 2.
Combining simple emission-line diagnostics with galaxy morphology we identify
a significant population of early-type LINERs at $0.35\leq z\leq 0.65$. In
that redshift range early-type LINERs constitute about 23% of all early-types
galaxies, a much larger fraction than E+A post-starburst galaxies. However,
our population synthesis models show that early-type LINERs contain
substantial populations of intermediate age stars that can easily explain the
observed line emission, as recently proposed by Cid-Fernandes et al. (2010).
This led us to conclude that most LINERs in our sample are in fact post-
starburst galaxies.
3. 3.
The red limit in the spectral energy distribution of emission-line galaxies at
$z\leq 0.65$ is typical of bulge-dominated spirals with moderate star
formation, and of early-type LINERs. Thus, early-type LINERs and E+As define
the “entrance gate” of the red sequence of passively evolving galaxies, while
bulge-dominated spirals have diluted star formation.
###### Acknowledgements.
EG thanks the hospitality of ESO and Universidad Catolica in Santiago during
the initial phase of this work. JME thanks the hospitality of Nanjing
University during the initial phase of this research. QGU would like to
acknowledge the financial support from the China Scholarship Council (CSC),
the National Natural Science Foundation of China under grants 10878010,
10221001, and 10633040, and the National Basic Research Program (973 program
No. 2007CB815405). HQU thanks partial support from FONDAP “Centro de
Astrofísica”. PZE acknowledge a studentship from CONICYT. We thank S. di
Serego Alighieri for reading a preliminary version of the manuscript and for
his suggestions, and R. Cid-Fernandes for fruitful discussions.
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|
# Image Inpainting Models are Effective Tools
for Instruction-guided Image Editing
Solution for GenAI Media Generation Challenge Workshop @ CVPR
Xuan Ju1,2, Junhao Zhuang1, Zhaoyang Zhang1∗, Yuxuan Bian1,2, Qiang Xu2, Ying
Shan1
1ARC Lab, Tencent PCG, 2The Chinese University of Hong Kong ∗Project Lead
https://github.com/TencentARC/BrushNet/tree/main/InstructionGuidedEditing
###### Abstract
This is the technique report for the winning solution of the CVPR2024 GenAI
Media Generation Challenge Workshop’s Instruction-guided Image Editing track.
Instruction-guided image editing has been largely studied in recent years. The
most advanced methods, such as SmartEdit and MGIE, usually combine large
language models with diffusion models through joint training, where the former
provides text understanding ability, and the latter provides image generation
ability. However, in our experiments, we find that simply connecting large
language models and image generation models through intermediary guidance such
as masks instead of joint fine-tuning leads to a better editing performance
and success rate. We use a 4-step process IIIE (Inpainting-based Instruction-
guided Image Editing): editing category classification, main editing object
identification, editing mask acquisition, and image inpainting. Results show
that through proper combinations of language models and image inpainting
models, our pipeline can reach a high success rate with satisfying visual
quality.
## 1 Introduction
With the rapid development of diffusion models, the field of text-guided image
generation [7, 19, 13, 15] has seen unprecedented progress in creating images
with superior quality [18], diversity [3], and adherence to text guidance
[14]. However, in image editing tasks, which provide a source image and an
editing instruction as input and expect a target image as output, we do not
observe such success. This implies that the language understanding ability and
image generation ability are not fully explored in editing tasks.
In the aim of applying image generation capabilities to image editing,
previous methods have attempted two strategies: (1) collecting paired “source
image-instruction-target image” data and fine-tuning diffusion models for
editing tasks (e.g., InstructPix2Pix [1] and InstructDiffusion [5]), and (2)
jointly fine-tuning Large Language Models (LLMs) and diffusion models to endow
the diffusion models with a stronger understanding of text (e.g., SmartEdit
[8] and MGIE [4]). For the former strategy, due to the difficulty of
collecting paired manually edited data, the training data are usually
generated by LLMs and inference-based image editing methods (e.g., Prompt-to-
Prompt [6] and Masactrl [2]). Due to the low success rate and unstable editing
quality of these inference-based image editing methods [11], the collected
dataset is usually noisy and unreliable, which lead to an unsatisfying
performance of the trained image editing model. For the latter strategy, a
joint training of the LLMs and diffusion model usually make it hard to fully
use the text understanding capability of LLMs. Although SmartEdit and MGIE
have achieved better performance than previous solutions, we find that they
still do not fully optimize the potential of LLMs and diffusion.
This technique report provides a different solution, connecting large language
models and image generation models simply through intermediary guidance (e.g.,
edit objects and masks), and is the winning solution for The GenAI Media
Generation Challenge (MAGIC). By detracting two models apart in an agent
architecture, we find it easier to fully leverage the capabilities of both.
Specifically, we use a 4-step process IIIE (Inpainting-based Instruction-
guided Image Editing): (1) editing category classification, (2) main editing
object identification, (3) editing mask acquisition, and (4) image inpainting.
Step (1)-(3) use LLMs and detection model to determine the editing type, edit
object, edit masks, and target prompt. Then, step (4) perform image editing in
the way of image inpainting, which fully use generation ability. In this way,
step (1)-(3) use LLMs to extract information in instruction and summarize them
to intermediary guidance that can be used by diffusion models.
Visual results show that the proposed IIIE substantially surpass previous
instruction-guided image editing methods and other solutions in MAGIC
considering visual quality and instruction faithfulness.
## 2 MAGIC
The MAGIC hosts two challenge tracks: (1) text-to-image generation and (2)
text-guided image editing. This technique report presents the solution for the
second track, text-guided image editing. We list the instructions here:
Guidelines For text-guided image editing, we test the capacity of the model to
change a given image’s contents based on some text instructions. The specific
type of instructions that we test for are the following: • Addition: Adding
new objects within the images. • Remove: Removing objects • Local: Replace
local parts of an object and later the object’s attributes, i.e., make it
smile • Global: Edit the entire image, i.e., let’s see it in winter • Inpaint:
Alter an object’s visual appearance without affecting its structure •
Background: Change the scene’s background
Evaluation Protocol We leverage both human and automated evaluations. In
human-based evaluations, we use human annotators. We mainly evaluate the
following aspects: • Edit Faithfulness - whether the edited image follows the
editing instruction • Content Preservation - whether the edited image
preserves the regions of the original image that should not be changed •
Overall Instruction Following - considering both edit faithfulness and content
preservation, whether the edited image is artifact-free, keeping the core
visual features of the original image, etc On automatic evaluation, similar to
the text to image track, we will leverage existing methods that have developed
automatic metrics to help in assessing the outputs of the image based on the
prompt and instruction. To determine winners, we use automatic evaluation to
help prune the total number of entries to 10 finalists. At 10, we would use
human annotation and evaluation to determine the final winners.
More information about the detailed workshop information can be found at the
official website page: https://gamgc.github.io/.
Figure 1: The pipeline of our proposed 4-step image editing process ITIE.
## 3 Method
Previous instruction-guided image editing methods include two categories: (1)
diffusion model finetuning [1, 5], and (2) jointly LLMs and diffusion model
[8, 4]. These methods either not include LLMs in the model, or use joint fine-
tuning of LLMs and diffusion models. Both of these strategies can not fully
unleash the capabilities of LLMs, leading to a weak understanding of
instructions. Consequently, these methods show low success rates and
unsatisfying results. Contrary to these methods, we find that a simple tool-
based combination of LLMs and diffusion models can lead to a much better
visual results, coming from a full utilization of the language understanding
ability of LLMs.
In this competition, we propose a 4-step process for instruction-based image
editing, IIIE (Inpainting-based Instruction-guided Image Editing), as shown in
Fig. 1. Firstly, we use GPT4-o to categorize the current editing instructions
into one of the editing categories: Local Edit, Background Edit, Global Edit,
Addition, and Remove. Local Edit includes local changes such as replacing
object (e.g., change a cat to a dog) and change the attribute of an object
(e.g., change color or texture). For example, Fig. LABEL:fig:teaser row 1 show
two examples of Local Edit. Background Edit means changing the background and
remain the main object unchanged. For example, Fig. LABEL:fig:teaser row 2
column 1 and row 4 column 2 show two examples of Background Edit. Global Edit
transfer the overall style of an image. For example, Fig. LABEL:fig:teaser row
4 column 1 show one example of Global Edit. Addition and Remove separately add
and remove object from an image. For example, Fig. LABEL:fig:teaser row 3 show
two examples of adding object, and row 2 column 2 shows one example of
removing object. We classify editing category to these 4 categories since they
can cover most editing instruction types, and requires different operations in
editing.
Then, in step 2, we find the main editing object by making further
conversations with GPT4-o based on the editing category. For example, editing
instruction “Make the horse into a unicorn” has a main object of “horse”. For
editing categories of global edit and addition that do not contain an editing
object, we leave the editing object blank. After that, we use Grounded-SAM
[17] combined with GPT4-o to obtain the editing mask in step 3. Specifically,
we generate image background as mask for Background Edit. For Local Edit and
Removing, we use the main object of step 2 as the input of Grounded-SAM to
locate the editing mask. For addition editing category, we use the visual
ability of GPT4-o and get a possible location of adding the object. For Global
editing, the mask is default as the whole image.
Finally, in step 4, we use image generation model to perform image editing
based on a target prompt generated by GPT4-o. For Background Edit, Local Edit,
Remove, and Addition, we use image inpainting model BrushNet [10] combined
with PowerPaint [21] to inpaint the masked region based on the target prompt.
For Global Edit, we use InfEdit [20] to change the global style. We find that
a direct utilization of the diffusion model by giving conditions of mask and
text prompt can fully leverage the capabilities of these models, thus leads to
high-quality generation results.
## 4 Experiment Results
Method | Rank | Edit | Content | Overall Instruction
---|---|---|---|---
Faithfulness | Preservation | Following
IIIE | 1 | 0.51 | 0.78 | 0.80
LEdits++ | 2 | 0.46 | 0.64 | 0.74
Tasvir | 3 | 0.40 | 0.49 | 0.62
Table 1: Comparison of IIIE and the other two winning solutions. The score is
calculated with an average of 1.2k images in MAGIC benchmark. A bigger score
means a better user preference. Bold stands for the best results.
Visualization comparison in Fig. LABEL:fig:teaser show a higher success rate
and visual quality of IIIE compared to previous methods. In the MAGIC
workshop, a benchmark of 1.2k images is used for evaluation and 3 winning
solutions are announced. The benchmark measure the quality of different
editing methods using user study on instruct following, edit fidelity, and
content preservation. 3 raters are involved in each job and the majority vote
is taken as the final results. Each image is rated with 1 or 0 for each
metric. Results of the three winning solution of this competition is shown in
Tab. 1. IIIE show better score on all three metrics compared to the other two
winning solutions. We have made our code and edited results publicly available
in the hope of our findings can offer some insights for relevant field [12, 9,
16].
In conclusion, in this technique report presents the winning solution for the
CVPR2024 GenAI Media Generation Challenge’s Instruction-guided Image Editing
track. We show that without cumbersome fine-tuning or training, a simple
combination of LLMs and text-to-image diffusion model can lead to a image
editing results with better performance and higher success rates than previous
methods.
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|
# Catenary and Mercator projection
Mikhail A. Akhukov
Huawei Nizhny Novgorod Research Center, Nizhny Novgorod, Russia
<EMAIL_ADDRESS>
& Vasiliy A. Es’kin
Department of Radiophysics, University of Nizhny Novgorod, Nizhny Novgorod,
Russia, 603950
and
Huawei Nizhny Novgorod Research Center, Nizhny Novgorod, Russia
<EMAIL_ADDRESS>
& Mikhail E. Smorkalov
Skolkovo Institute of Science and Technology, Moscow, Russia
and
Huawei Nizhny Novgorod Research Center, Nizhny Novgorod, Russia
<EMAIL_ADDRESS>Corresponding author: Mikhail Aleksandrovich Akhukov
<EMAIL_ADDRESS>
###### Abstract
The Mercator projection is sometimes confused with another mapping technique,
specifically the central cylindrical projection, which projects the Earth’s
surface onto a cylinder tangent to the equator, as if a light source is at the
Earth’s center. Accidentally, this misconception is rather close to a truth.
The only operation that the map needs is a free bending in a uniform
gravitational field if the map’s material is dense and soft enough to produce
a catenary profile. The north and south edges of the map should be parallel
and placed in the same plane at the appropriate distance. In this case, the
bent map been projected onto this plane gives the Mercator projection. This
property is rather curious, since it allows to make such a sophisticated one-
to-one mapping as the Mercator projection using simple tools available in the
workroom.
_Keywords_ Mercator projection, Central cylindrical projection, Gudermannian
function, Catenary
## 1 Introduction
A catenary is a curve that appears in the profile of an idealized freely
hanging heavy chain under its own weight, supported only at its ends in a
uniform gravitational field. The exact shape of the curve is described by the
hyperbolic cosine function.
The Mercator projection is a cylindrical map projection. It is a conformal
projection, which means that it preserves the angles. The formal equation for
the Mercator projection can be expressed in a number of ways. Within them,
there is also a variant that includes a hyperbolic cosine function (Osborne,
2016). This fact will be discussed in more details further.
The map in Mercator projection was published in 1569 by the Flemish
philosopher and mathematician Gerhardus Mercator. The map was originally
designed for navigation, as it maps loxodromes into the straight lines. The
loxodrome or rhumb line is a line on the sphere of a ship’s course in the
constant compass direction. This allows sailors to plot a straight course
between two points by following a constant compass bearing. The exact
calculation of Mercator projection requires logarithms and modern calculus
that had not yet been invented at that time.
Figure 1: To calculate the Mercator projection of a given point $N$ with
latitude $\varphi$, we first project the point $N$ to the point $B$ on the
vertical line $AB$, then move it vertically downward to the point $Q$. The
exact vertical coordinate of the point $Q$ is $\psi$, which is expressed by
the equation (1).
## 2 Mercator projection
The Mercator projection is derived from a differential equation, which is
based on the conception of equal scales and requires the integration of the
secant function. Having a latitude $\varphi$ (see figure 1), in case of a unit
sphere, the formal equation of the vertical coordinate of the Mercator map
projection is as follow:
$\psi:=\int\limits_{0}^{\varphi}\frac{dt}{\cos{t}}=\ln\left(\tan\left(\frac{\varphi}{2}+\frac{\pi}{4}\right)\right).$
(1)
This equation can be expressed in a number of ways using the Gudermannian
function ((Weisstein, 1999) page 778, (Yanpolskiy, 1960) page 47), which is
popular in calculus, because it relates circular ($\varphi$) and hyperbolic
($\psi$) angle measures as $\varphi=gd(\psi)$, and allows to establish a
number of relations between trigonometric and hyperbolic function directly
without the imaginary unit.
The theory of the Gudermannian function introduces a concept of hyperbolic
amplitude ((Yanpolskiy, 1960) page 47), that is a special angle constructed
for an arbitrary point belonging to a unit hyperbola. It is also known as the
gudermannian of the hyperbolic measure $\psi$ because it is expressed in terms
of the Gudermannian function: $\varphi=\operatorname{gd}(\psi)$.
Figure 2: The hyperbolic amplitude $\varphi$ is constructed for an arbitrary
point $M$ belonging to the unit hyperbola (blue curve), as shown here. Also
$\varphi$ is the latitude of the point $N$ figured out in the Mercator
projection (see figure 1 for comparison).
The theory of the Gudermannian function allows to establish a number of
relations (see figure 2):
$L:=|AB|=\tan(\varphi)=\sinh(\psi)=|MP|,$ (2)
$|OB|=\frac{1}{\cos(\varphi)}=\cosh(\psi)=|OP|.$ (3)
Using elementary trigonometry in case of $0\leq\varphi<\pi/2$ we can write:
$\tan\left(\frac{\varphi+\pi/2}{2}\right)=\frac{1+\sin{\varphi}}{\cos{\varphi}}=\frac{1}{\cos{\varphi}}+\sqrt{\frac{1}{\cos^{2}{\varphi}}-1}.$
(4)
Equation (4) allows to rewrite the right-hand side of equation (1) in the next
form:
$\psi=\ln\left(\tan\left(\frac{\varphi}{2}+\frac{\pi}{4}\right)\right)=\operatorname{arcosh}\left(\frac{1}{\cos(\varphi)}\right).$
(5)
See also equation (3) for comparison. For more details on the integration of
the secant function and its relation to the history of the Mercator projection
see Vis (2018).
Equation (5), via area hyperbolic cosine, implicitly introduces a catenary
oriented in the horizontal direction, when vertexes of the unit hyperbola and
area hyperbolic cosine touch each other, see figure 3.
Figure 3: After introduction of the hyperbolic amplitude $\varphi$ and
disclose its relation to the hyperbolic measure $\psi$ via the Gudermannian
function, the length of the catenary (solid magenta curve $AC$) introduced
implicitly via the area hyperbolic cosine as described by equation (5) is
equal to the length $L$ (solid green line $AB$) that is vertical component of
the central cylindrical map projection.
A key point in this work is next. We need to calculate the length of the area
hyperbolic cosine introduced in such a way as in the equation (5), as a
function of the hyperbolic measure $\psi$, which is the vertical component of
the Mercator map projection:
$L_{catenary}(\psi)=\\\
\int\limits_{0}^{\psi}\sqrt{1+\left(\frac{d\cosh(t)}{dt}\right)^{2}}dt=\\\
\int\limits_{0}^{\psi}\sqrt{1+\sinh^{2}(t)}dt=\\\
\int\limits_{0}^{\psi}\cosh(t)dt=\sinh(\psi).$ (6)
Using equation (2) we see that $L_{catenary}(\psi)=\tan(\varphi)=L$, where $L$
— is vertical component of the central cylindrical map projection. The
equality of two these lengths, the length of line segment $AB$ and the length
of catenary segment $AC$ allows to construct the Mercator map projection from
the central cylindrical map projection by bending the former on the catenary
profile and then project the bent map onto the plane. See figure 3 111All
figures are generated by the scripts in Python programming language (Van
Rossum and Drake Jr, 1995) using the matplotlib library (Hunter, 2007).
### 2.1 Bending parameter
Finally, we calculate the exact parameter of the bending for the map in the
central cylindrical projection to produce the map in the Mercator projection.
If the map in the central cylindrical projection has the plots of a number of
loxodromes, then the straightening of the loxodromes has to be observed from
high enough point of view, that will be an unavoidable sign of the Mercator
map projection.
Suppose, we have a map in the central cylindrical projection constructed
between $\alpha$ degrees of north and south latitudes and physical map’s
height is $H$. The only parameter we need is a distance $D$ between the north
and south edges of the map been placed parallel and in the same plane. Using
equation (2) in case of the unit sphere the distance $D_{unit~{}sphere}$ is
exactly $2\psi$: $D_{unit~{}sphere}=2\operatorname{arsinh}(\tan(\alpha))$. In
case of sphere with arbitrary radius $R$:
$R=\frac{H}{2\tan{\alpha}},$ (7)
we re-scale $D_{unit~{}sphere}$ by $R$, that gives the value of distance $D$:
$D=\frac{H}{\tan{\alpha}}\operatorname{arsinh}\left(\tan{\alpha}\right).$ (8)
## 3 Conclusion
We show that the map in the Mercator projection can be constructed from the
map in the central cylindrical projection as if some one has such map been
printed on dense and soft enough material allows it to hang freely to make the
catenary profile and then project the bent map onto the plane. Interestingly
speaking, if such map has a number of loxodromes, then a person standing high
enough can see that the loxodromes become straight lines when original map is
bent when, for example, the map is carried by two other persons if they hold
the map at the north and south edges.
This hypothetical situation could happen in real life in some kind of workroom
and serve as a basis for further study of this property — the straightening of
loxodromes printed on the map in the central cylindrical projection when this
map hangs freely as described above.
## References
* Osborne [2016] Peter Osborne. _The Mercator Projections_. Zenodo, February 2016. doi:10.5281/zenodo.35392. URL https://doi.org/10.5281/zenodo.35392.
* Weisstein [1999] Eric W Weisstein. _CRC concise encyclopedia of mathematics_. CRC press, 1999.
* Yanpolskiy [1960] Avraam Ruvimovich Yanpolskiy. _Giperbolicheskie funkcii_. Ripol Classic, 1960.
* Vis [2018] MJP Vis. History of the mercator projection. B.S. thesis, Utrecht University, The Netherlands, 2018.
* Van Rossum and Drake Jr [1995] Guido Van Rossum and Fred L Drake Jr. _Python tutorial_ , volume 620. Centrum voor Wiskunde en Informatica Amsterdam, The Netherlands, 1995\.
* Hunter [2007] J. D. Hunter. Matplotlib: A 2d graphics environment. _Computing in Science & Engineering_, 9(3):90–95, 2007.
|
# Approximation schemes for bounded distance problems on fractionally
treewidth-fragile graphs
Zdeněk Dvořák Supported by the ERC-CZ project LL2005 (Algorithms and
complexity within and beyond bounded expansion) of the Ministry of Education
of Czech Republic. email<EMAIL_ADDRESS>Charles University, Prague,
Czech Republic Abhiruk Lahiri Supported by ISF grant 822/18 and Ariel
University Post-doctoral fellowship. email<EMAIL_ADDRESS>Ariel
University
###### Abstract
We give polynomial-time approximation schemes for monotone maximization
problems expressible in terms of distances (up to a fixed upper bound) and
efficiently solvable in graphs of bounded treewidth. These schemes apply in
all fractionally treewidth-fragile graph classes, a property which is true for
many natural graph classes with sublinear separators. We also provide
quasipolynomial-time approximation schemes for these problems in all classes
with sublinear separators.
## 1 Introduction
In this paper, we consider optimization problems such as:
* •
Maximum $r$-Independent Set, $r\in\mathbb{Z}^{+}$: Given a graph $G$, the
objective is to find a largest subset $X\subseteq V(G)$ such that distance in
$G$ between any two vertices in $X$ is at least $r$.
* •
Maximum weight induced forest: Given a graph $G$ and an assignment
$w:V(G)\to\mathbb{Z}_{0}^{+}$ of non-negative weights to vertices, the
objective is to find a subset $X\subseteq V(G)$ such that $G[X]$ does not
contain a cycle and subject to that, $w(X)\coloneqq\sum_{v\in X}w(v)$ is
maximized.
* •
Maximum $(F,r)$-Matching, for a fixed connected graph $F$ and
$r\in\mathbb{Z}^{+}$: Given a graph $G$, the objective is to find a largest
subset $X\subseteq V(G)$ such that $G[X]$ can be partitioned into vertex-
disjoint copies of $F$ such that distance in $G$ between any two vertices
belonging to different copies is at least $r$.
To be precise, to fall into the scope of our work, the problem must satisfy
the following conditions:
* •
It must be a maximization problem on certain subsets of vertices of an input
graph, possibly with non-negative weights. That is, the problem specifies
which subsets of vertices of the input graph are _admissible_ , and the goal
is to find an admissible subset of largest size or weight.
* •
The problem must be defined in terms of distances between the vertices, up to
some fixed bound. That is, there exists a parameter $r\in\mathbb{Z}^{+}$ such
that for any graphs $G$ and $G^{\prime}$, sets $X\subseteq V(G)$ and
$X^{\prime}\subseteq V(G^{\prime})$, and a bijection $f:X\to X^{\prime}$, if
$\min(r,d_{G}(u,v))=\min(r,d_{G^{\prime}}(f(u),f(v)))$ holds for all $u,v\in
X$, then $X$ is admissible in $G$ if and only if $X^{\prime}$ is admissible in
$G^{\prime}$.
* •
The problem must be monotone (i.e., all subsets of an admissible set must be
admissible), or at least near-monotone (as happens for example for Maximum
$(F,r)$-Matching) in the following sense: There exists a parameter
$c\in\mathbb{Z}^{+}$ such that for any admissible set $A$ in a graph $G$,
there exists a system $\\{R_{v}\subseteq A:v\in A\\}$ of subsets of $A$ such
that every vertex belongs to $R_{v}$ for at most $c$ vertices $v\in A$, $v\in
R_{v}$ for each $v\in A$, and for any $Z\subseteq A$, the subset
$X\setminus\bigcup_{v\in Z}R_{v}$ is admissible in $G$.
* •
The problem must be tractable in graphs of bounded treewidth, that is, there
must exist a function $g$ and a polynomial $p$ such that given any graph $G$,
its tree decomposition of width $t$, an assignment $w$ of non-negative weights
to the vertices of $G$, and a set $X_{0}\subseteq X$, it is possible to find a
maximum-weight admissible subset of $X_{0}$ in time $g(t)p(|V(G)|)$.
Let us call such problems _$(\leq\\!r)$ -distance determined $c$-near-monotone
$(g,p)$-tw-tractable_. Note that a convenient way to verify these assumptions
is to show that the problem is expressible in _solution-restricted Monadic
Second-Order Logic_ ($\operatorname{\mathsf{MSOL}}$) _with bounded-distance
predicates_ , i.e., by a $\operatorname{\mathsf{MSOL}}$ formula with one free
variable $X$ such that the quantification is restricted to subsets and
elements of $X$, and using binary predicates $d_{1}$, …, $d_{r}$, where
$d_{i}(u,v)$ is interpreted as testing whether the distance between $u$ and
$v$ in the whole graph is at most $i$. This ensures that the problem is
$(\leq\\!r)$-distance determined, and $(g,O(n))$-tw-tractable for some
function $g$ by Courcelle’s metaalgorithmic result [5].
Of course, the problems satisfying the assumptions outlined above are
typically hard to solve optimally, even in rather restrictive circumstances.
For example, Maximum Independent Set is $\operatorname{\mathsf{NP}}$-hard even
in planar graphs of maximum degree at most $3$ and arbitrarily large (fixed)
girth [1]. Moreover, it is hard to approximate it within factor of $0.995$ in
graphs of maximum degree at most three [4]. Hence, to obtain polynomial-time
approximation schemes ($\operatorname{\mathsf{PTAS}}$), i.e., polynomial-time
algorithms for approximating within any fixed precision, further restrictions
on the considered graphs are needed.
A natural restriction that has been considered in this context is the
requirement that the graphs have sublinear separators (a set $S$ of vertices
of a graph $G$ is a _balanced separator_ if every component of $G\setminus S$
has at most $|V(G)|/2$ vertices, and a hereditary class $\mathcal{G}$ of
graphs has _sublinear separators_ if for some $c<1$, every graph
$G\in\mathcal{G}$ has a balanced separator of size $O(|V(G)|^{c})$). This
restriction still lets us speak about many interesting graph classes (planar
graphs [18] and more generally proper minor-closed classes [2], many geometric
graph classes [20], …). Moreover, the problems discussed above admit
$\operatorname{\mathsf{PTAS}}$ in all classes with sublinear separators or at
least in substantial subclasses of these graphs:
* •
Maximum Independent Set has been shown to admit $\operatorname{\mathsf{PTAS}}$
in graphs with sublinear separators already in the foundational paper of
Lipton and Tarjan [19].
* •
For any positive integer, Maximum $r$-Independent Set and several other
problems are known to admit $\operatorname{\mathsf{PTAS}}$ in graphs with
sublinear separators by a straightforward local search algorithm [16].
* •
All of the problems mentioned above (an more) are known to admit
$\operatorname{\mathsf{PTAS}}$ in planar graphs by a layering argument of
Baker [3]; this approach can be extended to some related graph classes,
including all proper minor-closed classes [6, 12].
* •
The problems also admit $\operatorname{\mathsf{PTAS}}$ in graph classes that
admit thin systems of overlays [11], a technical property satisfied by all
proper minor-closed classes and by all hereditary classes with sublinear
separators and bounded maximum degree.
* •
Bidimensionality arguments [7] apply to a wide range of problems in proper
minor-closed graph classes.
However, each of the outlined approaches has drawbacks. On one side, the local
search approach only applies to specific problems and does not work at all in
the weighted setting. On the other side of the spectrum, Baker’s approach is
quite general as far as the problems go, but there are many hereditary graph
classes with sublinear separators to which it does not seem to apply. The
approach through thin systems of overlays tries to balance these concerns, but
it is rather technical and establishing this property is difficult.
Another option that has been explored is via _fractional treewidth-fragility_.
For a function $f\colon\mathbb{Z}^{+}\times\mathbb{Z}^{+}\to\mathbb{Z}^{+}$
and a polynomial $p$, a class of graphs $\mathcal{G}$ is _$p$ -efficiently
fractionally treewidth-$f$-fragile_ if there exists an algorithm that for
every $k\in\mathbb{Z}^{+}$ and a graph $G\in\mathcal{G}$ returns in time
$p(|V(G)|)$ a collection of subsets $X_{1},X_{2},\dots X_{m}\subseteq V(G)$
such that each vertex of $G$ belongs to at most $m/k$ of the subsets, and
moreover, for $i=1,\ldots,m$, the algorithm also returns a tree decomposition
of $G\setminus X_{i}$ of width at most $f(k,|V(G)|)$. We say a class is _$p$
-efficiently fractionally treewidth-fragile_ if $f$ does not depend on its
second argument (the number of vertices of $G$). This property turns out to
hold for basically all known natural graph classes with sublinear separators.
In particular, a hereditary class $\mathcal{G}$ of graphs is efficiently
fractionally treewidth-fragile if
* •
$\mathcal{G}$ has sublinear separator and bounded maximum degree [9],
* •
$\mathcal{G}$ is proper minor-closed [8, 12], or
* •
$\mathcal{G}$ consists of intersection graphs of convex objects with bounded
aspect ratio in a finite-dimensional Euclidean space and the graphs have
bounded clique number, as can be seen by a modification of the argument of
Erlebach et al. [15]. This includes all graph classes with polynomial growth
[17].
In fact, Dvořák conjectured that every hereditary class with sublinear
separators is fractionally treewidth-fragile, and gave the following result
towards this conjecture.
###### Theorem 1 (Dvořák [10]).
There exists a polynomial $p$ so that the following claim holds. For every
hereditary class $\mathcal{G}$ of graphs with sublinear separators, there
exists a polynomial $q$ such that $\mathcal{G}$ is $p$-efficiently
fractionally treewidth-$f$-fragile for the function $f(k,n)=q(k\log n)$.
Moreover, Dvořák [9] observed that weighted Maximum Independent Set admits a
$\operatorname{\mathsf{PTAS}}$ in any efficiently fractionally treewidth-
fragile class of graphs. Indeed, the algorithm is quite simple, based on the
observation that for the sets $X_{1}$, …, $X_{m}$ from the definition of
fractional treewidth-fragility, at least one of the graphs $G\setminus X_{1}$,
…, $G\setminus X_{m}$ (of bounded treewidth) contains an independent set whose
weight is within the factor of $1-1/k$ from the optimal solution. A problem
with this approach is that it does not generalize to more general problems;
even for the Maximum $2$-Independent Set problem, the approach fails, since a
$2$-independent set in $G\setminus X_{i}$ is not necessarily $2$-independent
in $G$. Indeed, this observation served as one of the motivations behind more
restrictive (and more technical) concepts employed in [11, 12].
As our main result, we show that this intuition is in fact false: There is a
simple way how to extend the approach outlined in the previous paragraph to
all bounded distance determined near-monotone tw-tractable problems.
###### Theorem 2.
For every class $\mathcal{G}$ of graphs with bounded expansion, there exists a
function $h:\mathbb{Z}^{+}\times\mathbb{Z}^{+}\to\mathbb{Z}^{+}$ such that the
following claim holds. Let $c$ and $r$ be positive integers,
$g:\mathbb{Z}^{+}\to\mathbb{Z}^{+}$ and
$f:\mathbb{Z}^{+}\times\mathbb{Z}^{+}\to\mathbb{Z}^{+}$ functions and $p$ and
$q$ polynomials. If $\mathcal{G}$ is $q$-efficiently fractionally
treewidth-$f$-fragile, then for every $(\leq\\!r)$-distance determined
$c$-near-monotone $(g,p)$-tw-tractable problem, there exists an algorithm that
given a graph $G\in\mathcal{G}$, an assignment of non-negative weights to
vertices, and a positive integer $k$, returns in time
$h(r,c)|V(G)|+q(|V(G)|)\cdot p(|V(G)|)\cdot g(f(h(r,c)k,|V(G)|))$ an
admissible subset of $V(G)$ whose weight is within the factor of $1-1/k$ from
the optimal one.
Note that the assumption that $\mathcal{G}$ has bounded expansion is of little
consequence—it is true for any hereditary class with sublinear separators [14]
as well as for any fractionally treewidth-fragile class [9]; see Section 2 for
more details. The time complexity of the algorithm from Theorem 2 is
polynomial if $f$ does not depend on its second argument, and quasipolynomial
(exponential in a polylogaritmic function) if $f$ is logarithmic in the second
argument and $g$ is single-exponential (i.e., if $\log\log g(n)=O(\log n)$).
Hence, we obtain the following corollaries.
###### Corollary 3.
Let $c$ and $r$ be positive integers, $g:\mathbb{Z}^{+}\to\mathbb{Z}^{+}$ a
function and $p$ a polynomial. Every $(\leq\\!r)$-distance determined
$c$-near-monotone $(g,p)$-tw-tractable problem admits a
$\operatorname{\mathsf{PTAS}}$ in any efficiently fractionally treewidth-
fragile class of graphs.
We say a problem admits a quasipolynomial-time approximation schemes
($\mathsf{QPTAS}$) if there exist quasipolynomial-time algorithms for
approximating the problem within any fixed precision. Combining Theorems 1 and
2, we obtain the following result.
###### Corollary 4.
Let $c$ and $r$ be positive integers, $g:\mathbb{Z}^{+}\to\mathbb{Z}^{+}$ a
single-exponential function, and $p$ a polynomial. Every $(\leq\\!r)$-distance
determined $c$-near-monotone $(g,p)$-tw-tractable problem admits a
$\mathsf{QPTAS}$ in any hereditary class of graphs with sublinear separators.
The idea of the algorithm from Theorem 2 is quite simple: We consider the sets
$X_{1},\ldots,X_{m}$ from the definition of fractional
treewidth-$f$-fragility, extend them to suitable supersets $Y_{1}$, …,
$Y_{m}$, and argue that for $i=1,\ldots,m$, any admissible set in $G\setminus
X_{i}$ disjoint from $Y_{i}$ is also admissible in $G$, and that for some $i$,
the weight of the heaviest admissible set in $G\setminus X_{i}$ disjoint from
$Y_{i}$ is within the factor of $1-1/k$ from the optimal one. The construction
of the sets $Y_{1}$, …, $Y_{m}$ is based on the existence of orientations with
bounded outdegrees that represent all short paths, a result of independent
interest that we present in Section 2.
Let us remark one can develop the idea of this paper in further directions.
Dvořák proved in [13](via a substantially more involved argument) that every
monotone maximization problem expressible in first-order logic admits a
$\operatorname{\mathsf{PTAS}}$ in any efficiently fractionally treewidth-
fragile class of graphs. Note that this class of problems is incomparable with
the one considered in this paper (e.g., Maximum Induced Forest is not
expressible in the first-order logic, while Maximum Independent Set consisting
of vertices belonging to triangles is expressible in the first-order logic but
does not fall into the scope of the current paper).
Finally, it is worth mentioning that our results only apply to maximization
problems. We were able to extend the previous uses of fractional treewidth-
fragility by giving a way to handle dependencies over any bounded distance.
However, for the minimization problems, we do not know whether fractional
treewidth-fragility is sufficient even for the distance-$1$ problems. For a
simple example, consider the Minimum Vertex Cover problem in fractionally
treewidth-fragile graphs, or more generally in hereditary classes with
sublinear separators. While the unweighted version can be dealt with by the
local search method [16], we do not know whether there exists a
$\operatorname{\mathsf{PTAS}}$ for the weighted version of this problem.
## 2 Paths and orientations in graphs with bounded expansion
For $r\in\mathbb{Z}^{+}_{0}$, a graph $H$ is an _$r$ -shallow minor_ of a
graph $G$ if $H$ can be obtained from a subgraph of $G$ by contracting
pairwise vertex-disjoint connected subgraphs, each of radius at most $r$. For
a function $f\colon\mathbb{Z}^{+}\to\mathbb{Z}^{+}$, a class $\mathcal{G}$ of
graphs has _expansion bounded_ by $f$ if for all non-negative integers $r$,
all $r$-shallow minors of graphs from $\mathcal{G}$ have average degree at
most $f(r)$. A class has bounded expansion if its expansion is bounded by some
function $f$. The theory of graph classes with bounded expansion has been
developed in the last 15 years, and the concept has found many algorithmic and
structural applications; see [22] for an overview. Crucially for us, this
theory includes a number of tools for dealing with short paths. Moreover, as
we have pointed out before, all hereditary graph classes with sublinear
separators [14] as well as all fractionally treewidth-fragile classes [9] have
bounded expansion.
Let $\vec{G}$ be an orientation of a graph $G$, i.e, $uv$ is an edge of $G$ if
and only if the directed graph $\vec{G}$ contains at least one of the directed
edges $(u,v)$ and $(v,u)$; note that we allow $\vec{G}$ to contain both of
them at the same time, and thus for the edge $uv$ to be oriented in both
directions. We say that a directed graph $\vec{H}$ with the same vertex set is
a _$1$ -step fraternal augmentation of $\vec{G}$_ if
$\vec{G}\subseteq\vec{H}$, for all distinct edges $(x,y),(x,z)\in E(\vec{G})$,
either $(y,z)$ or $(z,y)$ is an edge of $\vec{H}$, and for each edge $(y,z)\in
E(\vec{H})\setminus E(\vec{G})$, there exists a vertex $x\in
V(\vec{G})\setminus\\{y,z\\}$ such that $(x,y),(x,z)\in E(\vec{G})$. That is,
to obtain $\vec{H}$ from $\vec{G}$, for each pair of edges $(x,y),(x,z)\in
E(\vec{G})$ we add an edge between $y$ and $z$ in one of the two possible
directions (we do not specify the direction, but in practice we would choose
directions of the added edges that minimize the maximum outdegree of the
resulting directed graph). For an integer $a\geq 0$, we say $\vec{F}$ is an
_$a$ -step fraternal augmentation of $\vec{G}$_ if there exists a sequence
$\vec{G}=\vec{G}_{0},\vec{G}_{1},\ldots,\vec{G}_{a}=\vec{F}$ where for
$i=1,\ldots,a$, $\vec{G}_{i}$ is a $1$-step fraternal augmentation of
$\vec{G}_{i-1}$. We say $\vec{F}$ is an $a$-step fraternal augmentation of an
undirected graph $G$ if $\vec{F}$ is an $a$-step fraternal augmentation of
some orientation of $G$. A key property of graph classes with bounded
expansion is the existence of fraternal augmentations with bounded outdegrees.
Let us remark that whenever we speak about an algorithm returning an $a$-step
fraternal augmentation $\vec{H}$ or taking one as an input, this implicitly
includes outputing or taking as an input the whole sequence of $1$-step
fraternal augmentations ending in $\vec{H}$.
###### Lemma 5 (Nešetřil and Ossona de Mendez [21]).
For every class $\mathcal{G}$ with bounded expansion, there exists a function
$d:\mathbb{Z}^{+}_{0}\to\mathbb{Z}^{+}$ such that for each $G\in\mathcal{G}$
and each non-negative integer $a$, the graph $G$ has an $a$-step fraternal
augmentation of maximum outdegree at most $d(a)$. Moreover, such an
augmentation can be found in time $O(d(a)|V(G)|)$.
As shown already in [21], fraternal augmentations can be used to succintly
represent distances between vertices of the graph. For the purposes of this
paper, we need a more explicit representation by an orientation of the
original graph (without the additional augmentation edges). By a _walk_ in a
directed graph $\vec{G}$, we mean a sequence $W=v_{0}v_{1}v_{2}\ldots v_{b}$
such that for $i=1,\ldots,b$, $(v_{i-1},v_{i})\in E(\vec{G})$ or
$(v_{i},v_{i-1})\in E(\vec{G})$; that is, the walk does not have to respect
the orientation of the edges. The walk $W$ is _inward directed_ if for some
$c\in\\{0,\ldots,b\\}$, we have $(v_{i},v_{i+1})\in E(\vec{G})$ for
$i=0,\ldots,c-1$ and $(v_{i},v_{i-1})\in E(\vec{G})$ for $i=c+1,\ldots,b$. For
a positive integer $r$, an orientation $\vec{G}$ of a graph $G$ _represents
$(\leq\\!r)$-distances_ if for each $u,v\in V(G)$ and each
$b\in\\{0,\ldots,r\\}$, the distance between $u$ and $v$ in $G$ is at most $b$
if and only if $\vec{G}$ contains an inward-directed walk of length at most
$b$ between $u$ and $v$. Note that given such an orientation with bounded
maximum outdegree for a fixed $r$, we can determine the distance between $u$
and $v$ (up to distance $r$) by enumerating all (constantly many) walks of
length at most $r$ directed away from $u$ and away from $v$ and inspecting
their intersections.
Our goal now is to show that graphs from classes with bounded expansion admit
orientations with bounded maximum outdegree that represent
$(\leq\\!r)$-distances. Let us define a more general notion used in the proof
of this claim, adding to the fraternal augmentations the information about the
lengths of the walks in the original graph represented by the added edges. A
_directed graph with $(\leq\\!r)$-length sets_ is a pair $(\vec{H},\ell)$,
where $\vec{H}$ is a directed graph and $\ell$ is a function assigning a
subset of $\\{1,\ldots,r\\}$ to each _unordered_ pair $\\{u,v\\}$ of vertices
of $\vec{H}$, such that if neither $(u,v)$ nor $(v,u)$ is an edge of
$\vec{H}$, then $\ell(\\{u,v\\})=\emptyset$. We say that $(\vec{H},\ell)$ is
an _orientation_ of a graph $G$ if $G$ is the underlying undirected graph of
$\vec{H}$ and $\ell(\\{u,v\\})=\\{1\\}$ for each $uv\in E(G)$. We say that
$(\vec{H},\ell)$ is an _$(\leq\\!r)$ -augmentation_ of $G$ if
$V(\vec{H})=V(G)$, for each $uv\in E(G)$ we have $1\in\ell(\\{u,v\\})$, and
for each $u,v\in V(G)$ and $b\in\ell(\\{u,v\\})$ there exists a walk of length
$b$ from $u$ to $v$ in $G$. Let $(\vec{H}_{1},\ell_{1})$ be another directed
graph with $(\leq\\!r)$-length sets. We say $(\vec{H}_{1},\ell_{1})$ is a
_1-step fraternal augmentation_ of $(\vec{H},\ell)$ if $\vec{H}_{1}$ is a
$1$-step fraternal augmentation of $\vec{H}$ and for all distinct $u,v\in
V(\vec{H})$ and $b\in\\{1,\ldots,r\\}$, we have $b\in\ell_{1}(\\{u,v\\})$ if
and only if $b\in\ell(\\{u,v\\})$ or there exist $x\in
V(\vec{H})\setminus\\{u,v\\}$, $b_{1}\in\ell(\\{x,u\\})$, and
$b_{2}\in\ell(\\{x,v\\})$ such that $(x,u),(x,v)\in E(\vec{H})$ and
$b=b_{1}+b_{2}$. Note that a $1$-step fraternal augmentation of an
$(\leq\\!r)$-augmentation of a graph $G$ is again an $(\leq\\!r)$-augmentation
of $G$. The notion of an $a$-step fraternal augmentation of a graph $G$ is
then defined in the natural way, by starting with an orientation of $G$ and
peforming the $1$-step fraternal augmentation operation $a$-times. Let us now
restate Lemma 5 in these terms (we just need to maintain the edge length sets,
which can be done with $O(a^{2})$ overhead per operation).
###### Lemma 6.
Let $\mathcal{G}$ be a class of graphs with bounded expansion, and let
$d:\mathbb{Z}^{+}_{0}\to\mathbb{Z}^{+}$ be the function from Lemma 5. For each
$G\in\mathcal{G}$ and each non-negative integer $a$, we can in time
$O(a^{2}d(a)|V(G)|)$ construct a directed graph with $(\leq\\!a+1)$-length
sets $(\vec{H},\ell)$ of maximum outdegree at most $d(a)$ such that
$(\vec{H},\ell)$ is an $a$-step fraternal augmentation of $G$.
Let $(\vec{H},\ell)$ be an $(\leq\\!r)$-augmentation $(\vec{H},\ell)$ of a
graph $G$. For $b\leq r$, a _length $b$ walk_ in $(\vec{H},\ell)$ is a tuple
$(v_{0}v_{1}\ldots v_{t},b_{1},\ldots,b_{t})$, where $v_{0}v_{1}\ldots v_{t}$
is a walk in $\vec{H}$, $b_{i}\in\ell(\\{v_{i-1},v_{i}\\}$ for $i=1,\ldots,t$,
and $b=b_{1}+\ldots+b_{t}$. Note that if there exists a length $b$ walk from
$u$ to $v$ in $(\vec{H},\ell)$, then there also exists a walk of length $b$
from $u$ to $v$ in $G$. We say that $(\vec{H},\ell)$ _represents
$(\leq\\!r)$-distances_ in $G$ if for all vertices $u,v\in V(G)$ at distance
$b\leq r$ from one another, $(\vec{H},\ell)$ contains an inward-directed
length $b$ walk between $u$ and $v$. Next, we show that this property always
holds for sufficient fraternal augmentations.
###### Lemma 7.
Let $G$ be a graph and $r$ a positive integer and let $(\vec{H},\ell)$ be a
directed graph with $(\leq\\!r)$-length sets. If $(\vec{H},\ell)$ is obtained
as an $(r-1)$-step fraternal augmentation of $G$, then it represents
$(\leq\\!r)$-distances in $G$.
###### Proof.
For $b\leq r$, consider any length $b$ walk $W=(v_{0}v_{1}\ldots
v_{t},b_{1},\ldots,b_{t})$ in an $(\leq\\!r)$-augmentation
$(\vec{H}_{1},\ell_{1})$ of $G$, and let $(\vec{H}_{2},\ell_{2})$ be a
$1$-step augmentation of $(\vec{H}_{1},\ell_{1})$. Note that $W$ is also a
length $b$ walk between $v_{0}$ and $v_{t}$ in $(\vec{H}_{2},\ell_{2})$.
Suppose that $W$ is not inward-directed in $(\vec{H}_{1},\ell_{1})$, and thus
there exists $i\in\\{1,\ldots,t-1\\}$ such that
$(v_{i},v_{i-1}),(v_{i},v_{i+1})\in E(\vec{H}_{1})$. By the definition of
$1$-step fraternal augmentation, this implies
$b_{i}+b_{i+1}\in\ell_{2}(v_{i-1},v_{i+1})$, and thus $(v_{0}\ldots
v_{i-1}v_{i+1}\ldots v_{t},b_{1},\ldots,b_{i}+b_{i+1},\ldots b_{t})$ is a
length $b$ walk from $v_{0}$ to $v_{t}$ in $(\vec{H}_{2},\ell_{2})$.
Let $(\vec{G}_{0},\ell_{0})$, …, $(\vec{G}_{r-1},\ell_{r-1})$ be a sequence of
$(\leq\\!r)$-augmentations of $G$, where $(\vec{G},\ell_{0})$ is an
orientation of $G$, $(\vec{G}_{r-1},\ell_{r-1})=(\vec{H},\ell)$, and for
$i=1,\ldots,r-1$, $(\vec{G}_{i},\ell_{i})$ is a $1$-step fraternal
augmentation of $(\vec{G}_{i-1},\ell_{i-1})$. Let $u$ and $v$ be any vertices
at distance $b\leq r$ in $G$, and let $P$ be a shortest path between them.
Then $P$ naturally corresponds to a length $b$ walk $P_{0}$ in
$(\vec{G}_{0},\ell_{0})$. For $i=1,\ldots,r-1$, if $P_{i-1}$ is inward-
directed, then let $P_{i}=P_{i-1}$, otherwise let $P_{i}$ be a length $b$ walk
in $(\vec{G}_{i},\ell_{i})$ obtained from $P_{i-1}$ as described in the
previous paragaph. Since each application of the operation decreases the
number of vertices of the walk, we conclude that $P_{r-1}$ is an inward-
directed length $b$ walk between $u$ and $v$ in $(\vec{H},\ell)$. Hence,
$(\vec{H},\ell)$ represents $(\leq\\!r)$-distances in $G$. ∎
Next, let us propagate this property back through the fraternal augmentations
by orienting some of the edges in both directions. We say that
$(\vec{H},\ell)$ is an _$a$ -step fraternal superaugmentation_ of a graph $G$
if there exists an $a$-step fraternal augmentation $(\vec{F},\ell)$ of $G$
such that $V(\vec{F})=V(\vec{H})$, $E(\vec{F})\subseteq E(\vec{H})$ and for
each $(u,v)\in E(\vec{H})\setminus E(\vec{F})$, we have $(v,u)\in E(\vec{F})$.
We say that $(\vec{F},\ell)$ is a _support_ of $(\vec{H},\ell)$.
###### Lemma 8.
Let $G$ be a graph and $r$ a positive integer and let $(\vec{H},\ell)$ be an
$(\leq\\!r)$-augmentation of $G$ of maximum outdegree $\Delta$ representing
$(\leq\\!r)$-distances. For $a\geq 1$, suppose that $(\vec{H},\ell)$ is an
$a$-step fraternal superaugmentation of $G$. Then we can in time
$O(r^{2}\Delta|V(G)|)$ obtain an $(a-1)$-step fraternal superaugmentation of
$G$ representing $(\leq\\!r)$-distances, of maximum outdegree at most
$(r+1)\Delta$.
###### Proof.
Let $(\vec{F},\ell)$ be an $a$-step fraternal augmentation of $G$ forming a
support of $(\vec{H},\ell)$, obtained as a $1$-step fraternal augmentation of
an $(a-1)$-step fraternal augmentation $(\vec{F}_{1},\ell_{1})$ of $G$. Let
$(\vec{H}_{1},\ell_{1})$ be the $(a-1)$-step fraternal superaugmentation of
$G$ obtained from $(\vec{F}_{1},\ell_{1})$ as follows:
* •
For all distinct vertices $y,z\in V(G)$ such that $(y,z),(z,y)\in E(\vec{H})$,
$(y,z)\in E(\vec{F}_{1})$, and $(z,y)\not\in E(\vec{F}_{1})$, we add the edge
$(z,y)$.
* •
For each edge $(y,z)\in E(\vec{H})$ and integer
$b\in\ell(\\{y,z\\})\setminus\ell_{1}(\\{y,z\\})$, we choose a vertex $x\in
V(G)\setminus\\{y,z\\}$ such that $(x,y),(x,z)\in E(\vec{F}_{1})$ and
$b=b_{1}+b_{2}$ for some $b_{1}\in\ell_{1}(\\{x,y\\})$ and
$b_{2}\in\ell_{1}(\\{x,z\\})$, and add the edge $(y,x)$. Note that such a
vertex $x$ and integers $b_{1}$ and $b_{2}$ exist, since $b$ was added to
$\ell(\\{y,z\\})$ when $(\vec{F},\ell)$ was obtained from
$(\vec{F}_{1},\ell_{1})$ as a $1$-step fraternal augmentation.
Each edge $(y,x)\in E(\vec{H}_{1})\setminus E(\vec{H})$ arises from an edge
$(y,z)\in E(\vec{H})$ leaving $y$ and an element
$b\in\ell(\\{y,z\\})\setminus\ell_{1}(\\{y,z\\})$, and each such pair
contributes at most one edge leaving $y$. Hence, the maximum outdegree of
$\vec{H}_{1}$ is at most $(r+1)\Delta$.
Consider a length $b$ inwards-directed walk $(v_{0}v_{1}\ldots
v_{t},b_{1},\ldots,b_{t})$ in $\vec{H}$, for any $b\leq r$. Then $\vec{H}$
contains a length $b$ inwards-directed walk from $v_{0}$ to $v_{t}$ obtained
by natural edge replacements: For any edge $(y,z)\in E(\vec{H})$ of this walk
and $b^{\prime}\in\ell_{i}(\\{y,z\\})$, the construction described above
ensures that if $(y,z)\not\in E(\vec{H}_{1})$ or
$b^{\prime}\not\in\ell_{1}(\\{y,z\\})$, then there exists $x\in
V(G)\setminus\\{y,z\\}$ such that $(y,x),(x,z)\in E(\vec{H}_{1})$ and
$b^{\prime}=b^{\prime\prime}+b^{\prime\prime\prime}$ for some
$b^{\prime\prime}\in\ell_{1}(\\{x,y\\})$ and
$b^{\prime\prime\prime}\in\ell_{1}(\\{x,z\\})$, and we can replace the edge
$(y,z)$ in the walk by the edges $(y,x)$ and $(x,z)$ of $E(\vec{H}_{1})$.
Since $\vec{H}$ represents $(\leq\\!r)$-distances in $G$, this transformation
shows that so does $\vec{H}_{1}$. ∎
We are now ready to prove the main result of this section.
###### Lemma 9.
For any class $\mathcal{G}$ with bounded expansion, there exists a function
$d^{\prime}:\mathbb{Z}^{+}\to\mathbb{Z}^{+}$ such that for each
$G\in\mathcal{G}$ and each positive integer $r$, the graph $G$ has an
orientation with maximum outdegree at most $d^{\prime}(r)$ that represents
$(\leq\\!r)$-distances in $G$. Moreover, such an orientation can be found in
time $O(r^{2}d^{\prime}(r)|V(G)|)$.
###### Proof.
Let $d$ be the function from Lemma 5, and let
$d^{\prime}(r)=(r+1)^{r-1}d(r-1)$. By Lemma 6, we obtain an $(r-1)$-step
fraternal augmentation $(\vec{H},\ell)$ of $G$ of maximum outdegree at most
$d(r-1)$. By Lemma 7, $(\vec{H},\ell)$ represents $(\leq\\!r)$-distances in
$G$. Repeatedly applying Lemma 8, we obtain a $0$-step fraternal
superaugmentation $(\vec{G},\ell_{0})$ of $G$ of maximum outdegree at most
$d^{\prime}(r)$ representing $(\leq\\!r)$-distances. Clearly, $\vec{G}$ is an
orientation of $G$ of maximum outdegree at most $d^{\prime}(r)$ representing
$(\leq\\!r)$-distances. ∎
## 3 Approximation schemes
Let us now prove Theorem 2. To this end, let us start with a lemma to be
applied to the sets arising from fractional treewidth-fragility.
###### Lemma 10.
Let $\vec{G}$ be an orientation of a graph $G$ with maximum outdegree
$\Delta$. Let $A$ be a set of vertices of $G$ and for a positive integer $c$,
let $\\{R_{v}:v\in A\\}$ be a system of subsets of $A$ such that each vertex
belongs to at most $c$ of the subsets. For $X\subseteq V(G)$ and a positive
integer $r$, let $D_{\vec{G},r}(X)$ be the union of the sets $R_{v}$ for all
vertices $v\in V(G)$ such that $\vec{G}$ contains a walk from $v$ to $X$ of
length at most $r$ directed away from $v$. For a positive integer $k$, let
$X_{1}$, …, $X_{m}$ be a system of subsets of $V(G)$ such that each vertex
belongs to at most $\frac{m}{c(\Delta+1)^{r}k}$ of the subsets. For any
assignment $w$ of non-negative weights to vertices of $G$, there exists
$i\in\\{1,\ldots,m\\}$ such that $w(A\setminus
D_{\vec{G},r}(X_{i}))\geq(1-1/k)w(A)$.
###### Proof.
For a vertex $z\in A$, let $B(z)$ be the set of vertices reachable in
$\vec{G}$ from vertices $v\in A$ such that $z\in R_{v}$ by walks of length at
most $r$ directed away from $v$. Note that $|B(z)|\leq c(\Delta+1)^{r}$ and
that for each $X\subseteq V(G)$, we have $z\in D_{\vec{G},r}(X)$ if and only
if $B(z)\cap X\neq\emptyset$.
Suppose for a contradiction that for each $i$ we have $w(A\setminus
D_{\vec{G},r}(X_{i}))<(1-1/k)w(A)$, and thus $w(D_{\vec{G},r}(X_{i}))>w(A)/k$.
Then
$\displaystyle\frac{m}{k}w(A)$
$\displaystyle<\sum_{i=1}^{m}w(D_{\vec{G},r}(X_{i}))=\sum_{i=1}^{m}\sum_{z\in
D_{\vec{G},r}(X_{i})}w(z)=\sum_{i=1}^{m}\sum_{z\in A:B(z)\cap
X_{i}\neq\emptyset}w(z)$ $\displaystyle\leq\sum_{i=1}^{m}\sum_{z\in
A}w(z)|B(z)\cap X_{i}|=\sum_{z\in A}w(z)\sum_{i=1}^{m}|B(z)\cap X_{i}|$
$\displaystyle=\sum_{z\in A}w(z)\sum_{x\in B(z)}|\\{i\in\\{1,\ldots,m\\}:x\in
X_{i}\\}|\leq\sum_{z\in A}w(z)\sum_{x\in B(z)}\frac{m}{c(\Delta+1)^{r}k}$
$\displaystyle=\sum_{z\in A}w(z)|B(z)|\frac{m}{c(\Delta+1)^{r}k}\leq\sum_{z\in
A}w(z)\frac{m}{k}=\frac{m}{k}w(A),$
which is a contradiction. ∎
Next, let us derive a lemma on admissibility for $(\leq\\!r)$-distance
determined problems.
###### Lemma 11.
For a positive integer $r$, let $\vec{G}$ be an orientation of a graph $G$
representing $(\leq\\!r)$-distances. For a set $X\subseteq V(G)$, let
$Y_{\vec{G},r}(X)$ be the set of vertices $y$ such that $\vec{G}$ contains a
walk from $y$ to $X$ of length at most $r$ directed away from $y$. For any
$(\leq\\!r)$-distance determined problem, a set $B\subseteq V(G)\setminus
Y_{\vec{G},r}(X)$ is admissible in $G$ if and only if it is admissible in
$G-X$.
###### Proof.
Since the problem is $(\leq\\!r)$-distance determined, it suffices to show
that $\min(r,d_{G}(u,v))=\min(r,d_{G-X}(u,v))$ holds for all $u,v\in B$.
Clearly, $d_{G}(u,v)\leq d_{G-X}(u,v)$, and thus it suffices to show that if
the distance between $u$ and $v$ is $G$ is $b\leq r$, then $G-X$ contains a
walk of length $b$ between $u$ and $v$. Since $\vec{G}$ represents
$(\leq\\!r)$-distances, there exists an inward-directed walk $P$ of length $b$
between $u$ and $v$ in $\vec{G}$. Since $u,v\not\in Y_{\vec{G},r}(X)$, we have
$V(P)\cap X=\emptyset$, and thus $P$ is also a walk of length $b$ between $u$
and $v$ in $G-X$. ∎
We are now ready to prove the main result.
###### Proof of Theorem 2.
Let $d^{\prime}$ be the function from Lemma 9 for the class $\mathcal{G}$. Let
us define $h(r,c)=c(d^{\prime}(r)+1)^{r}$. The algorithm is as follows. Since
$\mathcal{G}$ is $q$-efficiently fractionally treewidth-$f$-fragile, in time
$q(|V(G)|)$ we can find sets $X_{1},\ldots,X_{m}\subseteq V(G)$ such that each
vertex belongs to at most $\frac{m}{h(r,c)k}$ of them, and for each $i$, a
tree decomposition of $G-X_{i}$ of width at most $f(h(r,c)k,|V(G)|)$. Clearly,
$m\leq q(|V(G)|)$. Next, using Lemma 9, we find an orientation $\vec{G}$ of
$G$ that represents $(\leq\\!r)$-distances. Let $Y_{\vec{G},r}$ be defined as
in the statement of Lemma 11. Since the problem is $(g,p)$-tw-tractable
problem, for each $i$ we can in time $p(|V(G)|)\cdot g(f(h(r,c)k,|V(G)|))$
find a subset $A_{i}$ of $V(G)\setminus Y_{\vec{G},r}(X_{i})$ admissible in
$G-X_{i}$ of largest weight. By Lemma 11, each of these sets is admissible in
$G$; the algorithm return the heaviest of the sets $A_{1}$, …, $A_{m}$.
As the returned set is admissible in $G$, it suffices to argue about its
weight. Let $A$ be a heaviest admissible set in $G$. Let $\\{R_{v}\subseteq
A:v\in A\\}$ be the system of subsets from the definition of $c$-near-
monotonicity, and let $D_{\vec{G},r}$ be defined as in the statement of Lemma
10. By the definition of $c$-near-monotonicity, for each $i$ the set
$A\setminus D_{\vec{G},r}(X_{i})$ is admissible in $G$. Since $v\in R_{v}$ for
each $v\in A$, we have $Y_{\vec{G},r}(X_{i})\subseteq D_{\vec{G},r}(X_{i})$,
and thus by Lemma 11, $A\setminus D_{\vec{G},r}(X_{i})$ is also admissible in
$G-X_{i}$, and by the choice of $A_{i}$, we have $w(A_{i})\geq w(A\setminus
D_{\vec{G},r}(X_{i}))$. By Lemma 10, we conclude that for at least one $i$, we
have $w(A_{i})\geq(1-1/k)w(A)$, as required. ∎
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# Dual CFT on Nariai limit for Kerr-Sen-dS black holes
Muhammad Fitrah Alfian Rangga Sakti<EMAIL_ADDRESS>Piyabut Burikham
<EMAIL_ADDRESS>High Energy Physics Theory Group, Department of Physics,
Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
###### Abstract
In this work, we study the Kerr-Sen-de Sitter black hole (BH) in the Nariai
limit where the event and cosmological horizon coincide. We show that the
near-horizon Kerr-Sen-de Sitter black hole in Nariai limit is a fiber over
AdS2 with an appropriate coordinate transformation, instead of fiber over dS2.
Hence, we can compute the associated central charge and CFT temperature by
using the Kerr/CFT method. It is remarkably exhibited that through Cardy’s
growth of states, the Bekenstein-Hawking entropy on cosmological horizon is
reproduced. Moreover, we show that the radial equation of the quantum scalar
field in $J$\- and $Q$-pictures on this charged rotating background in Nariai
limit can be portrayed in quadratic Casimir operator form with $SL(2,R)\times
SL(2,R)$ isometry. We also compute the corresponding thermodynamic quantities
from CFT to find the absorption cross-section and real-time correlator in
$J$-picture. In $Q$-picture, we do not find a well-defined CFT description. We
then extend the study of quantum scalar field in Nariai limit for Kerr-Newman-
dS black hole solution and show that the hidden conformal symmetry on this
black hole’s background in in $J$\- and $Q$-pictures is well-defined.
††preprint: APS/123-QED
## I Introduction
Establishing the holographic correspondence between de Sitter space (dS) and
quantum theory is a significant challenge in theoretical physics. The
holographic duality, most notably captured by the Anti-de Sitter/Conformal
Field Theory (AdS/CFT) correspondence, has provided profound insights into the
quantum theory of gravity. However, extending this framework to dS space has
proven to be more problematic. One of the main difficulties in the holographic
description of dS space arises from the absence of a global timelike Killing
vector and spatial infinity. In the AdS/CFT correspondence, the asymptotic
behavior of AdS space allows for a well-defined boundary at infinity, which
plays a crucial role in the holographic interpretation. This boundary serves
as a holographic screen, encoding the dynamics of the gravitational theory in
terms of a dual quantum field theory living on the boundary. In contrast, dS
space does not possess a similar asymptotic structure or boundary at infinity.
As a result, the standard techniques used in the AdS/CFT correspondence cannot
be straightforwardly applied to dS space. It is an intriguing and challenging
problem that requires a deeper understanding of the quantum nature of gravity,
as well as the dynamics of dS space itself.
Inspite of the challenges coming from the asymptotically de Sitter spacetime,
this solution may provide a correct interpretation to explain our physical
world in the context of cosmological model. The standard model of cosmology,
whose details are explained extensively e.g. in [1], is surprisingly
consistent with this de Sitter solution. The observed accelerated expansion of
our universe, supported by astronomical observations such as the cosmic
microwave background radiation and the distribution of galaxies, suggests that
our universe could be described by a dS spacetime [2, 3]. In particular, the
connection between the cosmological constant and the behavior of smaller
objects, such as black holes, is also of great interest. The presence of a
non-zero cosmological constant affects the configurations and properties of
black holes. In fact, the cosmological constant can influence the formation,
growth, and evaporation of black holes. Moreover, one can study the quantum
gravity from black hole’s properties through the gravitational waves
observations [4, 5, 6]. Hence, in the future, we expect that the gravitational
wave observations may give us more precise information about quantum structure
of black holes including when the cosmological constant is present.
Regarding the black hole solutions, one of the central challenges in quantum
gravity is to understand the microscopic origin of black hole entropy, which
is given by the famous Bekenstein-Hawking formula stating that the entropy is
one quarter of the black hole’s event horizon area. Remarkably, the AdS/CFT
correspondence provides a potential framework for addressing this problem. It
is assumed that the partition function of conformal field theory to be
identical to black hole’s partition function. This relation was firstly proven
by computing the entropy the extremal Kerr black hole [7]. That study is
inspired from the success of the investigation of the asymptotic symmetries of
Banados-Teitelboim-Zanelli (BTZ) black hole which contains two-dimensional
(2D) local conformal algebra. CFT techniques can then be used to calculate the
state degeneracy [8] leading to the agreement of CFT entropy with the
Bekenstein-Hawking entropy.
In the following, we consider a class of charged rotating black hole solution
in Einstein-Maxwell-Dilaton-Axion (EMDA) supergravity theory and Einstein-
Maxwell theory, namely dyonic Kerr-Sen-dS (KSdS) and dyonic Kerr-Newman-dS
(KNdS) black holes, respectively. Both black holes possess more than two
horizons because of the existence of the positive cosmological constant. The
largest horizon is the cosmological horizon for which the first thermodynamics
law on this horizon resembles the first thermodynamic law on black hole’s
horizon. In the cosmological horizon, one can also study the thermal spectrum
with non-zero entropy that is also proportional to the horizon area, likewise
in the event horizon. The existence of three horizons leads to different
limits. When the Cauchy and event horizons coincide, one can obtain an
extremal black hole. When the cosmological and event horizon coincide, the
resulting metric will be Nariai solution. Furthermore, when all three horizons
coincide, the ultracold solution will be produced.
The first holographic description of the cosmological entropy of the Nariai
solution obtained from Kerr-dS and KNdS solutions has been investigated in
Ref. [9]. The Nariai solution possesses $SL(2,R)\times U(1)$ isometry where
the spacetime metric is a fiber over dS2. We want to show that in Nariai
limit, we can write the solution with similar isometry, although with the
spacetime metric as a fiber over AdS2 by choosing different sign on the radial
near-horizon coordinate. This metric form on Nariai limit denotes the similar
form with near-horizon extremal metric [10, 11, 12, 13]. Furthermore, we want
to study the scalar wave equation on the geometry when we take the Nariai
limit. As has been studied for generic Kerr solution [14], we exhibit that in
Nariai limit, the scalar wave equation possesses $SL(2,R)\times SL(2,R)$
isometry. This isometry happens to appear in scalar wave equation of some
black hole solutions [15, 16, 17, 18] and also for higher-spin field [19, 20,
21]. Remarkably, one can also compute the absorption cross-section and real-
time correlator from 2D CFT of the solution on Nariai limit. We want to prove
that for both KNdS and KSdS black holes in Nariai limit, the conformal
symmetry can be shown on the spacetime metric directly and on the scalar wave
equation. Moreover, CFT dual will be constructed trough the matching between
the entropy, absorption cross-section, and the real-time correlator. We will
consider two different pictures, $J$\- and $Q$ pictures. $J$\- picture denotes
the CFT description when the scalar probe is neutral while $Q$-picture denotes
the CFT description when the scalar probe is electrically charged.
We organize the paper as follows. In Sec. II, we carry out the KSdS solution
and its thermodynamic quantities on the event and cosmological horizons. In
Sec. III, we show the ”fin” diagrams which portray the horizons and the
extremal limits. In the next section, we study the Nariai limit on KSdS
solution and find the entropy using the Kerr/CFT correspondece. In Sec. V, we
study the conformal symmetry on the scalar wave equation in $J$\- and
$Q$-pictures in the background of KSdS solution in Nariai limit. In Sec. VI,
we further extend the study of the scalar field on KNdS black hole in Nariai
limit. In Sec. VII, we summarize our findings for the whole paper.
## II Dyonic Kerr-Sen-de Sitter solution and its thermodynamics
The dyonic KSdS black hole is the exact solution to EMDA supergravity theory
with positive cosmological constant. The line element of KSdS spacetime reads
as [46]
$ds^{2}=-\frac{\Delta}{\varrho^{2}}\hat{X}^{2}+\frac{\varrho^{2}}{\Delta}d\hat{r}^{2}+\frac{\varrho^{2}}{\Delta_{\theta}}d\theta^{2}+\frac{\Delta_{\theta}\sin^{2}\theta}{\varrho^{2}}\hat{Y}^{2},\
$ (1)
where
$\hat{X}=d\hat{t}-a\sin^{2}\theta\frac{d\hat{\phi}}{\Xi},~{}\hat{Y}=ad\hat{t}-(\hat{r}^{2}-d^{2}-k^{2}+a^{2})\frac{d\hat{\phi}}{\Xi},\
$ (2)
$\Delta=(\hat{r}^{2}-d^{2}-k^{2}+a^{2})\left(1-\frac{\hat{r}^{2}-d^{2}-k^{2}}{l^{2}}\right)-2m\hat{r}+p^{2}+q^{2},\
\ $ (3)
$\Delta_{\theta}=1+\frac{a^{2}}{l^{2}}\cos^{2}\theta,~{}\Xi=1+\frac{a^{2}}{l^{2}},~{}\varrho^{2}=\hat{r}^{2}-d^{2}-k^{2}+a^{2}\cos^{2}\theta.\
$ (4)
Note that $m,a,d,k,p,q,l$ are the parameters of mass, spin, dilaton charge,
axion charge, magnetic charge, electric charge, and dS length, respectively.
The KSdS solution above is obtained from the gauged dyonic Kerr-Sen black hole
solution [46, 45] via analytic continuation [28]. There is an important
relation between charges of KSdS solution given by
$d=\frac{p^{2}-q^{2}}{2m},~{}~{}~{}~{}~{}k=\frac{pq}{m}.\ $ (5)
Therefore, when the magnetic charge vanishes, the axion charge also vanishes
while when $p=q$, the dilaton charge will vanish. One also can write
$d^{2}+k^{2}=\left(\frac{p^{2}+q^{2}}{2m}\right)^{2}.\ $ (6)
KSdS black hole possesses three positive horizons. The three positive horizons
are inner ($r_{-}$), outer/event ($r_{+}$) and cosmological ($r_{c}$) horizons
where $r_{c}\geq r_{+}\geq r_{-}\geq 0$. We can study the thermodynamics on
its horizons. Within this paper, we emphasize the study on thermodynamics on
the event horizon and cosmological horizon. On the event horizon, the KSdS
black hole satisfies the thermodynamic relation
$dM=T_{H}dS_{BH}+\Omega_{H}dJ+\Phi_{H}dQ+\Psi_{H}dP+Vd\mathcal{P},$ (7)
with the following quantities
$M=\frac{m}{\Xi},~{}~{}~{}J=\frac{ma}{\Xi},~{}~{}~{}Q=\frac{q}{\Xi},~{}~{}~{}P=\frac{p}{\Xi},\
$ (8)
$T_{H}=\frac{r_{+}(l^{2}-2r_{+}^{2}+2d^{2}+2k^{2}-a^{2})-ml^{2}}{2\pi(r_{+}^{2}-d^{2}-k^{2}+a^{2})l^{2}},$
(9)
$S_{BH}=\frac{\pi}{\Xi}(r_{+}^{2}-d^{2}-k^{2}+a^{2}),~{}~{}~{}\Omega_{H}=\frac{a\Xi}{r_{+}^{2}-d^{2}-k^{2}+a^{2}},$
(10)
$\Phi_{H}=\frac{q(r_{+}+d-p^{2}/m)}{r_{+}^{2}-d^{2}-k^{2}+a^{2}},~{}\Psi_{H}=\frac{p(r_{+}+d-p^{2}/m)}{r_{+}^{2}-d^{2}-k^{2}+a^{2}},$
(11) $V=\frac{4}{3}r_{+}S_{BH},~{}~{}~{}\mathcal{P}=-\frac{3}{8\pi l^{2}}.$
(12)
where those are physical mass, angular momentum, electric charge, magnetic
charge, Hawking temperature, Bekenstein-Hawking entropy, angular velocity,
electric potential, magnetic potential, volume and pressure. We can also
consider the thermodynamic quantities on the cosmological horizon $r_{c}$.
Those thermodynamic quantities on $r_{c}$ are given as follows
$M_{c}=-\frac{m}{\Xi},~{}~{}~{}J_{c}=-\frac{ma}{\Xi},~{}~{}~{}Q_{c}=-\frac{q}{\Xi},~{}~{}~{}P_{c}=-\frac{p}{\Xi},\
$ (13)
$T_{c}=\frac{r_{c}(2r_{c}^{2}-2d^{2}-2k^{2}+a^{2}-l^{2})+ml^{2}}{2\pi(r_{c}^{2}-d^{2}-k^{2}+a^{2})l^{2}},$
(14)
$S_{c}=\frac{\pi}{\Xi}(r_{c}^{2}-d^{2}-k^{2}+a^{2}),~{}~{}~{}\Omega_{c}=\frac{a\Xi}{r_{c}^{2}-d^{2}-k^{2}+a^{2}},$
(15)
$\Phi_{c}=\frac{q(r_{c}+d-p^{2}/m)}{r_{c}^{2}-d^{2}-k^{2}+a^{2}},~{}\Psi_{c}=\frac{p(r_{c}+d-p^{2}/m)}{r_{c}^{2}-d^{2}-k^{2}+a^{2}},$
(16) $V_{c}=\frac{4}{3}r_{c}S_{c},~{}~{}~{}\mathcal{P}_{c}=\frac{3}{8\pi
l^{2}}.$ (17)
These thermodynamic quantities are obtained by considering the black hole’s
event horizon as the boundary [26]. This is in contrast with the previous
thermodynamic quantities on the black hole’s event horizon where the
cosmological horizon is considered as the boundary. For this black hole, we
consider the cosmological constant as dynamical variable as a consequence of
considering mass of the black hole as enthalpy of the spacetime [27, 28].
## III Structure of Dyonic Kerr-Sen-de Sitter Spacetime
In order to gain insight into the structure of dyonic KSdS spacetime, we
explore the horizon solutions by plotting the “fin” diagram by slicing through
the parameter space with 4 planes; (a) $p=0$, (b) $q=0$, (c) $p=1$, (d) $q=1$,
for $a=0.5,l=10$ as shown in Fig. 1. The zeroes of metric function
$\Delta/\varrho^{2}$ give four roots for the horizons,
$\hat{r}=r_{i},~{}(i=1-4)$. We observe the continuity of the real part of
horizon ${\rm Re}[r_{i}]$ connecting solutions of merging horizons. In
addition, there is a ring singularity at $\varrho=0$ or
$\hat{r}=\sqrt{d^{2}+k^{2}}=\displaystyle{\frac{p^{2}+q^{2}}{2m}}\equiv
r_{s}$.
For zero-charge spinning KSdS spacetime at fixed $a,l$, there is an extremal
horizon where both inner (Cauchy) horizon $r_{-}$ and outer (black hole)
horizon $r_{+}$ emerge at the critical $m$, above which $r_{-}$ and $r_{+}$
move away from one another. Finally at certain mass, $r_{+}$ will merge with
cosmological horizon $r_{c}$ and the outer physical spacetime region
disappears leaving the naked singularity at $\hat{r}=0$.
When either electric ($q$) or magnetic ($p$) charge is turned on, the
spacetime structure becomes more complicated as depicted in Fig. 1. Region I
contains an outer horizon, naked singularity and cosmological horizon where
$r_{+}<r_{s}<r_{c}$. Region II, IV, and VII are spacetime with naked
singularity and cosmological horizon and no black holes. Region III and VI are
spacetime with black holes and cosmological horizon, there exists singulairty
hidden behind the Cauchy horizon. Region V intriguingly has black hole region
with inner and outer horizon behind the naked singularity and cosmological
horizon at the furthest.
The boundary lines between each Region where extremal horizons occur are
categorized as follows.
1. (i)
I and II: $r_{-}=r_{+}<r_{s}$,
2. (ii)
II and III+VI: $r_{-}=r_{+}>r_{s}$,
3. (iii)
III+VI and IV: $r_{+}=r_{c}>r_{s}$, Nariai BH,
4. (iv)
II and V: $r_{-}=r_{+}<r_{s}$,
5. (v)
I and V: $r_{-}=0,r_{+}<r_{s}$,
6. (vi)
IV and VII: $r_{1}=r_{4}\equiv r_{c}>r_{s}$,
7. (vii)
II and IV, VII: $r_{1}=r_{2}\equiv r_{c}>r_{s}$.
The Nariai limit of all cases occurs in the high mass region where
$r_{+}=r_{c}$, the boundary line between region III+VI and IV. On the other
hand, the boundary line between region II and III+VI represents extremal
solutions where $r_{-}=r_{+}$. In comparison to the “shark fin” figure in Ref.
[29], the dependence of $d,k$ on $m$ and $a$ distort the fin shape in the
small mass and charge region in KSdS case. Interestingly, similar kinds of
“flag” diagram where extremal structures are explored for generalized
spherically symmetric metric are also presented in Ref. [30, 31].
|
---|---
(a) $p=0$ KSdS | (b) $q=0$ KSdS
|
(c) $p=1$ KSdS | (d) $q=1$ KSdS
Figure 1: Horizon structure of KSdS spacetime for $a=0.5,l=10$, color region
represents parameter space with the horizon $r_{i}\geq 0$ for $i=1-4$. Region
I consists of an outer horizon behind a naked singularity and cosmological
horizon. Region II, IV, and VII are spacetime with naked singularity and
cosmological horizon. Region III and VI are black hole spacetime with
singularity behind the Cauchy horizon, event horizon, and cosmological
horizon. Region V contains naked singularity and cosmological horizon, and
inner and outer horizon of black hole locating behind a naked singularity.
## IV Charged Rotating Nariai/CFT Correspondence
The rotating Nariai solutions from the Kerr-dS and KNdS solutions haves been
obtained in Ref. [9]. In this section, we will exhibit that this rotating
solution in Nariai limit can also be expressed in the metric form with a fiber
over AdS2 as the common near-horizon metric form in the Kerr/CFT
correspondence [10, 11, 12, 13]. Another purpose of this section is to give a
more detailed derivation of the central charge, temperature and entropy for
solution in Nariai limit obtained from KSdS black hole. Furthermore, we will
also study the massless scalar wave equation on this rotating background in
Nariai limit which may possesses $SL(2,R)\times SL(2,R)$ isometry.
### IV.1 Geometry of Nariai Limit
To find the rotating geometry in Nariai limit, we will assume two different
limits which are Nariai limit and near-horizon limit. The parameter that
parameterizes the Nariai limit is defined by
$\varepsilon=\frac{r_{c}-r_{+}}{\lambda r_{0}}.$ (18)
$r_{0}$ is a scaling constant that we define as
$r_{0}^{2}=r_{+}^{2}-d^{2}-k^{2}+a^{2}$ for KSdS solution. When
$\varepsilon\rightarrow 0$, we can obtain $r_{c}=r_{+}$. Furthermore, the
near-horizon coordinate transformations are given by [22]
$\hat{r}=r_{+}+\lambda
r_{0}r,~{}~{}\hat{t}=\frac{r_{0}}{\lambda}t,~{}~{}\hat{\phi}=\phi+\frac{\Omega_{c}r_{0}}{\lambda}t.\
$ (19)
The constant $\lambda$ is to parameterize the near-horizon limit.
$\lambda\rightarrow 0$ denotes the near-horizon limit. Using those coordinate
transformations, one can find the charged rotating solution in Nariai limit as
follows
$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\Gamma(\theta)\left(-r(r-\epsilon)dt^{2}+\frac{dr^{2}}{r(r-\epsilon)}+\alpha(\theta)d\theta^{2}\right)$
(20) $\displaystyle+\gamma(\theta)\left(d\phi+erdt\right)^{2},\ $
where the metric functions are given by
$\Gamma(\theta)=\frac{\varrho_{+}^{2}}{\upsilon},~{}~{}\alpha(\theta)=\frac{\upsilon}{\Delta_{\theta}},~{}~{}\gamma(\theta)=\frac{r_{0}^{4}\Delta_{\theta}\sin^{2}\theta}{\varrho_{+}^{2}\Xi^{2}},\
$ (21)
$\varrho_{+}^{2}=r_{+}^{2}-d^{2}-k^{2}+a^{2}\cos^{2}\theta,~{}~{}e=\frac{2ar_{+}\Xi}{r_{0}^{2}\upsilon}.$
(22)
Note that we already approximate the function $\Delta$ as
$\Delta\simeq\upsilon(r-r_{c})(r-r_{+}),\ $ (23)
where $\upsilon=1-(6r_{+}^{2}-d^{2}-k^{2}+a^{2})/l^{2}$. The rotating solution
in Nariai limit (20) is a fiber over AdS2. This fact can obviously be seen
from the factor on time-like and radial coordinates of the metric in Nariai
limit. One can also write the spacetime metric (20) in Poincaré coordinates by
applying the following coordinate transformations [41]
$\displaystyle t$ $\displaystyle=$
$\displaystyle-\frac{2}{\varepsilon}\text{log}\frac{y}{\sqrt{\tau^{2}y^{2}-1}},$
(24) $\displaystyle r$ $\displaystyle=$
$\displaystyle\frac{\varepsilon}{2}(1+\tau y),$ (25) $\displaystyle\phi$
$\displaystyle=$ $\displaystyle\varphi+\frac{1}{2}\text{log}\frac{\tau
y+1}{\tau y-1}.\ $ (26)
From those transformations, one can find
$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\Gamma(\theta)\left(-y^{2}d\tau^{2}+\frac{dy^{2}}{y^{2}}+\alpha(\theta)d\theta^{2}\right)$
(27) $\displaystyle+\gamma(\theta)\left(d\varphi+eyd\tau\right)^{2},\ $
This metric form is obviously similar with the common near-horizon metric that
is used in the Kerr/CFT correspondence. This metric possesses $SL(2,R)\times
U(1)$ isometry generated by the following vector fields
$\zeta_{0}=\partial_{\varphi},\ $ (28)
which denote the rotational $U(1)$ isometry and
$X_{1}=\partial_{\tau},~{}~{}~{}X_{2}=\tau\partial_{\tau}-y\partial_{y},\ $
(29)
$X_{3}=\left(\frac{1}{2y^{2}}+\frac{\tau^{2}}{2}\right)\partial_{\tau}-\tau
y\partial_{y}-\frac{e}{y}\partial_{\varphi},$ (30)
denoting $SL(2,R)$ isometry.
As we have mentioned in the beginning of this section, in Ref. [9] they manage
to show another form of charged rotating Nariai geometry from dyonic KNdS
solution that is a fiber over dS2 in Eq. (B.2) in their appendix, which is
given by
$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\Gamma(\theta)\left(-(1-r^{2})dt^{2}+\frac{dr^{2}}{1-r^{2}}+\alpha(\theta)d\theta^{2}\right)$
(31) $\displaystyle+\gamma(\theta)\left(d\phi+krdt\right)^{2}.\ $
For the detail forms of $\Gamma(\theta),\alpha(\theta),\gamma(\theta),k$, one
can see in the reference that we have mentioned. This metric can be found by
using the near-horizon coordinate transformations (2.13) in Ref. [9] then
using the coordinate changes (2.18). In that paper, they use
$\hat{r}=r_{c}-\lambda r_{c}r$ in Eq. (2.13), instead of using positive new
radial coordinate, that will result in metric with a fiber over dS2. This is
different with our result which is a fiber over AdS2. Hence, the charged
rotating geometry in Nariai limit can be portrayed in both fibers over dS2 and
AdS2, depending on the sign of the near-horizon radial coordinate
transformation we choose. In the other words, we can also obtain the charged
rotating solution of KSdS black hole in Nariai limit with a fiber over dS2
when we use $\hat{r}=r_{c}-\lambda r_{c}r$.
### IV.2 CFT Duals
For rotating Nariai geometry in Ref. [9], they basically apply the similar
Kerr/CFT method to study the relation between black hole’s thermodynamics
macroscopically and microscopically using 2D CFT. They propose the similar
fall-off conditions for the metric deviations although the spacetime structure
contains dS2 slice, unlike in the common dual CFT which contains AdS2 slice.
#### IV.2.1 Central charge
In the Kerr/CFT correspondence, the entropy is assumed to originate from
partition function of 2D CFT resulting in Cardy entropy formula. The main
ingredients for this formula are the central charges and temperatures. The
basic procedure to compute the central charges is to employ the approach of
Brown and Henneaux [32] where the asymptotic symmetry group (ASG) needs to
satisfy some certain boundary conditions. In this work, we consider the
similar boundary conditions for the metric deviations as given in Ref. [7],
$\displaystyle
h_{\mu\nu}\sim\left(\begin{array}[]{cccc}\mathcal{O}(r^{2})&\mathcal{O}\left(\frac{1}{r^{2}}\right)&\mathcal{O}\left(\frac{1}{r}\right)&\mathcal{O}(1)\\\
&\mathcal{O}\left(\frac{1}{r^{3}}\right)&\mathcal{O}\left(\frac{1}{r^{2}}\right)&\mathcal{O}\left(\frac{1}{r}\right)\\\
&&\mathcal{O}\left(\frac{1}{r}\right)&\mathcal{O}\left(\frac{1}{r}\right)\\\
&&&\mathcal{O}(1)\end{array}\right).$ (36)
in the basis $(t,r,\theta,\phi)$ for metric (20). These metric deviations are
subleading with respect to the background metric. These boundary conditions
are basically chosen to eliminate excitations above extremality. The
asymptotic symmetry of the general black hole family includes diffeomorphisms
$\xi$ that satisfy
$\delta_{\xi}g_{\mu\nu}=\mathcal{L}_{\xi}g_{\mu\nu}=\xi^{\sigma}(\partial_{\sigma}g_{\mu\nu})+g_{\mu\sigma}(\partial_{\nu}\xi^{\sigma})+g_{\sigma\nu}(\partial_{\mu}\xi^{\sigma}),\
$ (37)
where the metric deviation is denoted by $\delta_{\xi}g_{\mu\nu}=h_{\mu\nu}$.
The most general diffeomorphism symmetry that preserves such boundary
conditions (36) in the asymptotic infinity is generated by the following
Killing vector field
$\displaystyle\zeta$ $\displaystyle=$
$\displaystyle\left[c_{t}+\mathcal{O}\left(r^{-3}\right)\right]\partial_{t}+\left[-r\epsilon^{\prime}(\phi)+\mathcal{O}(1)\right]\partial_{r}$
(38)
$\displaystyle+\mathcal{O}\left(r^{-1}\right)\partial_{\theta}+\left[\epsilon(\phi)+\mathcal{O}\left(r^{-2}\right)\right]\partial_{\phi},\
$
where $c_{t}$ is an arbitrary constant and the prime $(^{\prime})$ denotes the
derivative with respect to $\phi$. This ASG contains one copy of the conformal
group of the circle which is generated by
$\displaystyle\zeta_{\epsilon}=\epsilon(\phi)\partial_{\phi}-r\epsilon^{\prime}(\phi)\partial_{r},$
(39)
that will be the part of the near-horizon metric in Nariai limit. We know that
the azimuthal coordinate is periodic under the rotation $\phi\sim\phi+2\pi$.
Hence, we may define $\epsilon_{\hat{n}}=-e^{-i\hat{n}\phi}$ and
$\zeta_{\epsilon}=\zeta_{\epsilon}(\epsilon_{\hat{n}})$. By the Lie bracket,
the symmetry generator (39) satisfies the Witt algebra,
$\displaystyle
i[\zeta_{\hat{m}},\zeta_{\hat{n}}]_{LB}=({\hat{m}}-{\hat{n}})\zeta_{{\hat{m}}+{\hat{n}}}.\
$ (40)
${\hat{m}},{\hat{n}}$ are just integers. The zero mode is the azimuthal
translation or $U(1)$ isometry, $\zeta_{0}=-\partial_{\phi}$.
The associated conserved charge is [33]
$\displaystyle Q_{\xi}=\frac{1}{8\pi}\int_{\partial\Sigma}k^{g}_{\zeta}[h;g].$
(41)
This given integral is over the boundary of a spatial slice. The contribution
of the metric tensor on the central charge is given explicitly by
$\displaystyle k^{g}_{\zeta}[h;g]$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\epsilon_{\rho\sigma\mu\nu}\bigg{\\{}\zeta^{\nu}D^{\mu}h-\zeta^{\nu}D_{\lambda}h^{\mu\lambda}$
(42)
$\displaystyle+\frac{h}{2}D^{\nu}\zeta^{\mu}-h^{\nu\lambda}D_{\lambda}\zeta^{\mu}+\zeta_{\lambda}D^{\nu}h^{\mu\lambda}$
$\displaystyle\left.+\frac{h^{\lambda\nu}}{2}\left(D^{\mu}\zeta_{\lambda}+D_{\lambda}\zeta^{\mu}\right)\right\\}dx^{\rho}\wedge
dx^{\sigma}.\ $
We should note that the last two terms in Eq. (42) vanish for an exact Killing
vector and an exact symmetry, respectively. The charge $Q_{\zeta}$ generates
symmetry through the Dirac brackets. The ASG possesses algebra which is given
by the Dirac bracket algebra of the following charges [33]
$\displaystyle\\{Q_{\zeta},Q_{\bar{\zeta}}\\}_{DB}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\int
k^{g}_{\zeta}\left[\mathcal{L}_{\bar{\zeta}}g;g\right]$ (43) $\displaystyle=$
$\displaystyle Q_{[\zeta,\bar{\zeta}]}+\frac{1}{8\pi}\int
k^{g}_{\zeta}\left[\mathcal{L}_{\bar{\zeta}}\bar{g};\bar{g}\right].\ \ $
By using (39) and upon quantization, we can transform the Dirac bracket
algebra into a commutation relation that allows us to interpret the classical
central charge as a quantum central charge of the dual CFT. For the
quantization, we replace the classical charges $Q_{\zeta}$ by their quantum
counterpart $L_{{\hat{n}}}$,
$\displaystyle Q_{\zeta}\equiv L_{{\hat{n}}}-x\delta_{{\hat{n}},0},$ (44)
so that we obtain the conserved charges algebra in quantum form, such that
$\displaystyle\left[L_{\hat{m}},L_{\hat{n}}\right]=({\hat{m}}-{\hat{n}})L_{{\hat{m}}+{\hat{n}}}+\frac{c_{L}}{12}\hat{m}({\hat{m}}^{2}-1)\delta_{{\hat{m}}+{\hat{n}},0}.$
(45)
$x$ is a free parameter to scale the central charge on the term $\sim\hat{m}$
(see Appendix B). The relation (40) and (44) have been used to find the
Virasoro algebra above. For the charged rotating solution in Nariai limit
obtained from KSdS solution, the corresponding central charge is then
$\displaystyle c_{L}$ $\displaystyle=$
$\displaystyle\frac{3e}{\Xi}\int^{\pi}_{0}d\theta\sqrt{\Gamma(\theta)\alpha(\theta)\gamma(\theta)}$
(46) $\displaystyle=$
$\displaystyle\frac{12ar_{+}}{1-\frac{6r_{+}^{2}-d^{2}-k^{2}+a^{2}}{l^{2}}}.\
$
The main difference between this central charge and the Nariai solution from
KNdS black hole [9] is the presence of dilaton and axion charges.
#### IV.2.2 Temperature
In the Nariai limit where $r_{c}\rightarrow r_{+}$, the variation of the
entropy can be expressed completely in terms of a variation of the angular
momentum, electric charge, magnetic charge, and the pressure. It takes the
following form
$\displaystyle
dS_{BH}=\frac{dJ}{T_{L}}+\frac{dQ}{T_{q}}+\frac{dP}{T_{p}}+\frac{d\mathcal{P}}{T_{\mathcal{P}}},\
$ (47)
where $T_{L},T_{q},T_{p},T_{\mathcal{P}}$ are the left-moving temperature,
conjugate temperature of electric charge, conjugate temperature of magnetic
charge, and conjugate temperature of pressure, respectively. Hence, we can
derive that
$\displaystyle T_{L}=-\frac{\partial T_{H}/\partial
r_{+}}{\partial\Omega_{H}/\partial
r_{+}}\bigg{|}_{r_{+}=r_{c}}\frac{\upsilon(r_{+}^{2}-d^{2}-k^{2}+a^{2})}{4\pi
ar_{+}\Xi}.\ $ (48)
Since we already take $r_{c}\rightarrow r_{+}$, so the right-moving
temperature is equal to zero which is proportional to the Hawking temperature
on the cosmological horizon. However, in the next section, we will see that
$T_{R}\sim\varepsilon$ generally which is fairly small.
#### IV.2.3 Cardy Entropy
We have calculated the central extension and the corresponding temperature
which are the main ingredients in Cardy entropy. So, the main upshot of this
section is to provide the derivation of Cardy entropy [7] for the charged
rotating solution in Nariai limit. Firstly, Cardy entropy formula is given as
follows
$S_{CFT}=\frac{\pi^{2}}{3}\left(c_{L}T_{L}+c_{R}T_{R}\right).$ (49)
The right-moving part is exactly zero, so only left-moving part contributes.
On the cosmological horizon, by using above formula, we find the following
entropy for charged rotating solution in Nariai limit
$S_{CFT}=\frac{\pi}{\Xi}(r_{+}^{2}-d^{2}-k^{2}+a^{2})=S_{BH}=S_{c}.\ $ (50)
This result exhibits that the Kerr/CFT correspondence can be applied for
charged rotating solution in Nariai limit obtained from KSdS black hole which
precisely possesses $SL(2,R)\times U(1)$ isometry like the near-horizon
extremal black hole. The entropy on the event horizon is now similar with the
entropy on the cosmological horizon since we take $r_{c}\rightarrow r_{+}$.
So, for an observer in the visible region, the total entropy in Nariai limit
is [34]
$S_{tot}=S_{c}+S_{BH}=2S_{c}.\ $ (51)
Note that this is the total entropy for charged rotating solution in Nariai
limit with non-vanishing dyonic charge. So, when $p=0$, it is just the total
entropy for electrically charged one. Furthermore, when all electromagnetic
charges vanish, it will be the total entropy of rotating Nariai solution
obtained from Kerr-dS solution [9]. For the entropy on the event horizon for
extremal KSdS black hole, we calculate in Appendix A where we also prove the
result in Ref. [45] on KSdS solution that is mentioned therein. In case of
asymptotically AdS spacetime, it has been carried out in Ref. [45]. For
vanishing cosmological constant and $p$, this result recovers the result in
Ref. [47].
## V Scattering of Scalar Field and Hidden Conformal Symmetry
In this section, we will further study the scattering of massless scalar field
on the background of charged rotating Nariai solution in $J$ (angular
momentum)- and $Q$ (electric charge)-pictures. We will also compute the
absorption cross-section as well as the real-time correlator on the
corresponding background. Yet, first we will show that there are hidden
conformal symmetries on this solution in Nariai limit. Lastly, we also
consider $Q$-picture where the scalar probe is assumed to be electrically
charged.
### V.1 Scalar wave equation
To explore the hidden conformal symmetries in $J$-picture, we assume a
massless neutral scalar field in the background of charged rotating solution
in Nariai limit. The massless scalar wave equation for the scalar probe is
given by
$\nabla_{\alpha}\nabla^{\alpha}\Phi=0.$ (52)
We know that the charged rotating solution in Nariai limit is conserved under
time-like and azimuthal translations. Hence, we can separate the coordinates
in the scalar wave equation as
$\Phi(\hat{t},\hat{r},\theta,\hat{\phi})=\mathrm{e}^{-i\omega\hat{t}+in\hat{\phi}}R(\hat{r})S(\theta),$
(53)
where $\omega$ and $n$ are the asymptotic energy and angular momentum of the
scalar field. Plugging Eq. (53) into Eq. (52), leads to two differential
equations i.e., the angular $S(\theta)$ and radial $R(\hat{r})$ wave
functions,
$\displaystyle\left[\frac{1}{\sin\theta}\partial_{\theta}(\sin\theta\partial_{\theta})-\frac{n^{2}\Xi^{2}}{\Delta_{\theta}\sin^{2}\theta}\right]S(\theta)$
$\displaystyle+\left[\frac{2an\omega\Xi-a^{2}\omega^{2}\sin^{2}\theta}{\Delta_{\theta}}\right]S(\theta)=-K_{l}S(\theta),\
$ (54)
$\bigg{[}\partial_{\hat{r}}(\Delta\partial_{\hat{r}})+\frac{\left[(\hat{r}^{2}-d^{2}-k^{2}+a^{2})\omega-
an\Xi\right]^{2}}{\Delta}-K_{l}\bigg{]}R(\hat{r})=0,\ $ (55)
where $K_{l}$ is the separation constant. Note that we still use the generic
KSdS metric (1). These equations are identical with the wave equations in the
Kerr-dS solutions. We do not consider the backreaction of the scalar field in
this case.
Firstly, we will study the radial equation (55) at the near region, which is
defined by $\omega\hat{r}\ll 1$. Furthermore, we assume that the frequency of
the scalar field to be very small $\omega M\ll 1$. Consequently, we also
impose $\omega a\ll 1,\,\omega d\ll 1,$ and $\omega k\ll 1$. On the other
hand, we also have $\omega q\ll 1,$ and $\omega p\ll 1$. Since we have
approximated $\Delta$ as (23), we may set up the radial equation in a suitable
form for exploring its hidden conformal symmetry. The radial equation (55)
reduces to
$\displaystyle\partial_{\hat{r}}\left[(\hat{r}-r_{c})(\hat{r}-r_{+})\partial_{\hat{r}}\right]R(\hat{r})+\frac{r_{c}-r_{+}}{\hat{r}-r_{c}}AR(\hat{r})=0,$
$\displaystyle+\left[\frac{r_{c}-r_{+}}{\hat{r}-r_{+}}B+C\right]R(\hat{r})=0,$
(56)
where
$A=\frac{\left[(r_{c}^{2}-d^{2}-k^{2}+a^{2})\omega-
an\Xi\right]^{2}}{\upsilon^{2}(r_{c}-r_{+})^{2}},\ $ (57)
$B=-\frac{\left[(r_{+}^{2}-d^{2}-k^{2}+a^{2})\omega-
an\Xi\right]^{2}}{\upsilon^{2}(r_{c}-r_{+})^{2}},~{}~{}~{}C=\frac{-K_{l}}{\upsilon}.\
$ (58)
It is worth noting that the angular wave equation does not have $SL(2,R)\times
SL(2,R)$ isometry, yet $SU(2)\times SU(2)$ isometry [39]. Hence, we will not
consider to investigate further the angular part.
In order to reveal the hidden symmetries of Eq. (56), we need to perform the
following conformal coordinate transformations [14, 41]
$\displaystyle\omega^{c}=\sqrt{\frac{\hat{r}-r_{c}}{\hat{r}-r_{+}}}e^{2\pi
T_{R}\hat{\phi}+2n_{R}\hat{t}},$
$\displaystyle\omega^{+}=\sqrt{\frac{\hat{r}-r_{c}}{\hat{r}-r_{+}}}e^{2\pi
T_{L}\hat{\phi}+2n_{L}\hat{t}},$
$\displaystyle\hat{y}=\sqrt{\frac{r_{c}-r_{+}}{\hat{r}-r_{+}}}e^{\pi(T_{L}+T_{R})\hat{\phi}+(n_{L}+n_{R})\hat{t}}.\
$ (59)
Then we may define three locally conformal operators in terms of the new
conformal coordinates $\omega^{c},\,\omega^{+}$ and $\hat{y}$ as
$\displaystyle H_{1}=i\partial_{c},$ $\displaystyle
H_{-1}=i\left(\omega^{c2}\partial_{c}+\omega^{c}\hat{y}\partial_{\hat{y}}-\hat{y}^{2}\partial_{+}\right),$
$\displaystyle
H_{0}=i\left(\omega^{c}\partial_{c}+\frac{1}{2}\hat{y}\partial_{\hat{y}}\right),$
(60)
as well as
$\displaystyle\bar{H}_{1}=i\partial_{+},$
$\displaystyle\bar{H}_{-1}=i\left(\omega^{+2}\partial_{+}+\omega^{+}\hat{y}\partial_{\hat{y}}-\hat{y}^{2}\partial_{c}\right),$
$\displaystyle\bar{H}_{0}=i\left(\omega^{+}\partial_{+}+\frac{1}{2}\hat{y}\partial_{\hat{y}}\right).$
(61)
Note that we have used
$\partial_{c}=\partial/\partial\omega^{c},\partial_{+}=\partial/\partial\omega^{+}$.
The set of operators (V.1) satisfies the $SL(2,R)$ Lie algebra
$\displaystyle\left[H_{0},H_{\pm 1}\right]=\mp iH_{\pm
1},~{}~{}~{}\left[H_{-1},H_{1}\right]=-2iH_{0},$ (62)
while a similar $SL(2,R)$ algebra exists for the set of operators (V.1). From
every set of operators, we can construct the quadratic Casimir operator as
given by
$\displaystyle\mathcal{H}^{2}$ $\displaystyle=$
$\displaystyle\bar{\mathcal{H}}^{2}=-H_{0}^{2}+\frac{1}{2}(H_{1}H_{-1}+H_{-1}H_{1})$
(63) $\displaystyle=$
$\displaystyle\frac{1}{4}(\hat{y}^{2}\partial_{\hat{y}}^{2}-y\partial_{\hat{y}})+\hat{y}^{2}\partial_{c}\partial_{+}.$
We have found that the radial equation (V.1) could be re-written in terms of
the $SL(2,R)$ quadratic Casimir operator as
$\mathcal{H}^{2}R(r)=\bar{\mathcal{H}}^{2}R(r)=-CR(r)$ where, in $J$-picture,
we should identify the constants as
$n_{L}=-\frac{\upsilon}{2(r_{c}+r_{+})},~{}~{}~{}n_{R}=0,$ (64)
$T_{L}=\frac{\upsilon(r_{c}^{2}+r_{+}^{2}-2d^{2}-2k^{2}+2a^{2})}{4\pi
a(r_{c}+r_{+})\Xi},~{}~{}~{}T_{R}=\frac{\upsilon(r_{c}-r_{+})}{4\pi a\Xi}.$
(65)
$T_{R,L}$ are identified as the CFT temperatures that emerge as a result of
the spontaneously symmetry breaking of the partition function on
$SL(2,R)\times SL(2,R)$ theory to the partition function of $U(1)\times U(1)$
CFT. One can see that the periodic identification of the azimuthal coordinate
$\hat{\phi}\sim\hat{\phi}+2\pi$ causes the $SL(2,R)\times SL(2,R)$ symmetry to
spontaneously break down to $U(1)\times U(1)$ symmetry by temperatures
$T_{R},T_{L}$,
$\omega^{c}\sim e^{4\pi^{2}T_{R}\omega^{c}},~{}\omega^{+}\sim
e^{4\pi^{2}T_{L}\omega^{+}},~{}\hat{y}\sim e^{2\pi^{2}(T_{L}+T_{L})\hat{y}}.$
(66)
This identification is generated by the $SL(2,R)\times SL(2,R)$ group element,
$e^{-i4\pi^{2}T_{R}H_{0}-i4\pi^{2}T_{L}\bar{H}_{0}}$.
After finding the temperatures (65), we can also compute conjugate charges
$E_{L},E_{R}$ for this charged rotating solution. These conjugate charges can
be obtained from the entropy via [41]
$\delta S_{CFT}=\frac{\delta E_{L}}{T_{L}}+\frac{\delta E_{R}}{T_{R}}.$ (67)
In order to compute the conjugate charges, we consider the first law of
thermodynamics for the charged rotating solution in Nariai limit (7) with the
quantities (13)-(17). Since we consider the neutral scalar field, we have
$\delta Q=\delta P=\delta\mathcal{P}=0$. Now we can find the conjugate charges
via $\delta S_{BH}=\delta S_{CFT}$. Using the identification $\delta M$ as
$\omega$ and $\delta J$ as $n$ yields to the identification of $\delta
E_{L,R}$ as $\omega_{L,R}$. Hence, in $J$-picture, we obtain that
$\displaystyle\omega_{L}=\frac{r_{c}^{2}+r_{+}^{2}-2d^{2}-2k^{2}+2a^{2}}{2a\Xi}\omega,~{}\omega_{R}=\omega_{L}-n.\
$ (68)
So, we find the left and right frequencies of 2D CFT for the charged rotating
solution in Nariai limit for KSdS black hole in $J$-picture. For the study of
the scalar field in KSAdS black hole, one can see in Ref. [18].
### V.2 Absorption cross-section in $J$-picture
In the previous section, we have derived one remarkable result of the dual
CFT, i.e. the entropy. Another realization of this duality is the equivalence
of the absorption cross-section of the scalar probe. In investigating the
absorption cross-section, we need to consider the asymptotic region. Since we
have approximated $\Delta$ in near-horizon region, this approximation in the
asymptotic region will break down, except we consider the Nariai limit. This
is similar in the near-extremal case of the scalar wave equation [15, 16, 20,
38, 42]. Beside the Nariai limit on (18), we consider the near-horizon
coordinate transformations (19). We also consider the scalar probe with
frequencies around the superradiant bound
$\omega=n\Omega_{H}+\hat{\omega}\frac{\lambda}{r_{0}},$ (69)
where $\Omega_{H}$ are given by (10). We can re-write the radial equation
(V.1) by
$\left(\partial_{r}\left[r\left(r-\varepsilon\right)\right]\partial_{r}+\frac{A_{s}}{r-\varepsilon}+\frac{B_{s}}{r}+C_{s}\right)R(r)=0,$
(70)
where
$\displaystyle
A_{s}=\frac{(\hat{\omega}+2nr_{+}\varepsilon\Omega_{H})^{2}}{\upsilon^{2}\varepsilon},~{}~{}~{}B_{s}=-\frac{\hat{\omega}^{2}}{\upsilon^{2}\varepsilon},~{}~{}~{}C_{s}=C_{s}(\hat{\omega}),\
$
and $C_{s}$ is the new separation constant that is dependent on the frequency
$\hat{\omega}$ that can be obtained by solving the angular wave equation. We
then apply the coordinate transformation $z=(r-\varepsilon)/r$. In this new
radial coordinate, the radial equation (70) becomes
$\left[z(1-z)\partial_{z}^{2}+(1-z)\partial_{z}+\frac{\hat{A_{s}}}{z}+\hat{B_{s}}+\frac{C_{s}}{1-z}\right]R(z)=0,$
(71)
where
$\displaystyle\hat{A_{s}}=\frac{(\hat{\omega}+2nr_{+}\varepsilon\Omega_{H})^{2}}{\upsilon^{2}\varepsilon^{2}},~{}~{}~{}\hat{B_{s}}=-\frac{\hat{\omega}^{2}}{\upsilon^{2}\varepsilon^{2}}.\
$
The ingoing solution to differential equation (71) is also given by
hypergeometric function
$R(z)=z^{-i\sqrt{\hat{A_{s}}}}(1-z)^{1+h}~{}_{2}F_{1}(a_{s},b_{s};c_{s};z),\ $
(72)
with the parameters $a_{s}=1+h-i(\sqrt{\hat{A}_{s}}+\sqrt{-\hat{B}_{s}})$,
$b_{s}=1+h-i(\sqrt{\hat{A}_{s}}-\sqrt{-\hat{B}_{S}})$, and
$c_{s}=1-2i\sqrt{\hat{A}_{s}}$. For this superradiant case, the relation
between $h$ and $C_{s}$ is given by
$h=\frac{1}{2}\left(-1+\sqrt{1-4C_{s}}\right).$ (73)
For the asymptotic region of the radial coordinate $\hat{r}$ (or equivalently
$r\gg\varepsilon$), where $z\sim 1$, the solution (72) reduces to
$\displaystyle R(y)\sim D_{0}r^{h}+D_{1}r^{-1-h},\ $ (74)
where
$D_{0}=\frac{\Gamma(c_{s})\Gamma(1+2h)}{\Gamma(a_{s})\Gamma(b_{s})},~{}~{}~{}D_{1}=\frac{\Gamma(c_{s})\Gamma(-1-2h)}{\Gamma(c_{s}-a_{s})\Gamma(c_{s}-b_{s})}.\
$ (75)
The conformal weight of the scalar field is equal to $h+1$.
For the coefficient (75), we find the absorption cross-section of the scalar
fields as
$P_{abs}\sim\left|D_{0}\right|^{-2}\sim\sinh\left({2\pi\hat{A}_{s}^{1/2}}\right){\left|{\Gamma\left(a_{s}\right)}{\Gamma\left(b_{s}\right)}\right|^{2}}.$
(76)
To be more supporting the correspondence between the charged rotating solution
in Nariai limit and 2D CFT, we show that the absorption cross-section for the
scalar fields (76) can be obtained from the absorption cross-section in a 2D
CFT [14]
$\displaystyle P_{abs}$ $\displaystyle\sim$
$\displaystyle{T_{L}}^{2h_{L}-1}{T_{R}}^{2h_{R}-1}\sinh\left({\frac{{\omega_{L}}}{{2{T_{L}}}}+\frac{{\omega_{R}}}{{2{T_{R}}}}}\right)$
(77) $\displaystyle\times$
$\displaystyle\left|{\Gamma\left({h_{L}+i\frac{{\omega_{L}}}{{2\pi{T_{L}}}}}\right)}{\Gamma\left({h_{R}+i\frac{{\omega_{R}}}{{2\pi{T_{R}}}}}\right)}\right|^{2}.\
$
The agreement between (76) and (77) can be shown when we choose proper left
and right frequencies $\omega_{L},\omega_{R}$. Since in the previous
subsection we have found the CFT temperatures and frequencies, we can directly
calculate the similar quantities for the frequency of the scalar field in the
near-superradiant bound by neglecting the second-order and higher corrections
from $\lambda$. Hence, in terms of Nariai limit parameter and in near-
superradiant bound, we obtain
$T_{L}=\frac{\upsilon}{4\pi\Omega_{H}r_{+}},~{}~{}~{}T_{R}=\frac{\upsilon\lambda
r_{0}}{4\pi a\Xi}\varepsilon.$ (78)
When we take the Nariai limit $\varepsilon\rightarrow 0$ the right-moving
temperature will vanish. Then we recover the result for $T_{R}$ and $T_{L}$ in
the previous section. The frequencies are now
$\omega_{L}=n,~{}~{}~{}\omega_{R}=\frac{r_{0}}{a\Xi}\left(\hat{\omega}+nr_{+}\varepsilon\Omega_{c}\right),\
$ (79)
The conformal weights in CFT are given by
$h_{L}=h_{R}=h+1.\ $ (80)
This gives rise to another nontrivial evidence to support the Kerr/CFT
correspondence for black holes in Nariai limit.
### V.3 Real-time correlator in $J$-picture
Furthermore, one can also compute the real-time correlator. The asymptotic
behaviors of the scalar field with ingoing boundary condition on the
background of charged rotating solution in Nariai limit (74) indicate that two
coefficients possess different roles where $D_{1}$ indicates the response and
$D_{0}$ indicates the source. Hence, the two-point retarded correlator is
simply [43, 44]
$G_{R}\sim\frac{D_{1}}{D_{0}}=\frac{\Gamma(-1-2h)}{\Gamma(1+2h)}\frac{\Gamma(a_{s})\Gamma(b_{s})}{\Gamma(c_{s}-a_{s})\Gamma(c_{s}-b_{s})}.\
$ (81)
From Eq. (81), it is easy to check that
$G_{R}\sim\frac{\Gamma(h_{L}-i\frac{\omega_{L}}{2T_{L}})\Gamma(h_{R}-i\frac{\omega_{R}}{2T_{R}})}{\Gamma(1-h_{L}-i\frac{\omega_{L}}{2T_{L}})\Gamma(1-h_{R}-i\frac{\omega_{R}}{2T_{R}})}.$
(82)
Then by using the relation $\Gamma(z)\Gamma(1-z)=\pi/\sin(\pi z)$, we can
write above two-point retarded correlator into
$\displaystyle G_{R}$ $\displaystyle\sim$ $\displaystyle\sin\left(\pi
h_{L}+i\frac{\omega_{L}}{2T_{L}}\right)\sin\left(\pi
h_{R}+i\frac{\omega_{R}}{2T_{R}}\right)$ (83)
$\displaystyle\bigg{|}\Gamma\left(h_{L}+i\frac{\omega_{L}}{2T_{L}}\right)\Gamma\left(h_{R}+i\frac{\omega_{R}}{2T_{R}}\right)\bigg{|}^{2}.$
Since $h_{L},h_{R}$ are integers, so we have
$\displaystyle\sin\left(\pi
h_{L}+i\frac{\omega_{L}}{2T_{L}}\right)\sin\left(\pi
h_{R}+i\frac{\omega_{R}}{2T_{R}}\right)$
$\displaystyle=(-1)^{h_{L}+h_{R}}\sin\left(i\frac{\omega_{L}}{2T_{L}}\right)\sin\left(i\frac{\omega_{R}}{2T_{R}}\right).\
$ (84)
In CFT, the Euclidean correlator is given by
$\displaystyle G_{E}(\omega_{EL},\omega_{ER})\sim
T_{L}^{2h_{L}-1}T_{R}^{2h_{R}-1}e^{i\omega_{EL}/2T_{L}}e^{i\omega_{ER}/2T_{R}}$
$\displaystyle\times\bigg{|}\Gamma\left(h_{L}+\frac{\omega_{EL}}{2T_{L}}\right)\Gamma\left(h_{R}+\frac{\omega_{EL}}{2T_{R}}\right)\bigg{|}^{2}.$
(85)
Where we can define the Euclidean frequencies $\omega_{EL}=i\omega_{L}$, and
$\omega_{ER}=i\omega_{R}$. It is important noting that $G_{E}$ corresponds to
the values of retarded correlator $G_{R}$. The retarded Green function $G_{R}$
is analytic on the upper half complex $\omega_{L,R}$-plane. The value of
$G_{R}$ along the positive imaginary $\omega_{L,R}$-axis gives the following
correlator
$G_{E}(\omega_{EL},\omega_{ER})=G_{R}(i\omega_{L},i\omega_{R}),~{}~{}~{}\omega_{EL,ER}>0.$
(86)
At finite temperature, $\omega_{EL,ER}$ should take discrete values of the
Matsubara frequencies, given by
$\omega_{EL}=2\pi m_{L}T_{L},~{}~{}~{}\omega_{ER}=2\pi m_{R}T_{R},$ (87)
where $m_{L}$ and $m_{R}$ are half integers for fermionic modes and integers
for bosonic modes. At these certain frequencies, the gravity computation for
correlator (83) matches precisely with CFT result trough Eq. (86) up to a
numerical normalization factor.
### V.4 Hidden conformal symmetry in $Q$-Picture
In the previous subsections, we have shown the existence of hidden conformal
symmetry on the $J$-picture with the background charged rotating black hole in
Nariai limit. In this subsection, we will further study the scattering of
massless scalar field on the background of charged rotating black hole
solution in Nariai limit in $Q$-picture. On the other hand, we will consider
charged scalar field on the background solution. For some black hole
backgrounds, $Q$-picture has been explored well to exhibit the existence of
the conformal symmetries [15, 16, 37, 38, 35, 36]. Nonetheless, the
$Q$-picture is not well-defined for non-extremal Kerr-Sen black holes [35]. In
this section, we will investigate the $Q$-picture when the cosmological
constant exists. Since we will only consider $Q$-picture, for simplicity, we
may assume that $p=0$, so that $k=0$.
The hidden conformal symmetries can be explored by assuming a massless charged
scalar probe in the background of charged rotating black hole solution in
Nariai limit. It is given by
$(\nabla_{\alpha}-igA_{\alpha})(\nabla^{\alpha}-igA^{\alpha})\Phi=0,$ (88)
where $g$ is the electric charge of the scalar probe. In $Q$-picture, in
addition to the modes of asymptotic energy and angular momentum, we need to
add the charge $g$ as the eigenvalue of the operator $\partial_{\chi}$ that
denotes the additional internal direction $\chi$ to four dimensions. Actually,
$g$ has a natural geometrical interpretation as the radius of extra circle
when the the black hole solution is considered to be embedded into 5D. In
fact, this coordinate possesses similar $U(1)$ symmetry like the azimuthal
coordinate. However, so far, we do not find that 4D Kerr-Sen(-dS) black hole
can be uplifted to 5D solution. It is different with Kerr-Newman(-dS) black
hole that can be uplifted to 5D [22]. Here, we will just first assume the
fifth coordinate to reveal the conformal symmetries of the charged probe in
$Q$-picture. The ansatz in $Q$-picture is given by
$\Phi(\hat{t},\hat{r},\theta,\hat{\phi},\chi)=\mathrm{e}^{-i\omega\hat{t}+in\hat{\phi}+ig\chi}R(\hat{r})S(\theta).$
(89)
Beside the low-frequency assumption, we need to assume also that the probe’s
charge to be very small ($gq\ll 1$). By plugging Eq. (89) into Eq. (88) and
assuming small charge and low frequency, in the near-horizon region, we can
find the radial equation
$\displaystyle\partial_{\hat{r}}\left[(\hat{r}-r_{c})(\hat{r}-r_{+})\partial_{\hat{r}}\right]R(\hat{r})+\frac{r_{c}-r_{+}}{\hat{r}-r_{c}}AR(\hat{r})=0,$
$\displaystyle+\left[\frac{r_{c}-r_{+}}{\hat{r}-r_{+}}B+C\right]R(\hat{r})=0,$
(90)
where
$B=-\frac{\left[(r_{+}^{2}-d^{2}+a^{2})\omega-an\Xi-
gq(r_{+}+d)\right]^{2}}{\upsilon^{2}(r_{c}-r_{+})^{2}},\ $ (91)
$A=\frac{\left[(r_{c}^{2}-d^{2}+a^{2})\omega-an\Xi-
gq(r_{c}+d)\right]^{2}}{\upsilon^{2}(r_{c}-r_{+})^{2}},~{}~{}~{}C=\frac{-K_{l}}{\upsilon}.\
$ (92)
In $Q$-picture, we take $n=0$. As the previous computation for $J$-picture, we
can identify that
$n_{L}=-\frac{\upsilon(r_{c}+r_{+})}{4\left[(r_{c}+d)(r_{+}+d)-a^{2}\right]},\
$ (93)
$n_{R}=-\frac{\upsilon(r_{c}-r_{+})}{4\left[(r_{c}+d)(r_{+}+d)-a^{2}\right]},$
(94) $T_{L}=\frac{\upsilon(r_{c}^{2}+r_{+}^{2}-2d^{2}+2a^{2})}{4\pi
g\left[(r_{c}+d)(r_{+}+d)-a^{2}\right]},\ $ (95)
$T_{R}=\frac{\upsilon(r^{2}_{c}-r^{2}_{+})}{4\pi
g\left[(r_{c}+d)(r_{+}+d)-a^{2}\right]}.$ (96)
The temperatures (96) are irregular. We cannot check directly that
$(r_{c}+d)(r_{+}+d)-a^{2}=0$ because we do not have the explicit analytical
form of $r_{c},r_{+}$. However, we can easily check that the form of CFT
temperatures is similar with the case when the cosmological constant vanishes.
So, when $1/l^{2}=0$, one can obtain
$(r_{+}+d)(r_{-}+d)-a^{2}=0,\ $ (97)
where
$r_{\pm}=M\pm\sqrt{M^{2}+d^{2}-a^{2}-q^{2}}.$ (98)
This is the reason that $Q$\- picture is not well-defined for Kerr-Sen black
hole. With the similar form of temperatures, we conclude also that $Q$-picture
for Kerr-Sen-dS black hole in Nariai limit is not well-defined.
## VI Hidden Conformal Symmetry on Nariai limit for Kerr-Newman-de Sitter
solution
In the previous sections, we have carried out the calculation of the dual CFT
for charged rotating black hole in Nariai limit obtained from KSdS solution
and also the hidden conformal symmetry of the scalar probe in the black hole’s
background. In this section, we will study the scattering of scalar field on
Nariai limit for KNdS solution both in $J$\- and $Q$-pictures. Yet, we will
revisit the computation of the entropy from CFT for KNdS solution in Nariai
limit. The KNdS black hole solution is given by the following metric [22, 23]
$\displaystyle
ds^{2}=-\frac{\Delta}{\varrho^{2}}X^{2}+\frac{\varrho^{2}}{\Delta}d\hat{r}^{2}+\frac{\varrho^{2}}{\Delta_{\theta}}d\theta^{2}+\frac{\Delta_{\theta}\sin^{2}\theta}{\varrho^{2}}Y^{2},\
$ (99)
where
$X=d\hat{t}-\frac{a\sin^{2}\theta}{\Xi}d\hat{\phi},~{}~{}~{}Y=ad\hat{t}-\frac{(\hat{r}^{2}+a^{2})}{\Xi}d\hat{\phi},\
$ (100)
$\Delta_{\theta}=1+\frac{a^{2}}{l^{2}}\cos^{2}\theta,~{}~{}\Xi=1+\frac{a^{2}}{l^{2}},~{}~{}~{}\varrho^{2}=\hat{r}^{2}+a^{2}\cos^{2}\theta,\\\
$ (101)
$\Delta=(\hat{r}^{2}+a^{2})\left(1-\frac{\hat{r}^{2}}{l^{2}}\right)-2m\hat{r}+p^{2}+q^{2}.\
$ (102)
The parameters $a$, $m$, $p$, $q$, and $l$ are spin, mass, magnetic charge,
electric charge, and de Sitter radius, respectively. The electromagnetic
potential and its dual are given by
$\displaystyle\textbf{A}=-\frac{q\hat{r}}{\varrho^{2}}X-\frac{p\cos\theta}{\varrho^{2}}Y,~{}~{}~{}\textbf{B}=-\frac{p\hat{r}}{\varrho^{2}}X+\frac{q\cos\theta}{\varrho^{2}}Y,$
(103)
At the event horizon, the thermodynamic quantities are given by [22, 23]
$M=\frac{m}{\Xi},~{}~{}~{}J=\frac{ma}{\Xi},~{}~{}~{}Q=\frac{q}{\Xi},~{}~{}~{}P=\frac{p}{\Xi},$
(104)
$T_{H}=\frac{r_{+}(l^{2}-2r_{+}^{2}-a^{2})-ml^{2}}{2\pi(r_{+}^{2}+a^{2})l^{2}},$
(105) $S_{BH}=\frac{\pi}{\Xi}(r_{+}^{2}+a^{2}),$ (106)
$\Omega_{H}=\frac{a\Xi}{r_{+}^{2}+a^{2}},~{}~{}\Phi_{H}=\frac{qr_{+}}{r_{+}^{2}+a^{2}},~{}~{}\Psi_{H}=\frac{pr_{+}}{r_{+}^{2}+a^{2}},$
(107) $V=\frac{4}{3}r_{+}S_{BH},~{}~{}~{}\mathcal{P}=-\frac{3}{8\pi l^{2}}.$
(108)
where those are physical mass, angular momentum, electric charge, magnetic
charge, Hawking temperature, Bekenstein-Hawking entropy, angular velocity,
electric potential, magnetic potential, thermodynamic volume and pressure.
These thermodynamic quantities also satisfy the relation (7). Furthermore, one
can find the similar relation for thermodynamics on the cosmological horizon
with the following quantities [24, 25]
$M_{c}=-\frac{m}{\Xi},~{}~{}~{}J_{c}=-\frac{ma}{\Xi},~{}~{}~{}Q_{c}=-\frac{q}{\Xi},~{}~{}~{}P_{c}=-\frac{p}{\Xi},$
(109)
$T_{c}=\frac{r_{c}(2r_{c}^{2}+a^{2}-l^{2})+ml^{2}}{2\pi(r_{c}^{2}+a^{2})l^{2}},$
(110) $S_{c}=\frac{\pi}{\Xi}(r_{c}^{2}+a^{2}),$ (111)
$\Omega_{c}=\frac{a\Xi}{r_{c}^{2}+a^{2}},~{}~{}\Phi_{c}=\frac{qr_{c}}{r_{c}^{2}+a^{2}},~{}~{}\Psi_{c}=\frac{pr_{c}}{r_{c}^{2}+a^{2}},$
(112) $V_{c}=\frac{4}{3}r_{c}S_{c},~{}~{}~{}\mathcal{P}_{c}=\frac{3}{8\pi
l^{2}}.$ (113)
These thermodynamic quantities are obtained also by considering the event
horizon of the black hole as the boundary. Likewise the KSdS solution, the
cosmological constant can be assumed as a dynamical quantity.
### VI.1 Nariai limit on Kerr-Newman-de Sitter black hole revisited
On the Nariai limit, likewise the KSdS solution, we can find the geometry
which is fiber over AdS2. Using the similar transformations (18) and (19) on
the spacetime metric (1), we can obtain
$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\Gamma(\theta)\left(-r(r-\epsilon)dt^{2}+\frac{dr^{2}}{r(r-\epsilon)}+\alpha(\theta)d\theta^{2}\right)$
(114) $\displaystyle+\gamma(\theta)\left(d\phi+erdt\right)^{2},\ $
where the metric functions are given by
$\Gamma(\theta)=\frac{\varrho_{+}^{2}}{\upsilon},~{}~{}\alpha(\theta)=\frac{\upsilon}{\Delta_{\theta}},~{}~{}\gamma(\theta)=\frac{r_{0}^{4}\Delta_{\theta}\sin^{2}\theta}{\varrho_{+}^{2}\Xi^{2}},\
$ (115)
$\varrho_{+}^{2}=r_{+}^{2}+a^{2}\cos^{2}\theta,~{}~{}e=\frac{2ar_{+}\Xi}{r_{0}^{2}\upsilon}.$
(116)
and now we have $r_{0}^{2}=r_{+}^{2}+a^{2}$ and
$\upsilon=1-(6r_{+}^{2}+a^{2})/l^{2}$. When the spin vanishes, the spacetime
becomes AdS${}_{2}\times S^{2}$.
Even we have different slice in the asymptotic region, by using similar
analysis of ASG, we can obtain the exact central charge as found in [9]. It is
precisely given by
$c_{L}=\frac{12ar_{+}}{1-\frac{6r_{+}^{2}+a^{2}}{l^{2}}}.\ $ (117)
The main different of this central charge with that of the solution in Nariai
limit obtained from KSdS solution is the presence of the dilaton and axion
charges. The temperature can be calculated directly as the previous result for
Nariai limit on KNdS solution. We can find that
$\displaystyle T_{L}=\frac{\upsilon(r_{+}^{2}+a^{2})}{4\pi
ar_{+}\Xi},~{}~{}T_{R}=0.\ $ (118)
We have obtained the corresponding central charge and the CFT temperature for
the solution in Nariai limit obtained from KNdS black hole. On the
cosmological horizon, by using Cardy formula, we find the following entropy
for the charged rotating black hole from KNdS black hole in Nariai limit,
$S_{CFT}=\frac{\pi}{\Xi}(r_{+}^{2}+a^{2})=S_{BH}=S_{c}.\ $ (119)
So, in this case, the total entropy will be $2S_{c}$.
### VI.2 Hidden conformal symmetry in $J$-picture
For the charged rotating solution in Nariai limit obtained from KNdS black
hole, we also assume the massless neutral scalar probe. We also consider the
low-frequency limit for the scalar field to exhibit the conformal symmetry.
For the small frequency, we have $\omega M\ll 1$, $\omega a\ll 1,\,\omega q\ll
1,$ and $\omega p\ll 1$. The radial equation is then given by
$\displaystyle\partial_{\hat{r}}\left[(\hat{r}-r_{c})(\hat{r}-r_{+})\partial_{\hat{r}}\right]R(\hat{r})+\frac{r_{c}-r_{+}}{\hat{r}-r_{c}}AR(\hat{r})=0,$
$\displaystyle+\left[\frac{r_{c}-r_{+}}{\hat{r}-r_{+}}B+C\right]R(\hat{r})=0,$
(120)
where
$A=\frac{\left[(r_{c}^{2}+a^{2})\omega-
an\Xi\right]^{2}}{\upsilon^{2}(r_{c}-r_{+})^{2}},\ $ (121)
$B=-\frac{\left[(r_{+}^{2}+a^{2})\omega-
an\Xi\right]^{2}}{\upsilon^{2}(r_{c}-r_{+})^{2}},~{}~{}~{}C=\frac{-K_{l}}{\upsilon}.\
$ (122)
The investigation of hidden conformal symmetry for KNdS solution is also done
in Ref. [40], however, the author does not consider Nariai limit. The
conformal symmetry can be revealed using the conformal coordinates
transformations (59) which similarly result in $SL(2,R)\times SL(2,R)$
isometry for the set of operators (V.1) and (V.1). Using the similar
computation, we can calculate the temperatures. In this case, we obtain
$n_{L}=-\frac{\upsilon}{2(r_{c}+r_{+})},~{}~{}~{}n_{R}=0,$ (123)
$T_{L}=\frac{\upsilon(r_{c}^{2}+r_{+}^{2}+2a^{2})}{4\pi
a(r_{c}+r_{+})\Xi},~{}~{}~{}T_{R}=\frac{\upsilon(r_{c}-r_{+})}{4\pi a\Xi}.$
(124)
Note that those quantities are different to those of the Nariai solution
obtained from KSdS solution since there exist dilaton and axion charges. When
we take $\varepsilon\rightarrow 0$, the temperatures (124) reduce to (118).
After computing the CFT temperatures (124), we will compute the conjugate
charges $E_{L},E_{R}$ for this solution. These conjugate charges can be
obtained from the entropy via Eq. (67) by considering the first law of
thermodynamics for the charged rotating black hole solution (7) with the
quantities (109)-(113). For the neutral scalar field, we have $\delta Q=\delta
P=\delta\mathcal{P}=0$. Using again the identification $\delta M$ as $\omega$
and $\delta J$ as $n$ yields to the identification of $\delta E_{L,R}$ as
$\omega_{L,R}$. We obtain the following left and right frequencies,
$\omega_{L}=\frac{r_{c}^{2}+r_{+}^{2}+2a^{2}}{2a\Xi}\omega,~{}~{}~{}\omega_{R}=\omega_{L}-n.\
$ (125)
These are the left and right frequencies of 2D CFT for the charged rotating
solution for KNdS solution in Nariai limit.
Regarding the absorption cross-section and real-time correlator, we need to
employ the radial equation (120). Similarly with KNdS case, the approximation
of $\Delta$ in the asymptotic region will break down, except we consider the
Nariai limit which is identical to near-extremal limit. We also consider the
scalar probe with frequencies around the superradiant bound (69). With the
similar lengthy computation, we find the absorption cross-section of the
scalar fields as given in Eq. (76) which is precisely similar to (77) for the
charged rotating solution in Nariai limit obtained from KNdS solution. Once
again, the agreement between (76) and (77) for this solution can be shown when
we choose proper left and right frequencies $\omega_{L},\omega_{R}$ as given
in Eq. (125). In the superradiant bound, we also find the form of temperatures
(78) and the frequencies (79) with the conformal weights (80). Yet, we have to
note again that in the solution coming from Einstein-Maxwell theory, there is
no contribution from $d,k$ on $r_{+}$. We can also compute the real-time
correlator. This is given by Eq. (82). This further exhibits that the Kerr/CFT
correspondence for KNdS black hole in Nariai limit is valid likewise the
solution from KSdS black hole.
### VI.3 Hidden conformal symmetry in $Q$-picture
In exploring the hidden conformal symmetries on the wave equation, again, we
assume a massless charged scalar probe in the background (99) in Nariai limit
as given by Eq. (88). By plugging Eq. (89) into Eq. (88) and assuming small
charge and low frequency, in the near-horizon region of (99) in Nariai limit,
we can find the radial equation
$\displaystyle\partial_{\hat{r}}\left[(\hat{r}-r_{c})(\hat{r}-r_{+})\partial_{\hat{r}}\right]R(\hat{r})+\frac{r_{c}-r_{+}}{\hat{r}-r_{c}}AR(\hat{r})=0,$
$\displaystyle+\left[\frac{r_{c}-r_{+}}{\hat{r}-r_{+}}B+C\right]R(\hat{r})=0,$
(126)
where
$B=-\frac{\left[(r_{+}^{2}+a^{2})\omega-an\Xi-
gqr_{+}\right]^{2}}{\upsilon^{2}(r_{c}-r_{+})^{2}},\ $ (127)
$A=\frac{\left[(r_{c}^{2}+a^{2})\omega-an\Xi-
gqr_{c}\right]^{2}}{\upsilon^{2}(r_{c}-r_{+})^{2}},~{}~{}~{}C=\frac{-K_{l}}{\upsilon}.\
$ (128)
For this charged rotating solution obtained from KNdS black hole in Nariai
limit, in $Q$-picture, we can identify that
$n_{L}=-\frac{\upsilon(r_{c}+r_{+})}{4\left(r_{c}r_{+}-a^{2}\right)},~{}n_{R}=-\frac{\upsilon(r_{c}-r_{+})}{4\left(r_{c}r_{+}-a^{2}\right)},$
(129) $T_{L}=\frac{\upsilon(r_{c}^{2}+r_{+}^{2}+2a^{2})}{4\pi
g\left(r_{c}r_{+}-a^{2}\right)},~{}T_{R}=\frac{\upsilon(r^{2}_{c}-r^{2}_{+})}{4\pi
g\left(r_{c}r_{+}-a^{2}\right)}.$ (130)
These CFT temperatures are regular, unlike to that of KSdS solution in Nariai
limit.
Since there exists electric charge of the scalar probe, in order to satisfy
the entropy relation (67), we need the conjugate of the electric charge,
namely the chemical potential ($\mu_{L,R}$), so that
$E_{L,R}=\hat{\omega}_{L,R}=\omega_{L,R}-\mu_{L,R}q_{L,R}.\ $ (131)
So, in $Q$-picture, the charged scalar probe on the charged rotating solution
obtained from KNdS black hole in Nariai limit is related to the following CFT
frequencies,
$\displaystyle\omega_{L}$ $\displaystyle=$
$\displaystyle\frac{(r_{c}+r_{+})(r_{c}^{2}+r_{+}^{2}+2a^{2})}{2g(r_{c}r_{+}-a^{2})}\omega,$
$\displaystyle\mu_{L}$ $\displaystyle=$
$\displaystyle\frac{r_{c}^{2}+r_{+}^{2}+2a^{2}}{2(r_{c}r_{+}-a^{2})},~{}~{}~{}q_{L}=q,$
$\displaystyle\omega_{R}$ $\displaystyle=$
$\displaystyle\omega_{L}-\frac{2a(r_{c}+r_{+})}{2g(r_{c}r_{+}-a^{2})}n,$
$\displaystyle\mu_{R}$ $\displaystyle=$
$\displaystyle\frac{(r_{c}+r_{+})^{2}}{2(r_{c}r_{+}-a^{2})},~{}~{}~{}q_{R}=q.\
$ (132)
One can apply the frequencies (131), temperatures (130), and conformal weights
(80) in order to compute the absorption cross-section and real-time
correlator. Note that to study the superradiant bound in $Q$-picture, we have
assume the frequency near the superradiant bound
$\omega=n\Omega_{H}+g\Phi_{H}+\hat{\omega}\frac{\lambda}{r_{0}}$.
## VII Summary
In this work, we have shown the horizon solutions and extremal limits in Fig.
1. The Nariai limit could be achieved in some values of parameters. We then
have carried out the calculation of the entropy for KSdS solutions in Nariai
limit parameterized by a constant $\varepsilon$. When $\varepsilon\rightarrow
0$, the cosmological horizon and event horizon coincide. This solution in
Nariai limit can be written in the metric form with AdS2 structure. Hence, we
have shown that, in the Nariai limit, we could also portray the solution as a
fiber over AdS2, instead of a fiber over dS2 as exhibited in Ref. [9]. We have
computed the corresponding central charge and CFT temperature where the right-
moving temperature is proportional to $\varepsilon$ denoting that it will
vanish when $\varepsilon\rightarrow 0$. It is found that by employing Cardy
entropy formula, the Bekenstein-Hawking entropy on the cosmological constant
is reproduced. It denotes that the charged rotating solutions from KSdS black
hole in Nariai limit is holographically dual with 2D CFT.
To further support the CFT dual on KSdS black hole in Nariai limit, we have
investigated the neutral ($J$-picture) and charged ($Q$-picture) massless
scalar probes on that background. Similarly with generic rotating black hole,
we could exhibit the conformal symmetries on the radial wave equation. With
the appropriate locally conformal coordinate transformations, it has been
shown that the radial equation possesses $SL(2,R)\times SL(2,R)$ isometry.
This is similar with AdS3 space. The periodic identification of azimuthal
coordinate portrays the spontaneous symmetry breaking from $SL(2,R)\times
SL(2,R)$ to $U(1)\times U(1)$ by the left- and right-moving temperatures. In
$J$-picture, the temperatures produced on this symmetry breaking are precisely
similar with the temperatures that are obtained when the conformal symmetry
appears directly on the spacetime metric. Hence, if we employ the central
charges with the given temperatures, we can again reproduce the Bekenstein-
Hawking entropy on the cosmological horizon. Moreover, we also have computed
the absorption cross-section and the real-time correlator that correspond with
the KSdS solution in Nariai limit. However, in $Q$-picture, we coud not find a
well-defined CFT description because the temperatures are irregular.
We have also extended the calculation to the KNdS black hole in Nariai limit
both in $J$\- and $Q$-pictures. The CFT description in both pictures are well-
defined. The results from gravity calculation are also exactly in agreement
with the result from 2D CFT. So, this calculation is another proof that the
black hole solution in Nariai limit is holographically dual to 2D CFT.
## Appendix A Entropy on the Event Horizon for Extremal KSdS solution
We will briefly derive the entropy of extremal KSdS solution in this section.
The entropy of the gauged dyonic Kerr-Sen black hole solution or KSAdS
solution has been calculated previously in Ref. [45]. For the asymptotically
de Sitter solution, we just need to follow the computation given in Ref. [45]
where therein, we have argued on the central charge for KSdS solution by
taking $l^{2}\rightarrow-l^{2}$. It has been noted in Ref. [45] that in the
extremal case of KSAdS solution, the mass can have more than two branches as
it should be similar to KSdS solution. Let start directly from the near-
horizon extremal form of KSdS solution,
$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\Gamma(\theta)\left(-r^{2}dt^{2}+\frac{dr^{2}}{r^{2}}+\alpha(\theta)d\theta^{2}\right)$
(133) $\displaystyle+\gamma(\theta)\left(d\phi+erdt\right)^{2},\ $
where the metric functions are given by
$\Gamma(\theta)=\frac{\varrho_{+}^{2}}{\upsilon},~{}~{}~{}\alpha(\theta)=\frac{\upsilon}{\Delta_{\theta}},~{}~{}~{}\gamma(\theta)=\frac{r_{0}^{4}\Delta_{\theta}\sin^{2}\theta}{\varrho_{+}^{2}\Xi^{2}},\
$ (134)
$\varrho_{+}^{2}=r_{+}^{2}-d^{2}-k^{2}+a^{2}\cos^{2}\theta,~{}~{}~{}e=\frac{2ar_{+}\Xi}{r_{0}^{2}\upsilon}.\
$ (135) $\upsilon=1-\frac{6r_{+}^{2}-2d^{2}-2k^{2}+a^{2}}{l^{2}}.$ (136)
In this case, we can precisely derive the corresponding central charge as
$\displaystyle c_{L}$ $\displaystyle=$
$\displaystyle\frac{3e}{\Xi}\int^{\pi}_{0}d\theta\sqrt{\Gamma(\theta)\alpha(\theta)\gamma(\theta)}$
(137) $\displaystyle=$
$\displaystyle\frac{12ar_{+}}{1-\frac{6r_{+}^{2}-d^{2}-k^{2}+a^{2}}{l^{2}}}.\
$
Note that although this central charge looks identical with that of in Nariai
limit, both central charges are differnt because we are considering different
limit on the horizons. When the magnetic charge vanishes, one can obtain the
central charge that corresponds to non-dyonic KSAdS solution. Furthermore,
when the cosmological constant vanishes, the central charge recovers the
central charge of Kerr-Sen black hole [47, 35]. For the CFT temperature, we
just need to follow Ref. [45]. For KSdS solution, we can find
$\displaystyle T_{L}=-\frac{\partial T_{H}/\partial
r_{+}}{\partial\Omega_{H}/\partial
r_{+}}\bigg{|}_{ex}=\frac{\upsilon(r_{+}^{2}-d^{2}-k^{2}+a^{2})}{4\pi
ar_{+}\Xi}.\ $ (138)
By implementing Cardy entropy formula and inserting (137) and (138) into it,
we successfully derive
$S_{CFT}=\frac{\pi}{\Xi}(r_{+}^{2}-d^{2}-k^{2}+a^{2}).\ $ (139)
This is precisely the Bekenstein-Hawking entropy of the KSdS black hole on the
event horizon. This result shows that we can change $l^{2}$ on those of KSAdS
solution into $-l^{2}$ by assuming analytical continuation in order to find
the result on those of KSdS solution. This is identical with the KNAdS(dS)
case [22].
## Appendix B Virasoro algebra
The Dirac bracket of two classical conserved charges are given by
$\displaystyle\left\\{Q_{\zeta_{m}},Q_{\zeta_{n}}\right\\}=Q_{[\zeta_{m},\zeta_{n}]}+K[\zeta_{m},\zeta_{n}],$
(140)
where $K[\xi,\zeta]$ is the central term. Using the quantum version of charge
$Q_{\xi}$ (44), from Eq. (140) we can find
$\displaystyle[L_{m},L_{n}]$ $\displaystyle=$ $\displaystyle
i\\{Q_{\zeta_{m}},Q_{\zeta_{n}}\\}$ $\displaystyle=$ $\displaystyle
i\left(Q_{[\zeta_{m},\zeta_{n}]}+K[\zeta_{m},\zeta_{n}]\right)$
$\displaystyle=$
$\displaystyle(m-n)L_{m+n}-2mx\delta_{m+n}+iK[\zeta_{m},\zeta_{n}].\ $
In order to find the Virasoro algebra, we need to find the form of term
$K[\zeta_{m},\zeta_{n}]$. By comparing above equation with Virasoro algebra,
$\left[L_{m},L_{n}\right]=({m}-{n})L_{m+n}+\frac{c_{L}}{12}m(m^{2}-1)\delta_{m+n,0},$
(142)
we obtain
$K[\zeta_{m},\zeta_{n}]=-i\frac{c_{L}}{12}m\left(m^{2}-1+\frac{24x}{c_{L}}\right)\delta_{m+n,0}.$
(143)
So, the central charge is determined by the coefficient $m^{3}$ in term
$K[\zeta_{m},\zeta_{n}]$. The term linear in $m$ is not important because $x$
is a free parameter to scale the last term in the bracket of Eq. (143).
Acknowledgments
The authors thank the reviewer for the fruitful suggestions on this
manuscript. This work is supported by the Second Century Fund (C2F),
Chulalongkorn University, Thailand. P. B. is supported in part by National
Research Council of Thailand (NRCT) and Chulalongkorn University under Grant
N42A660500. This research has received funding support from the NSRF via the
Program Management Unit for Human Resources & Institutional Development,
Research and Innovation [grant number B39G660025].
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11institutetext: INAF, Osservatorio Astrofisico di Catania, Via S. Sofia 78,
95123 Catania, Italy 22institutetext: Observatoire Astronomique de
l’Université de Genève, Chemin Pegasi 51, CH-1290 Versoix, Switzerland
33institutetext: ETH Zurich, Department of Physics, Wolfgang-Pauli-Strasse 2,
CH-8093 Zurich, Switzerland 44institutetext: Cavendish Laboratory, JJ Thomson
Avenue, Cambridge CB3 0HE, UK 55institutetext: Physikalisches Institut,
University of Bern, Gesellschaftsstrasse 6, 3012 Bern, Switzerland
66institutetext: Instituto de Astrofisica e Ciencias do Espaco, Universidade
do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal 77institutetext:
Department of Astronomy, Stockholm University, AlbaNova University Center,
10691 Stockholm, Sweden 88institutetext: Centre for Exoplanet Science, SUPA
School of Physics and Astronomy, University of St Andrews, North Haugh, St
Andrews KY16 9SS, UK 99institutetext: Aix Marseille Univ, CNRS, CNES, LAM, 38
rue Frédéric Joliot-Curie, 13388 Marseille, France 1010institutetext: Space
sciences, Technologies and Astrophysics Research (STAR) Institute, Université
de Liège, Allée du 6 Août 19C, 4000 Liège, Belgium 1111institutetext: Center
for Space and Habitability, University of Bern, Gesellschaftsstrasse 6, 3012
Bern, Switzerland 1212institutetext: Instituto de Astrofisica de Canarias,
Via Lactea s/n, 38200 La Laguna, Tenerife, Spain 1313institutetext:
Departamento de Astrofisica, Universidad de La Laguna, Astrofísico Francisco
Sanchez s/n, 38206 La Laguna, Tenerife, Spain 1414institutetext: Institut de
Ciencies de l’Espai (ICE, CSIC), Campus UAB, Can Magrans s/n, 08193
Bellaterra, Spain 1515institutetext: Institut d’Estudis Espacials de
Catalunya (IEEC), Gran Capità 2-4, 08034 Barcelona, Spain 1616institutetext:
Admatis, 5. Kandó Kálmán Street, 3534 Miskolc, Hungary 1717institutetext:
Depto. de Astrofisica, Centro de Astrobiologia (CSIC-INTA), ESAC campus, 28692
Villanueva de la Cañada (Madrid), Spain 1818institutetext: Departamento de
Fisica e Astronomia, Faculdade de Ciencias, Universidade do Porto, Rua do
Campo Alegre, 4169-007 Porto, Portugal 1919institutetext: Space Research
Institute, Austrian Academy of Sciences, Schmiedlstrasse 6, A-8042 Graz,
Austria 2020institutetext: Université Grenoble Alpes, CNRS, IPAG, 38000
Grenoble, France 2121institutetext: INAF, Osservatorio Astronomico di Padova,
Vicolo dell’Osservatorio 5, 35122 Padova, Italy 2222institutetext: Université
de Paris Cité, Institut de physique du globe de Paris, CNRS, 1 Rue Jussieu,
F-75005 Paris, France 2323institutetext: ESTEC, European Space Agency,
Keplerlaan 1, 2201AZ, Noordwijk, NL 2424institutetext: Institute of Planetary
Research, German Aerospace Center (DLR), Rutherfordstrasse 2, 12489 Berlin,
Germany 2525institutetext: INAF, Osservatorio Astrofisico di Torino, Via
Osservatorio, 20, I-10025 Pino Torinese To, Italy 2626institutetext: Centre
for Mathematical Sciences, Lund University, Box 118, 221 00 Lund, Sweden
2727institutetext: Astrobiology Research Unit, Université de Liège, Allée du
6 Août 19C, B-4000 Liège, Belgium 2828institutetext: Centre Vie dans
l’Univers, Faculté des sciences, Université de Genève, Quai Ernest-Ansermet
30, CH-1211 Genève 4, Switzerland 2929institutetext: Leiden Observatory,
University of Leiden, PO Box 9513, 2300 RA Leiden, The Netherlands
3030institutetext: Department of Space, Earth and Environment, Chalmers
University of Technology, Onsala Space Observatory, 439 92 Onsala, Sweden
3131institutetext: Dipartimento di Fisica, Universita degli Studi di Torino,
via Pietro Giuria 1, I-10125, Torino, Italy 3232institutetext: Department of
Astrophysics, University of Vienna, Türkenschanzstrasse 17, 1180 Vienna,
Austria 3333institutetext: Science and Operations Department - Science
Division (SCI-SC), Directorate of Science, European Space Agency (ESA),
European Space Research and Technology Centre (ESTEC), Keplerlaan 1, 2201-AZ
Noordwijk, The Netherlands 3434institutetext: Konkoly Observatory, Research
Centre for Astronomy and Earth Sciences, 1121 Budapest, Konkoly Thege Miklós
út 15-17, Hungary 3535institutetext: ELTE Eötvös Loránd University, Institute
of Physics, Pázmány Péter sétány 1/A, 1117 Budapest, Hungary
3636institutetext: German Aerospace Center (DLR), Institute of Optical Sensor
Systems, Rutherfordstraße 2, 12489 Berlin 3737institutetext: IMCCE, UMR8028
CNRS, Observatoire de Paris, PSL Univ., Sorbonne Univ., 77 av. Denfert-
Rochereau, 75014 Paris, France 3838institutetext: Institut d’astrophysique de
Paris, UMR7095 CNRS, Université Pierre & Marie Curie, 98bis blvd. Arago, 75014
Paris, France 3939institutetext: Astrophysics Group, Lennard Jones Building,
Keele University, Staffordshire, ST5 5BG, United Kingdom 4040institutetext:
Institute of Optical Sensor Systems, German Aerospace Center (DLR),
Rutherfordstrasse 2, 12489 Berlin, Germany 4141institutetext: Dipartimento di
Fisica e Astronomia ”Galileo Galilei”, Universita degli Studi di Padova,
Vicolo dell’Osservatorio 3, 35122 Padova, Italy 4242institutetext: Department
of Physics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United
Kingdom 4343institutetext: Zentrum für Astronomie und Astrophysik, Technische
Universität Berlin, Hardenbergstr. 36, D-10623 Berlin, Germany
4444institutetext: Institut fuer Geologische Wissenschaften, Freie
Universitaet Berlin, Maltheserstrasse 74-100,12249 Berlin, Germany
4545institutetext: Department of Astrophysics, University of Vienna,
Tuerkenschanzstrasse 17, 1180 Vienna, Austria 4646institutetext:
Physikalisches Institut, University of Bern, Sidlerstrasse 5, 3012 Bern,
Switzerland 4747institutetext: Université de Liège, Allée du 6 Août 19C, 4000
Liège, Belgium 4848institutetext: ELTE Eötvös Loránd University, Gothard
Astrophysical Observatory, 9700 Szombathely, Szent Imre h. u. 112, Hungary
4949institutetext: MTA-ELTE Exoplanet Research Group, 9700 Szombathely, Szent
Imre h. u. 112, Hungary 5050institutetext: Institute of Astronomy, University
of Cambridge, Madingley Road, Cambridge, CB3 0HA, United Kingdom
5151institutetext: Institute for Theoretical Physics and Computational
Physics, Graz University of Technology, Petersgasse 16, 8010 Graz, Austria
# Constraining the reflective properties of WASP-178 b using CHEOPS
photometry.††thanks: The CHEOPS program ID is CH_PR100016.††thanks: The CHEOPS
photometric data used in this work are only available in electronic form at
the CDS via anonymous ftp to cdsarc.cds.unistra.fr (130.79.128.5) or via
https://cdsarc.cds.unistra.fr/cgi-bin/qcat?J/A+A/
I. Pagano
${}^{\href{https://orcid.org/0000-0001-9573-4928}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
11 G. Scandariato
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11 V. Singh
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11 M. Lendl
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22 D. Queloz
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3344 A. E. Simon
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55 S. G. Sousa
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66 A. Brandeker
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77 A. Collier Cameron
${}^{\href{https://orcid.org/0000-0002-8863-7828}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
88 S. Sulis
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99 V. Van Grootel
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1010 T. G. Wilson
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88 Y. Alibert
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111155 R. Alonso
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12121313 G. Anglada
${}^{\href{https://orcid.org/0000-0002-3645-5977}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
14141515 T. Bárczy
${}^{\href{https://orcid.org/0000-0002-7822-4413}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
1616 D. Barrado Navascues
${}^{\href{https://orcid.org/0000-0002-5971-9242}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
1717 S. C. C. Barros
${}^{\href{https://orcid.org/0000-0003-2434-3625}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
661818 W. Baumjohann
${}^{\href{https://orcid.org/0000-0001-6271-0110}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
1919 M. Beck
${}^{\href{https://orcid.org/0000-0003-3926-0275}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
22 T. Beck 55 W. Benz
${}^{\href{https://orcid.org/0000-0001-7896-6479}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
551111 N. Billot
${}^{\href{https://orcid.org/0000-0003-3429-3836}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
22 X. Bonfils
${}^{\href{https://orcid.org/0000-0001-9003-8894}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2020 L. Borsato
${}^{\href{https://orcid.org/0000-0003-0066-9268}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2121 C. Broeg
${}^{\href{https://orcid.org/0000-0001-5132-2614}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
551111 G. Bruno
${}^{\href{https://orcid.org/0000-0002-3288-0802}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
11 L. Carone 1919 S. Charnoz
${}^{\href{https://orcid.org/0000-0002-7442-491X}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2222 C. Corral van Damme 2323 Sz. Csizmadia
${}^{\href{https://orcid.org/0000-0001-6803-9698}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2424 P. E. Cubillos 25251919 M. B. Davies
${}^{\href{https://orcid.org/0000-0001-6080-1190}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2626 M. Deleuil
${}^{\href{https://orcid.org/0000-0001-6036-0225}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
99 A. Deline 22 L. Delrez
${}^{\href{https://orcid.org/0000-0001-6108-4808}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
27271010 O. D. S. Demangeon
${}^{\href{https://orcid.org/0000-0001-7918-0355}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
661818 B.-O. Demory
${}^{\href{https://orcid.org/0000-0002-9355-5165}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
111155 D. Ehrenreich
${}^{\href{https://orcid.org/0000-0001-9704-5405}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
222828 A. Erikson 2424 A. Fortier
${}^{\href{https://orcid.org/0000-0001-8450-3374}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
551111 L. Fossati
${}^{\href{https://orcid.org/0000-0003-4426-9530}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
1919 M. Fridlund
${}^{\href{https://orcid.org/0000-0002-0855-8426}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
29293030 D. Gandolfi
${}^{\href{https://orcid.org/0000-0001-8627-9628}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
3131 M. Gillon
${}^{\href{https://orcid.org/0000-0003-1462-7739}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2727 M. Güdel 3232 M. N. Günther 2323 Ch. Helling 19195151 S. Hoyer
${}^{\href{https://orcid.org/0000-0003-3477-2466}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
99 K. G. Isaak
${}^{\href{https://orcid.org/0000-0001-8585-1717}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
3333 L. L. Kiss 34343535 E. Kopp 3636 K. W. F. Lam
${}^{\href{https://orcid.org/0000-0002-9910-6088}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2424 J. Laskar
${}^{\href{https://orcid.org/0000-0003-2634-789X}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
3737 A. Lecavelier des Etangs
${}^{\href{https://orcid.org/0000-0002-5637-5253}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
3838 D. Magrin
${}^{\href{https://orcid.org/0000-0003-0312-313X}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2121 P. F. L. Maxted
${}^{\href{https://orcid.org/0000-0003-3794-1317}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
3939 C. Mordasini 551111 M. Munari
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11 V. Nascimbeni
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2121 G. Olofsson
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77 R. Ottensamer 3232 E. Pallé
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12121313 G. Peter
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4040 G. Piotto
${}^{\href{https://orcid.org/0000-0002-9937-6387}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
21214141 D. Pollacco 4242 R. Ragazzoni
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21214141 N. Rando 2323 H. Rauer
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242443434444 C. Reimers 4545 I. Ribas
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14141515 M. Rieder 46461111 N. C. Santos
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661818 D. Ségransan
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22 A. M. S. Smith
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2424 M. Stalport 4747 M. Steller
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1919 Gy. M. Szabó 48484949 N. Thomas 55 S. Udry
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22 J. Venturini
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22 N. A. Walton
${}^{\href{https://orcid.org/0000-0003-3983-8778}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
5050
###### Abstract
Context. Multiwavelength photometry of the secondary eclipses of extrasolar
planets is able to disentangle the reflected and thermally emitted light
radiated from the planetary dayside. This leads to the measurement of the
planetary geometric albedo $A_{g}$, which is an indicator of the presence of
clouds in the atmosphere, and the recirculation efficiency $\epsilon$, which
quantifies the energy transport within the atmosphere.
Aims. In this work we aim to measure $A_{g}$ and $\epsilon$ for the planet
WASP-178 b, a highly irradiated giant planet with an estimated equilibrium
temperature of 2450 K.
Methods. We analyzed archival spectra and the light curves collected by CHEOPS
and TESS to characterize the host WASP-178, refine the ephemeris of the system
and measure the eclipse depth in the passbands of the two respective
telescopes.
Results. We measured a marginally significant eclipse depth of 70$\pm$40 ppm
in the TESS passband and statistically significant depth of 70$\pm$20 ppm in
the CHEOPS passband.
Conclusions. Combining the eclipse depth measurement in the CHEOPS
($\lambda_{\rm eff}=6300\leavevmode\nobreak\ \AA$) and TESS ($\lambda_{\rm
eff}=8000\leavevmode\nobreak\ \AA$) passbands we constrained the dayside
brightness temperature of WASP-178 b in the 2250-2800 K interval. The
geometric albedo 0.1¡$\rm A_{g}$¡0.35 is in general agreement with the picture
of poorly reflective giant planets, while the recirculation efficiency
$\epsilon>$0.7 makes WASP-178 b an interesting laboratory to test the current
heat recirculation models.
###### Key Words.:
techniques: photometric – planets and satellites: atmospheres – planets and
satellites: detection – planets and satellites: gaseous planets – planets and
satellites: individual: WASP-178 b
## 1 Introduction
In the last few years we have gained access to a detailed characterization of
exoplanets. Ground-based and space-born instrumentation have progressed such
to allow the analysis of the atmosphere of exoplanets, in terms of
thermodynamic state and chemical composition (Sing et al. 2016; Giacobbe et
al. 2021). In particular, current photometric facilities allow to observe the
secondary eclipse of giant exoplanets in close orbits (Stevenson et al. 2017;
Lendl et al. 2020; Wong et al. 2020; Singh et al. 2022). In this research
area, CHEOPS (Benz et al. 2021) is bringing a valuable contribution given its
ultra-high photometric accuracy capabilities (Lendl et al. 2020; Deline et al.
2022; Hooton et al. 2022; Brandeker et al. 2022; Parviainen et al. 2022;
Scandariato et al. 2022; Demory et al. 2023).
The depth of the eclipse quantifies the brightness of the planetary dayside
with respect to its parent star. Depending on the temperature of the planet
and the photometric band used for the observations, the eclipse depth provides
insight into the reflectivity and energy redistribution of the atmosphere.
WASP-178 b (HD 134004 b) is a hot Jupiter discovered by Hellier et al. (2019)
and independently announced by Rodríguez Martínez et al. (2020) as KELT-26 b.
It orbits an A1 IV-V dwarf star (Table 1) at a distance of 7 stellar radii:
these features place WASP-178 b among the planets that receive the highest
energy budget from their respective host stars. It is thus an interesting
laboratory to test atmospheric models in the presence of extreme irradiation.
In this paper we analyze the eclipse depths measured using the data collected
by the CHEOPS and TESS space telescopes. It is organized as follows. Sect. 2
describes the data acquisition and reduction, while in Sect. 3 we describe how
we derive the stellar radius, mass and age. In Sect. 4 we update the orbital
solution of WASP-178 b, put an upper limit on the eclipse depth using TESS
data and get a significant detection using CHEOPS photometry. Finally, in
Sect. 5 we discuss the implication of the extracted eclipse signal in terms of
geometric albedo and atmospheric recirculation efficiency.
Table 1: Stellar and system parameters. Parameter | Symbol | Units | Value | Ref.
---|---|---|---|---
V mag | | | 9.95 | Hellier et al. (2019)
Spectral Type | | | A1 IV-V | Hellier et al. (2019)
Effective temperature | $\rm T_{eff}$ | K | $9350\pm 150$ | Hellier et al. (2019)
Surface gravity | $\log g$ | — | $4.35\pm 0.15$ | Hellier et al. (2019)
Metallicity | [Fe/H] | — | $0.21\pm 0.16$ | Hellier et al. (2019)
Projected rotational velocity | $v\sin{i}$ | km/s | $8.2\pm 0.6$ | Hellier et al. (2019)
Stellar radius | $\rm R_{\star}$ | $\rm R_{\sun}$ | $1.722\pm 0.020$ | this work
Stellar mass | $\rm M_{\star}$ | $\rm M_{\sun}$ | $2.169_{-0.089}^{+0.083}$ | this work
Stellar age | $t_{\star}$ | Gyr | $0.05_{-0.05}^{+0.06}$ | this work
Radial velocity semi-amplitude | $\rm K_{RV}$ | m/s | $139\pm 9$ | Hellier et al. (2019)
## 2 Observations and data reduction
### 2.1 TESS observations
TESS (Ricker et al. 2014) observed the WASP-178 system in sector 11 (from 2019
April 23 to 2019 May 20) with a 30 min cadence and in sector 38 (from 2020
April 29 to 2020 May 26) with a 2 min cadence. Rodríguez Martínez et al.
(2020) claimed a modulation with period of 0.369526 days in the photometry of
sector 11, interpreting it as a $\delta$ Scuti pulsation mode. Later,
Lothringer et al. (2022) found out that the TESS photometry is heavily
contaminated by the background eclipsing binary ASASSN-V J150908.07-424253.6,
located at a projected distance of 50.4″ from the target, whose orbital period
matches the periodicity in the photometry of WASP-178. In order to optimize
the photometric extraction and avoid the background variable contamination, we
extracted the Light Curve (LC) corresponding to each pixel in the aperture
mask defined by the TESS pipeline, and we excluded the pixels for which the
periodogram shows a peak at the same orbital period of the binary system. We
thus re-extracted the photometry by retrieving the calibrated Full Frame
Imagess and Target Pixel Files for sector 11 and 38 respectively. We used a
custom extraction pipeline combined with the default quality bitmask. The
extracted LCs were then background-corrected after determining the sky level
using custom background masks on the FFIs and TPFs. A principal component
analysis was then conducted on the pixels in these background masks across all
frames in order to measure the flux contribution of scattered light in the
TESS cameras. We detrended the data using these principal components as a
linear model. Lastly, we further corrected for any photometric trends due to
spacecraft pointing jitter by retrieving the Co-trending Basis Vectorss and
two-second cadence engineering quaternion measurements for the specific
cameras WASP-178 was observed in. For each sector we computed the mean of the
quaternions over the length of a science observation, i.e. 30 min for the FFIs
and 2 min for the TPFs. We subsequently used these vectors along with the CBVs
to detrend the TESS photometry in a similar manner as was done in Delrez et
al. (2021).
We clip out the photometry taken before BJD 2459334.7 and in the windows
2458610–2458614.5 and 2459346–2459348: these data present artificial trends
due to the momentum dumps of the telescope. We also visually identified and
excluded a short bump in the LC between BJD 2459337.6 and 2459337.9, most
likely an instrumental artifact or some short term photometric variability
feature. The final LCs cover 7 and 8 transits for sector 11 and 38
respectively.
### 2.2 CHEOPS observations
CHEOPS (Benz et al. 2021) observed the WASP-178 system during six secondary
eclipses of the planet WASP-178 b with a cadence of 60 s. The aim is the
measurement of the eclipse depth and derivation of the brightness of the
planet in the CHEOPS passband (3500–11000 Å). Each visit is $\sim$12 hr long,
scheduled in order to bracket the eclipse and equally long pre- and post-
eclipse photometry. The logbook of the observations, which are part of the
CHEOPS Guaranteed Time Observation (GTO) program, is summarized in Table 2.
Table 2: Logbook of the CHEOPS observations of WASP-178. The filekey is the unique identifier associated with each dataset processed by the CHEOPS DRP. Filekey | Start time ]UT] | Visit duration [hr] | Exposure time [s] | N. frames | Efficiency [%]
---|---|---|---|---|---
PR100016_TG014201_V0200 | 2021-04-05 12:49:30 | 11.54 | 60.0 | 427 | 61.7
PR100016_TG014202_V0200 | 2021-04-15 13:54:09 | 12.14 | 60.0 | 473 | 64.9
PR100016_TG014203_V0200 | 2021-05-02 06:56:09 | 11.54 | 60.0 | 465 | 67.2
PR100016_TG014204_V0200 | 2021-05-22 08:52:09 | 11.54 | 60.0 | 431 | 62.3
PR100016_TG014205_V0200 | 2021-05-28 23:49:09 | 11.44 | 60.0 | 415 | 60.5
PR100016_TG014206_V0200 | 2022-05-21 22:33:49 | 11.54 | 60.0 | 445 | 64.3
The data were reduced using version 13 of the CHEOPS Data Reduction Pipeline
(DRP) (Hoyer et al. 2020). This pipeline performs the standard calibration
steps (bias, gain, non-linearity, dark current and flat fielding) and corrects
for environmental effects (cosmic rays, smearing trails from nearby stars, and
background) before the photometric extraction.
As for the case of TESS LCs, the aperture photometry is significantly
contaminated by ASASSN-V J150908.07-424253. To decontaminate the LC of
WASP-178 we performed the photometric extraction using a modified version of
the package PIPE111https://github.com/alphapsa/PIPE (Brandeker et al. 2022;
Morris et al. 2021; Szabó et al. 2021), upgraded in order to compute the
simultaneous Point Spread Function (PSF) photometry of the target and the
background contaminant.
The extracted LCs present gaps due to Earth occultations which cover $\sim$40%
of the visits. The exposures close to the gaps are characterized by high value
of the background flux, due to stray-light from Earth. The corresponding flux
measurements are thus affected by a larger photometric scatter. To avoid these
low quality data, we applied a 5$\sigma$ clipping to the background
measurements: this selection criterion removes less than 20% of the data with
the highest background counts. Finally, for a better outlier rejection, we
smoothed the data with a Savitzky-Golay filter, computed the residuals with
respect to the filtered LC and 5$\sigma$-clipped the outliers. This last
rejection criterion excludes a handful of data points in each LC.
Finally, we normalize the unsmoothed LCs by the median value of the
photometry. These normalized LCs are publicly available at CDS PUT LINK.
## 3 Stellar radius, mass and age
We use a Infra-Red Flux Method (IRFM) in a Markov Chain Monte Carlo (MCMC)
approach to determine the stellar radius of WASP-178 (Blackwell & Shallis
1977; Schanche et al. 2020). We downloaded the broadband fluxes and
uncertainties from the most recent data releases for the following bandpasses:
Gaia G, GBP, and GRP, 2MASS J, H, and K, and WISE W1 and W2 (Skrutskie et al.
2006; Wright et al. 2010; Gaia Collaboration et al. 2021). Then we matched the
observed photometry with synthetic photometry computed in the same bandpasses
by using the theoretical stellar Spectral Energy Distributions corresponding
to stellar atmospheric parameters (Table 1). The fit is performed in a
Bayesian framework and, to account for uncertainties in stellar atmospheric
modeling, we averaged the atlas (Kurucz 1993; Castelli & Kurucz 2003) and
phoenix (Allard 2014) catalogs to produce weighted averaged posterior
distributions. This process yields a $R_{\star}=1.722\pm 0.020\,R_{\odot}$.
Assuming the $T_{\mathrm{eff}}$, [Fe/H], and $R_{\star}$ listed in Table 1 as
input parameters, we also computed the stellar mass $M_{\star}$ and age
$t_{\star}$ by using two different sets of stellar evolutionary models. In
detail, we employed the isochrone placement algorithm (Bonfanti et al. 2015,
2016) and its capability of interpolating within pre-computed grids of
PARSEC222PAdova and TRieste Stellar Evolutionary Code:
http://stev.oapd.inaf.it/cgi-bin/cmd v1.2S (Marigo et al. 2017) for retrieving
a first pair of mass and age estimates. A second pair of mass and age values,
instead, was computed by CLES (Code Liègeois d’Évolution Stellaire; Scuflaire
et al. 2008), which generates the best-fit evolutionary track of the star by
entering the input parameters into the Levenberg-Marquadt minimisation scheme
as described in Salmon et al. (2021). After carefully checking the mutual
consistency of the two respective pairs of estimates through the
$\chi^{2}$-based criterion broadly discussed in Bonfanti et al. (2021), we
finally merged the outcome distributions and we obtained
$M_{\star}=2.169_{-0.089}^{+0.083}\,M_{\odot}$ and $t_{\star}=50_{-50}^{+60}$
Myr.
## 4 Light curve analysis
### 4.1 TESS photometry
To compare in a homogeneous way the LCs of sectors 11 and 38, we rebinned the
photometry of sector 38 to 30 min. The standard deviation of the TESS LCs,
after transits and secondary eclipses are clipped, is of $\sim$300 ppm for
both sectors, and in both cases the photometric uncertainty can account for
only $\sim$70% of the variance. This indicates that there is some noise in the
LCs due to astrophysical signals and/or instrumental leftovers. To investigate
if the unexplained variance is related to periodic signals, for both sectors
we compute the Generalized Lomb-Scargle (GLS) periodogram (Zechmeister &
Kürster 2009) of the out-of-transit and out-of-eclipse photometry (Fig. 1).
For sector 11 we did not find any significant periodic signal, while for
sector 38 there is a strong peak at frequency $\rm\nu=0.304\pm
0.003\leavevmode\nobreak\ d^{-1}$ with False Alarm Probability (FAP) lower
than 0.1%. The amplitude of the corresponding sinusoidal signal is $90\pm 10$
ppm.
Figure 1: Out-of-transit and out-of-eclipse 30 min cadence TESS photometry of
WASP-178 during sector 11 (left panel) and sector 38 (central panel). In each
panel, the dashed vertical lines mark the planetary transit, while the solid
blue line is a smoothing of the data points to emphasize the correlated noise.
The right panel shows the GLS periodogram of sector 11 and 38 (top and bottom
box respectively). In each box, we report the bootstrap-computed 0.1% and 1%
FAP levels (horizontal dashes) and the planetary orbital period (vertical red
dotted line).
The periodicity detected in sector 38 is consistent within uncertainties with
the planetary orbital period. Nonetheless, we exclude that this signal is of
planetary origin because it remained undetected in sector 11 and is not
coherent from orbit to orbit in sector 38. This signal might be due to stellar
rotation, and is consistent with the typical amplitude reported by Balona
(2011) for A-type stars. Its periodicity would correspond to the $v\sin
i_{\star}$ in Table 1 if the inclination of the stellar rotation axis is $\sim
15^{\circ}$. This speculation supports the hypothesis of Rodríguez Martínez et
al. (2020) that the star is seen nearly pole-on. A significant misalignment
between the stellar rotation axis and the planetary orbit axis is not
surprising, both because of its young age (no time for realignment to occur),
and the stellar temperature. As a matter of fact, hot stars (Teff¿6200 K) are
observed to often have large misalignments, which is thought to be due to
their lack of a convective zone (Winn et al. 2010), needed to tidally align
the planetary orbit with the stellar spin axis. Furthermore, Albrecht et al.
(2021) found that misaligned orbits are most often polar, or close to polar,
as seems to be the case of WASP-178 b. We expand the discussion on the
photometric variability and the orientation of the stellar rotation axis in
Appendix B.
The fact that the period of the correlated noise is similar to the orbital
period of WASP-178 b makes it difficult to extract the complete planetary
Phase Curve (PC) signal. We thus first tried a simpler and more robust
approach to analyze the planetary transits and eclipses (Sect. 4.1.1), then we
attempted a more complex analysis framework aimed at retrieving the full PC of
WASP-178 b (Sect. 4.1.2).
#### 4.1.1 Fit of transits and eclipses
We computed the ephemeris of the planet by trimming segments of the LCs
centered on the transit events (7 transits in sector 11 and 8 transits in
sector 8) and as wide as 3 times the transit duration. To further constrain
the ephemeris of WASP-178 b, we included in our analysis the WASP-South
photometry (2006 May – 2014 Aug) and EulerCAM I-band photometry (2018 Mar 26)
presented in Hellier et al. (2019). Also for the case of the WASP-South
photometry, we trimmed the LC around the transits and kept the 24 intervals
containing more than 20 data points.
We fit the data using the same Bayesian approach described in Scandariato et
al. (2022). In summary, it consists in the fit of a model - in a likelihood-
maximization framework - which includes the transits, a linear term for each
transit to detrend against stellar/instrumental systematics and a jitter term
to fit the white noise not included in the photometric uncertainties. The
transit profile is formalized using the quadratic limb darkening (LD) law
indicated by Mandel & Agol (2002) with the reparametrization of the LD
coefficients suggested by Kipping (2013). For TESS sector 11 the model is
rebinned to the same 30 min cadence of the data. The likelihood maximization
is performed with MCMC using the python emcee package version 3.1.3 (Foreman-
Mackey et al. 2013), using a number of samplings long enough to ensure
convergence. We used flat priors for all the fitting parameters but the
stellar density, for which we used the Gaussian prior $N(0.43,0.02)$ given by
the stellar mass and radius in Table 1. We also used the Gaussian priors for
the Limb Darkening (LD) coefficients given by the LDTk package
333https://github.com/hpparvi/ldtk. For simplicity, we use the same LD
coefficients for the three datasets. This is motivated by the fact that the
WASP-South photometry is not accurate enough to constrain the LD profile, and
that the TESS passband is basically centered on the standard I band, thus the
same LD profile is expected for the TESS and EulerCAM LCs. Since the model
fitting is computationally demanding, we ran the code in the HOTCAT computing
infrastructure (Bertocco et al. 2020; Taffoni et al. 2020).
The result of the model fitting is listed in Table 3. The orbital solution we
derived is consistent with previous studies (Hellier et al. 2019; Rodríguez
Martínez et al. 2020) within 5$\sigma$. The best fit model, corresponding to
the Maximum A Posteriori (MAP) parameters, is over-plotted to the phase-folded
data in Fig 2, where we rebinned the model and the LC of sector 38 to the same
30 min cadence of sector 11 (we do not show the WASP-South and EulerCAM
photometry not to clutter the plot). In Fig. 9 we show the corner plot of the
system’s parameters from the fit of the transit LCs.
Table 3: Model parameters for the fit of the TESS WASP-South and EulerCAM
data.
Jump parameters | Symbol | Units | MAP | C.I.aa$a$Uncertainties expressed in parentheses refer to the last digit(s). | Prior
---|---|---|---|---|---
Time of transit | $T_{0}$ | BJDTDB-2400000 | 56612.6581 | 56612.6581(3) | $U$(56612.6, 56612.7)
Orbital frequency | $\nu_{\rm orb}$ | days-1 | 0.29896856 | 0.29896855(3) | $U$(0.2989,0.2990)
Stellar density | $\rho_{\star}$ | $\rho_{\sun}$ | 0.45 | 0.44(1) | $N$(0.43,0.02)
Radii ratiobb$b$Fitting WASP-South, EulerCAM and TESS data altogether. | $R_{p}/R_{\star}$ | — | 0.1124 | 0.1125(2) | $U$(0.05,0.12)
Radii ratiocc$c$Fitting TESS sector 11 only. | $R_{p}/R_{\star}$ | — | 0.1109 | 0.1108(4) | $U$(0.05,0.12)
Radii ratiodd$d$Fitting TESS sector 38 only. | $R_{p}/R_{\star}$ | — | 0.1141 | 0.1141(4) | $U$(0.05,0.12)
Impact parameter | b | — | 0.52 | 0.51(1) | $U$(0.01,0.9)
First LD coef.bb$b$Fitting WASP-South, EulerCAM and TESS data altogether. | q1 | — | 0.147 | 0.147(8) | $N$(0.133,0.014)
Second LD coef.bb$b$Fitting WASP-South, EulerCAM and TESS data altogether. | q2 | — | 0.344 | 0.34(2) | $N$(0.333,0.023)
Secondary eclipse depth | $\delta_{\rm ecl}$ | ppm | 70 | 70(40) | $U$(0,400)
Derived parameters | Symbol | Units | MAP | C.I. |
Planetary radiusbb$b$Fitting WASP-South, EulerCAM and TESS data altogether. | Rp | RJ | 1.88 | 1.88(2) | including the stellar radius uncertainty
Planetary radiuscc$c$Fitting TESS sector 11 only. | Rp | RJ | 1.86 | 1.85(2) | including the stellar radius uncertainty
Planetary radiusdd$d$Fitting TESS sector 38 only. | Rp | RJ | 1.91 | 1.91(2) | including the stellar radius uncertainty
Orbital period | Porb | day | 3.3448332 | 3.3448333(4) |
Transit duration | T14 | hr | 3.488 | 3.488(8) |
Scaled semi-major axis | $a/R_{\star}$ | — | 7.20 | 7.19(6) |
Orbital inclination | $i$ | degrees | 85.8 | 85.8(1) |
444
Since the TESS LCs of sector 11 and 38 show different levels of variability,
we investigated any seasonal dependence of the apparent planet-to-star radius
ratio. We used the same Bayesian framework as above, where we fix the
ephemeris of WASP-178. The planet-to-star radius ratio we derived is
0.1108$\pm$0.0004 for sector 11 and 0.1141$\pm$0.0004 for sector 38. The
difference is thus 0.0033$\pm$0.0006, that is we found different transit
depths with a 5.5$\sigma$ significance. In particular, we remark that the
planet looks larger in sector 38, where the rotation signal is stronger. We
thus speculate that WASP-178 was in a low variability state during sector 11,
while one year later ($\sim$80 stellar rotations) during sector 38 the stellar
surface hosted dark spots in co-rotation with the star. This hypothesis is
consistent with Hümmerich et al. (2018), according to which magnetic
chemically peculiar stars may show complex photometric patterns due to surface
inhomogeneities.
We used a similar framework to extract the secondary eclipse signal of
WASP-178: we simultaneously fit segments of the TESS LCs centered on the
planetary eclipses keeping fixed the ephemeris of WASP-178 to the values
listed in Table 3. We did not attempt the disjoint analysis of the two sectors
as the expected eclipse depth is of the order of $\sim$100 ppm (see below),
and the photometric precision of the LCs is not good enough to appreciate
differences in the eclipse depth with enough statistical evidence. The only
free planetary parameter of the model is thus the eclipse depth, which turned
out to be $\delta_{\rm ecl}=70\pm 40$ ppm (MAP: 70 ppm). The detrended phase-
folded eclipses are shown in Fig 2 together with the MAP model.
Figure 2: Best fit of the transits and eclipses observed by TESS. Left -
Detrended and phase-folded planetary transits (top panel). The photometry of
sector 11 and 38 are shown with different colors to emphasize the difference
in transit depth (the photometry of sectors 11 and 38 is systematically offset
respectively upwards and downwards with respect to the bestfit transit
profile). The solid blue line is the best fit model, while the cyan lines
represent 100 models corresponding to random samples of the MCMC fit. For the
sake of comparison, the photometry of sector 38 and the theoretical models are
rebinned to the same 30 min cadence of sector 11. The black dots represent the
rebinned photometry. The corresponding O-C diagram is shown in the bottom
panel. Right - Same as in the left panel, but centered on the eclipse. For
clarity, we do not mark the two sectors with different colors.
#### 4.1.2 Fit of the phase curve
As a more advanced data analysis, we jointly fit the two TESS sectors to
extract the full planetary PC (for a homogeneous analysis, we rebinned sector
38 to the same 30 min cadence of sector 11). The fitting model now includes
the transits, secondary eclipses, the planetary PC, a Gaussian Process (GP) to
fit the correlated noise in the data and, for each TESS sector, a long-term
linear trend and a jitter term. Given the system parameters listed in Table 1,
the expected amplitude of ellipsoidal variations (Morris 1985) and Doppler
boosting (Barclay et al. 2012) in the planetary PC are 2.5 ppm and 1 ppm
respectively, out of reach for the TESS photometry. Hence, for the sake of
simplicity, we do not include them in the model fitting.
We jointly fit the two TESS sectors, starting with the simplest model where
the out of transit PC is flat (that is, we assumed that the planetary PC is
not detectable) and the putative stellar rotation is modeled as GP with the
SHO kernel555Provided by the celerite2 python package version 0.2.1 (Foreman-
Mackey et al. 2017; Foreman-Mackey 2018), indicated for quasi-periodic
signals. Checking the residuals of the fit we notice that some aperiodic
correlated noise is left. We thus increased the complexity of the GP model by
adding a Matérn3/2 kernel to capture the remaining correlated noise adding
only two free parameters to the model. This composite GP model turned out to
ensure better convergence to the fit of the model.
Finally, we also included the planetary PC. Assuming that the planet is
basically composed of two homogeneous dayside and nightside, the PC is in
principle the combination of three components: the PC due to reflected light
(with amplitude $A_{\rm refl}$), the PC of the planetary night side (with
amplitude $A_{\rm n}$) and the PC of the planetary day side, whose amplitude
$A_{\rm d}$ is parameterized such to be larger than $A_{\rm n}$ by an
increment $\delta_{\rm d}$ (this parametrization avoids the nonphysical case
of a planetary night side brighter than the dayside). The three terms are
modeled assuming Lambert’s cosine law.
For the reflected PC, the amplitude can be expressed as:
$A_{\rm refl}=A_{\rm g}\left(\frac{R_{\rm p}}{a}\right)^{2},$ (1)
where $A_{\rm g}$ is the planetary geometric albedo (Seager 2010). Assuming a
Lambertian reflective planetary surface, the geometric albedo is fixed by the
Bond albedo $A_{\rm B}$ by $A_{\rm g}$=$\frac{2}{3}A_{\rm B}$. In the
optimistic case of a perfectly reflecting body ($A_{\rm B}$=1), the maximum
amplitude for the reflected PC is thus $A_{\rm refl}=167\pm 5$ ppm, where we
have used the system parameters in Table 3.
With a slight re-adaptation of the formalism in Seager (2010), the amplitude
thermal component $A_{\rm therm}$ can be expressed as:
$A_{\rm therm}=\left(\frac{R_{\rm
p}}{R_{\star}}\right)^{2}\frac{\int\eta(\lambda)B(\lambda,T)d\lambda}{\int\eta(\lambda)I_{\star}(\lambda,T_{\rm
eff})d\lambda},$ (2)
where $\eta(\lambda)$ is the optical throughput of the telescope and
$I_{\star}(\lambda,T_{\rm eff})$ is the expected stellar intensity computed
with the NextGen model (Hauschildt et al. 1999) corresponding to the effective
temperature $T_{\rm eff}$. For the planetary thermal emission we lack the
infrared information which can constrain the emission spectrum. We thus assume
a black body spectrum $B(\lambda,T)$ at temperature $T$.
The amplitude of the thermal emission from the dayside and the nightside
depends on the respective temperatures, which we estimated following Cowan &
Agol (2011). The expected temperature of the substellar point of WASP-178 b
is:
$T_{\rm 0}=T_{\rm eff}\sqrt{\frac{R_{\star}}{a}}=3530\pm
60\leavevmode\nobreak\ {\rm K},$ (3)
where we used the stellar effective temperature in Table 1 and the scaled
semi-major axis in Table 3.
The nightside temperature $T_{\rm n}$ and the dayside temperature $T_{\rm d}$
depend on the Bond albedo $A_{B}$ and heat recirculation coefficient
$\epsilon$:
$\displaystyle T_{\rm n}$ $\displaystyle=T_{\rm 0}\left(1-A_{\rm
B}\right)^{1/4}\left(\frac{\epsilon}{4}\right)^{1/4}$ (4) $\displaystyle
T_{\rm d}$ $\displaystyle=T_{\rm 0}\left(1-A_{\rm
B}\right)^{1/4}\left(\frac{2}{3}-\frac{5}{12}\epsilon\right)^{1/4}$ (5)
The highest dayside temperature, corresponding to $A_{\rm B}$=0 and no heat
recirculation ($\epsilon=0$), is:
$T_{\rm d}=T_{\rm 0}\left(\frac{2}{3}\right)^{1/4}=3190\pm
60\leavevmode\nobreak\ {\rm K},$ (6)
while the maximum nightside temperature, obtained in the case of full energy
recirculation ($\epsilon=1$), is:
$T_{\rm n}=T_{\rm 0}\left(\frac{1}{4}\right)^{1/4}=2500\pm
40\leavevmode\nobreak\ {\rm K}.$ (7)
These maximum temperatures lead to the upper limit on the dayside ($A_{\rm
d}$) and nightside ($A_{\rm n}$) thermal emission amplitude of:
$A_{\rm d}=225\pm 27\leavevmode\nobreak\ {\rm ppm};A_{\rm n}=48\pm
7\leavevmode\nobreak\ {\rm ppm}.$ (8)
In Fig. 3 we plot the upper limits for the three PCs discussed so far. The
figure shows that the three components may reach amplitudes of comparable
orders of magnitude, meaning that none of them can in principle be neglected
in the extraction of the planetary PC. We note that the two components
belonging to the dayside (reflection and thermal emission) have the same
shape, thus it is not possible to disentangle them. To simplify the model and
avoid the degeneracy between dayside reflection and emission, we thus
artificially set $A_{\rm refl}=0$ and let $A_{\rm d}$ absorb the whole signal
belonging to the planetary dayside.
Figure 3: Upper limits on the reflected PC, dayside thermal PC and nightside
thermal PC for WASP-178 b.
The Akaike Information Criterion (AIC, Burnham & Anderson 2002) favors the
model that includes the planetary PCs. The simpler model with flat out-of-
transit PC has a relative likelihood of 55% and cannot be rejected. The
posterior distributions of the common parameters (system ephemeris and planet-
to-star radii ratio) between the two models do not differ significantly.
For both models, the SHO GP has a frequency $\nu\sim$0.30 d-1, an amplitude
$\sim$70 ppm and a timescale $\sim$11 d. Frequency $\nu$ and the amplitude of
the quasi-periodic GP are consistent with the frequency and amplitude of the
periodic signal found in the periodogram (Sect. 4.1). While the correspondence
between the orbital period of WASP-178 b and the periodicity of the GP
suggests that the red noise in the data is of planetary origin, we do not have
other strong evidence that support this hypothesis. On the contrary, the
planetary origin seems odd for the reasons discussed in Sect. 4.1. The
aperiodic red noise fitted as a Matérn3/2 GP has an amplitude $\sim$150 ppm
and a timescale of $\sim$50 min. Despite the aperiodic red noise evolves on
timescales shorter than the transit duration ($T_{\rm 14}$=3.53 hr), the
retrieved posterior distributions of the orbital parameters are similar to the
ones obtained in Sect. 4.1.1.
Regarding the retrieval of the planetary PCs, we derived an upper limit of
$A_{\rm n}<$174 ppm at the 99.9% confidence level, while for the dayside we
derived a 1$\sigma$ confidence band of 110$\pm$40 ppm which includes both the
reflection and thermal emission components. We remark that the $A_{\rm d}$
parameter corresponds in our model to the eclipse depth $\delta_{\rm ecl}$ and
is consistent within uncertainties with the estimate reported in Table 3.
The fit of the full TESS LC thus confirms the results previously obtained in
Sect. 4.1.1. Given the similarities between the results obtained with the two
approaches, we preferred the former one as it is less prone to interference
between the data detrending and the extraction of the orbital parameters and
the eclipse depth.
### 4.2 CHEOPS photometry
To analyze the CHEOPS LCs we use the same approach as for the fit of the
eclipses observed by TESS (Sect. 4.1.1), with an additional module in the
fitting model that takes into account the systematics in the CHEOPS data. As a
matter of fact, CHEOPS photometry is affected by variable contamination from
background stars in the field of view (e.g. Lendl et al. 2020; Deline et al.
2022; Hooton et al. 2022; Wilson et al. 2022; Scandariato et al. 2022). This
variability is due to the interplay of the asymmetric PSF with the rotation of
the field of view (Benz et al. 2021). PIPE by design uses nominal magnitudes
and coordinates of the stars in the field to fit the stellar PSFs, but
residual correlated noise is present in the LCs due to inaccuracies in the
assumptions. This signal is phased with the roll angle of the CHEOPS satellite
and, in the case of WASP-178, is clearly visible in the GLS periodogram of the
data together with its harmonics. To remove this signal, we included in our
algorithm a module which fits independently for each visit the harmonic
expansions of the telecope’s orbital period and its harmonics. A posteriori,
we found that the roll angle phased modulation is adequately suppressed (that
is, there is no peak in the periodograms at the frequencies in the harmonic
series) if we include in the model the fundamental harmonic and its first two
harmonics.
We also noticed that the photometry is significantly correlated with the
coordinates of the centroid of the PSF on the detector. To decorrelate against
this instrumental jitter we thus included in the model a bi-linear function of
the centroid coordinates.
Finally, to take into account a white noise not included in the formal
photometric uncertainties, we add to our model a diagonal GP kernel of the
form:
$k(t_{i},t_{j})=j_{v}^{2}\delta_{i,j},$ (9)
with an independent jitter term $j_{v}$ for each CHEOPS visit.
As for the fit of the TESS data (Sect. 4.1.1), we search the bestfit
parameters through likelihood-maximization in a MCMC framework, and we find
$\rm\delta_{ecl}=70\pm 20$ ppm. The ephemeris of the planet were locked to the
MAP values listed in Table 3, which guarantee an uncertainty on the eclipse
time less than 23 s for each CHEOPS visit. The phase-folded data detrended
against stellar and instrumental correlated noise are shown in Fig. 4.
Figure 4: Same as the right panel in Fig. 2 for the eclipses of WASP-178
observed by CHEOPS.
## 5 Discussion
In the previous sections we have analyzed the TESS and CHEOPS PCs of WASP-178
b in order to extract its planetary eclipse depth $\delta_{\rm ecl}$,
obtaining respectively $70\pm 40$ ppm and $70\pm 20$ ppm. These two
measurements take into account both the reflection from the planetary dayside
and its thermal emission. With the assumptions discussed below, the combined
information on the eclipse depth from two different instruments allows to
disentangle these two contributions. For a given telescope with passband
$\eta(\lambda)$ we can combine Eq. 1 and Eq. 2 obtaining:
$\delta_{\rm ecl}=A_{\rm g}\left(\frac{R_{\rm
p}}{a}\right)^{2}+\left(\frac{R_{\rm
p}}{R_{\star}}\right)^{2}\frac{\int\eta(\lambda)B(\lambda,T_{\rm
d})d\lambda}{\int\eta(\lambda)I_{\star}(\lambda,T_{\rm eff})d\lambda},$ (10)
where $T_{\rm d}$ is the planetary dayside temperature.
Solving for $A_{\rm g}$, and indicating with the subscripts $C$ (for CHEOPS)
and $T$ (for TESS) the wavelength-dependent quantities, we obtain:
$\begin{cases}A_{\rm
g}^{C}=\left(\frac{a}{R_{\star}}\right)^{2}\left[\frac{\delta_{\rm
ecl}^{C}}{\left(\frac{R_{\rm
p}}{R_{\star}}\right)^{2}}-\frac{\int\eta^{C}(\lambda)B(\lambda,T_{\rm
d})d\lambda}{\int\eta^{C}(\lambda)I_{\star}(\lambda,T_{\rm
eff})d\lambda}\right],\\\ A_{\rm
g}^{T}=\left(\frac{a}{R_{\star}}\right)^{2}\left[\frac{\delta_{\rm
ecl}^{T}}{\left(\frac{R_{\rm
p}}{R_{\star}}\right)^{2}}-\frac{\int\eta^{T}(\lambda)B(\lambda,T_{\rm
d})d\lambda}{\int\eta^{T}(\lambda)I_{\star}(\lambda,T_{\rm
eff})d\lambda}\right].\end{cases}$ (11)
In the most general case, $A_{g}^{C}$ and $A_{g}^{T}$ differ according to the
planetary reflection spectrum. Since we do not have any spectroscopic analysis
of the dayside of WASP-178 b, we define the proportionality coefficient
$\alpha=A_{g}^{T}/A_{g}^{C}$ to account for differences between the two
passbands. Solving Eq. 11 for $T_{\rm d}$ we thus obtain the implicit
function:
$\frac{\alpha\delta_{\rm ecl}^{C}-\delta_{\rm ecl}^{T}}{\left(\frac{R_{\rm
p}}{R_{\star}}\right)^{2}}+\frac{\int\eta^{T}(\lambda)B(\lambda,T_{\rm
d})d\lambda}{\int\eta^{T}(\lambda)I_{\star}(\lambda)d\lambda}-\alpha\frac{\int\eta^{C}(\lambda)B(\lambda,T_{\rm
d})d\lambda}{\int\eta^{C}(\lambda)I_{\star}(\lambda)d\lambda}=0.$ (12)
We initially assumed the simplest scenario of a gray albedo spectrum
($\alpha$=1) in the spectral range covered by CHEOPS and TESS (the respective
bandpasses are shown in Fig. 6). In this scenario, we expect that the
contribution of reflection to the eclipse depth is the same in the CHEOPS and
TESS passbands. We also expect that the thermal emission from a $T\simeq$3000
K black body has a larger contribution in the TESS passband, because it favors
redder wavelengths compared with CHEOPS (see Fig. 6).
To estimate the dayside temperature $T_{\rm d}$ that best explains our best
observations together with its corresponding uncertainty, we randomly
extracted 10 000 steps from the Monte Carlo chains obtained in the fit of the
transit and eclipse LCs (Sect. 4.1.1 and 4.2). We also generated 10 000
samples of the stellar effective temperature using a normal distribution with
mean and standard deviations as reported in Table 1. For each of the 10 000
samples we thus numerically solved Eq. 12 using the fsolve method of the scipy
Python package, thus obtaining 10 000 samples of $T_{\rm d}$. Then, we plugged
all the samples back into Eq. 11 to derive $A_{g}^{C}$(=$A_{g}^{T}$).
The root finding algorithm failed for $\sim$50% of the samples. These failures
correspond to the combination of parameters that prevent Eq. 12 from having a
$T_{\rm d}$ root in the domain of real numbers. This happens in particular for
the samples where $\delta_{\rm ecl}^{T}<\delta_{\rm ecl}^{C}$ that, as
discussed above, are not consistent with the gray albedo hypothesis. We thus
reject these samples and show in Fig. 7 the $A_{g}^{C}$–$T_{\rm d}$ density
map of the remaining ones. Our results indicate $A_{g}^{C}=0.2\pm 0.1$ and
$T_{\rm d}=2400\pm 300$ K, consistent with the temperature-pressure profile
derived by (Lothringer et al. 2022) at pressure higher that 1$\rm\mu$bar by
means of ultraviolet transmission spectroscopy. Our result also follows the
general trend of $A_{\rm g}$ increasing with $T_{\rm d}$ indicated by Wong et
al. (2021) (Fig. 5).
Figure 5: Adaptation of Fig. 10 in Wong et al. (2021) including our analysis
of WASP-178 b (in blue). The green squares indicate the systems from the first
and second year of the TESS primary mission. The black circles indicate the
Kepler-/CoRoT-band geometric albedos for the targets that were observed by
those missions.
By definition, the geometric albedo $A_{\rm g}$ refers to the incident light
reflected back to the star at a given wavelength (or bandpass). Integrating at
all angles, the spherical albedo $A_{\rm S}$ is related to $A_{\rm g}$ by the
phase integral $q$: $A_{\rm S}$=$q$$A_{\rm g}$ (see for example Seager 2010).
Depending on the scattering law, exoplanetary atmospheres have 1¡$q$¡1.5
(Pollack et al. 1986; Burrows & Orton 2010). Unfortunately, for the reasons
explained in Sect. 4.1.2, we could not extract a robust phase curve for
WASP-178 b, hence we could not place any better constraint on $q$. In the
following we thus consider the two limiting scenarios $A_{S}^{\rm min}=A_{g}$
and $A_{S}^{\rm max}=1.5A_{g}$.
The Bond albedo $A_{\rm B}$ is computed as the average of $A_{\rm S}$ weighted
over the incident stellar spectrum:
$A_{\rm B}=\frac{\int_{0}^{\infty}A_{\rm
S}(\lambda)I_{\star}(\lambda)d\lambda}{\int_{0}^{\infty}I_{\star}(\lambda)d\lambda}.$
(13)
The conversion into $A_{\rm B}$ thus relies on the measurement of $A_{\rm S}$
across the stellar spectrum. In this study we only covered the optical and
near-infrared part of the spectrum, as shown in Fig. 6, and we lack the
necessary information in the UV, mid, and far infrared domains. Following the
approach of Schwartz & Cowan (2015), we explore the scenario of minimum Bond
albedo $A_{\rm B}^{\rm min}$ obtained through Eq. 13 assuming $A_{\rm
S}$=$A_{\rm S}^{\rm min}$ in the spectral range covered by CHEOPS and TESS and
$A_{\rm S}$=0 otherwise. The opposite limiting case assumes $A_{\rm
S}$$(\lambda)=A_{\rm S}^{\rm max}$ at all wavelength, which leads to $A_{\rm
B}^{\rm max}=A_{\rm S}^{\rm max}$. To compute the integrals in Eq. 13, we used
the synthetic spectrum in the BT-Settl library corresponding to the parameters
of WASP-178 (Allard et al. 2012).
Figure 6: BT-Settl synthetic stellar spectrum for WASP-178 together with the
CHEOPS and TESS passbands.
Under the hypothesis that $\alpha=1$, we obtained samples of $A_{\rm
g}^{C}=A_{\rm g}^{T}=$$A_{\rm g}$, which now translate into samples of $A_{\rm
B}^{\rm min}$ and $A_{\rm B}^{\rm max}$ with the assumptions discussed above.
Moreover, inverting Eq. 5 yields to:
$\epsilon=\frac{1}{5}\left[8-\left(\frac{T_{\rm d}}{T_{\rm
eff}}\right)^{4}\left(\frac{a}{R_{\star}}\right)^{2}\frac{12}{1-A_{\rm
B}}\right].$ (14)
Equation 14 indicates that $\epsilon$ is a decreasing function of $A_{\rm B}$.
Plugging in the samples of $T_{\rm d}$, Teff, $a/R_{\star}$ and alternatively
$A_{\rm B}^{\rm min}$ and $A_{\rm B}^{\rm max}$, we obtained the corresponding
samples $\epsilon^{\rm max}$ and $\epsilon^{\rm min}$.
The albedo–recirculation density maps of the two scenarios are shown in the
right panel of Fig. 7. Unsurprisingly, we find that the upper limit on $A_{\rm
B}$ is larger ($\sim$0.6) in the scenario where all the factors concur to push
up the reflectivity of the atmosphere (high $q$ and maximum $A_{\rm S}$), and
it decreases to $\sim$0.3 in the opposite scenario of minimum reflectivity.
The recirculation coefficient $\epsilon$ tends towards large values,
eventually exceeding 1 due to measurement uncertainties, in both scenarios.
While $\epsilon>1$ is physically meaningless, still the posterior distribution
indicates a high level of atmospheric energy recirculation in both cases of
minimum and maximum albedo ($\epsilon^{\rm max}=1.0\pm 0.3$ and $\epsilon^{\rm
min}=0.8\pm 0.3$ respectively). Following different atmospheric models (e.g.,
Perez-Becker & Showman 2013; Komacek & Showman 2016; Schwartz et al. 2017;
Parmentier & Crossfield 2018), WASP-178 b is in a temperature regime where
heat recirculation of zonal winds is suppressed and recirculation efficiency
is expected to be low. Nonetheless, Zhang et al. (2018) collected
observational evidence that Hot Jupiters with irradiation temperatures similar
to the one of WASP-178 b are characterized by efficient day-to-night
recirculation. Using the global circulation models of Kataria et al. (2016),
they ascribed this efficiency to the presence of zonal winds, that also
explain the eastward offset of the PC of the same planets. While the
theoretical predictions on the offset (and the recirculation efficiency
correspondingly) likely overestimate the truth (see Fig. 15 in Zhang et al.
2018), still there is an indication that zonal winds might indeed explain the
energy transfer from the dayside to the nightside of WASP-178 b.
Unfortunately, as discussed in Sect.4.1.2, it is difficult to disentangle the
planetary phase curve and the stellar variability. By consequence, any
measurement of the phase curve offset is precluded. Another possibility is
that ionized winds flow from the dayside, where temperatures higher that 2500
K leads to complete dissociation of H2, to the nightside, where lower
temperatures allow the molecular recombination and the consequent energy
release. This scenario is supported by several recent studies (Bell & Cowan
2018; Tan & Komacek 2019; Mansfield et al. 2020; Helling et al. 2021, 2023).
Figure 7: Left panel - $A_{g}^{C}$–$T_{\rm d}$ density maps in the case of
$\alpha=1$. Right panel - Albedo–recirculation density maps assuming maximum
$A_{\rm B}$ (red contours) and minimum $A_{\rm B}$ (blue contours).
According to the synthetic models computed by Sudarsky et al. (2000), the
albedo spectrum of highly irradiated giant planets is expected to show
molecular absorption bands by H2O at around 1 $\mu m$ and longer wavelength.
Since the passband of TESS is more sensitive in the near-infrared than CHEOPS,
it is plausible to assume that the geometric albedo in the TESS passband is
lower than for CHEOPS. In order to assess how the assumption on $\alpha$
affects our results, we re-run our analysis assuming an extreme $\alpha=0.5$.
The results are shown in Fig. 8: we find almost the same posterior
distribution for dayside temperature, albedo and recirculation, indicating
that the most important source of uncertainties are not the assumptions we
make but the measurement uncertainties on the eclipse depth extracted from the
CHEOPS and TESS LCs.
Figure 8: Same as in Fig. 7 in the case $\alpha=0.5$.
## 6 Summary and conclusion
In this work we have analyzed the space-borne photometry of the WASP-178
system obtained with TESS and CHEOPS. For both telescopes we have tailored the
photometric extraction in order to avoid the strong contamination by a
background eclipsing binary. We have found evidence that the stellar host is
rotating with a period of $\sim$3 days, consistent with the orbital period of
its HJ, and that it is seen in a pole-on geometry.
The similarity between the stellar rotation period and the orbital period of
the planet, coupled with the quality of the TESS data, does not allow a robust
analysis of the full PC of the planet, nor we could put strong constraints on
the planetary nightside emission. Nonetheless, focusing on the transit and
eclipse events, we could update the ephemeris of the planet and measure an
eclipse depth of 70$\pm$40 ppm and 70$\pm$20 ppm in the TESS and CHEOPS
passband respectively.
The joint analysis of the eclipse depth measured in the two passbands allowed
us to constrain the temperature in the 2250–2750 K range, consistent with the
temperature-pressure profile derived by (Lothringer et al. 2022) at pressure
higher that 1$\rm\mu$bar. Moreover, we could also constrain the optical
geometric albedo $A_{\rm g}$¡0.4, which fits the general increasing albedo
with stellar irradiation indicated by Wong et al. (2021).
Finally, we found indication of an efficient atmospheric heat recirculation of
WASP-178 b. If confirmed, this evidence challenges the models that predict a
decreasing heat transport by zonal winds as the equilibrium temperature
increases (e.g., Perez-Becker & Showman 2013; Komacek & Showman 2016; Schwartz
et al. 2017; Parmentier & Crossfield 2018). Conversely, WASP-178 b is an
interesting target to test atmospheric models where heat transport is granted
by ionized winds from the dayside to the nightside, where the recombination of
H2 takes place (e.g. Mansfield et al. 2020; Helling et al. 2021, 2023).
Additional observations of WASP-178 b are needed to better constrain its
atmospheric recirculation and rank current competing models. The WASP-178
system is going to be observed again by TESS in May 2023, but this is not
expected to add much to the two sectors analyzed in this work. Conversely,
dedicated observations from larger space observatories might provide helpful
insights of the physical and chemical equilibrium of the atmosphere of
WASP-178 b.
###### Acknowledgements.
We thank the referee, N. B. Cowan, for his valuable comments and suggestions.
CHEOPS is an ESA mission in partnership with Switzerland with important
contributions to the payload and the ground segment from Austria, Belgium,
France, Germany, Hungary, Italy, Portugal, Spain, Sweden, and the United
Kingdom. The CHEOPS Consortium would like to gratefully acknowledge the
support received by all the agencies, offices, universities, and industries
involved. Their flexibility and willingness to explore new approaches were
essential to the success of this mission. IPa, GSc, VSi, LBo, GBr, VNa, GPi,
and RRa acknowledge support from CHEOPS ASI-INAF agreement n. 2019-29-HH.0. ML
acknowledges support of the Swiss National Science Foundation under grant
number PCEFP2_194576. This work was also partially supported by a grant from
the Simons Foundation (PI Queloz, grant number 327127). S.G.S. acknowledge
support from FCT through FCT contract nr. CEECIND/00826/2018 and POPH/FSE
(EC). ABr was supported by the SNSA. ACCa and TWi acknowledge support from
STFC consolidated grant numbers ST/R000824/1 and ST/V000861/1, and UKSA grant
number ST/R003203/1. V.V.G. is an F.R.S-FNRS Research Associate. YAl
acknowledges support from the Swiss National Science Foundation (SNSF) under
grant 200020_192038. We acknowledge support from the Spanish Ministry of
Science and Innovation and the European Regional Development Fund through
grants ESP2016-80435-C2-1-R, ESP2016-80435-C2-2-R, PGC2018-098153-B-C33,
PGC2018-098153-B-C31, ESP2017-87676-C5-1-R, MDM-2017-0737 Unidad de Excelencia
Maria de Maeztu-Centro de Astrobiología (INTA-CSIC), as well as the support
of the Generalitat de Catalunya/CERCA programme. The MOC activities have been
supported by the ESA contract No. 4000124370. S.C.C.B. acknowledges support
from FCT through FCT contracts nr. IF/01312/2014/CP1215/CT0004. XB, SC, DG, MF
and JL acknowledge their role as ESA-appointed CHEOPS science team members.
This project was supported by the CNES. The Belgian participation to CHEOPS
has been supported by the Belgian Federal Science Policy Office (BELSPO) in
the framework of the PRODEX Program, and by the University of Liège through an
ARC grant for Concerted Research Actions financed by the Wallonia-Brussels
Federation. L.D. is an F.R.S.-FNRS Postdoctoral Researcher. This work was
supported by FCT - Fundação para a Ciência e a Tecnologia through national
funds and by FEDER through COMPETE2020 - Programa Operacional Competitividade
e Internacionalizacão by these grants: UID/FIS/04434/2019, UIDB/04434/2020,
UIDP/04434/2020, PTDC/FIS-AST/32113/2017 & POCI-01-0145-FEDER- 032113,
PTDC/FIS-AST/28953/2017 & POCI-01-0145-FEDER-028953, PTDC/FIS-AST/28987/2017 &
POCI-01-0145-FEDER-028987, O.D.S.D. is supported in the form of work contract
(DL 57/2016/CP1364/CT0004) funded by national funds through FCT. B.-O. D.
acknowledges support from the Swiss State Secretariat for Education, Research
and Innovation (SERI) under contract number MB22.00046. This project has
received funding from the European Research Council (ERC) under the European
Union’s Horizon 2020 research grant agreement No 724427). It has also been
carried out in the frame of the National Centre for Competence in Research
PlanetS supported by the Swiss National Science Foundation (SNSF). DE
acknowledges financial support from the Swiss National Science Foundation for
project 200021_200726. and innovation programme (project Four Aces. MF and CMP
gratefully acknowledge the support of the Swedish National Space Agency (DNR
65/19, 174/18). DG gratefully acknowledges financial support from the CRT
foundation under Grant No. 2018.2323 “Gaseousor rocky? Unveiling the nature of
small worlds”. M.G. is an F.R.S.-FNRS Senior Research Associate. MNG is the
ESA CHEOPS Project Scientist and Mission Representative, and as such also
responsible for the Guest Observers (GO) Programme. MNG does not relay
proprietary information between the GO and Guaranteed Time Observation (GTO)
Programmes, and does not decide on the definition and target selection of the
GTO Programme. SH gratefully acknowledges CNES funding through the grant
837319. KGI is the ESA CHEOPS Project Scientist and is responsible for the ESA
CHEOPS Guest Observers Programme. She does not participate in, or contribute
to, the definition of the Guaranteed Time Programme of the CHEOPS mission
through which observations described in this paper have been taken, nor to any
aspect of target selection for the programme. This work was granted access to
the HPC resources of MesoPSL financed by the Region Ile de France and the
project Equip@Meso (reference ANR-10-EQPX-29-01) of the programme
Investissements d’Avenir supervised by the Agence Nationale pour la Recherche.
PM acknowledges support from STFC research grant number ST/M001040/1. IRI
acknowledges support from the Spanish Ministry of Science and Innovation and
the European Regional Development Fund through grant PGC2018-098153-B- C33, as
well as the support of the Generalitat de Catalunya/CERCA programme. GyMSz
acknowledges the support of the Hungarian National Research, Development and
Innovation Office (NKFIH) grant K-125015, a a PRODEX Experiment Agreement No.
4000137122, the Lendület LP2018-7/2021 grant of the Hungarian Academy of
Science and the support of the city of Szombathely. NAW acknowledges UKSA
grant ST/R004838/1. NCS acknowledges funding by the European Union (ERC,
FIERCE, 101052347). Views and opinions expressed are however those of the
author(s) only and do not necessarily reflect those of the European Union or
the European Research Council. Neither the European Union nor the granting
authority can be held responsible for them. KWFL was supported by Deutsche
Forschungsgemeinschaft grants RA714/14-1 within the DFG Schwerpunkt SPP 1992,
Exploring the Diversity of Extrasolar Planets. In this work we use the python
package PyDE available at https://github.com/hpparvi/PyDE. This research has
made use of the SVO Filter Profile Service (http://svo2.cab.inta-
csic.es/theory/fps/) supported from the Spanish MINECO through grant
AYA2017-84089.
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## Appendix A Posterior distributions of the model parameters from the fit of
the TESS transit light curves
Figure 9: Corner plot of the MCMC chains of planetary parameters from the fit
of the TESS transits (see Sect. 4.1.1). In each plot, the solid blue lines
mark the MAP values.
## Appendix B Analysis of the TESS transits in high cadence
In Sect. 4.1 we analyzed the TESS photometry at low cadence (30 min), and we
found a significant difference between the transit depth between sectors 11
and 38. We speculated that the difference in transit depth is due to
photospheric variability, which manifests also as a quasi-periodic signal in
the LC of sector 38. This hypothesis is supported by the growing evidence that
A-type stars show rotational signals due to photospheric inhomogeneities
(e.g., Balona 2011; Böhm et al. 2015; Sikora et al. 2020). This scenario is
further supported by the fact that the WASP-178 is classified as an Am star by
Hellier et al. (2019).
To test if the visible hemisphere of the star hosts dark spots, we searched
for transit anomalies in the TESS LCs, that are localized bumps in the
residuals of the transit fits. If present, they indicate that the planet’s
projection on the stellar surface crosses a darker area compared to the
quiescent photosphere (see for example Béky et al. 2014; Scandariato et al.
2017). To this purpose, the low cadence photometry analyzed in Sect. 4.1 is of
little help, as a finer time sampling is needed. We thus compared the high
cadence (2 min) TESS photometry of sector 38 with the bestfit model discussed
in Sect. 4.1.2, computed using $R_{\rm p}/R*$=0.1141 (see Sect. 4.1.1). This
comparison does not show any bump inside the transits (Figs. 10–13), and we
concluded that in the eight transits observed in sector 38 there is no spot-
crossing event.
We also remark that, in contrast with Rodríguez Martínez et al. (2020), we did
not detect any systematic asymmetry in the transit profile. The stellar flux
distribution along the transit path is symmetric with respect to the transit
center. This either indicates that the star does not show any gravity
darkening, which is unlikely if the stellar rotation period of $\sim$3.2 days
is confirmed, or supports the hypothesis that the star is in a pole-on
geometry, which leads to a radially symmetric flux distribution over the
visible stellar hemisphere.
Figure 10: Detrended short cadence LC of the first (left panel) and second
(right panel) transit observed in TESS sector 38. In each panel, the top plot
shows the model computed with the MAP parameters in Table 3 as a blue solid
line. The bottom plots show the residuals of the short cadence photometry with
respect to the planetary model shown in the top panel. As a guide, we plot
with a blue line the smoothing of the residuals obtained with a Savitzky-Golay
filter. In all plots, the vertical dashed lines mark the first and fourth
contacts.
Figure 11: Detrended short cadence LC of the third (left panel) and fourth
(right panel) transit observed in TESS sector 38. Details are the same as in
Fig. 10.
Figure 12: Detrended short cadence LC of the fifth (left panel) and sixth
(right panel) transit observed in TESS sector 38. Details are the same as in
Fig. 10.
Figure 13: Detrended short cadence LC of the seventh (left panel) and eighth
(right panel) transit observed in TESS sector 38. Details are the same as in
Fig. 10.
*[LC]: Light Curve
*[LCs]: Light Curve
*[FFIs]: Full Frame Images
*[TPFs]: Target Pixel File
*[CBVs]: Co-trending Basis Vectors
*[DRP]: Data Reduction Pipeline
*[PSF]: Point Spread Function
*[IRFM]: Infra-Red Flux Method
*[MCMC]: Markov Chain Monte Carlo
*[GLS]: Generalized Lomb-Scargle
*[FAP]: False Alarm Probability
*[PC]: Phase Curve
*[LD]: Limb Darkening
*[MAP]: Maximum A Posteriori
*[GP]: Gaussian Process
*[PCs]: Phase Curve
*[PSFs]: Point Spread Function
*[HJ]: Hot Jupiter
|
# Representations of Materials for Machine Learning111Accepted for publication
in Annual Review of Materials Research Volume 53,
https://www.annualreviews.org/.
James Damewood Department of Materials Science and Engineering, Massachusetts
Institute of Technology, 77 Massachusetts Avenue, Cambridge, USA, 02129
Jessica Karaguesian Department of Materials Science and Engineering,
Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge,
USA, 02129 Center for Computational Science and Engineering, Massachusetts
Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, USA, 02139
Jaclyn R. Lunger Department of Materials Science and Engineering,
Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge,
USA, 02129 Aik Rui Tan Department of Materials Science and Engineering,
Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge,
USA, 02129 Mingrou Xie Department of Materials Science and Engineering,
Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge,
USA, 02129 Department of Chemical Engineering, Massachusetts Institute of
Technology, 77 Massachusetts Avenue, Cambridge, MA, USA, 02139 Jiayu Peng
Department of Materials Science and Engineering, Massachusetts Institute of
Technology, 77 Massachusetts Avenue, Cambridge, USA, 02129 Rafael Gómez-
Bombarelli<EMAIL_ADDRESS>Department of Materials Science and Engineering,
Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge,
USA, 02129
###### Abstract
High-throughput data generation methods and machine learning (ML) algorithms
have given rise to a new era of computational materials science by learning
relationships among composition, structure, and properties and by exploiting
such relations for design. However, to build these connections, materials data
must be translated into a numerical form, called a representation, that can be
processed by a machine learning model. Datasets in materials science vary in
format (ranging from images to spectra), size, and fidelity. Predictive models
vary in scope and property of interests. Here, we review context-dependent
strategies for constructing representations that enable the use of materials
as inputs or outputs of machine learning models. Furthermore, we discuss how
modern ML techniques can learn representations from data and transfer chemical
and physical information between tasks. Finally, we outline high-impact
questions that have not been fully resolved and thus, require further
investigation.
###### Contents
1. 1 INTRODUCTION
2. 2 STRUCTURAL FEATURES FOR ATOMISTIC GEOMETRIES
1. 2.1 Local Descriptors
2. 2.2 Global Descriptors
3. 2.3 Topological Descriptors
3. 3 LEARNING ON PERIODIC CRYSTAL GRAPHS
4. 4 CONSTRUCTING REPRESENTATIONS FROM STOICHIOMETRY
5. 5 DEFECTS, SURFACES, AND GRAIN BOUNDARIES
6. 6 TRANSFERABLE INFORMATION BETWEEN REPRESENTATIONS
7. 7 GENERATIVE MODELS FOR INVERSE DESIGN
8. 8 DISCUSSION
1. 8.1 Trade-offs of Local and Global Structural Descriptors
2. 8.2 Prediction from Unrelaxed Crystal Prototypes
3. 8.3 Applicability of Compositional Descriptors
4. 8.4 Extensions of Generative Models
## 1 INTRODUCTION
Energy and sustainability applications demand the rapid development of
scalable new materials technologies. Big data and machine learning (ML) have
been proposed as strategies to rapidly identify “needle-in-the-haystack”
materials that have the potential for revolutionary impact.
High-throughput experimentation platforms based on robotized laboratories can
increase the efficiency and speed of synthesis and characterization. However,
in many practical open problems, the number of possible design parameters is
too large to be analyzed exhaustively. Virtual screening somewhat mitigates
this challenge by using physics-based simulations to suggest the most
promising candidates, reducing the cost but also the fidelity of the
screens[1, 2, 3].
Over the past decade, hardware improvements, new algorithms, and the
development of large-scale repositories of materials data [4, 5, 6, 7, 8, 9]
have enabled a new era of ML methods. In principle, predictive ML models can
identify and exploit nontrivial trends in high-dimensional data to achieve
accuracy comparable with or superior to first-principles calculations, but
with orders of magnitude reduction in cost. In practice, while a judicious
model choice is helpful in moving towards this ideal, such ML methods are also
highly dependent on the numerical inputs used to describe systems of
interest—the so-called representations. Only when the representation is
composed of a set of features and descriptors from which the desired physics
and chemistry are emergent can the promise of ML be achieved.
Thus, the problem that materials informatics researchers must answer is: how
can we best construct this representation? Previous works have provided
practical advice for constructing materials representations [10, 11, 12, 13,
14], namely that: (1) the similarity/difference between two data points should
match the similarity/difference between representations of those two data
points, (2) the representation should be applicable to the entire materials
domain of interest, (3) the representation should be easier to calculate than
the target property.
Representations should reflect the degree of similarity between data points
such that similar data have similar representations and as data points become
more different their representations diverge. Indeed, the definition of
similarity will depend on the application. Consider, as an example, a
hypothetical model predicting the electronegativity of an element, excluding
Nobel gases. One could attempt to train the model using atomic number as
input, but this representation violates the above principle, as atoms with a
similar atomic number can have significantly different electronegativities
(e.g. fluorine and sodium), forcing the model to learn a sharply varying
function whose changes appear at irregular intervals. Alternatively, a
representation using period and group numbers would closely group elements
with similar atomic radii and electron configurations. Over this new domain,
the optimal prediction will result in a smoother function that is easier to
learn.
The approach used to extract representation features from raw inputs should be
feasible over the entire domain of interest—all data points used in training
and deployment. If data required to construct the representation is not
available for a particular material, ML screening predictions cannot be made.
Finally, for the ML approach to remain a worthwhile investment, the
computational cost of obtaining representation features and descriptors for
new data should be smaller than that of obtaining the property itself through
traditional means, either experimentally or with first-principles
calculations. If, for instance, accurately predicting a property calculated by
density functional theory (DFT) with ML requires input descriptors obtained
from DFT on the same structure and at the same level of theory, the machine
learning model does not offer any benefit.
A practicing materials scientist will notice a number of key barriers to
forming property-informative representations that satisfy these criteria.
First, describing behavior often involves quantifying structure-to-property
relationships across length scales. The diversity of possible atomistic
structure types considered can vary over space groups, supercell size, and
disorder parameters. This challenge motivates researchers to develop flexible
representations capable of capturing local and global information based on
atomic positions. Beyond this idealized picture, predicting material
performance relies upon understanding the presence of defects, the
characteristics of the microstructure, and reactions at interfaces. Addressing
these concerns requires extending previous notions of structural similarity or
developing new specialized tools. Furthermore, atomistic structural
information is not available without experimental validation or extensive
computational effort [15, 16]. Therefore, when predictions are required for
previously unexplored materials, models must rely on more readily available
descriptors such as those based on elemental composition and stoichiometry.
Lastly, due to experimental constraints, datasets in materials science can
often be scarce, sparse, and restricted to relatively few and self-similar
examples. The difficulty in constructing a robust representation in these
scenarios has inspired strategies to leverage information from high-quality
representations built for closely related tasks through transfer learning.
In this review, we will analyze how representations of solid-state materials
(Figure 1) can be developed given constraints on the format, quantity, and
quality of available data. We will discuss the justification, benefits, and
trade-offs of different approaches. This discussion is meant to highlight
methods of particular interest rather than provide exhaustive coverage of the
literature. We will discuss current limitations and open problems whose
solutions would have high impact. In summary, we intend to provide readers
with an introduction to current state of the field and exciting directions for
future research.
Figure 1: Summary of representations for perovskite SrTiO3. Top Left. 2D cross
section of Voronoi decomposition. Predictive features can be constructed from
neighbors and geometric shape of cells [17]. Middle Left. Crystal graph of
SrTiO3 constructed assuming periodic boundary conditions and used as input to
graph neural networks [18]. Bottom Left. Compositional data including
concentrations and easily accessible atomic features including
electronegativities and atomic radii [19]. Data taken from Reference [20]. Top
Right. Deviations on a pristine bulk structure induced by an oxygen vacancy to
predict formation energy [21]. Middle Right. Representations can be learned
from large repositories using deep neural networks. The latent physical and
chemical information can be leveraged in related but data-scare tasks. Bottom
Right. Training of generative models capable of proposing new crystal
structures by placing atoms in discretized volume elements [22, 23, 24, 25].
## 2 STRUCTURAL FEATURES FOR ATOMISTIC GEOMETRIES
Simple observations in material systems (e.g. higher ductility of face-
centered cubic metals compared to body-centered cubic metals) have made it
evident that material properties are highly dependent on crystal
structure—from coordination and atomic ordering to broken symmetries and
porosity. For a computational material scientist, this presents the question
of how to algorithmically encode information from a set of atoms types
(${a_{1},a_{2},a_{3},...}$), positions (${x_{1},x_{2},x_{3},...}$), and
primitive cell parameters into a feature set that can be effectively utilized
in machine learning.
For machine learning methods to be effective, it is necessary that the
machine-readable representation of a material’s structure fulfills the
criteria as outlined in the introduction [10, 11, 12, 13, 14]. Notably, scalar
properties (such as heat capacity or reactivity) do not change when
translations, rotations, or permutations of atom indexing are applied to the
atomic coordinates. Therefore, to ensure representations reflect the
similarities between atomic structures, the representations should also be
invariant to those symmetry operations.
### 2.1 Local Descriptors
One strategy to form a representation of a crystal structure is to
characterize the local environment of each atom and consider the full
structure as a combination local representations. This concept was applied by
Behler and Parinello[26], who proposed the atom-centered symmetry functions
(ACSF). ACSF descriptors (Figure 2a) can be constructed using radial,
$G_{i}^{1}$, and angular, $G_{i}^{2}$, symmetry functions centered on atom i,
$G_{i}^{1}=\sum^{\text{neighbors}}_{j\neq
i}e^{-\eta(R_{ij}-R_{s})^{2}}f_{c}(R_{ij})$ (1)
$G_{i}^{2}=2^{1-\zeta}\sum^{\text{neighbors}}_{j,k\neq
i}(1+\lambda\cos\theta_{ijk})^{\zeta}e^{-\eta(R_{ij}^{2}+R_{ik}^{2}+R_{jk}^{2})}f_{c}(R_{ij})f_{c}(R_{ik})f_{c}(R_{jk})$
(2)
with the tunable parameters $\lambda$, $R_{s}$, $\eta$, and $\zeta$. $R_{ij}$
is the distance between the central atom $i$ and atom $j$, and $\theta_{ijk}$
corresponds to the angle between the vector from the central atom to atom $j$
and the vector from the central atom to atom $k$. The cutoff function $f_{c}$
screens out atomic interactions beyond a specified cutoff radius and ensures
locality of the atomic interactions. Because symmetry functions rely on
relative distances and angles, they are rotationally and translationally
invariant. Local representations can be constructed from many symmetry
functions of the type $G_{i}^{1}$ and $G_{i}^{2}$ with multiple settings of
tunable parameters to probe the environment at varying distances and angular
regions. With the set of localized symmetry functions, neural networks can
then predict local contributions to a particular property and approximate
global properties as the sum of local contributions. The flexibility of this
approach allows for modification of the $G_{i}^{1}$ and $G_{i}^{2}$ functions
[27, 28] or higher capacity neural networks for element-wise prediction [28].
In search of a representation with fewer hand-tuned parameters and a more
rigorous definition of similarity, Bartok et al. [12] proposed a rotationally
invariant kernel for comparing environments based on local atomic density.
Given a central atom, the Smooth Overlap of Atomic Positions (SOAP) defines
the atomic density function $\rho(\mathbf{r})$ as a sum of Gaussian functions
centered at each neighboring atom within a cutoff radius (Figure 2b). The
choice of Gaussian function is motivated by the intuition that representations
should be continuous such that small changes in atomic positions should result
in correspondingly small changes in the metric between two configurations.
With a basis of radial functions $g_{n}(\mathbf{r})$ and spherical harmonics
$Y_{lm}(\theta,\phi)$, $\rho(\mathbf{r})$ for central atom $i$ can be
expressed as:
$\rho_{i}(\mathbf{r})=\sum_{j}\exp{-\frac{|\mathbf{r}-\mathbf{r}_{ij}|^{2}}{2\sigma^{2}}}=\sum_{nlm}c_{nlm}g_{n}(\mathbf{r})Y_{lm}(\mathbf{\hat{r}})$
(3)
and the kernel can be computed[12, 29]:
$K(\rho,\rho^{\prime})=\mathbf{p}(\mathbf{r})\cdot\mathbf{p}^{\prime}(\mathbf{r})$
(4) $\mathbf{p}(\mathbf{r})\equiv\sum_{m}c_{nlm}(c_{n^{\prime}lm})^{*}$ (5)
where $c_{nlm}$ are the expansion coefficients in Equation 3. In practice,
$\mathbf{p}(\mathbf{r})$ can be used as a vector descriptor of the local
environment and is also referred to as a power spectrum [12]. SOAP has
demonstrated extraordinary versatility for materials applications both as a
tool for measuring similarity [30] and as a descriptor for machine learning
algorithms [31]. Furthermore, the SOAP kernel can be used to compare densities
of different elements by adding an additional factor that provides a
definition for similarity between atoms, where for instance, atoms in the same
group could have higher similarity [29]. The mathematical connections between
different local atomic density representations including ACSFs and SOAP are
elucidated by a generalized formalism introduced by Willatt et al. [32],
offering a methodology through which the definition of new variants can be
clarified.
Instead of relying on the density of nearby atoms, local representations can
be derived from a Voronoi tessellation of a crystal structure. The Voronoi
tessellation segments space into cells such that each cell contains one atom
and all points in space such that atom A is the closest atom are contained in
the same cell as atom A (Figure 2c). From these cells, Ward et al. [17]
identified a set of descriptive features including an effective coordination
number computed using the area of the faces, the lengths and volumes of nearby
cells, ordering of the cells based on elements, and atomic properties of
nearest neighbors weighted by the area of the intersecting face. When combined
with compositional features [19], their representation results in better
performance on predictions of formation enthalpy for ICSD than partial radial
distribution functions [33] (Figure 1 in Reference [17]). In subsequent work,
these descriptors have facilitated the prediction of experimental heat
capacities in MOFs [34]. Similarly, Isayev et al. [35] replaced faces of the
Voronoi tessellation with virtual bonds and separated the resulting framework
into sets of linear (up to four atoms) and shell-based (up to nearest
neighbors) fragments. Additional features related to the atomic properties of
constituent elements were associated with each fragment, and the resulting
vectors were concatenated with attributes of the supercell. In addition to
demonstrating accurate predictive capabilities, models could be interpreted
through the properties of the various fragments. For instance, predictions of
band gap could be correlated with the difference in ionization potential in
two-atom linear fragments, a trend that could be exploited to design
material’s properties through tuning of composition[35].
Figure 2: (a) Examples of radial, $G^{1}_{i}$, and angular, $G^{2}_{i}$,
symmetry functions from the local atom-centered symmetry function descriptor
proposed by Behler and Parinello[26]. (b) In the Smooth Overlap of Atomic
Positions (SOAP) descriptor construction, the atomic neighborhood density of a
central atom is defined by a sum of Gaussian functions around each neighboring
atom. A kernel function can then be built to compare the different
environments by computing the density overlap of the atomic neighborhood
functions. Figure is reprinted from reference [36]. (c) Voronoi tessellation
in two and three dimensions. Yellow circles and spheres show particles while
the blue lines divide equidistantly the space between two neighboring
particles. Polygonal spaces encompassed by the blue lines are the Voronoi
cells. Figure is reprinted from reference [37]. (d) Illustration of a Coulomb
matrix where each element in the matrix shows Coulombic interaction between
the labeled particles in the system on the left. Diagonal elements show self-
interactions. (e) The births and deaths of topological holes in a point cloud
(left) are recorded on a persistence diagram (right). Persistent features lie
far from the parity line and indicate more significant topological features.
Feature is reprinted from reference [38].
### 2.2 Global Descriptors
Alternatively, to more explicitly account for interactions beyond a fixed
cutoff, atom types and positions can be encoded into a global representation
that reflects geometric and physical insight. Inspired by the importance of
electrostatic interactions in chemical stability, Rupp et al. [39] proposed
the Coulomb matrix (Figure 2d), which models the potential between electron
clouds:
$M_{i,j}=\begin{cases}Z_{i}^{2.4}&\text{for }i=i\\\
\frac{Z_{i}Z_{j}}{|r_{i}-r_{j}|}&\text{for }i\neq j\end{cases}$ (6)
Due to the fact that off-diagonal elements are only dependent on relative
distances, Coulomb matrices are rotation and translation invariant. However,
the representation is not permutation invariant since changing the labels of
the atoms will rearrange the elements of the matrix. While originally
developed for molecules, the periodicity of crystal structures can be added to
the representation by considering images of atoms in adjacent cells, replacing
the $\frac{1}{|r_{i}-r_{j}|}$ dependence with another function with the same
small distance limit and periodicity that matches the parent lattice, or using
an Ewald sum to account for long range interactions [11]. BIGDML [40] further
improved results by restricting predictions from the representation to be
invariant to all symmetry operations within the space group of the parent
lattice and demonstrated effective implementations on tasks ranging from H
interstitial diffusion in Pt to phonon density of states. While this approach
has been able to effectively model long-range physics, these representations
rely on a fixed supercell and may not be able to achieve the same chemical
generality as local environments [40].
Global representations have also been implemented with higher-order tensors.
Partial radial distribution functions (PRDF) are 3D non-permutation invariant
matrices $g_{\alpha\beta r}$ whose elements correspond to the density of
element $\beta$ in the environments of element $\alpha$ at radius $r$ [33].
The many-body tensor representation (MBTR) provides a more general framework
[10] that can quantify k-body interactions and account for chemical similarity
between elements. The MBTR is translationally, rotationally, and permutation
invariant and can be applied to crystal structures by only summing over atoms
in the primitive cell. While MBTR exhibited better performance than SOAP or
Coulomb matrices for small molecules, its accuracy may not extend to larger
systems [10].
Another well-established method for representing crystal structures in
materials science is the cluster expansion. Given a parent lattice and a
decoration $\mathbf{\sigma}$ defining the element that occupies each site,
Sanchez et al. sought to map this atomic ordering to material properties and
proposed evaluating the correlations between sites through a set of cluster
functions. Each cluster function $\Phi$ is constructed from a product of basis
functions ${\phi}$, over a subset of sites [41]. To ensure the representation
is appropriately invariant, symmetrically equivalent clusters are grouped into
classes denoted by $\alpha$. The characteristics of the atomic ordering can be
quantified by averaging cluster functions over the decoration
$<\Phi_{\alpha}>_{\mathbf{\sigma}}$, and properties $q$ of the configuration
can be predicted as:
$q(\mathbf{\sigma})=\sum_{\alpha}J_{\alpha}m_{\alpha}<\Phi_{\alpha}>_{\sigma}$
(7)
where $m_{\alpha}$ is a multiplicity factor that accounts for the rate of
appearance of different cluster types, and $J_{\alpha}$ are parameters
referred to as effective cluster interactions that must be determined from
fits to data [42]. While cluster expansions have been constructed for decades
and provided useful models for configurational disorder and alloy
thermodynamics [42], cluster expansions assume the structure of the parent
lattice and models cannot generally be applied across different crystal
structures [43, 44]. Furthermore, due to the increasing complexity of
selecting cluster functions, implementations are restricted to binary and
ternary systems without special development [45]. Additional research has
extended the formalism to continuous environments (Atomic Cluster Expansion)
by treating $\mathbf{\sigma}$ as pairwise distances instead of site-
occupancies and constructing ${\phi}$ from radial functions and spherical
harmonics [46]. The Atomic Cluster Expansion framework has provided a basis
for more sophisticated deep learning approaches [47].
### 2.3 Topological Descriptors
Topological data analysis (TDA) has found favor over the past decade in
characterizing structure in complex, high-dimensional datasets. When applied
to the positions of atoms in amorphous or crystalline structures, topological
methods reveal underlying geometric features that inform behavior in
downstream predictions such as phase changes, reactivity, and separations. In
particular, persistent homology (PH) is able to identify significant
structural descriptors that are both machine readable and physically
interpretable. The data can be probed at different length scales (formally
called filtrations) by computing a series of complexes that each include all
sets of points where all pairwise distances are less than the corresponding
length [48]. Analysis of complexes by homology in different dimensions reveals
holes or voids in the data manifold, which can be described by the range of
length scales they are observed at (persistences), as well as when they are
produced (births) and disappear (deaths). Emergent features with significant
persistence values are less likely to be caused by noise in the data or as an
artifact of the chosen length scales. In practice, multiple persistences,
births, and deaths produced from a single material can be represented together
by persistent diagrams (Figure 2e) or undergo additional feature engineering
to generate a variety of descriptors as machine learning inputs [49].
While persistent homology has been applied to crystal structures in the Open
Quantum Material Database [50], the method is particularly useful in the
analysis of porous materials. The identified features (births, deaths,
persistences) hold direct physical relevance to traditional structural
features used to describe the pore geometries. For instance, persistent 2D
deaths represent the largest sphere that can be included inside the pores of
the materials. Krishnapriyan et al. has showed that these topological
descriptors outperform traditional structural descriptors when predicting
carbon dioxide adsorption under varying conditions for metal-organic
frameworks [38], as did Lee et al. for zeolites for methane storage capacities
[51]. Representative cycles can trace the topological features back to the
atoms that are responsible for the hole or the void, creating a direct
relationship between structure and predicted performance (Figure 4 in
Reference [38]). Similarity methods for comparing barcodes can then be used to
identify promising novel materials with similar pore geometries for targeted
applications.
A caveat is that PH does not inherently account for system size and is thus
size-dependent. The radius cutoff, or the supercell size, needs to be
carefully considered to encompass all significant topological features and
allow comparison across systems of interest. In the worst case scenario, the
computation cost per filtration for a structure is $O(N^{3})$, where $N$ is
the number of sets of points defining a complex. Although the cost is
alleviated by the sparsity of the boundary matrix [52], the scaling is poor
for structures whose geometric features exceed unit cell lengths. The benefit
of using PH features to capture more complex structural information has to be
carefully balanced with the cost of generating these features.
## 3 LEARNING ON PERIODIC CRYSTAL GRAPHS
In the previous section, we described many physically-inspired descriptors
that characterize materials and can be used to efficiently predict properties.
The use of differentiable graph-based representations in convolutional neural
networks, however, mitigates the need for manual engineering of descriptors
[53, 54]. Indeed, advances in deep learning and the construction of large-
scale materials databases [4, 5, 6, 7, 8, 9] have made it possible to learn
representations directly from structural data. From a set of atoms
${a_{1},a_{2},...}$ located at positions ${x_{1},x_{2},x_{3},...}$, materials
can be converted to a graph $G(V,E)$ defined as the set of atomic nodes $V$
and the set of edges $E$ connecting neighboring atoms. Many graph-based neural
network architectures were originally developed for molecular systems, with
edges representing bonds. By considering periodic boundary conditions and
defining edges as connections between neighbors within a cutoff radius,
graphical representations can be leveraged for crystalline systems. The
connectivity of the crystal graph thus naturally encodes local atomic
environments [18].
When used as input to machine learning algorithms, the graph nodes and edges
are initialized with an associated set of features. Nodal features can be as
simple as a one-hot vector of the atomic number or can explicitly include
other properties of the atomic species (e.g. electronegativity, group,
period). Edge features are typically constructed from the distance between the
corresponding atoms. Subsequently, a series of convolutions parameterized by
neural networks modify node and/or edge features based on the current state of
their neighborhood (Figure 3a). As the number of convolutions increases,
interactions from further away in the structure can propagate, and graph
features become tuned to reflect the local chemical environment. Finally, node
and edge features can be pooled to form a single vector representation for the
material [53, 55].
Crystal Graph Convolution Neural Networks (CGCNN) [18] and Materials Graph
Networks (MEGNet) [56] have become benchmark algorithms capable of predicting
properties across solid-state materials domains including bulk, surfaces,
disordered systems, and 2D materials [57, 58]. Atomistic Line Graph Neural
Network (ALIGNN) extended these approaches by including triplet three-body
features in addition to nodes and edges and exhibited superior performance to
CGCNN over a broad range of regression tasks including formation energy,
bandgap, and shear modulus [59]. Other variants have used information from
Voronoi polyhedra to construct graphical neighborhoods and augment edge
features [60] or initialized node features based on the geometry and electron
configuration of nearest neighbor atoms [61].
Figure 3: (a) General architecture of graph convolutional neural networks for
property prediction in crystalline systems. Three-dimensional crystal
structure is represented as a graph with nodes representing atoms and edges
representing connections between nearby atoms. Features (e.g. nodal, edge,
angular) within local neighborhoods are convolved, pooled into a crystal-wide
vector, then mapped to the target property. Figure adapted from [62]. (b)
Information loss in graphs built from pristine structures. Geometric
distortions of ground-state crystal structures are captured as differing edge
features in graphical representations. This information is lost in graphs
constructed from corresponding unrelaxed structures. (c) Graph-based models
can struggle to capture periodicity-dependent properties, such as cell lattice
parameters. $R^{2}$ scores presented here were reported by Gong et al. for
lattice parameter, a, predictions in short and long 1D single carbon chain toy
structures. Figure adapted from [63]. (d) Ability of graphical representations
to distinguish toy structures. Assuming a sufficiently small cutoff radius,
the invariant representation—using edge lengths and/or angles—cannot
distinguish the two toy arrangements, while the equivariant representation
with directional features can. Figure adapted from [64].
While these methods have become widespread for property prediction, graph
convolution updates based only on the local neighborhood may limit the sharing
of information related to long-range interactions or extensive properties.
Gong et al. demonstrated that these models can struggle to learn materials
properties reliant on periodicity, including characteristics as simple as
primitive cell lattice parameters (Figure 3c)[63]. As a result, while graph-
based learning is a high-capacity approach, performance can vary substantially
by the target use case. In some scenarios, methods developed primarily for
molecules can be effectively implemented “out-of-the-box” with the addition of
periodic boundary conditions, but especially in the case of long-range
physical phenomena, optimal results can require specialized modeling.
Various strategies to account for this limitation have been proposed. Gong et
al. found that if the pooled representation after convolutions was
concatenated with human-tuned descriptors, errors could be reduced by $90\%$
for related predictions, including phonon internal energy and heat capacity
[63]. Algorithms have attempted to more explicitly account for long-range
interactions by modulating convolutions with a mask defined by a local basis
of Gaussians and a periodic basis of plane waves [65], employing a unique
global pooling scheme that could include additional context such as
stoichiometry [66], or constructing additional features from the reciprocal
representation of the crystal [67]. Other strategies have leveraged
assumptions about the relationships among predicted variables, such as
representing phonon spectra using a Gaussian mixture model [68].
Given the promise and flexibility of graphical models, improving the data-
efficiency, accuracy, generalizability, and scalability of these
representations are active areas of research. While our previous discussion of
structure-based material representations relied on the invariance of scalar
properties to translation and rotation, this characteristic does not continue
to hold for higher-order tensors. Consider a material with a net magnetic
moment. If the material is rotated $180^{\circ}$ around an axis perpendicular
to the magnetization, the net moment then points in the opposite direction.
The moment was not invariant to the rotation but instead, transformed
alongside the operation in an equivariant manner [69]. For a set of
transformations described by group $G$, equivariant functions $f$ satisfy
$g*f(x)=f(g*x)$ for every input $x$ and every group element $g$ [69, 70].
Recent efforts have shown that by introducing higher-order tensors to node and
edge features (Figure 3d) and restricting the update functions such that
intermediate representations are equivariant to the group E3 (encompassing
translations, rotations, and reflections in $R^{3}$), models can achieve
state-of-the-art accuracy on benchmark datasets and even exhibit comparable
performance to structural descriptors in low-data ($\sim$100 datapoints)
regimes [71, 69, 64]. Further accuracy improvements can be made by explicitly
considering many-body interactions beyond edges [72, 73, 47]. Such models,
developed for molecular systems, have since been extended to solid-state
materials and shown exceptional performance. Indeed, Chen et al. trained an
equivariant model to predict phonon density of states and was able to screen
for high heat capacity targets [74], tasks identified to be particularly
challenging for baseline CGCNN and MEGNet models [63]. Therefore, equivariant
representations may offer a more general alternative to the specialized
architectures described above.
A major restriction of these graph-based approaches is the requirement for the
positions of atomic species to be known. In general, ground-state crystal
structures exhibit distortions that allow atoms to break symmetries, which are
computationally modeled with expensive DFT calculations. Graphs generated from
pristine structures lack representation of relaxed atomic coordinates (Figure
3b) and resulting model accuracy can degrade substantially [75, 76]. These
graph-based models are therefore often most effective at predicting properties
of systems for which significant computational resources have already been
invested, thus breaking advice (3) from Section 1. As a result, their
practical usage often remains limited when searching broad regions of crystal
space for an optimal material satisfying a particular design challenge.
Strategies have therefore been developed to bypass the need for expensive
quantum calculations and use unrelaxed crystal prototypes as inputs. Gibson et
al. trained CGCNN models on datasets composed of both relaxed structures and a
set of perturbed structures that map to the same property value as the fully
relaxed structure. The data-augmentation incentivizes the CGCNN model to
predict similar properties within some basin of the fully relaxed structure
and was demonstrated to improve prediction accuracy on an unrelaxed test set
[76]. Alternatively, graph-based energy models can be used to modify unrelaxed
prototypes by searching through a fixed set of possibilities [77] or using
Bayesian optimization [78] to find structures with lower energy. Lastly,
structures can be relaxed using cheap surrogate model (e.g. a force field)
before a final prediction is made. The accuracy and efficiency of such a
procedure will fundamentally rely on the validity and compositional
generalizability of the surrogate relaxation approach [75].
## 4 CONSTRUCTING REPRESENTATIONS FROM STOICHIOMETRY
The phase, crystal system, or atomic positions of materials are not always
available when modeling materials systems, rendering structural and graphical
representations impossible to construct. In the absence of this data, material
representations can also be built purely from stoichiometry (the concentration
of the constituent elements) and without knowledge of the geometry of the
local atomistic environments. Despite their lack of structural information and
apparent simplicity, these methods provide unique benefits for materials
science researchers. First, descriptors used to form compositional
representations such as common atomic properties (e.g. atomic radii,
electronegativity) do not require computational overhead and can be readily
found in existing databases [19]. In addition, effective models can often be
built using standard algorithms for feature selection and prediction that are
implemented in freely available libraries [79], increasing accessibility to
non-experts when compared with structural models. Lastly, when used as tools
for high-throughput screening, compositional models identify a set of
promising elemental concentrations. Compared with the suggestion of particular
atomistic geometries, stoichiometric approaches may be more robust, as they
make weaker assumptions about the outcomes of attempted syntheses.
Compositional-based rules have long contributed to efficient materials design.
Hume-Rothery and Linus Pauling designed rules for determining the formation of
solid solutions and crystal structures that include predictions based on
atomic radii and electronic valence states [80, 81]. However, many exceptions
to their predictions can be found [82].
Machine learning techniques offer the ability to discover and model
relationships between properties and physical descriptors through statistical
means. Meredig et al. demonstrated that a decision tree ensemble trained using
a feature set of atomic masses, positions in the periodic table, atomic
numbers, atomic radii, electronegativities, and valence electrons could
outperform a conventional heuristic on predicting whether ternary compositions
would have formation energies $<100$ meV/atom [83]. Ward et al. significantly
expanded this set to 145 input properties, including features related to the
distribution and compatibility of the oxidation states of constituent atoms
[19]. Their released implementation, MagPie, can be a useful benchmark or
staring point for the development of further research methods [79, 84, 85].
Furthermore, if a fixed structural prototype (e.g. elapsolite) is assumed,
these stoichiometric models can be used to analyze compositionally-driven
variation in properties [86, 87].
Even more subtle yet extremely expressive low-dimensional descriptors can be
obtained by initializing a set with standard atomic properties and computing
successive algebraic combinations of features, with each calculation being
added to the set and used to compute higher order combinations in the next
round. While the resulting set will grow exponentially, compressive sensing
can then be used to identify the most promising descriptors from sets that can
exceed $10^{9}$ possibilities [88, 89]. Ghiringhelli et al. found descriptors
that could accurately predict whether a binary compound would form in a
zincblende or rocksalt structure [90], and Bartel et al. identified an
improved tolerance factor $\tau$ for the formation of perovskite systems [91]
(Table 1). While these approaches do not derive their results from a known
mechanism, they do provide enough interpretability to enable the extraction of
physical insights for the screening and design of materials.
Table 1: Example Descriptors Determined through Compressive Sensing Descriptor | Prediction | Variables
---|---|---
| | IP-Ionization Potential
$\frac{IP(B)-EA(B)}{r_{p}(A)^{2}}$ | Ordering in AB Compound | EA-Electron Afinity
| | $r_{p}$-Radius of Maximum Density of p-Orbital
$\frac{r_{X}}{r_{B}}-n_{A}(n_{A}-\frac{r_{A}/r_{B}}{ln[r_{A}/r_{B}]})$ | Stability of ABX3 Perovskite | $n_{Y}$-Oxidation State of Y
| | $r_{y}$-Ionic Radius of Y
When large datasets are available, deep neural networks tend to outperform
traditional approaches, and that is also the case for compositional
representations. The size of modern materials science databases have enabled
the development of information-rich embeddings that map elements or
compositions to vectors as well as the testing and validation of deep learning
models. Chemically meaningful embeddings can be constructed by counting all
compositions in which that element appeared in the Materials Project [92] or
learned through the application of natural language processing to previously
reported results in the scientific literature [93]. These data-hungry methods
were able to demonstrate that their representations could be clustered based
on atomic group [92] and could be used to suggest new promising compositions
based on similarity with the best known materials [93]. The advantages of
training deep learning algorithms with large datasets are exemplified by
ElemNet, which only uses a vector of fractional stoichiometry as input.
Despite its apparent simplicity, when $>3,000$ training points where
available, ElemNet performed better than a MagPie-based model at predicting
formation enthalpies [94].
While the applicability of ElemNet is limited to problem domains with
$O(10^{3})$ datapoints, more recent methods have significantly reduced this
threshold. ROOST [95] represented each composition as a fully-connected graph
with nodes as elements, and properties were predicted using a message-passing
scheme with an attention mechanism that relied on the stoichiometric fraction
of each element. ROOST substantially improved on ElemNet, achieving better
performance than MagPie in cases with only hundreds of training examples.
Meanwhile, CrabNet [96] forms element-derived matrices as a sum of embeddings
of each element’s identity and stoichiometric fraction. This approach achieves
similar performance to ROOST by updating the representation using self-
attention blocks. The fractional embedding can take log-scale data as input
such that even dopants in small concentrations can have a significant effect
on predictions. Despite the inherent challenges of predicting properties
purely from composition, these recent and significant modeling improvements
suggest that continued algorithmic development could be an attractive and
impactful direction for future research projects.
Compositional models have the advantage that they can suggest new systems to
experimentalists without requiring a specific atomic geometry and, likewise,
can learn from experimental data without necessitating an exact crystal
structure [97]. Owing to their ability to incorporate experimental findings
into ML pipelines and provide suggestions with fewer experimental requirements
(e.g. synthesis of a particular phase), compositional models have become
attractive methods for materials design. Zhang el al. trained a compositional
model using atomic descriptors on previous experimental data to predict
Vicker’s harness and validated their model by synthesizing and testing eight
metal disilicides [97]. Oliynik et al. identified new Heusler compounds, while
also verifying their approach on negative cases where they predicted a
synthesis would fail [87]. Another application of their approach enabled the
prediction of the crystal structure prototype of ternary compounds with
greater than 96% accuracy. By training their model to predict the probability
associated with each structure, they were able to experimentally engineer a
system (TiFeP) with multiple competing phases [98].
While researchers have effectively implemented compositional models as methods
for materials design, their limitations should be considered when selecting a
representation for ML studies. Fundamentally, compositional models will only
provide a single prediction for each stoichiometry regardless of the number of
synthesizeable polymorphs. While training models to only predict properties of
the lowest-energy structure is physically justifiable [99], extrapolation to
technologically relevant meta-stable systems may still be limited.
Additionally, graph-based structural models such as CGCNN [18] or MEGNet [56]
generally outperform compositional models [84]. Therefore, composition models
are most practically applicable when atomistic resolution of materials is
unavailable, and thus structural representations cannot be effectively
constructed.
## 5 DEFECTS, SURFACES, AND GRAIN BOUNDARIES
Mapping the structure of small molecules and unit cells to materials
properties has been a reasonable starting point for many applications of
materials science modeling. However, materials design often requires
understanding of larger length scales beyond the small unit cell, such as in
defect and grain boundary engineering, and in surface science [100]. In
catalysis, for example, surface activity is highly facet dependent and cannot
be modeled using the bulk unit cell alone. It has been shown that the (100)
facet of RuO2, a state-of-the-art catalyst for the oxygen evolution reaction
(OER), has an order of magnitude higher current for OER than the active site
on the thermodynamically stable (110) facet [101]. Similarly, small unit cells
are not sufficient for modeling transport properties, where size, orientation,
and characteristics of grain boundaries play a large role. In order to apply
machine learning to practical materials design, it is therefore imperative to
construct representations that can characterize environments at the relevant
length scales.
Figure 4: (a) Point defect properties are learned from a representation of the
pristine bulk structure and additional relevant information on conduction and
valence band levels. (b) Surface properties are learned from a combination of
pristine bulk structure representation, miller index, and density of states
information. (c) Local environments of atoms near a grain boundary versus
atoms in the pristine bulk are compared to learn grain boundary properties.
Figure 4a adapted from [102].
Defect engineering offers a common and significant degree of freedom through
which materials can be tuned. Data science can contribute to the design of
these systems as fundamental mechanisms are often not completely understood
even in long-standing cases such as carbon in steels [103]. Dragoni et al.
[31] developed a Gaussian Approximate Potential (GAP) [104] using SOAP
descriptors for face-centered cubic iron that could probe vacancies,
interstitials, and dislocations, but their model was confined to a single
phase of one element and required DFT calculations incorporating $O(10^{6})$
unique environments to build the interpolation.
Considering that even a small number of possible defects significantly
increases combinatorial complexity, a general approach for predicting
properties of defects from pristine bulk structure representations could
accelerate computation by orders of magnitude (Figure 4a). For example, Varley
et al. observed simple and effective linear relationships between vacancy
formation energy and descriptors derived from the band structure of the bulk
solid [102]. While their model only considered one type of defect, their
implementation limits computational expense by demonstrating that only DFT
calculations on the pristine bulk were required [102]. Structure- and
composition-aware descriptors of the pristine bulk have additionally been
shown to be predictive of vacancy formation in metal oxides [105, 106] and
site/antisite defects in AB intermetallics [107]. To develop an approach that
can be used over a broad range of chemistries and defect types, Frey et al.
formed a representation by considering relative differences in characteristics
(atomic radii, electronegativity, etc.) of the defect structure compared to
the pristine parent[21]. Furthermore, because reference bulk properties could
be estimated using surrogate ML models, no DFT calculations were required for
prediction of either formation energy or changes in electronic structure [21].
We also note that in some cases it may be judicious to design a model that
does not change significantly in the presence of defects. For these cases,
representations based on simulated diffraction patterns are resilient to site-
based vacancies or displacements [108].
Like in defect engineering, machine learning for practical design of catalyst
materials requires representations beyond the single unit cell. Design of
catalysts with high activity crucially depends on interactions of reaction
intermediates with materials surfaces based on the Sabatier principle, which
argues that activity is greatest when intermediates are bound neither too
weakly nor too strongly [109]. From a computational perspective, determining
absorption energies involves searches over possible adsorption active sites,
surface facets, and surface rearrangements, leading to a combinatorial space
that can be infeasible to exhaustively cover with DFT. Single dimension
descriptors based on electronic structure have been established that can
predict binding strengths and provide insight on tuning catalyst compositions,
such as metal d-band center for metals [110] and oxygen 2p-band center for
metal oxides [111]. Additional geometric approaches include describing the
coordination of the active site (generalized coordination number in metals,
adjusted generalized coordination number in metal oxides) [112]. Based on the
success of these simple descriptors, machine learning models have been
developed to learn binding energy using the density of states and geometric
descriptors of the pristine bulk structure as features (Figure 4b) [113].
However, these structural and electronic descriptors are often not
generalizable across chemistries [110, 114], limiting the systems over which
they can be applied and motivating the development of more sophisticated
machine learning techniques. To reduce the burden on high-throughput DFT
calculations, active learning with surrogate models using information from
pure metals and active-site coordination has been used to identify alloy and
absorbate pairs that have the highest likelihood of producing near-optimal
binding energies [115]. Furthermore, when sufficient data ($>10,000$ examples)
is available, modifications of graph-convolutional models have also predicted
binding energies with high accuracy even in datasets with up to 37 elements,
enabling discovery without detailed mechanistic knowledge [114]. To generalize
these results, the release of Open Catalyst 2020 and its related competitions
[6, 9] has provided both over one million DFT energies for training new models
and a benchmark through which new approaches can be evaluated [75]. While
significant advancements have been made, state-of-the-art models still exhibit
high-errors for particular absorbates and non-metallic surface elements,
constraining chemistry over which effective screening can be conducted [75].
Furthermore, the complexity of the design space relevant for ML models grows
considerably when accounting for interactions between absorbates and different
surface facets [116].
Beyond atomistic interactions, the mechanical and thermal behavior of
materials can be significantly modulated by processing conditions and the
resulting microstructure. Greater knowledge of local distortions introduced at
varying grain boundary incident angles would give computational materials
scientists a more complete understanding of how experimentally chosen
chemistries and synthesis parameters will translate into device performance.
Strategies to quantify characteristics of grain boundary geometry have
included reducing computational requirements by identifying the most promising
configurations with virtual screening [117], estimating grain boundary free
volume as a function of temperature and bulk composition [118], treating the
microstructure as a graph of nodes connected across grain boundaries [119,
54], and predicting the energetics, and hence feasiblity, of solute
segregation [120]. While the previous approaches did not include features
based on the constituent atoms and were only benchmarked on systems with up to
three elements, recent work has demonstrated that the excess energy of the
grain boundary relative to the bulk can be approximated across compositions
with five variables defining its orientation and the bond lengths within the
grain boundary (Figure 4c)[121].
Further research has tried to map local grain boundary structure to function.
Algorithmic approaches to grain boundary structure classification have been
developed (see for example VoroTop [122]), but such approaches typically rely
on expert users and do not provide a continuous representation that can
smoothly interpolate between structures [123]. To eliminate these challenges,
Rosenbrock et al. proposed computing SOAP descriptors for all atoms in the
grain boundary, clustering vectors into classes, and identifying grain
boundaries through its local environment classes. The representation was not
only predictive of grain boundary energy, temperature-dependent mobility, and
shear coupling but also provides interpretable effects of particular
structures within the grain boundary [124]. A related approach computed SOAP
vectors relative to the bulk structure when analyzing thermal conductivity
[125]. Representations based on radial and angular structure functions can
also quantify the mobility of atoms within a grain boundary [126]. When
combined, advancing models for grain boundary stability as well as structure
to property relationships opens the door for functional design of grain
boundaries.
## 6 TRANSFERABLE INFORMATION BETWEEN REPRESENTATIONS
Applications of machine learning to materials science are limited by the scope
of compositions and structures over which algorithms can maintain sufficient
accuracy. Thus, building large-scale, diverse datasets is the most robust
strategy to ensure trained models can capture the relevant phenomena. However,
in most contexts, materials scientists are confronted with sparsely
distributed examples. Ideally, models can be trained to be generalizable and
exhibit strong performance across chemistries and configurations even with few
to no data points in a given domain. In order to achieve this, representations
and architectures must be chosen such that models can learn to extrapolate
beyond the space observed in the training set. Effective choices often rely on
inherent natural laws or chemical features that are shared between the
training set and extrapolated domain such as physics constraints [127, 128],
the geometric [129, 130] and electronic [131, 132] structure of local
environments, and positions of elements in the periodic table [133, 134]. For
example, Li et al. were able to predict absorption energies on high entropy
alloy surfaces after training on transition metal data by using the
coordination number and electronic properties of neighbors at the active site
[129]. While significant advancements have been made in the field,
extrapolation of machine learning models across materials spaces typically
requires specialized research methods and is not always feasible.
Likewise, it is not always practical for a materials scientist to improve
model generality by just collecting more data. In computational settings, some
properties can only be reliably estimated with more expensive, higher levels
of theory, and for experimentalists, synthetic and characterization challenges
can restrict throughput. The deep learning approaches that have demonstrated
exceptional performance over a wide range of tests cases discussed in this
review can require at least $~{}10^{3}$ training points, putting them
seemingly out for the realm of possibility for many research projects.
Instead, predictive modeling may fall back on identifying relationships
between a set of human-engineered descriptors and target properties.
Alternatively, the hidden, intermediate layers of deep neural networks can be
conceptualized as a learned vector representation of the input data. While
this representation is not directly interpretable, it must still contain
physical and chemical information related to the prediction task, which
downstream layers for the network utilize to generate model outputs. Transfer
learning leverages these learned representations from task A and uses them in
the modeling of task B. Critically, task A can be chosen to be one for which a
large number of data points are accessible (e.g. prediction all DFT formation
energies in the Materials Project), and task B can be of limited size (e.g.
predicting experimental heats of formation of a narrow class of materials). In
principle, if task A and task B share an underlying physical basis (the
stability of the material), the features learned when modeling task A may be
more informationally-rich than a human-designed representation [135]. With
this more effective starting point, subsequent models for task B can reach
high accuracy with relatively few new examples.
The most straightfoward methods to implement transfer learning in the
materials science community follow a common procedure: (1) train a neural
network model to predict a related property (task A) for which $>O(1,000)$
data points are available (pretraining), (2) fix parameters of the network up
a chosen depth $d$ (freezing), and (3) given the new dataset for task B,
either retrain the remaining layers, where parameters can be initialized
randomly or from the task A model (finetuning), or treat the output of the
model at depth d as in input representation to another ML algorithm (feature
extraction) [136, 137]. The robustness of this approach has been demonstrated
across model classes including those using composition only (ElemNet [135,
137], ROOST [95]), crystal graphs (CGCNN) [138], and equivariant convolutions
(GemNet) [139]. Furthermore, applications of task B, range from experimental
data [135, 95] to DFT-calculated surface absorption energies [139].
The sizes of the datasets for task A and task B will determine the
effectiveness of a transfer learning approach in two ways. First, the quality
and robustness of the representation learned for task A will increase as the
number of observed examples (the size of dataset A) increases. Secondly, as
the size of dataset B decreases, data becomes too sparse for a ML model to
learn a reliable representation alone and prior information from the solution
to task A can provide an increasingly useful method to interpolate between the
few known points. Therefore, transfer learning typically exhibits the greatest
boosts in performance when task A has orders of magnitude more data than task
B [135, 138].
In addition, the quality of information sharing through transfer learning
depends on the physical relationship between task A and task B. Intuitively,
the representation from task A provides a better guide for task B if the tasks
are closely related. For example, Kolluru et al. demonstrated that transfer
learning from models trained on the Open Catalyst Dataset [6] exhibited
significantly better performance when applied to absorption of new species
than energies of less-related small molecules [139]. While it is difficult to
choose the optimal task A for a given task B a priori, shotgun transfer
learning [136] has demonstrated that the best pairing can be chosen
experimentally by empirically validating a large pool of possible candidates
and selecting top performers.
The depth $d$ from which features should be extracted from task A to form a
representation can also be task dependent. Kolluru et al. provided evidence
that to achieve optimal performance more layers of the network should be
allowed to be retrained in step (3) as the connection between task A and task
B becomes more distant [139]. Gupta et al. arrived at a similar conclusion
that the early layers of deep neural networks learned more general
representations and performed better in cross-property transfer learning
[137]. Inspired by this observation that representations at different neural
network layers contain information with varying specificity to a particular
prediction task, representations for transfer learning that combine
activations from multiple depths have been proposed [139, 140].
When tasks are sufficiently different, freezing neural network weights may not
be the optimal strategy and instead representations for task B can include
predictions for task A as descriptors. For instance, Cubuk et al. observed
that structural information was critical to predict Li conductivity but was
only available for a small set of compositions for which crystal structures
were determined. By training a separate surrogate model to predict structural
descriptors from composition and using those approximations in subsequent Li
conductivity models, the feasible screening domain was expanded by orders of
magnitude [141]. Similarly, Greenman et al. [142] used $O(10,000)$ TD-DFT
calculations to train a graph neural network whose estimates could be used as
an additional descriptor for a model predicting experimental peaks in
absorption spectra. Representations have also been sourced from the output of
generative models. Kong et al. trained a Generative Adversial Network (GAN) to
sample electronic density of states (DOS) given a particular material
composition. Predictions of absorption spectra of a particular composition
were improved by concatenating stoichiometric data with the average DOS
sampled from the generative model [143].
## 7 GENERATIVE MODELS FOR INVERSE DESIGN
While, in principle, machine learning methods can significantly reduce the
time required to compute materials properties, and material scientists can
employ these models to screen for a set of target systems by rapidly
estimating the stability and performance, the space of feasible materials
precludes a naive global optimization strategy in most cases. Generative
models including Variational Autoencoders (VAE) [144, 1], Generative
Adversarial Networks (GAN) [145, 146], and diffusion models [147, 148] can be
trained to sample from a target distribution and have proved to be capable
strategies for optimization in high-dimensional molecular spaces [1, 149].
While some lessons can be drawn from the efforts of researchers in the
computational chemistry community, generative models face unique challenges
for proposing crystals [150, 151]. First, the diversity of atomic species
increases substantially when compared with small organic molecules. In
addition, given a composition, a properly defined crystal structure requires
both the positions of the atoms within the unit cell as well as the lattice
vectors and angles that determine the systems periodicity. This definition is
not unique, and the same material can be described after rotations or
translations of atomic coordinates as well as integer scaling of the original
unit cell. Lastly, many state-of-the-art materials for catalysis (e.g.
zeolites, metal organic frameworks) can have unit cells including $>100$ of
atoms, increasing the dimensionality of the optimization problem [150, 151].
Figure 5: Approaches for crystal structure generative models. (Left) Initial
models based on voxel representations defined positions of atoms by
discretizing space into finite volume elements but were not applied generally
over the space of crystal structures [23, 22, 24, 25]. (Center) Restricting
the generation process to be invariant to permutation, translation, and
rotations, through an appropriately constrained periodic decoder (PGNNDec)
results in sampling structures exhibiting more diversity and stability.
(Right) When features of the material can be assumed, such as a finite number
of possible topologies connecting substructures, the dimensionality of the
problem can be substantially reduced and samples over larger unit cell
materials can be generated. Figures on left, center, and right are adapted
from [23], [152], and [153], respectively.
One attempt to partially address the challenges of generative modeling for
solid materials design is a voxel representation [150], in which unit cells
are divided into volume elements and models are built using techniques from
computer vision. Hoffman represented unit cells using a density field that
could be further segmented into atomic species and was able to generate
crystals with realistic atomic spacings. However, atoms could be mistakenly
decoded into other species with nearby atomic number and most of the generated
structures could not be stably optimized with a DFT calculation [22].
Alternate approaches could obtain more convincing results, but over a confined
region of material space [154]. iMatgen (Figure 5a) invertibly mapped all unit
cells into a cube with Gaussian-smeared atomic density and trained a VAE
coupled with a surrogate energy prediction. The model was able to rediscover
stable structures but was constrained over the space of Vanadium oxides [23].
A similar approach constructed a separate voxel representation for each
element and employed a GAN trained alongside an energy constraint to explore
the phases of Bi-Se [155]. In order to resolve some of Hoffman el al’s
limitations, Court et al. [24] reduced segmentation errors by augmenting the
representation with a matrix describing the occupation (0,1) of each voxel and
a matrix recording the atomic number of occupied voxels. Their model was able
to propose new materials that exhibited chemical diversity and could be
further optimized with DFT but restricted analysis to cubic systems. Likewise,
compositions of halide perovskites with optimized band gaps could be proposed
using a voxelized representation of a fixed perovskite prototype [25].
Voxel representations can be relaxed to continuous coordinates in order to
develop methods that are more comprehensively applicable over crystal space.
Kim et al. represented materials using a record of the unit cell as well as a
point cloud of fractional coordinates of each element. The approach proposed
lower energy structures than iMatgen for V-O binaries and was also applicable
over more diverse chemical spaces (Mg-Mn-O ternaries) [156]. Another
representation including atomic positions along with elemental properties
could be leveraged for inverse design over spaces that vary in both
composition and lattice structure. In a test case, the model successfully
generated new materials with negative formation energy and promising
thermoelectric power factor [154]. While these models have demonstrated
improvements in performance, they lack the translational, rotational, and
scale invariances of real materials and are restricted to sampling particular
materials classes [156, 152].
Recently, alternatives that account for these symmetries have been proposed.
Fung et al. proposed a generative model for rotationally and translationally
invariant atom-centered symmetry functions (ACSF) from which target structures
could be reconstructed [157]. Crystal Diffusion VAEs (Figure 5b) leveraged
periodic graphs and SE(3) equivariant message-passing layers to encode and
decode their representation in a translationally and rotationally invariant
way [152]. They also proposed a two step generation process during which they
first predicted the crystal lattice from a latent vector and subsequently
sampled the composition and atomic positions through Langevin dynamics.
Furthermore, they established well-defined benchmark tasks and demonstrated
that for inverse design their method was more flexible than voxel models with
respect to crystal system and more accurate than point cloud representations
at identifying crystals with low formation energy.
Scaling solid-state generative modeling techniques to unit cells with
$(10^{4})$ atoms would enable inverse design of porous materials that are
impossible to explore exhaustively but demonstrate exceptional technological
relevance. Currently, due to the high number of degrees of freedom, sampling
from these spaces requires imposing physical constraints in the modeling
process. Such restrictions can be implemented as post-processing steps or
integrated into the model representation. ZeoGAN [158] generated positions of
oxygen and silicon atoms in a 32x32x32 grid to propose new Zeolites. While
some of the atomic positions proposed directly from their model violated
conventional geometric rules, they could obtain feasible structures by
filtering out divergent compositions and repairing bond connectivity through
the insertion or deletion of atoms. Alternatively, Yao et al. designed
geometric constraints directly into the generative model by representing Metal
Organic Frameworks (MOFs) by their edges, metal/organic vertexes, and distinct
topologies (RFcodes) (Figure 5c) [153]. Because this representation is
invertible, all RFcodes correspond to a structurally possible MOFs. By
training a VAE to encode and decode this RFcode representation, they
demonstrated the ability to interpolate between structures and optimize
properties. In general, future research should balance more stable structure
generation against the possible discovery of new motifs and topologies.
## 8 DISCUSSION
In this review, we have introduced strategies for designing representations
for machine learning in the context of challenges encountered by materials
scientists. We discussed local and global structural features as well as
representations learned from atomic-scale data in large repositories. We noted
additional research that extends beyond idealized crystals to include the
effects of defects, surfaces, and microstructure. Furthermore, we acknowledged
that in practice the availability of data both in quality and quantity can be
limited. We described methods to mitigate this including developing models
based on compositional descriptors alone or leveraging information from
representations built for related tasks through transfer learning. Finally, we
analyzed how generative models have improved by incorporating symmetries and
domain knowledge. As data-based methods have become increasingly essential for
materials design, optimal machine learning techniques will play a crucial role
in the success of research programs. The previous sections demonstrate that
the choice of representation will be among these pivotal factors and that
novel approaches can open the door to new modes of discovery. Motivated by
these observations, we conclude by summarizing open problems with the
potential to have high impact on the field of materials design.
### 8.1 Trade-offs of Local and Global Structural Descriptors
Local structural descriptors including SOAP [12] have become reliable metrics
to compare environments with a specific cutoff radius, and when properties can
be defined through short-range interactions, have demonstrated strong
predictive performance. Characterizing systems based off local environments
allows models to extrapolate to cases where global representations may vary
substantially (e.g. an extended supercell of a crystal structure)[14] and
enables highly-scalable methods of computation that can extend the practical
limit of simulations to much larger systems [159]. However, Unke et al. notes
that the required complexity of the representation can grow quickly when
modeling systems with many distinct elements and the quality of ML predictions
will be sensitive to the selected hyperparameters, such as the characteristics
distances and angles in atom-centered symmetry functions[160]. Furthermore, it
is unclear if these high quality results extend to materials characteristics
that depend strongly on long-range physics or periodicity of the crystal. On
the other hand, recent global descriptors [40] can more explicitly model these
phenomena, but have not exhibited the same generality across space groups and
system sizes. Strategies exploring appropriate combinations of local and long-
range features [161] have the potential to break through these trade-offs to
provide more universal models for material property prediction.
### 8.2 Prediction from Unrelaxed Crystal Prototypes
If relaxed structures are required to form representations, the space over
which candidates can be screened is limited to those materials for which
optimized geometries are known. Impressively, recent work [162, 163] has shown
that ML force-fields, even simple models with relatively high errors, can be
used optimize structures and obtain converged results that are lower in energy
than those obtained using VASP [164]. Their benchmarking on the OC20 [6]
dataset and lower accuracy requirements suggest that the approach could be
generalizable across a wide-class of material systems and thus significantly
expand the availability of structural descriptors. Similarly, Chen et al.
demonstrated that a variant of MEGNET could perform high fidelity relaxations
of unseen materials with diverse chemistries and that leveraging the resulting
structures could improve downstream ML predictions of energy when compared
with unrelaxed inputs [165]. The strong performance of these approaches and
their potential to significantly increase the scale and effectiveness of
computational screening motivates high-value research questions concerning the
scale of data sets required for training, the generalizabiltiy over material
classes, and the applicability to prediction tasks beyond stability.
### 8.3 Applicability of Compositional Descriptors
Compositional descriptors are typically readily available as tabulated values,
but even state-of-the-art models do not perform as well as the best structural
approaches. However, there is some evidence that the scale of improvement when
including structural information is property dependent. System energies can be
conceptualized as a sum of site energies that are highly dependent on the
local environment, and graph neural networks provide significantly more robust
predictions of materials stability [84]. On the other hand, for properties
dependent on global features such as phonons (vibrations) or electronic band
structure (band gap) the relative improvement may not be as large [99, 166,
167]. Identifying common trends connecting tasks for which this difference is
the least significant would provide more intuition on which scenarios
compositional models are most appropriate. Furthermore, in some modeling
situations, structural information is available but only over a small fraction
of the dataset. To maximize the value of this data, more general strategies
involving transfer learning [141] or combining separate composition and
structural models [85] should be developed.
### 8.4 Extensions of Generative Models
Additional symmetry considerations and the implementation of diffusion-based
architectures led to generative models that improved significantly over
previous voxel approaches. While this strategy is a promising direction for
small unit cells, efforts pertaining to other parameters critical to material
performance including microstructure [168], dimensionality [169] and surfaces
[170] should also be pursued. In addition, research groups have side-stepped
some of the challenges of materials generation by designing approaches that
only sample material stoichiometry [171]. While this strategy limits the full
characterization of new materials through a purely computational pipeline,
there may be cases where they are sufficient to propose promising regions for
experimental analysis.
## DISCLOSURE STATEMENT
The authors are not aware of any affiliations, memberships, funding, or
financial holdings that might be perceived as affecting the objectivity of
this review.
## ACKNOWLEDGMENTS
JD was involved in the writing of all sections. AT and MX collaborated on the
writing and designed the figure for Atomistic Structure section, JK
collaborated on the writing and designed the figure for the Periodic Graph
section, JL collaborated on the writing and designed the figure for the
Defects, Surfaces, and Grain Boundaries section. JP provided valuable insights
for the organization and content of the article. RGB selected the topic and
focus of the review, contributed to the central themes and context, and
supervised the project. All authors participated in discussions and the
reviewing of the final article. The authors would like to thank Anna Bloom for
editorial contributions.
The authors acknowledge financial support from the Advanced Research Projects
Agency–Energy (ARPA-E), US Department of Energy under award number DE-
AR0001220. JD, MX and ART thank the National Defense Science and Engineering
Graduate Fellowship, the National Science Scholarship from Agency for Science,
Technology and Research, and Asahi Glass Company, respectively, for financial
support. RGB thanks the Jeffrey Cheah Chair in Engineering.
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|
[a]Wolfgang Unger
# Hamiltonian Lattice QCD from Strong Coupling Expansion
Pratitee Pattanaik
###### Abstract
We present generalizations of Hamiltonian Lattice QCD as derived from the
continuous time limit of strong coupling lattice QCD: we discuss the flavor
dependence and the effect of gauge corrections. This formalism can be applied
at finite temperature and baryon density as well as isospin density and allows
both for analytic and numeric investigations that are sign problem-free.
## 1 Introduction
The Hamiltonian formulation of lattice QCD has been discussed in detail in [1]
in the strong coupling limit for ${N_{f}}=1$. In contrast to Hamiltonian
formulations in the early days of lattice QCD [2] this formulation is based on
the dual representation that is obtained when integrating out the gauge links
first, and then the Grassmann-valued fermions [3]. The resulting dual degrees
of freedom are color singlets, such as mesons and baryons. In this
representation, the finite density sign problem is much milder, as it is re-
expressed in terms of the geometry of baryonic world-lines. The dual
representation of lattice QCD with staggered fermions has been studied in the
strong coupling limit both via mean-field theory [4] and Monte Carlo [5, 6]
and has been extended in both approaches to include gauge corrections [7, 8].
The Hamiltonian formulation is based on the continuous time limit of the dual
representation: at fixed bare temperature $aT=\frac{a}{a_{t}N_{t}}$, the
limits $a_{t}\rightarrow 0$ and $N_{t}\rightarrow\infty$ are taken
simultaneously [9]. This limit requires to determine the non-perturbative
relation between the bare anisotropy $\gamma$ and the physical anisotropy
$\frac{a}{a_{t}}\equiv\xi(\gamma)$, based on pion current fluctuations. A
conservation law for the pion current can be identified: if a quantum number
$\mathfrak{m}(x)$ is raised/lowered by a spatial dimer, then at the site
connected by the spatial dimer the quantum number is lowered/raised. This is a
direct consequence of the even/odd decomposition for staggered fermions. With
interactions derived from a high temperature series, the resulting partition
sum can be expressed in terms of a Hamiltonian that is composed of mesonic
annihilation and creation operators $\hat{J}^{\pm}$:
$\displaystyle Z_{\rm CT}(\mathcal{T},\mu_{\mathcal{B}})={\rm
Tr}_{\mathfrak{h}}\left[e^{(\hat{\mathcal{H}}+\hat{\mathcal{N}}\mu_{\mathcal{B}})/\mathcal{T}}\right],\qquad\hat{\mathcal{H}}=\frac{1}{2}\sum_{\langle\vec{x},\vec{y}\rangle}\left(\hat{J}^{+}_{\vec{x}}\hat{J}^{-}_{\vec{y}}+\hat{J}^{-}_{\vec{x}}\hat{J}^{+}_{\vec{y}}\right),\qquad\hat{\mathcal{N}}=\sum_{\vec{x}}\hat{\omega}_{x},$
$\displaystyle\hat{J}^{+}={\footnotesize\left(\begin{array}[]{cccc|cc}0&0&0&0&&\\\
\hat{v}_{\hskip
0.56003pt\text{\lx@text@tweaked{yoffset=-0.5ex}{\lx@mvs@LSteel}}\hskip
0.56003pt}&0&0&0&&\\\ 0&\hat{v}_{\hskip
0.56003pt\text{\lx@text@tweaked{yoffset=0.5ex}{\lx@mvs@TSteel}}\hskip
0.56003pt}&0&0&&\\\ 0&0&\hat{v}_{\hskip
0.56003pt\text{\lx@text@tweaked{yoffset=-0.5ex}{\lx@mvs@LSteel}}\hskip
0.56003pt}&0&&\\\ \hline\cr&&&&0&0\\\ &&&&0&0\\\
\end{array}\right)},\qquad\hat{J}^{-}=(\hat{J}^{+})^{T},\qquad\hat{\omega}={\footnotesize\left(\begin{array}[]{cccc|cc}0&0&0&0&&\\\
0&0&0&0&&\\\ 0&0&0&0&&\\\ 0&0&0&0&&\\\ \hline\cr&&&&1&0\\\ &&&&0&-1\\\
\end{array}\right)}$ (13)
with the local Hilbert space given by
$|\mathfrak{h}\rangle=|\mathfrak{m};\mathfrak{b}\rangle=|0,\pi,2\pi,3\pi;B^{+},B^{-}\rangle.$
The matrix elements $\hat{v}_{\hskip
0.70004pt\text{\lx@text@tweaked{yoffset=-0.5ex}{\lx@mvs@LSteel}}\hskip
0.70004pt}=1$ and $\hat{v}_{\hskip
0.70004pt\text{\lx@text@tweaked{yoffset=0.5ex}{\lx@mvs@TSteel}}\hskip
0.70004pt}=\frac{2}{\sqrt{3}}$ are computed from the local weights of the
corresponding meson vertices. Since Pauli saturation holds on the level of the
quarks and mesons have a fermionic substructure, the meson occupation numbers
$|\mathfrak{m}\rangle$ are also bounded from above. This results in a
particle-hole symmetry, leading to an SU(2) algebra which is
$d={N_{c}}+1$-dimensional:
$\displaystyle\hat{J}_{1}$
$\displaystyle=\frac{\sqrt{{N_{c}}}}{2}\left(\hat{J}^{+}+\hat{J}^{-}\right),\quad\hat{J}_{2}=\frac{\sqrt{{N_{c}}}}{2i}\left(\hat{J}^{+}-\hat{J}^{-}\right),$
$\displaystyle\hat{J}_{3}$
$\displaystyle=i[J_{1},J_{2}]=\frac{{N_{c}}}{2}[\hat{J}^{+},\hat{J}^{-}],\quad\hat{J}^{2}=\frac{{N_{c}}({N_{c}}+2)}{4},$
$\displaystyle\hat{J}_{3}\left|\frac{{N_{c}}}{2},\mathfrak{s}\right\rangle$
$\displaystyle=\mathfrak{s}\left|\frac{{N_{c}}}{2},\mathfrak{s}\right\rangle,$
$\displaystyle\hat{J}^{2}\left|\frac{{N_{c}}}{2}\mathfrak{s}\right\rangle$
$\displaystyle=\frac{{N_{c}}\left({N_{c}}+2\right)}{4}\left|\frac{{N_{c}}}{2},\mathfrak{s}\right\rangle,$
$\displaystyle[\hat{J}^{2},\hat{J}_{3}]$ $\displaystyle=0$ (14)
with $\mathfrak{m}\mapsto\mathfrak{s}=\mathfrak{m}-\frac{{N_{c}}}{2}$. The
${N_{f}}=1$ formulation at strong coupling has many similarities with full
QCD, and via Quantum Monte Carlo the grand-canonical and canonical phase
diagram could be determined [1]. The nuclear interactions are of entropic
nature: the nucleons, which are point-like in the strong coupling limit,
attract each other due to the modification of the pion bath that surrounds
them. However, pion exchange cannot be realized in the strong coupling limit
for ${N_{f}}=1$ since the Grassmann constraint does not allow for pions and
nucleons to overlap. This is clearly a lattice artifact of the strong coupling
limit, but this is overcome by formulations with ${N_{f}}>1$ and/or by
including gauge corrections.
The aim of the generalizations discussed in this proceedings is to allow for
Quantum Monte Carlo simulations (QMC) at both non-zero baryon and isospin
chemical potential, and possibly also strangeness chemical potential. We will
argue that in the strong coupling limit, the ${N_{f}}=2$ formulation remains
sign problem-free.
## 2 Hamiltonian formulation in the strong coupling limit for any $N_{f}$
Strong coupling lattice QCD with staggered fermions for ${N_{f}}>1$ admits
more than one baryon per site and the Grassmann constraint allows for pion
exchange between them, modifying nuclear interactions substantially. It has so
far only been studied via mean-field theory [10]. A Hamiltonian formulation
for ${N_{f}}=2$ allows for QMC and provides a more realistic scenario for
nuclear interactions and the phase diagram. It also compares better to the
strong coupling regime with Wilson fermions in a world-line formulation, as
discussed in the context of the 3-dim. Polyakov effective theory (for a
review, see [11]). Also for ${N_{f}}>1$ the suppression of spatial bonds
$\gamma^{-k}$, $k>2$ applies, hence the continuous time limit is well defined.
The first step to derive the Hamiltonian is to determine the local Hilbert
space $\mathbbm{H}_{\mathfrak{h}}$ via canonical sectors
$B\in\\{-{N_{f}},\ldots,{N_{f}}\\}$, i.e. we need to consider all possible
single-site quantum states $\mathfrak{h}$ in a non-interacting theory to
establish the basis of quantum states that generalize the ${N_{f}}=1$ states
$|\mathfrak{m}\rangle$ and $|\mathfrak{b}\rangle$. The static partition sum
for any ${N_{c}}$, ${N_{f}}$ is:
$\displaystyle Z_{\rm
stat}({N_{c}},{N_{f}})\sum_{B=-{N_{f}}}^{{N_{f}}}\prod_{a=0}^{N_{c}}\frac{a!(2{N_{f}}+a)!}{({N_{f}}+a+B)!(Nf+a-B)!}e^{B\mu_{B}/T}$
(15)
where the coefficient encode the number of hadronic states, i.e. the dimension
of $\mathfrak{h}$. For ${N_{c}}=3$ (separated in canonical sectors)
* •
for ${N_{f}}=1$: $d=[1,4,1]=6$,
* •
for ${N_{f}}=2$: $d=[1,20,50,20,1]=92$,
* •
for ${N_{f}}=3$: $d=[1,56,490,980,490,56,1]=2074$.
Unfortunately there is no general formula known including the isospin sectors,
required for $\mu_{I}\neq 0$. The individual states $\mathfrak{h}$ are also
required for constructing the flavored $\hat{J}^{\pm}$. To arrive at this
basis, we consider the ${\rm SU}(3)$ one-link integrals in terms of the
fermion matrix
$(\mathcal{M})_{ij}=\bar{\chi}^{\alpha}_{i}(x)\chi^{\alpha}_{i}(y)$ and
$(\mathcal{M}^{\dagger})_{kl}=\chi^{\beta}_{k}(y)\bar{\chi}^{\beta}_{l}(x)$,
valid also for ${N_{f}}\geq 1$:
$\displaystyle\mathcal{J}(\mathcal{M},\mathcal{M}^{\dagger})$
$\displaystyle=\int\limits_{{\rm SU}(3)}dUe^{{\rm
tr}[U\mathcal{M}^{\dagger}+U^{\dagger}\mathcal{M}]}=\sum_{B=-{N_{f}}}^{N_{f}}\sum_{n_{1},n_{2},n_{3}}C_{B,n_{1},n_{2},n_{3}}\frac{E^{B}}{|B|!}\prod_{i=1}^{3}\frac{X_{i}^{n_{i}}}{n_{i}!},\quad
E=\begin{cases}\det{\mathcal{M}}&B>0\\\ 1&B=0\\\
\det{\mathcal{M}^{\dagger}}&B<0\\\ \end{cases}$ $\displaystyle
C_{B,n_{1},n_{2},n_{3}}$
$\displaystyle=2\binom{n_{1}+2n_{2}+4n_{3}+2|B|+2}{n_{3}+|B|}\frac{|B|!}{(n_{1}+2n_{2}+3n_{3}+2|B|+2)!(n_{2}+2n_{3}+|B|+1)!}$
(16)
which in contrasts to [12] is now expressed in a more suitable basis involving
the baryon sectors $B$. The sum over $B$ and $n_{i}$ $(i=1,\ldots 3)$
terminates due to the Grassmann integration, depending on ${N_{f}}$. The
corresponding invariants $D$, $X_{i}$ can be evaluated as follows:
$\displaystyle X_{1}$ $\displaystyle={\rm
tr}[\mathcal{M}\mathcal{M}^{\dagger}]={\rm
Tr}[M_{x}M_{y}]=k_{U}+k_{D}+\ldots+k_{\pi^{+}}+k_{\pi^{-}}+\ldots$
$\displaystyle X_{2}$ $\displaystyle=\frac{1}{2}\left({\rm
tr}[\mathcal{M}\mathcal{M}^{\dagger}]-{\rm
tr}[(\mathcal{M}\mathcal{M}^{\dagger})^{2}]\right)=\frac{1}{2}\left({\rm
Tr}[M_{x}M_{y}]^{2}+{\rm Tr}[(M_{x}M_{y})^{2}]\right)=X_{1}^{2}-D_{2}$
$\displaystyle X_{3}$
$\displaystyle=\det[\mathcal{M}\mathcal{M}^{\dagger}]=\frac{1}{6}\left({\rm
Tr}[M_{x}M_{y}]^{3}+3{\rm Tr}[M_{x}M_{y}]{\rm Tr}[(M_{x}M_{y})^{2}]+2{\rm
Tr}[(M_{x}M_{y})^{3}]\right)=X_{1}^{3}-2X_{1}D_{2}+D_{3}$ $\displaystyle
D_{2}$ $\displaystyle=\frac{1}{2}\left({\rm Tr}[M_{x}M_{y}]^{2}-{\rm
Tr}[(M_{x}M_{y})^{2}]\right)=k_{U}k_{D}+k_{\pi^{+}}k_{\pi^{-}}+\ldots-(k^{(2)}_{\pi^{+}\pi^{-},UD}+k^{(2)}_{UD,\pi^{+}\pi^{-}}+\ldots)$
$\displaystyle D_{3}$ $\displaystyle=\frac{1}{6}\left({\rm
Tr}[M_{x}M_{y}]^{3}-3{\rm Tr}[M_{x}M_{y}]{\rm Tr}[(M_{x}M_{y})^{2}]+2{\rm
Tr}[(M_{x}M_{y})^{3}]\right)=\frac{1}{6}(3X_{1}D_{2}-X_{1}^{3})$
$\displaystyle E$ $\displaystyle=\det[\mathcal{M}]=\sum_{f\leq g\leq
h}B_{fgh}\quad(B>0)\quad\text{or}\quad
E=\det[\mathcal{M}^{\dagger}]=\sum_{f\leq g\leq h}\bar{B}_{fgh},\quad(B<0)$
(17)
where the invariants are expressed in terms of nearest neighbors
$(M_{x}M_{y})^{n}=(-1)^{n+1}(\mathcal{M}\mathcal{M}^{\dagger})^{n}$. To be
explicit, for ${N_{f}}=2$, the (anti-) baryons and flavored dimers in terms of
quarks are:
$\displaystyle B_{uud}$ $\displaystyle=\bar{u}\bar{u}\bar{d}_{x}uud_{y},$
$\displaystyle B_{udd}$ $\displaystyle=\bar{u}\bar{d}\bar{d}_{x}udd_{y},$
$\displaystyle B_{uuu}$ $\displaystyle=\bar{u}\bar{u}\bar{u}_{x}uuu_{y},$
$\displaystyle B_{ddd}$ $\displaystyle=\bar{d}\bar{d}\bar{d}_{x}ddd_{y},$
$\displaystyle\bar{B}_{uud}$ $\displaystyle=uud_{x}\bar{u}\bar{u}\bar{d}_{y},$
$\displaystyle\bar{B}_{udd}$ $\displaystyle=udd_{x}\bar{u}\bar{d}\bar{d}_{y},$
$\displaystyle\bar{B}_{uuu}$ $\displaystyle=uuu_{x}\bar{u}\bar{u}\bar{u}_{y},$
$\displaystyle\bar{B}_{ddd}$ $\displaystyle=ddd_{x}\bar{d}\bar{d}\bar{d}_{y},$
$\displaystyle k_{U}$ $\displaystyle=\bar{u}u(x)\bar{u}u(y),$ $\displaystyle
k_{D}$ $\displaystyle=\bar{d}d(x)\bar{d}d(y),$ $\displaystyle k_{\pi^{+}}$
$\displaystyle=\bar{u}d(x)\bar{d}u(y),$ $\displaystyle k_{\pi^{-}}$
$\displaystyle=\bar{d}u(x)\bar{u}d(y)$ $\displaystyle
k^{(2)}_{\pi^{+}\pi^{-},UD}$
$\displaystyle=\bar{u}d(x)\bar{d}u(x)\bar{u}u(y)\bar{d}d(y),$ $\displaystyle
k^{(2)}_{UD,\pi^{+}\pi^{-}}$
$\displaystyle=\bar{u}u(x)\bar{d}d(x)\bar{u}d(y)\bar{d}u(y).$ (18)
Every state of the local Hilbert space can be described by a set of
${N_{f}}^{2}$ charges: baryon number $B\in\\{-{N_{f}},\ldots{N_{f}}\\}$,
isospin number $I\in\\{-{N_{c}},\ldots{N_{c}}\\}$, diagonal flavor numbers
$U,D,S,\ldots\in\\{0,\ldots{N_{c}}\\}$ and for ${N_{f}}>2$ additional
generalizations of isospin
$K_{1},K_{2},\ldots\in\\{-{N_{c}},\ldots{N_{c}}\\}$. The one-link integral
$J_{k}^{(B)}$ from Eq. (16) of order $k=n_{1}+2n_{2}+3n_{3}$ can be evaluated
for various ${N_{f}}$. We show here the general result for SU(3) in terms of
$X=X_{1}$, $D_{2}$, $D_{3}$ and $E$: in black the ${N_{f}}=1$ contributions,
in blue the additional ${N_{f}}=2$ contributions and in red the additional
${N_{f}}=3$ contributions:
$\displaystyle J_{0}^{(0)}$ $\displaystyle=1,\qquad
J_{1}^{(0)}=\frac{1}{3}X,\qquad
J_{2}^{(0)}=\frac{1}{12}X^{2}-{\color[rgb]{0,0,1}{\frac{1}{24}{D_{2}}}},\qquad
J_{3}^{(0)}=\frac{1}{36}X^{3}-{\color[rgb]{0,0,1}{\frac{1}{24}X{D_{2}}}}+{\color[rgb]{1,0,0}{\frac{1}{60}{D_{3}}}},$
$\displaystyle J_{4}^{(0)}$
$\displaystyle=\frac{1}{720}\left({\color[rgb]{0,0,1}{\frac{37}{12}X^{4}-\frac{11}{2}X^{2}{D_{2}}+\frac{1}{6}{D_{2}}^{2}}}+{\color[rgb]{1,0,0}{\frac{7}{3}X{D_{3}}}}\right),$
$\displaystyle J_{5}^{(0)}$
$\displaystyle=\frac{1}{840}\left({\color[rgb]{0,0,1}{\frac{47}{120}X^{5}-\frac{31}{36}X^{3}{D_{2}}+\frac{1}{4}X{D_{2}}^{2}}}+{\color[rgb]{1,0,0}{\frac{1}{3}X^{2}{D_{3}}-\frac{1}{9}\,D_{2}D_{3}}}\right),$
$\displaystyle J_{6}^{(0)}$
$\displaystyle=\frac{1}{448}\left({\color[rgb]{0,0,1}{\frac{53}{2700}X^{6}-\frac{1}{18}X^{4}{D_{2}}+\frac{5}{144}X^{2}{D_{2}}^{2}+\frac{1}{45}{D_{3}}X^{3}-\frac{1}{6480}D_{2}^{3}}}-{\color[rgb]{1,0,0}{\frac{1}{40}XD_{2}{D_{3}}+\frac{1}{240}{D_{3}}^{2}}}\right),$
$\displaystyle J_{7}^{(0)}$
$\displaystyle=\frac{1}{25920}\left({\color[rgb]{1,0,0}{\frac{241}{2940}X^{7}-\frac{19}{70}D_{2}X^{5}+\frac{227}{1008}{D_{2}}^{2}X^{3}+\frac{1}{9}\,{D_{3}}X^{4}-\frac{13}{1008}D_{2}^{3}X-\frac{29}{168}D_{2}D_{3}X^{2}+\frac{1}{168}{D_{2}}^{2}D_{3}}}+{\color[rgb]{1,0,0}{\frac{11}{336}\,D_{3}^{2}X}}\right),$
$\displaystyle J_{8}^{(0)}$
$\displaystyle=\frac{1}{21772800}\left({\color[rgb]{1,0,0}{\frac{13259}{3360}\,X^{8}-\frac{149}{10}D_{2}X^{6}+\frac{631}{40}{D_{2}}^{2}X^{4}+\frac{121}{20}\,D_{3}X^{5}-\frac{403}{120}\,D_{2}^{3}X^{2}-\frac{143}{12}D_{2}D_{3}X^{3}+\frac{1}{240}\,D_{2}^{4}}}\right.$
$\displaystyle\left.\hskip
45.5244pt{\color[rgb]{1,0,0}{+\frac{11}{4}X{D_{2}}^{2}D_{3}+\frac{11}{5}X^{2}D_{3}^{2}-\frac{11}{20}D_{2}D_{3}^{2}}}\right),$
$\displaystyle J_{9}^{(0)}$
$\displaystyle=\frac{1}{17107200}\left({\color[rgb]{1,0,0}{\frac{63163}{423360}X^{9}-\frac{377}{588}X^{7}D_{2}+\frac{1429}{1680}X^{5}{D_{2}}^{2}+\frac{73}{280}X^{6}D_{3}-\frac{1663}{5040}X^{3}{D_{2}}^{3}-\frac{551}{840}X^{4}{D_{2}}{D_{3}}}}\right.$
$\displaystyle\hskip
45.5244pt\left.{\color[rgb]{1,0,0}{+\frac{17}{3360}X{D_{2}}^{4}+\frac{71}{210}X^{2}{D_{2}}^{2}D_{3}+\frac{13}{105}X^{3}{D_{3}}^{2}-\frac{1}{420}{D_{2}}^{3}{D_{3}}-\frac{13}{120}X{D_{2}}{D_{3}}^{2}+\frac{13}{1260}{D_{3}}^{3}}}\right),$
$\displaystyle J_{0}^{(1)}$ $\displaystyle=\frac{E}{6},\qquad
J_{1}^{(1)}=\frac{E}{6}\left({\color[rgb]{0,0,1}{\frac{1}{4}X}}\right),\qquad
J_{2}^{(1)}=\frac{E}{6}\left({\color[rgb]{0,0,1}{\frac{1}{24}X^{2}-\frac{1}{60}D_{2}}}\right),\qquad
J_{3}^{(1)}=\frac{E}{6}\left({\color[rgb]{0,0,1}{\frac{1}{144}X^{3}-\frac{1}{120}XD_{2}}}+{\color[rgb]{1,0,0}{\frac{1}{360}{D_{3}}}}\right),$
$\displaystyle J_{4}^{(1)}$
$\displaystyle=\frac{E}{6}{\color[rgb]{1,0,0}{\left(\frac{1}{1344}X^{4}-\frac{1}{840}X^{2}D_{2}+\frac{1}{20160}{D_{2}}^{2}+\frac{1}{2240}X{D_{3}}\right)}},$
$\displaystyle J_{5}^{(1)}$
$\displaystyle=\frac{E}{6}{\color[rgb]{1,0,0}{\left(\frac{143}{2419200}X^{5}-\frac{29}{241920}\,D_{2}X^{3}+\frac{1}{32256}D_{2}^{2}X+\frac{1}{23040}\,D_{3}X^{2}-\frac{1}{80640}D_{2}D_{3}\right)}},$
$\displaystyle J_{6}^{(1)}$
$\displaystyle=\frac{E}{6}{\color[rgb]{1,0,0}{\left(\frac{19}{4838400}X^{6}-\frac{221}{21772800}X^{4}{D_{2}}+\frac{241}{43545600}X^{2}{D_{2}}^{2}+\frac{11}{2903040}X^{3}D_{3}-\frac{1}{21772800}{D_{2}}^{3}-\frac{11}{3110400}X{D_{2}}{D_{3}}+\frac{11}{21772800}{D_{3}}^{2}\right)}},$
$\displaystyle J_{0}^{(2)}$
$\displaystyle={\color[rgb]{0,0,1}{\frac{E^{2}}{144}}},\qquad
J_{1}^{(2)}={\color[rgb]{0,0,1}{\frac{E^{2}}{144}}}\left({\color[rgb]{1,0,0}{\frac{1}{5}X}}\right),\qquad
J_{2}^{(2)}={\color[rgb]{0,0,1}{\frac{E^{2}}{144}}}\left({\color[rgb]{1,0,0}{\frac{1}{40}X^{2}-\frac{1}{120}D_{2}}}\right),\qquad
J_{3}^{(2)}={\color[rgb]{0,0,1}{\frac{E^{2}}{144}}}\left({\color[rgb]{1,0,0}{\frac{1}{360}X^{3}-\frac{1}{360}XD_{2}+\frac{1}{1260}D_{3}}}\right),$
$\displaystyle J_{0}^{(3)}$
$\displaystyle={\color[rgb]{1,0,0}{\frac{1}{8640}E^{3}}}$ (19)
The number of hadronic states in each conserved charge sector $(B,I,S,\ldots)$
can be obtained from the one-link integral by combining them in alternating
chains $J_{k}^{(B)}J_{(N_{f}-B){N_{c}}-k}^{(B)}$, with $k$ denoting the
mesonic occupation number $\mathfrak{m}_{k}$. To obtain the Hamiltonian, we
still have to integrate out the Grassmann variables. Here we consider the
chiral limit only. The Grassmann constraint then dictates that all quarks
$u,d,s,\ldots$ and anti-quarks $\bar{u},\bar{d},\bar{s},\ldots$ within mesons
or baryons appear exactly ${N_{c}}$ times. The Grassmann integral in the
chiral limit for U(3) (i.e. SU(3) with $B=0$) for a given site $x$ is
$\displaystyle I_{G}$ $\displaystyle=\int\prod_{\alpha}\prod_{f}[{\rm
d}\bar{f}_{\alpha}{\rm
d}f_{\alpha}]\prod_{f,g}(\bar{f}g)^{k_{\bar{f}g}}(-1)^{\frac{1}{2}\sum\limits_{f\neq
g}k_{\bar{f}g}}F_{N_{f}}({N_{c}},\\{k_{\bar{f}g}\\})$ (20)
with $k_{\bar{f}g}$ the sum of flavored dimers attached to site $x$, and with
$F_{N_{f}}({N_{c}})$ some ${N_{c}}$-dependent function, given the power
$k_{fg}$ of charged fluxes ($f\neq g$). The result for non-zero baryon number
$B\neq 0$ and details on $F_{N_{f}}({N_{c}})$ will be presented in a
forthcoming publication. On a given configuration, the Grassmann integration
simplifies due to flux conservation: for the non-diagonal pseudo-scalar mesons
$k_{\bar{f}g}=k_{\bar{g}f}$ such that all minus signs from the Grassmann
integration cancel. Only the non-trivial contribution of type
$k^{(2)}_{UD,\pi^{+}\pi^{-}}$ and its generalizations for ${N_{f}}>2$ induce
minus signs. Those link states can be resummed and diagonalized, e.g. the
following states have an eigenvalue 1:
$\displaystyle\pi_{0}^{2}$
$\displaystyle=\left(k_{U}k_{D}+\frac{1}{\sqrt{{N_{c}}}}k^{(2)}_{\pi^{+}\pi^{-},UD}\right),$
$\displaystyle\bar{\pi}_{0}^{2}$
$\displaystyle=\left(k_{\pi^{+}}k_{\pi^{-}}+\frac{1}{\sqrt{{N_{c}}}}k^{(2)}_{UD,\pi^{+}\pi^{-}}\right)$
(21)
Likewise all dimer-based states are resummed, as states are only
distinguishable on the quark level. This reduces the state space drastically
to the physical Hilbert state, e.g. for ${N_{f}}=2$: $B=0:340\mapsto 50$,
$B=\pm 1:152\mapsto 20$, $B=\pm 2:4\mapsto 1$, resulting in the local Hilbert
space $\mathbbm{H}_{\mathfrak{h}}$ as given in Tab. 1, classified by baryon
number $B$, isospin number $I$ and meson occupation number $\mathfrak{m}$. The
particle-hole symmetry generalizes to
$\mathfrak{m}\mapsto\mathfrak{s}=\mathfrak{m}-\frac{{N_{c}}}{2}({N_{f}}-|B|)$
as the meson raising and lowering operators fulfill an SU(2) algebra for each
meson charge $Q_{i}$. All states $\mathfrak{h}\in\mathbbm{H}_{\mathfrak{h}}$
for ${N_{f}}=2,3$ contribute with a weight 1, although some dimer-based link
weights are negative.
$B$ | $I$ | $\mathfrak{s}=\mathfrak{m}-\frac{3}{2}(2-|B|)$ | $\Sigma$
---|---|---|---
| | $-3$ | $-\frac{5}{2}$ | $-2$ | $-\frac{3}{2}$ | $-1$ | $-\frac{1}{2}$ | $\,0\,$ | $+\frac{1}{2}$ | $+1$ | $+\frac{3}{2}$ | $+2$ | $+\frac{5}{2}$ | $+3$ |
-2 | 0 | | | | | | | 1 | | | | | | | 1
-1 | $-\frac{3}{2}$ | | | | 1 | | 1 | | 1 | | 1 | | | | 4
-1 | $-\frac{1}{2}$ | | | | 1 | | 2 | | 2 | | 1 | | | | 6
-1 | $+\frac{1}{2}$ | | | | 1 | | 2 | | 2 | | 1 | | | | 6
-1 | $+\frac{3}{2}$ | | | | 1 | | 1 | | 1 | | 1 | | | | 4
0 | -3 | | | | | | | 1 | | | | | | | 1
0 | -2 | | | | | 1 | | 2 | | 1 | | | | | 4
0 | -1 | | | 1 | | 2 | | 4 | | 2 | | 1 | | | 10
0 | 0 | 1 | | 2 | | 4 | | 6 | | 4 | | 2 | | 1 | 20
0 | -1 | | | 1 | | 2 | | 4 | | 2 | | 1 | | | 10
0 | -2 | | | | | 1 | | 2 | | 1 | | | | | 4
0 | -3 | | | | | | | 1 | | | | | | | 1
1 | $-\frac{3}{2}$ | | | | 1 | | 1 | | 1 | | 1 | | | | 4
1 | $-\frac{1}{2}$ | | | | 1 | | 2 | | 2 | | 1 | | | | 6
1 | $+\frac{1}{2}$ | | | | 1 | | 2 | | 2 | | 1 | | | | 6
1 | $+\frac{3}{2}$ | | | | 1 | | 1 | | 1 | | 1 | | | | 4
2 | 0 | | | | | | | 1 | | | | | | | 1
$\Sigma$ | | 1 | 0 | 4 | 8 | 10 | 12 | 22 | 12 | 10 | 8 | 4 | 0 | 1 | 92
Table 1: All 92 possible quantum states for the ${N_{f}}=2$ Hamiltonian
formulation with ${\rm SU}(3)$ gauge group. The number of states are given for
the sectors specified baryon number $B$ and isospin number $I$, and
symmetrized meson occupation number
$\mathfrak{s}=\mathfrak{m}-\frac{{N_{c}}}{2}({N_{f}}-|B|)$. Note the mesonic
particle-hole symmetry $\mathfrak{s}\leftrightarrow-\mathfrak{s}$ which
corresponds to the shift symmetry by ${a_{\tau}}$.
Only single meson exchange is possible (multiple meson exchange becomes
resolved into single mesons in the continuous time limit), the resulting
interaction Hamiltonian has ${N_{f}}^{2}$ terms:
$\displaystyle\hat{\mathcal{H}}$
$\displaystyle=\frac{1}{2}\sum_{\langle\vec{x},\vec{y}\rangle}\sum_{Q_{i}}\left({\hat{J}_{Q_{i},\vec{x}}}^{+}{\hat{J}_{Q_{i},\vec{y}}}^{-}+{\hat{J}_{Q_{i},\vec{x}}}^{-}{\hat{J}_{Q_{i},\vec{y}}}^{+}\right),$
(22)
e.g. for ${N_{f}}=2$: $Q_{i}\in\\{\pi^{+},\pi^{-},\pi_{U},\pi_{D}\\}$ and for
${N_{f}}=3$:
$Q_{i}\in\\{\pi^{+},\pi^{-},K^{+},K^{-},K_{0},\bar{K_{0}},\pi_{U},\pi_{D},\pi_{S}\\}$.
For the transition $\mathfrak{h}_{1}\mapsto\mathfrak{h}_{2}$, the matrix
elements ${\langle\mathfrak{h}_{1}|Q_{i}|\mathfrak{h}_{2}\rangle}$ of
$\hat{J}_{Q_{i}}^{\pm}$ are determined from Grassmann integration and a square
root per participating link weight (in, hopping, out); only those matrix
elements are non-zero which are consistent with current conservation for all
$Q_{i}$. Some matrix elements are negative, but in the combination of a closed
charged meson loop the total weight remains positive. Some examples for
${N_{f}}=2$ are:
$\displaystyle{\langle\pi^{+},\pi^{-}|\pi^{+}|\pi_{U},\pi_{D}\pi^{-}\rangle}$
$\displaystyle=\frac{\sqrt{3}}{2},$
$\displaystyle{\langle\pi_{U},\pi^{-}|\pi^{+}|\pi_{U},2\pi^{-}\rangle}$
$\displaystyle=\frac{\sqrt{6}}{3},$ $\displaystyle{\langle
B_{uud}|\pi^{+}|B_{uud},\pi_{U},\pi_{D}\rangle}$
$\displaystyle=\frac{2}{\sqrt{5}},$ $\displaystyle{\langle
B_{uud}\pi^{+}|\pi^{+}|B_{uud},\pi_{D}\rangle}$
$\displaystyle=-\frac{\sqrt{3}}{6}$ (23)
There are 130 non-zero matrix elements for $B=0$ and 40 matrix elements for
$B=\pm 1$. The $B=\pm 2$ states do not allow for pion exchange. Only after
contraction of the matrix elements the resummation is carried out for the
internal hadronic states, resulting in positive weights. An important
application of the ${N_{f}}=2$ partition function is to determine the QCD
phase diagram with both finite baryon and isospin chemical potential [13, 14].
Our formulation is still sign-problem free in the continuous time limit. As we
have not yet performed dynamical simulations, we can only obtain analytic
results in a high temperature expansion of the partition sum $Z$, which is an
expansion around the static limit
$\displaystyle Z\left(\frac{\mu_{B}}{T},\frac{\mu_{I}}{T}\right)=$
$\displaystyle
2\cosh\frac{3\mu_{I}}{T}+8\cosh\frac{2\mu_{I}}{T}+20\cosh\frac{\mu_{I}}{T}+20$
$\displaystyle+2\cosh\frac{\mu_{B}}{T}\left(8\cosh{\frac{\frac{3}{2}\mu_{I}}{T}}+12\cosh\frac{\frac{1}{2}\mu_{I}}{T}\right)+2\cosh\frac{2\mu_{B}}{T}$
(24)
in the number of spatial mesons. This will be presented in a forthcoming
publication.
## 3 Leading order gauge corrections to the Hamiltonian formulation for
$N_{f}=1$
The QCD Partition can be expanded via the strong coupling expansion in
$\beta$:
$\displaystyle Z_{QCD}$ $\displaystyle=\int d\psi
d\bar{\psi}dUe^{S_{G}+S_{F}}=\int d\psi d\bar{\psi}Z_{F}{\langle
e^{S_{G}}\rangle}_{Z_{F}},$ $\displaystyle{\langle e^{S_{G}}\rangle}_{Z_{F}}$
$\displaystyle\simeq 1+{\langle
S_{G}\rangle}_{Z_{F}}+\mathcal{O}(\beta^{2})=1+\frac{\beta}{2{N_{c}}}\sum_{P}{\langle{\rm
tr}[U_{P}+U_{P}^{\dagger}]\rangle}_{Z_{F}}+\mathcal{O}(\beta^{2})$ (25)
Additional color singlet link states are due to plaquette excitations. For
discrete time lattices, the dual formulation has been extended beyond
$\mathcal{O}(\beta)$ in terms of a tensor network [8]. Here we want to
consider the leading order gauge corrections in a Hamiltonian formulation. On
anisotropic lattices, the anisotropy $\xi=\frac{a_{s}}{a_{t}}$ is a function
of two bare anisotropies $\gamma_{F}$ and
$\gamma_{G}=\sqrt{\frac{\beta_{t}}{\beta_{s}}}$, i.e
$\xi=\xi(\gamma_{F},\gamma_{G},\beta)$. However, in the continuous time limit
$a_{t}\rightarrow 0$ ($\xi\rightarrow\infty$) and for small $\beta$, spatial
plaquettes are suppressed over temporal plaquettes by
$(\gamma_{G}\gamma_{F})^{-2}$, hence only temporal plaquettes need to be
considered. They are of the same order as meson exchange, hence the
$\mathcal{O}(\beta)$ weights will contribute to the corresponding operators
$\hat{J}^{\pm}$. Note that $\hat{J}^{\pm}$ still has block-diagonal structure,
but they will also allows to couple to baryons. Fig. 1 shows the relevant
weights as computed in [8].
Figure 1: Tensors and their weights of $\mathcal{O}(\beta)$ that can be
incorporated into the Hamiltonian formulation. Blue: dimers, red: baryon flux,
green: plaquette flux due to gauge corrections.
## 4 Summary and Outlook
We have presented two extensions to the Hamiltonian formulation of lattice
QCD: (1) generalization from ${N_{f}}=1$ to ${N_{f}}>1$ in the strong coupling
limit, (2) $\mathcal{O}(\beta)$ gauge corrections for ${N_{f}}=1$. In a
forthcoming publication, we also address gauge corrections to the ${N_{f}}=2$
Hamiltonian. It turns out that - although not sign problem-free - the sign
problem is much milder than in the corresponding formulation at finite
${N_{\tau}}$ as for ${N_{\tau}}\rightarrow\infty$ only temporal plaquettes
contribute. Meson exchange between baryons can thus be either due to the gauge
corrections or the flavor content. All results presented here are valid in the
chiral limit. We have argued in [1] that also a Hamiltonian formulation at
finite quark mass $am_{q}$ is well-defined, which extends this approach to
staggered lattice QCD further. We are preparing Quantum Monte Carlo
simulations that will help to obtain the QCD phase diagram on the lattice in
the parameter space $(aT,a\mu_{B},a\mu_{I},am_{q},\beta)$ in lattice units or
in units of the baryon mass $am_{B}$.
This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) – project number 315477589 – TRR 211.“
## References
* [1] Marc Klegrewe and Wolfgang Unger. Strong Coupling Lattice QCD in the Continuous Time Limit. Phys. Rev. D, 102(3):034505, 2020.
* [2] John B. Kogut and Leonard Susskind. Hamiltonian Formulation of Wilson’s Lattice Gauge Theories. Phys. Rev. D, 11:395–408, 1975.
* [3] Pietro Rossi and Ulli Wolff. Lattice QCD With Fermions at Strong Coupling: A Dimer System. Nucl. Phys., B248:105–122, 1984.
* [4] N. Kawamoto and J. Smit. Effective Lagrangian and Dynamical Symmetry Breaking in Strongly Coupled Lattice QCD. Nucl. Phys., B192:100, 1981. [,556(1981)].
* [5] F. Karsch and K. H. Mutter. Strong Coupling QCD at Finite Baryon Number Density. Nucl. Phys., B313:541–559, 1989.
* [6] Philippe de Forcrand and Michael Fromm. Nuclear Physics from lattice QCD at strong coupling. Phys. Rev. Lett., 104:112005, 2010.
* [7] Philippe de Forcrand, Jens Langelage, Owe Philipsen, and Wolfgang Unger. Lattice QCD Phase Diagram In and Away from the Strong Coupling Limit. Phys. Rev. Lett., 113(15):152002, 2014.
* [8] Giuseppe Gagliardi and Wolfgang Unger. New dual representation for staggered lattice QCD. Phys. Rev. D, 101(3):034509, 2020.
* [9] Philippe de Forcrand, Wolfgang Unger, and Helvio Vairinhos. Strong-Coupling Lattice QCD on Anisotropic Lattices. Phys. Rev., D97(3):034512, 2018.
* [10] Neven Bilic, Frithjof Karsch, and Krzysztof Redlich. Flavor dependence of the chiral phase transition in strong coupling QCD. Phys. Rev., D45:3228–3236, 1992.
* [11] Owe Philipsen. Strong coupling methods in QCD thermodynamics. Indian J. Phys. 95, 1599–1611 (2021)
* [12] K. E. Eriksson, N. Svartholm, and B. S. Skagerstam. On Invariant Group Integrals in Lattice QCD. J. Math. Phys., 22:2276, 1981.
* [13] Yusuke Nishida. Phase structures of strong coupling lattice QCD with finite baryon and isospin density. Phys. Rev., D69:094501, 2004.
* [14] B. B. Brandt, G. Endrodi, and S. Schmalzbauer. QCD phase diagram for nonzero isospin-asymmetry. Phys. Rev. D, 97(5):054514, 2018.
|
# ZipLM: Inference-Aware Structured Pruning
of Language Models
Eldar Kurtic
IST Austria
<EMAIL_ADDRESS>
&Elias Frantar
IST Austria
<EMAIL_ADDRESS>
&Dan Alistarh
IST Austria & Neural Magic
<EMAIL_ADDRESS>
###### Abstract
The breakthrough performance of large language models (LLMs) comes with major
computational footprints and high deployment costs. In this paper, we progress
towards resolving this problem by proposing a novel structured compression
approach for LLMs, called ZipLM. ZipLM achieves state-of-the-art accuracy-vs-
speedup, while matching a set of desired target runtime speedups in any given
inference environment. Specifically, given a model, a dataset, an inference
environment, as well as a set of speedup targets, ZipLM iteratively identifies
and removes components with the worst loss-runtime trade-off. Unlike prior
methods that specialize in either the _post-training/one-shot_ or the _gradual
compression_ setting, and only for specific families of models such as BERT
(_encoder_) or GPT (_decoder_), ZipLM produces state-of-the-art compressed
models across all these settings. Furthermore, ZipLM achieves superior results
for a fraction of the computational cost relative to prior distillation and
pruning techniques, making it a cost-effective approach for generating an
entire family of smaller, faster, and highly accurate models, guaranteed to
meet the desired inference specifications. In particular, ZipLM outperforms
all prior $\textrm{BERT}_{\small{\textrm{base}}}\,$distillation and pruning
techniques, such as CoFi, MiniLM, and TinyBERT. Moreover, it matches the
performance of the heavily optimized MobileBERT model, obtained via extensive
architecture search, by simply pruning the baseline
$\textrm{BERT}_{\small{\textrm{large}}}\,$model. When compressing GPT2, ZipLM
outperforms DistilGPT2 while being 60% smaller and 30% faster. Our code is
available at: https://github.com/IST-DASLab/ZipLM.
## 1 Introduction
The high accuracy of modern language models from the Transformer family [1]
comes at the price of massive computational cost, which hinders their
practical adoption in resource-constrained settings. This has motivated the
development of _model compression_ techniques, which can be categorized into
_pruning_ [2], _quantization_ [3], and _distillation_ [4]. In this paper, we
focus on _structural compression_ , whose goal is to reduce model size by
removing entire sub-components, such as rows or columns from the model’s
weight matrices. The key advantage of structured pruning, relative to
unstructured pruning of individual weights, is that the model can be reshaped
to new dimensions, and the resulting computational savings can be leveraged on
any hardware, without specialized computational support. At the same time,
structured pruning introduces significant challenges. First, models are
usually highly-sensitive to structured compression, and most methods require
_gradual compression_ , including retraining cycles designed to allow the
model to recover accuracy. In addition, structural compression significantly
complicates the use of knowledge distillation [5], which is usually done via
manual or dynamic layer mapping [6, 7]. On the practical side, another
challenge is that most existing techniques do not provide _runtime speedup_
guarantees: the model is pruned to a fixed sparsity or FLOPS target, and then
must be evaluated in the target inference environment. If the pruned model
fails to meet the target inference specifications, the whole process must be
repeated from scratch.
#### Overview.
In this paper, we resolve these issues and provide a novel structured pruning
approach called ZipLM, which achieves state-of-the-art performance, both in
the _post-training/one-shot_ setting, where retraining is not desirable, as
well as in the popular _gradual compression_ setting, where retraining is
possible. We accomplish this via an inference-aware algorithm, which
successfully balances the loss-runtime trade-off at each pruning step. By
taking runtime into account, we avoid removing components that do not bring
significant speedup gains. Additionally, our algorithm provides speedup
guarantees for compressed models, a highly-desirable property in practical
applications.
We summarize our contributions as follows:
* •
We introduce a novel structured pruning approach, which unifies the saliency
criteria investigated by prior work–weight magnitude, activation impact, and
removal of linearly-redundant structures, while considering local (layer-wise)
and global correlations. We augment it to be _inference-aware_ , ensuring
desired latency or throughput in any given configuration.
* •
We complement the algorithm with a novel _layer-wise token-level distillation_
, which consistently boosts accuracy on small datasets and does not require
manual layer matching, circumventing a limitation of prior structured pruning
techniques.
* •
ZipLM is the first structured pruning approach that achieves state-of-the-art
results for both, _post-training/one-shot_ compression and _gradual pruning_
settings, while being applicable to both, BERT (_encoder_) and GPT (_decoder_)
language models, without any modifications.
* •
ZipLM is practical and efficient. For a set of desired speedups (e.g. 2x, 5x,
10x) in the target inference environment (e.g. batch-size=128, sequence-
length=384, device=V100), in a single run and under the same set of hyper-
parameters, it produces the entire family of compressed models, one for each
speedup target. Consequently, it leads to state-of-the-art results in _GPU-
based_ inference environments. Moreover, it is compatible with unstructured
pruning and quantization, leading to state-of-the-art results even for _CPU-
based_ environments.
## 2 Related Work
#### Distillation-based compression methods
focus on training a smaller student model to mimic the representations of a
larger teacher model. The “distance” between the representations of student
and teacher is often architecture-specific. MiniLM [8] uses a deep self-
attention mechanism to replicate the attention mechanism of the teacher, and
TinyBERT [6] employs a bespoke distillation mechanism, for a manually-picked
subset of layers. Both methods offer a very strong baseline, generally
outperforming other approaches, except for MobileBERT. MobileBERT [9] involves
first training a custom large BERT teacher model from scratch, and then
deviates from the standard architecture [10] by introducing heavily-optimized
components with reduced latency, whose combinations are decided in neural
architecture search (NAS)-like fashion. It achieves strong results in terms of
accuracy-per-parameter, at the cost of significant computational costs in the
search process. DistilBERT and DistilGPT2 [11] involve training a fixed
student obtained by removing every other layer from the teacher, while BERT-
PKD [12] employs incremental knowledge extraction through the distillation of
intermediate layers. Well-Read-Students [13] reduces the size of the standard
BERT architecture through principled downscaling of internal dimensions.
DynaBERT [14], on the other hand, distills knowledge to a student model that
is both depth- and width-adaptive.
#### Structural pruning methods
usually start from a large pre-trained model, and iteratively reduce the
dimensions of weight matrices. Block Movement Pruning [15] identifies and
removes redundant rectangular blocks of weights while following the movement
pruning intuition [16] that weights moving towards zero during fine-tuning
should be removed. FLOP [17] and Low-Rank [18] use matrix decomposition
techniques to progressively remove rank-1 components from factorized weight
matrices during training. BERT-of-Theseus [19] employs a similar approach, but
replaces entire submodules with smaller counterparts. Methods like LayerDrop
[20] and Poor Man’s BERT [21] address structured compression through various
layer-dropping techniques. LayerDrop uses structured layer-dropout
regularization to train a model resilient to sub-network selection during
inference, while Poor Man’s BERT explores a wide range of layer-dropping
strategies. The recent CoFi method [7] employs masks of different
granularities to jointly prune coarse and fine-grained submodules during fine-
tuning, combined with an optional customized distillation technique. CoFi is
the state-of-the-art _structural pruning_ method; relative to distillation
methods, CoFi outperforms MiniLM and TinyBERT, but not MobileBERT, in terms of
accuracy-vs-speedup.
#### Other compression methods
such as the ones that exploit dynamic forms of sparsity which appear at
runtime [22], or the ones that utilize lower bit-width representation of
weights and/or activations [23, 24] are complementary to our approach. We
demonstrate this in Section 5 where we apply quantization to obtain even
higher compression ratios for edge deployment environments like commodity
CPUs.
## 3 Method
Removing large structures like entire matrix columns or attention heads from a
language model quickly leads to severe accuracy degradation, from which it is
often difficult to recover even with extensive finetuning. This is why current
state-of-the-art approaches like Block Movement Pruning [15] or CoFi [7] opt
for integrating pruning directly into training (via sampling or differentiable
approximations), rather than performing it in the standard gradual pruning
fashion of discrete steps with finetuning in between. However, as we will
show, by designing a new highly accurate pruning algorithm which is able to
account for both local correlations of structures within single layers as well
as global correlations across layers, we can actually apply the gradual
pruning paradigm, with all its advantages, to improve significantly over the
current state-of-the-art.
### 3.1 The ZipLM Structured Pruning Algorithm (Local Correlations)
Most existing structured pruning criteria [25, 26] are based on one or two of
the following assumptions about saliency: structures with lower (average)
weight magnitude are easier to prune [27, 28], structures with small input
activations can be removed at little loss [29], and structures that are close
to a linear combination of other structures are the most redundant [30, 31].
We will now show how all these aspects can be jointly considered in a
principled manner via our new ZipLM technique.
#### Problem formulation.
Our approach starts from the idea of applying structured compression layer-
wise, in a way that allows the layer to preserve most of its output
characteristics. This setup is popular in the post-training quantization and
unstructured pruning literature [32, 33, 34], and can be implemented as
follows. We are given a small amount of calibration data, which we run through
the network, to obtain “reference” inputs and outputs for each layer. Then,
for each layer, given the calibration inputs $\mathbf{X}$ and the original
layer weights $\mathbf{{W}}$, we aim to find compressed weights
$\mathbf{\widehat{W}}$ respecting the compression constraint $\mathcal{C}$,
which best approximate the original output, measured via the squared error
metric. If we assume that the input and weight matrices have an appropriate
rectangular form, the problem can be formalized as:
$\text{argmin}_{\mathbf{\widehat{W}}}\,||\mathbf{\widehat{W}}\mathbf{X}-\mathbf{W}\mathbf{X}||_{2}^{2}\quad\text{subject
to}\quad\mathbf{\widehat{W}}\in\mathcal{C}.$ (1)
This objective can be decomposed across the rows of $\mathbf{W}$, leading to a
set of sparse linear regression problems, one per row. These row-wise problems
are independent, which forms the basis of related work [34]; yet, since we do
structured pruning, they become dependent, as we would like to prune the same
weight indices _across all rows_ , i.e. prune entire columns. Thus, finding
the optimal weights $\mathbf{\widehat{W}}\in\mathcal{C}$ is equivalent to
finding: 1) the optimal structure $\mathbf{S}$ of the desired shape to be
removed, which we assume to be applied across all rows, with corresponding
pruning mask $\mathbf{M_{S}}$, where pruned indices have value $1$ in the
mask, and others are $0$; and 2) the corresponding update
$\boldsymbol{\delta_{S}}$ to all of the remaining weights, optimally
compensating for the error caused by the removal of weights in $\mathbf{S}$.
#### Saliency scores and weight update.
Let $\mathbf{H}=\mathbf{X}\mathbf{X}^{\top}$ be the Hessian matrix for the
$\ell_{2}$-minimization problem in Equation 1, which is independent of the
weights. Define $\mathbf{W}_{i,\mathbf{M_{S}}}$ to be the subset of weights
under the mask $\mathbf{M_{S}}$ in row $i$, and by
$(\mathbf{H}^{-1})_{\mathbf{M_{S}},\mathbf{M_{S}}}$ the submatrix of the
inverse Hessian corresponding to the entries under the mask $\mathbf{M_{S}}$.
Then, we can obtain the optimal mask and weight update as follows:
$\text{argmin}_{\mathbf{S}}\,\sum_{i=0}^{d_{\text{row}}}\mathbf{W}_{i,\mathbf{M_{S}}}\cdot\left(\left(\mathbf{H}^{-1}\right)_{\mathbf{M_{S}},\mathbf{M_{S}}}\right)^{-1}\cdot\mathbf{W}^{\top}_{i,\mathbf{M_{S}}}$
(2)
$\boldsymbol{\delta_{S}}=-\mathbf{W}_{:,\mathbf{M_{S}}}\cdot\left(\left(\mathbf{H}^{-1}\right)_{\mathbf{M_{S}},\mathbf{M_{S}}}\right)^{-1}\cdot\left(\mathbf{H}^{-1}\right)_{\mathbf{M_{S}},:}$
(3)
We obtain this by extending the Optimal Brain Surgeon [35, 36] formulas for
solving Equation 1 to cover all $d_{\text{row}}$ weight matrix rows
simultaneously. Importantly, the subselection of the inverse Hessian
$((\mathbf{H}^{-1})_{\mathbf{M_{S}},\mathbf{M_{S}}})^{-1}$ is shared between
all rows. Further, since we generally consider only non-overlapping sets
$\mathbf{S}$ of the same size, we pay just
$O(d_{\text{col}}\cdot|\mathbf{M_{S}}|^{2})$ total cost for all extra
inversions. Since the number of structures in the mask $|\mathbf{M_{S}}|$ is
usually small, e.g. attention heads usually consist of 64 columns, the overall
cost of these inversions is low.
Simply selecting the structures to prune according to the criterion in
Equation 2 unifies the weight magnitude and activation influence criteria (via
the Hessian), but still ignores any correlations between structures. We
address this by pruning structures _one-at-a-time_ , while always applying
update $\boldsymbol{\delta_{S}}$ and fully recomputing $\mathbf{H}^{-1}$
relative to the remaining structures. For example, if there exist two
redundant structures $S_{1}$ and $S_{2}$, we will first drop $S_{1}$ and
update $S_{2}$ to compensate for this removal, at which point $S_{2}$ is no
longer easy to prune. Without this one-at-a-time removal, both structures
would have been incorrectly removed as they each individually seem easy to
prune according to Equation 2. Executing this strategy naively will require a
full $O(d_{\text{col}}^{3})$ recomputation of the inverse Hessian relative to
the remaining structures at each step, which would be very slow. However, this
can be avoided by removing the rows and columns corresponding to
$\mathbf{M_{S}}$ directly in the inverse with one step of Gaussian elimination
[34], applied block-wise to cover larger structures, as follows:
$\mathbf{H}^{-1}-\mathbf{H}^{-1}_{:,\mathbf{M_{S}}}\cdot\left(\left(\mathbf{H}^{-1}\right)_{\mathbf{M_{S}},\mathbf{M_{S}}}\right)^{-1}\cdot\mathbf{H}^{-1}_{\mathbf{M_{S}},:},$
(4)
which takes only $O(|\mathbf{M_{S}}|\cdot d_{\text{col}}^{2})$ time. We
provide complete pseudocode in Algorithm 1.
Algorithm 1 The ZipLM pruning algorithm. Given inverse Hessian
$\mathbf{H}^{-1}=(2\mathbf{X}\mathbf{X}^{\top}+\lambda\mathbf{I})^{-1}$, we
remove exactly $k$ structures from the corresponding weight matrix
$\mathbf{W}$.
$\mathbf{R}\leftarrow$ set of all possible structures
for $k$ times do
$\mathbf{S}\leftarrow\text{argmin}_{\mathbf{S}}\,\sum_{i=0}^{d_{\text{row}}}\mathbf{W}_{i,\mathbf{M_{S}}}\cdot((\mathbf{H}^{-1})_{\mathbf{M_{S}},\mathbf{M_{S}}})^{-1}\cdot\mathbf{W}^{\top}_{i,\mathbf{M_{S}}}$
$\boldsymbol{\delta_{S}}\leftarrow-\mathbf{W}_{:,\mathbf{M_{S}}}\cdot((\mathbf{H}^{-1})_{\mathbf{M_{S}},\mathbf{M_{S}}})^{-1}\cdot(\mathbf{H}^{-1})_{\mathbf{M_{S}},:}$
$\mathbf{W}\leftarrow\mathbf{W}+\boldsymbol{\delta_{S}}$
$\mathbf{H}^{-1}\leftarrow\mathbf{H}^{-1}-\mathbf{H}^{-1}_{:,\mathbf{M_{S}}}\cdot((\mathbf{H}^{-1})_{\mathbf{M_{S}},\mathbf{M_{S}}})^{-1}\cdot\mathbf{H}^{-1}_{\mathbf{M_{S}},:}$
$\mathbf{R}\leftarrow\mathbf{R}-\\{\mathbf{S}\\}$
end for
$\mathbf{W}\leftarrow\mathbf{W}\odot\mathbf{M_{R}}$
We utilize the fact that the values corresponding to pruned weights in
$\mathbf{W}$ and in the inverse Hessian $\mathbf{H}^{-1}$ do not affect any
subsequent calculations and can therefore be ignored even if they are not
exactly zero. However, in the end we have to prune them explicitly again by
multiplying with the overall mask to ensure that they are exactly zero. In a
practical implementation,
$((\mathbf{H}^{-1})_{\mathbf{M_{S}},\mathbf{M_{S}}})^{-1}$ should only be
computed once and reused when computing the corresponding sum across all rows.
#### Pruned structures.
Focusing on Transformers, we consider three types of structural removal:
dropping attention heads, shrinking the expanded intermediate dimension of the
fully-connected network (FC) layers, and removing entire residual parts, i.e.
attention or FC-modules. We implement this by dropping $d_{\text{head}}$
consecutive columns in the out-matrix of the attention block and individual
columns in the second linear layer of the feed-forward network. Once these
column-structures are zeroed out, corresponding rows in previous layers can be
safely removed without any output change. Crucially, by pruning e.g. columns
in the FC2 layer rather than equivalent rows in FC1, we can utilize the input
correlations via Hessian-information using the ZipLM pruner.
#### Novelty relative to existing Optimal Brain Surgeon (OBS) approaches.
The original framework [35], as well as modern efficient versions [37, 38,
34], have been explicitly developed for _unstructured pruning_ , i.e. removing
individual weights. It is nontrivial to extend them to structured pruning, as
this involves considering additional correlations, both within as well as
across multiple blocks (such blocks are usually employed for computational
tractability). For example, the state-of-the-art layer-wise approach of [34],
performs unstructured pruning by handling weight matrix rows separately, and
then greedily merging results. In contrast, we perform structured pruning
jointly across multiple rows, which is not only necessary for correctness but
additionally enables us to design an algorithm with a computational complexity
that is lower by a full factor of the hidden dimension size. Additionally,
structured pruning requires explicitly matching matrix shapes for consecutive
layers and a dedicated strategy for utilizing weight updates even when entire
blocks/rows are pruned.
### 3.2 Inference-Aware Structured Pruning (Global Correlations)
We now describe how to augment the algorithm to be _inference-aware_ , in the
sense that it accepts inference specifications, such as batch-size, sequence-
length, and speedup on the target hardware, as additional inputs to optimize
for.
#### Motivation.
The main benefit of inference-aware structured pruning is the fact that
pruning decisions are not guided purely by saliency scores, but instead by
loss-vs-speedup trade-offs associated with the removal of each component in
the model. Prior methods, e.g. [36, 15, 7] focus solely on pruning until a
specific sparsity threshold is reached, without taking into account the real-
world speedups corresponding to the compression threshold, which can vary
significantly between settings. For example, a 95% sparse BERT produced by
CoFi [7] has 12x speedup on a V100 GPU, but only 5x on an A100 GPU. With
existing methods, if real-world timings fail to meet the inference
requirements, the entire process has to be repeated with different sparsity
values until the target speedup is achieved, which is both time-consuming and
error-prone. An additional advantage of inference-awareness, which we showcase
in our GPT experiments in Section 4, is that it enables optimizing for
different real-world metrics, such as latency or throughput.
#### Runtime awareness.
We integrate runtime constraints via a latency table [39] for our target
inference environment, where we record the time to run an attention block,
including all overheads, with $0,\dots,N_{\text{heads}}-1$ heads pruned and
similarly for the fully-connected block with the intermediate dimension shrunk
by a factor of $0.9^{i}$, for $i=0,\dots,42$; in relative steps of $10\%$ up
until $\approx 99\%$ sparsity, following [40]. This allows rapid runtime
estimation for different per-layer sparsity configurations. We provide an
example of our latency table in Appendix E.
#### Finding the optimal sparsity configuration.
Ultimately, our goal is to find a per-layer-sparsity configuration that
satisfies a certain speedup-constraint while maximizing accuracy. A popular
paradigm of doing this [27, 41] is to produce a large number of pruned models
with different sparsity distributions across layers and then select the one,
satisfying a target constraint, with the highest accuracy. To make this
computationally feasible, it is crucial that pruning is cheap, yet accurate.
ZipLM treats each layer independently, which makes it possible to precompute a
database of several pruned versions with different sparsities for each layer.
The entire database can be produced in a single run, utilizing the algorithm’s
one-at-a-time nature. While our algorithm is compatible with various search
methods for finding layer-wise profiles [27, 42], we adapt the recent SPDY
approach [40].
#### Structured SPDY search.
The SPDY approach is designed for unstructured pruning and assigns a quadratic
prior to per-layer sensitivity of different sparsity levels. This is not valid
in our structured pruning scenario, since for instance it would suggest that
dropping a full layer is only slightly more difficult than pruning it to 99%
sparsity. Thus, using standard SPDY would lead the algorithm to explore a
large number of sub-optimal configurations, significantly wasting
computational resources. To alleviate this problem, for a structured sparsity
$s$, we introduce a better prior $p_{s}$ as the relative layer-wise squared
error incurred by pruning, defined as
$p_{s}=||\mathbf{\widehat{W}_{s}}\mathbf{X}-\mathbf{W}\mathbf{X}||_{2}/||\mathbf{W}\mathbf{X}||_{2}$,
which simply has a value of 1 for a fully dropped layer. Furthermore, the
original SPDY approach uses shrinking neighborhood search, which has high
variance in both runtime and solution quality for structured compression.
Therefore, we perform a fixed number of $1000$ steps, randomly mutating in
expectation 10% of the layer-wise sensitivity coefficients. Finally, we note
that any candidate evaluated by this procedure actually achieves the target
speedup, leading to significantly decreased search time. We validate our
approach in Appendix F, where we demonstrate that our speedup estimations are
indeed very accurate in practice. Specifically, real-world on-device
measurements deviate at most by 5.28% from their expected values.
### 3.3 Layer-wise Token Distillation
For structured pruning, it is common to apply _layer-wise distillation_
objectives to transfer intermediate representations. However, structured
pruning creates compatibility issues relative to the fixed teacher
architecture, leading most methods to develop customized distillation
strategies. A popular approach, introduced in [6] and improved by [7], solves
the problem via static [6] or dynamic [7] mapping of a subset of teacher
layers to a subset of student layers. Their main limitation is manual layer
selection, where making the “optimal” choice would require evaluating all
possible combinations, which can be very expensive. Another limitation is
shape-matching between intermediate layers, which is solved by introducing a
learnable linear transformation matrix attached to student outputs.
#### Our approach.
We address these challenges differently, by leveraging the fact that ZipLM
preserves the hidden dimension size, and propose to use distillation of
intermediate token representations across the entire model. The resulting
minimization objective consists of three components:
$\displaystyle\mathcal{L}(\theta^{\mathrm{s}},\theta^{\mathrm{t}}|x)=\lambda_{\mathrm{1}}\mathcal{L}_{\mathrm{task}}(\theta^{\mathrm{s}}|x)\,+\,\lambda_{\mathrm{2}}\mathcal{L}_{\mathrm{logit}}(\theta^{\mathrm{s}},\theta^{\mathrm{t}}|x)+\lambda_{3}\mathcal{L}_{\mathrm{token}}(\theta^{\mathrm{s}},\theta^{\mathrm{t}}|x),$
(5)
where $\theta^{\mathrm{s}}$ and $\theta^{\mathrm{t}}$ represent student and
teacher models respectively, $x$ are the inputs, $\mathcal{L}_{\mathrm{task}}$
is the loss associated with the task (e.g. cross-entropy for text-
classification), $\mathcal{L}_{\mathrm{logit}}$ is the KL-divergence between
output logits as described in [5], and $\mathcal{L}_{\mathrm{token}}$ is our
token-level distillation loss. Hidden tensors passed between consecutive
transformer layers are of constant shape $\mathbf{H}\in\mathbb{R}^{B\times
seq\times H}$, where $B$ stands for the batch-size, $seq$ for the sequence
length, and $H$ for the hidden size defined by the model architecture. This
tensor can be interpreted as a collection of $B\times seq$ vectors
$\mathbf{h}\in\mathbb{R}^{H}$, each carrying intermediate model
representations of input tokens $x$. We define the loss
$\mathcal{L}_{\mathrm{token}}$ as an Euclidean distance $\Delta$ between
vectors $\mathbf{h}$ corresponding to each non-padded token in the input
sequence, averaged over all unpruned layers. Formally, for a layer $k$, it is
defined as
$\mathcal{L}_{\mathrm{token}}^{k}=\frac{1}{\sum_{j=1}^{B\times
seq}\mathbbm{1}[j\notin\mathbf{P}]}\sum_{j=1}^{B\times
seq}\mathbbm{1}[j\notin\mathbf{P}]\cdot\Delta(\mathbf{h}^{\theta_{\mathrm{s}}},\mathbf{h}^{\theta_{\mathrm{t}}}),$
(6)
where $\mathbf{P}$ stands for the set of padding tokens. This formulation
encourages the student model to generate vector representations for each token
that are similar to those produced by the teacher model. In Appendix B, we
present ablation studies and comparisons for ZipLM and CoFi, with and without
their respective distillation objectives.
## 4 Experiments
#### Setup.
Given a pre-trained model, a dataset, and a set of desired speedups in a
target inference environment, we iteratively fine-tune and prune the model in
a structured way such that in the end we obtain a set of accurate compressed
models, one for each speedup target. We consider pruning of the standard
$\textrm{BERT}_{\small{\textrm{base}}}\,$and
$\textrm{BERT}_{\small{\textrm{large}}}\,$architectures, evaluating on dev-
sets of established benchmarks: SQuADv1.1 [43], and a subset of GLUE [44]
tasks: SST-2 [45], QNLI [44], MNLI [46], and QQP [47], selected to match
publicly-available checkpoints from prior work. For a precise comparison to
prior work [7], our inference environment is a single NVIDIA V100 16GB GPU,
batch size of 128, and sequence lengths of 384 and 128 for SQuAD and GLUE
tasks, respectively. In addition to encoder-based BERT models, we also
consider pruning of the decoder-based GPT2 model on the OpenWebTextCorpus
[48], for which we consider two inference environments: pruning for throughput
(batch-size=16, sequence-length=1024), and pruning for latency (batch-size=1,
a set of prompts with varying lengths). For illustration, our pipeline is
depicted in Figure 1. In Appendix H and I, we report exact values for all
results, as well as hyper-parameters for reproducibility.
Figure 1: Illustration of the ZipLM pipeline: 1) inference specifications, 2)
runtime benchmarking of candidates for pruning, 3) gradual structured pruning
until all speedup targets are met.
#### Baselines.
In the _gradual pruning_ setting, we explore the performance of ZipLM pruning
of BERT- and GPT2-family models, across a wide range of inference speedup
targets, ranging from 2x to 15x, in unit increments. This allows us to compare
the effectiveness of our approach against a diverse set of structured pruning
and distillation-based techniques, including state-of-the-art CoFi pruning,
competitive Block Movement Pruning, and distillation approaches including
TinyBERT, DistilBERT, DistilGPT2, MobileBERT, MiniLM, and DynaBERT.
Additionally, we include comparisons with other relevant methods. For
fairness, we follow [7] and report TinyBERT and DynaBERT results without data
augmentations. In the _post-training/one-shot_ setting, which does not allow
retraining, we demonstrate that ZipLM outperforms the prior state-of-the-art
approach of [49]. We evaluate inference speedups of all models in the same
environment, unless the models are not publicly available, in which case we
report speedups from their respective papers. We refer to ZipLM compressed
BERT models as ZipBERT, and to ZipLM compressed GPT2 models as ZipGPT2.
### 4.1 Gradual Structured Pruning
#### $\textrm{BERT}_{\small{\textrm{base}}}\,$results.
In Figure 2 we compare structured compression methods on the SQuADv1.1 task.
ZipLM outperforms both CoFi and TinyBERT, prior state-of-the-art techniques,
by 3 points in the F1 score at the same speedup factor, while at the same F1
score it is able to improve inference speedups by at least 60%. In Figure 3,
we extend this comparison to a subset of GLUE tasks and provide an exhaustive
overview of various structured compression techniques. Results on the other
four remaining GLUE tasks are provided in Appendix Figure 7. As can be
observed, distillation-based methods usually provide either one or a few
structurally-compressed models, due to the massive costs associated with
training from scratch for each new model. Relative to the most competitive
approaches, such as TinyBERT, CoFi, and MiniLM, ZipLM provides consistent
improvements in terms of both, accuracy and speedup, while providing
guarantees for each compressed model in terms of the expected speedup in the
target inference environment. Interestingly, on tasks like QQP and SST-2,
ZipLM is able to compress the $\textrm{BERT}_{\small{\textrm{base}}}\,$model
up to 6x and 10x speedups, respectively, while maintaining the accuracy of the
_dense_ model. In Appendix D, we provide additional comparisons against CoFi
on test-set results from the official GLUE evaluation server.
Figure 2: Structured compression of
$\textrm{BERT}_{\small{\textrm{base}}}\,$(left) and
$\textrm{BERT}_{\small{\textrm{large}}}\,$(right) on the SQuADv1.1 task.
Dashed horizontal lines represent full and 99% accuracy recovery of the
uncompressed model. Figure 3: Structured compression of
$\textrm{BERT}_{\small{\textrm{base}}}\,$on QNLI, MNLI, SST-2, and QQP tasks.
Dashed horizontal lines represent full and 99% accuracy recovery of the
uncompressed model.
#### $\textrm{BERT}_{\small{\textrm{large}}}\,$results.
To verify that our approach does not pertain only to the
$\textrm{BERT}_{\small{\textrm{base}}}\,$model, we apply ZipLM structured
pruning to the 3x larger $\textrm{BERT}_{\small{\textrm{large}}}\,$model on
the SQuADv1 task. In this setup, we compare against the only two approaches
that attempted to structurally compress this larger model, Block Movement
Pruning and distillation-based MobileBERT. As can be seen in Figure 2, ZipLM
is able to compress $\textrm{BERT}_{\small{\textrm{large}}}\,$up to 4x faster
inference while maintaining the F1 score of the uncompressed model. At the
same F1 score as the fastest Block Movement Pruning model (3x), ZipLM doubles
the inference speedup (6x). A result worth emphasizing is that ZipLM is even
able to match the performance of the highly optimized MobileBERT model by
simply compressing the baseline BERT architecture, without the many additional
optimizations and custom components used by MobileBERT. Specifically, some of
the module- and operator-level optimizations used by MobileBERT include:
bottleneck structures and carefully-balanced self-attention and feed-forward
modules, embedding layer factorization, a bespoke closed-source teacher model,
replacement of LayerNorm layers with lower-latency NoNorm layers, and
replacement of GELU activation functions with ReLU activations.
#### 99% recovery.
The MLPerf Benchmark [50] targets recovery of $>$99% of the baseline accuracy.
At this industry-defined threshold, ZipLM models set new state-of-the-art
performance across all of the considered datasets with the following
$\textrm{BERT}_{\small{\textrm{base}}}\,$inference speedups: 5x on the SQuADv1
task, 6x on QNLI and MNLI, and, surprisingly, 13x and 15x on SST-2 and QQP,
respectively. When compressing $\textrm{BERT}_{\small{\textrm{large}}}\,$on
the SQuADv1 task, ZipLM produces a 6x faster model at 99% recovery.
#### GPT2 results.
To validate that our approach does not only apply to encoder-based models, we
apply ZipLM structured pruning to the decoder-based GPT2 model. In addition to
this, to further demonstrate the inference-awareness property of our approach
and its importance for real-world applications, we consider two different
regimes: pruning for throughput and pruning for latency. An example
application for the former regime is a server-side deployment where the model
processes many queries at the same time, while an application for the latter
regime is a text-generation scenario where the model is used in an online
fashion to auto-complete user’s text.
For a fair comparison, we follow the DistilGPT2 setup [11] and prune the 124M
parameters GPT2 variant on the OpenWebTextCorpus dataset, followed by zero-
shot evaluations, without any fine-tuning, on the test-split of the WikiText
[51] dataset. Because of the enormous vocabulary size, the maximum achievable
speedup in the throughput regime for this model is roughly 3.5x. Thus, we run
ZipLM pruning to 1.5x, 2x, 2.5x, and 3x speedup targets. For the latency
regime, we report the median time to process sequences of various lengths when
generating text with Top-K sampling [52]. In Table 1, we present zero-shot
evaluations of the uncompressed GPT2 model which serves as a baseline relative
to the competing DistilGPT2 approach, and four variants of our ZipLM pruned
GPT2. In the pruning for throughput scenario, at similar speedup and decoder
size (1.6x-vs-1.5x and 42.5M-vs-47.3M), ZipGPT2 achieves significantly lower
perplexities relative to DistilGPT2. Further, at slightly better (lower)
perplexities, ZipGPT2 reduces the decoder size from 42.5M to only 26.5M
parameters (60% reduction) and improves speedup from 1.6x to 2.1x (30%
faster). In the pruning for latency scenario, at a similar speedup of
1.9x-vs-2.0x, ZipGPT2 reduces the decoder size by 3M params while providing
almost 2 points improvement in the zero-shot perplexity.
Table 1: Zero-Shot perplexity (PPL) of compressed GPT2 in two regimes: pruning
for throughput and pruning for latency. ∗GPT2 was trained by OpenAI [53] on a
much larger closed-source dataset and for significantly longer. The only
direct comparison is between DistilGPT2 and ZipGPT2.
Model | Pruning for throughput | Pruning for latency
---|---|---
Speedup | Decoder size | Wiki Text-103 PPL $\downarrow$ | Speedup | Decoder size | Wiki Text-103 PPL $\downarrow$
GPT2∗ | 1.0x | 85.0M | 28.5 | 1.0x | 85.0M | 28.5
DistilGPT2 | 1.6x | 42.5M | 43.0 | 1.9x | 42.5M | 43.0
ZipGPT2 (ours) | 1.5x | 47.3M | 35.4 | 1.6x | 48.7M | 37.8
2.1x | 26.5M | 41.5 | 2.0x | 39.2M | 41.2
2.7x | 14.0M | 50.4 | 2.2x | 26.6M | 49.0
3.3x | 15.7M | 72.1 | 2.5x | 20.7M | 55.0
Table 2: One-shot (post-training) structured pruning of
$\textrm{BERT}_{\small{\textrm{base}}}\,$on three downstream datasets and two
speedup targets.
Speedup | Kwon et al. [49] | ZipBERT base
---|---|---
SQuAD, F1 | |
1.5x 2.0x | 86.2 76.5 | 87.1 84.1
QQP, acc. | |
1.5x 2.0x | 89.5 83.9 | 89.7 84.8
MNLI, acc. | |
1.5x 2.0x | 82.8 78.1 | 83.0 78.2
### 4.2 On the Importance of Inference-Awareness
#### Depth vs. width pruning.
A particularly interesting illustration of the importance of inference-
awareness in the pruning algorithm is given by our GPT2 models running
directly in the PyTorch-HuggingFace framework, which can be used in two
different modes: batch-prediction (throughput-constrained) and text-generation
(latency-constrained). For the former, inputs are typically large, and
shrinking weight matrices is an effective way to achieve speedups. However,
for the latter, the inputs are much smaller, and the size of weight matrices
is no longer the primary bottleneck. In this scenario, the only way to achieve
substantial speedups is to completely drop some modules, which prior methods
cannot account for as they solely optimize for overall model sparsity.
However, with ZipLM, runtime measurements from the target inference
environment guide pruning decisions, allowing it to learn the best way to
compress the model for an optimal speedup-accuracy trade-off. Our GPT2
compression results in Table 1 clearly illustrate and support these
statements. Even though pruned for the same speedup target, the final
architectures of ZipGPT2 models are drastically different. For the throughput-
constrained scenario, the model’s depth was preserved but the matrix
dimensions were significantly reduced (roughly by a factor of 10) making the
corresponding multiplications with large input tensors much faster. In
contrast, for the latency-constrained scenario, the model’s width (shapes of
weight matrices) was mostly preserved but the depth was shrunk almost by a
factor of 4, making the forward pass with small inputs faster by reducing the
effective number of modules.
#### Inference device capabilities.
Incorporating capabilities of the inference device is another important aspect
for effective structured pruning which prior methods do not account for as
they solely optimize for higher sparsities. As noted in Section 3.2, this
reflects in larges discrepancies between speedups obtained on different
devices, e.g. a compressed model with 12x speedup on a V100 is only 5x faster
on an A100 GPU. This arises because the A100 GPU is significantly more
powerful and thus faster on the dense model; at the same time, it is highly
underutilized for small matrices, which significantly limits the speedups for
very high sparsity. To illustrate this, we have measured the speedup from
reducing the MLP size for both GPU types (see Table 3). As can be seen,
pruning to $\approx$90% sparsity (3072 $\rightarrow$ 302) gives $\approx$7x
speedup on a V100 but only $\approx$3x speedup on an A100. Such differences
are automatically captured by ZipLM, where pruning for sparsity is replaced by
pruning for speedup.
#### Pruning for speedup vs. pruning for sparsity.
In Figure 4 we compare results with ZipLM pruning when the target for pruning
is sparsity (like prior approaches) and when the target for pruning is speedup
(the ZipLM approach). Pruning for speedup brings significant improvements, up
to 10 points, especially at higher speedups where inference-awareness is very
important as the algorithm does not remove components that do not bring any
further speed and therefore helps preserving accuracy.
Figure 4: Ablation study for the impact of the pruning target: pruning for
sparsity (like prior approaches) versus pruning for speedup (the ZipLM
approach).
Table 3: Speedups from shrinking the intermediate size of MLPs in the FFN
section of a Transformer layer, on different GPUs.
| Speedup
---|---
MLP size | V100 | A100
3072 | 11.0x | 1.0x
1814 | 11.6x | 1.1x
1322 | 12.0x | 1.4x
1302 | 16.9x | 3.1x
1130 | 11.8x | 4.4x
1176 | 13.1x | 4.4x
1133 | 14.8x | 4.4x
### 4.3 Post-training/One-shot Structured Pruning
We now study the performance of ZipLM when applied purely in one-shot, without
any retraining. In this setting, we compare against the state-of-the-art
method of Kwon et al. [49] which combines several heuristics: Fisher-based
mask search, mask rearrangement, and mask tuning. Instead of heuristics, our
pruning framework utilizes direct end-to-end loss information to find the
optimal sparsity configuration. During the warm-start phase, [49] utilizes a
diagonal Fisher matrix to estimate the significance of heads and filters,
which discards correlations caused by off-diagonal elements. Although the
approach attempts to address this limitation by approximating correlations
within a single layer, it will not capture global dependencies. Furthermore,
the weights are adapted for layer-wise reconstruction at the very end of the
compression step, whereas our method does it continuously during the pruning
(please see Section 4 for the significance of doing this). For a fair
comparison, we apply the authors’ own implementation in latency-constrained
mode on the exact same model weights. Table 2 presents results on several
datasets and speedups, showing that ZipLM is even more accurate than the
approach designed and optimized specifically for the post-training/one-shot
pruning.
#### Sensitivity to calibration data.
Additionally, we have found that ZipLM is very robust to the amount of
calibration data. In Table 4 we present a sensitivity analysis with respect to
the number of calibration samples. We one-shot prune
$\textrm{BERT}_{\small{\textrm{base}}}\,$on the SQuADv1.1 task for two speedup
targets: 1.5x and 2.0x. In this setup, we compare results against Kwon et al.
[49], which uses 2048 samples by default. As can be seen from the table, ZipLM
outperforms prior state-of-the-art starting at only 32 samples. As we increase
the number of samples, the results improve, up to 2 points in F1 score.
## 5 Discussion and Extensions
#### CPU as an LLM-inference environment.
In Section 4 we have focused on various GPU-based inference environments as it
enabled us to conduct fair comparisons against prior structural compression
techniques. However, CPUs present another compelling inference environment
focused on edge deployment of LLMs. Therefore, we target the recently proposed
compound compression pipeline of [36], which involves three steps: structured
pruning, unstructured pruning, and quantization. We replace their structured
pruning approach based on layer dropping with ZipLM. As a result, at full
accuracy recovery, we are able to improve speedup from 3x to 13x, and at the
largest compression ratio from 30x to 50x. Due to space constraints, we
provide full results in Appendix A.
#### Computational efficiency.
Relative to distillation-based methods, structured pruning is an order of
magnitude more efficient in terms of GPU hours due to the massive costs
associated with pretraining from scratch for each compressed model [7, 9, 6].
For efficiency comparisons to CoFi, we consider the task of producing a full
family of compressed $\textrm{BERT}_{\small{\textrm{base}}}\,$models with
speedup targets ranging from 2x to 15x. In this setup, ZipLM requires only 115
epochs in total, whereas CoFi would require 560 epochs. Therefore, ZipLM is
_4.87 times more efficient_ than CoFi. In terms of end-to-end runtime, ZipLM
produces the entire family of compressed
$\textrm{BERT}_{\small{\textrm{base}}}\,$models on a single RTX A6000 GPU in
$\sim$35 hours on larger datasets (e.g. MNLI) and only $\sim$10 hours on
smaller ones (e.g. SST2). Finally, it is worth emphasizing that we have not
taken into account the cost of hyper-parameter tuning in the above
comparisons, but that this is very favorable to ZipLM: it uses a single set of
hyper-parameters to produce an entire family of compressed models while other
methods require hyper-parameter tuning for each model independently.
Figure 5: Scaling laws of structured pruning vs. distillation on the standard
BERT architecture.
Table 4: Sensitivity to the number of calibration samples.
| | F1 score at
---|---|---
Method | Num samples | 1.5x | 2.0x
ZipLM | 4 | 82.3 | 48.4
32 | 86.8 | 82.6
128 | 86.8 | 83.6
512 | 86.8 | 84.1
2048 | 87.1 | 84.1
4096 | 87.6 | 84.7
Kwon et al. | 2048 | 86.2 | 76.5
#### Scaling laws for structured pruning.
To further understand the accuracy-speedup trade-offs, we run ZipLM on larger
speedup ratios, up to 55x for $\textrm{BERT}_{\small{\textrm{large}}}\,$ and
75x for $\textrm{BERT}_{\small{\textrm{base}}}\,$. To the best of our
knowledge, this is the first result in literature demonstrating that such
extreme compression ratios are achievable with structured pruning without
model collapse. In Figure 5, we compare these results against distillation-
based downscaling of the BERT architecture [13]. The results clearly
demonstrate that each of the pruned models, based either on
$\textrm{BERT}_{\small{\textrm{large}}}\,$or
$\textrm{BERT}_{\small{\textrm{base}}}\,$, significantly outperforms
comparable pre-trained variants. An emergent behavior that can be observed is
that structurally pruned models tend to follow a linear scaling law, meaning
that the accuracy decreases linearly with the increase of the speedup ratio,
at a slope given by the original model. Fitting linearly via least squares
produces the following expressions for the accuracy-speedup relationship:
$\texttt{F1}_{\texttt{large}}\approx
92.1-0.3\times\texttt{speedup}_{\texttt{large}}$, and
$\texttt{F1}_{\texttt{base}}\approx
90.3-0.6\times\texttt{speedup}_{\texttt{base}}$. Thus, the rate of decrease in
accuracy for $\textrm{BERT}_{\small{\textrm{base}}}\,$is twice as large as
that of $\textrm{BERT}_{\small{\textrm{large}}}\,$, which can be attributed to
the presence of more redundant representations in the larger model, making it
more resilient to pruning. In Appendix G we provide additional analysis of the
structure of pruned models.
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## Appendix A Compound Compression for Edge Deployment
Deploying large language models in edge environments requires running
inference on low-power devices such as CPUs. Therefore, we follow the compound
compression approach from [36] which bundles together structured, unstructured
pruning, and quantization for efficient inference on CPUs. We start with ZipLM
structurally pruned models, and apply on top the state-of-the-art oBERT
unstructured pruning method [36] to 80% sparsity. After structured and
unstructured pruning, we apply quantization-aware-training (QAT) [54] to
quantize FP32 weights into INT8 representations. We benchmark these compound
compressed models by running inference in the DeepSparse [55] engine, on a
single-core of Intel Cascade Lake CPU. In this setting, we compare our results
against the compound compression pipeline of [36] which applies layer dropping
as a form of structured pruning. As can be seen from Figure 6, when we
substitute layer dropping with a principled structured pruning via ZipLM, the
resulting compound compressed models achieve very competitive latency-vs-
accuracy performance in the edge-inference regime. At full accuracy recovery,
ZipLM improves the speedup from 3x to 13x, while at the largest compression
ratio ZipLM improves the speedup from 30x to 50x.
Figure 6: Improvements in CPU-inference speedups for compound compressed
$\textrm{BERT}_{\small{\textrm{base}}}\,$models on the SQuADv1.1 task when
ZipLM is used for structured pruning. End-to-end latency indicated by the
dashed line.
## Appendix B Ablation Studies
In Table 5, we present ablation results for ZipLM and CoFi, with and without
their respective layer-wise distillation techniques. ZipLM outperforms CoFi in
all tasks when both methods use distillation, and in three out of four when
distillation is not used. For example, ZipLM outperforms CoFi with a
significant 3 point increase in F1 score on the SQuAD task in both setups.
Furthermore, when comparing ZipLM results with and without layer-wise
distillation, it can be observed that benefits are pronounced for low data
tasks, where accuracy improvements reach up to 2 points.
Table 5: Comparison of ZipLM and CoFi dev-set results, with and without layer-wise distillation. | SST-2 acc. | QNLI acc. | MNLI m-acc. | SQuAD F1
---|---|---|---|---
CoFi | 90.4 | 86.1 | 80.6 | 82.6
$\textrm{ZipBERT}_{\small{\textrm{base}}}$ | 91.7 | 88.6 | 81.7 | 85.7
CoFi w/o $\mathcal{L}_{\mathrm{layer}}$ | 91.1 | 85.1 | 79.7 | 82.5
$\textrm{ZipBERT}_{\small{\textrm{base}}}$ w/o $\mathcal{L}_{\mathrm{token}}$ | 89.2 | 86.5 | 81.2 | 85.7
## Appendix C Additional GLUE results
Due to space constraints, in Figure 3 we present results only on four GLUE
tasks. Therefore, for completeness, in Figure 7 we present results on the
remaining four GLUE tasks, namely: CoLA, MRPC, STS-B, and RTE. Results show
the same trends as the other four GLUE tasks, with large improvements for
ZipLM, especially at higher compression rates.
Figure 7: Structured compression of
$\textrm{BERT}_{\small{\textrm{base}}}\,$on CoLA, MRPC, STS-B, and RTE tasks.
Dashed horizontal lines represent full accuracy recovery of the uncompressed
model.
## Appendix D Additional Validation
Evaluating and comparing compressed models on the development set (dev-set) is
standard practice, as it enables comparisons with off-the-shelf results from
the literature. However, an implicit assumption behind such comparisons is
that all methods tune their hyper-parameters only on a subset of the dev-set
before evaluating and reporting results on all samples, which is not always
the case. Moreover, specifically-tuned hyper-parameters can lead to large
performance differences, especially when compressing LLMs [56]. To ensure that
there is no such “overfitting” on the dev-set, in Table 6 we compare ZipLM
against the prior state-of-the-art CoFi approach on _unseen test-set_ ,
obtained by submitting predictions to the official GLUE evaluation server. The
results show consistent improvements over CoFi, on both dev- and test-sets.
Table 6: Dev- and test-set comparison of $\textrm{ZipBERT}_{\small{\textrm{base}}}$ and CoFi models with comparable speedups. | dev-set | test-set
---|---|---
| CoFi | $\textrm{ZipBERT}_{\small{\textrm{base}}}$ | CoFi | $\textrm{ZipBERT}_{\small{\textrm{base}}}$
QNLI, acc. | 86.1 | 88.6 | 85.8 | 88.4
SST-2, acc. | 90.4 | 91.7 | 88.2 | 91.8
MNLI, m-acc. | 80.6 | 81.7 | 80.7 | 81.9
MNLI, mm-acc. | 80.7 | 82.0 | 79.9 | 80.6
SQuAD, F1 | 82.6 | 85.7 | N/A | N/A
## Appendix E Latency table used for ZipLM pruning
As described in Section 3.2, we record the time to run an attention block,
including all overheads, with $0,\dots,N_{\text{heads}}-1$ heads pruned
(pruning everything with runtime 0 is also considered) and similarly for the
fully-connected block with the intermediate dimension shrunk by a factor of
$0.9^{i}$, for $i=0,\dots,42$; in relative steps of $10\%$ up until $\approx
99\%$ sparsity, following [40]. In Table 7 we present an example of such a
latency table used in ZipLM pruning approach.
Table 7: An example of the latency table used by the ZipLM pruning approach. Intermediate size | Latency (ms) | Number of heads | Latency (ms)
---|---|---|---
3072 | 11.9 | 12 | 7.9
1814 | 7.4 | 10 | 6.7
1322 | 5.8 | 8 | 5.8
302 | 1.6 | 6 | 4.4
130 | 1.0 | 4 | 3.2
76 | 0.9 | 2 | 1.9
33 | 0.7 | 0 | 0
## Appendix F Speedup Evaluations
As shown in Figure 1, ZipLM is based on measuring runtimes of higher-level
modules, such as attention heads and fully connected matrices, rather than
low-level operators. This makes our approach independent of underlying
optimizations in different inference engines and frameworks, which usually
perform further optimizations such as operator-fusion. Our runtime lookup
table contains information about the runtime of a Transformer layer with
different numbers of attention heads, and various dimensions of the fully
connected matrices. This implies that we measure runtimes of a layer with 12
heads, 11 heads, 10 heads, and so on, as well as the runtimes of fully
connected matrices with hidden sizes ranging from 3072 to 0. We utilize this
information to guide pruning decisions.
To fully validate the ability of ZipLM to compress the model while satisfying
desired speedup constraints via the described approach, we provide the timing
results in Table 8, comparing the desired (target) speedup and the achieved
(measured) speedup for different models.
Table 8: Comparison of target (desired) inference speedups with achieved (on-device measured) speedups obtained with our ZipLM pruning approach. $\textrm{BERT}_{\small{\textrm{base}}}\,$on SQuADv1.1 | $\textrm{BERT}_{\small{\textrm{large}}}\,$on SQuADv1.1
---|---
Target speedup | Achieved speedup | Deviation | Target speedup | Achieved speedup | Deviation
2 | 1.98 | -1.00% | 2 | 2.01 | +0.50%
4 | 4.05 | +1.25% | 4 | 4.05 | +1.25%
6 | 6.16 | +2.67% | 6 | 6.09 | +1.50%
8 | 8.25 | +3.12% | 8 | 8.27 | +3.37%
10 | 10.36 | +3.60% | 10 | 10.33 | +3.30%
12 | 12.31 | +2.58% | 12 | 12.46 | +3.83%
14 | 14.33 | +2.35% | 14 | 14.74 | +5.28%
As can be seen from the Table 8, the deviation between the desired (target)
and the achieved (measured) speedup is at most 5.28%. This confirms that our
approach indeed provides reliable runtime information to guide the pruning
decisions.
## Appendix G Structure of Pruned Models
Through a comprehensive examination of ZipLM pruned BERT models across all
datasets considered in Section 4, we aim to identify trends in the pruning of
key components of the Transformer layer, namely attention heads and
intermediate size, needed to achieve a specific speedup target. As illustrated
in Figure 8, we observe that the intermediate size is pruned at a higher rate
relative to attention heads, which aligns with the fact that the intermediate
size dictates the dimensions of the two large linear layers in the feed-
forward part of the Transformer block. For instance, to attain a 2x speedup,
roughly 60% of the intermediate size and 40% of the attention heads need to be
removed. Additionally, in Figure 9, we visualize the entire encoder size
needed to reach a specific speedup target. Interestingly, we find that 15x
faster models retain on average only 2% of intermediate size and 6% of
attention heads which amounts to only 2.9M parameters overall, while at the
same time recovering more than 95% of the uncompressed model’s accuracy (see
Figure 3).
Figure 8: Percentage of pruned attention heads and intermediate size to reach
a specific speedup target with ZipLM. Figure 9: Encoder size vs. speedup
factor of ZipLM pruned $\textrm{BERT}_{\small{\textrm{base}}}\,$models,
averaged over all considered datasets in Section 4.
Additionally, in Figures 10, 11, 12, 13 we visualize the number of remaining
heads and intermediate size across all Transformer layers and various speedup
targets on a subset of GLUE datasets.
Figure 10: Remaining number of attention heads and intermediate size across
all layers of the ZipLM compressed
$\textrm{BERT}_{\small{\textrm{base}}}\,$model at various speedups and MNLI
dataset.
Figure 11: Remaining number of attention heads and intermediate size across
all layers of the ZipLM compressed
$\textrm{BERT}_{\small{\textrm{base}}}\,$model at various speedups and QNLI
dataset.
Figure 12: Remaining number of attention heads and intermediate size across
all layers of the ZipLM compressed
$\textrm{BERT}_{\small{\textrm{base}}}\,$model at various speedups and QQP
dataset.
Figure 13: Remaining number of attention heads and intermediate size across
all layers of the ZipLM compressed
$\textrm{BERT}_{\small{\textrm{base}}}\,$model at various speedups and SST-2
dataset.
## Appendix H Experiments - Additional Results
In Table 9 we report accuracy and model size of ZipLM pruned models visualized
in Section 4, in Figures 2 and 3.
Table 9: Accuracy and model size for ZipLM pruned models in Section 4. | $\textrm{BERT}_{\small{\textrm{base}}}\,$ | $\textrm{BERT}_{\small{\textrm{large}}}\,$
---|---|---
| QNLI | MNLI | SST2 | QQP | SQuADv1 | SQuADv1
Speedup | Acc. | Encoder size (M) | Acc. | Encoder size (M) | Acc. | Encoder size (M) | Acc. | Encoder size (M) | F1 | Encoder size (M) | F1 | Encoder size (M)
32x | 91.4 | 38.0 | 84.8 | 38.5 | 93.4 | 38.7 | 91.3 | 37.8 | 89.1 | 37.3 | 91.6 | 141.1
33x | 91.1 | 23.8 | 84.8 | 23.5 | 93.4 | 24.1 | 91.3 | 23.8 | 88.6 | 23.4 | 91.4 | 188.3
34x | 90.9 | 16.9 | 84.0 | 17.1 | 93.0 | 17.2 | 91.3 | 16.8 | 88.0 | 16.8 | 91.1 | 163.1
35x | 90.8 | 12.5 | 84.0 | 13.5 | 93.0 | 13.5 | 91.1 | 13.0 | 87.5 | 13.0 | 90.8 | 148.5
36x | 90.4 | 39.5 | 83.5 | 10.5 | 93.0 | 11.0 | 91.1 | 10.2 | 86.7 | 10.4 | 90.2 | 139.1
37x | 89.8 | 38.0 | 83.2 | 38.8 | 93.0 | 39.0 | 90.9 | 38.1 | 86.1 | 38.7 | 89.9 | 132.7
38x | 89.2 | 36.4 | 83.1 | 37.5 | 93.0 | 37.6 | 90.9 | 36.8 | 85.7 | 37.5 | 89.7 | 127.5
39x | 89.1 | 35.7 | 82.8 | 36.3 | 93.0 | 36.7 | 90.8 | 35.8 | 85.3 | 36.2 | 89.3 | 123.8
10x | 88.6 | 34.9 | 82.7 | 35.4 | 93.0 | 35.7 | 90.8 | 34.9 | 84.2 | 35.3 | 89.1 | 120.9
11x | 88.6 | 34.0 | 82.5 | 34.7 | 92.7 | 34.9 | 90.7 | 34.3 | 83.8 | 34.7 | 88.8 | 118.4
12x | 87.8 | 33.6 | 81.7 | 34.1 | 91.7 | 34.2 | 90.6 | 34.1 | 83.2 | 34.0 | 88.4 | 116.4
13x | 87.6 | 33.2 | 81.3 | 33.5 | 91.7 | 33.8 | 90.6 | 33.7 | 82.5 | 33.4 | 87.9 | 114.9
14x | 87.4 | 32.8 | 81.2 | 33.3 | 91.7 | 33.6 | 90.3 | 33.3 | 81.7 | 33.2 | 87.7 | 113.7
15x | 87.2 | 32.6 | 80.8 | 32.9 | 90.7 | 33.2 | 90.3 | 32.9 | 81.4 | 32.9 | 87.6 | 112.5
## Appendix I Hyper-parameters for Reproducibility
To facilitate reproducibility, we conduct experiments in the open-source
Transformers library [57], and use publicly available datasets [58]. We plan
to open-source our entire framework which supports one-shot and gradual
structured pruning via SparseML [59], making it very easy to experiment with
other models and datasets. In addition to our code, we plan to open-source all
of our compressed models via the popular HuggingFace Hub. In Table 10 we
report hyper-parameters used to produce our ZipLM pruned models in Section 4.
Because of the excessive memory overhead, we don’t make use of any kind of
knowledge distillation when pruning the GPT2 model. Following insights from
DistilGTP2, we hypothesize that this can further improve our results. We
follow [60] and disable dropout regularization while pre-training ZipGPT2
models at OpenWebTextCorpus dataset.
Table 10: Hyper-parameters used for gradual ZipLM runs in Section 4. | $\textrm{BERT}_{\small{\textrm{base}}}\,$ | $\textrm{BERT}_{\small{\textrm{large}}}\,$ | GPT2
---|---|---|---
batch-size | 16 SQuADv1 32 GLUE | 128
max-seq-length | 384 SQuADv1 128 GLUE | 1024
finetune before pruning | 3 epochs | 50k steps
finetune in-between pruning steps | 8 epochs | 10 epochs | 2 epochs
LR schedule in-between pruning steps | linear decay | linear decay
initial LR | 8e-5 | 5e-5 | 1e-3
#calibration samples | 2048 | 512
speedup-targets | {2, 3, 4, 5, …, 15}x | {1.5, 2, 2.5, 3}x
knowledge distillation $\lambda_{1}$ | 0 | 1.0
knowledge distillation $\lambda_{2}$ | 1.0 SQuADv1 0.5 GLUE | 0
knowledge distillation $\lambda_{3}$ | 0.0 SQuADv1 0.5 GLUE | 0
weight-decay | 0.03 | 0.05 | 0
## Appendix J Broader Impact and Limitations
Our results contribute to the line of work on efficient language models. Thus,
it should help reduce the energy and monetary cost of inference over such
models, and allow them to be used without access to powerful hardware. While
this is a mainly positive outcome, it also reduces the cost of employing these
models for detrimental purposes, such as spam generation. Thus, this
significant cost reduction for inference should also be seen as further
motivation for methods to ensure safe usage of these models, such as
watermarking or alignment.
As any academic study, our work is not without its limitations. All of our
benchmarks are focused on English-language datasets and therefore our results
do not provide insights into compression effects for low-data languages.
Unfortunately, this limitation is inherent to all of the existing works on
compression due to the lack of standardized benchmarks. Given that our
structured pruning approach relies on a small sample of calibration data to
perform pruning decisions, we hypothesize that our approach should be able to
provide satisfying results in the low-data setup as well. At the moment we do
not have data to support these claims, but we see it as an opportunity for
future work.
|
# Cross-Validated Decision Trees with Targeted Maximum Likelihood Estimation
for Nonparametric causal mixtures analysis
David B. McCoy
Department of Environmental Health Sciences
University of California, Berkeley
Berkeley, CA 94704
<EMAIL_ADDRESS>
& Alan E. Hubbard
Department of Biostatistics
University of California, Berkeley
Berkeley, CA 94704
<EMAIL_ADDRESS>
& Alejandro Schuler
Department of Biostatistics
University of California, Berkeley
Berkeley, CA 94704
<EMAIL_ADDRESS>
& Mark J. van der Laan
Department of Biostatistics
University of California, Berkeley
Berkeley, CA 94704
<EMAIL_ADDRESS>
###### Abstract
Exposure to a mixture of chemicals such as drugs, pollutants and nutrients
occur in most realistic exposure or treatment situations. We can imagine that,
within this exposure space, there are arbitrary regions wherein we can measure
the covariate adjusted outcome within the region compared to the complimentary
exposure space. Ideally, it is most useful to estimate a causal estimand that
maximizes this mean difference. A statistical estimator that identifies
regions which maximize this difference and delivers the relevant effect
unbiasedly would be valuable to public health efforts which aim to understand
what levels of pollutants or drugs have the strongest effect. This estimator
would take in as input a vector of exposures $A$ which can be a variety of
data types (binary, multinomial, continuous), a vector of baseline covariates
$W$ and outcome $Y$ and outputs 1. a region of the exposure space that
attempts to optimize this maximum mean difference and 2. an unbiased estimate
of the causal effect comparing the average outcomes if every unit were forced
to self-select exposure within that region compared to the equivalent space
outside of that region (the average regional-exposure effect (ARE)). Rather
than the region of interest being of arbitrary shape, searching for
rectangular regions is most helpful for policy implications by helping
policymakers decide on what combination of thresholds to set on allowable
combinations of exposures. This is because rectangular regions can be
expressed as a series of thresholds, for example, $A_{1}\geq a_{1}$ while
$A_{2}\leq a_{2}$ where $a_{1},a_{2}$ are specific doses of exposures
$A_{1},A_{2}$ respectively. Non-parametric methods such as decision trees are
a useful tool to evaluate combined exposures by finding partitions in the
joint-exposure (mixture) space that best explain the variance in an outcome.
We present a methodology for evaluating the causal effects of a data-
adaptively determined mixture region using decision trees. This approach uses
K-fold cross-validation and partitions the full data in each fold into a
parameter-generating sample and an estimation sample. In the parameter-
generating sample, decision trees are applied to a mixed exposure to obtain
the region and estimators for our statistical target parameter. The region
indicator and estimators are then applied to the estimation sample where the
average regional-exposure effect is estimated. Targeted learning is used to
update our initial estimates of the ARE in the estimation sample to optimize
bias and variance towards our target parameter. This results in a plug-in
estimator for our data-adaptive decision tree parameter that has an
asymptotically normal distribution with minimum variance from which we can
derive confidence intervals. Likewise, our approach uses the full data with no
loss of power due to sample splitting. The open source software, CVtreeMLE, a
package in R, implements our proposed methodology. Our approach makes possible
the non-parametric estimation of the causal effects of a mixed exposure
producing results that are both interpretable and asymptotically efficient.
Thus, CVtreeMLE allows researchers to discover important mixtures of exposure
and also provides robust statistical inference for the impact of these
mixtures.
## 1 Introduction
In most environmental epidemiology studies, researchers are interested in how
a joint exposure affects an outcome. This is because, in most real world
exposure settings, an individual is exposed to a multitude of chemicals
concurrently or, a mixed exposure. Individuals are exposed to a range of
multi-pollutant chemical exposures from the environment including air
pollution, endocrine disrupting chemicals, pesticides, and heavy metals.
Because many of these chemicals may affect the same underlying biological
pathway which lead to a disease state, the toxicity of these chemicals can be
modified by simultaneous or sequential exposure to multiple agents. In these
mixed exposure settings, the joint impact of the mixture on an outcome may not
be equal to the additive effects of each individual agent. Mixed exposures may
have impacts that are greater than expected given the sum of individual
exposures or effects may be less than additive expectations if certain
exposures antagonize the affects of others. Likewise, the effects of a mixed
exposure may be different for subpopulations of individuals based on
environmental stressors, genetic, and psychosocial factors that may modify the
impact of a mixed exposure. Safe (1993); Kortenkamp (2007)
Causal inference of mixed exposures has been limited by reliance on parametric
models and, in most cases, by researchers considering only one exposure at a
time, usually estimated as a coefficient in a generalized linear regression
model (GLM). This independent assessment of exposures poorly estimates the
joint impact of a collection of the same exposures in a realistic exposure
setting. Given that most researchers simply add individual effects to estimate
the joint impact of an exposure, it is almost certain that the current
evidence of the total impact environmental toxicants have on chronic disease
is incorrectly estimated. The impact of using linear modeling is not limited
to just potential bias: in the case where linearity does not hold, it’s not
even clear what is being estimated.
The limitation in effective estimation of the joint effects of mixed exposure
is (in-part) due to the lack of robust statistical methods. There has been
some method development for estimation of joint effects of mixed exposures,
such as Weighted Quantile Sum Regression Keil et al. (2019), Bayesian Mixture
Modeling De Vocht et al. (2012), and Bayesian Kernel Machine Regression Bobb
et al. (2014). However, these mixture methods have strong assumptions built
into them, including directional homogeneity (e.g. all mixtures having a
positive effect), linear/additive assumptions and/or require information
priors. Many methods suffer from human bias due to choice of priors or poor
model fit. More flexible models remain more or less a black-box and describe
the mixture through a series of plots rather than with an interpretable
summary statistic Bobb et al. (2014). Given that the National Institute for
Environmental Health Sciences (NIEHS) has included the study of mixtures as a
key goal in its 2018-2023 strategic plan National Institute of Environmental
Health Sciences (2018) (NIEHS), it is imperative to develop new statistical
methods for mixtures that are less biased, rely less on human input, use
machine learning (ML) to model complex interactions, and are designed to
return an interpretable parameter of interest.
Decision trees are a useful tool for outcome prediction based on exposures
because they are fast, nonparametric (i.e. can discover and model interaction
effects), and interpretable Leo Breiman (1984). However, it is not immediately
clear how to adapt outcome prediction methods to inference about the effect of
some kind of hypothetical intervention on the mixture of exposures- especially
because in these settings we don’t have a particular intervention in mind.
Rather than leveraging decision trees for a simple prediction model, we
introduce a target parameter on top of the prediction model, which is the
average outcome within a fixed region of the exposure space. When an ensemble
of decision trees is applied to an exposure mixture, this coincides with a
leaf in the best fitting decision tree. By cross-estimating the average
outcome given exposure to this region which maximizes the outcome difference
we are able to build an estimator that is asymptotically unbiased with the
smallest variance for our causal parameter of interest. Previous work, in the
most naive approach, confidence intervals (CI) and hypothesis testing of
decision trees is done by constructing a $(1-\alpha)\times 100$% confidence
interval for a node mean $\bar{y}_{t}$ as
$\bar{y}_{t}\pm±z_{1-\alpha/2}(\frac{s_{t}}{\sqrt{n_{t}}})$ where
$\bar{y}_{t}$ is the node mean and $s_{t}$ is the standard deviation estimates
in the node. Of course, these CI intervals tend to be overly optimistic
because 1. decision trees are adaptive and greedy algorithms, meaning that
they have a tendency to overfit and 2. the target parameter, in this case the
node average, is estimated on the same data by which the node was created.
Because of this the estimated CIs are too narrow. The best approach is to use
an independent test set to derive inference for the expected outcome in each
leaf. However, this approach is costly if additional data is gathered or power
is greatly reduced if sample-splitting is done. Sampling splitting is done in
previous work for causal inference of decision trees using so-called "honest
estimation" for estimation of heterogeneous causal effects of a binary
treatment. This approach Athey and Imbens (2016) uses one part of the data for
constructing the partition nodes and and and another for estimating effects
within leaves of the partition. Our proposed approach follows a similar
sample-splitting technique where one part of the data is used to determine the
partition nodes and the other is used to estimate the parameter of interest;
however, we extend this technique to K-fold cross-validation where we rotate
through the full data. Additionally, rather than estimating heterogenous
treatment effects, we are interested in mapping a set of exposures that are of
a variety of data types (continuous, binary, multinomial) into a set of
partitioning rules using the best fitting decision tree from which we can
estimate the average regional-exposure effect, or the expected outcome
difference if all individuals were exposed to an exposure region compared to
if no individuals were exposed to this region.
In most research scenarios, the analyst is interested in causal inference for
an a priori specified treatment or exposure. However, in the evaluation of a
mixed exposure it is not known what mixture components, levels of these
components and combinations of these component levels contribute to the most
to a change in the outcome. In the ideal scenario, the analyst has knowledge
of the full, multidimensional dose-response curve $E[Y(A_{1},A_{2},...A_{k})]$
where $A$ are the exposures and $Y$ is the outcome. However, even in this
case, it is difficult to estimate and/or interpret this curve. Estimation is
hard because 1. we need unrealistic assumptions to get identifiability for the
full curve and 2. the curve isn’t pathwise differentiable which means there
aren’t any robust methods to build confidence intervals. Therefore, a sensible
approach is to instead categorize the joint exposure and compare averages
between categories as one would for a binary exposure. This approach is
helpful because we can define interpretable categories like
$(A_{1}>a_{1})\&(A_{2}<a_{2})$ where $a_{i}$ are specific values in $A$ (vs
complement of this space) which are of clear interest to policymakers.
Identification assumptions are also more transparent in this setting. However,
we don’t know a priori what the right categorization of the exposure space are
given some objective function. We have to use the data to tell us what regions
are determined given a predefined objective function. In our case, we want a
categorization that shows a maximal mean difference in outcomes. Regression
trees are a nice way to do this while respecting the fact that we want
interpretable rules like the above. The idea is to fit a kind of decision tree
to figure out what thresholds in the exposure space produce a maximal exposure
effect. As discussed, the result can be biased if we use the same data to
define the thresholds and to estimate the effects in each leaf. We solve that
problem by splitting the data, doing threshold estimation in one part and
regional-exposure effect estimation (given the fixed thresholds) in the other.
We can even redo the splits in a round-robin fashion (K-fold cross-validation)
to efficiently use all of the data. Lastly, once we have thresholds, we want
to get the best possible inference for the effect. We could always do a
difference in outcome means between the samples in each category/region, but
that estimate would be 1. biased by confounding and 2. have a large confidence
interval because we haven’t used covariates to soak up residual variance. Our
approach is thus to use a doubly-robust efficient estimator (TMLE) that
simultaneously addresses both these problems.
Building on prior work related to data-adaptive parameters Hubbard et al.
(2016) and cross-validated targeted minimum loss-based estimation (CV-TMLE)
Zheng and van der Laan (2010), our method, called CVtreeMLE, is a novel
approach for estimating the joint impact of a mixed exposure by using CV-TMLE
which guarantees consistency, efficiency, and multiple robustness despite
using highly flexible learners (ensemble machine learning) to estimate a data-
adaptive parameter. CVtreeMLE summarizes the effect of a joint exposure on the
outcome of interest by first doing an iterative backfitting procedure, similar
to general additive models Hastie and Tibshirani (1990), to fit $f(A)$, a
Super Learner van der Laan Mark et al. (2007) of decision trees, and $h(W)$,
an unrestricted Super Learner, in a semi-parametric model;
$E(Y|A,W)=f(A)+h(W)$, where $A$ is a vector of exposures and $W$ is a vector
of covariates. In many public health settings, the analyst is first interested
in a parsimonious set of thresholds focusing on the exposure space that best
explains some outcome across the whole population rather partitions that also
include baseline covariates. This additive model approach allows us to
identify partitioning nodes in the exposure space while flexibly adjusting for
covariates. In this way, we can data-adaptively find the best fitting decision
tree model which has the lowest cross-validated model error while flexibly
adjusting for covariates. This procedure is done to find partitions in the
mixture space which allows for an interpretable mixture contrast parameter,
"What is the expected difference in outcomes if all individuals were exposed
to this region of the mixed exposure vs. if no individuals were exposed?".
This approach easily extends to marginal case (partitions on individual
exposures) as well. Our approach for integrating decision trees as a data-
adaptive parameter with cross-validated targeted minimum loss-based estimation
(CV-TMLE) allows for flexible machine learning estimators to be used to
estimate nuisance parameter functionals while preserving desirable asymptotic
properties of our target parameter. We provide implementations of this
methodology in our free and open source software CVtreeMLE package, for the R
language and environment for statistical computing [R Core Team, 2022].
This manuscript is organized as follows, in Section 2.1 we give a background
of semi-parametric methodology, in section 2.2 we discuss the ARE target
parameter for a fixed exposure region and in 2.3 the assumptions necessary for
our statistical estimate to have a causal interpretation. In section 3 we
discuss estimation and inference of the ARE for a fixed region. In section 4
we discuss data-adaptively determining the region which maximizes the
W-controlled mean outcome difference. In section 5 we show how this requires
cross-estimation which builds from 2.2 for a fixed region ARE. In section 5 we
expand this to cross-estimation to k-fold CV and discuss methods for pooling
estimates across the folds. Lastly, in section 5.3, because we may have
different data-adaptively identified regions across the CV folds, we discuss
the union rule which pairs with the pooled estimates. In section 6 we discuss
simulations with two and three exposures and show our estimator is
asymptotically unbiased with a normally sampling distribution. In section 7 we
apply CVtreeMLE to the NIEHS mixtures workshop data and identify interactions
built into the synthetic data. In section 7.1 we compare CVtreeMLE to the
popular quantile sum g-computation method. In section 7.2 we apply CVtreeMLE
to NHANES data to determine if there is association between mixed metals and
leukocyte telomere length. Section 8 describes our CVtreeMLE software. We end
with a brief discussion of the CVtreeMLE method in Section 9.
## 2 The Estimation Problem
### 2.1 Setup and Notation
Our setting is an observational study with baseline covariates
($W\in\mathbb{R}^{p}$), multiple exposures ($A\in\mathbb{R}^{m}$), and a
single-timepoint outcome ($Y$). Let $O=(W,A,Y)$ denote the observable data. We
presume that there exists a potential outcome function $Y(a)$ (i.e. $Y(a)$ is
a random variable for each value of $a$) that generates the outcome that would
have obtained for each observation had exposure been forced to the value
$A=a$. These potential outcomes are unobserved but the observed outcome $Y$
corresponds to the potential outcome for the observed value $A$ of the
exposure, i.e. $Y=Y(A)$. Let $E[Y(a)|W=w]=\mu(a,w)$ denote the causal dose-
response curve for observations with covariates $w$ so that $E[\mu(a,W)]$
represents the average outcome we would observe if we forced treatment to $a$
for all observations.
We use $P_{0}$ to denote the data-generating distribution. That is, each
sample from $P_{0}$ results in a different realization of the data and if
sampled many times we would eventually learn the true $P_{0}$ distribution. We
assume our $O_{1},O_{2},...,O_{n}$ are iid draws of $O=(W,A,Y)\sim P_{0}$. We
decompose the joint density as
$p_{Y,A,W}(y,a,w)=p_{Y|A,W}(y,a,w)p_{A|W}(a,w)p_{W}(w)$ and make no
assumptions about the forms of these densities.
Compare this to to many methods which assume a parametric model which is one
where each probability distribution $P\in\mathcal{M}$ can be uniquely
described with a finite-dimensional set of parameters. Many methods assume $O$
to be identically distributed normal random variables, which means that the
model can be described by the mean and standard deviation. Models like GLMs
assume normal-linear relationships and assume
$Y=X\beta+\mathcal{N}(\mu,\sigma^{2})$. Thus, methods for mixtures that use
this approach have three parameters: the slope $\beta$, mean $\mu$ and
standard deviation $\sigma$ of the normally-distributed random noise. This
model assumes that the true relationship between $X$ and conditional mean of
$Y$ is additive and linear and that the conditional distribution of $Y$ given
$X$ is normal with a standard deviation that is fixed and doesn’t depend on
$X$. Of course, these are very strict assumptions especially in the case of
mixtures where exposures from a common source may be highly correlated, may
interact on the outcome in a non-additive way, and may have non-normal
distributions. As such, simply adding coefficients attached to variables in a
mixture to estimate the overall joint effect may be biased.
Our statistical target parameter, $\Psi(P_{0})$, is defined as a mapping from
the statistical model, $\mathcal{M}$, to the parameter space (i.e., a real
number) $\mathbb{R}$. That is, $\Psi$: $\mathcal{M}\rightarrow\mathbb{R}$. We
can think of this as, if $\Psi$ were given the true distribution $P_{0}$ it
would provide us with our true estimand of interest.
We can think of our observed data $(O_{1}\dots O_{n})$ as a (random)
probability distribution $P_{n}$ that places probability mass $1/n$ at each
observation $O_{i}$. Our goal is to obtain a good approximation of the
estimand $\Psi$, thus we need an estimator, which is an a-priori specified
algorithm that is defined as a mapping from the set of possible empirical
distributions, $P_{n}$ to the parameter space. More concretely, the estimator
is a function that takes as input the observed data, a realization of $P_{n}$,
and gives as output a value in the parameter space, which is the estimate,
$\hat{\Psi}(P_{n})$. Since the estimator $\hat{\Psi}$ is a function of the
empirical distribution $P_{n}$, the estimator itself is a random variable with
a sampling distribution. So, if we repeat the experiment of drawing $n$
observations we would every time end up with a different realization of our
estimate. We would like an estimator that is provably unbiased relative to the
true (unknown) target parameter and which has the smallest possible sampling
variance so that our estimation error is as small as it can be on average.
### 2.2 Defining the Differential Effect Given Regional Exposure
In problems with binary treatment $A\in\\{0,1\\}$ the standard counterfactual
model defines potential outcomes $Y(0)$ and $Y(1)$ describing what would
happen to each individual had they been forced onto either treatment. The
estimand of interest is most often the average treatment effect
$E[Y(1)]-E[Y(0)]$.
In our setting $A\in\mathbb{R}^{m}$ is continuous and we must thus define a
potential outcome $Y(a)$ for each of the infinite possible values of the
exposure. There is no singular, obvious “effect of treatment”.
In this work, we propose asking about the differential policy effect of
allowing self-selection within a region of the exposure space. For example, we
may be interested in the effect of a law that limits arsenic soil levels to a
certain level while also limiting cadmium to a different level. In this
setting, communities are still left to “self-select” exposures, but they must
conform to the legal limits. In order to estimate such an effect, we have to
specify how each individual would modify their “preference” for each exposure
level when presented with a more limited set of choices. In this work we
presume that relative self-selection preferences are preserved. For example,
if a person were twice as likely to choose level $a_{1}$ as they were to
choose $a_{2}$, that relative preference would continue to hold whether or not
$a_{3}$ were a legal option.111 Another way of looking at this is to define a
random variable $T$ that represents the activation of the policy (e.g. law)
that nominally forces the exposures into the desired region $\mathcal{A}$ and
ask about the ATE of $T$ on $Y$. In this view, we continue by presuming that
$T$ only affects $Y$ through the ”potential exposures” $A(t)$ and that the
relationship between counterfactual exposure levels $A(0)$ and $A(1)$ is:
$p_{A(1)|W}(a,w)=\frac{1_{\mathcal{A}}(a)}{\pi_{\mathcal{A}}(w)}p_{A(0)|W}(a,w)$
e.g., we assume that by enacting the “law” $T$ we are changing the
distribution of exposures a way that does not violate said law and that
maintains relative self-selection preferences, conditional on covariates.
Then, to estimate the ATE of enacting this policy it suffices to estimate the
ARE for $\mathcal{A}$. This is not always reasonable a reasonable assumption
to make! For example, with a single continuous exposure with
$\mathcal{A}=[\alpha,\infty]$, it could be more reasonable to assume that the
modified treatment takes the value $\tilde{A}=\text{max}(A,\alpha)$. This has
important and general implications for estimating the effect of “binary”
policies that are really thresholds on continuous exposures. This is a
special kind of modified treatment policy Hejazi et al. (2021, 2022); Iván
Díaz Muñoz and Mark van der Laan* (2012).
Formally, let $\mathcal{A}\subset\mathbb{R}^{m}$ denote a subset of the
exposure space. For example, presume that $\mathcal{A}$ represents dosages of
drugs that have been deemed safe for combination. Let $\mathcal{A}$ also
denote the binary random variable $1_{\mathcal{A}}(A)$ indicating whether an
observation conformed to the policy. Let
$\pi_{\mathcal{A}}(w)=P(A\in\mathcal{A}|W=w)$ be the probability that an
observation with covariates $w$ naturally self-selects treatment in the
requested region $\mathcal{A}$. We define the modified treatment variable
$\tilde{A}$ to represent the distribution of exposure once all exposures are
forced to be self-selected within the region $\mathcal{A}$. The modified
treatment has density:
$p_{\tilde{A}|W}(a,w)=\frac{1_{\mathcal{A}}(a)}{\pi_{\mathcal{A}}(w)}p_{A|W}(a,w)$
which preserves the relative self-selection preferences for each available
exposure and sets preferences for “outlawed” exposures to zero. Now we can
define the population expected outcome ($\mu$) if we were to impose this
policy:
$\displaystyle E[\mu(\tilde{A},W)]$
$\displaystyle=\int\mu(a,w)\,dP_{\tilde{A},W}$
$\displaystyle=\int\mu(a,w)\frac{1_{\mathcal{A}}(a)}{\pi_{\mathcal{A}}(w)}\,p_{A|W}(a,w)p_{W}(w)\,da\,dw$
$\displaystyle=\int_{w}\underbrace{\left[\int_{a\in\mathcal{A}}\mu(a,w)\frac{p_{A|W}(a,w)}{\pi_{\mathcal{A}}(w)}\
da\right]}_{Q(\mathcal{A}=1,w)}\ p_{W}(w)dw$
$\displaystyle=E[Q(\mathcal{A}=1,W)]$
What we have shown is that this parameter is a population average of some
function $m$ when $\mathcal{A}$ is forced to 1. For any value of $w$, $m(1,w)$
is a particular convexly weighted average of the causal dose-response curve
across the different exposure levels, thus collapsing it down to a single
number for each $w$.
In a similar fashion we define the population average outcome under the
complementary policy $\mathcal{A}^{c}$, which is $E[Q(\mathcal{A}=0,W)]$. With
this, we are ready to introduce our parameter of interest, which we call the
average regional effect (ARE):
$\psi^{*}=E[Q(\mathcal{A}=1,W)]-E[Q(\mathcal{A}=0,W)]$
representing the difference in average outcomes if we forced exposure to self-
selection within $\mathcal{A}$ vs. to self-selection within $\mathcal{A}^{c}$.
In most applications, it is not known a-priori what region $\mathcal{A}$
should be set. For example, we may not know how various chemicals or drugs
interact and how to set safe limits for all of them. Therefore $\mathcal{A}$
itself is in practice something that should be estimated in order to maximize
some objective. That said, for the purposes of establishing our theory we
should first imagine $\mathcal{A}$ as known and fixed but depends on the
original distribution of $A$. We will then show how we can choose what policy
to enact (i.e. choose $\mathcal{A}$) while also unbiasedly estimating its
effect.
### 2.3 Identification and Causal Assumptions
Our target parameter is defined on the causal data-generating process, so it
remains to show that we can define it only in reference to observable
quantities under certain assumptions. Standard conditioning arguments show
that
$\psi=E[E[Y|\mathcal{A}=1,W]-E[Y|\mathcal{A}=0,W]]$
identifies the causal effect as long as the following assumptions hold:
1. 1.
Conditional Randomization: $A\perp Y(a)\mid W$ for all $a$
2. 2.
Positivity: $P(\mathcal{A}=1|W)>0$ for all $w$
Our identification result shows that we can get at the causal ARE by
estimating an observable “ATE” under certain conditions. Our goal is now to
show how to efficiently estimate the observable ATE without imposing any
additional assumptions (e.g. linearity, normality, etc.). While our
identification assumptions may not always hold in all applications, we can at
least eliminate all model misspecification bias and minimize random variation.
Once we’ve established how to estimate the ARE for a fixed region, we’ll turn
our attention to the problem of finding a good region $\mathcal{A}$ and lastly
how to do that without incurring selection bias in estimating the ARE for that
region.
## 3 Estimating ARE with TMLE
In the previous sections we established that the causal ARE is equivalent to
the observable ATE $E[E[Y|\mathcal{A}=1,W]-E[Y|\mathcal{A}=0,W]]$ under
standard identifying assumptions. Therefore to estimate it all we need to do
is 1) create a new binary random variable
$\mathcal{A}_{i}=1_{\mathcal{A}}(A_{i})$ and 2) proceed as if we were
estimating the observable ATE from the observational data structure
$(Y,\mathcal{A},W)$.
There is an extensive literature on estimating the ATEs from observational
data Nichols (2007); Winship and Morgan (1999); Rubin (2006). Using split-
sample machine learning we can construct estimators that are provably unbiased
(modulo bias from any violations of identifying assumptions), have the minimum
possible sampling variance, and which are “doubly robust” Zheng and van der
Laan (2010); Zivich and Breskin (2021). Augmented inverse propensity-weighting
(AIPW) and targeted maximum likelihood (TMLE) are two established estimation
approaches that accomplish these goals. Although they are usually very similar
in practice, TMLE is often better for smaller samples Luque-Fernandez et al.
(2018); Smith et al. (2022); van der Laan and Rose (2011, 2018) and should
generally be preferred. In what follows we use the TMLE estimator of the ATE,
which we briefly describe here.
The TMLE estimator is inspired by the fact that if we knew the true
conditional mean $Q(\mathcal{A},W)=E[Y|\mathcal{A},W]$ we could estimate the
ATE with the empirical average $\frac{1}{n}\sum_{i}Q(1,W_{i})-Q(0,W_{i})$. Of
course, we do not know $Q$, but we can estimate it by regressing the outcome
$Y$ onto the exposure $\mathcal{A}$ and covariates $W$. However a detailed
mathematical analysis shows that we incur bias if we use our estimate
$\hat{Q}$ instead of the truth. This bias might decrease as sample size
increases, but it dominates relative to random variability, making it
impossible to establish p-values or confidence intervals. TMLE solves this
problem by computing a correction to the regression model $\hat{Q}$ that
removes the bias. In other words, it “targets” the estimate $\hat{Q}$ to the
parameter of interest (here the ATE). The process is as follows:
1. 1.
Use cross-validated ensembles of machine learning algorithms (a “super
learner”) to generate estimates of the conditional means of treatment:
$\hat{g}(\mathcal{A}=a,W)\approx P(\mathcal{A}=a|W)$ (i.e. propensity score)
and outcome: $\hat{Q}(\mathcal{A},W)\approx E[Y|\mathcal{A},W]$
2. 2.
Regress $Y$ (scaled to $[0,1]$) onto the “clever covariate”
$H_{i}=\frac{1_{1}(\mathcal{A}_{i})}{\hat{g}(1,W_{i})}-\frac{1_{0}(\mathcal{A}_{i})}{\hat{g}(0,W_{i})}$
using a logistic regression with a fixed offset term
$\text{logit}(\hat{Q}(\mathcal{A},X))$. The (rescaled) output of this is our
targeted regression model $\hat{Q}^{*}$
3. 3.
Compute the plug-in estimate using the targeted model:
$\hat{\psi}=\frac{1}{n}\sum_{i}\hat{Q}^{*}(1,W_{i})-\hat{Q}^{*}(0,W_{i})$
An estimated standard error for $\hat{\psi}$ is given by
$\hat{\sigma}^{2}=\frac{1}{n^{2}}\sum_{i}\left[\left(\frac{1_{1}(\mathcal{A}_{i})}{\hat{g}(1,W_{i})}-\frac{1_{0}(\mathcal{A}_{i})}{\hat{g}(0,W_{i})}\right)\bigg{(}Y_{i}-\hat{Q}^{*}(\mathcal{A}_{i},W_{i})\bigg{)}+\bigg{(}\hat{Q}^{*}(1,W_{i})-\hat{Q}^{*}(0,W_{i})\bigg{)}-\hat{\psi}\right]^{2}$
with corresponding 95% confidence interval $\hat{\psi}\pm 1.96\hat{\sigma}$.
Explaining why the targeting step takes the form of a logistic regression and
how the estimated standard error is derived are beyond the scope of this work.
van der Laan and Rubin (2006); van der Laan and Rose (2011, 2018) offer
explanations targeted to audiences with varying levels of mathematical
sophistication.
To obtain these estimates we need only to specify the ensemble of machine
learning algorithms used to estimate the propensity and initial outcome
regressions $\hat{g}$ and $\hat{Q}$. The theoretical guarantees hold as long
as a sufficiently rich library is chosen.
For estimating the ARE, we must also specify the region $\mathcal{A}$ so that
we can compute our binary “exposure” variable. The issue of course is that we
have been treating $\mathcal{A}$ as a known region, whereas in many
applications the important question is figuring out what guidelines to impose
in the first place. This is the focus of the next section.
## 4 Defining the Target Region
Thus far we have not focused on how we define the target region $\mathcal{A}$.
First, let’s think of $\mathcal{A}$ as nonparametrically defined as the
maximizer of some criterion, independent of an estimator. We can think of this
as any region on the exposure gradient that maximizes the outcome (can take
any shape). However, such a region isn’t interpretable. Therefore, it is
easier to constrain the optimization so $\mathcal{A}$ is a rectangle in the
exposure space. This is because these sections can be easily described using
$\geq$ and $\leq$ rules. This is also important from a public health
standpoint where these rules effectively are thresholds of exposures found to
have the most severe (or least severe) affects. For this purpose, regression
trees are an ideal estimator to get at such a region. Each decision tree
algorithm uses some objective function to split a node into two or more sub-
nodes. Of course, it is generally impossible to know a priori which learner
will perform best for a given prediction problem and data set. Decision trees
have many hyper-parameters such as the maximum depth, minimum samples in a
leaf, and criteria for splitting amongst others. As such, we need to find the
decision tree estimator that best fits the data given a set of nodes. We do
this by creating a library of decision tree estimators to be applied to the
exposure data and use cross-validation to select nodes based on the best
fitting decision tree. This CV selection of the best fitting decision tree
algorithm defines our exposure Super Learner $f(A)$ in our additive semi-
parametric model $E(Y|A,W)=f(A)+h(W)$. This additive model is needed because
we are interested in finding regions that maximize an outcome within only an
exposure space not including the baseline covariates.
### 4.1 Discovering Regions in Multiple Exposures using Ensemble Trees
To discover regions in multiple exposures and therefore discover interactions
in the exposure space, we use predictive learning via rule ensembles Friedman
and Popescu (2008). Thus, as part of the data-adaptive procedure the $f(A)$ is
a regression model constructed as a linear combinations of simple rules
derived from the exposure data. Each rule consists of a conjunction of a small
number of simple statements concerning the values of individual input
exposures. Machine learning using rule ensembles have not only been shown to
have predictive accuracy comparable to the best methods but also result in a
linear combination of interpretable rules. Prediction rules used in the
ensemble are logical if [conditions] then [prediction] statements, which in
our case the conditions are regions in the exposure space that are predictive
of the outcome. Learning ensembles have the structure:
$F(x)=a_{0}+\sum_{m=1}^{M}a_{m}f_{m}(x)$
where $M$ is the size of the ensemble (total number of trees) and each
ensemble member $f_{m}(x)$ is a different function of the input exposures $A$
derived from the training data in the cross-estimation procedure (discussed
later). Ensemble predictions from $F(x)$ are derived from a linear combination
of the predictions of each ensemble member with ${a_{m}}_{0}^{M}$ being the
parameters specifying the linear combination. Given a set of base learners,
trees constructed using the exposures, ${{f_{m}(x)}_{1}^{M}}$ the parameters
for the linear combination are obtained by a regularized linear regression
using the training data. Ideally, each tree in the ensemble is limited to
including only 2-3 exposures at a time which enhances interpretability. For
instance, given the noise and small sample size in most public health studies,
it is unlikely that signal is strong enough to detect interactions with 4 or
more variables. Not only that but trees with partitions across many variables
become less interpretable. Therefore, in our case we are interested in using
an ensemble algorithm that creates a linear combination of smaller trees but
also shows optimal prediction performance. To accomplish this goal we use the
PRE package Fokkema (2020a) which is similar to the original RuleFit algorithm
Friedman and Popescu (2008) with some enhancements including 1. unbiased
recursive partitioning algorithms, 2. complete implementation in R, 3.
capacity to handle many outcome types and 4. includes a random forest approach
to generating prediction rules in addition to bagging and boosting methods.
Here, the package PRE is fit to the exposure data in the training sample.
Mechanically the procedure is 1. generate an ensemble of trees using exposure
data, 2. fit a lasso regression using these trees to predict the outcome, 3.
extract the tree basis with nonzero coefficients, 4. store these basis as
rules which when evaluated on the exposure space demarcate an exposure region
$\mathcal{A}$. There may then be many exposure regions such as
$\mathcal{A}_{X_{1},X_{2}}$ which is a region including exposures
$X_{1},X_{2}$ or another region in the exposure space which uses variables
$X_{4},X_{5}$, $\mathcal{A}_{X_{4},X_{5}}$ etc. The ARE is then calculated for
each of these regions which are based off of trees found to be predictive in
the ensemble. In the case that multiple trees are included in the ensemble
which are composed of the same set of exposures, we select the tree with the
largest coefficient. This procedure is done in each fold of the cross-
validation procedure.
### 4.2 Discovering Regions in Single Exposures using Decision Trees
In addition to finding interactions in the exposure space, the analyst may
also be interested in identifying what exposures have a marginal impact and at
what levels the outcome changes the most in these exposures. To answer this
question we include a marginal tree fitting procedure which is very similar to
the method described in 4.1. Here, $f(A)$ is a Super Learner of decision trees
fit onto one exposure at a time. We then extract the rules determined from the
best fitting tree. Each terminal leaf demarcates a region in the exposure and
thus similarly we may have several $\mathcal{A}$ for 1 exposure which are the
regions found when creating the partitions which best explains the outcome.
Here, rather than calculating an ARE for each region we calculate an ARE
comparing each region to the reference region. The reference region is defined
as the region that captures the lowest values of $A$. For example, consider
our resulting decision tree when fit to variable $A_{1}$ resulted in terminal
leaves $A_{1}<0.6$, $A_{1}>0.6$ & $A_{1}<0.9$, $A_{1}>0.9$, in this case the
reference region would be $A_{1}<0.6$. We would then have two ARE estimates
for the two regions above the reference region. Mechanically, in the training
sample, we find the best fitting decision tree which finds partitions in one
exposure that best explains an outcome, these rules are evaluated as if
statements on the exposure to create $\mathcal{A}_{i}=1_{\mathcal{A}}(A_{i})$
for the respective exposure. We then subset the reference level out and row
bind it with each region above the reference region and pass that data to our
estimators of the ARE. This approach was chosen to give users a dose-response
type estimate for data-adaptively determined thresholds in the univariate
exposure space.
### 4.3 Iterative Backfitting
We need an algorithm that will allow us to fit $f(A)$ while controlling for
$W$ but not including $W$ in the partitions (trees). As such, we iteratively
backfit two Super Learners $f(A)$ a Super Learner of decision trees and $h(W)$
an unrestricted Super Learner applied to the covariates. However, both
algorithms need to use the same convergence criteria (here maximum likelihood
estimation). Thus, $f(A)$ uses an ensemble of regression trees and $h(W)$ uses
an ensemble of flexible MLE based algorithms (MARS, elastic net, Highly
Adaptive Lasso amongst others). The algorithm first initializes by getting
predictions from $f(A)$ and $h(W)$, that is, simply fitting a Super Learner to
the exposures and covariates separately and then getting predictions. Then we
begin fitting each algorithm offset by the predictions of the other. So at
iteration 1 we fit $f(A,\text{offset}=h(W)_{\text{iter 0}})$ and likewise
$h(W,\text{offset}=f(A)_{\text{iter 0}})$; where the offsets are predictions
of the models fit individually (without offset at iteration 0). The
predictions of these models without an offset then gives us $f(A)_{\text{iter
1}}$ and $h(W)_{\text{iter 1}}$. These predictions are then used as offsets at
iteration 2. This process continues until convergence where convergence is is
defined as the absolute mean difference between the two models being less than
some very small number $\delta$ where $\delta$ by default is 0.001. In this
way, for both where $A$ in $f(A)$ is a vector of exposures (resulting rules
include combinations of different exposure levels) and when $A$ is a single
exposure, we are able to identify cut-points in the exposure space while
controlling for $W$ in the additive model that converges in maximum
likelihood. We evaluate the best fitting decision tree onto the exposure space
which results in an indicator of the exposure region and calculate our k-fold
specific and pooled target parameters give this region.
### 4.4 Convergence to the True Region
The cross-validation selector is simply the recursive partitioning algorithm
which performed best in terms of cross-validated risk within the parameter-
generating sample. Assuming that a partition $\\{\mathcal{D}_{d}\\}d=1,...,D$
of the space $A=A_{1},...,A_{Z}$ exists with $D$ cells (or segments) such that
the $\mathcal{D}_{d}$ cells are used to define the $\mathcal{A}$ boundaries,
then the oracle selector as defined in Theorem 2 of van der Laan et al. (2008)
is the estimator, among the decision tree learners considered, which minimizes
risk under the true data-generating distribution $P_{0}$. That is, the oracle
selector is the best possible estimator given the set of candidate decision
tree learners considered; however, it of course depends on both the observed
data and $P_{0}$, and thus is unknown. Theorem 2 in van der Laan et al. (2008)
shows that the Super Learner performs as well (in terms of expected risk
difference) as the oracle selector, up to a typically second order term.
Therefore, as long as the number of candidate decision tree learners
considered is polynomial in sample size ($n^{x}$), the Super Learner is the
optimal learner. Given that the Super Learner performs asymptotically as well
(in the risk difference sense) as the oracle selector, which chooses the best
of the decision tree candidate learners, and given that, this class of
learners are restricted to algorithms which partition the mixture space into
$\mathcal{D}_{d}$ nodes of finite depth, it follows then that the selection of
$\mathcal{D}_{d}$ nodes used in the decision tree estimator selected by the
Super Learner converges asymptotically to the $\mathcal{D}_{d}$ nodes used by
the oracle selector under depth constraints. Here the finite tree depth is
needed because without tree depth limits, as sample size increases the best
fitting estimator will always be the decision tree with increased depth and
therefore there is no convergence to some true set of $\mathcal{D}_{d}$ nodes.
Likewise, for interpretability, we are interested in some small set of nodes,
or thresholds, per variable that is informative for public policy. That is to
say, we may limit the ensemble of decision trees to a depth of 3 for
interpretability. For example, we may want a more concise rule such as, "if
arsenic is greater than X and lead is greater than Y", or without monotonic
assumptions, "if arsenic is between X1 and X2 and lead is between Y1 and Y2".
Overall, under functional averaging theory developed and additive model
assumptions we can perhaps say there is some convergence to a true region
which best differentiates the values of the outcome as sample size goes to
infinity. That being said, we show simulations in the next section to measure
how good CVtreeMLE is at identifying the true ARE and true exposure region
built into a data-generating process as sample size increases.
## 5 K-fold Cross-Estimation
Of course, the mixture region used to estimate the ARE is not defined a
priori. If we were to use the same data to both identify the region and make
the ARE our estimates will be biased. Thus, for desirable asymptotic
properties to hold without additional assumptions, we need our conditional
means to be cross-estimated from the observed data. We split the data into
$P_{n_{-k}}$ (parameter-generating) and $P_{n_{k}}$ (estimation) samples.
These splits or folds are part of a k-fold cross-validation framework. K-fold
cross-validation involves: (i) ${1,...,n}$, observations, is divided into $K$
equal size subgroups, (ii) for each $k$, an estimation-sample, notationally
$P_{k}$, is defined by the k-th subgroup of size $n/K$, while the parameter-
generating sample, $P_{n_{-k}}$, is its complement. In this round robin manner
we rotate through our data and thus, in the case of $K=10$ get 10 difference
target parameter mappings $\mathcal{A}_{n}$, outcome estimators $Q_{n}$ and
propensity estimators $g_{n}$. We want one summary measure of the target
parameter found across the folds, such as the average.
With $P_{n_{-k}}$ we find thresholds in our exposure space (using the results
of a decision tree) which designates exposure region. Then given this exposure
region using the same $P_{n_{-k}}$ we train our $g_{n}$ and $Q_{n}$ estimators
which are needed for our TMLE update step to debias our initial estimates of
the ARE and give us an asymptotically unbiased estimator. We then plug-in our
$P_{n_{k}}$ to this unbiased estimator to get our ARE estimate in this
estimation sample.
Let $\bar{Q}_{n}$ denote a substitution estimator that plugs in the empirical
distribution with weight 1/n for each observation which approximates the true
conditional mean $\bar{Q}_{0}$ in $P_{0}$, this estimator, in our case is a
Super Learner, or ensemble machine learning algorithm, our substitution
estimator looks like:
$\Psi(Q_{P_{n_{k}}})=\frac{1}{V}\sum_{v=1}^{V}{\bar{Q}_{n_{-k}}(\mathcal{A}_{n_{-k}}=1,W_{v})-\bar{Q}_{n_{-k}}(\mathcal{A}_{n_{-k}}=0,W_{v})}$
Let’s focus first on the $k$ subscripts, we split data into $k\in 1...K$ non-
overlapping folds and fit $K$ different models. Thus, $\bar{Q}_{n_{-k}}$
denotes our outcome regression function fit when excluding the data for fold
$k$. $P_{n_{k}}$ denotes our estimation-data and $P_{n_{-k}}$ is the
parameter-generating sample, that is, our parameter-generating sample is used
to train our estimators and then we pass our estimation-data in to get
estimates. $\bar{Q}_{n_{-k}}$ then, in our case, is a Super Learner fit using
the parameter-generating data. Likewise, $\mathcal{A}_{n_{-k}}$ is a decision
tree fit using the parameter-generating data. $\Psi(Q_{P_{n_{k}}})$ then
indicates that we pass the estimation-sample data into our estimators trained
with the parameter-generating data; so here we first fit a decision tree to
the exposure space of the parameter-generating data, then apply the rules
found to the estimation-sample data to create an exposure region indicator.
Then using this exposure and the estimation-sample covariates, we feed this
into the outcome regression model trained on the parameter-sample data. We
then get predicted outcomes under different counterfactuals for a data-
adaptively determined exposure using our estimation-sample data. Our cross-
estimated TMLE estimator for this data-adaptively defined exposure produces an
unbiased, efficient substitution estimator of target parameters of a data-
generating distribution we are interested in. This estimator looks like:
$\Psi(Q_{P_{n_{k}}}^{\star})=\frac{1}{V}\sum_{v=1}^{V}\\{\bar{Q}_{n_{-k}}^{\star}(\mathcal{A}_{n_{-k}}=1,W_{v})-\bar{Q}_{n_{-k}}^{\star}(\mathcal{A}_{n_{-k}}=0,W_{v})\\}$
Here we can see the only change to our above equation is $\bar{Q}^{\star}$
which is the TMLE augmented estimate. This new function,
$f(\bar{Q}_{n_{-k}}^{\star}(A,W))=f(\bar{Q}_{n_{-k}}(A,W))+\epsilon_{n_{-k}}\cdot
h_{n_{-k}}(A,W)$, where $f(\cdot)$ is the appropriate link function (e.g.,
logit), $\epsilon_{n}$ is an estimated coefficient and $h_{n}(A,W)$ is a
"clever covariate" which is now cross-estimated. Here what we mean is that,
the initial estimates for the estimation-sample using models trained using the
parameter-generating data are updated through this so-called, least-favorable
submodel. The cross-estimated clever covariate looks like:
$h_{n_{-k}}(\mathcal{A},W)=\frac{\mathbbm{I}(\mathcal{A}_{n_{-k}}=1)}{g_{n_{-k}}(\mathcal{A}_{n_{-k}}=1|W)}-\frac{\mathbbm{I}({\mathcal{A}_{n_{-k}}=0)}}{g_{n_{-k}}(\mathcal{A}_{n_{-k}}=0|W)}$
Here, $g_{n_{-k}}(W)=\mathbb{P}(\mathcal{A}_{n_{-k}}=1\mid W)$, the propensity
score of the data-adaptively determined exposure region, is being estimated
using a Super Learner with the parameter-generating data. That is, in our
parameter generating sample we get the exposure region, and an estimator
$g_{n}$ we apply this exposure region to the estimation sample and then get
predictions for the probability of that exposure region indicator using the
estimation sample, we then plug these estimates into the above cross-estimated
clever covariate used in the TMLE update.
We can see that by using v-fold cross-validation, we can do better than
traditional sample splitting as v-fold allows us to make use of the full data
which results in tighter confidence intervals because our variance is
estimated over the full data. Similarly, our estimate is an average of the
v-fold specific estimates:
$\Psi_{n}(P)=Ave\\{\Psi_{P_{n_{-k}}}(P)\\}\equiv\frac{1}{V}\sum_{v=1}^{V}\Psi_{P_{n,-k}}(P)$
We do this in a pooled TMLE update manner where we stack the estimation-sample
estimates for each nuisance parameter and then do a pooled TMLE update across
all the initial estimates using clever covariates across all the folds to get
our estimate $\epsilon$ we then update our counterfactuals across all the
folds and take the average. More concretely, in each fold we have our initial
estimates from that fold from $Q_{n_{-k}}(Y|\mathcal{A},W)$ and the fold
specific clever covariate $h_{n_{-k}}(\mathcal{A}|W)$ of length $k$ for a fold
specific exposure found using $\mathcal{A}_{n_{-k}}$. We stack all the
$Q_{n_{-k}}$’s and $h_{n_{-k}}(\mathcal{A}|W)$’s together along with the
outcomes in each validation fold and do our fluctuation step:
$f(\bar{Q}_{n}^{\star}(\mathcal{A},W))=f(\bar{Q}_{n}(\mathcal{A},W))+\epsilon_{n}\cdot
h_{n}(\mathcal{A},W)$
Notice here the $k$ subscripts are removed, this is because we are using our
cross-estimates for all of $n$. Using the $\epsilon$ from this model, we then
update the counterfactuals across all the folds and take the difference for
our final ARE. In a similar fashion, we use the updated conditional means,
counterfactuals, and clever covariates to solve the IC across the whole
sample. By pooling the cross-estimates across the folds and then calculating
the SE for this pooled IC we are able to derive more narrow confidence
intervals compared to if we were to average the IC estimated in each of the
folds (because the IC is scaled by $n$ and not $n/K$). This pooled estimate
still provides us with proper intervals because all estimates in its
construction were cross-estimated.
An alternative to this pooled approach is to simply report the k-fold specific
estimates of the ARE and fold specific variance estimates for this ARE using
the fold specific IC. We do this as well. We do this because, if the exposure
region $\mathcal{A}$ identified in each region is highly variable, that is, if
the region that that maximizes the difference for sets of exposure variables
are very different across the folds, then interpreting the pooled ARE is
difficult. By calculating and providing both k-fold specific and pooled
results users can investigate how variable a pooled result is across the
folds.
### 5.1 Inverse-Variance Method for Combining K-fold Results
In addition to the pooled TMLE approach to aggregate k-fold specific data-
adaptive target estimates, we also calculate the inverse-variance method (IVM)
commonly used in meta-analyses. We call this method the k-fold harmonic mean.
Here each fold is given a weight defined as:
$w_{k_{i}}=\frac{1}{SE(\hat{\theta}_{k_{i}})^{2}}$
Which is simply an inverse of the standard error such that estimates with
smaller SE are given a higher weight. The inverse-variance pooled ARE across
the folds is given as:
$\hat{\theta}_{IVM}=\frac{\sum w_{k_{i}}\hat{\theta}_{k_{i}}}{\sum w_{k_{i}}}$
And lastly, the pooled SE is calculated as:
$SE(\hat{\theta}_{IVM})\frac{1}{\sqrt{\sum w_{k_{i}}}}$
For which confidence intervals and p-values are derived for the pooled IVM
estimate.
This pooled estimate is given because, in the event of high inconsistency of
the k-fold estimates in lower sample size, the confidence intervals from
pooled influence curve may not cover the true ARE if the pooled ARE was
applied to $P_{0}$. This is because the union rule attached to the pooled ARE
is a conservative rule which covers all observations across the folds
(discussed later). The IVM derived CIs are wider and provide better coverage
in the event of high inconsistency (which we show in simulations). We explain
rule stability metrics and establishing a union rule across the folds in the
next section.
### 5.2 Defining the Union Region
The pooled TMLE ARE is matched with a pooled region that encompasses all the
observation indicated by each fold specific regions. We group the trees across
the folds according to what variable sets the trees are composed of. That is,
a linear combination of tree ensembles is fit to each training sample specific
to the fold. There may be variability in where the partition is set for trees
with the same variable sets across the folds, or certain ensembles don’t use
certain variable sets at all in some folds but used in others. We need a
method of creating a pooled region and give stability metrics for how
consistently trees with a respective variable set are found in the cross-
validation procedure. For this we create a union region. There are other
possible ways of pooling the regions, such as averaging the partitions per
exposure variable across the folds. Here we choose a conservative approach.
This is the union region of the k-fold regions in the sense that, we create a
new region that is the OR combination of each k-fold specific tree. For three
folds and therefore three partitions say, $X_{1}<2$ & $X_{2}>5$, $X_{1}<2.3$ &
$X_{2}>5.2$ and $X_{1}<1.9$ & $X_{2}>5.3$, the union rule is $X_{1}<2$ &
$X_{2}>5$ OR $X_{1}<2.3$ & $X_{2}>5.2$ OR $X_{1}<1.9$ & $X_{2}>5.3$ forms the
rule: $X_{1}<2.3$ & $X_{2}>5$ because this region covers all the observations
indicated in the fold specific regions. For variables where the logic is $>$
we take the minimum value across the folds and likewise for $<$ we take the
maximum. This union region is conservative and sensitive to outlier partition
points found across the folds and therefore higher $K$ folds will lead to more
stable partitions if there is signal in the data. Additionally, the analyst
should investigate the fold specific regions to determine the interpretability
of the pooled region. If there is high variability or outliers, there may be
bias in the TMLE pooled estimate when compared to the expected difference in
outcomes if the respective pooled region was applied to the true population
$P_{0}$.
### 5.3 Stability Metrics
Given a pooled region, we simply give the proportion of folds trees with a
respective variable set are found across the folds. For example, consider a
study of mixed metals that uses CVtreeMLE and the results across three folds
are: 1. lead > 2.2 & arsenic > 1.3, 2. lead > 2.1 & arsenic > 1.2, 2. lead >
2.0 & arsenic > 1.1. Our pooled region is lead > 2.0 & arsenic > 1.1 because
this region contains all the fold specific regions. The stability metric here
is 100% because a tree with lead and arsenic was found in all three folds. If
however, this tree was only found in 2 of three folds, the stability metric is
67%.
## 6 Simulations
In this section, we demonstrate using simulations that our approach identifies
the correct exposure region which maximizes the difference in conditional
means and estimates the correct difference built into a DGP for this region.
### 6.1 Data-Generating Processes
Because a two dimensional exposure space is easier to visualize and describe
compared to higher dimensional spaces, we start by investigating a squared
dose-response relationship between two exposure variables where an interaction
occurs between the exposures when each meets a particular threshold value. We
extend simulations to the three dimensional case. In both 2-D and 3-D exposure
simulations there are specific outcome values generated for each subspace of
the mixture based on split points $\mathcal{D}_{d}$ but there exists one
region with the maximum outcome (the truth that we want). In both scenarios,
the goal is to determine if our data adaptive target parameter is targeting
the region that maximizes the conditional mean outcome for the given sample
and evaluate how CVtreeMLE approaches this desired oracle parameter as sample
size increases. To meet this goal, we construct a data-generating process
(DGP) where $Y$ is generated from a tree-structured covariate-adjusted
relationship of a mixture consisting of components, $A_{1},A_{2},A_{3},A_{n}$.
That is, generally in each simulation we generate exposure regions, where the
density of the region is driven by covariates and there is one region that has
the maximum difference compared to outside the respective region. More details
for each simulation are given below.
#### 6.1.1 Two-Dimensional Exposure Simulations
This DGP has the following characteristics, $O=(W,A,Y)$. $W$ are three
baseline covariates
$W_{1}\sim\mathcal{N}(\mu=37,\sigma=3),W_{2}\sim\mathcal{N}(\mu=20,\sigma=1),W_{3}\sim\mathcal{B}(\mu=0.5)$
Where $\mathcal{B}$ is a Bernoulli distribution and $\mathcal{N}$ is normal.
These distributions and values were chosen to represent a study with
covariates for age, BMI and sex. Our generated exposures were likewise created
to represent a chemical exposure quantized into 5 discrete levels. The values
and range of the outcome were chosen to represent common environmental health
outcomes such as telomere length or epigenetic expression.
We are interested in sampling observations into a 2-dimensional exposure grid.
Here a $5\times 5$ grid is based on combinations of two discrete exposure
levels with values 1-5. We want the number of observations in each of these
cells to be affected by covariates. To do this we define a conditional
categorical distribution $P\\{(A_{1},A_{2})=(a_{1},a_{2})|W=w\\}$ and sample
from it.
$P\\{(A_{1},A_{2})=(a_{k},a_{l})|W\\}=\frac{e^{W^{\top}\beta_{k,l}}}{1+\sum_{k,l}e^{W^{\top}\beta_{k,l}}}$
Here the $\beta$’s attached to each covariate were drawn from a normal
distribution with means 0.3, 0.4, 0.5 and 0.5 respectively all with a standard
deviation of 2. This then gives us 25 unique exposure regions with densities
dependent on the covariates. We then want to assign an outcome in each of
these regions based on main effects and interactions between the exposures. We
use the relationship
$Y=0.2A_{1}^{2}+0.5A_{1}A_{2}+0.5A_{2}^{2}+0.2*\text{age}+0.4*\text{sex}+\epsilon(0,0.1)$
Which indicates there is a slightly weaker squared effect for $A_{2}$ relative
to $A_{1}$ and a strong interaction between the exposures and confounding due
to age and sex. The resulting data distribution and generating process is
shown in Figure 1.
Figure 1: 2D Exposure Simulation
Of course, it is also possible to explore other dose-response relationships
(such as logarithmic) by changing the coefficient matrix.
##### Computing Ground Truth
The fact that our exposures are discrete in this simulation lets us easily
compute the ground-truth ARE for any region $\mathcal{A}$ because we can
explicitly compute the conditional mean function $m$
$\displaystyle m(\mathcal{A}=1,w)$
$\displaystyle=\int_{a\in\mathcal{A}}\mu(a,w)\frac{p_{A|W}(a,w)}{\pi_{\mathcal{A}}(w)}\
da$
$\displaystyle=\frac{\sum_{a\in\mathcal{A}}\mu(a,w)p_{A|W}(a,w)}{\sum_{a\in\mathcal{A}}p_{A|W}(a)}$
Therefore to approximate the ARE to arbitrary precision we can
1. 1.
Sample a large number of times (e.g. $b=100,000$) from the covariate
distribution to obtain $W_{\\{1,\dots i,\dots b\\}}$.
2. 2.
Compute the values $m(\mathcal{A}=1,W_{i})$ using the above formula. This is
possible because the functions $\mu$ and $p{A|W}$ are known for the data-
generating process 222If the exposure space were not discrete, this step would
require numerical approximation of an integral for each different value of $w$
which would be generally impractical.. In a similar fashion compute
$m(\mathcal{A}=0,w)$.
3. 3.
Compute
$\text{ARE}(\mathcal{A})=\frac{1}{b}\sum_{i}^{b}m(1,W_{i})-m(0,W_{i})$.
#### 6.1.2 Three-Dimensional Exposure Simulations
This DGS has the same general structure, $O=W,A,Y$. $W$ and baseline
covariates
$W_{1}=\mathcal{N}(\mu=37,\sigma=3),W_{2}=\mathcal{N}(\mu=20,\sigma=1),W_{3}=\mathcal{B}(\mu=0.5)$
In this 3D simulation we are interested in keeping the exposures continuous as
this is more realistic compared to the 2D simulation.
Here $A$ are three continuous mixtures from a multivariate normal
distribution:
$\displaystyle\begin{pmatrix}A_{1}\\\ A_{2}\\\ A_{3}\end{pmatrix}$
$\displaystyle\sim N\begin{bmatrix}\begin{pmatrix}0\\\ 0\\\ 0\\\
\end{pmatrix}\\!\\!,&\begin{pmatrix}1.0&0.5&0.8\\\ 0.5&1.0&0.7\\\
0.8&0.7&1.0\end{pmatrix}\end{bmatrix}$
We assign one partition point value to each exposure which creates 8 possible
regions in the $2x2x2$ 3D grid for which we want to assign outcomes. Just as
the first simulation we want the number of observations in each of the cells
in the "mixture cube" to be affected by covariates. To do this we define a
conditional categorical distribution
$P\\{(A_{1},A_{2},A_{3})=(a_{1},a_{2},a_{3})|W=w\\}$ and sample from it.
$P\\{(A_{1},A_{2},A_{3})=(a_{k},a_{l},a_{j})|W_{i}\\}=\frac{e^{W_{i}^{\top}\beta_{k,l,j}}}{1+\sum_{k,l,j}e^{W_{i}^{\top}\beta_{k,l,j}}}$
For each of these categories which defines a region in the exposure space we
need to assign exposure values while also preserving the local correlation
structure within that region. To do this, we convert the cumulative
distribution function of the exposures to a uniform distribution then back
transform this uniform distribution to the original exposure distribution with
bounds for each exposure region. So for instance, in the region where each
exposure is less than each threshold value, we back transform the uniform
distribution with the minimum value set as the minimum for each exposure and
max as the partition value for each exposure. These values then are attached
to the categorical variables generated which represent the mixture region.
This then generates continuous exposure values with a correlation structure in
each region.
The outcome $Y$ is then generated via a linear regression of the form:
$Y=\beta_{0}+\beta_{1}\mathbbm{1}(A=1)+\beta_{2}\mathbbm{1}(A=2)+...+\beta_{7}\mathbbm{1}(A=7)+\beta_{W_{1}}W_{1}+\beta_{W_{2}}W_{2}+\epsilon,\epsilon~{}N(0,\sigma)$
Where the $\beta_{j}$ are chosen so some mixture groups have a high mean, some
have a low mean and $\mathbbm{1}$ represent indicators of each of the possible
8 regions. Thus, the outcome in each region of the mixture cube is determined
by the $\beta$ assigned to that region. Given this formulation of a DGP it is
possible to then generate $Y$ by shifting the drivers or "hot spots" around
the mixture space, thereby simulating possible agonist and antagonistic
relationships. We could assign something like $\beta_{2}=2$ with all other
regions having a $\beta_{\neq 2}=0$. This then would mean the ARE in the true
DGP is 2. Likewise we could assign $\beta$’s in each region in which case the
truth by our definition is the region with the max ARE. The process for this
DGP is shown in Figure 2.
Figure 2: 3D Exposure Simulation
Overall, our 3D example is very similar to the 2D exposure simulation but we
aim to test CVtreeMLE in identifying thresholds used to generate an outcome in
a space of three continuous exposures. Also, because we keep the space of
possible outcomes relatively simple here, we simply generate individual
outcomes for each mixture subspace. This allows us to create situations where
only one region drives the outcome while the complementary space is 0 or there
is an outcome in each region and we are interested in identifying the region
with the maximum outcome. In each simulation we are interested in the
bias/variance of our estimates compared to the truth, the bias of our rule
compared to the true rule and the bias of our data-adaptive rule compared to
the expected ARE if that rule was applied to the true population. We discuss
this next.
##### Computing Ground Truth
Previously, in the discrete exposure case, we could directly estimate ground
truth by inverse weighting given the summed probability in the exposure region
multiplied by the outcome. This is not possible in the continuous case. To
make things simpler, we z-score standardize the covariates so the mean of each
covariate is 0. Therefore we can directly compute the mean in the region
indicated by the ground-truth rule and the mean outcome in the complementary
space and take the difference. This is the same as the max coefficient minus
the mean of the other coefficients in the linear model, this is the true ARE.
### 6.2 Evaluating Performance
The following steps breakdown how each simulation was tested to determine 1.
asymptotic convergence to the true mixture region used in the DGP, 2.
convergence to the true ARE based on this true region and 3. convergence to
the true data-adaptive ARE, that is CVtreeMLE’s ability to correctly estimate
the ARE if the data-determined rule was applied to the population. We do this
by:
1. 1.
To approximate $P_{0}$, we draw a very large sample (500,000) from the above
described DGP.
2. 2.
We then generate a random sample from this DGP of size $n$ which is broken
into $K$ equal size estimation samples of size $n_{k}=n/K$ with corresponding
parameter generating samples of size $n-n/K$.
3. 3.
At each iteration the parameter generating fold defines the region and is used
to create the necessary estimators. The estimation fold is used to get our
TMLE updated causal parameter estimate, we then do this for all folds.
4. 4.
For an iteration, we output the ARE estimates given pooled TMLE, k-fold
specific TMLE and the harmonic mean. The region identified in the fold is
applied to the large sample $P_{0}$ to estimate the data-adaptive bias.
Likewise, each estimate is compared to the ground-truth ARE and region.
For each iteration we calculate metrics for bias, variance, MSE, CI coverage,
and confusion table metrics for the true maximal region compared to the
estimated region. For each type of estimate (pooled TMLE, k-fold specific TMLE
estimates, and harmonic mean) we have bias when comparing our estimate to 1.
the ARE based on the true region in the DGP that maximizes the mean difference
and 2. the ARE when the data-adaptively determined region is applied to the
population. Therefore, when comparing to the true "oracle" region ARE we have:
1. 1.
$\psi^{0}_{\text{pooled tmle bias}}$: This is the bias of the pooled TMLE ARE
compared to the ground-truth ARE for the true region built into the DGP which
maximizes the mean difference in adjusted outcomes.
2. 2.
$\psi^{0}_{\text{mean v-fold tmle bias}}$: This is the bias of the mean k-fold
specific AREs compared to the ground-truth ARE for the true region built into
the DGP.
3. 3.
$\psi^{0}_{\text{harmonic mean v-fold tmle bias}}$ This is the bias of the
harmonic mean of k-fold specific AREs compared to the ground-truth ARE for the
true region built into the DGP.
The above bias metrics are each compared to the true ARE for the oracle region
in the DGP. We are also interested in the ARE if the data-adaptively
determined region, the region estimated to maximizes the difference in
outcomes in the sample data, were applied to $P_{0}$ the true population.
Therefore, there are also bias estimates for:
1. 1.
$\psi^{DA}_{\text{pooled tmle bias}}$: This is the bias of the pooled TMLE ARE
compared to the ARE of the union region across the folds applied to $P_{0}$.
2. 2.
$\psi^{DA}_{\text{mean v-fold tmle bias}}$: This is the bias of the mean
k-fold specific AREs compared to the mean ARE when all the k-fold specific
rules are applied to $P_{0}$.
3. 3.
$\psi^{DA}_{\text{harmonic mean k-fold tmle bias}}$ This is the bias of the
harmonic mean of k-fold specific AREs compared to the ARE of the union region
across the folds applied to $P_{0}$.
We multiple each bias estimate by $\sqrt{n}$ to ensure the rate of convergence
is at or faster than $\sqrt{n}$. For each ARE estimate we calculate the
variance and subsequently the mean-square error as:
$\text{MSE}=\text{bias}^{2}+\text{variance}$. MSE estimates were also
multiplied by $n$. For each ARE estimate we calculate the confidence interval
coverage of the true ARE parameter givent the oracle region and the ARE given
the data-adaptively determined region applied to $P_{0}$. For the TMLE pooled
estimates these are lower and upper confidence intervals based on the pooled
influence curve. For the k-fold specific coverage, we take the mean lower and
upper bounds. For the harmonic pooled coverage, we calculate confidence
intervals from the pooled standard error. In each case, we check to see if the
ground-truth rule ATE and data-adaptive rule ATE are within the interval.
Lastly, we compare the data-adaptively identified region to the ground-truth
region using the confusion table metrics for true positive, true negative,
false positive and false negative to determine whether, as sample size
increases, we converge to the true region.
These performance metrics were calculated at each iteration, where 50
iterations were done for each sample size $n=$ (200, 350, 500, 750, 1000,
1500, 2000, 3000, 5000). It was ensured that, for each data sample, at least
one observation existed in the ground-truth region to ensure confusion table
estimates could be calculated. CVtreeMLE was run with 5 fold CV (to speed up
calculations in the simulations) with default learner stacks for each nuisance
parameter and data-adaptive parameter. Our data-adaptive parameter for
interactions was the tree with the max ARE (positive coefficient) for each
variable set in the ensemble.
### 6.3 Default Estimators
As discussed, CVtreeMLE needs estimators for $\bar{Q}=E(Y|A,W)$ and
$g_{n}=P(A|W)$. CVtreeMLE has built in default algorithms to be used in a
Super Learner van der Laan Mark et al. (2007) that are fast and flexible.
These include random forest, general linear models, elastic net, and xgboost.
These are used to create Super Learners for both $\bar{Q}$ and $g_{n}$.
CVtreeMLE also comes with a default tree ensemble which is fit to the
exposures during the iterative backfitting procedure. These trees are built
from the partykit package Hothorn and Zeileis (2015) in R. By default we
include 7 trees in the tree Super Learner that have various levels for the
hyper-parameters alpha (p-value to partition on), max-depth (maximum depth of
the tree), bonferroni correction (whether to adjust alpha by bonferroni) and
min-size (minimum number of observations in terminal leaves). These trees are
used during the iterative backfitting in estimating partitions for each
individual exposure. For the rule ensemble, the predictive rule ensemble
package (pre) Fokkema (2020b) is used with default settings and 10-fold cross-
validation. Users can pass in their own libraries for these nuisance and data-
adaptive parameters. For these simulations, we use these default estimators in
each Super Learner.
### 6.4 Results
#### 6.4.1 CVtreeMLE Algorithm Identifies the True Region with Maximum ARE
First we describe results for identifying the true region built into the DGP.
It is obviously necessary for this to converge to the truth as sample size
increases in order for the $\psi^{0}$ estimates to be asymptotically unbiased.
Overall we find the tree algorithm identifies the true region in the DGP and
therefore provides results which have high-value for treatment policies.
Figure 3 shows metrics comparing observations covered by the estimated pooled
region to those indicated by the true region in the DGP for two discrete
exposures. From this figure it can be seen that, at around 1500 observations,
the pooled region is the true region. Figure 4 shows the confusion table
metrics comparing the data-adaptive pooled region to the oracle region in the
three continuous exposure scenario. As sample size increases, the false
positives approach 0 which is what we would desire in this continuous case.
From this, we see that in both instances of discrete and continuous exposures,
CVtreeMLE is able to identify the correct region in the exposure data which
has the maximum ARE. There is some small disparity in the discovered region
compared to the truth in the continuous case, this is because for false-
positives to perfectly match the true region, the tree search algorithm must
identify the exact set of continuous digits that delineate the region which is
very difficult. In our case, this region is is $M_{3}$ <= 2.5 & $M_{1}$ >=
0.99 & $M_{2}$ >= 2.0. In this three exposure case there is antagonism of
$M_{3}$. Given the exposures are continuous, it is likely that the tree search
algorithm gets very close but not absolutely exact to these boundaries. In the
two exposure case where the exposures were discretized finding the boundaries
is easier. As such, our future evaluation is focused more on the data-adaptive
estimates (comparing estimates to the ARE given applying the data-adaptive
rule to $P_{0}$). Ultimately, the data-adaptive target parameter theory only
holds for the data-adaptive parameter and not the parameter given an oracle
rule; however, we include both again to investigate how CVtreeMLE approaches
the oracle rule as sample size increases.
Figure 3: 2D Exposure Confusion Table Metrics of Rule Coverage Figure 4: Three
Exposure Confusion Table Metrics of Rule Coverage
#### 6.4.2 CVtreeMLE Unbiasedly Estimates the Data-Adaptive Parameter
Looking at the bias for the ARE estimate given two discrete exposures compared
to the data-adaptively discovered region applied to $P_{0}$ TMLE unbiasedly
estimates the data-adaptive parameter at root $n$ rates with good coverage.
Below, Figure 5 A shows the data-adaptive rule ARE bias
($\psi^{DA}_{\text{pooled tmle bias}}$, $\psi^{DA}_{\text{mean k-fold tmle
bias}}$, $\psi^{DA}_{\text{harmonic mean k-fold tmle bias}}$) and MSE (B).
Figure 5: 2D Exposure Bias and MSE
In Figure 5 A the data-adaptive rule ARE bias is larger for the pooled
estimates (pooled TMLE ARE and harmonic mean ARE compared to ARE if the pooled
rule was applied to $P_{0}$) compared to the average folds bias (mean k-fold
ARE compared to the mean of each k-fold rule applied to $P_{0}$). This is
because inconsistent rule estimates in lower sample sizes can bias the pooled
ARE compared to the pooled region. Consider a 3-fold situation where for
variables $X_{1}$ and $X_{2}$ the region was designated by $X_{1}>4\text{ \&
}X_{2}>4$, $X_{1}>4\text{ \& }X_{2}>4$, and $X_{1}>2\text{ \& }X_{2}>4$;
because $X_{1}>2\text{ \& }X_{2}>4$ is found in one of the folds, this is the
pooled region (as it covers observations for $X_{1}>4$) and thus (if the true
ARE for $X_{1}>4\text{ \& }X_{2}>4$ applied to $P_{0}$) is higher, our pooled
results would be biased to this higher ARE because two of three of our folds
have an ARE for this region. This bias converges to average k-fold bias at a
sample size of 1500. Effectively, once the trees across the folds stabilizes
there is less bias in the pooled estimate compared to the pooled region ATE.
This similar pattern is reflected in the pooled estimates MSE (given higher
bias in smaller samples). For the user, this indicates that, in smaller sample
sizes ($n$ < 1000) the analyst should look at fold specific results to ensure
the trees are close in the cut-off values in order to interpret the pooled
result. If not, k-fold specific results should be reported as these show very
low bias/MSE even in smaller sample sizes. The bias and MSE for all estimates
compared to the ground-truth rule ATE show an $1/sqrt(n)$ reduction as sample
size increases. In sum, as sample size increases the bias for all estimates
converge to 0 which which is necessary for our estimator to have valid
confidence intervals.
Figure 6: 3D Exposure Bias and MSE
Figure 6 A and C likewise show the asymptotic bias in the three continuous
exposure case. All estimates show bias decreasing when evaluated against the
ARE when the data-adaptively determined region is applied to $P_{0}$; however,
these estimates do not go to 0 exactly (at max sample size equal to 5000) as
the data-adaptive rule is still not exactly the true rule, which is expected.
Figure 7 A shows the confidence interval coverage for each estimate compared
to the data-adaptive region applied to $P_{0}$.
Figure 7: 2D Exposure Confidence Interval Coverage
For coverage of the ARE of the pooled rule applied to $P_{0}$, the CIs
calculated from the pooled k-fold standard errors showed coverage between 95%
- 100%. The pooled TMLE CIs showed poorer coverage at lower sample sizes, this
is likely due to the bias of the pooled ARE estimate compared to the pooled
region applied to $P_{0}$ paired with the more narrow confidence intervals
calculated across the full sample. The harmonic pooled k-fold CIs were wider
and thus covered the truth in this pooled setting. Coverage for the k-fold
specific CIs were almost always at or above 90% and converge to 95% at higher
sample sizes.
Figure 8: 3D Exposure CI Coverage
Figure 8 A shows the CI coverage of the data-adaptive rule in the three
exposures simulation. As expected, the average k-fold CI converges to 95%. The
pooled estimates are lower given the conservative pooled rule.
#### 6.4.3 CVtreeMLE Unbiasedly Estimates the Oracle Target Parameter
Now we look at comparing estimates to the true ARE given the oracle region in
the DGP. Figure 5 B and D show the bias and MSE for this comparison in the two
discrete exposures. As can be seen, both decrease at root $n$ rate for all
estimates. Figure 6 B and D likewise show this same rate of convergence for
the three continuous exposure case. Based on these simulations, CVtreeMLE
unbiasedly estimates the oracle target parameter at root $n$ rates. We next
look at the coverage. Figure 7 B shows coverage of the true ARE given the true
region. The CIs calculated from the harmonic pooled k-fold standard errors had
consistent 95% coverage, the k-fold specific CIs converged to 95% when sample
sizes reached 1500 and the pooled TMLE CIs converged to 75% coverage. The same
is shown for the three continuous exposures in Figure 8 B the inverse variance
CI converges to 95% for coverage of the true ATE with the mean k-fold slightly
lower around 82%. Table 2 gives the bias, SD, MSE, and coverage for sample
sizes 200, 1000, and 5000, comparing estimates to the data-adaptive truth.
#### 6.4.4 CVtreeMLE has a Normal Sampling Distribution for Valid Inference
For our estimator to have valid inference, we must ensure that the estimator
has a normal sampling distribution centered at 0 that gets more narrow as
sample size increases. To confirm this, we next examine the empirical
distribution of the standardized differences, ($\psi_{n}$ \-
$\psi^{0}$)/SE($\psi_{n}$), this is the ARE estimate bias compared to the true
ARE given the true region divided by the standard error of the estimates over
the iterations and $\psi_{n}$ \- $\psi^{DA}$)/SE($\psi_{n}$) which is the same
standardized difference but compared to the resulting ARE when the data-
adaptive region is applied to $P_{0}$. Figure 9 shows the sampling
distribution for each sample size with 50 iterations per sample size to
estimate the probability density distribution of the standardized bias
compared to the data-adaptive ARE. We see convergence to a mean 0 normal
sampling distribution as sample size increases for all estimates. Figure 9 A
shows the sampling distribution of the standardized bias of the mean k-fold
AREs compared to the ground-truth ARE. We can see that this sampling
distribution is quite tight around 0. Figures 9 B and C show the sampling
distribution for the harmonic mean and the pooled TMLE estimates which are
mirror reflections of each other. For both estimates, lower sample sizes (such
as in purple $n$ = 200) there is a wider spread of bias (estimates vary more
widely) with z-scores out to 2 or 4 but this distribution gets tighter as
sample size increases.
Figure 9: Bias Standardized by Standard Error Compared to ATE of Data-Adaptive
Rule
Likewise, Figure 10 A-C show the standardized bias of each estimate compared
to the ground-truth region ARE. All estimates generally follow the same
distribution and converge to a 0 mean normal distribution as sample size
increases.
N | Absolute Bias | SD | MSE | Coverage
---|---|---|---|---
200 | 0.574 | 2.058 | 4.565 | 1
1000 | 0.379 | 1.458 | 2.268 | 0.97
5000 | 0.140 | 1.058 | 1.138 | 0.97
Table 1: Simulation results for Estimating the Data-Adaptive ARE using the
Average k-fold Estimates
Table 1 shows the results of the simulations based on comparing the mean fold
estimated ARE to the mean ARE of data-adaptive rules applied to $P_{0}$. It
can be seen that the estimation is unbiased, and the coverage of confidence
intervals based IC-based estimates of the standard errors is slightly high.
Figure 11 shows the sampling distribution for each sample size for each type
of estimate in the three continuous exposures. We see each estimate converge
to a mean 0 normal sampling distribution as sample size increases with the
average k-fold estimate having a tighter distribution.
Figure 10: Bias Standardized by Standard Error Compared to ATE of True Rule
N | Absolute Bias | SD | MSE | Coverage
---|---|---|---|---
200 | 0.608 | 2.10 | 4.797 | 0.95
1000 | 0.382 | 1.437 | 2.210 | 0.95
5000 | 0.178 | 0.894 | 0.831 | 0.96
Table 2: Simulation results for Estimating the Data-Adaptive ARE using the
Average k-fold Estimates in Three Exposure Simulations Figure 11: Bias
Standardized by Standard Error Compared to ARE of Data-Adaptive Rule for Three
Exposures
## 7 Applications
### 7.1 NIEHS Synthetic Mixtures
The NIEHS synthetic mixtures data (found here on github) is a commonly used
data set to evaluated the performance of statistical methods for mixtures.
This synthetic data can be considered the results of a prospective cohort
study. The outcome cannot cause the exposures (as might occur in a cross-
sectional study). Correlations between exposure variables can be thought of as
caused by common sources or modes of exposure. The nuisance variable Z can be
assumed to be a potential confounder and not a collider. There are 7 exposures
($X_{1}-X_{7}$) which have a complicated dependency structure with a
biologically-based dose response function based on endocrine disruption. For
details the github page synthetic data key for data set 1 (used here) gives a
description as to how the data was generated. Largely, there are two exposure
clusters ($X_{1},X_{2},X_{3}$ and $X_{5},X_{6}$). And therefore, correlations
within these clusters are high. $X_{1},X_{2},X_{7}$ contribute positively to
the outcome; $X_{4},X_{5}$ contribute negatively; $X_{3}$ and $X_{6}$ do not
have an impact on the outcome which makes rejecting these variables difficult
given their correlations with cluster group members. This correlation and
effects structure is biologically plausible as different congeners of a group
of compounds (e.g., PCBs) may be highly correlated, but have different
biological effects. There are various agonistic and antagonistic interactions
that exist in the exposures. Table 3 gives a breakdown of the variable sets
and their relationships.
Variables | Interaction Type
---|---
X1 and X2 | Toxic equivalency factor, a special case of concentration addition (both increase Y)
X1 and X4 | Competitive antagonism (similarly for X2 and X4)
X1 and X5 | Competitive antagonism (similarly for X2 and X4)
X1 and X7 | Supra-additive (“synergy”) (similarly for X2 and X7)
X4 and X5 | Toxic equivalency factor, a type of concentration addition (both decrease y)
X4 and X7 | Antagonism (unusual kind) (similarly for X5 and X7)
Table 3: NIEHS Synthetic Data Interactions
Given these toxicological interactions we can expect certain statistical
interactions determined as cut-points for sets of variables from CVtreeMLE.
For example, we might expect a positive ARE attached to a rule for
$X_{1}>=x_{1}\text{ \& }X_{2}>=x_{2}$ where $x_{1},x_{2}$ are certain values
for the respective exposures because these two exposures both have a positive
impact on Y. Likewise, in the case for antagonistic relationships such as in
the case of $X_{2},X_{4}$, we would expect a positive ARE attached to a rule
$X_{2}>=x_{2}\text{ \& }X_{4}<=x_{4}$. This is because we might expect the
outcome to be highest in a region where $X_{2}$ is high and $X_{4}$ is low
given the antagonistic interaction.
The NIEHS data set has 500 observations and 9 variables. Z is a binary
confounder. Of course, in this data there is no ground-truth, like in the
above simulations, but we can gauge CVtreeMLE’s performance by determining if
the correct variable sets are used in the interactions and if the correct
variables are rejected. Because many machine learning algorithms will fail
when fit with one predictor (in our case this happens for g(Z)), we simulate
additional covariates that have no effects on the exposures or outcome but
prevent these algorithms from breaking.
We apply CVtreeMLE to this NIEHS synthetic data using 10-fold CV and the
default stacks of estimators used in the Super Learner for all parameters. We
select for trees with positive coefficients in the ensemble during the data-
adaptive estimation and therefore report results as positive AREs. We
parallelize over the cross-validation to test computational run-time on a
newer personal machine an analyst might be using.
Mixture ATE | Standard Error | Lower CI | Upper CI | P-value | P-value Adj | Vars | Union_Rule
---|---|---|---|---|---|---|---
8.24 | 0.56 | 7.14 | 9.34 | 0.00 | 0.00 | X1-X5 | X1 $>$= 0.267 & X5 $<$= 3.189
8.16 | 0.56 | 7.07 | 9.26 | 0.00 | 0.00 | X1-X7 | X1 $>$= 0.326 & X7 $>$= 0.22
6.68 | 0.62 | 5.46 | 7.89 | 0.00 | 0.00 | X2-X5 | X2 $>$= 0.602 & X5 $<$= 3.189
6.82 | 0.58 | 5.68 | 7.95 | 0.00 | 0.00 | X2-X7 | X2 $>$= 0.619 & X7 $>$= 1.171
7.29 | 0.51 | 6.29 | 8.29 | 0.00 | 0.00 | X5-X7 | X5 $<$= 3.269 & X7 $>$= 0.138
Table 4: NIEHS Synthetic Data Consistent Interaction Results
Table 4 shows the results from CVtreeMLE when applied to this NIEHS synthetic
data set using the aforementioned settings. We filter results to only
interactions that were found in all 10-folds, that is, trees with variable
sets found across all the folds and therefore have consistent "signal" in the
data. Let’s focus on the second row with variables $X_{1}$ and $X_{7}$. Table
3 shows that these two variables have a supra-additive or synergistic non-
additive relationship. The union rule for trees including these two variables
was $X1>=0.326\text{ \& }X7>=0.22$ meaning this rule covers all observations
indicated by the fold-specific rules. The mixture ARE is then interpreted as,
if all individuals were exposed to $X_{1}$ at levels at or greater than 0.326
and exposed to levels of $X_{7}$ at or greater than 0.22 the outcome would be
8.16 units greater compared to if all individuals were exposed to levels less
than these respective levels. The subsequent standard errors derived from the
pooled influence curve (column 2) are used to derive the confidence intervals
and p-values for hypothesis testing. Overall, comparing these statistical
interactions to the toxicological interactions listed CVtreeMLE identifies 5
of 9 interactions. The other interactions in the above table are interpreted
in the same way as the $X_{1}$ and $X_{7}$ interaction.
We next can investigate how consistent the results are across the folds by
looking at the k-fold specific results, this gives us a sense of how reliable
our ARE estimates are for the pooled rule. Let’s dig deeper into this $X_{1}$
and $X_{7}$ interaction. Table 5 shows the k-fold specific results for the
interactions found for the variables $X_{1}$ and $X_{7}$. Each row is the
results for each fold and the final row is the inverse variance weighted
pooled result, pooling estimates across the folds. Estimates show stability
across the folds with only one fold, fold 8, deviating from the trend. Cut-
points at $X_{1}$ were either at 0.991 or 0.998 with fold 8 having a lower
cut-point of 0.319. Likewise, $X_{7}$ was partitioned at 0.48 in most folds.
Each fold-specific result has valid inference however it is also necessary to
evaluate how consistent results were across the folds and thus determine if
partitions are stable. Here we see the $X1>=0.99\text{ \& }X7>=0.48$ partition
for these two variables is stable and found in 8 of the folds. The ARE
estimate for these rules ranges from 8-9 all with a significant effect.
CVtreeMLE also provides plots of k-fold estimates to more easily assess for
trends, Figure 12 gives an example of this plot for the interaction $X_{1}$
and $X_{7}$.
Mixture ATE | SE | Lower CI | Upper CI | P-Value | P-Value Adj | Mix Rule | Fold
---|---|---|---|---|---|---|---
8.09 | 1.03 | 6.08 | 10.10 | 0.00 | 0.00 | X1 $>$ 0.998 & X7 $>$ 0.48 | 1
7.12 | 1.63 | 3.92 | 10.32 | 0.00 | 0.00 | X1 $>$ 0.998 & X7 $>$ 2.108 | 2
8.38 | 1.51 | 5.41 | 11.34 | 0.00 | 0.00 | X1 $>$ 0.991 & X7 $>$ 0.48 | 3
9.42 | 2.23 | 5.05 | 13.80 | 0.00 | 0.00 | X1 $>$ 0.991 & X7 $>$ 0.441 | 4
8.81 | 1.76 | 5.36 | 12.26 | 0.00 | 0.00 | X1 $>$ 0.998 & X7 $>$ 0.439 | 5
8.43 | 1.53 | 5.42 | 11.43 | 0.00 | 0.00 | X1 $>$ 1.049 & X7 $>$ 0.482 | 6
8.84 | 1.84 | 5.23 | 12.44 | 0.00 | 0.00 | X1 $>$ 0.991 & X7 $>$ 0.413 | 7
5.20 | 2.69 | -0.07 | 10.48 | 0.05 | 0.53 | X7 $>$ 0.217 & X1 $>$ 0.319 | 8
8.41 | 1.04 | 6.36 | 10.46 | 0.00 | 0.00 | X1 $>$ 0.991 & X7 $>$ 0.482 | 9
8.94 | 1.39 | 6.22 | 11.67 | 0.00 | 0.00 | X1 $>$ 0.991 & X7 $>$ 0.482 | 10
8.30 | 5.49 | -2.45 | 19.05 | 0.13 | 0.13 | X1 $>$= 0.326 & X1 $<$= 4.687 & X7 $>$= 0.22 & X7 $<$= 4.886 | Pooled
Table 5: $X_{1}$ and $X_{7}$ k-fold Interaction Results Figure 12: K-fold
specific results for the interaction $X_{1}$ and $X_{7}$
Overall, CVtreeMLE is able to determine subspaces in the respective variables
that have the most impact on the endocrine disrupting outcome. Of note is the
fact that no interactions include the variables $X_{3}$ and $X_{6}$ both of
which have no impact on the outcome.
#### 7.1.1 Comparison to Existing Methods
Currently, quantile g-computation is a popular method for mixture analysis in
environmental epidemiology. The method yields estimates of the effect of
increasing all exposures by one quantile, simultaneously under linear model
assumptions. Quantile g-computation looks like:
$Y_{i}=\beta_{0}+\sum_{j=1}^{d}\beta_{j}X_{ji}^{q}+\beta Z_{i}+\epsilon_{i}$
Where $X^{q}$ are the quantized mixture components and $Z$ are the covariates.
Which works by first transforming mixture components into quantiles. Then the
negative and positive coefficients from a linear model for the mixture
components are summed to give a mixture ($\Psi$) summary measure which
characterizes the joint impact. There are many assumptions that should be
poignant after our discussion of mixtures. Firstly, quantiles may not
characterize the exposure-response relationship (could be non-monotonic) which
occurs in endocrine disrupting compounds. For interpretable weights and
mixture estimate $\Psi$, assumes additive relationship of quantiles ($\Psi$ is
just sum of $\beta$’s in front of mixture components). After our discussion,
in mixtures our main goal is model possible interactions in the data because
we expect exposures to have non-additive, possible non-monotonic, antagonistic
and agonistic relationships. Therefore, we should expect interactions in our
mixture data. In quantile g-computation, with the inclusion of interactions,
the proportional contribution of an exposure to the overall effect then varies
according to levels of other variables and therefore weights cannot be
estimated. Because we can never assume no interactions, quantile g-computation
then boils down to getting conditional expectations when setting mixtures to
quantiles through a linear model with interaction terms specified by the
analyst. After our discussion of mixtures this should feel incorrect. As we
argue, the important variables, relationships, and thresholds in a mixture are
all unknown to the analyst which makes this a data-adaptive target parameter
problem. Even testing quantile g-computation on the NIEHS data is difficult
because we don’t know what interactions to include a priori. The best we can
do is run it out of the box and with two-way interactions and compare results
to the ground-truth measures. Lastly, quantile g-computation does not flexibly
control for covariates.
We run quantile g-computation on the NIEHS data using 4 quantiles with no
interactions to investigate results using this model. The scaled effect size
(positive direction, sum of positive coefficients) was 6.28 and included
$X_{1},X_{2},X_{3},X_{7}$ and the scaled effect size (negative direction, sum
of negative coefficients) was -3.68 and included $X_{4},X_{5},X_{6}$. Compared
to the NIEHS ground-truth, $X_{3},X_{6}$ are incorrectly included in these
estimates. However the positive and negative associations for the other
variables are correct.
Next, because we expect interactions to exist in the mixture data, we would
like to assess for them but the question is which interaction terms to
include? Our best guess is to include interaction terms for all the exposures.
We do this and show results in Table 6.
| Estimate | Std. Error | Lower CI | Upper CI | Pr($>$$|$t$|$)
---|---|---|---|---|---
(Intercept) | 21.29 | 1.58 | 18.19 | 24.39 | 0.00
psi1 | 0.02 | 1.62 | -3.16 | 3.20 | 0.99
psi2 | 0.59 | 0.67 | -0.71 | 1.90 | 0.37
Table 6: Quantile G-Computation Interaction Results from NIEHS Synthetic Data
In Table 6 $\Psi_{1}$ is the summary measure for main effects and $\Psi_{2}$
for interactions. As can be seen, when including all interactions neither of
the estimates are significant. Of course this is to be expected given the
number of parameters in the model and sample size $n=500$. However, moving
forward with interaction assessment is difficult, if we were to assess for all
2-way interaction of 7 exposures the number of sets is 21 and with 3-way
interactions is 35. We’d have to run this many models and then correct for
multiple testing. Hopefully this example shows why mixtures are inherently a
data-adaptive problem and why popular methods such as this, although succinct
and interpretable, fall short even in a simple synthetic data set.
### 7.2 NHANES Data
Environmental chemical and metal exposure can affect telomere length, which is
a mediating pathway for adverse health outcomes including cancer. Studies on
the association of metals with leucocyte telomere length (LTL) are mainly
limited to single-metal effect Xia et al. (2022); Zota et al. (2015); Clarity
et al. (2021). Some research has investigated the overall joint associations
metal mixture with LTL using parameteric models like multiple linear
regression or quantile-sum g-computation Keil et al. (2019); Lai et al.
(2022). Since environmental stressors are known to disrupt telomere length
homeostasis which plays a crucial role in cellular aging and disease we
investigate the association of mixed metal exposure on LTL. Our desires are
two fold, 1. to show CVtreeMLE results applied to real-world data and 2. to
provide such data and data processing code with the open-source CVtreeMLE
package. As such, we develop a pipeline to download and clean National Health
and Nutrition Examination Survey (NHANES) dataset. We download and format the
relevant NHANES 1999–2002 dataset containing demographic data, disease
history, nine urine metals, and LTL. The demographic data used as possible
confounders ($W$) include age, gender, race, education level, marital status,
alcohol, smoking (cotinine) , body mass index (BMI), family poverty ratio
(PIR), fasting glucose, systolic and diastolic blood pressure, exercise and
birth country. Urine metal contained barium (Ba), cadmium (Cd), cobalt (Co),
cesium (Cs), molybdenum (Mo), lead (Pb), antimony (Sb), thallium (Tl) and
tungsen (W). The outcome is LTL. The number of observations in this test data
is 2510. The coding pipeline and data are available in the CVtreeMLE package.
We apply CVtreeMLE using the default learners in each stack. We use 10-fold CV
and set the max number of iterations in the iterative backfitting to 10 as
well. Because previous research has shown the exposure to metals shortens LTL,
we set the ATE direction to negative to select trees in the data-adaptive
procedure which have the minimum (negative) impact and thus return negative
ATEs for each fold.
Mixture ATE | Standard Error | Lower CI | Upper CI | P-value | P-value Adj | Vars | Union_Rule | %Fold
---|---|---|---|---|---|---|---|---
0.06 | 0.04 | -0.02 | 0.13 | 0.17 | 1.00 | cadmium-molybdenum | cadmium $>$= 0 & cadmium $<$= 0.715 & molybdenum $>$= 19.8 & molybdenum $<$= 436.8 | 0.80
-0.03 | 0.01 | -0.06 | -0.01 | 0.02 | 0.37 | cadmium-thallium | cadmium $>$= 0.027 & cadmium $<$= 36.777 & thallium $>$= 0.01 & thallium $<$= 0.38 | 1.00
Table 7: Consistent Pooled TMLE Results NHANES Metal Mixture-LTL
Table 7 shows the pooled TMLE ARE results for rules found in more than 75% of
the folds. Here we see rules including cadmium and thallium were found in all
the folds and rules including cadmium and molybdenum were found in 80% of the
folds. The cadmium-thallium interaction had a significant ARE of -0.03 and the
cadmium-molybdenum was borderline significant with an ARE of 0.06. These
results show that, exposure to high levels of cadmium >= 0.027 and low levels
of thallium <= 0.38 is associated with a reduced telomere length of 0.03
compared to exposure levels of cadmium levels lower than 0.027 and thallium
levels greater than 0.38. This result implies an antagonistic relationship
between cadmium and thallium. Likewise, telomere length was longer (0.06) for
those exposed to low levels of cadmium <= 0.715 and high levels of molybdenum
>= 19.8 compared to those exposed to the inverse exposure region for these two
metals.
Like the NIEHS synthetic data results, we can investigate the k-fold specific
results for these pooled results. Let’s look at the cadmium and thallium
interaction in each fold to see how stable the partition points were for each
metal.
ATE | SE | Lower CI | Upper CI | P-Value | P-Value Adj | Mix Rule
---|---|---|---|---|---|---
-0.03 | 0.06 | -0.15 | 0.09 | 0.67 | 1.00 | thallium $<$= 0.21 & cadmium $>$ 0.243
-0.04 | 0.03 | -0.10 | 0.03 | 0.24 | 1.00 | cadmium $>$ 0.295 & thallium $<$= 0.31
-0.03 | 0.03 | -0.09 | 0.03 | 0.32 | 1.00 | thallium $<$= 0.21 & cadmium $>$ 0.101
-0.05 | 0.04 | -0.13 | 0.02 | 0.14 | 1.00 | cadmium $>$ 0.097 & thallium $<$= 0.38
-0.03 | 0.03 | -0.09 | 0.03 | 0.37 | 1.00 | cadmium $>$ 0.295 & thallium $<$= 0.21
-0.03 | 0.05 | -0.12 | 0.06 | 0.54 | 1.00 | thallium $<$= 0.21 & cadmium $>$ 0.143
-0.04 | 0.08 | -0.20 | 0.12 | 0.62 | 1.00 | cadmium $>$ 0.29 & thallium $<$= 0.21
-0.04 | 0.05 | -0.14 | 0.06 | 0.44 | 1.00 | thallium $<$= 0.36 & cadmium $>$ 0.092
-0.02 | 0.06 | -0.13 | 0.10 | 0.76 | 1.00 | cadmium $>$ 0.027 & thallium $<$= 0.14
-0.03 | 0.04 | -0.11 | 0.05 | 0.52 | 1.00 | thallium $<$= 0.22 & cadmium $>$ 0.254
-0.03 | 0.16 | -0.34 | 0.27 | 0.83 | 0.83 | cadmium $>$= 0.027 & thallium $<$= 0.38
Table 8: K-fold specific results for cadmium-thallium interactions associated
with LTL
Table 8 shows the k-fold specific results for cadmium and thallium
interaction. This interaction was found in all the folds with an ARE ranging
from -0.02 to -0.05. None of the fold specific results were significant due to
the variance estimates being calculated on the 251 observations in each
validation fold, making standard errors high. However, we see consistent
partitioning of thallium between 0.14 and 0.38 and partitioning of cadmium
between 0.027 and 0.29. Overall, we see consistent cut-points across the folds
which indicates this interaction is stable. The last row in this table is the
inverse weighted pooled results. Here we can see that we gain much power by
using the pooled influence curve in the pooled TMLE procedure which is able to
borrow variance information across the folds because all estimates are cross-
estimated. Here, we can see the pooled estimated has much higher variance and
wider confidence intervals.
ATE | SE | Lower CI | Upper CI | P-Value | P-Value Adj | Mix Rule | fold
---|---|---|---|---|---|---|---
0.05 | 0.04 | -0.04 | 0.14 | 0.30 | 1.00 | molybdenum $>$ 55.2 & cadmium $<$= 0.384 | 2
0.04 | 0.10 | -0.15 | 0.23 | 0.68 | 1.00 | molybdenum $>$ 52.9 & cadmium $<$= 0.368 | 3
0.01 | 0.04 | -0.07 | 0.09 | 0.77 | 1.00 | cadmium $<$= 0.715 & molybdenum $>$ 19.7 | 4
0.11 | 0.14 | -0.16 | 0.37 | 0.42 | 1.00 | cadmium $<$= 0.35 & molybdenum $>$ 102.5 | 5
0.05 | 0.05 | -0.04 | 0.14 | 0.27 | 1.00 | cadmium $<$= 0.292 & molybdenum $>$ 57.2 | 6
0.02 | 0.22 | -0.41 | 0.46 | 0.92 | 1.00 | cadmium $<$= 0.124 & molybdenum $>$ 21.7 | 7
0.02 | 0.06 | -0.09 | 0.14 | 0.71 | 1.00 | cadmium $<$= 0.429 & molybdenum $>$ 44.5 | 8
0.14 | 0.28 | -0.42 | 0.69 | 0.63 | 1.00 | cadmium $<$= 0.131 & molybdenum $>$ 55.6 | 9
0.04 | 0.41 | -0.77 | 0.84 | 0.93 | 0.93 | cadmium $>$= 0 & cadmium $<$= 0.715 & molybdenum $>$= 19.8 & molybdenum $<$= 436.8 | Pooled
Table 9: K-fold specific results for cadmium-molybdenum interactions
associated with LTL
Lastly, we look at the cadmium-molybdenum interactions in Table 9 . As we can
see here, interactions are not found in every fold and the partition points
have a larger range although they all point in the same direction (low cadmium
and high molybdenum) and all fold specific results are positive. This makes
sense given that molybdenum processes proteins and genetic material like DNA
and helps break down drugs and toxic substances that enter the body.
Therefore, we would expect low cadmium and high molybdenum to be associated
with higher telomere length.
Overall, in this NHANES example, we show that in real world data, CVtreeMLE
can answer questions regarding expected outcomes under different exposure
levels of a mixture which are otherwise occult given the limitation of
existing methods.
## 8 Software
The development of asymptotically linear estimators for data-adaptive
parameters are critical for the field of mixed exposure statistics. However,
the development of open-source software which translates semi-parametric
statistical theory into well-documented functional software is a formidable
challenge. Such implementation requires understanding of causal inference,
semi-parametric statistical theory, machine learning, and the intersection of
these disciplines. The CVtreeMLE R package provides researchers with an open-
source tool for evaluating the causal effects of a mixed exposure using the
methodology described here. The CVtreeMLE package is well documented and
includes a vignette detailing semi-parametric theory for data-adaptive
parameters, examples of output, results with interpretations under various
real-life mixture scenarios, and comparison to existing methods. The NIEHS
synthetic data and the NHANES mixed metal exposure data are provided. The
NIEHS synethetic data application is used in the vignette of the package which
makes these results reproducible to any researcher and likewise the NHANES
data and code are provided for reproducibility. CVtreeMLE can run sequentially
or parallelized across folds using the furrr package Vaughan and Dancho
(2022). New statistical software using machine learning often presume the
availability of significant computational resources in order to run in a
timely manner. Here, our applications of NIEHS and NHANES were all run on a
personal macbook machine in under 30 minutes by utilizing parallelization and
using flexible yet efficient estimators. Of course, for the simulations high
performance computing was used to parallelize iteration over clusters. To-date
in scientific publication, the release of reproducible software is the
exception rather than the rule. In an effort to make robust statistical
software adopted in the future, rather than reliance of simple parameteric
models, we make CVtreeMLE available with clear, easily accessible, highly
detailed documentation of the coding methods. We also make all functions user-
accessible, and develop numerous tests and examples. Coding notebooks show
simulations of mixed exposure data and CVtreeMLE output with detailed
summaries of interpretation. Lastly, the CVtreeMLE package is well maintained
to ensure accessibility with ongoing improvements tested at each iteration.
The CVtreeMLE package has been made publicly available via GitHub. The pseduo-
code that describes the code which executes the described method is shown in
Figure 13.
Figure 13: CVtreeMLE Pseudo-Code
## 9 Discussion
In this paper we introduce a new method for estimating the effects of a mixed
exposure. Our approach treats ensemble decision trees as a data-adaptive
target parameter for which we estimate the average effects of exposure for
regions identified in the best fitting decision trees. This is done within a
cross-validated framework paired with targeted learning of our target
parameter which provides estimates that are asymptotically unbiased and have
the lowest variance for studies which satisfy the unconfoundedness and
positivity assumptions. Our proposed method provides valid confidence
intervals without restrictions on the number of exposure, covariates, or the
complexity of the data-generating process. Our method first partitions the
exposure space into subspaces or regions that best explains the outcome. The
output of our method is the exposure effect and respective confidence
intervals if all individual were exposed to the exposure region compared to if
all individual were not exposed to this region. Our approach has potentially
many important applications including identifying what combinations of drugs
lead to the most beneficial patient outcomes as well as finding what
combinations of pollution chemicals have the most deleterious outcomes on
public health. Our approach allows for "dredging with dignity" wherein
exposure regions can be discovered in the data which are not known a priori
and still provide unbiased estimates for the target parameter with valid
confidence intervals. This approach of course comes with some cost as
construction of a pooled region across the folds is rather ad hoc. This is the
main limitation in the proposed method and other alternatives may exist such
as using the average partitioning values of each exposure variable rather than
our union approach which is conservative. Our simulations with ground-truth,
NIEHS synethetic data and real-world data application show the robustness and
interpretability of our approach. In an effort to make adoption of semi-
parametric methods such as this more seamless we provide the CVtreeMLE R
package on github which is well documented for analysts to apply to their
respective data.
## 10 Appendix
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11institutetext: Paper ID 555611institutetext: King Abdullah University of
Science and Tehchnology (KAUST), Saudi Arabia 22institutetext: University of
Oxford, United Kingdom
22email<EMAIL_ADDRESS>
# On the Robustness of Quality Measures for GANs
Anonymous ECCV submission Motasem Alfarra 11 Juan C. Pérez 11 Anna Frühstück
11 Philip H. S. Torr 22
Peter Wonka 11 Bernard Ghanem 11
###### Abstract
This work evaluates the robustness of quality measures of generative models
such as Inception Score (IS) and Fréchet Inception Distance (FID). Analogous
to the vulnerability of deep models against a variety of adversarial attacks,
we show that such metrics can also be manipulated by additive pixel
perturbations. Our experiments indicate that one can generate a distribution
of images with very high scores but low perceptual quality. Conversely, one
can optimize for small imperceptible perturbations that, when added to real
world images, deteriorate their scores. We further extend our evaluation to
generative models themselves, including the state of the art network
StyleGANv2. We show the vulnerability of both the generative model and the FID
against additive perturbations in the latent space. Finally, we show that the
FID can be robustified by simply replacing the standard Inception with a
robust Inception. We validate the effectiveness of the robustified metric
through extensive experiments, showing it is more robust against
manipulation.111Code: https://github.com/R-FID-Robustness-of-Quality-Measures-
for-GANs
###### Keywords:
Generative Adversarial Networks, Perceptual Quality, Adversarial Attacks,
Network Robustness
## 1 Introduction
Figure 1: Does the Fréchet Inception Distance (FID) accurately measure the
distances between image distributions? We generate datasets that demonstrate
the unreliability of FID in judging perceptual (dis)similarities between image
distributions. The top left box shows a sample of a dataset constructed by
introducing imperceptible noise to each ImageNet image. Despite the remarkable
visual similarity between this dataset and ImageNet (bottom box), an extremely
large FID (almost 8000) between these two datasets showcases FID’s failure to
capture perceptual similarities. On the other hand, a remarkably low FID
(almost 1.0) between a dataset of random noise images (samples shown in the
top right box) and ImageNet illustrates FID’s failure to capture perceptual
dissimilarities.
Deep Neural Networks (DNNs) are vulnerable to small imperceptible
perturbations known as adversarial attacks. For example, while two inputs $x$
and $(x+\delta)$ can be visually indistinguishable to humans, a classifier $f$
can output two different predictions. To address this deficiency in DNNs,
adversarial attacks [11, 7] and defenses [20, 27] have prominently emerged as
active areas of research. Starting from image classification [28], researchers
also assessed the robustness of DNNs for other tasks, such as segmentation
[1], object detection [30], and point cloud classification [18]. While this
lack of robustness questions the reliability of DNNs and hinders their
deployment in the real world, DNNs are still widely used to evaluate
performance in other computer vision tasks, such as that of generation.
Metrics in use for assessing generative models in general, and Generative
Adversarial Networks (GANs) [10] in particular, are of utmost importance in
the literature. This is because such metrics are widely used to establish the
superiority of a generative model over others, hence guiding which GAN should
be deployed in real world. Consequently, such metrics are expected to be not
only useful in providing informative statistics about the distribution of
generated images, but also reliable and robust. In this work, we investigate
the robustness of metrics used to assess GANs. We first identify two
interesting observations that are unique to this context. First, current GAN
metrics are built on pretrained classification DNNs that are nominally trained
(i.e. trained on clean images only). A popular DNN of choice is the Inception
model [25], on which the Inception Score (IS) [22] and Fréchet Inception
Distance (FID) [12] rely. Since nominally trained DNNs are generally
vulnerable to adversarial attacks [7], it is expected that DNN-based metrics
for GANs also inherit these vulnerabilities. Second, current adversarial
attacks proposed in the literature are mainly designed at the instance level
(e.g. fooling a DNN into misclassifying a particular instance), while GAN
metrics are distribution-based. Therefore, attacking these distribution-based
metrics requires extending attack formulations from the paradigm of instances
to that of distributions.
In this paper, we analyze the robustness of GAN metrics and recommend
solutions to improve their robustness. We first attempt to assess the
robustness of the quality measures used to evaluate GANs. We check whether
such metrics are actually measuring the quality of image distributions by
testing their vulnerability against additive pixel perturbations. While these
metrics aim at measuring perceptual quality, we find that they are extremely
brittle against imperceptible but carefully-crafted perturbations. We then
assess the judgment of such metrics on the image distributions generated by
StyleGANv2 [15] when its input is subjected to perturbations. While the output
of GANs is generally well behaved, we still observe that such metrics provide
inconsistent judgments where, for example, FID favors an image distribution
with significant artifacts over more naturally-looking distributions. At last,
we endeavor to reduce these metrics’ vulnerability by incorporating robustly-
trained models.
We summarize our contributions as follows:
* •
We are the first to provide an extensive experimental evaluation of the
robustness of the Inception Score (IS) and the Fréchet Inception Distance
(FID) against additive pixel perturbations. We propose two instance-based
adversarial attacks that generate distributions of images that fool both IS
and FID. For example, we show that perturbations $\delta$ with a small budget
(i.e. $\|\delta\|_{\infty}\leq 0.01$) are sufficient to increase the FID
between ImageNet [8] and a perturbed version of ImageNet to $\sim 7900$, while
also being able to generate a distribution of random noise images whose FID to
ImageNet is $1.05$. We illustrate both cases in Figure 1.
* •
We extend our evaluation to study the sensitivity of FID against perturbations
in the latent space of state-of-the-art generative models. In this setup, we
show the vulnerability of both StyleGANv2 and FID against perturbations in
both its $z$\- and $w$\- spaces. We found that FID provides inconsistent
evaluation of the distribution of generated images compared to their visual
quality. Moreover, our attack in the latent space causes StyleGANv2 to
generate images with significant artifacts, showcasing the vulnerability of
StyleGANv2 to additive perturbations in the latent space.
* •
We propose to improve the reliability of FID by using adversarially-trained
models in its computation. Specifically, we replace the traditional Inception
model with its adversarially-trained counterpart to generate the embeddings on
which the FID is computed. We show that our robust metric, dubbed R-FID, is
more resistant against pixel perturbations than the regular FID.
* •
Finally, we study the properties of R-FID when evaluating different GANs. We
show that R-FID is better than FID at distinguishing generated fake
distributions from real ones. Moreover, R-FID provides more consistent
evaluation under perturbations in the latent space of StyleGANv2.
## 2 Related Work
GANs and Automated Assessment. GANs [10] have shown remarkable generative
capabilities, specially in the domain of images [14, 15, 4]. Since the advent
of GANs, evaluating their generative capabilities has been challenging [10].
This challenge spurred research efforts into developing automated quantitative
measures for GAN outputs. Metrics of particular importance for this purpose
are the Inception Score (IS), introduced in [22], and the Fréchet Inception
Distance (FID), introduced in [12]. Both metrics leverage the ImageNet-
pretrained Inception architecture [25] as a rough proxy for human perception.
The IS evaluates the generated images by computing conditional class
distributions with Inception and measuring (1) each distribution’s
entropy—related to Inception’s certainty of the image content, and (2) the
marginal’s entropy—related to diversity across generated images. Noting the IS
does not compare the generated distribution to the (real world) target
distribution, Heusel et al. [12] proposed the FID. The FID compares the
generated and target distributions by (1) assuming the Inception features
follow a Gaussian distribution and (2) using each distribution’s first two
moments to compute the Fréchet distance. Further, the FID was shown to be more
consistent with human judgement [24].
Both the original works and later research criticized these quantitative
assessments. On one hand, IS is sensitive to weight values, noisy estimation
when splitting data, distribution shift from ImageNet, susceptibility to
adversarial examples, image resolution, difficulty in discriminating GAN
performance, and vulnerability to overfitting [2, 22, 3, 29]. On the other
hand, FID has been criticized for its over-simplistic assumptions
(“Gaussianity” and its associated two-moment description), difficulty in
discriminating GAN performance, and its inability to detect overfitting [3,
19, 29]. Moreover, both IS and FID were shown to be biased to both the number
of samples used and the model to be evaluated [6]. In this work, we provide
extensive empirical evidence showing that both IS and FID are not robust
against perturbations that modify image quality. Furthermore, we also propose
a new robust FID metric that enjoys superior robustness.
Adversarial Robustness. While DNNs became the de facto standard for image
recognition, researchers found that such DNNs respond unexpectedly to small
changes in their input [26, 11]. In particular, various works [5, 20] observed
a widespread vulnerability of DNN models against input perturbations that did
not modify image semantics. This observation spurred a line of research on
adversarial attacks, aiming to develop procedures for finding input
perturbations that fool DNNs [7]. This line of work found that these
vulnerabilities are pervasive, casting doubt on the nature of the impressive
performances of DNNs. Further research showed that training DNNs to be robust
against these attacks [20] facilitated the learning of perceptually-correlated
features [13, 9]. Interestingly, a later work [23] even showed that such
learnt features could be harnessed for image synthesis tasks. In this work, we
show (1) that DNN-based scores for GANs are vulnerable against adversarial
attacks, and (2) how these scores can be “robustified” by replacing nominally
trained DNNs with robustly trained ones.
## 3 Robustness of IS and FID
To compare the output of generative models, two popular metrics are used: the
_Inception Score_ (IS) and the _Fréchet Inception Distance_ (FID). These
metrics depend only on the statistics of the distribution of generated images
in an ImageNet-pretrained Inception’s embedding space, raising the question:
_What do quality measures for generative models, such as IS and FID, tell us
about image quality?_
We investigate this question from the robustness perspective. In particular,
we analyze the sensitivity of these metrics to carefully crafted
perturbations. We start with preliminary background about both metrics.
### 3.1 Preliminaries
We consider the standard image generation setup where a generator
$G:\mathbb{R}^{d_{z}}\rightarrow\mathbb{R}^{d_{x}}$ receives a latent code
$z\in\mathbb{R}^{d_{z}}$ and outputs an image $x\in\mathbb{R}^{d_{x}}$. Upon
training, $G$ is evaluated based on the quality of the generated distribution
of images $\mathcal{D}_{G}$ by computing either the IS [22] or the FID [12].
Both metrics leverage an ImageNet-pretrained [8] InceptionV3 [25]. Salimans et
al. [22] proposed measuring the perceptual quality of the generated
distribution $\mathcal{D}_{G}$ by computing the IS as:
$\text{IS}(\mathcal{D}_{G})=\exp\left(\>\mathbb{E}_{x\sim\mathcal{D}_{G}}\left(\text{KL}\left(p(y|x)\>||\>p(y)\right)\right)\>\right),$
(1)
where $p(y|x)$ is the output probability distribution of the pretrained
Inception model. While several works have argued about the effectiveness of
the IS and its widely-used implementation [2], its main drawback is that it
disregards the relation between the generated distribution, $\mathcal{D}_{G}$,
and the real one, $\mathcal{D}_{R}$, used for training $G$ [12]. Consequently,
Heusel et al. proposed the popular FID, which involves the statistics of the
real distribution. In particular, FID assumes that the Inception features of
an image distribution $\mathcal{D}$ follow a Gaussian distribution with mean
$\mu_{\mathcal{D}}$ and covariance $\Sigma_{\mathcal{D}}$, and it measures the
squared Wasserstein distance between the two Gaussian distributions of real
and generated images. Hence, $\text{FID}(\mathcal{D}_{R},\mathcal{D}_{G})$, or
FID for short, can be calculated as:
$\text{FID}=\|\mu_{R}-\mu_{G}\|^{2}+\text{Tr}\left(\Sigma_{R}+\Sigma_{G}-2(\Sigma_{R}\Sigma_{G})^{\nicefrac{{1}}{{2}}}\right),$
(2)
where $._{R},._{G}$ are the statistics of the real and generated image
distributions, respectively, and $\text{Tr}(\cdot)$ is the trace operator.
Note that the statistics of both distributions are empirically estimated from
their corresponding image samples. In principle, FID measures how close
(realistic) the generated distribution $\mathcal{D}_{G}$ is to
$\mathcal{D}_{R}$. We remark that the FID is the _de facto_ metric for
evaluating image generation-related tasks. Therefore, our study focuses mostly
on FID.
We note here that both the IS and the FID are oblivious to $G$’s training
process and can be computed to compare two arbitrary sets of images
$\mathcal{D}_{R}$ and $\mathcal{D}_{G}$. In generative modeling, this is
typically a set of real images (photographs) and a set of generated images.
However, it is also possible to compare two sets of photographs, two sets of
generated images, manipulated photographs with real photographs, _etc_. This
flexibility allows us to study these metrics in a broader context next, where
no generative model is involved.
### 3.2 Robustness under Pixel Perturbations
We first address the question presented earlier in Section 3 by analyzing the
sensitivity of IS and FID to additive pixel perturbations. In particular, we
assume $\mathcal{D}_{R}$ to be either CIFAR10 [17] or ImageNet [8] and ask:
(i) can we generate a distribution of imperceptible additive perturbations
$\delta$ that deteriorates the scores for
$\mathcal{D}_{G}=\mathcal{D}_{R}+\delta$? Or, alternatively, (ii) can we
generate a distribution of low visual quality images, i.e. noise images, that
attain good quality scores? If the answer is yes to both questions, then FID
and IS have limited capacity for providing information about image quality in
the worst case.
#### 3.2.1 Good Images - Bad Scores
Figure 2: Sensitivity of Inception Score (IS) against pixel perturbations.
First row: real-looking images (sampled from
$\mathcal{D}_{G}=\mathcal{D}_{R}+\delta$) with a low IS (below 3). Second row:
random noise images with a high IS (over 135).
We aim at constructing a distribution of real-looking images with _bad_
quality measures, i.e. low IS or high FID. While both metrics are
distribution-based, we design instance-wise proxy optimization problems to
achieve our goal.
Minimizing IS. Based on Eq. (1), one could minimize the IS by having both the
posterior $p(y|x)$ and the prior $p(y)$ be the same distribution. Assuming
that $p(y)$ is a uniform distribution, we minimize the IS by maximizing the
entropy of $p(y|x)$. Therefore, we can optimize a perturbation $\delta^{*}$
for each real image $x_{r}\sim\mathcal{D}_{R}$ by solving the following
problem:
$\displaystyle\delta^{*}=\operatorname*{arg\,max}_{\|\delta\|_{\infty}\leq\epsilon}~{}\mathcal{L}_{\text{ce}}\left(p(y|x_{r}+\delta),\hat{y}\right),$
(3) $\displaystyle\text{s.t.
}\hat{y}=\operatorname*{arg\,max}_{i}~{}p^{i}(y|x_{r}+\delta),$
where $\mathcal{L}_{\text{ce}}$ is the Cross Entropy loss. We solve the
problem in Eq. (3) with 100 steps of Projected Gradient Descent (PGD) and zero
initialization. We then compile the distribution $\mathcal{D}_{G}$, where each
image $x_{g}=x_{r}+\delta^{*}$ is a perturbed version of the real dataset
$\mathcal{D}_{R}$. Note that our objective aims to minimize the network’s
confidence in predicting all labels for each $x_{g}$. In doing so, both
$p(y|x_{g})$ and $p(y)$ tend to converge to a uniform distribution, thus,
minimizing the KL divergence between them and effectively lowering the IS.
Note how $\epsilon$ controls the allowed perturbation amount for each image
$x_{r}$. Therefore, for small $\epsilon$ values, samples from
$\mathcal{D}_{G}$ and $\mathcal{D}_{R}$ are perceptually indistinguishable.
Table 1: Robustness of IS and FID against pixel perturbations. We assess the robustness of IS and FID against perturbations with a limited budget $\epsilon$ on CIFAR10 and ImageNet. In the last row, we report the IS and FID of images with carefully-designed random noise having a resolution similar to CIFAR10 and ImageNet. $\epsilon$ | CIFAR10 | ImageNet
---|---|---
| IS | FID | IS | FID
0.00 | 11.54 | 0.00 | 250.74 | 0.00
$5\times 10^{-3}$ | 2.62 | 142.45 | 3.08 | 3013.33
$0.01$ | 2.50 | 473.19 | 2.88 | 7929.01
random noise | 94.87 | 9.94 | 136.82 | 1.05
Maximizing FID. Next, we extend our attack setup to the more challenging FID.
Given an image $x$, we define
$f(x):\mathbb{R}^{d_{x}}\rightarrow\mathbb{R}^{d_{e}}$ to be the output
embedding of an Inception model. We aim to maximize the FID by generating a
perturbation $\delta$ that pushes the embedding of a real image away from its
original position. In particular, for each $x_{r}\sim\mathcal{D}_{R}$, we aim
to construct $x_{g}=x_{r}+\delta^{*}$ where:
$\delta^{*}=\operatorname*{arg\,max}_{\|\delta\|_{\infty}\leq\epsilon}~{}\left\|f(x_{r})-f(x_{r}+\delta)\right\|_{2}.$
(4)
In our experiments, we solve the optimization problem in Eq. (4) with 100 PGD
steps and a randomly initialized $\delta$ [20]. Maximizing this objective
indirectly maximizes FID’s first term (Eq. (2)), while resulting in a
distribution of images $\mathcal{D}_{G}$ that is visually indistinguishable
from the real $\mathcal{D}_{R}$ for small $\epsilon$ values.
Experiments. We report our results in Table 1. Our simple yet effective
procedure illustrates how both metrics are very susceptible to attacks. In
particular, solving the problem in Eq. (3) yields a distribution of noise that
significantly decreases the IS from 11.5 to 2.5 in CIFAR10 and from 250.7 to
2.9 in ImageNet. We show a sample from $\mathcal{D}_{G}$ in Figure 2, first
row. Similarly, our optimization problem in Eq. (4) can create imperceptible
perturbations that maximize the FID to $\approx$7900 between ImageNet and its
perturbed version (examples shown in Figure 1).
#### 3.2.2 Bad Images - Good Scores
While the previous experiments illustrate the vulnerability of both the IS and
FID against small perturbations (i.e. good images with bad scores), here we
evaluate if the converse is also possible, i.e. bad images with good scores.
In particular, we aim to construct a distribution of noise images (e.g. second
row of Figure 2) that enjoys good scores (high IS or low FID).
Maximizing IS. The IS has two terms: Inception’s confidence on classifying a
generated image, i.e. $p(y|x_{g})$, and the diversity of the generated
distribution of predicted labels, i.e. $p(y)$. One can maximize the IS by
generating a distribution $\mathcal{D}_{G}$ such that: (i) each
$x_{g}\sim\mathcal{D}_{G}$ is predicted with high confidence, and (ii) the
distribution of predicted labels is uniform across Inception’s output
$\mathcal{Y}$. To that end, we propose the following procedure for
constructing such $\mathcal{D}_{G}$. For each $x_{g}$, we sample a label
$\hat{y}\sim\mathcal{Y}$ uniformly at random and solve the problem:
$x_{g}=\operatorname*{arg\,min}_{x}~{}\mathcal{L}_{ce}(p(y|x),\hat{y}).$ (5)
In our experiments, we solve the problem in Eq. (5) with 100 gradient descent
steps and random initialization for $x$.
Minimizing FID. Here, we analyze the robustness of FID against such a threat
model. We follow a similar strategy to the objective in Eq. (4). For each
image $x_{r}\sim\mathcal{D}_{R}$, we intend to construct $x_{g}$ such that:
$x_{g}=\operatorname*{arg\,min}_{x}~{}\left\|f(x)-f(x_{r})\right\|_{2}$ (6)
with a randomly initialized $x$. In our experiments, we solve Eq. (6) with 100
gradient descent steps. As such, each $x_{g}$ will have a similar Inception
representation to a real-world image, i.e. $f(x_{g})\approx f(x_{r})$, while
being random noise.
##### Experiments.
We report our results in the last row of Table 1. Both the objectives in Eqs.
(5) and (6) are able to fool the IS and FID, respectively. In particular, we
are able to generate distributions of noise images with resolutions $32\times
32$ and $224\times 224$ (i.e. CIFAR10 and ImageNet resolutions) but with IS of
94 and 136, respectively. We show a few qualitative samples in the second row
of Figure 2. Furthermore, we generate noise images that have embedding
representations very similar to those of CIFAR10 and ImageNet images. This
lowers the FID of both datasets to 9.94 and 1.05, respectively (examples are
shown in Figure 1).
### 3.3 Robustness under Latent Perturbations
In the previous section, we established the vulnerability of both the IS and
FID against pixel perturbations. Next, we investigate the vulnerability
against perturbations in a GAN’s latent space. Designing such an attack is
more challenging in this case, since images can only be manipulated
indirectly, and so there are fewer degrees of freedom for manipulating an
image. To that end, we choose $G$ to be the state of the art generator
StyleGANv2 [14] trained on the standard FFHQ dataset [14]. We limit the
investigation to the FID metric, as IS is not commonly used in the context of
unconditional generators, such as StyleGAN. Note that we always generate $70$k
samples from $G$ to compute the FID.
Recall that our generator $G$ accepts a random latent vector
$z\sim\mathcal{N}(0,\text{I})$222The appendix presents results showing that
sampling $z$ from different distributions still yields good looking
StyleGANv2-generated images. and maps it to the more expressive latent space
$w$, which is then fed to the remaining layers of $G$. It is worthwhile to
mention that “truncating” the latent $w$ with a pre-computed
$\bar{w}$333$\bar{w}$ is referred to as the mean of the $w$-space. It is
computed by sampling several latents $z$ and averaging their representations
in the $w$-space. and constant $\alpha\in\mathbb{R}$ (i.e. replacing $w$ with
$\alpha w+(1-\alpha)\bar{w}$) controls both the quality and diversity of the
generated images [14].
Figure 3: Effect of attacking truncated StyleGANv2’s latent space on the
Fréchet Inception Distance (FID). We conduct attacks on the latent space of
StyleGANv2 and record the effect on the FID. We display the resulting samples
of these attacks for two truncation values, $\alpha=0.7$ (top row) and
$\alpha=1.0$ (bottom row). Despite the stark differences in realism between
the images in the top and bottom rows—i.e. the top row’s remarkable quality
and the bottom row’s artifacts—the FID to FFHQ reverses this ranking, wherein
the bottom row is judged as farther away from FFHQ than the top row.
##### Effect of Truncation on FID.
We first assess the effect of the truncation level $\alpha$ on both image
quality and FID. We set $\alpha\in[0.7,1.0,1.3]$ and find FIDs to be
$[21.81,2.65,9.31]$, respectively. Based on our results, we assert the
following observation: while the visual quality of generated images at higher
truncation levels, e.g. $\alpha=0.7$, is better and has fewer artifacts than
the other $\alpha$ values, the FID does not reflect this fact, showing lower
(better) values for $\alpha\in\\{1.0,1.3\\}$. We elaborate on this observation
with qualitative experiments in the appendix.
##### FID-Guided Sampling.
Next, we extend the optimization problem in Eq. (4) from image to latent
perturbations. In particular, we aim at constructing a perturbation
$\delta^{*}_{z}$ for each sampled latent $z$ by solving:
$\displaystyle\delta^{*}_{z}$
$\displaystyle=\operatorname*{arg\,max}_{\delta}~{}\left\|f(G(z+\delta))-f(x_{r})\right\|_{2}.$
(7)
Thus, $\delta^{*}_{z}$ perturbs $z$ such that $G$ produces an image whose
embedding differs from that of real image $x_{r}$. We solve the problem in Eq.
(7) for $\alpha\in\\{0.7,1.0\\}$.
##### Experiments.
We visualize our results in Figure 3 accompanied with their corresponding FID
values (first and second rows correspond to $\alpha=0.7\text{ and }1.0$,
respectively). While our attack in the latent space is indeed able to
significantly increase the FID (from 2.65 to 31.68 for $\alpha=1.0$ and 21.33
to 34.10 for $\alpha=0.7$), we inspect the results and draw the following
conclusions. (i) FID provides inconsistent evaluation of the generated
distribution of images. For example, while both rows in Figure 3 have
comparable FID values, the visual quality is significantly different. This
provides practical evidence of this metric’s unreliability in measuring the
performance of generative models. (ii) Adding crafted perturbations to the
input of a state of the art GAN deteriorates the visual quality of its output
space (second row in Figure 3). This means that GANs are also vulnerable to
adversarial attacks. This is confirmed in the literature for other generative
models such as GLOW [16, 21]. Moreover, we can formulate a problem similar to
Eq. (7) but with the goal of perturbing the $w$-space instead of the
$z$-space. We leave results of solving this formulation for different $\alpha$
values to the appendix.
##### Section Summary.
In this section, we presented an extensive experimental evaluation
investigating if the quality measures (IS and FID) of generative models
actually measure the perceptual quality of the output distributions. We found
that such metrics are extremely vulnerable to pixel perturbations. We were
able to construct images with very good scores but no visual content (Section
3.2.2), as well as images with realistic visual content but very bad scores
(Section 3.2.1). We further studied the sensitivity of FID against
perturbations in the latent space of StyleGANv2 (Section 3.3), allowing us to
establish the inconsistency of FID under this setup as well. Therefore, we
argue that such metrics, while measuring useful properties of the generated
distribution, lead to questionable assessments of the visual quality of the
generated images.
## 4 R-FID: Robustifying the FID
After establishing the vulnerability of IS and FID to perturbations, we
analyze the cause of such behavior and propose a solution. We note that, while
different metrics have different formulations, they rely on a pretrained
Inception model that could potentially be a leading cause of such
vulnerability. This observation suggests the following question:
_Can we robustify the FID by replacing its Inception
component with a robustly trained counterpart?_
We first give a brief overview of adversarial training.
### 4.1 Leveraging Adversarially Trained Models
Adversarial training is arguably the _de facto_ procedure for training robust
models against adversarial attacks. Given input-label pairs $(x,y)$ sampled
from a training set $\mathcal{D}_{tr}$, $\ell_{2}$-adversarial training solves
the following min-max problem:
$\min_{\theta}~{}\mathbb{E}_{(x,y)\sim\mathcal{D}_{tr}}\left[\max_{\|\delta\|_{2}\leq\kappa}\mathcal{L}\left(x+\delta,y;\theta\right)\right]$
(8)
for a given loss function $\mathcal{L}$ to train a robust network with
parameters $\theta$. We note that $\kappa$ controls the robustness-accuracy
trade-off: models trained with larger $\kappa$ tend to have higher robust
accuracy (accuracy under adversarial attacks) and lower clean accuracy
(accuracy on clean images). Since robust models are expected to resist pixel
perturbations, we expect such models to inherit robustness characteristics
against the attacks constructed in Section 3.2. Moreover, earlier works showed
that robustly-trained models tend to learn more semantically-aligned and
invertible features [13]. Therefore, we hypothesize that replacing the
pretrained Inception model with its robustly trained counterpart could
increase FID’s sensitivity to the visual quality of the generated distribution
(i.e. robust against attacks in Section 3.3).
To that end, we propose the following modification to the FID computation. We
replace the pretrained Inception model with a robustly trained version on
ImageNet following Eq. (8) with $\kappa\in\\{64,128\\}$. The training details
are left to the appendix. We refer to this alternative as R-FID, and analyze
its robustness against perturbations next.
Figure 4: Attacking R-FID with pixel perturbations. We attack two variants of R-FID ($\kappa=64$ and $\kappa=128$) and visualize samples from the resulting datasets. Attempting to fool these R-FIDs at the pixel level yields perturbations that correlate with semantic patterns, in contrast to those obtained when attempting to fool the standard FID (as shown in Figure 1). Table 2: R-FID against attacks in the pixel space. We study the robustness of R-FID against the adversarial attacks in Eq. 4. $\epsilon$ | CIFAR10 | ImageNet
---|---|---
| $\kappa=64$ | $\kappa=128$ | $\kappa=64$ | $\kappa=128$
$0.01$ | 1.5 | 0.3 | 21.0 | 4.5
$0.02$ | 20.7 | 7.8 | 293.8 | 92.1
$0.03$ | 46.4 | 19.7 | 657.9 | 264.6
### 4.2 R-FID against Pixel Perturbations
We first test the sensitivity of R-FID against additive pixel perturbations.
For that purpose, we replace the Inception with a robust Inception, and repeat
the experiments from Section 3.2.1 to construct real images with bad scores.
We conduct experiments on CIFAR10 and ImageNet with
$\epsilon\in\\{0.01,0.02,0.03\\}$ for the optimization problem in Eq. (4), and
we report the results in Table 2. We observe that the use of a robustly-
trained Inception significantly improves robustness against pixel
perturbations. Our robustness improvement for the same value of
$\epsilon=0.01$ is of 3 orders of magnitude (an FID of 4 for $\kappa=128$
compared to 7900 reported in Table 1). While both models consistently provide
a notable increase in robustness against pixel perturbations, we find that the
model most robust to adversarial attacks (i.e. $\kappa=128$) is also the most
robust to FID attacks. It is worthwhile to mention that this kind of
robustness is expected since our models are trained not to alter their
prediction under additive input perturbations. Hence, their feature space
should enjoy robustness properties, as measured by our experiments. In Figure
4 we visualize a sample from the adversarial distribution $\mathcal{D}_{G}$
(with $\epsilon=0.08$) when $\mathcal{D}_{R}$ is ImageNet. We observe that our
adversaries while aiming only at pushing the feature representation of samples
of $\mathcal{D}_{G}$ away from those of $\mathcal{D}_{R}$, are also more
correlated with human perception. This finding aligns with previous
observations in the literature, which find robustly-trained models have a more
interpretable (more semantically meaningful) feature space [13, 9]. We leave
the evaluation under larger values of $\epsilon$, along with experiments on
unbounded perturbations, to the appendix.
Table 3: Truncation’s effect on R-FID. We study how truncation affects R-FID against FFHQ (first two rows), and across different truncation levels (last two rows). $(\mathcal{D}_{G}(\alpha)$, $\mathcal{D}_{R})$ | 0.7 | 0.9 | 1.0
---|---|---|---
$\kappa=64$ | 98.3 | 90.0 | 88.1
$\kappa=128$ | 119.9 | 113.7 | 113.8
$(\mathcal{D}_{G}(\alpha_{i}),\mathcal{D}_{G}(\alpha_{j}))$ | (0.7, 1.0) | (0.7, 0.9) | (0.9, 1.0)
$\kappa=64$ | 10.5 | 4.9 | 0.48
$\kappa=128$ | 9.9 | 4.6 | 0.46
### 4.3 R-FID under Latent Perturbations
In Section 4.2, we tested R-FID’s robustness against pixel-level
perturbations. Next, we study R-FID for evaluating generative models. For
this, we follow the setup in Section 3.3 using an FFHQ-trained StyleGANv2 as
generator $G$.
Figure 5: Robustness of R-FID against perturbations in StyleGANv2 latent
space. We conduct attacks on two variants of R-FID ($\kappa=64$ on the left,
and $\kappa=128$ on the right) and two truncation values ($\alpha=0.7$ on the
top, and $\alpha=1.0$ on the bottom) by perturbing the latent space. We also
visualize samples from the generated distributions. For the pairs
$(\kappa,\alpha)\in\\{(64,0.7),(64,1.0),(128,0.7),(128,1.0)\\}$, we find
corresponding R-FID values of {128.1, 157.8, 126.6, 162,8}. In contrast to the
minimal changes required to fool the standard FID (Fig. 3), fooling the R-FID
leads to a dramatic degradation in visual quality of the generated images.
Effect of Truncation on R-FID. Here, we analyze the R-FID when the generator
is using different truncation levels. In particular, we choose
$\alpha\in\\{0.7,0.9,1.0\\}$ and report results in Table 3. We observe that
the robust Inception model clearly distinguishes the distribution generated by
StyleGANv2 from the FFHQ dataset, regardless of the truncation $\alpha$. In
this case, we obtain an R-FID of 113.8, substantially larger than the 2.6
obtained when the nominally-trained Inception model is used. This result
demonstrates that, while the visual quality of StyleGANv2’s output is
impressive, the generated image distribution is far from the FFHQ
distribution. We further evaluate if the R-FID is generally large between any
two distributions by measuring the R-FID between two distributions of images
generated at two truncation levels $(\alpha_{i},\alpha_{j})$. Table 3 reports
these results. We observe that (i) the R-FID between a distribution and itself
is $\approx 0$, e.g. R-FID = $10^{-3}$ at (1.0, 1.0). Please refer to the
appendix for details. (ii) The R-FID gradually increases as the image
distributions differ, e.g. R-FID at (0.9, 1.0) $<$ (0.7, 1.0). This
observation validates that the large R-FID values found between FFHQ and
various truncation levels are a result of the large separation in the
embedding space that robust models induce between real and generated images.
R-FID Guided Sampling. Next, we assess the robustness of the R-FID against
perturbations in the latent space of the generator $G$. For this purpose, we
conduct the attack proposed in Eq. (7) with $f$ now being the robustly-trained
Inception. We report results and visualize few samples in Figure 5. We make
the following observations. (i) While the R-FID indeed increases after the
attack, the relative increment is far less than that of the non-robust FID.
For example, R-FID increases by 44% at $\kappa=64$ and $\alpha=0.7$ compared
to an FID increase of 1000% under the same setup. (ii) The increase in R-FID
is associated with a significantly larger amount of artifacts introduced by
the GAN in the generated images. This result further evidences the
vulnerability of the generative model. However, it also highlights the changes
in the image distribution that are required to increase the R-FID. We leave
the $w$\- space formulation for the attack on the R-FID, along with its
experiments, to the appendix.
Section Summary. In this section, we robustified the popular FID by replacing
the pretrained Inception model with a robustly-trained version. We found this
replacement results in a more robust metric (R-FID) against perturbations in
both the pixel (Section 4.2) and latent (Section 4.3) spaces. Moreover, we
found that pixel-based attacks yield much more perceptually-correlated
perturbations when compared to the attacks that used the standard FID (Figure
2). Finally, we observed that changing R-FID values requires a more
significant and notable distribution shift in the generated images (Figure 5).
### 4.4 R-FID against Quality Degradation
Table 4: Sensitivity of R-FID against noise and blurring. We measure R-FID $(\kappa=128)$ between ImageNet and a transformed version of it under Gaussian noise and blurring. As $\sigma$ increases, the image quality decreases and R-FID increases. $\nicefrac{{\sigma_{N}}}{{\sigma_{B}}}$ | $\nicefrac{{0.1}}{{1.0}}$ | $\nicefrac{{0.2}}{{2.0}}$ | $\nicefrac{{0.3}}{{3.0}}$ | $\nicefrac{{0.4}}{{4.0}}$
---|---|---|---|---
Gaussian (N)oise | 16.65 | 61.33 | 128.8 | 198.3
Gaussian (B)lur | 15.54 | 54.07 | 78.67 | 89.11
At last, we analyze the effect of transformations that degrade image quality
on R-FID. In particular, we apply Gaussian noise and Gaussian blurring on
ImageNet and report the R-FID $(\kappa=128)$ between ImageNet and the degraded
version in Table 4. Results show that as the quality of the images degrades
(i.e. as $\sigma$ increases), the R-FID steadily increases. Thus, we find that
R-FID is able to distinguish a distribution of images from its degraded
version.
## 5 Discussion, Limitations, and Conclusions
In this work, we demonstrate several failure modes of popular GAN metrics,
specifically IS and FID. We also propose a robust counterpart of FID (R-FID),
which mitigates some of the robustness problems and yields significantly more
robust behavior under the same threat models.
Measuring the visual quality for image distributions has two components: (1)
the statistical measurement (e.g. Wasserstein distance) and (2) feature
extraction using a pretrained model (e.g. InceptionV3). A limitation of our
work is that we only focus on the second part (the pretrained model). As an
interesting avenue for future work, we suggest a similar effort to assess the
reliability of the statistical measurement as well, i.e. analyzing and finding
better and more robust alternatives to the Wasserstein distance.
Current metrics mainly focus on comparing the distribution of features. In
these cases, visual quality is only hoped to be a side effect and not directly
optimized for nor tested by these metrics. Developing a metric that directly
assesses visual quality remains an open problem that is not tackled by our
work but is recommended for future work.
Acknowledgments. This work was supported by the King Abdullah University of
Science and Technology (KAUST) Office of Sponsored Research (OSR) under Award
No. OSR-CRG2019-4033.
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## Appendix 0.A Sampling $z$ Outside Standard Gaussian
In this section, we check the effect of sampling the latent $z$ from
distributions other than the one used in training. In particular, instead of
sampling $z$ from a standard Gaussian distribution, we try the following
setups:
* •
$z\sim\mathcal{N}(\mu,I)$ where
$\mu\in\\{0.1,0.2,0.7,0.8,0.9,1.0,2.0,6.0,7.0\\}$.
* •
$z\sim\mathcal{N}(\mu,I)+\mathcal{U}[0,1]$ where $\mathcal{U}$ is a uniform
distribution.
* •
$z\sim\mathcal{U}[0,1]$
We report the results in Figures 6 and 7, setting the truncation to
$\alpha=0.5$. We observe that the effect of the distribution from which $z$ is
sampled has a minor effect on the quality of the generated output image from
StyleGAN. Therefore, we run our latent attack as an _unconstrained_
optimization.
Figure 6: Effect of shifting the mean of the Gaussian distribution on the
output visual quality. We notice that, for a truncation level $\alpha=0.5$,
shifting the mean of the Gaussian distribution from which we sample the latent
$z$ has a _very minor_ effect on the visual quality of the generated images.
Figure 7: Sampling from other distributions than standard Gaussian. We analyze
the effect of adding a random uniform vector to the sampled $z$ from a
standard Gaussian in the first row. In the second row, we sample $z$ from a
uniform distribution as opposed to the standard Gaussian. In both cases, and
for truncation level of $\alpha=0.5$, we note that StyleGANv2 is capable of
producing output images with good visual quality.
## Appendix 0.B Visualizing the Output of StyleGANv2 at Different Truncation
Levels
In Section 3.3, we argued that FID favours a distribution of images with more
artifacts. That is, FID values for a distribution of images generated with
truncation of $\alpha=0.7$ are worse than the ones for
$\alpha\in\\{1.0,1.3\\}$, while the latter suffer from significantly more
artifacts. We visualize some examples in Figure 8 for completeness.
Figure 8: Visualizing the output of StyleGANv2 at different truncation levels.
We observe that while outputs with $\alpha=0.7$ are more stable in terms of
visual quality, the FID for $\alpha\in\\{1.0,1.3\\}$ is better.
## Appendix 0.C Maximizing FID in the $w-$ Space
In Section 3.3, we showed the vulnerability of both the FID and StyleGANv2
against perturbations in the latent space $z$. One natural question that could
arise is whether this vulnerability propagated to the $w-$ space as well. To
that end, we replicate the setup in Section 3.3 with the following procedure:
for each $z_{i}\in\mathcal{N}(0,I)$, we map it to the $w-$ space and construct
the perturbation $\delta^{*}_{w}$ by solving the following optimization
problem:
$\displaystyle\delta^{*}_{w}$
$\displaystyle=\operatorname*{arg\,max}_{\delta}~{}\left\|f(\hat{G}(w+\delta))-f(x_{r})\right\|_{2}.$
(9)
We note here that $\hat{G}$ is a StyleGANv2 model excluding the mapping layers
from the $z-$ space to the $w-$ space. We solve the optimization problem in
(9) with 20 iterations of SGD and learning rate of 0.3. We note that the
number of iterations is set to a relatively small value compared to the
attacks conducted in the $z-$space for computational purposes.
We visualize the results in Figure 9. For a truncation value of $\alpha=1.0$,
the FID increases from 2.65 to 6.42. We note here that, similar to earlier
observations, the FID is providing inconsistent judgement by favouring a
distribution with larger artifacts (comparing Figure 9 with the first row of
Figure 8). Moreover, even with the small learning rate and number of
iterations, we observe the StyleGANv2 is vulnerable against manipulations in
the $w-$space.
Figure 9: Robustness of FID against perturbations in $w-$space. We analyze the
sensitivity of StyleGANv2 and FID against perturbations in the $w-$space. We
report an FID value of 6.4, as opposed to 2.65 without perturbations
$(\alpha=1.0)$.
## Appendix 0.D Training Details and Code
We conducted $\ell_{2}$ PGD adversarial training by solving the problem in
Equation (8). At each iteration, we compute the adversary using 2 steps of PGD
attack and random initialization with Gaussian noise. We train the network for
90 epochs with SGD optimizer and a learning rate of $0.1$. We drop the
learning rate by a factor of $10$ after each 30 epochs. We train on ImageNet’s
training set from scratch. We release our implementation and pre-trained
models at https://github.com/R-FID-Robustness-of-Quality-Measures-for-GANs.
## Appendix 0.E Attacking R-FID with Larger $\epsilon$
In Section 4.2, we tested the sensitivity of R-FID against pixel perturbations
that are limited by an $\epsilon$ budget. In the main paper, we reported the
results after attacking R-FID with a budget of
$\epsilon\in\\{0.01,0.02,0.03\\}$. For completeness, we conduct experiments
with $\epsilon\in\\{0.04,0.05,0.06,0.07,0.08\\}$ for the robust Inception
model trained with $\kappa=128$. We find R-FID values of
$\\{503.6,663.2,817.1,891,960.7\\}$, respectively. We note that, even under
the largest $\epsilon$ value we considered ($\epsilon=0.08$), the R-FID is
still one order of magnitude smaller than of FID when being attacked with
$\epsilon=0.01$. This provides further evidence to the effectiveness of R-FID
in defending against pixel perturbations.
##### Unbounded Perturbations.
Here we test the robustness of R-FID against noisy images. In Section 3.2.2,
we showed the sensitivity of FID in assigning good scores to noisy images. We
replicate our setup from Table 1 for ImageNet and conducted the attack on
R-FID. For the optimized noise images (noise images in this case should be
assigned low R-FID), we found the R-FID to be 340, significantly higher than
when attacking FID (Table 1 reports an FID of 1.05 for random noise images).
We note that while better metrics could be proposed in the future, we believe
that R-FID is a step towards a more reliable metri—more robust to both pixel
and latent perturbations).
## Appendix 0.F Effect of Truncation on R-FID
In Section 4.3, and specifically Table 3, we analyzed whether R-FID outputs
large values for any pair of distributions. We provided R-FID values for
distributions generated from StyleGANv2 with pairs of truncation values
$(\alpha_{i},\alpha_{j})$. For completeness, we report the results for the
rest of the pairs, including the R-FID between two splits of FFHQ dataset in
Table 5. We observe that the R-FID is very small for identical distributions
(e.g. two splits of FFHQ, or at the same truncation level (1.0, 1.0)).
Moreover, R-FID increases gradually as the distributions differ. This fact
confirms our earlier observation that R-FID better discriminates the generated
distribution from the real one.
Figure 10: Robustness of R-FID against perturbations in the $w-$space. We report R-FID (at $\alpha=1.0$) of 114.3, as opposed to 113.8 without perturbations. Table 5: R-FID between two distributions. We analyze the R-FID between distributions of images generated at different truncation levels. The last column is the R-FID between two non-overlapping splits of the FFHQ dataset. $(\mathcal{D}_{G}(\alpha_{i}),\mathcal{D}_{G}(\alpha_{j}))$ | (0.7, 1.0) | (0.7, 0.9) | (0.9, 1.0) | (1.0, 1.0) | $(\hat{\mathcal{D}}_{R},\hat{\mathcal{D}}_{R})$
---|---|---|---|---|---
$\kappa=64$ | 10.5 | 4.9 | 0.48 | 0.007 | 0.004
$\kappa=128$ | 9.9 | 4.6 | 0.46 | 0.008 | 0.006
## Appendix 0.G Maximizing R-FID in the $w-$ Space
We replicate our setup in Appendix 0.C to analyze the sensitivity of R-FID
against perturbations in the $w-$ space. To that end, we leverage our attack
in Equation (9) but replace the pretrained Inception with a robustly trained
version with $\kappa=128$. We visualize the results accompanied by the R-FID
value in Figure 10.
We draw the following observations: (i): The increase in the R-FID under the
same threat model is much smaller than the increase of FID (113.8
$\rightarrow$ 114.3 compared to 2.54 $\rightarrow 6.4$). That is, R-FID is
more robust than FID against latent perturbations in the $w-$space. (ii):
Changes in the R-FID are accompanied by significant changes in the visual
quality of the generated image from StyleGANv2. This is similar to the earlier
observation noted in Section 4.3. This constitutes further evidence about the
effectiveness of R-FID for providing a robust metric against manipulation.
## Appendix 0.H How Large is $\delta^{*}$
In Section 3.3, we constructed $\delta^{*}$ to perturb the latent code in an
unbounded fashion. While the random latent $z$ belongs to a standard normal
distribution, there are no bounds on how each latent $z$ should look like.
Nevertheless, we analyze the latent perturbation $\delta^{*}$ to better assess
the robustness under latent perturbations. To that end, we measure the
Wasserstein distance between the unperturbed and perturbed latent codes. We
found this value to be small ($\sim$0.07 on average across experiments). We
attribute such a small value to using a small step size and a moderate number
of iterations for solving the optimization problem
## Appendix 0.I Additional Comments on the Motivation
This work aims at characterizing the reliability of the metrics used to judge
generative models. Such metrics play a sensitive role in determining whether a
generative model is doing a better job than the other. Throughout our
assessment, we found that both IS and FID can be easily manipulated by
perturbing either the pixel or the latent space. That is, GAN designers could
potentially improve the scores of their generative model by simply adding
small imperceptible perturbations to the generated distribution of images or
latents. This makes the IS and FID less trustworthy, urging for more reliable
metrics. In this work, we also proposed one possible fix to increase the
reliability of FID, by replacing the pretrained InceptionV3 with a robustly
trained version. We note, at last, that while better metrics could appear in
the future, we conjecture that R-FID will be part of future solutions to this
problem.
Table 6: Robust Inception Score against pixel perturbations on CIFAR10. $\epsilon$ | 0.0 | $5\times 10^{-3}$ | $0.01$ | random noise
---|---|---|---|---
R-IS | 9.94 | 5.49 | 3.91 | 1.01
## Appendix 0.J Robust Inception Score (R-IS)
Finally and for completeness, we explore the robustness enhancements that the
robust model provide to the Inception Score (IS). Thus, we replicate the setup
from Table 1 and conduct pixel perturbations on CIFAR10 dataset. We report the
results for $\kappa=128$ in Table 6. We observe that the variation of R-IS
against pixel perturbations is much more stable than regular IS. For instance,
R-IS drops from 9.94 to 5.49 at $\epsilon=5\times 10^{-3}$ compared to IS
which drops from 11.54 to 2.62 for the same value of $\epsilon$. Moreover,
running the same optimization for constructing noise images with good IS does
not yield a good R-IS. This demonstrates an additional advantage of deploying
robust models in GANs quality measures.
## Appendix 0.K Additional Visualizations
In the main paper, and due to space constraints, we provided only six samples
from the analyzed distributions. For completeness and fairer qualitative
comparison, we show additional samples from each considered distribution. In
particular, we visualize the output of StyleGANv2 after attacking the latent
space by: (i): Maximizing FID with truncation $\alpha=0.7$ (Figure 11) (ii):
Maximizing FID with truncation $\alpha=1.0$ (Figure 12) (iii): Maximizing
R-FID $(\kappa=128)$ with truncation $\alpha=0.7$ (Figure 13) (iv): Maximizing
R-FID $(\kappa=128)$ with truncation $\alpha=1.0$ (Figure 14) (v): Maximizing
R-FID $(\kappa=64)$ with truncation $\alpha=0.7$ (Figure 15) (vi): Maximizing
R-FID $(\kappa=64)$ with truncation $\alpha=1.0$ (Figure 16).
Figure 11: Visualizing samples after attacking the latent space $z$ for
StyleGANv2 to maximize FID with truncation $\alpha=0.7$. Figure 12:
Visualizing samples after attacking the latent space $z$ for StyleGANv2 to
maximize FID with truncation $\alpha=1.0$. Figure 13: Visualizing samples
after attacking the latent space $z$ for StyleGANv2 to maximize R-FID with
$\kappa=128$ and truncation $\alpha=0.7$. Figure 14: Visualizing samples after
attacking the latent space $z$ for StyleGANv2 to maximize R-FID with
$\kappa=128$ and truncation $\alpha=1.0$. Figure 15: Visualizing samples after
attacking the latent space $z$ for StyleGANv2 to maximize R-FID with
$\kappa=64$ and truncation $\alpha=0.7$. Figure 16: Visualizing samples after
attacking the latent space $z$ for StyleGANv2 to maximize R-FID with
$\kappa=64$ and truncation $\alpha=1.0$.
|
# Pandemic Culture Wars: Partisan Asymmetries in the Moral Language of
COVID-19 Discussions
Ashwin Rao 1, 2, Siyi Guo 1, 2, Sze-Yuh Nina Wang 3, Fred Morstatter 2,
Kristina Lerman 2
###### Abstract
Effective response to the COVID-19 pandemic required coordinated adoption of
mitigation measures, like masking and quarantines, to curb virus’s spread.
However, political divisions that emerged early in the pandemic hindered
consensus on the appropriate response. To better understand these divisions,
our study examines a vast collection of COVID-19-related tweets. We focus on
five contentious issues: coronavirus origins, lockdowns, masking, education,
and vaccines. We describe a weakly supervised method to identify issue-
relevant tweets and employ state-of-the-art computational methods to analyze
moral language and infer political ideology. We explore how ideological
divisions and moral language shape conversations about these issues. Our
findings reveal ideological differences in issue salience and the dynamics of
moral language. We find that conservatives use more negatively-valenced moral
language than liberals, but moral language use by conservatives is less
persistent and appears to be driven by dynamics of the news cycle.
Furthermore, we find that political elites use moral rhetoric to a greater
extent than non-elites across most issues. Examining the evolution and
moralization on divisive issues can provide valuable insights into the
dynamics of COVID-19 discussions and assist policymakers in better
understanding the emergence of ideological divisions.
## Introduction
The COVID-19 pandemic presented a major challenge to US society, testing its
resilience and capacity to respond to a crisis. Slowing the rapidly spreading
illness required coordinated adoption of non-pharmaceutical interventions
recommended by public health experts, such as wearing a mask, staying home,
social distancing, and getting vaccines once they became available. However,
the response to the pandemic became politically polarized, with Republicans
and Democrats disagreeing both on the gravity of the crisis and the
appropriate measures to address it (Funk and Tyson 2020). Some Republican
leaders downplayed the severity of the virus and expressed skepticism about
the effectiveness of measures like mask-wearing and social distancing, while
some Democratic leaders advocated for more stringent measures like closing
schools, businesses, and even parks and beaches. Policy differences in
pandemic mitigation emerged at the state level as well. Republican-led states
lifted restrictions on businesses and public gatherings earlier than
Democratic-led states. When the COVID-19 vaccine was approved, many
Democratic-led states introduced vaccine and mask mandates for returning to
work and school, while some Republican-led states passed bills to ban such
mandates. These political divisions hampered effective pandemic response,
leading to more than a million deaths in US, one of the highest mortality
rates among advanced economies (Mordecai and Connaughton 2020).
To understand how the pandemic response became so polarized, we study a large
corpus of pandemic-related tweets (Chen et al. 2020), comprising over 270M
tweets from 2.1M users around United States. We focus on five highly
politicized and contentious wedge issues: coronavirus origins, lockdowns and
business closures, masking, online education, and vaccines. These issues have
been previously found to be salient during the pandemic (Schaeffer 2020;
Connaughton 2021; Rojas 2020; Luttrell and Trentadue 2023; Chan 2021; Pierri
et al. 2022; Rathje et al. 2022). We classify user ideology based on the
partisanship of information sources users share and also describe a method for
identifying issues raised in a tweet. These models enable us to label millions
of tweets and users, thereby allowing us to study polarized discussions at
scale.
We study online conversations about the pandemic from the lens of Moral
Foundations Theory (MFT) (Haidt and Graham 2007). This theory provides a
framework for understanding how moral values shape people’s political
attitudes and behaviors. MFT proposes that individuals’ values and judgments
can be described by five moral foundations: care/harm, fairness/cheating,
loyalty/betrayal, authority/subversion, and sanctity/degradation. Individuals
and groups vary in how much emphasis they place on each foundation. While both
liberals and conservatives consider the care and fairness foundations to be
important, conservatives are thought to endorse the latter three foundations
to a greater extent than liberals (Graham, Haidt, and Nosek 2009). Studying
online conversations from the perspective of MFT can provide insights into the
ways in which partisans use moral appeals to articulate and defend their
political positions. It can also help us understand how to frame messaging so
as to better appeal to the values of each group.
We organize our investigations around the following research questions:
1. 1.
What COVID-19 issues did liberals and conservatives discuss online?
2. 2.
How do liberals and conservatives differ in their use of moral language on
different issues?
3. 3.
Does moral language persist over time or quickly die out? What moral
foundations resonate with partisans on different issues?
4. 4.
How does moral language use differ between political elites and non-elites?
We propose and evaluate a weakly-supervised method to identify issue-relevant
tweets. This novel method extracts relevant phrases from publicly available
Wikipedia pages and then uses them to detect issues in the text of tweets. In
addition, we use state-of-the-art computational methods to detect moral
foundations in tweets (Guo, Mokhberian, and Lerman 2023) and to infer the
political ideology of users (Rao et al. 2021). These models enable us to track
partisan attention to polarized issues at scale from the start of the COVID-19
pandemic to November 2021.
We uncover important differences in the issues that liberals and conservatives
discuss over time, as well as their use of moral language over time. We also
unveil differences in the moral appeals of political elites and non-elites
across the ideological spectrum. Understanding the moralization on divisive
issues can provide valuable insights into the emergence of polarization and
help public health experts better tailor messaging to the values and concerns
of each group.
## Related Work
Table 1: Issue Detection: Wikipedia Articles Issue | Wikipedia Articles
---|---
Origins | Plandemic, Investigations into the origin of COVID-19, COVID-19 lab leak theory
Lockdowns | COVID-19 lockdowns, COVID-19 protests in the United States, U.S. federal government response to the COVID-19 pandemic, Protests against responses to the COVID-19 pandemic, Social impact of the COVID-19 pandemic in the United States, Stay-at-home order, Social distancing measures related to the COVID-19 pandemic, Quarantine
Masking | Face masks during the COVID-19 pandemic, Face masks during the COVID-19 pandemic in the United States, Maskne, Anti-mask sentiment
Education | Impact of the COVID-19 pandemic on education, Homeschooling during the COVID-19 pandemic, Impact of the COVID-19 pandemic on education in the United States
Vaccines | COVID-19 vaccination in the United States, COVID-19 vaccine, COVID-19 vaccine hesitancy in the United States, COVID-19 vaccine misinformation and hesitancy, Deaths of anti-vaccine advocates from COVID-19, Herman Cain Award
Baseline | COVID-19 pandemic, COVID-19 pandemic in the United States, Politics of the United States
#### Polarization
American society has been divided by “culture wars” (Hunter 1992), with
liberals and conservatives disagreeing on controversial issues (also known as
wedge issues) like abortion, women’s and LGBTQ+ rights (DiMaggio, Evans, and
Bryson 1996; Evans 2003; Abramowitz and Saunders 2008). This polarization was
further exacerbated by the COVID-19 pandemic. Partisans disagreed on the
virus’s existence and severity, its origins, mitigation strategies like social
distancing and masking, and vaccine mandates. Political ideology was found to
explain differences in compliance with health guidelines and mandates
(Gollwitzer et al. 2020; Grossman et al. 2020). A study of elite messaging on
Twitter at the onset of the pandemic (Green et al. 2020) found that Democratic
elites discussed the pandemic more frequently and emphasized its threats to
public health, while Republican elites focused on China and the pandemic’s
impact on businesses.
These divisions extend beyond political elites: nearly 3 out of 10 Americans
believed that COVID-19 was created in a lab, with 4 in 10 Republicans
endorsing this claim (Schaeffer 2020). Government efforts to restrict the
spread of COVID-19 through lockdowns were met with resistance in the form of
protests and demonstrations, which were largely driven by conservatives:
nearly $52\%$ of conservatives in the United States preferred to have fewer
COVID-19 restrictions as opposed to only $7\%$ of liberals (Connaughton 2021).
These divisions can have important behavioural consequences: following and
interacting with right-leaning political elites, influencers and hyper-
partisan media sources was found to be associated with vaccine hesitancy in
the US (Pierri et al. 2022; Rathje et al. 2022).
#### Moral Foundations
Differences in moral values may underlie these partisan divisions. We analyze
these through the framework of moral foundations theory (MFT), which proposes
five moral foundations of care, fairness, in-group loyalty, respect for
authority, and purity/sanctity (Haidt and Graham 2007). These foundations can
be further categorized into virtues (care, authority, fairness, loyalty &
purity) and vices (harm, subversion, cheating, betrayal & degradation). A
number of language analysis techniques have been used to quantify the moral
foundations in text data (e.g. (Graham and Haidt 2012; Garten et al. 2016;
Guo, Mokhberian, and Lerman 2023)), with past work suggesting that moral
appeals may increase message diffusion (Brady et al. 2017; Wang and Inbar
2022).
To comprehend and mitigate issue-based political divisions, it is crucial to
understand how individuals attach moral significance to their viewpoints
(Koleva et al. 2012). Some past work has done this in the context of COVID-19.
For example, (Díaz and Cova 2022) found that valuing the care foundation was
correlated with compliance with COVID-19 public health recommendations.
Similarly, (Chan 2021) found that endorsing the fairness and care moral
foundations predicted compliance with calls for staying-at-home, wearing
masks, and social distancing, while purity predicted compliance with face
masks and social distancing. Others have found that framing masking-related
health messaging using ideology-matched moral arguments was effective for
liberals, but not conservatives (Luttrell and Trentadue 2023). Moral values
can also be used to predict county-level vaccination rates: counties with
residents who prioritize moral concerns about purity had lower vaccination
rates while counties whose residents prioritized fairness and loyalty had
higher vaccination rates (Reimer et al. 2022).
Some initial work has assessed COVID-19 vaccine discourse on Twitter.
(Borghouts et al. 2023) found that liberal tweets about the COVID-19 vaccine
expressed care, fairness, liberty111Note that some researchers include a sixth
foundation, liberty/oppression. and authority moral foundations more than
conservative tweets, while oppression and harm were referenced more by
conservatives. Similarly, (Pacheco et al. 2022) found that care/harm was
associated with pro-vaccine sentiment whereas, liberty/oppression was
correlated with anti-vaccine attitudes. While vaccinations are a critical
polarizing issue in the discussion of COVID-19, no work as of yet has explored
differences in moral appeals across a broader range of contentious COVID-19
issues.
## Methods
### Data
We use a publicly available dataset (Chen et al. 2020) consisting of 1.4B
tweets about COVID-19 posted between January 21, 2020 and November 4, 2021.
These tweets contained one or more COVID-19-related keywords, such as
coronavirus, pandemic, and Wuhan, among others. We utilized Carmen (Dredze et
al. 2013), a geo-location identification technique for Twitter data to assign
tweets to locations within US. This method leverages tweet metadata, including
“place” and “coordinates” objects that encode location information, such as
country, city, and geographic coordinates. Additionally, the technique uses
mentions of locations in a user’s bio on Twitter to infer their location. A
manual review confirmed that this approach was effective in identifying a
user’s home state. As a result, we were left with 270 million tweets generated
by 2.1 million geo-located users in the United States.
(a) Origins (b) Lockdowns (c) Masking (d) Education (e) Vaccines
Figure 1: Wordclouds showing phrases identified from Wikipedia articles using
SAGE for each issue. We show the top-50 most frequently occurring phrases in
tweets. Bigrams and trigrams are connected by __ for better visualization.
### Issue Detection
Table 2: Sample tweets highlighting wedge issues in COVID-19 online discussions. Issue | Tweets
---|---
Origins | No matter what the Chinese Communist Party says, given the mounting evidence, the most likely origins for the China virus are the Wuhan labs studying bats and coronavirus.
Lockdowns | This is a GREAT idea. We’re all in this together. Take care of each other. #StayHome #TakeItSeriously #FlattenTheCurve #COVID19
Masking | We’re in the middle of a pandemic and y’all are still coughing and sneezing without covering your mouths? Come on now.
Education | More glimmers of hope as we “safely” move forward and open up Texas A&M University while containing #COVID19.
Vaccines | You are joking right? Zero sympathy for anti-vaxxers who quit their jobs rather than get vaccinated. They put us all at risk and make the pandemic prolonged for the world.
As the pandemic progressed, a set of polarizing issues such as origins of the
coronavirus, lockdowns and masking mandates, school closures and online
schooling and vaccines, emerged and grew to dominate conversations about
COVID-19 (Schaeffer 2020; Connaughton 2021; Rojas 2020; Pierri et al. 2022;
Rathje et al. 2022). Previous studies (Garimella et al. 2018; Brady et al.
2017) relied on a small set of manually-selected keywords and phrases to
harvest issue-relevant tweets. However, there is a lack of data on the
reliability of these methods, and a systematic framework to identify issue-
relevant conversations is largely missing. While issue detection has analogues
to topic-detection (Blei, Ng, and Jordan 2003; Grootendorst 2022), one cannot
control for the stochasticity of topics identified in traditional topic-
detection approaches. Although (Card et al. 2015) presented a corpus of
several thousand news articles annotated with frames on traditional policy
issues such as smoking, same-sex marriages, and immigration, there has not
been an attempt to propose a framework for identifying COVID-19 issues in
social media discourse. The lack of expertly annotated data on the issues we
explore in this work prevents us from employing supervised learning
techniques.
We define the origins issue to encompass discussions surrounding the origins
of the pandemic, including topics such as pangolins, gain of function
research, wet-markets, and bats. The lockdown issue comprises content
pertaining to early state and federal mitigation efforts, such as quarantines,
stay-at-home orders, business closures, reopening, and calls for social
distancing. The masking issue is defined by discussions on the use of face
coverings, mask mandates, mask shortages, and anti-mask sentiment. Education-
related content involves tweets about school closures, reopening of
educational institutions, homeschooling, and online learning during the
pandemic. The vaccines issue pertains to discussions about COVID-19 vaccines,
vaccine mandates, anti-vaccine sentiment, and vaccine hesitancy in the US.
We describe a weakly-supervised method to harvest relevant keywords from
Wikipedia pages discussing these issues (see Table 1). We use SAGE
(Eisenstein, Ahmed, and Xing 2011) to identify distinctive keywords and
phrases that are relevant to a specific issue. SAGE calculates the deviation
in log-frequencies of words from a baseline lexical distribution. As a
baseline, we use Wikipedia articles discussing general aspects of the pandemic
and politics in the US (see 1). We concatenate these pages for each issue.
Depending on how we tokenize the corpus, we can use SAGE to identify issue-
relevant n-grams. In this study, we restrict analysis to unigrams, bigrams and
trigrams. Using SAGE we then compare tokens from issue-relevant articles to
baseline tokens to identify keywords and phrases that uniquely define each
issue. We manually verify the keywords identified by SAGE to ensure precision
and relevance. Word clouds in Fig. 1 show the top 50 keywords/phrases
extracted from Wikipedia articles for each issue. We assess the relevance of a
tweet to a issue by the presence of these keywords and phrases. For example,
terms such as ‘Wuhan labs’ or ‘wet markets’ would indicate a tweet is about
the origins of the pandemic, while ‘cover your mouth’, ‘N-95 masks’ would
indicate a masking-related tweet. Examples of tweets discussing these issues
are shown in Table 2.
Table 3: Inter-rater Agreement and Evaluation of Issue Detection. Issue | Pairwise | Multi- | F1- | Support
---|---|---|---|---
| Cohen’s $\kappa$ | annotator | Score |
| | Fleiss’ $\kappa$ | |
Origins | $0.73\pm 0.06$ | 0.45 | 0.51 | 25
Lockdowns | $0.79\pm 0.04$ | 0.47 | 0.90 | 108
Masking | $0.92\pm 0.04$ | 0.55 | 0.83 | 101
Education | $0.71\pm 0.07$ | 0.50 | 0.74 | 54
Vaccines | $0.87\pm 0.05$ | 0.53 | 0.92 | 84
##### Validation
We evaluate issue detection on a subset of tweets, which we draw at random
with the constraint that there are at least five tweets for each combination
of issues and moral foundation. This gave us a test set of 784 tweets, which
were then labeled by five trained annotators. Table 3 shows that the mean
Cohen’s Kappa between each pair of annotators for different issues were all
above 0.7. The Multi-annotator Fleiss’ Kappa values are lower because it
relies on agreement among all five annotators rather than pairs of annotators.
Finally, we evaluate performance of issue detection method on this test set
and show that F1 scores for most issues are above 0.7 (Table 3), indicating
good model performance.
### Morality Detection
We capture the moral sentiments in tweets based on the Moral Foundations
Theory. We train our morality detection model on top of the transformer-based
pretrained language model BERT (Devlin et al. 2018). We train BERT with three
Twitter datasets, including a manually annotated COVID dataset (Rojecki,
Zheleva, and Levine 2021), the Moral Foundation Twitter Corpus dataset with
six different topics (Hoover et al. 2020), and a dataset of political tweets
published by US congress members (Johnson and Goldwasser 2018). We test our
model on a test subset from the COVID dataset and show the F1 scores from 10
random runs in Table 4. The method has good performance on the major
categories such as care and harm, although the performance inevitably varies
with the amount of support for different moral categories, which is also
observed in previous studies (Hoover et al. 2020; Trager et al. 2022).
However, by fusing an in-domain training set that is also about COVID-19,
along with other datasets consisting of various topics, we are improving the
model generalizability when applied to the target data (Guo, Mokhberian, and
Lerman 2023). We compare our method with a widely used dictionary-based method
Distributed Dictionary Representations (DDR) (Garten et al. 2018) and find our
method outperforms DDR for most moral categories (Refer Table 4).
Table 4: Evaluation of Morality Detection Method on Test Data. We show the mean and standard deviation of F1 scores over 10 random runs, and the median of support of 10 runs. Moral Foundation | DDR | Ours | Support
---|---|---|---
Care | $0.71\pm 0.02$ | $\mathbf{0.82\pm 0.02}$ | 239
Harm | $0.21\pm 0.04$ | $\mathbf{0.75\pm 0.03}$ | 111
Fairness | $0.09\pm 0.08$ | $\mathbf{0.24\pm 0.17}$ | 5
Cheating | $0.10\pm 0.12$ | $\mathbf{0.24\pm 0.18}$ | 7
Loyalty | $0.33\pm 0.08$ | $\mathbf{0.59\pm 0.06}$ | 56
Betrayal | $0.03\pm 0.02$ | $\mathbf{0.16\pm 0.07}$ | 11
Authority | $0.00\pm 0.00$ | $\mathbf{0.68\pm 0.09}$ | 42
Subversion | $0.20\pm 0.15$ | $\mathbf{0.35\pm 0.15}$ | 11
Purity | $\mathbf{0.47\pm 0.26}$ | $0.33\pm 0.11$ | 8
Degradation | $0.05\pm 0.04$ | $\mathbf{0.20\pm 0.35}$ | 2
### Ideology Detection
A number of methods identify the political ideology of social media users,
with recent works (Le et al. 2019; Nikolov, Flammini, and Menczer 2020;
Cinelli et al. 2021; Rao et al. 2021) leveraging URLs that users share in
tweets. We use the method discussed in (Rao et al. 2021) to estimate the
ideology of individual users. This method relies on Media Bias-Fact Check
(MBFC) (Check 2023) to compute ideology scores for individuals. MBFC provides
ideological leaning for over 6K Pay-Level Domains (PLDs) with leaning
categorized as Left/Hardline Liberal ($0$), Left-Center ($0.25$), Least-
Biased/Center ($0.5$), Right-Center ($0.75$), Right/Hardline Conservative
($1$). Each individual’s ideological leaning is estimated to be the weighted
average of the leanings of the URLs they share.
These scores are then binarized with a threshold of $\leq 0.4$ for liberals
($0$) and $\geq 0.6$ for conservatives ($+1$). The model described in (Rao et
al. 2021) then leverages a fastText embedding model pre-trained on Twitter
data (Pagliardini, Gupta, and Jaggi 2018), to generate embeddings for user’s
tweets over time. These embeddings serve as features to train a Logistic
Regression classifier to predict the binarized ideological leanings for all
users, including users who have not shared URLs from domains in the MBFC. We
were able to identify ideology for 2.1M US users comprising of 1.6M liberal
users and 500K conservatives.
##### Validation
We assess the level of agreement between the ideology scores estimated by the
methods proposed in (Rao et al. 2021) and (Barberá 2015), which uses the
follower graph to estimate user ideology. We discretize the continuous ideal
point estimates in (Barberá 2015) using a threshold of $<0$ for liberals and
$>0$ for conservatives, which we respectively label as $0$ and $+1$. We
identify approximately 35K users who appear in both datasets and calculate the
F1-score between the two models. The resulting F1-score of $0.75$ and Jaccard
overlap of $0.87$ indicates a high level of agreement between the two methods.
Figure 2: Dynamics of attention to issues. Daily fraction of (a) original
tweets, (b) retweets and (c) replies related to each issue. We use 7-day
rolling average to reduce noise. Major events are marked with vertical lines
of the same color as the issue and placed along the timeline. Figure 3:
Difference in the daily share of original tweets by liberals and conservatives
on issues. Positive (negative) values indicate that the issue is $x\%$ more
significant for liberals (resp. conservatives). Seven-day rolling average is
leveraged to reduce noise. Vertical lines mark significant events that brought
shifts in issue-related discussions.
(a) Origins (b) Lockdowns (c) Masking (d) Education (e) Vaccines
Figure 4: Partisan differences in framing each issue. Values show the log odds
ratio of phrases used by liberals (blue) and conservatives (red) in issue-
relevant tweets across the time period.
## Results
Figure 2 shows the daily share of tweets, retweets and replies discussing each
issue, smoothed using a 7-day rolling window, with vertical lines marking
major issue-related events. The five issues we monitored accounted for small
share of all messages at the beginning of the pandemic, but grew to dominate
all discussions. Lockdowns were among the first mitigation efforts carried out
by state governments, with California the first state in the mainland US to
issue a stay-at-home order on March 19, 2020. This date marks the start of an
increase in lockdown related discussions. Although the peak in the masking
discussions did not occur until the Black Lives Matter protests in June 2020,
Fig. 2 indicates an early upward trend on April 3rd, 2020, when the CDC
recommended face coverings to prevent infections. On July 8, 2020 President
Trump promised to cut funding to schools that did not reopen for in-person
instruction, which is associated with a spike in education related
discussions. Similarly, Pfizer-BioNTech’s announcement (Pfizer 2020) on
November 9, 2020 that their COVID-19 vaccine was highly effective in Phase 3
clinical trials created a spike in vaccine related tweets. A spike in the
discussions of COVID-19 origins is seen on May 23, 2021 when the Wall Street
Journal published an article citing an US Intelligence Report that three
researchers from the Wuhan Institute of Virology had sought hospital care in
November 2019 shortly before the outbreak (Gordon, Strobel, and Hinshaw 2021).
The story marked the first major report on the origins of the pandemic by a
major news organization. These results provide a qualitative check of
robustness of our issue detection approach, as well as insights into
importance of issues at different times. Fig. 2 also shows differences in the
engagement on different issues. For example, tweets about masking seemed to
garner more replies, suggesting that they may have generated more discussion
or controversy.
(a) Time series: Education (b) Auto-correlation: Education
Figure 5: Partisan asymmetries in the persistence of moral language. (a) Time
series of moral language used by liberal and conservative accounts in the
discussions of education, separated by moral foundation. We show only the
three more widely present foundations, and for benefits of visualization,
reflect the time series of conservative tweets about the x-axis. White line
shows the difference between the two groups, indicating which side mentions
the issue more. (b) Auto-correlation function (ACF), showing decay of the
correlation of the time series with its lagged version for different lags (in
days).
(a) ACF: Pre-Vaccine (b) ACF: Post-Vaccine
Figure 6: Persistence of moral language. Each cell of the heatmap shows the
minimum lag time in day required for the auto-correlation function to drop
below the upper confidence level when time series are split into the pre-
vaccine period (before 12/11/2020) and post-vaccine period.
(a) Origins (b) Lockdowns (c) Masking (d) Education (e) Vaccines
Figure 7: Partisan differences in moral language use along each issue. Values
indicate percentage of tweets shared by liberals (blue) and conservatives
(red) containing the specific moral foundation. Values for Care and Harm are
downscaled by a factor of two for better visualization. Figure 8: Moral
language differences between elites and non-elites. Boxplot shows the
differences in the share of daily tweets made by each group on the issues: (a)
origins, (b) lockdowns,(c) masking, (d) education and (e) vaccines.
### Issue Salience by Ideology
Next, we examine the relative prevalence of issues in tweets by conservatives
and liberals. Since the Twitter user population in our sample is ideologically
unbalanced, with three times as many liberals and conservatives, we cannot
directly compare the volume of messages posted by each side. Instead, we
compare the share of all messages posted by liberals and conservatives that
are devoted to each issue. Specifically, we calculate the difference $\Delta$
along an issue $i$ at time (day) $d$ as follows:
$\Delta(i,d)=(\frac{\tau_{l}(i,d)}{\tau_{l}(d)}-\frac{\tau_{c}(i,d)}{\tau_{c}(d)})$
where $\tau_{l}$ and $\tau_{c}$ represent the number of all original tweets
made by liberals and conservatives on that day respectively, and
$\tau_{l}(i,d)$ and $\tau_{c}(i,d)$ represent the number of original tweets
about the issue posted by each group.
Fig. 3 illustrates the difference in the share of issue-relevant original
tweets posted by liberals and conservatives. Positive (negative) values on the
y-axis indicate higher prevalence for liberals (conservatives). Vertical lines
mark events where significant shifts in conversations occurred. While liberals
discussed education more than conservatives, conservatives discussed lockdowns
and COVID-19 origins more than liberals. Early in the pandemic, masking was an
issue marginally more important to liberals, but after George Floyd’s killing,
it was more discussed among conservatives. Ideological divisions about masking
remained small, however, until the summer of 2021. We also see interesting
changes in vaccine-related discussions. While there were minimal differences
between liberals and conservatives prior to the vaccine release, we see that
liberals were discussing the vaccine more than conservatives after it was made
available in the US on December 11, 2020. However, following vaccine mandates
for federal workers on September 9, 2021 (Biden 2021), the issue became a
greater focus for conservatives, likely due to their outrage about the
mandates. After the Wall Street Journal’s report (Gordon, Strobel, and Hinshaw
2021) in May 2021 claiming that COVID-19 originated at the Wuhan Institute of
Virology, there was an increase in conservative’s share of COVID-19 origins
tweets. While we limit our discussion here to original tweets due to space
constraints, we see similar trends in our analysis of retweets and replies.
To characterize differences in liberal and conservative tweets about issues,
we identify adjectives that were used to modify issue-relevant phrases
(anchors) extracted earlier. We leverage SpaCy’s Dependency Parser to extract
dependency relations of the form: XX-$amod\rightarrow ANC$ and
XX–$amod\rightarrow YY\leftarrow amod$–$ANC$, where $XX$ refers to the
extracted adjective and $ANC$ refers to an issue-relevant phrase. We then pair
the extracted adjective with its corresponding anchor as a phrase of the form
$XX$_$ANC$ and compute the log-odds ratio of these phrases being expressed in
liberal and conservative tweets. Wordclouds in Fig. 4 depict the top 10 most-
likely used phrases in liberal (blue) and conservative (red) issue-discourse.
Conservatives employ conspiratorial phrases such as “batshit conspiracy”,
“fake plandemic”, and “military bioweapon” to discuss the origins issue. In
contrast, liberals use phrases like “likely bats” and “medical virology”,
suggesting uncertainty. When it comes to lockdowns, conservatives express
negative sentiment, framing them as “deadly”, “unconstitutional”, and
“stringent”. Liberals, however, approach the topic cautiously, using phrases
such as “premature reopening”, “safe reopening”, and “soft lockdowns”. Similar
patterns emerge in the discourse on masking, with conservatives deeming mask
mandates as “unconstitutional” and “useless”, while liberals focus on
“reusable masks” and “mandatory masking”. Concerning education, conservatives
express distrust in mitigation strategies employed by educational
institutions, using terms like “fascist education” and “unmasked students”,
while liberals highlight “hybrid learning”, “vulnerable students”, and “early
education.” On the issue of vaccines, conservatives exhibit skepticism,
referring to “experimental mRNA” and “dangerous vaccines”. In contrast,
liberals emphasize the benefits, discussing “free vaccinations” and “mobile
vaccines”.
### Persistence of Moral Language
The fluctuations in the daily volume of tweets, even when disaggregated by
issue (Fig. 2), obscure the dynamics and complexity of moral expression. To
better understand how partisans use moral appeals, we split original tweets on
a particular issue by moral foundation label and user ideology. Fig. 5 shows
the share of tweets posted by liberals and conservatives that express each
moral foundation in the discussions of education. To ease visualization, the
time series for conservatives tweets is reflected around the x-axis. In this
analysis, both virtues and vices of a moral foundation contribute to its
presence in a tweet. As with other issues, the care/harm foundation is the
most widely expressed dimension of morality, followed by authority/subversion
and fairness/cheating. The two remaining moral foundations are far less
common. The time series of moral expression displays complex dynamics, with
strong symmetry between liberals and conservatives suggesting that both groups
are driven by external events. However, there are also notable differences.
The care/harm language used by liberals shows characteristic weekly patterns
of activity, with dips on weekends, that are less visible in conservative
tweets. In contrast, conservative language shows more short-lived spikes that
dissipate as quickly as they appear.
To quantify differences in the dynamics of moral language, we compute the
auto-correlation function (ACF) of each time series. This function measures
the correlation between a time series and its own past values at different
lags. Fig. 5 shows this function at different lags from 0 to 60 days. By
design, ACF starts at one and decays at larger lags, becoming insignificant
when it drops below the confidence bound (horizontal line). For liberals, the
ACF values for the care/harm and fairness/cheating foundations have peaks at
lag 7, 14, 21, etc. days, a signature of weekly pattern in the time series.
The ACF for fairness/cheating decays quickly for conservatives, becoming
insignificant for lags larger than 10 days. The fast decaying ACF is
associated with short-lived spikes in the time series of moral language that
occur at irregular time intervals. This pattern suggests that conservatives
react to external events but their attention quickly fades. In contrast, a
slowly-decaying ACF for liberals suggests that the moral language resonates
within a community and continues to be used long after the precipitating
event.
We measure how long it takes for ACF to decay to insignificant values. Fig. 6
shows these values for different issues discussed by liberals and
conservatives. We split the time series into two periods: before the first
COVID-19 vaccine became available in the US (December 11, 2020) and after. The
interesting differences in the two heatmaps suggest changes in the moral
language. In general, the care/harm and authority/subversion foundations
resonate with both communities, taking a long time to decay. Moral language
tends to persist longer in the discussions of liberals compared to
conservatives. The exceptions to this observations are seen in the discussions
about vaccines, lockdowns and COVID-19 origins in the post-vaccine period.
Given that pandemic origins is a partisan issue of greater interest to
conservatives, we expect their discussions on this topic to persist longer,
consistently with the trends we observe in the heatmap. In the discussions of
lockdowns, care and fairness resonate more for conservatives, while loyalty
resonates more for liberals. Purity language is no longer resonant for either
community; it is largely driven by the 3–4 day news cycles. Another surprising
reversal is for conservative discussions of vaccines, where purity language
becomes far more resonant in the post-vaccine period. This likely reflects the
worries of conservatives to inject themselves with an “experimental vaccine”.
This is even more surprising as purity language becomes less resonant among
both liberals and conservatives in the post-vaccine period, making its
persistence among conservative discussions of vaccines that much more
remarkable. The findings suggest that informational campaigns aimed at
different groups should appeal to moral foundations that resonate within each
group.
Similar analysis of the moral language of retweets shows that ACF quickly
decays, with correlations losing significance after three to five days. This
suggests that moral language in retweets is driven largely by the news cycle.
### Partisan Differences in Moral Language
To study which moral values are important to liberals and conservatives, we
measure the share of issue-related tweets posted by each group that expresses
a certain moral foundation. We examine moral valence by separately calculating
the share for vices and virtues. The proportion $\rho(m,i)$ of original tweets
related to issue $i$ that express moral foundation vice or virtue $m$ is:
$\rho(m,i)=\frac{\tau(m,i)}{\tau(i)},$ where, $\tau(i)$ returns the number of
original tweets on issue $i$.
Figure 7 shows partisan differences in the proportions $\rho$. Across issues,
we find that care and harm the most prevalent moral virtues and vices,
followed by authority and subversion, with very few tweets expressing purity
and degradation. Conservatives express more vices than virtues. There are
other partisan differences in the use of moral language. For example, while
both appealed to subversion, betrayal, and cheating when discussing the
origins of COVID-19, liberals focus more on harm (see Fig. 7(a)). This could
reflect greater concern among liberals about the potential harms of these
speculations, as they led to a rise in anti-Asian hate crimes across the
nation (Gover, Harper, and Langton 2020).
Across lockdowns, masking, education, and vaccine issues (Figs. 7(b)-(e)),
conservatives expressed greater levels of harm and subversion than liberals,
while liberals expressed more care. This may indicate greater support for
mandates (e.g., school shutdowns and vaccine mandates) among liberals, and
opposition by conservatives, consistent with partisan divides on shutdowns,
social distancing, and masks (Deane, Parker, and Gramlich 2021).
Overall, we find similarities and interesting differences in the moral values
that liberals and conservatives appeal to when discussing COVID-19.
Conservatives appear to make more negatively valenced moral appeals when it
comes to lockdowns, masking, education, and vaccines.
### Differences in Moral Language Use by Elites and Non-Elites
Past research has shown that moral language used by political elites (Clifford
and Jerit 2013; Wang and Inbar 2021) in the US can influence the opinions and
moral reasoning of non-elites (Clayton et al. 2021). In this work, we compare
the moral language used by elites and non-elites. Political elites in US
include prominent members of Congress, politicians, journalists and media
pundits. Previous studies (Wojcieszak et al. 2022; Shugars et al. 2021) have
curated extensive lists of Twitter handles for over 4K US political elites. To
ensure a fair comparison with non-elites, we randomly sample non-elite
original tweets such that the number matches the number of tweets from elites
on a given day. The random sampling also ensures that tweets from non-elites
with higher overall activity are selected. We then calculate the daily
proportion of issue-related tweets that express a certain moral foundation.
Specifically, we calculate the proportion $\rho$ of original tweets relevant
to issue $i$ that express moral foundation $m$ on day $d$:
$\rho(m,i,d)=\frac{\tau(m,i,d)}{\tau(i,d)}$ where, $\tau$ returns the number
of original tweets. We calculate this separately for elites and non-elites. To
account for variation, we bootstrap non-elite tweets 100 times.
Figure 8((a)-(e)) compares moral language use by elites and non-elites across
issues. Boxplots show the distribution of the daily share of tweets in each
category. We do not show loyalty/betrayal and purity/degradation foundations,
as they were not substantially expressed by elites. The lower variance for
non-elites is an attribute of bootstrapping. We rely on the non-parametric
Mann-Whitney U Test to test the significance of our comparisons. All
differences are assessed for significance at $p<0.001$ unless otherwise
specified.
On the issue of origins (Fig. 8(a)), non-elites express more care, subversion
and cheating than elites. Conservative non-elites use more harm language than
conservative elites, a difference that is not significant for liberals.
Lockdowns, masking, and education tweets all show a pattern of elites using
more moral language than non-elites across most moral foundations, with
liberal elites and non-elites tending to refer more to care (Fig. 8(b-d)).
Discussions of vaccines deviate somewhat from these trends (Fig. 8(e)), with
conservative non-elites expressing more harm than both liberals and
conservative elites. Conservatives (both elites and non-elites) also express
more subversion than liberals.
## Conclusions
The COVID-19 pandemic presented a significant challenge to society. The
ensuing response to the pandemic quickly became polarized, with liberals and
conservatives disagreeing on the severity of the pandemic and appropriate
measures to address it. Sharp differences in issue positions were observed in
discussions relating to the origins of the pandemic, lockdowns and business
closures, masking mandates, disruptions to education, and vaccines. Previous
studies (Schaeffer 2020; Connaughton 2021; Rojas 2020; Luttrell and Trentadue
2023; Chan 2021; Pierri et al. 2022; Rathje et al. 2022) attest to the
salience of these issues during the pandemic. We use a massive corpus of over
200M tweets (Chen et al. 2020) to assess these ideological divisions.
We find that importance of these issues varies by political ideology. While
liberals dominated the discussion on education, tweets about COVID-19 origins
and lockdowns were much more prevalent among conservative. Tweets about the
vaccine were predominantly made by liberals once the vaccine became available,
but switched to being more important to conservatives after federal vaccine
mandate were announced. We also found strong differences in moral language
among liberals and conservatives. We found that tweets expressing the
care/harm and authority/subversion foundations were most persistent,
suggesting that these values resonated with both groups. Our time series
analysis of moral expression suggests that conservatives were more reactive,
driven by the dynamics of the news cycle. Our analysis revealed further
differences in the moral language of partisans. While conservatives relied on
negatively-valenced moral language, like harm and subversion, liberals
primarily leveraged care. We also found that political elites in the US tended
to use more moral language compared to non-elites, regardless of ideology. In
sum, our findings suggest that differences in the expressions of moral values
drove ideological divisions on different issues, fueling polarization.
However, it is worth noting that our methods have several limitations. We
infer geo-location of tweets based on tweet metadata and user bios’ which may
not be accurate. Expression of ideological preferences and moral appeals can
be both subtle and highly subjective. Methods leveraged to identify ideology
and moral foundations, while state-of-the-art, may suffer from inconsistencies
and biases given the task’s inherent complexity. While our method to identify
issue relevant tweets is highly scalable, less explicit references to issues
in tweets may be missed by this method. Future work could test a supervised
classifier that may more accurately identify all relevant tweets. While we
made an effort to combine available lists of US political elites on Twitter,
this list is by no means exhaustive.
### Broader perspective, ethics and competing interests
All data used in this study is publicly available (Chen et al. 2020; Check
2023). The study was deemed exempt from review by the Institutional Review
Board (IRB) at the University of Southern California, as it relied solely on
publicly available data. Our study also adheres to Twitter’s terms of service
(Twitter 2022). Tweet objects contain user information and this brings user
anonymity into question. Users can restrict their tweets by switching to a
private account or by deleting their tweets. Additionally, we preserve user
anonymity during analysis by relying on user IDs instead of screen names.
However, we acknowledge that many Twitter users may not be aware their data is
used for research purposes (Fiesler and Proferes 2018).In this article, we
present aggregated statistics to address further mitigate this concern. The
authors declare that there are no competing interests.
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# Hadron yield correlation and constituent quark degree of freedom in heavy
ion collisions
Rui-qin Wang Department of Physics, Qufu Normal University, Shandong 273165,
People’s Republic of China Feng-lan Shao<EMAIL_ADDRESS>Department
of Physics, Qufu Normal University, Shandong 273165, People’s Republic of
China Jun Song Department of Physics, Shandong University, Shandong 250100,
People’s Republic of China Qu-bing Xie Department of Physics, Shandong
University, Shandong 250100, People’s Republic of China
###### Abstract
Based on the assumption of the production of deconfined quark matter, we use a
quark combination model to systematically investigate hadron yields in heavy
ion collisions from RHIC $\sqrt{s_{NN}}=200,130,62.4$ GeV to SPS
$E_{beam}=158,80,40,30,20$ AGeV. We find that as the collision energy is
greater than or equal to 30 AGeV the yields of various hadrons, their
correlations, in particular, the observables
$A=\frac{\overline{\Lambda}~{}k^{-}~{}p}{\Lambda~{}k^{+}~{}\overline{p}}$ and
$B=\frac{\Lambda~{}k^{-}~{}\overline{\Xi}^{+}}{\overline{\Lambda}~{}k^{+}~{}\Xi^{-}}$
, are all reproduced; however, as the collision energy drops to 20 AGeV quark
combination fails. This indicates that the constituent quark degrees of
freedom represent a decisive factor in thermal hadron production above 30 AGeV
and seem to be invalid at 20 AGeV. In addition, hadron yields as well as
particle ratios at midrapidity in the most central Pb+Pb collisions at
$\sqrt{s_{NN}}=5.5$ TeV are predicted.
###### pacs:
25.75.Dw, 25.75.Gz, 25.75.Nq, 25.75.-q
## I INTRODUCTION
The question at which collision energy in heavy ion collisions the
deconfinement is first reached has attracted more and more attentions in
recent years Andronic2009PLB ; YGMa2009nuclex ; Hove1982PLB ; Gazdzicki ;
Gazdz ; GAGo99 ; Bugaev2003PLB ; Akkelin . The Beam Energy Scan programme of
NA49 experiment at the CERN-SPS has suggested a preliminary answer — around 30
AGeV20A30A2008PRC . The ongoing Beam Energy Scan programme of STAR
Collaboration at Brookhaven National Lab provides an opportunity to study it
in more detailstarESC ; besRHIC . Once the deconfined hot and dense quark
matter is produced in heavy ion collisions, the observables of various thermal
hadrons after hadronization, e.g. yields and momentum spectra etc, have some
correlations originated from early quark degrees of freedom. One of the most
typical examples is the elliptic flow (v2) of hadrons measured at RHIC
energies. As both v2 and transverse momentum($p_{T}$) are divided by the
constituent quark number of hadron, the rescaled v2 of various baryons and
mesons, which are just that of constituent quarks, almost coincide with each
other in the intermediate $p_{T}$ range AAdare2007PRL . If the hot and dense
quark matter is hadronized by quark (re-)combination/coalescenceFries2003PRL ;
Greco2003PRL ; FLShao2005PRC , as is commonly accepted, these correlations of
hadrons can be beautifully explained. In quark (re-)combination/coalescence
scenario, quarks and antiquarks are available in unbound state before
hadronization and they can coalesce freely into various hadrons, and thereby
these correlations from early quark degrees of freedom among different hadron
species are naturally formed. On the other hand, if the deconfined quark
matter is not produced at all in collisions, there is no free quarks and
antiquarks (much less their subsequent combination) and these so-called
“quark-level” correlations of hadrons maybe disappear or contort. Therefore,
we can study whether the deconfinement is achieved by investigating these
correlations among various hadrons produced in heavy ion collisions.
Hadron yield is one of the most significant observables from which one can
obtain a lot of important information on the hot nuclear matter produced at
the early stage of relativistic heavy ion collisions. At the high collision
energies RHIC and top SPS, the yields of different hadron species have shown
an explicit “quark-level” correlation in quark (re-)combination/coalescence
scenario A.Bialas1998PLB ; J.Zimanyi2000PLB ; FLShao2005PRC . In this paper,
we make an energy scan from RHIC energies $\sqrt{s_{NN}}=200,130,62.4$ GeV to
SPS energies $E_{beam}=158,80,40,30,20$ AGeV to study at which collision
energy this “quark-level” correlation of hadron yields first breaks. In
particular, we define two correlation quantities
$A=\frac{\overline{\Lambda}~{}k^{-}~{}p}{\Lambda~{}k^{+}~{}\overline{p}}$ and
$B=\frac{\Lambda~{}k^{-}~{}\overline{\Xi}^{+}}{\overline{\Lambda}~{}k^{+}~{}\Xi^{-}}$,
sensitive to quark degrees of freedom. The values of A and B are equal to one
in the framework of quark (re-)combination/coalesce, independent of models.
The deviation of A and B from one or not can be regarded as a possible signal
of deconfinement in heavy ion collisions. We apply a quark combination model,
which can exclusively describe hadron production and well reproduce the yields
and momentum spectra of final-state hadrons in relativistic heavy ion
collisionsFLShao2007PRC ; CEShao2009PRC ; JSong2009MPA ; DMWei2008MPA ;
YFWang2008CPC ; WHan2009PRC , to carry out the concrete calculations.
The paper is organized as follows. In Sec.II, we present the relations among
yields of various hadrons in quark (re-)comibination/coalescence scenario. In
Sec.III, we use the quark combination model to calculate the yields of various
hadrons, their yield ratios and correlation quantities
$A=\frac{\overline{\Lambda}~{}k^{-}~{}p}{\Lambda~{}k^{+}~{}\overline{p}}$ and
$B=\frac{\Lambda~{}k^{-}~{}\overline{\Xi}^{+}}{\overline{\Lambda}~{}k^{+}~{}\Xi^{-}}$
at midrapidity in the most central A+A collisions at different energies.
Sec.IV summaries our work.
## II hadron yields in quark (re-)combination/coalescence scenario
Let us start from the general inclusive formula of hadron production in quark
(re-)combination/coalescence scenario
$\displaystyle N_{M}$ $\displaystyle=$ $\displaystyle\int
dp_{1}dp_{2}F_{q\bar{q}}(p_{1},p_{2})\,\mathcal{R}_{M}(p_{1},p_{2})$ (1)
$\displaystyle N_{B}$ $\displaystyle=$ $\displaystyle\int
dp_{1}dp_{2}dp_{3}F_{qqq}(p_{1},p_{2},p_{3})\,\mathcal{R}_{B}(p_{1},p_{2},p_{3}).$
(2)
Here $F_{q\bar{q}}$ ($F_{qqq}$) is the joint quark-antiquark (three quark)
distribution. $\mathcal{R}_{M}$ ($\mathcal{R}_{B}$) is the combination
function which stands for the formation probability of quark antiquark (three
quarks) into a meson (baryon), dominated by chromodynamics. In sudden
approximation, it is equal to the overlap between two (three) quark wave
functions and the wave function of meson (baryon). Neglecting exotic (multi-
quark) states, mesons and baryons exhaust all fate of quarks and antiquarks.
One reaches the following relations : $\sum N_{M}+3\sum N_{B}=\sum N_{q}$ and
$\sum N_{M}+3\sum N_{\bar{B}}=\sum N_{\bar{q}}$, where $N_{q}$ ($N_{\bar{q}}$)
is the quark (antiquark) number of flavor $q$($\bar{q}$). Extracting $N_{q}$
and $N_{\bar{q}}$ from the joint two (three) quark distribution $F_{q\bar{q}}$
($F_{qqq}$) and putting them out of the integral, one has the following
schematic relations between hadron yields and quark numbers after integrating
over quark momenta
$N_{M(q\bar{q})}\propto C_{M}N_{q}N_{\bar{q}},\hskip
28.45274ptN_{B({qqq})}\propto C_{B}N_{q}N_{q}N_{q},$ (3)
and quark number conservation will fix the proportionality coefficient. The
effects of combination function on hadron yields are characterized with the
factors $C_{M}$ and $C_{B}$, and $C_{M}=C_{\overline{M}}$ and
$C_{B}=C_{\overline{B}}$ are assumed.
One realized method of quark number conservation during combination is adding
a factor $b_{q}$ for each quark flavor in above equations, as did in ALCOR
model alcor95 ,
$\displaystyle N_{M(q\bar{q})}$ $\displaystyle=$ $\displaystyle
C_{M}(b_{q}N_{q})(b_{\bar{q}}N_{\bar{q}}),$ (4) $\displaystyle N_{B({qqq})}$
$\displaystyle=$ $\displaystyle C_{B}(b_{q}N_{q})(b_{q}N_{q})(b_{q}N_{q}).$
(5)
Then $b_{q}$ can be uniquely determined by quark number conservation.
According to Eqs.(4) and (5), we have the following relations between hadrons
and the corresponding antihadrons
$\displaystyle\frac{\overline{p}}{p}$ $\displaystyle=$
$\displaystyle(\frac{b_{\overline{q}}\overline{q}}{b_{q}q})^{3},\hskip
45.52458pt\frac{k^{+}}{k^{-}}=(\frac{b_{q}q}{b_{\overline{q}}\overline{q}})(\frac{b_{\overline{s}}\overline{s}}{b_{s}s}),$
$\displaystyle\frac{\overline{\Lambda}}{\Lambda}$ $\displaystyle=$
$\displaystyle(\frac{b_{\overline{q}}\overline{q}}{b_{q}q})^{2}\frac{b_{\overline{s}}\overline{s}}{b_{s}s},\hskip
25.6073pt\frac{\Xi^{-}}{\overline{\Xi}^{+}}=(\frac{b_{s}s}{b_{\overline{s}}\overline{s}})^{2}\frac{b_{q}q}{b_{\overline{q}}\overline{q}}.$
Here, we use particle symbols stand for their numbers for short, $q$ for light
quark number and $s$ for strange quark number. Hiding the quark content in
hadron yield, we obtain the following interesting relations among different
hadron species
$\frac{\overline{\Lambda}~{}k^{-}}{\Lambda~{}k^{+}}=\frac{\overline{p}}{p},\hskip
28.45274pt\frac{\Lambda~{}k^{-}}{\overline{\Lambda}~{}k^{+}}=\frac{\Xi^{-}}{\overline{\Xi}^{+}}.$
We define correlation quantities $A$ and $B$ as follows
$A=\frac{\overline{\Lambda}~{}k^{-}~{}p}{\Lambda~{}k^{+}~{}\overline{p}}\hskip
28.45274ptB=\frac{\Lambda~{}k^{-}~{}\overline{\Xi}^{+}}{\overline{\Lambda}~{}k^{+}~{}\Xi^{-}}.$
(6)
If the quark matter exists and hadronizes via sudden (re-)combination/coalesce
in heavy ion collisions, A and B should be equal to one for directly produced
hadrons. It is a general result under the constraint of quark number
conservation, which is independent of specific models.
Different from the well-known recombination and coalescence models
Fries2003PRL ; Greco2003PRL , the quark combination model FLShao2005PRC ;
QBXie1988PRD is unique for its combination rule. The main idea of the
combination rule is to line up the (anti)quarks in a one-dimensional order in
phase space, e.g., in rapidity, and then let them combine into initial hadrons
one by one according to this orderFLShao2005PRC . Three (anti)quarks or a
quark-antiquark pair in the neighborhood form a (anti)baryon or a meson,
respectively. At last all quarks and antiquarks are combined into hadrons. The
relations between hadron yields and the corresponding quark numbers are easily
obtained CEShao2009PRC . With this rule, the model can give the yields and
momentum distributions of all hadrons (included in the model) in an event,
possessing some exclusive nature. The decay of the short-life resonances is
systematically taken into account in the model. The model has been realized in
Monte Carlo program and has described many properties of hadron production in
relativistic heavy ion collisions FLShao2007PRC ; CEShao2009PRC ; JSong2009MPA
; DMWei2008MPA ; YFWang2008CPC ; WHan2009PRC .
## III RESULTS AND DISCUSSIONS
In this section, we use the quark combination model to calculate the hadron
yields and their ratios as well as correlation quantities
$A=\frac{\overline{\Lambda}~{}k^{-}~{}p}{\Lambda~{}k^{+}~{}\overline{p}}$ and
$B=\frac{\Lambda~{}k^{-}~{}\overline{\Xi}^{+}}{\overline{\Lambda}~{}k^{+}~{}\Xi^{-}}$
at midrapidity in the most central A+A collisions at
$\sqrt{s_{NN}}=200,~{}130,~{}62.4$ GeV and
$E_{beam}=158,~{}80,~{}40,~{}30,~{}20$ AGeV. The predictions at LHC are also
presented. The necessary input of the model, i.e., quark distribution just
before hadronization, can be obtained by applying the hydrodynamics to
describe the evolution of hot and dense quark matter just before
hadronization, see Appendix A for details.
### III.1 Hadron yields, their ratios and correlation quantities A and B
Table 1: The calculated hadron yields $dN/dy$ at midrapidity in the most
central A+A collisions at different energies. The experimental data are taken
from Refs20A30A2008PRC ; 20A30A40A80A158A2006PRC ; 20A30A40A80A158A2008PRC ;
40A80A158A2004PRL ; 40A158A2005PRL ; 158A2002PRC ; 622009nuclex ; 1302002PRL ;
1302004PRL ; 1302004PRC ; 2002004PRC ; 2002007PRL .
| Au+Au 200 GeV | Au+Au 130 GeV | Au+Au 62.4 GeV | Pb+Pb 158 AGeV
---|---|---|---|---
| data | model | data | model | data | model | data | model
$\pi^{+}$ | $286.4\pm 24.2$ | 281.0 | $276\pm 3\pm 35.9$ | 267.2 | $233\pm 17$ | 227.4 | $170.1\pm 0.7\pm 9$ | 165.1
$\pi^{-}$ | $281.8\pm 22.8$ | 281.8 | $270\pm 3.5\pm 35.1$ | 270.5 | $237\pm 17$ | 233.5 | $175.4\pm 0.7\pm 9$ | 176.5
$k^{+}$ | $48.9\pm 6.3$ | 48.6 | $46.7\pm 1.5\pm 7.0$ | 45.2 | $37.6\pm 2.7$ | 38.3 | $29.6\pm 0.3\pm 1.5$ | 27.2
$k^{-}$ | $45.7\pm 5.2$ | 46.1 | $40.5\pm 2.3\pm 6.1$ | 42.4 | $32.4\pm 2.3$ | 32.2 | $16.8\pm 0.2\pm 0.8$ | 16.2
$p$ | $18.4\pm 2.6$ | 17.0 | $28.7\pm 0.9\pm 4.0$ | 25.7 | $29.0\pm 3.8$ | 29.1 | $29.6\pm 0.9\pm 2.9$ | 30.1
$\overline{p}$ | $13.5\pm 1.8$ | 12.5 | $20.1\pm 1.0\pm 2.8$ | 18.2 | $13.6\pm 1.7$ | 13.5 | $1.66\pm 0.17\pm 0.16$ | 1.91
$\Lambda$ | $16.7\pm 0.2\pm 1.1$ | 15.3 | $17.3\pm 1.8\pm 2.8$ | 14.5 | $14.9\pm 0.2\pm 1.49$ | 13.7 | $10.9\pm 1.0\pm 1.3$ | 13.3
$\overline{\Lambda}$ | $12.7\pm 0.2\pm 0.9$ | 12.1 | $12.7\pm 1.8\pm 2.0$ | 10.9 | $8.02\pm 0.11\pm 0.8$ | 7.28 | $1.62\pm 0.16\pm 0.2$ | 1.66
$\Xi^{-}$ | $2.17\pm 0.06\pm 0.19$ | 2.05 | $2.04\pm 0.14\pm 0.2$ | 1.93 | $1.64\pm 0.03\pm 0.014$ | 1.67 | $1.44\pm 0.10\pm 0.15$ | 1.18
$\overline{\Xi}^{+}$ | $1.83\pm 0.05\pm 0.20$ | 1.69 | $1.74\pm 0.12\pm 0.17$ | 1.53 | $0.989\pm 0.057\pm 0.057$ | 1.02 | $0.31\pm 0.03\pm 0.03$ | 0.23
$\Omega^{-}$ | | | | | | | $0.14\pm 0.03\pm 0.01$ | 0.11
$\overline{\Omega}^{+}$ | $0.53\pm 0.04\pm 0.04$ | 0.56 | $0.56\pm 0.11\pm 0.06$ | 0.52 | $0.356\pm 0.046\pm 0.014$ | 0.379 | $0.07\pm 0.02\pm 0.01$ | 0.04
$\chi^{2}/ndf$ | $2.8/8$ | $2.0/8$ | $2.1/8$ | $8.5/9$
| Pb+Pb 80 AGeV | Pb+Pb 40 AGeV | Pb+Pb 30 AGeV | Pb+Pb 20 AGeV
---|---|---|---|---
| data | model | data | model | data | model | data | model
$\pi^{+}$ | $132.0\pm 0.5\pm 7$ | 129.9 | $96.6\pm 0.4\pm 6$ | 97.8 | $83.0\pm 0.4\pm 4.2$ | 84.8 | $72.9\pm 0.3\pm 3.6$ | 73.6
$\pi^{-}$ | $140.4\pm 0.5\pm 7$ | 141.8 | $106.1\pm 0.4\pm 6$ | 110.1 | $96.5\pm 0.5\pm 4.8$ | 99.3 | $84.8\pm 0.4\pm 4.2$ | 85.0
$k^{+}$ | $24.6\pm 0.2\pm 1.2$ | 23.8 | $20.1\pm 0.3\pm 1.0$ | 19.4 | $21.2\pm 0.8^{+1.5}_{-0.9}$ | 21.6 | $16.4\pm 0.6\pm 0.4$ | 16.6
$k^{-}$ | $11.7\pm 0.1\pm 0.6$ | 11.9 | $7.58\pm 0.12\pm 0.4$ | 7.27 | $7.8\pm 0.1\pm 0.2$ | 7.3 | $5.58\pm 0.07\pm 0.11$ | 5.01
$p$ | $30.1\pm 1.0\pm 3.0$ | 31.5 | $41.3\pm 1.1\pm 4.1$ | 38.6 | $42.1\pm 2.0\pm 4.2$ | 37.7 | $46.1\pm 2.1\pm 4.6$ | 37.0
$\overline{p}$ | $0.87\pm 0.07\pm 0.09$ | 0.83 | $0.32\pm 0.03\pm 0.03$ | 0.36 | $0.16\pm 0.02\pm 0.02$ | 0.17 | $0.06\pm 0.01\pm 0.006$ | 0.07
$\Lambda$ | $13.5\pm 0.7\pm 1.0$ | 14.5 | $15.3\pm 0.6\pm 1.0$ | 14.7 | $14.7\pm 0.2\pm 1.2$ | 14.2 | $13.4\pm 0.1\pm 1.1$ | 11.5
$\overline{\Lambda}$ | $1.06\pm 0.08\pm 0.1$ | 0.95 | $0.42\pm 0.04\pm 0.04$ | 0.49 | $0.21\pm 0.02\pm 0.02$ | 0.20 | $0.10\pm 0.02\pm 0.01$ | 0.08
$\Xi^{-}$ | $1.22\pm 0.14\pm 0.13$ | 1.23 | $1.15\pm 0.11\pm 0.13$ | 1.10 | $1.17\pm 0.13\pm 0.13$ | 1.38 | $0.93\pm 0.13\pm 0.10$ | 0.92
$\overline{\Xi}^{+}$ | $0.21\pm 0.03\pm 0.02$ | 0.15 | $0.07\pm 0.01\pm 0.01$ | 0.08 | $0.05\pm 0.01\pm 0.01$ | 0.07 | $------$ | 0.02
$\Omega^{-}+\overline{\Omega}^{+}$ | $------$ | 0.13 | $0.10\pm 0.02\pm 0.02$ | 0.09 | $------$ | 0.14 | $------$ | 0.07
$\chi^{2}/ndf$ | $2.9/7$ | $3.0/8$ | $5.6/7$ | $15.3/6$
Table 1 shows the hadron density dN/dy at midrapidity in the most central A+A
collisions at RHIC energies 200, 130, 62.4 GeV and SPS energies 158, 80, 40,
30, 20 AGeV. The experimental data are taken from Refs20A30A2008PRC ;
20A30A40A80A158A2006PRC ; 20A30A40A80A158A2008PRC ; 40A80A158A2004PRL ;
40A158A2005PRL ; 158A2002PRC ; 622009nuclex ; 1302002PRL ; 1302004PRL ;
1302004PRC ; 2002004PRC ; 2002007PRL . The agreement between the calculation
results and the experimental data is good except at 20 AGeV where
$\chi^{2}/ndf$ is far greater than one. We note that ALCOR model alcor95 can
also describe well the hadron yields at top SPS energy, which gives a cross-
verification of quark combination. On the contrary, HIJING or HIJING/B model
which is in compliance with the hadronic scenario for the early evolution of
heavy ion collisions via a fragmentation hadronization, can not self-
consistently explain the data of multi-strange hadrons topor94 ; Csizmadia99 .
Fig. 1 shows the ratios of antihadrons to hadrons at midrapidity as the
function of collision energy. The filled triangles are the calculated results,
and the experimental data are taken from Refs20A30A2008PRC ;
20A30A40A80A158A2006PRC ; 20A30A40A80A158A2008PRC ; 40A80A158A2004PRL ;
158A2002PRC ; 622009nuclex ; 1302002PRL ; 1302004PRC ; 2002004PRC ; 2002007PRL
. These ratios are mainly influenced by the no-zero baryon number density. As
the collision energy increases, the nuclear transparency power becomes strong
and baryon number density at midrapidity becomes small. This results in a
rapid increase for ratios of $K^{-}/K^{+}$, $\overline{p}/p$,
$\overline{\Lambda}/\Lambda$ with the increasing collision energy.
$\pi^{-}/\pi^{+}$ follows a different pattern. At low collision energies
$\pi^{-}/\pi^{+}$ is slightly higher than one, while it is close to one at
high energies. This is caused by the asymmetry of decay contribution from
hyperons and anti-hyperons (e.g. $\Lambda\rightarrow p\,\pi^{-}$). As the
collision energy increases, the yields of hyperons are close to that of anti-
hyperons and their decay contributions to pion yields are almost the same and
therefore the ratio of $\pi^{-}/\pi^{+}$ is close to one.
Figure 1: (Color online)The yield ratios of antihadrons to hadrons at
midrapidity as the function of the collision energy. The filled symbols are
the calculated results, and the experimental data, open symbols with error
bar, are from Refs20A30A2008PRC ; 20A30A40A80A158A2006PRC ;
20A30A40A80A158A2008PRC ; 40A80A158A2004PRL ; 158A2002PRC ; 622009nuclex ;
1302002PRL ; 1302004PRC ; 2002004PRC ; 2002007PRL .
Fig. 2 shows the ratios of baryons to mesons $p/\pi$ and the strangeness
ratios $K/\pi$ at midrapidity as the function of collision energy. The filled
triangles (up and down) are the computed results, and the experimental data
are taken from Refs20A30A2008PRC ; 20A30A40A80A158A2006PRC ; 158A2002PRC ;
622009nuclex ; 1302002PRL ; 1302004PRC ; 2002004PRC . The big splits between
$p/\pi^{+}$ and $\overline{p}/\pi^{-}$ and between $K^{+}/\pi^{+}$ and
$K^{-}/\pi^{-}$ at low collision energies is due to the high baryon number
density. At 20 AGeV, the result of $p/\pi^{+}$ deviates seriously from the
data. This is probably because the participant nucleons are not broken
completely in collisions. These nucleon fragments deposited in midrapidity
region lead to the extra contribution of proton production besides those from
quark combination and lead to the excessively high ratio of $p/\pi^{+}$.
Figure 2: (Color online)The values of the relative production yields $p/\pi$
and $k/\pi$ at different energies. The triangles down are the numerical
results for $p/\pi^{+}$ and $k^{+}/\pi^{+}$, and the triangles up are the
numerical results for $\overline{p}/\pi^{-}$ and $k^{-}/\pi^{-}$. The
experimental data, open symbols with error bar, are from Refs20A30A2008PRC ;
20A30A40A80A158A2006PRC ; 158A2002PRC ; 622009nuclex ; 1302002PRL ; 1302004PRC
; 2002004PRC .
Following the experimental data in Table 1, A and B at different collision
energies are evaluated and the results are presented with squares in Fig.3. We
find that the data of A and B at RHIC energies are almost equal to one while
at SPS energies they deviate from one; particularly, at 20 AGeV the value of A
amounts to two, seriously deviating from one (the data for B are unavailable).
This is probably because the decay of resonances will blur A and B to a
certain degree. In order to explore the decay effect, we use the quark
combination model to compute the values of A and B for the directly produced
hadrons and the final-state hadrons, respectively. The dashed lines are the
results of directly produced hadrons. Just as analyzed above, the dashed lines
keep an invariant value of one for both A and B, independent of collision
energy. The very small fluctuations of dashed lines are due to the fact that
the rapidity distributions of the formed hadrons are slightly different from
those of quarks, which leads to a small amount of hadrons formed by
midrapidity quarks escape from the midrapidity region FLShao2007PRC . The
filled triangles down are model results of final-state hadrons. We see that as
collision energy is greater than 30 AGeV these triangles agree well with the
experimental data within statistical uncertainties. Removing the part of
resonance decay, the data of A and B for directly produced hadrons should be
equal to one, respectively. This suggests the existence of the “quark level”
correlation of directly produced hadrons at these collision energies. However,
at 20 AGeV the value of A calculated via quark combination for final state
hadrons seriously deviates the data. This in itself indicates the quark
degrees of freedom do not represent a decisive factor in hadron production at
20 AGeV.
Figure 3: (Color online)The correlation quantities A and B as the function of
collision energy. The experimental data, open symbols with error bar, are from
Refs20A30A2008PRC ; 20A30A40A80A158A2006PRC ; 20A30A40A80A158A2008PRC ;
40A80A158A2004PRL ; 158A2002PRC ; 622009nuclex ; 1302002PRL ; 1302004PRL ;
1302004PRC ; 2002004PRC ; 2002007PRL .
From the above analysis of hadron yields, hadron ratios and correlation
quantities A and B, we find that as the collision energy is greater than or
equal to 30 AGeV the quark combination can describe reasonably well all these
observables which indicates the existence of constituent quark degrees of
freedom in the energy region. When the collision energy drops to 20 AGeV,
however, the quark combination mechanism can not self-consistently describe
these quantities, in particular the correlation quantity A. Note that in order
to best fit the data of hadron yields, we have adjusted the strangeness
parameter $\lambda_{s}$ (see Table 4) to the quite high values at 20, 30 AGeV
which are far greater than the saturated values at higher collision energies
and are also greater than those at lower energies thermalreview . But even so
the quark combination can not self-consistently explain the data at 20 AGeV.
In addition, the high strangeness entangled with baryon density leads to the
Sawtooth-like shape for calculated final-state A and B at low SPS energies. In
fact this peak behavior of strangeness (i.e. $K^{+}/\pi^{+}$ ratio or
$\lambda_{s}$ determined mainly by the former ) at low SPS energies has been
interpreted in Ref Gazdzicki ; GAGo99 as a result of onset of deconfinement,
i.e. the result of strangeness carrier changing from strange hadrons to
strange quarks at the onset of deconfinement. The failure of quark combination
energy indicates the (partial) disappearance of constituent quark degrees of
freedom and closely relates to the onset of deconfinement around 30 AGeV
observed by NA49 Collaboration 20A30A2008PRC .
### III.2 Predictions at LHC
With the increasing collision energy, the strangeness $\lambda_{s}$ of the hot
and dense quark matter in Table 4 tends to be saturated and the squared sound
velocity $c_{s}^{2}$ approaches an ideal value $1/3$ and the baryon number
density $n_{0}$ decreases regularly. So we take $\lambda_{s}$=0.43,
$c_{s}^{2}=1/3$ and $n_{0}=0.0fm^{-3}$ at $\sqrt{s_{NN}}=5.5$ TeV. The initial
entropy density of the central point at the beginning of hydrodynamic
evolution is taken to be 256.5 /fm3 according to Eq. (13). In Tables 2 and 3,
we present our predictions of hadron yields and hadron ratios at LHC energy
($\sqrt{s_{NN}}=5.5$ TeV). Our computed particle ratios are consistent with
the results of the thermal model in RefAndronic2009PLB .
Table 2: The predicted $dN/dy$ of identified hadrons at midrapidity in the most central ($0-5\%$) Pb+Pb collisions at $\sqrt{s_{NN}}=5.5$ TeV. $\pi^{+}$ | $\pi^{-}$ | $k^{+}$ | $k^{-}$ | $p$ | $\overline{p}$ | $\Lambda$ | $\overline{\Lambda}$ | $\Xi^{-}$ | $\overline{\Xi}^{+}$ | $\Omega^{-}$ | $\overline{\Omega}^{+}$
---|---|---|---|---|---|---|---|---|---|---|---
435.5 | 435.5 | 71.6 | 71.6 | 34.6 | 34.6 | 20.5 | 20.5 | 2.9 | 2.9 | 0.4 | 0.4
Table 3: The predicted hadron ratios at midrapidity in the most central Pb+Pb collisions at $\sqrt{s_{NN}}=5.5$ TeV. $p/\pi^{+}$ | $\overline{p}/\pi^{-}$ | $k^{+}/\pi^{+}$ | $k^{-}/\pi^{-}$ | $\Lambda/\pi^{-}$ | $\Xi^{-}/\pi^{-}$ | $\Omega^{-}/\pi^{-}$
---|---|---|---|---|---|---
0.079 | 0.079 | 0.165 | 0.165 | 0.047 | 0.007 | 0.001
## IV Summary
In this paper, we used the quark combination model to make a systematical
study of the hadron yields in heavy ion collision in a broad collision energy
region. At the collision energies where the deconfined quark matter has been
created, the yields of various hadrons after hadronization have some
correlations inherited from the early quark degrees of freedom. We investigate
at which collision energy this “quark-level” correlation of hadron yields
first breaks. We apply the hydrodynamics to describe the evolution of
deconfined quark matter and to obtain the quark distribution just before
hadronization, then utilize the quark combination model to describe
hadronization. We find that as the collision energy is greater than or equal
to 30 AGeV, the quark combination well reproduce the yields of various
hadrons, their ratios and correlation quantities A and B; however, as the
collision energy drops to 20 AGeV, the mechanism can not self-consistently
describe these quantities. This indicates that the constituent quark degrees
of freedom represent a decisive factor in thermal hadron production above 30
AGeV and seem to be invalid at 20 AGeV. It is related to the onset of
deconfinement observed at collision energy 20-30 AGeV. Finally, we predict the
yields of various hadrons and their ratios in the most central Pb+Pb
collisions at $\sqrt{s_{NN}}=5.5$ TeV.
### ACKNOWLEDGMENTS
The authors thank Z. T. Liang, Q. Wang and G. Li for helpful discussions. R.
Q. Wang would like to thank L. G. Pang and J. Deng for fruitful discussions.
The work is supported in part by the National Natural Science Foundation of
China under the grant 10775089, 10947007 and 10975092.
## Appendix A Quark distribution just before hadronization
Table 4: The values of the initial baryon density $n_{0}$, strangeness factor $\lambda_{s}$ and squared sound velocity $c_{s}^{2}$ in central A+A collisions at different energies. Energy | 200GeV | 130GeV | 62.4GeV | 158AGeV | 80AGeV | 40AGeV | 30AGeV | 20AGeV
---|---|---|---|---|---|---|---|---
$n_{0}$ ($fm^{-3}$) | 0.30 | 0.34 | 0.74 | 1.31 | 1.46 | 1.64 | 1.74 | 1.56
$\lambda_{s}$ | 0.43 | 0.43 | 0.43 | 0.44 | 0.50 | 0.57 | 0.80 | 0.70
$c_{s}^{2}$ | $1/3.1$ | $1/3.4$ | $1/4.0$ | $1/6.0$ | $1/6.0$ | $1/6.0$ | $1/6.0$ | $1/6.0$
The quark distribution just before hadronization is needed for the quark
combination to describe the hadron production in relativistic heavy ion
collisions. Here, we use relativistic hydrodynamics Landau53 ; DirkH1995NPA ;
Kolb0305084nuth to describe the time-space evolution of the hot and dense
quark matter before hadronization. The evolution equation of hydrodynamics
follows from the local conservation laws for energy, momentum, and other
conserved charges, e.g. baryon number,
$\displaystyle\partial_{\mu}T^{\mu\nu}(x)=0,~{}~{}(\nu=0,1,2,3)$ (7)
$\displaystyle\partial_{\mu}j^{\mu}(x)=0,$ (8)
by inserting the ideal fluid decomposition
$\displaystyle
T^{\mu\nu}(x)=\bigg{(}e(x)+p(x)\bigg{)}u^{\mu}(x)u^{\nu}(x)-g^{\mu\nu}p(x),$
(9) $\displaystyle j^{\mu}(x)=n(x)u^{\mu}(x).$ (10)
Here, $u^{\mu}(x)=\gamma(1,v_{x},v_{y},v_{z})$ with
$\gamma=1/\sqrt{1-v_{x}^{2}-v_{y}^{2}-v_{z}^{2}}$ is the local four velocity
of a thermalized fluid cell; $e(x)$ is the energy density, $p(x)$ the
pressure, and $n(x)$ the conserved number density.
As the energy density of the fluid cell drops to 1.0 GeV/fm3 (a common
criteria of phase transition from Lattice QCDKarsch2002NPA ), we stop the
hydrodynamic evolution and let the constituent quarks and antiquarks freeze
out according to Cooper-Frye formalism Coopfrye74
$E\frac{dN_{i}}{d^{3}p}=\frac{dN_{i}}{dyp_{T}dp_{T}d\varphi}=\frac{g_{i}}{(2\pi)^{3}}\int_{\Sigma}f_{i}(p\cdot
u(x),x)p\cdot d^{3}\sigma(x),$ (11)
where $d^{3}\sigma(x)$ is the outward normal vector on the freeze-out surface
$\Sigma(x)$, $g_{i}$ the degeneracy factor of quarks ($g_{i}=6$). The phase-
space distribution $f$ in the formula is taken to be a local equilibrium
distribution,
$f_{i}(E,x)=\frac{1}{\exp[(E-\mu_{i}(x))/T(x)]+1}.$ (12)
We consider only the freeze-out of light and strange quarks and antiquarks.
The three chemical potentials $\mu_{q}$, $\mu_{\bar{q}}$ and
$\mu_{s}=\mu_{\bar{s}}$ at freeze-out can be determined uniquely by the global
conservation of energy and baryon number plus an ancillary constraint of
strangeness.
We study in this paper the hadron production in midrapidity region only. The
code of KolbKolb2000PRC ; Kolb2001NPA ; Kolb2003PRC for 2+1- dimensional
hydrodynamics with the longitudinal boost invariance is used to simulate the
evolution of quark system before hadronization.
These frozen-out quarks and antiquarks with momentum distributions in Eq. (11)
are hadronized by the quark combination modelFLShao2005PRC ; QBXie1988PRD . In
this method of hydrodynamics + quark combination, the quark combination model
is default setting while some inputs for hydrodynamics need fixing. The first
is initial entropy density $s_{0}$ of the central point at the beginning of
hydrodynamic evolution. The distribution of transverse entropy density is
determined by optical Glauber modelGlauber1959 . We apply the following
empirical formula of $s_{0}$entropy2008nuth as the function of collision
energy $\sqrt{s_{NN}}$
$s_{0}(\sqrt{s_{NN}})=6.99\times
10^{-3}\Big{(}312.5\log_{10}{\sqrt{s_{NN}}}-64.8\Big{)}^{3/2}.$ (13)
The second is the initial baryon number density $n_{0}$ at central point fixed
by the data of the net-proton rapidity density. The distribution of transverse
baryon number density is also determined by optical Glauber model. The third
is the equation of state $p=p(e)$ for the hot and dense quark matter which is
taken to be the simplest pattern $p=c_{s}^{2}e$. The squared sound velocity
$c_{s}^{2}$ is obtained by fitting the data of transverse momentum spectra of
protons. The fourth is the strangeness of hot and dense quark matter denoted
by the factor $\lambda_{s}=2\langle s\bar{s}\rangle/\langle
u\bar{u}+d\bar{d}\rangle$. In order to make a better description of strange
hadrons, we regard it as a free parameter in the present paper. The values of
$n_{0}$, $\lambda_{s}$ and $c_{s}^{2}$ in central A+A collisions at different
energies are shown in Table 4. We note that the extracted $\lambda_{s}$ around
30 AGeV are quite high which are consistent with the analytic results of
thermal model thermalreview .
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|
# Learning Relationships between Text, Audio, and Video via Deep Canonical
Correlation for Multimodal Language Analysis
Zhongkai Sun,1 Prathusha K Sarma,2 William A Sethares,1 Yingyu Liang1
1University of Wisconsin-Madison, 2Curai
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Work done while at UW-Madison
###### Abstract
Multimodal language analysis often considers relationships between features
based on text and those based on acoustical and visual properties. Text
features typically outperform non-text features in sentiment analysis or
emotion recognition tasks in part because the text features are derived from
advanced language models or word embeddings trained on massive data sources
while audio and video features are human-engineered and comparatively
underdeveloped. Given that the text, audio, and video are describing the same
utterance in different ways, we hypothesize that the multimodal sentiment
analysis and emotion recognition can be improved by learning (hidden)
correlations between features extracted from the outer product of text and
audio (we call this text-based audio) and analogous text-based video. This
paper proposes a novel model, the Interaction Canonical Correlation Network
(ICCN), to learn such multimodal embeddings. ICCN learns correlations between
all three modes via deep canonical correlation analysis (DCCA) and the
proposed embeddings are then tested on several benchmark datasets and against
other state-of-the-art multimodal embedding algorithms. Empirical results and
ablation studies confirm the effectiveness of ICCN in capturing useful
information from all three views.
## 1 Introduction
Human language communication occurs in several modalities: via words that are
spoken, by tone of voice, and by facial and bodily expressions. Understanding
the content of a message thus requires understanding all three modes. With the
explosive growth in availability of data, several machine learning algorithms
have been successfully applied towards multimodal tasks such as sentiment
analysis (?; ?), emotion recognition (?), image-text retrieval (?), and aiding
medical diagnose (?; ?) etc. Among multimodal language sentiment or emotion
experiments involving unimodal features (?; ?; ?; ?), it is commonly observed
that text based features perform better than visual or auditory modes. This is
plausible for at least three reasons: 1) Text itself contains considerable
sentiment-related information. 2) Visual or acoustic information may sometimes
confuse the sentiment or emotion analysis task. For instance: “angry” and
“excited” may have similar acoustic performances (high volume and high pitch)
even though they belong to opposite sentiments. Similarly, “sad” and
“disgusted” may have different visual features though they both belong to the
negative sentiment. 3) Algorithms for text analysis have a richer history and
are well studied.
Based on this observation, learning the hidden relationship between verbal
information and non-verbal information is a key point in multi-modal language
analysis. This can be approached by looking at different ways of combining
multi-modal features.
The simplest way to combine text (T), audio (A) and video (V) for feature
extraction and classification is to concatenate the A, V, and T vectors. An
alternative is to use the outer product (?; ?) which can represent the
interaction between pairs of features, resulting in 2D or 3D arrays that can
be processed using advanced methods such as convolutional neural networks
(CNNs) (?). Other approaches (?; ?; ?; ?) study multi-modal interactions and
intra-actions by using either graph or temporal memory networks with a
sequential neural network LSTM (?). While all these have contributed towards
learning multi-modal features, they typically ignore the hidden correlation
between text-based audio and text-based video. Individual modalities are
either combined via neural networks or passed directly to the final classifier
stage. However, it is obvious that attaching both audio and video features to
the same textual information may enable non-text information to be better
understood, and in turn the non-text information may impart greater meaning to
the text. Thus, it is reasonable to study the deeper correlations between
text-based audio and text-based video features.
This paper proposes a novel model which uses the outer-product of feature
pairs along with Deep Canonical Correlation Analysis (DCCA) (?) to study
useful multi-modal embedding features. The effectiveness of using an outer-
product to extract cross-modal information has been explored in (?; ?). Thus,
features from each mode are first extracted independently at the sentence (or
utterance) level and two outer-product matrices ($T\otimes V$ and $T\otimes
A$) are built for representing the interactions between text-video and between
text-audio. Each outer-product matrix is connected to a convolutional neural
network (CNN) for feature extraction. Outputs of these two CNNs can be
considered as feature vectors for text-based audio and text-based video and
should be correlated.
In order to better correlate the above text-based audio and text-based video,
we use Canonical Correlation Analysis (CCA) (?), which is a well-known method
for finding a linear subspace where two inputs are maximally correlated.
Unlike cosine similarity or Euclidean distance, CCA is able to learn the
direction of maximum correlation over all possible linear transformations and
is not limited by the original coordinate systems. However, one limitation of
CCA is that it can only learn linear transformations. An extension to CCA
named Deep CCA (DCCA) (?) uses a deep neural network to allow non-linear
relationships in the CCA transformation. Recently several authors (?; ?) have
shown the advantage of using CCA-based methods for studying correlations
between different inputs. Inspired by these, we use DCCA to correlate text-
based audio and text-based video. Text-based audio and text-based video
features derived from the two CNNs are input into a CCA layer which consists
of two projections and a CCA Loss calculator. The two CNNs and the CCA layer
then form a DCCA, the weights of the two CNNs and the projections are updated
by minimizing the CCA Loss. In this way, the two CNNs are able to extract
useful features from the outer-product matrices constrained by the CCA loss.
After optimizing the whole network, outputs of the two CNNs are concatenated
with the original text sentence embedding as the final multi-modal embedding,
which can be used for the classification.
We evaluate our approach on three benchmark multi-modal sentiment analysis and
emotion recognition datasets: CMU-MOSI (?), CMU-MOSEI (?), and IEMOCAP(?).
Additional experiments are presented to illustrate the performance of the ICCN
algorithm. The rest of the paper is organized as follows: Section 2 presents
related work, Section 3 introduces our proposed model and Section 4 describes
our experimental setup. Section 5 presents a discussion on the empirical
observations, Section 6 concludes this work.
## 2 Related Work
The central themes of this paper are related to learning (i) multi-modal
fusion embeddings and (ii) cross-modal relationships via canonical correlation
analysis (CCA).
Multi-modal fusion embedding: Early work (?) concatenates the audio, video and
text embeddings to learn a larger multi-modal embedding. But this may lead to
a potential loss of information between different modalities. Recent studies
on learning multi-modal fusion embeddings train specific neural network
architectures to combine all three modalities. In their work (?) propose
improvements to multi-modal embeddings using reinforcement learning to align
the multi-modal embedding at the word level by removing noises. A multi-modal
tensor fusion network is built in (?) by calculating the outer-product of
text, audio and video features to represent comprehensive features. However
this method is limited by the need of a large computational resources to
perform calculations of the outer dot product. In their work (?) developed an
efficient low rank method for building tensor networks which reduce
computational complexity and are able to achieve competitive results. A Memory
Fusion Network (MFN) is proposed by (?) which memorizes temporal and long-term
interactions and intra-actions between cross-modals, this memory can be stored
and updated in a LSTM. (?) learned multistage fusion at each LSTM step so that
the multi-modal fusion can be decomposed into several sub-problems and then
solved in a specialized and effective way. A multimodal transformer is
proposed by (?) that uses attention based cross-modal transformers to learn
interactions between modalities.
Cross-modal relationship learning via CCA: Canonical Correlation Analysis(CCA)
(?) learns the maximum correlation between two variables by mapping them into
a new subspace. Deep CCA (DCCA) (?) improves the performance of CCA by using
feedforward neural networks in place of the linear transformation in CCA.
A survey of recent literature sees applications of CCA-based methods in
analyzing the potential relationship between different variables. For example,
a CCA based model to combine domain knowledge and universal word embeddings is
proposed by (?). Models proposed by (?) use Deep Partial Canonical Correlation
Analysis (DPCCA), a variant of DCCA, for studying the relationship between two
languages based on the same image they are describing. Work by (?)
investigates the application of DCCA to simple concatenations of multimodal-
features, while (?) applied CCA methods to learn joint-representation for
detecting sarcasm. Both approaches show the effectiveness of CCA methods
towards learning potential correlation between two input variables.
## 3 Methodology
This section first introduces CCA and DCCA. Next, the interaction canonical
correlation network (ICCN), which extracts the interaction features of a CNN
in a DCCA-based network, is introduced. Finally, the whole pipe-line of using
this method in a multimodal language analysis task is described.
### CCA and DCCA
Given two sets of vectors $X\in\mathbb{R}^{n_{1}\times m}$ and
$Y\in{\mathbb{R}}^{n_{2}\times m}$, where $m$ denotes the number of vectors,
CCA learns two linear transformations $A\in\mathbb{R}^{n_{1}\times r}$ and
$B\in\mathbb{R}^{n_{2}\times r}$ such that the correlation between $A^{T}X$
and $B^{T}Y$ is maximized. Denote the covariances of $X$ and $Y$ as
$S_{11},S_{22}$, and the cross-covariance of $X,Y$ as $S_{12}$. The CCA
objective is
$\begin{split}A^{*},B^{*}&=\mathop{\arg\max}_{A,B}\textrm{corr}(A^{T}X,B^{T}Y)\\\
&=\mathop{\arg\max}_{A,B}\frac{A^{T}S_{12}B}{\sqrt{A^{T}S_{11}A}\sqrt{B^{T}S_{22}B}}.\end{split}$
(1)
The solution of the above equation is fixed and can be solved in multiple ways
(?; ?). One method suggested by (?) lets $U,S,V^{T}$ be the Singular Value
Decomposition ($SVD$) of the matrix $Z$ =
$S_{11}^{-\frac{1}{2}}S_{12}S_{22}^{-\frac{1}{2}}$. Then $A^{*},B^{*}$ and the
total maximum canonical correlation are
$\begin{split}A^{*}&=S_{11}^{-\frac{1}{2}}U\\\
B^{*}&=S_{22}^{-\frac{1}{2}}V\\\
\textrm{corr}(A^{*T}X,B^{*T}Y)&=\textrm{trace}(Z^{T}Z)^{\frac{1}{2}}.\end{split}$
(2)
One limitation of CCA is that it only considers linear transformations. DCCA
(?) learns non-linear transformations using a pair of neural networks. Let
$f,g$ denote two independent neural networks, the objective of DCCA is to
optimize parameters $\theta_{f},\theta_{g}$ of $f,g$ so that the canonical
correlation between the output of $f$ and $g$, denoted as $F_{X}=$
$f(X;\theta_{1})$ and $F_{Y}=$ $g(Y;\theta_{2})$, can be maximized by finding
two linear transformations $C^{*},D^{*}$. The objective of DCCA is
$\begin{split}\theta_{f}^{*},\theta_{g}^{*}&=\mathop{\arg\max}_{\theta_{f},\theta_{g}}\textrm{CCA}(F_{X},F_{Y})\\\
&=\mathop{\arg\max}_{\theta_{f},\theta_{g}}\textrm{corr}(C^{*T}F_{X},D^{*T}F_{Y}).\end{split}$
(3)
In order to update the parameters of $f,g$, a loss for measuring the canonical
correlation must be calculated and back-propagated. Let $R_{11},R_{22}$ be
covariances of $F_{X},F_{Y}$, the cross-covariance of $F_{X},F_{Y}$ as
$R_{12}$. Let $E=R_{11}^{-\frac{1}{2}}R_{12}R_{22}^{-\frac{1}{2}}$. According
to (2), the canonical correlation loss for updating $F_{x},F_{Y}$ can be
defined by
$\textrm{CCA Loss}=-\textrm{trace}(E^{T}E)^{\frac{1}{2}}.$ (4)
Networks $f(X;\theta_{f}),g(Y;\theta_{g})$’s parameters can be updated by
minimizing the CCA Loss (4) (i.e. maximize the total canonical correlation).
### Text Based Audio Video Interaction Canonical Correlation
Previous work of (?; ?) on multi-modal feature fusion has shown that the
outer-product is able to learn interactions between different features
effectively. Thus, we use the outer-product to represent text-video and text-
audio features. Given that outer-product and DCCA are applied at the utterance
(sentence)-level, we extract utterance level features for each uni-modal
independently in order to test the effectiveness of ICCN more directly. Let
$H_{t}\in\mathbb{R}^{d_{t}}$ be the utterance-level text feature embedding,
and $H_{v}\in\mathbb{R}^{d_{v}\times l_{v}},H_{a}\in\mathbb{R}^{d_{a}\times
l_{a}}$ be the video and audio input sequences. A 1D temporary convolutional
layer is used to extract local structure of the audio and video sequences, and
the outputs of the 1D-CNN are denoted as $H_{a1}\in\mathbb{R}^{d_{a1}\times
l_{a}},H_{v1}\in\mathbb{R}^{d_{v1}\times l_{v}}$. Next, two LSTMs process the
audio and video sequences. The final hidden state of each LSTM is used as the
utterance-level audio or video feature, denoted as
$H_{a2}\in\mathbb{R}^{d_{a2}},H_{v2}\in\mathbb{R}^{d_{v2}}$. Once each
utterance-level feature has been obtained, the text-based audio feature matrix
and text-based video feature matrix can then be learned using the outer-
product on $H_{t},H_{v2},H_{a2}$:
$\begin{split}H_{ta}&=H_{t}\otimes H_{a2},H_{ta}\in\mathbb{R}^{d_{t}\times
d_{a2}}\\\ H_{tv}&=H_{t}\otimes H_{v2},H_{tv}\in\mathbb{R}^{d_{t}\times
d_{v2}}.\end{split}$ (5)
In order to extract useful features from the outer-product matrices
$H_{ta},H_{tv}$, a Convolutional Neural Network is used as the basic feature
extractor. $H_{ta}$ and $H_{tv}$ are connected by multiple 2D-CNN layers with
max pooling. Outputs of the two 2D-CNNs are reshaped to 1D vector and then be
used as inputs to the CCA Loss calculation. 1D-CNN, LSTM, and 2D-CNN’s weights
are again updated using the back-propagation of the CCA Loss (4). Thus the two
2D-CNNs learn to extract features from $H_{tv}$ and $H_{ta}$ so that their
canonical correlation is maximized.
Figure 1: ICCN method for aligning text-based audio features and text-based
video features. Sentence level uni-modal features are extracted independently.
Outer-product matrices of text-audio and text-video are used as input to the
Deep CCA network. After learning the CNN’s weights using the CCA Loss, outputs
of the two CNNs are concatenated with the original text to form the multi-
modal embedding. This can be used as input to independent downstream tasks.
Algorithm 1 provides the pseudo-code for the whole Interaction Canonical
Correlation Network (ICCN).
0: Data $H_{t}\in\mathbb{R}^{d_{t}\times N}$, $H_{a}\in\mathbb{R}^{d_{a}\times
l_{a}\times N}$, $H_{v}\in\mathbb{R}^{d_{v}\times l_{v}\times N}$, $epoch$,
$\eta$
Initialize $W_{a}$ of ($\textrm{CNN1D}_{a},\textrm{LSTM}_{a}$, and
$\textrm{CNN2D}_{ta}$)
Initialize $W_{v}$ of ($\textrm{CNN1D}_{v},\textrm{LSTM}_{v}$, and
$\textrm{CNN2D}_{tv}$)
while $epoch>0$ do
$H_{a1}=\textrm{CNN1D}_{a}(H_{a})$
$H_{v1}=\textrm{CNN1D}_{v}(H_{v})$
$H_{a2}=\textrm{LSTM}{a}(H_{a1})$
$H_{v2}=\textrm{LSTM}{v}(H_{v1})$
$H_{tv}=H_{t}\otimes H_{v2}$, $H_{ta}=H_{t}\otimes H_{a2}$
$K_{tv}=\textrm{CNN2D}_{tv}(H_{tv})$
$K_{ta}=\textrm{CNN2D}_{ta}(H_{ta})$
Compute gradients $\nabla W_{v}$, $\nabla W_{a}$ of:
$\min\limits_{W_{v},W_{a}}\textrm{CCALoss}(K_{tv},K_{ta})$
Update:
$W_{v}\leftarrow W_{v}-\eta\nabla W_{v}$
$W_{a}\leftarrow W_{a}-\eta\nabla W_{a}$
$epoch\leftarrow epoch-1$
end while
$K_{tv},K_{ta}$
Algorithm 1 Interaction Canonical Correlation Network
### Pipe-line for Downstream Tasks
The ICCN method acts as a feature extractor. In order to test its performance,
an additional downstream classifier is also required. Uni-modal features can
be extracted using a variety of simple extraction schemes as well as by
learning features using more complex neural network based models such as a
sequential LSTM. Once uni-modal features for text, video, and audio have been
obtained, the ICCN can be used to learn text-based audio feature $K_{ta}$ and
text-based video feature $K_{tv}$. The final multi-modal embedding can be
formed as the concatenation of the text-based audio, the original text, and
the text-based video features, which are denoted as $[K_{ta};H_{t};K_{tv}]$.
This $[K_{ta};H_{t};K_{tv}]$ can then be used as an input to downstream
classifiers such as logistic regression or multilayer perceptron. Figure 1
shows the pipe-line using the ICCN for downstream tasks in our work.
## 4 Experiment Settings
This section describes the experimental datasets and baseline algorithms
against which ICCN is compared.
### Datasets
The proposed algorithm is tested using three public benchmark multi-modal
sentiment analysis and emotion recognition datasets: CMU-MOSI (?), CMU-MOSEI
(?), and IEMOCAP(?). Both CMU-MOSI and CMU-MOSEI’s raw features, and most of
the corresponding extracted features can be acquired from the CMU-
MultimodalSDK (?).
* •
CMU-MOSI: This dataset is a multi-modal dataset built on 93 Youtube movie
reviews. Videos are segmented to 2198 utterance clips, and each utterance
example is annotated on a scale of [-3, 3] to reflect sentiment intensity. -3
means an extremely negative and 3 means an extremely positive sentiment. This
data set is divided into three parts, training (1283 samples), validation (229
samples) and test (686 samples).
* •
CMU-MOSEI: This dataset is similar to the CMU-MOSI, but is larger in size. It
consists of 22856 annotated utterances extracted from Youtube videos. Each
utterance can be treated as an individual multi-modal example. Train,
validation, and test sets contain 16326, 1871, and 4659 samples respectively.
* •
IEMOCAP: This dataset contains 302 videos in which speakers performed 9
different emotions (angry, excited, fear, sad, surprised, frustrated, happy,
disappointed and neutral). Those videos are divided into short segments with
emotion annotations. Due to the imbalance of some emotion labels, we follow
experiments in previous papers (?; ?) where only four emotions (angry, sad,
happy, and neutral) are used to test the performance of the algorithm. Train,
validation, and test partitions contain 2717, 789, and 938 data samples
respectively.
### Multi-modal Features
The following uni-modal features are used prior to combinations,
* •
Text Features: For MOSI and MOSEI, we use a pre-trained transformer model BERT
(?) to extract utterance level text features, (many other approaches use Glove
word-level embeddings followed by a LSTM). The motivation behinds using BERT
is 1) BERT is the state-of-the-art in sentence encoding algorithms and has
demonstrated tremendous success in several downstream text applications such
as sentiment analysis, question-answering, semantic similarity tasks etc, 2)
using BERT simplifies the training pipe-line, with a large focus now towards
improving the performance of ICCN on a particular downstream task. We input
the raw text to the pre-trained uncased BERT-Base model (without fine-tuning).
Sentence encodings output from BERT are used as the text features. Each
individual text feature is of size 768. For IEMOCAP, we used InferSent(?) to
encode utterance level text. Since data is provided in the form of word
indices for GLOVE embeddings rather than raw text, we use InferSent; a BiLSTM
layer followed by a max-pooling layer to learn sentence embeddings.
* •
Audio Features: The audio feature is extracted by using COVAREP (?), which is
a public software used for extracting acoustic features such as pitch, volume,
and frequency. The CMU-MultimodalSDK provides COVAREP feature sequence for
every multi-modal example, the dimension of each frame’s audio feature is 74.
* •
Video Features: Facet(?) has been used for extracting facial expression
features such as action units and face pose. Similarly, every multimodal
example’s video feature sequence is also obtained from the CMU-MultimodalSDK.
The size of each frame’s video feature is 35.
### Baseline Methods
We consider a variety of baseline methods for multi-modal embedding
comparison. In order to focus on the multi-modal embedding itself, we input
each multi-modal embedding to the same downstream task classifier (or
regressor). Experimental comparisons are reported in two parts 1) we report
the effectiveness of DCCA over the simpler CCA based methods when used as
inputs to the ICCN and 2) we compare ICCN against newer utterance level
embeddings algorithms that learn features for a down stream task in an end-to-
end fashion. The following baselines are used in our comparisons,
* •
Uni-modal and Concatenation: This is the simplest baseline in which uni-modal
features are concatenated to obtain a multi-modal embedding.
* •
Linear CCA: CCA (?) considers linear transformations for different inputs. We
use the CCA to learn a new common space for audio and video modes, and combine
the learned audio and video features with the original text embedding. This is
because, (?) showed that using a CCA-based method to correlate audio and video
is more effective that correlating audio-text or video-text.
* •
Kernel-CCA: Kernel-CCA (?) introduces a nonlinearity via kernel maps. Kernel-
CCA can be used exactly like CCA.
* •
GCCA: Generalized CCA (?) learns a common subspace across multiple views. We
use GCCA in two ways: 1) use the GCCA output embedding directly and 2) combine
the GCCA output embedding with the original text embedding.
* •
DCCA: A Deep CCA based algorithm proposed by (?). Audio and video features are
simply concatenated and then be correlated with text features using DCCA.
Outputs of the DCCA are again concatenated with raw text, audio, and video
features to formulate the multimodal embedding.
In the proposed ICCN algorithm, text features are encoded by a pre-trained
BERT transformer or by InferSent. This is unlike most of the state-of-the-art
algorithms that obtain sentence level encodings by passing word embeddings
through variants of RNNs. However, since the idea is to compare modal
features, we also choose the following three state-of-the-art utterance-level
(i.e. sentence-level) fusion models (whose core algorithm is agnostic to the
text encoding architecture) as additional baselines. To make the comparison
fair, these methods use the same features as ours.
* •
TFN: Tensor Fusion Network (TFN) (?) combines individual modal’s embeddings
via calculating three different outer-product sub-tensors: unimodal, bimodal,
and trimodal. All tensors will then be flattened and used as a multi-modal
embedding vector.
* •
LMF: Low-Rank Multimodal Fusion (LMF) (?) learns the multimodal embedding
based the similar tensor processing of TFN, but with an additional low-rank
factor for reducing computation memory.
* •
MFM: Multimodal Factorization Model (MFM) (?) is consists of a discriminative
model for prediction and a generative model for reconstructing input data. A
comprehensive multimodal embedding is learned via optimizing the generative-
discriminative objective simultaneously.
Data View | CMU-MOSI | CMU-MOSEI
---|---|---
| Acc-2 | F-score | MAE | Acc-7 | Corr | Acc-2 | F-score | MAE | Acc-7 | Corr
Audio | 45.15 | 45.83 | 1.430 | 16.21 | 0.248 | 58.75 | 59.23 | 0.785 | 38.59 | 0.298
Video | 48.10 | 49.06 | 1.456 | 15.51 | 0.339 | 59.25 | 59.90 | 0.770 | 36.09 | 0.288
Text | 80.80 | 80.17 | 0.897 | 35.92 | 0.688 | 82.83 | 83.02 | 0.582 | 48.76 | 0.681
Text+Video | 81.00 | 80.91 | 0.920 | 35.11 | 0.676 | 82.86 | 83.01 | 0.581 | 47.92 | 0.674
Text+Audio | 80.59 | 80.56 | 0.909 | 35.08 | 0.672 | 82.80 | 82.96 | 0.582 | 49.02 | 0.689
Audio+Video+Text | 80.94 | 81.00 | 0.895 | 36.41 | 0.689 | 82.72 | 82.87 | 0.583 | 50.11 | 0.692
CCA | 79.45 | 79.35 | 0.893 | 34.15 | 0.690 | 82.94 | 83.06 | 0.573 | 50.23 | 0.690
KCCA | 79.82 | 79.91 | 0.889 | 34.76 | 0.689 | 83.05 | 83.14 | 0.574 | 50.09 | 0.692
GCCA | 62.50 | 62.15 | 1.403 | 17.29 | 0.533 | 75.12 | 75.46 | 0.653 | 45.33 | 0.602
GCCA+Text | 77.80 | 77.87 | 1.107 | 25.94 | 0.658 | 82.75 | 82.90 | 0.613 | 46.06 | 0.644
DCCA | 80.60∗ | 80.57∗ | 0.874 | 35.51 | 0.703 | 83.62∗ | 83.75∗ | 0.579 | 50.12 | 0.707
TFN | 80.82 | 80.77 | 0.901 | 34.94 | 0.698 | 82.57 | 82.09 | 0.593 | 50.21 | 0.700
LMF | 82.53 | 82.47 | 0.917 | 33.23 | 0.695 | 82.03 | 82.18 | 0.623 | 48.02 | 0.677
MFM | 81.72 | 81.64 | 0.877 | 35.42 | 0.706 | 84.40 | 84.36 | 0.568 | 51.37 | 0.717
ICCN | 83.07 | 83.02 | 0.862 | 39.01 | 0.714 | 84.18 | 84.15 | 0.565 | 51.58 | 0.713
Table 1: Results for experiments on CMU-MOSI and CMU-MOSEI. Best numbers are
in bold. For accuracy, F-score, and Correlation, higher is better. For mean
absolute error, lower is better. Results marked with $*$ are reported in
original papers. For TFN, LMF, and MFM, we re-did experiments with using our
features for a fair comparison.
### Ablation studies of ICCN
In order to analyze the usefulness of different components of the ICCN, we
consider the following two questions:
Question 1: Is using canonical correlation better than using other methods
like Cosine-Similarity?
Question 2: Is learning the interactions between text and video (or audio)
useful?
We design several experiments to address these two questions, First, we
replace the CCA Loss with Cosine-Similarity Loss while leaving other parts of
the ICCN unchanged. Second, instead of using the outer-product of audio (or
video) and text as input to the CCA Loss, we use audio and video directly as
the input to DCCA network. We compare different ICCN variants’ performance to
prove the usefulness of each component of the network.
### Evaluation Methods
To evaluate ICCN against previous baselines, the following performance metrics
as described in (?; ?; ?) are reported,
* •
On the CMU-MOSI and CMU-MOSEI we report four performance metrics, i) binary
accuracy, ii) F1-score iii) mean absolute error and iv) 7-class sentiment
level / Correlation with human labeling.
* •
On the IEMOCAP we used i) binary accuracy and ii) F1-score for evaluation.
Evaluations Details on CMU-MOSI, CMU-MOSEI: The original MOSI and MOSEI
datasets are labeled in the range [-3,3]. The author of the datasets suggests
a criterion for building binary labels: examples with label in [-3, 0) are
considered to have negative sentiment while examples with label in (0, 3] are
considered to have positive sentiment. 7-class sentiment level is also
calculated based on the label distribution in [-3,3]. The correlation of
predicted results with human labeling is also used as a criteria.
### Hyperparameter Tuning
A basic Grid-Search is used to tune hyperparameters, and the best
hyperparameter settings for the ICCN are chosen according to its performance
on the validation dataset. For ICCN, hyperparameters and tuning ranges are:
learning rate ($1e-5$–$1e-3$), mini-batch size ($128$–$512$), the number of
epoch ($10$–$100$), hidden dimensions of MLP ($64$–$512$), and output
dimension of the CCA projection ($30$–$100$). ReLU is used as the activation
function, RMSProp is used as the optimizer.
Whenever the training of the ICCN with a specific hypyerparameter setting has
finished, features learned from the ICCN are used as input to the same
downstream task models (a simple MLP is used in this work). Test results are
reported by using the best hyperparameter setting learned above.
Data View | IEMOCAP
---|---
Emotions | Happy | Angry | Sad | Neutral
| Acc-2 | F-score | Acc-2 | F-score | Acc-2 | F-score | Acc-2 | F-score
Audio | 84.03 | 81.09 | 85.49 | 84.03 | 82.75 | 80.26 | 63.08 | 59.24
Video | 83.14 | 80.36 | 85.91 | 83.27 | 81.19 | 80.35 | 62.30 | 58.19
Text | 84.80 | 81.17 | 85.12 | 85.21 | 84.18 | 83.63 | 66.02 | 63.52
Text+Video | 85.32 | 82.01 | 85.22 | 85.10 | 83.33 | 82.96 | 65.82 | 65.81
Text+Audio | 85.10 | 83.47 | 86.09 | 84.99 | 83.90 | 83.91 | 66.96 | 65.02
Audio+Video+Text | 86.00 | 83.37 | 86.37 | 85.88 | 84.02 | 83.71 | 66.87 | 65.93
CCA | 85.91 | 83.32 | 86.17 | 84.39 | 84.19 | 83.71 | 67.22 | 64.84
KCCA | 86.54 | 84.08 | 86.72 | 86.32 | 85.03 | 84.91 | 68.29 | 65.93
GCCA | 81.15 | 80.33 | 82.47 | 78.06 | 83.22 | 81.75 | 65.34 | 59.99
GCCA+Text | 87.02 | 83.44 | 88.01 | 88.00 | 84.79 | 83.26 | 68.26 | 67.61
DCCA | 86.99 | 84.32 | 87.94 | 87.85 | 86.03 | 84.36 | 68.87 | 65.93
TFN | 86.66 | 84.03 | 87.11 | 87.03 | 85.64 | 85.75 | 68.90 | 68.03
LMF | 86.14 | 83.92 | 86.24 | 86.41 | 84.33 | 84.40 | 69.62 | 68.75
MFM | 86.67 | 84.66 | 86.99 | 86.72 | 85.67 | 85.66 | 70.26 | 69.98
ICCN | 87.41 | 84.72 | 88.62 | 88.02 | 86.26 | 85.93 | 69.73 | 68.47
Table 2: Results for experiments on IEMOCAP. Best numbers are in bold. For
accuracy and F-score, higher is better. For TFN, LMF, and MFM, we re-did
experiments with using our features for a fair comparison.
Data View | CMU-MOSI | CMU-MOSEI
---|---|---
| Acc-2 | F-score | MAE | Acc-7 | Corr | Acc-2 | F-score | MAE | Acc-7 | Corr
ICCN | 83.07 | 83.02 | 0.862 | 39.01 | 0.714 | 84.18 | 84.15 | 0.565 | 51.58 | 0.713
ICCN1(no text) | 82.13 | 82.05 | 0.874 | 35.51 | 0.703 | 83.01 | 83.10 | 0.575 | 50.12 | 0.707
ICCN2(cos) | 82.32 | 82.27 | 0.876 | 36.01 | 0.702 | 82.98 | 82.90 | 0.575 | 50.63 | 0.700
ICCN3(no text + cos) | 81.49 | 81.58 | 0.889 | 35.77 | 0.692 | 82.59 | 82.73 | 0.578 | 50.21 | 0.696
Table 3: Ablation studies of ICCN on CMU-MOSI and CMU-MOSEI. ICCN1-3 denote
different variants, “no text” means applying DCCA to audio and video directly
instead of applying to the outer-product with text; “cos” means replacing CCA
Loss with Cosine-Similarity Loss;“no text + cca” means removing outer-product
with text and use Cosine-Similarity Loss.
## 5 Discussion of Empirical Results
This section presents and discusses results on the CMU-MOSI, CMU-MOSEI, and
IEMOCAP datasets.
### Performance on Benchmark Datasets
Tables 1 and 2 present results of experiments on the CMU-MOSI, CMU-MOSEI, and
IEMOCAP datasets.
* •
First, when compared with results of using uni-modal and simple
concatenations, ICCN outperforms all of them in all of the criteria. Note that
the performance of the text feature is always better than the performance of
the audio and video, and that a simple concatenation of text, video, and audio
does not work well. This shows the advance of highly-developed pre-trained
text features capable of improving the overall multimodal task performance.
However, it also shows the challenge of how to effectively combine such a
highly developed text feature with audio and video features.
* •
Second, ICCN also outperforms other CCA-based methods. The results of using
other CCA-based methods show that they cannot improve the multimodal
embedding’s performance. We argue this occurs because 1) CCA / KCCA / GCCA do
not exploit the power of neural network architectures so that their learning
capacities are limited. 2) Using DCCA without learning the interactions
between text-based audio and text-based video may sacrifice useful
information.
* •
Third, the ICCN still achieves better or similar results when compared with
other neural network based state-of-the-art methods (TFN, LMF, and MFM). These
results demonstrate the ICCN’s competitive performance.
### Results of Ablation Studies
Table 3 shows results of using variants of ICCN on CMU-MOSI and CMU-MOSEI
datasets.
First, using the CCA Loss performs better than using Cosine-Similarity Loss
with or without the outer-product. This is reasonable as the DCCA is able to
learn the hidden relationships (with the help of non-linear transformations)
but cosine-similarity is restrained by the original coordinates. To further
verify this, we also record changes of canonical correlation and cosine
similarity between text-based audio and text-based video (i.e., between the
two outputs of the CNNs in the ICCN) by using CCA Loss or Cosine Similarity
Loss for the ICCN with using the CMU-MOSI dataset. Curves in Figures 2 and 3
summarize the results of the experiments. Results show that maximizing the
canonical correlation by using the CCA Loss does not necessarily increase the
cosine similarity, and vice versa. This demonstrates that the canonical
correlation is a genuinely different objective function than cosine
similarity, and explains the different behaviors in the downstream
applications. CCA is capable of learning hidden relationships between inputs
that the cosine similarity does not see.
Second, learning the interactions between non-text and text performs better
than using audio and video directly. This also makes sense because audio and
video are more correlated when they are based on the same text, thus learning
text-based audio and text-based video performs better. In summary, Table 3
shows the usefulness of using a text-based outer-product together with DCCA.
Figure 2: Changes of mean canonical correlation and mean cosine similarity
between text-based audio and text-based video when training with CCA Loss: The
network learns to maximize the canonical correlation but the cosine similarity
isn’t affected. Figure 3: Changes of mean canonical correlation and cosine
similarity between text-based audio and text-based video when training with
Cosine Similarity Loss: convergence of cosine similarity doesn’t affect
canonical correlation.
## 6 Conclusion and Future Work
This paper has proposed the ICCN method, which uses canonical correlation to
analyze hidden relationships between text, audio, and video. Testing on a
multi-modal sentiment analysis and emotion recognition task shows that the
multi-modal features learned from the ICCN model can achieve state-of-the-art
performance, and shows the effectiveness of the model. Ablation studies
confirm the usefulness of different part of the network.
There is, of course, considerable room for improvement. Possible directions
include learning dynamic intra-actions in each model together with inter-
actions between different modes; learning the trade-off between maximum
canonical correlation and best downstream task performance; and developing an
interpretable end-to-end multi-modal canonical correlation model. In the
future, we hope to move forward in the development of multi-modal machine
learning.
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|
# Quantum Gravity in Flat Spacetime
Jarmo Mäkelä111Vaasa University of Applied Sciences, Wolffintie 30, 65200
Vaasa, Finland, email<EMAIL_ADDRESS>
###### Abstract
Inspired by Einstein’s Strong Principle of Equivalence we consider the effects
of quantum mechanics to the gravity-like phenomena experienced by an observer
in a uniformly accelerating motion in flat spacetime. Among other things, our
model of quantum gravity, derived from the first principles, predicts the
Unruh effect, and a discrete area spectrum for spacelike two-surfaces.
## 1 Introduction
When Albert Einstein began to formulate his general theory of relativity, he
used as a starting point his Strong Principle of Equivalence. In broad terms,
this principle states that it is impossible to decide, by means of local
measurements, whether one is in a gravitational field or in an accelerating
frame of reference. In this sense the effects of gravity are equivalent with
those of acceleration. [1, 2] For instance, if an observer finds all bodies in
his surroundings to fall with the same acceleration, the observer has no way
to tell, whether he is in a uniform gravitational field, caused by the
curvature of spacetime, or in a uniformly accelerating motion in flat
spacetime.
One of the greatest challenges of modern physics is to create the quantum
theory of gravity, which is supposed to bring together general relativity and
quantum mechanics. It appears to the author that in the attempts to quantize
gravity developed so far the Equivalence Principle has received insufficient
attention. It is possible that we should take the Equivalence Principle as the
starting point of quantum gravity in the same way as Einstein took it as the
starting point of his general relativity.
Inspired by these thoughts we shall, in this paper, quantize the effects of
gravity in flat spacetime from the point of view of an observer in a uniformly
accelerating motion. According to the Equivalence Principle quantization of
gravity in curved spacetime should, at least locally, be equivalent with the
quantization of gravity in flat spacetime from the point of view of an
acccelerating observer. We shall see that the resulting ”quantum gravity in
flat spacetime” may indeed be developed systematically, beginning from the
first principles.
In our approach we consider the gravity-like effects on a spacelike two-plane
at rest with respect to our accelerating observer, and it plays a role
somewhat similar to that of the event horizon of a black hole. The standard
rules of quantum mechanics applied to our model imply that the accelerating
plane has a discrete structure in the sense that it consists of a finite
number of separate constituents, all of them having an equally spaced area
spectrum. It also predicts the Unruh effect, according to which an observer in
an accelerating motion detects thermal radiation with a characteristic
temperature, which is proportional to the proper acceleration of the observer.
Unless otherwise stated, we shall always use the natural units, where
$\hbar=G=c=k_{B}=1$.
## 2 Action
Whenever we attempt to quantize gravity, we may use the classical Einstein-
Hilbert action
$S_{EH}:=\frac{1}{16\pi}\int_{M}R\sqrt{-g}\,d^{4}x$ (2.1)
as the starting point. In this equation $R$ is the Riemann curvature scalar,
and $g$ is the determinant of the metric tensor of spacetime. We have
integrated over the whole spacetime $M$. However, if the spacetime region
under consideration possesses a boundary, we must supplement the Einstein-
Hilbert action with the Gibbons-Hawking boundary term [3]
$S_{GH}:=\frac{1}{8\pi}\int_{\partial M}K\,dV,$ (2.2)
where $K$ is the trace of the exterior curvature tensor induced on the
boundary $\partial M$ of spacetime. $dV$ is the volume element on the
boundary, and we have integrated over the whole boundary. So the whole action
takes the form:
$S=S_{EH}+S_{GH}=\frac{1}{16\pi}\int_{M}R\sqrt{-g}\,d^{4}x+\frac{1}{8\pi}\int_{\partial
M}K\,dV.$ (2.3)
In flat spacetime the Riemann curvature scalar $R\equiv 0$, and we are left
with the Gibbons-Hawking boundary term only. Hence the action becomes to:
$S=\frac{1}{8\pi}\int_{\partial M}K\,dV.$ (2.4)
Even if spacetime were flat, the action may still be non-zero, if the boundary
of spacetime has been chosen appropriately. Because of the presence of the
potentially non-zero boundary term, the quantum-mechanical properties of the
gravity-like effects caused by a choice of a non-inertial frame of reference
are far from trivial.
In this paper we first pick up from the four-dimensional flat spacetime an
inertial frame of reference, where each point of spacetime is identified by
means of four coordinates $(T,X,Y,Z)$. The coordinate $T$ is timelike, and the
coordinates $X$, $Y$ and $Z$ spacelike. When written in terms of these
coordinates the line element of spacetime takes the flat Minkowski form:
$ds^{2}=-dT^{2}+dX^{2}+dY^{2}+dZ^{2}.$ (2.5)
We choose the boundary of spacetime to consist of five timelike hyperplanes,
where $X=X_{1}$, $Y=Y_{1}$, $Y=Y_{2}$, $Z=Z_{1}$ and $Z=Z_{2}$ such that
$Y_{2}>Y_{1}$ and $Z_{2}>Z_{1}$. In other words, the boundary of spacetime is
a three-dimensional, rectangular box, which proceeds in time, and where the
faces stay still in our frame of reference.
One of the faces of the box, however, is not assumed to stay still, but it is
assumed to accelerate with constant proper acceleration $a$ to the direction
of the positive $X$-axis. As a consequence, the $X$-coordinates of the ponts
of the face satisfy an equation:
$X^{2}-T^{2}=\frac{1}{a^{2}},$ (2.6)
which implies:
$X=\sqrt{T^{2}+\frac{1}{a^{2}}}.$ (2.7)
So the boundary of spacetime consists, as a whole, of five timelike
hyperplanes, and of a one timelike hypersurface, which is created, when a
spacelike two-plane accelerates along the $X$-axis.
The exterior curvature of all of the timelike hyperplanes is zero, and so they
do not contribute to the action. The exterior curvature of the hypersurface
created by the accelerating plane, however, is non-zero. To calculate the
trace of the exterior curvature tensor on that hypersurface we introduce the
Rindler coordinates to our spacetime. As it is well known, between the Rindler
coordinates $(t,x)$ and the coordinates $(T,X)$ there is the relationship: [4]
$\displaystyle t$ $\displaystyle=$
$\displaystyle\tanh^{-1}\left(\frac{T}{X}\right),$ (2.8a) $\displaystyle x$
$\displaystyle=$ $\displaystyle\sqrt{X^{2}-T^{2}}.$ (2.8b)
When written in terms of the Rindler coordinates, together with the
coordinates $Y$ and $Z$, the line element of spacetime takes the form:
$ds^{2}=-x^{2}\,dt^{2}+dx^{2}+dY^{2}+dZ^{2}.$ (2.9)
An observer with constant Rindler coordinate $x$, together with constant
coordinates $Y$ and $Z$, has constant proper acceleration $a$, which is
related to $x$ such that
$x=\frac{1}{a},$ (2.10)
which follows from Eqs. (2.6) and (2.8b). Eq. (2.8 a), in turn, implies that
the Rindler time coordinate $t$ gives the boost angle $\phi$ of an observer
with constant proper acceleration.
On the timelike hypersurface created, when a plane is in an accelerating
motion with constant proper acceleration $a$ the Rindler coordinate $x$ is a
constant. The only non-zero component of the exterior curvature tensor on that
hypersurface is:
$K_{tt}=\Gamma_{tt}^{x}=x,$ (2.11)
and therefore its trace is:
$K=g^{tt}K_{tt}=-\frac{1}{x}.$ (2.12)
The volume element on our hypersurface is:
$dV=x\,dY\,dZ\,dt,$ (2.13)
and so we find that the action in Eq. (2.7) takes, if the Rindler time
coordinate $t$ lies within an interval $[t_{i},t_{f}]$, the form:
$S=-\frac{1}{8\pi}A(t_{f}-t_{i}),$ (2.14)
where $A$ is the area of the accelerating plane. [5]
## 3 Hamiltonian
The bridge from the action to the Hamiltonian $H(q_{j},p_{j};t)$ of any system
is formed by the Hamilton-Jacobi equation:
$H(q_{j},\frac{\partial S}{\partial q_{j}};t)+\frac{\partial S}{\partial
t}=0,$ (3.1)
where $S$ is the principal function of the system. The quantities $q_{j}$ are
the coordinates of the configuration space, and the quantities
$p_{j}=\frac{\partial S}{\partial q_{j}}$ the corresponding coordinates of the
momentum space. The principal function of the system has been denoted by the
same symbol as its action for a very good reason: As it is well known, the
principal function agrees with the action integral calculated along the
classical path. Hence it follows that if we know the action $S$ calculated
along the classical path, the Hamiltonian of the system will be: [6]
$H=-\frac{\partial S}{\partial t}.$ (3.2)
Our derivation of the classical Hamiltonian of the gravitational field is
based on Eq. (3.2). Before proceeding to the use of Eq. (3.2), however, we
must decide, which time coordinate to use. The very idea of this paper is to
consider the effects of gravity from the point of view of a uniformly
accelerating observer. It is therefore sensible to take the proper time $\tau$
of the observer as the time coordinate. Eq. (2.9) implies that between the
proper time $\tau$ and the Rindler time $t$ there is the relationship:
$d\tau=x\,dt.$ (3.3)
Writing
$H=-\frac{\partial S}{\partial\tau},$ (3.4)
and using Eqs. (2.10) and (2.14) we therefore find:
$H=\frac{a}{8\pi}A.$ (3.5)
Eq. (3.5) gives the classical Hamiltonian of the gravitational field from the
point of view of an observer moving along a plane with area $A$ and proper
acceleration $a$. Remarkably, the Hamiltonian in Eq. (3.5) is identical to the
one obtained in several papers in spacetimes with horizon, from the point of
view of an observer with constant proper acceleration $a$, just outside of a
horizon with area $A$. [7-14] When the proper acceleration $a$ of the observer
tends to infinity, Eq. (2.10) implies that the Rindler coordinate $x$ tends to
zero, which means that in this limit our plane tends to a Rindler horizon of
flat spacetime.
We may always define a diffeoforphism $f$ from the unit square
$I^{2}:=[0,1]\times[0,1]$ to the accelerating plane, which we shall henceforth
denote by $B$. In general, this map $f:I^{2}\longrightarrow B$ is of the form:
$f(\chi^{1},\chi^{2})=(Y(\chi^{1},\chi^{2}),Z(\chi^{1},\chi^{2}))$ (3.6)
for all $\chi^{1},\chi^{2}\in[0,1]$, where the coordinates
$Y(\chi^{1},\chi^{2})$ and $Z(\chi^{1},\chi^{2})$ of the point $(Y,Z)$ on the
plane $B$ are functions of the coordinates $\chi^{1}$ and $\chi^{2}$ of the
point $\chi=(\chi^{1},\chi^{2})$ in $I^{2}$. Defining the function
$p(\chi):=p(\chi^{1},\chi^{2}):=\frac{1}{8\pi}\left|\begin{matrix}\frac{\partial
Y(\chi^{1},\chi^{2})}{\partial\chi^{1}}&\frac{\partial
Y(\chi^{1},\chi^{2})}{\partial\chi^{2}}\\\ \frac{\partial
Z(\chi^{1},\chi^{2})}{\partial\chi^{1}}&\frac{\partial
Z(\chi^{1},\chi^{2})}{\partial\chi^{2}}\end{matrix}\right|$ (3.7)
we may write the area of the plane $B$ as:
$A=8\pi\int_{I^{2}}p(\chi)\,d^{2}\chi,$ (3.8)
where we have integrated over the unit square. As a consequence, the
Hamiltonian of the gravitational field takes, from the point of view of our
accelerating observer, the form:
$H=a\int_{I^{2}}p(\chi)\,d^{2}\chi.$ (3.9)
It must be emphasized that the proper acceleration $a$ is not a dynamical
variable of the system, but just a parameter, which determines the observer.
We shall now take the quantities $p(\chi)$ associated with the points $\chi$
as the coordinates of the momentum space. The corresponding coordinates of the
configuration space we shall denote by $q(\chi)$. They obey the Hamiltonian
equations of motion:
$\dot{q}(\chi)=\frac{\delta H}{\delta p(\chi)}=a,$ (3.10)
where the dot means the derivative with respect to the proper time $\tau$ of
the accelerating observer. The general solution to Eq. (3.10) is:
$q(\chi)=a\tau+C(\chi),$ (3.11)
where $C(\chi)$ is an arbitrary constant function of the proper time $\tau$,
and depends on the coordinates $(\chi^{1},\chi^{2})$ of the point $\chi$ only.
Eqs. (2.10) and (3.3) imply that the quantity $a\tau$ may be identified as the
boost angle $\phi$ of the observer. Hence we may write Eq. (3.11) as:
$q(\chi)=\phi+C(\chi).$ (3.12)
Since our Hamiltonian in Eq. (3.10) does not involve $q(\chi)$, the
Hamiltonian equations of motion written for the variables $p(\chi)$ read as:
$\dot{p}(\chi)=-\frac{\delta H}{\delta q(\chi)}=0,$ (3.13)
which means that the quantities $p(\chi)$ are constants of motion of the
system.
## 4 Quantization
When we go over from the classical to the quantum-mechanical consideration of
the gravitational field in flat spacetime we must replace the classical
quantities $q(\chi)$ and $p(\chi)$ by the corresponding operators
$\hat{q}(\chi)$ and $\hat{p}(\chi)$ obeying the canonical commutation
relations:
$\displaystyle[\hat{q}(\chi),\hat{q}(\chi^{\prime})]=[\hat{p}(\chi),\hat{p}(\chi^{\prime})]$
$\displaystyle=$ $\displaystyle 0,$ (4.1a)
$\displaystyle[\hat{q}(\chi),\hat{p}(\chi^{\prime})]$ $\displaystyle=$
$\displaystyle i\delta^{2}(\chi,\chi^{\prime})$ (4.1b)
at all points $\chi,\chi^{\prime}\in I^{2}$. In Eq. (4.1b)
$\delta^{2}(\chi,\chi^{\prime})$ is the two-dimensional delta function in
$I^{2}$. When written in terms of these operators the classical Hamltonian $H$
in Eq. (3.9) becomes replaced by the corresponding Hamiltonian operator
$\hat{H}=a\int_{I^{2}}\hat{p}(\chi)\,d^{2}\chi,$ (4.2)
and therefore the time-independent Schrödinger equation
$\hat{H}|\psi\rangle=E|\psi\rangle$ (4.3)
written for the energy eigenstates $|\psi\rangle$ of the gravitational field
takes the form:
$a\int_{I^{2}}\hat{p}(\chi)|\psi\rangle=E|\psi\rangle.$ (4.4)
Identifying the energy $E$ of the gravitational field with the classical
Hamiltonian $H$ in Eq. (3.9) we observe that this equation reduces to:
$\int_{I^{2}}\hat{p}(\chi)|\psi\rangle=\int_{I^{2}}p(\chi)|\psi\rangle.$ (4.5)
Eq. (4.5) has an obvious solution, where $|\psi\rangle$ is the common
eigenstate of all of the operators $\hat{p}(\chi)$, which means that:
$\hat{p}(\chi)|\psi\rangle=p(\chi)|\psi\rangle$ (4.6)
for all $\chi\in I^{2}$. If the eigenstate $|\psi\rangle$ is normed to unity,
we must have:
$\langle\psi|\psi\rangle=1.$ (4.7)
We shall take Eqs. (4.6) and (4.7) as the postulates of our model, and see,
where this will take us.
In the position representation the energy eigenstates $|\psi\rangle$ are
expressed as functionals $\psi[q(\chi)]$ of the functions $q(\chi)$, and they
may be regarded as the wave functions of the gravitational field. The inner
product between the wave functions $\psi_{1}[q(\chi)]$ and $\psi_{2}[q(\chi)]$
is defined as:
$\langle\psi_{1}|\psi_{2}\rangle:=\int\psi_{1}^{*}[q(\chi)]\psi_{2}[q(\chi)]\,\mathcal{D}[q(\chi)],$
(4.8)
where $\mathcal{D}[q(\chi)]$ is an appropriate integration measure in the
space of the functions $q(\chi)$. The momentum operators $\hat{p}(\chi)$ are
expressed as functional differential operators:
$\hat{p}(\chi):=-i\frac{\delta}{\delta q(\chi)},$ (4.9)
and so Eq. (4.6) takes the form:
$-i\frac{\delta\psi[q(\chi)]}{\delta q(\chi)}=p(\chi)\psi[q(\chi)].$ (4.10)
Fortunately, this functional deifferential equation may be solved explicitly.
Its general solution is:
$\psi[q(\chi)]=\mathcal{N}\exp\left[i\int_{I^{2}}p(\chi)q(\chi)\,d^{2}\chi\right],$
(4.11)
where $\mathcal{N}$ is an arbitrary complex number.
As one may observe, the functional $\psi[q(\chi)]$ in Eq. (4.11) is uniquely
determined, up to the constant coefficient $\mathcal{N}$, by the function
$p(\chi)$, which gives, at each point $\chi$, the eigenvalue of the operator
$\hat{p}(\chi)$. If we denote by $p^{\prime}(\chi)$ another eigenvalue of
$\hat{p}(\chi)$, the corresponding eigenfunction is:
$\psi^{\prime}[q(\chi)]=\mathcal{N}^{\prime}\exp\left[i\int_{I^{2}}p^{\prime}(\chi)q(\chi)\,d^{2}\chi\right],$
(4.12)
and since eigenstates associated with different eigenvalues must be
orthogonal, we must have:
$\int\psi^{*}[q(\chi)]\psi^{\prime}[q(\chi)]\,\mathcal{D}[q(\chi)]=1,$ (4.13)
whenever
$p^{\prime}(\chi)\equiv p(\chi),$ (4.14)
and
$\int\psi^{*}[q(\chi)]\psi^{\prime}[q(\chi)]\,\mathcal{D}[q(\chi)]=0,$ (4.15)
if $p^{\prime}(\chi)$ differs from $p(\chi)$ at any point $\chi$. In Eq.
(4.12) $\mathcal{N}^{\prime}$ is an arbitrary complex number such that Eq.
(4.13) is satisfied.
We now divide the unit square $I^{2}$ in $N$ non-intersecting subsets $D_{j}$
$(j=1,2,\dots,N)$ such that $D_{j}\cap D_{k}=\emptyset$ for all $j\neq k$,
where $j,k\in\\{1,2,\dots,N\\}$, and
$I^{2}=D_{1}\cup D_{2}\cup\cdots\cup D_{N}.$ (4.16)
We shall assume that each subset $D_{j}$ has the same area
$\Delta\chi:=\frac{1}{N}.$ (4.17)
We shall also assume that in the limit, where $N$ tends to infinity, the
supremum of the diameters of the subsets $D_{j}$ tends to zero. Picking up
from each of the subsets $D_{j}$ a point $\chi_{j}$ we may write the integral
in the exponential in Eq. (4.11) as a Riemann integral:
$\int_{I^{2}}p(\chi)q(\chi)\,d^{2}\chi=\lim_{N\rightarrow\infty}\left(\sum_{j=1}^{N}p_{j}q_{j}\Delta\chi\right),$
(4.18)
where we have denoted:
$\displaystyle p_{j}$ $\displaystyle:=$ $\displaystyle p(\chi_{j}),$ (4.19a)
$\displaystyle q_{j}$ $\displaystyle:=$ $\displaystyle q(\chi_{j})$ (4.19b)
for all $j=1,2,\dots,N$. Hence we have:
$\psi[q(\chi)]=\mathcal{N}\lim_{N\rightarrow\infty}\left[\prod_{j=1}^{N}\exp(ip_{j}q_{j}\Delta\chi)\right].$
(4.20)
Eq. (4.20) enables us to define the integral on the left hand side of Eq.
(4.13):
$\int\psi^{*}[q(\chi)]\psi^{\prime}[q(\chi)]\,\mathcal{D}[q(\chi)]:=\mathcal{N}^{*}\mathcal{N}^{\prime}\lim_{N\rightarrow\infty}\left\\{\prod_{j=1}^{N}\int_{-L/2}^{L/2}\exp[i(p^{\prime}_{j}-p_{j})q_{j}\,\Delta\chi]\,dq_{j}\right\\},$
(4.21)
where we have denoted ${p^{\prime}}_{j}:=p^{\prime}(\chi_{j})$. As one may
see, we have introduced a new, positive number $L$. We might, of course, have
taken $L$ to infinity, and thereby integrate $q_{j}$ from the negative to the
positive infinity. Such a choice, however, would make the integral in Eq.
(4.21) infinite. Because of that we shall henceforth take $L$ to be a fixed
number. If we want Eqs. (4.13) and (4.15) to be satisfied, we must have, for
every $j=1,2,\dots,N$:
$p_{j}\Delta\chi=n_{j}\frac{2\pi}{L},$ (4.22)
where $n_{j}=0,\pm 1,\pm 2\,\dots$.
The diffeomorphism $f:I^{2}\longrightarrow B$ defined in Eq. (3.6) maps every
subset $D_{j}$ of $I^{2}$ to a certain piece, or constituent
$U_{j}=f(D_{j})$ (4.23)
of the accelerating plane $B$. These constituents have the properties:
$U_{j}\cap U_{k}=\emptyset$ (4.24)
for all $j\neq k$, and
$B=U_{1}\cup U_{2}\cup\cdots\cup U_{N}.$ (4.25)
In the limit, where the number $N$ of the constituents becomes very large, we
may view the quantity
$A_{j}:=8\pi p_{j}\Delta\chi$ (4.26)
as the area of the constituent $j$. According to Eq. (4.22) the possible area
eigenvalues of the constituent are:
$A_{j}=n_{j}\frac{16\pi^{2}}{L}.$ (4.27)
Obviously, negative area eigenvalues do not make sense, and therefore we shall
ignore the negative values of the quantum numbers $n_{j}$, and keep only the
non-negative ones. In other words, $n_{j}=0,1,2,\dots$ in Eq. (4.27).
As one may observe from Eq. (4.27), the areas of the constituents of the
accelerating plane $B$ have a discrete spectrum with an equal spacing. Unless
the plane has an infinite number of constituents with zero area, we are
therefore forced to conclude that the number of the constituents of the plane
is finite. This number we shall denote by $N$. The total area of the plane is
the sum of the areas of its constituents:
$A=A_{1}+A_{2}+\cdots+A_{N}.$ (4.28)
So we see that the postulates (4.6) and (4.7) of our model imply that our
plane necessarily has a discrete structure in the sense that it consists of a
finite number of separate constituents. The possible area eigenvalues of the
plane are of the form:
$A=\frac{16\pi^{2}}{L}(n_{1}+n_{2}+\cdots+n_{N}).$ (4.29)
The eigenfunctions associated with these area eigenvalues are of the form:
$\psi(q_{1},q_{2},\dots,q_{N})=\mathcal{N}\exp\left[\frac{i}{8\pi}(A_{1}q_{1}+A_{2}q_{2}+\cdots+A_{N}q_{N})\right].$
(4.30)
Hence the wave functions are no more functionals, but ordinary functions of a
finite number of variables $q_{1},q_{2},\dots,q_{N}$. The inner product
between the wave functions is defined as:
$\langle\psi_{1}|\psi_{2}\rangle:=\int_{-L/2}^{L/2}dq_{1}\int_{-L/2}^{L/2}dq_{2}\cdots\int_{-L/2}^{L/2}dq_{N}\,\psi^{*}_{1}(q_{1},q_{2},\dots,q_{N})\psi_{2}(q_{1},q_{2},\dots,q_{N}).$
(4.31)
and if we want the wave function $\psi(q_{1},q_{2},\dots,q_{N})$ to be normed
to unity, we may take:
$\mathcal{N}=\frac{1}{\sqrt{L^{N}}}.$ (4.32)
So we find:
$\psi(q_{1},q_{2},\dots,q_{N})=\frac{1}{\sqrt{L^{N}}}\exp\left[\frac{i}{8\pi}(A_{1}q_{1}+A_{2}q_{2}+\cdots+A_{N}q_{N})\right].$
(4.33)
Eqs. (3.5) and (4.29) imply that the energy eigenvalues of the gravitational
field from the point of view of our accelerating observer are:
$E=\frac{\alpha a}{8\pi}(n_{1}+n_{2}+\cdots n_{N}),$ (4.34)
where
$\alpha:=\frac{16\pi^{2}}{L}.$ (4.35)
As the reader may have noticed, quantization of the gravitational field in
flat spacetime from the point of view of an accelerating observer is, in our
model, very similar to the quantization of the system of free particles in
non-relativistic quantum mechanics. In such systems the coordinates of the
momentum space are constants of motion, and the coordinates of the
configuration space are, possibly up to additive constants, linear functions
of time, whereas in our system the momentum variables $p(\chi)$ are constants
, and the corresponding coordinates $q(\chi)$ of the configuration space
depend, up to additive constants, linearly on the proper time $\tau$ of the
observer. When quantizing the system of free particles in non-relativistic
quantum mechanics one must use box normalization for the wave function of the
system. When using the box normalization one assumes that the particles of the
system lie in a rectangular box. Finally, at the end of the calculations, the
edge kength of the box is taken to infinity. In our model we also applied a
sort of box normalization, where the coordinates of the confuguration space of
the system were assumed to lie within the interval $[-L/2,L/2]$. However, we
did not take the ”edge length” $L$ of the box to infinity, but we took $L$ to
be a fixed number. As a consequence, $L$ appears as a parameter of the model,
which must be fixed such that the predictions of the model agree with
experiments.
## 5 Unruh Effect
In Section 4 we completed, in our model, the quantization of gravity in flat
spacetime. The most important outcome of our model was that the area of an
accelerating plane has a discrete spectrum with equal spacing. We are now
prepared to go to the thermodynamical properties of gravity. As it is well
known, the thermodynamical properties of any system may be deduced from its
partition function
$Z(\beta):=\sum_{n}e^{-\beta E_{n}},$ (5.1)
where $\beta$ is the temperature parameter, and we have summed over the energy
eigenstates $n$ of the system. In what follows, we shall base our calculation
of the partition function on Eq. (4.34). We say that constituent $j$ is in
vacuum, if $n_{j}=0$; otherwise the constituent is in an excited state.
When obtaining an expression for the partition function of our system we shall
assume that at least one of the constituents of the accelerated plane is in an
excited state, and when the quantum states of the constituents are
interchanged, the quantum state of the system will also change. With these
assumptions the partition function takes the form:
$\begin{split}Z(\beta)=&\,\,\,\,\sum_{n_{1}=1}^{\infty}\exp\left(-\frac{\alpha\beta
a}{8\pi}n_{1}\right)\\\ &+\sum_{n_{1}=1}^{\infty}\exp\left(-\frac{\alpha\beta
a}{8\pi}n_{1}\right)\sum_{n_{2}=1}^{\infty}\exp\left(-\frac{\alpha\beta
a}{8\pi}n_{2}\right)\\\ &+\cdots\\\
&+\sum_{n_{1}=1}^{\infty}\exp\left(-\frac{\alpha\beta
a}{8\pi}n_{1}\right)\cdots\sum_{n_{N}=1}^{\infty}\exp\left(-\frac{\alpha\beta
a}{8\pi}n_{N}\right).\end{split}$ (5.2)
In the first term on the right hand side of Eq. (5.2) just one of the
constituents is in an excited state, and we have summed over all of those
states. In the second term two of the constituents are in excited states.
Finally, in the last term all of the $N$ constituents are in excited states.
Defining the characteristic temperature
$T_{C}:=\frac{\alpha a}{8\pi\ln(2)}$ (5.3)
we may write Eq. (5.2) as:
$\begin{split}Z(\beta)&=\frac{1}{2^{\beta T_{C}}-1}+\left(\frac{1}{2^{\beta
T_{C}}-1}\right)^{2}+\cdots+\left(\frac{1}{2^{\beta T_{C}}-1}\right)^{N}\\\
&=\frac{1}{2^{\beta T_{C}}-2}\left[1-\left(\frac{1}{2^{\beta
T_{C}}-1}\right)^{N}\right],\end{split}$ (5.4)
whenever $\beta T_{C}\neq 1$. When $\beta T_{C}=1$, we have:
$Z(\beta)=N.$ (5.5)
The energy of the gravitational field may be calculated from its partition
function $Z(\beta)$ as:
$E=-\frac{\partial}{\partial\beta}\ln[Z(\beta)],$ (5.6)
and if we define the energy per constituent as
$\bar{E}:=\frac{E}{N},$ (5.7)
Eqs. (5.4) and (5.6) imply:
$\bar{E}=\bar{E}_{1}+\bar{E}_{2},$ (5.8)
where we have denoted:
$\displaystyle\bar{E}_{1}$ $\displaystyle:=$
$\displaystyle\frac{1}{N}\frac{2^{\beta T_{C}}}{2^{\beta
T_{C}}-2}T_{C}\ln(2),$ (5.9a) $\displaystyle\bar{E}_{2}$ $\displaystyle:=$
$\displaystyle\frac{2^{\beta T_{C}}}{2^{\beta T_{C}}-1-(2^{\beta
T_{C}}-1)^{N+1}}T_{C}\ln(2),$ (5.9b)
whenever $\beta T_{C}\neq 1$. When $\beta T_{C}=1$, a calculation similar to
the one performed in Ref. [7] gives, in the large $N$ limit, the result:
$\bar{E}=T_{C}\ln(2).$ (5.10)
Eqs. (4.34), (5.3), (5.8) and (5.9) imply that the average
$\bar{n}:=\frac{n_{1}+n_{2}+\cdots n_{N}}{N}$ (5.11)
of the quantum numbers $n_{1},n_{2},\dots,n_{N}$ depends of the temperature
parameter $\beta$ as:
$\bar{n}=\frac{1}{N}\frac{2^{\beta T_{C}}}{2^{\beta T_{C}}-2}+\frac{2^{\beta
T_{C}}}{2^{\beta T_{C}}-1-(2^{\beta T_{C}}-1)^{N+1}}.$ (5.12)
So far everything has been quite exact, and no approximations have been made.
At this point we proceed to consider, what happens in the limit, where the
number $N$ of the constituents of the accelerating plane tends to infinity. In
this limit the first term on the right hand side of Eq. (5.12) will vanish,
and we have, in effect:
$\bar{n}=\frac{2^{\beta T_{C}}}{2^{\beta T_{C}}-1-(2^{\beta T_{C}}-1)^{N+1}}.$
(5.13)
When $\beta T_{C}>1$, which means that the temperature $T$ experienced by the
observer moving along the plane is less than the characteristic temperature
$T_{C}$ we have:
$\lim_{N\rightarrow\infty}(2^{\beta T_{C}}-1)^{N+1}=\infty,$ (5.14)
and so the constituents of the plane are, in effect, in vacuum. However, when
$\beta T_{C}<1$, which means that $T>T_{C}$, we find:
$\lim_{N\rightarrow\infty}(2^{\beta T_{C}}-1)^{N+1}=0,$ (5.15)
and so we may write, as an excellent approximation:
$\bar{n}=\frac{2^{\beta T_{C}}}{2^{\beta T_{C}}-1}.$ (5.16)
An interesting property of this expression is that, when $T$ tends to $T_{C}$
from the right hand side, which means that $\beta T_{C}$ tends to $1$ from the
left hand side, we have:
$\bar{n}=2.$ (5.17)
So we find that at the characteristic temperature $T_{C}$ the accelerating
plane performs a phase transition, where the constituents of the plane jump,
in average, from the vacuum to the second excited states. The latent heat per
constituent associated with this phase transition is
$\bar{L}=2T_{C}\ln(2).$ (5.18)
Since the constituents of the plane are effectively in the vacuum, when
$T<T_{C}$, and jump to the second excited states at the characteristic
temperature $T_{C}$, we may view the characteristic temperature $T_{C}$,
defined in Eq. (5.3), as the lowest possible temperature experienced by the
accelerating observer. Interestingly, we find that if we choose:
$\alpha=4\ln(2),$ (5.19)
then:
$T_{C}=\frac{a}{2\pi},$ (5.20)
which exactly agrees with the Unruh temperature [15]
$T_{U}:=\frac{a}{2\pi}$ (5.21)
experienced by the observer. In this sense we have been able to derive the
Unruh effect from our model of quantum gravity in flat spacetime: An
accelerating observer detects thermal radiation with a temperature, which is
proportional to the proper acceleration $a$ of the observer. That radiation is
produced, when the constituents of the plane perform quantum jump from the
excited states to the vacuum.
## 6 Concluding Remarks
In this paper we have constructed, step by step, a model of quantum gravity in
flat spacetime from the point of view of a unformly accelerating observer. An
observer of this kind detects phenomena identical to those in a uniform
gravitational field in his surroundings, and we considered those effects
quantum-mechanically. Among other things, our nodel of quantum gravity
predicted that spacetime has a discrete structure in the sense that the plane
at rest with respect to the accelerating observer consists of separate
constituents, each of which has an area, which is an integer times a certain
fundamental area. As a consequence, the area eigenvalues of the accelerating
plane are of the form:
$A=\alpha(n_{1}+n_{2}+\cdots n_{N}),$ (6.1)
where the quantum numbers $n_{j}$ are non-negative integers for all
$j=1,2,\dots,N$, and $N$ is the number of the constituents of the plane. We
found that with the choice
$\alpha=4\ln(2)$ (6.2)
for the parameter $\alpha$ of our model an observer with constant proper
acceleration $a$ detects thermal radiation with certain minimum temperature,
which agrees with the Unruh temperature
$T_{U}=\frac{a}{2\pi}$ (6.3)
of the observer. In this sense our model of quantum gravity in flat spacetime
implies the Unruh effect.
Basically, the results of the paper were outcomes of our decision to
supplement the standard Einstein-Hilbert action with the standard Gibbons-
Hawking boundary term. We first picked up a rectangular box on a spacelike
hypersurface of the flat Minkowski spacetime spacetime, where the time
coordinate is a constant, and then put one of its faces into a uniformly
accelerating motion, while the other faces were kept at rest in the flat
Minkowski system of coordinates. When the faces of the box proceeded in the
flat spacetime, a three-dimensional boundary was created for spacetime. It
turned out that the trace of the exterior curvature tensor on the three-
dimensional timelike hypersurface created by the accelerating plane was non-
zero, while the exterior curvature tensor on the timelike hypersurfaces
created by the other faces of the box vanished identically. As a consequence,
the Gibbons-Hawking boundary term was non-zero, even though the Einstein-
Hilbert action vanished. Thus, the presence of the Gibbons-Hawking boundary
term produced non-zero gravitational action, despite of the fact that
spacetime was flat.
From the action we obtained the classical Hamiltonian with respect to the
accelerating observer. The discrete structure of the accelerating plane
followed from the straightforward replacement of the classical Hamiltonian by
the corresponding Hamiltonian operator according to the standard rules of
quantum mechanics. The coordinates $p_{j}$ of the momentum space agree, up to
the coefficient $\frac{1}{8\pi}$, with the areas of the constituents of the
plane, whereas the corresponding coordinates $q_{j}$ of the configuration
space agree, up to additive constants, with the boost angle of the observer.
In the process we were compelled to introduce a box normalization for the wave
function of the gravitational field, where the coordinates $q_{j}$ of the
configuration space are confined to live inside of a box with edge length $L$.
222Not to be confused with the edge length of the accelerating box in flat
spacetime! Defining a parameter $\alpha$ in terms of the edge length $L$ as:
$\alpha=\frac{16\pi^{2}}{L}$ (6.4)
we obtained the area spectrum in Eq. (6.1).
As we have seen, quantization of gravity in flat spacetime from the point of
view of an accelerating observer is really very simple and straightforward,
and may be carried out explicitly. Nevertheless, there are still some
shortcomings in our procedure. For instance, our model of quantum gravity
predicted for the areas of the constituents of the accelerating plane both
positive and negative eigenvalues, and we excluded the negative eigenvalues
”by hand”. An even more serious problem in our approach is the presence of an
undetermined parameter, the edge length $L$ of the box used in the
normalization of the wave function. Eqs. (6.2) and (6.4) imply that if we
choose
$L=\frac{4\pi^{2}}{\ln(2)}\approx 57.0,$ (6.5)
our model predicts the Unruh effect. The close relationship between the
variables $q_{j}$ and the boost angle $\phi$ of the observer suggests that the
edge length $L$ sets the bound for $\phi$ such that, for symmetry reasons,
$-\frac{L}{2}\leq\phi\leq\frac{L}{2}$. Whether such interpetation is valid, is
still an open question, as well as are its possible consequences.
Taken as a whole, our paper highlights an aspect of the potential quantum
theory of gravity, which so far has received very little attention: The
quantum states of the gravitational field are not absolute, but relative, and
they depend on the state of motion of the observer. For instance, an
accelerating observer in flat spacetime detects gravity-like effects, which
may be quantized by means of the standard rules of quantum mechanics, whereas
inertial observers detect no effects of gravity whatsoever. It is to be hoped
that the results obtained this paper would pave the way for a proper quantum
theory of gravitation in curved spacetime.
## References
* [1] A. Einstein, Jahrb. Radioaktivität Elektronik 4 (1907) 414.
* [2] A. Einstein, Ann. Phys. 35 (1911) 898.
* [3] G. W. Gibbons, S. W. Hawking, Phys. Rev. D15 (1977) 2752.
* [4] See, for example, R. M. Wald, General Relativity (The University of Chicago Press, Chicago 1984).
* [5] There are still conributions to the boundary term emerging from the edges of our box. Those contributions, however, may be cancelled, if necessary, by means of an addition of an appropriate, constant boundary term.
* [6] See, for example, H. Goldstein, Classical Mechanics (Second Edition) (Addison-Wesley, 1980).
* [7] J. Mäkelä, Entropy 13 (2011) 1324.
* [8] J. Mäkelä, Int. J. Mod. Phys. D23 (2014) 1450001.
* [9] J. Mäkelä, Phys. Rev. D87 (2013) 104040.
* [10] J. Mäkelä, Phys. Rev. D91 (2015) 124050.
* [11] J. Mäkelä, Phys. Rev. D93 (2016) 084002.
* [12] J. Mäkelä, Int. J. Mod. Phys. D28 (2019) 1950129.
* [13] J. Mäkelä, Phys. Rev. D103 (2021) 027002.
* [14] J. Mäkelä, Int. J. Mod. Phys. D31 (2022) 2250082.
* [15] W. G. Unruh, Phys. Rev. D14 (1976) 5670.
|
# Activation of a supercooled liquid confined in a nanopore.
Felix Mercier Laboratoire de Photonique d’Angers EA 4464, Université
d’Angers, Physics Department, 2 Bd Lavoisier, 49045 Angers, France Gaetan
Delhaye Laboratoire de Photonique d’Angers EA 4464, Université d’Angers,
Physics Department, 2 Bd Lavoisier, 49045 Angers, France Victor Teboul
<EMAIL_ADDRESS>Laboratoire de Photonique d’Angers EA 4464,
Université d’Angers, Physics Department, 2 Bd Lavoisier, 49045 Angers, France
###### Abstract
It is well stablished that confinement of supercooled liquids in nano-pores
induces various effects as a strong modification of the dynamics and a
layering of the local structure. In this work we raise the issue as how these
confinement effects are modified when the liquid is out of equilibrium. To
answer that question, we use molecular dynamics simulations to investigate the
effect of confinement on a supercooled liquid activated by the periodic
folding of a molecular motor. We find that the motor’s opening angle controls
the activation of the medium and use that result to study the effect of
different activations on the confined supercooled liquid. We observe an
increase of the activation effect on the dynamics when the medium is confined.
We find that the confinement slowing down dependence with the pore radius
depends on the activation of the liquid. We argue that these findings result
from a modification of dynamic correlation lengths with activation. In this
picture the activation permits to control the liquid correlation length and
dynamics inside the pore. Studying the local structure we observe a
modification of the layering organization induced by the activation. We also
find that the mobility inside the pore depends on the layers, being larger
where the local density is small.
dynamic heterogeneity,glass-transition
###### pacs:
64.70.pj, 61.20.Lc, 66.30.hh
## I Introduction
Liquids and solids constituted or comprising molecules like molecular motors
that are able to move by themselves are called activeactive1 ; active5 ;
active-confi . An example is cells of living biological organisms that are
constituted of active soft materials. Active matter opens a window to the
statistical physics of non equilibrium systems and to biological physics
making it a fascinating new domain of research that develops rapidlyactive1 ;
active2 ; active3 ; active4 ; active5 ; active6 ; active7 ; active8 ; active9
; active10 ; active11 ; active12 ; active13 ; active14 ; active15 ; active16 ;
active17 ; active18 ; active-confi . The different statistical physics
associated with non equilibrium systems, induces a number of new phenomena, as
for example non-equilibrium phase transitionsactive4 .
When liquids are cooled fast enough to remain liquid below their melting
temperature $T_{m}$, they are subject to a large increase of their viscosity.
The viscosity below $T_{m}$ then increases at least exponentially when the
temperature drops, leading eventually to a glass at the glass-transition
temperature $T_{g}$. That dramatic increase of the viscosity and the
associated decrease of most dynamical properties like the diffusion
coefficient, strangely appears without significant structural modifications of
the liquid. The nature of that transition from the liquid to the glass, known
as the long standing glass-transition problem, is still puzzling
scientistsanderson ; gt0 ; gt1 ; gt2 ; dh0 ; c3 ; c4 ; md16 . As phase
transitions are usually associated with the divergence of correlation lengths
and the apparition of cooperative mechanisms, increasing correlation lengths
have been searched extensively in supercooled liquids. Confinement of the
liquid inside nano-pores has been seen as a way to cutoff the correlation
lengths and obtain indirect informations on them by studying the associated
dynamical modifications in the liquid. For that fondamental reason as well as
multiple practical applications, confinement has been the subject of a number
of studies and is still a subject of active researchconff1 ; conff2 ; conff3 ;
conff4 ; conff5 ; conff6 ; conf-1 ; conf0 ; conf1 ; conf2 ; conf3 ; conf4 ;
conf5 ; conf6 ; confine ; conf7 ; active-confi . Indeed, in supercooled
liquids, confinement induces various effects. As suggested by the theoretical
perspective described above, confinement in most cases induces a slowing down
of the dynamics reminiscent of the slowing down induced by a temperature
decrease. It also induces a structural organization called layering due to the
fixation of the local structure in front of the immobile walls of the pore. In
this work we raise the question to how the confinement effects are modified
for active matter. In order to answer that question, we use a simple
butterfly-like flat molecular motormotoro1 ; motoro2 ; motoro3 ; motoro4 ;
motoro5 ; motoro6 ; motoro7 ; motoro8 ; motoro9 ; motoro10 ; motoro11 (Figure
1), to activate with its periodic folding the supercooled liquid surrounding
it, confined in nano-pores of various radii (Figures 2 and 3). Experimentally
our motor can be seen as a simplification of the azobenzene molecule or
derivatives, which undergoes a photo-isomerization processaz1 ; az2 ; az3 ;
az4 ; az5 ; az6 ; az7 ; az8 ; az9 ; cage ; md16 ; az10 when subject to a
light stimulus. We study the effect of different activations induced by
different opening angles of our motor (Figure 1). We observe an increase of
the activation effect on the dynamics when the medium is confined. Studying
the local structure we observe a modification of the layering organization
induced by the activation. We also find that the motility is dependent on the
layers, being larger where the local density is small.
Figure 1: (color online) Picture of the folded and unfolded motor molecule.
Through this work we use the parameter $d_{z}$ to tune and quantify the
activation of the medium by the motor’s folding. We validate that choice at
posteriori in the excitation density section. Figure 2: (color online)
Snapshots of simulations for pores of radii (a) $R=5$Å, (b) $R=8$Å, (c)
$R=10$Å, and (d) $R=12.5$Å. The colors are arbitrary. $T=500K$. The
simulations use $8$ pores of radii varying from $5$ to $12.5$Å separated by
steps of $1$ Å ($1.5$ Å for the last step). Figure 3: (color online) Snapshot
of a simulation observed from the side to show the motor. The pore radius is
$R=10$Å. $T=500K$.
## II Calculation
In addition to theoretical and experimental methods, simulations in particular
molecular dynamics and Monte Carlo simulationsmd1 ; md2 ; md2b ; md4 together
with model systemsms1 ; ms2 ; ms3 ; ms4 ; ms5 are now widely used to unravel
unsolved problems in condensed matter and complex systems physicskeys ; md3 ;
md4b ; md6 ; md7 ; md8 ; md9 ; ee2 ; md11 ; md12 ; md13 ; md14 ; md15 ; ee1 ;
finite1 ; u1 . The reader will find details on our simulation procedure in
previous paperspccp ; prefold ; ariane , however for convenience we will
resume it here. Our simulations use one motor molecule (see Figure 1 for its
description) imbedded inside a medium constituted of $1760$ linear molecules,
in a parallelepiped box $31.1$ Å wide and $62.2$ Å high. After aging the
simulation box during $10ns$ at the temperature of studies, we create the pore
by suddenly freezing molecules located at a distance larger than $R$ from the
center axis of our simulation box. This procedure is intended to minimize the
structure modification of the liquid by the confinement. After their sudden
freezing, the glassy molecules of the pores are no more allowed to move,
leading to pure elastic interactions with the confined liquid. Consequently
the liquid temperature is not modified by the interactions with the pore
walls. After that procedure, as the dynamics changes with the confinement, the
liquid is again aged during $10ns$ before the beginning of any study.
We integrate the equations of motion using the Gear algorithm with a
quaternion decompositionmd1 and a time step $\Delta t=10^{-15}s$. When the
motor is active our simulations are out of equilibrium, because the motor’s
folding periodically release energy into the system. However our system is in
a steady state and does not age as explained below. For that purpose we use
the Berendsen thermostatberendsen to evacuate from the system the energy
created by the motor’s folding. Notice that the thermostat applies to the
medium’s molecules but not to the motor. Thus the motor’s cooling results only
from the molecular interactions with the medium’s molecules. We use the NVT
canonic thermodynamic ensemble as approximated by Berendsen thermostat (see
ref.finite2 for an evaluation of the effect of the thermostat on our
calculations), and periodic boundary conditions. The molecules of the medium
(host)ariane are constituted of two rigidly bonded atoms ($i=1,2$) at the
fixed interatomic distance $l_{h}$$=1.73$Å. These atoms interact with atoms
of other molecules with the following Lennard-Jones potentials:
$V_{ij}=4\epsilon_{ij}((\sigma_{ij}/r)^{12}-(\sigma_{ij}/r)^{6})$ (1)
with the parametersariane : $\epsilon_{11}=\epsilon_{12}=0.5KJ/mol$,
$\epsilon_{22}=0.4KJ/mol$, $\sigma_{11}=\sigma_{12}=3.45$Å,
$\sigma_{22}=3.28$Å. The mass of the motor is $M=540g/mole$ (constituted of
$18$ atoms, each one of mass $30g/mole$) and the mass of the host molecule is
$m=80g/mole$ ($2$ atoms with a mass of $40g/mole$ each). We model the motor
with $18$ atoms in a rectangular shape constituted of two rows of $9$ rigidly
bonded atoms. The width of the swimmer is $L_{s}=4.4$Å and its length
$l_{s}=15.4$Å. The length of the host molecule is $l_{h}=5.09$Å and its width
$L_{h}=3.37$Å. The motor’s atoms interact with the medium’s atoms using mixing
rules and a Lennard-Jones interatomic potential on each atom of the motor,
defined by the parameters: $\epsilon_{33}=1.88KJ/mol$, $\sigma_{33}=3.405$Å.
We use the following mixing rules mix1 ; mix2 :
$\epsilon_{ij}=(\epsilon_{ii}.\epsilon_{jj})^{0.5};\sigma_{ij}=(\sigma_{ii}.\sigma_{jj})^{0.5}$
(2)
for the interactions between the motor and the host atoms. Our medium is a
fragile liquidfragile1 ; fragile2 that falls out of equilibrium in bulk
simulations below $T=340K$, i.e. $T=340$ K is the smallest temperature for
which we can equilibrate the bulk liquid when the motor is not active.
Consequently the bulk medium above that temperature behaves as a viscous
supercooled liquid in our simulations while below it behaves as a solid (as
$t_{simulation}<\tau_{\alpha}$). Notice however that when active, the motor’s
motions induce a fluidization of the medium around itmd16 ; flu1 ; flu2 ; flu3
; flu4 ; carry ; flu5 ; rate , an effect that has been experimentally
demonstratedflu1 ; flu2 ; flu3 ; flu4 ; flu5 with azobenzene photo-
isomerizing molecules embedded in soft matter. We evaluate the glass
transition temperature $T_{g}$ in the bulk to be slightly smaller
$T_{g}\approx 250K$. Notice that as they are modeled with Lennard-Jones atoms,
the host and motor potentials are quite versatile. Due to that property, a
shift in the parameters $\epsilon$ will shift all the temperatures by the same
amount, including the glass-transition temperature and the melting temperature
of the material. Each folding is modeled as continuous, using a constant
quaternion variation, with a folding time $\tau_{f}=0.3ps$. The total cycle
period is constant and fixed at $\tau_{p}=400ps$. The folded and unfolded part
of the cycle have the same duration equal to $\tau_{p}/2=200ps$ including the
$0.3ps$ folding or unfolding. Our system, while out of equilibrium, is in a
steady state and is not aging. That behavior is obtained because the energy
released by the motor into the medium is small enough and the time lapse
between two stimuli large enough for the system to relax before a new stimuli
appears. In other words we are in the linear response regimepccp . Through
this work we use the mean square displacement to obtain informations on the
molecules mobility and diffusion inside the pore. The mean square displacement
is defined asmd1 :
$\displaystyle{<r^{2}(t)>={1\over N.N_{t_{0}}}\sum_{i,t_{0}}\mid{{\bf
r}_{i}(t+t_{0})-{\bf r}_{i}(t_{0})}}\mid^{2}$ (3)
From the time evolution of the mean square displacement we then calculate the
diffusion coefficient $D$ for diffusive displacements using the Stokes-
Einstein equation:
$\displaystyle{\lim_{t\to\infty}<r^{2}(t)>=6Dt}$ (4)
Finally we will use the Non Gaussian parameter $\alpha_{2}(t)$ to measure the
extent of cooperative motions inside the medium.
$\displaystyle{\alpha_{2}(t)=\frac{3<r^{4}(t)>}{5<r^{2}(t)>^{2}}-1}$ (5)
Unless otherwise specified the results presented in this work correspond to a
temperature $T=500K$ at which our model liquid is supercooled. Notice however
that our system is a model system with Lennard-Jones interactions only.
Therefore, due to the properties of the Lennard-Jones potential, one can shift
the temperatures by any factor $\alpha$ ($T^{\prime}=\alpha T$) creating a new
material. That new material will have all the $\epsilon$ parameters of the LJ
potentials (including for the motor) modified in the same way
$\epsilon^{\prime}=\alpha\epsilon$. The results corresponding to that new
material will have to be translated from the results of our work by a shift of
time $t^{\prime}=t/\alpha^{0.5}$. Similarly, the size of the atoms can be
scaled by a factor $\gamma$ ($\sigma^{\prime}=\gamma\sigma$) provided that all
the distances of the system are scaled by the same factor. Our results will in
that case have to be translated by a shift of distance $\gamma$
($r^{\prime}=\gamma r$) and a shift of time $\gamma$ ($t^{\prime}=\gamma t$).
In that way our simulations are valid for a large number of materials although
approximately, and experimentalists can adjust our data to their system of
concern.
## III Results and discussion
### III.1 Diffusion inside the pore
Figure 4: (color online) Mean square displacement of the medium molecules
located at a distance $r<10$ Å from the motor at time $t=0$, for different
pore radii $R$. The activation parameter $d_{z}$ is different for each Figure.
$Z$ is the direction of the pore axis and $z$ the direction perpendicular to
the motor’s plane in the motor’s molecular frame. In Figure (a) $d_{z}=0$
meaning that there is no activation and the motor acts as a passive probe.
Figures 4a 4b 4c and 4d show the mean square displacement of the confined
medium for different pore radii ranging from the Bulk to a pore radius $R=4$Å,
each Figure corresponding to a different activation quantified by the motion
of the arms $d_{z}$ (Figure 1). For a purely passive motor molecule (Figure
4a), the pore effect is important. As the pore size decreases, the diffusion
inside the medium decreases (i.e. the viscosity increases) leading for pores
of radii smaller than 11Å to a solid non diffusive medium. This confinement
effect is due to the decrease of the probability of molecules near the wall to
escape the cage created by the molecules surrounding them, as part of their
cage is created by the pore wall and thus never opens. Notice that, this well
established confinement effect conf-1 ; conf0 ; conf1 ; conf2 ; conf3 ; conf4
; conf5 ; conf6 ; confine ; conf7 , displays a number of similarities with the
effect of a decrease in temperature in supercooled liquids. As expected, when
the motor activates the medium in Figures 4b to 4d, the diffusion inside the
pores increases. However, surprisingly it increases more in the small pores
(see for example the rapid increase for $R=5$Å), than in the larger ones,
leading to a decrease of the difference in diffusions. Eventually, for the
larger activation investigated in Figure 4d, the diffusion inside the
different pores is rather similar. Thus we observe in Figure 4 that the
activation progressively suppresses the effect of confinement.
Figure 5: (color online) Diffusion coefficient of the medium molecules for
different pore radii $R$ and activation parameters $d_{z}$.
The evolution of the diffusion coefficient in Figure 5 resumes that behavior.
The upper circles on the Figure show the effect of the larger activation by
the motor on the medium’s diffusion for various pore radii. The diffusion
coefficients appear roughly constant for that large activation, even
increasing slightly when the pore radius decreases. As discussed before, the
activation of the medium inside the pore washes out the slowing down induced
by the confinement. For an activation slightly weaker the second rank of
circles from the top show a constant diffusion for the different pores. When
the activation decreases the pore slowing down effect increases but is still
much smaller in the Figure than for the non-activated medium ($d_{z}=0$).
### III.2 Activation effect
Figure 5 shows that the diffusion coefficients follow exponentially decreasing
laws with the inverse of the pore radius $1/R$.
$\displaystyle{D=D_{0}exp(-{\zeta}/R)}$ (6)
Where $\zeta$ is a correlation length that depends on the activation parameter
$d_{z}$. As a result, one can explain the decrease of the pore effect with the
activation, by the decrease of that correlation length of the medium when it
is activated. For our largest activation the correlation length $\zeta$ tends
to zero and the pore effect disappears.
Figure 6: (color online) Mean square displacement of the medium molecules for
various activation parameters $d_{z}$; (a) in the bulk ($R=\infty$); (b) for a
pore of radius $R=5$Å. The activation dependence appears much larger in the
pore.
Figures 6(a) and (b) compare the effect of the activation in the bulk and in a
pore of radius $R=5$Å. Figure 6 shows a much larger activation effect on the
medium’s displacements inside the small pore than in the bulk. For a time
lapse of $10ns$ the mean square displacement in the pore varies indeed from
$10^{-1}$Å2 to $4.10^{1}$Å2, that is by a factor $400$ depending on the
activation, while in the bulk it varies by a mere factor $3$. This result
agrees well with the findings of previous section that the diffusion for large
activation doesn’t depend on the pore size. As the pore radius decreases, the
slowing down by the pore wall increases, but the activation effect also
increases compensating that slowing down. Notice that as the pore radius
decreases, the relative number of motor to free molecules increases due to the
geometry. While results we show take only into account molecules a distance
$r<10$Å apart from the motor, reducing that geometrical effect, it doesn’t
eliminate it. Geometry appears consequently as a possible cause for the
increase of the activation impact on the diffusion for small pores. In order
to conclude on that possibility we calculated the mean square displacements at
a smaller distance $r<5$Å from the motor. The results (not shown) were very
similar to the results of Figure 6, showing that the increase of the
activation effect on diffusion for small pores, is not arising from the
geometry (i.e. from the mean distance to the motor).
It is interesting to point out however that the increase of the effect of
activation when the pore size decreases doesn’t increase the diffusion to
values larger than bulk values but tends instead to reach the bulk values from
below. That point comes in support of the decrease of the correlation lengths
explanation.
### III.3 Cooperative motions
Figure 7: (color online) Non Gaussian parameter
$\alpha_{2}(t)=\frac{3<r^{4}(t)>}{5<r^{2}(t)>^{2}}-1$ around the motor
($r<10$Å) for various activation parameters $d_{z}$ and for a pore of radius
$R=5$Å, at $T=500K$.
Spontaneous (thermal)dh0 and activation-induced cooperative motionsmd16 ; c3
; c4 are present in supercooled liquids and have been well documented.
Because cooperativity is directly related to the physical correlation length
of the liquid, its increase or decrease can give us information on that
correlation length. Therefore, to shed some light on the correlation length
explanation, we now study the extent of cooperative motions in our system. For
that purpose we display in Figure 7 the Non Gaussian parameter $\alpha_{2}(t)$
evolution with the activation. Cooperative motions in supercooled liquids
induce a tail in the Van Hove self correlation function, leading to its
departure from the Gaussian shape. For that reason the departure from the
Gaussian shape of the Van Hove, measured by the Non Gaussian coefficient, has
been used extensively as a measure of the cooperative motions.
In Figure 7 on the right $\alpha_{2}(t)$ shows peaks at times corresponding to
multiple of the motor’s folding period, that are the result of the excitations
induced by the motor’s folding on the medium. The activation-induced
cooperativity as measured by $\alpha_{2}(t)$ decreases when we increase the
activation parameter $d_{z}$ from $3.3$Å to $4.9$Å. That result comes again in
support of the decrease of the medium’s correlation length with the
activation. Notice however that as expected when the activation parameter is
null the induced cooperativity disappears as there is no more activation. This
result suggests an increase of the cooperativity for small activations and an
optimum.
### III.4 Excitation density
To better understand previous results, we now turn our attention to the
density of excitations inside the pores. Following Keys et al. keys we define
excitations as elementary diffusion processes. In our work, excitations are
molecules that move in a time lapse $\Delta t=10ps$ a distance larger than
$\Delta r=1$ Å from their previous position. Figure 8 shows the excitation
density for various pore radii and excitation processes. We find in Figure 8
that the density of excitation increases with the activation whatever the pore
radius. That result validates at posteriori our choice of the activation
parameter as the displacement of the motor’s arm $d_{z}$.
Interestingly the maximum excitation density is not always located at the pore
center. This effect could be due to layering inside the pores. Layering
induces oscillations of the density inside the pores and low densities promote
excitations as cages are then loose, letting molecules escape easier. We will
test that interpretation in Figure 9.
Figure 8 shows that when the pore size decreases the excitation density
decreases much less rapidly for large activations. Therefore the excitation
density is larger than expected for small pores when the activation is on, in
agreement with the activation effect described in a previous section. That
result suggests that the larger than expected diffusion observed for small
pores (the activation effect) is due to a larger density of excitations that
is a larger number of mobile molecules as opposed to larger motions of some
molecules.
Figure 8: (color online) Excitation density $\rho_{excitation}$ as a function
of the distance $r$ from the pore central axis, for different pore radii $R$
and activation parameters $d_{z}$. Excitations are here defined as medium’s
molecules motion larger than $\Delta r=1$ Å occuring during a time lapse
$\Delta t=10ps$.
### III.5 Structural organization
We will now investigate the local structure inside the pores. In Figure 9 we
display the local density inside the pores as a function of the distance $r$
from the pore center. That is, $r$ is the radius of cylindrical coordinates,
$z$ being the pore axis. The Figure shows a clear layering effect, that we
highlight with a line when the activation is off. The local density is in most
cases as expected maximum at the center of the pore ($r=0$) and minimum around
the wall. However this is not always the case. For instance for a pore of
$R=5$Å radius, the maximum density is not located in the center of the pore,
and around the wall for large activations the density is not null. For a pore
of $R=8$Å radius, there is a maximum density at the pore center but it
corresponds to a thin peak, and we observe a wider maximum apart from the
center.
As expected, the layering explains the excitation density maxima shifted from
the center of the pore, as the local density is actually smaller at the
location of excitation density maxima. Moreover an unexpected effect appears,
the layering changes with the activation. Not only it decreases slightly but
the oscillations locations are also modified. Work is in progress to better
understand that interesting effect.
Figure 9: (color online) Density $\rho$ of medium molecules inside the pore as
a function of the distance $r$ from the pore central axis. The Figures show
that result for different pore radii $R$ and activation parameters $d_{z}$.
The values are normalized by the average density inside the pore $\rho_{0}$.
The dashed line displays the layering when the motor is not active. The
figures show that the layering is strongly modified with the activation.
## IV Conclusion
In supercooled liquids, nano-confinement modifies the dynamical properties of
the liquid usually inducing an important slowing down of the dynamics.
Confinement also induces a structural organization called layering. In this
work we investigated how these confinement effects are modified when the
liquid is out of equilibrium. Results show that the dynamical slowing down
induced by the confinement is strongly modified by the activation, leading to
a diffusion that may not depend anymore on the pore diameter. In agreement
with that finding our results show that the activation effect is much larger
upon confinement than in the bulk, increasing rapidly when the pore size
decreases. We interpret these effects as arising from an activation induced
decrease of the dynamical correlation lengths of the liquid. In agreement with
that picture we observe a decrease of the cooperative motions upon activation.
While the structure of the liquid is unchanged, we observe a decrease of the
layering inside the pore but also a qualitative modification of the layering
with the activation. Eventually we observe a relation between the layering and
the elementary diffusion processes called excitations, a result strongly
suggesting that the layering triggers an increase of the mobility in the less
dense parts of the pore.
Conflict of interest
There are no conflict of interest to declare.
Data availability
The data that support the findings of this study are available from the
corresponding author upon reasonable request.
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|
Original Article Journal Section Cuneyt Gurcan Akcora, Departments of Computer
Science and Statistics, University of Manitoba, Winnipeg, MB R3T-2N2, Canada
<EMAIL_ADDRESS>NSF DMS 1925346, NSF ECCS 2039716, RGPIN-2020-05665
# Blockchain Networks: Data Structures of Bitcoin, Monero, Zcash, Ethereum,
Ripple and Iota
Cuneyt Gurcan Akcora Departments of Statistics and Computer Science,
University of Manitoba, Winnipeg, MB R3T-2N2, Canada Yulia R. Gel Department
of Mathematical Sciences, University of Texas at Dallas, Richardson, TX75080,
USA Murat Kantarcioglu Department of Computer Science, University of Texas
at Dallas, Richardson, TX75080, USA
###### Abstract
Blockchain is an emerging technology that has enabled many applications, from
cryptocurrencies to digital asset management and supply chains. Due to this
surge of popularity, analyzing the data stored on blockchains poses a new
critical challenge in data science.
To assist data scientists in various analytic tasks on a blockchain, in this
tutorial, we provide a systematic and comprehensive overview of the
fundamental elements of blockchain network models. We discuss how we can
abstract blockchain data as various types of networks and further use such
associated network abstractions to reap important insights on blockchains’
structure, organization, and functionality.
###### keywords:
_blockchain_ , _bitcoin and litecoin_ , _ethereum_ , _monero_ , _zcash_ ,
_ripple_ , _iota_
## 1 Introduction
On October 31, 2008, an unknown person called Satoshi Nakamoto posted a white
paper titled “Bitcoin: A Peer-to-Peer Electronic Cash System” to the
Cyberpunks mailing list. In eight pages [1], Satoshi explained the network,
transactions, incentives, and other building blocks of a digital currency that
he called Bitcoin.
Bitcoin solves the problem of sending and receiving digital currency on the
world wide web. However, the idea of a digital currency is as old as the
Internet. For similar purposes, traditional banks and Internet companies have
created online payment services, such as Paypal, Visa, and Master. However, a
trusted entity intermediates currency flows in these solutions and updates
user balances as transactions are processed. Blockchain removes the trusted
entity and provides a framework to process transactions and maintain user
balances correctly and securely.
Bitcoin and blockchain have been used interchangeably in the past, but Bitcoin
is just one financial application among many other use cases of blockchain
technology.
Blockchain stores a limited number of transactions (i.e., coin transfers) in a
data structure called a block, which is in turn stored in a public ledger. We
may consider the ledger as a notebook that has information to calculate user
balances written in it. Blockchain represents a user with a blockchain address
as a fixed-length string of characters (e.g., 1aw345….). A transaction can be
as simple as sending bitcoins from one address to another, along with the
required digital signatures. Transaction size increases if more addresses are
involved. Blocks contain executed transactions. Block sizes (which determine
the number of transactions in a block) are usually small (e.g., 1MB in Bitcoin
that allows approximately 3000 transactions).
Blockchain uses an underlying peer-to-peer network to transmit blocks and
proposed transactions between blockchain users worldwide. We will elaborate on
“users” in later sections; however, users of the first blockchain, Bitcoin,
are ordinary web users who can join the network by downloading and installing
an application called a wallet.
A copy of the ledger is stored locally at every participant (i.e., node) of
the peer-to-peer Bitcoin network. Every user is supposed to check its
blockchain copy to learn about user balances. Bitcoin ledger is extended by
appending a new block to the end of the chain every 10 minutes through a
process that is called mining. The mining is where transaction
approval/verification and coin creation occur, and blockchain removes
financial institutions from this process and allows ordinary users to serve in
their role. Mining is achieved by solving a cryptographic puzzle by trial and
error. A valid solution is a proof that the miner has completed some work and
effort to find the answer to the puzzle. The answer itself is an integer
called nonce that once appended to the end of block content, the hash of the
content satisfies a predetermined difficulty goal. The nonce and the resulting
hash are known as proof-of-work, which has its roots in a research article
[2].
Mining of each block is an open competition worldwide. Any user may compete to
solve the puzzle to earn a handsome sum called the block reward, creating new
bitcoins. This process is similar to fiat money printing; however, coins are
created in predefined quantities (50 BTC in 2009, the amount halves every four
years) at each block and given to the miner. Mining is, by purpose, designed
to be difficult so that miners will not litter the blockchain with blocks.
However, the difficulty is adaptive and depends on the number of miners and
their computing power. Furthermore, Bitcoin uses an adaptive difficulty level
to reduce the frequency of block mining. The blockchain peer-to-peer network
participants will receive the latest block in time and be aware of its
transactions. Awareness prevents malicious users from spending the same coin
multiple times. For example, Bitcoin updates the mining difficulty to create a
ten-minute gap between two blocks on average, and Ethereum aims at 12-15
seconds per block.
Figure 1: Block structure. We show transactions with squares within blocks.
The canonical blockchain is the main chain that contains blocks and continues
to grow in time. New blocks are expected to be mined on the top (rightmost
part) of this blockchain. Non-transactional block data, such as block hash or
nonce, are not shown in this figure, but edges indicate parents. Blocks that
we depict in gray are called stale blocks because they do not appear in the
canonical blockchain. At the right end of the blockchain, two blocks are
competing to be the $19$th block. Both blocks are valid, but eventually, the
canonical blockchain will include only one of them, and the other will be a
stale block. The long transactions at the top of each block are coinbase
transactions (rewards of miners) which contain the sum of newly minted coins
(in red) and transaction fees (in green). Block 15 has no transaction other
than the coinbase one.
As nodes compete worldwide, miners create candidate blocks and append them to
the chain. The competition is resolved by adopting the candidate that appears
on the longest blockchain (see Figure 1). As miners append more blocks to the
longest chain, the earlier blocks are deemed less likely to change, i.e., data
in earlier blocks are considered final.
In Figure 1 a blockchain has 19 blocks (the latest block, $b_{19}$ has two
competing candidates). The longest chain is called the canonical path, and
blocks on the chain are deemed valid (i.e., $b_{11}$ to $b_{19}$). Gray boxes
show stale blocks that have failed to be a part of the blockchain. An arrow
shows a parent-child relationship between blocks. For example, $b_{14}$ is the
parent of $b_{15}$ ($b_{15}$ is mined later than $b_{14}$).
Blockchains proliferate, and the ecosystem is in evolution. A diverse set of
blockchains have proposed novel mechanisms in storing data and representing
users, coin transfers, smart-contract operations, and more. As a result,
mining blockchain data requires domain expertise which may seem daunting to
data scientists.
In this survey, we take a holistic view and define salient characteristics of
six significant blockchains in terms of their data structures: Bitcoin, ZCash,
Monero, Ethereum, Ripple, and IOTA. As Bitcoin is the first blockchain, we
first teach the UTXO blockchains and design choices of Nakamoto that have
significantly affected how new blockchains function and store data. The
Ethereum section covers the basics of smart contracts, tokens, and other
decentralized finance constructs. In the Ripple and IOTA sections, we teach
their use cases beyond cryptocurrency and platform aspects. We conclude the
survey with an overview of graph types found in these blockchains.pears on the
longest blockchain
## 2 A Taxonomy of Blockchains and Transaction Networks
After Bitcoin’s popularity, various improvements to blockchain have been
suggested and implemented in other digital currencies. Bitcoin, so far, has
been very conservative in changes to its core technology. A few common points
stand out between all blockchain implementations in mining, coding
capabilities, and data scalability solutions.
We can broadly classify blockchains as i) private or public blockchains and
ii) currency or platform blockchains. A third discussion point is about first
vs. second layer technologies that determine how much data blockchains should
store on-chain.
Private, Consortium vs. Public. Most blockchains, such as Bitcoin and
Ethereum, are permissionless. They store data publicly, and any user can mine
a block (Ethereum 2.0, when implemented, will change the mining algorithm).
Anyone can join the Blockchain network and download blocks to view the
transactions in them. When used by companies, a public blockchain would reveal
sensitive corporate data to outsiders. Instead, private blockchains, such as
Hyperledger fabric by IBM [3], have been developed to restrict block mining
rights or data access to one participant (i.e., private blockchain) or a few
verified participants (i.e., consortium blockchain).
Currency vs Platform. Bitcoin and other cryptocurrencies store coin
transactions in blocks as data. However, blockchain is oblivious to the type
of stored data, which can be multimedia files, weblogs, digitized books, and
any other data that we can fingerprint by using a hash function. A hash
function takes an unlimited length of data and creates a fixed length (e.g.,
256 bits in SHA-256) representation using mathematical operations.
In 1997, Nick Szabo, an American computer scientist, had envisioned embedding
what he called smart contracts as “contractual clauses in the hardware and
software to make a breach of contract expensive” [4]. Almost 20 years later,
Blockchain enthusiasts saw a path to implement this vision. Blockchain
platforms Neo (2014), Nem (2015), Ethereum (2015), and Waves (2016) have
stored and run software code, called smart contracts, on a blockchain. Smart
contracts are written in Turing-complete language. However, the block gas
limit restricts what transactions can include in the code. For example, on
Ethereum, a smart contract cannot be arbitrarily long to encode an algorithm.
As a result, blockchain opponents argue that the Ethereum gas limit prevents
Turing-complete smart contracts.
The benefits of storing code on a blockchain are multi-fold. Every blockchain
network node can read smart contracts, and transactions can execute contracts
by sending call messages to them, and all blockchain nodes will run the
contract with the message. These aspects mean that smart contracts allow
unstoppable, unmodifiable, and publicly verifiable code execution as
transactions between blockchain participants. Any blockchain classification
can be flexed to allow public-private functionalities. For example,
Hyperledger Fabric uses smart contracts in permissioned settings. The Ripple
credit network is a permissioned blockchain that restricts block mining but
stores data publicly.
First vs. Second Layer Technologies. Over time, blockchains started to run
into scalability issues due to limited block sizes and increased user
participation. Blockchains developed initial solutions, such as
SegregatedWitness,111https://en.bitcoin.it/wiki/Segregated_Witness to leave
some of the encryption signatures and other non-transactional data out of
blocks to make space for more transactions. Scalability efforts have
culminated in second layer solutions, such as the Lightning Network [5], where
the network executes most of the transactions off-the-blockchain. The first
layer (i.e., the blockchain itself) only stores a summary of transactions that
occur on the second layer. Second layer networks, such as the Lightning
Network, are a hot research area in network analysis that can be the topic of
an extensive review article on its own. Due to space limitations, this
manuscript will only teach blockchain networks that we can extract from the
first layer, i.e., the blockchain itself.
Blockchains have come to contain various data such as sensor messages on IOTA,
software code on Ethereum, and international banking transactions on Ripple.
We give an outline by considering two major blockchain types: cryptocurrencies
and platforms. Cryptocurrencies are designed to store financial transactions
between addresses. A Blockchain platform stores financial transactions as well
but furthermore stores software code in smart contracts.
Blockchains can further be categorized into two broad categories in terms of
transaction type: account-based (e.g., Ethereum) and unspent transaction
output (UTXO)-based (e.g., Bitcoin, Litecoin) blockchains. Traditionally
cryptocurrencies are UTXO-based, whereas platforms are account-based. The
difference between UTXO and account-based models has a profound impact on
blockchain networks. In the first type of account-based blockchains, an
account (i.e., address) can spend a fraction of its coins and keep the
remaining balance. An analogy to account-based blockchains is a bank account
that makes payments and keeps the remaining balance in the account. A
transaction has exactly one input and one output address (an address is a
unique identifier on the blockchain transaction network). We may use an
address to receive and send coins multiple times. The resulting network is
similar to traditional social networks, which implies that we can apply social
network analysis tools directly to account networks. In turn, the second type
(i.e., UTXO-based blockchains), such as Bitcoin, are the earliest and most
valuable blockchains. Bitcoin constitutes around 45-60% of total
cryptocurrency market capitalization. Litecoin has approximately 2%
capitalization. On UTXO based blockchains, nodes (i.e., addresses) are mostly
one-time-use only, complicating network analysis. See Section 8 for issues
that complicate network analyses.
###### Remark 2.1.
A blockchain platform uses smart contracts to implement complex transactions,
resulting in a diverse set of networks. Traditionally, blockchain platforms
have followed an account-based blockchain model. For this reason, we use the
terms account blockchain and blockchain platform interchangeably. However, in
theory, a platform can also employ a UTXO transaction model. Similarly, a
cryptocurrency could use an account-based transaction model. The reader must
discern the distinction between account and UTXO based blockchain networks.
Table 1: Blockchain types. The private/public columns indicate block mining permissions. IOTA and Ripple are maintained by consortiums that own the mining rights. Many projects (such as Ripple and IOTA) developed smart contract functionality many years after their launch. Although IOTA uses a directed acyclic graph instead of a canonical blockchain, the IOTA transaction model is UTXO based. Privacy coins Monero and Zcash use UTXO models with cryptographic security. | UTXO | Account | DAG | Platform | Cryptocurrency | Private | Public
---|---|---|---|---|---|---|---
Bitcoin | | | | | | |
ZCash | | | | | | |
Monero | | | | | | |
Ripple | | | | | | |
Ethereum | | | | | | |
IOTA | | | | | | |
Table 1 shows the focus of this manuscript and lists the blockchains that we
will cover. Specifically, we teach two projects that differ from
cryptocurrencies and platforms in critical aspects: Ripple and IOTA. Ripple is
a credit network that predates blockchain fame. Ripple keeps a ledger of
transactions where participants can trade user-issued currencies along with
the native cryptocurrency of the Ripple, which is called XRP. The Ripple
network moves currencies across the world, and financial institutions mainly
use it. Recently, Ripple has also implemented smart contracts (called Hooks).
We include Ripple in this manuscript since Ripple uses key blockchain
technologies, such as hash-based identities and smart contracts, although not
a blockchain by definition. IOTA Tangle is a directed acyclic graph of
transactions. However, the underlying transaction model is UTXO based. IOTA
relegates mining to each transaction’s creator and does not incur a
transaction fee. Although IOTA has its associated cryptocurrency, mainly
Internet-of-Things devices use IOTA to share and store data in industrial
applications.
## 3 Bitcoin, Monero and ZCash: UTXO Networks
A cryptocurrency transaction is a construct that consumes one or more outputs
of previous transactions and creates one or more outputs. As Bitcoin is a good
representative of cryptocurrencies, we will use its data to teach this
section. Note that Litecoin data is in the same format as Bitcoin, and in the
past, we could have parsed them by using the same software library (e.g.,
Bitcoin4J). We will conclude the section with Monero and Zcash networks.
Figure 2: A coinbase transaction $t_{1}$ with two outputs. Note that $t_{1}$
has no input address. Compare this transaction with the ordinary spending
transaction of Figure 3.
Bitcoin stores sequential block data in blk*.dat files on the disk. For
example, block1 is serialized and appended into the blk00000.dat file. Next,
Bitcoin appends a magic byte to separate block 1 from the upcoming block 2 in
the dat file. For example, the Bitcoin mainnet uses 0xD9B4BEF9 as the magic
byte. As the number and size of transactions in a block determine the block
size, each dat file may contain a different number of blocks.
Each block contains one coinbase transaction and zero or more spending
transactions. A coinbase transaction is the first transaction of the block and
contains the block reward plus a sum of fees from the block’s transactions, if
any. Note that a transaction may leave nothing as a transaction fee but still
get mined in the block. The Bitcoin protocol sets the mining reward. If a
miner records the block reward higher than the set amount, the network will
reject the block. However, if the miner sets the block reward less than the
set amount, the block will be accepted, and the miner will have lost bitcoins.
In reality, these bitcoins are lost to the protocol, and the total bitcoin
supply decreases from 21 million bitcoins.
Figure 3: A spending transaction that consumes a previous output and creates
two new outputs.
A coinbase transaction has no input but one or more outputs (Figure 2). In a
sense, in a coinbase transaction, the miner can write itself a check in the
amount of mining reward plus transaction fees. A coinbase transaction
increases the total bitcoin supply by the mining reward amount. This is where
new coins are created in Bitcoin. For this reason, a coinbase transaction
lists no input. All other transactions must list one or more inputs that are
being spent. Usually, a coinbase transaction has a single output address, and
the address belongs to the miner. Some mining pools pay their miners in
coinbase transactions. Such transactions may have hundreds of output addresses
where each address belongs to an individual pool member. In Figure 2 the
coinbase transaction has two outputs that total 1.270B satoshis, out of which
1.25B (12.5 bitcoins) are the mining reward, whereas 20M are the sum of
transaction fees. Satoshi is the subunit of bitcoin (as Cent is the subunit of
US dollar), and one bitcoin contains 100 million satoshis. As Figure 2 shows,
an output has two useful information: amount and address. An address can next
use the received output of $o_{1}$ in a spending transaction. Figure 3 shows
the associated spending transaction that consumes $o_{1}$ and creates two new
outputs.
A look at Figure 3 reveals an interesting point about how Bitcoin records
transactions. Transaction $t_{2}$ gives the consumed output with its previous
transaction id (the hash of $t_{1}$, which created this output) and the output
index. From this information, we can deduce that we are spending the zeroth
output of $t_{1}$, but this does not tell us the amount or address ($a_{1}$).
By only linking the previous transaction $t_{1}$, we can learn the
corresponding amount and address.
From Figure 2 we see that the zeroth output of $t_{1}$ holds 1B satoshis. We
will denote the satoshi amount in an output with $A()$, i.e., $A(o_{1})=1B$.
Next, in Figure 3 500M and 495M satoshis are sent to $a_{3}$ and $a_{4}$. The
difference between $A(o_{1})-[A(o_{3})+A(o_{4})]=5M$ satoshis are implicitly
left (by the transaction creator) as the transaction fee. The miner’s job is
to validate that output amounts are greater than or equal to the input
amounts. Furthermore, the miner must observe all previous blocks and ensure
that another transaction has not already spent $o_{1}$. If the miner makes a
mistake and includes an invalid (e.g., already spent output, too great an
output amount) transaction, the network will reject the miner’s block.
In practice, starting from an output, it is possible to go back in time and
trace the lineage of a coin to reach a coinbase transaction. For example, the
output linked graph in Figure 4 shows the third transaction $t_{3}$, which has
two inputs. The output $o_{6}$ which has two lineages: $o_{1}\rightarrow
o_{4}\rightarrow o_{6}$ and $o_{2}\rightarrow o_{6}$. However, the lineage can
be obscured by mixing schemes [6].
Figure 4: Bitcoin transaction network where we link transaction inputs to
previous transaction outputs.
As well as multiple outputs, a Bitcoin transaction allows multiple inputs. The
owner of each input must sign its portion of the transaction to prove an
authorized coin transfer. A many-to-many transaction is an interesting data
type that we do not encounter in banking and finance; multiple addresses merge
their coins and send them to multiple receiving addresses. For the output, a
Bitcoin address can be created for free and shared with the sender. Output
addresses of a transaction need not belong to the same real-life entity:
$a_{5}$ and $a_{6}$ may belong to people who have never met in real life.
The same reasoning is less likely to apply to input addresses. There can be
multiple scenarios considering the inputs.
* •
The user behind $a_{2}$ signs the input of $t_{3}$ and sends the partially
signed transaction to the user behind $a_{4}$, who signs its input and finally
sends the transaction to the mempool. This scenario implies that the users are
different people but know and communicate with each other.
* •
The user behind $a_{2}$ signs the input of $t_{3}$ and sends the partially
signed transaction to a public forum where other users can add their inputs,
and the transaction is finally sent to the mempool by one of them. This
scenario implies that the users are different people, do not know but
communicate with each other.
* •
A single user owns both $a_{2}$ and $a_{4}$. The user signs both inputs and
sends the transaction to the mempool.
The tree cases mean that we can never be sure about the joint ownership of
inputs. However, in practice, most of the time, all input addresses belong to
the same user. We can use this information to link addresses across
transactions.
###### Remark 3.1.
When a transaction sends coins to an address, the address appears in the
transaction, but the public key of the receiver address is unknown. The
receiver discloses their public key only when spending the received coins, and
any user in the network can hash the key to verify that the hash equals the
address. The matching hash is additional proof that the assets belong to the
receiver. In a P2PSH (pay to script hash) address (which starts with ’3’), the
owner must show the script whose hash equals the address.
### 3.1 Graph Rules for UTXO Blockchains
Before modeling the Bitcoin network, we emphasize three graph rules that
restrict how transactions can transfer coins. These rules are due to Bitcoin
design choices made by Satoshi Nakamoto. For clarity, we will replace outputs
with the associated addresses and work with the toy graph shown in Figure 5.
Figure 5: A blockchain network with six transactions and 12 addresses.
Addresses $a_{1},a_{2},a_{3},a_{4}$ receive coins from earlier transactions
that are not shown here. Transaction $t_{5}$ uses a previously used address
$a_{1}$ in edge $e_{1}$. Edge $e_{2}$ denotes an address reuse where the
change is sent back to the sender address $a_{10}$. Dashed edge $e_{3}$ cannot
be created with the given transactions, because $a_{5}$ receives only a single
output in $t_{1}$. In this network, outputs of $t_{5}$ (to addresses $a_{1}$
and $a_{11}$) and $t_{6}$ (to addresses $a_{12}$ and $a_{10}$) remain unspent.
1. 1.
Balance Rule: Bitcoin transactions consume and create outputs. Outputs are
indivisible units; we can spend each output in one transaction only.
Consequently, we must spend coins received from one output in a single
transaction. Any amount that we do not send to an output address is considered
a transaction fee, and the miner collects the change as the transaction fee.
To spend a portion of the coins in output and keep the change, we must create
a new address and send the remaining balance to this new address. Another
option is to use our existing address as one of the output addresses and re-
direct the balance (as done in edge $e_{2}$ in Figure 5). As a community
practice, reuse of the spender’s address (i.e., address reuse) is discouraged.
As a result, most address nodes appear in the graph two times: first when they
receive coins and second when they spend them.
2. 2.
Source Rule: We can merge input coins from multiple transactions and spend
them in a single transaction (e.g., the address $a_{10}$ receives coins from
$t_{3}$ and $t_{4}$ to spent in $t_{5}$ and $t_{6}$ in Fig. 5). However,
$a_{10}$ could have spent all its coins in a single transaction as well.
3. 3.
Mapping Rule: In a transaction the input-output address mappings are not
explicitly recorded. For instance, consider the transaction $t_{2}$ in Fig. 5.
The output to address $a_{6}$ may come from either $a_{3}$ or $a_{4}$. We can
make an analogy with lakes where in-flowing rivers (inputs) bring water
(coins) to the lake (transaction), and outgoing emissaries (outputs) take the
water (coin) out.
Besides the three rules, in our research, we have encountered the following
Bitcoin spending practices:
* •
A transaction may list the same address in multiple outputs. We have
encountered transactions where all outputs have the same address. As the
address must be recorded more than once, this (spam) practice unnecessarily
increases the transaction size.
* •
In time, an address may receive outputs from multiple transactions. Typically,
an address collects all outputs and spends them at once. In rare cases, an
address can receive outputs again after it has spent its coins. For example,
edge $e_{1}$ in Figure 5 brings an output to $a_{1}$ after $a_{1}$ has spent
its coins in $t_{1}$.
* •
Address reuse, as shown in $e_{2}$, is widespread. Although the community
frowns upon address reuse, users are still reluctant to create a new address
for change. Hierarchically deterministic wallets facilitate automated address
creation (https://en.bitcoin.it/wiki/Deterministic_wallet). However, address
reuse did not abate.
* •
Community practice is to wait for six confirmations (blocks) after receiving a
transaction output before spending it. The practice protects bitcoin buyers
against history reversion attacks. Thereby, we must spend coins of an output
in a minimum of six blocks after they are received ($6\times 10=60$ minutes
later). A day sees 144 blocks; a coin should move at most 24 times (in 24
blocks). In reality, however, once a transaction reaches the mempool, it can
be considered final, and absent a double-spending, its coins can be used in
another transaction immediately, even in the same block.
* •
It is possible to have $n$ ordered transactions in a single block, where
$1<x<y<n$ and $t_{y}$ consumes the outputs generated by $t_{x}$. As a result,
some coins move in the Blockchain network more than 144 times a day.
###### Remark 3.2.
In theory, Monero, Dash, and ZCash privacy coins are UTXO blockchains; their
Blockchain networks are similar to Bitcoin’s. In practice, privacy coins
employ cryptographic techniques to hide node and edge attributes in the
blockchain network. For example, ZCash hides information in its shielded pool,
whereas Monero adds decoy UTXOs to the input UTXO set. This section explains
how to model the blockchain network when the node and edge attributes are
public (as in Bitcoin).
Data scientists model Bitcoin transaction networks as address or transaction
graphs. Address graph omits transactions, and the transaction graph omits
addresses. Both approaches are influenced by traditional social network
analysis, which employs graphs with one node type only. Additionally, one can
extract chainlet substructures from the blockchain network. In the following
sections, we will cover these graph models.
### 3.2 UTXO Transaction graph
Transaction graphs omit address nodes from the transaction network and create
edges among transactions only. Figure 6 shows the transaction graph of the
network shown in Figure 5. The most important aspect of the transaction graph
is that a node can appear only once. There will be no future edges that reuse
a transaction node.
The transaction graph contains far fewer nodes than the network it models. We
can immediately observe a few drawbacks from Figure 6. By omitting addresses,
we lose the information that $t_{5}$ and $t_{1}$ are connected by $a_{1}$. The
address reuse of $a_{10}$ is hidden in the transaction graph as well.
Additionally, unspent transaction outputs are not visible; we cannot know how
many outputs are there in $t_{5}$ and $t_{6}$. Similarly, if $t_{3}$ had an
unspent output, we would not learn this information from the graph. In
Bitcoin, many outputs stay unspent for years; the transaction graph will
ignore all of them.
Figure 6: Transaction graph representation of the Blockchain network in Figure
5.
The advantages of the transaction graph are multiple. First, we may be more
interested in analyzing transactions than addresses. For example, anti-money
laundering tools aim at detecting mixing transactions, and once they are
found, we can analyze the involved addresses next. Many chain analysis
companies focus their efforts on identifying e-crime transactions. Second, the
graph order (node count) and size (edge count) are smaller, which is better
for large-scale network analysis. The reduction is because, on UTXO networks,
transaction nodes are typically less than half the number of address nodes.
For example, Bitcoin contains 400K-800K unique daily addresses but 200K-400K
transactions only. As we will explain in the next section, the address graph
contains many more edges than the transaction graph.
### 3.3 UTXO Address Graph
The address graph is the most commonly used graph model for UTXO networks. The
address graph omits transactions and creates edges between addresses only.
Address nodes may appear multiple times, which implies that addresses may
create new transactions or receive coins from new transactions in the future.
Address graphs are larger than transaction graphs in node and edge counts. As
the mapping rule states, a UTXO transaction does not explicitly create an edge
between input and output addresses in the blockchain transaction network. When
omitting the intermediate transaction node, we cannot know how to connect
input-output address pairs. As a result, data scientists create an edge
between every pair. For example, in Figure 7 we create six edges between
inputs ($a_{3},~{}a_{4}$) and outputs ($a_{6},~{}a_{7},~{}a_{8}$). If there
are few addresses in the transaction, this may not be a big problem. However,
large transactions can easily end up creating millions of edges. For example,
the highest number of inputs in a Bitcoin transaction was 20000 (821 on
Litecoin), whereas the highest number of outputs in Bitcoin was 13107 (5094 on
Litecoin). The address graph approach will have to create one million edges
for a transaction with one thousand inputs and one thousand outputs.
Graph size is not the only problem. The address graph loses the association of
input or output addresses. For example, the address graph in Figure 7 loses
the information that edges $a_{3}$ and $a_{4}$ were used in a single
transaction; address graph edges would be identical if the addresses had used
two separate transactions to transfer coins to $a_{6}~{}a_{7}$ and $a_{8}$. We
can solve this issue by adding an attribute (e.g., transaction id) to the
edge; however, this requires additional edge features.
Address graph edge weighting is done as follows. Consider a transaction $t$
with its input addresses $I$ and output addresses $O$. We will denote the
amount an address $a_{x}$ sends to or receives from $t$ as $w_{x}$. Amount of
coins in an address is denoted as $A(a_{x})$. The edge between an input
address $a_{i}\in I$ and an output address $a_{j}\in O$ is assigned an edge
weight as
$w_{i\rightarrow j}=A(a_{i})\times\dfrac{A(a_{j})}{\sum_{a_{x}\in
O}{A(a_{x})}}.$
###### Example 3.3.
Consider a transaction $t$ with input addresses $I=\\{a_{1},a_{2}\\}$ and
amounts as $A(a_{1})=1$, $A(a_{2})=3$ bitcoin. The transaction has two output
addresses $O=\\{a_{3},a_{4}\\}$ and their associated amounts as
$A(a_{3})=0.9$, $A(a_{4})=2$ bitcoin. The transaction fee is left as the
difference between inputs and outputs: $1+3-(0.9+2)$ bitcoin. When creating
the address graph, $w_{2\rightarrow 3}$ is weighted as $3\times(0.9/2.9)$,
whereas $w_{2\rightarrow 4}=3\times(2/2.9)$.
Compared to the transaction graph, the address graph loses less information
from the Blockchain network. For example, the address graph does not fail to
record past address reuse (edge from $a_{9}$ to $a_{1}$) and change address
reuse (as self-loop at $a_{10}$). Furthermore, unspent output addresses remain
visible in the graph.
###### Remark 3.4.
Address reuse: A blockchain address can be used in multiple transactions as a
coin sender or receiver. However, address reuse is discouraged. Ideally, a
Bitcoin user should create a new address each time it receives bitcoins in a
transaction. Address clustering methods aim at linking multiple addresses to
identify real-life entities behind the addresses.
Figure 7: Address graph representation of the Blockchain network in Figure 5.
Once we create the address graph, we may be inclined to run traditional
network science tools and algorithms on it. However, some of these methods are
ineffective in blockchain networks. We can count address clustering, motif
analysis, and core decomposition among these methods. UTXO networks are sparse
and devoid of closed triangles. Furthermore, address reuse is discouraged on
UTXO blockchains as a community practice; most addresses appear in two
transactions (i.e., receiving and spending coins). As a result, off-the-shelf
Data Science algorithms and software libraries are relatively inefficient on
address graphs.
For example, network motif analysis [7] aims at finding repeating subgraphs of
specific orders (usually three nodes). Searching for motifs in Figure 7 will
try to find shapes without considering how specific addresses appear together
as inputs or outputs. Addresses $a_{3}~{}a_{6}$, and $a_{4}$ will most likely
never form a triangle. Also, a triangle between $a_{6},~{}a_{7}$, and $a_{8}$
is very unlikely. As a second example, motifs do not consider which addresses
can be active. For instance, once $a_{3}$ spends its output in a transaction
without receiving a new output, it will never create another edge in the
future. Keeping such nodes in memory and searching for their future edges are
unnecessary.
Past address reuse is discouraged, but motifs do not use this information and
search for (non-existent) edges to past addresses in a large address graph. As
a third example, consider that many coins are moved in the network more than
100 times a day, and each time they are received in newly-created addresses.
The resulting graph will be huge, but there will be almost no closed edge
triangles.
Due to these issues, we argue that we should model UTXO networks as a forward
branching tree rather than networks. More importantly, Data Scientists should
develop tools to incorporate domain practices such as aversion to address-
reuse.
### 3.4 Monero and Zcash Transaction Networks
Monero and Zcash are two UTXO based cryptocurrencies that hide transaction
details from the public. For this reason, Monero and Zcash are called privacy
coins.
#### 3.4.1 Monero Networks
Figure 8: Monero transaction network. We show the actual transaction addresses
with red nodes and decoy UTXO addresses with blue nodes. Each ring surrounds
nodes in light blue. Dashes indicate the remaining UTXOs that are linked to
previous transaction outputs. Note that each input ring has exactly ten decoys
and one actual input. However, outputs are listed individually and not in a
ring. Address $a_{1}$ is truly spent in $t_{1}$, but listed as a ring member
in $t_{5}$ as well. By only looking at transaction inputs, one cannot learn
which transaction spends the coins at $a_{1}$. Since the RingCT update, Monero
has hidden the UTXO amount and address as well. The reader should note how
difficult, if not impossible, it would be to link addresses to transactions in
such a graph.
Monero (created in 2014) uses ring signatures to hide inputs of a transaction.
In a ring signature scheme [8], any one of the $m$ members of the ring can
sign a document. Any third party can publicly verify the signature but cannot
deduce which ring participant signed it. Monero treats each transaction input
as a document that ring members will sign. The process to create a transaction
is carried out as follows:
1. 1.
Choose one or more of your UTXOs that you want to spend in a transaction.
2. 2.
For each UTXO that you plan to spend, choose 10 foreign UTXOs (i.e., ring
member UTXOs that do not necessarily belong to you) that you will use as
decoys.
3. 3.
Prepare the ring signature by using the private key of your address.
4. 4.
After signing each input, forward the transaction to the Peer-to-Peer network.
The reader should note that decoy UTXOs can belong to any other user, and the
transaction creator does not need the consent of these users to include their
UTXOs as decoys in a ring. Choosing the best decoy UTXOs to hide the identity
of the actual input UTXO is an active research area.
Monero initially allowed using 0-decoys, i.e., not adding any decoys to the
ring. In this case, a Monero transaction was equivalent to a Bitcoin
transaction where all information is public. Monero also allowed a user to set
the ring size, which could be very big. Privacy research has shown the
drawbacks of this liberal approach [9, 10]. First, 0-decoy transactions
jeopardize the privacy of other blockchain users [10] since a 0-decoy
transaction input reveals that no other transaction could have spent it. As a
result, we can remove the input from the rings of future transactions. Second,
we can use a meta-analysis on similar ring sizes and link transactions of the
same user.
Monero banned 0-decoy transactions in 2016, incremented the mandatory ring
size from 5 to 7 and eventually to 11. With the RingCT update, Monero has also
hidden UTXO addresses and amounts. As a result, very few pieces of
transactions are visible to the public, making it an ideal privacy coin.
Figure 8 shows a Monero address-transaction graph. Note that although inputs
list 11 member rings, outputs are listed individually. However, the receiving
address and received amount are hidden in the UTXO. Monero uses commitments
and range-proofs to make sure that amounts are spent correctly (See Chapter 4
in [11]).
#### 3.4.2 Zcash Networks
Zcash uses zero-knowledge proofs to hide information about some transactions.
In cryptography, a zero-knowledge proof is a method by “which one party can
prove to another party that they know a value $x$, without conveying any
information apart from the fact that they know the value $x$”.
222https://z.cash/technology/
We can broadly categorize Zcash transactions into shielded and public
transactions. Public transactions are identical to Bitcoin transactions where
amounts, addresses, and all other pieces of information are public. Shielded
transactions hide all transaction information from the public by using zero-
knowledge proofs. Addresses that are used in public transactions are called
t-addresses and start with the letter “t” such as
t1K2UQ5VzGHGC1ZPqGJXXSocxtjo5s6peSJ. Shielded transactions use shielded (also
called private) addresses that start with the letter “z”.
Figure 9: A Zcash address-transaction graph that contains six transactions.
Amounts on edges are coins (we assume zero transaction fees). Although we show
the shielded pool nodes and edges here, in reality, the pool is not visible to
the public at all. Transactions which involve $z$-addresses are
computationally costly to create and validate. As a result, only around 10% of
Zcash transactions are shielded [12].
With $t$ and $z$ addresses, there are five Zcash transactions types. We will
show these types by using Figure 9, and explain them as follows:
1. 1.
$t$-to-$t$ (public) transactions: all transactions details are public ($t_{1}$
in Figure 9).
2. 2.
$t$-to-$z$ (shielding) transactions: $t$ address and its sending amount are
public, $z$ address is encrypted ($t_{2}$ in Figure 9).
3. 3.
$z$-to-$t$ (de-shielding) transactions: $t$ address and amount it receives are
public, $z$ address is encrypted ($t_{6}$ in Figure 9).
4. 4.
$z$-to-$z$ (private) transactions: the addresses, transaction amount, and the
memo field are all encrypted and not publicly visible, i.e., $t$ (without an
index) in the shielded pool of Figure 9.
5. 5.
$tz$-to-$tz$ (mixed) transactions: $z$-addresses are involved, but there are
public inputs or outputs in the transaction ($t_{3}$ and $t_{5}$ in Figure 9).
A few design choices have made an impact on Zcash transactions. First, the
Zcash protocol includes a consensus rule that coinbase rewards must be sent to
a shielded address, and typically these coins are forwarded to $t$-addresses
very soon. Second, Zcash initially used a zero-knowledge scheme that supported
at most two hidden inputs and two hidden outputs. As a result, including more
than two input or output addresses required more cryptographic work, which was
costly. A newer scheme called sapling does not have this limitation.
Both Monero and Zcash are much more costly than Bitcoin in terms of
computational costs and resource usage. Monero transactions store ten decoys
for each actual input, which considerably increases transaction size on disk,
and Zcash transaction validation takes too much memory and time. As a result,
online exchanges avoid storing balances of $z$-addresses. However, Monero and
Zcash both use UTXO models that we can analyze by graph analysis tools
developed for Bitcoin.
### 3.5 Chainlets for UTXO Transaction Networks
Subgraph encoding on the Blockchain network is an alternative to address and
transaction graph approaches.
In traditional graphs, we consider nodes and edges the building blocks because
nodes are created in time and may establish new edges at various time points.
However, UTXo transactions create multiple edges at once. That is, we can
think of a transaction itself as a building block of the network. With its
nodes and edges, a transaction represents an immutable decision that is
encoded as a substructure on the UTXO network. Rather than using individual
edges or nodes, we can use this substructure as the building block in network
analysis. If we consider multiple transactions and their connections through
addresses, we may extend the substructure idea to blockchain subgraphs by
considering multiple transactions. We use the term chainlet to refer to such
subgraphs [13].
Consider a UTXO network with transaction and address nodes. The node set
$V=\\{\text{address},\text{transaction}\\}$. An edge $e\in E$ connects an
address $a_{x}$ to a transaction $t_{y}$ (i.e., $a_{x}\rightarrow t_{y}$) or a
transaction $t_{y}$ to an address $a_{z}$ (i.e., $t_{y}\rightarrow a_{z}$).
This implies that there are no edges between two nodes of the same type.
A UTXO chainlet $\mathcal{G^{\prime}}=(V^{\prime},E^{\prime},B)$ is a subgraph
of $\mathcal{G}$, if $V^{\prime}\subseteq V$ and $E^{\prime}\subseteq E$. If
$\mathcal{G^{\prime}}=(V^{\prime},E^{\prime},B)$ is a subgraph of
$\mathcal{G}$ and $E^{\prime}$ contains all edges $e_{u,v}\in E$ such that
$(u,v)\in V^{\prime}$, then $G^{\prime}$ is called an induced subgraph of $G$.
Figure 10: Examples of the first order Merge ($\mathbb{C}_{2\rightarrow 1}$),
Transition ($\mathbb{C}_{2\rightarrow 2}$) and Split
($\mathbb{C}_{2\rightarrow 3}$) $k=1$ chainlets from the UTXO network of
Figure 5. An address can appear in multiple chainlets, each time spending or
receiving a different UTXO. Transaction nodes can appear in a single chainlet
only.
Let $k$-chainlet $\mathcal{G}_{k}=(V_{k},E_{k},B)$ be a subgraph of
$\mathcal{G}$ with $k$ nodes of type “transaction”. If there exists an
isomorphism between $\mathcal{G}_{k}$ and $\mathcal{G}^{\prime}$,
$\mathcal{G}^{\prime}\in\mathcal{G}$, we say that there exists an occurrence
of $\mathcal{G}_{k}$ in $\mathcal{G}$. A $\mathcal{G}_{k}$ is called a
blockchain $k$-chainlet.
As a starting point, we can focus on the first order chainlet ($k=1$), which
consists of a single transaction node and the address nodes. The first order
chainlet is, by definition, a substructure of the network. We denote a
chainlet of $x$ inputs and $y$ outputs with $\mathbb{C}_{x\rightarrow y}$. A
natural classification of first-order chainlets can be made regarding the
number of inputs $x$ and outputs $y$ since there is only one transaction
involved. We can quickly identify three main types of first-order chainlets.
If the transaction merges input UTXOs, it will have a higher number of inputs
than outputs. We call these merge chainlets, i.e., $\mathbb{C}_{x\rightarrow
y}$ such that $x>y$, which show an aggregation of coins into fewer addresses.
Two other classes of chainlets are transition and split chainlets with $x=y$
and $x<y$, respectively, as shown in Figure 10. We refer to these three
chainlet types as the aggregate chainlets.
Figure 11 visualizes the percentage of aggregate Bitcoin chainlets in time.
For example, the transition chainlets are those $\mathbb{C}_{x\rightarrow x}$
for $x\geq 1$. Figure 11 shows that the Bitcoin network, starting as an
unknown project, stabilized only after summer 2011. From 2014 and on-wards,
the split chainlets continued to rise steadily, compared to merge and
transition chainlets. Spam attacks on the Bitcoin blockchain, which created
too many transactions on the mempool to slow down transaction processing and
force a block size increase, were visible around late 2015.
Figure 11: Percentage of daily aggregate chainlets in Bitcoin. Splits
constitute around 75% of all transactions.
Extending this discussion, higher-order chainlets, as shown in Figure 12, can
be classified in terms of their shapes.
Figure 12: Two second order ($k=2$) chainlets from the Blockchain network of
Figure 5.
### 3.6 Occurrence and Amount Information in Chainlets
Chainlets provide two lenses to look at the transaction network; we can
consider amounts or counts of chainlets.
For a given time granularity, such as one day, we can take snapshots of the
blockchain network and extract chainlets from it. From the Blockchain network
snapshot for a given granularity (e.g., daily), we mine two information for
each chainlet type: amount to store volume of coin transfers by using the
chainlet and occurrence to store instances (i.e., counts) of the chainlet.
Figure 13: A UTXO transaction network with amounts. Transaction fees range
from 0.02 to 0.1 coins.
For example, Figure 13 contains six transactions and 12 addresses. However, if
we look at the first order chainlets, we find five unique chainlets:
Transaction $t_{1}$ creates a split chainlet $\mathbb{C}_{2\rightarrow 1}$.
Transaction $t_{2}$ creates split chainlet $\mathbb{C}_{2\rightarrow 3}$.
Transactions $t_{3}$ and $t_{6}$ create split chainlets
$\mathbb{C}_{1\rightarrow 2}$. Transaction $t_{4}$ creates merge chainlet
$\mathbb{C}_{3\rightarrow 1}$. Transaction $t_{5}$ creates transition chainlet
$\mathbb{C}_{2\rightarrow 2}$.
From these chainlets, we can create two $3\times 3$ matrices to hold
occurrence and amount information as
$O=\begin{bmatrix}0&2&0\\\ 1&1&1\\\
1&0&0\end{bmatrix},~{}A=\begin{bmatrix}0&3.58&0\\\ 1.9&1.75&3\\\
2.8&0&0\end{bmatrix}.$
where $\mathcal{O}[i,j]$ and $\mathcal{A}[i,j]$ store the number and
transferred amount of chainlets of the type $\mathbb{C}_{i\rightarrow j}$,
respectively, where $i\geq 1$ and $j\geq 1$. For example, transactions $t_{3}$
and $t_{6}$ are stored in $\mathcal{O}[{1,2}]=2$. The amounts output in these
transactions are $\mathcal{A}[1,2]=1+0.8+1+1.78=3.58$.
###### Remark 3.5.
A coinbase transaction has no input address but $\geq 1$ output addresses. If
we plan to use the occurrence matrix $\mathcal{O}$ to store coinbase
transactions, we need to extend the matrix with $i=0$ as well. Coinbase
transactions would then be stored in $\mathcal{O}[{0,j}]$.
For the transaction network in Figure 13, $i=j=3$ suffices to store the
matrices since transactions have at most three input addresses and three
output addresses. However, in a real blockchain matrix, dimensions can easily
reach thousands. UTXO blockchains restrict the number of input and output
addresses in a transaction by limiting the block size (1MB in Bitcoin), but
the number of inputs and outputs can still reach thousands. As a result, we
can have large chainlets (e.g., $\mathbb{C}_{1000\rightarrow 200}$,
$\mathbb{C}_{901\rightarrow 200}$ or $\mathbb{C}_{100\rightarrow 951}$).
Consider the case where we choose to create $1000\times 1000$ to store the
occurrence and amount matrices with 1 million cells. Having so many cells is
neither useful nor practical since most transaction types will not exist in
the network. In turn, corresponding matrix cells in occurrence and amount
would hold many zero values, implying that the matrices would be very sparse.
As an alternative, we can choose a suitable value for matrix dimensions.
For the optimal matrix dimension $N$, we have analyzed the history of Bitcoin
and Litecoin. We have found that % 91.38 of Bitcoin and % 91.27 of Litecoin
chainlets have $N$ of 5 (i.e., $\mathbb{C}_{x\rightarrow y}$ s.t., $x<5$ and
$y<5$) in average for daily snapshots. This value reaches % 98.10 and % 96.14
for $N$ of 20, for the respective coins. We have selected $N$ of 20 since it
can distinguish a sufficiently large number (i.e., 400) of chainlets and still
offers a dense matrix.
The choice of $N$ requires a strategy to deal with transactions whose
dimensions are bigger than $N$. We record these chainlets in the last
columns/rows of the matrices, which are defined as follows:
###### Definition 3.6.
Amount Matrix. We denote the total amount of coins transferred by a chainlet
$\mathbb{C}_{x\rightarrow y}$ in a graph snapshot as
$\mathbb{A}(\mathbb{C}_{x\rightarrow y})$. Amount of coins transferred by
chainlets in the graph snapshot are stored as an $N\times N$-matrix
$\mathcal{A}$ such that for $i\leq N,j\leq N$
$\mathcal{A}[i,j]=\left\\{\begin{array}[]{l
l}\mathbb{A}(\mathbb{C}_{i\rightarrow j})&\quad\text{if $i<N$ and $j<N$, }\\\
\sum\limits_{z=N}^{\infty}\mathbb{A}(\mathbb{C}_{i\rightarrow
z})&\quad\text{if $i<N$ and $j=N$,}\\\
\sum\limits_{y=N}^{\infty}\mathbb{A}(\mathbb{C}_{y\rightarrow
j})&\quad\text{if $i=N$ and $j<N$,}\\\
\sum\limits_{y=N}^{\infty}\sum\limits_{z=n}^{\infty}\mathbb{A}(\mathbb{C}_{y\rightarrow
z})&\quad\text{if $i=N$ and $j=N$.}\\\ \end{array}\right.$
###### Definition 3.7.
Occurrence Matrix. We denote the total number of chainlet
$\mathbb{C}_{x\rightarrow y}$ in a graph snapshot as
$\mathbb{O}(\mathbb{C}_{x\rightarrow y})$. Chainlet counts obtained from the
graph snapshot are stored as an $N\times N$-matrix $\mathcal{O}$ such that for
$i\leq N,j\leq N$
$\mathcal{O}[i,j]=\left\\{\begin{array}[]{l
l}\mathbb{O}(\mathbb{C}_{i\rightarrow j})&\quad\text{if $i<N$ and $j<N$, }\\\
\sum\limits_{z=N}^{\infty}\mathbb{O}(\mathbb{C}_{i\rightarrow
z})&\quad\text{if $i<N$ and $j=N$,}\\\
\sum\limits_{y=N}^{\infty}\mathbb{O}(\mathbb{C}_{y\rightarrow
j})&\quad\text{if $i=N$ and $j<N$,}\\\
\sum\limits_{y=N}^{\infty}\sum\limits_{z=N}^{\infty}\mathbb{O}(\mathbb{C}_{y\rightarrow
z})&\quad\text{if $i=N$ and $j=N$.}\\\ \end{array}\right.$
###### Example 3.8.
Consider the following example matrices
$O=\begin{bmatrix}0&2&1\\\ 1&1&1\\\
1&0&3\end{bmatrix},~{}A=\begin{bmatrix}0&3.58&0.5\\\ 1.9&1.75&3\\\
2.8&0&4\end{bmatrix}.$
If we define $N=2$ for this example, $\mathcal{O}[1,3]$,$\mathcal{O}[2,3]$,
$\mathcal{O}[3,1]$, $\mathcal{O}[3,3]$ and
$\mathcal{A}[1,3]$,$\mathcal{A}[2,3]$, $\mathcal{A}[3,1]$, $\mathcal{A}[3,3]$
values must be stored inside the $2\times 2$ matrices. The updated matrices
are
$O=\begin{bmatrix}0&2+1\\\ 1+1&1+1+3\\\
\end{bmatrix},~{}A=\begin{bmatrix}0&3.58+0.5\\\ 1.9+2.8&1.75+3+4\\\
\end{bmatrix}.$
Extreme Chainlets: In occurrence and amount matrices, choosing an $N$ value,
such as $N=20$, means that a chainlet with more than 20 inputs/outputs (i.e.,
$\mathbb{C}_{x\rightarrow y}$ s.t., $x\geq 20$ or $y\geq 20$) is recorded in
the $N$-th row or column. That is, we aggregate chainlets with large
dimensions that would otherwise fall outside matrix dimensions. We use the
term extreme chainlets to refer to these aggregated chainlets on the $N$-th
row and column.
Extreme chainlets are large transactions that may involve more than 10
thousand addresses and are also large in coin amounts. We see two behavior in
extreme chainlets. In the first case, chainlets are of the type
$\mathbb{C}_{i<20\rightarrow j>20}$; less than 20 input addresses (usually one
or two) sell coins to more than 20 addresses. An example is the Bitcoin
transaction a79b970c17d97557357ec0661a2b9de44724440e1c635e1b603381c53ece725d
in 2018. These extreme chainlets are split-chainlets which may indicate
selling behavior.
In the second case, chainlets are of the type $\mathbb{C}_{i>20\rightarrow
j<20}$. This chainlet type is useful in finding large coin buys. Usually, an
online exchange collects coins of many individual sellers and creates an
extreme chainlet that has few (usually one or two) output addresses.
Figure 14: Percentage of Bitcoin chainlets for $N=6$. Dark colors indicate
higher percentages. The first row shows the percentages of one input
chainlets. For example, 8.45% of all Bitcoin transactions are 1-to-1
chainlets, whereas 57.04% are 1-to-2 chainlets.
Extreme chainlets capture large coin movements. In our research, we have found
that information from extreme chainlets shows utility in volatility, risk, and
price prediction in cryptocurrencies [14, 15].
## 4 Ethereum: Account Networks
Account blockchains, such as Ethereum, do not use the UTXO data structure of
Bitcoin. Unlike UTXO coin transactions involving as few as two or as many as
thousands of addresses, coin transactions on account blockchains involve only
two addresses: sender and receiver.
Many design choices behind account blockchains originate from the most popular
account-based blockchain Ethereum. In this section, we will teach account
networks by using Ethereum as our data source. However, the reader will find
similar, if not the same, network and graph models in other account
blockchains.
A major difference between UTXO and account blockchains is the type and
variety of networks. UTXO networks are transaction and lightning networks,
whereas, on account blockchains, we can observe the following networks:
1. 1.
Coin transaction network. Similar to the UTXO transaction network, this
network is created from the coin (ether) transfers between addresses. Network
edges only carry the native currency (coin) of the blockchain.
2. 2.
Token transaction networks. Asset trading networks that are created by
internal smart contract transactions.
3. 3.
Trace network. Interactions between all address types. The name trace implies
that a transaction triggers a cascade of calls to smart contracts or
externally owned addresses.
In this section, we discuss the three networks and their building blocks
separately. We begin by first outlining network node and edge types.
It is useful to re-consider the Ethereum address types. In account networks,
we classify addresses into two node types and one special address.
1. 1.
Externally owned address (EOA) has a private key. An EOA is managed by a real-
life entity such as an investor or a centralized exchange. An entity may
create and use multiple EOAs at the same time. Typically, an EOA is used over
a long time in many transactions.
2. 2.
Smart contract address does not have a private key. We can only distinguish
Smart contract addresses by searching for smart contract code at the address.
3. 3.
Null address 0x000.. node that has two use cases. First, a smart contract is
created by using the NULL address as the receiver. Second, the NULL address is
used to dispose of crypto assets. Any coin or asset sent to the address cannot
be reclaimed and considered burned. The address may have a private key, but
finding it by trial is considered impossible.
Figure 15 visualizes the three-node symbols that we will use to teach account
networks.
Figure 15: Three address types and corresponding node shapes in account
networks.
### 4.1 Account Transaction Network
The account transaction network contains coin transfers between the three node
types. The network edges always start from an EOA because a contract address
or a NULL address cannot initiate a transaction. However, a smart contract can
send coins to an EOA when a trace initiates the transfer (we will cover traces
shortly). Specifically, transaction network edges are from i) EOA to EOA, ii)
EOA to contract, iii) EOA to NULL address. iv) contract to EOA.
Network edges may have i) coin amount, ii) account nonce, iii) gas price, and
iv) timestamp features.
The coin amount in an edge is in subunits (Wei). Account nonce is a number
that orders transactions initiated by an EOA. Miners must mine transactions of
an address in nonce order. For example, in Table 2 $a_{1}$ creates four
transactions with nonce values 0 to 3. A future transaction of $a_{1}$ with
nonce 5 has to wait because the transaction of nonce 4 has not been mined.
Nonce order ensures that the network cannot have out-of-order or missing
edges. However, miners can mine multiple transactions from an EOA in the same
block according to the nonce order.
Figure 16: A toy network of five nodes and edges with ether amounts (in wei)
from four blocks. Edge features are given in Table 2. The address $a_{1}$ owns
300 weis from an earlier transaction (not shown). Transaction fee is one wei.
Although we mention transaction time as an edge feature, transactions do not
have timestamps themselves. The timestamp comes from the timestamp of the
block that contains the transaction. Transactions of a block are ordered in
the block by the miner, and online explorers record the order as the block
index. However, even the miner cannot truly know when a transaction was
created.
Another edge feature could be the used gas field of the transaction. However,
the feature is not useful for the transaction network because the gas cost
equals the base fee (currently 21000 gas) for coin transfers. The input data
field of the transaction is empty for coin transfer transactions.
Figure 16 shows an example account transaction network. Table 2 gives the edge
table of this network.
We easily model the account transaction network as a directed and weighted
multigraph (i.e., multiple edges exist between node pairs). Although address
creation is cheap, address reuse is not discouraged on Ethereum, and a node
may appear in multiple blocks and send and receive coins for an extended
period. Furthermore, smart contracts have permanent addresses. These factors
help us track node behavior and network dynamics in time.
block height | from | to | amount (wei) | nonce | block index | timestamp
---|---|---|---|---|---|---
10646423 | $a_{1}$ | $a_{2}$ | 100 | 0 | 1 | Aug-12-2020 05:11:17 PM +UTC
10646423 | $a_{1}$ | $a_{2}$ | 200 | 1 | 2 | Aug-12-2020 05:11:17 PM +UTC
10646424 | $a_{2}$ | $a_{3}$ | 249 | 0 | 1 | Aug-12-2020 05:11:18 PM +UTC
10646424 | $a_{2}$ | $a_{1}$ | 49 | 1 | 2 | Aug-12-2020 05:11:18 PM +UTC
10646425 | $a_{1}$ | NULL | 24 | 2 | 1 | Aug-12-2020 05:11:26 PM +UTC
10646426 | $a_{1}$ | $a_{4}$ | 24 | 3 | 1 | Aug-12-2020 05:11:43 PM +UTC
Table 2: Network edges and their features of some transaction from blocks
10646423 to 10646426.
### 4.2 Token Transaction Networks
A token is created by deploying a smart contract where the token’s features
and business logic are defined. Any blockchain participant can create a token
and facilitate its trade. The token’s contract defines meta attributes about
the token, such as its symbol, total token supply, or decimals. Only the
address of a token is unique in the blockchain; multiple tokens can have the
same symbol, which creates confusion in trades.
Currently, there are more than a hundred thousand tokens on Ethereum [16]. We
consider each token to have its network and set of traders.
A token’s current supply is the number of token instances created by the smart
contract. If the instances are given unique ids, the token is considered non-
fungible (as in the ERC 721 standard); each instance has its characteristics,
owner, and price. Otherwise, one token instance will be equal to any other;
the token is considered to be fungible (as in the ERC 20 standard). In this
section, we will consider networks of fungible tokens, but we can easily
extend our analysis to the non-fungible case.
A token transaction network has EOA, NULL, and smart contract addresses as
nodes. We outline the following three types of transactions that a Data
Scientist must know to analyze token networks.
* •
The creation transaction that assigns an address for the token initializes its
smart contract and state variables.
* •
A trade transaction that moves some tokens between addresses.
* •
A management transaction that can only be initiated by the smart contract
creator (or any address that the owner specifies). The transaction may delete
the contract or forward its balance (in ether or token) to another address. We
will study management transactions in Section 4.3.
token | block height | from | to | amount (Gwei) | tx fee (Gwei) | input data
---|---|---|---|---|---|---
Binance: BNB Token | 3978343 | 0x00c5…454 | 0xb8c7…d52 | 0 | 32643080 | […..]
ChainLink: LINK Token | 4281611 | 0xf550…780 | 0x5149…6ca | 0 | 21895728 | […..]
Tether USD | 4634748 | 0x3692…d57 | 0xdac1…ec7 | 0 | 12683176 | […..]
Table 3: Contract creation transactions for three popular tokens. The input
data field contains the contract code. The from address is the owner of the
smart contract, which resides at the to address. The Ethereum protocol
deterministically computes the address from i) the owner’s address and ii) the
account nonce when the transaction is created (by the owner).
Table 3 lists the creation transactions of three popular tokens on Ethereum.
The from field lists the contract creator; this address is the owner of the
token. Contract creation is expensive, and the transaction fee increases with
more elaborate contracts. The contract code appears in the input data field of
the transaction.
###### Remark 4.1.
Usually, the transaction that creates a smart contract carries no ether (as
shown with 0 amounts in Table 3). The actual payload is the smart contract
code written in the input data field of the transaction.
Token networks evolve with changing user balances that we denote as edges
between addresses. However, token transfers are internal transactions which
are not broadcast to the network in the form of an ordinary Ethereum
transaction. In other words, what we call a token trade is, in reality, an
update of balances in smart contract variables of the token. It is as simple
as changing the values of two keys in a hashmap.
(a) Two transactions are mined in blocks.
(b) Account transaction network records an edge for transferring 100 wei
(c) Token transaction network records an edge for transferring two tokens
Figure 17: A trade between addresses $a_{1}$ and $a_{2}$ for the tokens issued
by $a_{3}$. Transaction and token networks capture partial views from the
transactions given in (17(a)). The zero amount transaction is omitted in the
token transaction network, whereas the transfer of two tokens are not captured
in an Ethereum transaction network.
We will explain the observed transactions and states with the help of Figure
17. A token trade involves the following steps.
1. 1.
Address $a_{2}$ owns tokens that are issued by $a_{3}$. $a_{2}$ could have
received tokens through various channels. For example, $a_{2}$ may have bought
them in an earlier transaction, or the tokens might have been given to $a_{2}$
by the owner of $a_{3}$.
2. 2.
A price per token is agreed upon between $a_{1}$ or $a_{2}$ by any off-the-
blockchain channel or through a clause in the contract (e.g., the contract
sets token price as 50 Wei).
3. 3.
$a_{1}$ pays $a_{2}$ an ether amount (100 Wei in Figure 17(a)) by creating a
transaction, where $a_{2}$ is the receiver and the input data field of the
transaction is blank. Here $a_{1}$ sends the transaction to the network to be
mined.
4. 4.
$a_{2}$ downloads the latest blocks and sees that 100 Weis have been sent to
its address from $a_{1}$. In turn, $a_{2}$ uses the conversion rate and
decides to send 2 tokens to $a_{1}$.
5. 5.
$a_{2}$ creates a transaction where the receiver is $a_{3}$, and the
transferred ether amount is 0. However, the input data lists $a_{1}$ as the
recipient of two tokens. Typically the input data is a call to the transfer
function of the smart contract at $a_{3}$ with two parameters:
to_address:$a_{1}$ and amount:$2$. $a_{2}$ sends this transaction to the
network to be mined.
6. 6.
At every node of the Ethereum network, the Ethereum Virtual Machine executes
the transaction of step 5. The smart contract at $a_{3}$ decreases the balance
of $a_{2}$ by two tokens and increases the balance of $a_{1}$ by two tokens.
This balance update, which is an internal transaction, is recorded as an edge
from $a_{2}$ to $a_{3}$ in the token network of Figure 17(c), but not sent to
the network as a transaction. This step fails if $a_{2}$ does not own two
tokens.
7. 7.
$a_{1}$ downloads and observes the transaction from $a_{2}$ to $a_{3}$. The
node at $a_{1}$ can run the transaction in its local Ethereum Virtual Machine
to ensure that tokens are transferred without any error.
As Figure 17 shows, transaction and token networks have different views from
the two mined transactions. Without running a virtual machine and executing
the transactions, we cannot observe the internal transactions nor create the
token transaction network.
Token networks attract trades and traders every day. As Ethereum addresses
(both EOA and contract) are reused in multiple days, a token network may see
the same node trading in multiple days. However, daily token networks are
sparse and consist of disconnected components, and very few traders appear in
a token’s network every day.
Some traders appear in networks of multiple tokens, making token networks an
ideal setting for studying multi-layer networks. Each token network
constitutes a layer with edges and nodes, and nodes overlap between layers.
### 4.3 Trace Network
Ethereum stores an ecosystem of addresses, smart contracts, and decentralized
organizations. In transaction and token networks, we studied financial
relationships between addresses. This section now shifts our focus to
relationships, call-dependencies, inheritances, and other interactions between
Ethereum addresses. A trace network stores these relationships where nodes are
EOA and smart contract addresses, and edges are interactions between EOA-
contract, contract-EOA, and contract-contract pairs.
An EOA creates a transaction that is directed to the address of a smart
contract. The smart contract can execute a function and terminate or call
functions of other smart contracts. The contract can also move coins to EOAs.
The trace can be extended until the transaction gas limit is exhausted. All
the addresses that are involved in this call/interaction chain create a trace.
In graph terms, a trace is a hyper-edge (i.e., an edge that connects more than
two nodes).
A trace can involve every operation that a smart contract can execute. For
example, a trace can create a smart contract, call a smart contract function,
or delete a smart contract. As such, we can label parts of the trace with the
executed operation.
Figure 18: Transactions and message calls. The account 0x0..ow initiates the
first transaction (1st step) whose amount is forfeited as it is less than 1
ether. The forfeited amount is deposited to the contract at 0x0..25 for the
owner (0x0..1e) of the smart contract to withdraw it in the third step. Next,
the account creates another transaction (4th step) that successfully converts
2 ethers to 6 tokens.
###### Example 4.2.
We construct a trace network from a toy scenario given in Figure 18. We show
addresses with their first and last characters (e.g., 0x0..57). Figure 18
contains three smart contracts (0x0..57, 0x0..a4 and 0x0..25) and two
externally owned accounts (0x0..ow and 0x0..1e). Edges denote transactions or
message calls. Transactions are created by externally owned accounts
explicitly and mined in blocks, whereas message calls are not.
In this example, a contract acts as a salesman for a cryptoasset: 0x0..a4
receives ethers from addresses, and stores their amount of token in its
storage. The contract has a rule that the minimum deposit amount is 1 ether.
0x0..ow is not aware of this rule at first and creates a transaction that
sends 0.2 ethers to 0x0..a4, which calls a deposit function at 0x0..25 that
seizes the amount and stores it in the address. 0x0..1e can observe 1 in the
blockchain as a transaction, but 2 is a message call that requires 0x0..1e to
execute the contract call to discover.
Usually contracts have withdraw functions that allow the contract creator to
remove the deposited ethers from the contract. In 3 0x0..1e makes a contract
call transaction to withdraw these ethers.
0x0..ow learns about the minimum amount rule, and sees that its 0.2 ethers are
forfeited. 0x0..ow makes yet another attempt to buy tokens by sending 2 ethers
again in transaction 4. This time, 0x0..a4 accepts the amount, and
DELEGATECALLS a conversion function from 0x0..57. which directly stores a
state variable that records account 0x0..57 has 6 tokens (three for each
ether).
So far, three transactions ( 1, 4 and 3) have been initiated by EOAs and mined
in blocks. We observe the following three traces from the three transactions:
1. 1.
Trace 1: 0x0..ow $\xrightarrow{\leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\hbox
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$
0x0..a4$\xrightarrow{\leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,.5,0}\pgfsys@color@rgb@fill{1}{.5}{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,.5,0}\pgfsys@color@rgb@fill{1}{.5}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
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2. 2.
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3. 3.
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0x0..25$\xrightarrow{\leavevmode\hbox to14.18pt{\vbox
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Note that 6 creates a state change (an internal transaction) but it is not
recorded in the trace network.
Figure 19: A hyper-graph from Ethereum traces. Three hyper-edges connect five
nodes. Nodes that appear on a trace are all considered neighbors. A finer
grained hyper graph can add edge features based on interaction types between
node pairs.
Placing all these traces (hyper-edges) together, we obtain the hypergraph
shown in Figure 19.
###### Remark 4.3.
The Ethereum virtual machine executes every transaction in order and executes
calls accordingly. There is no concurrency in Ethereum transaction execution;
EVM does not make two calls simultaneously. When a contract calls two
contracts sequentially, EVM will make the second call only after the first
contract call (and any further contract calls it makes) ends.
In theory, traces can reach large sizes in the shape of trees with many
branches. However, the transaction fee grows with additional calls and
prevents the creation of huge traces. Note that even when a transaction sets a
high gas limit to pay a big transaction fee, the transaction will not be mined
if the gas used exceeds the Ethereum block gas limit (currently 12.5K). If a
trace creates a call loop, this may deplete smart contract balances before the
transaction gas limit is reached.
We can understand smart contract behavior by trace network analysis. For
example, we can study the network to discover unexpected call branches in
smart contracts.
## 5 Ripple: Credit Networks
Ripple (www.ripple.com) and Stellar (www.stellar.org) are two credit networks
that closely resemble the ancient Hawala system (in Arabic, hawala means to
transfer or trust). The main idea of Hawala is to allow a money sender to use
connections of people who trust each other to make a payment in a distant
geographical location. Merchants have historically used these types of
networks to transfer money between countries [17].
We model credit networks as directed, weighted graphs that we build from trust
lines between address pairs. A trust line is a directed edge between two
addresses, which implies that the source address trusts the target address. We
can add weight (i.e., money amount) to the edge to show the limits of that
trust. In a credit network built from trust lines, a directed edge is a
promise by the source node that it will let the target node use a loan amount
in a future transaction. Trust lines can be deleted or updated for amounts in
time.
It is helpful to explain a few confusion points in Ripple to a blockchain
researcher. Ripple uses the term ledger instead of a block. Transactions have
multiple types and may involve financial constructs (e.g., checks), user-
issued currencies (e.g., USD), and path-based settlements. The reader should
remember that Ripple contains elaborate business logic in its building blocks
(e.g., transactions). Before we delve into Ripple, we will use a few scenarios
to explain how we can use a network of trust lines to make a payment.
###### Remark 5.1.
Academic articles have conflicting views on how to represent a trust line as a
directed graph edge. Here, we follow the notation used in the official Ripple
documentation, i.e., the edge is from the lender (source) to the borrower
(target). The lender trusts the borrower.
Figure 20: Two Ripple trust lines between Charlie, Bob, and Alice. Charlie has
created a trust line for Alice for 500USD. Alice has already used 5USD from
the trust line. The amount below both edges, 500 USD, is the maximum limit
(set by Charlie and Bob) that Alice can use.
Figure 20 shows two trust lines between Alice, Charlie, and Bob. Each edge has
two edge features. First, a value below the edge (500 USD for both edges) is
the maximum credit amount (known as the limit) that Charlie and Bob trust with
Alice that Alice can use in future transactions. Second, a value above the
edge (5 and 15 USD) is the balance amount that Alice has already used with
Charlie and Bob. For example, Alice can use $500-15=$485 in a future
transaction by using the trust line from Bob.
Users may create the Ripple trust line due to two reasons. First, Alice may
have paid 500 USD to Charlie and Bob each in an off-the-Ripple transaction.
Second, Charlie and Bob may have trusted Alice based on her reputation and
created trust lines for her to use. In practice, the amounts are almost always
due to the first reason, i.e., offline deposits. By default, Ripple limits are
lower-bounded by 0 and upper-bounded by $\infty$. A high limit creates credit
risks for the lender. We make a Ripple trust graph from such trust lines. We
can delete an existing trust line by setting its trust limit to zero. We can
use trust lines to make payments to third addresses over a path that traverses
individual trust lines in the same currency, such as USD. Traveling trust
lines is known as rippling. Figure 21 will help the reader learn basic
rippling across trust lines.
(a) Trust lines. Before Sarah sends any currency, the Ripple trust graph
includes five nodes and five trust lines.
(b) Payment. After the rippling, owed amounts are updated and shown in red
rectangles.
Figure 21: An example of the Ripple trust graph before (left) and after
(right) Sarah uses rippling to send 50 USD to Bob. On the (left) network,
Sarah has two paths that connect her to Bob: 1) Bob $\rightarrow$ Alice
$\rightarrow$ Sarah and 2) Bob $\rightarrow$ John $\rightarrow$ Tim
$\rightarrow$ Sarah. However, in the 1st path that traverses two trust lines,
the trust line between Sarah and Alice has a max limit of 20 USD (2 USD of
which is already used), which is less than the 50 USD that Sarah wants to
send. As a result, Sarah cannot use the first path to send all 50 USD. The
second path is longer than the first because it traverses three trust lines;
however, the limits allow the 50 USD transfer. Sarah uses the second path for
rippling. Owed amounts (values above edges) are increased by the transferred
amount: 50 USD. Sarah needs to consider updated (right) values in future
transactions; she can no longer send 50 USD to Bob because the trust line John
$\rightarrow$ Tim has a current capacity of USD 100 - USD 75 = USD 25 only.
The updated (increased) amounts imply that after the payment, Sarah owes 50
USD to Tim, who owes 75 USD to John, who owes 65 USD to Bob.
### 5.1 Ripple Networks
On cryptocurrencies, a single type of coin, such as bitcoin, is traded between
addresses. Similarly, Ripple has its currency called XRP with a drop sub-unit
(1 XRP = 1 million drops). However, Ripple also allows users to issue
currencies and create trust lines in the issued currencies.
Ripple officially recognizes two main currency types: i) Ripple (abbreviated
as XRP) and ii) user-issued currencies. Academic articles further divide user-
issued currencies into country currencies (e.g., US dollar, European Euro,
Japanese Yen), cryptocurrencies (e.g., Bitcoin), and fictional currencies such
as tokens. For example, in Figure 20, USD amount is a user-issued currency.
User-issued digital tokens are fictional currencies with no outside backing.
Traders must be aware that such tokens have no inherent value.
XRP is issued natively and has no issuer. All other currencies are represented
as “currency.issuer” in transactions. We represent a currency with i) three
characters such as USD, EUR, or ii) less frequently by a 40 character string.
USD issued by address1 is USD.address1 on the Ripple ledger (also called the
XRP Ledger). Multiple issuers can issue a currency: USD.address1 and
USD.address2 are considered two different currencies, even though they are
both USD currencies. However, in rippling user issued currencies of the same
denomination (e.g., USD) are considered to be the same currency. Trust in the
issuer determines the fungibility of a user-issued currency. If Alice does not
trust address1 but trusts address2, 1 USD.address1 will not be worth 1
USD.address2 for Alice. Accordingly, Alice may refuse to participate in
rippling transactions that bring her USD.addres2 amounts.
Each Ripple address needs to store a non-trivial XRP amount as a reserve that
it cannot spend; otherwise, Ripple considers the address deleted. The reserve
requirement (currently 20 XRP) discourages multiple address creation and
usage. If a security breach occurs, a user may change its address; however,
best practice reuses the same address for multiple transactions.
###### Info 1
A currency issuer may set a transfer fee for every transaction in that
currency between other addresses. The fee is similar to earning a commission
for every transaction in the currency. The issuer of a popular currency thus
accumulates transfer fees, which reduces the amount of offline liability it
must hold to exist as an entity in the Ripple network.
The issuer can also freeze all transactions of its currency (XRP cannot be
frozen by anyone). However, holders of the currency can still redeem the
currency by returning it to the issuer and getting paid outside of the XRP
Ledger. However, in real life, the issuer may have already gone bankrupt or
may ignore redeeming requests. Every address on the ledger must make its own
trust decisions by considering these issues and create/allow trust lines
accordingly.
Figure 22: Ripple address types.
We outline Ripple Ledger address types in Figure 22 as a gateway, cold wallet
(issuing address), hot wallet (operational addresses), market maker, and
wallets (i.e., ordinary traders). Gateways are the currency issuers of the
Ripple network and link the XRP Ledger to the rest of the world. All network
nodes can buy Ripple or any other issued currency by paying money (e.g., US
dollars) to the gateway (outside the Ripple ledger). The following steps,
taken verbatim from the Ripple documentation, explain the six necessary steps.
1. 1.
A customer sends money to a gateway’s offline accounts. This could be fiat
money, Bitcoin, or any other asset not native to the XRP Ledger.
2. 2.
The gateway takes custody of the money and records it.
3. 3.
The gateway issues a balance in the XRP Ledger, denominated in the same
currency, to an address belonging to the customer. This is done by creating a
trust line to the customer’s Ripple address.
4. 4.
The customer uses the issued currency in the XRP Ledger, such as by sending
cross-currency payments or trading in the decentralized exchange.
5. 5.
A customer (not necessarily the one who deposited the money initially) sends
the issued currency to the gateway’s XRP Ledger address.
6. 6.
The gateway confirms the customer’s identity who sent the balance in the XRP
Ledger funds and gives the corresponding amount of money outside the XRP
Ledger to that customer.
An institution may use both cold and hot wallets on the XRP ledger. Connected
to the Internet, a hot wallet signs ordinary transactions, and the cold wallet
stores currencies offline and is used less frequently.
Figure 23: A Ripple trust graph. The market has trust lines to two nodes. Note
that it is the gateway and market that have credit lines to the users. If the
balance (edge weight above the line) is non-zero, the node has already used
the credit to make a payment to another node. Assume that market and gateway
are low accounts. Below lines, we show low and high limits, respectively.
A market maker is an address that converts a currency to another and takes a
conversion fee through its offer, which is broadcast to the ledger in an offer
create transaction. The offer may immediately consume earlier offers and gain
the market maker what currency it desires. If the offer is not fully consumed,
it is stored in the ledger as an offer object that can be used in future
payments or consumed by future offers (of other addresses). Offers must be
updated in real-time as currency conversion rates change (for country
currencies, the conversion rate must be updated according to international
exchange rates). A standing exchange offer is canceled in an offer cancel
transaction.
We will teach Ripple networks in two subsections: Ripple trust graph and
Ripple payment graph. The trust graph stores trust lines and balances. The
payment graph utilizes the trust graph to process transactions.
### 5.2 Ripple trust graph
We create the Ripple trust graph from trust lines issued by addresses. In our
basic examples of Figures 20 and 21, trust lines were not reciprocal. However,
in Ripple, both addresses can create a trust line to each other in a currency.
Furthermore, two addresses can have trust lines in multiple currencies.
Ripple creates a single RippleState object to connect two accounts per
currency, sorts the account addresses numerically, and labels the numerically
lower address as low account and the other as the high account. Assume that
Alice and Bob have trust lines in USD and EUR amounts. Ripple creates two
RippleState objects for Alice and Bob; one for EUR and one for USD.
The object has high and low limit fields to record the limit from high
$\rightarrow$ low and low $\rightarrow$ high trust lines. However, the net
balance is a single value shared by the addresses, and Ripple stores the net
balance of the trust line from the low account’s perspective.
###### Remark 5.2.
So far, we have used trust lines between Ripple users (e.g., John
$\rightarrow$ Sarah) in our payment scenarios. However, ordinary users do not
issue currencies on Ripple, which means that actual trust lines connect
currency issuer-user pairs (gateway $\rightarrow$ Sarah).
We can check the currency issuer in a RippleState with balance. If the balance
is positive, the high account is the issuer. If the balance is negative, the
low account is the issuer. Often, the issuer has its limit set to 0, and the
other account has a positive limit, but this is not reliable because limits
can be increased or decreased without affecting an existing balance.
The Ripple trust graph is a directed, weighted multi-graph where an edge has
four features: currency name (i.e., type of the edge), balance, low limit, and
high limit. Multiple currency edges can connect node pairs. For example, in
Figure 23 the gateway and rf1…Jpn have trust lines in USD and EUR currencies.
We give the graph data of this figure in Table 4 where we have not excluded 0
high limits.
low | high | currency | balance | low limit | high limit
---|---|---|---|---|---
gateway | rSA…Adw | USD | 250 | 500 | 0
gateway | rf1…Jpn | USD | 0 | 300 | 0
gateway | rf1…Jpn | EUR | 0 | 30 | 0
market | rf1…Jpn | EUR | 50 | 500 | 0
market | r4Y…b3n | USD | 0 | 370 | 0
Table 4: Ripple trust graph data. Gateway and market are assumed to be low
accounts, whereas user addresses are high accounts. High limit values are 0
because high accounts are all user addresses.
A trust line deactivates in three cases. First, the lender may remove the
trust line by updating trust parameters in a TrustSet transaction. Second, the
lender may freeze the trust line temporarily. Third, the currency issuer can
freeze all transactions of the currency (except for the redeeming
transaction). In two cases, a node may hold a balance on a trust line greater
than the limit, that is, when the node acquires more of that currency through
trading and when the node decreases the limit on the trust line. The average
node degree is less than four in the trust graph, and the clustering
coefficient is around 0.10.
### 5.3 Ripple payment graph
The Ripple trust graph shares edge features with the Ripple payment graph.
Additionally, the payment graph stores a node feature called reserve (i.e.,
XRP balance of the node). We cannot use the reserve amount in transactions. In
addition to the base reserve, the reserve amount increases for every object
(such as a trust line) that we create. The reserve increase applies only to
the node extending trust, not to the node receiving it.
We create the Ripple payment graph from 1) direct payments, 2) path-based
settlements, and 3) a few other financial constructs. We will teach these
payment types before giving a graph model for the payment graph.
#### 5.3.1 Direct payments
A direct payment is an XRP transfer between two Ripple addresses and does not
require a trust line. Regardless of what trust lines exist in the trust graph,
two ripple addresses can send to and receive XRP from each other.
The XRP balance of an address must not fall below the total required reserve.
Otherwise, the address is not allowed to create some types of transactions.
When an address’s XRP balance falls below the reserve, the address may issue
OfferCreate transactions to buy more XRP or other currencies on its existing
trust lines. However, these transactions cannot create credit that we can use
to buy XRP to satisfy the reserve requirement (i.e., when the account
balance$<$ the required reserve). For example, the address may pay a
transaction fee and create an offer for XRP on the ledger. Once a buyer
consumes the offer, the address will receive its XRP payment and satisfy the
reserve requirement.
When an address is below the reserve requirement, it can send new OfferCreate
transactions to acquire more XRP or other currencies on its existing trust
lines. These transactions cannot create new trust lines, nor Offer objects in
the ledger, so they can only execute trades that consume existing Offers.
#### 5.3.2 Path-based settlements
A path-based settlement transaction connects a source and a destination node
pair via a network path through rippling. For example, in Figure 23 the node
rsA…AdW can make a payment to rf1…Jpn through the gateway. We can make a
payment by consuming exchange offers along the path. For example, if the
sender wants to make a payment in USD, the path can use trust lines that
follow USD$\rightarrow$EUR$\rightarrow$USD, where the arrow indicates a
currency exchange. Traversing exchanges is called bridging. If the transaction
offers a better cost, currency$\rightarrow$XRP and XRP$\rightarrow$currency
conversions can automatically be executed on the path as well. This is called
auto-bridging.
A path-based settlement can explicitly write one or more paths (called a
pathset) in a payment transaction. All paths in a pathset must start and end
with the same currency, and the sender can also leave path selection to
Ripple.
Pathfinding is difficult because user XRP balances and trust lines change
every few seconds as new ledgers (i.e., blocks of transactions) are validated.
As a result, Ripple is not designed to compute the absolute best path. The
simplest possible path to connect the steps of the transaction is called the
default path. In addition to the paths written in the transaction, Ripple can
choose to use the default path. The sender must set the tfNoDirectRipple flag
to avoid the default path and force Ripple to use the paths of the pathset.
In a path-based settlement, we can set a few fields of the transaction to
restrict path selection. These are the Account (sender), Destination
(receiver), Amount (currency and amount to be delivered), and SendMax
(currency and amount to be sent, if specified) variables of the transaction.
The variables shape the path in these ways:
* •
The Account (sender) is set to be the first step of the path.
* •
If the SendMax variable is set to be a currency issuer, the issuer must be the
second step of the path.
* •
If the Amount variable is set to be a currency issuer, the issuer must be the
second-to-last step of the path.
* •
The path ends at the Destination address.
The actual cost of a path-based payment can change between submission and
execution of a transaction based on updated information (e.g., the limit of a
trust line) from transactions.
A path-based payment can fail for multiple reasons; first, the receiving
address may not have the reserve XRP amount. Ripple checks whether the payment
would deliver enough XRP to meet the reserve requirement in such a case. If
not, the payment fails. Next, the receiving address may have limitations on
receiving payments, such as Deposit Authorization or RequireDest (We will
cover these constructs shortly). 333Currency issuers set such rules. Third,
the paths that the sender specifies may have dried up (i.e., trust line limits
are exceeded).
A rippling transaction can redistribute credit from a more trusted to a less
trusted issuer without the specific consent of the involved address’ owner
[18]. Gateway and market addresses should allow rippling; however, ordinary
addresses may benefit from disallowing rippling on their trust lines.
Since 2013, Ripple has allowed setting a no_ripple flag to disable rippling
transactions involving the trust line. Since March 2015, the flag has been set
to true by default; users must opt-in to allow rippling. Furthermore, the
defaultRipple flag enables rippling among all the wallet’s trust lines.
Researchers in 2016 had found that most paths are short [18]: 15.1% of
transactions are direct payments. 52.7% of transactions use a single
intermediate address and represent the interactions of gateways with their
users following the hot-cold wallet mechanism. 32% of transactions use two or
more intermediate addresses. In the network, the average path length was 6.83,
whereas the diameter was 13.
#### 5.3.3 Financial Concepts
Ripple adopts and uses the following concepts from traditional finance:
offers, checks, and escrows.
Offer. Offers are orders to exchange two currencies (XRP or issued currencies
in any combination). Furthermore, we can trade the same currencies issued by
different issuers in offers. An offer is created in an OfferCreate transaction
by the sender, and Ripple tries to match the offer by using existing offers
fully. If a matching offer is available, the currency balances of both offer
senders are updated accordingly. If the offer is not fully consumed (for the
requested amount), the offer is written as an offer object to the ledger for
the remaining amount. Future transactions can consume the remaining part of
the offer. Offers are also used on path-based settlements to convert one
currency to another. This auto-bridging of currencies happens automatically on
any OfferCreate transaction.
###### Example 5.3.
An offer $o_{2}$ that proposes exchanging 10 USD for 7 EUR ($o_{2}$ sender has
10 USD) will consume an existing offer $o_{1}$ that proposes exchanging 7 EUR
for 10 USD ($o_{1}$ sender has 7 EUR). Balances of both senders will be
updated; $o_{2}$ sender will have 7 EUR whereas $o_{1}$ sender will have 10
USD.
If the exchange rate favors the sender of $o_{2}$, it will sell less currency.
For example, if $o_{1}$ offers 7 EUR for 9 USD, $o_{2}$ will take the order
and save 1 USD; the sender of $o_{2}$ buys 7 EUR with 9 USD and keeps its 1
USD.
Offers are typically used to create cross-currency payments which convert one
currency to another by consuming Offers. A cross-currency payment involves two
or more currencies; at least one of the currencies must be non-XRP. Cross-
currency payments are fully atomic; either the payment fully executes or it
fails.
Note that an address can buy user-issued currency through an offer. In that
case, the user is deemed to hold an amount of the issued currency, which
creates a trust line to the currency issuer. This new trust line is written to
the ledger in an object, which increases the reserve XRP requirement of the
offer’s owner. For this reason, if the owner does not have enough XRP as a
reserve, the offer will be considered unfunded and fail.
Offers are explicit instructions to acquire certain issued currencies, so they
can go beyond limits set in a trust line.
Check. A check is a deferred payment that its intended recipient can cash out.
Checks record sender, receiver, amount information, and expiration date. The
receiver can partially cash a check, and only at the cashing time must the
sender have the required amount of the currency in its account. Otherwise, the
check fails. Differing from bank checks, both the sender and receiver can
cancel a Ripple check before it is cashed out. Checks can send either XRP or
issued currencies.
Checks are helpful for financial institutions to avoid receiving unwanted
payments and comply with financial regulations. If a user wants to deposit an
amount, it can write a check for a Ripple address of the financial institution
and expect the institution to cash it. The receiving address can fully or
partially cash or refuse the check.
Ripple allows rejecting all incoming payments by default, which we can achieve
by enabling Deposit Authorization on an address by setting the asDepositAuth
flag in an AccountSet transaction. Our address can receive payments from i)
pre-authorized addresses or 2) Escrow, Payment Channels, or Checks.
Escrow. Escrows are conditional payments that set aside XRP and deliver it
when certain conditions are met. The conditions can be time-based unlocks and
crypto-conditions. Escrows can expire if not finished in time.
Escrows lock up XRP to be used by a specific receiving address. The escrowed
XRP cannot be spent elsewhere nor used or destroyed until the escrow has been
successfully finished or canceled. After the expiration time, the unused XRP
will return to the sender.
Payment Channel. Similar to Bitcoin, Ripple uses Payment Channels to set aside
XRP that can be used in high-volume microtransactions without recording all
channel transactions in ledgers.
Payment Channels and Escrow can use XRP only, and an address can send XRP to
itself through Escrow. However, an address cannot use Payment Channels to send
XRP to itself. Checks are more constringent; an address cannot use checks to
send XRP nor issued currency to itself.
Partial Payment. The Amount field of a Payment transaction specifies the exact
amount to deliver after charging exchange rates and transfer fees. However,
the trust graph may not allow the exact amount to be delivered due to trust
line limits, exchange rates, and transfer fees. Rather than failing the
transaction altogether, the sender can set the Partial Payment flag
(tfPartialPayment) to automatically reduce the starting amount to a
deliverable amount.
Partial payments are notorious because they have been used to exploit naive
integrations with the XRP Ledger to steal money from exchanges and gateways.
Assume that an address sends 100 USD to a gateway with the partial payment
flag set. The delivered amount (e.g., 30 USD) may be much less than the
intended amount. The gateway should not use the amount field to redeem the
money. Otherwise, it will pay the address 100 USD and lose 70 USD. Partial
Payments have the following limitations: a partial payment cannot provide XRP
to satisfy the reserve requirement of an address. Also, direct XRP-to-XRP
payments cannot be partial payments. With all the financial concepts and path-
based settlements, 79% of transactions involve user-issued currencies, whereas
21% transfer the native cryptocurrency XRP.
## 6 IOTA: Tangle Networks
The IOTA blockchain uses a Proof-of-Work algorithm for mining a transaction,
and the concept of a block does not exist. The approach is markedly different
from other blockchains, where miners mine blocks of transactions rather than
individual transactions.
We will cover IOTA networks in two sections: the IOTA Tangle graph and the
IOTA transaction graph. As a starting point for both graph types, we will
discuss the seed, subseed, private key, difficulty, and hashing concepts that
IOTA uses in address and transaction.
IOTA uses two types of addresses; one-time addresses and Merkle root
addresses. One-time addresses send and receive payments. In a sense, one-time
addresses are similar to ordinary Bitcoin addresses. Merkle root addresses are
used in special applications such as Masked Message Authentication, allowing
data channels with publishers and subscribers.
###### Remark 6.1.
This section will teach one-time addresses because IOTA is phasing out some of
the applications that use Merkle root addresses. For example, Masked Message
Authentication is being replaced with IOTA Streams. We do not yet know how
Merkle root addresses will be used in the new technologies.
IOTA uses a concept of seed which is a string of 81 characters. Each seed
character is generated from a tryte, which in turn is three characters (trits)
that can take -1, 0, or 1 values. The use of trits, rather than bits that can
take 0 or 1, is quite unusual in the blockchain. IOTA defends the practice by
arguing that “ternary computing is considered to be more efficient as it can
represent data in three states rather than just two”. For example,
$\\{0,0,0\\}$, $\\{0,-1,1\\}$ and $\\{1,0,1\\}$ are three example trytes. As
each trit can take three values, a tryte of three trits can take $3^{3}$
possible values. IOTA uses the number 9 and the uppercase letters A-Z (27
characters in total) to map a tryte to an easy-to-read character. Table 5
shows three example trytes and their character mappings.
Decimal number | Tryte = 3 Trits | Tryte-encoded character
---|---|---
0 | 0, 0, 0 | 9
1 | 1, 0, 0 | A
2 | -1, 1, 0 | B
Table 5: IOTA uses trytes and encodes each tryte in a character from a 27
character alphabet for ease of use.
A second concept is called an index, which can take values between 0 and
9,007,199,254,740,991. We use the index to create multiple addresses from the
same seed. The hash value of seed+index is called the subseed. IOTA uses the
KERL hash function, which is the Ternary version of the Keccak-384 hashing
algorithm. The hash produces a 243 trit long (81 trytes) subseed.
###### Info 2
In cryptography, a sponge function or sponge construction is a class of
algorithms with a finite internal state that takes an input bitstream of any
length and produces an output bitstream of any desired length. Wikipedia.
IOTA uses three security levels to determine the length of the private key: 1,
2, and 3. The private key is computed by absorbing and squeezing the subseed
in a sponge function 27 times for each security level. The default security
level is 2, which produces a private key of 4374 trytes. Longer private keys
bring better security; however, they increase the signature length of a
transaction (see Figure 24).
Figure 24: IOTA address creation by using index and security parameters. An
address is created by hashing the computed private key. Note that we compute a
different address for each difficulty. A seed can create $9^{57}$ addresses.
Signature length causes efficiency problems in transactions. IOTA uses empty
transactions that carry neither currency nor a message to store the signature
of an address that has security level 2 or 3. Assume that for a security level
2 address $a_{i}$, we create a (regular) transaction $t_{0}$ and prepare its
signature, which will be 4374 trytes. However, signatureMessageFragment field
of a single transaction such as $t_{0}$ can contain only 2187 trytes.
Consequently, we need to create another empty transaction $t_{1}$ to store the
second fragment of the signature. A data structure called bundle links
transactions $t_{0}$ and $t_{1}$ to each other, and the transactions are
submitted to the Tangle (i.e., the directed acyclic graph) together. The
bundle hash is created from individual transactions of the bundle. As such,
any change in any transaction invalidates the bundle hash. For security level
3, we would need to create two empty transactions. Considering that each
transaction requires its POW computations and takes up disk storage, the
bundling scheme reduces the efficiency of IOTA. We will cover bundle usages in
both the tangle graph and transaction graph.
The private key is split into 81-tryte segments where each segment is hashed
26 times. Afterward, segment hashes are hashed together to create an 81 tryte
long IOTA address. Sometimes, a checksum of 9 trytes is appended to the
address resulting in 90 trytes. An example IOTA address is
OGMMQJUDMNNYSOAXMJWAMNAJPHWMGVAY9UWBXRGTXXVEDIEWSNYRNDQY99NDJQB9QQBPCRRNFAIUPGPLZ.
###### Info 3
One-time signatures are public-key signature schemes that have the property
that we can use the signatures to sign one single message only. On the other
hand, such schemes are usually highly efficient and easy to implement in very
constrained devices since they only require the implementation of hash
functions and not any advanced arithmetic operations [19].
IOTA uses the Winternitz one-time signature scheme to generate digital
signatures [19]. Winternitz has the advantage of being robust, easy to
compute, and resistant to attacks from quantum computers. On the downside,
Winternitz reveals a part of the private key whenever the key signs a
transaction. For example, if the address signs two transactions, the signing
process reveals more than 50% of its private key. As a result, an address
should sign a single transaction only to minimize risk. The address can
receive coins from multiple transactions, but it should spend all of them at
one attempt. If the address leaves a balance or receives future payments after
it has spent its coins, malicious users can attempt to predict the address
private key by using brute-force attacks. As a result, address reuse in IOTA
is possible but not recommended.
### 6.1 IOTA tangle graph
An IOTA transaction validates and lists two past transactions, which are
called tips. Afterward, the transaction is mined with a POW that is
considerably easier to solve than the Bitcoin POW. This way, IOTA distributes
the mining task to individual users who do not need powerful POW mining
devices, which makes IOTA an ideal blockchain for low-powered Internet-of-
Thing devices. The resulting transaction dependency network is called the
Tangle, as shown in Figure 26.
Figure 25: An IOTA transaction with its two linked past transactions. Edges to
tips have dashed and solid fills (as used in official IOTA documentation)
depending on their type. Tip types are used to mark information in certain
transactions such as in bundles.
Each new transaction must select two existing transactions and list them as
trunk and branch. This process is called tip selection, which is at the heart
of IOTA security. Although theoretically, a new transaction could be spending
the coins received from one of its tip transactions, the new transaction does
not have to have any relation with tips. We can select any previous
transaction as a tip.
Multiple new transactions can select an existing transaction as a tip. A
transaction $t$ validates its trunk and branch tip directly. Future
transactions that use $t$ as a tip confirm the trunk and branch tips of $t$
indirectly.
Unfortunately, confirmations by individual transactions do not prevent double-
spending attacks on IOTA. In Figure 26, assume that $t_{1}$ and $t_{2}$ spend
the same IOTA coins in an address $a_{1}$. Each double-spending transaction
can reach a different set of users, who select either $t_{1}$ or $t_{2}$ as
one of their tips. Note that users would not select both $t_{1}$ and $t_{2}$
because they spend the same coins. In traditional blockchains, the block miner
would choose to include either $t_{1}$ or $t_{2}$, and the network would be
safe from the double-spending problem.
Through an entity known as the Coordinator, the IOTA foundation creates and
uses milestone transactions, which protect the Tangle against double-spending
attacks. Milestones arrive every minute and act as anchor points where the
confirmed subtangle is considered to be valid and in consensus. The
coordinator address is hard-coded in the IOTA software so that all tangle
participants can locate and trust milestone transactions. In this aspect, IOTA
is a mix of private and public blockchains because although the Tangle is open
to the public, the consensus on the Tangle state is guaranteed by the IOTA
foundation. In the future, IOTA is aiming to transition to a coordinator-free
POW scheme known as Coordicide (i.e., killing the coordinator).
Figure 26: IOTA tangle graph with milestone transactions $m_{0}$ and $m_{1}$.
Assume that transactions $t_{1}$ and $t_{2}$ double spend the same IOTA coins.
As $m_{1}$ indirectly confirms $t_{1}$, all transactions (marked with a cross
sign) that confirm $t_{2}$ are building on an invalid subgraph. As a result,
these transactions will not be used as tips in future transactions. If a
transaction such as Self1 is not confirmed in time, we can create a new zero
value transaction Self2 that confirms Self1. New transactions will use Self2
as a tip so that Self1 will be indirectly confirmed by them. Consequently, we
hope that Self1 will be indirectly confirmed by a future milestone
transaction.
In Figure 26, we show two milestones $m_{0}$ and $m_{1}$ which indirectly
confirm $t_{1}$ and invalidate $t_{2}$. The reason is due to the fact that
tangle users will see $t_{1}$ being confirmed by $m_{1}$, and the users can
mark $t_{2}$ as double-spending the coins used in $t_{1}$.
tx hash | epoch value | value | bundle | tag | address | branch tip | trunk tip
---|---|---|---|---|---|---|---
FPX..999 | 1603985801 | 0 | XTR..QZX | COR..VU | COR..CJZ | EJR..999 | RBL..999
EJR..999 | 1603985790 | 0 | DKR..NF9 | COR..UM | COR..CJZ | MNF..999 | NWX..999
RBL..999 | 1603985788 | 0 | PZY..SP9 | COR..UL | COR..CJZ | DPW..999 | YBD..999
Table 6: Graph edge list for the Tangle for three zero-value transactions with
the same address (we shorten addresses and tx hashes for clarity). A value 0
implies zero-value (message) transactions where the address is the transaction
creator. In value transactions, the address may belong to the transaction
sender or receiver. The first transaction
FPXNJVTEVMSAQCUMGRTLGLLNZQOLOUHWXNNYZAXBBEIDXOTTXT9PIRKKBGNTKHPMATIQUHSMNELV99999
has the next two transactions as its tips. Note that these transactions do not
belong to the same bundle. The transaction tag (attachmentTag field) is used
to label transactions. The epoch value is an integer that counts the number of
milliseconds since Jan 1, 1970.
### 6.2 IOTA transaction graph
The tangle graph is extended by appending new transactions that confirm tip
transactions. We classify IOTA transactions into value and zero-value
transactions.
Zero-value transactions: IOTA creates a zero-value transaction (see Table 27)
for two purposes. First, the transaction may store signature fragments for
addresses that use security levels 2 or 3. Second, the transaction may store
any data such as sensor data, voting results, or encrypted messages. In both
types, data is stored in signatureMessageFragment or attachmentTag fields. All
other fields of the transaction are trivial.
Figure 27: Bundle structure. Two Bitcoin UTXO transactions (top) and their
corresponding IOTA bundles (bottom). Dashed edges denote trunk tips, whereas
solid edges show branch edges. Tips of bundles ($t_{0},\ldots,t_{3}$) are past
transactions that are unrelated to these IOTA bundles. We show trunk tips with
dashed edges. The figure assumes that security level=2 for addresses, which
requires creating one zero-value transaction (in gray color) to store the
second fragment of each input address signature. For the 3rd level, there
would be two zero-value transactions for each input address. We give bundle
indices (e.g., $0/4$) next to shapes. The left bundle has five transactions,
whereas the right bundle has four.
Value transactions: IOTA adopts the UTXO transaction model with a few
modifications. UTXO model assumes that a transaction has one or more inputs
and one or more outputs. For each input, IOTA creates an input transaction
which lists an address and the withdrawn token value. For each output, IOTA
creates an output transaction with an address and a value. The value signs can
distinguish input and output transactions: negative for input and positive for
output transactions. For the input transaction of an address with security
level 1, the address signature is stored in the signatureMessageFragment field
of the input transaction. If the address security level is 2 or 3, additional
zero-value transactions are created to store second and third key fragments.
Output transactions do not require any address signatures.
transaction | address | time | type | value
---|---|---|---|---
AXI..99 | OOC..H9X | 1603951001 | input | -142 998 000
MVP..99 | EM9..SYW | 1603951006 | output | 1 000
EKG..99 | NBN..GJA | 1603950941 | output | 142 997 000
Table 7: One input and two outputs transactions from bundle SGEAO9WIHW
HEEWZRQIVCNIS HBFBE ZCTEYTKTSRUBLXKLD ETKG9ISSUGBOQGFFTDLD GGFBNEIYPYUBEK9C.
IOTA tokens are transferred from the first row address to the next two row
addresses. As the input transaction uses security level 2, the bundle contains
an additional zero-value transaction (omitted here) to store the second
fragment of the signature of OOC..H9X.
In Table 7 we show a UTXO transaction and its input and output transactions on
IOTA. We chose to show the “type” field to help the reader, but we can as well
omit the field because the value can distinguish transaction type. The table
omits zero-value transactions that are required to store signatures. In Figure
27, we show two UTXO transactions with all required IOTA bundle transactions.
In the figure, note that zero-value transactions are indexed just after their
input transactions; $a_{1}$ signature is stored in index $1/4$. However, the
order of the input-output transactions is not important in a bundle; output
transaction of address $a_{5}$ could also be given at index $0/4$.
As Figure 27 clearly shows, an IOTA bundle uses far more pointers and stores
many more data pieces than a UTXO transaction. This inefficiency, coupled with
low POW difficulty, results in a fast-flowing and large Tangle, which is too
costly to store.
IOTA uses snapshots to limit tangle size periodically. A snapshot resets all
transaction history but saves a list of address balances. Only IOTA permanodes
store all IOTA history. Snapshots occur every few months, and IOTA users are
expected to observe the snapshot and manage their IOTA usage accordingly.
### 6.3 IOTA Streams
IOTA Streams is a framework for building cryptographic messaging protocols on
IOTA in the Rust programming language. An older version of Streams is known as
the Masked Authenticated Messaging, which allowed publishers to attach
periodic data (e.g., home temperature readings every hour) as zero value
transactions to the Tangle. Streams has a native protocol for attaching
messages to the Tangle, but users can extend Streams to send messages from
other mediums, such as HTTP URLs.
IOTA Streams allows attaching data in public, private, and subscriber-only
modes. In the private case, we can only read Streams messages if the
encryption key is known. In the subscriber case, we use a subscription key in
addition to the private key. Streams messages contain a reference for the next
(future) message in the Tangle. If the message is encrypted or subscriber-
only, we cannot locate the future message in the Tangle. Even if we locate the
message, we cannot decrypt its content without the private and subscriber
keys. Streams is still in development while its prior-form, the Masked
Authentication Messaging, is widely used in the Tangle.
We can model the Streams graph as a hypergraph where nodes are transactions,
and directed edges between nodes indicate the shared stream. Note that Streams
messages may use previous Streams messages as tips, but this is unnecessary
for a well-formed stream. Streams transactions may be value or zero-value
transactions. In the zero-value case, Streams transactions do not have to be
confirmed by any other IOTA transaction.
## 7 Tools for Blockchain Data Analytics
We can access blockchain data easily by downloading wallet software and
connecting to the P2P network. Recently, Google has implemented ETL libraries
(https://github.com/blockchain-etl) to help parse Bitcoin, Monero, ZCash,
Ethereum, and a few other blockchains. However, blockchain sizes have become
prohibitively large to run data analytics in a single machine. For example,
the BlockSci tool [20] requires around 60GB of memory to load the Bitcoin
transaction graph.
Many blockchain analytics companies offer REST APIs for blockchain data.
Examples are etherscan.io, blockcypher.com, and infura.io. However, APIs are
either costly or allow limited access to the API. We recommend running your
wallet and using the Blockchain-etl libraries of Google to extract blockchain
data.
Most tools and algorithms for blockchain data are related to e-crime or
financial (e.g., price, investor) analytics. For example, Barjasic et al [21]
“analyze the time-series of minute price returns on the Bitcoin market through
the statistical models of the generalized autoregressive conditional
heteroskedasticity family”. From ransomware payment detection [22] to
sextortion discovery [23], transaction network analysis has proven useful to
study blockchain address importance and to cluster blockchain addresses.
Network flow algorithms [24], random walks [25] and Petri nets [26] are the
main unsupervised methods in this line of research. For example, starting from
Bitcoin addresses of potential interest, Egret [23] analyzes neighborhood
subgraphs in terms of path length and confluence to detect suspicious Bitcoin
flow and other wallet addresses controlled by malicious actors. A rather
curious case of clustering research involves identifying heuristics that can
link multiple addresses used by a real-life entity. Although heuristics are
error-prone, they are widely used for blockchain data analytics (e.g., see
[27, 28, 29])). Another emerging promising approach for ransomware payment
detection is topological data analysis (TDA) [30, 31]. TDA systematically
infers qualitative and quantitative geometric and topological structures of
blockchain transaction graphs at multiple resolutions [32, 33]. As a result,
TDA allows us to capture subtler patterns in the transaction graphs, including
changes in chainlet dynamics, which are often associated with illicit or
malicious activity and which are inaccessible with more conventional methods
based on various forms of information aggregation [22, 34].
## 8 Limitations of Data Science on Blockchains
Having covered blockchain data structures, we now turn our attention to the
difficulties faced by data scientists when mining blockchain data.
First of all, many exchanges create off-channel transactions and do not store
them on the blockchain, and this means that many transactions are executed
without being recorded on the blockchain. All of this off-chain transaction
information is not directly accessible to data scientists.
A further concern is to integrate, model, and query data arriving from
hundreds of blockchains. The Bitcoin blockchain itself produces approximately
300K transactions per day, arriving from 400K-800K user addresses (see
blockchain.com/en/charts), in a temporal resolution scale of 10 minutes. Some |
# Increasing Fault Tolerance and Throughput with Adaptive Control Plane in
Smart Factories
Cao Vien Phung and Admela Jukan Technische Universität Braunschweig, Germany
Email: {c.phung<EMAIL_ADDRESS>
###### Abstract
Future smart factories are expected to deploy an emerging dynamic Virtual
Reality (VR) applications with high bandwidth wireless connections in the THz
communication bands, where a factory worker can follow activities through
360°video streams with high quality resolution. THz communications, while
promising as a high bandwidth wireless communication technology, are however
known for low fault tolerance, and are sensible to external factors. Since THz
channel states are in general hard to estimate, what is needed is a system
that can adaptively react to transceiver configurations in terms of coding and
modulation. To this end, we propose an adaptive control plane that can help us
configure the THz communication system. The control plane implements a
workflow algorithm designed to adaptively choose between various coding and
modulation schemes depending on THz channel states. The results show that an
adaptive control plane can improve throughput and signal resolution quality,
with theoretically zeroed bit error probability and a maximum achievable
throughput in the scenarios analayzed.
## I Introduction
Future smart factory scenarios are envisioned with the downlink applications
using high bandwidth wireless links able to provide high throughput and fault
tolerance. A typical scenario, as illustrated in Fig. 1a, includes several
robots, a factory worker is using an augmented or virtual reality (VR) and
high quality resolution device (as a receiver), and a Base Station (BS) (as a
transmitter). The high data rate of communications in Gb/s is today achievable
with Terahertz (THz) frequency band ($0.3-10$ THz). A system that can combine
THz communications, and virtual- and augmented reality will enable several
advanced features in the manufacturing process, such as the real-time
placement of images onto the real working environment to see the proper
instructions on how to assemble a particular product or a component, hands-
free [1]. Such applications in general are expected to improve the
manufacturing efficiency, decrease the training time of factory works, and
enhance the safety.
To guarantee high throughput and fault tolerance, the system needs to be able
to adaptively respond to changed transmission and applications conditions.
First, this is necessary since VR users move, leading to the outdated channel
states, making static system configuration obsolete. Second, and more
critically, THz band is sensitive to molecular absorption [2], leading to an
easily deteriorated Signal-to-Noise Ratio (SNR), i.e., a high Bit Error Rate
(BER). Thus, to provide the desired robustness, the adoption and development
of the adequate coding schemes are subject of ongoing research [3]. In regard
to coding, a few FEC codes with redundant data are under discussion for THz
communication to protect original data against losses, such as Low Density
Parity Check (LDPC), RS or Polar code [4, 5]. To increase the transmission
performance, data can also be transferred with either a higher modulation
formats, or over longer paths, which also is a major challenge in THz
spectrum. Previous work analyzed time-varying channel modeling and tracking
for THz indoor communications [6]. Papers [7] and [8] analyzed to this end
different mobility scenarios in THz indoor communications.
In this paper, we conceptualize the need to provide an adaptive system
configuration in response to mobility and channel state changes, with the aim
of improving throughput and fault tolerance. To this end we propose data and
control plane architectures, akin to what is known in high speed networks,
e.g., [9]. In the control plane, we design a workflow feedback loop algorithm
which can adaptively update the channel states and generate adaptive coding
and modulation THz system configurations. In the data plane, we consider four
modulation schemes: BPSK, QPSK, 8PSK and 16QAM. For coding, the system can
choose between and Multidimensional Parity Check (MDPC) [10] and Reed-Solomon
(RS) codes [11]. The reasons for the above selections is because these
modulations and coding schemes are well known and can be analyzed, though our
approach is applicable to other schemes as they emerge. The results show an
adaptive control plane can choose the best coding and modulation schemes with
the lowest transmission overhead, and the highest code rate and throughput,
therefore improving the system performance with theoretically zeroed bit error
probability and a maximum achievable throughput in the scenarios analayzed.
(a)
(b)
Figure 1: (a) An augmented reality smart factory application with wireless
communications in THz band; (b) A THz control plane architecture with feedback
loop workflow protocol.
## II System Design
### II-A THz data plane
The THz data plane includes the transmitter (Tx), also referred to as THz base
station, and the receiver (Rx) module, here corresponding to the VR device. In
our model we assume that Rx (VR) can move, whereas Tx (BS) is static. At the
Transmitter (Tx) (THz base station), the data source with a long bit stream is
split into substreams. Each substream is referred to as a coding generation of
$K$ bits, whereby the coding generation includes a set of bits coded and
decoded together. Based on channel condition, $K$ can be adaptively chosen and
all Ks are temporarily stored at the coding cache for the coding process. A
coding generation of $K$ bits generates $R$ redundant bits from the coding
process, where $R$ bits are used to support for correcting error bits from $K$
bits. We consider two coding schemes: RS code, and MDPC($n$D/$m$L) code with
$n$ dimensions (D) and the same length (L) of $m$ bits for each dimension,
whereby the original bits stay unchanged after coding, meaning that only
redundant coded bits are generated by coding. The control plane can choose one
of these two schemes adaptively, based on an algorithm.
If MDPC($n$D/$m$L) code is chosen, the encoding process for
$R=(m+1)^{n}-m^{n}$ parity bits [10] generated from $K=m^{n}$ bits can be
performed as follows: A special case of MDPC($n$D/$m$L) code with n = 2 has
column and row parity bits in the last row and column of the two-dimensional
matrix, respectively. The original data bits are put at the remaining columns
and rows of the matrix. Data bits in the column of the matrix and in the row
of the matrix are secured by one column parity bit in the last row and by one
row parity bit in the last column, respectively. The parity check bit on check
bits for securing column and row parity bits is at the bottom right corner of
the matrix. The construction of MDPC($n$D/$m$L) code with $n\geq 3$ is a
multidimensional hypercube combined from basic three-dimensional parity check
cubes. A parity check cube of three dimensions is a set of the two-dimensional
parity matrices layered into a third dimension and the parity bits across the
layers of data bits belong to the last layer.
If RS code is chosen, the related coding mechanism as well known from [11] can
be performed as follows: The original data $M(X)$ with the size of $K$ bits is
operated on the coding process with the modulo-$g(X)$ function to create the
redundant data $CK(X)$, i.e., $CK(X)=X^{\frac{R}{s}}\cdot M(X)\;mod\;g(X)$,
whereby $X^{\frac{R}{s}}$ represents the displacement shift, $s$ is the coding
symbol size in bits, $R$ is the size of redundant coded information bits and
$g(X)$ represents the generator. Therefore, the RS code word
$C(X)=X^{\frac{R}{s}}\cdot M(X)+CK(X)$ includes the redundant data $CK(X)$
appended systematically onto the original data $M(X)$, i.e., $K$ original bits
are increased to $K+R$ bits during the coding process. $R+K$ bits are divided
into symbols of $s$ bits each and all coding operations are performed via
finite field $\mathbb{F}_{2^{s}}$. The maximum codeword length of RS code is
$\frac{K+R}{s}=2^{s}-1$. If $R$ is fixed and $\frac{K+R}{s}<2^{s}-1$, then $Z$
zero padding symbols can be added into $\frac{K}{s}$ original symbols by the
encoder so that the codeword length $\frac{K+R}{s}+Z=2^{s}-1$ can be achieved,
and $\frac{R}{s}$ redundant coded symbols are generated. These zero padding
symbols will not be sent out of transmission channels, but the decoder needs
to use them for the decoding process. After the coding process, $K+R$ data
bits are modulated to generate $X$ modulation symbols. The control plane can
choose one of the four modulation schemes: BPSK, QPSK, 8PSK and 16QAM,
depending on the THz channel state. After the modulation process, $X$
modulation symbols are sent over the THz channel.
At the Receiver (Rx) side (VR device), we collect $X$ symbols to demodulate.
$X$ modulation symbols are demodulated to re-create $K$ original data bits and
$R$ redundant bits, for decoding. $K+R$ bits will be then delivered to the
decoding block. As large amount of data arrives at the same time, a receiving
buffer is necessary to temporally store all arriving data for the decoding
process, whereby the decoded generations are serialized. If MDPC($n$D/$m$L)
code is used, the iterative decoding method can be applied [10] to correct the
data bits with the highest error probability in each decoding iteration. All
parity bits and their related data bits are checked in each decoding
iteration, and then we will count a failed dimension marker (FDM). The FDM of
the arbitrary bit is set to $k$, if that bit discovers a parity check error in
the $k$ dimension. If the FDM reaches the highest value less than $2$, then
the decoding process stops. Otherwise, all bits are inverted with the highest
FDM value. In the next step, the iteration number limit is checked to decide
whether the decoding process stops or all steps are repeated in the next
iteration, based on the predefined maximum iteration. Finally, if
$FDM_{max}=0$ after the decoding process, the frame is considered error-free.
Otherwise, we face an uncorrectable error.
If RS code [11] is chosen, the data of RS code is decoded as follows. The
syndrome components are calculated from the received word at the first stage.
The second stage is to calculate the error-locator word from the syndrome
components. At the third stage, the error locations are calculated from the
error-locator numbers which are from the error-locator word. The fourth stage
is for calculating the error values from the syndrome components and the
error-locator numbers. Finally, we calculate the decoded code word from the
received word, the error locations, and the error values.
Figure 2: BW 8.64 GHz at center frequency 300.24 GHz.
### II-B THz Control Plane
The control plane module includes the two main components referred to as
Transmitter Controller and Adaptive Configuration Module. In between, a
messaging protocol (feedback messaging protocol) is used for communications
over a wireless channel. The latter can be either in THz or out of THz band,
and is not critical in the overall design due to its low bandwidth required.
Adaptive configuration module generates new system configurations when the
channel state changes. This module collects the necessary measurements (e.g.,
BER) used for estimating the receiver performance as well as channel state
information. Such measurements are not foreseen at the transmitter module,
here THz BS, but can be easily added to the system. For instance, transmitter
can measure the noise levels and calibrate the transmitting power as the noise
increases (e.g., due to equipment aging or heat). The arrow labelled with the
message "Retrieve BER value $BER_{m}$" illustrated that the measured BER value
is periodically updated from the receiver module to the adaptive configuration
module. The messaging arrows labelled as "Set demodulation scheme" and "Set
decoding scheme: RS($K,s$)/MDPC($K$)" illustrate the actuation messaging to
inform the data plane when the best demodulation and decoding schemes are
chosen, respectively. This messages are sent out after the new system
configurations are generated by the adaptive configuration module.
It is important to note that a buffer needs to be defined sufficiently large
for the system to work and store the messages about the system states. Let us
define a buffer $B$ with an enough size for $N$ messages. This buffer is used
to save measured BER values $BER_{m}$ (before decoding) sent by the receiver.
$BER_{m}$ after buffering is denoted to be $BER_{f}$. The $BER_{f}s$ stored in
the buffer are used to anticipate the status of the receiver (e.g., moving or
standing idly at a certain position), or the transmission distance. For
example, changed $BER_{f}s$ mean that it is likely that the receiver is
moving. Consider the BER values of bandwidth $8.64$ GHz at center frequency
$300.24$ GHz over different transmission distances and modulation schemes
conducted using the Simulator for Mobile Networks (SiMoNe) in Fig. 2 from
[12], e.g., if we have $BER_{m}$ or $BER_{f}$ is approximately equal to
$0.0579$, then the transmission distance is estimated to be $d=20$ m with
using BPSK, whereby $BER_{m}$ and $BER_{f}$ can be deduced for the current
transmission distance and previous transmission distance, respectively. At the
beginning, we store the value of $0$ in the buffer. The values $BER_{m}$ are
periodically updated from the receiver module at a constant time interval $t$.
We assume that a small moving speed of a pedestrian is at $1$ m/s, i.e., a
footstep of worker with $0.5$ m will take $0.5$ s. So, to estimate the moving
of receiver, we recommend setting the constant time interval $t=0.5$ s for
periodically updating $BER_{m}$.
In Fig. 2, on the other hand, the fluctuations of channel state are continuous
when the receiver is moving to different transmission distances. Hence, the
adaptive configuration module only generates the new configurations, when the
receiver is predicted to stop at a certain position. This can be explained as
follows: The channel state changes so fast, when the receiver is moving. So,
the new configurations have not been able to actuate yet, and are already
obsolete. Also, too frequent channel state update can cause an extremely large
overhead in each transmitted frame. At the beginning, let us assume that we
set up the system using RS code $(224,240)$ and the main THz channel using
16QAM, and they are known by transmitter, receiver and adaptive configuration
module.
### II-C Adaptive Algorithm
To generate new states, an Algorithm 1 is used. For line 1, the Selection
function includes $BER_{m}$ periodically updated by the receiver module. For
line 2-line 3, we compare $BER_{m}$ with the other previous BER values
$BER_{f}$s stored in the buffer. If there exists at least a difference of
$BER_{m}$ and any $BER_{f}$ stored in the buffer $B$, where that difference is
more than or equal to $\varepsilon$, then it is likely that the receiver is
moving, whereby $\varepsilon>0$. For example, based on BER values over
different distances and modulation schemes in Fig. 2, if the difference
between $BER_{m}$ and $BER_{f}$ is at least $\varepsilon\approx 7.646\cdot
10^{-8}$ for BPSK, $\varepsilon\approx 6.649\cdot 10^{-9}$ for QPSK,
$\varepsilon\approx 9.375\cdot 10^{-7}$ for 8PSK and $\varepsilon\approx
2.992\cdot 10^{-7}$ for 16QAM for bandwidth $8.64$ GHz at center frequency
$300.24$ GHz, then the receiver is predicted to be moving. When the receiver
is moving, $BER_{m}$s will be continuously changed. Therefore, $BER_{f}$s
previously stored in the buffer will be removed, and then the new $BER_{m}$ is
added into the buffer. The reason for the deletion is that when the receiver
is moving, $BER_{f}$s become outdated. If
$\left|BER_{m}-BER_{f}\right|<\varepsilon$, $\forall BER_{f}\in B$ (line 4),
then it is likely that the receiver stops at a certain position. However, no
new configuration is generated because we are not totally sure that whether
the receiver keeps walking or not, i.e., we need more $BER_{m}$ updates to
give a decision of generating new configurations. If the buffer $B$ is not
full (line 5), then we add the new $BER_{m}$ into it (line 6). We keep adding
new $BER_{m}$s, which satisfy the condition of line 4, until the buffer is
full (line 7).
For line 8-10, the adaptive configuration module starts generating new
adaptive configurations because highly likely the receiver stops moving. Based
on $BER_{m}$ received with the current modulation scheme used by the
transmitter and the BER database over different transmission distances and
modulation schemes stored in this module, we can approximately predict the
current transmission distance between the transmitter and receiver (line 8).
Based on the current distance of THz transmission channel and the BER database
over different transmission distances and modulation schemes stored in this
module, we can approximately predict the bit error probability $p_{e}$, if the
transmitter uses modulation scheme $e\in\\{$ BPSK, QPSK, 8PSK,16QAM $\\}$ at
that transmission distance of main THz channel (line 9). Next, we discuss line
10. First of all, we consider the number of error bits $t_{MDPC}$ corrected by
MDPC code and the number of error symbols $t_{RS}$ corrected by RS code:
$\text{ }\left\\{\begin{matrix}t_{MDPC}=2^{n-1}-1,&\\\
t_{RS}=\frac{R}{2s}&,\end{matrix}\right.$ (1)
whereby $t_{V}$ ($V\in$ {MDPC code, RS code}) is assumed to be fixed. On the
other hand, we consider two parameters: Code rate $R_{F}$ and coding system
throughput $TH$ defined to be the number of useful data bits transferred by
the THz transmission system to a certain destination per unit of time. These
two ones with formulas are expressed by:
$\text{ }\left\\{\begin{matrix}R_{F}=\frac{K}{K+R},&\\\ TH=R_{F}\cdot
D\cdot(1-P_{re}^{V}),&\end{matrix}\right.$ (2)
where $D$ is the data transmission rate and $P_{re}^{V}$ is the residual bit
error probability after decoding. With fixed value $t_{MDPC}$ (Eq. (1)) and
$p_{e}$ from four modulation schemes considered from line 9, we can choose $4$
optimized values of $K=m^{n}$ original bits with $R=(m+1)^{n}-m^{n}$ redundant
bits from $4$ modulation schemes for MDPC code, where $m,n\in\mathbb{N}$, so
that they satisfy the eqution as:
$(m+1)^{n}\cdot p_{e}\leq t_{MDPC}.$ (3)
At the same time, similarly, with fixed value $t_{RS}$ (Eq. (1)) and $p_{e}$
from four modulation schemes considered from line 9, we can choose $4$
optimized values of $K$ original bits with $R$ redundant bits from $4$
modulation schemes for RS code, so that they satisfy the eqution as:
$\text{ }\left\\{\begin{matrix}\frac{K+R}{s}\cdot P_{s}\leq
t_{RS}\;\;;\;\;K\geq 0&\\\ 2^{s-1}\leq\frac{K}{s}+\frac{R}{s}\leq
2^{s}-1&,\end{matrix}\right.$ (4)
where $P_{s}=1-(1-p_{e})^{s}$ is the symbol error probability on the main THz
channel and the maximum codeword length of RS code is $\frac{K+R}{s}=2^{s}-1$.
Note that the values of $K$, $R$ or $s$ chosen from Eq. (3) for MDPC code and
Eq. (4) for RS code over different modulation schemes, the total number of
error bits/symbols occurred is lower than or equal to the ability of
correcting bit/symbol errors by MDPC($n$D/$m$L) and RS code, when $K+R$ bits
or $\frac{K+R}{s}$ symbols are sent over the main THz channel with the bit
error probability $p_{e}$ or symbol error probability $P_{s}$, with the aim of
optimizing the fault tolerance system. Theoretically, with this approach, the
residual bit error probability $P_{re}^{V}$ after decoding is approximately
equal to $0$. From eight optimized values of $K$ original bits/symbols with
$R$ redundant bits/symbols of MDPC and RS code over four modulation schemes,
we can choose the best coding and modulation scheme with the highest code rate
$R_{F}$ and throughput $TH$, based on Eq.(2). As the adaptive configurations
are chosen, the THz data transmission can achieve high throughput and code
rate with an optimized system fault tolerance. If $BER_{m}$ arrives when the
buffer of adaptive configuration module is full and
$\left|BER_{m}-BER_{f}\right|<\varepsilon$, $\forall BER_{f}\in B$ (line 12),
then we only remove the oldest $BER_{f}$ stored in the buffer, and then add
$BER_{m}$ into it. In this case, we do not generate any new system
configurations because the receiver does not still change its position, i.e.,
the current channel state is similar to the previous one with the adaptive
configurations already generated from line 7 to line 10. Note that if MDPC
code is chosen, this module generates the configuration: MDPC($K$), whereby
$K$ is the size of original input data. Else if, RS code is chosen, this
module generates the configuration: RS($K,s$), whereby $s$ is the size of
coding symbol in bits.
#### II-C1 Analysis of algorithm complexity
In this subsection, we calulate the overhead for generating a new adaptive
configuration, when a $BER_{m}$ is updated at the adaptive configuration
module. As analyzed from line 5-line 10 in Algorithm 1, this module only
generates new configurations, if the new $BER_{m}$ arrives at the time when
the buffer $B$ stores $N-1$ $BER_{f}$ messages. Assume that a computation for
the process of generating adaptive configurations is counted as one unit and
we ignore the simple processes of removing or storing from the bufffer. We
need $N-1$ units to compare $BER_{m}$ and $BER_{f}$ at line 2. At line 8, we
need one unit for predicting the transmission distance of VR user. At line 9,
we need $M=4$ units for predicting the bit error probability for four
modulation schemes. Finally, for each coding scheme, we need $M\cdot 3=12$
units at line 10 because we have $M=4$ values of bit error probability $p_{e}$
applied for $3$ equations (Eqs. (2), (3), (4)) to find the best coding and
throughput with the highest code rate and throughput. As a result, in general,
we need $N+M(1+3C)$ units for generating a new system configuration, where $M$
is the number of modulation schemes considered and $C$ is the number of coding
schemes considered.
Algorithm 1 Adaptive Workflow Configuration Algorithm
1:function selection($BER_{m}$)
2: if $\exists$ $\left|BER_{m}-BER_{f}\right|\geq\varepsilon$, $\forall
BER_{f}\in B$ then
3: Remove all $BER_{f}$s, and buffering $BER_{m}$;
4: else
5: if $B$ is not full then
6: Add $BER_{m}$ into the buffer;
7: if $B$ is full then
8: Based on $BER_{m}$ $\rightarrow$ transmission distance.
9: Based on transmission distance $\rightarrow$ bit error probabilities
$p_{e}$, if using BPSK, QPSK, 8PSK or 16QAM.
10: Using $p_{e}$ from four modulation schemes, $t_{V}$ fixed and Eqs. (2),
(3), (4) $\rightarrow$ chosen coding and modulation scheme with the highest
$R_{F}$ and $TH$.
11: end if
12: else
13: Remove the oldest $BER_{f}$, and buffer $BER_{m}$;
14: end if
15: end if
16:end function
(a) Transmission distance.
(b) Code rate.
(c) Transmission overhead.
Figure 3: BER vs. distance without coding, transmission distance randomly
chosen, code rate and transmission overhead.
(a) Chosen coding scheme.
(b) Chosen modulation scheme.
(c) Throughput.
Figure 4: Theoretical results. For chosen coding scheme, $1$ and $2$ denote
chosen RS code and MDPC code, respectively. For chosen modulation scheme,
$4,5,6$ and $7$ represent chosen BPSK, QPSK, 8PSK and 16QAM, respectively..
## III Performance Evaluation
In this section, we show the performance results in the smart factory
applications shown, considering THz data and control planes proposed. All data
including over the THz channel, which has bandwidth $8.64$ GHz at center
frequency $300.24$ GHz. For the BER values, we make use of the data shown in
Fig. (2). These BER values are used to estimate the transmission distance
between VR user and THz BS and bit error probability $p_{e}$ over different
modulations under the Algorithm 1. To estimate the moving of VR user, we
assume setting the constant time interval $t=0.5$ s for periodically updating
$BER_{m}$ from VR user to the adaptive configuration module. Note that at the
beginning, we store temporally the value $0$ in the buffer and use RS code
$(224,240)$ with the modulation scheme of 16QAM. The adaptive configuration
module only generates the new configurations, when VR user stops at a certain
position. As discussed in Algorithm 1, based on the database of BER values
from Fig. 2, we define $\varepsilon\approx 7.646\cdot 10^{-8}$ for BPSK,
$\varepsilon\approx 6.649\cdot 10^{-9}$ for QPSK, $\varepsilon\approx
9.375\cdot 10^{-7}$ for 8PSK and $\varepsilon\approx 2.992\cdot 10^{-7}$ for
16QAM to predict whether the VR user is moving or stops at a certain place.
The data transmission rate $D$ for BPSK, QPSK, 8PSK and 16QAM is $7.04$ Gbps,
$14.08$ Gbps, $21.12$ Gbps and $28.16$ Gbps, respectively.
Table I: List of main values applied for evaluation. Values | Meaning
---|---
$t_{MDPC}=1$ | Number of bits can be corrected by MDPC code.
$t_{RS}=1$ | Number of symbols can be corrected by RS code.
$N=4$ | Maximum number of messages stored in buffer $B$.
$t=0.5$ | Time for periodically updating $BER_{m}$ in second.
$\varepsilon\approx 7.646\cdot 10^{-8}$ | Likely VR user is moving if using BPSK.
$\varepsilon\approx 6.649\cdot 10^{-9}$ | Likely VR user is moving if using QPSK.
$\varepsilon\approx 9.375\cdot 10^{-7}$ | Likely VR user is moving if using 8PSK.
$\varepsilon\approx 2.992\cdot 10^{-7}$ | Likely VR user is moving if using 16QAM.
$D=7.04$ | Data transmission rate for BPSK in Gbps.
$D=14.08$ | Data transmission rate for QPSK in Gbps.
$D=21.12$ | Data transmission rate for 8PSK in Gbps.
$D=28.16$ | Data transmission rate for 16QAM in Gbps.
Fig. 3a shows the different standing positions (transmission distances) of VR
user around $60600$ s, where they are randomly chosen between $0.5$ m and
$20.0$ m in steps of $0.5$ m from the transmitter. When VR user moves between
two positions, the channel state changes. Assume that based on $BER_{m}$
periodically updated from the VR user (line 8 in Algorithm 1). Based on the
channel states in Fig. 3a, our adaptive algorithm can choose the best coding
scheme (Fig. 4a) and modulation scheme (Fig. 4b) with the lowest transmission
overhead (Fig. 3c), the highest code rate (Fig. 3b) and throughput (Fig. 4c).
Assume that the time when the VR user is idly standing at any position is
randomly chosen in $[3,4,5,6,7]$ minutes. We assume $R$ redundant bits are
generated from $K$ original bits so that the number of bits can be corrected
to be $t_{MDPC}=1$ bit for MDPC code and to be $t_{RS}=1$ symbol for RS code.
The reason for this assumption is because with a low redundant coding data,
the coding and decoding time is optimized with a low latency, while it can
still satisfy the optimized fault tolerance with high code rate and
throughput, as discussed in section II-C. Assume that the processing time for
the feedback controller and adaptive configuration module is small and
ignored.
Fig. 3b, Fig. 3c, Fig. 4a, Fig. 4b and Fig. 4c theoretically show the code
rate, transmission overhead, chosen coding and modulation scheme and
throughput, respectively, where the throughput is observed in the time
interval, when VR user idly stands at a specific position. $1$ and $2$ denote
the chosen coding scheme, respectively. $4$, $5$, $6$ and $7$ denote the
chosen modulation scheme of BPSK, QPSK, 8PSK and 16QAM, respectively. Based on
the standing positions of VR user predicted in Fig. 3a (line 8 in Algorithm
1), we can approximately estimate four bit error probabilities $p_{e}$ from
the database of BER values over different transmission distances and
modulation schemes in Fig. 2, stored at the adaptive configuration module, if
THz BS uses BPSK, QPSK, 8PSK or 16QAM (line 9 in Algorithm 1). With four bit
error probabilities from four modulation schemes if used by THz BS (line 9 in
Algorithm 1), $t_{MDPC}=1$ bit and $t_{RS}=1$ symbol, we will find four values
of $K$ for MDPC code satisfying Eq. (3) and four values of $K$ and $s$ for RS
code satisfying Eq. (4). Theoretically, if we can choose a reasonable value of
input data $K$ (section II-C), then the residual bit error probability
$P_{re}^{V}\approx 0$ after decoding. Therefore, the throughput is equivalent
to the information bit rate, i.e., based in Eq. (2), $TH=R_{F}\cdot D$. With
$R$ redundant bits deduced from $t_{MDPC}=1$ bit and $t_{RS}=1$ symbol by Eq.
(1), and eight values of $K$ are previouly found, all of them are applied from
Eq. (2). As a result, the adaptive configuration module will find the best
coding scheme (Fig. 4a) and modulation scheme (Fig. 4b) with the highest code
rate (Fig. 3b) and throughput (Fig. (4c)). The transmission overhead in Fig.
(3c) is given by: $\theta=1-R_{F}$, where $R_{F}$ is code rate.
For code rate in Fig. 3b, $0.25\leq R_{F}\leq 1$, i.e., $0\leq\theta\leq 0.75$
in Fig. 3c; the longer the transmission distance, the lower the code rate For
the chosen coding scheme in Fig. 4a, RS code performs better than MDPC. For
the chosen modulation scheme in Fig. 4b, 16QAM is the best, while BPSK is the
lowest selection because its data transmission rate is the lowest. We see that
no modulation scheme 8PSK is chosen in our scenario. This can be explained
since BER of 8PSK and 16QAM are quite similar, while the data transmission
rate of 16QAM is higher than that of 8PSK. Therefore Algorithm 1 always
chooses the coding and modulation scheme with the code rate and throughput as
the highest. As discussed above, with the suitable code rates or transmission
overheads chosen from the best coding and modulation schemes in Fig. 3b and
Fig. 3c over different channel states, we can theoretically improve the system
fault tolerance with the residual bit error probability $P_{re}^{V}\approx 0$.
Also, since $P_{re}^{V}\approx 0$, the throughput can achieve the information
bit rate $TH=R_{F}\cdot D$, and due to the flexibility in choosing the best
coding and modulation schemes with the highest code rate, sometime we can
theoretically get a maximum throughput of $28.16$ Gbps in Fig. 4c, equivalent
to data transmission rate of 16QAM.
## IV Conclusion
We proposed a THz data and control plane system architecture for VR
applications in smart factories. The results shows that an adaptive design
system can improve throughput and fault tolerance, whereby the residual bit
error probability is $P_{re}^{V}\approx 0$ after decoding and we get can
theoretically get a maximum throughput of $28.16$ Gbps, equivalent to data
transmission rate of 16QAM. Future work needs to address algorithm computation
offloading and its response time.
## Acknowledgment
The authors acknowledge the financial support by the Federal Ministry of
Education and Research, Germany, program "Souveran. Digital. Vernetzt."
project 6G-RIC, 16KISK031 and partial support by the DFG Project Nr.
JU2757/12-1.
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|
# M Subdwarf Research<EMAIL_ADDRESS>Atmospheric Parameters and Kinematics
Shuo Zhang CAS Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Beijing 100101, China University of Chinese Academy of
Sciences, Beijing 100049, China Department of Astronomy, School of Physics,
Peking University, Beijing 100871, P. R. China Kavli institute of Astronomy
and Astrophysics, Peking University, Beijing 100871, P. R. China A-Li Luo CAS
Key Laboratory of Optical Astronomy, National Astronomical Observatories,
Beijing 100101, China University of Chinese Academy of Sciences, Beijing
100049, China School of Information Management & Institute for Astronomical
Science, Dezhou University, Dezhou 253023, China Department of Physics and
Astronomy, University of Delaware, Newark, DE 19716, USA Georges Comte Aix-
Marseille Univ, CNRS, CNES, LAM, Laboratoire d’Astrophysique de Marseille,
Marseille, France Rui Wang CAS Key Laboratory of Optical Astronomy, National
Astronomical Observatories, Beijing 100101, China University of Chinese
Academy of Sciences, Beijing 100049, China Yinbi Li CAS Key Laboratory of
Optical Astronomy, National Astronomical Observatories, Beijing 100101, China
Bing Du CAS Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Beijing 100101, China Wen Hou CAS Key Laboratory of Optical
Astronomy, National Astronomical Observatories, Beijing 100101, China Li Qin
CAS Key Laboratory of Optical Astronomy, National Astronomical Observatories,
Beijing 100101, China School of Information Management & Institute for
Astronomical Science, Dezhou University, Dezhou 253023, China John Gizis
Department of Physics and Astronomy, University of Delaware, Newark, DE 19716,
USA Jian-Jun Chen CAS Key Laboratory of Optical Astronomy, National
Astronomical Observatories, Beijing 100101, China Xiang-Lei Chen CAS Key
Laboratory of Optical Astronomy, National Astronomical Observatories, Beijing
100101, China University of Chinese Academy of Sciences, Beijing 100049,
China Yan Lu CAS Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Beijing 100101, China School of Information Management &
Institute for Astronomical Science, Dezhou University, Dezhou 253023, China
Yi-Han Song CAS Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Beijing 100101, China Hua-Wei Zhang Department of Astronomy,
School of Physics, Peking University, Beijing 100871, P. R. China Kavli
institute of Astronomy and Astrophysics, Peking University, Beijing 100871, P.
R. China Fang Zuo CAS Key Laboratory of Optical Astronomy, National
Astronomical Observatories, Beijing 100101, China University of Chinese
Academy of Sciences, Beijing 100049, China
###### Abstract
Applying the revised M subdwarf classification criteria discussed in Paper i@
to LAMOST DR7, combining the M subdwarf sample from Savcheva et al, a new M
subdwarf sample was constructed for further study. The atmospheric parameters
for each object were derived fitting with the PHOENIX grid, combining with
Gaia DR2, the relationship between the gravity and metallicity were explored
according to the locus both in the color-absolute magnitude diagram and the
reduced proper motion diagram. Objects that have both the largest gravity and
the lowest metallicity are located away from the main-sequence cloud and may
be considered as the intrinsic M subdwarfs, which can be classified as
luminosity class<EMAIL_ADDRESS>Another group of objects whose spectra show typical M
subdwarf characters have lower gravity and relatively moderate metal
deficiency and occupy part of the ordinary M dwarf region in both diagrams.
The Galactic $U$, $V$, $W$ space velocity components and their dispersion show
that the local Galactic halo population sampled in the solar neighborhood is
represented by objects of high gravity and an inconspicuous bimodal
metallicity distribution, with a fraction of prograde orbits. The other M
subdwarfs seem to partly belong to the thick disk component with a significant
fraction of thin disk moderately metal-poor objects intricately mixed with
them. However, the selection effects, especially the favored anti-center
direction of investigation in the LAMOST sub-sample, but also contamination by
multiplicity and parameter coupling could play important roles and need to be
further investigated.
Stellar astronomy — Stellar types — Late-type stars— M stars, Stellar
kinematics, Atmospheric parameters
††journal: ApJ††software: astropy (Astropy Collaboration et al., 2013),
matplotlib (Barrett et al., 2005), pandas(McKinney et al., 2010), numpy
(Oliphant, 2007), scipy (Virtanen et al., 2019)
## 1 Introduction
M subdwarfs are usually recognized as the oldest members of the low-mass
stellar populations and metal-poor counterparts of the late-type M dwarfs.
With their extremely long nuclear-burning lifetimes, subdwarfs are supposed to
represent early generations of star formation and are thus potential tracers
of Galactic dynamic and chemical evolution history (Savcheva et al., 2014).
Despite their scarcity in the solar neighborhood, M subdwarfs are supposed to
comprise the majority of stars in the Milky Way stellar halo (Bochanski et
al., 2013), while a number of them also populate the thick disk, whose
population is supposed to be older than that of the thin disk. Additionally,
the cool atmospheres of M subdwarfs provide conditions for studying molecule
and dust formation as well as radiative transfer in metal-poor environments
(Allard et al., 1997).
In many early investigations, M subdwarfs are considered as metal-poor main-
sequence dwarfs (e.g. Lépine et al. 2007; Rajpurohit et al. 2014) solely with
some noteworthy kinematic and spectral properties: red subdwarfs, exhibiting
large space velocities relative to the Sun, kinematically associated with the
Galactic halo (Gizis, 1997) and thick disk, display shallower TiO absorption
bands in the optical spectra than the ordinary red dwarfs dominating the
Galactic thin disk. Historically, they were usually searched based on their
low luminosity combined with large proper motion (Sandage, 1969; Jones, 1972;
Hartwick et al., 1984; Sandage & Kowal, 1986; Gizis, 1997; Lépine et al.,
2003, 2007; Smith et al., 2009; Jao et al., 2011) and characteristic spectral
features (Mould, 1976; Mould & McElroy, 1978; Ake & Greenstein, 1980; Lépine &
Scholz, 2008; Savcheva et al., 2014; Bai et al., 2016; Zhang et al., 2019;
Hejazi et al., 2020). The research of metal-poor subdwarfs also extends to the
late M-type and L-type in recent years (e.g. Zhang et al. 2017b, a, 2018;
Zhang 2019).
On the other hand, some studies have suggested that subdwarfs should have a
luminosity classification independent of dwarfs—vi@ because of 1$\sim$2
magnitude less luminous than the main sequence dwarfs in the H-R diagram,
forming an independent sequence, and more or less showing larger surface
gravity than the dwarfs (Jao et al., 2008). Results from Kesseli et al. (2019)
show that ultra subdwarfs can be as much as five times smaller than their
solar-metallicity counterparts for a given temperature. However, subdwarf
identification in most of the researches depends much more on the metallicity
rather than the gravity, and whether gravity constrain for subdwarfs is
necessary still remains an open question.
Characterizing M subdwarfs in terms of atmospheric parameters has long been a
challenge. Like M dwarfs, the opacity sources are dominated by the molecular
absorption bands such as TiO, VO, MgH, FeH and CaH in the optical, as well as
H${}_{\text{2}}$O and CO in the infrared, each of which has millions of
spectral lines. These complex band structures leave no window to access the
true continuum and only allow the strongest atomic lines such as Cai@, Nai@,
Ki@ to show out at low spectral resolution. In addition, dust clouds form with
decreasing temperature, the stellar atmosphere becomes more complicated, and
the spectra show increasingly diatomic and triatomic molecules (Rajpurohit et
al., 2016). Thus, the traditional technique of estimating the effective
temperature ($T_{\text{eff}}$) of a star based on black body approximation and
broadband photometry is not applicable to cool M subdwarfs in which the true
continuum is covered by a wide variety of molecular absorbers(Allard et al.,
1997). It is also difficult to disentangle the effect of reduced metal
abundance from that of the increasing surface gravity or the temperature
decrease because these three parameters affect the pressure structure of the
photosphere in a similar way (Jao et al., 2008).
Since accurate metallicities for most M subdwarfs are difficult to measure,
proxies for metallicity have been introduced. The most widely used
metallicity-tracer parameter $\zeta$, (a parameter derived from CaH2, CaH3,
and TiO5 molecular bandhead indices originally used by Gizis 1997), was
introduced by Lépine et al. (2007) and revised by Dhital et al. (2012) and
Zhang et al. (2019). With the parameter $\zeta$ or revised definition, two
large samples of M subdwarfs are spectroscopically identified by Savcheva et
al. (2014) and Zhang et al. (2019) using Sloan Digital Sky Survey (SDSS; York
et al. 2000) and the Large Sky Area Multi-Object Fiber Spectroscopic Telescope
(LAMOST; Cui et al. 2012) respectively. According to the decreasing order of
$\zeta$, the M subdwarfs can be divided into three metallicity subclasses: M
subdwarfs (sdM), extreme metal-poor M subdwarfs (esdM), and ultra metal-poor M
subdwarfs (usdM).
The first model grid based on the PHOENIX radiative transfer code (Allard &
Hauschildt, 1995) has been used by several authors to compare synthetic
spectra to observed ones and to the broad-band colors: Dahn et al. (1995);
Leggett et al. (1996); Jones et al. (1996). Gizis (1997) was thus able to
constrain the metallicity scale of subdwarfs from $-$2 dex for esdM to $-$1 to
$-$1.5 for sdM, while the stars not classified as subdwarfs were spanning the
range $-$1 dex to 0. However, this pioneering model grid was far from being
optimized because of the lack of theoretical molecular opacities, especially
for the very important species CaH.
The NextGen model grid (Hauschildt et al., 1999) brought a number of
improvements and was in turn used by e.g. Woolf & Wallerstein (2005) who
demonstrated the $\alpha$ enrichment of halo metal-poor M stars. Several
papers using NextGen models to interpret spectra of cool M dwarfs and
subdwarfs were subsequently published (e.g.Leggett et al. 2000; Lépine et al.
2004) while Burgasser & Kirkpatrick (2006) compared the performances of
NextGen models and AmesCond models on the spectrum of an esdM star. Progress
in modeling led the PHOENIX-based computations to the BT-Settl series of
grids, the last generation of which proved highly successful in reproducing
the features of the observed spectra in detail. Comparisons of low mass stars
between a few high-resolution spectroscopic spectra and theoretical synthetic
ones show that the latest PHOENIX models have already reached a
milestone.(Rajpurohit et al., 2014, 2018a, 2018b).
Gaia Data Release 2 (Gaia Collaboration et al., 2018b) has provided astrometry
and photometry on more than one billion stars. As a result, a large number of
subdwarfs that were out of the reach of parallax studies now have accurate
available trigonometric parallaxes permitting distance determination and hence
their placement in the Hertzsprung-Russell diagram, while their 3D kinematics
can be studied thanks to high accuracy proper motions from Gaia and radial
velocity from spectroscopy.
M subdwarfs have been found both at high Galactic latitudes, as expected for
halo population members, and at much lower latitudes in fields located towards
the Galactic anti-center, suggesting that one could use them to study the mix
of various population components in the Solar neighborhood. It becomes thus
especially interesting to explore the links between their atmospheric
parameters, especially gravity and metallicity, and their kinematic
properties. For this, we need a reasonably accurate estimation of the
atmospheric parameters of a large sample of subdwarfs for which we have
accurate photometry and astrometry and the same data on a control sample of
ordinary M dwarfs. The present work, based only on low-resolution
spectroscopy, has its limitations and must be considered as a preliminary
attempt towards this goal.
The paper is organized as follows. In Section 2, we briefly summarize the
observational material that will be used in Sections 4 and 5. In Section 3, we
analyze the properties of the PHOENIX-based BT-Settl CIFIST model grid to
evaluate its performance by studying spectral indices of BT-Settl synthetic
spectra compared with those measured on real-world template spectra of M
dwarfs and subdwarfs. Then, we fit the template spectra with the synthetic
grid to get the atmospheric parameter ranges for each template spectra. In
section 4, we derive the atmospheric parameters for spectra of a sample of M
dwarf and subdwarf from the best-fit PHOENIX BT-Settl synthetic spectra. This
leads to the construction of a cleaned genuine subdwarf sample, a normal dwarf
sample and a third population whose properties point towards non-genuine
subdwarfs but rather metal-deficient dwarfs. In section 5, we explore the
links between the distribution of the atmospheric parameters of the stars
belonging to these three samples and their photometric and kinematic
properties as provided by the Gaia Data Release 2, using color-absolute
magnitude diagram, reduced proper motion diagram and 3D Galactic motion
distributions. Finally, conclusions and summary are provided in Section 6.
## 2 Observational data
The low-resolution spectra of M dwarfs and subdwarfs are from LAMOST DR7 and
the SDSS DR7, and the astrometric data is from Gaia DR2.
LAMOST is a reflecting Schmidt telescope located in Xinglong Station of
National Astronomical Observatory, China (40∘N, 105∘E) with a mean aperture of
4.3 meters and a field of view of 5∘. The observable sky area extends from
$-$10∘ to +90∘ declination. 4000 optical fibers positioned on the focal plane
yield a high spectrum acquisition rate per night. Until March 2020, DR7
published more than 10 million low-resolution spectra (R$\sim$1800) covering
3800$-$9000$\rm\AA$ (Luo et al., 2015) and 2.17 million medium resolution
spectra (R$\sim$7500) covering 3700$-$5900$\rm\AA$ and 5700$-$9000$\rm\AA$
(Liu et al., 2020).
The SDSS began regular survey operations in 2000, and has progressed through
several phases until now: SDSS-i@ (2000$-$2005), SDSS-ii@ (2005$-$2008), SDSS-
iii@ (2008$-$2014), and SDSS-iv@ (2014$-$2020). The SDSS DR7 contains over 1.6
million low-resolution spectra in total, including 930,000 galaxies, 120,000
quasars, and 460,000 stars. It was collected by a dedicated wide-field 2.5 m
telescope at Apache Point Observatory (Gunn et al., 2006). The telescope
employed a drift-scan technique, imaging the sky in $u$,$g$,$r$,$i$,$z$ wide
bands along five camera columns (Gunn et al., 1998). The fiber-fed
spectrographs acquired 640 spectra (R$\sim$2000) simultaneously, and the
spectral range is 3800$-$9200 $\rm\AA$.
The proper motions, trigonometric parallaxes, and photometry used in this work
come from Gaia Data Release 2 made available in April 2018 by European Space
Agency. Gaia DR2 contains the position and brightness information of more than
1.69 billion stars, the parallax and proper-motion measurements for 1.33
billion stars (Gaia Collaboration et al., 2018a; Lindegren et al., 2018; Luri
et al., 2018), the color information for 1.38 billion stars (Gaia
Collaboration et al., 2018a), the radial velocities for more than 7 million
stars (Katz et al., 2019), etc. The mean parallax error is up to 0.7 mas for
sources having G=20 mag, and the proper motion uncertainty is up to 0.2 and
1.2 mas yr-1 for G = 17 mag and G = 20 mag respectively.
## 3 PHOENIX BT-Settl Model Analysis
### 3.1 PHOENIX BT-Settl Model Grid
Classical theoretical model calculation codes include ATLAS code from Kurucz
(1973) and Castelli & Kurucz (2004), MARCS code from Gustafsson et al. (1975,
2008), PHOENIX code from Allard & Hauschildt (1995), and so on. In the
atmosphere of very low mass stars, various molecular absorption (each with
hundreds of thousands to millions of spectral lines) and the presence of
numerous condensates make accurate modeling extremely complicated; the
convection zone extends to the most outer photosphere layer, which means that
the evolution of the model strictly depends on the precise treatment of
surface boundary (Allard et al., 2013). It is worth celebrating that in recent
years, the PHOENIX code differs from previous methods by computing the
opacities during the model execution (or on-the-fly), considering non-local
thermodynamic equilibrium effects, etc., and combining with the latest solar
abundance (Asplund et al., 2009; Caffau et al., 2011). The atmospheric model,
especially for modeling very low mass stars such as M-type stars, brown
dwarfs, and even Jupiter-like planets, has made breakthrough progress.
The latest version PHOENIX BT-Settl models represent decisive progress
compared to the previous model grids (Allard et al., 2001) in different
aspects, including updated molecular line lists, better solar abundance
estimates and treatment of convection, though residual incompleteness of
opacities and a limitation of the mixing length theory formalism still exist.
Further studies on a complete, comprehensive and uniform grid of models are
addressing these shortcomings and will better reproduce observational
constraints (Allard, 2016).
The synthetic spectra generated by BT-Settl model atmospheres (Allard et al.,
2012, 2013; Baraffe et al., 2015) have been compared with observed spectra and
their consistency verified in a number of recent M dwarf/subdwarf studies
(e.g., Rajpurohit et al. 2014, 2016, 2018a, 2018b).
In this work, we use the version BT-Settl CIFIST2011 model atmosphere which is
valid across the entire parameter range (Allard et al., 2011, 2012). A part of
pre-calculated grid as list in 1 is used in the analysis and the following
parameter measurement process.
The models are calculated with effective temperatures from $T_{\text{eff}}$ =
400 to 8000 K in 100 K steps, surface grativity ranging from log $g$ = 0.0 to
6.0 in steps of 0.5 dex, and metallicity from [M/H]=$-$2.5 to +0.5 in steps of
0.5 dex (M stands for all elements heavier than H and He; assuming [X/H] =
[Fe/H] for most elements and so [Fe/H] denotes the overall metallicity). Alpha
element (O, Ne, Mg, Si, S, Ar, Ca, and Ti) enhancement is taken into account
as follows: for [M/H] = 0.0, no enhancement; for [M/H]=$-$0.5, a value of
[$\alpha$/M]=+0.2 has been used; for [M/H] $\leq-$1.0, [$\alpha$/M]=+0.4. Note
that in PHOENIX the [$\alpha$/M] is similar to [$\alpha$/Fe] since it is
computed as the change in dex of the abundance of alpha elements relative to
the non-alpha average, and the heaviest is Fe.
Table 1: Parameter Space of the Grid Used in this Work Variable | Range | Step size
---|---|---
T${}_{\text{eff}}$ | 2700 $-$ 4500 | 100 K
log g | 0 $-$ 6.0 | 0.5 dex
$[\text{M/H}]$ | $-$2.5 $-$ +0.5 | 0.5 dex
$[\alpha/\text{M}]$ | 0 $-$ +0.4 | 0.2 dex
### 3.2 Comparison of Models with Real-world Template Spectra in the
“canonical” diagram
In this section, we firstly compare M-type template spectra from Zhong et al.
(2015) with BT-Settl models. The templates, assembled from SDSS low-resolution
(R$\sim$2000) spectra and classified using Lépine et al. (2007, 2013)
calibration, are spanning the spectral subtypes K7.0 to M8.5 and covering
every half-subtype across the range.
The relevant pre-calculated synthetic spectra111https://phoenix.ens-
lyon.fr/Grids/BT-Settl/CIFIST2011/SPECTRA/ generated by PHOENIX code have been
convolved down to the spectral resolution of SDSS, using a series of Gaussian
kernels with parameters sigma and corresponding FWHM listed in Table 2. To
determine the sigma value of a convolution kernel, we fit gaussians profiles
to night sky lines (assuming they have a natural width much smaller than the
spectral psf). In the red range, clean lines of OH molecule are numerous, in
the blue the 5577$\rm\AA$ is also good, as well as some mercury lines, except
5461$\rm\AA$ which has fine structure. Na D lines of city lights are avoided
because they have a very large width (high pressure lamps).
Table 2: The gaussian kernels with the following sigmas and corresponding fwhm used for convolving synthetic spectra. Central $\rm\lambda(\rm\AA)$ | $\rm\sigma$ | Pixels of 0.5 $\rm\AA$ | FWHM
---|---|---|---
4200 | 1.05 | 2.10 | 4.935
5100 | 1.08 | 2.16 | 5.075
6000 | 1.25 | 2.50 | 5.875
6900 | 1.43 | 2.86 | 6.720
7800 | 1.40 | 2.80 | 6.580
8700 | 1.39 | 2.78 | 6.535
Figure 1: The “canonical” diagram classically used to identify subdwarfs from
dwarfs: spectral composite index CaH2+CaH3 versus TiO5. Models are plotted
with the following codes : continuous lines: log $g$ = 4.5 models, dashed
lines: log $g$ = 5.0 models, and dotted lines: log $g$ = 5.5 models. Black :
solar metallicity, blue: $-$1.0 dex, and magenta: $-$2.0 dex. The small
2-digit labels indicate $T_{\text{eff}}$ points (e.g. “32” means 3200 K).
Observed template spectra: black dots represent normal main-sequence dwarfs,
subdwarf templates colored in blue are for “sdM” subclass, green for “esdM”
and magenta for “usdM” subclass. The M0v@ to M7v@ “standard dwarf” spectral
type sequence is labeled on all graphs.
Figure 1 is the “canonical” [TiO5, CaH2+CaH3] diagram classically used by many
authors to identify and spectroscopically separate the subdwarfs from the
normal dwarfs. The spectral indices were defined initially by Reid et al.
(1995). The boundaries of the d/sd/esd/usd metallicity classification scheme
are also plotted. This diagram may represent more or less a metallicity-
gravity diagram, since TiO5 is a metallicity (and alpha) indicator while
CaH2+CaH3 is rather a gravity indicator, and both are naturally strongly
$T_{\text{eff}}$ dependent. The subdwarfs are supposed to order along
temperature sequences, while the fan-shape of their distribution reflects
their atmospheric metal deficiencies.
The normal dwarf template sequence is well-ordered and fits quite well with
the model family {log $g$ = 4.5, [Fe/H] = 0.0} track, although with the
coolest types (M6$\sim$M7) being a bit offset, as it would be expected from an
insufficient value of the model gravity.This is an important point and needs
to be further discussed. All late-type M dwarfs are expected to have higher
surface gravity than early M dwarfs due to the near 1-to-1 mass radius
relation for low-mass stars. Note that between metallicities of 0.0 and
$-$1.0, the displacement of the model track is quite small even at the lowest
$T_{\text{eff}}$, making the estimates of moderate metal deficiencies rather
difficult on this diagram.
The subdwarf templates also draw quite well-defined and well separated
sequences with increasing slopes from sd to usd, the latter one being almost
vertical at a constant TiO5 index of almost null absorption. The coolest types
of dwarfs and sd are somehow confused, this questions the validity of the
templates themselves which are built from spectra either in insufficient
number or too noisy to be really representative of the genuine subtype
definition.
The metallicity of the sd subclass does not comply with a $-$1.0 value, it
must be more deficient, or affected by a higher gravity, or both. The model
family {log $g$ = 5.0, [Fe/H] = $-$2.0} could provide a reasonable fit for
usdM0$\sim$3, but the track moves towards significant values of the TiO5
opacity for later subtypes, thus the real stars must be more deficient. The
model family {log $g$ = 5.5, [Fe/H] = $-$2.0} track yields more or less an
outer envelope for the subdwarfs, implying that most of the right-hand
distribution of their template representative points is driven more by large
metal deficiencies than by high gravity.
### 3.3 Fitting the dM and sdM Templates with the Model
Using template spectra rather than hundreds of individual spectra to compare
with the models enables to smooth accidental differences due to noise and
uncertainties in individual classification, and in principle gives access to
high signal-to-noise data. However, the final quality of a template depends on
the number of individual spectra co-added to build it. We shall hereafter use
the COMPLETE collection of templates of Zhong et al. (2015), i.e. the
interpolated 221 spectra in total spanning 12 subclasses (from dMr to usdMp)
and 18 subtypes (from M0 to M8.5).
The parameter grid of the synthetic spectra is shown in Table 1. Note that the
grid is not uniform: the value log $g$ = 6.0 only pertains for
$T_{\text{eff}}\geq$ 3000 K, and the [$\alpha$/M] ratio is dependent on the
metallicity.
The spectral fitting region extends from 6000$\rm\AA$ to 8800$\rm\AA$ with a
step size of 1$\rm\AA$ which covers most prominent spectral features used for
classification. Before the fitting process, the synthetic spectra were
smoothed at the resolution of the observed spectra.
In the fitting process, we set the spectral flux as a function of
$T_{\text{eff}}$, log $g$, and [M/H], and assume:
$F_{\text{o},\lambda}=F_{\text{m},\lambda}\times
P_{\text{n},\lambda}+\epsilon_{\lambda},$ (1)
where $F_{\text{o},\lambda}$ is the flux of the observed spectrum at
wavelength $\lambda$, $F_{\text{m},\lambda}$ is the flux of the synthetic
spectrum generated by PHOENIX model wavelength $\lambda$,
P${}_{\text{n},\lambda}$ is an n-order polynomial factor used to correct the
inaccuracies of the continuum of the observed spectrum (here we set n = 4
based on the optimal experimental results), $\epsilon_{\lambda}$ is the
difference between the synthetic spectrum.
The equation above can be solved by minimizing the loss function
L=$\sum_{\lambda=1}^{\text{N}}(F_{\text{m},\lambda}\times$
P${}_{\text{n},\lambda}-F_{\text{o},\lambda})^{2}$ using the least square
method to obtain the coefficient matrix solution of P${}_{\text{n},\lambda}$.
Then we obtain the best-fit synthetic spectrum by minimizing the $\chi^{2}$
distance,
$\chi^{2}\ =\
\frac{\sum_{i\text{=1}}^{N}[(F_{\text{o}_{i}}-F^{\prime}_{\text{m}_{i}})^{2}/\epsilon_{i}^{2}]}{N}$
(2)
where $F^{\prime}_{\text{m}_{i}}$ is the flux of the $i^{\text{th}}$ pixel of
the synthetic spectrum corrected by polynomial, $\epsilon_{i}$ is the flux
difference of the $i^{\text{th}}$ pixel, and $N$ is the number of pixels
involved in the calculation.
For each given template spectrum, we calculate the $\chi^{2}$ values when
matching all the synthetic spectra in the grid and then interpolate between
the parameters by synthesizing linear combinations of top-five best-fit
synthetic spectra, as the same way in Yee et al. (2017).
Figure 2: Template spectra of subtype dM0, sdM2, esdM4 and usdM6 with their
corresponding synthetic spectra and fitting residuals. Figure 3: The
residuals between template spectra of different spectral subtypes and their
best-fit synthetic spectra. Black, blue, purple, and red represent subclass
dM, sdM, esdM, and usdM respectively. The error bar is the standard deviation
of the difference between the flux of the template spectrum and the flux of
the synthetic spectrum. The number beside each node represents the number of
spectra used to construct the template spectrum. Figure 4: The distributions
of estimated atmospheric parameters of the templates spectra of different
subclasses. The colors represent the same as in Figure 3.
The fitting results of four template spectra of different spectral subtypes
and metallicity subclasses (dM0, sdM2, esdM4 and usdM6) with their
corresponding estimated parameters are shown in Figure 2 as examples. The
shape of the residual spectra indicate that the errors mainly come from the
region of 6000$-$7000Å, where there remain some incompleteness in line lists
and/or missing/incomplete opacities on some molecules used in the modeling
(CaOH, CN, CaH, etc.). Besides, the far red region of the observed spectra
could be contaminated by telluric absorption.
Figure 3 shows the residual levels of the fitting results of all the
templates. The residual level of the fitting results gradually increases from
$\sim$5% for early type M stars to 10% for the late types: both spectral
quality and the number of objects for template building decrease dramatically
for the latest types.
Finally, the distributions of the estimated results of log $g$ and [Fe/H] of
the templates above are shown in Figure 4. The dispersions are large for both
the two parameters, with the basic trend being a clear decrease of the average
metallicity from sd to usd accompanied by a more confuse increase of gravity.
Note that when using the complete set of templates (221 in total), we can meet
spectra of slightly atypical characteristics, as for instance illustrated in
the top panel of Figure 2 in which a dM0 is metal-poor and 100 K cooler than a
standard dM0 of solar metallicity.
## 4 Sample Construction and Atmospheric Parameter Estimates
To estimate the atmospheric parameters $T_{\text{eff}}$, log $g$ and [M/H] of
M subdwarfs from low-resolution spectra and investigate the trends of these
parameters with the kinematic properties of the stars, we need a sufficiently
large sample of subdwarfs spanning the broadest possible range of parameters.
In Paper i@, we had explored LAMOST DR4 in which 2,791 M subdwarf were
identified. Extending the search to LAMOST DR7, a total of 4,767 visually
selected candidates are retrieved including the previously obtained DR4 ones.
Combining this sample with SDSS objects published by Savcheva et al. (2014),
we submitted these spectra to a least-squares based fitting process with
PHOENIX synthetic spectra. The determination of atmospheric parameters is not
always possible with a reasonable accuracy because the error budget of the
fitting process depends on the signal-to-noise (S/N) of the spectra. Thus, the
final combined LAMOST + SDSS sample of stars with known atmospheric parameters
will be reduced with respect to the initial numbers.
### 4.1 Sample Construction and Individual Spectra Fitting
Because of magnitude limitation, most M dwarfs of LAMOST range in spectral
type before M3, and only a few cooler and fainter spectra of low S/N are
available. The data of SDSS DR7 covers a more extended parameter space, with
cooler and lower metallicity objects. The strategies of two surveys point to
different fields, i.e. SDSS at high Galactic latitude favoring halo and thick
disk objects detection, while LAMOST survey has in priority been targeted at
regions of the Galactic anti-center. Therefore, in this study we combine the
subdwarfs from SDSS DR7 (SEGUE-I/II) and LAMOST DR7 to construct a consistent
sample of M subdwarfs, more representative of the total population of these
stars.
First, it is useful to discuss three points on which the processing of the
LAMOST and SDSS spectra sets will differ: (1) selecting criteria, (2)
spectroscopic resolution, and (3) spectral data reduction.
(1) Selecting criteria.
Several works have been conducted to identify M subdwarfs from SDSS and LAMOST
low resolution spectral data sets (West et al., 2004; Lépine & Scholz, 2008;
Bochanski et al., 2013; Savcheva et al., 2014; Zhang et al., 2016; Bai et al.,
2016), Paper i@, and the two largest samples published were from Savcheva et
al. (2014) and Paper i@ respectively.
The selecting conditions differ for these two latter samples: the principal
separator used is the standard parameter $\zeta$ initially introduced by
Lépine et al. (2007) based on a combination of CaH2, CaH3 and TiO5 spectral
indices:
$\zeta_{\text{CaH}/\text{TiO}}=\frac{1-\text{TiO5}}{1-[\text{TiO5}]_{Z_{\odot}}}$
(3)
in which the [TiO5]${}_{Z_{\odot}}$ is a third order polynomial fit of the
TiO5 spectral index as a function of the CaH2+CaH3 index as:
$\begin{split}[\text{TiO5}]_{Z_{\odot}}\ &=\ a\\\ +&b\times(\text{CaH2}\ +\
\text{CaH3})\\\ +&c\times(\text{CaH2}\ +\ \text{CaH3})^{2}\\\
+&d\times(\text{CaH2}\ +\ \text{CaH3})^{3}\end{split}$ (4)
so that $\zeta$ = 1 for solar metallicity and $\zeta$ = 0 for the most metal-
poor objects where TiO5 absorption becomes undetectable. The coefficients $a$,
$b$, $c$, $d$ derived for SDSS objects by Lépine et al. (2007) led to the
calibration $\zeta_{\text{L07}}$ which was later revised by Lépine et al.
(2013) (hereafter $\zeta_{\text{L13}}$). The condition for a star to belong to
the subdwarf population was set as $\zeta_{\text{L07}}<$0.825 and
$\zeta_{\text{L13}}<$0.825
Since $\zeta$ was found to be dependent on the M dwarfs reference sample,
instrumental setup and various factors influential in the reduction process
(Lépine et al., 2013), we proposed a new calibration adapted to LAMOST spectra
in Paper i@ (hereafter $\zeta_{\text{Z19}}$), and we suggested the condition
for LAMOST subdwarfs being set as $\zeta_{\text{Z19}}<$0.75 with an additional
screening condition based on the CaH1 index.
(2) Spectroscopic resolution.
Along the wavelength of a spectrum from SDSS or LAMOST, the physical
resolution (depending on the grating$\times$focal length of
camera$\times$pixel size of the detector) varies in a complicated way. For
LAMOST, when spectra are extracted and rebinned in constant log-wavelength
steps by the 1D pipeline, this variation is +/- “dissolved” into another
variation. Additionally, though in theory, all spectra obtained from the same
instrument are supposed to have an identical resolving power, in practice for
a large-scale survey project, the resolution changes slightly at different
observation dates, due to imperfect instrument stability.
In the actual calculation, assuming a gaussian instrumental profile, the
resolution R of a spectrum at wavelength $\lambda$ can be derived from
$\lambda$/$\Delta\lambda$, where $\Delta\lambda$ is the full width at half
maximum (FWHM) of the spectral line broadening at that wavelength. The
Gaussian kernel $\sigma$ = FWHM/2.355 used to convolve the synthetic spectrum
is derived from $\sqrt{{\sigma_{\text{o}}}^{2}-{\sigma_{\text{m}}}^{2}}$,
where $\sigma_{\text{m}}$ corresponds to the synthetic spectrum, and
$\sigma_{\text{o}}$ corresponds to the observed spectrum. Since the resolution
of the synthetic spectra is extremely high, we set $\sigma_{\text{m}}$ as 0
here.
Prior to the fitting process, the observed spectra are shifted to rest-frame
based on the measured radial velocities. The fitting range was set to
6000$-$8800Å, in which the pixels marked by “bad” flags and the spectral
intervals heavily contaminated by telluric absorption (7210$-$7350Å,
7560$-$7720Å, and 8105$-$8240Å) were masked. We divided the region
6000$-$8800$\rm\AA$ into 7 bands evenly and used a dedicated convolution
kernel for each 400$\rm\AA$ band.
For SDSS and LAMOST spectra, we adopted different convolution kernels
calculated as follows. For an SDSS spectrum, we computed the mean value of the
$\sigma$ in each band by the “dispersion” value of each pixel stored in the
fits file. For a LAMOST spectrum, we calculated the mean value of the FWHM of
the arc lamp spectrum in each band, obtained on the same day for wavelength
calibration purpose.
(3) Data reduction.
In the fitting process described above, we first correct the continuum of an
observed spectrum using a fourth-order polynomial and then choose the best-fit
synthetic spectrum based on the chi-square minimum principle. We find that the
continuum slopes of some observed spectra of LAMOST set differ largely from
their corresponding corrected spectra. This deformation is mainly caused by
flux calibration problems (Du et al., 2016) and would affect the accuracy of
the spectral indices. To control this unwanted bias, we measure the spectral
indices of each polynomial-corrected observed spectrum, and calculate the
difference between them and the original indices measured on the uncorrected
spectrum. We then construct the distribution of each index differences, as
shown in Figure 5. Although the offset of a single index is generally small,
the error on the $\zeta$ parameter composed of multiple indices may be
considerably enlarged. Hence, targets with at least one spectral index
difference beyond 3$\sigma$ of the corresponding difference distribution are
removed from the final subdwarf sample.
Figure 5: Each subdiagram shows a spectral index error distribution for the
LAMOST subdwarfs. The errors are the differences between the values measured
from the original spectra minus the values measured from the continuum-
corrected spectra. They reflect the deformation level of the local
pseudocontinuum, due to imperfect flux calibration. The bottom right panel
shows the distribution of the derived parameter $\zeta$ values with a scatter
0.133 although the scatters of each of the indices (CaH2, CaH3 and TiO5) used
to calculate $\zeta$ are quite smaller.
We summarize the specific solution to build the combined LAMOST + SDSS
subdwarf sample in seven steps as follows:
(1) Select 4,767 visually inspected subdwarfs from LAMOST DR7.
(2) Recalculate the indices of the SDSS subdwarfs from Savcheva et al. (2014)
with the radial velocities recorded in the catalog (see Paper i@ for more
comments).
(3) Calculate the resolution of each spectrum using the profile of
corresponding arc-lamp spectrum (LAMOST) or the “dispersion” value stored in
the fits file (SDSS) and get the gaussian convolution kernels.
(4) Smooth the synthetic spectra to fit every observed spectrum and estimate
the atmospheric parameters. Figure 6 shows the fitting result of a LAMOST
extreme subdwarf in the upper panel and an SDSS subdwarf spectrum showing
H$\alpha$ emission in the lower panel. The spectral bands contaminated by
telluric absorptions are masked.
(5) For each index, measure its value on each observed spectrum and on its
corresponding continuum-corrected spectrum, and get their difference.
(6) Exclude the objects with one or more spectral index differences beyond
3$\sigma$ of the corresponding difference distribution.
(7) Combine the two samples above, and apply the selecting cuts by removing
the objects of $\zeta_{\text{L07}}\geq$0.825, $\zeta_{\text{L13}}\geq$0.825,
and $\zeta_{\text{Z19}}\geq$0.75.
Finally, a sample of 3,131 subdwarfs is assembled, 1,722 of which have
reliable kinematic properties. To provide better understanding, we summarize
in Table 3 the various subsamples used along this work.
Table 3: The various stellar samples used in the present work.
(1) | LAMOST M dwarfs (2) | LAMOST M subdwarfs initial sample | SDSS M subdwarfs initial sample (3) | LAMOST+SDSS combined subdwarf sample with atmospheric param. before CaOH/CaH1 cut | LAMOST+SDSS green-marked “subdwarfs” | LAMOST+SDSS “genuine” subdwarfs after CaOH/CaH1 cut (4) | LAMOST M giants (5)
---|---|---|---|---|---|---|---
N | $\sim$70,000 | 4,767 | 2,780 | 3,131 (LAMOST: 1,852) (SDSS: 1,279) | 855 | 2,276 | 7,200
N${}_{\text{Gaia}}$ | 35,382 | - | - | 1,722 (LAMOST: 1,160) (SDSS: 562) | 526 | 1,196 | -
* •
* •
Notes to Table 3:
* •
(1) The numbers corresponding to N${}_{\text{Gaia}}$ means the objects (N)
have available data from Gaia DR2 and can be used for kinematic analysis. The
flag is set as 1 if all of the following conditions are true:
a. phot_bp_mean_flux_over_error $>$ 10
b. phot_rp_mean_flux_over_error $>$ 10
c. phot_g_mean_flux_over_error $>$ 10
d. ResFlag = 1 $\&$ ModFlag = 1 (from Bailer-Jones et al. 2018)
e. astrometric_excess_noise_sig $\leq$ 2
f. ruwe $\leq$ 1.4
* •
(2) A visually identified sample from Zhang et al. (2019) for comparison.
* •
(3) Savcheva et al. (2014), the spectral indices are recalculated using the
radial velocities published in the catalog.
* •
(4) i.e. using the Equation 5 in the present paper.
* •
(5) A visually identified sample from Zhang et al. (2019) for comparison.
Figure 6: Upper panel: a LAMOST extreme subdwarf spectrum and its best-fit
synthetic spectrum. The spectral bands contaminated by telluric absorptions
are masked. Lower panel: an SDSS subdwarf spectrum showing H$\alpha$ emission
and its best-fit synthetic spectrum. Figure 7: The M subdwarf sample
constructed in this work, combining the low-resolution spectra from SDSS (blue
dots) and LAMOST (red dots) data sets. The M dwarfs used for comparison (black
dots) are visually confirmed from LAMOST data (Zhang et al., 2019). The bottom
right internal subdiagram shows the spectral type distributions of the two
sub-samples.
We must underline that we haven’t used the CaH1 index as a filter until now.
For comparison, we also estimate the atmospheric parameters of a M dwarf
sample which contains around 70,000 visually inspected spectra from LAMOST
(Zhang et al., 2019). Figure 7 shows the combined LAMOST+SDSS sample of
subdwarfs together with the comparison sample of LAMOST M dwarfs. The LAMOST
sample contains a large number of early-type and medium metal-poor stars,
while the SDSS sample covers a broader range of spectral types and metal
abundances.
### 4.2 Results of Parameter Estimation
#### 4.2.1 Signal-to-Noise impact on quality
Statistically, the fit residuals between the observed spectra and the
synthetic spectra vary with signal-to noise ratio (S/N). Figure 8 shows the
distribution of fitting residuals for subdwarfs from the two surveys. The left
panel shows the LAMOST subdwarfs mainly covering M0$-$M3 while the SDSS
subdwarfs on the right panel cover M0 to M8. The residual levels stabilize
below 10% after the S/N becomes larger than $\sim$30\. The later-type
subdwarfs from both samples show higher residual levels because of an average
lower spectrum quality.
Figure 8: The spectral fitting residual of subdwarfs as a function of the
signal-to-noise in $i$ band. Left panel: LAMOST subdwarfs, right panel: SDSS
subdwarfs.
#### 4.2.2 Comparison with literature results
In order to estimate the accuracy of the derived parameters, we cross-match
the entire LAMOST DR7 M-type stars with objects having known parameters from
literature. As a result, we find a total of 269 observations for 167 stars
studied with high or medium spectral resolution in 9 papers, and the
comparison is shown in Figure 9. Although a systematic bias in the effective
temperature appears from around 3700K to the cool end, it generally follows a
1-to-1 trend with the literature values. The metallicity also seems to follow
a 1-to-1 trend, while the gravity values show a larger range than the
literature ones. Note that we have masked several telluric absorption bands
where gravity-sensative K and Na lines are located in the fitting process, the
accuracy of our estimated gravity values is therefore affected somehow. The
distributions of the differences (measurements from this work $-$ literature
values) for the three parameters are shown in Figure 9. The bias and scatter
are 165 K and 170 K for $T_{\text{eff}}$, 0.03 dex and 0.32 dex for log $g$,
and $-$0.17 dex and 0.41 dex for [Fe/H], respectively.
Figure 9: The upper panels show the comparison of atmospheric parameters of
167 M stars estimated from 269 LAMOST spectra (including multi-observations)
with the ones compiled from literature. Parameter values from different
literature are shown in different colors as listed in legend. Error estimates
provided are shown as corresponding error bars. The uncertainties for the
values from LAMOST spectra are the upper values of internal errors determined
from signal-to-ratio and $\sqrt{\chi^{2}}$. The bottom panels show the
distribution of differences along with the corresponding mean value and
standard deviation.
## 5 Sample Properties Analysis
In this section, we analyze the properties of the subdwarf sample by comparing
them with the dwarf sample on spectral index diagrams and Gaia DR2 H-R
diagram, and provide a preliminary account of their kinematic properties from
the reduced proper motion diagram and in the 3D galactic motion system.
In Paper i@, we suggested a screening condition associated with the CaH1 index
be combined with the classical $\zeta$ condition to identify a subdwarf via
its low-resolution spectrum. To show the importance of gravity and gravity-
dependent index—CaH1 for identifying a “genuine” subdwarf, we divide the
subdwarf sample selected above into two groups by the separator curve
(equation 9 in Paper i@) on the [CaOH, CaH1] index diagram:
CaH1 $\displaystyle<0.4562\times\text{CaOH}^{3}-0.1977\times\text{CaOH}^{2}$
(5) $\displaystyle-0.01899\times\text{CaOH}-0.7631$
The objects that do not meet Equation (5) are marked in green in the following
figures.
For further analysis, we retrieve proper motions, and photometric magnitudes
in $G$, $BP$, and $RP$ bands of these stars from Gaia Data Release 2, and
estimated distance from Bailer-Jones et al. (2018). To ensure reliability, we
set a series of filter conditions as listed in the notes to Table 3, and
finally 1,722 objects among the 3,131 selected in Section 4 meet the
conditions, 526 of which are green-marked “subdwarfs”.
Figure 10: The gravity and metallicity distributions of the green-marked
“subdwarfs” vs. the other sdMs. The two upper panels show the distribution of
spectral indices of four groups of subdwarfs: green-marked “subdwarfs” (the
subdwarfs that don’t meet Equation 5), sdMs, esdMs, and usdMs (the remaining
subdwarfs classified into to categories according to the $\zeta$ values). The
sdMs, esdMs and usdMs are colored in blue, purple and magenta respectively.
The bottom left and middle panels show the comparison of green-marked
“subdwarfs” with the sdMs. The bottom right panel compares the total proper-
motion distribution of the green-marked “subdwarfs” with that of the other
sdMs.
### 5.1 Spectral Index Diagram
As shown in Figure 10, the upper left panel shows the green marked
“subdwarfs”. According to the $\zeta$ values, we also divide the remaining
subdwarfs into sdM/esdM/usdM subclasses on the [TiO5, CaH2+CaH3] diagram, as
can be seen in the upper right panel, and the objects in the three subclasses
are plotted as blue, purple and magenta points respectively. Most of the green
marked “subdwarfs” appear located in the “sdM” region, which is colored in
blue. Therefore, we compare the gravity and metallicity of the two groups and
display those distributions in the bottom left and middle panels. As shown in
the bottom middle panel, the green objects share similar metal abundance
distribution with the blue labeled ones, however, their gravities are
systematically lower than the latter ones by some 0.2 dex.
With the available proper-motions of these targets from Gaia DR2, we also
compare the total proper motions of the two groups in bottom right panel of
Figure 10. The proper-motions of green marked “subdwarfs” are overall smaller
than those of the sdMs. Generally speaking, large proper motion is a classical
property of subdwarfs, though some objects may have relatively small proper
motion due to the 3D motion direction. Proper motions of esdMs and usdMs are
larger than sdMs (not shown in the figure), which means that the green ones do
behave more like dwarfs than subdwarfs.
Figure 11: M-type stars with different gravity shown on the spectral index
diagram CaH1 vs. CaOH. Color coding is the same as in the upper left panel of
Figure 10. To illustrate the sensitivity to gravity of this diagram, an M
giant sample (7,200 objects visually identified from Zhang et al. 2019 and
colored in yellow) and the M dwarf sample are added.
The gravity differences can be seen more clearly in Figure 11, in which we add
the LAMOST M dwarfs and a new sample of M giants onto the [CaOH, CaH1] index
diagram. The giants were also spectroscopically identified by visual
inspection (Zhang et al., 2019). Giants, and dwarfs subdwarfs are classified
as luminosity class iii@, v@ respectively and Jao et al. (2008) have proposed
that subdwarfs be classified as vi@, this sequence tracing the gravity
variation trend. Therefore, Figure 11 clearly illustrates the gravity
segregation between the luminosity classes of giants/dwarfs/subdwarfs thanks
to the sensitivity of CaH1 index to log $g$ especially when compared to CaOH
index.
### 5.2 H-R Diagram
Using estimated distance by Bailer-Jones et al. (2018), we derive the absolute
magnitudes M${}_{\text{G}}$, M${}_{\text{BP}}$ and M${}_{\text{RP}}$ of the
dwarfs and subdwarfs. Since most of the M-type stars we analyze here are
located within 1 kpc, we choose to ignore the effect of extinction in the
calculation. Figure 12 shows the stars on Gaia color-absolute magnitude
diagram [BP$-$RP, M${}_{\text{RP}}$]. On this diagram, which is an
observational Hertzsprung-Russel diagram, the gray dots represent the dwarf
sample, the green dots represent the “subdwarfs” that do not meet Equation
(5), and the red dots represent the remaining subdwarfs (all subclasses
together). The green-marked “subdwarfs” fall almost completely in the dwarf
region, and thus once again verify the conclusion that they behave more like
dwarfs than subdwarfs.
Figure 12: All the subdwarfs and dwarfs on the Gaia color-magnitude diagram.
The absolute magnitudes in RP band are derived from the Gaia DR2 apparent
magnitudes and estimated distances from Bailer-Jones et al. (2018). Most of
the green-marked “subdwarfs” fall into the main sequence region populated by
the M dwarfs plotted as black dots, and the remaining subdwarfs plotted as red
dots lie in part below the main sequence with a large dispersion, but some are
also distributed across the main sequence region. Figure 13: After the
removal of the green-marked “subdwarfs”, the remaining samples are shown on
the Gaia color-absolute magnitude diagram again. On the left panel, subdwarf
are color-coded for different metallicities, while the same ones on the right
panel are color-coded for different gravities. Note that the subdwarfs with
log $g>$ 5.5 or log $g<$ 4.5 (38 objects in total) are not plotted on the
right panel to make the color bar more tight.
Besides, we find that a number of Equation (5) selected subdwarfs (red dots)
also overlap with dwarfs on this diagram. This point will be developed below.
Figure 13 shows the metal abundance and gravity variation of these stars on
the Gaia color-magnitude diagram. The subdwarfs falling in the dwarf region
are the ones with the highest metal abundances or the smallest surface
gravities. In other words, stars that are away from the main sequence have
both large gravity and low metal abundance, and they should be the “genuine”
subdwarfs.
### 5.3 Kinematics
#### 5.3.1 Reduced Proper Motion diagram
Aside from the H-R diagram, another useful tool for effectively separating
different stellar populations is the reduced proper motion (RPM; Jones 1972)
diagram (e.g. Salim & Gould 2002; Subasavage et al. 2005; Lépine et al. 2007;
Faherty et al. 2009), which uses photometric information (color) and proper
motions of the stars. Even without knowledge of the distance to the stars, the
RPM diagram can be used to constrain kinematic properties of large stellar
samples, because a given stellar population whose members are orbiting in the
Galactic potential has a characteristic space velocity distribution reflected
in a combination of luminosity and transverse motion. The definition of the
reduced proper motion $H$ is
$H=m+5log\ \mu+5=M+5log\ T$ (6)
where $m$ is the apparent magnitude of a star in a given photometric band,
$\mu$ is the annual total proper motion (in arcsec yr-1), $M$ is the absolute
magnitude, and $T$ is the tangential velocity (in a.u. yr-1).
Since the dwarfs belong to the thin disk population and the subdwarfs are
claimed to belong to the thick disk and halo populations, the comparison
between blue to red or optical to near-infrared colors and the reduced proper
motions can in principle relatively clearly distinguish the Population i@
dwarfs and Population ii@ subdwarfs : (1) given an absolute magnitude,
subdwarfs are bluer in color; (2) subdwarfs being members of a population
spanning a spatial volume more extended out of the Galactic plane with more
eccentric orbits, they have higher average tangential velocities (Salim &
Gould, 2002; Lépine et al., 2007). The RPM diagram combines these two features
so that the subdwarfs and main sequence stars fall into two separate regions.
This method has long been used to separate high-velocity subdwarfs from dwarfs
(Jones, 1972; Yong & Lambert, 2003a, b).
Figure 14: After the removal of the green-marked “subdwarfs”, the remaining
samples are shown on the reduced proper motion diagram. The color coding is
the same as in Figure 12. The fiducial separation line between Cloud A and
Cloud B objects is drawn (see text).
Figure 14 shows the RPM diagram for our stars, using Gaia DR2 color and proper
motions. A clear similarity with the information contained in the H-R diagram
of Figure 14 can be noted. Although the selection biases of the combined
LAMOST+SDSS subdwarf sample are poorly known (mix of pointing directions,
different limiting depth of surveys, velocity ellipsoid sampling, density
distribution sampling, spatial inhomogeneities, etc.), basic trends can be
outlined. First, the subdwarfs falling below the region occupied by the main-
sequence dwarfs have low metal abundances and higher gravities. Second,
remaining subdwarfs appear clearly divided in two clouds (noted A and B in the
following) with a quite clean separation: Cloud A is not distinct from the
lower part of the main-sequence dwarf cloud, while Cloud B is centered around
a half magnitude below the center of Cloud A. The fact that Cloud A and B are
especially distinct in the RPM diagram imply that these two clouds sample two
distinct populations whose kinematic behaviors are different.
#### 5.3.2 Galactic velocity components
Using the usual textbook transformation formula, we compute the three-
dimensional space velocity components $U$, $V$, $W$ in the right-handed
Galactic frame: the $U$ and $V$ are contained in the Galactic plane, $U$
points toward the Galactic center, while $V$ is the velocity component in the
direction of the solar motion. The $W$ component is perpendicular to the
Galactic plane and points towards the North Galactic pole. $U$, $V$, $W$ for
our targets are determined using the radial velocities, coordinates, proper
motions and distances derived from the parallaxes. The components are reduced
to the Local Standard of Rest (LSR; Fuchs et al. 2009) by correcting from the
solar motion assumed to be ($U_{\odot}$, $V_{\odot}$, $W_{\odot}$)=[11.1,
12.24, 7.25] km s-1 (Schönrich et al., 2010).
Figure 15: Velocity component distribution of dwarfs and subdwarfs with
different gravities in the Galactic 3D motion system ($U$, $V$, $W$). From
left to right, the abscissa represents the velocity component of $U$, $V$, and
$W$ respectively. The ordinate represents the distance from Galactic plane.
Except for the green-marked “subdwarfs”, the remaining subdwarfs are divided
into groups with different gravities and distinguished by different colors.
Each dwarf bin contains 7,000 stars and each subdwarf bin contains 100 stars.
The node and the error bar show the mean value and the standard deviation of
each bin respectively. Figure 16: Velocity component distribution of dwarfs
and subdwarfs with different metallicities in the Galactic 3D motion system
($U$, $V$, $W$). Except for the green-marked “subdwarfs”, the remaining
subdwarfs are divided into groups with different metallicities and
distinguished by different colors. The meaning of the axes and the number of
stars contained in each bin are the same as in Figure 15.
In Figures 15 and 16, we divide our remaining subdwarfs into different gravity
and metallicity groups and compare them with the green-marked “subdwarfs” and
dwarfs. Each dwarf bin contains 7,000 objects and each subdwarf bin contains
100 objects. As expected, the green-marked “subdwarfs” behave mostly like
field dwarfs with only a slightly larger dispersion in all the velocity
components. In addition, the yellow bins in Figure 15 representing the
subdwarf group with the lowest gravity and the light blue bins in Figure 16
representing the subdwarf group with the highest metallicity have nearly the
same behaviors in $V$ as the dwarfs and the green-marked “subdwarfs”.
To further explore the kinematic behavior of the full subsamples, we draw
several histograms in Figure 17, in which the dwarfs, green-marked
“subdwarfs”, Cloud A and Cloud B subdwarfs are plotted separately. The mean
values and standard deviations are also shown on each panel, except for
$|$z$|$, the absolute value of the distance to the Galactic plane, for which
the quartiles are computed. The separation between Cloud A and Cloud B objects
is defined by the fiducial line drawn on Figure 14. Several points emerge:
Figure 17: Histograms of some characteristics for the various sub-populations
of dwarfs and subdwarfs for which Gaia astrometry is available. From left to
right, log $g$, [Fe/H], $U$, $V$, $W$ space velocity components and $|$Z$|$.
From top to bottom: (a) all spectroscopically identified M dwarfs, (b) green-
marked “subdwarfs”, (c) Cloud A subdwarfs (see Figure 14 for definition), (d)
Cloud B subdwarfs, (e) Extended Cloud A (see text for definition), (f) Disk
outliers (spectroscopically identified M dwarfs whose total space velocity
w.r.t. the LSR is larger than that of 99 % of the population, see Figure 19
for their location on Toomre diagram). The $\mu$ and $\sigma$ on each panel
are indicated for the mean value and standard deviation respectively. For the
$|$Z$|$ distributions, the quartiles are indicated.
(a) The gravities increase systematically from dwarfs (mean close to 5.0, in
agreement with the results from Section 3) to Cloud B subdwarfs (mean around
5.2).
(b) Metallicity decreases systematically from dwarfs (mean around $-$0.4 dex)
to Cloud B subdwarfs. The metallicity of the dwarfs, quite surprisingly,
appears systematically poorer than solar, with a distribution tail down to
$-$1\. dex. Although the error assessment and control is difficult in the
spectra fitting process, we recall that the comparison in Subsection 4.2 with
the sample of 167 stars having a metallicity measurement in the literature did
not reveal any major systematic deviation.
Figure 18: The bimodal distribution of metallicity of the halo candidates,
which are selected from Cloud B subdwarfs with $v_{tot}\geq$220 km s-1. The
mean values and standard deviations of each Gaussian fit are also shown on the
diagram.
(c) As shown in Figure 18, the halo candidates in Cloud B subdwarfs exhibit an
inconspicuous bimodal distribution of metallicity, with two sub-populations,
one around [Fe/H] = $-$1 dex and the other around $-1.5$ dex.
(d) The mean values and dispersions of the $U$, $V$, and $W$ components agree
quite well with the dwarf sample belonging to the thin disk, with a velocity
lag of $V$ = $-$10 km s-1. The green-marked and Cloud A objects have more
velocity dispersion suggesting possible membership of the thick disk, but
their mean $V$ value significantly deviates from the usually adopted $-$60 km
s-1. Cloud B objects are obviously dominated by halo members, with a mean $V$
largely retrograde and considerable dispersion.
(e) The quartiles of the $|$z$|$ distributions show that the dwarf sample is
the one which is the most concentrated around the Galactic plane, while the
Cloud B contains the most diffuse population out of this plane. These
histograms also show that the LAMOST+SDSS surveys sample the M dwarf
population inside scale height of the thin disk above the Galactic plane (600
pc) (Jurić et al., 2008), while the subdwarf one is sampled only up to less
than 1 kpc (one scale height of the thick disk or less) (Gilmore & Reid, 1983;
Veltz et al., 2008; Carollo et al., 2010).
(f) The striking similarity of all the indicators (except for a marginal
difference in log $g$ ) between the green-marked objects and the Cloud A ones
imply that they are members of the same stellar population as defined
kinematically, only differing by the CaH1 and CaOH indices. In the following,
these two groups will be considered as a unique subsample which will be
designated as Extended Cloud A.
#### 5.3.3 The Toomre diagram
Trying to push forward the analysis, we have plotted in Figure 19 the Toomre
diagram for all our stars.
The dwarf cloud appears slightly asymetric, with an extension towards negative
$V$ velocity component. This is not unexpected since 2 or 3 - component disk
(thin, thick and intermediate) model populations (Schönrich & Binney 2009 and
references herein) predict such an asymmetry. The Extended Cloud A group seems
to be centered more or less coincident with the dwarf cloud, while Cloud B
objects are scattered across a large area of the diagram.
1. 1.
Halo objects.
To select halo objects, a stringent condition on their total velocity (Bonaca
et al., 2017) which would avoid pollution by possibly numerous thick disk
stragglers (Smith et al., 2009), is:
$|V-V_{\text{LSR}}|\geq 220\ \text{km s}^{-1}$ (7)
which translates, which the adopted velocity components of the LSR, ($U$ = 0,
$V$ = 220, $W$ = 0 km s-1) into:
$v_{\text{tot}}=\sqrt{U^{2}+V^{2}+W^{2}}\geq 220\ \text{km s}^{-1}$ (8)
where $v_{\text{tot}}$ is the total space velocity with respect to the LSR.
Figure 19: Left panel: Toomre diagram for all stars in the present study.
Black points: all spectroscopically identified M dwarfs, green circles: the
green-marked “subdwarfs”, blue circles: Cloud A subdwarfs, red circles: Cloud
B subdwarfs. The black dashed curve encloses 99 % of the M dwarf population:
all black points outside this curve are considered as representative of
“outliers”. The dashed red circle is our adopted limit $v_{\text{tot}}$ = 220
km s-1 for halo population assignment. The dashed magenta vertical line
separates retrograde orbits on its left from prograde orbits on its right.
Right panel: the M dwarfs with thin disk kinematics have been removed. The
green-marked “subdwarfs” and Cloud A subdwarfs are merged into a single group:
Extended Cloud A. The M dwarfs outliers have been separated based on their
total space velocity $v_{\text{tot}}$ w.r.t. the LSR. The Cloud B are color-
coded in four categories based on their metallicity and $v_{\text{tot}}$.
This selection provides a kinematic sample of the local halo haunting the
Solar Neighbourhood, which contains essentially stars from Cloud B, (220
objects), with a handful from Extended Cloud A, (9 objects) accompanied by
targets considered as outliers from the dwarf cloud (71 objects)(see below).
Its kinematic characteristics are in fair agreement with those of already
identified halo population stars selected across widely more extended volumes
and using more luminous targets (Smith et al., 2009; Bond et al., 2010;
Bochanski et al., 2013): $<$$U$$>$ = 5 km s-1 and $\sigma_{U}$ = 173 km s-1,
$<$$V$$>$ = $-$248 km s-1 and $\sigma_{V}$ = 77 km s-1, and $<$$W$$>$ = 6 km
s-1 and $\sigma_{W}$ = 89 km s-1. With the adopted value of the LSR velocity,
the fiducial line at $V$ = $-$220 km s-1 separates the stars which have
retrograde orbits from those who have prograde ones. A striking fact is the
quite high fraction of prograde objects (258 stars or 40%) observed in the
present local halo sample, thus the strongly negative value of $<$$V$$>$
quoted above does not account completely for a complex orbital distribution.
The metallicity of this halo sample, in spite of the kinematical cut applied,
remains bimodal. Previous studies, mostly based on F-G-K stars in which
metallicity is derived from atomic lines have already found a substantial
metal-rich component in the halo population, (and the metallicity distribution
of globular clusters is one historic and well-documented example (see e.g.
Zinn 1985; Harris 1996) but the metal-rich peak is generally found around
[Fe/H] = $-$0.6 dex: Bonaca et al. (2017). The prograde objects are also
reputedly more metal-rich than the retrograde ones, implying that they could
belong to different subpopulations, the metal-rich one containing inner halo
objects while the metal-poor one is made of outer halo objects, these two
subpopulations having different formation histories and different space
distributions. In the present local halo sample, we do not find such a
segregation, the number of prograde metal-poor objects remaining important
while many higher metallicity stars are on obviously retrograde orbits, as
shown in the right panel of Figure 19.
2. 2.
Thin disk dwarfs:
Since some spectroscopically identified dwarfs are found to kinematically
belong to the local halo, we could expect some continuity in the dwarf high
velocity distribution tail, and therefore a number of them would also be
kinematically mixed with the Extended Cloud A population and need to be
disentangled.
To select thin disk dwarfs, a condition of type $v_{\text{tot}}$ = constant is
not completely satisfactory because of the asymmetry of the dwarf cloud in the
Toomre diagram. A correct way to proceed would be to compute for each star its
probability of membership to one of the three populations - halo, thick disk
and thin disk as proposed e.g. by Bensby et al. (2003) but we prefer to
postpone such a calculation because the selection effects in the present
sample are not correctly assessed. We choose a simplified and empirical
procedure as follows: we assume that the distribution of the dwarf stars in
the Toomre diagram is consistent with an incomplete ellipse, and that the
isodensity contours of this ellipse trace constant probability level of a star
to belong to its parent population. We select the isodensity contour
containing 99% of the dwarf cloud and fit an ellipse to this contour. All
stars found outside are considered as “outliers” from the thin disk dwarf
population.
The kinematically “purified” dwarf sample shows typical velocity component
values of the thin disk population (Edvardsson et al., 1993; Reddy et al.,
2003): $<$$U$$>$ = 0 km s-1 and $\sigma_{U}$ = 44 km s-1, $<$$V$$>$ = $-$13 km
s-1 and $\sigma_{V}$ = 30 km s-1, and $<$$W$$>$ = 0 km s-1 and $\sigma_{W}$ =
26 km s-1, and its stars have exclusively prograde orbits as expected. The
metallicity, as outlined above, is in average slightly deficient, the
histogram showing three peaks, one solar or slightly super-solar, the
principal one around $-$0.4 dex and the third one around $-$1 dex. Since the
selection effects are not under control, we doubt that this three-component
would be significant.
3. 3.
Thick disk population and its elusive limits
The Extended Cloud A occupies a large area, somewhat fan-shaped vertically, in
the Toomre diagram, broadly covering the dwarf cloud and merging beyond it
until the halo region, with practically all orbits prograde. The shape of the
$|$Z$|$ histogram, with its peak displaced from that of the thin disk dwarfs
and its quartiles systematically larger shows that this population is clearly
thicker in $z$. The average metal content is poorer than that of the thin disk
dwarfs at [Fe/H] = $-$0.74 dex with a [Fe/H] distribution having a long metal-
poor tail extending to $-$2.5 dex. We suspect that the Extended Cloud A
population is an intricate mix of objects really belonging to a thick disk
component with pollution by an important contribution of thin disk “mild”
subdwarfs very difficult to disentangle, with unknown selection effects
playing in background. A possible way to separate these contributions is the
detailed computation of individual orbits and study of the $z_{\text{max}}$
and eccentricities, but such a work is far beyond the scope of the present
study.
In the Toomre diagram, the Extended Cloud A population is mixed with a
fraction of the Cloud B population, corresponding to stars that preferentially
belong to the high metallicity peak of the bimodal [Fe/H] distribution of
Cloud B in Figure 17. These stars likely belong to the halo for a part,
because our halo selection kinematic cut is very stringent, and to the thick
disk for another part, but their respective assessment to the two components
would probably also need a detailed orbital calculation.
The [Fe/H] distribution of the outliers from the dwarf cloud show that they
span the same broad metallicity range as the principal dwarf population,
peaking at $-$0.6 dex, only slightly more metal-poor than the ordinary disk
dwarfs. Their distribution in $|$Z$|$ is clearly more extended, making them
probable members of the thick disk. Their velocity dispersions $\sigma_{U}$ =
119 km s-1, $\sigma_{V}$ = 90 km s-1, $\sigma_{W}$ = 75 km s-1 suggest that
they contain a majority of objects belonging to the thick disk with some
pollution by an inner halo fraction.
To conclude, the combined subdwarf sample extracted from LAMOST DR7 and SDSS
DR7 contains a mix of objects belonging to various components: clear halo
members with an implicit bimodal metallicity and considerable orbital mixing
between retrograde and prograde cases, and an important thick disk fraction
merging both into halo and thin disk. The spectroscopically selected M dwarfs
themselves, whose quasi-totality belong to the standard thin disk, contain a
small fraction (less than 1% of the total) of outliers belonging both to the
thick disk and the halo. It is quite unclear if the present situation is
dependent or not on the selection effects, especially the favoured directions
of investigation towards the Galactic anticenter in the LAMOST survey.
## 6 Summary and Conclusion
In the most widely used classification system, M-type main sequence stars are
classified into dM/sdM/esdM/usdM subclasses corresponding to different metal
abundances according to the CaH2, CaH3 and TiO5 molecular bands in the low-
resolution spectrum. The metal-poor subclasses are named as subdwarfs, which
have typical kinematic characteristics of thick disk and halo population.
However, in the present work, we suggest the “genuine” subdwarfs belonging to
population ii@ be classified as luminosity class vi@ on the H-R diagram. For
these stellar objects, having larger surface gravity is a necessary condition
as important as “being metal deficient”, and the three spectral indices in
classical classification system can not be fully competent for selecting these
subdwarfs. This conclusion is drawn by a series of experiments including the
comparison of synthetic spectra with the templates, measuring atmospheric
parameters for a subdwarf sample and combine with kinematics analysis, etc).
According to the spectral index CaOH-CaH1 relationship and the empirical
relationship on RPM diagram, we divide a spectroscopically identified
subdwarfs into multiple groups and analyze the multi-population properties in
this sample from their multi-dimensional parameters.
The sample including subdwarfs from both LAMOST and SDSS combined with a
control group of dwarfs selected from LAMOST, and estimate their atmospheric
parameters fitting with the PHOENIX grid. The technical details of the whole
fitting procedure of individual spectra is developed, including the use of
different convolution kernels to smooth the synthetic spectra and the
pseudocontinuum flux correction specifically implemented for the LAMOST data.
The residuals of the fitting procedure can be used to determine the precision
which depend on the S/N of the observed spectra. The accuracy of the process
is evaluated from 167 M-type stars with measured atmospheric parameters from
literature which have LAMOST spectra: and the bias and scatter of the
comparison of parameters are 165$\pm$170 K for $T_{\text{eff}}$, 0.03$\pm$0.32
dex for gravity, and -0.17$\pm$0.41 dex for metallicity. A systematic bias in
effective temperature exists for $T_{\text{eff}}$ lower than 3700 K while the
gravity and metallicity both appear in fair agreement. The effective
temperature scale produced by these detailed fits is reasonably consistent
with those published by Rajpurohit et al. (2014) for the subdwarfs and
Rajpurohit et al. (2018a) for the dwarfs. The following analysis are mainly
dependent on the latter two parameters.
For the sample of subdwarfs and dwarfs, apparent magnitudes, broad-band
colors, and proper-motions are obtained from cross-matching these two sets
with Gaia DR2, and the distances are from Bailer-Jones et al. (2018). The
objects are selected so as to have acceptable quality flags in photometry,
distance and accuracy. The spectroscopic “subdwarfs” do not entirely lie below
the main-sequence but in an extended region which overlaps that occupied by
the M dwarfs in the color-absolute magnitude diagram, while the objects that
lie away (and below) from the main sequence are the ones with both larger
gravity and lower metal abundance. They are the “genuine” subdwarfs, for which
the luminosity class vi@ ((Jao et al., 2008)) is justified. Alternatively,
some “subdwarfs” with gravity similar as that of the dwarfs or only slightly
higher also behave like dwarfs in many aspects although they are medium metal-
deficient through the definition using the CaH and TiO spectral indices, we
have been able to sort out a significant subsample of objects which, in spite
of average low metallicity, appear intricately mixed with the ordinary main-
sequence dwarfs and also sharing their location in a reduced proper motion
diagram. These “subdwarfs” should perhaps better be classified as “metal-poor
dwarfs”.
A provisional kinematic analysis of these samples show that the sdM population
lying below the main-sequence cloud is dominated by halo stars, in which an
implicit bimodal metallicity distribution appears, with a peak around $-$1.0
dex and a second peak around $-$1.5 dex. This group, which samples the
galactic halo only in a very local volume (typically less than 1 kpc radius),
contains a mix of objects of prograde and retrograde orbits, as revealed by
their location in the Toomre diagram, without clear trend with the
metallicity. The other subdwarfs, including those which we prefer to consider
as “metal-poor dwarfs” appear as a mixed population of thick disk and thin
disk members. If “lying below the main sequence” and “associated with halo
kinematics” (large negative $V$ component and large dispersions in $U$, $V$
and $W$) define intrinsic features of the genuine subdwarfs belonging to
Population ii@ stars, then both large gravity (average log $g$ = 5.2 to 5.4)
$and$ low metal abundance should be essential attributes of them. The fact
that the two properties must exist simultaneously needs to be underlined: we
have only found a handful of halo objects among the other subdwarfs, even
those which have a metallicity lower than $-$1.0 dex.
The ordinary dwarfs populate in their vast majority the Galactic thin disk, as
expected, with an average metallicity slightly lower than solar. A small
fraction (less than 1%) form a set of kinematic outliers, halo and thick disk
probable members. Finally, CaH1 index is a good indicator of gravity
experimentally, which behaves better in separating subdwarfs from dwarfs than
the composite index CaH2+CaH3.
Though the contamination by low ratio of multiplicity (12.5$\pm$1.9%: Ziegler
et al. 2015) and the parameter coupling effect would influence the accuracy of
parameters measurement, a preliminary indicative conclusion can still be drawn
from the analysis results above. Besides, accurate discrimination of subdwarfs
has to be revisited with more information such as high-resolution spectra and
multi-band data, and the estimates of parameters also need a more complete
model grid (an independent alpha enrichment such as the analysis in Hejazi et
al. 2020, more complete opacity and line lists, etc). The sub-population
membership assessment of the subdwarfs found in numbers in the large area
spectroscopic surveys, in turn needs careful selection effects control and
detailed orbital calculation.
ZS thanks Dr. Yan. H.L. for the thoughtful discussion about smoothing the
synthetic spectra, and Dr. Allard, F. for her helpful private communication
about PHOENIX models. This work is supported by National Key R&D Program of
China(No. 2019YFA0405502), National Science Foundation of China (No.
U1931209). Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber
Spectroscopic Telescope, LAMOST) is a National Major Scientific Project built
by the Chinese Academy of Sciences. Funding for the project has been provided
by the National Development and Reform Commission. LAMOST is operated and
managed by the National Astronomical Observatories, Chinese Academy of
Sciences.
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Table 4: Data description of the subdwarf catalog.
Column Name | Unit | Description
---|---|---
ID | - | SpecObjID for SDSS objects and obsid for LAMOST objects
specname | - | Name of spectrum
ra | degrees | Right Ascension (J2000)
dec | degrees | Declination (J2000)
l | degrees | Galactic longitude
b | degrees | Galactic latitude
rv | km s-1 | Heliocentric radial velocity
Teff | K | Estimated effective temperature
logg | dex | Estimated surface gravity
FeH | dex | Estimated overall metallicity
snri | - | Average spectrum signal to noise ratio in i band
spt | - | Spectral subtype
CaOH | - | Spectral index of CaOH band
CaH1 | - | Spectral index of CaH1 band
CaH2 | - | Spectral index of CaH2 band
CaH3 | - | Spectral index of CaH3 band
TiO5 | - | Spectral index of TiO5 band
ZetaL07 | - | The value of $\zeta$ parameter calibrated by Lépine et al. (2007)
ZetaD12 | - | The value of $\zeta$ parameter calibrated by Dhital et al. (2012)
ZetaL13 | - | The value of $\zeta$ parameter calibrated by Lépine et al. (2013)
ZetaZ19 | - | The value of $\zeta$ parameter calibrated by Zhang et al. (2019)
source | - | The source of spectrum (lamost/sdss)
parallax | mas | Gaia DR2 parallax
parallax_error | mas | Standard error of Gaia DR2 parallax
pmra | mas yr-1 | Gaia DR2 proper motion in right ascension direction
pmra_error | mas yr-1 | Standard error of Gaia DR2 proper motion in right ascension direction
pmdec | mas yr-1 | Gaia DR2 proper motion in declination direction
pmdec_error | mas yr-1 | Standard error of Gaia DR2 proper motion in declination direction
astrometric_gof_al | - | Goodness of fit statistic of model wrt along-scan observations
astrometric_excess_noise_sig | - | Significance of excess noise
phot_g_mean_flux_over_error | - | G-band mean flux divided by its error
phot_g_mean_mag | mag | Gaia DR2 G-band mean magnitude
phot_bp_mean_flux_over_error | - | Integrated BP mean flux divided by its error
phot_bp_mean_mag | mag | Gaia DR2 integrated BP mean magnitude
phot_rp_mean_flux_over_error | - | Integrated RP mean flux divided by its error
phot_rp_mean_mag | mag | Gaia DR2 integrated RP mean magnitude
bp_rp | mag | Gaia DR2 BP-RP colour.
ruwe | - | Renormalized Unit Weight Error associated to each Gaia source
rest | pc | Estimated distance from Bailer-Jones et al. (2018), based on Gaia DR2
ResFlag | - | The Result Flag from Bailer-Jones et al. (2018)
ModFlag | - | The Modality Flag from Bailer-Jones et al. (2018)
flag | - | Set as 1 for the final sample used in kinematic part, see table 3 for details
$V_{total}$ | km s-1 | Total spacial velocity
$Z$ | pc | Vertical distance away from Galactic plane
$U$ | km s-1 | Velocity component points towards the galactic center
$V$ | km s-1 | The tangential velocity component rotating around galactic center
$W$ | km s-1 | Velocity component points towards the north galactic pole
Note. — The complete catalog can be accessed online222http://paperdata.china-
vo.org/work/subdwarf-catalog.csv.
|
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# The rational hull of modules
gangyong lee Gangyong Lee, Department of Mathematics Education, Chungnam
National University Daejeon 34134, Republic of Korea e-mail:
<EMAIL_ADDRESS>
###### Abstract.
In this paper, we provide several new characterizations of the maximal right
ring of quotients of a ring by using the relatively dense property. As a ring
is embedded in its maximal right ring of quotients, we show that the
endomorphism ring of a module is embedded into that of the rational hull of
the module. In particular, we obtain new characterizations of rationally
complete modules. The equivalent condition for the rational hull of the direct
sum of modules to be the direct sum of the rational hulls of those modules
under certain assumption is presented. For a right $H$-module $M$ where $H$ is
a right ring of quotients of a ring $R$, we provide a sufficient condition to
be $\text{End}_{R}(M)=\text{End}_{H}(M)$. Also, we give a condition for the
maximal right ring of quotients of the endomorphism ring of a module to be the
endomorphism ring of the rational hull of a module.
###### 2020 Mathematics Subject Classification:
Primary 16D70; 16S50, Secondary 16D50
Key Words: rational hull, injective hull, maximal right ring of quotients
## 1\. Introduction
The theory of rings of quotients has its origin in the work of Ø. Ore [11] and
K. Asano [2] on the construction of the total ring of fractions, in the 1930’s
and 40’s. But the subject did not really develop until the end of the 1950’s,
when a number of important papers appeared (by R.E. Johnson [6], Y. Utumi
[15], A.W. Goldie [5], J. Lambek [8] et al). In particular, Johnson(1951),
Utumi(1956), and Findlay $\&$ Lambek(1958) have studied the maximal right ring
of quotients of a ring which is an extended ring of the base ring. For the
well-known example, the maximal right ring of quotients of integers is
rational numbers. It is the same as the injective hull of integers. For a
commutative ring, the classical right ring of quotients of a ring is to its
total quotient ring as the maximal right ring of quotients of a ring is to its
complete ring of quotients.
As we know, the study of the rational hull of a module is the same as that of
the maximal right ring of quotients in a different way. Also, like every
module has the injective hull, it is known that every module has the rational
hull in [4, Theorem 2.6]. Now, we introduce the definition of the rational
hull of a module and present its well-known results, briefly. Let $M$ be a
right $R$-module and $T=\text{End}_{R}(E(M))$. Put
$\widetilde{E}(M)=\\{x\in E(M)|\,\vartheta(M)=0~{}\text{with}~{}\vartheta\in
T~{}\Rightarrow\
\vartheta(x)=0\\}=\displaystyle\bigcap_{\begin{subarray}{c}M\subseteq\text{Ker}\vartheta\\\
\vartheta\in
T\end{subarray}}\text{Ker}\vartheta=\mathbf{r}_{E(M)}\left(\mathbf{l}_{T}(M)\right).$
Then $\widetilde{E}(M)$ is the unique maximal rational extension of $M$. We
call it the _rational hull_ of $M$. Also, it is known that
$\mathbf{r}_{E(M)}(J(T))\leq\mathbf{r}_{E(M)}\left(\mathbf{l}_{T}(M)\right)=\widetilde{E}(M)$
because $\mathbf{l}_{T}(M)\subseteq J(T)$ where $J(T)=\\{\alpha\in
T\,|\,\text{Ker}\alpha\leq^{\text{ess}}E(M)\\}$ is a Jacobson radical of a
ring $T$. Note that the maximal right ring of quotients of $R$ is
$Q(R)=\mathbf{r}_{E(R)}(\mathbf{l}_{H}(R))$ where $H=\text{End}_{R}(E(R))$
(see [8, Proposition 2]).
After the necessary background history, results, and notations in this
section, we provide several characterizations of the rational hull of a module
in Section 2 (see Theorem 2.3 and Corollary 2.10). In addition,
characterizations of rationally complete modules are presented. As a
corollary, we obtain several new characterizations of the maximal right ring
of quotients of a ring. In particular, we show that the endomorphism ring of a
module is embedded into that of the rational hull of the module as the
inherited property of its maximal right ring of quotients (see Theorem 2.15).
Our focus, in Section 3, is on the question of when is the rational hull of
the direct sum of modules the direct sum of the rational hulls of those
modules. For $M=\bigoplus_{k\in\Lambda}M_{k}$, we prove that
$\widetilde{E}(M)=\bigoplus_{k\in\Lambda}\widetilde{E}(M_{k})$ if and only if
$M_{i}$ is $M_{j}$-dense in $\widetilde{E}(M_{i})$ for all $i,j\in\Lambda$
when either $R$ is right noetherian or $|\Lambda|$ is finite (see Theorem
3.6). In the last section, we obtain a condition to be
$\text{End}_{R}(M)=\text{End}_{H}(M)$ where $H$ is a right ring of quotients
of a ring $R$ (Theorem 4.1). This condition is called the _relatively dense
property_ to a module. Also, we provide a sufficient condition for the maximal
right ring of quotients of the endomorphism ring of a module to be the
endomorphism ring of the rational hull of a module (see Theorem 4.5).
Throughout this paper, $R$ is a ring with unity and $M$ is a unital right
$R$-module. For a right $R$-module $M$, $S=\text{End}_{R}(M)$ denotes the
endomorphism ring of $M$; thus $M$ can be viewed as a left $S$\- right
$R$-bimodule. For $\varphi\in S$, $\text{Ker}\varphi$ and $\text{Im}\varphi$
stand for the kernel and the image of $\varphi$, respectively. The notations
$N\leq M$, $N\leq^{\text{ess}}M$, $N\leq^{\text{den}}M$ or $N\leq^{\oplus}M$
mean that $N$ is a submodule, an essential submodule, a dense submodule, or a
direct summand of $M$, respectively. By $E(M)$, $\widehat{M}$, and
$\widetilde{E}(M)$ we denote the injective hull, the quasi-injective hull, and
the rational hull of $M$, respectively, and $T=\text{End}_{R}(E(M))$. $Q(R)$
denotes the maximal right ring of quotients of $R$. The direct sum of
$\Lambda$ copies of $M$ is denoted by $M^{(\Lambda)}$ where $\Lambda$ is an
arbitrary index set. $\mathsf{CFM}_{\mathbb{N}}(F)$ denotes the
$\mathbb{N}\times\mathbb{N}$ column finite matrix ring over a field $F$. By
$\mathbb{Q}$, $\mathbb{Z}$, and $\mathbb{N}$ we denote the set of rational,
integer, and natural numbers, respectively. $\mathbb{Z}_{n}$ denotes the
$\mathbb{Z}$-module $\mathbb{Z}/n\mathbb{Z}$. For $x\in M$, $x^{-1}K=\\{r\in
R\,|\,xr\in K\\}\leq R_{R}$ with a right $R$-submodule $K$ of $M$. We also
denote $\mathbf{r}_{M}(I)=\\{m\in M\,|\,Im=0\\}$ for $I\leq S$ and
$\mathbf{l}_{S}(N)=\\{\varphi\in S\,|\,\varphi N=0\\}$ for $N\leq M$.
We give some properties of dense submodules. Recall that a submodule $N$ of
$M$ is said to be _dense_ in $M$ if for any $x,0\neq y\in M,$ there exists
$r\in R$ such that $xr\in N$ and $0\neq yr$.
###### Proposition 1.1 ([3, Proposition 1.3.6]).
Let $N\leq M$ be right $R$-modules. Then the following conditions are
equivalent:
1. (a)
$N$ is dense in $M$;
2. (b)
$\emph{Hom}_{R}(M/N,E(M))=0$;
3. (c)
for any submodule $P$ such that $N\leq P\leq M$, $\emph{Hom}_{R}(P/N,M)=0$.
###### Proposition 1.2 ([7, Proposition 8.7]).
Let $L,N$ be submodules of a right $R$-module $M$:
1. (i)
If $L\leq^{\emph{den}}M$ and $N\leq^{\emph{den}}M$ then $L\cap
N\leq^{\emph{den}}M$.
2. (ii)
Let $L\leq V\leq M$. Then $L\leq^{\emph{den}}M$ if and only if
$L\leq^{\emph{den}}V$ and $V\leq^{\emph{den}}M$.
###### Proposition 1.3 ([3, Proposition 1.3.7]).
Let $M$ be a right $R$-module and $M\leq V\leq{E}(M)$. Then
$M\leq^{\emph{den}}V$ if and only if $V\leq\widetilde{E}(M)$.
We remind of some important characterizations of the rational hull of a
module.
###### Proposition 1.4.
The following hold true for a right $R$-module $M$ and
$T=\emph{End}_{R}(E(M))$:
1. (i)
_([9, Exercises 5])_ $\widetilde{E}(M)=\\{x\in
E(M)|\,~{}\vartheta|_{M}=1_{M}~{}\emph{with}~{}\vartheta\in T\ ~{}\Rightarrow\
\vartheta(x)=x\\}$.
2. (ii)
_([7, Proposition 8.16])_ $\widetilde{E}(M)=\\{x\in E(M)\,|\,~{}\forall y\in
E(M)\\!\setminus\\!\\{0\\},~{}y\cdot x^{-1}M\neq 0\\}$.
## 2\. The rational hull of a module
As the injective hull of a module $M$ is the minimal injective module
including $M$, the next result shows that the rational hull of a module $M$ is
the minimal rationally complete module including $M$. Recall that a right
$R$-module $M$ is said to be _rationally complete_ if it has no proper
rational (or dense) extensions, or equivalently $\widetilde{E}(M)=M$. Thus,
the rational hull $\widetilde{E}(M)$ of a module $M$ is rationally complete.
###### Theorem 2.1.
The following conditions are equivalent for right $R$-modules $M$ and $F$:
1. (a)
$F$ is maximal dense over $M$;
2. (b)
$F$ is rationally complete, and is dense over $M$;
3. (c)
$F$ is minimal rationally complete, and is essential over $M$.
Note that a right $R$-module $F$ is exactly the rational hull of a module $M$
if $F$ satisfies any one of the above equivalent conditions.
###### Proof.
(a)$\Rightarrow$(b) From Proposition 1.3, it is easy to see that $F$ has no
proper dense extension. So, $F$ is a rationally complete module.
(b)$\Rightarrow$(c) Let $F^{\prime}$ be rationally complete such that $M\leq
F^{\prime}\leq F$. Since $M\leq^{\text{den}}F$, from Proposition 1.2(ii)
$M\leq^{\text{den}}F^{\prime}\leq^{\text{den}}F$. Thus, from Proposition 1.3
$F\leq^{\text{den}}\widetilde{E}(F^{\prime})=F^{\prime}$ because $F^{\prime}$
is rationally complete. Therefore $F=F^{\prime}$. (c)$\Rightarrow$(a) Let $F$
be minimal rationally complete over $M$. Since $F$ is essential over $M$,
$M\leq F\leq E(M)$. Since $M\leq^{\text{den}}\widetilde{E}(M)$,
$\text{Hom}_{R}(\widetilde{E}(M)/M,E(M))=0$. Also, since $E(F)=E(M)$,
$\text{Hom}_{R}(\widetilde{E}(M)/M,E(F))=0$. From [7, Theorem 8.24], an
inclusion map $\iota:M\rightarrow F$ extends to
$\rho:\widetilde{E}(M)\rightarrow F$ as $F$ is rationally complete (see also
Proposition 2.13). Note that $\rho$ is a monomorphism. Since
$\widetilde{E}(M)$ is rationally complete and $F$ is minimal,
$\widetilde{E}(M)=F$. ∎
The next example shows that the condition “essential over $M$” in Theorem
2.1(c) is not superfluous.
###### Example 2.2.
Let $M=\mathbb{Z}$ and $F=\mathbb{Z}_{(p)}\oplus\mathbb{Z}_{p}$ be right
$\mathbb{Z}$-modules where $\mathbb{Z}_{(p)}$ is the localization of
$\mathbb{Z}$ at the prime ideal $(p)$. It is easy to see that $M$ is not
essential in $F$, so $F$ is not a rational hull of $M$. In fact, $F$ is
minimal rationally complete over $M$: From [7, Example 8.21], $F$ is
rationally complete because $F$ is the rational hull of
$L=\mathbb{Z}\oplus\mathbb{Z}_{p}$. It is enough to show that $F$ is minimal
over $M$: Let $K$ be a rationally complete module such that $M\leq K\leq F$.
Hence $1=\text{u.dim}(M)\leq\text{u.dim}(K)\leq\text{u.dim}(F)=2$. Assume that
$\text{u.dim}(K)=1.$ Then $M\leq^{\text{ess}}K$, and hence $K$ is nonsingular
since $M$ is nonsingular. Thus $M\leq^{\text{den}}K$, which implies that
$K\cong\mathbb{Q}$ since $K$ is rationally complete and
$\widetilde{E}(M)=\mathbb{Q}$. It follows that $\mathbb{Q}$ can be embedded
into $F=\mathbb{Z}_{(p)}\oplus\mathbb{Z}_{p}$, a contradiction. Therefore,
$\text{u.dim}(K)=2.$ Then $K\leq^{\text{ess}}F$, and hence
$K\cap\mathbb{Z}_{p}\neq 0$. Thus $\mathbb{Z}_{p}\leq K$, which implies that
$L=\mathbb{Z}\oplus\mathbb{Z}_{p}\leq K$. Note that $L\leq^{\text{den}}F$
since $F=\widetilde{E}(L)$. Hence $K\leq^{\text{den}}F$, so that $K=F$ due to
the fact that $K$ is rationally complete.
We provide another characterization for the rational hull of a module using
the relatively dense property. A right ideal $I$ of a ring $R$ is called
_relatively dense to a right $R$-module_ $M$ (or $M$-_dense_) in $R$ if for
any $r\in R$ and $0\neq m\in M$, $m\cdot r^{-1}I\neq 0$. It is denoted by
$I\leq^{\text{den}}_{M}R$.
###### Theorem 2.3.
Let $M$ be a right $R$-module. Then $\widetilde{E}(M)=\\{x\in
E(M)\,|\,x^{-1}M\leq^{\emph{den}}_{M}R\\}$.
###### Proof.
Let $x\in\widetilde{E}(M)$ be arbitrary. Consider a right ideal $x^{-1}M\leq
R$. Let $0\neq m\in M$ and $r\in R$. Since
$M\leq^{\text{den}}\widetilde{E}(M)$, there exists $s\in R$ such that $ms\neq
0$ and $(xr)s=x(rs)\in M$, that is, $rs\in x^{-1}M$. Hence
$x^{-1}M\leq^{\text{den}}_{M}R$.
For the reverse inclusion, let $x\in E(M)$ such that
$x^{-1}M\leq^{\text{den}}_{M}R$. For an arbitrary nonzero element $0\neq y\in
E(M)$, it suffices to show that $y\cdot x^{-1}M\neq 0$. As
$M\leq^{\text{ess}}E(M)$, $0\neq yr\in M$ for some $r\in R$. Since
$x^{-1}M\leq^{\text{den}}_{M}R$, there exists $s\in R$ such that $yrs\neq 0$
and $rs\in x^{-1}M$. Hence $0\neq yrs\in y\cdot x^{-1}M$. Therefore
$x\in\widetilde{E}(M)$. ∎
The next definition was shown in [4, pp79] as $N\leq M(K)$, so we call a
submodule $N$ relatively dense to a module $K$ in a module $M$. (For details,
see [17].)
###### Definition 2.4.
A submodule $N$ of a right $R$-module $M$ is said to be _relatively dense to a
right $R$-module_ $K$ (or $K$-_dense_) in $M$ if for any $m\in M$ and $0\neq
x\in K$, $x\cdot m^{-1}N\neq 0$, denoted by $N\leq^{\text{den}}_{K}M$. Note
that $N$ is $M$-dense in $M$ if and only if $N$ is dense in $M$.
We provide some characterizations of the relative density property. One can
compare the following characterizations to Proposition 1.1. The equivalence
(a)$\Leftrightarrow$(c) in the following proposition is provided by [4, pp79].
###### Proposition 2.5.
The following are equivalent for right $R$-modules $M,K$ and $N\leq M$:
1. (a)
$N$ is $K$-dense in $M$;
2. (b)
$\emph{Hom}_{R}(M/N,E(K))=0$;
3. (c)
for any submodule $P$ such that $N\leq P\leq M$, $\emph{Hom}_{R}(P/N,K)=0$.
###### Proof.
(a)$\Rightarrow$(b) Assume that there exists
$0\neq\alpha\in\text{Hom}_{R}(M,E(K))$ with $\alpha N=0$. Since $\alpha M\cap
K\neq 0$, there exist $x\in M$ and $0\neq y\in K$ such that $\alpha(x)=y$.
Since $N$ is $K$-dense in $M$, there exists $r\in R$ such that $xr\in N$ and
$0\neq yr$. However, $0=\alpha(xr)=\alpha(x)r=yr\neq 0$, a contradiction.
Hence $\text{Hom}_{R}(M/N,E(K))=0$.
(b)$\Rightarrow$(c) Assume that for any submodule $P$ such that $N\leq P\leq
M$, there exists $0\neq\eta\in\text{Hom}_{R}(P/N,K)$. Then by the injectivity
of $E(K)$, we can extend $\eta$ to a nonzero homomorphism from $M/N$ to
$E(K)$, a contradiction. Hence $\text{Hom}_{R}(P/N,K)=0$.
(c)$\Rightarrow$(a) Assume that $y\cdot x^{-1}N=0$ for some $x\in M$ and
$0\neq y\in K$. We define $\gamma:N+xR\rightarrow K$ given by
$\gamma(n+xr)=yr$ for $n\in N$ and $r\in R$. It is easy to see that $\gamma$
is a well-defined $R$-homomorphism vanishing on $N$. Since $N\leq N+xR\leq M$,
by hypothesis $0=\gamma(x)=y\neq 0$, a contradiction. Thus $N$ is $K$-dense in
$M$. ∎
We obtain another characterization of the relative density property related to
homomorphisms.
###### Proposition 2.6.
Let $M,K$ be right $R$-modules. Then a submodule $N$ is $K$-dense in $M$ if
and only if $\mathbf{l}_{H}(N)=0$ where $H=\emph{Hom}_{R}(M,E(K))$.
###### Proof.
Suppose $N$ is $K$-dense in $M$. Assume that $0\neq\varphi\in H$ such that
$\varphi N=0$. Then there exists $m\in M\setminus N$ such that $\varphi(m)\neq
0$. Since $\varphi(m)\in E(K)$, $0\neq\varphi(m)r\in K$ for some $r\in R$.
Hence there exists $s\in R$ such that $mrs\in N$ and $\varphi(m)rs\neq 0$
because $N\leq^{\text{den}}_{K}M$. That yields a contradiction that
$0\neq\varphi(m)rs=\varphi(mrs)\in\varphi N=0$. Therefore
$\mathbf{l}_{H}(N)=0$. Conversely, assume that $x\cdot m^{-1}N=0$ for some
$0\neq x\in K$ and $m\in M$. We define $\gamma:N+mR\rightarrow E(K)$ by
$\gamma(n+mt)=xt$ for $n\in N$ and $t\in R$. Clearly, $\gamma$ is a nonzero
$R$-homomorphism vanishing on $N$. Also, there exists
$\overline{\gamma}:M\rightarrow E(K)$ such that
$\overline{\gamma}|_{N+mR}=\gamma$. Since $0=\overline{\gamma}N$,
$0\neq\overline{\gamma}\in\mathbf{l}_{H}(N)$, a contradiction. Therefore
$x\cdot m^{-1}N\neq 0$. ∎
If $M=R$, the following result is directly provided.
###### Corollary 2.7 ([14, Proposition 1.1]).
Let $K$ be a right $R$-module and $I$ be a right ideal of a ring $R$. Then $I$
is $K$-dense in $R$ if and only if $\mathbf{l}_{E(K)}(I)=0$.
###### Proposition 2.8.
Let $K$ be a right $R$-module and $I$ be an ideal of a ring $R$. Then
$\mathbf{l}_{K}(I)=0$ if and only if $\mathbf{l}_{E(K)}(I)=0$.
###### Proof.
Since one direction is trivial, we need to show the other direction. Suppose
$\mathbf{l}_{K}(I)=0$. Assume that $\mathbf{l}_{E(K)}(I)\neq 0$. Then there
exists $0\neq x\in E(K)$ such that $xI=0$. Also, $0\neq xr\in K$ for some
$r\in R$ because $K\leq^{\text{ess}}E(K)$. Since $xrI\subseteq xI=0$, $0\neq
xr\in\mathbf{l}_{K}(I)$, a contradiction. Therefore $\mathbf{l}_{E(K)}(I)=0$.
∎
###### Corollary 2.9 ([3, Proposition 1.3.11(iv)]).
Let $I$ be an ideal of a ring $R$. Then $I\leq^{\emph{den}}R_{R}$ if and only
if $\mathbf{l}_{R}(I)=0$.
###### Proof.
The proof also follows from Corollary 2.7 and Proposition 2.8. ∎
Using Theorem 2.3 and Corollary 2.7, we obtain another characterization for
the rational hull of a module. Also, using the characterization of the
relatively dense property, new characterization for the rational hull of a
module is provided.
###### Corollary 2.10.
Let $M$ be a right $R$-module. Then the following statements hold true:
1. (i)
_([14, Proposition 1.4(b)])_ $\widetilde{E}(M)=\\{x\in
E(M)\,|\,\mathbf{l}_{E(M)}(x^{-1}M)=0\\}$.
2. (ii)
$\widetilde{E}(M)=\\{x\in E(M)\,|\,\emph{Hom}_{R}(R/x^{-1}M,E(M))=0\\}$.
###### Proof.
It directly follows from Theorem 2.3, Corollary 2.7, and Proposition 2.5. ∎
Several new characterizations for the maximal right ring of quotients of a
ring are provided as the following.
###### Theorem 2.11.
Let $R$ be a ring. Then the following statements hold true:
1. (i)
A right ideal $I$ is dense in $R$ if and only if $\mathbf{l}_{E(R)}(I)=0$.
2. (ii)
$Q(R)=\\{x\in E(R)\,|\,x^{-1}R\leq^{\emph{den}}R\\}$.
3. (iii)
$Q(R)=\\{x\in E(R)\,|\,\mathbf{l}_{E(R)}(x^{-1}R)=0\\}$.
4. (iv)
$Q(R)=\\{x\in E(R)\,|\,\emph{Hom}_{R}(R/x^{-1}R,E(R))=0\\}$.
We give characterizations for a rationally complete module.
###### Theorem 2.12.
The following conditions are equivalent for a right $R$-module $M$:
1. (a)
$M$ is a rationally complete module;
2. (b)
$\\{\overline{x}\in
E(M)/M\,|\,\mathbf{l}_{\text{E}(\text{M})}(\mathbf{r}_{\text{R}}(\overline{x}))=0\\}=\overline{0}$;
3. (c)
For any $I\leq^{\emph{den}}_{M}R$, $\varphi\in\emph{Hom}_{R}(I,M)$ can be
uniquely extended to $\widetilde{\varphi}\in\emph{Hom}_{R}(R,M)$.
###### Proof.
Take $A:=\\{\overline{x}\in
E(M)/M\,|\,\mathbf{l}_{\emph{E}(\emph{M})}(\mathbf{r}_{\text{R}}(\overline{x}))=0\\}$.
(a)$\Rightarrow$(b) Assume that $x\in E(M)\setminus M$ such that
$\overline{x}\in A$. From Corollary 2.7,
$\mathbf{r}_{\text{R}}(\overline{x})\leq^{\text{den}}_{M}R$. Since
$\mathbf{r}_{\text{R}}(\overline{x})=x^{-1}M$,
$x^{-1}M\leq^{\text{den}}_{M}R$. Hence from Theorem 2.3
$x\in\widetilde{E}(M)=M$ because $M$ is rationally complete, a contradiction.
Therefore $A=\overline{0}$. (b)$\Rightarrow$(c) Assume to the contrary of the
condition (c). For $I\leq^{\text{den}}_{M}R$, since $M\subseteq E(M)$, there
exists $\varphi\in\text{Hom}_{R}(I,M)$ such that
$\widetilde{\varphi}\in\text{Hom}_{R}(R,E(M))$,
$\widetilde{\varphi}|_{I}=\varphi$, and $\widetilde{\varphi}(1)\notin M$.
Since $\overline{0}\neq\widetilde{\varphi}(1)+M\in E(M)/M$ and
$I\subseteq\mathbf{r}_{\text{R}}(\widetilde{\varphi}(1)+M)\leq^{\text{den}}_{M}R$,
$\mathbf{l}_{\emph{E}(\emph{M})}(\mathbf{r}_{\text{R}}(\widetilde{\varphi}(1)+M))=0$
from Corollary 2.7, a contradiction that $A=\overline{0}$. Therefore
$\varphi\in\text{Hom}_{R}(I,M)$ is extended to
$\widetilde{\varphi}\in\text{Hom}_{R}(R,M)$. For the uniqueness, the proof is
similar to that of Proposition 2.13. (c)$\Rightarrow$(a) Assume that $M$ is
not rationally complete. Then there exists $x\in\widetilde{E}(M)\setminus M$
such that $x^{-1}M\leq^{\text{den}}_{M}R$ from Theorem 2.3. Define
$\varphi:x^{-1}M\rightarrow M$ given by $\varphi(r)=xr$. By hypothesis,
$\widetilde{\varphi}(1)=x1=x\in M$, a contradiction. Therefore $M$ is
rationally complete. ∎
Next, as a ring is embedding into its maximal right ring of quotients, we
provide the relationship between the endomorphism rings of a module and its
rational hull.
###### Proposition 2.13.
Let $M$ and $K$ be right $R$-modules. For any $N\leq^{\emph{den}}_{K}M$,
$\varphi\in\emph{Hom}_{R}(N,K)$ is uniquely extended to
$\widetilde{\varphi}\in\emph{Hom}_{R}(M,\widetilde{E}(K))$ and
$\widetilde{\varphi}|_{N}=\varphi$. In addition, $\varphi
N\leq^{\emph{den}}\widetilde{\varphi}M$.
###### Proof.
(Existence) Let $\varphi\in\text{Hom}_{R}(N,K)$ be arbitrary. Then there
exists $\widetilde{\varphi}\in\text{Hom}_{R}(M,E(K))$ such that
$\widetilde{\varphi}|_{N}=\varphi$. Since $\widetilde{\varphi}$ induces a
surjection from $M/N$ to
$(\widetilde{\varphi}M+\widetilde{E}(K))/\widetilde{E}(K)$ and
$\text{Hom}_{R}(M/N,E(K))=0$ (see Proposition 2.5),
$\text{Hom}_{R}\left(\frac{\widetilde{\varphi}M+\widetilde{E}(K)}{\widetilde{E}(K)},E(K)\right)=0$.
Hence $\widetilde{E}(K)\leq^{\text{den}}\widetilde{\varphi}M+\widetilde{E}(K)$
by Proposition 1.1. As $\widetilde{E}(K)$ is rationally complete,
$\widetilde{\varphi}M\subseteq\widetilde{E}(K)$.
(Uniqueness) Suppose $\widetilde{\varphi}$ and $\widetilde{\psi}$ are in
$\text{Hom}_{R}(M,\widetilde{E}(K))$ such that
$\widetilde{\varphi}|_{N}=\widetilde{\psi}|_{N}$. It is enough to show that
$\widetilde{\varphi}=\widetilde{\psi}$. Assume that
$\widetilde{\varphi}(x)\neq\widetilde{\psi}(x)$ for some $x\in M$. Take $0\neq
y=(\widetilde{\varphi}-\widetilde{\psi})(x)\in\widetilde{E}(K)$. Thus, there
exists $r\in R$ such that $0\neq yr\in K$. Since $N\leq^{\text{den}}_{K}M$,
there exists $s\in R$ such that $xrs\in N$ and $yrs\neq 0$. This yields a
contradiction that $0\neq
yrs=(\widetilde{\varphi}-\widetilde{\psi})(xrs)=(\widetilde{\varphi}|_{N}-\widetilde{\psi}|_{N})(xrs)=0$.
Therefore $\widetilde{\varphi}=\widetilde{\psi}$.
In addition, let $x_{1}\in\widetilde{\varphi}M$ and $0\neq
x_{2}\in\widetilde{\varphi}M$. Then
$\widetilde{\varphi}(m_{1})=x_{1},\widetilde{\varphi}(m_{2})=x_{2}$ for some
$m_{1},m_{2}\in M$. As $\widetilde{\varphi}M\subseteq\widetilde{E}(K)$, $0\neq
x_{2}r\in K$ for some $r\in R$. Since $N\leq^{\text{den}}_{K}M$ and $m_{1}r\in
M$, there exists $s\in R$ such that $m_{1}rs\in N$ and $0\neq x_{2}rs$. Thus
$x_{1}rs=\widetilde{\varphi}(m_{1}rs)\in\varphi N$ and $0\neq x_{2}rs$.
Therefore $\varphi N\leq^{\text{den}}\widetilde{\varphi}M$. ∎
Note that the dense property implies the essential property, however the
relatively dense property does not imply the essential property in general:
See
$\mathbb{Z}_{p}\leq^{\text{den}}_{\mathbb{Z}}\mathbb{Z}_{p}\oplus\mathbb{Z}_{p}$
but $\mathbb{Z}_{p}\nleq^{\text{ess}}\mathbb{Z}_{p}\oplus\mathbb{Z}_{p}$ as a
$\mathbb{Z}$-module. However, Proposition 2.13 shows that $\varphi
N\leq^{\text{den}}\widetilde{\varphi}M$ when $N\leq^{\text{den}}_{K}M$ for any
$\varphi\in\text{Hom}_{R}(N,K)$. As a corollary, we have a generalized result
of Theorem 2.12((a)$\Rightarrow$(b)).
###### Corollary 2.14.
Let $M$ be a right $R$-module. If $K$ is rationally complete, then for any
$N\leq^{\emph{den}}_{K}M$, $\varphi\in\emph{Hom}_{R}(N,K)$ is uniquely
extended to $\widetilde{\varphi}\in\emph{Hom}_{R}(M,K)$ and
$\widetilde{\varphi}|_{N}=\varphi$.
###### Theorem 2.15.
Let $M$ be a right $R$-module. Then $\emph{End}_{R}(M)$ is considered as a
subring of $\emph{End}_{R}(\widetilde{E}(M))$.
###### Proof.
Since $M\leq^{\text{den}}\widetilde{E}(M)$, from Proposition 2.13
$\varphi\in\text{End}_{R}(M)$ can be uniquely extended to
$\widetilde{\varphi}\in\text{End}_{R}(\widetilde{E}(M))$ because
$\text{End}_{R}(M)\subseteq\text{Hom}_{R}(M,\widetilde{E}(M))$. Thus we have a
one-to-one correspondence between $\text{End}_{R}(M)$ and
$\\{\widetilde{\varphi}\in\text{End}_{R}(\widetilde{E}(M))\,|\,\widetilde{\varphi}|_{M}={\varphi}\in\text{End}_{R}(M)\\}$
given by $\Omega(\varphi)=\widetilde{\varphi}$. We need to check that $\Omega$
is a ring homomorphism.
(i) Since
$\Omega(\varphi+\psi)|_{M}=(\widetilde{\varphi+\psi})|_{M}=\varphi+\psi=\Omega(\varphi)|_{M}+\Omega(\psi)|_{M}=(\Omega(\varphi)+\Omega(\psi))|_{M}$,
from the uniqueness of Proposition 2.13 we have
$\Omega(\varphi+\psi)=\Omega(\varphi)+\Omega(\psi)$.
(ii) Since
$\Omega{(\varphi\circ\psi)}|_{M}=(\widetilde{\varphi\circ\psi})|_{M}=\varphi\circ\psi=\Omega(\varphi)|_{M}\circ\Omega(\psi)|_{M}=(\Omega(\varphi)\circ\Omega(\psi))|_{M}$
because $\Omega(\varphi)|_{M}\leq M$, from the uniqueness of Proposition 2.13
we have $\Omega(\varphi\circ\psi)=\Omega(\varphi)\circ\Omega(\psi)$.
Thus $\text{End}_{R}(M)$ is isomorphic to a subring of
$\text{End}_{R}(\widetilde{E}(M))$. Therefore we consider $\text{End}_{R}(M)$
as a subring of $\text{End}_{R}(\widetilde{E}(M))$. ∎
We conclude this section with results for the rational hulls of quasi-
continuous modules and quasi-injective modules.
###### Theorem 2.16.
The following statements hold true for a module $M$:
1. (i)
If $M$ is a quasi-continuous module then $\widetilde{E}(M)$ is a quasi-
continuous module.
2. (ii)
If $M$ is a quasi-injective module then $\widetilde{E}(M)$ is a quasi-
injective module.
###### Proof.
(i) Let $T=\text{End}_{R}(E(\widetilde{E}(M)))=\text{End}_{R}(E(M))$. From
[10, Theorem 2.8], we need to show that
$f\widetilde{E}(M)\leq\widetilde{E}(M)$ for all idempotents $f^{2}=f\in T$:
Assume that $f\widetilde{E}(M)\nleq\widetilde{E}(M)$ for some idempotent
$f^{2}=f\in T$. Then there exists $x\in\widetilde{E}(M)$ such that
$f(x)\notin\widetilde{E}(M)$. Thus, there exists $g\in T$ such that $gM=0$ and
$gf(x)\neq 0$. Since $gf(x)\in E(M)$, there exists $r\in R$ such that $0\neq
gf(xr)\in\widetilde{E}(M)$. Thus, as $M\leq^{\text{den}}\widetilde{E}(M)$ and
$xr\in\widetilde{E}(M)$, there exists $s\in R$ such that $0\neq gf(xrs)$ and
$xrs\in M$. Note that $fM\leq M$ for all idempotents $f^{2}=f\in T$ because
$M$ is quasi-continuous. However, $0\neq gf(xrs)\in gfM\leq gM=0$, a
contradiction. Therefore $\widetilde{E}(M)$ is a quasi-continuous module.
(ii) The proof is similar to that of part (i) by using [10, Corollary 1.14]. ∎
###### Remark 2.17 ([1, Theorem 5.3]).
The rational hull of every extending module is an extending module.
Note that if $M$ is an injective module then $M=\widetilde{E}(M)$ (see [7,
Examples 8.18(1)]). The next examples exhibit that the converses of Theorem
2.16 and Remark 2.17 do not hold true.
###### Example 2.18.
(i) Consider $\mathbb{Z}$ as a $\mathbb{Z}$-module. Then
$\widetilde{E}(\mathbb{Z})=\mathbb{Q}$ is (quasi-)injective, while
$\mathbb{Z}$ is not quasi-injective.
(ii)([10, Example 2.9]) Consider a ring $R=\left(\begin{smallmatrix}F&F\\\
0&F\end{smallmatrix}\right)$ where $F$ is a field. Then
$\widetilde{E}(R_{R})=\left(\begin{smallmatrix}F&F\\\
F&F\end{smallmatrix}\right)$ is injective (hence, quasi-continuous), while
$R_{R}$ is not quasi-continuous.
(iii) Consider $\mathbb{Z}^{(\mathbb{N})}$ as a $\mathbb{Z}$-module. Then
$\widetilde{E}(\mathbb{Z}^{(\mathbb{N})})=\mathbb{Q}^{(\mathbb{N})}$ is
injective (hence, extending), while $\mathbb{Z}^{(\mathbb{N})}$ is not
extending.
###### Corollary 2.19.
The maximal right ring of quotients of a quasi-continuous ring is also a
quasi-continuous ring.
###### Remark 2.20 ([7, Exercises 13.8]).
The maximal right ring of quotients of a simple (resp., prime, semiprime) ring
is also a simple (resp., prime, semiprime) ring.
###### Open Question 1.
Is the rational hull of a continuous module always a continuous module?
## 3\. Direct sum of rational hulls of modules
As we know, the injective hull of the direct sum of two modules is the direct
sum of the injective hulls of each module without any condition. However, the
rational hull case is different from the injective hull case. In this section,
we discuss the condition for the rational hull of the direct sum of two
modules to be the direct sum of the rational hulls of those modules. The next
example shows that the rational hull of the direct sum of two modules is not
the direct sum of the rational hulls of each module, in general.
###### Example 3.1.
Consider $M=\mathbb{Z}\oplus\mathbb{Z}_{p}$ as a $\mathbb{Z}$-module where $p$
is prime. Then $\widetilde{E}(\mathbb{Z})=\mathbb{Q}$ and
$\widetilde{E}(\mathbb{Z}_{p})=\mathbb{Z}_{p}$. However, by [7, Example 8.21]
$\widetilde{E}(M)=\mathbb{Z}_{(p)}\oplus\mathbb{Z}_{p}\neq\mathbb{Q}\oplus\mathbb{Z}_{p}$
where
$\mathbb{Z}_{(p)}=\\{\frac{m}{n}\in\mathbb{Q}\,|\,{m,n}\in\mathbb{Z},(n,p)=1\\}$.
Hence $M$ is not a dense submodule of $\mathbb{Q}\oplus\mathbb{Z}_{p}$: For
$(\frac{1}{p},\overline{0})$ and
$0\neq(0,\overline{1})\in\mathbb{Q}\oplus\mathbb{Z}_{p}$, there is no
$n\in\mathbb{Z}$ such that
$n(\frac{1}{p},\overline{0})\in\mathbb{Z}\oplus\mathbb{Z}_{p}$ and
$n(0,\overline{1})\neq 0$.
###### Proposition 3.2.
Let $M=\bigoplus_{k\in\Lambda}M_{k}$ where $M_{k}$ be a right $R$-module and
$\Lambda$ is any index set. If either $R$ is right noetherian or $|\Lambda|$
is finite, then
$\widetilde{E}(M)\leq\bigoplus_{k\in\Lambda}\widetilde{E}(M_{k})$.
###### Proof.
Suppose $0\neq m\in\widetilde{E}(M)$. Since $\widetilde{E}(M)\subseteq
E(M)=\oplus_{k\in\Lambda}E(M_{k})$ because $R$ is right noetherian or
$|\Lambda|$ is finite, there exists $\ell\in\mathbb{N}$ such that
$m\in\oplus_{i=1}^{\ell}E(M_{i})$. Thus, $m=(m_{1},\dots,m_{\ell})$ where
$m_{i}\in E(M_{i})$. Since $(0,\dots,0,y_{i},0,\dots,0)\cdot m^{-1}M\neq 0$
for all $0\neq y_{i}\in E(M_{i})$ and $m^{-1}M=m_{1}^{-1}M_{1}\cap\cdots\cap
m_{\ell}^{-1}M_{\ell}$, $y_{i}\cdot m_{i}^{-1}M_{i}\neq 0$ for all $0\neq
y_{i}\in E(M_{i})$. Thus, $m_{i}\in\widetilde{E}(M_{i})$ for all $1\leq
i\leq\ell$ from Proposition 1.4. So,
$m=(m_{1},\dots,m_{\ell})\in\oplus_{i=1}^{\ell}\widetilde{E}(M_{i})\subseteq\oplus_{k\in\Lambda}\widetilde{E}(M_{k})$.
Therefore $\widetilde{E}(M)\leq\oplus_{k\in\Lambda}\widetilde{E}(M_{k})$. ∎
###### Remark 3.3.
Example 3.1 illustrates Proposition 3.2 because $R=\mathbb{Z}$ is a noetherian
ring, that is,
$\widetilde{E}(\mathbb{Z}\oplus\mathbb{Z}_{p})=\mathbb{Z}_{(p)}\oplus\mathbb{Z}_{p}\lneq\mathbb{Q}\oplus\mathbb{Z}_{p}=\widetilde{E}(\mathbb{Z})\oplus\widetilde{E}(\mathbb{Z}_{p})$.
However, Example 3.7 shows that the condition “either $R$ is right noetherian
or $|\Lambda|$ is finite” is not superfluous because
$\widetilde{E}(\oplus_{k\in\Lambda}\mathbb{Z}_{2})=\prod_{k\in\Lambda}\mathbb{Z}_{2}\gneq\oplus_{k\in\Lambda}\mathbb{Z}_{2}=\oplus_{k\in\Lambda}\widetilde{E}(\mathbb{Z}_{2})$
with a non-noetherian ring
$R=\langle\oplus_{k\in\Lambda}\mathbb{Z}_{2},1\rangle$.
To get the reverse inclusion of Proposition 3.2, first we provide the
properties of the relatively dense property.
###### Lemma 3.4.
Let $N\leq M$ and $K_{i}$ be right $R$-modules for all $i\in\Lambda$. Then $N$
is $K_{i}$-dense in $M$ for all $i\in\Lambda$ if and only if $N$ is
$\bigoplus_{i\in\Lambda}K_{i}$-dense in $M$ if and only if $N$ is
$\bigoplus_{i\in\Lambda}\widetilde{E}(K_{i})$-dense in $M$.
###### Proof.
Let $P$ be any submodule such that $N\leq P\leq M$. Since $N$ is $K_{i}$-dense
in $M$, $\text{Hom}_{R}(P/N,K_{i})=0$ for all $i\in\Lambda$ from Proposition
2.5. Consider the sequence
$0\rightarrow\oplus_{i\in\Lambda}K_{i}\rightarrow\prod_{i\in\Lambda}K_{i}$.
Then we have
$0\rightarrow\text{Hom}_{R}(P/N,\oplus_{i\in\Lambda}K_{i})\rightarrow\text{Hom}_{R}(P/N,\prod_{i\in\Lambda}K_{i})\cong\prod_{i\in\Lambda}(\text{Hom}_{R}(P/N,K_{i})=0$.
Thus $\text{Hom}_{R}(P/N,\oplus_{i\in\Lambda}K_{i})=0$. Therefore $N$ is
$\oplus_{i\in\Lambda}K_{i}$-dense in $M$ from Proposition 2.5. Conversely,
since $\text{Hom}_{R}(P/N,\oplus_{i\in\Lambda}K_{i})=0$,
$\text{Hom}_{R}(P/N,K_{i})=0$ for each $i\in\Lambda$. Hence $N$ is
$K_{i}$-dense in $M$ for all $i\in\Lambda$.
For the second equivalence, since $E(\widetilde{E}(K_{i}))=E(K_{i})$, from
Proposition 2.5 it is easy to see that $N$ is $K_{i}$-dense in $M$ if and only
if $N$ is $\widetilde{E}(K_{i})$-dense in $M$, for all $i\in\Lambda$. After
using the first equivalence, we have the second equivalence. ∎
Using Lemma 3.4, we obtain a characterization for $\oplus_{k\in\Lambda}N_{k}$
to be a dense submodule of $\oplus_{k\in\Lambda}M_{k}$ where $N_{i}$ is a
submodule of $M_{i}$ for each $i\in\Lambda$.
###### Proposition 3.5.
Let $N_{i}\leq M_{i}$ be right $R$-modules for all $i\in\Lambda$ where
$\Lambda$ is any index set. Let $N=\bigoplus_{k\in\Lambda}N_{k}$ and
$M=\bigoplus_{k\in\Lambda}M_{k}$. Then $N\leq^{\emph{den}}M$ if and only if
$N_{i}$ is $M_{j}$-dense in $M_{i}$ for all $i,j\in\Lambda$.
###### Proof.
Suppose $N\leq^{\text{den}}M$. Then $N$ is $M$-dense in $M$ by the definition.
From Lemma 3.4 $N$ is $M_{j}$-dense in $M$ for all $j\in\Lambda$. Let
$x_{i}\in M_{i}$ and $0\neq y_{j}\in M_{j}$ be arbitrary for each
$i,j\in\Lambda$. Since $(0,\dots,0,x_{i},0,\dots)\in M$ and $0\neq y_{j}\in
M_{j}$, there exists $r\in R$ such that $(0,\dots,0,x_{i},0,\dots)r\in N$ and
$y_{j}r\neq 0$. Since $x_{i}r\in N_{i}$ and $y_{j}r\neq 0$, $N_{i}$ is
$M_{j}$-dense in $M_{i}$ for all $i,j\in\Lambda$.
Conversely, suppose $N_{i}$ is $M_{j}$-dense in $M_{i}$ for all
$i,j\in\Lambda$. From Lemma 3.4, $N_{i}$ is $\oplus_{k\in\Lambda}M_{k}$-dense
in $M_{i}$ for all $i\in\Lambda$. Let $x\in M$ and $0\neq y\in M$ be
arbitrary. Then there exists $\ell\in\mathbb{N}$ such that
$x=(x_{1},\dots,x_{\ell})\in\oplus_{k=1}^{\ell}M_{k}\leq M$. Since $N_{1}$ is
$M$-dense in $M_{1}$, there exists $r_{1}\in R$ such that $x_{1}r_{1}\in
N_{1}$ and $0\neq yr_{1}\in M$. Also, since $N_{2}$ is $M$-dense in $M_{2}$,
there exists $r_{2}\in R$ such that $x_{2}r_{1}r_{2}\in N_{2}$ and $0\neq
yr_{1}r_{2}\in M$. By the similar processing, we have $r=r_{1}r_{2}\cdots
r_{\ell}\in R$ such that $xr\in\oplus_{k=1}^{\ell}N_{k}\leq N$ and $yr\neq 0$.
Therefore $N\leq^{\text{den}}M$. ∎
From Propositions 3.2 and 3.5, we have a characterization for the rational
hull of the direct sum of modules to be the direct sum of the rational hulls
of each module.
###### Theorem 3.6.
Let $M=\bigoplus_{k\in\Lambda}M_{k}$ where $M_{k}$ is a right $R$-module and
$\Lambda$ is any index set. If either $R$ is right noetherian or $|\Lambda|$
is finite, then $\widetilde{E}(M)=\bigoplus_{k\in\Lambda}\widetilde{E}(M_{k})$
if and only if $M_{i}$ is $M_{j}$-dense in $\widetilde{E}(M_{i})$ for all
$i,j\in\Lambda$.
###### Proof.
Suppose $\widetilde{E}(M)=\oplus_{k\in\Lambda}\widetilde{E}(M_{k})$. Since
$M\leq^{\text{den}}\oplus_{k\in\Lambda}\widetilde{E}(M_{k})$, from Proposition
3.5 $M_{i}$ is $\widetilde{E}(M_{j})$-dense in $\widetilde{E}(M_{i})$ for all
$i,j\in\Lambda$. Thus, $M_{i}$ is $M_{j}$-dense in $\widetilde{E}(M_{i})$ for
all $i,j\in\Lambda$ from Lemma 3.4.
Conversely, suppose $M_{i}$ is $M_{j}$-dense in $\widetilde{E}(M_{i})$ for all
$i,j\in\Lambda$. Then $M_{i}$ is $\widetilde{E}(M_{j})$-dense in
$\widetilde{E}(M_{i})$ for all $i,j\in\Lambda$ from Lemma 3.4. Thus, from
Proposition 3.5 $M\leq^{\text{den}}\oplus_{k\in\Lambda}\widetilde{E}(M_{k})$.
Hence $\oplus_{k\in\Lambda}\widetilde{E}(M_{k})\leq\widetilde{E}(M)$ from
Proposition 1.3. Also, from Proposition 3.2
$\widetilde{E}(M)\leq\oplus_{k\in\Lambda}\widetilde{E}(M_{k})$. Therefore
$\widetilde{E}(M)=\oplus_{k\in\Lambda}\widetilde{E}(M_{k})$. ∎
The next examples show that the condition “$R$ is right noetherian or
$|\Lambda|$ is finite” in Theorem 3.6 is not superfluous.
###### Example 3.7.
(i) Let $R=\langle\oplus_{k\in\Lambda}\mathbb{Z}_{2},1\rangle$ and
$M=\oplus_{k\in\Lambda}M_{k}$ where $M_{k}=\mathbb{Z}_{2}$. Note that $R$ is
not noetherian. Since $\mathbb{Z}_{2}$ is an injective $R$-module,
$\widetilde{E}(\mathbb{Z}_{2})=\mathbb{Z}_{2}$. Thus $M_{i}$ is $M_{j}$-dense
in $\widetilde{E}(M_{i})$ for all $i,j\in\Lambda$. However,
$\widetilde{E}(\oplus_{k\in\Lambda}\mathbb{Z}_{2})=\prod_{k\in\Lambda}\mathbb{Z}_{2}\gneq\oplus_{k\in\Lambda}\mathbb{Z}_{2}=\oplus_{k\in\Lambda}\widetilde{E}(\mathbb{Z}_{2})$.
(ii) Let $R=\\{(a_{k})\in\prod_{k\in\Lambda}\mathbb{Z}\,|\,a_{k}~{}\text{is
eventually constant}\\}$ and $M=\oplus_{k\in\Lambda}\mathbb{Z}$. Note that $R$
is not noetherian. Then $\widetilde{E}(\mathbb{Z})=\mathbb{Q}$ and
$\mathbb{Z}$ is $\mathbb{Z}$-dense in $\widetilde{E}(\mathbb{Z})$. However,
$\widetilde{E}(\oplus_{k\in\Lambda}\mathbb{Z})=\prod_{k\in\Lambda}\mathbb{Q}\gneq\oplus_{k\in\Lambda}\mathbb{Q}=\oplus_{k\in\Lambda}\widetilde{E}(\mathbb{Z})$.
The next example illustrates Theorem 3.6.
###### Example 3.8.
Consider $M=\mathbb{Z}\oplus\mathbb{Z}_{p}$ as a $\mathbb{Z}$-module where $p$
is prime. Then $\mathbb{Z}_{p}$ is $\mathbb{Z}$-dense in
$\widetilde{E}(\mathbb{Z}_{p})=\mathbb{Z}_{p}$, but $\mathbb{Z}$ is not
$\mathbb{Z}_{p}$-dense in $\mathbb{Q}$ because for
$\frac{1}{p}\in\mathbb{Q},\overline{1}\in\mathbb{Z}_{p}$, there is no element
$t\in\mathbb{Z}$ such that $t\frac{1}{p}\in\mathbb{Z}$ and $t\overline{1}\neq
0$. Thus, from Theorem 3.6
$\widetilde{E}(M)=\mathbb{Z}_{(p)}\oplus\mathbb{Z}_{p}\lneq\mathbb{Q}\oplus\mathbb{Z}_{p}=\widetilde{E}(\mathbb{Z})\oplus\widetilde{E}(\mathbb{Z}_{p})$.
(See Example 3.1 for details.)
###### Corollary 3.9.
Let $M$ be a right $R$-module. If either $R$ is right noetherian or $\Lambda$
is a finite index set, then
$\widetilde{E}(M^{(\Lambda)})=(\widetilde{E}(M))^{(\Lambda)}$.
###### Corollary 3.10.
Let $\\{M_{k}\\}_{k\in\Lambda}$ be a class of rationally complete right
$R$-module for any index set $\Lambda$. If either $R$ is right noetherian or
$|\Lambda|$ is finite, then $M=\bigoplus_{k\in\Lambda}M_{k}$ is rationally
complete.
###### Proof.
Since $\widetilde{E}(M_{i})=M_{i}$, $M_{i}$ is $M_{j}$-dense in
$\widetilde{E}(M_{i})$ for all $i,j\in\Lambda$. From Theorem 3.6,
$\widetilde{E}(M)=\oplus_{k\in\Lambda}\widetilde{E}(M_{k})=\oplus_{k\in\Lambda}M_{k}=M$.
Therefore $\oplus_{k\in\Lambda}M_{k}$ is rationally complete. ∎
###### Proposition 3.11 ([14, Proposition 1.9]).
Let $\\{S_{i}\\}_{i\in\Lambda}$ be a set of nonisomorphic simple modules,
representing all singular simple modules. Then every module containing the
module $P=\bigoplus_{i\in\Lambda}S_{i}$ is rationally complete.
## 4\. The endomorphism ring of a module over a right ring of quotients of a
ring
In this section, we obtain some condition to be
$\text{End}_{R}(M)=\text{End}_{H}(M)$ where $H$ is a right ring of quotients
of a ring $R$. Recall that an extension ring $H$ of a ring $R$ is called a
_right ring of quotients_ of $R$ if for any two elements $x\neq 0$ and $y$ of
$H$, there exists an element $r\in R$ such that $xr\neq 0$ and $yr\in R$.
###### Theorem 4.1.
Let $M$ be a right $H$-module where $H$ is a right ring of quotients of a ring
$R$. If $R$ is $M_{R}$-dense in $H_{R}$ then
$\emph{End}_{R}(M)=\emph{End}_{H}(M)$.
###### Proof.
Since $\text{End}_{H}(M)\subseteq\text{End}_{R}(M)$, it suffices to show that
$\text{End}_{R}(M)\subseteq\text{End}_{H}(M):$ Let
$\varphi\in\text{End}_{R}(M)$ be arbitrary. Assume that
$\varphi\notin\text{End}_{H}(M)$. Then there exist $m\in M,t\in H$ such that
$\varphi(mt)-\varphi(m)t\neq 0.$ Since $R$ is $M_{R}$-dense in $H_{R}$, there
exists $r\in R$ such that $(\varphi(mt)-\varphi(m)t)r\neq 0$ and $tr\in R$.
Hence
$0\neq(\varphi(mt)-\varphi(m)t)r=\varphi(mt)r-\varphi(m)(tr)=\varphi(mtr)-\varphi(mtr)=0$,
a contradiction. Therefore $\text{End}_{R}(M)=\text{End}_{H}(M)$. ∎
###### Remark 4.2.
(i) $R$ is always $E(R)$-dense in $H_{R}$ where $H$ is a right ring of
quotients of $R$. For, let $x\in H_{R}$ and $0\neq y\in E(R)$. Since
$H\leq^{\text{ess}}E(R)_{R}$, there exists $s\in R$ such that $0\neq ys\in H$.
Also, $xs\in H$. Since $R\leq^{\text{den}}H_{R}$, there exists $t\in R$ such
that $xst\in R$ and $0\neq yst$. Therefore $R$ is $E(R)$-dense in $H_{R}$.
(ii) If $M$ is a nonsingular $R$-module, then $R$ is $M_{R}$-dense in $H_{R}$:
For, let $0\neq m\in M$ and $t\in H$ be arbitrary. Take $t^{-1}R=\\{r\in
R\,|\,tr\in R\\}$ a right ideal of $R$. Note that
$t^{-1}R\leq^{\textrm{ess}}R_{R}$. Since
$t^{-1}R\nleq^{\text{ess}}\mathbf{r}_{R}(m)$, there exists $r\in t^{-1}R$ and
$r\notin\mathbf{r}_{R}(m)$. Thus, $tr\in R$ and $mr\neq 0$. Therefore $R$ is
$M_{R}$-dense in $H_{R}$.
(iii) If $M$ is a submodule of a projective right $H$-module, then $R$ is
$M_{R}$-dense in $H_{R}$. For, let $P$ be a projective right $R$-module
including $M$, that is, $M\leq P$ where $P\leq^{\oplus}H^{(\Lambda)}$ with
some index set $\Lambda$. Then there is a right $R$-module $K\leq E(P)$ such
that $E(P)=E(M)\oplus K.$ Since $R\leq^{\text{den}}H_{R}$, we get that $R$ is
$H^{(\Lambda)}$-dense in $H_{R}$ from Lemma 3.4. Hence $R$ is $P$-dense in
$H_{R}$. Thus $\text{Hom}_{R}(H/R,E(P))=0$ from Proposition 2.5. Since
$\text{Hom}_{R}(H/R,E(P))\cong\text{Hom}_{R}(H/R,E(M))\oplus\text{Hom}_{R}(H/R,K)$,
we obtain $\text{Hom}_{R}(H/R,E(M))=0$. It follows that $R$ is $M_{R}$-dense
in $H_{R}.$
###### Corollary 4.3.
Let $M$ be a projective right $H$-module where $H$ is a right ring of
quotients of $R$. Then $\emph{End}_{R}(M)=\emph{End}_{H}(M)$.
The next example illustrates Corollary 4.3.
###### Example 4.4.
Let $Q=\prod_{n=1}^{\infty}\mathbb{Z}_{2}$ and $R=\\{(a_{n})\in Q~{}|~{}a_{n}$
is eventually constant$\\}$. Then $Q$ is a maximal right ring of quotients of
$R$. Hence from Theorem 4.1, End${}_{R}(Q^{(\Lambda)})$=
End${}_{Q}(Q^{(\Lambda)})=\mathsf{CFM}_{\Lambda}(Q)$.
###### Theorem 4.5.
Let $M$ be a finitely generated free $R$-module with $S=\emph{End}_{R}(M)$. If
either $R$ is right noetherian or $\Lambda$ is any finite index set, then
$\emph{End}_{R}\left(\widetilde{E}(M^{(\Lambda)})\right)=\mathsf{CFM}_{\Lambda}\left(Q(S)\right)$.
###### Proof.
Let $M=R^{(n)}$ for some $n\in\mathbb{N}$. From Corollary 3.9,
$\widetilde{E}(R^{(n)})=\widetilde{E}(R)^{(n)}=Q(R)^{(n)}$ as
$\widetilde{E}(R)=Q(R)$. Hence
$\text{End}_{R}\left(\widetilde{E}(M^{(\Lambda)})\right)=\text{End}_{R}\left(\widetilde{E}(M)^{(\Lambda)}\right)=\text{End}_{R}\left((Q(R)^{(n)})^{(\Lambda)}\right)=\text{End}_{Q(R)}\left((Q(R)^{(n)})^{(\Lambda)}\right)=\mathsf{CFM}_{\Lambda}\left(\text{End}_{Q(R)}(Q(R)^{(n)})\right)=\mathsf{CFM}_{\Lambda}\left(\mathsf{Mat}_{n}(Q(R))\right)$
from Theorem 4.1. Therefore
$\text{End}_{R}\left(\widetilde{E}(M^{(\Lambda)})\right)=\mathsf{CFM}_{\Lambda}\left(Q(\text{End}_{R}(M))\right)$
because $\mathsf{Mat}_{n}(Q(R))=Q(\mathsf{Mat}_{n}(R))$ by [15, 2.3] and
$\text{End}_{R}(M)=\mathsf{Mat}_{n}(R)$. ∎
The next result is generalized from [15, 2.3].
###### Corollary 4.6.
Let $M$ be a finitely generated free $R$-module. Then
$Q(\emph{End}_{R}(M))=\emph{End}_{R}(\widetilde{E}(M))$.
The following example shows that the above result can not be extended to flat
modules. This example also shows that $R$ is not $M_{R}$-dense in $Q_{R}$
where $Q$ is a right ring of quotients of $R$.
###### Example 4.7.
Let $Q=\prod_{n=1}^{\infty}\mathbb{Z}_{2}$, $R=\\{(a_{n})\in Q~{}|~{}a_{n}$ is
eventually constant$\\}$, and $I=\\{(a_{n})\in
Q~{}|~{}a_{n}=0~{}\textrm{eventually}\\}$. Note that $Q=Q(R)$. Let $M=Q/I$,
which is a flat $Q$-module but not projective. We claim that
End${}_{Q}(M)\subsetneq$ End${}_{R}(M)$. Indeed, define $f:M\to M$ via
$f[(a_{1},a_{2},\dots,a_{n},a_{n+1},\dots)+I]=(a_{1},0,a_{2},0,\dots,a_{n},0,a_{n+1},0,\dots)+I,$
for any $\overline{a}=a+I=(a_{1},a_{2},\dots,a_{n},a_{n+1},\dots)+I\in M$. It
is easy to see that $f(\overline{a}+\overline{b})$ = $f(\overline{a})$ \+
$f(\overline{b})$ for any $\overline{a}$, $\overline{b}\in M$. Meanwhile, for
any $r=(r_{1},r_{2},\dots,r_{n},r_{n+1},\dots)\in R$, we have
$(a+I)r=ar+I=\left\\{\begin{array}[]{ll}(0,\dots,0,a_{n},a_{n+1},\dots)+I,&\mbox{if
}r_{n}=r_{n+1}=\dots=1;\\\ (0,\dots,0,0,0,\dots)+I,&\mbox{if
}r_{n}=r_{n+1}=\dots=0.\end{array}\right.$
Note that $a+I=(0,0,\dots,0,a_{n},a_{n+1},\dots)+I$ for some $n\in\mathbb{N}$.
One can easily see that $f[(a+I)r]$ = $[f(a+I)]r$ for all $a\in Q$, $r\in R$.
This shows $f\in$ End${}_{R}(M)$. However, for $q=(0,1,0,1,\dots)=q^{2}\in Q$,
we have $[f(q+I)]q=0+I$ while $f[(q+I)q]=f(q+I)\neq 0+I$. This means
$f\not\in$ End${}_{Q}(M)$. Thus, End${}_{Q}(M)\subsetneq$ End${}_{R}(M)$. Note
that $R$ is not $M_{R}$-dense in $Q$. For, let $q\in Q\setminus R$ and
$m=1+I\in M$. Since $(1+I)r=0+I$ for all $r\in R\setminus{1}$, it has to be
$r=1$ to get $mr\neq 0+I$. However, $qr\notin R$.
Recall that a module $M$ is said to be _polyform_ if every essential submodule
of $M$ is a dense submodule.
###### Lemma 4.8.
A module $M$ is polyform iff $\widetilde{E}(M)$ is a polyform quasi-injective
module.
###### Proof.
Let $X$ be essential in $\widetilde{E}(M)$. Then $X\cap M\leq^{\text{ess}}M$.
Hence $X\cap M$ is a dense submodule of $M$ because $M$ is polyform. Since
$X\cap M\leq^{\text{den}}M\leq^{\text{den}}\widetilde{E}(M)$, $X\cap
M\leq^{\text{den}}\widetilde{E}(M)$. Thus $X$ is a dense submodule of
$\widetilde{E}(M)$ from Proposition 1.2(ii). Therefore $\widetilde{E}(M)$ is a
polyform module. In addition, $\widehat{M}$ is also a polyform module from
[16, 11.1]. Since $M\leq^{\text{ess}}\widehat{M}$,
$M\leq^{\text{den}}\widehat{M}$. Thus
$\widetilde{E}(M)=\widetilde{E}(\widehat{M})$. Since the rational hull of a
quasi-injective module is also quasi-injective from Theorem 2.16,
$\widetilde{E}(M)$ is a quasi-injective module. Therefore $\widetilde{E}(M)$
is a polyform quasi-injective module.
Conversely, let $N$ be any essential submodule of $M$. Then $N$ is also
essential in $\widetilde{E}(M)$. Hence $N$ is a dense submodule of
$\widetilde{E}(M)$ as $\widetilde{E}(M)$ is polyform. So $N$ is a dense
submodule of $M$. Therefore $M$ is polyform. ∎
We show from Theorem 2.15 that there is a canonical embedding of the ring
$\text{End}_{R}(M)$ into the ring $\text{End}_{R}(\widetilde{E}(M))$. Next, we
obtain a condition when $\text{End}_{R}(M)$ and
$\text{End}_{R}(\widetilde{E}(M))$ are isomorphic. It is a generalization of
[7, Exercises 7.32].
###### Proposition 4.9.
If $M$ is a quasi-injective module then
$\emph{End}_{R}(M)\stackrel{{\scriptstyle\Omega}}{{\cong}}\emph{End}_{R}(\widetilde{E}(M))$.
In particular, if $M$ is a polyform module, then the converse holds true.
###### Proof.
In the proof of Theorem 2.15, we only need to show that
$\Omega:\text{End}_{R}(M)\rightarrow\text{End}_{R}(\widetilde{E}(M))$ given by
$\Omega(\varphi)=\widetilde{\varphi}$, is surjective: Let
$\psi\in\text{End}_{R}(\widetilde{E}(M))$ be arbitrary. Then there exists
$\widehat{\psi}\in\text{End}_{R}(E(M))$ such that
$\widehat{\psi}|_{\widetilde{E}(M)}=\psi$. Since $\widehat{\psi}M\leq M$ as
$M$ is quasi-injective, $\widehat{\psi}|_{M}=\psi|_{M}\in\text{End}_{R}(M)$.
Thus, $\Omega(\psi|_{M})=\psi$, which shows that $\Omega$ is surjective.
In addition, suppose that $M$ is a polyform module. Then from Lemma 4.8,
$\widetilde{E}(M)$ is quasi-injective. Thus, for any
$\vartheta\in\text{End}_{R}(E(M))$,
$\vartheta\widetilde{E}(M)\leq\widetilde{E}(M)$. Since
$\vartheta|_{\widetilde{E}(M)}\in\text{End}_{R}(\widetilde{E}(M))$ and
$\text{End}_{R}(M)\stackrel{{\scriptstyle\Omega}}{{\cong}}\text{End}_{R}(\widetilde{E}(M))$,
there exists $\varphi\in\text{End}_{R}(M)$ such that
$\Omega(\varphi)=\vartheta|_{\widetilde{E}(M)}$. Also by Theorem 2.15,
$\vartheta|_{M}=\varphi$. Thus, $\vartheta M=\varphi M\leq M$. Therefore $M$
is a quasi-injective module. ∎
###### Corollary 4.10.
If $M$ is a quasi-injective module, then
$Q(\emph{End}_{R}(M))\cong\emph{End}_{R}(\widetilde{E}(M))$.
###### Proof.
Since $M$ is a quasi-injective module $\text{End}_{R}(M)$ is a right self-
injective ring. So, $Q(\text{End}_{R}(M))=\text{End}_{R}(M)$. Thus,
$Q(\text{End}_{R}(M))\cong\text{End}_{R}(\widetilde{E}(M))$ by Proposition
4.9. ∎
Remark that if $M$ is a quasi-injective module then $\widetilde{E}(M)$ is a
quasi-injective module from Theorem 2.16 and
$\text{End}_{R}(M)\cong\text{End}_{R}(\widetilde{E}(M))$ from Proposition 4.9.
However, the next example shows that there exists a quasi-injective module $M$
such that $M\neq\widetilde{E}(M)$.
###### Example 4.11.
Let $R=\left(\begin{smallmatrix}F&F\\\ 0&F\end{smallmatrix}\right)$ and
$M=\left(\begin{smallmatrix}0&0\\\ 0&F\end{smallmatrix}\right)$ where $F$ is a
field. Then $M$ is a quasi-injective $R$-module. However,
$\widetilde{E}(M)=E(M)=\left(\begin{smallmatrix}0&0\\\
F&F\end{smallmatrix}\right)$ because $M$ is nonsingular. Thus $M$ is a quasi-
injective $R$-module such that $M\lneq\widetilde{E}(M)$ and
$\text{End}_{R}(M)\cong\left(\begin{smallmatrix}0&0\\\
0&F\end{smallmatrix}\right)\cong\text{End}_{R}(\widetilde{E}(M))$.
Because $\widetilde{E}(M)=E(M)$ for a right nonsingular module $M$, we have
the following well-known results as a consequence of Proposition 4.9.
###### Corollary 4.12 ([7, Exercises 7.32]).
For any nonsingular module $M$, the following statements hold true:
1. (i)
there is a canonical embedding $\Omega$ of the ring $\emph{End}_{R}(M)$ into
the ring $\emph{End}_{R}(E(M))$.
2. (ii)
$M$ is a quasi-injective $R$-module if and only if $\Omega$ is an isomorphism.
###### Corollary 4.13.
Let $M$ be a right $H$-module where $H$ is a right ring of quotients of a ring
$R$. The following statements hold true:
1. (i)
If $M$ is a nonsingular $R$-module then $\emph{End}_{R}(M)=\emph{End}_{H}(M)$.
2. (ii)
If $M$ is a submodule of a projective $H$-module, then
$\emph{End}_{R}(M)=\emph{End}_{H}(M)$.
3. (iii)
If $M$ is a nonsingular quasi-injective $R$-module then
$\emph{End}_{R}(M)\cong\emph{End}_{R}(E(M))$ and
$\emph{End}_{H}(M)\cong\emph{End}_{H}(E(M))$.
4. (iv)
If $M$ is a quasi-injective $R$-module and is a submodule of a projective
$H$-module then $\emph{End}_{R}(M)\cong\emph{End}_{R}(\widetilde{E}(M))$ and
$\emph{End}_{H}(M)\cong\emph{End}_{H}(\widetilde{E}(M))$.
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|
# A Complexity Dichotomy in Spatial Reasoning
via Ramsey Theory
Manuel Bodirsky and Bertalan Bodor Institut für Algebra, Fakultät für
Mathematik, TU Dresden and Department of Algebra, Faculty of Mathematics and
Physics, Charles University, Prague
###### Abstract.
Constraint satisfaction problems (CSPs) for first-order reducts of finitely
bounded homogeneous structures form a large class of computational problems
that might exhibit a complexity dichotomy, P versus NP-complete. A powerful
method to obtain polynomial-time tractability results for such CSPs is a
certain reduction to polynomial-time tractable finite-domain CSPs defined over
$k$-types, for a sufficiently large $k$. We give sufficient conditions when
this method can be applied and apply these conditions to obtain a new
complexity dichotomy for CSPs of first-order expansions of the basic relations
of the well-studied spatial reasoning formalism RCC5. We also classify which
of these CSPs can be expressed in Datalog. Our method relies on Ramsey theory;
we prove that RCC5 has a Ramsey order expansion.
The authors have received funding from the European Research Council (Grant
Agreement no. 681988, CSP-Infinity).
###### Contents
1. 1 Introduction
2. 2 Countably Categorical Structures
3. 3 Ramsey Theory
4. 4 Oligomorphic Clones
5. 5 Canonical Functions
1. 5.1 Behaviours
2. 5.2 Canonicity
3. 5.3 Canonisation
4. 5.4 Complexity
6. 6 Results
1. 6.1 The Unique Interpolation Property
2. 6.2 Other Characterisations of the UIP
7. 7 Proof of the Extension Lemma
1. 7.1 Diagonal Interpolation
2. 7.2 Rich subsets
3. 7.3 Interpolation invariance
4. 7.4 Proofs of main results
8. 8 Verification of the UIP
9. 9 First-order expansions of the basic relations of RCC5
1. 9.1 A Ramsey order expansion of $\mathfrak{R}$
2. 9.2 Polymorphisms of $\mathfrak{R}$ that are canonical with respect to $(\mathfrak{R},\prec)$
3. 9.3 Independent substructures
4. 9.4 Uniformly continuous minor-preserving maps
1. 9.4.1 Cyclic polymorphisms
2. 9.4.2 Factoring $\prec$
3. 9.4.3 Restricting to $\smash{{}^{\bot}_{\prec}}$
4. 9.4.4 Canonical pseudo-cyclic polymorphisms
5. 9.5 Verifying the UIP
6. 9.6 Proof of the complexity classification
10. 10 Expressibility in Datalog and Extensions
11. 11 Future Work
## 1\. Introduction
The _Constraint satisfaction problem (CSP)_ of a relational structure
$\mathfrak{B}$ is the computational problem of deciding whether a given finite
structure $\mathfrak{A}$ has a homomorphism to $\mathfrak{B}$. If the
structure $\mathfrak{B}$ is finite, then $\operatorname{CSP}(\mathfrak{B})$ is
in P or NP-complete; this was conjectured by Feder and Vardi [FV99] and proved
by Bulatov [Bul17] and, independently, by Zhuk [Zhu17]. Both approaches use
concepts and methods from universal algebra; in particular, they use that the
computational complexity of $\operatorname{CSP}(\mathfrak{B})$ is fully
determined by the set $\operatorname{Pol}(\mathfrak{B})$ of _polymorphisms_ of
$\mathfrak{B}$, which is the set of homomorphisms from $\mathfrak{B}^{n}$ to
$\mathfrak{B}$ for $n\in{\mathbb{N}}$. In fact,
$\operatorname{CSP}(\mathfrak{B})$ is NP-complete if
$\operatorname{Pol}(\mathfrak{B})$ has a minor-preserving map to the clone
$\operatorname{Proj}$ of projections on a two-element set, and is in P
otherwise.
Some of the results about finite structures can be lifted to countably
infinite structures whose automorphism group satisfies a certain finiteness
condition, called _oligomorphicity_ : the requirement is that for every
$n\in{\mathbb{N}}$ the componentwise action of the automorphism group on
$n$-tuples has only finitely many orbits. Examples of such structures arise
systematically in model theory: every homogeneous structure $\mathfrak{B}$
with a finite relational signature is of this type; such structures will be
called _finitely homogeneous_. An additional finiteness condition is to
require that the class of finite substructures of $\mathfrak{B}$ is described
by finitely many forbidden substructures, in which case $\mathfrak{B}$ is
called _finitely bounded_. The class of reducts of finitely bounded
homogeneous structures is a huge generalisation of the class of all finite
structures.
There is also a generalisation of the Feder-Vardi dichotomy conjecture, which
is still open, to reducts of finitely bounded homogeneous structures. In fact,
there is a known NP-hardness condition for the CSPs of such structures which
is conjectured to be at the border between polynomial-time tractable and NP-
complete CSPs [BPP21, BOP18, BKO+17]. The _infinite-domain tractability
conjecture_ states that the CSP for every structure that does not satisfy the
mentioned hardness condition is in P (details can be found in Section 4).
There are many classes of infinite-domain structures where the infinite-domain
tractability conjecture has been verified. Often, these classes consist of the
_first-order reducts_ of some fixed underlying structure $\mathfrak{B}$. By a
_first-order reduct of $\mathfrak{B}$_ we mean a reduct of the expansion of
$\mathfrak{B}$ by all relations that are first-order definable in
$\mathfrak{B}$. For example, the infinite-domain tractability conjecture has
been verified for
1. (1)
all CSPs of structures preserved by all permutations [BK08], which is
precisely the class of first-order reducts of pure sets (structures with no
relations);
2. (2)
all CSPs of structures with a highly set-transitive automorphism group [BK09b]
(i.e., for all finite subsets $X,Y$ with $|X|=|Y|$ there exists an
automorphism which maps $X$ to $Y$), which is precisely the class of first-
order reducts of unbounded dense linear orders;
3. (3)
all CSPs of first-order reducts of the homogeneous universal poset [KP18];
4. (4)
all CSPs of first-order reducts of the binary branching C-relation [BJP17];
5. (5)
all CSPs of first-order reducts of homogeneous graphs [BMPP19];
6. (6)
all CSPs of first-order reducts of _unary structures_ , i.e., structures with
a signature that consists of finitely many unary relation symbols [BM18];
7. (7)
all CSPs expressible in MMSNP [BMM18]; MMSNP is a fragment of existential
second-order logic introduced by Feder and Vardi [FV99].
In some, but not in all cases above the polynomial-time tractability results
for $\operatorname{CSP}(\mathfrak{B})$ can be obtained by reducing
$\operatorname{CSP}(\mathfrak{B})$ to $\operatorname{CSP}(\mathfrak{T})$ where
$\mathfrak{T}$ is a certain finite structure whose domain is the set of
complete $k$-types of $\mathfrak{B}$, for a sufficiently large $k$; this
method works if the _clone of canonical polymorphisms of $\mathfrak{B}$_ does
not have a minor-preserving map to the projections [BM18]. In this case,
$\operatorname{Pol}(\mathfrak{T})$ does not have a minor-preserving map to
$\operatorname{Proj}$, and hence satisfies the condition for the algorithms of
Bulatov and of Zhuk. The polynomial-time tractability of
$\operatorname{CSP}(\mathfrak{T})$ then implies the polynomial-time
tractability of $\operatorname{CSP}(\mathfrak{B})$, by a reduction from
[BM18]. The publications cited above show that the canonical polymorphisms
determine the complexity of the CSPs in (1), (3), (5), (6), and (7), but not
for the CSPs in (2) and (4).
We introduce a property, which is implicit in [BM18, BMM18], and which we call
the _unique interpolation property (UIP)_. We present equivalent
characterisations of the unique interpolation property; in particular, we show
that it suffices to verify the unique interpolation property for binary
polymorphisms. For our characterisation we need an additional assumption,
which holds in all the mentioned examples that appear in (1)-(7) above, and
which makes the connection between CSPs and universal algebra particularly
strong: we require that $\mathfrak{B}$ is a reduct of a finitely homogeneous
_Ramsey structure_ $\mathfrak{A}$. While the assumption to be Ramsey is quite
strong, the assumption that a finitely homogeneous structure is a reduct of a
finitely homogeneous Ramsey structure is weak: in fact, it might be that
_every_ finitely bounded homogeneous structure has a finitely bounded
homogeneous Ramsey expansion (see [Bod15, BPT13]).
We use our general results to prove a new complexity dichotomy for qualitative
spatial reasoning. A fundamental formalism in spatial reasoning is RCC5
[Ben94], which involves five binary relations defined on some general set of
_regions_ ; a formal definition can be found in Section 9. Renz and Nebel
[RN01] showed that the CSP for RCC5 is NP-complete, and Nebel [Neb95] showed
that the CSP for the five basic relations of RCC5 is in P (via a reduction to
2SAT111The proof of Nebel was formulated for RCC8 instead of RCC5, but the
result for RCC5 can be shown analogously.). Renz and Nebel [RN01] have
extended the polynomial-time tractability result to a superclass of the basic
relations, and they showed that their expansion is _maximal_ in the sense that
every larger subset of the RCC5 relations has an NP-hard CSP. Drakengren and
Jonsson [JD97] classified the computational complexity of the CSP for all
subsets of the RCC5 relations. We classify the complexity of the CSP for
expansions of the basic relations of RCC5 by first-order definable relations
of arbitrary finite arity, solving the essential part of Problem 4 in [BJ17].
In particular, we verify the tractability conjecture for these CSPs, and show
that the tractable cases can be solved by reducing them to polynomial-time
tractable finite-domain CSPs using the reduction from [BM18]. For our method,
we need a homogeneous Ramsey expansion of RCC5; to prove the Ramsey property
of the expansion that we construct we apply a recent Ramsey transfer result of
Mottet and Pinsker [MP21], using the well-known fact that a certain ordered
version of the atomless Boolean algebra has the Ramsey property [KPT05].
The results mentioned so far have been announced in the proceedings of LICS
2021 [BB21a], but most proofs were omitted because of space restrictions. We
additionally show that if the condition of the infinite-domain tractability
conjecture applies to a first-order expansion $\mathfrak{B}$ of the basic
relations of RCC5, then $\operatorname{CSP}(\mathfrak{B})$ can be solved by
Datalog [FV99] (in polynomial time); otherwise, it cannot be expressed in the
much stronger _fixed-point logic with counting (FPC)_ [ABD09].
### Related Work
There exists another powerful approach to CSP complexity classification under
similar conditions, the theory of _smooth approximations_ [MP22, MNPW21]. The
theory of smooth approximations also gives a sufficient condition for the UIP
in certain cases (see the discussion at the end of Section 3.2 in [MP22]); but
the machinery and the general statements there do not seem to directly relate
to the statements that we prove here. The smooth approximations approach
proved to be very efficient in re-deriving existing classifications and
proving new classifications where a surprisingly large proportion of the work
follows from the general results of smooth approximations, and only relatively
few arguments are needed to adapt to the classes under consideration. However,
we do not know how to apply their approach to derive our spatial reasoning
classification. We believe that also the general results presented here will
be useful in other settings; one example of a different setting where the
present approach has been applied is the class of monadically stable
$\omega$-categorical structures, for which the second author confirmed the
infinite-domain tractability conjecture in his PhD thesis [Bod22], using
results of this article. In contrast to the present article, the smooth
approximations approach also provides a systematic approach to Datalog
expressibility [MNPW21]; but also here we do not know how to obtain the
particular results about Datalog expressibility in spatial reasoning from
their general theorems.
### Outline
In Section 2, Section 3, and Section 4 we introduce some fundamental notions
and results from model theory, Ramsey theory, and universal algebra,
respectively, that we need to state our results. In Section 3 we also need a
consequence of the Ramsey property of an $\omega$-categorical structure
$\mathfrak{A}$ for the existence of independent elementary substructures of
$\mathfrak{A}$ (Theorem 3.2); this result might be of independent interest in
model theory. The formal definition of the unique interpolation property and
our first general main result about the UIP with an application to complexity
classification is presented in Section 6.1. We then present two results that
facilitate the verification of the UIP in concrete settings (Section 6.2). The
applications of the general results to the spatial reasoning formalism RCC5
are in Section 9, and the classification of the respective CSPs in Datalog in
Section 10.
## 2\. Countably Categorical Structures
All structures considered in this text are assumed to be relational unless
stated otherwise. If $\mathfrak{A}$ is a structure, then $A$ denotes the
domain of $\mathfrak{A}$. A structure $\mathfrak{A}$ is called _$\omega$
-categorical_ if the set of all first-order sentences that hold in
$\mathfrak{A}$ has exactly one countable model up to isomorphism. This concept
has an equivalent characterisation based on the automorphism group
$\operatorname{Aut}(\mathfrak{A})$ of $\mathfrak{A}$. For $l\in{\mathbb{N}}$,
the _$l$ -orbit_ of $a=(a_{1},\dots,a_{l})\in A^{l}$ in
$\operatorname{Aut}(\mathfrak{A})$ is the set
$\\{\alpha(a)\colon\alpha\in\operatorname{Aut}(\mathfrak{A})\\}\text{ where
}\alpha(a)\coloneqq(\alpha a_{1},\dots,\alpha a_{l}).$
We write ${\mathcal{O}}_{l}(\mathfrak{A})$ for the set of $l$-orbits of
$\operatorname{Aut}(\mathfrak{A})$. Engeler, Svenonius, and Ryll-Nardzewski
(see, e.g., Theorem 6.3.1 in [Hod97]) proved that a countable structure is
$\omega$-categorical if and only if $\operatorname{Aut}(\mathfrak{A})$ is
_oligomorphic_ , i.e., if ${\mathcal{O}}_{l}(\mathfrak{A})$ is finite for
every $l\in{\mathbb{N}}$. In fact, $a,b\in A^{l}$ have the same $l$-orbit if
and only if they satisfy the same first-order formulas over $\mathfrak{A}$; we
write $\operatorname{typ}^{\mathfrak{A}}(a)$ for the set of all first-order
formulas satisfied by $a$ over $\mathfrak{A}$, called the _type of $a$ over
$\mathfrak{A}$_. For $B\subseteq A$, we write
$\operatorname{typ}^{\mathfrak{A}}(a/B)$ for the set of all first-order
formulas with parameters from $B$ (i.e., formulas over an expanded signature
that also contains constant symbols denoting elements from $B$) satisfied by
$a$ over $\mathfrak{A}$.
Homogeneous structures. Many examples of $\omega$-categorical structures arise
from structures $\mathfrak{A}$ that are _homogeneous_. A relational structure
is _homogeneous_ if every isomorphism between finite substructures of
$\mathfrak{A}$ can be extended to an automorphism of $\mathfrak{A}$. It
follows from the above that if $\mathfrak{A}$ is finitely homogeneous, then
$\mathfrak{A}$ is $\omega$-categorical. If $a\in A^{n}$, we write
$\operatorname{qf-typ}^{\mathfrak{A}}(a)$ for the set of all quantifier-free
formulas that hold on $a$ over $\mathfrak{A}$. Clearly, in homogeneous
structures $\mathfrak{A}$ the quantifier-free type of $a$ determines the type
of $a$. Also note that every $\omega$-categorical structure can be turned into
a homogeneous $\omega$-categorical structure by expanding it by all first-
order definable relations.
Every homogeneous structure $\mathfrak{A}$ is uniquely given (up to
isomorphism) by its _age_ , i.e., by the class
$\operatorname{Age}(\mathfrak{A})$ of all finite structures that embed into
$\mathfrak{A}$ (see, e.g., Lemma 6.1.4 in [Hod97]). Conversely, every class
$\mathcal{C}$ of structures with finite relational signature which is an
_amalgamation class_ , i.e., is closed under isomorphism, substructures, and
which has the _amalgamation property_ , is the age of a homogeneous structure,
which we call the _Fraïssé-limit of ${\mathcal{C}}$_ (see, e.g., Theorem 6.1.2
in [Hod97]). We say that $\mathfrak{B}$ is _finitely bounded_ if there exists
a finite set of finite structures $\mathcal{F}$ such that the age of
$\mathfrak{B}$ equals the class of all finite structures $\mathfrak{A}$ such
that no structure from $\mathcal{F}$ embeds into $\mathfrak{A}$.
Model-complete cores. An $\omega$-categorical structure $\mathfrak{C}$ is
_model-complete_ if every self-embedding
$e\colon\mathfrak{C}\hookrightarrow\mathfrak{C}$ preserves all first-order
formulas. An $\omega$-categorical structure $\mathfrak{C}$ is called a _core_
if every endomorphism of $\mathfrak{C}$ is a self-embedding of $\mathfrak{C}$.
If there is a homomorphism from a structure $\mathfrak{A}$ to a structure
$\mathfrak{B}$ and vice versa, then $\mathfrak{A}$ and $\mathfrak{B}$ are
called _homomorphically equivalent_. Every $\omega$-categorical structure
$\mathfrak{A}$ is homomorphically equivalent to a model-complete core, which
is unique up to isomorphism, and again $\omega$-categorical, and which is
called _the_ model-complete core of $\mathfrak{A}$ [Bod07, BHM12].
## 3\. Ramsey Theory
For $\tau$-structures $\mathfrak{A},\mathfrak{B}$ we write
${\mathfrak{B}\choose\mathfrak{A}}$ for the set of all embeddings of
$\mathfrak{A}$ into $\mathfrak{B}$. If
$\mathfrak{A},\mathfrak{B},\mathfrak{C}$ are $\tau$-structures and
$r\in{\mathbb{N}}$ then we write
$\mathfrak{C}\to(\mathfrak{B})^{\mathfrak{A}}_{r}$
if for every function
$\chi\colon{\mathfrak{C}\choose\mathfrak{A}}\to\\{1,\dots,r\\}$ (also referred
to as a _colouring_ , where $\\{1,\dots,r\\}$ are the different colours) there
exists $e\in{\mathfrak{C}\choose\mathfrak{B}}$ such that $\chi$ is constant on
$\left\\{e\circ f\colon
f\in{\mathfrak{B}\choose\mathfrak{A}}\right\\}\subseteq{\mathfrak{C}\choose\mathfrak{A}}.$
###### Definition 1.
A class of finite $\tau$-structures has the _Ramsey property_ if for all
$\mathfrak{A},\mathfrak{B}$ and $r\in{\mathbb{N}}$ there exists $\mathfrak{C}$
such that $\mathfrak{C}\to(\mathfrak{B})^{\mathfrak{A}}_{r}$. A homogeneous
structure has the Ramsey property if its age has the Ramsey property.
Let $r\in{\mathbb{N}}$. We write $\mathfrak{C}\to(\mathfrak{B})_{r}$ if for
every $\chi\colon C^{|B|}\to\\{1,\dots,r\\}$ there exists
$e\colon\mathfrak{B}\hookrightarrow\mathfrak{C}$ such that if
$\operatorname{qf-typ}^{\mathfrak{B}}(s)=\operatorname{qf-
typ}^{\mathfrak{B}}(t)$ for $s,t\in B^{|B|}$, then $\chi(e(s))=\chi(e(t))$.
The Ramsey property has the following well-known consequence (see, e.g.,
Proposition 2.21 in [Bod15]).
###### Lemma 3.1.
Let $\mathcal{C}$ be a Ramsey class. Then for every $r\in{\mathbb{N}}$ and
$\mathfrak{B}\in\mathcal{C}$ there exists $\mathfrak{C}\in\mathcal{C}$ such
that $\mathfrak{C}\to(\mathfrak{B})_{r}$.
Another important consequence of the Ramsey property is presented in Section
5.3. We also need the following consequence of the Ramsey property (in the
proof of Proposition 7.3), which appears to be new. Let $\mathfrak{D}$ be a
structure and let $A,B\subseteq D$. We say that $A$ is _independent from $B$
in $\mathfrak{D}$_ if for all $\bar{a},\bar{b}\in A^{n}$, if
$\operatorname{typ}^{\mathfrak{D}}(\bar{a})=\operatorname{typ}^{\mathfrak{D}}(\bar{b})$,
then
$\operatorname{typ}^{\mathfrak{D}}(\bar{a}/B)=\operatorname{typ}^{\mathfrak{D}}(\bar{b}/B)$.
Two substructures $\mathfrak{A},\mathfrak{B}$ of $\mathfrak{D}$ are called
_independent_ if $A$ is independent from $B$ in $\mathfrak{D}$ and $B$ is
independent from $A$ in $\mathfrak{D}$. A substructure $\mathfrak{B}$ of a
$\tau$-structure $\mathfrak{A}$ is called _elementary_ if the identity mapping
from $\mathfrak{B}$ to $\mathfrak{A}$ preserves all first-order
$\tau$-formulas.
###### Theorem 3.2.
Let $\mathfrak{A}$ be a countable homogeneous $\omega$-categorical Ramsey
structure. Then $\mathfrak{A}$ contains two independent elementary
substructures.
In the proof of Theorem 3.2 we use the following consequence of the
compactness theorem of first-order logic.
###### Proposition 3.3.
Let $\mathfrak{A}$ be a countable $\omega$-categorical homogeneous structure.
Then the following are equivalent.
1. (1)
$\mathfrak{A}$ contains two independent elementary substructures.
2. (2)
For all $\mathfrak{B}_{1},\mathfrak{B}_{2}\in\operatorname{Age}(\mathfrak{A})$
there exists embeddings
$e_{i}\colon\mathfrak{B}_{i}\hookrightarrow\mathfrak{A}$, for $i\in\\{1,2\\}$,
such that $e_{1}(B_{1})$ and $e_{2}(B_{2})$ are independent in $\mathfrak{A}$.
###### Proof.
The forward implication is immediate from the definitions. For the converse
implication, let $a_{1},a_{2},\dots$ be an enumeration of $A$ and let
$B=\\{b_{1},b_{2},\dots\\}$ and $C=\\{c_{1},c_{2},\dots\\}$ be sets of new
constant symbols. Let $\Phi$ be the set of all sentences that express that for
all $n\in{\mathbb{N}}$, $\bar{b},\bar{b}^{\prime}\in B^{n}$, and
$\bar{c},\bar{c}^{\prime}\in C^{n}$
* •
if
$\operatorname{typ}^{\mathfrak{A}}(\bar{b})=\operatorname{typ}^{\mathfrak{A}}(\bar{b}^{\prime})$
then
$\operatorname{typ}^{\mathfrak{A}}(\bar{b},\bar{c})=\operatorname{typ}^{\mathfrak{A}}(\bar{b}^{\prime},\bar{c})$.
* •
if
$\operatorname{typ}^{\mathfrak{A}}(\bar{c})=\operatorname{typ}^{\mathfrak{A}}(\bar{c}^{\prime})$
then
$\operatorname{typ}^{\mathfrak{A}}(\bar{c},\bar{b})=\operatorname{typ}^{\mathfrak{A}}(\bar{c}^{\prime},\bar{b})$.
For $n\in{\mathbb{N}}$ let $\phi_{n}(x_{1},\dots,x_{n})$ be a formula that
expresses that
$\operatorname{typ}^{\mathfrak{A}}(x_{1},\dots,x_{n})=\operatorname{typ}^{\mathfrak{B}}(a_{1},\dots,a_{n})$.
By assumption, all finite subsets of
$T\coloneqq\operatorname{Th}(\mathfrak{A})\cup\Phi\cup\\{\phi_{n}(b_{1},\dots,b_{n})\wedge\phi_{n}(c_{1},\dots,c_{n}):n\in{\mathbb{N}}\\}$
are satisfiable. By compactness of first-order logic, it follows that $T$ has
a model $\mathfrak{A}^{\prime}$. By the downward Löwenheim-Skolem theorem, we
may assume that $\mathfrak{A}^{\prime}$ is countably infinite. The reduct of
$\mathfrak{A}^{\prime}$ with the same signature as $\mathfrak{A}$ satisfies
$\operatorname{Th}(\mathfrak{A})$, and since $\mathfrak{A}$ is
$\omega$-categorical this reduct is isomorphic to $\mathfrak{A}$, so that we
may assume that $\mathfrak{A}^{\prime}$ is an expansion of $\mathfrak{A}$. Let
$\mathfrak{B}$ be the substructure of $\mathfrak{A}$ induced by the constants
from $B$. Since
$\mathfrak{A}^{\prime}\models\\{\phi_{n}(b_{1},\dots,b_{n}):n\in{\mathbb{N}}\\}$
and by the homogeneity of $\mathfrak{A}$ we have that $\mathfrak{B}$ is an
elementary substructure of $\mathfrak{A}$. Likewise, the substructure
$\mathfrak{C}$ of $\mathfrak{A}$ induced by the constants from $C$ is an
elementary substructure of $\mathfrak{A}$. Since
$\mathfrak{A}^{\prime}\models\Phi$ the two substructures $\mathfrak{B}$ and
$\mathfrak{C}$ are independent. ∎
To find independent sets, we use the Ramsey property via the following lemma.
###### Lemma 3.4.
Let $\mathfrak{A}$ be a countable homogeneous $\omega$-categorical Ramsey
structure, $C\subseteq A$ finite, and
$\mathfrak{B}\in\operatorname{Age}(\mathfrak{A})$. Then there exists
$\mathfrak{D}\in\operatorname{Age}(\mathfrak{A})$ such that for every
$e\colon\mathfrak{D}\hookrightarrow\mathfrak{A}$ there exists
$f\colon\mathfrak{B}\hookrightarrow\mathfrak{A}$ such that $f(B)\subseteq
e(D)$ and for all $\bar{a},\bar{b}\in B^{|B|}$, if
$\operatorname{typ}^{\mathfrak{A}}(f(\bar{a}))=\operatorname{typ}^{\mathfrak{A}}(f(\bar{b}))$
then
$\operatorname{typ}^{\mathfrak{A}}(f(\bar{a})/C)=\operatorname{typ}^{\mathfrak{A}}(f(\bar{b})/C)$.
###### Proof.
Let $C=\\{c_{1},\dots,c_{n}\\}$. Let $r$ be the number of types of
$|B|+n$-tuples in $\mathfrak{A}$. Since $\mathfrak{A}$ is homogeneous Ramsey
there exists $\mathfrak{D}\in\operatorname{Age}(\mathfrak{A})$ such that
$\mathfrak{D}\to(\mathfrak{B})_{r}$. We claim that $\mathfrak{D}$ satisfies
the statement of the lemma. Let
$e\colon\mathfrak{D}\hookrightarrow\mathfrak{A}$ be an embedding. We color
$e(D)^{|B|}$ as follows: the colour of $t\in e(D)^{|B|}$ is the orbit of
$(t_{1},\dots,t_{|B|},c_{1},\dots,c_{n})$. There are at most $r$ such orbits.
Since $\mathfrak{D}\to(\mathfrak{B})_{r}$ we get that there exists an
$f\colon\mathfrak{B}\hookrightarrow\mathfrak{A}$ such that $f(B)\subseteq
e(D)$ and if $\bar{a},\bar{b}\in B^{|B|}$ are such that
$\operatorname{typ}^{\mathfrak{A}}(f(\bar{a}))=\operatorname{typ}^{\mathfrak{A}}(f(\bar{b}))$,
then
$\operatorname{typ}^{\mathfrak{A}}(f(\bar{a})/C)=\operatorname{typ}^{\mathfrak{A}}(f(\bar{b})/C)$.
∎
###### Proof of Theorem 3.2.
We use Proposition 3.3. Let
$\mathfrak{B}_{1},\mathfrak{B}_{2}\in\operatorname{Age}(\mathfrak{A})$; we may
assume that $\mathfrak{B}_{1}$ and $\mathfrak{B}_{2}$ are substructures of
$\mathfrak{A}$. Then Lemma 3.4 implies that there exists
$\mathfrak{D}_{1}\in\operatorname{Age}(\mathfrak{A})$ such that for every
embedding $e_{1}\colon\mathfrak{D}_{1}\hookrightarrow\mathfrak{A}$ there
exists an embedding $f_{1}\colon\mathfrak{B}_{1}\hookrightarrow\mathfrak{A}$
such that $f_{1}(B_{1})\subseteq e_{1}(D_{1})$ and for all $n\in{\mathbb{N}}$,
$b,b^{\prime}\in(B_{1})^{n}$
(1) $\displaystyle\text{ if
}\operatorname{typ}^{\mathfrak{A}}(f_{1}(b))=\operatorname{typ}^{\mathfrak{A}}(f_{1}(b^{\prime}))$
$\displaystyle\text{ then
}\operatorname{typ}^{\mathfrak{A}}(f_{1}(b)/B_{2})=\operatorname{typ}^{\mathfrak{A}}(f_{1}(b^{\prime})/B_{2}).$
We may assume that $\mathfrak{D}_{1}$ is a substructure of $\mathfrak{A}$.
Another application of Lemma 3.4 gives us an embedding
$f_{2}\colon\mathfrak{B}_{2}\hookrightarrow\mathfrak{A}$ such that for all
$n\in{\mathbb{N}}$, $b,b^{\prime}\in(B_{2})^{n}$, if
$\operatorname{typ}^{\mathfrak{A}}(f_{2}(b))=\operatorname{typ}^{\mathfrak{A}}(f_{2}(b^{\prime}))$
then
$\operatorname{typ}^{\mathfrak{A}}(f_{2}(b)/D_{1})=\operatorname{typ}^{\mathfrak{A}}(f_{2}(b^{\prime})/D_{1})$.
By the homogeneity of $\mathfrak{A}$, there exists
$\alpha\in\operatorname{Aut}(\mathfrak{A})$ extending $f_{2}$. The property of
$\mathfrak{D}_{1}$ implies that there exists an embedding
$f_{1}\colon\mathfrak{B}_{1}\hookrightarrow\mathfrak{A}$ such that
$f_{1}(B_{1})\subseteq\alpha^{-1}(D_{1})$ such that (1) holds for all
$b,b^{\prime}\in(B_{1})^{n}$.
We claim that $f_{1}(B_{1})$ and $B_{2}$ are independent. Clearly,
$f_{1}(B_{1})$ is independent from $B_{2}$. To show that $B_{2}$ is
independent from $f_{1}(B_{1})$, let $n\in{\mathbb{N}}$ and
$b,b^{\prime}\in(B_{2})^{n}$ be such that
$\operatorname{typ}^{\mathfrak{A}}(b)=\operatorname{typ}^{\mathfrak{A}}(b^{\prime})$.
Then
$\operatorname{typ}^{\mathfrak{A}}(\alpha(b))=\operatorname{typ}^{\mathfrak{A}}(\alpha(b^{\prime}))$,
and hence that
$\operatorname{typ}^{\mathfrak{A}}(\alpha(b)/D_{1})=\operatorname{typ}^{\mathfrak{A}}(\alpha(b^{\prime})/D_{1})$
by the choice of $f_{2}$. This in turn implies that
$\operatorname{typ}^{\mathfrak{A}}(b/\alpha^{-1}(D_{1}))=\operatorname{typ}^{\mathfrak{A}}(b^{\prime}/\alpha^{-1}(D_{1}))$
which implies the statement since $f_{1}(B_{1})\subseteq\alpha^{-1}(D_{1})$. ∎
## 4\. Oligomorphic Clones
In this section we introduce fundamental concepts and results about
polymorphism clones of finite and countably infinite $\omega$-categorical
structures that are needed for the statement of our results.
The _$k$ -th direct power_ of a structure $\mathfrak{A}$ (also called the
_categorical power_), denoted by $\mathfrak{A}^{k}$. It is the
$\tau$-structure with domain $A^{k}$ such that
$((a_{1,1},\dots,a_{1,k}),\dots,(a_{n,1},\dots,a_{n,k}))\in
R^{\mathfrak{A}^{k}}$
if and only if $(a_{1,i},\dots,a_{n,i})\in R^{\mathfrak{A}}$ for every
$i\in\\{1,\dots,k\\}$. A _polymorphism_ of $\mathfrak{A}$ is a homomorphism
from a finite direct power $\mathfrak{A}^{k}$ to $\mathfrak{A}$. The set of
all polymorphisms of $\mathfrak{A}$, denoted by
$\operatorname{Pol}(\mathfrak{A})$, forms a _clone (over $A$)_, i.e., it is
closed under composition and contains for every $k$ and $i\leq k$ the
projection $\pi^{k}_{i}$ of arity $k$ that always returns the $i$-th argument.
Moreover, $\operatorname{Pol}(\mathfrak{A})$ is closed with respect to the
_topology of pointwise convergence_ , i.e., if $f\colon A^{k}\to A$ is such
that for every finite $F\subseteq A$ there exists
$g\in\operatorname{Pol}(\mathfrak{A})$ such that
$g(a_{1},\dots,a_{k})=f(a_{1},\dots,a_{k})$ for all $a_{1},\dots,a_{k}\in F$,
then $f\in\operatorname{Pol}(\mathfrak{A})$. If ${\mathscr{S}}$ is a set of
operations over $A$, then $\overline{\mathscr{S}}$ denotes the smallest closed
set that contains ${\mathscr{S}}$. If ${\mathscr{S}}$ is a set of operations,
we write ${\mathscr{S}}^{(k)}$ for the set of operation in ${\mathscr{S}}$ of
arity $k$. The clone of all projections on $\\{0,1\\}$ is denoted by
$\operatorname{Proj}$.
###### Definition 2.
A function $\zeta\colon{\mathscr{C}}_{1}\to{\mathscr{C}}_{2}$ between two
clones over a countably infinite set $A$ is called
* •
_minor-preserving_ if
$\zeta\big{(}f(\pi^{k}_{i_{1}},\dots,\pi^{k}_{i_{n}})\big{)}=\zeta(f)(\pi^{k}_{i_{1}},\dots,\pi^{k}_{i_{n}})$
for all $f\in{\mathscr{C}}^{(n)}_{1}$ and
$i_{1},\dots,i_{n}\in\\{1,\dots,k\\}$.
* •
_uniformly continuous_ if and only if for any finite $G\subseteq A$ there is a
finite $F\subseteq A$ such that for all $n\in{\mathbb{N}}$ and
$f,g\in{\mathscr{C}}_{1}^{(n)}$, if $f|_{F^{n}}=g|_{F^{n}}$ then
$\zeta(f)|_{G^{n}}=\zeta(g)|_{G^{n}}$.
The set ${\mathscr{O}}_{A}\coloneqq\bigcup_{k\in{\mathbb{N}}}(A^{k}\to A)$ of
all operations on $A$ can be equipped with a complete metric such that the
closed clones with respect to this metric are precisely the polymorphism
clones of relational structures with domain $A$, and such that the uniformly
continuous maps are precisely the maps that are uniformly continuous with
respect to this metric in the standard sense, justifying our terminology; see,
e.g., [Bod21]. Now suppose that $\mathfrak{B}$ is a finite structure with a
finite relational signature; the following results imply that the existence of
a minor-preserving map from $\operatorname{Pol}(\mathfrak{B})$ to
$\operatorname{Proj}$ describes the border between containment in P and NP-
completeness.
###### Theorem 4.1 (follows from [BKJ05, BOP18]).
Let $\mathfrak{B}$ be a structure with a finite domain and a finite relational
signature. If there is a minor-preserving map
$\operatorname{Pol}(\mathfrak{B})\to\operatorname{Proj}$ then
$\operatorname{CSP}(\mathfrak{B})$ is NP-hard.
###### Theorem 4.2 ([MM08]).
Let $\mathfrak{B}$ be a structure with a finite domain. If there is no minor-
preserving map $\operatorname{Pol}(\mathfrak{B})\to\operatorname{Proj}$ then
$\operatorname{Pol}(\mathfrak{B})$ contains a _weak near unanimity (WNU)
operation_ , i.e., an operation $f$ of some arity $k\geq 2$ that satisfies for
all $a,b\in B$ that
$f(a,\dots,a,b)=f(a,\dots,b,a)=\dots=f(b,a,\dots,a).$
###### Theorem 4.3 (Bulatov [Bul17], Zhuk [Zhu17]).
Let $\mathfrak{B}$ be a structure with a finite domain and a finite relational
signature. If $\mathfrak{B}$ has a WNU polymorphism, then
$\operatorname{CSP}(\mathfrak{B})$ is in P.
The hardness result in Theorem 4.1 can be generalised to a large class of
countable structures $\mathfrak{B}$.
###### Theorem 4.4 ([BOP18]).
Let $\mathfrak{B}$ be a countable $\omega$-categorical structure with finite
relational signature such that $\operatorname{Pol}(\mathfrak{B})$ has a
uniformly continuous minor-preserving map to $\operatorname{Proj}$. Then
$\operatorname{CSP}(\mathfrak{B})$ is NP-hard.
The present paper is motivated by the following conjecture from [BPP21], in a
reformulation from [BKO+17], which generalises the Bulatov-Zhuk dichotomy
(Theorem 4.3).
###### Conjecture 4.5.
Let $\mathfrak{B}$ be a reduct of a countable finitely bounded homogeneous
structure such that there is no uniformly continuous minor-preserving map from
$\operatorname{Pol}(\mathfrak{B})$ to $\operatorname{Proj}$. Then
$\operatorname{CSP}(\mathfrak{B})$ is in P.
## 5\. Canonical Functions
Some of the strongest partial results towards Conjecture 4.5 is based on the
concept of _canonical_ polymorphisms. Let $\mathfrak{A}$ be an
$\omega$-categorical structure. An operation $f\colon A^{k}\to A$ is called
_canonical over ${\mathfrak{A}}$_ if for all $n\in{\mathbb{N}}$ and
$a_{1},\dots,a_{k},b_{1},\dots,b_{k}\in A^{n}$, if
$\operatorname{typ}^{\mathfrak{A}}(a_{i})=\operatorname{typ}^{\mathfrak{A}}(b_{i})$
for all $i\in\\{1,\dots,k\\}$, then
$\operatorname{typ}^{\mathfrak{A}}(f(a_{1},\dots,a_{k}))=\operatorname{typ}^{\mathfrak{A}}(f(b_{1},\dots,b_{k}))$
(the function is applied to the tuples componentwise). Note that if
$\mathfrak{A}$ is a finitely homogeneous structure, and $m$ is the maximal
arity of (the relations of) $\mathfrak{A}$, then $f$ is canonical with respect
to $\mathfrak{A}$ if and only if it is satisfies the condition above for
$n=m$. The set of all canonical operations over $\mathfrak{A}$ is denoted by
$\operatorname{Can}(\mathfrak{A})$; note that
$\operatorname{Can}(\mathfrak{A})$ is a clone and closed.
### 5.1. Behaviours
To conveniently work with canonical functions and with functions that locally
resemble canonical functions, we need the concept of a _behaviour_ of
functions $f\colon A^{k}\to A$ over a structure $\mathfrak{A}$, following
[BP11, BP21]. In order to have some flexibility when using the terminology, we
work in the setting of _partial functions_ from $A^{k}\to A$, i.e., functions
that are only defined on a subset of $A^{k}$.
###### Definition 3 (Behaviours).
Let $n,k\in{\mathbb{N}}$. Then an _$n$ -behaviour over $\mathfrak{A}$ (of a
$k$-ary partial map from $A^{k}$ to $A$)_ is a partial function from
${\mathcal{O}_{n}(\mathfrak{A})}^{k}$ to ${\mathcal{O}}_{n}(\mathfrak{A})$. A
_behaviour over $\mathfrak{A}$_ is a partial function $\mathcal{B}$ from
$\bigcup_{i=1}^{\infty}{\mathcal{O}_{i}(\mathfrak{A})}^{k}$ to
$\bigcup_{i=1}^{\infty}{\mathcal{O}}_{i}(\mathfrak{A})$ so that for every $i$
we have
$\mathcal{B}({\mathcal{O}}_{i}(\mathfrak{A})^{k})\subseteq{\mathcal{O}}_{i}(\mathfrak{A})$.
An $n$-behaviour (behaviour) is called _complete_ if it is defined on all of
${\mathcal{O}}_{n}(\mathfrak{A})^{k}$ (on all of
$\bigcup_{i=1}^{\infty}{\mathcal{O}_{i}(\mathfrak{A})^{k}}$).
Note that every $n$-behaviour is in particular a behaviour.
###### Definition 4 (Behaviours of functions).
Let $\mathfrak{A}$ be a structure and $k\in{\mathbb{N}}$. A partial function
$f\colon A^{k}\to A$ _realises_ a behaviour $\mathcal{B}$ over $\mathfrak{A}$
if for all $a^{1}_{1},\dots,a^{k}_{n}\in A$, if $O_{i}$ is the orbit of
$(a^{i}_{1},\dots,a^{i}_{n})$ for $i\in\\{1,\dots,k\\}$, then
${\mathcal{B}}(O_{1},\dots,O_{k})$ is the orbit of
$(f(a^{1}_{1},\dots,a^{k}_{1}),\dots,f(a^{1}_{n},\dots,a^{k}_{n}))$.
If $S\subseteq A^{k}$ and $\mathcal{B}$ is a behaviour over $\mathfrak{A}$,
then we say that $f\colon A^{k}\to A$ _realises $\mathcal{B}$ on $S$_ if the
restriction $f|_{S}$ realises $\mathcal{B}$. The following proposition can be
shown by a standard compactness argument.
###### Proposition 5.1.
Let $\mathfrak{A}$ be an $\omega$-categorical structure and
$k,n\in{\mathbb{N}}$. Then an $n$-behaviour over $\mathfrak{A}$ is realised by
some $f\in\operatorname{Pol}(\mathfrak{A})^{(k)}$ if and only if it is
realised by some $f\in\operatorname{Pol}(\mathfrak{A})^{(k)}$ on $F^{k}$ for
every finite subset $F\subseteq A$.
### 5.2. Canonicity
Let $n,k\in{\mathbb{N}}$. A partial function $f\colon A^{k}\to A$ is called
_$n$ -canonical over $\mathfrak{A}$_ if it realises a complete $n$-behaviour
over $\mathfrak{A}$, and _canonical over $\mathfrak{A}$_ if it realises some
complete behaviour over $\mathfrak{A}$; note that this is consistent with the
definition given in Section 4. If $S\subseteq A$, then we say that $f$ is _(
$n$-) canonical on $S$_ if the restriction $f|_{S}$ is ($n$-) canonical.
Suppose that $\mathfrak{B}$ is a homogeneous relational structure with maximal
arity $m$. If $f\colon B\to C$ realises some complete $m^{\prime}$-behaviour
over $(\mathfrak{B},\mathfrak{C})$ for $m^{\prime}=\max(m,2)$, then $f$ is
canonical over $(\mathfrak{B},\mathfrak{C})$ (in other words, in this
situation a complete $m$-behaviour uniquely determines a complete behaviour;
the requirement $m^{\prime}\geq 2$ is necessary so that the
$m^{\prime}$-behaviour allows, for instance, to distinguish constant functions
from injective functions). Hence, if $\mathfrak{B}$ is finitely homogeneous,
then there are only finitely many complete behaviours for functions of arity
$k$.
Note that the binary relation $\\{(u,\alpha(u))\colon u\in
A^{n},\alpha\in\operatorname{Aut}(\mathfrak{A})\\}$ is a congruence of
$\operatorname{Can}(\mathfrak{A})$, and so there is a uniformly continuous
clone homomorphism $\xi^{\mathfrak{A}}_{n}$ from
$\operatorname{Can}(\mathfrak{A})$ to a clone whose domain
${\mathcal{O}}_{n}(\mathfrak{A})$ is the (finite) set of congruence classes,
and which we denote by
$\xi^{\mathfrak{A}}_{n}(\operatorname{Can}(\mathfrak{A}))$. If
${\mathscr{C}}\subseteq\operatorname{Can}(\mathfrak{A})$, then we write
$\xi^{\mathfrak{A}}_{n}({\mathscr{C}})$ for the subclone of
$\xi^{\mathfrak{A}}_{n}(\operatorname{Can}(\mathfrak{A}))$ induced by the
image of $\mathscr{C}$ under $\xi^{\mathfrak{A}}_{n}$. Note that if
$f\in\operatorname{Can}(\mathfrak{A})$, then $g\colon A^{k}\to A$ has the same
behaviour as $f$ over $\mathfrak{A}$ if and only if $f$ interpolates $g$ and
$g$ interpolates $f$ over $\mathfrak{A}$.
###### Lemma 5.2 (Proposition 6.6 in [BPP21]).
Let $\mathfrak{A}$ be a finitely homogeneous structure. Let $m$ be the maximal
arity of $\mathfrak{A}$ and suppose that
$f,g\in\operatorname{Can}(\mathfrak{A})$ are such that
$\xi^{\mathfrak{A}}_{m}(f)=\xi^{\mathfrak{A}}_{m}(g)$. Then there are
$e_{1},e_{2}\in\overline{\operatorname{Aut}(\mathfrak{A})}$ such that
$e_{1}\circ f=e_{2}\circ g$.
The following proposition can be shown by a compactness argument.
###### Lemma 5.3.
Let $\mathfrak{B}$ be a first-order reduct of an $\omega$-categorical
structure $\mathfrak{A}$. Let $k,n,m\in{\mathbb{N}}$ and let
${\mathcal{B}}_{1},\dots,{\mathcal{B}}_{m}$ be $n$-behaviours over
$(\mathfrak{A}^{(k)},\mathfrak{A})$. Suppose that for every finite $F\subseteq
A$ and $i\in\\{1,\dots,m\\}$ there exist $\alpha_{i,1},\dots,\alpha_{i,k}$ and
$f\in\operatorname{Pol}(\mathfrak{B})$ such that
$f(\alpha_{i,1},\dots,\alpha_{i,k})$ realises ${\mathcal{B}}_{i}$ on $F^{k}$.
Then there exist
$e_{1,1},\dots,e_{m,k}\in\overline{\operatorname{Aut}(\mathfrak{A})}$ and
$f\in\operatorname{Pol}(\mathfrak{B})$ such that for every
$i\in\\{1,\dots,m\\}$ the operation $f(e_{i,1},\dots,e_{i,k})$ realises
${\mathcal{B}}_{i}$ on all of $A^{k}$. Moreover, if
$I\subseteq\\{1,\dots,k\\}$ and $\alpha_{i,j}=\operatorname{id}$ for all $j\in
I$ and every finite $X\subseteq A$ then we can guarantee
$e_{i,j}=\operatorname{id}$ for all $j\in I$.
### 5.3. Canonisation
The following is from [BPT13]; also see [BP21].
###### Lemma 5.4 (Canonisation lemma).
Let $\mathfrak{A}$ be a countable $\omega$-categorical homogeneous Ramsey
structure and let $f\colon A^{k}\to A$. Then there is a canonical function
over $\mathfrak{A}$ which is interpolated by $f$ over
$\operatorname{Aut}(\mathfrak{A})$.
In some situations it is practical to use a finite version of the canonisation
lemma (e.g. in the proof of Lemma 7.5 or Lemma 8.1), which can be derived
easily from Lemma 5.4 by a compactness argument, or directly from Lemma 3.1.
###### Lemma 5.5.
Let $\mathfrak{A}$ be a homogeneous $\omega$-categorical Ramsey structure and
$B\subseteq A$ finite. Then there exists a finite $C\subseteq A$ such that for
every $f\colon A^{k}\to A$ there are
$\alpha_{1},\dots,\alpha_{k}\in\operatorname{Aut}(\mathfrak{A})$ such that
$\alpha_{1}(B)\subseteq C,\dots,\alpha_{k}(B)\subseteq C$ and
$f(\alpha_{1},\dots,\alpha_{k})$ is canonical on $B^{k}$.
### 5.4. Complexity
The following theorem is essentially from [BM18] and uses [Bul17, Zhu17];
since we are not aware of a formulation of the result in the form below, which
is the form that is most useful for the purposes of this article, we added a
guide to the literature in the proof.
###### Theorem 5.6 ([BM18]).
Let $\mathfrak{B}$ be a reduct of a countable finitely bounded homogeneous
structure $\mathfrak{A}$ such that there is no uniformly continuous minor-
preserving map from
$\operatorname{Pol}(\mathfrak{B})\cap\operatorname{Can}(\mathfrak{A})$ to
$\operatorname{Proj}$. Then $\operatorname{CSP}(\mathfrak{B})$ is in P.
###### Proof.
Let $m$ be the maximal arity of the relations of $\mathfrak{B}$. If
$\xi_{m}^{\mathfrak{A}}$ has a (uniformly continuous) minor-preserving map to
$\operatorname{Proj}$ then so has $\operatorname{Pol}(\mathfrak{B})$ by
composing the maps, contrary to our assumptions. Therefore,
$\xi_{m}^{\mathfrak{A}}$ contains a _Siggers operation_ [Sig10]. Lemma 5.2
implies that
$\operatorname{Pol}(\mathfrak{B})\cap\operatorname{Can}(\mathfrak{A})$ has a
_Siggers operation modulo $\overline{\operatorname{Aut}(\mathfrak{A})}$_
[BP20]. The polynomial-time tractability of $\operatorname{CSP}(\mathfrak{B})$
then follows from Corollary 4.13 in [BM18], which uses [Bul17, Zhu17]. ∎
For many classes of reducts of finitely bounded homogeneous structures, the
conditions given in Theorem 4.4 and in Theorem 5.6 cover all the cases. The
next section presents a condition from [BM18] that turned out to be highly
relevant to understand this phenomenon.
## 6\. Results
In this section we state the main results of the paper that we then prove in
Section 7 and in Section 8. The applications to RCC5 in spatial reasoning can
be found in Section 9.
### 6.1. The Unique Interpolation Property
Let $\mathscr{M}$ be a set of function from $A$ to $A$ and let $f,g\colon
A^{k}\to A$ be operations. We say that _$f$ interpolates $g$ over
${\mathscr{M}}$_ if for every finite $F\subseteq A$ there are
$u,v_{1},\dots,v_{k}\in{\mathscr{M}}$ such that
$g(a_{1},\dots,a_{k})=u(f(v_{1}(a_{1}),\dots,v_{k}(a_{k})))$ for all
$a_{1},\dots,a_{k}\in F$.
###### Definition 5.
Let $\mathfrak{A}$ be a structure and let $\mathscr{C}$ be an operation clone
that contains $\operatorname{Aut}(\mathfrak{A})$. A function $\zeta$ defined
on $\mathscr{D}\subseteq\mathscr{C}$ has the _unique interpolation property
(UIP) with respect to ${\mathscr{C}}$ over $\mathfrak{A}$_ if
$\zeta(g)=\zeta(h)$ for all $g,h\in\mathscr{D}$ that are interpolated by the
same operation $f\in{\mathscr{C}}$ over $\operatorname{Aut}(\mathfrak{A})$.
The main result of this section, Lemma 6.1, gives a sufficient condition for
the existence of an extension of a minor-preserving map defined on the
canonical polymorphisms of a structure $\mathfrak{B}$ to a uniformly
continuous minor-preserving map defined on all of
$\operatorname{Pol}(\mathfrak{B})$. The statement is inspired by Theorem 5.5
in [BM18], and similar statements played an important role in [BMM18].
However, the original proof in [BM18] contains a mistake (in the proof the
mashup theorem, Theorem 5.5 in [BM18], more specifically in the claim that
$\phi$ is a minor-preserving map) and fixing this with a general approach is
one of the contributions of the present article.
###### Lemma 6.1 (The extension lemma).
Let $\mathfrak{B}$ be a reduct of a countable finitely homogeneous Ramsey
structure $\mathfrak{A}$ and let $\mathscr{D}$ be an operation clone. Suppose
that
$\zeta\colon\operatorname{Pol}(\mathfrak{B})\cap\operatorname{Can}(\mathfrak{A})\to\mathscr{D}$
is minor-preserving and has the UIP with respect to
$\operatorname{Pol}(\mathfrak{B})$ over $\mathfrak{A}$. Then $\zeta$ can be
extended to a uniformly continuous minor-preserving map
$\tilde{\zeta}\colon\operatorname{Pol}(\mathfrak{B})\to\mathscr{D}$.
The extension lemma motivates the study of the UIP in the context of
complexity classification of CSPs, and we obtain the following in Section 7.4.
###### Corollary 6.2.
Let $\mathfrak{C}$ be a reduct of a finitely bounded homogeneous structure
$\mathfrak{B}$, which is itself a reduct of a finitely homogeneous Ramsey
structure $\mathfrak{A}$. Suppose that whenever there is a uniformly
continuous minor-preserving map from
$\operatorname{Pol}(\mathfrak{C})\cap\operatorname{Can}(\mathfrak{B})$ to
$\operatorname{Proj}$ then there is also a minor-preserving map from
$\operatorname{Pol}(\mathfrak{C})\cap\operatorname{Can}(\mathfrak{A})$ to
$\operatorname{Proj}$ which has the UIP with respect to
$\operatorname{Pol}(\mathfrak{C})$ over $\mathfrak{A}$. Then
$\operatorname{CSP}(\mathfrak{C})$ is in P or NP-complete.
### 6.2. Other Characterisations of the UIP
To verify the UIP, the following characterisation of the UIP is helpful, since
it allows us to focus on binary polymorphisms. Let ${\mathscr{C}}$ be an
operation clone and let ${\mathscr{M}}\subseteq{\mathscr{C}}^{(1)}$. If
$v_{1},\dots,v_{k}\in{\mathscr{M}}$ we write $f(v_{1},\dots,v_{k})$ for the
operation that maps $(a_{1},\dots,a_{k})$ to
$f(v_{1}(a_{1}),\dots,v_{k}(a_{k}))$. We say that a function $\zeta$ defined
on $\mathscr{C}$ is _$\mathscr{M}$ -invariant_ if for all
$f\in{\mathscr{C}}^{(k)}$ and $u,v_{1},\dots,v_{k}\in\mathscr{M}$ we have
(2) $\displaystyle\zeta(f)=\zeta\big{(}u(f(v_{1},\dots,v_{k}))\big{)}.$
It follows from known results (see Proposition 7.7) that if $\mathfrak{A}$ is
a finitely homogeneous structure and
$\mathscr{C}\subseteq\operatorname{Can}(\mathfrak{A})$ is a closed clone, then
there exists a uniformly continuous minor-preserving map from $\mathscr{C}$ to
$\operatorname{Proj}$ if and only if there exists a minor-preserving map
$\zeta\colon{\mathscr{C}}\to\operatorname{Proj}$ which is
$\overline{\operatorname{Aut}(\mathfrak{A})}$-invariant.
###### Theorem 6.3 (Binary UIP verification).
Let $\mathfrak{A}$ be a countable finitely homogeneous Ramsey structure and
let $\mathfrak{C}$ be a reduct of $\mathfrak{A}$. Let $\mathscr{C}$ be the
clone of all polymorphisms of $\mathfrak{C}$ that are canonical over
$\mathfrak{A}$ and let $\zeta\colon{\mathscr{C}}\to\operatorname{Proj}$ be an
$\overline{\operatorname{Aut}(\mathfrak{A})}$-invariant minor-preserving map.
Then the following are equivalent.
1. (1)
$\zeta$ has the UIP with respect to $\operatorname{Pol}(\mathfrak{C})$ over
$\mathfrak{A}$.
2. (2)
for all $f\in\operatorname{Pol}(\mathfrak{C})^{(2)}$ and
$u_{1},u_{2}\in{\overline{\operatorname{Aut}(\mathfrak{A})}}$, if
$f(\operatorname{id},u_{1})$ and $f(\operatorname{id},u_{2})$ are canonical
over $\mathfrak{A}$ then
$\zeta(f(\operatorname{id},u_{1}))=\zeta(f(\operatorname{id},u_{2})).$
Theorem 6.3 can be strengthened further. We have already seen that every
homogeneous $\omega$-categorical Ramsey structure contains two independent
elementary substructures. The idea of the following theorem is that it
suffices to verify the UIP on such a pair of substructures.
###### Theorem 6.4.
Let $\mathfrak{A}$ be a countable finitely homogeneous Ramsey structure and
let $\mathfrak{C}$ be a reduct of $\mathfrak{A}$. Let ${\mathscr{C}}$ be the
clone of all polymorphisms of $\mathfrak{C}$ that are canonical over
$\mathfrak{A}$ and let $\zeta\colon{\mathscr{C}}\to\operatorname{Proj}$ be an
$\overline{\operatorname{Aut}(\mathfrak{A})}$-invariant minor-preserving map.
Then the following are equivalent.
1. (1)
$\zeta$ has the UIP with respect to $\operatorname{Pol}(\mathfrak{C})$ over
$\mathfrak{A}$.
2. (2)
for every pair $(\mathfrak{A}_{1},\mathfrak{A}_{2})$ of independent elementary
substructures of $\mathfrak{A}$ such that for all
$f\in\operatorname{Pol}(\mathfrak{C})^{(2)}$,
$u_{1},u_{2}\in{\overline{\operatorname{Aut}(\mathfrak{A})}}$ with
$u_{1}(A)\subseteq A_{1},u_{2}(A)\subseteq A_{2}$, if
$f(\operatorname{id},u_{1})$ and $f(\operatorname{id},u_{2})$ are canonical
over $\mathfrak{A}$ then
$\zeta(f(\operatorname{id},u_{1}))=\zeta(f(\operatorname{id},u_{2})).$
## 7\. Proof of the Extension Lemma
This section contains the proof of Lemma 6.1; Corollary 6.2 about the
complexity of CSPs is then an easy consequence.
### 7.1. Diagonal Interpolation
In our proofs we need the concept of _diagonal interpolation_ , which is a
more restricted form of interpolation, and was already used in [BMM18].
###### Definition 6.
Let $\mathscr{M}$ be a set of functions from $A$ to $A$. We say that $f\colon
A^{k}\to A$ _diagonally interpolates $g\colon A^{k}\to A$ over
${\mathscr{M}}$_ if for every finite $F\subseteq A$ there are
$u,v\in{\mathscr{M}}$ such that
$g(a_{1},\dots,a_{k})=u(f(v(a_{1}),\dots,v(a_{k})))$ for all
$a_{1},\dots,a_{k}\in F$.
Diagonal interpolation is well-behaved with respect to minor identities.
###### Lemma 7.1.
Let $\mathscr{M}$ be a set of functions from $A$ to $A$, let $f\colon A^{k}\to
A$ be an operation, and let $\pi=(\pi^{n}_{i_{1}},\dots,\pi^{n}_{i_{k}})$ be a
vector of $k$ projections, each of arity $n$. Suppose that $f$ diagonally
interpolates $g$ over $\mathscr{M}$. Then $f\circ\pi$ diagonally interpolates
$g\circ\pi$ over ${\mathscr{M}}$.
###### Proof.
By assumption $g\in\overline{\\{u(f(v,\dots,v))\colon u,v\in\mathscr{M}\\}}$.
Since composition with $\pi$ is continuous,
$\displaystyle g\circ\pi$
$\displaystyle\in\overline{\\{u(f(v,\dots,v))\circ\pi\colon
u,v\in\mathscr{M}\\}}$
$\displaystyle=\overline{\\{u(f\circ\pi)(v,\dots,v)\colon
u,v\in\mathscr{M}\\}}$
and thus $f\circ\pi$ diagonally interpolates $g\circ\pi$ over ${\mathscr{M}}$.
∎
We want to stress that the innocent-looking Lemma 7.1 fails for interpolation
instead of diagonal interpolation: the following example shows that Lemma 7.1
fails for interpolation instead of diagonal interpolation.
###### Example 7.2.
Let $\mathfrak{A}$ be a structure with countable domain and two disjoint
infinite unary relations $U_{1}$ and $U_{2}$. Let $f\colon A^{3}\to A$ be an
injective function such that for every $i\in\\{1,2\\}$
* •
$f(a,b,c)\in U_{i}$ if $a\in U_{i}$ and $b\neq c$,
* •
$f(a,b,c)\in U_{i}$ if $b\in U_{i}$ and $b=c$.
Let $e_{1},e_{2}$ be two self-embeddings of $\mathfrak{A}$ with disjoint
images and let $g\colon A^{3}\to A$ be given by
$g(x,y,z)\coloneqq f(x,e_{1}(y),e_{2}(z)).$
Then $g$ is injective and for each $i\in\\{1,2\\}$ we have
$g(U_{i},A,A)\subseteq U_{i}$. Hence, $g$ is canonical and
$\xi_{1}(g)=\pi^{3}_{1}$. We claim that $f^{\prime}\coloneqq
f\circ(\pi^{2}_{1},\pi^{2}_{2},\pi^{2}_{2})$ does not interpolate
$g^{\prime}\coloneqq g\circ(\pi^{2}_{1},\pi^{2}_{2},\pi^{2}_{2})$. For each
$i\in\\{1,2\\}$ we have $f^{\prime}(A,U_{i})\subseteq U_{i}$, so $f^{\prime}$
is canonical and $\xi_{1}(f^{\prime})=\pi^{2}_{2}$. On the other hand
$\xi_{1}(g^{\prime})=\xi_{1}(g)(\pi^{2}_{1},\pi^{2}_{2},\pi^{2}_{2})=\pi^{3}_{1}(\pi^{2}_{1},\pi^{2}_{2},\pi^{2}_{2})=\pi^{2}_{1}$
which proves the claim.
The next lemma plays an important role in the proof of Lemma 6.1.
###### Lemma 7.3.
Let $\mathfrak{B}$ be a reduct of a countable finitely homogeneous Ramsey
structure $\mathfrak{A}$. Let $f\in\operatorname{Pol}(\mathfrak{B})$. Then
there exists $g\in\operatorname{Pol}(\mathfrak{B})$ that diagonally
interpolates both $f$ and an operation in $\operatorname{Can}(\mathfrak{A})$
over $\operatorname{Aut}(\mathfrak{A})$.
###### Proof.
Let $k$ be the arity of $f$. We first show a local version of the statement,
and then derive the statement by a compactness argument. The local version is
that for every finite $X\subseteq A$ there exists
$g\in\operatorname{Pol}(\mathfrak{B})$ and
$e\in\overline{\operatorname{Aut}(\mathfrak{A})}$ such that $g(e,\dots,e)$ is
canonical on $X^{k}$ over $\mathfrak{B}$ and $g$ agrees with $f$ on $X^{k}$.
Let $c=(c_{1},\dots,c_{n})$ be such that $\\{c_{1},\dots,c_{n}\\}=X$. By
Theorem 3.2 there are two independent elementary substructures
$\mathfrak{C}_{1}$ and $\mathfrak{C}_{2}$ of $\mathfrak{A}$. Since
$\mathfrak{A}$ is $\omega$-categorical, for $i\in\\{1,2\\}$ there exists an
isomorphism $e_{i}$ from $\mathfrak{A}$ to $\mathfrak{C}_{i}$; by the
homogeneity of $\mathfrak{A}$ we have
$e_{i}\in\overline{\operatorname{Aut}(\mathfrak{A})}$. The restriction of
$e_{1}$ to $X$ can be extended to an automorphism
$\delta\in\operatorname{Aut}(\mathfrak{A})$. Then $\delta^{-1}(C_{1})$ and
$\delta^{-1}(C_{2})$ induce independent elementary substructures of
$\mathfrak{A}$, and $X\subseteq\delta^{-1}(C_{1})$. So we may assume without
loss of generality that $X\subseteq C_{1}$.
Since $\mathfrak{A}$ is Ramsey, $f(e,\dots,e)$ interpolates a canonical
function over $\operatorname{Aut}(\mathfrak{A})$; so there are
$\delta_{1},\dots,\delta_{k}\in\operatorname{Aut}(\mathfrak{A})$ such that
$f(e\circ\delta_{1},\dots,e\circ\delta_{k})$ is canonical on $X^{k}$. In other
words, for $Y_{i}\coloneqq e\circ\delta_{1}(X)$, the map $f$ is canonical on
$Y_{1}\times\cdots\times Y_{k}$ over $\mathfrak{A}$. Since $X\subseteq C_{1}$
and $Y_{i}\subseteq C_{2}$, the tuples
$e\circ\delta_{1}(c),\dots,e\circ\delta_{k}(c)$ all have the same type over
$c$. Hence, there exist $\alpha_{i}\in\operatorname{Aut}(\mathfrak{A})$ such
that $\alpha_{i}(Y_{1})=Y_{i}$ and $\alpha(c)=c$. Define $g\coloneqq
f(\alpha_{1},\dots,\alpha_{k})$. Then $f$ and $g$ agree on $X^{k}$. We claim
that $h\coloneqq g(e\delta_{1},\dots,e\delta_{1})$ is canonical on $X^{k}$
over $\mathfrak{A}$. This follows from $f$ being canonical on
$\displaystyle Y_{1}\times\cdots\times Y_{k}$
$\displaystyle=\alpha_{1}(Y_{1})\times\cdots\times\alpha_{k}(Y_{1})$
$\displaystyle=\alpha_{1}e\delta_{1}(X)\times\cdots\alpha_{k}e\delta_{k}(X)$
since $h$ equals $f(\alpha_{1}e\delta_{1},\dots,\alpha_{k}e\delta_{k})$ on
$X^{k}$.
To show how the local version implies the statement of the lemma, let $\sigma$
be the signature of $\mathfrak{B}$ and let $\mathfrak{C}$ be an expansion of
$\mathfrak{B}$ by countably many constants such that every element of
$\mathfrak{C}$ is named by a constant symbol.
Let $\rho$ be the signature of $\mathfrak{C}$ together with a new a $k$-ary
function symbol $g$. Consider the $\rho$-theory $T$ consisting of the union of
the following first-order sentences:
1. (1)
$\operatorname{Th}(\mathfrak{C})$;
2. (2)
A first-order sentence which asserts that $g$ preserves all relations from
$\sigma$;
3. (3)
For all $l\in{\mathbb{N}}$ and constant symbols
$c_{0},c_{1},\dots,c_{k}\in\rho$ such that
$f(c_{1}^{\mathfrak{C}},\dots,c_{k}^{\mathfrak{C}})=c_{0}^{\mathfrak{C}}$ the
sentence $g(c_{1},\dots,c_{k})=c_{0}$.
4. (4)
for all constant symbols $c_{1},\dots,c_{l}\in\sigma$ the sentence
$\psi_{c_{1},\dots,c_{l}}$ which expresses that there exist
$y_{1},\dots,y_{l}$ such that
$\operatorname{typ}^{\mathfrak{A}}(y_{1},\dots,y_{l})=\operatorname{typ}^{\mathfrak{A}}(c_{1},\dots,c_{l})$
and $g$ is canonical on $\\{y_{1},\dots,y_{l}\\}^{k}$.
It follows from the basic facts about $\omega$-categorical structures from
Section 2 that such sentences exist.
Claim. Every finite subset $S$ of $T$ has a model. Let $X$ be the (finite) set
of all constants $c$ such that the respective constant symbol appears in a
sentence of $S$ or in $\\{c_{1},\dots,c_{l}\\}$ for some
$\psi_{c_{1},\dots,c_{l}}\in S$. Let $d_{1},\dots,d_{n}$ be some enumeration
of $X$. Then $\psi_{d_{1},\dots,d_{n}}$ implies every sentence in $S^{\prime}$
from item (4). By the local version of the statement there exists
$g\in\operatorname{Pol}(\mathfrak{B})$ and
$e\in\overline{\operatorname{Aut}(\mathfrak{A})}$ such that
* •
$g(e,\dots,e)$ is canonical on $X^{k}$ over $\mathfrak{A}$, and
* •
$g$ agrees with $f$ on $X^{k}$.
Let $\mathfrak{D}$ be the $\rho$-expansion of $\mathfrak{C}$ where
$g^{\mathfrak{D}}\coloneqq g$. Then $\mathfrak{D}$ satisfies $S$. This is
clear for the sentences from item (1), (2), and (3). Finally, since
$g(e,\dots,e)$ is canonical on $X^{k}$ over $\mathfrak{A}$ the sentences of
$S^{\prime}$ from item (4) can be satisfied by setting $y_{i}\coloneqq
e(c_{i})$.
By the compactness theorem of first-order logic, $T$ has a model
$\mathfrak{D}$. By the downward Löwenheim-Skolem theorem we may also assume
that $\mathfrak{D}$ is countable. Then the $\sigma$-reduct of $\mathfrak{D}$
satisfies $\operatorname{Th}(\mathfrak{B})$ and hence is isomorphic to
$\mathfrak{B}$ by the $\omega$-categoricity of $\mathfrak{B}$. We may
therefore identify the element of $\mathfrak{D}$ with the elements of
$\mathfrak{B}$ along this isomorphism and henceforth assume that
$\mathfrak{D}$ is an expansion of $\mathfrak{A}$. Because of the sentences
under (2) we have that $g^{\mathfrak{D}}$ is a polymorphism of $\mathfrak{B}$.
The map $e\colon B\to B$ given by $c^{\mathfrak{C}}\mapsto c^{\mathfrak{D}}$
is an elementary self-embedding. The sentences under (3) imply that
$g(e,\dots,e)$ equals $e(f)$, so $g$ diagonally interpolates $f$. Finally, the
sentences under (4) imply that $g$ diagonally interpolates some function from
$\operatorname{Can}(\mathfrak{B})$. ∎
### 7.2. Rich subsets
To prove the existence of canonical functions it is useful to introduce a
notion of _rich_ substructures of a structure; this will be needed in the full
version of the proofs for Lemma 6.1, but also explicitly in Section 9 (e.g.,
in Lemma 9.18) when we apply the general results to the spatial reasoning
formalism RCC5.
###### Definition 7.
Let $\mathfrak{A}$ be a countable finitely homogeneous structure and let $m$
be the maximal arity of $\mathfrak{A}$. Let
${\mathscr{C}}\subseteq{\mathscr{O}}_{A}$ be a closed clone containing
$\operatorname{Aut}(\mathfrak{A})$ and let $k\in{\mathbb{N}}$. Then we say
that $X\subseteq A$ is _$k$ -rich with respect to ${\mathscr{C}}$_ if
* •
every $m$-orbit of $\mathfrak{A}^{(k)}$ contains a tuple from $(X^{k})^{m}$;
* •
every behaviour over $\mathfrak{A}$ which is realised on $X^{k}$ by some
operation in $\mathscr{C}$ is also realised on $A^{k}$ by some operation in
$\mathscr{C}$.
Note that if $X$ is $k$-rich then every behaviour which is realisable on
$X^{k}$ is complete. It is clear from the definition that if $X\subseteq A$ is
$k$-rich, and $X\subseteq Y$, then $Y$ is also $k$-rich.
###### Lemma 7.4.
Let $\mathfrak{A}$ be a countable finitely homogeneous structure. Let
$\mathscr{C}$ be a closed clone containing $\operatorname{Aut}(\mathfrak{A})$
and let $k\in{\mathbb{N}}$. Then there exists a finite $X\subseteq A$ which is
$k$-rich with respect to ${\mathscr{C}}$.
###### Proof.
Let $m$ be the maximal arity of $\mathfrak{A}$. The structure
$\mathfrak{A}^{(k)}$ is $\omega$-categorical and hence has finitely many
$m$-orbits. Therefore, we can choose a finite subset $X_{0}$ of $A$ such that
every orbit of $m$-tuples in $\operatorname{Aut}(\mathfrak{A}^{(k)})$
intersects $(X_{0}^{k})^{m}$. Let $X_{0}\subset X_{1}\subset\cdots$ be finite
subsets of $A$ such that $\bigcup_{i=1}^{\infty}X_{i}=A$. Let
$\mathcal{B}_{i}$ denote the (finite) set of all $m$-behaviours which are
realised on $X^{k}_{i}$ by some operation of ${\mathscr{C}}$. Then
${\mathcal{B}}_{0}\supseteq{\mathcal{B}}_{1}\supseteq\cdots$ and hence there
exists an $N$ such that ${\mathcal{B}}_{N}={\mathcal{B}}_{N+1}=\cdots$. Then
Proposition 5.1 implies that every behaviour in ${\mathcal{B}}_{N}$ is
realised on $A^{k}$ by some operation in ${\mathscr{C}}$. This proves that
$X_{N}$ is $k$-rich: the first item is satisfied because $X_{0}\subseteq
X_{N}$, and the second item because every behaviour that is realised in
$X^{k}_{N}$ is also realised on $A^{k}$. ∎
###### Definition 8.
Let $\mathfrak{A}$ and $\mathscr{C}$ be as in Definition 7, let $\eta$ be a
function defined on ${\mathscr{C}}\cap\operatorname{Can}(\mathfrak{A})$, let
$F$ be $k$-rich with respect to ${\mathscr{C}}$ for some $k\geq 1$, and let
$f\in\mathscr{C}^{(k)}$. Suppose that $f$ has on $F^{k}$ the same complete
behaviour as $g\in\operatorname{Can}(\mathfrak{A})$. Then we will also write
$\eta(f|_{F^{k}})$ for $\eta(g)$.
### 7.3. Interpolation invariance
It has been shown in [BP15] (see Theorem 6.4 in [BOP18]) that if
$\mathscr{C}\subseteq{\mathscr{O}_{B}}$ is a closed clone and $\mathscr{G}$ an
oligomorphic permutation group over the same base set $B$, then every
$\mathscr{G}$-invariant continuous map $\zeta$ defined on $\mathscr{C}$ is
uniformly continuous. In Lemma 7.5 we present a sufficient condition for
uniform continuity which does not require that the map $\zeta$ we start from
is continuous, for the special case that
$\mathscr{G}=\operatorname{Aut}(\mathfrak{C})$ for a finitely homogeneous
Ramsey structure $\mathfrak{C}$.
###### Definition 9.
Let $\mathscr{C}$ be a closed operation clone on a set $B$ and let
$\mathscr{M}$ be a set of functions from $B$ to $B$. A function $\zeta$
defined on ${\mathscr{C}}$ is called _interpolation invariant over
$\mathscr{M}$_ if $\zeta(f)=\zeta(g)$ whenever $f\in{\mathscr{C}}$
interpolates $g\in{\mathscr{C}}$ over $\mathscr{M}$.
Clearly, interpolation invariance over $\mathscr{M}$ implies
${\mathscr{M}}$-invariance; see 2.
###### Lemma 7.5.
Let $\mathfrak{B}$ be a first-order reduct of a countable finitely homogeneous
Ramsey structure $\mathfrak{A}$. Let $\zeta$ be a function defined on
$\operatorname{Pol}(\mathfrak{B})$ which is interpolation invariant over
$\operatorname{Aut}(\mathfrak{A})$. Then $\zeta$ is uniformly continuous.
###### Proof.
Let $m$ be the maximal arity of $\mathfrak{A}$. Suppose for contradiction that
$\zeta$ is not uniformly continuous. Then there exists a $k\in{\mathbb{N}}$ so
that for every finite $X\subseteq A$ there exist
$f_{X},g_{X}\in\operatorname{Pol}(\mathfrak{B})$ such that
$(f_{X})|_{X^{k}}=(g_{X})|_{X^{k}}$, but $\zeta(f_{X})\neq\zeta(g_{X})$. By
Lemma 7.4, there exists a finite $A_{0}\subseteq A$ which is $k$-rich with
respect to $\operatorname{Pol}(\mathfrak{B})$.
Claim. For every finite $F\subseteq A$ that contains $A_{0}$ there exists an
$h\in\operatorname{Pol}(\mathfrak{B})^{(k)}$ and
$h^{\prime}\in\operatorname{Pol}(\mathfrak{B})^{(k)}\cap\operatorname{Can}(\mathfrak{A})$
such that $h$ and $h^{\prime}$ have the same complete behaviour on $F^{k}$ and
$\zeta(h)\neq\zeta(h^{\prime})$. By Lemma 5.5 there exists a finite
$C\subseteq A$ such that for every $h\colon A^{k}\to A$ there are
$\alpha_{1},\dots,\alpha_{k}\in\operatorname{Aut}(\mathfrak{A})$ such that
$\alpha_{1}(F)\subseteq C,\dots,\alpha_{k}(F)\subseteq C$ and
$h(\alpha_{1},\dots,\alpha_{k})$ is canonical on $F^{k}$. In particular, this
holds for the operations $f_{C},g_{C}$ introduced above, so that
$f^{\prime}\coloneqq f_{C}(\alpha_{1},\dots,\alpha_{k})$ and
$g^{\prime}\coloneqq g_{C}(\alpha_{1},\dots,\alpha_{k})$ are canonical on
$F^{k}$. We have $(f_{C})|_{C^{k}}=(g_{C})|_{C^{k}}$ and
$\zeta(f_{C})\neq\zeta(g_{C})$, and $\zeta(f_{C})=\zeta(f^{\prime})$ and
$\zeta(g_{C})=\zeta(g^{\prime})$ because $f_{C}$ interpolates $f^{\prime}$ and
$g_{C}$ interpolates $g^{\prime}$ over $\operatorname{Aut}(\mathfrak{A})$.
Since $A_{0}\subseteq F$ is $k$-rich there exist
$h^{\prime}\in\operatorname{Pol}^{(k)}(\mathfrak{B})\cap\operatorname{Can}(\mathfrak{A})$
such that $h^{\prime}$ has the same complete behaviour as $f^{\prime}$ on
$F^{k}$, and therefore also the same complete behaviour as $g^{\prime}$ on
$F^{k}$. Hence, we must have $\zeta(h^{\prime})\neq\zeta(f^{\prime})$ or
$\zeta(h^{\prime})\neq\zeta(g^{\prime})$, and hence either
$(f^{\prime},h^{\prime})$ or $(g^{\prime},h^{\prime})$ provide us witnesses
for the claim.
Let $A_{0}\subseteq A_{1}\subseteq\cdots$ be finite such that
$\bigcup_{l\in{\mathbb{N}}}A_{l}=A$. By the claim above, for each
$l\in{\mathbb{N}}$ we can find a function
$h\in\operatorname{Pol}(\mathfrak{B})$ and
$h^{\prime}\in\operatorname{Pol}(\mathfrak{B})^{(k)}\cap\operatorname{Can}(\mathfrak{A})$
such that $h$ and $h^{\prime}$ have the same $m$-behaviour ${\mathcal{H}}_{l}$
on $F^{k}$ and $\zeta(h)\neq\zeta(h^{\prime})$. By Lemma 5.4 the operation $h$
interpolates an operation $g$ with a complete behaviour ${\mathcal{B}}_{l}$
over $\mathfrak{A}$, and $\zeta(g)=\zeta(h)\neq\zeta(h^{\prime})$. Since there
are finitely many $m$-behaviours over $\mathfrak{A}$, there exists
$({\mathcal{H}},{\mathcal{B}})$ such that
$({\mathcal{H}}_{l},{\mathcal{B}}_{l})=({\mathcal{H}},{\mathcal{B}})$ for
infinitely many $l\in{\mathbb{N}}$.
Lemma 5.3 shows that there exists $f\in\operatorname{Pol}(\mathfrak{B})$ and
$e_{1},e^{\prime}_{1},\dots,e_{k},e_{k}^{\prime}\in\overline{\operatorname{Aut}(\mathfrak{A})}$
such that $f\circ(e_{i},\dots,e_{k})$ has behaviour ${\mathcal{H}}$ and
$f\circ(e^{\prime}_{i},\dots,e^{\prime}_{k})$ has behaviour ${\mathcal{B}}$.
Hence, $f$ interpolates two functions that $\zeta$ maps to different values,
contradicting interpolation invariance of $\zeta$. ∎
###### Lemma 7.6.
Let $\mathfrak{A}$ be a finitely related homogeneous structure and let
$\mathscr{C}$ be a subclone of $\operatorname{Can}(\mathfrak{A})$. Let
$\zeta\colon{\mathscr{C}}\to\operatorname{Proj}$ be
$\overline{\operatorname{Aut}(\mathfrak{A})}$-invariant. Then $\zeta$ is
uniformly continuous.
###### Proof.
Let $m$ be the maximal arity of $\mathfrak{A}$, and let $X\subseteq A$ be
finite so that every orbit of $m$-tuples contains a witness in $X^{m}$. Let
$k\in{\mathbb{N}}$ and $f,g\in{\mathscr{C}}^{(k)}$ be such that
$f|_{X^{k}}=g|_{X^{k}}$. Then $\zeta(f)=\zeta(g)$ by the choice of $X$. By
Lemma 5.2, there are
$e_{1},e_{2}\in\overline{\operatorname{Aut}(\mathfrak{A})}$ such that
$e_{1}\circ f=e_{2}\circ g$. The
$\overline{\operatorname{Aut}(\mathfrak{A})}$-invariance of $\zeta$ implies
that $\zeta(f)=\zeta(e_{1}\circ f)=\zeta(e_{2}\circ g)=\zeta(g)$, proving the
uniform continuity of $\zeta$. ∎
The following can be obtained by combining known results in the literature.
###### Proposition 7.7.
Let $\mathfrak{B}$ be finitely related homogeneous with maximal arity
$m\in{\mathbb{N}}$. Let $\mathfrak{C}$ be a first-order reduct of
$\mathfrak{B}$ which is a model-complete core. Suppose that
${\mathscr{C}}\coloneqq\operatorname{Pol}(\mathfrak{C})\subseteq\operatorname{Can}(\mathfrak{B})$.
Then the following are equivalent.
1. (1)
There is no uniformly continuous minor-preserving map from
${\mathscr{C}}\to\operatorname{Proj}$.
2. (2)
There is no $\overline{\operatorname{Aut}(\mathfrak{B})}$-invariant minor-
preserving map ${\mathscr{C}}\to\operatorname{Proj}$.
3. (3)
$\xi^{\mathfrak{B}}_{m}({\mathscr{C}})$ has no minor-preserving map to
$\operatorname{Proj}$.
4. (4)
$\xi^{\mathfrak{B}}_{m}({\mathscr{C}})$ has a cyclic operation.
###### Proof.
$(1)\Rightarrow(2)$: An immediate consequence of Lemma 7.6.
$(2)\Rightarrow(3)$: If $\xi^{{\mathfrak{C}}}_{m}({\mathscr{C}})$ has a minor-
preserving map to $\operatorname{Proj}$, then we can compose it with
$\xi^{{\mathfrak{C}}}_{m}$ and obtain a
$\overline{\operatorname{Aut}(\mathfrak{B})}$-invariant minor-preserving map
from ${\mathscr{C}}$ to $\operatorname{Proj}$, and $(2)$ does not hold.
$(3)\Rightarrow(4)$ has been shown in [BOP18, BK12].
$(4)\Rightarrow(1)$. Since ${\mathfrak{C}}$ is finitely homogeneous, it has
less than doubly exponential growth. Therefore, $\mathfrak{B}$ has less than
doubly exponential growth as well, and the statement follows from results in
[BKO+19]. ∎
### 7.4. Proofs of main results
###### Proof of Lemma 6.1.
Let $f\in\operatorname{Pol}(\mathfrak{B})^{(k)}$. Since $\mathfrak{A}$ is a
finitely homogeneous Ramsey structure, we can apply the canonisation lemma
(Lemma 5.4), which implies that $f$ interpolates a function
$g\in\operatorname{Can}(\mathfrak{A})$ over
$\operatorname{Aut}(\mathfrak{A})$. Define $\bar{\zeta}(f)\coloneqq\zeta(g)$;
this is well-defined because $\zeta$ has the UIP with respect to
$\operatorname{Pol}(\mathfrak{B})$. Note that if
$f\in\operatorname{Can}(\mathfrak{A})$, then $\bar{\zeta}(f)=\zeta(f)$, so
$\bar{\zeta}$ extends $\zeta$.
Claim 1. $\bar{\zeta}$ is
$\overline{\operatorname{Aut}(\mathfrak{A})}$-invariant. Suppose that
$f,g\in\operatorname{Pol}(\mathfrak{B})$ are such that $f$ interpolates $g$
over $\operatorname{Aut}(\mathfrak{A})$. Lemma 5.4 implies that $g$
interpolates over $\operatorname{Aut}(\mathfrak{A})$ an operation
$h\in\operatorname{Can}(\mathfrak{A})$. Then $f$ interpolates $h$ over
$\operatorname{Aut}(\mathfrak{A})$, too, and we have
$\bar{\zeta}(f)=\zeta(h)=\bar{\zeta}(g)$.
Claim 2. $\bar{\zeta}$ is uniformly continuous. This follows from Claim 1 by
Lemma 7.5.
Claim 3. $\bar{\zeta}$ is minor-preserving. Arbitrarily choose
$f\in\operatorname{Pol}(\mathfrak{B})^{(k)}$ and a vector
$\pi=(\pi^{n}_{i_{1}},\dots,\pi_{i_{k}}^{n})$ of projections of arity
$n\in{\mathbb{N}}$. We have to show that
$\bar{\zeta}(f\circ\pi)=\bar{\zeta}(f)\circ\pi$. By Lemma 7.3, there exists
$g\in\operatorname{Pol}(\mathfrak{B})^{(k)}$ that diagonally interpolates $f$
and diagonally interpolates
$h\in\operatorname{Pol}(\mathfrak{B})^{(k)}\cap\operatorname{Can}(\mathfrak{A})$.
We obtain
$\displaystyle\bar{\zeta}(f\circ\pi)$
$\displaystyle=\bar{\zeta}(g\circ\pi)\quad\quad\quad\quad\text{(Lem.~{}\ref{lem:diag-
minor} and interpol.-inv.\ of $\bar{\zeta}$)}$
$\displaystyle=\bar{\zeta}(h\circ\pi)\quad\quad\quad\quad\text{(Lem.~{}\ref{lem:diag-
minor} and interpol.-inv.\ of $\bar{\zeta}$)}$
$\displaystyle=\zeta(h\circ\pi)\quad\quad\quad\quad\text{($h\circ\pi\in\operatorname{Pol}(\mathfrak{B})\cap\operatorname{Can}(\mathfrak{A})$)}$
$\displaystyle=\zeta(h)\circ\pi\quad\quad\quad\quad\text{($\zeta$ is minor
preserving)}$
$\displaystyle=\bar{\zeta}(h)\circ\pi=\bar{\zeta}(g)\circ\pi=\bar{\zeta}(f)\circ\pi.$
Claim 2 and 3 imply the statement of the theorem. ∎
###### Proof of Corollary 6.2.
Let $\mathfrak{C}\in{\mathcal{C}}$. First suppose that
$\operatorname{Pol}(\mathfrak{C})\cap\operatorname{Can}(\mathfrak{B})$ does
not have a uniformly continuous minor-preserving map to the projections.
Theorem 5.6 implies that $\operatorname{CSP}(\mathfrak{C})$ is in P. Now
suppose that
$\operatorname{Pol}(\mathfrak{C})\cap\operatorname{Can}(\mathfrak{B})$ has a
uniformly continuous minor-preserving map to $\operatorname{Proj}$. Then the
assumptions imply that there is also a uniformly continuous minor-preserving
map
$\zeta\colon\operatorname{Pol}(\mathfrak{C})\cap\operatorname{Can}(\mathfrak{A})\to\operatorname{Proj}$
that has the UIP with respect to $\operatorname{Pol}(\mathfrak{C})$ over
$\mathfrak{A}$. Lemma 6.1 shows that $\zeta$ can be extended to a uniformly
continuous minor-preserving map
$\bar{\zeta}\colon\operatorname{Pol}(\mathfrak{C})\to\operatorname{Proj}$.
Then $\operatorname{CSP}(\mathfrak{C})$ is NP-hard by Theorem 4.4. Since CSPs
of reducts of finitely bounded structures are in NP, this concludes the proof.
∎
Note that the proof of Corollary 6.2 shows that Conjecture 4.5 holds for all
structures $\mathfrak{C}\in{\mathcal{C}}$.
## 8\. Verification of the UIP
In this section we show that if a minor-preserving map to the clone of
projections does not have the UIP, then this is witnessed by binary operations
of a very special form, proving Theorem 6.3 and its strengthening Theorem 6.4.
We need the following ‘higher-dimensional checker board’ canonisation lemma.
###### Lemma 8.1.
Let $\mathfrak{A}$ be a countable finitely homogeneous Ramsey structure and
$f\colon A^{k}\to A$. Suppose that $f$ interpolates over $\mathfrak{A}$ the
operations $h_{1},\dots,h_{m}\in\operatorname{Can}(\mathfrak{A})$. Then for
every finite $X\subseteq A$ there exist
$\alpha_{1,1},\dots,\alpha_{m,k}\in\operatorname{Aut}(\mathfrak{A})$ such that
* •
for every $i\in\\{1,\dots,m\\}$ the operation
$f(\alpha_{i,1},\dots,\alpha_{i,k})$ has the same behaviour as $h_{i}$ on
$X^{k}$, and
* •
for all $u_{1},\dots,u_{k}\in\\{1,\dots,m\\}$ the operation
$f(\alpha_{u_{1},1},\dots,\alpha_{u_{k},k})$ is canonical on $X^{k}$.
###### Proof.
Let $u^{1},\dots,u^{p}$ be an enumeration of $\\{1,\dots,m\\}^{k}$. We show by
induction on $q\in\\{1,\dots,p\\}$ that for every finite $X\subseteq A$ there
exist $\alpha_{1,1},\dots,\alpha_{m,k}\in\operatorname{Aut}(\mathfrak{A})$
such that for every $i\in\\{1,\dots,m\\}$ the operation
$f(\alpha_{i,1},\dots,\alpha_{i,k})$ has the same behaviour as $h_{i}$ on
$X^{k}$, and for all $l\in\\{1,\dots,q\\}$ the operation
$f(\alpha_{u^{l}_{1},1},\dots,\alpha_{u^{l}_{k},k})$ is canonical on $X^{k}$.
For $q=p$ we obtain the statement of the lemma.
If $q=0$ the statement of the claim follows from the assumptions of the lemma.
For the inductive step, we assume that the statement holds for $q-1$ and prove
the statement for $q$. Let $X\subseteq A$ be finite. Lemma 5.5 asserts the
existence of a finite $C\subseteq A$ such that for every $f\colon A^{k}\to A$
there are $\beta_{1},\dots,\beta_{k}\in\operatorname{Aut}(\mathfrak{A})$ such
that $\beta_{1}(X)\subseteq C,\dots,\beta_{k}(X)\subseteq C$ and
$f(\beta_{1},\dots,\beta_{k})$ is canonical on $X^{k}$. We apply the induction
hypothesis to $C$, and obtain $\gamma_{1,1},\dots,\gamma_{m,k}$ such that for
every $i\in\\{1,\dots,m\\}$ the operation $f(\gamma_{i,1},\dots,\gamma_{i,k})$
has the same behaviour as $h_{i}$ on $C^{k}$ and
$f(\gamma_{u^{l}_{1},1},\dots,\gamma_{u^{l}_{k},k})$ is canonical on $C^{k}$
for every $l\in\\{1,\dots,q-1\\}$. Let $f^{\prime}\coloneqq
f(\gamma_{u^{q}_{1},1},\dots,\gamma_{u^{q}_{k},k})$. The property of $C$
implies that there are
$\beta_{1},\dots,\beta_{k}\in\operatorname{Aut}(\mathfrak{A})$ such that
$\beta_{1}(X)\subseteq C,\dots,\beta_{k}(X)\subseteq C$ and
$f^{\prime}(\beta_{1},\dots,\beta_{k})$ is canonical on $X^{k}$. For
$i\in\\{1,\dots,m\\}$ and $j\in\\{1,\dots,k\\}$ define
$\alpha_{i,j}\coloneqq\gamma_{i,j}\circ\beta_{j}$. Observe
$\alpha_{i,j}(X)\subseteq\gamma_{i,j}(C)$ and hence
* •
$f(\alpha_{u^{l}_{1},1},\dots,\alpha_{u^{l}_{k},k})$ is canonical on $X^{k}$
for $l\in\\{1,\dots,q-1\\}$: this follows from the inductive assumption that
$f(\gamma_{u^{l}_{1},1},\dots,\gamma_{u^{l}_{k},k})$ is canonical on $C^{k}$;
* •
$f(\alpha_{u^{q}_{1},1},\dots,\alpha_{u^{q}_{k},k})$ is canonical on $X^{k}$:
this follows from the property of $\beta_{1},\dots,\beta_{k}$ that
$f^{\prime}(\beta_{1},\dots,\beta_{k})=f(\alpha_{u^{q}_{1},1},\dots,\alpha_{u^{q}_{k},k})$
is canonical on $X^{k}$;
* •
for every $i\in\\{1,\dots,m\\}$, the operation
$f(\alpha_{i,1},\dots,\alpha_{i,k})$ has the same behaviour as $h_{i}$ on
$X^{k}$ because for every $j\in\\{1,\dots,k\\}$ and
$f(\gamma_{i,1},\dots,\gamma_{i,k})$ has the same behaviour as $h_{i}$ on
$C^{k}$ by the inductive assumption.
This concludes the proof that $\alpha_{1,1},\dots,\alpha_{m,k}$ satisfy the
inductive statement. ∎
We introduce useful notation for the proofs of Theorem 6.3 and Theorem 6.4. If
$f\colon B^{k}\to B$, $u\colon B\to B$, and $\ell\in\\{1,\dots,k\\}$, we write
$f^{u}_{\ell}$ for the $k$-ary operation defined by
$(x_{1},\dots,x_{k})\mapsto
f(x_{1},\dots,x_{\ell-1},u(x_{\ell}),x_{\ell+1},\dots,x_{k}).$
###### Proof of Theorem 6.3.
The forward implication is trivial. For the converse implication, suppose that
$\zeta$ does not have the UIP with respect to
$\operatorname{Pol}(\mathfrak{C})$ over $\mathfrak{A}$. That is, there are
operations $g,g_{1},g_{2}\in{\mathscr{C}}^{(k)}$ such that $g$ interpolates
both $g_{1}$ and $g_{2}$ over $\operatorname{Aut}(\mathfrak{A})$ and
$\zeta(g_{1})\neq\zeta(g_{2})$. By Lemma 7.4 there exists a finite
$X_{0}\subseteq A$ which is $k$-rich with respect to
$\operatorname{Pol}(\mathfrak{B})$. By Lemma 8.1 applied to $g$ and $X_{0}$
there exist
$\alpha_{1,1},\dots,\alpha_{2,k}\in\operatorname{Aut}(\mathfrak{A})$ such that
* •
for all $u_{1},\dots,u_{k}\in\\{1,2\\}$ the operation
$g(\alpha_{u_{1},1},\dots,\alpha_{u_{k},k})$ is canonical on $X_{0}^{k}$ over
$\mathfrak{A}$, and
* •
for $i\in\\{1,2\\}$ the operation $g(\alpha_{i,1},\dots,\alpha_{i,k})$ has the
same behaviour as $g_{i}$ on $X_{0}^{k}$.
Let $i\in\\{1,\dots,k\\}$ be the smallest index such that
$\displaystyle\zeta\big{(}g(\alpha_{1,1},\dots,\alpha_{1,i-1},\alpha_{1,i},\alpha_{2,{i+1}},\dots,\alpha_{2,k})\big{)}$
$\displaystyle\neq\;$
$\displaystyle\zeta\big{(}g(\alpha_{1,1},\dots,\alpha_{1,i-1},\alpha_{2,i},\alpha_{2,{i+1}},\dots,\alpha_{2,k})\big{)}.$
We know that such an index exists since
$\zeta\big{(}g(\alpha_{1,1},\dots,\alpha_{1,k})\big{)}=\zeta(g_{1})\neq\zeta(g_{2})=\zeta\big{(}g(\alpha_{2,1},\dots,\alpha_{2,k})\big{)}.$
Let $h\coloneqq
g(\alpha_{1,1},\dots,\alpha_{1,i-1},\operatorname{id},\alpha_{2,{i+1}},\dots,\alpha_{2,k})$.
So we have shown the following.
Claim. For every finite $X\subseteq A$ that contains $X_{0}$ there exist
$i\in\\{1,\dots,k\\}$, $h\in\operatorname{Pol}(\mathfrak{B})^{(k)}$,
$\alpha,\beta\in\operatorname{Aut}(\mathfrak{A})$, and
$h,h^{\prime}\in\operatorname{Pol}(\mathfrak{B})^{(k)}\cap\operatorname{Can}(\mathfrak{A})$
such that
* •
$f^{\alpha}_{\ell}$ has the same behaviour as $h$ on $X^{k}$,
* •
$f^{\beta}_{\ell}$ has the same behaviour as $h^{\prime}$ on $X^{k}$, and
* •
$\zeta(h)\neq\zeta(h^{\prime})$.
Let $X_{0}\subset X_{1}\subset\cdots$ be finite subsets of $A$ such that
$\bigcup_{i=0}^{\infty}X_{i}=A$. Then the claim applied to $X=X_{i}$, for
$i\in{\mathbb{N}}$, asserts the existence of $f_{i}$, $\alpha_{i}$,
$\beta_{i}$, and $\ell_{i}$ such that $(f_{i})_{\ell_{i}}^{\alpha_{i}}$ and
$(f_{i})_{\ell_{i}}^{\beta_{i}}$ are canonical on $X_{i}^{k}$ and
$\zeta((f_{i})^{\alpha_{i}}_{\ell_{i}})\neq\zeta((f_{i})^{\beta_{i}}_{\ell_{i}})$.
By thinning out the sequences $(f_{i})_{i\in\mathbb{N}}$,
$(\alpha_{i})_{i\in{\mathbb{N}}}$, $(\beta_{i})_{i\in{\mathbb{N}}}$, and
$(\ell_{i})_{i\in{\mathbb{N}}}$ we can assume that all $\ell_{i}$ are equal,
say $\ell$, and that the complete behaviour ${\mathcal{B}}_{1}$ of
$f_{i}^{\alpha_{i}}$ on $X_{0}$ and ${\mathcal{B}}_{2}$ of $f_{i}^{\beta_{i}}$
on $X_{0}$ does not depend on $i$. Note that we must have
${\mathcal{B}}_{1}\neq{\mathcal{B}}_{2}$. By Lemma 5.3 (and in particular the
statement at the end starting with “moreover”) there exist
$g\in\operatorname{Pol}(\mathfrak{B})$ and
$u,v\in\overline{\operatorname{Aut}(\mathfrak{A})}$ such that $g^{u}_{\ell}$
has behaviour ${\mathcal{B}}_{1}$ on all of $A$, and $g^{v}_{\ell}$ has
behaviour ${\mathcal{B}}_{2}$ on all of $A$. So they are canonical over
$\mathfrak{A}$ and $\zeta(g^{u}_{\ell})\neq\zeta(g^{v}_{\ell})$. Suppose that
$\zeta(g^{u}_{\ell})=\pi^{k}_{r}$ and $\zeta(g^{v}_{\ell})=\pi^{k}_{s}$. Let
$f(x,y)\coloneqq f(\underbrace{y,\dots,y}_{r-1},x,y,\dots,y).$
Then note that
(3) $\displaystyle\zeta\big{(}f(\pi^{2}_{1},u(\pi^{2}_{2}))\big{)}=\;$
$\displaystyle\zeta\big{(}g(\underbrace{u(\pi^{2}_{2}),\dots,u(\pi^{2}_{2})}_{r-1},\pi^{2}_{1},u(\pi^{2}_{2}),\dots,u(\pi^{2}_{2}))\big{)}$
(4) $\displaystyle=\;$
$\displaystyle\zeta\big{(}g^{u}_{\ell}(\underbrace{\pi^{2}_{2},\dots,\pi^{2}_{2}}_{r-1},\pi^{2}_{1},\pi^{2}_{2},\dots,\pi^{2}_{2})\big{)}$
(5) $\displaystyle=\;$
$\displaystyle\pi^{k}_{r}(\underbrace{\pi^{2}_{2},\dots,\pi^{2}_{2}}_{r-1},\pi^{2}_{1},\pi^{2}_{2},\dots,\pi^{2}_{2})$
(6) $\displaystyle=\;$ $\displaystyle\pi^{2}_{1}$
where equality of (3) and (4) holds since $\zeta$ is
$\overline{\operatorname{Aut}(\mathfrak{A})}$-invariant, and equality of (4)
and (6) holds since $\zeta$ is minor-preserving. Similarly, since $r\neq s$
$\displaystyle\zeta\big{(}f(\pi^{2}_{1},v(\pi^{2}_{2}))\big{)}$
$\displaystyle=\pi^{k}_{s}(\underbrace{\pi^{2}_{2},\dots,\pi^{2}_{2}}_{r-1},\pi^{2}_{1},\pi^{2}_{2},\dots,\pi^{2}_{2})=\pi^{2}_{2}$
Therefore, $f,u,v$ show that item $(2)$ from the statement does not hold,
concluding the proof that $(2)\Rightarrow(1)$. ∎
We finally prove Theorem 6.4, which shows that it suffices to verify the UIP
on independent elementary substructures.
###### Proof of Theorem 6.4.
The forward implication holds trivially. We now show the converse,
$(2)\Rightarrow(1)$. It suffices to verify $(2)$ in Theorem 6.3. Let
$f\in\operatorname{Pol}(\mathfrak{C})^{(2)},u_{1},u_{2}\in\overline{\operatorname{Aut}(\mathfrak{A})}$
be such that $f^{u_{1}}_{2},f^{u_{2}}_{2}\in\operatorname{Can}(\mathfrak{A})$.
Let $F\subseteq A$ be 2-rich with respect to
$\operatorname{Pol}(\mathfrak{C})$. We have to verify that $f^{u_{1}}_{2}$ and
$f^{u_{2}}_{2}$ have the same complete behaviour on $F^{2}$.
Let $(\mathfrak{A}_{1},\mathfrak{A}_{2})$ be a pair of independent elementary
substructures of $\mathfrak{A}$; such a pair exists by Theorem 3.2. For
$j\in\\{1,2\\}$ let $e_{j}$ be an embedding of $\mathfrak{A}$ into
$\mathfrak{A}_{j}$. Since $\mathfrak{A}$ is homogeneous there exists
$\epsilon\in\operatorname{Aut}(\mathfrak{A})$ such that $e_{1}$ and $\epsilon$
agree on $u_{1}(F)\cup u_{2}(F)$. By Lemma 5.4 applied to
$f(\operatorname{id},\epsilon^{-1}e_{2})$ there are
$\beta_{1},\beta_{2}\in\operatorname{Aut}(\mathfrak{A})$ such that
$f(\beta_{1},\epsilon^{-1}e_{2}\beta_{2})$ is canonical on $F^{2}$. Let
$v_{i}\coloneqq\epsilon u_{i}$ and $w\coloneqq e_{2}\beta_{2}$. Note that
$v_{i}(F)\subseteq A_{1}$ and $w(F)\subseteq A_{2}$. Let $h\coloneqq
f(\beta_{1},\epsilon^{-1})\in\operatorname{Pol}(\mathfrak{C})$, and note that
$h_{2}^{v_{i}}=f(\beta_{1},\epsilon^{-1}\epsilon u_{i})=f(\beta_{1},u_{i})$
is canonical since $f_{2}^{u_{i}}=f(\operatorname{id},u_{i})$ is canonical,
and that
$h_{2}^{w}=f(\beta_{1},\epsilon^{-1}e_{2}\beta_{2})$
is canonical on $F^{2}$ as we have seen above. We thus apply (2) two times to
obtain that $\zeta(h_{2}^{v_{1}})=\zeta(h_{2}^{w})=\zeta(h_{2}^{v_{2}})$.
Finally, note that
$\zeta(f(\operatorname{id},u_{i}))=\zeta(f(\beta_{1},u_{i}))=\zeta(f(\beta_{1},\epsilon^{-1}\epsilon
u_{i}))=\zeta(h_{2}^{v_{i}}).$
We conclude that
$\zeta(f(\operatorname{id},u_{1}))=\zeta(f(\operatorname{id},u_{2}))$. ∎
## 9\. First-order expansions of the basic relations of RCC5
RCC5 is a fundamental formalism for spatial reasoning, and may be viewed as an
$\omega$-categorical structure $\mathfrak{R}$ (see, e.g., [BC09, BJ17]). There
are many equivalent ways of formally introducing RCC5; we follow the
presentation in [BC09] and refer to [BJ17] for other definitions and their
equivalence. Let $\mathfrak{S}$ be the structure with domain $S\coloneqq
2^{\mathbb{N}}\setminus\\{\emptyset\\}$, i.e., the set of all non-empty
subsets of the natural numbers ${\mathbb{N}}$. The signature of $\mathfrak{S}$
consists of the five binary relation symbols
$\mathtt{EQ},\mathtt{PP},\mathtt{PPI},\mathtt{DR},\mathtt{PO}$ where for
$x,y\subseteq\mathbb{N}$ we have
(7) $\displaystyle(x,y)\in\mathtt{EQ}$ $\displaystyle\text{ iff }x=y,$ “$x$
and $y$ are equal” (8) $\displaystyle(x,y)\in\mathtt{PP}$ $\displaystyle\text{
iff }x\subset y,$ “$x$ is strictly contained in $y$” (9)
$\displaystyle(x,y)\in\mathtt{PPI}$ $\displaystyle\text{ iff }x\supset y,$
“$x$ strictly contains $y$” (10) $\displaystyle(x,y)\in\mathtt{DR}$
$\displaystyle\text{ iff }x\cap y=\emptyset,$ “$x$ and $y$ are disjoint” (11)
$\displaystyle(x,y)\in\mathtt{PO}$ $\displaystyle\text{ iff }x\not\subset
y\wedge y\not\subset x\wedge x\cap y\neq\emptyset,$ “$x$ and $y$ properly
overlap”.
Note that by definition every pair $(x,y)\in S^{2}$ is contained in exactly
one of the relations
$\mathtt{DR}^{\mathfrak{S}},\mathtt{PO}^{\mathfrak{S}},\mathtt{PP}^{\mathfrak{S}},\mathtt{PPI}^{\mathfrak{S}},\mathtt{EQ}^{\mathfrak{S}}$.
Note that the structure $\mathfrak{S}$ is not $\omega$-categorical; however,
$\operatorname{Age}(\mathfrak{S})$ is an amalgamation class (Theorem 30 in
[BC09]), and hence there exists a countable homogeneous structure
$\mathfrak{R}$ with the same age as $\mathfrak{S}$. We refer to the relations
$\mathtt{EQ}^{\mathfrak{R}},\mathtt{PP}^{\mathfrak{R}},\mathtt{PPI}^{\mathfrak{R}},\mathtt{DR}^{\mathfrak{R}},\mathtt{PO}^{\mathfrak{R}}$
as the _basic relations of RCC5_.
The _composition_ of two binary relations $R_{1}$ and $R_{2}$ is the binary
relation $R_{1}\circ R_{2}\coloneqq\big{\\{}(x,y):\exists
z\big{(}R_{1}(x,z)\wedge R_{2}(z,y)\big{)}\big{\\}}$. The _converse_
(sometimes also called _inverse_) of a relation $R$ is the relation
$\\{(y,x)\colon(x,y)\in R\\}$, and denoted by $R^{\smile}$. The converse of
$\mathtt{PP}$ is $\mathtt{PPI}$, and $\mathtt{EQ}^{\mathfrak{R}}$,
$\mathtt{DR}^{\mathfrak{R}}$, and $\mathtt{PO}^{\mathfrak{R}}$ are their own
converse. The full binary relation containing all pairs of elements of
$\mathfrak{R}$ is denoted by $\bf 1$. It is straightforward to verify that the
relations
$\mathtt{EQ}^{\mathfrak{R}},\mathtt{PP}^{\mathfrak{R}},\mathtt{PPI}^{\mathfrak{R}},\mathtt{DR}^{\mathfrak{R}},\mathtt{PO}^{\mathfrak{R}}$
compose as shown in Table 1.
###### Lemma 9.1.
$\mathfrak{R}$ is finitely bounded.
###### Proof.
A finite structure embeds into $\mathfrak{R}$ if and only if all three-element
substructures of the structure embed into $\mathfrak{R}$. For details we refer
to [BJ17], implication $2\Rightarrow 3$ in Proposition 15. ∎
###### Remark 9.2.
As a _relation algebra_ , RCC5 is given by the composition table and the data
about the converses. Our structure $\mathfrak{R}$ introduced above is a
_representation_ of RCC5. We do not introduce relation algebras formally,
because they will not be needed in the following, and rather refer to [BJ17].
∘ DR PO PP PPI EQ DR 1 PP∪DR∪PO PP∪DR∪PO DR DR PO PPI∪DR∪PO 1 PP∪PO PPI∪PO PO
PP DR PP∪DR∪PO PP 1 PP PPI PPI∪DR∪PO PPI∪PO 1 ∖DR PPI PPI EQ DR PO PP PPI EQ
Figure 1. The composition table for the relations of $\mathfrak{R}$.
A _first-order expansion_ of a structure $\mathfrak{A}$ is an expansion of
$\mathfrak{A}$ by relations that are first-order definable over
$\mathfrak{A}$.
###### Theorem 9.3.
For every first-order expansion $\mathfrak{C}$ of $\mathfrak{R}$ either
* •
$\operatorname{Pol}(\mathfrak{C})\cap\operatorname{Can}(\mathfrak{R})$ does
not have a uniformly continuous minor-preserving map to $\operatorname{Proj}$,
in which case $\operatorname{CSP}(\mathfrak{C})$ is in P, or
* •
$\operatorname{Pol}(\mathfrak{C})$ has a uniformly continuous minor-preserving
map to $\operatorname{Proj}$, in which case $\operatorname{CSP}(\mathfrak{C})$
is NP-complete.
We prove Theorem 9.3 using Corollary 6.2 and hence first need to introduce a
Ramsey expansion $(\mathfrak{R};\prec)$ of $\mathfrak{R}$ (Section 9.1); we
use a recent Ramsey transfer result of Mottet and Pinsker [MP21]. Every first-
order expansion $\mathfrak{C}$ of $\mathfrak{R}$ is clearly a model-complete
core. We verify the remaining assumption in Corollary 6.2 in two steps.
Assuming that there exists a uniformly continuous minor-preserving map from
$\operatorname{Pol}(\mathfrak{C})\cap\operatorname{Can}(\mathfrak{R})$ to
$\operatorname{Proj}$, we construct Section 9.4 either
* •
a minor-preserving map
$\eta\colon\operatorname{Pol}(\mathfrak{C})\cap\operatorname{Can}(\mathfrak{R},\prec)\to\operatorname{Proj}$
which arises from the action of the canonical polymorphisms on the two
relations $\mathtt{PP}$ and $(\mathtt{DR}\cup\mathtt{PO})\,\cap\prec$, or
* •
a minor-preserving map
$\rho\colon\operatorname{Pol}(\mathfrak{C})\cap\operatorname{Can}(\mathfrak{R},\prec)\to\operatorname{Proj}$
which arises from the action of the canonical polymorphisms on the two
relations $\mathtt{DR}\,{\cap}\,{\prec}$ and $\mathtt{PO}\,{\cap}\,{\prec}$.
In the second step, we prove that if such a map
$\eta,\rho\colon\operatorname{Pol}(\mathfrak{C})\cap\operatorname{Can}(\mathfrak{R},\prec)\to\operatorname{Proj}$
exists, then it has the UIP with respect to $\operatorname{Pol}(\mathfrak{C})$
over $(\mathfrak{R},\prec)$ (Section 9.5); here we use Theorem 6.4. The
statement then follows from Corollary 6.2.
### 9.1. A Ramsey order expansion of $\mathfrak{R}$
The structure $\mathfrak{R}$ is not Ramsey, but it has a homogeneous Ramsey
expansion by a linear order. Let $\mathcal{C}$ be the class of all expansions
of structures from $\operatorname{Age}(\mathfrak{S})$ with the signature
$\\{\mathtt{EQ},\mathtt{DR},\mathtt{PO},\mathtt{PP},\mathtt{PPI},\prec\\}$
such that $\prec$ denotes a linear extension of $\mathtt{PP}$.
###### Proposition 9.4.
The class $\mathcal{C}$ defined above is an amalgamation class.
###### Proof.
It is clear from the definition that $\mathcal{C}$ is closed under
isomorphisms and substructures. It is well-known that in order to prove the
amalgamation property, it suffices to verify the 1-point amalgamation property
(see, e.g., [Bod21]): for all structures
$\mathfrak{A},\mathfrak{B}_{1},\mathfrak{B}_{2}\in{\mathcal{C}}$ such that
$A=B_{1}\cap B_{2}$ and $B_{i}=A\cup\\{b_{i}\\}$, for $i\in\\{1,2\\}$ and
$b_{1}\neq b_{2}$, there exists $\mathfrak{C}\in{\mathcal{C}}$ with
$C=B_{1}\cup B_{2}=A\cup\\{b_{1},b_{2}\\}$ such that $\mathfrak{B}_{1}$ and
$\mathfrak{B}_{2}$ are substructures of $\mathfrak{C}$. It is easy to see that
such a structure $\mathfrak{C}$ can be determined by specifying which
relations of $\mathfrak{C}$ contain the pair $(b_{1},b_{2})$.
Bodirsky and Chen [BC09] (also see [Bod21]) proved that there exists
$R\in\\{\mathtt{DR},\mathtt{PO},\mathtt{PP},\mathtt{PPI}\\}$ such that
* •
if we add the pair $(b_{1},b_{2})$ to $R^{\mathfrak{C}}$ and the pair
$(b_{2},b_{1})$ to $(R^{\mathfrak{C}})^{\smile}$, then the
$\\{\mathtt{EQ},\mathtt{DR},\mathtt{PO},\mathtt{PP},\mathtt{PPI}\\}$-reduct of
the resulting structure $\mathfrak{C}$ is in
$\operatorname{Age}(\mathfrak{S})$, and
* •
$(b_{1},b_{2})\in\mathtt{PP}^{\mathfrak{C}}$ only if there exists $a\in A$
such that $(b_{1},a)\in\mathtt{PP}^{\mathfrak{B}_{1}}$ and
$(a,b_{2})\in\mathtt{PP}^{\mathfrak{B}_{2}}$.
If $R$ equals $\mathtt{PP}$ we add $(b_{1},b_{2})$ to $\prec^{\mathfrak{C}}$;
if $R$ equals $\mathtt{PPI}$ we add $(b_{2},b_{1})$ to $\prec^{\mathfrak{C}}$.
If $R$ is from $\\{\mathtt{DR},\mathtt{PO}\\}$, then we add $(b_{1},b_{2})$ or
$(b_{2},b_{1})$ to $\prec^{\mathfrak{C}}$ according to an order-amalgam of the
$\\{\prec\\}$-reducts of $\mathfrak{B}_{1}$ and $\mathfrak{B}_{2}$ over
$\mathfrak{A}$. In each of these cases, $\prec^{\mathfrak{C}}$ is a linear
order that extends $\mathtt{PP}^{\mathfrak{C}}$, and hence $\mathfrak{C}$ is
in ${\mathcal{C}}$. ∎
One can check by an easy back-and-forth argument that the
$\\{\mathtt{EQ},\mathtt{DR},\mathtt{PO},\mathtt{PP},\mathtt{PPI}\\}$-reduct of
the Fraïssé-limit of ${\mathcal{C}}$ is isomorphic to $\mathfrak{R}$; hence,
we may denote this Fraïssé-limit by $(\mathfrak{R},\prec)$. To show that
$(\mathfrak{R},\prec)$ has the Ramsey property, we use a recent Ramsey
transfer result.
###### Definition 10 (Mottet and Pinsker [MP21]).
Let $\mathscr{G}$ be a permutation group on a set $X$. A function $g\colon
X\to X$ is called _range-rigid with respect to $\mathscr{G}$_ if for every
$k\in{\mathbb{N}}$, every orbit of $k$-tuples in the range of $g$ is preserved
by $g$.
Mottet and Pinsker [MP21] proved that if $\mathfrak{B}$ is a homogeneous
structure and $g\colon B\to B$ is range-rigid with respect to
$\operatorname{Aut}(\mathfrak{B})$, then the age of the structure induced by
$\mathfrak{B}$ on the image of $g$ has the amalgamation property; we denote
the Fraïssé-limit of this class by $\mathfrak{B}_{g}$.
###### Theorem 9.5 (Lemma 10 in [MP21]).
Let $\mathfrak{B}$ be a homogeneous structure and let $g\colon B\to B$ be
range-rigid with respect to $\operatorname{Aut}(\mathfrak{B})$. If
$\mathfrak{B}$ is Ramsey, then so is $\mathfrak{B}_{g}$.
Let $\mathfrak{A}=(A;\cap,\cup,\overline{\cdot},0,1)$ be the countable
atomless Boolean algebra [Hod97], which we view as a subalgebra of
$(2^{\mathbb{N}};\cap,\cup,\overline{\cdot},0,1)$. We use the usual shortcuts:
* •
$x\subseteq y$ stands for $x\cap y=x$,
* •
$x\subset y$ stands for $x\subseteq y$ and $x\neq y$, and
* •
$x+y$ stands for $(x\cap\overline{y})\cup(y\cap\overline{x})$.
Let $\mathfrak{A}^{\prime}$ be the ($\omega$-categorical) relational structure
with the same domain as $\mathfrak{A}$ carrying all relations that are first-
order definable in $\mathfrak{A}$. Kechris, Pestov, and Todorčević described a
linear order $\prec$ on $A$ so that the expanded structure
$(\mathfrak{A}^{\prime},\prec)$ is a homogeneous $\omega$-categorical Ramsey
structure [KPT05]. The age of $(\mathfrak{A}^{\prime},\prec)$ can be described
as follows. If $\mathfrak{F}$ is a finite substructure of $\mathfrak{A}$, then
$\mathfrak{F}$ is a finite Boolean algebra and there exists an enumeration
$a_{1},\dots,a_{n}$ of the atoms of $\mathfrak{F}$ such that for all $u,v\in
F$ with $u=\bigcup_{i=1}^{n}(d_{i}\cap a_{i})$ and
$v=\bigcup_{i=1}^{n}(e_{i}\cap a_{i})$ for $d,e\in\\{0,1\\}^{n}$ we have
$u\prec v$ if and only if there exists some $j\in\\{1,\dots,n\\}$ such that
$d_{j}<e_{j}$ and $d_{i}=e_{i}$ for all $i>j$; such an ordering is called an
_antilexicographical ordering_.
Let $\mathfrak{B}$ be the relational structure with domain $B\coloneqq
A\setminus\\{0\\}$ and for all $k,l\geq 1$ the relation
$\displaystyle
R_{k,l}\coloneqq\big{\\{}(a_{1},\dots,a_{k},b_{1},\dots,b_{l})\mid\bigcap_{i=1}^{k}a_{i}\subseteq\bigcup_{j=1}^{l}b_{j}\big{\\}}$
and for all $k\geq 1$ and $e,d\in\\{0,1\\}^{k}$ the relation
$\displaystyle
O_{k,d,e}\coloneqq\big{\\{}(a_{1},\dots,a_{k})\mid\bigcap_{i=1}^{k}(a_{i}+d_{i})\prec\bigcap_{i=1}^{k}(a_{i}+e_{i})\big{\\}}.$
Note that $x\prec y$ has the definition $O_{2,(0,1),(1,0)}(x,y)$ in
$\mathfrak{B}$, which holds precisely if
$x\cap\overline{y}\prec{\overline{x}}\cap y$.
###### Lemma 9.6.
The structure $\mathfrak{B}$ is homogeneous.
###### Proof.
Let $i$ be an isomorphism between finite induced substructures of
$\mathfrak{B}$. We claim that $i$ preserves all quantifier-free formulas of
$\mathfrak{A}$. To see this, it suffices to show that for every Boolean term
$f$ of arity $n$ and every Boolean term $g$ of arity $k$, the function $i$
preserves the formula $f(x_{1},\dots,x_{n})=g(y_{1},\dots,y_{k})$. Since
$f(x_{1},\dots,x_{n})=g(y_{1},\dots,y_{k})$ if and only if
$f(x_{1},\dots,x_{n})+g(y_{1},\dots,y_{k})=0$, it suffices to show the
statement for the special case $f(x_{1},\dots,x_{n})=0$. Every Boolean term
can be rewritten as a finite union of finite intersections of arguments and
complements of arguments. Therefore, $f(x_{1},\dots,x_{n})=0$ is equivalent to
a conjunction of formulas of the form $x_{i_{1}}\cap\cdots\cap
x_{i_{r}}\subseteq x_{j_{1}}\cup\cdots\cup x_{j_{s}}$. These formulas are
preserved by $i$, because $i$ preserves $R_{r,s}$.
We claim that $i$ also preserves all quantifier-free formulas of
$(\mathfrak{A},\prec)$. The definition of $\prec$ implies that it suffices to
show that $i$ preserves $\prec$ for the atoms of the Boolean algebra generated
by the domain $\\{a_{1},\dots,a_{m}\\}$ of $i$. Let $b$ and $c$ be two such
atoms with $b\prec c$. Then $b=\bigcap_{i=1}^{m}(a_{i}+d_{i})$ and
$c=\bigcap_{i=1}^{m}(a_{i}+e_{i})$ for some $d,e\in\\{0,1\\}^{m}$, and hence
$i(b)\prec i(c)$, because $i$ preserves the relation $O_{m,d,e}$. By the
homogeneity of $(\mathfrak{A},\prec)$, the map $i$ has an extension to an
automorphism of $(\mathfrak{A},\prec)$. The restriction of this map to $B$ is
an automorphism of $\mathfrak{B}$, concluding the proof. ∎
###### Lemma 9.7.
The structure $\mathfrak{B}$ is Ramsey.
###### Proof.
The structure $(\mathfrak{A}^{\prime},\prec)$ and its homogeneous substructure
$\mathfrak{A}^{\prime\prime}$ with domain $B$ have topologically isomorphic
automorphism groups; since the Ramsey property only depends on the topological
automorphism group (see, e.g., [KPT05]), it follows that
$\mathfrak{A}^{\prime\prime}$ is Ramsey. It is easy to see that
$\mathfrak{A}^{\prime\prime}$ and $\mathfrak{B}$ have the same automorphism
group, even as permutation groups, which implies the statement. ∎
Consider the
$\\{\mathtt{EQ},\mathtt{DR},\mathtt{PO},\mathtt{PP},\mathtt{PPI}\\}$-structure
with domain $A$ whose relations are defined over $\mathfrak{A}$ by the
expressions in (7), (8), (9), (10), (11); let $\mathfrak{D}$ be the
substructure of this structure with domain $B$.
###### Lemma 9.8.
There exists a self-embedding $e$ of $(\mathfrak{D},\prec)$ such that
1. (1)
for all $k,l\in{\mathbb{N}}$ and $a_{1},\dots,a_{k},b_{1},\dots,b_{l}\in B$ we
have that
$(e(a_{1}),\dots,e(a_{k}),e(b_{1}),\dots,e(b_{l}))\in R_{k,l}$
if and only if $(a_{i},b_{j})\in\mathtt{PP}\cup\mathtt{EQ}$ or
$(a_{i},a_{j})\in\mathtt{DR}$ for some $i,j\leq\max(k,l)$, and
2. (2)
for every $k\in{\mathbb{N}}$ and $d,e\in\\{0,1\\}^{k}$ there exists a
quantifier-free formula $\phi_{k,d,e}$ in the signature of
$(\mathfrak{D},\prec)$ that defines $O_{k,d,e}$.
The function $e$ is range-rigid with respect to
$\operatorname{Aut}(\mathfrak{B})$.
###### Proof.
First note that any function $e$ that satisfies the first statement of the
lemma must be range-rigid. By the homogeneity of $\mathfrak{B}$, orbits of
$k$-tuples in $\mathfrak{B}$ can be described by conjunctions of relations of
the form $R_{k,l}$ and of the form $O_{k,d,e}$. Therefore, every orbit of
$k$-tuples in the range of $e$ has by items (1) and (2) of the statement a
quantifier-free definition over the relations of $(\mathfrak{D},\prec)$. Since
$e$ preserves these relations, $e$ preserves all orbits of $k$-tuples in the
range of $e$.
To prove the first part of the statement, by a standard compactness argument
it suffices to prove that for every finite substructure $\mathfrak{F}$ of
$\mathfrak{B}$ there exists a finite substructure $(\mathfrak{G},\prec)$ of
$(\mathfrak{A},\prec)$ and a map $f\colon F\to G\setminus\\{0\\}$ that
preserves the relations of $(\mathfrak{D},\prec)$ and satisfies the property
of $e$ formulated in items (1) and (2) in the statement. Let
$\displaystyle I\coloneqq\big{\\{}$
$\displaystyle\\{a_{1},\dots,a_{k}\\}\mid\;k\in{\mathbb{N}},a_{1},\dots,a_{k}\in
F,(a_{i},a_{j})\in\mathtt{PO}^{\mathfrak{F}}\text{ for all }1\leq i<j\leq
k\\}\big{\\}}$
and let $f\colon F\to 2^{I}$ be the map given by
$f(a)\coloneqq\\{X\in I\mid\exists a^{\prime}\in X.\,a^{\prime}\subseteq
a\\}.$
Define the relation $\sqsubset$ on $I$ so that $X\sqsubset Y$ holds for
$X,Y\in I$ if and only if the smallest element of $X+Y=(X\cup
Y)\setminus(X\cap Y)$ with respect to $\prec$ is contained in $X$.
Claim 1. $\sqsubset$ defines a linear order on $I$. It is clear from the
definition of $\sqsubset$ that for all $X,Y\in I$ exactly one of $X=Y$,
$X\sqsubset Y$, and $Y\sqsubset X$ holds. To show that $\sqsubset$ is
transitive it suffices to show that $\sqsubset$ does not contain a directed
3-cycle. Let $X,Y,Z\in I$ be pairwise distinct. Let $a$ be the smallest
element in $(X\cup Y\cup Z)\setminus(X\cap Y\cap Z)$ and suppose that $a\in
X\setminus Y$, so that $X\sqsubset Y$.
* •
If $a\in Z$ then $a$ is the the smallest element in $Z+Y$, and thus
$Z\sqsubset Y$.
* •
If $a\notin Z$ then $a$ is the smallest element in $X+Z$, and thus $X\sqsubset
Z$.
In either case the restriction of $\sqsubset$ to $X,Y,Z$ does not form a
directed 3-cycle. By symmetry, the assumption that $x\in X\setminus Y$ can be
made without loss of generality; this proves the claim.
Let $\mathfrak{G}$ be the Boolean algebra on $2^{I}$, and let $\prec$ denote
the linear order on $G$ such that $S\prec T$ if the largest element of the
symmetric difference $S+T$ with respect to the linear order $\sqsubset$ is
contained in $T$. Note that $\prec$ is an antilexicographic linear order on
$G$ and hence $(\mathfrak{G};\prec)$ embeds into $(\mathfrak{A};\prec)$. Also
note that the image of $f$ lies in $G\setminus\\{0\\}$.
Claim 2. Every $X\in I$ equals the largest element of $I$ with respect to
$\sqsubset$ which is contained in $\bigcap_{a\in X}f(a)$. By construction,
$X\in f(a)$ for all $a\in X$, that is, $X\in\bigcap_{a\in X}f(a)$. Assume that
there exists $Y\in\bigcap_{a\in X}f(a)$. We have to show that $Y=X$ or
$Y\sqsubset X$. If $X\subseteq Y$ then we are done. Otherwise, let $a$ be the
smallest element of $X\setminus Y$ with respect to $\prec$. Since $Y\in f(a)$
there exists some $a^{\prime}\in Y$ such that $a^{\prime}\subseteq a$ by the
definition of $f(a)$. Since $a\notin Y$, it follows that $a^{\prime}\neq a$.
So $a^{\prime}\subset a$ and thus $a$ and $a^{\prime}$ cannot both be in $X$.
Since $a\in X$ we conclude that $a^{\prime}\notin X$. Hence, $a^{\prime}\in
Y\setminus X$. Since $\prec$ extends $\subset$ it follows that
$a^{\prime}\prec a$. Since $a$ was chosen to be minimal in $X\setminus Y$ with
respect to $\prec$ this implies that $Y\sqsubset X$.
To prove that $f$ preserves the relations of $\mathfrak{D}$, it suffices to
show that $f$ preserves $\mathtt{PP}$, $\mathtt{DR}$, and $\mathtt{PO}$.
* •
Suppose that $a,b\in F$ satisfy $\mathtt{PP}(a,b)$ in $\mathfrak{F}$. Let us
assume that $X\in f(a)$ for some $X\in I$. By definition we know that
$a^{\prime}\in X$ for some $a^{\prime}\subseteq a$. Then we also have
$a^{\prime}\subseteq b$, and thus by definition $X\in f(b)$. This concludes
the proof that $f(a)\subseteq f(b)$. On the other hand $\\{b\\}\in
f(b)\setminus f(a)$, thus in fact $\mathtt{PP}(f(a),f(b))$ holds in
$\mathfrak{G}$.
* •
Suppose that $a,b\in F$ satisfy $\mathtt{DR}(a,b)$ in $\mathfrak{F}$. If $X\in
f(a)$ then by definition there exists $a^{\prime}\in X$ such that
$a^{\prime}\subseteq a$. Since $X\in I$ and $a^{\prime}\cap b=0$ this implies
that $b^{\prime}\notin X$ for every $b^{\prime}\subseteq b$. Therefore,
$X\notin f(b)$. This implies that $\mathtt{DR}(f(a)\cap f(b))$ holds in
$\mathfrak{G}$.
* •
Suppose that $a,b\in F$ satisfy $\mathtt{PO}(a,b)$ in $\mathfrak{F}$. Then
$\\{a\\}\in f(a)\setminus f(b)$, $\\{b\\}\in f(b)\setminus f(a)$, and
$\\{a,b\\}\in\gamma(a)\cap\gamma(b)$. Therefore, $\mathtt{PO}(f(a),f(b))$
holds in $\mathfrak{G}$.
We now show that $f$ preserves $\prec$. Let $a,b\in F$ be such that $a\prec
b$. Then by definition $\\{a\\}\sqsubset\\{b\\}$. Claim 2 implies that the
largest elements contained in $f(a)$ and $f(b)$ are $\\{a\\}$ and $\\{b\\}$,
respectively. It follows that $f(a)\prec f(b)$.
To prove that $f$ satisfies items (1) and (2) of the statement, the following
notation will be convenient. For $X\subseteq F$, we write $M(X)$ for the
minimal elements in $S$ with respect to $\subset$, i.e.,
$M(X)\coloneqq\\{a\in X\mid\neg\exists b\in X.\,b\subset a\\}.$
Note that for every $a\in X$ there exists an $a^{\prime}\in M(X)$ such that
$a^{\prime}\subseteq a$, because $X\subseteq F$ is finite.
Claim 3. Let $a_{1},\dots,a_{k},b_{1},\dots,b_{l}\in F$. Then the following
are equivalent.
1. (1)
$(f(a_{1}),\dots,f(a_{k}),f(b_{1}),\dots,f(b_{l}))\notin R_{k,l}$;
2. (2)
$c\coloneqq\bigcap_{i=1}^{k}f(a_{i})\cap\bigcap_{i=1}^{l}\overline{f(b_{i})}\neq\emptyset$;
3. (3)
$a_{i}\cap a_{j}\neq 0$ and $a_{i}\not\subseteq b_{j}$ for all $1\leq
i<j\leq\max(k,l)$.
4. (4)
The $\sqsubset$-maximal element of
$c=\bigcap_{i=1}^{k}f(a_{i})\cap\bigcap_{i=1}^{k}\overline{f(b_{i})}$ is
$M(\\{a_{1},\dots,a_{k}\\})$.
(1) and (2) are equivalent by definition.
(2) implies (3): if there are $i,j\in\\{1,\dots,k\\}$ such that $a_{i}\cap
a_{j}=\emptyset$, then $f(a_{i})\cap f(a_{j})=\emptyset$ since $f$ preserves
$\mathtt{DR}$, and hence $c=\emptyset$. If there are $i\leq k$ and $j\leq l$
such that $a_{i}\subseteq b_{j}$ then $a\cap\overline{b_{j}}=\emptyset$ and
hence $c=\emptyset$.
(3) implies (4): Let $X\coloneqq M(\\{a_{1},\dots,a_{k}\\})$. Then by
assumption $X\in I$, and by Claim 2 we know that $X$ is the maximal element of
$\bigcap_{i=1}^{k}f(a_{i})$ with respect to $\sqsubset$. Suppose for
contradiction that $X\in f(b_{j})$ for some $j\in\\{1,\dots,l\\}$. The
definition of $f$ implies that there exists $b_{j}^{\prime}\subseteq b_{j}$
such that $b_{j}^{\prime}\in X$. In other words, $b_{j}^{\prime}=a_{i}$ for
some $i\in\\{1,\dots,k\\}$, and therefore we have $a_{i}\subseteq b_{j}$, a
contradiction.
(4) implies (2): trivial.
Note that the equivalence of (1) and (3) in Claim 3 immediately implies that
$f$ satisfies item (1) of the statement. To prove that $f$ satisfies item (2),
let $a_{1},\dots,a_{k}\in F$ and $d,e\in\\{0,1\\}^{k}$. Let $X\coloneqq
M(\\{a_{i}\mid d_{i}=0\\})$ and $Y\coloneqq M(\\{a_{i}\mid e_{i}=0\\})$. We
will prove that $(f(a_{1}),\dots,f(a_{k}))\in O_{k,d,e}$ if and only if
$\bigcap_{i=1}^{k}(f(a_{i})+e_{i})\neq 0$ and either
1. (1)
$\bigcap_{i=1}^{k}(f(a_{i})+d_{i})=0$, or
2. (2)
$\bigcap_{i=1}^{k}(f(a_{i})+d_{i})\neq 0$ and $X\sqsubset Y$.
If $\bigcap_{i=1}^{k}(f(a_{i})+e_{i})=0$ then clearly
(12)
$\displaystyle\bigcap_{i=1}^{k}(f(a_{i})+d_{i})\prec\bigcap_{i=1}^{k}(f(a_{i})+e_{i})$
is false and hence $(f(a_{1}),\dots,f(a_{k}))\notin O_{k,d,e}$ and we are
done. So suppose that $\bigcap_{i=1}^{k}(f(a_{i})+e_{i})\neq 0$. If
$\bigcap_{i=1}^{k}(f(a_{i})+d_{i})=0$ then $(\ref{eq:order})$ holds. In this
case $(f(a_{1}),\dots,f(a_{k}))\in O_{k,d,e}$ and we are again done. So
suppose that $\bigcap_{i=1}^{k}(f(a_{i})+d_{i})\neq 0$.
Then by Claim 3 we know that $X,Y\in I$ and $X$ and $Y$ are the
$\sqsubset$-maximal elements in $\bigcap_{i=1}^{k}(f(a_{i})+d_{i})$ and
$\bigcap_{i=1}^{k}(f(a_{i})+e_{i})$, respectively. Thus, if $X\sqsubset Y$
then $\bigcap_{i=1}^{k}(f(a_{i})+d_{i})\prec\bigcap_{i=1}^{k}(f(a_{i})+e_{i})$
and so $(f(a_{1}),\dots,f(a_{k}))\in O_{k,d,e}$ and we are done. If
$Y\sqsubset X$ then
$\bigcap_{i=1}^{k}(f(a_{i})+e_{i})\prec\bigcap_{i=1}^{k}(f(a_{i})+d_{i})$ and
hence $(f(a_{1}),\dots,f(a_{k}))\notin O_{k,d,e}$ and we are also done. If
$X=Y$, let $i\in\\{1,\dots,k\\}$. If $d_{i}=0$ then by the definition of $X$
there exists an $a_{j}\in X$ such that $a_{j}\subseteq a_{i}$. Then $a_{j}\in
Y$ and thus $e_{j}=0$. Since $\bigcap_{l=1}^{k}(f(a_{l})+e_{l})\neq 0$ is
follows that in particular
$f(a_{j})\cap(f(a_{i})+e_{i})=(\gamma(a_{j})+e_{j})\cap(f(a_{i})+e_{i})\neq\emptyset$.
Since $f$ preserves $\mathtt{DR}$ this implies that $a_{j}\cap(a_{i}\cap
e_{i})\neq 0$. Since $a_{j}\subseteq a_{i}$ this is only possible if
$e_{i}=0$. Similarly one can show that $e_{i}=0$ implies $d_{i}=0$. Therefore,
$d=e$, and thus $O_{k,d,e}=\emptyset$. In particular,
$(f(a_{1}),\dots,f(a_{k}))\notin O_{k,d,e}$.
Claim 3 implies that $\bigcap_{i=1}^{k}(f(a_{j})+e_{i})\neq 0$ and
$\bigcap_{i=1}^{k}(f(a_{i})+d_{i})=0$ can be defined by quantifier-free
$\\{\mathtt{PP},\mathtt{EQ},\mathtt{DR}\\}$-formulas. Note that condition (2)
above can be expressed (assuming for simplicity of notation that
$X=\\{a_{i}\mid d_{i}=0\\}$ and $Y=\\{a_{i}\mid e_{i}=0\\}$; the general case
is similar) by the quantifier-free $\\{\prec\\}$-formula
$\bigvee_{i:d_{i}=0,e_{i}=1}\bigwedge_{j:d_{j}\neq e_{j}}a_{i}\prec a_{j}$
which concludes the proof. ∎
###### Proposition 9.9.
$\operatorname{Age}(\mathfrak{D},\prec)=\operatorname{Age}(\mathfrak{R},\prec)$.
###### Proof.
Note that $\mathfrak{D}$ has the same age as $\mathfrak{S}$ and hence the same
age as $\mathfrak{R}$. Also, $\prec$ is a linear extension of
$\mathtt{PP}^{\mathfrak{D}}$, and hence every finite substructure of
$(\mathfrak{D},\prec)$ is also a substructure of $(\mathfrak{R},\prec)$.
Conversely, let $(\mathfrak{F},\prec)$ be a finite substructure of
$(\mathfrak{R},\prec)$. Let $g$ be an embedding of $\mathfrak{F}$ into
$\mathfrak{D}$. Let $u_{1},\dots,u_{k}$ be an enumeration of $F$ such that
$u_{1}\prec\cdots\prec u_{k}$. Let $v_{1},\dots,v_{k}\in B$ be such that
$(v_{i},g(u_{j}))\in\mathtt{DR}^{\mathfrak{D}}$ for all
$i,j\in\\{1,\dots,k\\}$ and such that
$(v_{i},v_{j})\in\mathtt{DR}^{\mathfrak{D}}$ for $i\neq j$. Then we define
$f\colon F\to B$ by
$b(u)\coloneqq g(u)\cup\bigcup_{(u_{i},u)\in\mathtt{PP}^{\mathfrak{F}}}v_{i}.$
Note that $v_{1},\dots,v_{k}$ are atoms in the Boolean algebra generated by
the elements $g(F)\cup\\{v_{1},\dots,v_{n}\\}$ in $\mathfrak{A}$, and let
$w_{1},\dots,w_{\ell}$ be the other atoms. We may assume that $\prec$ is
defined on this Boolean algebra according the enumeration
$w_{1},\dots,w_{\ell},v_{1},\dots,v_{k}$ of the atoms. We prove that $f$ is an
embedding of $(\mathfrak{F},\prec)$ into $(\mathfrak{D},\prec)$. Let
$u,u^{\prime}\in F$. If $(u,u^{\prime})\in\mathtt{PP}^{\mathfrak{F}}$, then
$g(u)\subset g(v)$ and
$\\{i\colon(u_{i},u)\in\mathtt{PP}^{\mathfrak{F}}\\}\subset\\{i\colon(u_{i},u^{\prime})\in\mathtt{PP}^{\mathfrak{F}}\\}$
by the transitivity of $\mathtt{PP}^{\mathfrak{F}}$, so
$(b(u),b(u^{\prime}))\in\mathtt{PP}^{\mathfrak{D}}$. It is also clear that $b$
preserves $\mathtt{EQ}$, $\mathtt{PPI}$, $\mathtt{DR}$. To see that $b$
preserves $\mathtt{PO}$, note that if
$(u,u^{\prime})\in\mathtt{PO}^{\mathfrak{F}}$, then
$(g(u),g(u^{\prime}))\in\mathtt{PO}^{\mathfrak{D}}$, so $g(u)\cap
g(u^{\prime})$, $g(u)\cap\overline{g(u^{\prime})}$, and $\overline{g(u)}\cap
g(u^{\prime})$ are non-empty. Note that $g(u)\cap g(u^{\prime})\subseteq
b(u)\cap b(u^{\prime})$, $g(u)\cap\overline{g(u^{\prime})}\subseteq
b(u)\cap\overline{b(u^{\prime})}$, and $\overline{g(u)}\cap
g(u^{\prime})\subseteq\overline{b(u)}\cap b(u^{\prime})$, so
$(b(u),b(u^{\prime}))\in\mathtt{PO}^{\mathfrak{D}}$.
To prove that $b$ preserves $\prec$, let $i,j\in\\{1,\dots,k\\}$ be such that
$i<j$ and $u_{i}\prec u_{j}$. Then
$\mathtt{PP}^{\mathfrak{D}}(v_{j},b(u_{j}))$,
$\mathtt{DR}^{\mathfrak{D}}(v_{j},b(u_{i}))$, and for every
$m\in\\{j+1,\dots,k\\}$ we have $\mathtt{DR}^{\mathfrak{D}}(v_{m},b(u_{i}))$
and $\mathtt{DR}^{\mathfrak{D}}(v_{m},b(u_{j}))$. Indeed, if
$\mathtt{DR}^{\mathfrak{D}}(v_{m},b(u_{i}))$ does not hold, then
$\mathtt{PP}^{\mathfrak{D}}(v_{m},b(u_{i}))$ and hence
$\mathtt{PP}^{\mathfrak{F}}(u_{m},u_{i})$ by the definition of $b$. This in
turn implies that $u_{m}\prec u_{i}$ and hence $m<i$, a contradiction. By the
definition of $\prec$ on $A$ this implies that $b(u_{i})\prec b(u_{j})$ and
finishes the proof of the claim. ∎
Every relation of $(\mathfrak{D},\prec)$ has a quantifier-free definition in
the homogeneous structure $\mathfrak{B}$; let $(\mathfrak{C},\prec)$ be the
$\\{\mathtt{EQ},\mathtt{DR},\mathtt{PO},\mathtt{PP},\mathtt{PPI},\prec\\}$-structure
defined over $\mathfrak{B}_{e}$ via these quantifier-free definitions.
###### Proposition 9.10.
$(\mathfrak{C},\prec)$ is homogeneous.
###### Proof.
It suffices to show that $(\mathfrak{C},\prec)$ and $\mathfrak{B}_{e}$ are
quantifier-free interdefinable. By definition, every relation of
$(\mathfrak{C},\prec)$ has a quantifier-free definition in $\mathfrak{B}_{e}$.
Conversely, let $R$ be a relation of $\mathfrak{B}_{e}$. If $R$ is of the form
$R_{k,l}$ then item (1) of Lemma 9.8 implies that $R$ has a quantifier-free
definition over $(\mathfrak{C},\prec)$. If $R$ is of the form $O_{k,d,e}$ then
item (2) of Lemma 9.8 implies that $R$ has a quantifier-free definition over
$(\mathfrak{C},\prec)$. ∎
###### Corollary 9.11.
$(\mathfrak{C},\prec)$ and $(\mathfrak{R},\prec)$ are isomorphic.
###### Proof.
It suffices to prove that the homogeneous structures $(\mathfrak{C},\prec)$
and $(\mathfrak{R},\prec)$ have the same age. By Proposition 9.9, the age of
$(\mathfrak{R},\prec)$ equals the age of $(\mathfrak{D},\prec)$. ∎
###### Theorem 9.12.
$(\mathfrak{R},\prec)$ is Ramsey.
###### Proof.
We apply the Ramsey transfer (Theorem 9.5) to the homogeneous Ramsey structure
$\mathfrak{B}$ from Lemma 9.6 and the map $e$ from Lemma 9.8 which is range-
rigid with respect to $\operatorname{Aut}(\mathfrak{B})$. Theorem 9.5 states
that the homogeneous structure $\mathfrak{B}_{e}$ is Ramsey. The structure
$\mathfrak{B}_{e}$ has the same automorphism group as the homogeneous
structure $(\mathfrak{R};\prec)$; it follows that $(\mathfrak{R};\prec)$ is
Ramsey as well. ∎
### 9.2. Polymorphisms of $\mathfrak{R}$ that are canonical with respect to
$(\mathfrak{R},\prec)$
From now on, we identify the symbols
$\mathtt{EQ},\mathtt{PP},\mathtt{PPI},\mathtt{DR},\mathtt{PO}$ with the
respective relations of $\mathfrak{R}$, and we write $\succ$ for the converse
of $\prec$. Note that $\mathtt{PP}\cup\mathtt{DR}\cup\mathtt{PO}$ and
$\mathtt{PPI}\cup\mathtt{DR}\cup\mathtt{PO}$ are primitively positively
definable in $\mathfrak{R}$, since they are entries in Table 1. Hence, their
intersection $\mathtt{DR}\cup\mathtt{PO}$ is primitively positively definable
in $\mathfrak{R}$, too. We also write $\bot$ instead of
$\mathtt{DR}\cup\mathtt{PO}$, and $\smash{{}^{\bot}_{\prec}}$ for the relation
$\bot\,\cap\prec$.
The composition table for the binary relations with a first-order definition
over $(\mathfrak{R};\prec)$ can be derived from the composition table of
$\mathfrak{R}$ (Table 1) using the following lemma.
###### Lemma 9.13.
Let
$R_{1},R_{2}\in\\{\mathtt{EQ},\mathtt{PP},\mathtt{PPI},\mathtt{DR},\mathtt{PO}\\}$
and let $O_{1},O_{2}$ be two orbits of pairs of $(\mathfrak{R};\prec)$ such
that $O_{i}\subseteq R_{i}$. Then
$O_{1}\circ O_{2}=\begin{cases}(R_{1}\circ R_{2})\,\cap\prec&\text{ if
}O_{1},O_{2}\subseteq\;\prec\\\ (R_{1}\circ R_{2})\,\cap\succ&\text{ if
}O_{1},O_{2}\subseteq\;\succ\\\ R_{1}\circ R_{2}&\text{ otherwise.
}\end{cases}$
###### Proof.
It is clear that $O_{1}\circ O_{2}\subseteq R_{1}\circ R_{2}$ and that if
$O_{1},O_{2}\subseteq\;\prec$ then
$O_{1}\circ O_{2}\subseteq\,\prec\circ\prec\,=\,\prec.$
So the $\subseteq$-containment in the statement of the lemma holds in the
first case, and by similar reasoning also in the other two cases. The reverse
containment in the first two cases is also clear since the homogeneity of
$(\mathfrak{R},\prec)$ implies that the expression in the statement on the
right describes an orbit of pairs in $\operatorname{Aut}(\mathfrak{R},\prec)$
(we have already seen that it is non-empty). To show the equality in the third
case, let $(x,z)\in R_{1}\circ R_{2}$. If $R_{1}\circ R_{2}$ equals
$\mathtt{PP}$ (or $\mathtt{PPI}$) then $x\prec z$ (or $z\succ x$), and again
$R_{1}\circ R_{2}$ is an orbit of pairs in
$\operatorname{Aut}(\mathfrak{R},\prec)$ and the equality holds. If
$R_{1}\circ R_{2}$ equals $\mathtt{EQ}$ then the statement is clear, too.
Otherwise, let $x^{\prime},y^{\prime},z^{\prime}\in R$ be such that
$(x^{\prime},y^{\prime})\in O_{1}$ and $(y^{\prime},z^{\prime})\in O_{2}$. If
$(x^{\prime},z^{\prime})$ lies in the same orbit as $(x,z)$ in
$\operatorname{Aut}(\mathfrak{R},\prec)$ then $(x,z)\in O_{1}\circ O_{2}$.
Otherwise, consider the structure induced by $(\mathfrak{R},\prec)$ on
$\\{x^{\prime},y^{\prime},z^{\prime}\\}$; we claim that if we replace the
tuple $(x^{\prime},z^{\prime})$ in $\prec$ by the tuple
$(z^{\prime},x^{\prime})$, the resulting structure still embeds into
$(\mathfrak{R};\prec)$. By assumption, $O_{1}\subseteq\,\prec$ and
$O_{2}\subseteq\,\succ$, or $O_{1}\subseteq\,\succ$ and
$O_{2}\subseteq\,\prec$, so the modified relation $\prec$ is still acyclic.
Moreover, since $R_{1}\circ R_{2}\in\\{\mathtt{DR},\mathtt{PO}\\}$, the
modified relation $\prec$ is still a linear extension of $\mathtt{PP}$ and
hence in $\operatorname{Age}(\mathfrak{R};\prec)$. The homogeneity of
$(\mathfrak{R};\prec)$ implies that $(x^{\prime},z^{\prime})$ lies in the same
orbit as $(x,z)$, and hence $(x,z)\in O_{1}\circ O_{2}$. This concludes the
proof that $R_{1}\circ R_{2}\subseteq O_{1}\circ O_{2}$. ∎
###### Corollary 9.14.
Let $O\subseteq\;\prec$ be an orbit of pairs in
$\operatorname{Aut}(\mathfrak{R};\prec)$. Then $\mathtt{PP}\subseteq O\circ
O$.
###### Proof.
If $O=\mathtt{PP}$, then $O\circ O=O=\mathtt{PP}$. If
$O=(\mathtt{DR}\;\cap\prec)$ then
$\displaystyle O\circ O$
$\displaystyle=(\mathtt{DR}\circ\mathtt{DR})\,\cap\prec$ (Lemma 9.13)
$\displaystyle={\bf 1}\,\cap\prec$ (table in Figure 1)
$\displaystyle=\;\prec\,.$
Similarly we may compute that $\mathtt{PP}$ is contained in
$(\mathtt{PO}\,{\cap}\,{\prec})\circ(\mathtt{PO}\,{\cap}\,{\prec})$. ∎
The following general observations are useful when working with operations
that preserve binary relations over some set $B$. If $f\colon B^{k}\to B$ is
an operation that preserves
$R_{1},\dots,R_{k},R^{\prime}_{1},\dots,R_{k}^{\prime}\subseteq B^{2}$ then
$\displaystyle f(R_{1}\circ R_{1}^{\prime},\dots,R_{k}\circ R^{\prime}_{k})$
(13) $\displaystyle\subseteq\;$ $\displaystyle f(R_{1},\dots,R_{k})\circ
f(R^{\prime}_{1},\dots,R^{\prime}_{k}).$
Also note that
(14) $\displaystyle
f(R_{1}^{\smile},\dots,R_{k}^{\smile})=f(R_{1},\dots,R_{k})^{\smile}.$
###### Lemma 9.15.
Let
$f\in\operatorname{Pol}(\mathfrak{R})\cap\operatorname{Can}(\mathfrak{R},\prec)$.
Then $f$ preserves $\prec$ and $\smash{{}^{\bot}_{\prec}}$.
###### Proof.
Let $a_{1},\dots,a_{k},b_{1},\dots,b_{k}$ be elements of $\mathfrak{R}$ such
that $a_{i}\prec b_{i}$ for all $i\in\\{1,\dots,k\\}$ and
$f(b_{1},\dots,b_{k})\preceq f(a_{1},\dots,a_{k})$. For $i\in\\{1,\dots,k\\}$,
let $O_{i}$ be the orbit of $(a_{i},b_{i})$ in
$\operatorname{Aut}(\mathfrak{R},\prec)$. We then have
$\displaystyle f(\mathtt{PP},\dots,\mathtt{PP})$ $\displaystyle\subseteq
f(O_{1}\circ O_{1},\dots,O_{k}\circ O_{k})$ (Corollary 9.14)
$\displaystyle\subseteq f(O_{1},\dots,O_{k})\circ f(O_{1},\dots,O_{k})$ (13)
$\displaystyle\subseteq(\succeq\circ\succeq)$ (canonicity)
$\displaystyle=\;\succeq$
which is a contradiction to $f$ being a polymorphism of $\mathfrak{R}$. Since
$\bot$ is primitively positively definable in $\mathfrak{R}$, it also follows
that every polymorphism of $\mathfrak{R}$ which is canonical with respect to
$(\mathfrak{R},\prec)$ preserves $\smash{{}^{\bot}_{\prec}}$. ∎
###### Lemma 9.16.
Let $f\in\operatorname{Pol}(\mathfrak{R})$ be canonical over $\mathfrak{R}$ or
over $(\mathfrak{R},\prec)$. Then $f$ preserves $1\setminus\mathtt{EQ}$.
###### Proof.
Let $k$ be the arity of $f$. First suppose that $f$ is canonical over
$(\mathfrak{R},\prec)$. Let $O_{1},\dots,O^{k}$ be orbits of pairs of
$\operatorname{Aut}(\mathfrak{R};\prec)$ that are distinct from $\mathtt{EQ}$.
Suppose for contradiction that $f(O_{1},\dots,O_{k})=\mathtt{EQ}$. Then
$\displaystyle f(O_{1}\circ O_{1},\dots,O_{k}\circ O_{k})$
$\displaystyle\subseteq f(O_{1},\dots,O_{k})\circ\cdots\circ
f(O_{1},\dots,O_{k})$
$\displaystyle\subseteq\mathtt{EQ}\circ\cdots\circ\mathtt{EQ}=\mathtt{EQ}.$
Since $O_{i}\circ O_{i}$ contains $\mathtt{PP}$ or $\mathtt{PPI}$ for every
$i\in\\{1,\dots,k\\}$ by Corollary 9.14, it follows that
$f(P_{1},\dots,P_{k})\subseteq\mathtt{EQ}$ for some
$P_{1},\dots,P_{k}\in\\{\mathtt{PP},\mathtt{PPI}\\}$. Note that
$f(P_{1}^{\smile},\dots,P_{k}^{\smile})=\mathtt{EQ}^{\smile}=\mathtt{EQ}$ by
14. Since $\mathtt{PP}\subseteq\mathtt{PP}\circ\mathtt{PPI}$ and
$\mathtt{PP}\subseteq\mathtt{PPI}\circ\mathtt{PP}$ we obtain that
$f(\mathtt{PP},\mathtt{PP})\subseteq\mathtt{EQ}\circ\mathtt{EQ}=\mathtt{EQ}$,
a contradiction to $f(\mathtt{PP},\mathtt{PP})\subseteq\mathtt{PP}$. We
conclude that $f$ preserves $1\setminus\mathtt{EQ}$. If $f$ is canonical over
$\mathfrak{R}$ instead of $(\mathfrak{R},\prec)$, then the statement can be
shown similarly, using the table in Figure 1 instead of Corollary 9.14. ∎
### 9.3. Independent substructures
To verify the UIP property, we will use Theorem 6.4 and therefore need certain
pairs $(\mathfrak{A}_{1},\mathfrak{A}_{2})$ of independent elementary
substructures of $(\mathfrak{R},\prec)$.
###### Lemma 9.17.
There are elementary substructures $\mathfrak{A}_{1}$ and $\mathfrak{A}_{2}$
of $(\mathfrak{R},\prec)$ such that for all $a_{1}\in A_{1}$ and $a_{2}\in
A_{2}$ we have $(a_{1},a_{2})\in\mathtt{DR}$ and $a_{1}\prec a_{2}$; in
particular, $\mathfrak{A}_{1}$ and $\mathfrak{A}_{2}$ are independent.
###### Proof.
By the homogeneity of $(\mathfrak{R},\prec)$ it suffices to show that for
every structure
$(\mathfrak{B},\prec)\in\operatorname{Age}(\mathfrak{R},\prec)$ there are
embeddings $e_{1},e_{2}\colon\mathfrak{B}\to(\mathfrak{R},\prec)$ such that
for all $b_{1},b_{2}\in B$ we have $(e_{1}(a_{1}),e_{2}(a_{2}))\in\mathtt{DR}$
and $e_{1}(b_{1})\prec e_{2}(b_{2})$. Choose an embedding $f$ of
$\mathfrak{B}$ into the structure $\mathfrak{S}$ from the definition of
$\mathfrak{R}$; so $f(b)\subseteq{\mathbb{N}}$ for each $b\in B$. Then
$f_{1}\colon b\mapsto\\{2n\colon n\in f(b)\\}$ and $f_{2}\colon
b\mapsto\\{2n+1\colon n\in f(b)\\}$ are two embeddings of $\mathfrak{B}$ into
$\mathfrak{S}$ such that $(f_{1}(b_{1}),f_{2}(b_{2}))\in\mathtt{DR}$ for all
$b_{1},b_{2}\in B$. Let $\mathfrak{B}^{\prime}$ be the substructure of
$\mathfrak{S}$ with domain $B^{\prime}\coloneqq f_{1}(B)\cup f_{2}(B)$. For
all $b_{1},b_{2}\in B$ with $b_{1}\prec b_{2}$, define the linear order
$\prec$ on $B^{\prime}$ by $f_{1}(b_{1})\prec f_{1}(b_{2})$,
$f_{2}(b_{1})\prec f_{2}(b_{2})$, $f_{1}(b_{1})\prec f_{2}(b_{2})$. Then it is
straightforward to check that $\prec$ extends $\mathtt{PP}$, and hence
$(\mathfrak{B}^{\prime},\prec)\in\operatorname{Age}(\mathfrak{R},\prec)$,
which concludes the proof. ∎
Independent substructures are used in the proof of the following lemma that
plays an important role when verifying the UIP later.
###### Lemma 9.18.
Let $\mathfrak{C}$ be a first-order expansion of $\mathfrak{R}$ and let
$\zeta\colon\operatorname{Pol}(\mathfrak{C})\cap{\operatorname{Can}(\mathfrak{R},\prec)}\to\operatorname{Proj}$
be a clone homomorphism which does not have the UIP with respect to
$\operatorname{Pol}(\mathfrak{C})$. Then for every finite 2-rich finite subset
$F$ of the domain of $\mathfrak{R}$ there exist
$f\in\operatorname{Pol}(\mathfrak{C})^{(2)}$ and
$\alpha_{1},\alpha_{2}\in\operatorname{Aut}(\mathfrak{R};\prec)$ such that
* •
$f$ is canonical on $F\times\alpha_{1}(F)$ and on $F\times\alpha_{2}(F)$ with
respect to $(\mathfrak{R},\prec)$,
* •
$\zeta(f(\operatorname{id},\alpha_{1})|_{F^{2}})\neq\zeta(f(\operatorname{id},\alpha_{2})|_{F^{2}})$
(recall Definition 8),
* •
for all $a,b\in F$, if $(a,b)\in\mathtt{PP}\cup\mathtt{EQ}$, then
$(\alpha_{1}(a),\alpha_{2}(b))\in\mathtt{PP}$, and
* •
for all $a,b\in F$, if $(a,b)\in\;\bot$ then
$(\alpha_{1}(a),\alpha_{2}(b))\in\;\bot$.
###### Proof.
Let $\mathfrak{A}_{1},\mathfrak{A}_{2}$ be the two independent elementary
substructures of $(\mathfrak{R},\prec)$ from Lemma 9.17. Since $\zeta$ does
not have the UIP with respect to $\operatorname{Pol}(\mathfrak{C})$, by
Theorem 6.4 there exists $f^{\prime}\in\operatorname{Pol}(\mathfrak{C})^{(2)}$
and $u_{1},u_{2}\in\overline{\operatorname{Aut}(\mathfrak{A})}$ such that for
$i\in\\{1,2\\}$ the image $\Im(u_{i})$ of $u_{i}$ is contained in $A_{i}$, the
operation $f^{\prime}(\operatorname{id},u_{i})$ is canonical with respect to
$(\mathfrak{R},\prec)$, and
$\zeta(f^{\prime}(\operatorname{id},u_{1}))\neq\zeta(f^{\prime}(\operatorname{id},u_{2}))$.
By Lemma 5.5 there exists a finite subset $X$ of the domain of $\mathfrak{R}$
such that for all $g\in\operatorname{Pol}(\mathfrak{R})^{(2)}$ there exist
$\beta_{1},\beta_{2}\in\operatorname{Aut}(\mathfrak{R},\prec)$ with
$\beta_{1}(F),\beta_{2}(F)\subseteq X$ such that $g$ is canonical on
$\beta_{1}(F)\times\beta_{2}(F)$. For $i\in\\{1,2\\}$, let
$\epsilon_{i}\in\operatorname{Aut}(\mathfrak{R},\prec)$ be such that
$\epsilon_{i}(X)\subseteq\Im(u_{i})$. We may view the substructure of
$\mathfrak{R}$ induced by $\epsilon_{1}(X)\cup\epsilon_{2}(X)$ as a
substructure of $\mathfrak{S}$. By the homogeneity of $\mathfrak{R}$ we may
also assume that the substructure of $\mathfrak{S}$ on
$\epsilon_{1}(X)\cup\epsilon_{2}(X)\cup\\{\epsilon_{1}(x)\cup\epsilon_{2}(x)\colon
x\in X\\}$ has an embedding $e$ into $\mathfrak{R}$ such that
$e(\epsilon_{i}(x))=\epsilon_{i}(x)$ for all $x\in X$ and $i\in\\{1,2\\}$.
Define
$\epsilon\colon X\to\mathfrak{R}\text{ by }\epsilon(x)\coloneqq
e\big{(}\epsilon_{1}(x)\cup\epsilon_{2}(x)\big{)}.$
Let $a,b\in X$. Note that
1. (1)
if $(a,b)\in\mathtt{PP}\cup\mathtt{EQ}$ then
$\displaystyle\epsilon_{i}(a)=e(\epsilon_{i}(a))$ $\displaystyle\subset
e\big{(}\epsilon_{1}(a)\cup\epsilon_{2}(a)\big{)}$ $\displaystyle\subseteq
e\big{(}\epsilon_{1}(b)\cup\epsilon_{2}(b)\big{)}=\epsilon(b)$
and hence $(\epsilon_{i}(a),\epsilon(b))\in\mathtt{PP}$;
2. (2)
if $(a,b)\in\;\bot$ then $(\epsilon_{i}(a),\epsilon_{i}(b))\in\;\bot$. Since
$\epsilon_{i}(a)\in A_{i}$ it follows that $\epsilon_{i}(a)$ is disjoint from
$\epsilon_{j}(b)$ for $j\neq i$. This implies that
$\epsilon_{i}(a)\setminus\epsilon(b)=\epsilon_{i}(a)\setminus\big{(}\epsilon_{1}(b)\cup\epsilon_{2}(b)\big{)}=\epsilon_{i}(a)\setminus\epsilon_{i}(b)\neq\emptyset$
and that for $j\neq i$ we have
$\epsilon_{j}(b)=\epsilon_{j}(b)\setminus\epsilon_{i}(a)\subseteq\epsilon(b)\setminus\epsilon_{i}(a).$
In particular, $\epsilon(b)\setminus\epsilon_{i}(a)\neq\emptyset$. We obtained
that the sets $\epsilon_{i}(a)\setminus\epsilon(b)$ and
$\epsilon(b)\setminus\epsilon_{i}(a)$ are both non-empty. Therefore
$(\epsilon_{i}(a),\epsilon(b))\in\;\bot$.
We define an order $\prec^{\prime}$ on
$\epsilon_{1}(X)\cup\epsilon_{2}(X)\cup\epsilon(X)$ by setting
$a\prec^{\prime}b$ if one of the following holds.
* •
$a,b\in\epsilon_{1}(X)\cup\epsilon_{2}(X)$ and $a\prec b$;
* •
$a\in\epsilon_{1}(X)\cup\epsilon_{2}(X)$ and $b\in\epsilon(X)$;
* •
$a,b\in\epsilon(X)$ and $\epsilon^{-1}(a)\prec\epsilon^{-1}(b)$.
Then it is easy to see that $\prec^{\prime}$ defines a partial order that
extends $\mathtt{PP}$; let $\prec^{\prime\prime}$ be a linear order that
extends $\prec^{\prime}$. By the definition of $(\mathfrak{R};\prec)$ there
exists an automorphism $\gamma$ of $\mathfrak{R}$ that maps
$\prec^{\prime\prime}$ to $\prec$. This shows that we may assume that
$\epsilon$ preserves $\prec$ (otherwise, replace $\epsilon$ by
$\gamma\circ\epsilon$).
By the definition of $X$ there are
$\beta_{1},\beta_{2}\in\operatorname{Aut}(\mathfrak{R},\prec)$ such that
$\beta_{1}(F)\subseteq X$, $\beta_{2}(F)\subseteq X$, and
$f^{\prime}(\operatorname{id},\epsilon)$ is canonical on
$\beta_{1}(F)\times\beta_{2}(F)$ over $(\mathfrak{R},\prec)$. Since
$\epsilon_{i}\beta_{2}(F)\subseteq\epsilon_{i}(X)\subseteq\Im(u_{i})$
for $i\in\\{1,2\\}$ we have that
$\zeta(f^{\prime}|_{\beta_{1}(F)\times\epsilon_{1}\beta_{2}(F)})\neq\zeta(f^{\prime}|_{\beta_{1}(F)\times\epsilon_{2}\beta_{2}(F)})$.
Then for some $i\in\\{1,2\\}$ we have that
$\zeta(f^{\prime}|_{\beta_{1}(F)\times\epsilon_{i}\beta_{2}(F)})\neq\zeta(f^{\prime}|_{\beta_{1}(F)\times\epsilon\beta_{2}(F)})$.
We claim that $f\coloneqq f^{\prime}(\beta_{1},\operatorname{id})$,
$\alpha_{1}\coloneqq\epsilon_{i}\circ\beta_{2}$ and
$\alpha_{2}\coloneqq\epsilon\circ\beta_{2}$ satisfy the conclusion of the
lemma:
* •
$f$ is canonical on $F\times\alpha_{i}(F)$, for $i\in\\{1,2\\}$, because
$f^{\prime}(\operatorname{id},\epsilon)$ is canonical on
$\beta_{1}(F)\times\beta_{2}(F)$.
* •
for all $(a,b)\in F^{2}\cap(\mathtt{PP}\cup\mathtt{EQ})$ we have that
$(\beta_{2}(a),\beta_{2}(b))\in X^{2}\cap(\mathtt{PP}\cup\mathtt{EQ})$ and by
(1) we obtain
$(\alpha_{1}(a),\alpha_{2}(b))=(\epsilon_{i}(\beta_{2}(a)),\epsilon(\beta_{2}(b)))\in\mathtt{PP}$.
* •
for all $(a,b)\in F^{2}\cap\bot$ we have that $(\beta_{2}(a),\beta_{2}(b))\in
X^{2}\cap\bot$ and by (2) we obtain
$\displaystyle(\alpha_{1}(a),\alpha_{2}(b))$
$\displaystyle=(\epsilon_{i}(\beta_{2}(a)),\epsilon(\beta_{2}(b)))\in\;\bot.\qed$
### 9.4. Uniformly continuous minor-preserving maps
Let $\mathfrak{B}$ be a first-order expansion of $\mathfrak{R}$. In this
section we show that if there exists a uniformly continuous minor-preserving
map from
$\operatorname{Pol}(\mathfrak{B})\cap\operatorname{Can}(\mathfrak{R})$ to
$\operatorname{Proj}$, then a specific minor-preserving map $\eta$ (introduced
in Section 9.4.2) or $\rho$ (introduced in Section 9.4.3) is a uniformly
continuous minor-preserving map from
$\operatorname{Pol}(\mathfrak{B})\cap\operatorname{Can}(\mathfrak{R},\prec)\to\operatorname{Proj}$.
For this task, we need some more results from finite universal algebra that
will be recalled in the following (Section 9.4.1).
#### 9.4.1. Cyclic polymorphisms
An operation $f\colon A^{k}\to A$ is
* •
_idempotent_ if $f(x,\dots,x)=x$ for every $x\in A$.
* •
_cyclic_ if $k\geq 2$ and $f(x_{1},\dots,x_{k})=f(x_{2},\dots,x_{k},x_{1})$ |
# Mean-field limits for non-linear Hawkes processes with excitation and
inhibition
by P. Pfaffelhuber, S. Rotter and J. Stiefel
_Albert-Ludwigs University Freiburg_
###### Abstract
We study a multivariate, non-linear Hawkes process $Z^{N}$ on the complete
graph with $N$ nodes. Each vertex is either excitatory (probability $p$) or
inhibitory (probability $1-p$). We take the mean-field limit of $Z^{N}$,
leading to a multivariate point process $\bar{Z}$. If $p\neq\tfrac{1}{2}$, we
rescale the interaction intensity by $N$ and find that the limit intensity
process solves a deterministic convolution equation and all components of
$\bar{Z}$ are independent. In the critical case, $p=\tfrac{1}{2}$, we rescale
by $N^{1/2}$ and obtain a limit intensity, which solves a stochastic
convolution equation and all components of $\bar{Z}$ are conditionally
independent.
†† _AMS 2000 subject classification._ 60G55 (Primary) , 60F05 (Secondary).††
_Keywords and phrases._ Multivariate Hawkes process; Volterra equation; spike
train
## 1 Introduction
In [19, 21], Hawkes processes were introduced as self-excitatory point
processes. Today, they are used in various fields of applications including
seismology [33, 17], interactions in social networks [41, 31], finance [14,
20] and neuroscience [18, 27]. In the classical univariate, linear Hawkes
process, the firing rate at time $t$ is a linear function of
$I_{t}:=\sum_{i}\varphi(t-T_{i})$, where the $T_{i}$’s are previous jump
times. In this case, since rates cannot become negative, $\varphi\geq 0$ is
required, leading to a self-excitatory process.
Our main motivation to study Hawkes processes comes from the neurosciences. In
a graph, vertices model neurons, whereas the (directed) edges are synapses
linking the neurons. A point process indexed by the vertices models action
potentials or spike trains of electrical impulses. Communication via synapses
leads to correlated point processes such that each spike in one neuron
influences the rate by which a neighboring vertex fires. In this field, it is
known that neurons cannot only excite others, but inhibition is another
important factor (see e.g. [30]). The main goal of this paper is to obtain
limit results for the multivariante, non-linear Hawkes process, where the
firing rate at time $t$ is $h(I_{t})$ and $\varphi\leq 0$ can occur as well,
which we interpret as inhibition.
Nonlinear Hawkes processes have been studied to some extent in the past
decades. [4, 32] focus on ergodic properties using Lipschitz conditions of the
transfer function $h$. In [39, 40], central limit theorems and large deviation
results for a univariate nonlinear Hawkes process are given. The case of
inhibition in a multivariate setting was studied in [6] using a thinning
process representation, while [11] uses renewal techniques to establish limit
theorems in the univariate setting.
Mean-field models have frequently been applied in the life sciences; see e.g.
[36] where particular applications in neuroscience are discussed. Mean-field
limits of nonlinear Hawkes processes were first studied by [12]. This has been
extended to age-dependent nonlinear Hawkes processes by [8, 7], and to
specific models including multiple classes [13]. In addition to the law of
large numbers in the mean-field model, a central limit theorem is proved in
[7, 23]. Another branch of research studies a mean-field limit of a spatially
extended (geometric) Hawkes process, showing a law of large numbers [9] and
central limit theorem [10]. Here, a first proof of the neural field equation
could be obtained.
In the present paper, our approach is a combination and extension of [8],[23]
and [15], with a special focus on models including excitatory and inhibitory
neurons/vertices. We assume that each vertex/neuron is either excitatory or
inhibitory, i.e. excites or inhibits all of its neighbors. We will denote by
$p$ the fraction of excitatory vertices/neurons and distinguish the critical
case $p=1/2$ from the non-critical one. For the latter, we obtain in Theorem 1
a classical mean-field result, i.e. by rescaling the interaction intensity by
$N$, a deterministic limit of the intensity and independent point processes
driven by this intensity arises. We also provide a central limit result for
the intensity. In Theorem 2, we are dealing with the critical case. Here, we
rescale the interaction intensity by $N^{1/2}$, leading to a limiting
intensity which is itself stochastic, and drives conditionally independent
point processes. We will discuss connections of our findings to previous work
in Remarks 3.4 and 3.6.
The mean-field model we discuss here is unrealistic for applications in
neuroscience for various reasons. Above all, not all neurons are connected.
Sparse graphs are one possibility for a more realistic model [5], while
multiple classes of differently interconnected neurons are another option
[13]. For a detailed discussion on implications of our findings for
neuroscience, see Section 4.
## 2 Model and main results
We use the following general model for a non-linear Hawkes process:
###### Definition 2.1 (Multi-variate, non-linear Hawkes process).
Let $\mathbb{G}=(\mathbb{V},\mathbb{E})$ be some finite, directed graph, and
write $ji\in\mathbb{E}$ if $j\to i$ is an edge in $\mathbb{G}$. Consider a
family of measurable, real-valued functions $(\varphi_{ji})_{ji\in\mathbb{E}}$
and a family of real-valued, non-negative functions
$(h_{i})_{i\in\mathbb{V}}$. Then, a point process $Z=(Z^{i})_{i\in\mathbb{V}}$
(with state space $\mathbb{N}_{0}^{\mathbb{V}}$) is a multi-variate non-linear
Hawkes process with interaction kernels $(\varphi_{ji})_{ji\in\mathbb{E}}$ and
transfer functions $(h_{i})_{i\in\mathbb{V}}$, if $Z^{i},Z^{j}$ do not jump
simultaneously for $i\neq j$, almost surely, and the compensator of $Z^{i}$
has the form $(\int_{0}^{t}\lambda_{s}^{i}ds)_{t\geq 0}$ with
$\lambda_{t}^{i}:=\lambda_{t}^{i}(Z_{s<t}):=h_{i}\Big{(}\sum_{j:ji\in\mathbb{E}}\int_{0}^{t-}\varphi_{ji}(t-s)dZ_{s}^{j}\Big{)},\qquad
i\in\mathbb{V}.$
We need some mimimal conditions such that the multivariate, non-linear Hawkes
process is well-defined (i.e. exists). As mentioned in [12], Remark 5, the law
of the non-linear Hawkes process is well-defined, provided that the following
assumption holds.
###### Assumption 2.2.
All interaction kernels $\varphi_{ji},ji\in\mathbb{E}$ are locally integrable,
and all transfer functions $(h_{i})_{i\in\mathbb{V}}$ are Lipschitz
continuous.
###### Remark 2.3 (Interpretation and initial condition).
1. 1.
If $dZ_{s}^{j}=1$, we call $\varphi_{ji}(t-s)dZ_{s}^{j}$ the influence of the
point at time $s$ in vertex $j$ on vertex $i$.
2. 2.
Consider the case of monotonically increasing transfer functions. If
$\varphi_{ji}\leq 0$, we then say that vertex $j$ inhibits $i$, since any
point in $Z^{j}$ decreases the jump rate of $Z^{i}$. Otherwise, if
$\varphi_{ji}\geq 0$, we say that $j$ excites $i$.
3. 3.
In our formulation, we have $Z_{0}^{i}=0,i\in\mathbb{V}$, with the consequence
that the $dZ_{u}^{j}$-integral in (5.5) could also be extended to $-\infty$
without any change. We note that it would also be possible to use some initial
condition, i.e. some (fixed) $(Z_{t}^{i})_{i\in\mathbb{V},t\leq 0}$, and
extend the integral to the negative reals.
Let us now come to the mean-field model, where we now fix some basic
assumptions. Note that we will show convergence for large graphs, i.e. all
processes come with a scaling parameter $N$, which determines the size of the
graph.
###### Assumption 2.4 (Mean-field setting).
Let
1. 1.
$\mathbb{G}_{N}=(\mathbb{V}_{N},\mathbb{E}_{N})$ be the complete graph on $N$
vertices, i.e. $\mathbb{V}_{N}=\\{1,...,N\\}$ and $ji,ij\in\mathbb{E}$ for all
$i,j\in\mathbb{V}$;
2. 2.
$h_{i}=h$ for all $i\in\mathbb{V}_{N}$ where $h\geq 0$ is bounded, $h$ and
$\sqrt{h}$ are Lipschitz with constant $h_{Lip}$;
3. 3.
$\varphi_{ji}=\theta_{N}U_{j}\varphi$ for all $j,i\in\mathbb{V}_{N}$, where
$U_{1},U_{2},...$ are iid with $\mathbb{P}(U_{1}=1)=1-\mathbb{P}(U_{1}=-1)=p$,
$\varphi\in\mathcal{C}_{b}^{1}([0,\infty))$, the set of bounded continuously
differentiable functions with bounded derivative, and
$\theta_{N}\in\mathbb{R}$.
The form of $\varphi_{ji}$ implies that node $j$ is exciting all other nodes
with probability $p$, and inhibiting all other nodes with probability $1-p$.
By the law of large numbers, we have that
$\displaystyle\frac{1}{N}\sum_{j=1}^{N}U_{j}\xrightarrow{N\to\infty}2p-1$
(2.1) almost surely, and by the central limit theorem,
$\displaystyle\frac{1}{\sqrt{N}}\sum_{j=1}^{N}(U_{j}+1-2p)=:W_{N}\xRightarrow{N\to\infty}W\sim
N(0,4p(1-p)),$ (2.2)
where $\Rightarrow$ denotes weak convergence. For convergence, this suggests
to use $\theta_{N}=\tfrac{1}{N}$ for $p\neq\tfrac{1}{2}$ and
$\theta_{N}=\tfrac{1}{\sqrt{N}}$ for $p=\tfrac{1}{2}$.
## 3 Results on the mean-field model
Our main goal is to give a limit result on the family $Z^{N}$, the
multivariate, non-linear Hawkes process on the graph $\mathbb{G}_{N}$ with
interaction kernels and transfer functions as given in Assumption 2.4. We
distinguish the cases $p\neq\tfrac{1}{2}$ (Theorem 1) and $p=\tfrac{1}{2}$
(Theorem 2). In both cases, the limit compensator is given by
$(\int_{0}^{t}h(I_{s})ds)_{t\geq 0}$, where $I$ is the weak limit of $I^{N}$
(see (3.1) and (3.9)), and we find the following limit results: For
$p\neq\tfrac{1}{2}$, we have that $(I_{t})_{t\geq 0}$ follows a linear,
deterministic convolution equation, and all components of the limit of $Z^{N}$
are independent. For $p=\tfrac{1}{2}$, the critical case, the process
$(I_{t})_{t\geq 0}$ is the unique solution of a stochastic convolution
equation and all components of $Z^{N}$ are conditionally independent given
$I$. Below, we denote by $\Rightarrow$ weak convergence in
$\mathcal{D}_{\mathbb{R}^{n}}([0,\infty))$, the space of cadlag paths, which
is equipped with the Skorohod topology; see e.g. Chapter 3 in [16]. The proof
of the following result can be found in Section 5.4.
###### Theorem 1 (Mean-field limit of multi-variate non-linear Hawkes
processes, $p\neq\tfrac{1}{2}$).
Let Assumption 2.4 hold with $p\neq\tfrac{1}{2}$ and
$\theta_{N}=\tfrac{1}{N}$. Let $Z^{N}=(Z^{N,1},...,Z^{N,N})$ be the
multivariate, non-linear Hawkes process from Definition 2.1, and
$\displaystyle
I^{N}(t):=\frac{1}{N}\sum_{j=1}^{N}\int_{0}^{t-}U_{j}\varphi(t-s)dZ_{s}^{j,N}.$
(3.1)
1. 1.
Then, $I^{N}\xrightarrow{N\to\infty}I$, uniformly on compact time intervals in
$L^{2}$, where $I=(I_{t})_{t\geq 0}$ is the unique solution of the integral
equation
$\displaystyle I_{t}=(2p-1)\int_{0}^{t}\varphi(t-s)h(I_{s})ds.$ (3.2)
2. 2.
For all $n=1,2,...$,
$(Z^{N,1},...,Z^{N,n})\xRightarrow{N\to\infty}(\bar{Z}^{1},...,\bar{Z}^{n}),$
where $\bar{Z}^{1},...,\bar{Z}^{n}$ are independent and $\bar{Z}^{i}$ is a
simple point process with intensity at time $t$ given by $h(I_{t})$,
$i=1,...,n$. It is possible to build $\bar{Z}^{1},...,\bar{Z}^{n}$, such that
the convergence is almost surely (in Skorohod-distance).
3. 3.
Assume that $h\in\mathcal{C}^{1}(\mathbb{R})$ and that $h^{\prime}$ is bounded
and Lipschitz. Then $\sqrt{N}(I^{N}-I)\xRightarrow{N\to\infty}K$, where $K$ is
the unique (strong) solution of
$\displaystyle K_{t}$ $\displaystyle=\int_{0}^{t}\varphi(t-s)dG_{s},$ (3.3)
$\displaystyle G_{t}$
$\displaystyle=\int_{0}^{t}(Wh(I_{s})+(2p-1)h^{\prime}(I_{s})K_{s})ds+\int_{0}^{t}\sqrt{h(I_{s})}dB_{s},$
where $W$ is given by (2.2) and $B$ is a Brownian motion independent from $W$.
###### Remark 3.1.
The form of (3.2) tells us that $I$ follows a linear Volterra convolution
equation [3]. Turning into a differential equation, we write, using Fubini,
$\displaystyle\frac{dI}{dt}$
$\displaystyle=\frac{d}{dt}(2p-1)\Big{(}\int_{0}^{t}\int_{s}^{t}\varphi^{\prime}(r-s)h(I_{s})drds+\int_{0}^{t}\varphi(0)h(I_{s})ds\Big{)}$
$\displaystyle=(2p-1)\Big{(}\int_{0}^{t}\varphi^{\prime}(t-s)h(I_{s})ds+\varphi(0)h(I_{t})\Big{)}.$
In particular, the special choice of $\varphi(s)=e^{-\lambda s}$ gives
$\displaystyle\frac{dI}{dt}$
$\displaystyle={\color[rgb]{0,0,0}-\lambda(2p-1)\int_{0}^{t}\varphi(t-s)h(I_{s})ds+(2p-1)h(I_{t})=-\lambda
I_{t}+(2p-1)h(I_{t}),}$
i.e. $I$ follows some ordinary differential equation in this case.
While Theorem 1 is concerned with convergence of the limit intensity of the
multivariate, non-linear Hawkes process, we are also in the situation to study
convergence of the average intensity of $Z^{N}$. The proof of the next
corollary is found in Section 5.5.
###### Corollary 3.2.
Let $Z^{N}=(Z^{N,1},...,Z^{N,N})$ be as in Theorem 1, and
$\bar{Z}=(\bar{Z}^{1},\bar{Z}^{2},...)$ be as in Theorem 1.2. Then,
$\displaystyle\frac{1}{N}\sum_{j=1}^{N}\Big{(}Z^{N,j}-\int_{0}^{.}h(I^{N}_{s})ds\Big{)}\xrightarrow{N\to\infty}0$
(3.4) and
$\displaystyle\frac{1}{N}\sum_{j=1}^{N}\Big{(}Z^{N,j}-\bar{Z}^{j}\Big{)}\xrightarrow{N\to\infty}0$
(3.5)
in probability, uniformly on compact intervals. Moreover,
$\displaystyle\frac{1}{\sqrt{N}}\sum_{j=1}^{N}\Big{(}Z^{N,j}-\int_{0}^{.}h(I_{s}^{N})ds\Big{)}\xRightarrow{N\to\infty}\int_{0}^{.}\sqrt{h(I_{s})}d\widetilde{B}_{s}$
(3.6)
for some Brownian motion $\widetilde{B}$, and (writing
$h(I):=(h(I_{t}))_{t\geq 0}$)
$\displaystyle\sqrt{N}\big{(}h(I^{N})-h(I)\big{)}\xRightarrow{N\to\infty}h^{\prime}(I)K.$
(3.7)
Let $B,\widetilde{B}$ be correlated Brownian motions with
$\mathbf{E}[B_{t}\widetilde{B}_{t}]=(2p-1)t,t\geq 0$. Use $B$ in the
definition of $K$ in (3.3). Then,
$\displaystyle\frac{1}{\sqrt{N}}\sum_{j=1}^{N}\Big{(}Z^{N,j}-\int_{0}^{.}h(I_{s})ds\Big{)}\xRightarrow{N\to\infty}\int_{0}^{.}h^{\prime}(I_{s})K_{s}ds+\int_{0}^{.}\sqrt{h(I_{s})}d\widetilde{B}_{s}.$
(3.8)
###### Remark 3.3 (Correlation between $B$ and $\widetilde{B}$).
Let us briefly discuss the correlated Brownian motions appearing in (3.8).
Clearly, the left hand sides of (3.6) and (3.7) sum to the left hand side of
(3.8). The limits $\int_{0}^{.}\sqrt{h(I_{s})}d\widetilde{B}_{s}$ and $K$
appearing on the right hand sides of (3.6) and (3.7) are weak limits of sums
of compensated point processes. While in (3.6), we sum over all point
processes in the system, $I^{N}$ in (3.7) distinguishes between nodes with
different signs $U_{j}$. Hence, the correlation is positive for the proportion
$p$ of point processes with positive sign, and negative for the proportion
$1-p$ of point processes with negative sign, summing to $p-(1-p)=2p-1$. For
more details, see the proof in Section 5.5.
###### Remark 3.4 (Connections of Theorem 1 to previous work).
Connections of Theorem 1.1 and Theorem 1.2 to [8, 23]. First note that 1. and
2. of Theorem 1 are a particular case of [8], Theorem 4.1. In their notation,
we have choosen $H_{ij}=U_{j}h$ as interaction functions, which satisfy their
condition (9), as stated in their remark 2.1 on the synaptic weights. We
reformulate the result to introduce 3. of of Theorem 1, which does not appear
in [8], and to allow for a comparison to the result and proof of our Theorem
2.
Note further that Theorem 1.1 implies that the compensator
$(\Lambda_{t})_{t\geq 0}$ of each of the independent limiting processes
$(\bar{Z}^{i}_{t})_{t\geq 0}$ is
$\Lambda_{t}=\int_{0}^{t}h(I_{s})ds=\int_{0}^{t}h\Big{(}(2p-1)\int_{0}^{s}\varphi(s-r)h(I_{r})dr\Big{)}ds=\int_{0}^{t}h\Big{(}(2p-1)\int_{0}^{s}\varphi(s-r)d\Lambda_{r}\Big{)}ds.$
See also (8) of[12] and (5), (7), (8) of [23] for the connection to (3.2),
both for the special case $p=1$ and $W=0$. In contrast to our setting, recall
that [23] require $\varphi(0)=0$.
Connections of Theorem 1.3 to [7], [23]. Equation (5.10) from Theorem 5.6 in
[7] gives an expression for the fluctuation of age dependent Hawkes processes.
In the case where the transfer function becomes independent of the age,
$\Psi(s,x)\equiv\Psi(x)$ in their notation, we discover our equation (3.3) in
the special case $p=1$ and $W=0$. As Corollary 2.2 of [23] is a special case
of this Theorem 5.6, (33) in that corollary is precisely (3.3), and the
convergence of that Corollary for the special case $p=1$ (as well as $W=0$ and
$\varphi(0)=0$) coincides with the statement of Theorem 1.3.
Connections of Corollary 3.2 to [23]. The convergence in (3.7) generalizes
Theorem 2.4 of [23] to the case $p\neq 1$, $W\neq 0$ and $\varphi(0)\neq 0$.
Moreover, the convergence in (3.8) generalizes Theorem 2.1 of [23]: the right
hand side of (3.8) for $p=1$, $W=0$ has $B=\widetilde{B}$ and hence equals $G$
from (3.3). Using integration by parts and $\varphi(0)=0$ we obtain
$\displaystyle 0$
$\displaystyle=\varphi(0)G_{t}-\varphi(t)G_{0}=\int_{0}^{t}\varphi(t-s)dG_{s}-\int_{0}^{t}G_{s}\varphi^{\prime}(t-s)ds.$
Therefore we can reshape $G$ from (3.3),
$\displaystyle G_{t}$
$\displaystyle=\int_{0}^{t}h^{\prime}(I_{u})\int_{0}^{u}\varphi(u-s)dG_{s}du+\int_{0}^{t}\sqrt{h(I_{s})}dB_{s}$
$\displaystyle=\int_{0}^{t}h^{\prime}(I_{u})\int_{0}^{u}\varphi^{\prime}(u-s)G_{s}dsdu+\int_{0}^{t}\sqrt{h(I_{s})}dB_{s}$
$\displaystyle=\int_{0}^{t}G_{s}\int_{s}^{t}\varphi^{\prime}(u-s)h^{\prime}(I_{u})duds+\int_{0}^{t}\sqrt{h(I_{s})}dB_{s},$
to obtain their (24).
We now turn to the case $p=\tfrac{1}{2}$, where we can rescale the intensity
by $\sqrt{N}$ rather than $N$. This is more involved since the limit of
$I^{N}$ turns out to be a stochastic process. For the proof of the following
Theorem, see Section 5.3.
###### Theorem 2 (Mean-field limit of multi-variate non-linear Hawkes
processes, $p=\tfrac{1}{2}$).
Let Assumption 2.4 hold with $p=\tfrac{1}{2}$ and
$\theta_{N}=\tfrac{1}{\sqrt{N}}$. Let $Z^{N}=(Z^{N,1},...,Z^{N,N})$ be the
multivariate, non-linear Hawkes process fom Definition 2.1, and
$\displaystyle
I^{N}(t):=\frac{1}{\sqrt{N}}\sum_{j=1}^{N}\int_{0}^{t-}U_{j}\varphi(t-s)dZ_{s}^{j,N}.$
(3.9)
1. 1.
For all $w\in\mathbb{R}$, and a Brownian motion $B$, the stochastic integral
equation
$\displaystyle
I_{t}=I_{0}+w\int_{0}^{t}\varphi(t-s)h(I_{s})ds+\int_{0}^{t}\varphi(t-s)\sqrt{h(I_{s})}dB_{s}$
(3.10)
has a unique strong solution.
2. 2.
Let $W\sim N(0,1)$, $B$ be an independent Brownian motion and $I$ be the
solution of (3.10) with $w$ replaced by $W$. Then, for all $n=1,2,...$,
$(I^{N},Z^{N,1},...,Z^{N,n})\xRightarrow{N\to\infty}(I,\bar{Z}^{1},...,\bar{Z}^{n})$,
where $\bar{Z}^{1},...,\bar{Z}^{n}$ are conditionally independent given $I$
and $\bar{Z}^{i}$ is a simple point process with intensity at time $t$ given
by $h(I_{t})$, $i=1,...,n$.
Again, we discuss convergence of the average intensity of $Z^{N}$. The proof
is in Section 5.5.
###### Corollary 3.5.
Let $Z^{N}=(Z^{N,1},...,Z^{N,N})$ be as in Theorem 2, and
$(I,\bar{Z}^{1},\bar{Z}^{2},...)$ be as in Theorem 2.2. Then, (3.4), (3.5),
(3.6) hold.
We stress that the results corresponding to (3.7) and (3.8) do not hold here
since $I^{N}\Rightarrow I$, so no further upscaling is possible.
###### Remark 3.6 (Some connections to [15]).
Our model and the statements of Theorem 2.2 should be compared to the findings
in [15]. In their model, the size of interactions is given by a centered
probability measure, so at each spike of a neuron, the potential given to
other neurons in the system may change. In contrast, in our model, the
potential given to other neurons is fixed for each neuron, but varies from
neuron to neuron. The latter case seems more relevant in neuroscience; see the
discussion in Section 4. Still, results from Theorem 2 can be compare to
Theorem 1.4(ii) and Theorem 1.7. of [15], where the same type of convergence
is shown, but the interaction kernel $\varphi$ is an exponential function. As
we see in Remark 3.7.2 below, this special choice makes the intensity $I$ a
Markov process. However, as the intensity of both models is still comparable,
we find (3) from [15] in our Remark 3.7.2 below by inserting $w=0$.
###### Remark 3.7 (Some properties of $I$).
1. 1.
If $\varphi^{\prime}$ exists, it is not hard to see that $I_{t}$ is a
semimartingale, since
$\displaystyle\int_{0}^{t}\varphi(t-u)\sqrt{h(I_{u})}dB_{u}$
$\displaystyle=\int_{0}^{t}\int_{0}^{s}\varphi^{\prime}(t-s)\sqrt{h(I_{u})}dB_{u}ds+\int_{0}^{t}\varphi(0)\sqrt{h(I_{u})}dB_{u}$
by the stochastic Fubini Theorem [3, Theorem 4.A], where the first term on the
right hand side has finite variation and the second is a (local) martingale.
This calculation also shows that $I$ has finite variation provided that
$\varphi(0)=0$.
2. 2.
Consider the special case $\varphi(s)=e^{-\lambda s}$. Then, a solution of
(3.10) also solves
$dI=(wh(I)-\lambda I)dt+\sqrt{h(I)}dB$
and in particular is a Markov process. In order to see this, we write
$\displaystyle I_{t}-I_{0}=e^{-\lambda t}\Big{(}w\int_{0}^{t}e^{\lambda
s}h(I_{s})ds+\int_{0}^{t}e^{\lambda s}\sqrt{h(I_{s})}dB_{s}\Big{)}.$
Using integration by parts (on the processes $A_{t}:=e^{-\lambda t}$, which
satisfies $dA=-\lambda Adt$, and the term $C_{t}$ in brackets), we find that
$\displaystyle I_{t}-I_{0}$
$\displaystyle=A_{t}C_{t}=\int_{0}^{t}A_{s}dC_{s}+\int_{0}^{t}C_{s}dA_{s}$
$\displaystyle=w\int_{0}^{t}h(I_{s})ds+\int_{0}^{t}\sqrt{h(I_{s})}dB_{s}-\lambda\int_{0}^{t}A_{s}C_{s}ds$
$\displaystyle=\int_{0}^{t}(wh(I_{s})-\lambda
I_{s})ds+\int_{0}^{t}\sqrt{h(I_{s})}dB_{s}.$
## 4 Implications for neuroscience
The human brain is very large. It comprises 86 billion neurons, along with
other types of cells [24]. Each individual neuron makes contact with hundreds
or thousands of other neurons, forming a very complex network. It should be
thought of as a highly integrated circuit, made of biological parts, which are
capable of processing electro-chemical signals and transmitting them to
distant neurons with high fidelity. Neurons use electrical impulses (“action
potentials” or “spikes”) to communicate with other neurons in the same
network. Sequences of these impulses evolve in time (“spike trains”),
involving the collective activity of many interacting neurons. These
multivariate signals are believed to reflect the dynamic “computations”
performed by these networks. The biological “purpose” of these processes
covers a wide range from immediate control of bodily functions, like
activating a muscle, to abstract information processing, like making a complex
decision [30].
Neurons communicate with other neurons in a network by activating specialized
functional contacts between cells (“synapses”). In the most common type of
synapses, diffusing chemicals are used for signaling (“neurotransmitters”). In
a nutshell, each spike of the sender neuron triggers a transient change in
excitability of the receiver neuron. This can either increase or decrease the
tendency of the receiver neuron to fire an action potential. We speak of
“excitation” and “inhibition”, respectively. A hallmark of normal brain
function is a robust balance of excitation and inhibition. Brain diseases, in
contrast, are often linked with a marked imbalance of these two forces [38].
A whole zoo of mathematical models exist to mimic the dynamical processes of
synaptic interaction in networks of spiking neurons. Most of them are
difficult to analyze, due to the discrete nature of pulse trains and an
essential nonlinearity in the spike generation mechanism. From a mathematical
point of view, the Hawkes process seems like a very reasonable choice as it
can directly reflect several important biological features. First, as a
stochastic point process it is by construction based on pulse-coded signaling.
Second, the use of interaction kernels automatically reflects the causal
nature of communication in a physical system with recurrent loops, allowing
only spikes in the past to influence the future evolution of the network.
Third, as a multivariate process with coupled components it is able to capture
and consistently describe the collective activity dynamics.
The linear Hawkes process [22] has been employed as a mathematical model of
networks of spiking neurons, as it can directly reflect excitatory synaptic
interactions between spiking neurons using positive interaction kernels [28].
An important shortcoming of the linear model is that inhibitory synaptic
interaction cannot be included in a consistent way. Negative interaction
kernels are not allowed, as they might yield a negative point process
intensity. To enable the study of networks with excitation-inhibition balance,
however, negative interaction kernels must be admitted [35]. A nonlinear link
function can fix this deficit, and effective descriptions of nonlinear
biological neurons combining linear synaptic interaction with a nonlinear link
function are actually well-known in computational neuroscience. They have been
called cascade models [25], and from the viewpoint of statistics they lead to
generalized linear models [34]. For that reason, the nonlinear Hawkes process
is a natural extension of Hawkes’ classical model, greatly extending its range
of applications.
The specific model described in this paper can be further generalized,
allowing for a different interaction kernel for each pair of nodes in the
network. This includes networks with arbitrary graph topology, setting kernels
to zero where a synapse is absent [35]. In brain networks, the amplitude of
individual interaction kernels (“synaptic weight”) is thought to change as a
result of learning. The promise for the biologist is to be able to investigate
networks, the structure of which is either designed for a specific purpose, or
reflects previous experience and memory. The challenge for the mathematician
is to manage the lack of symmetry in these networks, and a genuinely
multivariate description becomes necessary.
There is an ongoing debate about the significance of scaling synaptic weights
for networks of infinite size [37, 2]. As the number of neurons in brain
networks is typically quite high, such limit may represent a meaningful
approximation to large brains. The difficulty is that large brains also have
to accommodate constraints with regard to the number of possible connections,
imposed by the distance of neurons in space and by the volume occupied by
fibers. So it might be more fruitful to rephrase the question a bit [1]: Which
aspect of the input to a neuron from within the network is more important: Is
it the mean drift, or is it the amplitude of the fluctuations? If there is a
perfect balance between excitation and inhibition, the drift is zero, and the
fluctuations take over and drive the neuron in a stochastic fashion. For the
slightest imbalance of the inputs, however, keeping the drift bounded in very
large networks forces the fluctuations to be very small, and the dynamics
becomes more deterministic. This is where biophysical intuition is in line
with the result of mathematical analysis presented in this paper.
## 5 Proofs
We start off in Subsection 5.1 with some preliminary results on convergence of
Poisson processes and convolution equations. We proceed in 5.2 with a
reformulation of the multivariate linear Hawkes process using a time-change
equation. Then, we prove Theorem 2 in Sebsection 5.3 and Theorem 1 in
Subsection 5.4. We give the proof of Theorem 2 first because it is more
involved than the proof of Theorem 1, mainly since the limit process $I$ is
stochastic.
### 5.1 Preliminairies
We need the following (rather standard) convergence results for Poisson
processes.
###### Lemma 5.1 (Convergence of Poisson processes).
Let $Y$ be a unit rate Poisson process, defined on some probability space
$(\Omega,\mathcal{F},\mathbb{P})$. Then, for all $t>0$,
$\displaystyle\sup_{0\leq s\leq
t}\frac{1}{N}(Y(Ns)-Ns)\xrightarrow{N\to\infty}0$ (5.1)
almost surely (and in $L^{2}$). In addition,
$\displaystyle\Big{(}\frac{1}{\sqrt{N}}(Y(Nt)-Nt)\Big{)}_{t\geq
0}\xRightarrow{N\to\infty}B$ (5.2)
for some Brownian motion $B$. We can extend $(\Omega,\mathcal{F},\mathbb{P})$
by a Brownian motion $B$ such that, for all $t\geq 0$,
$\displaystyle\sup_{0\leq s\leq
t}\Big{|}\frac{1}{\sqrt{N}}(Y(Ns)-Ns)-B_{s}\Big{|}\xrightarrow{N\to\infty}0$
(5.3)
almost surely and in $L^{2}$.
###### Proof.
For (5.1), we use Doob’s martingale inequality for the martingale
$(Y(s)-s)_{s\geq 0}$ and the fact that the fourth centered moment of a Poisson
random variable with parameter $\lambda$ is $3\lambda^{2}+\lambda$ in order to
see
$\displaystyle\mathbb{E}\Big{[}\sup_{0\leq s\leq
t}\Big{(}\frac{1}{N}(Y(Ns)-Ns)\Big{)}^{4}\Big{]}\leq\Big{(}\frac{4}{3}\Big{)}^{4}\mathbb{E}\Big{[}\Big{(}\frac{1}{N}(Y(Nt)-Nt)\Big{)}^{4}\Big{]}={\color[rgb]{0,0,0}\Big{(}\frac{4}{3N}\Big{)}^{4}}\big{(}3(Nt)^{2}+Nt\big{)},$
which is summable. Using Borel-Cantelli, almost-sure convergence follows.
Next, (5.2) follows by an application of Donsker’s theorem ([26] Ch VII, Cor.
3.11). By Skorohod’s Theorem, we can extend our probability space such that
this convergence is almost surely with respect to Skorohod distance, and by
continuity of $B$ it is equivalent to uniform convergence, i.e. (5.3) holds
almost surely for all $t\geq 0$. For the $L^{2}$-convergence in (5.3), it
suffices to show the uniform integrability of $\sup_{0\leq s\leq
t}\Big{(}\frac{1}{\sqrt{N}}(Y(Ns)-Ns)-B_{s}\Big{)}^{2}$. Using Doob’s and
Minkovski’s inequalities, we compute
$\displaystyle\mathbb{E}\Big{[}\sup_{0\leq s\leq
t}\Big{(}\frac{1}{\sqrt{N}}(Y(Ns)-Ns)-B_{s}\Big{)}^{4}\Big{]}^{1/4}$
$\displaystyle\leq\frac{4}{3}\mathbb{E}\Big{[}\Big{(}\frac{1}{\sqrt{N}}(Y(Nt)-Nt)-B_{t}\Big{)}^{4}\Big{]}^{1/4}$
$\displaystyle\leq\frac{4}{3}\Big{(}\mathbb{E}\Big{[}\Big{(}\frac{1}{\sqrt{N}}(Y(Nt)-Nt)\Big{)}^{4}\Big{]}^{1/4}+\mathbb{[}(B_{t})^{4}]^{1/4}\Big{)}$
$\displaystyle=\frac{4}{3}((3t^{2}+t/N)^{1/4}+(3t^{2})^{1/4})$
showing the desired uniform integrability and $L^{2}$-convergence follows. ∎
In order to bound the value of a convolution equation by its integrator we
need the following
###### Lemma 5.2.
Let $J$ be the sum of an Itô process with bounded coefficients and a càdlàg
pure-jump process, and let $\varphi\in\mathcal{C}_{b}^{1}([0,\infty))$. Then
$\displaystyle\sup_{0\leq s\leq
t}\Big{(}\int_{0}^{s}\varphi(s-r)dJ_{r}\Big{)}^{2}\leq
C(\varphi,t)\cdot\sup_{0\leq s\leq t}J_{s}^{2}.$ (5.4)
###### Proof.
Wlog, we have $J_{0}=0$. By [3, Theorem 4.A] and Fubini’s theorem for
Lebesgue-integrals we can apply the Stochastic Fubini Theorem to $J$, hence
$\displaystyle\sup_{0\leq s\leq
t}\Big{(}\int_{0}^{s}\varphi(s-r)dJ_{r}\Big{)}^{2}$ $\displaystyle=\sup_{0\leq
s\leq
t}\Big{(}\int_{0}^{s}\Big{(}\varphi(0)+\int_{r}^{s}\varphi^{\prime}(s-u)du\Big{)}dJ_{r}\Big{)}^{2}$
$\displaystyle\leq 2\,||\varphi||^{2}\cdot\sup_{0\leq s\leq
t}J_{\color[rgb]{0,0,0}s}^{2}+2\sup_{0\leq s\leq
t}\Big{(}\int_{0}^{s}\int_{0}^{u}\varphi^{\prime}(s-u)dJ_{r}du\Big{)}^{2}$
$\displaystyle=2\,||\varphi||^{2}\cdot\sup_{0\leq s\leq
t}J_{s}^{2}+2\sup_{0\leq s\leq
t}\Big{(}\int_{0}^{s}\varphi^{\prime}(s-u)J_{u}du\Big{)}^{2}$
$\displaystyle\leq
2\,(||\varphi||^{2}+t||\varphi^{\prime}||^{2})\cdot\sup_{0\leq s\leq
t}J_{s}^{2}.$
∎
### 5.2 Reformulation of Hawkes processes
Alternative descriptions of non-linear Hawkes processes have been given in the
literature. Above all, the construction using a Poisson random measures is
widely used; see e.g. Proposition 3 in [12]. Here, we rely on the following
construction using time-change equations (see e.g. Chapter 6 of [16]), which
we give here without proof.
###### Lemma 5.3.
Let $\mathbb{G}=(\mathbb{V},\mathbb{E})$, $(\varphi_{ji})_{ji\in\mathbb{E}}$
and $(h_{i})_{i\in\mathbb{V}}$ be as in Definition 2.1, and let Assumption 2.2
hold. A point process $Z=(Z^{i})_{i\in\mathbb{V}}$ is a multivariate, non-
linear Hawkes process with interaction kernels
$(\varphi_{ji})_{ji\in\mathbb{E}}$ and transfer functions
$(h_{i})_{i\in\mathbb{V}}$, if and only if it is the weak solution of the
time-change equations
$\displaystyle
Z_{t}^{i}=Y_{i}\Big{(}\int_{0}^{t}\lambda_{s}^{i}ds\Big{)}=Y_{i}\Big{(}\int_{0}^{t}h_{i}\Big{(}\sum_{j:ji\in\mathbb{E}}\int_{0}^{s-}\varphi_{ji}(s-u)dZ_{u}^{j}\Big{)}ds\Big{)},$
(5.5)
where $(Y_{i})_{i\in\mathbb{V}}$ is a family of independent unit rate Poisson
processes.
Under Assumption 2.4, the time-change equations from this lemma read, with
independent unit rate Poisson processes $Y_{1},...,Y_{N}$,
$\displaystyle Z_{t}^{N,i}$
$\displaystyle=Y_{i}\Big{(}\int_{0}^{t}h\Big{(}\sum_{j=1}^{N}\int_{0}^{s-}\theta_{N}U_{j}\varphi(s-u)dZ_{u}^{N,j}\Big{)}ds\Big{)}.$
(5.6) We rewrite this as $\displaystyle Z_{t}^{N,i}$
$\displaystyle=Y_{i}\Big{(}\int_{0}^{t}h(I_{s}^{N})ds\Big{)}$ (5.7) with
$\displaystyle I_{s}^{N}$
$\displaystyle=\int_{0}^{s-}\varphi(s-u)dJ_{u}^{N}\quad\text{ and }\quad
J_{u}^{N}=\theta_{N}\sum_{j=1}^{N}U_{j}Z_{u}^{N,j}.$ (5.8)
Now, we rewrite $J^{N}$ further. Observe that $\sum_{j:U_{j}=1}Y_{j}$ is a
Poisson process with rate
$\tfrac{1}{2}\sum_{j=1}^{N}(U_{j}+1)=pN+\tfrac{1}{2}\sqrt{N}W_{N}$ (recall
$W_{N}$ from (2.2)) and $\sum_{j:U_{j}=-1}Y_{j}$ is a Poisson process with
rate $(1-p)N-\tfrac{1}{2}\sqrt{N}W_{N}$. Hence we have that, for two
independent unit rate Poisson processes $Y^{+}$ and $Y^{-}$,
$\displaystyle J_{t}^{N}$
$\displaystyle=\theta_{N}\sum_{j=1}^{N}U_{j}Y_{j}\Big{(}\int_{0}^{t}h(I_{s}^{N})ds\Big{)}$
(5.9)
$\displaystyle=\theta_{N}Y^{+}\Big{(}\big{(}pN+\tfrac{1}{2}\sqrt{N}W_{N}\big{)}\int_{0}^{t}h(I_{s}^{N})ds\Big{)}-\theta_{N}Y^{-}\Big{(}\big{(}(1-p)N-\tfrac{1}{2}\sqrt{N}W_{N}\big{)}\int_{0}^{t}h(I_{s}^{N})ds\Big{)}.$
We extend our probability space such that the rescaled Poisson processes
$Y^{+},Y^{-}$ converge to Brownian motions $B^{+},B^{-}$ almost surely (as in
(5.3)). Consider the Brownian motion $\hat{B}=(\hat{B}(t))_{t\geq 0}$ with
$\hat{B}_{t}=B^{+}_{pt}-B^{-}_{(1-p)t}$. Since
$\Big{(}\hat{B}\Big{(}\int_{0}^{t}h(I_{s}^{N})ds\Big{)}\Big{)}_{t\geq 0}$ is a
continuous martingale, we can extend the probability space by another Brownian
motion $B^{N}$, such that
$\displaystyle\hat{B}\Big{(}\int_{0}^{t}h(I_{s}^{N})ds\Big{)}=\int_{0}^{t}\sqrt{h(I_{s}^{N})}dB_{s}^{N}\quad\text{for
}t\geq 0,$ (5.10)
by [29, Theorem 16.10].
### 5.3 Proof of Theorem 2
First, strong existence and uniqueness for (3.10) follows from [3, Theorem
3.A]. For the convergence result, we work on a probability space where the
convergence $W_{N}\to W$ from (2.2) holds almost surely. We will show in Steps
1 and 2 convergence of $J^{N}$ to $J$ with
$\displaystyle
J_{t}=W\int_{0}^{t}h(I_{s})ds+\int_{0}^{t}\sqrt{h(I_{s})}dB_{s}$ (5.11)
and from $I^{N}$ to $I$ from (3.10) with $w$ replaced by $W$. Step 3 then
gives the desired convergence of $(I^{N},Z^{N,1},...,Z^{N,n})$. We use
processes $\widetilde{J}^{N}$ and $\widetilde{I}^{N}$ which are unique strong
solutions of
$\displaystyle\widetilde{J}^{N}_{t}$
$\displaystyle=W_{N}\int_{0}^{t}h\big{(}\widetilde{I}_{s}^{N}\big{)}ds+\int_{0}^{t}\sqrt{h(\widetilde{I}_{s}^{N})}dB_{s}^{N},$
$\displaystyle\widetilde{I}_{s}^{N}$
$\displaystyle=\int_{0}^{s}\varphi(s-r)d\widetilde{J}^{N}_{r},$
where $B^{N}$ is the Brownian motion defined by (5.10). Again, strong
existence and uniqueness follows from [3, Theorem 3.A]. In Step 1, we will
show that $J^{N}-\widetilde{J}^{N}\to 0$ and $I^{N}-\widetilde{I}^{N}\to 0$
uniformly on compact sets in $L^{2}$, conditional on
$(U_{i})_{i\in\mathbb{N}}$. In Step 2, we show the remaining conditional
convergence $\widetilde{I}^{N}-I\to 0$ and conclude $I^{N}-I\to 0$ in
probability.
Step 1: Convergence $J^{N}-\widetilde{J}^{N}\to 0$ and
$I^{N}-\widetilde{I}^{N}\to 0$.
Recall $Y^{+}$ and $Y^{-}$ from Section 5.2. For
$\theta_{N}=\tfrac{1}{\sqrt{N}}$ and $p=\tfrac{1}{2}$ in (5.9) we get that,
$J_{t}^{N}=\frac{1}{\sqrt{N}}Y^{+}\Big{(}\tfrac{N+\sqrt{N}W_{N}}{2}\int_{0}^{t}h(I_{s}^{N})ds\Big{)}-\frac{1}{\sqrt{N}}Y^{-}\Big{(}\tfrac{N-\sqrt{N}W_{N}}{2}\int_{0}^{t}h(I_{s}^{N})ds\Big{)}.$
We write
$\displaystyle J_{t}^{N}-\widetilde{J}_{t}^{N}$
$\displaystyle=A_{t}^{N}+W_{N}C_{t}^{N}+D_{t}^{N}\quad\text{with}$ (5.12)
$\displaystyle A_{t}^{N}$
$\displaystyle=\frac{1}{\sqrt{N}}Y^{+}\Big{(}\frac{N+\sqrt{N}W_{N}}{2}\int_{0}^{t}h(I_{s}^{N})ds\Big{)}-\frac{1}{\sqrt{N}}Y^{-}\Big{(}\frac{N-\sqrt{N}W_{N}}{2}\int_{0}^{t}h(I_{s}^{N})ds\Big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-
W_{N}\int_{0}^{t}h(I_{s}^{N})ds-\hat{B}\Big{(}\int_{0}^{t}h(I_{s}^{N})ds\Big{)},$
$\displaystyle C_{t}^{N}$
$\displaystyle=\int_{0}^{t}h(I_{s}^{N})ds-\int_{0}^{t}h(\widetilde{I}_{s}^{N})ds,$
$\displaystyle D_{t}^{N}$
$\displaystyle=\int_{0}^{t}\sqrt{h(I_{s}^{N})}dB^{N}_{s}-\int_{0}^{t}\sqrt{h(\widetilde{I}_{s}^{N})}dB^{N}_{s}.$
We make the disclaimer, that we will use $C>0$ for a variable, which only
depends on $t,h,\varphi$, and which might change from line to line. We write
$\mathbb{E}_{U}$ for the conditional expectation with respect to
$U=(U_{i})_{i\in\mathbb{N}}$. The function $h$ is bounded, therefore, for some
unit rate Poisson process $Y$,
$\displaystyle\mathbb{E}{{}_{U}}[\sup_{0\leq s\leq t}(A_{s}^{N})^{2}]$
$\displaystyle\leq\mathbb{E}{{}_{U}}\Big{[}\sup_{0\leq s\leq
t||h||}\Big{(}\frac{1}{\sqrt{N}}Y^{+}\Big{(}\frac{N+\sqrt{N}W_{N}}{2}s\Big{)}-\frac{1}{\sqrt{N}}Y^{-}\Big{(}\frac{N-\sqrt{N}W_{N}}{2}s\Big{)}$
(5.13)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-
W_{N}s-\hat{B}(s)\Big{)}^{2}\Big{]}$ $\displaystyle\leq
C\cdot\Big{(}\mathbb{E}{{}_{U}}\Big{[}\sup_{0\leq s\leq
t||h||}\Big{(}\frac{1}{\sqrt{N}}(Y^{+}(Ns/2)-Ns/2)-B^{+}(s/2)\Big{)}^{2}\Big{]}$
$\displaystyle\qquad+\mathbb{E}{{}_{U}}\Big{[}\sup_{0\leq s\leq
t||h||}\Big{(}\frac{1}{\sqrt{N}}(Y^{-}(Ns/2)-Ns/2)-B^{-}(s/2)\Big{)}^{2}\Big{]}$
$\displaystyle\qquad+\mathbb{E}{{}_{U}}\Big{[}\sup_{0\leq
s\leq|W_{N}|t||h||}\Big{(}\frac{1}{\sqrt{N}}(Y(\sqrt{N}s))-s\Big{)}^{2}\Big{]}\Big{)}$
$\displaystyle{\color[rgb]{0,0,0}\leq
o(1)+C\cdot\mathbb{E}_{U}\Big{[}\Big{(}\frac{1}{\sqrt{N}}(Y(\sqrt{N}|W_{N}|t||h||s))-|W_{N}|t||h||s\Big{)}^{2}\Big{]}}$
$\displaystyle{\color[rgb]{0,0,0}\leq
o(1)+C\frac{1}{\sqrt{N}}W_{N}}\xrightarrow{N\to\infty}0$
almost surely, by Lemma 5.1 and Doob’s inequality. Then, using Jensen,
$\displaystyle\mathbb{E}{{}_{U}}[\sup_{0\leq s\leq t}(W_{N}C_{s}^{N})^{2}]$
$\displaystyle{\color[rgb]{0,0,0}\leq\sup_{N}W_{N}^{2}}\mathbb{E}{{}_{U}}\Big{[}\sup_{0\leq
s\leq
t}\Big{(}\int_{0}^{s}h(I_{r}^{N})-h(\widetilde{I}_{r}^{N})dr\Big{)}^{2}\Big{]}$
$\displaystyle\leq{\color[rgb]{0,0,0}\sup_{N}W_{N}^{2}}th_{Lip}^{2}\int_{0}^{t}\mathbb{E}{{}_{U}}[(I_{s}^{N}-\widetilde{I}_{s}^{N})^{2}]ds$
Note that $\sup_{N}W_{N}^{2}<\infty$, as $W_{N}$ converges to $W$, which is
finite almost surely. By Doob’s inequality and Ito’s isometry,
$\displaystyle\mathbb{E}{{}_{U}}[\sup_{0\leq s\leq t}(D_{s}^{N})^{2}]$
$\displaystyle\leq
C\cdot\mathbb{E}{{}_{U}}[(D_{t}^{N})^{2}]=C\cdot\mathbb{E}{{}_{U}}\Big{[}\Big{(}\int_{0}^{t}\sqrt{h(I_{s}^{N})}dB^{N}_{s}-\int_{0}^{t}\sqrt{h(\widetilde{I}_{s}^{N})}dB^{N}_{s}\Big{)}^{2}\Big{]}$
$\displaystyle=C\cdot\mathbb{E}{{}_{U}}\Big{[}\int_{0}^{t}\big{(}\sqrt{h(I_{s}^{N})}-\sqrt{h(\widetilde{I}_{s}^{N})}\big{)}^{2}ds\Big{]}$
$\displaystyle\leq
C\cdot\int_{0}^{t}\mathbb{E}{{}_{U}}[(I_{s}^{N}-\widetilde{I}_{s}^{N})^{2}]ds.$
In order to bound $\sup_{0\leq s\leq t}(W_{N}C_{s}^{N})^{2}$ and $\sup_{0\leq
s\leq t}(D_{s}^{N})^{2}$ by $\sup_{0\leq s\leq
t}(J_{s}^{N}-\widetilde{J}_{s}^{N})^{2}$, apply Lemma 5.2 to
$J_{s}^{N}-\widetilde{J}_{s}^{N}$. We obtain
$\displaystyle\mathbb{E}{{}_{U}}\big{[}\sup_{s\leq
t}(J_{s}^{N}-\widetilde{J}_{s}^{N})^{2}\big{]}$ $\displaystyle\leq
C\cdot\mathbb{E}{{}_{U}}[\sup_{0\leq s\leq
t}(A_{s}^{N})^{2}]+C\cdot\int_{0}^{t}\mathbb{E}{{}_{U}}[(I_{s}^{N}-\widetilde{I}_{s}^{N})^{2}]ds$
$\displaystyle\leq C\cdot\mathbb{E}{{}_{U}}[\sup_{0\leq s\leq
t}(A_{s}^{N})^{2}]+C\cdot\int_{0}^{t}\mathbb{E}{{}_{U}}[\sup_{r\leq
s}(J_{r}^{N}-\widetilde{J}_{r}^{N})^{2}]ds.$
By the Gronwall inequality and (5.13), we conclude that, for all $t\geq 0$,
$\displaystyle\mathbb{E}{{}_{U}}\big{[}\sup_{s\leq
t}(J_{s}^{N}-\widetilde{J}_{s}^{N})^{2}\big{]}$
$\displaystyle\xrightarrow{N\to\infty}0$ and with Lemma 5.2 also
$\displaystyle\mathbb{E}{{}_{U}}\big{[}\sup_{s\leq
t}(I_{s}^{N}-\widetilde{I}_{s}^{N})^{2}\big{]}$
$\displaystyle\xrightarrow{N\to\infty}0,$
almost surely.
Step 2: Convergence $\widetilde{I}^{N}-I\to 0$
Since $B^{N}$ is a Brownian motion, we have that, in distribution,
$\displaystyle\widetilde{I}_{t}^{N}$
$\displaystyle=W_{N}\int_{0}^{t}\varphi(t-s)h(\widetilde{I}_{s}^{N})ds+\int_{0}^{t}\varphi(t-s)\sqrt{h(\widetilde{I}_{s}^{N})}dB_{s}^{N},$
$\displaystyle I_{t}$
$\displaystyle=W\int_{0}^{t}\varphi(t-s)h(I_{s})ds+\int_{0}^{t}\varphi(t-s)\sqrt{h(\widetilde{I}_{s})}dB_{s}^{N}.$
Therefore, on this probability space,
$\displaystyle\widetilde{I}_{t}^{N}-I_{t}$
$\displaystyle=E_{t}^{N}+F_{t}^{N}+G_{t}^{n}\text{ with}$ $\displaystyle
E_{t}^{n}$ $\displaystyle:=(W_{N}-W)\int_{0}^{t}\varphi(t-s)h(I_{s})ds,$
$\displaystyle F_{t}^{N}$
$\displaystyle:=W_{N}\int_{0}^{t}\varphi(t-s)(h(\widetilde{I}_{s}^{N})-h(I_{s}))ds,$
$\displaystyle G_{t}^{N}$
$\displaystyle:=\int_{0}^{t}\varphi(t-s)\Big{(}\sqrt{h(\widetilde{I}_{s}^{N})}-\sqrt{h(I_{s})}\Big{)}dB_{s}^{N}.$
Since, again using some constant $C<\infty$, which can change from line to
line,
$\displaystyle\mathbb{E}{{}_{U}}[\sup_{0\leq s\leq t}(E_{s}^{N})^{2}]$
$\displaystyle\leq
C\cdot{\color[rgb]{0,0,0}(W_{N}-W)^{2}}\xrightarrow{N\to\infty}0,$
$\displaystyle\mathbb{E}{{}_{U}}[\sup_{0\leq s\leq t}(F_{s}^{N})^{2}]$
$\displaystyle\leq
C\cdot{\color[rgb]{0,0,0}\sup_{N}W_{N}^{2}}\int_{0}^{t}\mathbb{E}{{}_{U}}[\sup_{0\leq
r\leq s}(\widetilde{I}_{r}^{N}-I_{r})^{2}]ds,$
$\displaystyle\mathbb{E}{{}_{U}}[\sup_{0\leq s\leq t}(G_{s}^{N})^{2}]$
$\displaystyle\leq C\cdot\int_{0}^{t}\mathbb{E}{{}_{U}}[\sup_{0\leq r\leq
s}(\widetilde{I}_{r}^{N}-I_{r})^{2}]ds,$ we find that
$\displaystyle\mathbb{E}{{}_{U}}[\sup_{0\leq s\leq
t}(\widetilde{I}_{s}^{N}-I_{s})^{2}]$ $\displaystyle\leq
C\cdot{\color[rgb]{0,0,0}(W_{N}-W)^{2}}+C\cdot\int_{0}^{t}\mathbb{E}{{}_{U}}[\sup_{0\leq
r\leq s}(\widetilde{I}_{r}^{N}-I_{r})^{2}]ds,$
and the Gronwall inequality again implies that $\mathbb{E}{{}_{U}}[\sup_{0\leq
s\leq t}(\widetilde{I}_{s}^{N}-I_{s})^{2}]\xrightarrow{N\to\infty}0$, almost
surely. Combining Step 1 and Step 2 we obtain $\mathbb{E}_{U}[\sup_{0\leq
s\leq t}(I_{s}^{N}-I_{s})^{2}]\xrightarrow{N\to\infty}0$. We deduce
$\mathbb{E}_{U}[1\wedge\sup_{0\leq s\leq
t}(I_{s}^{N}-I_{s})^{2}]\xrightarrow{N\to\infty}0$ and, by dominated
convergence, $\mathbb{E}[1\wedge\sup_{0\leq s\leq
t}(I_{s}^{N}-I_{s})^{2}]\xrightarrow{N\to\infty}0$. Hence we have shown that
$\sup_{0\leq s\leq t}(I_{s}^{N}-I_{s})^{2}\to 0$ in probability.
Step 3: Convergence of $(I^{N},Z^{N,1},...,Z^{N,n})$
The predictable quadratic variation of the compensated process
$Z^{N,i}_{t}-\int_{0}^{t}h(I_{s}^{N})ds$ is given by $\langle
Z^{N,i}-\int_{0}^{\cdot}h(I_{s}^{N})ds\rangle_{t}=\int_{0}^{t}h(I_{s}^{N})ds$.
By [26][Chapter VI, Corollary 3.33, Theorem 4.13], tightness of
$(Z^{N,1},...,Z^{N,n})$ follows from convergence of $\sum_{i\leq n}\langle
Z^{N,i}-\int_{0}^{\cdot}h(I_{s}^{N})ds\rangle_{t}=n\int_{0}^{t}h(I_{s}^{N})ds$.
We already know that $I^{N}\Rightarrow I$, hence $(I^{N},Z^{N,1},...,Z^{N,n})$
is tight. To identify the limit, let us try to extend the state space such
that the Hawkes process becomes Markovian. Recall that each $Z^{N,i}$ jumps at
time $t$ at rate $h(I^{N}(t))$ with
$I^{N}(t)=\int_{0}^{t}\varphi(t-s)dJ^{N}_{s}$ and $J_{s}^{N}$ as given in
(5.8). The process $J^{N}$ jumps at time $t$ from $x$ to $x+\theta_{N}$ at
rate $h\Big{(}\int_{0}^{t}\varphi(t-s)dJ^{N}_{s}\Big{)}\cdot u^{+}$ with
$u^{+}:=\sum_{j}1_{U_{j}=1}$ and to $x-\theta_{N}$ at rate
$h\Big{(}\int_{0}^{t}\varphi(t-s)dJ^{N}_{s}\Big{)}\cdot u^{-}$ with
$u^{-}=\sum_{j}1_{U_{j}=-1}=N-u^{+}$. Note that the process $J^{N}$ satisfies,
for smooth $f$,
$\displaystyle\frac{\mathbb{E}\Big{[}f(J^{N}_{t+\varepsilon})-f(J^{N}_{t})|\mathcal{F}_{t}\Big{]}}{\varepsilon}$
$\displaystyle=h(I_{t}^{N})u^{+}\Big{[}f(J^{N}_{t}+\theta_{N})-f(J^{N}_{t})\Big{]}+h(I_{t}^{N})u^{-}\Big{[}f(J^{N}_{t}-\theta_{N})-f(J^{N}_{t})\Big{]}+o(\varepsilon)$
$\displaystyle=f^{\prime}(J^{N}_{t})\cdot\Big{(}h(I_{t}^{N})\underbrace{(u^{+}_{N}-u^{-}_{N})\theta_{N}}_{=W_{N}}\Big{)}+\tfrac{1}{2}f^{\prime\prime}(J^{N}_{t})\cdot
h(I_{t}^{N})\underbrace{(u^{+}_{N}+u^{-}_{N})\theta_{N}^{2}}_{=1}+o(\varepsilon)$
$\displaystyle=f^{\prime}(J^{N}_{t})\cdot
h(I_{t}^{N})W_{N}+\tfrac{1}{2}f^{\prime\prime}(J^{N}_{t})\cdot
h(I_{t}^{N})+o(\varepsilon).$
We obtain a similar expression if we ignore the jumps of $Z^{N,i},i=1,...,n$,
i.e. if we replace $J^{N}$ by $J^{N(n)}=\theta_{N}\sum_{j>n}U_{j}Z^{N,j}$. In
this case we have to replace $W_{N}$ by $W_{N(n)}=\theta_{N}\sum_{j>n}U_{j}$
and $(u^{+}_{N}+u^{-}_{N})\theta_{N}^{2}$ by $(N-n)/N$. In the limit
$N\to\infty$ we obtain the same expression, as $W_{N(n)}\to W$ and $(N-n)/N\to
1$. From now on write $J_{[0,t]}$ for the path of a process $J$ on the
interval $[0,t]$ and
$\displaystyle G_{J}^{N(n)}f(J_{[0,t]}^{\color[rgb]{0,0,0}N})$
$\displaystyle=f^{\prime}(J_{t}^{\color[rgb]{0,0,0}N})\cdot
h(I_{t}^{\color[rgb]{0,0,0}N})W_{N(n)}+\tfrac{N-n}{2N}f^{\prime\prime}(J_{t}^{\color[rgb]{0,0,0}N})\cdot
h(I_{t}^{\color[rgb]{0,0,0}N})$ $\displaystyle G_{J}f(J_{[0,t]})$
$\displaystyle=f^{\prime}(J_{t})\cdot
h(I_{t})W+\tfrac{1}{2}f^{\prime\prime}(J_{t})\cdot h(I_{t})$
The generator $G^{N}$ of the Markov process
$\big{(}J^{N}_{[0,t]},Z^{N,1}_{t},...,Z^{N,n}_{t}\big{)}$ with domain
$\mathcal{D}$ consisting of functions of the form
$f_{0,...,n}(J_{[0,t]},k_{1},...,k_{n})=f_{0}(J_{t})f_{1}(k_{1})...f_{n}(k_{n})$,
where we suppose the functions $f_{1},...,f_{n}$ to be bounded functions on
$\mathbb{N}$, and $f_{0}$ to be a smooth function on $\mathbb{R}$ with bounded
second derivative, is given by
$\displaystyle G^{N}f_{0,..,n}(J_{[0,t]}^{\color[rgb]{0,0,0}N},Z^{N,1}_{t}$
$\displaystyle,...,Z^{N,n}_{t})=G^{N(n)}_{J}f_{0}(J_{[0,t]}^{\color[rgb]{0,0,0}N})\prod_{i=1}^{n}f_{i}(Z^{N,i}_{t})$
$\displaystyle+h(I_{t})\sum_{j=1}^{n}\big{(}f_{0}(J_{t}+\theta_{N}U_{j})f_{j}(Z^{N,j}_{t}+1)-f_{0}(J_{t})f_{j}(Z^{N,j}_{t})\big{)}\prod_{i\neq
j}f_{i}(Z^{N,i}_{t}).$
By [16][Chapter 4, Proposition 1.7],
$\displaystyle
f_{0}(J^{N}_{t})\prod_{i=1}^{n}f_{i}(Z^{N,i}_{t})-\int_{0}^{t}G^{N}f_{0}...f_{n}(J_{[0,s]}^{\color[rgb]{0,0,0}N},Z^{N,1}_{s},...,Z^{N,n}_{s})ds$
is a martingale with respect to the filtration generated by $J,Z^{N,i}$. It is
equivalent that
$\displaystyle\mathbb{E}\Big{[}\Big{(}f_{0}(J^{N}_{t})\prod_{i=1}^{n}f_{i}(Z^{N,i}_{t})-f_{0}(J^{N}_{s})\prod_{i=1}^{n}f_{i}(Z^{N,i}_{s})-\int_{s}^{t}G^{N}f_{0}...f_{n}(J_{[0,r]}^{\color[rgb]{0,0,0}N},Z^{N,1}_{r},...,Z^{N,n}_{r})ds\Big{)}$
$\displaystyle\prod_{j=1}^{m}g_{j}(J_{[0,t_{j}]}^{\color[rgb]{0,0,0}N},Z^{N,1}_{t_{j}},...,Z^{N,n}_{t_{j}})\Big{]}=0,$
(5.14)
for all $m\in\mathbb{N},0\leq t_{1}\leq...\leq t_{m}\leq s<t$ and continuous
functions $g_{j}$. To identify the limit, assume
$\big{(}J^{N},Z^{N,1},...,Z^{N,n}\big{)}\Rightarrow\big{(}\bar{J},\bar{Z}^{1},...,\bar{Z}^{n}\big{)}$
in distribution with respect to Skorohod distance. By continuity of
projection, $J^{N}\Rightarrow\bar{J}$, hence $\bar{J}=J$. In Step 1 and 2 we
have already shown the stronger convergence $J^{N}\to J$ and $I^{N}\to I$ with
respect to uniform topology. As $G^{N}f$ converges to some $Gf$,
$\theta_{N}\to 0$ and the evaluation at timepoints $t$ is continuous with
respect to uniform topology, we can pass to the limit in (5.14) and obtain
that
$\displaystyle
f_{0}(J_{t})\prod_{i=1}^{n}f_{i}(\bar{Z}^{i}_{t})-\int_{0}^{t}Gf_{0,...,n}(J_{s},\bar{Z}^{1}_{s},...,\bar{Z}^{n}_{s})ds$
(5.15)
is a martingale for any $f_{0,...,n}\in\mathcal{D}$, where
$\displaystyle Gf_{0,..,n}(J_{[0,t]},Z^{1}_{t},...,Z^{n}_{t})=$ $\displaystyle
G_{J}f_{0}(J_{[0,t]})\prod_{i=1}^{n}f_{i}(Z^{i}_{t})$
$\displaystyle+h(I_{t})f_{0}(J_{t})\sum_{j=1}^{n}\big{(}f_{j}(Z^{j}_{t}+1)-f_{j}(Z^{j}_{t})\big{)}\prod_{i\neq
j}f_{i}(Z^{i}_{t}).$
It remains to read off the distribution of $(J,\bar{Z}^{1},...,\bar{Z}^{n})$
from (5.15). Choose $f_{0}=1$ and $f_{i}(k)=\exp(-u_{i}k)$ for $u_{j}\geq 0$.
Then
$\displaystyle 0=$
$\displaystyle\mathbb{E}\Big{[}f_{0}(J_{t})\prod_{i=1}^{n}f_{i}(\bar{Z}^{i}_{t})-\int_{0}^{t}Gf_{0}...f_{n}(J_{s},\bar{Z}^{1}_{s},...,\bar{Z}^{n}_{s})ds\Big{]}$
$\displaystyle=$
$\displaystyle\mathbb{E}\big{[}\prod_{i=1}^{n}e^{-u_{i}\bar{Z}^{i}_{t}}-\int_{0}^{t}h(I_{s})\sum_{j=1}^{n}\big{(}e^{-u_{j}}-1\big{)}\prod_{i=1}^{n}e^{-u_{i}\bar{Z}^{i}_{s}}ds\big{]}$
$\displaystyle=$
$\displaystyle\mathbb{E}\Big{[}\mathbb{E}\big{[}\prod_{i=1}^{n}e^{-u_{i}\bar{Z}^{i}_{t}}|I\big{]}-\int_{0}^{t}h(I_{s})\sum_{j=1}^{n}\big{(}e^{-u_{j}}-1\big{)}\mathbb{E}\big{[}{\color[rgb]{0,0,0}\prod_{i=1}^{n}}e^{-u_{i}\bar{Z}^{i}_{s}}|I\big{]}ds\Big{]},$
where, in the last equality, we have interchanged the $ds$-integral and
conditional expectation with respect to $I$, and used the $I$-measurability of
$h(I)$. The function in brackets is nonnegative, hence
$\displaystyle\mathbb{E}\Big{[}\prod_{i=1}^{n}e^{-u_{i}\bar{Z}^{i}_{t}}|I\Big{]}=\prod_{j=1}^{n}\exp\Big{(}(e^{-u_{j}}-1)\int_{0}^{t}h(I_{s})ds\Big{)}.$
As claimed, given $I$ the processes $\bar{Z}^{i}$ are independent poisson
processes with intensity at time $t$ given by $h(I_{t})$.
### 5.4 Proof of Theorem 1
Proof of 1.:
Define
$\displaystyle J_{t}=(2p-1)\int_{0}^{t}h(I_{s})ds,\text{ such that
}I_{t}=\int_{0}^{t}\varphi(t-s)dJ_{s}.$ (5.16)
For $\theta_{N}=\tfrac{1}{N}$ in (5.9) we get that,
$\displaystyle J_{t}^{N}-J_{t}$
$\displaystyle=\frac{1}{N}\Big{(}Y^{+}\Big{(}\big{(}pN+\tfrac{1}{2}\sqrt{N}W_{N}\big{)}\int_{0}^{t}h(I_{s}^{N})ds\Big{)}-Y^{+}\Big{(}pN\int_{0}^{t}h(I_{s}^{N})ds\Big{)}\Big{)}$
$\displaystyle\quad-\frac{1}{N}\Big{(}Y^{-}\Big{(}\big{(}(1-p)N-\tfrac{1}{2}\sqrt{N}W_{N}\big{)}\int_{0}^{t}h(I_{s}^{N})ds\Big{)}-Y^{-}\Big{(}(1-p)N\int_{0}^{t}h(I_{s}^{N})ds\Big{)}\Big{)}$
$\displaystyle\qquad\qquad+\frac{1}{N}Y^{+}\Big{(}pN\int_{0}^{t}h(I_{s}^{N})ds\Big{)}-p\int_{0}^{t}h(I_{s}^{N})ds$
$\displaystyle\qquad\qquad-\frac{1}{N}Y^{-}\Big{(}(1-p)N\int_{0}^{t}h(I_{s}^{N})ds\Big{)}+(1-p)\int_{0}^{t}h(I_{s}^{N})ds$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+(2p-1)\int_{0}^{t}h(I_{s}^{N})-h(I_{s})ds,$
so
$\displaystyle\sup_{0\leq s\leq t}(J_{s}^{N}-J_{s})^{2}$
$\displaystyle\leq\varepsilon_{N}+C\cdot\Big{(}\sup_{0\leq s\leq
t||h||}\Big{(}\frac{1}{N}Y^{+}\Big{(}pNs\Big{)}-ps\Big{)}^{2}+\Big{(}\frac{1}{N}Y^{-}\Big{(}(1-p)Ns\Big{)}-(1-p)s\Big{)}^{2}\Big{)}$
$\displaystyle\qquad\qquad\qquad+(2p-1)^{2}h_{Lip}\cdot\sup_{0\leq s\leq
t}\int_{0}^{s}(I_{r}^{N}-I_{r})^{2})ds$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\leq\varepsilon_{N}+C\cdot\int_{0}^{t}\sup_{0\leq
r\leq s}(I_{r}^{N}-I_{r})^{2}ds$ (5.17)
with $\varepsilon_{N}\xrightarrow{N\to\infty}0$ (by Lemma 5.1). By the
Gronwall inequality and Lemma 5.2 we can conclude
$\displaystyle\sup_{0\leq s\leq
t}\Big{(}J_{s}^{N}-J_{s}\Big{)}^{2}\xrightarrow{N\to\infty}0\qquad\text{and}\qquad\sup_{0\leq
s\leq t}\Big{(}I_{s}^{N}-I_{s}\Big{)}^{2}\xrightarrow{N\to\infty}0.$ (5.18)
Proof of 2.:
Define $\bar{Z}^{i}_{t}=Y_{i}\big{(}\int_{0}^{t}h(I_{s})ds\big{)}$, where
$Y_{1},...,Y_{N}$ are independent Poisson process as in (5.7). Fix
$\omega\in\Omega$, such that $\sup_{0\leq s\leq
t}|I_{s}^{N}(\omega)-I_{s}|\rightarrow 0$. As $h$ is Lipschitz,
$\int_{0}^{t}h(I_{s}^{N}(\omega))ds\rightarrow\int_{0}^{t}h(I_{s})ds$, hence
for any point of continuity of $t\mapsto\bar{Z}^{i}_{t}(\omega)$ we can
conclude $Z^{i}_{t}(\omega)\rightarrow\bar{Z}^{i}_{t}(\omega)$. As the points
of continuity are dense in $[0,\infty)$, convergence in Skorohod-distance
follows from [26][Theorem 2.15 c)(ii)].
Proof of 3.:
First, strong existence and uniqueness for (3.3) follows from [3]. The proof
proceeds similarly to the proof of Theorem 2, Step 1 and 2. Wlog, we assume
that $W_{N}\to W$ almost surely and in $L^{2}$. Recall the pair $(G,K)$ from
(3.3) (for some Brownian motion $B$). Further, we define the auxiliary
processes $\widetilde{G}^{N},\widetilde{K}^{N}$ by (recall from (5.10) the
Definition of the Brownian motion $B^{N}$)
$\displaystyle\widetilde{K}^{N}_{t}$
$\displaystyle=\int_{0}^{t}\varphi(t-s)d\widetilde{G}^{N}_{s}$ (5.19)
$\displaystyle\widetilde{G}^{N}_{t}$
$\displaystyle=\int_{0}^{t}Wh(I_{s})+(2p-1)h^{\prime}(I_{s})\widetilde{K}^{N}(s)ds+\int_{0}^{t}\sqrt{h(I_{s})}dB^{N}_{s}.$
In our proof, we will first show convergence of
$\sqrt{N}(J^{N}-J)-\widetilde{G}^{N}\rightarrow 0$ and
$\sqrt{N}(I^{N}-I)-\widetilde{K}^{N}\rightarrow 0$. We then replace $B^{N}$ by
$B$ and find $\widetilde{K}^{N}\Rightarrow K$, since
$(\widetilde{K}^{N},\widetilde{G}^{N})\sim(K,G)$. The process
$\sqrt{N}(J^{N}-J)-\widetilde{G}^{N}$ is described by the following equation:
$\displaystyle\sqrt{N}(J^{N}_{t}-J_{t})-\widetilde{G}^{N}_{t}=A^{N}_{t}+C^{N}_{t}+D^{N}_{t}+(2p-1)E^{N}_{t}$
with $\displaystyle A_{t}^{N}$
$\displaystyle=\frac{1}{\sqrt{N}}Y^{+}\Big{(}\Big{(}pN+\frac{\sqrt{N}W_{N}}{2}\Big{)}\int_{0}^{t}h(I_{s}^{N})ds\Big{)}-\frac{1}{\sqrt{N}}Y^{-}\Big{(}\Big{(}(1-p)N-\frac{\sqrt{N}W_{N}}{2}\Big{)}\int_{0}^{t}h(I_{s}^{N})ds\Big{)}$
$\displaystyle\qquad\qquad\qquad\qquad-\sqrt{N}(2p-1)\int_{0}^{t}h(I_{s}^{N})ds-
W_{N}\int_{0}^{t}h(I_{s}^{N})ds-\hat{B}\Big{(}\int_{0}^{t}h(I_{s}^{N})ds\Big{)},$
$\displaystyle C_{t}^{N}$
$\displaystyle=W_{N}\int_{0}^{t}h(I_{s}^{N})ds-W\int_{0}^{t}h({I}_{s})ds,$
$\displaystyle D_{t}^{N}$
$\displaystyle=\int_{0}^{t}\sqrt{h(I_{s}^{N})}dB^{N}_{s}-\int_{0}^{t}\sqrt{h(I_{s})}dB^{N}_{s},$
$\displaystyle E_{t}^{N}$
$\displaystyle=\int_{0}^{t}\sqrt{N}\big{(}h(I^{N}_{s})-h(I_{s})\big{)}-h^{\prime}(I_{s})\widetilde{K}^{N}_{s}ds.$
As in the proof of Theorem 2, Step 1, we can show that $C^{N},D^{N}\to 0$ (as
we already know that $I^{N}\to I$ and $W_{N}\rightarrow W$) uniformly on
compact time intervals in $L^{2}$. The uniform $L^{2}$-convergence $A^{N}\to
0$ follows from Lemma 5.1. For $E^{N}$ we obtain
$\displaystyle
E^{N}_{t}=\int_{0}^{t}\big{(}\sqrt{N}(I^{N}_{s}-I_{s})-\widetilde{K}^{N}_{s}\big{)}h^{\prime}(I^{N}_{s})+\sqrt{N}R_{1}h(I^{N}_{s},I_{s})ds,$
where $R_{1}h(I^{N}_{s},I_{s})$ is the first order remainder in Taylor’s
formula. Using the Peano form of the remainder we obtain for some
$\xi^{N}_{s}$ between $I^{N}_{s}$ and $I_{s}$
$\displaystyle\Big{(}\sqrt{N}R_{1}h(I^{N}_{s},I_{s})\Big{)}^{2}$
$\displaystyle=N\Big{(}(h^{\prime}(\xi^{N}_{s})-h^{\prime}(I_{s}))^{2}(I^{N}_{s}-I_{s})^{2}\Big{)}$
$\displaystyle\leq(h^{\prime}_{Lip})^{2}N(I^{N}_{s}-I_{s})^{4}.$
Therefore
$\displaystyle\mathbb{E}\Big{[}\sup_{0\leq s\leq
t}\big{(}E^{N}_{s}\big{)}^{2}\Big{]}$ $\displaystyle\leq
2\mathbb{E}\Big{[}\sup_{0\leq s\leq
t}\Big{(}\int_{0}^{s}\big{(}\sqrt{N}(I^{N}_{u}-I_{u})-\widetilde{K}^{N}_{u}\big{)}h^{\prime}(I^{N}_{u})du\Big{)}^{2}\Big{]}$
$\displaystyle\qquad\qquad+2\mathbb{E}\Big{[}\sup_{0\leq s\leq
t}\Big{(}\int_{0}^{t}\big{(}\sqrt{N}R_{1}h(I^{N}_{u},I_{u})\big{)}du\Big{)}^{2}\Big{]}$
$\displaystyle\leq
2t||h^{\prime}||^{2}\int_{0}^{t}\mathbb{E}\Big{[}\big{(}\sqrt{N}(I^{N}_{s}-I_{s})-\widetilde{K}^{N}_{s}\big{)}^{2}\Big{]}ds$
$\displaystyle\qquad\qquad+2t^{2}(h^{\prime}_{Lip})^{2}\mathbb{E}\Big{[}\sup_{0\leq
s\leq t}N(I^{N}_{s}-I_{s})^{4}\Big{]}.$
We can show convergence of the second term similarly to the proof of 1., as
$\displaystyle N\cdot\mathbb{E}\Big{[}\sup_{0\leq s\leq
t}\Big{(}\frac{1}{N}(Y^{\pm}(Ns)-Ns)\Big{)}^{4}\Big{]}$
$\displaystyle\leq\Big{(}\frac{4}{3}\Big{)}^{4}\frac{1}{N^{3}}\mathbb{E}\Big{[}\Big{(}(Y^{\pm}(Nt)-Nt)\Big{)}^{4}\Big{]}$
$\displaystyle=\Big{(}\frac{4}{3}\Big{)}^{4}\Big{(}\frac{3t^{2}}{N}+\frac{t}{N^{2}}\Big{)}\xrightarrow{N\to\infty}0.$
For the first term we apply Lemma 5.2 in order to obtain
$\displaystyle\mathbb{E}\Big{[}\sup_{0\leq s\leq
t}\Big{(}\big{(}\sqrt{N}(I^{N}_{s}-I_{s})-\widetilde{K}^{N}_{s}\big{)}\Big{)}^{2}\Big{]}\leq
C\cdot\mathbb{E}\Big{[}\sup_{0\leq s\leq
t}\Big{(}\big{(}\sqrt{N}(J^{N}_{s}-J_{s})-\widetilde{G}^{N}_{s}\big{)}\Big{)}^{2}\Big{]}.$
Summing up the results on $A^{N},C^{N},D^{N}$ and $E^{N}$ we conclude
$\displaystyle\mathbb{E}\Big{[}\sup_{0\leq s\leq
t}\big{(}\sqrt{N}(J^{N}_{s}-J_{s})-\widetilde{G}^{N}_{s}\big{)}^{2}\Big{]}\leq\varepsilon_{N}+C\int_{0}^{t}\mathbb{E}\Big{[}\sup_{0\leq
u\leq
s}\big{(}\sqrt{N}(J^{N}_{u}-J_{u})-\widetilde{G}^{N}_{u}\big{)}^{2}\Big{]}ds$
for $\varepsilon_{N}\to 0$ and by the Gronwall inequality
$\displaystyle\mathbb{E}\Big{[}\sup_{0\leq s\leq
t}\big{(}\sqrt{N}(J^{N}_{s}-J_{s})-\widetilde{G}^{N}_{s}\big{)}^{2}\Big{]}\xrightarrow{N\to\infty}0.$
Applying Lemma 5.2 it follows that
$\displaystyle\mathbb{E}\Big{[}\sup_{0\leq s\leq
t}\big{(}\sqrt{N}(I^{N}_{s}-I_{s})-\widetilde{K}^{N}_{s}\big{)}^{2}\Big{]}\xrightarrow{N\to\infty}0.$
### 5.5 Proof of Corollaries 3.2 and 3.5
###### Proof of Corollary 3.2.
Recall $Y^{+}$ and $Y^{-}$ from Section 5.2, and which are used frequently in
the proof of Theorem 1. For (3.4), using (5.7) and Lemma 5.1, for the Poisson
process $Y=(Y(t))_{t\geq 0}$, given by $Y(t)=Y^{+}(pt)+Y^{-}((1-p)t)$,
$\displaystyle\sup_{0\leq t\leq T}$
$\displaystyle\Big{|}\frac{1}{N}\sum_{j=1}^{N}\Big{(}Z_{t}^{N,j}-\int_{0}^{t}h(I_{s}^{N})ds\Big{)}\Big{|}=\sup_{0\leq
t\leq
T}\Big{|}\frac{1}{N}Y\Big{(}N\int_{0}^{t}h(I_{s}^{N})ds\Big{)}-\int_{0}^{t}h(I_{s}^{N})ds\Big{|}$
(5.20) $\displaystyle\leq\sup_{0\leq t\leq
T||h||}\Big{|}\frac{1}{N}(Y(Nt)-Nt)\Big{|}=0.$
For (3.5), by the law of large numbers,
$\frac{1}{N}\sum_{j=1}^{N}\bar{Z}^{j}_{T}\xrightarrow{N\to\infty}\int_{0}^{T}h(I_{t})dt$,
as well as $h(I^{N})\xrightarrow{N\to\infty}h(I)$ in $L^{2}$, uniformly on
compact time intervals by Theorem 1.1, therefore
$\displaystyle\frac{1}{N}\sum_{j=1}^{N}\Big{(}Z^{N,j}-\bar{Z}^{j}\Big{)}$
$\displaystyle=\frac{1}{N}\sum_{j=1}^{N}\Big{(}Z^{N,j}-\int_{0}^{.}h(I_{s}^{N})ds\Big{)}$
$\displaystyle\qquad\qquad-\frac{1}{N}\sum_{j=1}^{N}\Big{(}\bar{Z}^{j}-\int_{0}^{.}h(I_{s})ds\Big{)}+\int_{0}^{.}(h(I_{s}^{N})-h(I_{s}))ds\xrightarrow{N\to\infty}0.$
For (3.6), we write, with the same Poisson process as in (5.20), and with the
Brownian motions $B^{\prime}$ arising as limit of the compensated Poisson
process $Y$, and some $\widetilde{B}$,
$\displaystyle\frac{1}{\sqrt{N}}$
$\displaystyle\sum_{j=1}^{N}\Big{(}Z^{N,j}_{t}-\int_{0}^{t}h(I_{s}^{N})ds\Big{)}=\frac{1}{\sqrt{N}}\Big{(}Y\Big{(}N\int_{0}^{t}h(I_{s}^{N})ds\Big{)}-N\int_{0}^{t}h(I_{s}^{N})ds\Big{)}$
(5.21)
$\displaystyle=B^{\prime}\Big{(}\int_{0}^{t}h(I_{s})ds\Big{)}+o(1)=\int_{0}^{t}\sqrt{h(I_{s})}d\widetilde{B}_{s}+o(1).$
Next, for (3.7), with $K$ as in Theorem 1.3,
$\displaystyle\sqrt{N}(h(I_{t}^{N})-h(I_{t}))\Big{)}=\sqrt{N}h^{\prime}(I_{t})(I_{t}^{N}-I_{t})+o(1)=h^{\prime}(I_{t})K_{t}+o(1).$
Last, we note that the left hand side of (3.8) is a sum of the left hand sides
of (3.6) and (3.7). We obtain the result by summing the limits of these
equations, once we determine the correlation structure of the martingales
$\int_{0}^{.}\sqrt{h(I_{s})}dB_{s}$ in (3.3) and
$\int_{0}^{.}\sqrt{h(I_{s})}d\widetilde{B}_{s}$ in (3.6) on a joint
probability space. We will show that joint weak convergence of (3.6) and (3.7)
holds if
$\displaystyle\mathbb{E}[\int_{0}^{t}\sqrt{h(I_{s})}dB_{s}\int_{0}^{t}\sqrt{h(I_{s})}d\widetilde{B}_{s}]=(2p-1)\int_{0}^{t}h(I_{s})ds,$
(5.22)
implying the result. Recall from Section 5.2 the limits $B^{+}$ and $B^{-}$ of
the rescaled Poisson processes $Y^{+}$ and $Y^{-}$. By the construction of $Y$
above (5.20), we find that $B^{\prime}_{t}=B^{+}_{pt}+B^{-}_{(1-p)t}$ is the
limit of the rescaled Poisson process $Y$. In addition,
$\hat{B}_{t}=B^{+}_{pt}-B^{-}_{(1-p)t}$ arises above (5.10). Clearly, by
independence of $B^{+}$ and $B^{-}$, we have
$\mathbb{E}[\hat{B}_{t}B_{t}^{\prime}]=\mathbb{E}[(B_{pt}^{+}+B_{(1-p)t}^{-})(B_{pt}^{+}-B_{(1-p)t}^{-})]=(2p-1)t$.
From the proof of Theorem 1.3, we know that the convergence in this result is
even in probability if we exchange $K$ by $\widetilde{K}^{N}$ from (5.19).
Hence, using $I^{N}\to I$, we find for the integral appearing in (3.3), on a
joint probability space, and taking (5.10) into account,
$\int_{0}^{t}\sqrt{h(I_{s})}dB_{s}=\lim_{N\to\infty}\int_{0}^{t}\sqrt{h(I^{N}_{s})}dB^{N}_{s}=\lim_{N\to\infty}\hat{B}\Big{(}\int_{0}^{t}h(I^{N}_{s})ds\Big{)}=\hat{B}\Big{(}\int_{0}^{t}h(I_{s})ds\Big{)}.$
From this and (5.21), we can thus write
$\displaystyle\mathbb{E}\Big{[}\int_{0}^{t}\sqrt{h(I_{s})}dB_{s}\int_{0}^{t}\sqrt{h(I_{s})}d\widetilde{B}_{s}\Big{]}$
$\displaystyle=\mathbb{E}\Big{[}\hat{B}\Big{(}\int_{0}^{t}h(I_{s})ds\Big{)}\cdot
B^{\prime}\Big{(}\int_{0}^{t}h(I_{s})ds\Big{)}\Big{]}$
$\displaystyle=(2p-1)\int_{0}^{t}h(I_{s})ds$
and we are done. ∎
###### Proof of Corollary 3.5.
Actually, the proof of (3.4), (3.5) and (3.6) is literally the same as in the
proof above. ∎
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|
# Supplement: ``A mushy source for the geysers of Enceladus''
Colin R. Meyer<EMAIL_ADDRESS>Thayer School of Engineering,
Dartmouth College, Hanover, NH 03755 Jacob J. Buffo Thayer School of
Engineering, Dartmouth College, Hanover, NH 03755 Francis Nimmo Department
of Earth & Planetary Sciences, University of California, Santa Cruz, CA 95064
USA Andrew J. Wells Atmospheric, Oceanic & Planetary Physics, Department of
Physics, University of Oxford, OX1 3PU, UK Samuel Boury Courant Institute of
Mathematical Sciences, New York University, New York, NY 10012 Tara C.
Tomlinson Thayer School of Engineering, Dartmouth College, Hanover, NH 03755
Jamie R. G. Parkinson Atmospheric, Oceanic & Planetary Physics, Department of
Physics, University of Oxford, OX1 3PU, UK Geoffrey M. Vasil School of
Mathematics, University of Edinburgh, EH9 3FD, UK
## 1 Fracture depths
It is unclear how easy it is to produce a fracture spanning the entire ice
shell of Enceladus. Due to the limited tidal stresses ($\sim 100$ kPa), when
compared to the overburden pressure ($\sim 600$ kPa at the base of a 6 km ice
shell), it is hard for near-surface fractures to extend downwards more than
1–2 km (Smith-Konter and Pappalardo, 2008; Olgin et al., 2011; Walker et al.,
2021). This is less than the estimated minimum shell thickness of 6 km at the
south pole (Hemingway et al., 2018). Fractures extending upwards from the base
of the shell are less affected by overburden pressure because they would be
water-filled (Crawford and Stevenson, 1988; Shibley and Laughlin, 2021).
However, ice at the base of the shell is more ductile and thus harder to
fracture than ice at the surface (Schulson and Duval, 2009).
The current paradigm for the source of geyser material is that fractures
extend down to the subsurface ocean (Hurford et al., 2007). To get an idea for
the depth of the tiger stripe fractures, we balance the extensional stress
required to open a fracture with the overburden ice pressure acting to close
it (i.e. the zero-stress Nye (1955) criterion for closely spaced glacier
crevasses). Estimates for the extensional stress are on the order of $\tau\sim
1$ bar (Nimmo et al., 2007, 2011). Gravity on Enceladus is $g=0.113$ m s-2
(Hansen et al., 2006, 2008) and the salty ice density is about
$\rho_{\textrm{ice}}=940$ kg m-3 (Timco and Frederking, 1996). The fracture
close-off depth $h_{\textrm{top}}$ is given as
$h_{\textrm{top}}\sim\frac{\tau}{\rho_{\textrm{ice}}g}\sim 1~{}\mathrm{km},$
(1)
which is an estimate for the likely depth of surface fractures and agrees with
more sophisticated calculations (Smith, 1976; Broberg, 1999; Smith-Konter and
Pappalardo, 2008; Olgin et al., 2011). The fact that the surface fractures are
only likely to propagate to $\sim 1~{}\mathrm{km}$ depth is important because
$\sim 1~{}\mathrm{km}$ is shallower than the minimum predicted shell thickness
of $\sim 6-30~{}\mathrm{km}$ (Hemingway et al., 2018), which casts doubt on
the idea that the fractures penetrate from the surface down to the ocean.
Moreover, it would be a considerable feat for a fracture to extend up from the
ocean (i.e. a basal crevasse) to within $\sim 1~{}\mathrm{km}$ of the surface,
based on the extensional stress just mentioned. A water-filled basal crevasse
does lead to a larger close-off height $h_{\textrm{bottom}}$, given by
$h_{\textrm{bottom}}\sim\frac{\tau}{\left(\rho_{\textrm{water}}-\rho_{\textrm{ice}}\right)g}\sim
10\mbox{ km},$ (2)
where we use a value of seawater density, $\rho_{\textrm{water}}=1030$ kg m-3.
This is sufficient to fracture the shell if (i) the fracture emanates up from
the bottom and (ii) if the shell is on the lower end of thickness range (i.e.
$\sim 6-30~{}\mathrm{km}$ Hemingway et al., 2018). However, the lower shell is
likely ductile and less prone to fracturing (Nimmo, 2004; Schulson and Duval,
2009). On the other hand, there are additional sources of stress available to
open cracks which likely exceed the tidal stresses. These include the effects
of shell thickening (Rudolph et al., 2022) and flexural stresses from plume
loading (Hemingway et al., 2020). Thus, opening of through-going fractures
appears possible but not certain. At the same time, the dike mechanism
discussed in the main text could allow for the initiation of downward-
propagating through-going fractures that could be maintained by turbulent
water flow and shear heating. We discuss the dike mechanism further in section
5.
## 2 Analytical temperature solution
Here we demonstrate that the leading order thermal structure can be understood
from steady state conduction, with reactive flow causing subsequent
modifications to the porosity structure, composition and shape of the
inclusion. We consider a fracture of depth $h$ in a shell of thickness $H$
with a shear heat flux of $\mu\rho_{i}ghu$. So long as the latent heat sink of
partial melting and heat transport by interstitial flow are relatively small,
then the steady state temperature $T$ around the fracture is given by the
solution to Laplace's equation, i.e.
$\nabla^{2}T=0,$ (3)
where we treat the thermal conductivity as temperature independent to
facilitate analytical progress. Equation (3) is subject to the boundary
conditions
$\displaystyle-k\frac{\partial T}{\partial x}$ $\displaystyle=$ $\displaystyle
0~{}~{}~{}\mbox{on}~{}~{}~{}x=0~{}(z<h)~{}~{}~{}\mbox{and}~{}~{}~{}x=L,$ (4)
$\displaystyle-k\frac{\partial T}{\partial x}$ $\displaystyle=$
$\displaystyle\mu\rho_{i}ghu~{}~{}~{}\mbox{on}~{}~{}~{}x=0~{}(z\geq h),$ (5)
$\displaystyle-k\frac{\partial T}{\partial z}$ $\displaystyle=$ $\displaystyle
r_{1}+r_{2}(T-T_{s})~{}~{}~{}\mbox{on}~{}~{}~{}z=H,$ (6) $\displaystyle T$
$\displaystyle=$ $\displaystyle T_{e}~{}~{}~{}\mbox{on}~{}~{}~{}z=0.$ (7)
The surface condition is a linearized radiative condition, with respect to the
background temperature $T_{s}$ of space near Enceladus (see table 1), rather
than the full black-body radiation condition as used by Abramov and Spencer
(2009). This choice for the surface boundary condition allows for an
analytical solution and is the condition used in SOFTBALL. To derive the
linearized condition, we start from the surface energy balance where
conductive heat fluxes from the ice balance net black-body radiation as
$-k\frac{\partial T}{\partial z}=\epsilon\sigma\left(T^{4}-T_{s}^{4}\right),$
(8)
where $\epsilon$ is the emissivity and $\sigma$ is the Stefan-Boltzmann
constant (values in table 1). Expanding the temperature as a small deviation
from the space temperature, i.e. $T=T_{s}+\tilde{T}$, where $\tilde{T}\ll
T_{s}$, then we can use the binomial expansion and discard terms of second
order and higher, to find that
$-k\frac{\partial T}{\partial z}=4\epsilon\sigma T_{s}^{3}(T-T_{s}),$ (9)
which is in the form of the boundary condition above, with $r_{1}=0$ and
$r_{2}=4\epsilon\sigma T_{s}^{3}$.
We nondimensionalized the problem, by taking
$T=T_{e}+\Delta T\theta,~{}~{}~{}x=H\tilde{x},~{}~{}~{}z=H\tilde{z},$
which yields
$\tilde{\nabla}^{2}\theta=0,$ (10)
subject to the boundary conditions
$\displaystyle-\frac{\partial\theta}{\partial\tilde{x}}$ $\displaystyle=$
$\displaystyle
0~{}~{}~{}\mbox{on}~{}~{}~{}\tilde{x}=0~{}(\tilde{z}<\delta)~{}~{}~{}\mbox{and}~{}~{}~{}\tilde{x}=\lambda,$
(11) $\displaystyle-\frac{\partial\theta}{\partial\tilde{x}}$ $\displaystyle=$
$\displaystyle
F~{}~{}~{}\mbox{on}~{}~{}~{}\tilde{x}=0~{}(\tilde{z}\geq\delta),$ (12)
$\displaystyle-\frac{\partial\theta}{\partial\tilde{z}}$ $\displaystyle=$
$\displaystyle
G+b\left(\theta-\theta_{s}\right)~{}~{}~{}\mbox{on}~{}~{}~{}\tilde{z}=1,$ (13)
$\displaystyle\theta$ $\displaystyle=$ $\displaystyle
0~{}~{}~{}\mbox{on}~{}~{}~{}\tilde{z}=0.$ (14)
Here we define the parameters as
$\delta=\frac{h}{H},~{}~{}~{}\lambda=\frac{L}{H},~{}~{}~{}F=\frac{\mu\rho_{i}ghHu}{k_{o}\Delta
T},~{}~{}~{}G=\frac{Hr_{1}}{k_{o}\Delta
T},~{}~{}~{}b=\frac{Hr_{2}}{k_{o}},~{}~{}~{}\theta_{s}=\frac{T_{s}-T_{e}}{\Delta
T}.$
Due to convergence issues in the SOFTBALL simulations, we could not use large
values for $b$, and so the value quoted in table 1 is artificially lower than
what we would calculate from the parameters. The main effect of this
difference is on the surface temperature profile and not the size, shape, or
liquid volume of the mushy zone. Asymptotically, our analytical solutions
shows that large values of $b$ pin the surface temperature to the space
temperature, except very near the crack, where there is a narrow boundary
layer region of rapid temperature change. The boundary layer becomes thinner
as $b$ increases. Additionally, since $G=0$ here, we disregard it in
subsequent calculations.
To simplify the problem in the vertical direction, we write the total solution
$\theta$ as a sum of a particular solution $\theta_{I}$ to the inhomogeneous
problem and the homogenous solution $\theta_{H}$, i.e.
$\theta=\theta_{I}(\tilde{z})+\theta_{H}(\tilde{x},\tilde{z}).$
Solving first for the inhomogeneous part, we have that
$\frac{\partial^{2}\theta_{I}}{\partial\tilde{x}^{2}}+\frac{\partial^{2}\theta_{I}}{\partial\tilde{z}^{2}}=0\longrightarrow\frac{\partial^{2}\theta_{I}}{\partial\tilde{z}^{2}}=0,$
so
$\theta_{I}=A\tilde{z}+B.$
Using the boundary conditions, we find that
$B=0\ \ \ \mbox{ and }\ \ \ A=\frac{b\theta_{s}}{1+b}.$
For large values of $b$, the slope $A$ is approximately $\theta_{s}$, which is
consistent with the asymptotic description above, that the surface temperature
mostly stays close to the space temperature.
Now we can examine the homogeneous problem using separation of variables, i.e.
$\theta_{H}(\tilde{x},\tilde{z})=f(\tilde{x})g(\tilde{z}),~{}~{}~{}\mbox{which
implies}~{}~{}~{}f^{\prime\prime}g+fg^{\prime\prime}=0.$
Dividing through and choosing the separation constant $\alpha^{2}$, we find
the two ODEs
$f^{\prime\prime}=\alpha^{2}f\ \ \ \mbox{ and }\ \ \
g^{\prime\prime}=-\alpha^{2}g.$
Solving the vertical equation first, we have that
$g=C\sin(\alpha\tilde{z})+D\cos(\alpha\tilde{z}).$
With $g=0$ at $\tilde{z}=0$, we know that $D=0$. The homogeneous surface
boundary condition is
$-g^{\prime}=bg\ \ \ \mbox{ {at} }\ \ \ \tilde{z}=1,$
which results in the transcendental eigenvalue condition
$-\alpha\cos(\alpha)=b\sin(\alpha).$ (15)
This yields a full set of distinct eigenvalues $\alpha_{n}$ for $n=1,2,3,...$,
which can be computed numerically. In the horizontal direction, we have that
$f=De^{\alpha\tilde{x}}+Ee^{-\alpha\tilde{x}}.$
Using the no heat flux boundary condition at $\tilde{x}=\lambda$, we have that
$-f^{\prime}=0\ \ \ \mbox{ {at} }\ \ \ \tilde{x}=\lambda\ \ \ \mbox{ {implies}
}\ \ \ E=De^{2\alpha\lambda}.$
Thus, the full solution with unknown coefficients is given by
$\theta=\frac{b\theta_{s}}{1+b}\tilde{z}+\sum_{n=1}^{\infty}{c_{n}\left[e^{\alpha_{n}\tilde{x}}+e^{2\alpha_{n}\lambda}e^{-\alpha_{n}\tilde{x}}\right]\sin(\alpha_{n}\tilde{z})}.$
(16)
We find the values for $c_{n}$ by using orthogonality and integrating over the
heat flux condition at $\tilde{x}=0$, which gives
$\int_{\delta}^{1}{F\sin(\alpha_{m}\tilde{z})~{}d\tilde{z}}=\sum_{n=1}^{\infty}{c_{n}\alpha_{n}\left[1-e^{2\alpha_{n}\lambda}\right]\int_{0}^{1}{\sin(\alpha_{n}\tilde{z})\sin(\alpha_{m}\tilde{z})~{}d\tilde{z}}}.$
(17)
The integral on right is zero for $\alpha_{n}\neq\alpha_{m}$, using the
transcendental eigenvalue condition from equation (15) implying orthogonality
of the eigenfunctions $\sin(\alpha_{n}\hat{z})$. When $\alpha_{n}=\alpha_{m}$,
we have that
$\int_{0}^{1}{\sin(\alpha_{m}\tilde{z})\sin(\alpha_{m}\tilde{z})~{}d\tilde{z}}=\frac{1}{2}-\frac{\sin(2\alpha_{m})}{4\alpha_{m}}.$
We evaluate the integral on the left as
$\int_{\delta}^{1}{F\sin(\alpha_{m}\tilde{z})~{}d\tilde{z}}=-\frac{F}{\alpha_{m}}\left[\cos(\alpha_{m})-\cos(\alpha_{m}\delta)\right].$
Putting it all together gives,
$c_{n}=-\frac{4F}{\left[2\alpha_{n}^{2}-\alpha_{n}\sin(2\alpha_{n})\right]}\frac{\left[\cos(\alpha_{n})-\cos(\alpha_{n}\delta)\right]}{1-e^{2\alpha_{n}\lambda}}.$
(18)
We show the analytical solution in figure 1 and find that a mushy zone will
form along the crack face in regions with $T>T_{e}$. These results are similar
to what we find in the main text as well as Hammond (2020), Kalousová et al.
(2016), and Stevenson (1996). This analytical solution also allows us to
predict the size of the mushy zone, by looking for the contour where
$T=T_{e}$.
Figure 1: Analytical solution for the temperature around a heated crack with
$F=260$.
Figure 2: Liquid volume in the mushy zone for $F=500$: (a) evolution to a
steady state and average effective heat flux as well as (b) rate of liquid
volume production, which can support the entire flux observed by the Cassini
spacecraft from a single tiger stripe fracture.
## 3 Mushy zone physics
To analyze the effect of shear heating on a fracture in an icy shell, we use
SOFTBALL (SOlidification, Flow and Thermodynamics in Binary ALLoys), a mushy
layer code with adaptive mesh refinement. The github repository for this open-
source code is: https://github.com/jrgparkinson/mushy-layer (accessed 8 April
2023). This state-of-the-art code solves the equations of mushy layer theory,
as described below, in two-dimensional geometries (Wells et al., 2019;
Parkinson et al., 2020). The code uses a finite volume discretization and
adaptive time steps. In our simulations we use a spatial scale of $N_{x}=512$
grid points in the horizontal direction and $N_{z}=256$ in the vertical
direction. This gives a resolution of $\sim 40$ meters in both the horizontal
and vertical directions. The time steps adjust and are on the order of $\sim
30$ years. The simulations take on the order of a day to run on the linux
system (2-16 cores) at Dartmouth College.
Conservation of mass in a mushy layer dictates that
$\frac{\partial\left(\phi\rho\right)}{\partial
t}=M_{f}~{}~{}~{}\mbox{and}~{}~{}~{}\frac{\partial\left[\left(1-\phi\right)\rho\right]}{\partial
t}+\boldsymbol{\nabla}\cdot\boldsymbol{q}=-M_{f},$ (19)
where the solid and liquid densities $\rho$ are approximated as equal (using
an extended form of the Boussinesq approximation; Parkinson et al., 2020), the
solid fraction is $\phi$, the fluid flux relative to the solid is
$\boldsymbol{q}$, and the rate of mass phase change per volume is $M_{f}$. The
sum of equations (19) yields $\boldsymbol{\nabla}\cdot\boldsymbol{q}=0$, i.e.
a solenoidal fluid flow field. Here we assume that the porous solid matrix is
stationary. Fluid flux is a function of pressure gradients and gravity, which
can be described by Darcy's law, as
$\boldsymbol{q}=-\frac{\Pi}{\eta}\left(\nabla
p+\rho_{b}g\boldsymbol{z}\right),$ (20)
where $\eta$ is the liquid viscosity, $\rho_{b}$ is the density of liquid
brine (which is a function of the brine concentration and temperature),
$\boldsymbol{z}$ is the vertical unit vector, and $\Pi$ is the solid-fraction-
dependent permeability (Tait and Jaupart, 1992; Katz and Worster, 2008), given
by
$\Pi=\left[\frac{12}{d^{2}}+\frac{1}{K_{0}\phi^{3}}\right]^{-1}.$ (21)
Following Katz and Worster (2008), SOFTBALL uses a permeability regularization
for pure liquid regions (i.e. as $\phi\rightarrow 1$) such that there is a
background permeability and Darcy's law is solved everywhere, rather solving
Navier-Stokes in the pure fluid region (see Parkinson et al., 2020; Buffo et
al., 2021). The ratio of the background permeability $d^{2}/12$ to the
permeability prefactor $K_{0}$ is the reluctance $\mathtt{R}=d^{2}/(12K_{0})$.
Conservation of energy in the mushy layer is given by
$\overline{\rho c_{p}}\frac{\partial T}{\partial
t}+\rho_{b}c_{b}\boldsymbol{q}\cdot\boldsymbol{\nabla}T=\boldsymbol{\nabla}\cdot\left(\overline{k}\boldsymbol{\nabla}T\right)+\mathscr{L}M_{f},$
(22)
where $\overline{\rho c_{p}}$ and $\overline{k}$ are the solid-fraction-
weighted functions of ice and brine properties for heat capacity and thermal
conductivity, respectively. The latent heat is $\mathscr{L}$. We write a
conservation law for the concentration of brine, i.e.
$\left(1-\phi\right)\frac{\partial C}{\partial
t}+\boldsymbol{q}\cdot\boldsymbol{\nabla}C=\boldsymbol{\nabla}\cdot\left(\overline{D}\boldsymbol{\nabla}C\right)+\frac{\left(C-C_{s}\right)}{\rho}M_{f},$
(23)
where $\overline{D}$ is the phase-weighted salt diffusivity arising from
transport through the interstitial liquid.
The evolution of temperature and concentration are coupled in the mushy
region, as shown in figure 1 in the main text, where the ice and liquid phases
are at an equilibrium temperature given by the liquidus. This imposes a
constraint on the evolution equations (22) and (23) in mushy regions, where
the concentration and temperature are tied together by the liquidus
relationship $T=T_{L}(C)$. In idealized form, the liquidus relationship is
given as
$T_{L}=T_{m}-mC,$ (24)
where $T_{m}$ is the pure substance melting temperature and $m$ is the
liquidus slope. This constraint no longer applies when $T<T_{e}$, where
$M_{f}=0$, $\boldsymbol{q}=\boldsymbol{0}$, $\phi=1$, (22) becomes the heat
equation for thermal conduction and salt conservation (23) implies constant
bulk salinity.
## 4 Liquid volume and rates of production
In the main text, we show steady state temperature and porosity fields. We
treat the rate of shear heating as constant, given that the thermal diffusion
timescale is much longer than the tidal cycle (Nimmo and Gaidos, 2002; Nimmo
et al., 2007). Here, in figure 2(left), we show the approach to steady state
for the liquid volume in the mushy zone. Initially, there is no mushy zone and
therefore no liquid volume. Then a mushy zone forms and the liquid volume
increases rapidly. Due to the temperature field reaching a steady state and
the porosity-weakening of the fault friction saturates causing the heat flux
$F(1-\overline{\phi})$ to saturate in figure 2(left). In all simulations, the
local value of $\phi$ is used to calculate the shear heating rate. Here we
show the average value over fracture as an aggregate measure of the effect of
reducing the shear heating flux. From the curve of liquid volume with time, we
take the derivative to find the rate of liquid production, as shown in figure
2(right). This shows the same three regimes: no initial liquid volume, growth
of a mushy zone, and approach to steady state.
\begin{overpic}[width=138.76157pt]{pltU_tempcontour180.pdf}
\put(33.0,75.0){$F=180$}\end{overpic}\begin{overpic}[width=138.76157pt]{pltU_tempcontour240.pdf}
\put(33.0,75.0){$F=240$}\end{overpic}\begin{overpic}[width=138.76157pt]{pltU_tempcontour420.pdf}
\put(33.0,75.0){$F=420$}\end{overpic}\begin{overpic}[width=138.76157pt]{pltU_porosity180.pdf}
\end{overpic}\begin{overpic}[width=138.76157pt]{pltU_porosity240.pdf}
\end{overpic}\begin{overpic}[width=138.76157pt]{pltU_porosity420.pdf}
\end{overpic}
Figure 3: Temperature and porosity plots for three different values of the
nondimensional heat flux: (left) $F=180$, (middle) $F=240$, and (right)
$F=420$. Note that porosity is shown with different axis than the temperature
plots. Similar to the main text, we show vertical velocity arrows with
logarithmic magnitude with the largest arrows representing about 1 mm/year.
The observed geyser ejection rate for all four tiger stripes is $\sim$0.2 m3
s-1 (Hansen et al., 2006, 2008). Following a step change in heating and for
the range of heat fluxes we used in these simulations, we find that the
maximum rate of liquid volume production is greater than $0.2$ m3 s-1
($F=500$), which is sufficient to explain the geyser flux from all of the
tiger stripe fractures. Figure 2 shows that the melt rate is equal to the
observed geyser ejection rate initially and stays at a comparable magnitude
for $\sim$200 thousand years. At later times, the melt rate reduces and melt
volume saturates because a region of large porosity develops and weakens the
frictional heating on the crack. However, venting of liquid from the crack
will limit porosity growth, and enable sustained heating and melting rates.
Figure 4: Flow velocity in the mushy zone for $F=500$: (left) vertical
velocity, showing buoyant fluid rising along the crack and (right) horizontal
velocity, demonstrating the buoyant plume is stable at $\mathtt{Ra_{C}}=200$
(Boury et al., 2021) with periodic structures in the outer region of the mushy
zone.
The fact that peak melt rates exceeds the observed geyser flux is an exciting
demonstration of the viability of the shear heating model for the genesis of
plumes on Enceladus and other icy satellites. While the simple configuration
with steady heating doesn't account for the inherent complexities of the
transient evolution, our impulse-response calculations likely elucidate the
order of magnitude and timescale of heating. In figure 3, we show the steady
state temperature and porosity for three increasing nondimensional flux
values, $F=180$, $F=240$, and $F=420$. These figures mirror figures 2(right)
and 3(right) in the main text, where we show the temperature and porosity
structure for $F=500$. For $F=260$, the maximum rate of liquid volume
production is on the order of $0.05$ m3 s-1, which is less than rate required
to indefinitely produce plumes with the observed ejection rate, but the mushy
zone liquid reservoir could sustain geysers for tens of thousands of years.
This suggests that there is a parallel to the Enceladus heat signal, which may
be an indication of oscillations in thermal activity and hence plume activity
(Spencer and Nimmo, 2013; Nimmo et al., 2018). In other words, the liquid
volume production rate by shear heating may allow for a large reservoir to
develop periodically, leading to geysers that transiently exhausts the supply.
Lastly, in our simulations, we reduce the shear heating rate as liquid is
generated in the vicinity of the fracture. If we were to remove the liquid as
geyser material, however, the shear heating could stay high, leading to
additional liquid volume production. We currently do not remove the liquid
from our simulation domain and leave these simulations to future work.
Within the mushy zone, there is convection of the pore fluid and the fluid
near the fracture rises buoyantly, as shown in the main text and in figures 4
and 5. The intensity of convection is controlled by the thermal and
compositional mushy Rayleigh numbers,
$\displaystyle\mathtt{Ra_{T}}=\frac{\rho_{w}g\alpha\Delta
TK_{0}h}{\kappa_{o}\eta},$ (25)
$\displaystyle\mathtt{Ra_{C}}=\frac{\rho_{w}g\beta\Delta
CK_{0}h}{\kappa_{o}\eta},$ (26)
respectively. Boury et al. (2021) show that the rising plume along the
fracture is unstable, with the development of the instability here resulting
in rapid variation in horizontal velocity leading to numerical convergence
issues in SOFTBALL (figure 5). To find solutions that converge in a reasonable
amount of time for our grid resolution, we follow Boury et al. (2021) and
restrict the Rayleigh numbers to values below the instability threshold,
$\mathtt{Ra_{C}}\approx 205$ for our system and use $\mathtt{Ra_{C}}=200$ for
the results we present. There is a large uncertainty in the permeability
prefactor $K_{0}$ and the Rayleigh numbers that we present use a value of
$K_{0}$ that is on the lower end for sea ice simulations (Polashenski et al.,
2017; Buffo et al., 2018; Boury et al., 2021). The low permeability limits
convection and allows the simulations to converge numerically. In figure 5, we
compare three increasing Rayleigh numbers
$\mathtt{Ra_{C}}=200,~{}400,~{}\mbox{and}~{}800$, where the latter two
simulations do not converge. The higher Rayleigh numbers show that the flow-
and-phase-change instability leads to fine-scale structures that ultimately
approach the grid resolution, affecting the pattern of the horizontal velocity
field. The porosity field, however, is similar across the three Rayleigh
numbers, suggesting that the results for total volume and peak melt rate would
approximately hold as the Rayleigh number increased.
\begin{overpic}[width=138.76157pt]{pltU_horizvel500_Ra200.pdf}\put(33.0,76.0){$\mathtt{Ra_{C}}=200$}\end{overpic}\begin{overpic}[width=138.76157pt]{pltU_horizvel500_Ra400.pdf}\put(33.0,76.0){$\mathtt{Ra_{C}}=400$}\put(22.0,11.0){not
converged}\end{overpic}\begin{overpic}[width=138.76157pt]{pltU_horizvel500_Ra800.pdf}\put(33.0,76.0){$\mathtt{Ra_{C}}=800$}\put(22.0,11.0){not
converged}\end{overpic}
Figure 5: Increasing instability with stronger convection for $F=500$: (left)
$\mathtt{Ra_{C}}=200$ showing a stable interior plume and an onset of
instability in the region around the liquid inclusion. (middle)
$\mathtt{Ra_{C}}=400$ showing an onset of instability in the interior plume.
(right) $\mathtt{Ra_{C}}=800$ showing an unstable interior plume and a stable
outer region that becomes unstable at the edge of the mushy zone. The three
porosity plots, however, are similar.
Near the fracture, shear heating melts the ice and dissolves the salt.
Therefore, the pore fluid in this region has a higher salinity (figure 6) than
the surrounding shell. Here we plot the bulk salinity $C$, which is given as
$C=\phi C_{\ell}+(1-\phi)C_{s},$ (27)
where $\phi$ is the porosity, $C_{\ell}$ is the liquid salinity, and $C_{s}$
is the solid salinity. Starting from a step change in heating, the evolution
of the bulk salinity is shown in figure 7. First, shear heating warms the
salty ice above the eutectic temperature generating a permeable zone. Since
the brine salinity and temperature are connected through the liquidus
constraint within the mushy zone, colder and saltier brine along the top then
sinks and the resulting convective desalination reduces the bulk salinity when
compared to the background value. Next, the sinking brine cannot escape and
collects near the base of the porous inclusion along the fracture, generating
a salt-rich zone with porosity near unity. Convection continues within the
mushy zone. The convective desalination leaves behind regions of high bulk
salinity in the outer parts of the porous inclusion, that are trapped due to
the reduction of permeability in the neighbouring desalinated ice which now
has low porosity. Due to the partial melting within the mushy zone, the
chemical composition of the geyser ejecta sourced from the shear heating model
that we propose here will be different from geyser ejecta sourced from the
ocean. In our simulations, the salinity in the fluid around the crack is
higher than the background shell salinity. The shell likely froze from the
underlying ocean and will have a salinity structure that varies with depth
(Buffo et al., 2021).
Figure 6: Salinity for the mushy zone with $F=500$. (left) Bulk salinity $C$,
showing that the fluid within the interstices is saltier than the surrounding
shell, which froze from the underlying ocean. (right) Liquid salinity
$C_{\ell}$ showing the increase in salinity with depth in the mushy zone.
High-salinity channels form at an oblique angle, due to the competition
between melting and gravity drainage (Buffo et al., 2023).
## 5 Dike propagation
In our steady state simulations, a region of salty brine develops along the
fracture with a porosity approaching a value of $\phi\approx 1$. Here we do
not allow the liquid to escape as geysers, therefore the liquid volume can
build up within the mushy zone. The question then arises as to whether the
brine pocket can hydraulically fracture through the ice shell and into the
underlying ocean, thereby generating a dike that opens a surface-to-base
connection. We analyze this possibility through a scaling analysis, following
Lister (1990) and Kalousová et al. (2016). For $F=500$, we take the half width
of brine region to be $a\sim 200$ m, the breadth into the page as $w\sim 500$
km, and the height as $\ell\sim 4$ km, we have two options for the balance of
gravity. Either, viscous stresses resist propagation or the fracture toughness
$K_{I}\sim 100$ kPa m1/2 resists opening. Starting with the case of viscous
stress, we have that
$\phi\Delta\rho ga\ell w\sim\mu\frac{V}{a^{2}}a\ell w,$ (28)
where $\phi\sim 0.25$ is a representative average porosity in the brine
pocket; $\Delta\rho\sim 90$ kg m-3 is the density difference between the brine
and the surrounding ice; $\mu\sim 10^{14}$ Pa$\cdot$s is the brine-saturated
temperate ice viscosity; and $V$ is the rate of dike propagation. Canceling
terms, we find that the rate of propagation scales as
$V\sim\frac{\phi\Delta\rho ga^{2}}{\mu}.$ (29)
The time $t_{\textrm{dike}}$ for a dike to propagate a distance $\ell$, then
scales as
$t_{\textrm{dike}}\sim\frac{\ell}{V}\sim\frac{\mu\ell}{\phi\Delta\rho
ga^{2}}\sim 110~{}\mbox{kyr}.$ (30)
This timescale is on the same order of magnitude as the time to steady state
in our shear heating calculations, indicating that the brine pocket could form
and then propagate as a dike within that time frame, if the porosity grows to
order 1 without significant venting of the liquid from the crack.
Figure 7: Bulk salinity evolution with $F=500$, showing the drainage and brine
concentration along the fracture: (left) 541 kyr. (middle) 865 kyr. (right)
13,000 kyr.
If the balance is instead between gravity and the fracture toughness, we have
that
$\phi\Delta\rho ga\ell w\sim\frac{K_{I}}{\ell^{1/2}}aw,$ (31)
which determines whether the fracture will propagate or not. Simplifying and
plugging in numbers, we have that
$\phi\Delta\rho g\ell^{3/2}\sim 700~{}\mbox{kPa
m${}^{1/2}$}~{}~{}>~{}~{}K_{I}\sim 100~{}\mbox{kPa m${}^{1/2}$}.$ (32)
Thus, we find that $\phi\Delta\rho g\ell^{3/2}$ is greater than the ice
fracture toughness $K_{I}$ and much larger than values measured in brine-
saturated sea ice, e.g. $\sim 50$ kPa m1/2 (Schulson and Duval, 2009),
demonstrating that the ice will not provide significant resistance to
fracture.
In both cases, we see that dike propagation is possible if the porosity
reaches order 1 over an extended region, before the liquid is vented. The time
scale for diking is on the same order of magnitude as the for our simulations
to reach a steady state and the ice fracture toughness is not large enough to
prevent fracture. These scaling calculations spawn a few ideas. An internal
dike is interesting because there could be periodic brine pocket generation,
leading to opening a connection between surface and ocean, then Kite and Rubin
(2016)-style eruptions. Or the dike could drain all of the fluid internally,
leading to a stronger fault with larger shear heating, and a periodic mushy
zone source for the geysers. However, by not allowing for interstitial liquid
to escape, our estimates for the liquid brine volume are artificially high,
meaning that a region with porosity approaching 1 may never form within the
mushy zone. In this case, there would be no prospect for a dike and the
material generated would only emanate out as geysers. We leave additional
analysis exploring these ideas to future work.
parameters | |
---|---|---
specific heat of ice | $c_{i}$ | 2000 J kg-1 K-1
specific heat of ocean | $c_{o}$ | 4000 J kg-1 K-1
ice density | $\rho_{i}$ | 940 kg m-3
seawater density | $\rho_{w}$ | 1030 kg m-3
density coefficient for temperature | $\beta_{T}$ | 2.1$\times 10^{-4}$ kg m-3 K-1
density coefficient for salt | $\beta_{C}$ | 7.7$\times 10^{-4}$ kg m-3 ppt-1
thermal conductivity of ice | $k_{i}$ | 2.0 W m-1 K-1
thermal conductivity of ocean | $k_{o}$ | 0.6 W m-1 K-1
latent heat of fusion | $\mathscr{L}$ | 335000 J kg-1
water viscosity | $\eta$ | 2$\times 10^{-3}$ Pa s
salt diffusivity | $\kappa_{s}$ | 2$\times 10^{-9}$ m2 s-1
thermal diffusivity | $\kappa_{o}$ | 1.5$\times 10^{-7}$ m2 s-1
Stefan-Boltzmann constant | $\sigma$ | $5.67\times 10^{-8}$ W m-2 K-4
shell salt composition | $C_{i}$ | 35 ppt
eutectic composition | $C_{e}$ | 230 ppt
liquidus slope | $m$ | -0.0913 K ppt-1
eutectic temperature | $T_{e}$ | 252 K
surface temperature | $T_{s}$ | 75 K
gravity on Enceladus | $g$ | 0.113 m s-2
fracture depth | $h$ | 5 km
height scale | $H$ | 10 km
permeability prefactor | $K_{0}$ | 3.2$\times 10^{-13}$ m2
Hele-Shaw cell spacing | $d$ | 5$\times 10^{-5}$ m
time scale | $t_{c}$ | 2.2$\times 10^{7}$ years
slip velocity | $u$ | 3.4$\times$10-6 m s-1
temperature scale | $\Delta T$ | 18 K
emissivity | $\epsilon$ | 1
coefficient of friction | $\mu$ | 0.3
Stefan number | $\mathtt{St}$ | 4.7
Prandtl number | $\mathtt{Pr}$ | 12
Lewis number | $\mathtt{Le}$ | 73
mushy compositional Rayleigh number | $\mathtt{Ra_{C}}$ | 200
mushy thermal Rayleigh number | $\mathtt{Ra_{T}}$ | 2
reluctance | $\mathtt{R}$ | 9.6
shear heating | $F$ | 500
linearized radiation, nondimensional $r_{1}$ | $G$ | 0
linearized radiation, nondimensional $r_{2}$ | $b$ | 100
linearized radiation, nondimensional $T_{s}$ | $\theta_{s}$ | -10
Table 1: Parameters used in the SOFTBALL numerical simulations, grouped by
constants, variables, nondimensional parameters. Definitions follow Parkinson
et al. (2020). Values from Nimmo and Gaidos (2002), Nimmo et al. (2007),
Spencer and Nimmo (2013) as well as Buffo et al. (2021).
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|
# Gravitational wave fossils in nonlinear regime: halo tidal bias and
intrinsic alignments from gravitational wave separate universe simulations
Kazuyuki Akitsu<EMAIL_ADDRESS>School of Natural Sciences, Institute for
Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA Yin Li Center for
Computational Astrophysics & Center for Computational Mathematics, Flatiron
Institute, 162 5th Avenue, New York, NY 10010, USA Teppei Okumura Institute
of Astronomy and Astrophysics, Academia Sinica, No. 1, Sec. 4, Roosevelt Road,
Taipei 10617, Taiwan Kavli Institute for the Physics and Mathematics of the
Universe (WPI), UTIAS, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa,
Chiba 277-8583, Japan
###### Abstract
We investigate impacts of long-wavelength gravitational waves (GWs) on
nonlinear structure formation by utilizing the tidal separate universe
simulations. Based on the equivalence of a long-wavelength GW to a uniform
tidal field in a local frame, we provide a way to incorporate a long-
wavelength GW into the tidal separate universe simulation as an effective
anisotropic expansion. This methodology enables us to study effects of GWs on
large-scale structure efficiently. We measure the anisotropic imprint in the
local power spectrum from the tidal separate universe simulations with GWs,
which corresponds to the scalar-scalar-tensor bispectrum in squeezed limit or
the so-called power spectrum response to GWs. We also detect the halo tidal
bias induced by GWs from the response of the halo-matter cross-power spectrum
to GWs, as well as the linear shape bias (or the linear alignment coefficient)
induced by GWs from the one-point function of the halo ellipticity. In
contrast to the case of the tidal field induced by scalar perturbations, we
discover that the wavenumber dependence of the temporal evolution of GWs
naturally causes these biases to be scale-dependent. We also find that this
scale dependence is well approximated by the second-order density induced by
the coupling between scalar and tensor perturbation. This highlights that the
structure formation, especially the process to determine the halo shape, is
nonlocal in time. Our findings lay the foundation for predicting the impact of
GWs on large-scale structure.
## I Introduction
Gravitational waves (GWs) serve as vital means to observe the universe. In
particular, since long-wavelength GWs in the Gpc-Mpc range are unlikely to
originate from astrophysical events, they are thought to have a cosmological
origin (e.g., inflation Starobinsky (1979); Sato (1981); Guth (1981); Linde
(1982); Maggiore (2000)), making them an important probe in cosmology. In
spite of the various experiments trying to detect the primordial $B$-mode
signal in the polarization of the Cosmic Microwave Background (CMB) Komatsu
_et al._ (2011); Akrami _et al._ (2020); Ade _et al._ (2018); Adachi _et
al._ (2022); Abazajian _et al._ (2016), which is one of the most powerful
methods to hunt for long-wavelength cosmological GWs Zaldarriaga and Seljak
(1997); Kamionkowski _et al._ (1997), such GWs have not been observed yet.
Compared to the numerous studies for the effect of GWs on the perturbations of
the CMB, that on large-scale structure (LSS) of the universe has not received
as much attention. As large-scale structure is dominantly sourced by scalar
perturbations, there is a long history for the study on how the scalar
perturbations have shaped LSS, including the linear and nonlinear perturbation
theory Dodelson (2003); Suto and Sasaki (1991); Makino _et al._ (1992);
Bernardeau _et al._ (2002); Baumann _et al._ (2012) and the $N$-body
simulation Springel (2005); Potter _et al._ (2016); Springel _et al._
(2021); Angulo and Hahn (2021). On the other hand, there are fewer studies of
the effect of the tensor perturbations (GWs) on nonlinear structure formation,
as we list below.
There are two types of effects of GWs on LSS observables: the dynamical effect
and the projection effect. The former refers to the effect of GWs on nonlinear
structure formation itself, whereas the latter refers to the effect of GWs on
the light path emitted from distant galaxies to us. In other words, the
dynamical effect would be observed by a comoving observer in a local frame,
while the projection effect comes from the mapping of observables from the
galaxy’s local frame to our local frame (at the earth), which includes the
Sachs-Wolfe effect and the gravitational lensing effect caused by GWs. Both
effects have been studied by means of the perturbation theory; Refs. Kaiser
and Jaffe (1997); Dodelson _et al._ (2003); Yoo _et al._ (2009); Jeong _et
al._ (2012); Schmidt and Jeong (2012a); Jeong and Schmidt (2012) formulated
the projection effect on the galaxy clustering and the galaxy shape (shear) by
GWs, while Ref Schmidt and Jeong (2012a) pointed out that long-wavelength GWs
can also contribute the intrinsic alignments of galaxy shapes and Refs. Masui
and Pen (2010); Dai _et al._ (2013); Schmidt _et al._ (2014) computed the
second order matter density contrast induced by the coupling between scalar
perturbations and long-wavelength tensor perturbations (GWs). For the
projection effect the perturbative treatment would be adequate. For the
dynamical effect, however, the perturbation theory breaks down in nonlinear
scales and thus the nonlinear nature of LSS requires $N$-body simulations with
GWs to capture the fully nonlinear impact of GWs on structure formation.
Furthermore, given that the biased tracers of LSS such as halos are themselves
nonlinear objects, such simulations are necessary to understand their biases
to GWs even in the linear regime.
An $N$-body simulation with GWs is generally challenging because the usual
$N$-body simulations are based on Newtonian gravitational dynamics in an
expanding background whereas GWs are a purely general-relativistic effect. The
most straightforward way to introduce GWs into $N$-body simulations is to
develop a general-relativistic cosmological simulation Adamek _et al._ (2013,
2016a, 2016b), though Refs. Adamek _et al._ (2016a, b) only considered the
second-order (induced) tensor perturbations by scalar perturbations.111
Technically the induced tensor perturbations computed in Refs. Adamek _et
al._ (2016a, b) involve non-GWs contributions, which are tensor modes but not
propagating waves. See e.g., Ref. Domènech and Sasaki (2021) for detailed
discussion on this issue. Also, since GWs are much smaller than scalar
perturbations, it is difficult to single out the effect of GWs on structure
formation from that of the scalar perturbations in this sort of simulations.
In this paper, we utilize a separate universe approach to circumvent this
issue on $N$-body simulations with GWs. In the separate universe approach, the
influence of a long-wavelength perturbation is absorbed into the cosmic
expansion observed in the local frame, thereby the local expansion becomes
different from the global one. Accordingly, nonlinear structure formation in
the local region responds to this difference in the background expansions.
Using this technique, the response to the long-wavelength perturbation can be
accurately measured in $N$-body simulations Sirko (2005); Gnedin _et al._
(2011); Baldauf _et al._ (2011); Li _et al._ (2014); Wagner _et al._
(2015); Lazeyras _et al._ (2016); Li _et al._ (2016); Baldauf _et al._
(2016). Recently, the anisotropic extension of the separate universe
simulation was developed in Refs. Stücker _et al._ (2021); Masaki _et al._
(2020); Akitsu _et al._ (2021a). They considered a long-wavelength tidal
perturbation sourced by the long-wavelength scalar tides
$\propto\left(\partial_{i}\partial_{j}-\frac{1}{3}\delta^{\textsc{k}}_{ij}\right)\Phi$
Dai _et al._ (2015a); Akitsu _et al._ (2017); Akitsu and Takada (2018);
Akitsu _et al._ (2019); Li _et al._ (2018); Taruya and Akitsu (2021), and
the perturbation was absorbed into the local background, making the local
cosmic expansion anisotropic.
Taking advantage of the equivalence of a long-wavelength GW to a long-
wavelength tidal field in a local region, we apply this tidal separate
universe simulation to measure the impact of GWs on structure formation. In
contrast to the scalar case, the time evolution of GWs is scale-dependent (or
wavenumber-dependent). As a result, when mimicking the long-wavelength GW as
the local anisotropic expansion, the anisotropic expansion rate would be
different depending on the wavenumber of GWs, and so is the response of large-
scale structure, as considered for isotropic scale-dependent long-wavelength
perturbation in Refs. Hu _et al._ (2016); Chiang _et al._ (2016, 2018). The
purpose of this paper is to generalize the tidal separate universe simulation
for GWs in order to study the scale-dependent responses for long-wavelength
GWs of different wavelengths.
The remainder of this paper provides a way to implement long-wavelength GWs
into the tidal separate universe simulation and presents imprints of GWs on
large-scale structure measured from newly developed simulations. After giving
a brief review about the local coordinates in the presence of GWs and
perturbative results in Sec. II, we construct the tidal separate universe with
GWs in Sec. III. In Sec. IV, we measure the power spectrum responses for
matter auto-, halo-matter cross-, and halo auto-power spectra, which is
related to the scalar-scalar-tensor bispectrum in squeezed limit. Sec. V and
Sec. VI are devoted to the measurements of the halo tidal bias and linear
alignment coefficient (or the linear shape bias) from GWs, respectively. We
discuss possible observables in large-scale structure to probe GWs in Sec.
VII. Throughout this paper we adopt cosmological parameters consistent with
Planck result: $\Omega_{\rm r0}=4.1577\times 10^{-5}$, $\Omega_{\rm
m0}=0.3089$, $\Omega_{\Lambda 0}=0.6911$, $H_{0}=67.74$ Ade _et al._ (2016).
## II Local frame and the perturbative results
Here we first introduce the local coordinates in the presence of long-
wavelength GWs. We then briefly summarize the derivation of the second-order
density perturbations induced by the coupling between scalar and tensor
perturbations (GWs), following Ref. Schmidt _et al._ (2014). We employ the
Lagrangian perturbation formalism, which can be straightforwardly used to
construct the tidal separate universe in the next section.
### II.1 Long-wavelength gravitational waves in the conformal Fermi
coordinates
Figure 1: Left panels from top to bottom: Transfer function of GWs
$\mathcal{T}(a;k_{L})$, growth coefficient $\alpha(a;k_{L})$ and dilation
coefficient $\beta(a;k_{L})$ in Eq. (29) as a function of scale factor $a$ for
various wavenumbers $k_{L}$. Right panels from top to bottom: Same as the left
panels but as a function of wavenumber of GWs, $k_{L}$, at various redshifts.
Note that the functional shape of $\beta(a;k_{L})$ is equivalent to that of
$\mathcal{T}(a;k_{L})$ because
$\beta(a;k_{L})=-\frac{1}{2}\left[\mathcal{T}(a;k_{L})-1\right]$.
In the cosmological context, GWs are defined as the trace-free transverse
components in the perturbed FLRW metric,
$\displaystyle
ds^{2}=a^{2}[-d\eta^{2}+(\delta^{\textsc{k}}_{ij}+h_{ij})dx^{i}dx^{j}],$ (1)
where $a$ is the scale factor, $\eta$ is the conformal time,
$\delta^{\textsc{k}}_{ij}$ is Kronecker’s delta, and $h_{ij}$ is GWs
satisfying $\tensor{h}{{}^{i}_{i}}=0$ and $\partial^{i}h_{ij}=0$. We use
$k_{L}$ to denote the wavenumber of GWs in what follows since we focus on the
long-wavelength GWs in this paper. We introduce the transfer function of GWs,
$\mathcal{T}(\eta;k_{L})$, through
$\displaystyle h_{ij}(\eta;{\bf k}_{L})=\mathcal{T}(\eta;k_{L})h^{\rm
ini}_{ij}({\bf k}_{L}),$ (2)
where $h_{ij}^{\rm ini}(k_{L})$ denotes the primordial value of GWs, i.e.,
$h_{ij}^{\rm ini}(k_{L})=h_{ij}(0;k_{L})$. The transfer function
$\mathcal{T}(\eta;k_{L})$ obeys
$\displaystyle\mathcal{T}^{\prime\prime}(\eta;k_{L})+2\mathcal{H}\mathcal{T}^{\prime}(\eta;k_{L})+k_{L}^{2}\mathcal{T}(\eta;k_{L})=0,$
(3)
with ${}^{\prime}=\differential/\differential\eta$, and $\mathcal{H}=aH$ being
the conformal Hubble paramter. The top-left panel of Fig. 1 shows
$\mathcal{T}(\eta;k_{L})$ as a function of the scale factor for various
wavenumbers of GWs. Similarly, the top-right panel shows that as a function of
the wavenumbers of GWs at various redshifts. For any $k_{L}$,
$\mathcal{T}(\eta;k_{L})$ remains unity when $k_{L}\eta\ll 1$, which means
that GWs are frozen before they enter the horizon.
For later convenience, we expand GWs by the helicity basis:
$\displaystyle h_{ij}(\eta;{\bf k}_{L})=\sum_{\lambda=\pm
2}h_{(\lambda)}(\eta;{\bf k}_{L})e^{(\lambda)}_{ij}(\hat{k}_{L}),$ (4)
where $e_{ij}^{(\pm 2)}\equiv e_{i}^{(\pm)}e_{j}^{(\pm)}$ and ${\bf
e}^{(\pm)}\equiv({\bf e}_{1}\mp i{\bf e}_{2})/\sqrt{2}$ with $\\{{\bf
k}_{L},{\bf e}_{1},{\bf e}_{2}\\}$ being an orthonormal set. The helicity
basis satisfies $e^{(\lambda)}_{ij}e^{ij}_{(-\lambda)}=1$,
$e^{(\lambda)}_{ij}e^{ij}_{(\lambda)}=0$, and
$h_{(\lambda)}=e^{ij}_{(-\lambda)}h_{ij}$. The power spectrum of GWs for each
helicity mode is defined as
$\displaystyle\langle h_{(\lambda)}({\bf k}_{L};\eta)h^{*}_{(\lambda)}({\bf
k}^{\prime}_{L};\eta^{\prime})\rangle=(2\pi)^{3}\delta_{\rm D}^{(3)}({\bf
k}_{L}-{\bf k}^{\prime}_{L})P_{h_{(\lambda)}}(k_{L};\eta,\eta^{\prime}),$ (5)
which is related to the primordial power spectrum $P_{h_{(\lambda)}}(k_{L})$
through
$\displaystyle
P_{h_{(\lambda)}}(k_{L};\eta,\eta^{\prime})=\mathcal{T}(\eta;k_{L})\mathcal{T}(\eta^{\prime};k_{L})P_{h_{(\lambda)}}(k_{L}).$
(6)
For unpolarized GWs, the two power spectra has equal power, i.e.,
$P_{h_{(+2)}}(k)=P_{h_{(-2)}}(k)\equiv P_{h}(k)/2$ with $P_{h}(k)$ being the
total power spectrum of GWs. For chiral GWs, the chiral parameter $\chi(k)$
defined as
$\displaystyle\chi(k)\equiv\frac{P_{(+2)}(k)-P_{(-2)}(k)}{P_{h}(k)},$ (7)
measures the degree of the parity-breaking in GWs. The total power spectrum of
GWs is often characterized via
$\displaystyle\frac{k^{3}P_{h}(k)}{2\pi^{2}}\equiv
rA_{s}\left(\frac{k}{k_{*}}\right)^{n_{T}}$ (8)
with $r$ being the scalar-tensor ratio, $A_{s}$ being the amplitude for the
primordial curvature perturbations at the pivot scale $k_{*}$, and $n_{T}$
being the tensor tilt.
To investigate the physical effects of long-wavelength GWs on scalar
perturbations at smaller scales, consider a local region centered at the
timelike geodesic of a comoving observer. In this local patch we can construct
the so-called conformal Fermi coordinates (CFC), which is an extension of the
Fermi normal coordinates Manasse and Misner (1963), developed in Refs. Schmidt
_et al._ (2014); Dai _et al._ (2015b). The metric of this coordinates,
$g_{\mu\nu}^{F}$, takes the form of the FLRW metric along the central geodesic
with leading-order corrections of ${\cal O}(x^{2}_{F})$. We are interested in
the interaction between long-wavelength GWs and non-relativistic matter. In
this case $g_{00}^{F}$ encodes all the relevant impact from the long-
wavelength perturbations because the dynamics of non-relativistic matter is
solely determined by the usual Newtonian potential in $g_{00}^{F}$. One can
show that given the global coordinates Eq. (1), $g_{00}^{F}$ is computed as
$\displaystyle g_{00}^{F}=-a^{2}\left[1+\tau_{ij}x_{F}^{i}x_{F}^{j}\right],$
(9)
with
$\displaystyle\tau_{ij}(\eta;k_{L})=-\frac{1}{2}\left[a^{-1}\left(ah_{ij}^{\prime}\right)^{\prime}\right]=-\frac{1}{2}\left(h_{ij}^{\prime\prime}+\mathcal{H}h_{ij}^{\prime}\right)\equiv
T(\eta;k_{L})h^{\rm ini}_{ij}$ (10)
representing the effective tidal field induced by the long-wavelength GWs in
the local region. The derivation is summarized in App. A. It is worth noting
that $\tau_{ij}$ takes effect only after GWs cross the horizon as implied by
the time derivative on $h_{ij}$. Note that $x_{F}^{i}$ corresponds to the
comoving distance, unlike the physical (proper) distance in the Fermi normal
coordinates. In what follows, we work in this frame and drop the subscript $F$
in $x_{F}^{i}$, namely, denote $x^{i}$ as $x_{F}^{i}$.
### II.2 Second order density induced by the interaction between GWs and
scalar perturbations
The equation of motion of a matter particle in the local frame is 222Strictly
speaking, this is justified by considering the geodesic equation in the local
frame.
$\displaystyle\frac{\differential^{2}r_{i}}{\differential
t^{2}}=\frac{1}{a^{2}}\left[\frac{\differential^{2}}{\differential\eta^{2}}-\mathcal{H}\frac{\differential}{\differential\eta}\right]r_{i}=-\frac{\partial}{\partial
r_{i}}\left(\Phi_{\rm iso}+\phi\right),$ (11)
where $r_{i}=ax_{i}$ and we split the gravitational potential into a
background potential $\Phi_{\rm iso}$ and a peculiar potential $\phi$. The
subscript “iso” in $\Phi$ stands for the potential sourced only by the usual
isotropic background,
$\displaystyle\Phi_{\rm iso}\equiv\frac{2}{3}\pi G\bar{\rho}_{\rm
m}r^{2}-\frac{\Lambda}{6}r^{2},$ (12)
where $\bar{\rho}_{\rm m}$ is the mean density of matter. The peculiar
potential $\phi$ includes the potential sourced by local inhomogeneities
$\phi_{s}$ as well as the effective tidal field sourced by the long-wave
gravitational wave, $\phi=\phi_{s}+\frac{1}{2}\tau_{ij}x^{i}x^{j}$. In the
next sections, the effective tidal potential is absorbed into the back ground
potential, $\Phi=\Phi_{\rm iso}+\frac{1}{2}\tau_{ij}x^{i}x^{j}$ (see Eq. (33)
below). The potential sourced by local inhomogeneities, $\phi_{s}$, satisfies
the Poisson equation
$\displaystyle\nabla_{r}^{2}\phi_{s}=4\pi G\bar{\rho}_{\rm
m}\delta=\frac{3}{2}\frac{\Omega_{\rm m}(\eta)\mathcal{H}^{2}}{a^{2}}\delta,$
(13)
where $\delta=\rho/\bar{\rho}_{\mathrm{m}}-1$ denotes the overdensity field.
We can also decompose the left-hand side of Eq. (11) into the background and
peculiar parts, resulting in
$\displaystyle\mathcal{H}^{\prime}x_{i}$
$\displaystyle=-\frac{\partial}{\partial x_{i}}\Phi_{\rm iso},$ (14)
$\displaystyle x_{i}^{\prime\prime}+\mathcal{H}x_{i}^{\prime}$
$\displaystyle=-\frac{\partial}{\partial
x_{i}}\left(\phi_{s}+\frac{1}{2}\tau_{\ell m}x^{\ell}x^{m}\right).$ (15)
One can verify that Eq. (14) with Eq. (12) gives the usual Friedmann equation.
Eq. (15) can be regarded as the evolution equation for the displacement field
$\Psi_{i}$, which relates Lagrangian position $q_{i}$ to Eulerian position
$x_{i}$ via $x_{i}=q_{i}+\Psi_{i}$. Before shell-crossing the Jacobian
determinant of this mapping gives the overdensity as
$\displaystyle\delta=\left|\frac{\partial x_{i}}{\partial
q_{j}}\right|^{-1}-1=\left|\delta^{\textsc{k}}_{ij}+\frac{\partial\Psi_{i}}{\partial
q_{j}}\right|^{-1}-1.$ (16)
At linear order this reduces to $\delta^{(1)}=-\partial\Psi^{(1)}_{i}/\partial
q_{i}$. We split the linear displacement into the one sourced by pure scalar
contributions and that sourced by $\tau_{ij}$, each of which satisfies
$\displaystyle\Psi^{(1)\prime\prime}_{s,i}+\mathcal{H}\Psi^{(1)\prime}_{s,i}$
$\displaystyle=-\frac{3}{2}\Omega_{m}(\eta)\mathcal{H}^{2}\frac{\partial_{q}^{i}}{\partial^{2}_{q}}\delta^{(1)},$
(17)
$\displaystyle\Psi^{(1)\prime\prime}_{t,i}+\mathcal{H}\Psi^{(1)\prime}_{t,i}$
$\displaystyle=-\frac{1}{2}\partial_{q}^{i}\left[\tau_{kl}q^{k}q^{l}\right],$
(18)
where $\Psi^{(1)}_{s,i}$ and $\Psi^{(1)}_{t,i}$ denote the linear displacement
caused by the scalar and tensor perturbations, respectively, and we have used
Eq. (13) and the fact that $x_{i}$ can be replaced by $q_{i}$ at linear order.
The non-decaying solutions for $\Psi^{(1)}_{s,i}$ and $\Psi^{(1)}_{t,i}$ are
$\displaystyle\Psi^{(1)}_{s,i}(\eta)$
$\displaystyle=-\frac{D(\eta)}{D(\eta_{0})}\frac{\partial_{q}^{i}}{\partial_{q}^{2}}\delta^{(1)}(\eta_{0}),$
(19) $\displaystyle\Psi^{(1)}_{t,i}(\eta;k_{L})$
$\displaystyle=\frac{1}{2}\left[\mathcal{T}(\eta;k_{L})-1\right]h^{\rm
ini}_{ij}(k_{L})q^{j}\equiv-\beta(\eta;k_{L})h_{ij}^{\rm ini}(k_{L})q^{j},$
(20)
where we have used the boundary condition $\lim_{\eta\to
0}\mathcal{T}(\eta;k_{L})=1$ and have introduced the quantity
$\beta(\eta;k_{L})$ to characterize the linear displacement due to the tensor
perturbations, and $D(\eta)$ represents the linear growth function, which
follows
$\displaystyle
D^{\prime\prime}(\eta)+\mathcal{H}D^{\prime}(\eta)-\frac{3}{2}\Omega_{\rm
m}(\eta)\mathcal{H}^{2}D(\eta)=0.$ (21)
From this result one can confirm that GWs do not induce the linear density:
$\delta^{(1)}_{t}=-\partial\Psi^{(1)}_{t,i}/\partial q_{i}\propto h_{i}^{\
i}=0$, as expected.
At second order, however, GWs do affect the density field as well as the
displacement field. Expanding Eq. (16) up to second order leads to
$\displaystyle\delta^{(1)}({\bf x}({\bf q}))+\delta^{(2)}_{st}({\bf x}({\bf
q}))=$ $\displaystyle\delta^{(1)}({\bf
q})-\left.\frac{\partial\Psi^{(2)}_{st,i}}{\partial q_{i}}\right|_{{\bf
q}}+\left.\frac{\partial\Psi^{(1)}_{s,i}}{\partial
q_{j}}\frac{\partial\Psi^{(1)}_{t,j}}{\partial q_{i}}\right|_{{\bf q}}$
$\displaystyle=$ $\displaystyle\delta^{(1)}({\bf x})-\Psi_{t,i}^{(1)}({\bf
x})\partial_{x}^{i}\delta^{(1)}({\bf
x})-\left.\frac{\partial\Psi^{(2)}_{st,i}}{\partial q_{i}}\right|_{{\bf
q}={\bf x}}+\left.\frac{\partial\Psi^{(1)}_{s,i}}{\partial
q_{j}}\frac{\partial\Psi^{(1)}_{t,j}}{\partial q_{i}}\right|_{{\bf q}={\bf
x}},$ (22)
where we focus on the second-order density arising from the coupling between
the scalar and tensor modes, and $\delta_{st,i}^{(2)}$ and $\Psi_{st,i}^{(2)}$
respectively denote the second-order density and displacement induced by the
coupling. Namely, we omit the second-order contributions from the auto-
coupling between scalars (irrelevant) or tensors (subdominant). In deriving
the first equality above, we have used the fact that $\Psi^{(1)}_{t,i}$ is
divergence free. Taking the divergence of Eq. (15) with respect to ${\bf q}$
we obtain
$\displaystyle\psi_{st}^{(2)\prime\prime}+{\cal
H}\psi_{st}^{(2)\prime}=-\nabla_{x}^{2}\phi_{s}-\frac{\partial\Psi^{(1)}_{t,j}}{\partial
q_{i}}\frac{\partial^{2}\phi_{s}}{\partial x_{i}\partial
x_{j}}-\frac{\partial\Psi^{(1)}_{s,j}}{\partial q_{i}}\tau_{ij},$ (23)
where we have defined $\psi^{(2)}_{st}\equiv\partial\Psi_{st,i}^{(2)}/\partial
q_{i}$ and used
$\partial_{q}^{i}=\partial_{x}^{i}+\partial_{q}^{i}\Psi_{j}\partial_{q}^{j}\simeq\partial_{x}^{i}+\partial_{q}^{i}\Psi_{j}\partial_{x}^{j}$.
Rewriting $\phi_{s}$ in terms of $\delta$ by using Eq. (13) and using Eq. (22)
yield
$\displaystyle\psi_{st}^{(2)\prime\prime}+\mathcal{H}\psi_{st}^{(2)\prime}=\frac{3}{2}\Omega_{\rm
m}\mathcal{H}^{2}\left[-\frac{\partial\Psi^{(2)}_{st,i}}{\partial
q_{i}}+\frac{\partial\Psi^{(1)}_{s,i}}{\partial
q_{j}}\frac{\partial\Psi^{(1)}_{t,j}}{\partial
q_{i}}\right]-\frac{3}{2}\Omega_{\rm
m}\mathcal{H}^{2}\frac{\partial\Psi^{(1)}_{t,j}}{\partial
q_{i}}\frac{\partial\Psi^{(1)}_{s,i}}{\partial
q_{j}}-\frac{\partial\Psi^{(1)}_{s,j}}{\partial q_{i}}\tau_{ij}.$ (24)
Finally the equation for $\psi_{st}^{(2)}$ is found to be
$\displaystyle\psi_{st}^{(2)\prime\prime}+\mathcal{H}\psi_{st}^{(2)\prime}-\frac{3}{2}\Omega_{\rm
m}\mathcal{H}^{2}\psi_{st}^{(2)}=\left(\frac{\partial^{i}_{q}\partial^{j}_{q}}{\partial_{q}^{2}}\delta^{(1)}(\eta)\right)\tau_{ij}.$
(25)
We can write the solution as
$\displaystyle\psi_{st}^{(2)}({\bf
q},\eta;k_{L})=D_{st}^{(2)}(\eta;k_{L})\left(\frac{\partial^{i}_{q}\partial^{j}_{q}}{\partial_{q}^{2}}\delta^{(1)}({\bf
q},\eta_{0})\right)h_{ij}^{\rm ini}(k_{L}),$ (26)
where the time-dependent part of $\psi_{st}^{(2)}({\bf q},\eta;k_{L})$, which
we write $D_{st}^{(2)}(\eta;k_{L})$, satisfies
$\displaystyle
D_{st}^{(2)\prime\prime}(\eta;k_{L})+\mathcal{H}D_{st}^{(2)\prime}(\eta;k_{L})-\frac{3}{2}\Omega_{\rm
m}(\eta)\mathcal{H}^{2}D_{st}^{(2)}(\eta;k_{L})=\frac{D(\eta)}{D(\eta_{0})}\cdot\left[-\frac{1}{2a}\left(a\mathcal{T}^{\prime}(\eta;k_{L})\right)^{\prime}\right].$
(27)
We can solve this equation numerically given the cosmological parameters but
here we derive the analytic solution assuming the matter domination for the
later convenience. In the matter-dominated era where we have $\Omega_{\rm
m}=1$, $\mathcal{H}=2/\eta$ and $D(\eta)\propto a\propto\eta^{2}$, we can
write the solution using Green’s function:
$\displaystyle D_{st}^{(2)}(\eta;k_{L})=$
$\displaystyle\frac{1}{D(\eta_{0})}\int_{0}^{\eta}\differential\tilde{\eta}~{}\frac{1}{5}\left[\frac{\eta^{2}}{\tilde{\eta}}-\frac{\tilde{\eta}^{4}}{\eta^{3}}\right]D(\tilde{\eta})\left[-\frac{1}{2a}\left[a\mathcal{T}^{\prime}(\tilde{\eta};k_{L})\right]^{\prime}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{5}\frac{D(\eta)}{D(\eta_{0})}\left[\beta(\eta)+4\int_{0}^{\eta}\differential\tilde{\eta}\left(\frac{\tilde{\eta}}{\eta}\right)^{5}\beta^{\prime}(\tilde{\eta};k_{L})\right].$
(28)
Finally we find the second order density induced by the interaction between
the scalar and tensor perturbation as
$\displaystyle\delta^{(2)}_{st}({\bf x},\eta;k_{L})=$ $\displaystyle
h_{ij}^{\rm ini}\left[\beta(\eta;k_{L})x^{j}\partial_{x}^{i}\delta^{(1)}({\bf
x},\eta)-D_{st}^{(2)}(\eta;k_{L})\left(\frac{\partial^{i}_{x}\partial^{j}_{x}}{\partial_{x}^{2}}\delta^{(1)}({\bf
x},\eta_{0})\right)+\beta(\eta;k_{L})\left(\frac{\partial^{i}_{x}\partial^{j}_{x}}{\partial_{x}^{2}}\delta^{(1)}({\bf
x},\eta)\right)\right]$ $\displaystyle=$ $\displaystyle h_{ij}^{\rm
ini}\left[\alpha(\eta;k_{L})\frac{\partial^{i}_{x}\partial^{j}_{x}}{\partial_{x}^{2}}+\beta(\eta;k_{L})x^{j}\partial_{x}^{i}\right]\delta^{(1)}({\bf
x},\eta),$ (29)
where $\beta$ is defined in Eq. (20) and $\alpha$ is defined as
$\displaystyle\alpha(\eta;k_{L})\equiv$
$\displaystyle-\frac{D(\eta_{0})}{D(\eta)}D_{st}^{(2)}(\eta;k_{L})+\beta(\eta;k_{L})$
$\displaystyle=$
$\displaystyle\frac{4}{5}\left(\beta(\eta;k_{L})-\int_{0}^{\eta}\differential\tilde{\eta}\left(\frac{\tilde{\eta}}{\eta}\right)^{5}\beta^{\prime}(\tilde{\eta};k_{L})\right).$
(30)
Eq. (29) holds not only during the matter domination but also the entire
history of the universe as long as we solve Eq. (27) numerically, whereas the
second equality of Eq. (30) holds only for the matter domination. The first
term in the last line $(\propto\alpha)$ represents the changes in the short-
mode amplitude, called the growth effect, by the coupling between the scalar
tidal field and long-wavelength GWs. The second term in the last line
$(\propto\beta)$ represents the coordinate shift induced by the long-
wavelength GWs, known as the dilation effect. The power spectrum of the
density field in the presence of the long-wavelength GWs then becomes
$\displaystyle P_{\rm mm}({\bf k_{S}},\eta|h_{ij}(k_{L}))=P_{\rm
lin}(k_{S})\left[1+\hat{k}_{S}^{i}\hat{k}_{S}^{j}h_{ij}^{\rm
ini}(k_{L})\left(2\alpha(\eta;k_{L})-\beta(\eta;k_{L})\frac{\partial\ln{P_{\rm
lin}(k_{S},\eta)}}{\partial\ln{k_{S}}}\right)\right].$ (31)
We show $\mathcal{T}(k_{L};\eta)$, $\alpha(k_{L};\eta)$ and
$\beta(k_{L};\eta)$ as a function of the scale factor for various wavenumbers
of GWs (the left panels) and as a function of the wavemunber of GWs at various
redshifts at the right and left panels of Fig.1, respectively. To plot these
functions, we numerically integrated Eqs. (3) and (27) without assuming the
matter domination. In the limit of $k_{L}\eta\to 0$, it is obvious that there
is no physical effect from GWs, i.e., $\alpha(\eta;k_{L})\to 0$ and
$\beta(\eta;k_{L})\to 0$, since GWs are frozen on the super-horizon scales. On
the other hand, taking the limit $k_{L}\eta\gg 1$ is more interesting because
in this limit $\mathcal{T}(\eta;k_{L})\to 0$ but $\alpha(\eta;k_{L})$ and
$\beta(\eta;k_{L})$ do not vanish. In other words, even long after GWs have
decayed away, their impact on the growth and displacement remains, which is
sometimes called the “fossil” effect Masui and Pen (2010).
## III Tidal separate universe with gravitational waves
In this section, we describe how to incorporate long-wavelength GWs into the
simulation background with the help of the tidal separate universe simulation
technique developed in Ref. Akitsu _et al._ (2021a) (see also Refs. Stücker
_et al._ (2021); Masaki _et al._ (2020)). We focus on differences arising
from GWs and refer the readers to Ref. Akitsu _et al._ (2021a) about the
details of the implementation. In this section we do not employ Einstein’s
summation convention to avoid confusions.
### III.1 Anisotropic background
In the tidal separate universe simulation we introduce anisotropic scale
factors by absorbing the long-wavelength tidal perturbations into the
background. In general, reflecting that the tidal perturbations are expressed
by the $3\times 3$ symmetric matrix $\tau_{ij}$ the anisotropic scale factors
are also written by the $3\times 3$ symmetric matrix $a_{ij}$, which relates
the physical coordinate $r_{i}$ to the comoving coordinate $x_{i}$ as
$r_{i}=\sum_{j}a_{ij}x_{j}$. However, we can always rotate the simulation
coordinates to align with the eigenvectors of $\tau_{ij}$, leaving only the
diagonal components non-zero so that
$\tau_{ij}=\tau_{i}\delta^{\textsc{k}}_{ij}$ and
$a_{ij}=a_{i}\delta^{\textsc{k}}_{ij}$. Now we have different scale factors
for each axis and characterize these differences by $\Delta_{i}$ defined via
$\displaystyle a_{i}=a(1+\Delta_{i}),$ (32)
with $a$ being the global scale factor. While the equation of motion remains
the same as in Sec. II.2, now we want to absorb the long-wavelength effective
tidal potential induced by GWs into the background potential so that
$\displaystyle\Phi=\Phi_{\rm
iso}+\frac{1}{2}\sum_{i}\tau_{i}x_{i}^{2}=\frac{2}{3}\pi G\bar{\rho}_{\rm
m}r^{2}-\frac{\Lambda}{6}r^{2}+\frac{1}{2}\sum_{i}\tau_{i}x_{i}^{2}.$ (33)
The background and peculiar equations in an anisotropic background become
$\displaystyle\frac{1}{a^{2}}\left[a_{i}^{\prime\prime}-\mathcal{H}a_{i}^{\prime}\right]x_{i}=-\frac{1}{a_{i}}\frac{\partial}{\partial
x_{i}}\Phi,$ (34)
$\displaystyle\frac{1}{a^{2}}\left[a_{i}x_{i}^{\prime\prime}+2a_{i}^{\prime}x_{i}^{\prime}-\mathcal{H}a_{i}x_{i}\right]=-\frac{1}{a_{i}}\frac{\partial}{\partial
x_{i}}\phi_{s}.$ (35)
Subtracting the isotropic background, which is determined by Eq. (14), from
Eq. (34) and linearizing it in $\Delta_{i}$ we find
$\displaystyle\Delta^{\prime\prime}_{i}(\eta;k_{L})+\mathcal{H}\Delta_{i}^{\prime}(\eta;k_{L})=\frac{1}{2}a^{-1}(\eta)\left[a(\eta)h_{i}^{\prime}(\eta;k_{L})\right]^{\prime}.$
(36)
Note that anisotropic scale factors depend on the wavenumber of GWs as a
consequence of the wavenumber dependence of the transfer function of GWs,
$\mathcal{T}(\eta;k_{L})$, while they do not for the case of scalar large-
scale tidal field where the linear growth of scalar perturbations is
independent of their wavenumbers, as characterized by $D(\eta)$. Note also
that the source term for the anisotrpic scale factors on the right-hand side
is non-zero only when $h_{i}^{\prime}(\eta;k_{L})$ does not vanish; in other
words long-wavelength GWs induces the anisotropic scale factors only after GWs
enters the horizon as expected. In fact, integrating Eq. (36) twice yields
$\displaystyle\Delta_{i}(\eta;k_{L})=\frac{h^{\rm
ini}_{i}(k_{L})}{2}\left[\mathcal{T}(\eta;k_{L})-1\right]=-h^{\rm
ini}_{i}(k_{L})\beta(\eta;k_{L}),$ (37)
which goes to zero when $\eta\to 0$ because $\lim_{\eta\to
0}\mathcal{T}(\eta;k_{L})=1$.
The appearance of the function $\beta(\eta;k_{L})$ defined in Eq. (20) is
expected and this result can be understood in a more intuitive way. What is
specifically done in the separate universe construction is to absorb the
displacement caused by the long-wavelength perturbation into the background
expansion while keeping the physical distance unchanged. In other words, we
introduce the local scale factor $a_{i}$ to satisfy
$\displaystyle a_{i}x_{i}=a(x_{i}+\Psi_{i}^{\rm long}),$ (38)
which implies $\Delta_{i}x_{i}=\Psi_{i}^{\rm long}$. Given the displacement
caused by long-wavelength GWs in Eq. (20), this immediately leads to
$\Delta_{i}(\eta;k_{L})=-h_{i}^{\rm ini}(k_{L})\beta(\eta;k_{L})$.
We note that this matching only works for the non-relativistic matter. In
other words, the effect of long-wavelength GWs can be captured by the
anisotropic expansion only when we focus on non-relativistic particles that do
not care about $g_{0i}$ and $g_{ij}$ components in the metric. For example,
the method presented here is not useful in order to study the impact of long-
wavelentgh GWs on the radiation perturbations. However, this treatment is
sufficient to study the influence on dark matter particles that we are
interested in and consistent with the usual Newtonian $N$-body method.
### III.2 Initial conditions
The background anisotropy induces a correction to the 2LPT solution in the
isotropic background as discussed in Ref.Akitsu _et al._ (2021a). The
correction depends on $\Delta_{i}(\eta)$, which is different for the scalar
tidal field and GWs. Here we derive this correction induced by the background
anisotropy governed by long-wavelength GWs.
The equation for the displacement in the anisotropic background can be
obtained by combining Eq. (35) with Eq. (13) as
$\displaystyle\sum_{ij}\left|\frac{\partial{\bf x}}{\partial{\bf
q}}\right|\left[\delta_{ij}+\Psi_{i,j}\right]^{-1}\left[\Psi_{i,j}^{\prime\prime}+(\mathcal{H}+2\Delta_{i}^{\prime})\Psi_{i,j}^{\prime}\right]=\frac{3}{2}\Omega_{\rm
m}\mathcal{H}^{2}\left(\left|\frac{\partial{\bf x}}{\partial{\bf
q}}\right|-1\right),$ (39)
where we adopt the notation $\Psi_{i,j}\equiv\partial_{q}^{j}\Psi_{i}$ in this
subsection. Taking $\Delta_{i}=0$ results in the usual master equation for the
LPT. We introduce the correction $\epsilon^{(1)}_{i}$ of order ${\cal
O}(\delta^{(1)}\Delta_{i})$ with $\delta^{(1)}$ being short-wavelength modes
in the simulations:
$\displaystyle\Psi_{i}=\Psi_{i}^{(1)}+\Psi_{i}^{(2)}+\epsilon^{(1)}_{i},$ (40)
where $\Psi_{i}^{(1)}+\Psi_{i}^{(2)}$ is the usual 2LPT solution in the
isotropic background. In the following we also introduce the potentials such
that $\Psi_{i}^{(1)}=-\psi^{(1)}_{,i}$ and
$\epsilon_{i}^{(1)}=-\epsilon^{(1)}_{,i}$ for the convenience. The equation
for $\epsilon^{(1)}$ can be found as
$\displaystyle\sum_{i}\epsilon_{,ii}^{(1)\prime\prime}+\mathcal{H}\sum_{i}\epsilon_{,ii}^{(1)\prime}-\frac{3}{2}\Omega_{\rm
m}\mathcal{H}^{2}\sum_{i}\epsilon_{,ii}^{(1)}=-2\sum_{i}\Delta_{i}^{\prime}\psi_{,ii}^{(1)\prime}.$
(41)
Going to Fourier space and decomposing $\epsilon^{(1)}({\bf k})$ as
$\epsilon^{(1)}({\bf k})=\sum_{i}\hat{k}_{i}^{2}\varepsilon_{i}^{(1)}({\bf
k})$, this equation can be rewritten as
$\displaystyle\varepsilon_{i}^{(1)\prime\prime}+\mathcal{H}\varepsilon_{i}^{(1)\prime}-\frac{3}{2}\Omega_{\rm
m}\mathcal{H}^{2}\varepsilon_{i}^{(1)}=-2\Delta_{i}^{\prime}\psi^{(1)\prime}.$
(42)
Notice that $\varepsilon_{i}^{(1)}$ is different from $\epsilon_{i}^{(1)}$. We
can derive the matter-dominated solution for $\varepsilon_{i}^{(1)}$ for
initial-condition generation. Since Green’s function for this equation is the
same as Eq. (27) the solution is
$\displaystyle\varepsilon_{i}^{(1)}(\eta)=$
$\displaystyle\int_{0}^{\eta}\differential\tilde{\eta}~{}\frac{1}{5}\left[\frac{\eta^{2}}{\tilde{\eta}}-\frac{\tilde{\eta}^{4}}{\eta^{3}}\right]\cdot\left(-2\Delta_{i}^{\prime}(\tilde{\eta})\psi^{(1)\prime}(\tilde{\eta})\right)$
$\displaystyle=$
$\displaystyle-\frac{2}{5}\frac{\psi^{(1)}(\eta)}{D(\eta)}\frac{\Delta_{i}(\eta)}{\beta(\eta)}\int_{0}^{\eta}\differential\tilde{\eta}~{}\left[\frac{\eta^{2}}{\tilde{\eta}}-\frac{\tilde{\eta}^{4}}{\eta^{3}}\right]\cdot
D^{\prime}(\tilde{\eta})\beta^{\prime}(\tilde{\eta})$ $\displaystyle=$
$\displaystyle-\frac{4}{5}\psi^{(1)}(\eta)h_{i}^{\rm
ini}\left[\beta(\eta)-\int_{0}^{\eta}\differential\tilde{\eta}~{}\left(\frac{\tilde{\eta}}{\eta}\right)^{5}\beta^{\prime}(\tilde{\eta})\right]$
$\displaystyle=$ $\displaystyle\psi^{(1)}(\eta)h_{i}^{\rm ini}\alpha(\eta),$
(43)
where we have used $D(\eta)\propto\eta^{2}$ and $\Delta_{i}(\eta)=h_{i}^{\rm
ini}\beta(\eta)$ and $\alpha(\eta)$ is introduced in Eq. (29). Thus
$\displaystyle\epsilon^{(1)}(\eta)=\psi^{(1)}(\eta)\alpha(\eta)\sum_{i}h_{i}^{\rm
ini}\hat{k}_{i}^{2}.$ (44)
This implies that the linear growth function has a direction-dependent
modulation:
$\displaystyle D(\eta,{\bf
k})=D(\eta)\left[1+\alpha(\eta;k_{L})\sum_{i}h_{i}^{\rm
ini}\hat{k}_{i}^{2}\right],$ (45)
which is consistent with Eq. (29). Using Eq. (44), the correction to the
velocity can be computed as
$\displaystyle\epsilon_{i}^{(1)\prime}=$
$\displaystyle\left(f_{\alpha}+f_{1}\right)\mathcal{H}\epsilon_{i}^{(1)},$
(46)
with $f_{\alpha}\equiv\differential\ln\alpha/\differential\ln a$ and
$f_{1}\equiv\differential\ln D/\differential\ln a$. We implement these
modifications in 2LPTIC Crocce _et al._ (2006) and generate the initial
conditions at $z_{i}=99$.
### III.3 Simulations
Figure 2: Fractional anisotropic scale factor in the $x$-axis
$\Delta_{x}(a;k_{L})=a_{x}(a;k_{L})/a-1$ for various wavenumbers of GWs
$k_{L}$ as a function of the scale factor $a$ in the case of $h^{\rm
ini}_{ij}={\rm diag}(0.1,-0.1,0)$. The vertical dashed line represents the
starting redshift $z_{\rm ini}=99$.
We perform $N$-body simulations in the tidal backgrounds with $1024^{3}$
particles in $500~{}\mathrm{Mpc}/h$ boxes. The details of modifications of the
$N$-body code based on Gadget-2 Springel (2005) in the tidal background is
described in Ref. Akitsu _et al._ (2021a). One important additional
modification to the code was made in the drift operator, which is discussed in
Appendix C.
After rotating $\tau_{ij}$ to align its eigenvectors with the simulation axis,
the remaining degrees of freedom can be completely characterized by two
parameters, which we can parametrize as $\tau_{\rm
e}\equiv-(\tau_{1}-\tau_{2})/2$ and $\tau_{\rm
p}=-\tau_{3}+(\tau_{1}+\tau_{2})/2$. Taking into account the transverse
condition of GWs, we cannot consider $\tau_{\rm p}$-type tides for the tidal
separate universe with GWs unlike the case of the scalar tidal field in Ref.
Akitsu _et al._ (2021a). Hence in this paper we only consider $\tau_{\rm
e}$-type tides for the background anisotropy. In other words, we consider GWs
propagating along $z$ direction with $+$ mode polarization:
$\displaystyle h^{\rm ini}_{ij}=\begin{pmatrix}\pm\epsilon&0&0\\\
0&\mp\epsilon&0\\\ 0&0&0\end{pmatrix},$ (47)
where we choose $\epsilon=0.1$. Since we aim to measure the response of large-
scale structure to the long-wavelength GWs, the results should not be
dependent on the choice of the direction of the propagation and the
polarization basis.
In order to investigate effects of GWs over a wide range of wavenumbers, we
run tidal separate universe simulations with various wavenumbers of GWs:
$\displaystyle
k_{L}=\left\\{0.0001,\,0.0002,\,0.0005,\,0.001,\,0.002,\,0.005,\,0.01,\,0.02,\,0.05,\,0.1,\,0.2\right\\}\,[h/{\rm
Mpc}].$ (48)
These different wavenumbers of GWs give rise to the different time evolution
of the local anisotropic scale factors, through which long-wavelength GWs
affect the simulated small-scale structure formation. Some examples of the
time evolution of $\Delta_{x}$ are shown in Fig. 2. Given the box size of
$500~{}{\rm Mpc}/h$, some wavenumbers are larger than the fundamental mode in
the simulation: $k_{\rm F}=0.013~{}h/{\rm Mpc}$. For such larger wavemubers,
we cannot neglect the curvature of the long-wavelength modes and treat GW as a
unifiom tidal field over the whole simulation box and the approximation is
violated. Still, these GWs are longer-mode than the halo formation scale so GW
can be seen as a uniform tide in that local region. Therefore we can study the
impact of GWs on halos, in particular, the response of the halo shape as
discussed in Sec. VI, where we come back to this issue again.
For references, we also run fiducial simulations with the isotropic
background, and tidal separate universe simulations induced by the scalar
$\tau_{\rm e}$-type tidal field. These reference simulations share the same
parameters (including random seeds) as those used in the GW tidal separate
universe simulations. For each type of simulations (fiducial and tidal
separate universe with GWs and scalar tides), we run four realizations,
amounting to 100 simulations in total.
We use the AHF code Knollmann and Knebe (2009) to identify dark matter halos
in the simulations by spherical overdensity (SO) regions 200 times as dense as
the mean matter density. We need to identify SO halos in the _global_
coordinates on the _isotropic_ background, i.e., $a_{i}x_{i}/a$, while AHF by
default uses the local simulation coordinates $x_{i}$ on the anisotropic
background. To this end, we modify the AHF code to rescale the coordinate
correspondingly in distance computations.
## IV Anisotropic power spectrum response: tensor fossils in nonlinear regime
In this section, as a first example of imprints of GWs on large-scale
structure, we present the anisotropic impact on the matter auto-, matter-halo
cross-, and halo auto-power spectra by long-wavelength GWs measured from our
$N$-body simulations.
### IV.1 Growth-dilation decomposition
As we derived in Sec. II.2, long-wavelength GWs leave the anisotropic imprint
in the matter power spectrum in a given realization of GWs through the
nonlinear interaction of tidal fields. This tidal response consists of two
different contributions as in Eq. (31); the term proportional to $\alpha$ that
modulates the amplitude of the power spectrum and the term proportional to
$\beta$ that modulates the scales. In terms of the tidal separate universe
introduced in Sec. III.1, the former known as the growth effect describes the
changes in the amplitude of short-mode fluctuations measured in the local
anisotropic background, while the latter known as the dilation effect stems
from the anisotropic expansion of the local background with respect to the
global one. Specifically, the response of the power spectrum to long-
wavelength GWs can be decomposed as
$\displaystyle\left.\frac{\differential{\ln P_{\cal G}}}{\differential{h^{\rm
ini}_{ij}}}\right|_{{\bf k}_{\cal G}}=\left.\frac{\differential{\ln P_{\cal
L}}}{\differential{h^{\rm ini}_{ij}}}\right|_{{\bf k}_{\cal G}}=$
$\displaystyle\left.\frac{\partial\ln P_{\cal L}}{\partial h^{\rm
ini}_{ij}}\right|_{{\bf k}_{\cal L}}+\left.\frac{\partial\ln P_{\cal
L}}{\partial\ln k_{{\cal L},i^{\prime}}}\right|_{h_{ij}^{\rm
ini}}\left.\frac{\differential{\ln k_{{\cal L},i^{\prime}}}}{\partial h^{\rm
ini}_{ij}}\right|_{{\bf k}_{\cal G}}$ $\displaystyle\equiv$
$\displaystyle\,\hat{k}_{i}\hat{k}_{j}\left[R^{\rm GW}_{\rm
growth}(k;k_{L})+R^{\rm GW}_{\rm dilation}(k;k_{L})\right],$ (49)
where in the first line the power spectra defined with respect to the global
isotropic background and the local anisotropic background are denoted by
$P_{\cal G}$ and $P_{\cal L}$, respectively, and correspondingly the
wavenumbers by ${\bf k}_{\cal G}$ and ${\bf k}_{\cal L}$. The physical scale
should be unchanged in these two coordinates, $a_{i}x_{{\cal L},i}=ax_{{\cal
G},i}$, which implies $k_{{\cal L},i}=k_{{\cal G},i}(1+\Delta_{i})$. In the
first equality we have used that the variances must be conserved in the
coordinate transformation: $P_{\cal G}\,\differential^{3}{\bf k}_{\cal
G}=P_{\cal L}\,\differential^{3}{\bf k}_{\cal L}$, together with
$|\differential^{3}{\bf k}_{\cal L}/\differential^{3}{\bf k}_{\cal
G}|=\prod_{i=1}^{3}(1+\Delta_{i})=1$ at leading order. In the second line, We
relabel both $k_{\cal G}$ and $k_{\cal L}$ as $k$, since the responses $R^{\rm
GW}_{\rm growth}$ and $R^{\rm GW}_{\rm dilation}$ are already first order in
$h^{\rm ini}_{ij}$ (or $\Delta_{i}$) and hence here we do not need to
distinguish $k_{\cal G}$ and $k_{\cal L}$. Notice that we distinguish $k_{L}$
from $k_{\cal L}$; the former represents the wavenumber of GWs. Notice also
that we have defined the response with respect to the $h_{\rm ini}$, not
$h(z)={\cal T}(z)h_{\rm ini}$, which allows for simple computations of
observables in terms of the primordial tensor mode amplitude. Namely the
cosmology dependence other than the initial amplitude of GWs factorizes out in
the response function.
The power spectrum of biased tracers in the presence of long-wavelength GWs
then has a generic form,
$\displaystyle P_{XY}({\bf
k}|h_{ij}(k_{L}))=P_{XY}(k)\left[1+\hat{k}^{i}\hat{k}^{j}h_{ij}^{\rm
ini}(k_{L})\left(R^{\rm GW}_{{\rm growth};XY}(k;k_{L})+R^{\rm GW}_{{\rm
dilation};XY}(k;k_{L})\right)\right],$ (50)
where $X$ and $Y$ represent tracers being considered. For the matter auto-
power spectrum ($X=Y={\rm m}$), this corresponds to the nonlinear extension of
Eq. (31). We will also consider the matter-halo cross-power spectrum ($X={\rm
m}$ and $Y={\rm h}$) and the halo auto-power spectrum ($X=Y={\rm h}$) below.
Moreover, using these GW power spectrum responses, we can write down the
$X$-$Y$-GWs (scalar-scalar-tensor) bispectrum in squeezed limit as
$\displaystyle\lim_{k_{L}\to
0}B_{XYh_{(\lambda)}}(k,k^{\prime},k_{L})=\hat{k}^{i}\hat{k}^{j}e^{(\lambda)}_{ij}\left[R^{\rm
GW}_{{\rm growth};XY}(k;k_{L})+R^{\rm GW}_{{\rm
dilation};XY}(k;k_{L})\right]P_{XY}(k)P_{h_{(\lambda)}}(k_{L}),$ (51)
where we have defined the $X$-$Y$-GW bispectrum via $\langle X({\bf k})Y({\bf
k}^{\prime})h_{(\lambda)}({\bf k}_{L})\rangle=(2\pi)^{3}\delta^{(3)}_{\rm
D}({\bf k}+{\bf k}^{\prime}+{\bf
k}_{L})B_{XYh_{(\lambda)}}(k,k^{\prime},k_{L})$ and neglected the primordial
contribution.333 Although here we have neglected the primordial scalar-scalar-
tensor bispectrum, its contribution to the local observables appears only at
the order of ${\mathcal{O}}(k_{L}/k)^{2}$ and thus is subdominant for the
single-field inflation (see Ref. Pajer _et al._ (2013)).
Making use of the growth-dilation decomposition, we can go a little further
than the perturbative result. Because the dilation effect purely captures the
coordinate transformation, using Eq. (37), we can obtain the non-perturbative
result as
$\displaystyle R^{\rm GW}_{{\rm
dilation};XY}(k;k_{L})=\beta(k_{L})\frac{\partial\ln P_{XY}(k)}{\partial\ln
k}.$ (52)
Eqs. (31) and (52) are different because former perturbative result involves
the linear power spectrum while the latter involves the nonlinear power
spectrum and can be applied to a non-linear regime. This means that we can
compute the dilation piece without running simulations even in the nonlinear
regime given the slope of the nonlinear power spectrum. On the other hand, the
growth piece comes from the dynamical effect where we cannot extend the
perturbative result in the nonlinear regime. We thus need to rely on
simulations in order to calibrate the growth response in the nonlinear regime.
Hence, in this paper we focus on measuring the growth term from the tidal
separate universe simulations with GWs. By comparing the measurement of the
growth term of the matter auto-power spectrum in simulations with the
perturbation theory prediction at quasi-nonlinear scales, we can validate our
methodology using the relation:
$\displaystyle\lim_{k\ll k_{\rm NL}}R^{\rm GW}_{\rm
growth;mm}(k;k_{L})=2\alpha(k_{L}).$ (53)
Note that since we assume GWs are long-wavelength modes compared with scalar
perturbations the perturbative result is valid only for the range of $k_{L}\ll
k\ll k_{\rm NL}$.
### IV.2 Growth response of the matter auto-power spectrum from simulations
In the presence of GWs of $\tau_{\rm e}$-type configuration (Eq. (47)), the
matter auto-power spectrum takes a form of
$\displaystyle P_{\rm mm}({\bf k}|h_{\rm e}(k_{L}))=P_{\rm
mm}(k)\left[1+\frac{2}{3}R^{\rm GW}_{\rm growth;mm}(k;k_{L})\left({\cal
L}_{2}(\hat{k}_{1})-{\cal L}_{2}(\hat{k}_{2})\right)h^{\rm ini}_{\rm
e}\right],$ (54)
where $P_{\rm mm}(k)$ is the nonlinear matter power spectrum in the isotropic
background, ${\cal L}_{2}(x)$ is Legendre polynomial of order two,
$\hat{k}_{1}$ and $\hat{k}_{2}$ represent the $x$ and $y$ components of
$\hat{k}$ respectively, and $h_{\rm e}^{\rm ini}\equiv-\left(h_{11}^{\rm
ini}-h_{22}^{\rm ini}\right)/2=\pm\epsilon$. We can estimate the growth
response by taking the quadrupoles of the power spectrum along both $x$ and
$y$ axes:
$\displaystyle P^{\ell_{\rm e}=2}_{\rm mm}(k|h_{\rm e}(k_{L}))$
$\displaystyle\equiv P^{\ell_{x}=2}_{\rm mm}(k)-P^{\ell_{y}=2}_{\rm mm}(k)$
(55) $\displaystyle=2P_{\rm mm}(k)R^{\rm GW}_{\rm growth;mm}(k;k_{L})h^{\rm
ini}_{\rm e},$ (56)
where $P^{\ell_{i}=2}_{\rm mm}(k)$ ($i=\\{x,y\\}$) are defined as
$\displaystyle P^{\ell_{x}=2}_{XY}(k)$ $\displaystyle\equiv
5\int\frac{\differential^{2}\hat{{\bf k}}}{4\pi}P_{XY}({\bf k}){\cal
L}_{2}(\hat{k}_{1}),$ (57) $\displaystyle P^{\ell_{y}=2}_{XY}(k)$
$\displaystyle\equiv 5\int\frac{\differential^{2}\hat{{\bf
k}}}{4\pi}P_{XY}({\bf k}){\cal L}_{2}(\hat{k}_{2}).$ (58)
This leads to the estimator for $R_{\rm growth;mm}(k;k_{L})$ as
$\displaystyle R^{\rm GW}_{\rm growth;mm}(k;k_{L})=\frac{P^{\ell_{\rm
e}=2}_{\rm mm}(k|h^{\rm ini}_{\rm e}=+\epsilon)-P^{\ell_{\rm e}=2}_{\rm
mm}(k|h^{\rm ini}_{\rm e}=-\epsilon)}{4\epsilon P_{\rm mm}(k)}.$ (59)
Figure 3: Growth response of matter auto-power spectrum to GWs, $R^{\rm
GW}_{\rm growth}(k_{L};k)$, as a function of the short-wavenmer $k$ at various
wavenumvers of GWs (left) and various redshifts (right), measured from the
simulations. The top-left and bottom-left panels show the responses at $z=0.5$
$z=2$, respectively. The top-right and bottom-right panels show the responses
at $k_{L}=0.0002~{}h/{\rm Mpc}$ and $k_{L}=0.001~{}h/{\rm Mpc}$, respectively.
The dashed lines with each different color represent the predictions from the
perturbation theory at each wavenumber and redshift.
Figure 4: Growth response of matter auto-power spectrum to GWs, $R^{\rm
GW}_{\rm growth}(k_{L};k)$, as a function of the wavenumber of GWs $k_{L}$ at
$z=2,1,0.5,$ and $0$ (from the left to the right), measured from the
simulations. The upper and lower rows show the results for different $k$,
$k=0.376~{}h/{\rm Mpc}$ and $k=0.881~{}h/{\rm Mpc}$, respectively. The orange
line depicts the “rescaled growth response” to the scalar tides,
$\frac{7}{4}\alpha(k_{L})R_{\rm growth}^{\rm scalar}(k)$ with $R_{\rm
growth}^{\rm scalar}(k)$ measured from the tidal separate universe simulations
with the scalar tides.
We measure the growth response of the matter auto-power spectrum to GWs,
$R^{\rm GW}_{\rm growth;mm}(k;k_{L})$, from the simulations which share the
same initial random phase to reduce the sample variance. Since $R^{\rm
GW}_{\rm growth;mm}$ depends on the wavenumbers of the shote-mode, $k$, and of
GWs, $k_{L}$, as well as redshift $z$, it is hard to show all the dependences
in one figure. Thus, let us start by showing the measured growth response as a
function of $k$ for various $k_{L}$ at $z=0.5$ and $z=2$ in the upper- and
lower-left panels of Fig. 3, respectively. We then show it for
$k_{L}=0.0002h/{\rm Mpc}$ and $k_{L}=0.001h/{\rm Mpc}$ at various redshifts in
the upper- and lower-right panels, respectively. In each panel, the
corresponding perturbation theory predictions, Eq. (53), are shown in the
dashed lines. First, onc can see the excellent agreement between the measured
and predicted growth responses on large scales for all $k_{L}$, which verifies
that our methodology to incorporate GWs in simulations works correctly.
Second, the measured response deviates from the perturbation prediction on
nonlinear scales, in particular for larger $k_{L}$. Although the way it
deviates depends on $k_{L}$, the overall trend is similar among different
$k_{L}$ values; (i) at the smallest scales ($k\gtrsim 2\ h/{\rm Mpc}$) the
growth response decreases for all $k_{L}$ and redshifts. (ii) At earlier
redshift, the growth response is slightly enhanced compared to the
perturbation theory while at lower redshift it is largely suppressed. Third,
the redshift dependence of the growth response is clearer as shown in the
right panels. The agreement between the perturbation theory and the simulation
becomes worse at lower redshifts. At $z=2$ these two are in good agreement up
to $k\sim 0.2\ h/{\rm Mpc}$, while at $z=0$ the simulation results start to
differ from the perturbation theory around $k\sim 0.04\ h/{\rm Mpc}$.
These tendencies we found for the tensor mode above are actually very similar
to those for the case of the scalar tidal field which had been extensively
studied (see App.B and Refs. Akitsu _et al._ (2021a); Stücker _et al._
(2021); Masaki _et al._ (2020) for details). Given the similarity of the
behavior of the growth response on nonlinear scales between the scalar and
tensor cases, it is natural to ask how similar are these two quantitatively.
Let us finish this subsection by answering this question. For this purpose, we
consider a “rescaled growth response” to the scalar tides,
$\frac{7}{4}\alpha(\eta;k_{L})R^{\rm scalar}_{\rm growth;mm}(k)$, where the
scalar tidal response $R^{\rm scalar}_{\rm growth;mm}(k)$ is introduced in Eq.
(96) in App. B. Since $R^{\rm scalar}_{\rm growth;mm}(k)$ approaches $8/7$ at
the large-scale limit (see Eq. (95)), this scalar tidal response matches the
tensor tidal response in the large scale limit (Eq. 53),
$\displaystyle\lim_{k\to 0}\frac{7}{4}\alpha(\eta;k_{L})R^{\rm scalar}_{\rm
growth;mm}(k)=2\alpha(\eta;k_{L}).$ (60)
In Fig.4 we compare this “rescaled growth response” to the scalar tides with
the measured tensor tidal response on nonlinear scales as a function of
$k_{L}$ for various redshifts. The upper and lower panels show the results for
$k=0.376\ h/{\rm Mpc}$ and $k=0.881\ h/{\rm Mpc}$, respectively. Overall, the
rescaled response to the scalar tides captures the general feature of the
growth response to GWs on nonlinear scales. Such agreement can be seen for
both the two wavenumbers at all the redshifts when $k_{L}\lesssim
10^{-3}~{}h/{\rm Mpc}$. On the other hand, for $k_{L}\gtrsim 10^{-3}~{}h/{\rm
Mpc}$ the response to GWs has greater values than the rescaled response,
except for some $k_{L}$ at $z=0$. This difference gets larger at higher
redshifts and larger $k$. These can be attributed to the different time
evolution of the anisotropic scale factors in the scalar tide and GWs cases.
The anisotropic scalar factor induced by the scalar tidal field grows
monotonically in time, following the linear growth rate $D(\eta)$ at leading
order, $\Delta^{\rm scalar}_{i}(\eta)\propto D(\eta)$ (Eq. (94)), whereas that
induced by GWs has the time dependence described by $\beta(\eta;k_{L})$, which
is generally monotonous in time for $k_{L}\lesssim 10^{-3}~{}h/{\rm Mpc}$
while not so for $k_{L}\gtrsim 10^{-3}~{}h/{\rm Mpc}$ (see Fig. 2).
Specifically, for $k_{L}\gtrsim 10^{-3}~{}h/{\rm Mpc}$ the anisotropic scale
factors from GWs reach their asymptotic value at early redshift, when the
anisotropic scale factors from scalar tides are still tiny, which causes the
tidal response to be stronger in these $k_{L}$.
Figure 5: Growth response of the halo-matter cross-power spectrum to GWs,
$R^{\rm GW;hm}_{\rm growth}(k_{L};k)$, as a function of $k$ for various
$k_{L}$ (left), various redshifts (center), and various halo masses (right),
measured from the simulations. The blue dotted line in the right panel shows
the growth response of the matter auto-power spectrum. Figure 6: Similar to
Fig. 5 but the growth response of the halo auto-power spectrum to GWs, $R^{\rm
GW;hh}_{\rm growth}(k_{L};k)$ as a function of $k$ for various $k_{L}$ (left),
various redshifts (center), and various halo masses (right), measured from the
simulations. The upper and lower sets show the results normalized by the halo
auto-power spectrum with and without a shot noise, respectively. For the
latter, the shot noise contribution is subtracted assuming the Poisson
distribution. In the lower-right panel, we do not show the result for the mass
bin, $10^{14}M_{\odot}/h<M_{\rm vir}<10^{14.5}M_{\odot}/h$ because it is too
noisy.
### IV.3 Growth responses of the halo-matter cross- and halo auto-power
spectra from simulations
The halo-matter cross- and halo auto-power spectra in a local region are also
affected by GWs. The growth responses of these power spectra to GWs can be
estimated from the simulations in the same way as the matter auto-power
spectrum,
$\displaystyle R^{\rm GW}_{\rm growth;XY}(k;k_{L})=\frac{P^{\ell_{\rm
e}=2}_{\rm XY}(k|h^{\rm ini}_{\rm e}=+\epsilon)-P^{\ell_{\rm e}=2}_{\rm
XY}(k|h^{\rm ini}_{\rm e}=-\epsilon)}{4\epsilon P_{\rm XY}(k)},$ (61)
with $XY\in\\{{\rm hm},{\rm hh}\\}$.
In Fig. 5 we show the growth response of the halo-matter cross-power spectrum
to GWs as a function of the wavenumber of the short modes. The left and middle
panels focus on the fixed halo mass ($10^{12}M_{\odot}/h<M_{\rm
vir}<10^{12.5}M_{\odot}/h$), and compares the growth responses for various
wavenumbers of GWs at $z=0.5$ in the left panel and various redshfits at
$k_{L}=0.001~{}h/{\rm Mpc}$ in the middle panel. As for the
$k_{L}$-dependence, the response tends to be greater for larger $k_{L}$,
similar to the case of the matter auto response. On the other hand, the
redshift dependence is slightly different from the matter auto case. For the
halo-matter cross-power spectrum, the growth response at late redshifts ($z=0$
and $z=0.5$) gets slightly enhanced compared to the linear regime and persists
up to $k\lesssim 1~{}h/{\rm Mpc}$, whereas the matter auto response starts to
decrease around $k\simeq 0.1~{}h/{\rm Mpc}$. This can be clearly seen in the
right panel of Fig. 5, which directly compares the growth response of the
matter auto power-spectrum to that of the halo-matter cross-power spectrum
with various halo masses at $z=0.5$ and $k_{L}=0.001~{}h/{\rm Mpc}$. First, it
turns out that the scale-dependence ($k$-dependence) of the growth response
varies with the halo mass. The enhancement of the growth response around
$k\simeq 1~{}h/{\rm Mpc}$ is greater for less massive halos. This is mainly
due to the normalization. Here we define the response with respect to the
halo-matter cross-power spectrum in the fiducial simulation. However if we
define the response with respect to the matter auto-power spectrum, the
response of more massive halos gets more amplified in the nonlinear regime.
Second, the difference of the responses between the matter auto and halo-
matter cross cases stems from the halo biases. In particular at largest scales
it should be explained by the halo tidal bias induced by GWs, which we
investigate in the next section.
Next, let us focus on the growth response of the halo auto-power spectrum. To
obtain it, for the denominator of Eq. (61) we use measurements of the halo
auto-power spectrum with and without the shot noise contribution. For the
latter, the shot noise contribution is subtracted assuming the Poisson
distribution, $1/\bar{n}_{\rm h}$. The upper and lower rows of Fig. 6 show the
resulting response of the halo auto-power spectrum as a function of $k$ with
and without the shot noise, respectively. As with Fig. 5, the left and middle
columns show the responses with the fixed mass range
$10^{12}M_{\odot}/h<M_{\rm vir}<10^{12.5}M_{\odot}/h$ for various wavenumber
of GWs at $z=0.5$ and for $k_{L}=0.001~{}h/{\rm Mpc}$ at various redshfits,
respectively, while the right columns show the result with various halo masses
for $k_{L}=0.001~{}h/{\rm Mpc}$ at $z=0.5$. Overall, the $k_{L}$, redshift,
and halo mass dependencies of the responses for the halo auto spectra both
with and without the shot noise are similar to that for the halo-matter cross
spectrum. In the upper set of Fig. 6, the responses approach zero on small
scales where the halo auto-power spectra are dominated by shot noises. On the
contrary, the peaky features around $k\sim 2~{}h/{\rm Mpc}$ in the lower set
reflects the non-Poissonian behavior of the halo shot noise due to, e.g., the
exclusion effect Smith _et al._ (2007); Hamaus _et al._ (2010); Baldauf _et
al._ (2013). Therefore, the simple Poissonian shot-noise removal leads to
zero-crossing or negative, unphysical halo auto-power spectrum on small
scales, making the responses peaky. Nonetheless, up to $k\sim 0.5~{}h/{\rm
Mpc}$, where the shot noise contribution is still small, the responses are not
suppressed unlike the matter auto case. These trends seen in the response of
the halo-matter cross- and halo auto-power spectra are the same as those in
the scalar tidal field case (see App. B).
## V Halo tidal bias induced by GWs
Figure 7: Halo tidal bias induced by GWs as a function of halo mass for
various wavenumber of GWs at $z=0.5$ (the left panel) and at various redsfhits
for $k_{L}=0.001~{}h/{\rm Mpc}$ (the right panel).
As discussed in the previous section, the long-wavelength GWs can affect the
halo density field. In the perturbative regime, this effect should be
characterized in terms of the halo bias. When only scalar perturbations are
considered, which is the standard setup, the halo density field can be
expanded up to the second order as
$\displaystyle\delta_{\rm
h}=b_{1}\delta+\frac{1}{2}b_{2}\delta^{2}+b_{s^{2}}s^{2},$ (62)
where $s^{2}=s_{ij}s^{ij}$ with
$s_{ij}\equiv(\partial_{i}\partial_{j}/\partial^{2}-\delta^{\textsc{k}}_{ij}/3)\delta$.
Here $b_{s^{2}}$ is called the tidal bias that captures the effect of the
tidal fields on the halo density field. Since long-wavelength GWs are locally
indistinguishable from tidal fields induced by the scalar perturbations, it is
natural to expect that there is a tidal bias induced by GWs as well. Then, at
the linear order of GWs, the halo density field should acquire the following
contribution from GWs,
$\displaystyle\delta_{\rm h}^{(2)}(\eta)\supset b^{\rm
GW}_{s^{2}}(\eta;k_{L})s^{ij}(\eta)h_{ij}^{\rm ini}(k_{L}),$ (63)
where $b_{s^{2}}^{\rm GW}$ is the tidal bias coefficient induced by the
coupling between tidal fields induced by GWs and scalar density perturbations.
Note that we define $b_{s^{2}}^{\rm GW}$ with respect to the GWs at the
_initial_ epoch instead of the _same_ time, unlike the biases defined with
respect to the scalar perturbations. Because the GWs are additional degrees of
freedom to the scalar adiabatic perturbations, we expect the tidal bias to
depend on the wavenumber of GWs. Collecting the second order pieces in the
matter density field and halo density field, we find the tree-level halo-
matter-GWs bispectrum in the squeezed limit as
$\displaystyle\lim_{k_{L}\to 0}B_{{\rm
hm}h_{(\lambda)}}(k,k^{\prime},k_{L})=\hat{k}^{i}\hat{k}^{j}e^{(\lambda)}_{ij}\left[2b_{1}\alpha(\eta;k_{L})+2b_{s^{2}}^{\rm
GW}(\eta;k_{L})\right]P_{\rm lin}(k)P_{h_{(\lambda)}}(k_{L}),$ (64)
where we have omitted the dilation piece. This implies that the local halo-
matter power spectrum in the presence of long-wavelength GWs is
$\displaystyle P_{\rm hm}({\bf
k}|h_{ij}(k_{L}))=\left[b_{1}+\left[2b_{1}\alpha(\eta;k_{L})+2b^{\rm
GW}_{s^{2}}(\eta;k_{L})\right]k^{i}k^{j}h_{ij}^{\rm ini}\right]P_{\rm
lin}(k).$ (65)
Thus, we can estimate $b_{s^{2}}^{\rm GW}$ from the growth response of the
halo-matter power spectrum to GWs, which is presented in the previous section.
Although there are several ways to estimate $b_{s^{2}}^{\rm GW}$ from the
growth response of the halo-matter power spectrum, we use the following way to
reduce the uncertainty of $b_{1}$ and the sample variance. We first define the
local linear bias estimator as a ratio of the halo-matter and matter auto-
power spectra including the quadrupoles measured in the simulations with GWs,
$\displaystyle\hat{b}^{\ell_{x,y}}_{1}(k;\pm\epsilon)\equiv\frac{P^{\ell=0}_{\rm
hm}(k;\pm\epsilon)+P^{\ell_{x,y}=2}_{\rm hm}(k;\pm\epsilon)}{P^{\ell=0}_{\rm
mm}(k;\pm\epsilon)+P^{\ell_{x,y}=2}_{\rm
mm}(k;\pm\epsilon)}=\frac{b_{1}\pm(2b_{1}\alpha(\eta;k_{L})+2b_{s^{2}}^{\rm
GW})\epsilon}{1\pm 2\alpha(\eta;k_{L})\epsilon}\simeq\tilde{b}_{1}(k)\pm
2\tilde{b}_{s^{2}}^{\rm GW}(k)\epsilon,$ (66)
Therefore using this local linear bias, the estimator for $b_{s^{2}}^{\rm GW}$
is now
$\displaystyle\tilde{b}^{\rm
GW}_{s^{2}}(k)=\frac{1}{2}\left[\frac{\hat{b}_{1}^{\ell_{x}}(k;+\epsilon)-\hat{b}_{1}^{\ell_{x}}(k;-\epsilon)}{4\epsilon}+\frac{\hat{b}_{1}^{\ell_{y}}(k;+\epsilon)-\hat{b}_{1}^{\ell_{y}}(k;-\epsilon)}{4\epsilon}\right].$
(67)
We estimate $b_{s^{2}}^{\rm GW}$ utilizing the $\chi^{2}$ statistic, defined
as $\chi^{2}=\sum_{k=k_{\rm min}}^{k_{\rm max}}[b_{s^{2}}^{\rm
GW}-\tilde{b}_{s^{2}}^{\rm GW}(k)]^{2}/\sigma^{2}_{b_{s^{2}}^{\rm GW}}(k)$,
where $\sigma^{2}_{b_{s^{2}}^{\rm GW}}(k)$ is the variance of
$\tilde{b}_{s^{2}}^{\rm GW}(k)$ at each $k$-bin measured from simulations.
Adopting $k_{\rm max}=0.08~{}h/{\rm Mpc}$, we obtain the best-fitting value of
$b_{s^{2}}^{\rm GW}$ and its uncertainty by minimizing $\chi^{2}$. We restrict
the measurement of $b_{s^{2}}^{\rm GW}$ to a range of $k_{L}\leq
0.002~{}h/{\rm Mpc}$ because we rely on the perturbative results, which are
valid only when $k_{L}\ll k$.
Figure 8: Upper panels: Halo tidal bias induced by GWs as a function of
wavenumber of GWs with halo mass of $10^{13}~{}M_{\odot}/h<M_{\rm
vir}<10^{13.5}~{}M_{\odot}/h$. Lower panels: Same as upper panels but with
halo mass of $10^{14}~{}M_{\odot}/h<M_{\rm vir}<10^{14.5}~{}M_{\odot}/h$. From
the left to right , we show the results at $z=1$, $0.5$ and $0$. The blue
points with error bars are the measurement from the simulations. The orange
line represents the ansatz $b_{s^{2}}^{\rm
GW}(k_{L};M)=\frac{7}{4}b_{s^{2}}^{\rm scalar}(M)\alpha(k_{L})$ with the
shaded region being the error coming from $b_{s^{2}}^{\rm scalar}(M)$.
Fig. 7 shows the tidal bias from GWs determined in this way as a function of
halo mass. In the left panel we plot the result for various wavenumbers of GWs
at $z=0.5$, and in the right panel we plot the result for various redshifts at
$k_{L}=0.001~{}h/{\rm Mpc}$. Regardless of the wavenumber of GWs and redshift,
more massive halos have greater absolute values of $b_{s^{2}}^{\rm GW}$, which
has the same trend as the usual tidal bias induced by the scalar tidal fields.
In addition to the mass dependence, the redshift dependence is also similar to
the scalar tidal bias case; the absolute value of the tidal bias is larger at
higher redshift. On the other hand, in contrast to the case of the scalar
tidal bias, the tidal bias from GWs has a particular wavenumber dependence as
shown in the left panel of Fig. 7. This wavenumber-dependence (or scale-
dependence) inherits from the wavenumber-dependent transfer function of GWs
while the growth function for the scalar density fluctuations is independent
of the wavenumber (Eq. (21)). To look at this scale dependence in more detail,
Fig. 8 displays $b_{s^{2}}^{\rm GW}$ as a function of $k_{L}$ for several
redshifts and halo masses. We compare the measurements with the following
ansatz:
$\displaystyle b_{s^{2}}^{\rm
GW}(\eta;k_{L})=\frac{7}{4}\alpha(\eta;k_{L})b^{\rm scalar}_{s^{2}}(\eta),$
(68)
where $b_{s^{2}}^{\rm scalar}$ is the tidal bias induced by the scalar tides,
introduced in Eq. (62) (see also App. B). This ansatz is motivated by the fact
that (i) the tidal bias term stems from the coupling of tidal fields, which is
given by $s^{ij}(z)s_{ij}(z)$ for the scalar tide case ($b_{s^{2}}^{\rm
scalar}$) while $s^{ij}(z)h^{\rm ini}_{ij}$ for the tensor tide case
($b_{s^{2}}^{\rm GW}$), and (ii) the second-order matter density induced by
tidal fields is $\frac{4}{7}s^{ij}(z)s_{ij}(z)$ for the scalar tides case
while $\alpha(z;k_{L})s^{ij}(z)h^{\rm ini}_{ij}(k_{L})$ for the tensor tides
case. Despite the large error bars, overall the measurements are well
explained by this ansatz. This result suggests that we do not have to
introduce a new free bias parameter for GWs at leading order once
$b_{s^{2}}^{\rm scalar}$ is known.
## VI Intrinsic alignments induced by GWs
Figure 9: Linear shape bias induced by GWs as a function of halo mass. The
left and right panels show $b_{K}^{\rm GW}$ for various wavenumber of GWs at
$z=0.5$ and for various redshifts at $k_{L}=0.001~{}h/{\rm Mpc}$,
respectively.
As we discussed so far, the tidal fields contribute to the density field only
at the second order because tidal fields are a tensor while density fields are
a scalar. Conversely, the tidal fields should contribute to tensor quantities
at linear order. One well-known observable of tensor quantities in large-scale
structure is the intrinsic alignments of halo (or galaxy) shapes Catelan _et
al._ (2001); Hirata and Seljak (2004). The deviation of the halo intrinsic
shape from sphere is characterized by ellipticity at the lowest order. In
other words, the halo shape can be described by the trace-free rank-2 tensor
in three-dimensional space, $\gamma_{ij}$.
The linear alignment model of the intrinsic alignment relates the halo shape
with the tidal field as follows:
$\displaystyle\gamma_{ij}(\eta)$ $\displaystyle=b^{\rm
scalar}_{K}(\eta)s_{ij}(\eta),$ (69)
where $b^{\rm scalar}_{K}$ is the linear alignment coefficient or the linear
shape bias, which captures the sensitivity or the response of halo or galaxy
shape to the tidal field. Because long-wavelength GWs are locally equivalent
to the tidal field, we naturally expect that the halo shapes are also aligned
by GWs, namely
$\displaystyle\gamma_{ij}(\eta;k_{L})=b_{K}^{\rm GW}(\eta;k_{L})h^{\rm
ini}_{ij}(k_{L}),$ (70)
where we have introduced the linear shape bias $b_{K}^{\rm GW}$ for GWs, in
analogy with the tidal bias, $b^{\rm scalar}_{K}$. While $b^{\rm scalar}_{K}$
is defined as a response of the halo shape with respect to the scalar tidal
field at the same time, $b^{\rm scalar}_{K}$ is defined as that with respect
to GWs at the _initial_ time when the GWs are frozen, as in the tidal bias
case.
Figure 10: Linear shape bias induced by GWs as a function of the wavenumber
of GWs for the halos with $10^{13.5}~{}M_{\odot}/h<M_{\rm
vir}<10^{14}~{}M_{\odot}/h$. From the left to the right panels we show the
results at $z=2,1,0.5,$ and $0$, respectively. The orange line depicts the
ansatz introduced in Eq. (75), $b_{K}^{\rm
GW}(k_{L};M,z)=\frac{7}{4}\alpha(k_{L},z)b_{K}^{\rm scalar}(M,z)$, and is not
a fitting. The orange dashed region corresponds to the 1-$\sigma$ error of
$b_{K}^{\rm scalar}$.
The linear shape bias introduced above can be efficiently measured in the
tidal separate universe simulation Akitsu _et al._ (2021a); Stücker _et al._
(2021), as the linear halo bias can be obtained precisely in the isotropic
separate universe simulation Li _et al._ (2016); Baldauf _et al._ (2016);
Lazeyras _et al._ (2016). In this paper, we define the halo shape by its
reduced inertia tensor,
$\displaystyle J_{ij}=\sum_{n=1}^{N_{\rm p}}\frac{x_{n,i}x_{n,j}}{x_{n}^{2}},$
(71)
where $N_{\rm p}$ is the number of particles in the halo and $x_{n,i}$ is the
$i$-th component of the particle location with respect to the halo center.
Note that here $x$ represents the distance measured in the isotropic
background, i.e., the physical coordinates. As $\gamma_{ij}$ is regarded as
the trace-free part of $J_{ij}$, the linear alignment model states that this
halo shape can be written as
$\displaystyle
J_{ij}=J_{0}\left[\frac{1}{3}\delta^{\textsc{k}}_{ij}+\gamma_{ij}\right]=J_{0}\left[\frac{1}{3}\delta^{\textsc{k}}_{ij}+b_{K}K_{ij}\right],$
(72)
where $J_{0}$ is the normalization, for which we use the trace part of
$J_{ij}$: $J_{0}={\rm Tr}[J_{ij}]$, and $K_{ij}$ is the tidal field of either
$s_{ij}$ or $h_{ij}^{\rm ini}$. Therefore, we can measure $b_{K}$ as a
response of the one-point function of the ellipticity to the tidal field.
Specifically, introducing the following quantity
$\displaystyle J_{\rm e}\equiv\frac{J_{11}-J_{22}}{2},$ (73)
$b_{K}$ can be obtained as
$\displaystyle b_{K}(M,z)=\frac{J_{\rm e}(M,z|K_{\rm e}^{\rm
ini}=+\epsilon)-J_{\rm e}(M,z|K_{\rm e}^{\rm ini}=-\epsilon)}{2\epsilon
J_{0}(M,z)},$ (74)
where $K_{\rm e}^{\rm ini}=h_{\rm e}^{\rm ini}=\left(h_{11}^{\rm
ini}-h_{22}^{\rm ini}\right)/2$ for the tensor tides and $K_{\rm e}^{\rm
ini}=\left(s_{11}(z)-s_{22}(z)\right)/2$ for the scalar tides.
Fig. 9 shows $b_{K}^{\rm GW}$ as a function of halo mass for various
wavenumbers of GWs at $z=0.5$ (the left panel) and for $k_{L}=0.001~{}h/{\rm
Mpc}$ at various redshifts (the right panel). First, we find that GWs
influence the halo shape, implying that indeed intrinsic alignments can be
induced by GWs. Compared with the measurement of $b_{s^{2}}^{\rm GW}$, the
measurement of $b_{K}^{\rm GW}$ has much greater signal-to-noise ratio as we
use the one-point function to measure $b_{K}^{\rm GW}$ while $b_{s^{2}}^{\rm
GW}$ is measured from the large-scale limit of the power spectrum responses.
Second, the halo-mass and redshift dependences of $b_{K}^{\rm GW}$ are similar
to those of $b_{K}^{\rm scalar}$; that is, massive halos tend to more strongly
align with GWs and the strength of the alignment at fixed halo mass decreases
as the redshift gets smaller (see Fig. 16 and Ref. Akitsu _et al._ (2021a)
for details).444 Note, however, that for the tensor case $b_{K}^{\rm GW}(z)$
is defined with respect to the _initial_ amplitude of GWs and it thus
represents the strength of the alignment with respect to $h_{ij}^{\rm ini}$,
while for the scalar tides case, $b_{K}^{\rm scalar}(z)$ represents the
strength of the alignment with respect to $s_{ij}(z)$. Taking this difference
into account, however, does not change the trend of the redshift dependence
though, since $s_{ij}\propto D$ that increases monotonically with time.
Third, the unique feature in the case of GWs is that $b_{K}^{\rm GW}$ is
wavenumber-dependent (or scale-dependent), as in the cases with the power
spectrum response and the halo tidal bias. Fig. 10 plots $b_{K}^{\rm GW}$ as a
function of the wavenumber of GWs for the halos with
$10^{13.5}~{}M_{\odot}/h<M_{\rm vir}<10^{14}~{}M_{\odot}/h$ at $z=2,1,0.5,0$
to directly demonstrate this scale dependence. The scale dependence is clearly
seen because of the high $S/N$.
Figure 11: Ratio of the linear shape bias induced by GWs to the one induced
by scalar tidal fields as a function of the wavenumber of GWs for various halo
masses. From the left to the right, we show the results at $z=2,1,0.5,0$,
respectively. The orange line depicts the ansatz introduced in Eq. (75),
$b_{K}^{\rm GW}(k_{L};M,z)/b_{K}^{\rm
scalar}(M,z)=\frac{7}{4}\alpha(k_{L},z)$, and is not a fitting.
In Fig. 10 we compare the measurements with the following ansatz for the
intrinsic alignment from GWs Schmidt _et al._ (2014),555 The original ansatz
considered in Ref. Schmidt _et al._ (2014) is $b_{K}^{\rm
GW}(\eta;k_{L})=\frac{7}{2}\alpha(\eta;k_{L})b_{K}^{\rm scalar}(\eta)$ (see
their Eq. (74)). However, the prefactor should be $7/4$ given the factor of
two that comes from exchanging long-mode and short-mode in the second-order
coupling
($\delta^{(2)}=\frac{2}{7}s_{ij}s_{ij}\to\delta^{(2)}=\frac{4}{7}s^{\rm
long}_{ij}s^{\rm short}_{ij}$).
$\displaystyle b_{K}^{\rm
GW}(\eta;k_{L})=\frac{7}{4}\alpha(\eta;k_{L})b_{K}^{\rm scalar}(\eta).$ (75)
It is natural to assume that the halo shape is determined by the local tidal
environment around the halo, which is affected by long-wavelength tidal field
via nonlinear mode-coupling. The influence of the long-wavelength tides on the
small-scalar tides is captured by the response function that is discussed in
Sec. IV. In particular, in Fig. 4 we examine the relation between the response
function to the scalar tides and GWs and find that $R_{\rm growth;mm}^{\rm
GW}(k;k_{L})=\frac{7}{4}\alpha(\eta;k_{L})R_{\rm growth;mm}^{\rm scalar}(k)$
is a good approximation even in the nonlinear regime. As these responses serve
as the amplitude of the small-scale tides induced by the large-scale tides, we
expect that $\gamma^{\rm scalar}_{ij}\propto R_{\rm growth;mm}^{\rm
scalar}s_{ij}$ and $\gamma^{\rm GW}_{ij}\propto R_{\rm growth;mm}^{\rm
GW}h^{\rm ini}_{ij}$, leading to the above ansatz. The ansatz of Eq. (75) is
shown by the orange lines in Fig. 10. Remarkably, the measurements are well
explained by this prediction for all the redshifts. Fig. 11 investigates if
this agreement holds for all halo mass by displaying the ratio of the linear
shape biases, $b_{K}^{\rm GW}/b_{K}^{\rm scalar}$, which is equal to
$\frac{7}{4}\alpha$ in the ansatz regardless of halo mass. It turns out that
the trend seen in Fig. 10 remains the same for all halo masses. One important
point that follows from this agreement is that the process to determine the
halo shape is not local in time. For, if the halo shape responds to the tidal
field locally in time, meaning that the halo shape is related to the
instantaneous tides: $\gamma^{\rm scalar}_{ij}(\eta)\propto s_{ij}(\eta)$ and
$\gamma^{\rm GW}_{ij}(\eta)\propto\tau_{ij}(\eta)$, we expect $b_{K}^{\rm
GW}(\eta)=b_{K}^{\rm scalar}(\eta)T(\eta;k_{L})/a(\eta)$ with $T$ defined in
Eq. (10); however this is not the case.
Let us finish this section by considering possible reasons that cause the
difference between the measurements and the ansatz. First, the deviation is
also observed around $k_{L}\sim 10^{-3}~{}h/{\rm Mpc}$ at $z=0$. This might be
inherited from the large difference in the response functions to the scalar
and tensor tides shown in the upper row of Fig. 4, which implies that
relatively large-scale tides are more responsible for halo shapes than halo-
scale tides. Second, the deviation of the measurements from the ansatz gets
larger as $k_{L}$ increases. This trend is also consistent with the responses
of the matter power spectrum in Fig. 4. At the same time, this could be partly
because the approximation we employ in this paper is no longer valid for these
large $k_{L}$. In other words, GWs with larger-$k_{L}$ cannot be seen as a
uniform tidal field even in the halo formation region, given that the
Lagrangian halo radius $R_{M}=\left(4\pi\bar{\rho}_{\rm m}/3M_{\rm
vir}\right)^{1/3}$ is close to the wavelength of GWs. Specifically, ignoring
the curvature of GWs can leads the corrections to $b_{K}^{\rm GW}$, which
scales as ${\cal O}(k_{L}^{2}R_{M}^{2})$. In the case of $M_{\rm
vir}=1.0\times 10^{13}M_{\odot}/h$ and $k_{L}=0.1~{}h/{\rm Mpc}$, for example,
the correction is about $(0.1\cdot 3)^{2}\sim 0.1$, which cannot be
negligible. In order to accurately study the impact of such GWs, a different
technique is necessary.
We also note that the results for high-$k_{L}$ GWs can be verified by
analyzing the standard $N$-body simulation with staring an anisotropic initial
power spectrum because the effect of high-$k_{L}$ GWs is almost encoded in the
initial conditions. As shown in Fig. 2, the anisotropic scale factors induced
by high-$k_{L}$ GWs ($k_{L}\gtrsim 10^{-2}~{}h/{\rm Mpc}$) reach their
asymptote already at $z_{\rm ini}$ and thus their effect in the late time can
be absorbed into the overall time-independent rescaling of scale factors,
implying that it does not have a physical effect on structure formation. In
other words, for high-$k_{L}$ GWs halos are formed in absence of a large-scale
tidal field but with anisotropic small-scale modes, while for low-$k_{L}$ GWs
halos are formed in an evolving tidal field, in analogy to the difference in
$b_{1}$ (the response to the large-scale density field) and $b_{\phi}$ (the
response to the change of $\sigma_{8}$) in the density case Baldauf _et al._
(2011); Desjacques _et al._ (2018). This study is beyond our paper and we
leave it for future work.
## VII Discussion
In this paper, we have quantified the impact of GWs on large-scale structure
by means of the tidal separate universe simulation. To the best of our
knowledge, this is the first study for the effect of GWs on nonlinear
structure formation using $N$-body simulations. We found that GWs indeed
influence nonlinear structure formation in both the clustering statistics and
the intrinsic alignment of halo shapes. Our main finding is that the impact of
GWs on large-scale structure can be well described by combining the impact of
the scalar tides with the perturbation theory. Specifically, the halo tidal
bias and linear shape bias induced by GWs, $b_{s^{2}}^{\rm GW}$ and
$b_{K}^{\rm GW}$ respectively, can be approximated by $b_{X}^{\rm
GW}(k_{L})=\frac{7}{4}\alpha(k_{L})b_{X}^{\rm scalar}$ where
$X=\\{s^{2},K\\}$.
Figure 12: Shape auto-power spectra from GWs at $z=1$ assuming $r=0.1$. The
blue, orange, and green lines show the monopole of the $E$-mode auto-, the
monopole of the $B$-mode auto-, and the dipole of the $EB$ cross-power
spectra, respectively. For the $EB$ cross power spectrum we also assume a
maximally parity-violating case, i.e., $\chi(k)=1$. The blue dashed line
depicts the shape noise.
Let us discuss possible observables that could be used to probe GWs from LSS.
As GWs affect halo shapes at linear order shown in Sec. VI, the simplest probe
would be the shape correlation, namely the intrinsic alignment from GWs
Schmidt and Jeong (2012a); Schmidt _et al._ (2014); Biagetti and Orlando
(2020). Although the shape correlation is primarily affected by the scalar
perturbations, we can use the $E/B$-decomposition to distinguish the GWs
contribution from the scalar contribution at linear order. Under the flat-sky
approximation and assuming the line-of-sight direction, $\hat{n}$, is parallel
to the $z$-axis, we can define $E$-mode and $B$-mode via
$\displaystyle E({\bf k},\hat{n})\pm iB({\bf k},\hat{n})\equiv{{}_{\pm
2}\gamma}({\bf k},\hat{n})e^{\mp 2i\phi_{k}},$ (76)
where
$\displaystyle{{}_{\pm 2}\gamma}({\bf k},\hat{n})\equiv
m_{\mp}^{i}(\hat{n})m_{\mp}^{j}(\hat{n})\gamma_{ij}({\bf k}),$ (77)
with ${\bf m}_{\pm}\equiv(1,\mp i,0)/\sqrt{2}$. At linear order, the scalar
perturbations only induce $E$-mode with vanishing $B$-mode Hirata and Seljak
(2004). On the other hand, GWs generate both $E$-mode and $B$-mode as
$\displaystyle E({\bf k}_{L},\hat{n})$ $\displaystyle=b_{K}^{\rm
GW}(k_{L})\frac{1}{8}(1+\mu_{L}^{2})\sum_{\lambda}h_{(\lambda)}({\bf k}_{L}),$
(78) $\displaystyle B({\bf k}_{L},\hat{n})$ $\displaystyle=-b_{K}^{\rm
GW}(k_{L})\frac{i}{2}\mu_{L}\sum_{\lambda}\frac{\lambda}{2}h_{(\lambda)}({\bf
k}_{L}),$ (79)
where we have used Eq. (70) and $\mu_{L}\equiv\hat{k}_{L}\cdot\hat{n}$. Note
that here we project the halo shapes onto two-dimensional plane but do not
project their position; we can combine photometric and spectroscopic surveys
to get the projected shapes and their three-dimensional positions (e.g.,
Okumura and Taruya, 2020; Kurita _et al._ , 2021; Akitsu _et al._ , 2021b;
Kurita and Takada, 2022). Defining the power spectra for $E$\- and $B$-modes
via
$\displaystyle\langle X({\bf k})Y^{*}({\bf
k}^{\prime})\rangle\equiv(2\pi)^{3}\delta^{(3)}_{\rm D}({\bf k}-{\bf
k}^{\prime})P_{XY}({\bf k}),$ (80)
we obtain
$\displaystyle P_{EE}(k_{L},\mu_{L};z)$
$\displaystyle=\frac{1}{64}(1+\mu_{L}^{2})^{2}(b_{K}^{\rm
GW}(z))^{2}P_{h}(k_{L})\simeq\frac{49}{1024}(1+\mu_{L}^{2})^{2}\alpha^{2}(k_{L};z)(b_{K}^{\rm
scalar}(z))^{2}P_{h}(k_{L}),$ (81) $\displaystyle P_{BB}(k_{L},\mu_{L};z)$
$\displaystyle=\frac{1}{4}\mu_{L}^{2}(b_{K}^{\rm
GW}(z))^{2}P_{h}(k_{L})\simeq\frac{49}{64}\mu_{L}^{2}\alpha^{2}(k_{L};z)(b_{K}^{\rm
scalar}(z))^{2}P_{h}(k_{L}),$ (82)
and $P_{EB}=0$ for unpolarized GWs, where we have used Eq. (75). Considering
the chiral GWs, there appears a non-vanishing $EB$ correlation as
$\displaystyle P_{EB}(k_{L},\mu;z)$
$\displaystyle=\frac{i}{16}\mu_{L}(1+\mu_{L}^{2})(b_{K}^{\rm
GW}(z))^{2}\chi(k_{L})P_{h}(k_{L})\simeq
i\frac{49}{256}\mu_{L}(1+\mu_{L}^{2})\alpha^{2}(k_{L};z)(b_{K}^{\rm
scalar}(z))^{2}\chi(k_{L})P_{h}(k_{L}),$ (83)
where $\chi$ is defined in Eq. (7). Notice that $P_{h}(k_{L})$ is the
primordial power spectrum of the tensor mode.
In Fig. 12, we plot the lowest order moment of the multipoles of these spectra
at $z=1$, i.e., the monopole of the $EE$ and $BB$ spectra and the dipole of
the $EB$ spectrum, assuming $b_{K}^{\rm scalar}=0.1$, $r=0.1$, and $\chi(k)=1$
as a demonstration. On large scales, the suppression comes from the behaviour
of $\alpha(k_{L})$ while on small scales the shape obeys a power-law with
$P(k_{L})\propto k_{L}^{-3}$ because $\alpha(k_{L})$ does not change much. We
also display the shape noise as the dashed line, assuming
$\sigma^{2}_{\gamma}=0.2$ and $\bar{n}_{g}=5.0\times 10^{-4}~{}(h/{\rm
Mpc})^{3}$. As is evident, for all cases the shape noise contribution is much
greater than the expected signals, meaning that the detection of GWs using
these spectra is very challenging. There should also be the contributions of
the scalar-mode both in $EE$ (at linear order) and $BB$ (at one-loop order),
though they are absent in the $EB$ spectrum Biagetti and Orlando (2020).
666Ref. Biagetti and Orlando (2020) argued that the shape noise is absent in
the $EB$ power spectrum, making it a cleaner probe of the chiral GWs. Although
this is true at the signal level, the covariance of the $EB$ spectrum includes
the $EE$ and $BB$ spectra with the shape noise. Thus even for the $EB$ power
spectrum the detectability is limited by the shape noise. Still, this sort of
probes allows us to put the upper limit on the amplitude of GWs that are
generated after the recombination or the reionization, which is not
constrained by the CMB. Specifically, we could obtain the 1-$\sigma$ error on
$r$ of order $\sigma(r)\sim 10^{3}$ at $k\sim 10^{-3}~{}h/{\rm Mpc}$, implying
the total energy spectrum of GWs, $\Omega_{\rm GW}(k)$, could be constrained
as $\Omega_{\rm GW}(k)\lesssim 10^{-8}$ at $k\sim 10^{-3}~{}h/{\rm Mpc}$ from
the current galaxy surveys.
The density-density-shape bispectrum also involves the GWs contribution. For
instance, the tree-level bispectrum of $\delta_{\rm h}$-$\delta_{\rm h}$-$B$
in squeezed limit ($k_{L}\to 0$) is found to be
$\displaystyle\lim_{k_{L}\to 0}iB^{\rm grav.}_{{\rm hh}B}(k,k_{L};z)=$
$\displaystyle\frac{1}{2}b_{1}(z)b_{K}^{\rm
GW}(k_{L};z)\left[\left(2b_{1}(z)\alpha(k_{L};z)+2b^{\rm
GW}_{s^{2}}(k_{L};z)+b_{1}(z)\beta(k_{L};z)\frac{\partial}{\partial\ln{k}}\right)P_{\rm
lin}(k;z)\right]$ $\displaystyle\hskip
241.84842pt\times\mu_{L}\hat{k}^{i}\hat{k}^{j}\sum_{\lambda}\frac{\lambda}{2}e^{(\lambda)}_{ij}(\hat{k}_{L})P_{h_{(\lambda)}}(k_{L})$
(84) $\displaystyle\simeq$ $\displaystyle\frac{7}{8}b_{1}(z)b_{K}^{\rm
scalar}(z)\alpha(k_{L};z)\left[\left(2(b_{1}(z)+\frac{7}{4}b^{\rm
scalar}_{s^{2}}(z))\alpha(k_{L};z)+b_{1}(z)\beta(k_{L};z)\frac{\partial}{\partial\ln{k}}\right)P_{\rm
lin}(k;z)\right]$ $\displaystyle\hskip
241.84842pt\times\mu_{L}\hat{k}^{i}\hat{k}^{j}\sum_{\lambda}\frac{\lambda}{2}e^{(\lambda)}_{ij}(\hat{k}_{L})P_{h_{(\lambda)}}(k_{L}),$
(85)
where we neglect the contributions from the projection effect Schmidt and
Jeong (2012b) and the redshift-space distortion Kaiser (1987). Notice again
that here $P_{h_{(\lambda)}}(k_{L})$ is the primordial power spectrum and all
the redshift dependence is encoded in the bias coefficients ($b_{K}^{\rm
GW}(k_{L};z)$ and $b^{\rm GW}_{s^{2}}(k_{L};z)$), $\alpha(k_{L};z)$, and
$\beta(k_{L};z)$. The superscript “grav.” emphasises that this bispectrum is
induced by the gravitational interaction in the late time universe. In other
words, there could be an additional contribution from the primordial universe,
e.g., the scalar-scalar-tensor non-Gaussianity Endlich _et al._ (2013);
Domènech _et al._ (2017). In principle we can directly observe the scalar-
scalar-tensor non-Gaussianity by looking at this density-density-shape
bispectrum.
Another promising observable to probe GWs from LSS is the quadrupolar
anisotropic imprint in the local density power spectrum discussed in Sec. IV
and Sec. V. One can construct the optimal quadratic estimator for GWs by using
the anisotropic imprint in the local power spectrum Masui and Pen (2010);
Jeong and Kamionkowski (2012); Dimastrogiovanni _et al._ (2014); Dai _et
al._ (2013); Masui _et al._ (2017). Our result can be used to increase
$k_{\rm max}$ of the estimator in this method, allowing for the improved
detectability. As we demonstrated for the first time how GWs affect the biased
tracer, a more realistic estimator for the biased tracer can be available.
Taking the cross correlation with the CMB would also be valuable to improve
the detectability Dodelson (2010); Alizadeh and Hirata (2012); Chisari _et
al._ (2014); Philcox and Johnson (2022). We leave the detailed investigations
on these possibilities for future work.
Let us conclude by mentioning one potentially significant effect of GWs on LSS
observables. In this paper, we focused on the effect of GWs on the cold dark
matter (CDM) perturbations and ignored the effect on the photon-baryon fluid
in the early universe. However, given that the biased tracers such as galaxies
and halos trace the CDM-baryon perturbation, this could leave a distinctive
signature in LSS observables as well. We will investigate it in future work.
###### Acknowledgements.
We thank Giovanni Cabass, William Coulton, and Matias Zaldarriaga for
insightful discussions and Fabian Schmidt for valuable comments on the draft.
KA is supported by JSPS Overseas Research Fellowships. TO acknowledges support
from the Ministry of Science and Technology of Taiwan under Grants Nos. MOST
110-2112-M-001-045- and 111-2112-M-001-061- and the Career Development Award,
Academia Sinina (AS-CDA-108-M02) for the period of 2019 to 2023. Numerical
computation was carried out on the Helios and the Typhon cluster at the
Institute for Advanced Study and Popeye-Simons cluster at San Diego
Supercomputer Center. The Flatiron Institute is supported by the Simons
Foundation.
## Appendix A From global FLRW coordinates to local CFC coordinates
Here we outline the mapping from the global perturbed FLRW metric to the local
CFC coordinates. More detailed discussion can be found in Refs. Pajer _et
al._ (2013); Schmidt _et al._ (2014); Dai _et al._ (2015b). We start from
the perturbed FLRW metric,
$\displaystyle ds^{2}=a^{2}(\eta)\left[-(1+h_{00}(\eta,{\bf
x}))d\eta^{2}+\left(\delta^{\textsc{k}}_{ij}+h_{ij}(\eta,{\bf
x})\right)dx^{i}dx^{j}\right],$ (86)
where we neglect $h_{0i}$ component. The coordinate transformation is given by
$\displaystyle x^{0}$
$\displaystyle=x_{F}^{0}+\frac{1}{2}\int_{0}^{x^{0}_{F}}h_{00}(\tilde{\eta})\differential\tilde{\eta}+v_{i}x_{F}^{i}-\frac{1}{4}h^{\prime}_{ij}x_{F}^{i}x_{F}^{j},$
(87) $\displaystyle x^{i}$
$\displaystyle=x_{F}^{i}+v^{i}(x^{0}_{F}-\eta_{F})-\frac{1}{2}h^{i}_{\
j}x_{F}^{j}-\frac{1}{4}\left[h^{i}_{\ j,k}+h^{i}_{\ k,j}-h_{jk}^{\ \
,i}\right]x_{F}^{j}x_{F}^{k},$ (88)
where $v_{i}$ is the coordinate velocity of the central geodesic and
$h_{\mu\nu}$ is evaluated along the central geodesics. The CFC metric can be
obtained by using the transformation law of the metric,
$g_{\mu\nu}^{F}(\eta_{F},{\bf x}_{F})=g_{\alpha\beta}(\eta,{\bf x})(\partial
x^{\alpha}/\partial x^{\mu}_{F})(\partial x^{\beta}/\partial x^{\nu}_{F})$,
yielding
$\displaystyle g_{00}^{F}$
$\displaystyle=-a_{F}^{2}(\eta_{F})\left[1-\frac{1}{2}\left(h_{00,ij}+\mathcal{H}h^{\prime}_{ij}+h^{\prime\prime}_{ij}\right)x_{F}^{i}x_{F}^{j}\right],$
(89) $\displaystyle a_{F}(x^{0}_{F})$
$\displaystyle=a\left(\eta=\eta_{F}+\frac{1}{2}\int_{0}^{\eta_{F}}h_{00}(\tilde{\eta})\differential\tilde{\eta}\right).$
(90)
In Eq. (89), setting $h_{00}=0$ coincides to the main text (Eqs. (9)-(10)). On
the other hand, the case where $h_{00}=-2\Phi_{L}$ and $h_{ij}=0$ corresponds
to the usual tidal separate universe simulation picture (see App. B).
## Appendix B The tidal responses to the scalar tides from tidal separete
universe simulations
Figure 13: Growth response of matter auto-power spectrum to the scalar tides,
$R^{\rm scalar;mm}_{\rm growth}(k)$ for various redshifts, measured from the
simulations. The blue dashed line corresponds to the perturbation theory
prediction, $R^{\rm scalar}_{\rm growth;mm}=8/7$ Akitsu _et al._ (2017).
In this appendix, we summarize the tidal responses to the scalar tidal field.
These include not only the tidal response of the matter auto-power spectrum
and the linear shape bias, which are already presented in Refs. Akitsu _et
al._ (2021a); Stücker _et al._ (2021), but also the tidal responses of halo-
matter and halo auto-power spectra and the halo tidal bias.
Figure 14: Growth responses involving the halo density field as a function of
$k$. In the left panels the responses for various resfhits are plotted while
in the right panels the responses for various halo masses are plotted. The top
panels show the response of the halo-matter cross-power spectrum, $R^{\rm
scalar}_{\rm hm;growth}(k)$, The middle panels show the response of the halo
auto-power spectrum, $R^{\rm scalar}_{\rm hh;growth}(k)$, normalized by the
halo auto-power spectrum with the shot noise, and the bottom panels show the
response of the halo auto-power spectrum, $R^{\rm scalar}_{\rm hh;growth}(k)$,
normalized by the halo auto-power spectrum without the shot noise.
First, we sketch the construction of the tidal separate universe in the scalar
tides case, focusing on the difference to the GWs case, although we refer the
reader to Ref. Akitsu _et al._ (2021a) for details. The difference between
the GWs case and scalar-tides case in the induced tidal field in the local
region, $\tau_{ij}$, results in the different initial conditions and
anisotropic scale factors. In the scalar-tides case, the induced tides become
$\displaystyle\tau_{ij}(\eta)=-\frac{1}{2}h_{00,ij}$
$\displaystyle=\Phi_{L,ij}$ (91) $\displaystyle=\frac{1}{2}\Omega_{\rm
m}(\eta)\mathcal{H}^{2}\delta_{L}(\eta)\delta^{\textsc{k}}_{ij}+\frac{3}{2}\Omega_{\rm
m}(\eta)\mathcal{H}^{2}s_{L,ij}(\eta),$ (92)
where we have used Poisson equation and decomposed into the trace (large-scale
overdensity, $\delta_{L}$) and the traceless (large-scale pure tidal field,
$s_{L,ij}$). The construction of the tidal separate universe with the scalar
tides corresponds to replacing Eq. (10) with Eq. (92) and repeating the
analysis in Sec. II.2, Sec. III.1, and Sec. III.2. Focusing on the pure tidal
mode ($s_{L,ij}$), the equation that governs the evolution of the anisotropic
scale factors, $\Delta_{i}$, is now
$\displaystyle\Delta^{\prime\prime}_{i}(\eta)+\mathcal{H}\Delta_{i}^{\prime}(\eta)=-\frac{3}{2}\Omega_{\rm
m}(\eta)\mathcal{H}^{2}s_{L,i}(\eta),$ (93)
whose solution is given by
$\displaystyle\Delta_{i}(\eta)=-s_{L,i}(\eta_{0})\frac{D(\eta)}{D(\eta_{0})}.$
(94)
The perturbative prediction of the growth response is
$\displaystyle\lim_{k\to 0}R_{\rm mm;growth}^{\rm scalar}(k)$
$\displaystyle=\frac{8}{7},$ (95)
where the response with respect to the scalar tides is defined through
$\displaystyle\left.\frac{\differential{\ln P_{\cal
G}}}{\differential{s_{L,ij}}}\right|_{{\bf k}_{\cal
G}}=\left.\frac{\differential{\ln P_{\cal
L}}}{\differential{s_{L,ij}}}\right|_{{\bf k}_{\cal G}}=$
$\displaystyle\left.\frac{\partial\ln P_{\cal L}}{\partial
s_{L,ij}}\right|_{{\bf k}_{\cal L}}+\left.\frac{\partial\ln P_{\cal
L}}{\partial\ln k_{{\cal
L},\ell}}\right|_{s_{L,ij}}\left.\frac{\differential{\ln k_{{\cal
L},\ell}}}{\partial s_{L,ij}}\right|_{{\bf k}_{\cal G}}$ $\displaystyle\equiv$
$\displaystyle\,\hat{k}_{i}\hat{k}_{j}\left[R^{\rm scalar}_{\rm
growth}(k)+R^{\rm scalar}_{\rm dilation}(k)\right],$ (96)
which is analogue to Eq. (49). One important difference of Eq. (96) from Eq.
(49) is that in Eq. (96) we define the response with respect to the scalar
tides at the same epoch rather than the initail epoch. As a result, the
perturbative prediction remains the same for all redshifts. Also note that
these results are independent of the wavenumber of long-modes.
Fig. 13 shows the tidal response of the matter auto-power spectrum to the
scalar tidal field as a function of $k$ for various redshifts, measured from
tidal separate universe simulation with the scalar tides. This should be
contrasted with Fig. 3 in the main text. In Fig. 14 we show the tidal
responses of the halo-matter cross-power spectrum (the top panels), the halo
auto-power spectrum with and without the short noise (the middle and the
bottom panels respectively) for various redshifts (the left panels) and for
various halo masses (the right panels). This is analogue to Figs. 5-6 in the
main text. The differences of the responses for various redshifts and halo
masses on large scales should be explained by the halo biases (the combination
of $b_{1}(z;M)$ and $b^{\rm scalar}_{s^{2}}(z;M)$). The peaky feature appeared
in the halo auto response normalized by the halo auto-power without the shot
noise is due to the non-Poissonian feature of the shot noise term and thus not
physical (see the discussion in the last paragraph in Sec. IV.3).
Figure 15: Halo tidal bias measured from the halo-matter power spectrum
response. The left: The tidal bias as a function of halo mass at $z=0.5$. The
right: The tidal bias as a function of the linear bias $b_{1}$ from various
redshifts and halo masses. The black dashed line displays the Lagrangian
local-in-matter-density (LLIMD) prediction: $b_{s^{2}}^{\rm
scalar}=-\frac{2}{7}(b_{1}-1)$. Figure 16: Linear shape bias induced by the
scalar tides as a function of halo mass for various redshifts.
We measure the halo tidal bias $b^{\rm scalar}_{s^{2}}$ in the same way as Eq.
(66) and Eq. (67) in the main text. The result is presented in Fig. 15 where
the left panel shows $b_{s^{2}}^{\rm scalar}$ at $z=0.5$ for various halo
masses and the right panel shows $b_{s^{2}}^{\rm scalar}$ as a function of
$b_{1}$. We compare the result with the Lagrangian local-in-matter-density
(LLIMD) prediction Desjacques _et al._ (2018): $b_{s^{2}}^{\rm
scalar}=-\frac{2}{7}(b_{1}-1)$, which is plotted as the black-dashed line. In
general the LLIMD prediction fails to capture the behaivour of $b_{s^{2}}^{\rm
scalar}$, in particular at high-mass end, which is consistent with Refs.
Lazeyras and Schmidt (2018); Abidi and Baldauf (2018).
In Fig. 16 we show the linear shape bias (or the linear alignment coefficient)
induced by the scalar tides, $b_{K}^{\rm scalar}$, (introduced in Eq. (69)) as
a function of halo mass for various redshifts. The estimator for $b_{K}^{\rm
scalar}$ used here is the same as Eq. (74). Fig. 16 is a counterpart of Fig.
9.
## Appendix C The modification in the drift operator
Figure 17: Ratio of the approximate drift integral to the exact one as a
function of the scale factor. Left: the ratio for different wavenumber of GWs
with the same starting redshift $z_{\rm ini}=99$. Right: the ratio for
different starting redshifts with the same wavenumber of GWs
$k_{L}=0.02~{}h/{\rm Mpc}$.
In the tidal separate universe simulation, the drift operator changes from the
usual $N$-body simulation as
$\displaystyle
x_{i;n+1}=x_{i;n}+\frac{P_{i}}{m}\int^{a_{n+1}}_{a_{n}}\frac{\differential
a}{a_{i}^{2}\mathcal{H}}\equiv x_{i;n}+\frac{P_{i}}{m}~{}{\cal D}^{\rm
exact}_{i;n},$ (97)
where $P_{i}$ is the conjugate momenta of $x_{i}$, $i=x,y,z$ and $n$
represents the time step. In the original Gadget-2 code Springel (2005),
instead of computing the drift integral in Eq. (97) at each time step, first
it prepares the following table
$\displaystyle{\cal D}_{i}[j]\equiv\int^{a[j]}_{a_{\rm
ini}}\frac{\differential a}{a_{i}^{2}\mathcal{H}},$ (98)
where $1\leq j\leq N_{\rm drift}$ with $N_{\rm drift}$ being the length of the
table. $a[j]$ is the $j$-th scale factor that is sampled equally spaced in
logarithm from $a_{\rm ini}$ to $a=1$, regardless of the actual time step. The
actual drift integral at each time step is then evaluated by linearly
interpolating this drift table as
$\displaystyle{\cal D}^{\rm approx.}_{i;n}=$
$\displaystyle\int^{a_{n+1}}_{a_{\rm ini}}\frac{\differential
a}{a_{i}^{2}\mathcal{H}}-\int^{a_{n}}_{a_{\rm ini}}\frac{\differential
a}{a_{i}^{2}\mathcal{H}}$ $\displaystyle\simeq$ $\displaystyle\ {\cal
D}_{i}[j+1]+\left(a_{n+1}-a[j+1]\right)\left({\cal D}_{i}[j+2]-{\cal
D}_{i}[j+1]\right)$ $\displaystyle-{\cal
D}_{i}[j]-\left(a_{n}-a[j]\right)\left({\cal D}_{i}[j+1]-{\cal
D}_{i}[j]\right)$ (99)
with $j=\lfloor n\rfloor$.
This prescription works well for a monotonic integrand, which is the case for
the usual cosmological simulation. We found, however, that this approximation
for the drift operator breaks down for the tidal separate universe simulation
with GWs where the integrand oscillates. Fig. 17 compares the drift factor
evaluated by the above procedure and the direct calculation of the integral in
Eq. (97). Although the default length of the drift table is 1000 we increased
it to $N_{\rm drift}=200000$. Even with this large table, the approximated
drift factor fails to capture the exact result. Therefore we modified the
drift operator so that at each step the drift integral is directly evaluated
without using the drift table or interpolation. This modification is
particularly important for $k_{L}\gtrsim 0.02~{}h/{\rm Mpc}$ when $z_{\rm
ini}=99$. In fact, without this modification the results suffers from the
artifact.
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|
# A discrete Darboux–Lax scheme for integrable difference equations
X. Fisenko<EMAIL_ADDRESS>S. Konstantinou-Rizos
<EMAIL_ADDRESS>P. Xenitidis<EMAIL_ADDRESS>School of
Mathematics, Computer Science and Engineering, Liverpool Hope University, L19
9JD Liverpool, UK
###### Abstract
We propose a discrete Darboux–Lax scheme for deriving auto-Bäcklund
transformations and constructing solutions to quad-graph equations that do not
necessarily possess the 3D consistency property. As an illustrative example we
use the Adler–Yamilov type system which is related to the nonlinear
Schrödinger (NLS) equation [19]. In particular, we construct an auto-Bäcklund
transformation for this discrete system, its superposition principle, and we
employ them in the construction of the one- and two-soliton solutions of the
Adler–Yamilov system.
PACS numbers: 02.30.Ik, 02.90.+p, 03.65.Fd.
Mathematics Subject Classification 2020: 37K60, 39A36, 35Q55, 16T25.
Keywords: Darboux transformations, Bäcklund transformations, quad-graph
equations, partial
Keywords: difference equations, integrable lattice equations, 3D-consistency,
soliton solutions.
## 1 Introduction
It has become understood over the past few decades that integrable systems of
partial difference equations (P$\Delta$Es) are interesting in their own right,
see for instance [15] and references therein. On the one hand, they may model
various natural phenomena, as well as processes in industry and the IT sector.
On the other hand, they have many interesting algebro-geometric properties [6,
15], and are related to many important equations of Mathematical Physics such
as the Yang–Baxter equation and the tetrahedron equation [1, 9, 16, 25].
Moreover, they can be derived from the discretisation of nonlinear partial
differential equations (PDEs). One such approach is provided by the Darboux
transformations of integrable nonlinear PDEs of evolution type [19, 30];
namely, the resulting relations from the permutability of two Darboux
transformations can be interpreted as a P$\Delta$E.
In this paper, we focus on a special class of P$\Delta$Es, the so-called quad-
graph equations, or systems thereof. Quad-graph systems are equations of
P$\Delta$Es defined on an elementary quadrilateral of the two-dimensional
lattice. In particular, they are systems of the form
$Q(f_{00},f_{10},f_{01},f_{11};\alpha,\beta)=0,$ (1)
where $Q$ is a function of its arguments $f_{ij}$, $i,j=0,1$, and may also
depend on parameters $\alpha$ and $\beta$. Schematically, equation (2) can be
represented on the square, where $f_{00}$, $f_{10}$, $f_{01}$ and $f_{11}$ are
placed on vertices of the square and the parameters $\alpha$, $\beta$ are
placed on the edges as in Figure 2. Equation (2) can be thought as a partial
difference equation (P$\Delta$E) by identifying $f$ with a function of two
discrete variables $n,m\in\mathbb{Z}$, and $f_{ij}$ with its shifts in the $n$
and $m$ direction, i.e. $f_{ij}=f(n+i,m+j)$.
.8 (0 0) (1 0) (1 1) (0 1) (0 0)
(0 0) f:0.0 r:0.05 (1 0) f:0.0 r:0.05 (0 1) f:0.0 r:0.05 (1 1) f:0.0 r:0.05
(-.1 -.3) $f$ (.9 -.3) $f_{10}$ (-.1 1.1) $f_{01}$ (.9 1.1) $f_{11}$ (0.44
-.2) $\alpha$ (-.15 .45) $\beta$
Figure 1: Quad–graph equation.
Quad-graph equations have attracted the interest of many researchers in the
field of discrete integrable systems, see [15] for a review. This led to the
developement of methods for solving them, e.g. [2, 13, 15, 24, 26],
classifying them, e.g. [3, 7, 28], analysing their integrability properties,
e.g. [22, 27, 21, 29], and relating them to the theory of Yang–Baxter and
tetrahedron maps, e.g. [11, 25]. Of particular interest are the so-called
3D-consistent equations which can be extended to the three-dimensional lattice
in a consistent way. For such equations a Lax representation and a Bäcklund
transformation can be constructed in a systematic manner by employing the 3D
consistency of the system [15].
In this paper, we propose a discrete Darboux–Lax scheme for deriving Bäcklund
transformations and constructing solutions for quad-graph equations which are
not necessarily 3D consistent but have a Lax representation. More precisely,
we employ gauge-type transformations of the Lax pair and demonstrate how they
give rise to Bäcklund transformations for the discrete system. As an
illustrative example we use the Adler–Yamilov system which was derived as a
discretisation of the NLS equation via a Darboux transformation in [19].
The paper is organised as follows. In the next section we provide all the
necessary definitions for the text to be self-contained. More specifically,
after fixing our notation, we give the definitions of the integrability of a
system of P$\Delta$Es in the sense of the existence of a Lax pair, and of the
Bäcklund transformation. In Section 3 we present the discrete Darboux–Lax
scheme for constructing Bäcklund transformations and solving integrable
P$\Delta$Es which do not necessarily possess the 3D consistency property. In
Section 4 we apply our results to the Adler–Yamilov system given in [19] and
derive the one- and two-soliton solutions by using the associated Bäcklund
transformation and the corresponding Bianchi diagram. Finally, the closing
section contains a summary of the obtained results and a discussion on how
they can be extended and generalised.
## 2 Integrability of difference equations
Let us start this section by introducing our notation. In what follows we deal
with systems of equations which involve unknown functions depending on the two
discrete variables $n$ and $m$. The dependence of a function $f=f(n,m)$ on
these variables will be denoted with indices in the following way.
$f_{ij}=f(n+i,m+j),\quad{\mbox{for all }}i,j\in{\mathbb{Z}}$
We also denote by $\mathcal{S}$ and $\mathcal{T}$ the shift operators in the
$n$ and $m$ direction, respectively. Their action on a function $f$ is defined
as
${\mathcal{S}}^{k}(f_{00})=f_{k0},\quad{\mathcal{T}}^{\ell}(f_{00})=f_{0\ell}.$
We denote vectors with bold face letters, e.g.
$\textbf{f}_{00}=(f_{00}^{(1)},f_{00}^{(2)},\ldots f_{00}^{(k)})$. For systems
of equations we also use bold letters. In particular, a system of quad-graph
equations will be denoted by
$\textbf{Q}({\textbf{f}}_{00},\textbf{f}_{10},\textbf{f}_{01},\textbf{f}_{11})=0.$
(2)
Finally, we denote matrices with roman uppercase letters. For instance
$\rm{L}(\mathbf{f}_{00},\mathbf{f}_{10};\alpha,\lambda)$ denotes a matrix with
elements depending on $\mathbf{f}_{00}$, $\mathbf{f}_{10}$, $\alpha$ and
$\lambda$. The semicolon in the arguments of the matrix is used to separate
the fields $(\mathbf{f}_{00},\mathbf{f}_{10})$ from the parameters $\alpha$,
$\lambda$. By $\lambda$ we denote the spectral parameter throughout the text.
With our notation, let ${\rm{{L}}}(\mathbf{f}_{00},\mathbf{f}_{10};\lambda)$
and ${\rm{M}}(\mathbf{f}_{00},\mathbf{f}_{01};\lambda)$ be two $k\times k$
invertible matrices which depend on a function $\bf{f}$ and the spectral
parameter $\lambda$.111Matrices ${\rm{L}}$, ${\rm{M}}$ may also depend on
parameters but, as they do not play any role in our discussion in this and the
following section, we suppress this dependence. Let also $\Psi=\Psi(n,m)$ be
an auxiliary $k\times k$ matrix, and consider the following overdetermined
linear system.
${\cal{S}}(\Psi)={\rm{L}}(\mathbf{f}_{00},\mathbf{f}_{10};\lambda)\Psi,\qquad{\cal{T}}(\Psi)={\rm{M}}(\mathbf{f}_{00},\mathbf{f}_{01};\lambda)\Psi.$
(3)
For given $\bf{f}$, this system has a solution $\Psi$ provided that the two
equations are consistent, i.e. the compatibility condition
${\cal{T}}\left({\cal{S}}(\Psi)\right)={\cal{S}}\left({\cal{T}}(\Psi)\right)$
holds. The latter condition can be written explicitly as
${\rm{L}}(\mathbf{f}_{01},\mathbf{f}_{11};\lambda){\rm{M}}(\mathbf{f}_{00},\mathbf{f}_{01};\lambda)={\rm{M}}(\mathbf{f}_{10},\mathbf{f}_{11};\lambda){\rm{L}}(\mathbf{f}_{00},\mathbf{f}_{10};\lambda).$
(4)
If the above equation holds if and only if $\bf{f}$ satisfies (2), then we say
that system of P$\Delta$Es (2) is _integrable_ , system (3) is a _Lax pair_
for (2), and equation (4) is called a _Lax representation_ for (2). Moreover,
matrices ${\rm{{L}}}(\mathbf{f}_{00},\mathbf{f}_{10};\lambda)$ and
${\rm{M}}(\mathbf{f}_{00},\mathbf{f}_{01};\lambda)$ are referred to as _Lax
matrices_ , and without loss of generality we assume that they have constant
determinants.
We close this section by giving the definition of Bäcklund transformation for
quad-graph equations. Such transformations are related to the notion of
integrability as it will become evident in the next section where we explore
their connection to Lax pairs via the Darboux–Lax scheme.
###### Definition 2.1.
Let
${\bf{Q}}[{\bf{f}}]:={\bf{Q}}({\bf{f}}_{00},{\bf{f}}_{10},{\bf{f}}_{01},{\bf{f}}_{11})=0$
and
${\bf{P}}[{\bf{g}}]:={\bf{P}}({\bf{g}}_{00},{\bf{g}}_{10},{\bf{g}}_{01},{\bf{g}}_{11})=0$
be two systems of quad-graph equations. Let also
$\mathcal{B}(\mathbf{f}_{00},\mathbf{f}_{10},\mathbf{f}_{01},\mathbf{g}_{00},\mathbf{g}_{10},\mathbf{g}_{01};\varepsilon)=0$
(5)
be a system of P$\Delta$Es. If system $\mathcal{B}=0$ can be integrated for
$\bf{g}$ provided that $\bf{f}$ is a solution of $\mathbf{Q}[{\bf{f}}]=0$, and
the resulting ${\bf{g}}(n,m)$ is a solution to $\mathbf{P}[{\bf{g}}]=0$, and
vice versa, then system (5) is called a (hetero-) Bäcklund transformation for
equations ${\bf{Q}}[{\bf{f}}]=0$ and ${\bf{P}}[{\bf{g}}]=0$. If
$\mathbf{Q}[{\bf{a}}]=\mathbf{P}[{\bf{a}}]$, then (5) is called an auto-
Bäcklund transformation for equation ${\bf{Q}}[{\bf{f}}]=0$.
## 3 Discrete Darboux–Lax scheme
It is well known that for quad-graph systems which possess the 3D consistency
property a Lax representation can be derived algorithmically [23, 5, 8, 15]
and a Bäcklund transformation can be constructed systematically [2, 15].
However, there do exist quad-graph systems which have a Lax pair but do not
possess the 3D consistency property. For this kind of systems we propose here
a scheme for constructing Darboux and Bäcklund transformations. It should be
emphasized here that this scheme works for _any_ system of difference
equations irrespectively of their 3D consistency.
For the integrable quad-graph system
$\mathbf{Q}(\mathbf{f}_{00},\mathbf{f}_{10},\mathbf{f}_{01},\mathbf{f}_{11})=0\quad\Longleftrightarrow\quad{\rm{L}}(\mathbf{f}_{01},\mathbf{f}_{11};\lambda){\rm{M}}(\mathbf{f}_{00},\mathbf{f}_{01};\lambda)={\rm{M}}(\mathbf{f}_{10},\mathbf{f}_{11};\lambda){\rm{L}}(\mathbf{f}_{00},\mathbf{f}_{10};\lambda),$
(6)
we define the discrete Darboux transformation as follows.
###### Definition 3.1.
A discrete Darboux transformation for the integrable P$\Delta$E (6) is a
gauge-like, spectral parameter-dependent transformation that leaves Lax
matrices $\rm{L}$ and $\rm{M}$ covariant. That is, a transformation which
involves an invertible matrix $\rm{B}$ such that
$\displaystyle{\rm{L}}(\mathbf{f}_{00},\mathbf{f}_{10};\lambda)\longmapsto{\rm{L}}(\tilde{\mathbf{f}}_{00},\tilde{\mathbf{f}}_{10};\lambda)={\cal{S}}({\rm{B}}){\rm{L}}(\mathbf{f}_{00},\mathbf{f}_{10};\lambda){\rm{B}}^{-1},$
(7a)
$\displaystyle{\rm{M}}(\mathbf{f}_{00},\mathbf{f}_{01};\lambda)\longmapsto{\rm{M}}(\tilde{\mathbf{f}}_{00},\tilde{\mathbf{f}}_{01};\lambda)={\cal{T}}({\rm{B}}){\rm{M}}(\mathbf{f}_{00},\mathbf{f}_{01};\lambda){\rm{B}}^{-1}.$
(7b)
A consequence of the above definition is the following proposition.
###### Proposition 3.2.
The Darboux transformation maps fundamental solutions of the linear system
${\cal{S}}(\Psi)={\rm{L}}(\mathbf{f}_{00},\mathbf{f}_{10};\lambda)\Psi,\quad{\cal{T}}(\Psi)={\rm{M}}(\mathbf{f}_{00},\mathbf{f}_{01};\lambda)\Psi,$
(8)
to fundamental solutions of the linear system
${\cal{S}}(\tilde{\Psi})={\rm{L}}(\tilde{\mathbf{f}}_{00},\tilde{\mathbf{f}}_{10};\lambda)\tilde{\Psi},\quad{\cal{T}}(\tilde{\Psi})={\rm{M}}(\tilde{\mathbf{f}}_{00},\tilde{\mathbf{f}}_{01};\lambda)\tilde{\Psi},$
(9)
via the relation $\tilde{\Psi}={\rm{B}}\Psi$.
###### Proof.
Let $\Psi=\Psi(n,m)$ be a fundamental solution of the linear problem (8). We
set $\tilde{\Psi}={\rm{B}}\Psi$ and then we shift in the $n$ direction to find
that
${\cal{S}}(\tilde{\Psi})={\cal{S}}({\rm{B}}){\cal{S}}(\Psi)\stackrel{{\scriptstyle\eqref{linear-
sys-1}}}{{=}}{\cal{S}}({\rm{B}}){\rm{L}}(\mathbf{f}_{00},\mathbf{f}_{10};\lambda)\Psi\stackrel{{\scriptstyle\eqref{Darboux-
def-L}}}{{=}}{\rm{L}}(\tilde{\mathbf{f}}_{00},\tilde{\mathbf{f}}_{10};\lambda){\rm{B}}\Psi={\rm{L}}(\tilde{\mathbf{f}}_{00},\tilde{\mathbf{f}}_{10};\lambda)\tilde{\Psi}.$
Similarly, starting with $\tilde{\Psi}={\rm{B}}\Psi$ we shift in the $m$
direction and employ (8) and (7b) to find that
${\cal{T}}(\tilde{\Psi})={\rm{M}}(\tilde{\mathbf{f}}_{00},\tilde{\mathbf{f}}_{01};\lambda)\tilde{\Psi}$.
Moreover, the solution $\tilde{\Psi}$ is fundamental, since $\Psi$ is
fundamental and $\det(\tilde{\Psi})=\det{\rm{B}}\det{\Psi}\neq 0$. ∎
Using the above definition of the discrete Darboux transformation and
corresponding Darboux matrix, we propose the following approach for the
construction of a Darboux matrix and Bäcklund transformation, as well as for
the derivation of the superposition principle for the Bäcklund transformation.
* •
We start by assuming an initial form for matrix $\rm{B}$. The simplest
assumption we can make is that matrix ${\rm{B}}$ depends linearly on the
spectral parameter, i.e.
${\rm{B}}=\lambda{\rm{B}}^{(1)}+{\rm{B}}^{(0)},$ (10)
where matrices ${\rm{B}}^{(1)}$ and ${\rm{B}}^{(0)}$ do not depend on
$\lambda$.
* •
We determine the elements of these two matrices by employing equations (7)
written as
${\rm{L}}(\tilde{\mathbf{f}},\tilde{\mathbf{f}}_{10};\lambda){\rm{B}}={\cal{S}}({\rm{B}}){\rm{L}}(\mathbf{f},\mathbf{f}_{10};\lambda),\quad{\rm{M}}(\tilde{\mathbf{f}},\tilde{\mathbf{f}}_{01};\lambda){\rm{B}}={\cal{T}}({\rm{B}}){\rm{M}}(\mathbf{f},\mathbf{f}_{01};\lambda).$
(11)
In our calculations we also have to take into account that $\det({\rm{B}})$ is
a constant, an obvious consequence of (11) and our assumption that the Lax
matrices have constant determinants.
* •
The derived Darboux matrix will depend in general on the ‘old’ and the ‘new’
fields $\bf{f}$ and $\tilde{\bf{f}}$, as well as the spectral parameter
$\lambda$, and a parameter $\varepsilon$. It may also depend on some
auxilliary function, a potential, $g(n,m)$. That is,
${\rm{B}}=\lambda{\rm{B}}^{(1)}(\mathbf{f}_{00},\tilde{\mathbf{f}}_{00},g;\varepsilon)+{\rm{B}}^{(0)}(\mathbf{f}_{00},\tilde{\mathbf{f}}_{00},g;\varepsilon).$
* •
In the construction of the Darboux matrix (10) there will be some algebraic
relations that define the Darboux matrix elements, as well as some difference
equations for its elements. These difference equations will be of the form
$\mathcal{B}^{(n)}(\mathbf{f}_{00},\mathbf{f}_{10},\tilde{\mathbf{f}}_{00},\tilde{\mathbf{f}}_{10},g;\varepsilon)=0,\qquad\mathcal{B}^{(m)}(\mathbf{f}_{00},\mathbf{f}_{01},\tilde{\mathbf{f}}_{00},\tilde{\mathbf{f}}_{01},g;\varepsilon)=0,$
(12)
and constitute the $n$\- and the $m$-part, respectively, of an auto-Bäcklund
transformation that relates the ‘old’ and the ‘new’ fields. In what follows,
we will denote this transformation simply with
${\mathcal{B}}(\mathbf{f},\tilde{\mathbf{f}},g;\varepsilon)=0$.
* •
The Bianchi commuting diagram, aka superposition principle, for the auto-
Bäcklund transformation (12) follows from the permutation of four Darboux
matrices according to the diagram in Figure 2.
.6 0.01 t:F (0 0) (1.5 1.5) (3 0) (1.5 -1.5) (0 0) (0 0) f:0.0 r:0.08 (1.5
1.5) f:0.0 r:0.08 (3 0) f:0.0 r:0.08 (1.5 -1.5) f:0.0 r:0.08 (.8 .8) (.85 .85)
(2.25 .75) (2.3 .7) (.8 -.8) (.85 -.85) (2.25 -.75) (2.3 -.7) (-0.3 -0.1)
$\bf{f}$ (1.4 1.65) $\tilde{\bf{f}}$ (3.15 -.1)
$\tilde{\hat{\bf{f}}}=\hat{\tilde{\bf{f}}}$ (1.4 -1.85) $\hat{\bf{f}}$ (-0.75
0.9)${\rm{B}}(\mathbf{f},\tilde{\mathbf{f}},g_{1};\varepsilon_{1},\lambda)$
(-0.78
-1.1)${\rm{B}}(\mathbf{f},\hat{\mathbf{f}},g_{2};\varepsilon_{2},\lambda)$
(2.2
0.9)${\rm{B}}(\tilde{\mathbf{f}},\hat{\tilde{\mathbf{f}}},g_{12};\varepsilon_{2},\lambda)$
(2.3
-1.1)${\rm{B}}(\hat{\mathbf{f}},\tilde{\hat{\mathbf{f}}},g_{21};\varepsilon_{1},\lambda)$
Figure 2: Bianchi commuting diagram. It should be noted that $g_{12}\neq
g_{21}$.
More precisely, starting with a solution $\bf{f}$ of (6) we can construct two
new solutions $\tilde{\bf{f}}$ and $\hat{\bf{f}}$ using the Bäcklund
transformations
${\mathcal{B}}(\mathbf{f},\tilde{\mathbf{f}},g_{1};\varepsilon_{1})=0$ and
${\mathcal{B}}(\mathbf{f},\hat{\mathbf{f}},g_{2};\varepsilon_{2})=0$,
respectively. Then, we can use the Bäcklund transformation with initial
solution $\tilde{\bf{f}}$, a new potential $g_{12}$ and parameter
$\varepsilon_{2}$ to derive a new solution $\hat{\tilde{\mathbf{f}}}$, i.e.
${\mathcal{B}}(\tilde{\mathbf{f}},\hat{\tilde{\mathbf{f}}},g_{12};\varepsilon_{2})=0$.
In the same fashion, we can start with $\hat{\bf{f}}$, a new potential
$g_{21}$ and parameter $\varepsilon_{1}$ to derive solution
$\tilde{\hat{\mathbf{f}}}$, i.e.
${\mathcal{B}}(\hat{\mathbf{f}},\tilde{\hat{\mathbf{f}}},g_{21};\varepsilon_{1})=0$.
By requiring $\tilde{\hat{\mathbf{f}}}=\hat{\tilde{\bf{f}}}$, according to
Figure 2, we can construct this solution algebraically and it follows from the
commutativity of the corresponding Darboux matrices,
${\rm{B}}(\tilde{\mathbf{f}},\hat{\tilde{\mathbf{f}}},g_{12};\varepsilon_{2},\lambda){\rm{B}}(\mathbf{f},\tilde{\mathbf{f}},g_{1};\varepsilon_{1},\lambda)={\rm{B}}(\hat{\mathbf{f}},\tilde{\hat{\mathbf{f}}},g_{21};\varepsilon_{1},\lambda){\rm{B}}(\mathbf{f},\hat{\mathbf{f}},g_{2};\varepsilon_{2},\lambda).$
(13)
## 4 The Adler–Yamilov system
From the discussion in the previous section it is already obvious that
multidimensional consistency is not essential for this method of derivation of
Bäcklund transformations. In this section, we demonstrate the application of
our scheme using as an illustrative example the Adler–Yamilov system related
to the nonlinear Schrödinger equation [19].
The Adler–Yamilov system can be written as
$p_{10}-p_{01}-\frac{\alpha-\beta}{1+p_{00}q_{11}}p_{00}=0,\qquad
q_{10}-q_{01}+\frac{\alpha-\beta}{1+p_{00}q_{11}}q_{11}=0,$ (14)
where $\alpha$, $\beta$ are complex parameters. Moreover, using matrix
${\rm{L}}(f,g;a,\lambda)=\lambda{\rm{L}}^{(1)}+{\rm{L}}^{(2)}(f,g;a)=\lambda\begin{pmatrix}1&0\\\
0&0\end{pmatrix}+\begin{pmatrix}a+fg&f\\\ g&1\end{pmatrix},$
a Lax pair for (14) can be written as
$\displaystyle{\cal{S}}(\Psi)={\rm{L}}(p_{00},q_{10};\alpha,\lambda)\Psi=\left(\lambda\begin{pmatrix}1&0\\\
0&0\end{pmatrix}+\begin{pmatrix}\alpha+p_{00}q_{10}&p_{00}\\\
q_{10}&1\end{pmatrix}\right)\Psi,$ (15a)
$\displaystyle{\cal{T}}(\Psi)={\rm{L}}(p_{00},q_{01};\beta,\lambda)\Psi=\left(\lambda\begin{pmatrix}1&0\\\
0&0\end{pmatrix}+\begin{pmatrix}\beta+p_{00}q_{01}&p_{00}\\\
q_{01}&1\end{pmatrix}\right)\Psi.$ (15b)
### 4.1 The discrete Darboux–Lax scheme for the Adler–Yamilov system
We start with our choice (10) for the initial form of the Darboux matrix
$\rm{B}$,
${\rm{B}}=\lambda{\rm{B}}^{(1)}+{\rm{B}}^{(0)}=\lambda\begin{pmatrix}f^{(1)}_{00}&f^{(2)}_{00}\\\
f^{(3)}_{00}&f^{(4)}_{00}\end{pmatrix}+\begin{pmatrix}g^{(1)}_{00}&g^{(2)}_{00}\\\
g^{(3)}_{00}&g^{(4)}_{00}\end{pmatrix},$ (16)
and the determining equations (11), which now become
$\displaystyle\left(\lambda{\rm{L}}^{(1)}+{\rm{L}}^{(2)}(\tilde{p}_{00},\tilde{q}_{10};\alpha)\right)\left(\lambda{\rm{B}}^{(1)}+{\rm{B}}^{(0)}\right)=\left(\lambda{\cal{S}}\left({\rm{B}}^{(1)}\right)+{\cal{S}}\left({\rm{B}}^{(0)}\right)\right)\left(\lambda{\rm{L}}^{(1)}+{\rm{L}}^{(2)}(p_{00},q_{10};\alpha)\right),$
(17a)
$\displaystyle\left(\lambda{\rm{L}}^{(1)}+{\rm{L}}^{(2)}(\tilde{p}_{00},\tilde{q}_{01};\beta)\right)\left(\lambda{\rm{B}}^{(1)}+{\rm{B}}^{(0)}\right)=\left(\lambda{\cal{T}}\left({\rm{B}}^{(1)}\right)+{\cal{T}}\left({\rm{B}}^{(0)}\right)\right)\left(\lambda{\rm{L}}^{(1)}+{\rm{L}}^{(2)}(p_{00},q_{01};\beta)\right).$
(17b)
Since all the matrices involved in (17) are independent of $\lambda$, we
collect the coefficients of the different powers of the spectral parameter.
The $\lambda^{2}$ terms yield equations
${\rm{L}}^{(1)}{\rm{B}}^{(1)}={\cal{S}}\left({\rm{B}}^{(1)}\right){\rm{L}}^{(1)},\quad{\rm{L}}^{(1)}{\rm{B}}^{(1)}={\cal{T}}\left({\rm{B}}^{(1)}\right){\rm{L}}^{(1)},$
which lead to
$f^{(1)}_{00}=c_{1}\in{\mathbb{R}},\quad f^{(2)}_{00}=f^{(3)}_{00}=0.$ (18)
The $\lambda$ terms in relations (17) are
${\rm{L}}^{(1)}{\rm{B}}^{(0)}+{\rm{L}}^{(2)}(\tilde{p}_{00},\tilde{q}_{10};\alpha){\rm{B}}^{(1)}={\cal{S}}\left({\rm{B}}^{(1)}\right){\rm{L}}^{(2)}(p_{00},q_{10};\alpha)+{\cal{S}}\left({\rm{B}}^{(0)}\right){\rm{L}}^{(1)},$
${\rm{L}}^{(1)}{\rm{B}}^{(0)}+{\rm{L}}^{(2)}(\tilde{p}_{00},\tilde{q}_{01};\beta){\rm{B}}^{(1)}={\cal{T}}\left({\rm{B}}^{(1)}\right){\rm{L}}^{(2)}(p_{00},q_{01};\beta)+{\cal{T}}\left({\rm{B}}^{(0)}\right){\rm{L}}^{(1)},$
which in view of (18) imply
$f^{(4)}_{00}=c_{2}\in{\mathbb{R}},\quad
g^{(2)}_{00}=c_{1}p_{00}-c_{2}\tilde{p}_{00},\quad
g^{(3)}_{00}=c_{1}\tilde{q}_{00}-c_{2}q_{00},$ (19)
$\left({\cal{S}}-1\right)\left(g^{(1)}_{00}\right)=c_{1}\left(\tilde{p}_{00}\tilde{q}_{10}-p_{00}q_{10}\right),\quad\left({\cal{T}}-1\right)\left(g^{(1)}_{00}\right)=c_{1}\left(\tilde{p}_{00}\tilde{q}_{01}-p_{00}q_{01}\right).$
(20)
The $\lambda$ independent terms,
${\rm{L}}^{(2)}(\tilde{p}_{00},\tilde{q}_{10};\alpha){\rm{B}}^{(0)}={\cal{S}}\left({\rm{B}}^{(0)}\right){\rm{L}}^{(2)}(p_{00},q_{10};\alpha),\quad{\rm{L}}^{(2)}(\tilde{p}_{00},\tilde{q}_{01};\beta){\rm{B}}^{(0)}={\cal{T}}\left({\rm{B}}^{(0)}\right){\rm{L}}^{(2)}(p_{00},q_{01};\beta),$
(21)
determine $g^{(4)}_{00}$ and provide us with the corresponding auto-Bäcklund
transformation. Specifically, the $(2,2)$-elements of the above relations
yield
$\left({\cal{S}}-1\right)\left(g^{(4)}_{00}\right)=c_{2}\left(p_{00}q_{10}-\tilde{p}_{00}\tilde{q}_{10}\right),\quad\left({\cal{T}}-1\right)\left(g^{(4)}_{00}\right)=c_{2}\left(p_{00}q_{01}-\tilde{p}_{00}\tilde{q}_{01}\right).$
(22)
The remaining entries of (21) in view of (20) and (22) become
$\displaystyle
c_{1}\left(p_{10}+p_{00}(\alpha+p_{00}q_{10})\right)-c_{2}\left(\tilde{p}_{10}+\tilde{p}_{00}(\alpha+\tilde{p}_{00}\tilde{q}_{10})\right)=g^{(4)}_{00}\tilde{p}_{00}-g^{(1)}_{00}p_{00},$
(23a) $\displaystyle
c_{1}\left(p_{01}+p_{00}(\beta+p_{00}q_{01})\right)-c_{2}\left(\tilde{p}_{01}+\tilde{p}_{00}(\beta+\tilde{p}_{00}\tilde{q}_{01})\right)=g^{(4)}_{00}\tilde{p}_{00}-g^{(1)}_{00}p_{00},$
(23b)
$\displaystyle\tilde{q}_{10}=\frac{c_{1}\tilde{q}_{00}-c_{2}(q_{00}-\alpha
q_{10})-g^{(4)}_{00}q_{10}}{c_{1}(\alpha+p_{00}q_{10})-c_{2}\tilde{p}_{00}q_{10}-g^{(1)}_{00}},\quad\tilde{q}_{01}=\frac{c_{1}\tilde{q}_{00}-c_{2}(q_{00}-\beta
q_{01})-g^{(4)}_{00}q_{01}}{c_{1}(\beta+p_{00}q_{01})-c_{2}\tilde{p}_{00}q_{01}-g^{(1)}_{00}},$
(23c)
which play the role of the Bäcklund transformation.
Finally we require the determinant of the Darboux matrix $\rm{B}$, which in
view of (18) and (19) can be written as
$\det\left({\rm{B}}\right)=c_{1}c_{2}\lambda^{2}+\left(c_{2}g^{(1)}_{00}+c_{1}g^{(4)}_{00}\right)\lambda+g^{(1)}_{00}g^{(4)}_{00}-\left(c_{1}p_{00}-c_{2}\tilde{p}_{00}\right)\left(c_{1}\tilde{q}_{00}-c_{2}q_{00}\right)$,
to be constant. This requirement implies the relations
$c_{2}g^{(1)}_{00}+c_{1}g^{(4)}_{00}=\kappa,\quad
g^{(1)}_{00}g^{(4)}_{00}-\left(c_{1}p_{00}-c_{2}\tilde{p}_{00}\right)\left(c_{1}\tilde{q}_{00}-c_{2}q_{00}\right)=\varepsilon,\quad\kappa,\varepsilon\in{\mathbb{R}}.$
(24)
It should be noted that the first relation in (24) may also be viewed as a
consequence of (20) and (22).
Summarizing, so far we have shown that the Darboux matrix $\rm{B}$ has the
form
${\rm{B}}=\lambda{\rm{B}}^{(1)}+{\rm{B}}^{(0)}=\lambda\begin{pmatrix}c_{1}&0\\\
0&c_{2}\end{pmatrix}+\begin{pmatrix}g^{(1)}_{00}&c_{1}p_{00}-c_{2}\tilde{p}_{00}\\\
c_{1}\tilde{q}_{00}-c_{2}q_{00}&g^{(4)}_{00}\end{pmatrix},$ (25)
where potentials $g^{(1)}$ and $g^{(4)}$ are determined by (20), (22) and
(24), and the Bäcklund transformation is given by (23).
We consider now two cases: (i) $c_{1}=0$ and $c_{2}\neq 0$, and (ii)
$c_{1}\neq 0$ and $c_{2}=0$.
#### First case: $c_{1}=0$ and $c_{2}\neq 0$
If $c_{1}=0$ and $c_{2}\neq 0$, then we can choose $c_{2}=1$ without loss of
generality. Then equations (20) imply that $g^{(1)}_{00}$ is a constant and we
choose $g^{(1)}_{00}=1$.222It means we choose $\kappa=1$ in the first relation
of (24). Moreover, the second relation in (24) implies
$g^{(4)}_{00}=\tilde{p}_{00}q_{00}+\varepsilon,\quad\varepsilon\in{\mathbb{R}}.$
(26)
With these choices, the Darboux matrix becomes
${\rm{B}}(q_{00},\tilde{p}_{00};\varepsilon)=\lambda\begin{pmatrix}0&0\\\
0&1\end{pmatrix}+\begin{pmatrix}1&-\tilde{p}_{00}\\\
-q_{00}&\varepsilon+\tilde{p}_{00}q_{00}\end{pmatrix},$ (27)
and relations (23) become
$\displaystyle\tilde{p}_{10}=p_{00}+\frac{\alpha-\varepsilon}{1+\tilde{p}_{00}q_{10}}\tilde{p}_{00},\quad\tilde{q}_{10}=q_{00}-\frac{\alpha-\varepsilon}{1+\tilde{p}_{00}q_{10}}q_{10},$
(28a)
$\displaystyle\tilde{p}_{01}=p_{00}+\frac{\beta-\varepsilon}{1+\tilde{p}_{00}q_{01}}\tilde{p}_{00},\quad\tilde{q}_{01}=q_{00}-\frac{\beta-\varepsilon}{1+\tilde{p}_{00}q_{01}}q_{01}.$
(28b)
In view of the above choices and system (28), relations (22) hold identically.
System (28) is an auto-Bäcklund transformation for the Adler–Yamilov system
(14). Indeed, if we shift equations (28a) in the $m$ direction, and equations
(28b) in the $n$ direction, respectively, then it can be readily verified that
the two expressions for $\tilde{p}_{11}$ and the two expressions for
$\tilde{q}_{11}$ coincide modulo the Adler–Yamilov system (14). Conversely, we
rearrange the above system for $p_{00}$, $q_{10}$ and $q_{01}$,
$\displaystyle
q_{10}=\frac{q_{00}-\tilde{q}_{10}}{\alpha-\varepsilon-\tilde{p}_{00}(q_{00}-\tilde{q}_{10})},\quad
p_{00}=\tilde{p}_{10}-\tilde{p}_{00}\left(\alpha-\varepsilon-\tilde{p}_{00}(q_{00}-\tilde{q}_{10})\right),$
(29a) $\displaystyle
q_{01}=\frac{q_{00}-\tilde{q}_{01}}{\beta-\varepsilon-\tilde{p}_{00}(q_{00}-\tilde{q}_{01})},\quad
p_{00}=\tilde{p}_{01}-\tilde{p}_{00}\left(\beta-\varepsilon-\tilde{p}_{00}(q_{00}-\tilde{q}_{01})\right).$
(29b)
If we shift the first equation in (29a) in the $m$ direction and the first
equation in (29b) in the $n$ direction, then the resulting expressions for
$q_{11}$ coincide provided that $\tilde{p}$ and $\tilde{q}$ satisfy system
(14). Moreover, subtracting the two expressions for $p_{00}$ in (29) we end up
with
$\tilde{p}_{10}-\tilde{p}_{01}-\tilde{p}_{00}\left(\tilde{p}_{00}(\tilde{q}_{10}-\tilde{q}_{01})+\alpha-\beta\right)=0,$
which holds on solutions of (14).
Finally, according to Figure 2 and relation (13), the superposition principle
for the auto-Bäcklund transformation (28) follows from
${\rm{B}}(\tilde{q}_{00},\hat{\tilde{p}}_{00};\varepsilon_{2}){\rm{B}}(q_{00},\tilde{p}_{00};\varepsilon_{1})={\rm{B}}(\hat{q}_{00},\tilde{\hat{p}}_{00};\varepsilon_{1}){\rm{B}}(q_{00},\hat{p}_{00};\varepsilon_{2}),$
and can be written as
$\hat{\tilde{p}}_{00}=-\,\frac{\tilde{p}_{00}-\hat{p}_{00}}{\varepsilon_{1}-\varepsilon_{2}+(\tilde{p}_{00}-\hat{p}_{00})q_{00}},\quad\tilde{q}_{00}-\hat{q}_{00}=\left(\varepsilon_{1}-\varepsilon_{2}+(\tilde{p}_{00}-\hat{p}_{00})q_{00}\right)q_{00}.$
(30)
#### Second case: $c_{1}\neq 0$ and $c_{2}=0$
If $c_{2}=0$ and $c_{1}\neq 0$, we choose $c_{1}=1$, relations (22) imply that
$g^{(4)}_{00}=1$, and the determinant of $\rm{B}$ implies that
$g^{(1)}_{00}=p_{00}\tilde{q}_{00}+\varepsilon$. In view of these choices, we
arrive at a Darboux matrix which can be written in terms of matrix (27) as
${\rm{C}}(p_{00},\tilde{q}_{00};\varepsilon)=(\lambda+\varepsilon)\left({\rm{B}}(\tilde{q}_{00},p_{00};\varepsilon)\right)^{-1},$
(31)
whereas relations (23) yield actually system (29) with the roles of new and
old fields interchanged, i.e. system (29) accompanied by the interchange
$(p,q)\leftrightarrow(\tilde{p},\tilde{q})$. Thus in this case we end up with
the inverse of the Darboux and Bäcklund transformations we derived previously.
### 4.2 Derivation of soliton solutions
We employ the auto-Bäcklund transformation (28) and its superposition
principle (30) in the derivation of soliton solutions of the Adler–Yamilov
system (14).
Figure 3: The one soliton solution of the Adler–Yamilov system and the
potential $1/g_{00}$. In both cases $\alpha=8$, $\beta=4$, $\varepsilon=1$ and
$c=-2$.
More precisely, we start with the solution333This solution can be constructed
starting with the zero solution $p_{00}=q_{00}=0$ and using the second
transformation we discussed in subsection 4.1 with $\varepsilon=0$.
$p_{00}=0,\quad q_{00}=\alpha^{-n}\beta^{-m}.$ (32)
With this seed solution, the first equation in (28a) and the first one in
(28b) become
$\tilde{p}_{10}=\frac{\alpha-\varepsilon}{1+\tilde{p}_{00}\alpha^{-n-1}\beta^{-m}}\tilde{p}_{00},\quad\tilde{p}_{01}=\frac{\beta-\varepsilon}{1+\tilde{p}_{00}\alpha^{-n}\beta^{-m-1}}\tilde{p}_{00}.$
(33)
We can linearise these Ricatti equations by setting
$\tilde{p}_{00}=\alpha^{n}\beta^{m}/g_{00}$,
$(\alpha-\varepsilon)g_{10}=\alpha
g_{00}+1,\quad(\beta-\varepsilon)g_{01}=\beta g_{00}+1.$
The general solution of this linear system is
$g_{00}=\frac{\alpha^{n}}{(\alpha-\varepsilon)^{n}}\frac{\beta^{m}}{(\beta-\varepsilon)^{m}}c-\frac{1}{\varepsilon},$
(34)
where $c\in{\mathbb{R}}$ is the arbitrary constant of integration, and thus
$\tilde{p}_{00}=\frac{\varepsilon\,\alpha^{n}\beta^{m}(\alpha-\varepsilon)^{n}(\beta-\varepsilon)^{m}}{c\,\varepsilon\,\alpha^{n}\beta^{m}-(\alpha-\varepsilon)^{n}(\beta-\varepsilon)^{m}}.$
(35a) Using the seed solution (32) and the updated potential (35a), we can use
either the equation for $\tilde{q}_{10}$ in (28a) or the equation for
$\tilde{q}_{01}$ in (28b) to determine $\tilde{q}_{00}$. Both ways lead to
$\tilde{q}_{00}=\frac{c\,\varepsilon^{2}}{c\,\varepsilon\,\alpha^{n}\beta^{m}-(\alpha-\varepsilon)^{n}(\beta-\varepsilon)^{m}}.$
(35b)
This two-parameter family of solutions yields the one-soliton solution of
(14): even though both functions in (35) diverge, their product represents a
soliton. This interpretation is motivated by the relation of the Adler–Yamilov
system to the nonlinear Schrödinger equation [19], and it is evident from the
plot of the product $|\tilde{p}_{00}\tilde{q}_{00}|$. We also plot the
potential $1/g_{00}$ which is a kink. See Figure 3.
Figure 4: The two-soliton solution of the Adler–Yamilov system: solution (37)
with $\alpha=8$, $\beta=4$, $\varepsilon_{1}=1$, $c_{1}=-2$,
$\varepsilon_{2}=3$ and $c_{2}=8$. It should be noted that this solution
requires the combination of a non-singular solution, i.e. the pair
$(\tilde{p},\tilde{q})$ which corresponds to the one-soliton solution (35),
and a singular one which is $(\hat{p},\hat{q})$.
Having constructed the two-parameter family of solutions (35), we may use it
along with the superposition principle (30) to determine a third solution and
in particular the two-soliton solution of system (14). Starting with the same
seed solution, the two solutions $(\tilde{p}_{00},\tilde{q}_{00})$ and
$(\hat{p}_{00},\hat{q}_{00})$ involved in (30) follow from (35) by replacing
parameters $(\varepsilon,c)$ with $(\varepsilon_{1},c_{1})$ and
$(\varepsilon_{2},c_{2})$, respectively. In order to make the presentation
more comprehensible, let us introduce the shorthand notation
$\delta_{0}:=\alpha^{n}\beta^{m},\quad\delta_{1}:=(\alpha-\varepsilon_{1})^{n}(\beta-\varepsilon_{1})^{m},\quad\delta_{2}:=(\alpha-\varepsilon_{2})^{n}(\beta-\varepsilon_{2})^{m}.$
In terms of this notation, the seed solution is $p_{00}=0$,
$q_{00}=1/\delta_{0}$, and the two solutions we described can be written as
$\tilde{p}_{00}=\frac{\varepsilon_{1}\,\delta_{0}\delta_{1}}{c_{1}\,\varepsilon_{1}\,\delta_{0}-\delta_{1}},\leavevmode\nobreak\
\leavevmode\nobreak\
\tilde{q}_{00}=\frac{c_{1}\,\varepsilon_{1}^{2}}{c_{1}\,\varepsilon_{1}\,\delta_{0}-\delta_{1}},\quad\mbox{and}\quad\hat{p}_{00}=\frac{\varepsilon_{2}\,\delta_{0}\delta_{2}}{c_{2}\,\varepsilon_{2}\,\delta_{0}-\delta_{2}},\leavevmode\nobreak\
\leavevmode\nobreak\
\hat{q}_{00}=\frac{c_{2}\,\varepsilon_{2}^{2}}{c_{2}\,\varepsilon_{2}\,\delta_{0}-\delta_{2}},$
(36)
respectively. With these formulae at our disposal, the first relation in (30)
yields
$\hat{\tilde{p}}_{00}=\frac{\varepsilon_{1}\varepsilon_{2}\delta_{0}\left(c_{1}\delta_{2}-c_{2}\delta_{1}\right)+(\varepsilon_{1}-\varepsilon_{2})\delta_{1}\delta_{2}}{c_{1}c_{2}\varepsilon_{1}\varepsilon_{2}(\varepsilon_{1}-\varepsilon_{2})\delta_{0}-c_{1}\varepsilon_{1}^{2}\delta_{2}+c_{2}\varepsilon_{2}^{2}\delta_{1}}.$
(37a) To find $\hat{\tilde{q}}_{00}$ we work in the same way we derived (35b)
and according to the Bianchi diagram. More precisely, we can use the second
equation either in (28a) or in (28b) with $\tilde{q}$ replaced by
$\hat{\tilde{q}}$, $(p,q)$ replaced by $(\tilde{p},\tilde{q})$ given in (36),
and parameter $\varepsilon$ replaced by $\varepsilon_{2}$ (alternatively, we
can replace $(p,q)$ with $(\hat{p},\hat{q})$ given in (36), and parameter
$\varepsilon$ with $\varepsilon_{1}$), to find
$\hat{\tilde{q}}_{00}=\frac{c_{1}c_{2}\varepsilon_{1}^{2}\varepsilon_{2}^{2}(\varepsilon_{1}-\varepsilon_{2})}{c_{1}c_{2}\varepsilon_{1}\varepsilon_{2}(\varepsilon_{1}-\varepsilon_{2})\delta_{0}-c_{1}\varepsilon_{1}^{2}\delta_{2}+c_{2}\varepsilon_{2}^{2}\delta_{1}}.$
(37b)
This four-parameter family of solutions yields the two-soliton solution of
(14) in the same way we interpreted (35) as the one-soliton solution of the
Adler–Yamilov system. See the plots of the product
$|\hat{\tilde{p}}_{00}\hat{\tilde{q}}_{00}|$ in Figure 4.
## 5 Conclusions
In this paper we proposed a new method for deriving Bäcklund transformations
and constructing solutions for nonlinear integrable P$\Delta$Es which admit
Lax representation but do not necessarily possess the 3D consistency property.
Specifically, in our approach we consider Darboux transformations which leave
the given Lax pair covariant, and by construction lead to Bäcklund
transformations for the corresponding discrete system. The permutability of
four Darboux matrices according to the Bianchi diagram in Figure 2 leads to
the nonlinear superposition principle of the related Bäcklund transformation.
Moreover, the latter transformation and its superposition principle can be
used in the construction of interesting solutions to P$\Delta$Es starting from
some simple ones. As an illustrative example we used the Adler–Yamilov system
(14). For this system we constructed Darboux and corresponding Bäcklund
transformations. With the use of transformation (28) and its superposition
principle (30) we constructed the one- and two-soliton solutions starting with
the seed solution $p_{00}=0,q_{00}=\alpha^{-n}\beta^{-m}$.
In the illustrative example we considered in Section 4, the Lax representation
(6) involves matrices $\rm{L}$ and $\rm{M}$ which have the same form, see
(15). The natural question arises as to whether our method can be employed to
the case of integrable P$\Delta$Es with Lax representation (6) where matrices
${\rm{L}}$ and $\rm{M}$ do not have the same form. The answer to this question
is positive, the corresponding transformations may involve auxiliary functions
(potentials), and this derivation is similar to the generic construction we
presented in subsection 4.1, see Darboux matrix (25) and relations (20), (22),
(23)and (24).
Moreover our considerations can be extended to the generalized symmetries of
the discrete system. Generalized symmetries are (integrable) differential-
difference equations involving shifts in one lattice direction and are
compatible with the P$\Delta$E. Their Lax pair is semi-discrete and its
discrete part coincides with the one of the two equations of the fully
discrete Lax pair (3), see for instance [12]. This relation allows us to
extend the Darboux and Bäcklund transformations for the P$\Delta$E to
corresponding ones for the differential-difference equations and employ the
Bäcklund transformation and its superposition principle in the construction of
solutions for the symmetries. We will demonstrate this method and its
extensions in our future work in details using the Hirota KdV equation as an
illustrative example, as well as systems appeared in [4, 10, 22].
In fact our results can be used and extended in various ways.
1. 1.
Apply our method to construct solutions to all the NLS type equations derived
in [19].
In [19] we classified Darboux transformations related to NLS type equations
and constructed integrable discretisations of the latter, namely integrable
systems of nonlinear P$\Delta$Es. By employing the discrete Darboux–Lax scheme
we proposed in Section 3, one could derive Bäcklund transformations and
construct soliton solutions to these nonlinear P$\Delta$Es.
2. 2.
Study the solutions of the associated PDEs.
The Adler–Yamilov system (14) constitutes a discretisation of the NLS equation
via its Darboux transformation [19]. In this paper, we constructed soliton
solutions to this system, so one could consider the continuum limits of these
solutions to construct solutions to the NLS equation. This procedure could be
applied to other NLS type equations which appeared in [19].
3. 3.
Study the corresponding Yang–Baxter maps.
In [18, 20] matrix refactorisation problems of Darboux matrices for integrable
PDEs were considered in order to derive solutions to the Yang–Baxter equation
and the entwining Yang–Baxter equation. Since the generator of Yang–Baxter
maps is a matrix refactorisation problem (13) it makes sense to understand how
Bäcklund transformations for P$\Delta$Es are related to Yang–Baxter maps. In
our future work, we plan to show that Yang–Baxter and entwining Yang–Baxter
maps are superpositions of Bäcklund transformations of P$\Delta$Es.
4. 4.
Extend the results to the case of discrete systems on a 3D lattice.
One can extend the results employed in this paper to the case of 3D lattice
integrable systems. It is expected that the superposition of Bäcklund
transformations related to these systems are solutions to the tetrahedron
equation.
5. 5.
Extend the results to the case of Grassmann algebras.
Grassmann extensions of Darboux transformations were employed in the
construction of noncommutative versions of discrete systems, see for instance
[14, 30] . However it was realised in [17] that quad-graph systems may lose
their 3D consistency property in the Grassmann extension. The method we
presented here could be generalised and employed in the construction of
Bäcklund transformations and the derivation of solutions to Grassmann extended
quad-graph systems which appeared in the literature.
## 6 Acknowledgements
Xenia Fisenko’s work was supported by the Ministry of Research and Higher
Education (Regional Mathematical Centre “Centre of Integrable Systems,”
Agreement No. 075-02-2021-1397). Sotiris Konstantinou-Rizos’s work was funded
by the Russian Science Foundation (grant No. 20-71-10110).
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|
# comp-syn: Perceptually Grounded Word Embeddings with Color
Bhargav Srinivasa Desikan
University of Chicago
Knowledge Lab
<EMAIL_ADDRESS>
Tasker Hull
Psiphon Inc, Toronto
<EMAIL_ADDRESS>
Ethan O. Nadler
Stanford University
KIPAC & Department of Physics
<EMAIL_ADDRESS>
Douglas Guilbeault
University of California, Berkeley
Haas Business School
<EMAIL_ADDRESS>
Aabir Abubaker Kar
University of Chicago
Knowledge Lab
<EMAIL_ADDRESS>
Mark Chu
Columbia University
<EMAIL_ADDRESS>
Donald Ruggiero Lo Sardo
Sony CSL Paris
<EMAIL_ADDRESS>
###### Abstract
Popular approaches to natural language processing create word embeddings based
on textual co-occurrence patterns, but often ignore embodied, sensory aspects
of language. Here, we introduce the Python package comp-syn, which provides
grounded word embeddings based on the perceptually uniform color distributions
of Google Image search results. We demonstrate that comp-syn significantly
enriches models of distributional semantics. In particular, we show that (1)
comp-syn predicts human judgments of word concreteness with greater accuracy
and in a more interpretable fashion than word2vec using low-dimensional
word–color embeddings, and (2) comp-syn performs comparably to word2vec on a
metaphorical vs. literal word-pair classification task. comp-syn is open-
source on PyPi and is compatible with mainstream machine-learning Python
packages. Our package release includes word–color embeddings for over 40,000
English words, each associated with crowd-sourced word concreteness judgments.
## 1 Introduction
The embodied cognition paradigm seeks to ground semantic processing in bodily,
affective, and social experiences. This paradigm explains how sensory
information contributes to semantic processing, either through metaphors
involving references to sensory experience [Lakoff and Turner, 1989, Gallese
and Lakoff, 2005] or through the simulation of sensory experience in mental
imagery [Bergen, 2012]. For example, color is pervasive in linguistic
metaphors (e.g., “her bank accounts are in the _red_ ”), and primes a range of
affective and interpretative responses [Mehta and Zhu, 2009, Elliot and Maier,
2014].
Extant methods in computational linguistics are limited in their ability to
advance the embodied cognition paradigm due to their focus on text, which
often precludes multi-modal analyses of images and other forms of sensory
data—including color—involved in human meaning-making activities.
Distributional semantics is a particularly prominent approach, wherein textual
co-occurrence patterns are used to embed words in a high-dimensional space;
the resulting “distance” between word embeddings correlates with semantic
similarity [Landauer and Dumais, 1997]. Although leveraging neural networks
[Mikolov et al., 2013] and syntactic information [Levy and Goldberg, 2014] has
led to recent progress, popular models like word2vec lack firm grounding in
their use of sensory information. Moreover, high-dimensional word embeddings
created by neural networks are difficult to interpret [Şenel et al., 2018] and
require long training periods [Ji et al., 2019]. Thus, it remains difficult to
reconcile models of distributional semantics with theories of embodied
semantics.
To address these limitations, _multi-modal_ approaches to distributional
semantics combine text and image data. However, these models continue to infer
semantic associations using high-dimensional word-plus-image embeddings; their
interpretability therefore remains an issue [Bruni et al., 2014, Socher et
al., 2014, Lu et al., 2019]. With respect to color, multi-modal models operate
in standard colorspaces like RGB that are not perceptually uniform
[International Commission On Illumination, 1978] and embed color in a manner
that combines it with complex, multidimensional spatial information.
Here, we introduce a novel word embedding method based on color that is
explicitly interpretable with respect to theories of embodied cognition. We
build on work which shows that color distributions in online images reflect
both affective and semantic similarities among words in abstract domains
(e.g., in the domain of academic disciplines), while also characterizing human
judgments of concept concreteness [Brysbaert et al., 2014, Guilbeault et al.,
2020]. We present the Python package comp-syn111Short for “Computational
Synaesthesia.”, which allows users to explore word–color embeddings based on
the perceptually uniform color distributions of Google Image search results.
We provide embeddings for a set of 40,000 common English words, and we
benchmark the performance of our model using crowd-sourced human concreteness
ratings [Brysbaert et al., 2014]. We show that comp-syn complements the
performance of text-based distributional semantics models by providing an
interpretable embedding that both (1) predicts human judgments of concept
concreteness, and (2) distinguishes metaphorical and literal word pairs.
## 2 Python Package: comp-syn
comp-syn is an open-source Python package available on GitHub and downloadable
through PyPi. The code follows Python best practices and uses industry
standard packages for scientific computing, facilitating easy integration with
Python code bases; we provide complete details in the Supplementary Material.
comp-syn creates word–color embeddings by computing the perceptually uniform
$J_{z}A_{z}B_{z}$ [Safdar et al., 2017] color distributions of their
corresponding top $100$ Google Image search results. We represent these
distributions by their mean and (optionally) standard deviation in $8$ evenly-
segmented $J_{z}A_{z}B_{z}$ bins, yielding 8 to 16-dimensional word–color
embeddings. The details of our code, image searches and methods for generating
word–color embeddings are described in the Appendices.
We use Google rather than a curated image database such as ImageNet [Deng et
al., 2009] because popular datasets are heavily biased toward concrete
objects, which limits their applicability in abstract semantic domains.
Moreover, Google Image search results reflect content that users interact with
most frequently [Jing and Baluja, 2008], underscoring their relevance for
connecting distributional properties of words and images to human semantic
processing. To ensure that our approach is entirely unsupervised, we do not
select particular features of images when measuring their color distributions.
This approach avoids importing pre-determined semantic notions into our
analysis and connects our method to cognitive theories that attribute
aesthetic relevance to recognizable features in both the background and
foreground of images [Riley, 1995, Elliot and Maier, 2014, Guilbeault et al.,
2020].
Model | Embedding based on | Dimension | Concreteness prediction | Metaphor task
---|---|---|---|---
comp-syn | Perceptually uniform color distributions | $8$–$16$ | Linear: $R^{2}=0.96$ Nonlinear: $\ln\mathcal{L}=86$ | $92\%$ test set accuracy
word2vec | Textual co-occurrence | $\sim 300$ | Linear: $R^{2}=0.17$ Nonlinear: $\ln\mathcal{L}=76$ | $95\%$ test set accuracy
Table 1: Comparison of comp-syn and word2vec and summary of our main results.
## 3 Results
Table 1 summarizes our main results, which we now describe in turn.
Figure 1: The semantic coherence of comp-syn. (A) Word pair color similarity
in comp-syn, measured using the Jensen-Shannon divergence between their
perceptually uniform color distributions, correlates with word2vec cosine
similarity ($N>10^{6}$ pairs). This holds for both concrete (blue) and
abstract (magenta) word pairs, rated according to crowd-sourced human
judgments. Data represent average color similarity in $40$ equally-sized bins
of word2vec similarity. (B) Word pair similarity vs. human concreteness
judgments ($N>10^{7}$ pairs). comp-syn similarity monotonically decreases with
word pair concreteness, while word2vec similarity is a nonlinear function of
word pair concreteness. Data represent average word pair similarity in
word2vec (gray) and comp-syn (black) in $40$ equally-sized bins of summed word
pair concreteness.
### 3.1 Comparison to word2vec Similarity
We begin by demonstrating that pairwise word similarity is highly correlated
in comp-syn and word2vec, illustrating the semantic coherence of our color
embeddings. We calculate pairwise distances between each of the 40,000 words
in our dataset and 500 randomly-selected words from the same set, yielding
over $10^{7}$ distinct pairs. We compare the cosine similarity between word
pairs in the Mikolov et al. [Mikolov et al., 2013] word2vec model with the
Jensen-Shannon (JS) divergence between $J_{z}A_{z}B_{z}$ distributions in
comp-syn. Fig. 1A shows that JS divergence in comp-syn is significantly
correlated with cosine similarity in word2vec ($p<0.00001$, $\mathrm{JT}=773$,
Jonckheere-Terpstra test). This implies that our low-dimensional,
interpretable embedding captures aspects of the key information contained in a
widely-used distributional semantics model.
### 3.2 Relation to Human Concreteness Judgments
Next, we examine the relation between our word–color embeddings and human
concreteness judgments [Brysbaert et al., 2014]. We label word pairs with
summed concreteness ratings in the highest (lowest) quartile of our data as
“concrete” (“abstract”). Fig. 1A shows that abstract word pairs are more
similar in colorspace than concrete word pairs, even at fixed word2vec
similarity ($p=0.001$, $\mathrm{DF}=77$, Dickey-Fuller test). Moreover, as
shown in Fig. 1B, comp-syn captures concreteness judgments in a more
interpretable fashion than word2vec. In particular, because abstract words are
more similar in colorspace (on average), there is a monotonic relationship
between color similarity and concreteness; indeed, a linear model of comp-syn
similarity accounts for nearly all of the variance in word pair concreteness
($R^{2}=0.96$). On the other hand, word2vec similarity is a nonlinear function
of word pair concreteness. Although nonlinear functions of word2vec similarity
also predict concreteness well ($R^{2}>0.99$), comp-syn versions of the same
nonlinear models are significantly more accurate (log-likelihood difference
$\Delta\ln\mathcal{L}\sim 10$ in favor of comp-syn) and less complex (Bayesian
information criterion $\Delta\mathrm{BIC}\sim 20$, also in favor of comp-syn).
To qualitatively explore the relationship between our word–color embeddings
and human concreteness judgment, Fig. 3A shows the perceptually uniform color
distributions associated with some of the most and least concrete words in our
corpus. These _colorgrams_ illustrate that concrete words (e.g., “pyramid”)
are often associated with color distributions that are peaked in specific
regions of colorspace, while abstract words (e.g., “concept”) feature more
variegated color distributions. The spatial and textural features of the
images reflect these properties, and exploring the relationship between these
aspects of _colorgrams_ , is an interesting avenue for future study.
Figure 2: Classifying metaphorical vs. literal adjective-noun pairs with comp-
syn. (A) Test set classification accuracy for comp-syn (magenta) and word2vec
(gray) as a function of PCA embedding dimension. (B) Average similarity of
metaphorical and literal adjective-noun pairs in word2vec (left panel) and
comp-syn (right panel). Error bars indicate $95\%$ confidence intervals.
### 3.3 Metaphor Pair Analysis
The results above suggest that our word–color embeddings encode complementary
information about concept concreteness relative to purely textual embeddings.
This raises the question of what can be learned from cases in which word2vec
and comp-syn provide conflicting similarity predictions when evaluated on the
same pair of words. Here, we address this question by demonstrating that comp-
syn significantly enriches metaphorical word pair classification, which often
requires extensive manual tagging due to subtle uses of both sensory and
abstract features [Lakoff and Johnson, 2008, Bethard et al., 2009, Indurkhya
and Ojha, 2013, Dodge et al., 2015, Winter, 2019].
We trained a gradient-boosted tree classifier implemented via XGBoost [Chen
and Guestrin, 2016] to label adjective-noun pairs as either metaphorical or
literal, using over $8000$ word pairs from Tsvetkov et al. [Tsvetkov et al.,
2014] and Gutiérrez et al. [Gutiérrez et al., 2016]. This dataset encompasses
a statistically representative range of metaphorical and literal contexts for
each adjective [Gutiérrez et al., 2016]. To compare embeddings, we compressed
word2vec’s 300-dimensional word vector differences using PCA to match the
dimensionality of comp-syn, following Bolukbasi et al. [Bolukbasi et al.,
2016]. Fig. 2A shows that a classifier trained using only word2vec achieves a
limiting test set accuracy of $95\%$, compared to $92\%$ for comp-syn.
Importantly, comp-syn outperforms word2vec at low embedding dimensions,
indicating that it captures semantic content in an interpretable fashion. This
analysis does not demonstrate either model’s best-case performance on this
task; rather, it highlights the complementary information provided by comp-
syn.
Strikingly, word2vec and comp-syn distances provide qualitatively different
information when distinguishing literal and metaphorical adjective-noun pairs.
Fig. 2B shows that literal adjective-noun pairs are more similar than
metaphorical pairs in word2vec ($p<0.001$, Wilcoxon rank sum); the reverse
holds in comp-syn, where literal pairs are significantly _less_ similar than
metaphorical pairs ($p<0.001$, Wilcoxon rank sum). This is a consequence of
the fact that images associated with concrete words are more variable in
colorspace [Guilbeault et al., 2020]. In this way, comp-syn reveals
differences in color similarity between literal and metaphorical adjective-
noun pairs that are of interest for cognitive theory [Indurkhya and Ojha,
2013]. Particularly, our findings suggest that metaphors can exploit color
similarities between words that are dissimilar in textual embeddings, which
may help facilitate cognitive processing of semantic relations among concepts
from distinct domains [Guilbeault et al., 2020].
Qualitative inspection of specific adjective-noun word pairs highlights some
notable differences between textual and word–color embeddings. Fig. 3B
provides visual representations of the color distributions for word pairs in
our metaphorical vs. literal dataset that are most and least similar in comp-
syn. Interestingly, while metaphorical pairs are more similar than literal
pairs in comp-syn, the _least_ similar metaphorical pairs explicitly invoke
color, e.g., “deep orange”. Algorithmically, this is due to the fact that
comp-syn embeddings associated with color terms are unusually coherent. On the
other hand, the color distributions associated with the most similar pairs in
word2vec (e.g., “bushy beard”) often noticeably contrast, while word2vec’s
least similar pairs (e.g., “rough customer”) do not strongly invoke color.
These findings point to an important direction for future research enabled by
the grounded nature of comp-syn: how do linguistic metaphors leverage sensory
information to characterize colorspace itself (e.g., in the use of spatial
information in the popular metaphor “deep purple”)?
Figure 3: (A) Examples of the most and least concrete terms, visualized using
our word–color embedding method. (B) Examples of the most and least similar
adjective-noun pairs according to comp-syn word–color embeddings. We represent
each term using a _colorgram_ , i.e., a composite image produced by averaging
the perceptually uniform colors of pixels across Google Image search results.
## 4 Conclusion and Future Work
We have presented comp-syn, a new Python package that provides perceptually
grounded color-based word embeddings. These embeddings are interpretable with
respect to theories of embodied cognition because (1) comp-syn represents
color in a fashion that emulates human perception, and (2) comp-syn leverages
Google Image search results that human users dynamically interact with and
produce. By linking comp-syn with human concreteness judgments for 40,000
common English words, our package provides a multi-modal playground for
exploring grounded semantics. We demonstrated that comp-syn enriches popular
distributional semantics models in both word concreteness prediction and
metaphorical word-pair classification. A myriad of comp-syn applications
await, including color-based classification of text genres and the
characterization of sensory imagery in everyday language.
## Acknowledgements
The authors gratefully acknowledge the support of the Complex Systems Summer
School hosted at the Institute of American Indian Arts and the Santa Fe
Institute, where this project was initiated. D.G. gratefully acknowledges
financial support from the Institute on Research and Innovation in Science.
This research received support from the National Science Foundation (NSF)
under grant no. NSF DGE-1656518 through the NSF Graduate Research Fellowship
received by E.O.N. D.G. and T.H. acknowledge intellectual support from the
Institute for Advanced Learning (IAL) in Ontario, Canada.
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## Appendix A comp-syn Code
comp-syn is an open source Python package, available on GitHub and
downloadable through PyPi at https://github.com/comp-syn/comp-syn. It uses
NumPy [Walt et al., 2011] and scipy [Jones et al., 2001] for computational
purposes, matplotlib [Hunter, 2007] for visualisations, and Pillow [Clark,
2018] to load and manipulate images in Python. We provide documentation and
extensive examples of code use through Jupyter/IPython notebooks [Pérez and
Granger, 2007].
With a focus on community-based development, our code is PEP8 compliant,
heavily commented and documented, includes continuous code integration and
testing via Travis CI, and uses GitHub as a platform for issue tracking and
ideation. comp-syn is structured so that it is easily compatible with
contemporary deep learning, natural language processing and computational
linguistics Python packages, and particularly with NumPy and SciPy as standard
tools for scientific computing [Virtanen et al., 2020].
comp-syn includes the following modules:
* •
datahelper.py \- loader functions for scraping and organising image data;
* •
analysis.py \- analysis functions for computing word–color embeddings;
* •
logger.py \- logger functions;
* •
visualisation.py \- visualisation functions;
* •
vectors.py \- word–color embedding loader and manager;
* •
wordnet-functions.py \- WordNet functions to facilitate web scraping and
linking with concreteness measures.
## Appendix B Data Distribution
Along with the software, we also release word–color embeddings for the nearly
40,000 English language words. Our corpus of word–color embeddings is linked
to crowd-sourced human judgments of concept concreteness from Brysbaert et al.
[Brysbaert et al., 2014], and terms used for analysis are also linked to
WordNet [Miller, 1998]. These embeddings can easily be downloaded and loaded
into Python pipelines for various natural language processing tasks, including
jointly training or enhancing distributional semantics models. The data is
stored in the json file concreteness-color-embeddings.json and can be loaded
using the vectors.py module class LoadVectorsFromDisk. Once loaded, each
Vector object has the following information based on the top $100$ Google
Image results for each word:
* •
rgb-dist \- the average RGB distribution, computed in $8$ evenly-segmented
bins;
* •
jzazbz-dist \- the average perceptually uniform $J_{z}A_{z}B_{z}$
distribution, computed in $8$ evenly-segmented bins;
* •
jzazbz-dist-std \- the standard deviation over the perceptually uniform
$J_{z}A_{z}B_{z}$ distribution, computed in $8$ evenly-segmented bins;
* •
colorgram \- a composite image created by pixel-wise averaging over the top
$100$ images, representing the word’s “average image”;
* •
rgb-vector \- the mean RGB coordinates, averaged over the top $100$ images;
* •
jzazbz-vector \- the mean $J_{z}A_{z}B_{z}$ coordinates, averged over the top
$100$ images
* •
concreteness-mean \- the mean of the concreteness scores for the word from
[Brysbaert et al., 2014];
* •
concreteness-sd \- the standard deviation of the concreteness scores for the
word from [Brysbaert et al., 2014].
## Appendix C Methods
### C.1 Google Image Search
To generate word–color embeddings, we collect the top $100$ Google Image
search results for each of the 40,000 terms in our analysis. The Google Image
searches were run from 10 servers running in a commercial datacenter in New
York, USA. These servers were created and used only for this experiment, so
that results would not be overly-personalized. Additional search parameters
were included as query strings safe=off&site=&tbm=isch&source=hp&gs_l=img.
### C.2 Word–Color Embeddings
We use the PIL Python module to convert each image into an $m\times n\times 3$
array of sRGB values, where $m$ and $n$ are the intrinsic image dimensions.
For computational efficiency, we then compress each image into an anti-aliased
$300\times 300\times 3$ array. Next, we transform sRGB pixel values into their
counterparts in the perceptually uniform $J_{z}A_{z}B_{z}$ colorspace.222comp-
syn efficiently computes color embeddings in multiple colorspaces if desired,
including more traditional (non-perceptually uniform) options such as RGB and
HSV. Unlike in standard colorspace, Euclidean distances in $J_{z}A_{z}B_{z}$
coordinates linearly correspond to differences in human color perception
[Safdar et al., 2017]. Moreover, our use of the $J_{z}A_{z}B_{z}$ colorspace
rather than a standard colorspace like RGB increases the semantic coherence of
our word–color embeddings [Guilbeault et al., 2020].
We measure the color distribution of each image in 8 evenly-segmented
$J_{z}A_{z}B_{z}$ subvolumes spanning the range of $J_{z}A_{z}B_{z}$
coordinates that maps to all possible RGB tuples: $J_{z}\in[0,0.167]$,
$A_{z}\in[-0.1,0.11]$, $B_{z}\in[-0.156,0.115]$. Next, we average
$J_{z}A_{z}B_{z}$ distributions over all $100$ images for each term to obtain
an aggregate, 8-dimensional color mean embedding. We also compute the standard
deviation over the $100$ images in each $J_{z}A_{z}B_{z}$ subvolumes to obtain
an aggregate, 8-dimensional color variability embedding. These color mean and
variability embeddings can be concatenated to create a 16-dimensional
embedding. The details of our compression, binning, and averaging steps do not
affect our results [Guilbeault et al., 2020].
To compare word–color embeddings, we use the Jensen-Shannon (JS) divergence to
measure the similarity of aggregate $J_{z}A_{z}B_{z}$ distributions. In
particular, for $J_{z}A_{z}B_{z}$ distributions $C_{i}$ and $C_{j}$ associated
with terms $i$ and $j$, the JS divergence is given by
$D_{\mathrm{JS}}(C_{1}~{}||~{}C_{2})\equiv\frac{1}{2}\left[D_{\mathrm{KL}}(C_{1}~{}||~{}\bar{C}_{12})+D_{\mathrm{KL}}(C_{2}~{}||~{}\bar{C}_{12})\right],$
(1)
where $D_{\rm{KL}}$ is the Kullback-Leibler divergence and
$\bar{C}_{12}\equiv(C_{1}+C_{2})/2$. The JS divergence is a measure of the
distance between two color distributions, such that lower values correspond to
more similar distributions in perceptually uniform colorspace. We choose this
metric because it is a well-defined distance measure that satisfies the
triangle inequality and allows us to avoid undefined values associated with
empty $J_{z}A_{z}B_{z}$ bins. Terms with relatively high mutual JS divergences
usually exhibit _colorgrams_ with perceptibly different average colors.
## Appendix D Best Practices for Usage
We caution that our word–color embeddings are _distributions_ rather than
_vectors_. Thus, their components are positive semidefinite, and different
embeddings must be compared using similarity measures designed for
distributions such as JS divergence. On the other hand, word2vec embeddings
are vectors with components that can be positive or negative, and are compared
using similarity measures designed for vectors such as cosine similarity.
Importantly, unlike word2vec embeddings, our word–color embeddings cannot be
composed by vector addition. We are exploring algebraic techniques for word
composition in colorspace; these techniques must respect the underlying
mathematical structure of $J_{z}A_{z}B_{z}$ (and other) colorspaces, which are
not closed under standard binary operations like addition or multiplication.
|
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| Patrick M. Jensen was born in Copenhagen, Denmark, in 1994. He received his
B.Sc.Eng degree in 2017 and M.Sc.Eng degree in 2019, both in applied
mathematics, at the Technical University of Denmark (DTU), Kgs. Lyngby,
Denmark. He is currently pursuing a Ph.D. in 3D image analysis at the Visual
Computing group at the Department of Applied Mathematics and Computer Science,
Technical University of Denmark. His research interests lie in 3D image
segmentation.
---|---
| Niels Jeppesen is an image analysis and machine learning specialist at
FORCE Technology with a Ph.D. degree in image analysis of 3D structures from
the Department of Applied Mathematics and Computer Science, Technical
University of Denmark (DTU), Kgs. Lyngby, Denmark. His research interests lie
in min-cut/max-flow algorithms and quantitative analysis of structures in 3D
images. He applies these methods for automated quality control of structures
and materials, in particular, in the wind turbine industry.
---|---
| Anders Bjorholm Dahl is professor in 3D image analysis, and a head of the
Section for Visual Computing at the Department of Applied Mathematics and
Computer Science, Technical University of Denmark (DTU), Kgs. Lyngby, Denmark.
He is heading The Center for Quantification of Imaging Data from MAX IV,
focusing on quantitative analysis of 3D images. His research is focused on
image segmentation and its applications.
---|---
| Vedrana Andersen Dahl is an associate professor at the Department of
Applied Mathematics and Computer Science, Technical University of Denmark
(DTU), Kgs. Lyngby, Denmark. Her primary research interest is in the use of
geometric models for the analysis of volumetric data. This includes volumetric
segmentation and methods based on deformable meshes. She developed analysis
tools with applications in material science, industrial inspection, and
biomedicine.
---|--- |
# Topological properties of P.A. random graphs with edge-step functions
Caio Alves1 1 Institute of Mathematics, , University of Leipzig –
Augustusplatz 10, 04109 Leipzig
e-mail<EMAIL_ADDRESS>, Rodrigo Ribeiro2 2 PUC Chile, Av.
Vicuña Mackenna 4860, Macul, La Florida, Región Metropolitana, Chile.
e-mail<EMAIL_ADDRESS>and Rémy Sanchis3 3Departamento de Matemática,
Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627 C.P. 702 CEP
30123-970 Belo Horizonte-MG, Brazil
e-mail<EMAIL_ADDRESS>
(Date:
1 Institute of Mathematics, University of Leipzig
2 PUC Chile, Pontificia Universidad Católica de Chile.
3 Departamento de Matemática, Universidade Federal de Minas Gerais. )
###### Abstract.
In this work we investigate a preferential attachment model whose parameter is
a function $f:\mathbb{N}\to[0,1]$ that drives the asymptotic proportion
between the numbers of vertices and edges of the graph. We investigate
topological features of the graphs, proving general bounds for the diameter
and the clique number. Our results regarding the diameter are sharp when $f$
is a regularly varying function at infinity with strictly negative index of
regular variation $-\gamma$. For this particular class, we prove a
characterization for the diameter that depends only on $-\gamma$. More
specifically, we prove that the diameter of such graphs is of order $1/\gamma$
with high probability, although its vertex set order goes to infinity
polynomially. Sharp results for the diameter for a wide class of slowly
varying functions are also obtained. The almost sure convergence for the
properly normalized logarithm of the clique number of the graphs generated by
slowly varying functions is also proved.
_Keywords_ : complex networks; cliques; preferential attachment; concentration
bounds; diameter; scale-free; small-world
MSC 2010 subject classifications. Primary 05C82; Secondary 60K40, 68R10
## 1\. Introduction
P. Ërdos and A. Rényi in their seminal paper [17] introduced the random graph
model that now carries their name in order to solve combinatorial problems.
However, the theory of Random Graphs as a whole has proven to be a useful tool
for treating concrete problems as well. Any discrete set of entities whose
elements interact in a pairwise fashion may be seen as a graph: the vertices
represent the entities, and the edges, the possible interactions. This
approach is nowadays intuitive and very fruitful. In the scenario where there
exists some randomness on the interactions among the entities, random graphs
became the natural tool to represent abstract or real phenomena.
From a mathematical/statistical point of view, the Ërdos-Rényi model – and
many others related to it – is homogeneous, in the sense that its vertices are
statistically indistinguishable. However, the empirical findings of the
seminal work of A. Bárabasi and R. Álbert [5] suggested that many real-world
networks are non-homogeneous. They observed that such graphs were scale-free,
i.e., their degree sequence had a power-law distribution. The authors proposed
a mechanism – known as preferential attachment – that could explain the
emergence of such highly skewed distributions. Roughly speaking, the idea is
that some sort of popularity drives the interaction among the entities.
Motivated mainly by these empirical findings, nowadays Preferential Attachment
models (PA-models for short) constitutes a well known class of random graph
models investigated from both theoretical and applied perspectives. Recently,
the preferential attachment mechanism has been generalized in many ways and
combined with other rules of attachment, such as spatial proximity [19] and
fitness of vertices [14]. It also arises naturally even in models where it is
not entirely explicit such as the deletion-duplication models [4, 26], in
which vertices’ degree still evolve according to the PA-rule. Furthermore, the
PA-models provide an interesting and natural environment for other random
processes, such as bootstrap percolation, contact process and random walks,
see [3, 11, 20] for recent examples of random processes whose random media is
sampled from some PA-model.
When dealing with PA-models, there exists a set of natural questions that
arises. They concern the empirical degree distribution, the order of the
diameter and the robustness of the network. Their interest relies on modeling
purposes and on the implications for the graph’s combinatorial structures.
In this paper we address the two latter topics on a PA-model which is a
modification of the BA-model. The kind of result we pursuit is to show that
some graph properties hold asymptotically almost surely (a.a.s). Given a
sequence of random graphs $\\{G_{t}\\}_{t\in\mathbb{N}}$, we say that a graph
property $\mathcal{P}$ holds a.a.s, if
${\mathbb{P}}\left(G_{t}\in\mathcal{P}\right)=1-o(1)$
i.e., the probability of observing such property increases to $1$ as $t$ goes
to infinity. For instance, $\mathcal{P}$ may be the set of graphs having
diameter less than the logarithm of the total number of vertices.
In order to offer a clearer discussion of our results, we introduce the model
in the next subsection, then we discuss separately the properties which we
want the graph to satisfy a.a.s, as well as the associated motivation.
### 1.1. Preferential attachment model with an edge-step function
The model we investigate here in its generality was proposed in [2] and
combines the traditional preferential attachment rule with a function called
edge-step function that drives the growth rate of the vertex set.
The model has one parameter $f$ which is a real non-negative function with
domain given by $\mathbb{N}$ and bounded by one on the $L_{\infty}$-norm.
Without loss of generality and to simplify the expressions we deal with, we
start the process from an initial graph $G_{1}$ consisting in one vertex and
one loop. The model evolves inductively and at each step the next graph is
obtained by performing one of the two stochastic operations defined below on
the previous one:
* •
Vertex-step \- Add a new vertex $v$ and add an edge $\\{u,v\\}$ by choosing
$u\in G$ with probability proportional to its degree. More formally,
conditionally on $G$, the probability of attaching $v$ to $u\in G$ is given by
$P\left(v\rightarrow u\middle|G\right)=\frac{\text{degree}(u)}{\sum_{w\in
G}\text{degree}(w)}.$
* •
Edge-step \- Add a new edge $\\{u_{1},u_{2}\\}$ by independently choosing
vertices $u_{1},u_{2}\in G$ according to the same rule described in the
vertex-step. We note that both loops and parallel edges are allowed.
The model alternates between the two types of operations according to a
sequence $\\{Z_{t}\\}_{t\geq 1}$ of independent random variables such that
$Z_{t}\stackrel{{\scriptstyle\tiny d}}{{=}}\mathrm{Ber}(f(t))$. We then define
inductively a random graph process $\\{G_{t}(f)\\}_{t\geq 1}$ as follows:
start with $G_{1}$. Given $G_{t}(f)$, obtain $G_{t+1}(f)$ by either performing
a vertex-step on $G_{t}(f)$ when $Z_{t}=1$ or performing an edge-step on
$G_{t}(f)$ when $Z_{t}=0$.
Given $f$, its partial sum is an important quantity for us and we reserve the
letter $F$ to denote it, i.e., $F$ is a function defined as
(1.1) $F(t):=1+\sum_{s=2}^{t}f(s).$
Observe that the edge-step function $f$ is intimately related to the growth of
the vertex set. If we let $V_{t}$ denote the number of vertices added up to
time $t$, then
(1.2) $V_{t}=1+\sum_{s=2}^{t}Z_{s}\approx F(t),$
since the sequence of random variables $(Z_{s})_{s\geq 1}$ is independent.
Thus, abusing from the notation for a brief moment, we may write
$\frac{\mathrm{d}V_{t}}{\mathrm{d}t}=f(t).$
When the proper machinery has been settled, we will discuss in Section 8 that
some regularity should be imposed on $f$ in order to avoid some pathological
behaviors. For now, we define a list of conditions we may impose on $f$ at
different points of the paper in order to get the proper results. For
instance, we say $f$ satisfies condition $(\mathrm{D})$ if it is non-
increasing. We define the further conditions:
(D0) $f\text{ is non-increasing and }\lim_{t\to\infty}f(t)=0;$ (S)
$\sum_{s=1}^{\infty}\frac{f(s)}{s}<\infty;$ (Lκ)
$\sum_{s=t^{1/13}}^{t}\frac{f(s)}{s}<(\log t)^{\kappa},\text{ for all
}t\in\mathbb{N}\text{ and some }\kappa\in(0,1);$ (RVγ) $\exists\gamma,\text{
such that }\forall
a>0,\;\lim_{t\to\infty}\frac{f(at)}{f(t)}=\frac{1}{a^{\gamma}}.$
We must point out that for modeling purposes, conditions (D) and (D0) may be
desirable. For instance, in the context of social networks, these conditions
assure that the rate at which new individuals join the network is decreasing
as the size of the network increases. Whereas, conditions (S) and (Lκ) are
related to the order of the maximum degree of $G_{t}(f)$. In [2], the authors
point out that the maximum degree at time $t$ should be of order
(1.3) $t\cdot\exp\left\\{-\frac{1}{2}\sum_{s=2}^{t}\frac{f(s)}{s-1}\right\\}.$
A function satisfying condition (RVγ) is called regularly varying at infinity
and the exponent $\gamma$ is called the index of regular variation. Functions
in this class are well-studied in mathematics in many contexts and a variety
of asymptotic results for them and their integrals is known due mainly to the
theory developed by Karamata, see [8] for a complete reference.
In general, we may say that this paper investigates how sensitive some graph
observables are to changes of $f$ and aims at a general characterization of
such observables for a class of functions $f$ that is as wide as possible.
### 1.2. Robustness and large cliques
As said before, one of the main questions concerns the robustness of the
network or how vulnerable the network is to spread of a disease [6] or to
deliberate/random attacks [10]. Regarding the spread of rumors of diseases
some graph substructures play important roles, such as stars, triangles and
cliques (complete subgraphs). The latter is also related to the robustness of
the network, since the existence of a large clique may let the network less
vulnerable to attacks aiming at edge-deletion.
In [7], the authors give lower and upper bounds for the clique number in a
uncorrelated PA-model. Their bounds are polynomial on the number of vertices.
In the same direction, in [21] the authors prove some sort of phase transition
on the clique number of a random graph $G(n,\alpha)$ on $n$ vertices and
degree sequence obeying a power-law with exponent $\alpha$. For their model,
when $\alpha>2$ the clique number is of constant size, whereas for heavy
tails, when $0<\alpha<2$, $\omega(G(n,\alpha))$ is a power of $n$.
In this matter, we prove the existence a.a.s of large cliques whose order
depends essentially on the definite integral of $f$. The result is formally
stated on the theorem below
###### Theorem 1 (Large cliques).
Let $f$ be an edge-step function satisfying condition (D0). Then, for every
$\delta\in(0,1)$, there exist a positive constant $C$ depending on $\delta$
only such that
${\mathbb{P}}\left(\exists K_{n}\subset G_{t}(f),\textit{ such that }n\geq
CF(t^{\frac{1-\delta}{2}})\right)=1-o(1),$
where $K_{n}$ denotes a subgraph isomorphic to the complete graph with $n$
vertices.
For edge-step functions which are also slowly varying varying the above
theorem can provide the right order of largest clique of $G_{t}(f)$, i.e., the
so-called clique number, denoted by $\omega(G_{t}(f))$. In the more general
case of a regularly varying function $f$, a non-sharp result can be obtained.
We summarize this in the Corollary below
###### Corollary 1.4 (Clique number for regular varying functions).
let $f$ be an edge-step function satisfying conditions (D0) and (RVγ), for
$\gamma\in[0,1)$. Then,
1. (a)
for any $\varepsilon>0$ and $t$ sufficiently large
${\mathbb{P}}\left(t^{(1-\gamma)\frac{1-\varepsilon}{2}}\leq\omega(G_{t}(f))\leq
7t^{\frac{1}{2}}\right)\geq 1-\frac{1}{\log(t)};$
2. (b)
For $f$ under (RV0), we have
$\lim_{t\to\infty}\frac{\log\omega(G_{t}(f))}{\log
F(t^{\frac{1}{2}})}=1,\;\text{almost surely.}$
### 1.3. Shaping the diameter
Another topological property of graphs which is also related to spread of
rumors and connectivity of networks is the diameter, that is, the maximal
graph distance between two vertices of said graph. Originally, investigating
the diameter of real-world networks, the authors in [25] observed that,
although coming from different contexts, those networks usually have diameter
of order less than the logarithm of the number of verices, the so-called
small-world phenomena.
In this paper we also address the issue of determining the order of the
diameter. Our main goals in this subject are to obtain a characterization for
the diameter imposing conditions on $f$ as weak as possible and also to obtain
regimes for the diameter arbitrarily small but still preserving the
scalefreeness of the graph.
In order to slow the growth of the diameter of PA-models, two observables play
important roles: the maximum degree and the proportion of vertices with low
degree. The former tends to concentrate connections on vertices with very high
degree which acts in the way of shortening the diameter, since they attract
connections to them. Whereas the latter, acts in the opposite way. In [28] and
[18], the authors have shown that in the configuration model with power-law
distribution the diameter order is extremely sensitive to the proportion of
vertices with degree 1 and 2.
One way to reduce the effect of low degree vertices on the diameter is via
affine preferential attachment rules, i.e., introducing a parameter $\delta$
and choosing vertices with probability proportional to their degree plus
$\delta$. In symbols, conditionally on $G_{t}$, we connect a new vertex
$v_{t+1}$ to an existing one $u$ with probability
${\mathbb{P}}\left(v_{t+1}\rightarrow
u\middle|G_{t}\right)=\frac{\text{degree}(u)+\delta}{\sum_{w\in
G_{t}}(\text{degree}(w)+\delta)}.$
By taking a negative $\delta$, the above rule increases the influence of high
degree vertices and indeed decreases drastically the diameter’s order. For
instance, for positive $\delta$ the diameter of $G_{t}$ is at least $\log(t)$,
whereas for $\delta<0$ the diameter of $G_{t}$ is at most $\log\log t$. See
[15] for several results on the diameter of different combinations for the
affine preferential attachment rule.
Diminishing the effect of low degree vertices is not enough to break the
growth of the diameter completely. The reason for that is, despite their low
degree, these vertices exist in large amount. Even the existence of a vertex
with degree close to $t$ at time $t$ may not be enough to freeze the
diameter’s growth. In [23] the authors have proven that the maximum degree of
a modification of the BA-model is of order $t$ at time $t$. However, the
authors believe that this is not enough to obtain a diameter of order
$\log\log t$, the reason being that this large hub still has to compete with a
large number of low degree vertices.
#### 1.3.1. General bounds for the diameter
As said before, our goal is to develop bounds for the diameter of $G_{t}(f)$
with $f$ as general as possible. Under the condition of monoticity, we prove
the following lower bound
###### Theorem 2 (Lower bound on the diameter).
Let $f$ be an edge-step function satisfying condition $(\mathrm{D})$. Then
(1.5)
${\mathbb{P}}\left(\mathrm{diam}(G_{t}(f))\geq\frac{1}{3}\left(\frac{\log
t}{\log\log t}\wedge\frac{\log t}{-\log f(t)}\right)\right)=1-o(1).$
For the upper bound, by a coupling argument, we are able to prove that the
diameter of the Barabási-Albert random tree is the ceiling for the diameter
generated by any $f$. Requiring more information on $f$, we prove upper bounds
that, for a broad class of functions, are of the same order of the lower
bounds given by the previous theorem. This is all summarized in the Theorem
below.
###### Theorem 3 (Upper bound on the diameter).
Let $f$ be an edge-step function. Then
1. (a)
$\mathrm{diam}(G_{t}(f))$ is at most the diameter of the Barabási-Albert
random tree, i.e.,
${\mathbb{P}}\left(\mathrm{diam}(G_{t}(f))\leq\log t\right)=1-o(1).$
2. (b)
if $f$ also satisfies condition (S) then there exists a positive constant
$C_{1}$ such that
${\mathbb{P}}\left(\mathrm{diam}(G_{t}(f))\leq 2+6\left(\frac{\log
t}{-\log\left(\sum_{s=t^{\frac{1}{13}}}^{t}\frac{f(s)}{s-1}\right)}\wedge\frac{\log
t}{\log\log t}\right)\right)\geq 1-C_{1}t^{-144^{-1}};$
3. (c)
if $f$ satisfies condition (Lκ) then there exists a positive constant $C_{2}$
such that
${\mathbb{P}}\left(\mathrm{diam}(G_{t}(f))\leq 2+\frac{6}{1-\kappa}\frac{\log
t}{\log\log t}\right)\geq 1-C_{2}t^{-144^{-1}}.$
#### 1.3.2. The class regularly varying functions
In [2], the authors prove a characterization of the empirical degree
distribution of graphs generated by $f$ satisfying condition (RVγ), for
$\gamma\in[0,1)$. More specifically, they prove that the degree distribution
of such graphs obeys a power law distribution whose exponent depends only on
the index of regular variation $-\gamma$.
A byproduct of our general bounds is a similar characterization for the
diameter. For edge-step functions satisfying conditions (D0) and (RVγ) for
$\gamma\in(0,\infty)$ the graphs generated by such functions have constant
diameter and its order depends only on the index of regular variation
$-\gamma$. We state this result in the theorem below
###### Theorem 4 (Diameter of regularly varying functions).
Let $f$ be an edge-step function satisfying conditions (D0) and (RVγ), for
$\gamma\in(0,\infty)$. Then,
${\mathbb{P}}\left(\frac{1}{4\gamma}\leq\mathrm{diam}(G_{t}(f))\leq\frac{100}{\gamma}+2\right)=1-o(1).$
#### 1.3.3. The class of slowly varying functions
The case when $\gamma=0$ is richer in terms of possible orders of the diameter
and does not admit a nice characterization as the one we obtain for positive
$\gamma$. In this settings, we present another consequence of our bounds for
particular subclasses of the class of slowly varying functions. Let us first
define the subclass of functions and later state how our results fit these
specific classes.
(1.6) $L:=\left\\{f\text{ is an edge-step function such that
}f(t)=\frac{1}{\log^{\alpha}(t)},\text{ for some }\alpha>0\right\\};$ (1.7)
$E:=\left\\{f\text{ is an edge-step function such that
}f(t)=e^{-\log^{\alpha}(t)},\text{ for some }\alpha\in(0,1)\right\\}.$
It is straightforward to verify that functions belonging to the set above
defined are slowly varying. For functions belonging to the two subclasses $L$
and $E$, our results have the following consequences, verifiable through
elementary calculus,
###### Corollary 1.8.
Let $f$ be an edge-step function.
1. (a)
if $f$ belongs to $L$, with $\alpha\leq 1$, then
${\mathbb{P}}\left(\frac{1}{3}\frac{\log t}{\log\log
t}\leq\mathrm{diam}(G_{t}(f))\leq\frac{8}{\alpha}\frac{\log t}{\log\log
t}\right)=1-o(1);$
2. (b)
if $f$ belongs to $L$, with $\alpha>1$, then
${\mathbb{P}}\left(\frac{1}{3\alpha}\frac{\log t}{\log\log
t}\leq\mathrm{diam}(G_{t}(f))\leq\frac{7}{\alpha-1}\frac{\log t}{\log\log
t}\right)=1-o(1);$
3. (c)
if $f$ belongs to $E$, then
${\mathbb{P}}\left(C_{\alpha}^{-1}(\log
t)^{1-\alpha}\leq\mathrm{diam}(G_{t}(f))\leq C_{\alpha}(\log
t)^{1-\alpha}\right)=1-o(1),$
for some $C_{\alpha}\geq 1$.
### 1.4. Comparing $G_{t}(f)$ and $G_{t}(h)$
If two edge-step functions are “close” to each other in some sense, then one
should expect that the random processes they generate should be “close” as
well. We make precise this intuition in the theorem below. In words, if $f$
and $h$ are close in the $L_{1}(\mathbb{N})$-norm (denoted by
$\|\cdot\|_{1}$), then $\mathrm{Law}(\\{G_{t}(f)\\}_{t\geq 1})$ is close to
$\mathrm{Law}(\\{G_{t}(h)\\}_{t\geq 1})$ in the total variation distance,
denoted by $\mathrm{dist}_{TV}(\cdot,\cdot)$.
###### Theorem 5.
Consider $f$ and $h$ two edge-step functions. We have
(1.9) $\mathrm{dist}_{TV}\left(\mathrm{Law}(\\{G_{t}(f)\\}_{t\geq
1}),\mathrm{Law}(\\{G_{t}(h)\\}_{t\geq 1})\right)\leq\|f-h\|_{1}.$
In particular, if $(f_{n})_{n\in\mathbb{N}}$ is a sequence of edge-step
functions, then
(1.10) $f_{n}\stackrel{{\scriptstyle
L_{1}(\mathbb{N})}}{{\longrightarrow}}f\implies\mathcal{L}((G_{t}(f_{n}))_{t\geq
1})\stackrel{{\scriptstyle\mathrm{dist}_{TV}}}{{\longrightarrow}}\mathcal{L}((G_{t}(f))_{t\geq
1}).$
The above theorem may be read as a perturbative statement. It assures that we
may add a small noise $\epsilon=\epsilon(t)$ to an edge-step function $f$ and
still obtain the same process up to an error of at most $\|\epsilon\|_{1}$ in
total variation distance.
### 1.5. Main technical ideas
In order to prove the existence of some given subgraph in the (affine) BA-
random graphs a key ingredient is usually to use the fact that two given
vertices $v_{i}$ and $v_{j}$ may be connected only at one specific time-step,
since (assuming $i<j$) the model’s dynamic only allows $v_{j}$ to connect to
$v_{i}$ at the moment in which $v_{j}$ is created. This property facilitates
the computation of the probability of the occurrence of a given subgraph and
decreases the combinatorial complexity of the arguments. In [9, 16] the
authors estimate the number of triangles and cherries (paths of length $3$) on
the (affine) BA-model and their argument relies heavily on this feature of the
model. In our case, however, the edge-step prevents an application of such
arguments, since a specific subgraph may appear at any time after the vertices
have been added.
Another difficulty in our setup is the degree of generality we work with. Our
case replaces the parameter $p\in(0,1]$ in the models investigated in [1, 12,
13] by any non-negative real function $f$ with $\|f\|_{\infty}\leq 1$. The
introduction of such function naturally increases the complexity of any
analytical argument one may expect to rely on and makes it harder to discover
threshold phenomena. This is the reason why in our work the Karamata’s Theory
of regularly varying functions is crucial in order to prove sharper results.
In order to overcome the issues presented above, more specifically to prove
Theorem 1 and Theorem 5, we construct an auxiliary process that we call the
doubly-labeled random tree process, $\\{\mathcal{T}_{t}\\}_{t\geq 1}$. In
essence, this process is a realization of the traditional BA-model (obtained
in our settings choosing $f\equiv 1$) where each vertex has two labels
attached to it. We then show in Proposition 2.1 how to generate $G_{t}(f)$
from $\mathcal{T}_{t}$ using the information on those labels. This procedure
allows us to generate $G_{t}(f)$ and $G_{t}(h)$, for two distinct functions
$f$ and $h$, from exactly the same source of randomness. The upshot is that an
edge-step function may be seen as a map from the space of doubly-labeled trees
to the space of (multi)graphs – therefore it makes sense to use the notation
$f(\mathcal{T}_{t})$. Furthermore, this map has the crucial property of being
monotonic (in a way we make precise latter). Roughly speaking, if $f\leq h$,
then certain monotonic graph observables respect this order, so if $\zeta$ is
such an observable than
$\zeta(f(\mathcal{T}_{t}))\leq\zeta(h(\mathcal{T}_{t}))$. In Proposition 2.5
we give important examples of suitable monotonic graph observables, the
diameter being one of them. Our machinery then allows us to transpose some
results about graphs generated for functions in a particular regime to another
just by comparing the functions themselves. We use known results about cliques
when $f$ is taken to be a constant less than one to propagate this result down
other regimes of functions.
For the proof of Theorem 2, we apply the second moment method on the number of
isolated paths. This approach demands correlation estimations for the
existence of two such paths in $G_{t}(f)$, which we do only under the
assumption of $f$ being monotonic. For Theorem 3, we apply a lower bound for
the degree of earlier vertices which is obtained by estimation of negative
moments of a given vertex’s degree, and then show that, under conditions (S)
or (Lκ), long paths of younger vertices are unlikely and older vertices are
all very close in graph distance. Finally, using results from the Karamata’s
theory of regularly varying functions, we verify that this broad class of
functions satisfies our assumptions, proving Theorem 4.
### 1.6. Organization
In Section 2 we introduce the doubly-labeled random tree process and prove the
main results about it and as a consequence of this results, we obtain Theorem
5. In Section 3 we explore the theory developed in the previous section to
prove the existence of large cliques, i.e., Theorem 1. Then, in Section 4, we
prove technical estimates for the degree of a given vertex, which is needed
for the upper bound on the diameter. Section 5 is devoted to the general lower
bound for the diameter, i.e., for the proof of Theorem 2. We prove the upper
bound for the diameter, Theorem 3, in Section 6. Finally, in Section 7 we show
how our results fit on the class of the regularly varying functions and
subclasses of slowly varying functions. We end the paper at Section 8 with
some comments on the affine version of our model and a brief discussion on
what may happen to the model if some regularity conditions are dropped.
### 1.7. Notation
We let $V(G_{t}(f))$ and $E(G_{t}(f))$ denote the set of vertices and edges of
$G_{t}(f)$, respectively. Given a vertex $v\in V(G_{t}(f))$, we will denote by
$D_{t}(v)$ its degree in $G_{t}(f)$. We will also denote by $\Delta D_{t}(v)$
the _increment_ of the discrete function $D_{t}(v)$ between times $t$ and
$t+1$, that is,
$\Delta D_{t}(v)=D_{t+1}(v)-D_{t}(v).$
When necessary in the context, we may use $D_{G}(v)$ to denote the degree of
$v$ in the graph $G$.
Given two sets $A,B\subseteq V(G_{t}(f))$, we let $\\{A\leftrightarrow B\\}$
denote the event where there exists an edge connecting a vertex from $A$ to a
vertex from $B$. We denote the complement of this event by
$\\{A\nleftrightarrow B\\}$. We let $\mathrm{dist}(A,B)$ denote the graph
distance between $A$ and $B$, i.e. the minimum number of edges that a path
that connects $A$ to $B$ must have. When one of these subsets consists of a
single vertex, i.e. $A=\\{v\\}$, we drop the brackets from the definition and
use $\\{v\leftrightarrow B\\}$ and $\mathrm{dist}(v,B)$, respectively.
For $t\in\mathbb{N}$, we let $[t]$ denote the set $\\{1,\dots,t\\}$.
Regarding constants, we let $C_{1},C_{2},\dots$ and $c,c_{1},c_{2},\dots$ be
positive real numbers that do not depend on $t$ whose values may vary in
different parts of the paper. The dependence on other parameters will be
highlighted throughout the text.
Since our model is inductive, we use the notation $\mathcal{F}_{t}$ to denote
the $\sigma$-algebra generated by all the random choices made up to time $t$.
In this way we obtain the natural filtration
$\mathcal{F}_{0}\subset\mathcal{F}_{1}\subset\dots$ associated to the process.
## 2\. The doubly-labeled random tree process
In this section we introduce a stochastic process
$\\{\mathcal{T}_{t}\\}_{t\geq 1}$ that provides a grand coupling between the
random graphs $\\{G_{t}(f)\\}_{t\geq 1}$ for every edge-step function $f$.
The process $\\{\mathcal{T}_{t}\\}_{t\geq 1}$ is essentially a realization of
the Barabási-Albert random tree where each vertex has two labels: an earlier
vertex chosen according to the preferential attachment rule and an independent
uniform random variable. The label consisting in the earlier vertex can be
seen as a “ghost directed edge”, we later use these random labels to collapse
subsets of vertices into a single vertex in order to obtain a graph with the
same distribution as $G_{t}(f)$ for any prescribed function
$f:\mathbb{N}\to[0,1]$.
We begin our process with a graph $\mathcal{T}_{1}$ consisting as usual in a
single vertex and a single loop connecting said vertex to itself. We then
inductively construct the labeled graph $\mathcal{T}_{t+1}$ from
$\mathcal{T}_{t}$ in the following way:
Figure 1. A sample of the process $\\{\mathcal{T}_{t}\\}_{t\geq 1}$ up to time
$6$ without the uniform labels. The dashed lines indicate the label
$\ell(v_{j})$ taken by each vertex $v_{j}$.
* (i)
We add to $\mathcal{T}_{t}$ a vertex $v_{t+1}$;
* (ii)
We tag $v_{t+1}$ with a random label $\ell(v_{t+1})$ chosen from the set
$V(\mathcal{T}_{t})$ with the preferential attachment rule, that is, with
probability of choosing $u\in V(\mathcal{T}_{t})$ proportional to the degree
of $u$ in $\mathcal{T}_{t}$;
* (iii)
Independently from the step above, we add an edge $\\{w,v_{t+1}\\}$ to
$E(\mathcal{T}_{t})$ where $w\in V(\mathcal{T}_{t})$ is also randomly chosen
according to the preferential attachment rule, see Figure 1.
We then finish the construction by tagging each vertex $v_{j}\in
V(\mathcal{T}_{t})$ with a second label consisting in an independent random
variable $U_{j}$ with uniform distribution on the interval $[0,1]$, as shown
in Figure 2. We note that only the actual edges contribute to the degree taken
in consideration in the preferential attachment rule, the tags are not
considered.
Figure 2. The graph $\mathcal{T}_{6}$, now each vertex receives an independent
uniform random variable.
We now make precise the notation $f(\mathcal{T}_{t})$ which indicates that an
edge-step function $f$ may also be seen as a function that maps a doubly-
labeled tree to a (multi)graph. Our goal is to define this map in such way
that $f(\mathcal{T}_{t})\stackrel{{\scriptstyle d}}{{=}}G_{t}(f)$.
In order to do so, let us fix such a function $f$. Given $v_{j}\in
V(\mathcal{T}_{t})$, we compare $U_{j}$ to $f(j)$. If $U_{j}\leq f(j)$, we do
nothing. Otherwise, we collapse $v_{j}$ onto its label $\ell(v_{j})$, that is,
we consider the set $\\{v_{j},\ell(v_{j})\\}$ to be a single vertex with the
same labels as $\ell(v_{j})$. We then update the label of all vertices $v$
such that $\ell(v)=v_{j}$ to $\\{v_{j},\ell(v_{j})\\}$. This procedure is
associative in the sense that the order of the vertices on which we perform
this operation does not affect the final resulting graph, as long as we
perform it for all the vertices of $\mathcal{T}_{t}$. We let then
$f(\mathcal{T}_{t})$ be the (multi)graph obtained when this procedure has run
over all the $t$ vertices of $\mathcal{T}_{t}$. We refer to Figure 3 as an
illustration of the final outcome.
In the next proposition we prove that $f(\mathcal{T}_{t})$ is indeed
distributed as $G_{t}(f)$.
Figure 3. The figure shows how one can sample the distribution of $G_{6}(f)$
using the labeled graph $\mathcal{T}_{6}$.
###### Proposition 2.1.
Let $\mathcal{T}_{t}$ be the doubly-labeled tree above defined and $f$ an
edge-step function. Then,
$f(\mathcal{T}_{t})\stackrel{{\scriptstyle d}}{{=}}G_{t}(f).$
###### Proof.
We first observe that the associativity of the collapsing operation and the
independence of the sequence $(U_{j})_{j\geq 1}$ from the previous operations
imply that we can glue together the vertex $v_{t+1}$ to $\ell(v_{t+1})$
whenever $U_{t+1}>f(t+1)$ right after we complete step (iii) of the above
construction by induction. The resulting graph has either a new vertex
$v_{t+1}$ with an edge $\\{w(v_{t+1}),v_{t+1}\\}$ or an edge
$\\{w(v_{t+1}),\ell(v_{t+1})\\}$ with the exact same probability distribution
as the $(t+1)$-th step in the construction of the graph $(G_{t}(f))_{t\geq
1}$. By induction, both random graphs have the same distribution.
In the light of the above discussion and Proposition 2.1, from now on, we will
tacitly assume that all process $\\{G_{t}(f)\\}_{t\geq 1}$ for all edge-step
functions are on the same probability space provided by our previous results.
A straightforward consequence of Proposition 2.1 is Theorem 5 which roughly
speaking states that edge-step functions close to each other in the
$L_{1}(\mathbb{N})$-norm generates essentially the same processes. Once we
have the machinery provided by Proposition 2.1, the proof of this fact becomes
a simple application of the union bound.
###### Proof of Theorem 5.
Fixed two edge-step functions $f$ and $h$, by Proposition 2.1 we have that
(2.2) $\mathrm{Law}(\\{G_{t}(f)\\}_{t\geq
1})=\mathrm{Law}(\\{f(\mathcal{T}_{t})\\}_{t\geq 1}).$
Thus
(2.3) $\begin{split}\mathrm{dist}_{TV}\left(\mathrm{Law}(\\{G_{t}(f)\\}_{t\geq
1}),\mathrm{Law}(\\{G_{t}(h)\\}_{t\geq
1})\right)&\leq{\mathbb{P}}\left(\\{f(\mathcal{T}_{t})\\}_{t\geq
1}\neq\\{h(\mathcal{T}_{t})\\}_{t\geq 1}\right)\\\
&\leq\sum_{i=1}^{\infty}{\mathbb{P}}\left(U_{i}\in(f(i)\wedge h(i),f(i)\vee
h(i))\right)\\\ &=\sum_{i=1}^{\infty}|f(i)-g(i)|\\\ &=\|f-g\|_{1},\end{split}$
and the above equation immediately implies (1.10).
The next step is to use the coupling provided by the auxiliary process
$\\{\mathcal{T}_{t}\\}_{t\geq 1}$ to compare graph observable of graphs
generated by different edge-step functions. This method will allow us to
transport results we have obtained for a fixed $f$ to other edge-step
functions by comparing them with $f$. To do this we introduce the notion of
increasing (decreasing) graph observable.
###### Definition 1 (Monotone graph observable).
We say that a (multi)graph observable $\zeta$ is increasing if, given two
edge-step functions $f$ and $h$, we have that
(2.4) $f(s)\leq h(s),\forall s\leq
t\implies\zeta(f(\mathcal{T}_{t}))\leq\zeta(h(\mathcal{T}_{t}))\;\textit{a.s.}$
When the second inequality in (2.4) holds with “$\geq$”, we say $\zeta$ is
decreasing.
A first example of an increasing observable is the total number of vertices.
Indeed, since every vertex $v_{j}$ of $\mathcal{T}_{t}$ remain preserved under
$f$ whenever its $U_{j}$ label is less than $f(j)$, it follows that every
$f$-preserved $v_{j}$ is $h$-preserved as well.
The next proposition states that the maximum degree is a decreasing observable
whereas the diameter, in which we are interested in, is an increasing one.
###### Proposition 2.5.
Let $f$ and $h$ be two edge-step functions satisfying the relation $f(s)\leq
h(s)$ for all $s\in(1,\infty)$. Then,
1. (a)
the maximum degree is a _decreasing_ graph observable. More precisely, if we
let $D_{\textit{max}}(G)$ denote the maximum degree of a (multi)graph $G$,
then
${\mathbb{P}}\left(\forall t\in\mathbb{N},\;D_{\textit{max}}(G_{t}(f))\geq
D_{\textit{max}}(G_{t}(h))\right)=1;$
2. (b)
the diameter is an _increasing_ graph observable. More precisely,
${\mathbb{P}}\left(\forall
t\in\mathbb{N},\;\mathrm{diam}(G_{t}(f))\leq\mathrm{diam}(G_{t}(h))\right)=1$
###### Proof.
The proofs follow from an analysis of the action of $f$ over
$\mathcal{T}_{t}$.
Proof of part (a): We first point out that if $v_{s}$ is a vertex of
$\mathcal{T}_{t}$ whose $U_{s}$-label is less than $f(s)$, then
(2.6) $D_{G_{t}(f)}(v_{s})\geq D_{G_{t}(h)}(v_{s}).$
To see why the above inequality is true, notice that the degree of $v_{s}$ in
$G_{t}(f)$ is the total number of solid edges (see Figure 2) incident on
$v_{s}$ in $\mathcal{T}_{t}$ plus the total number of vertices whose
$\ell$-label points to $v_{s}$ and whose $U$-label is greater than $f$.
Formally,
(2.7) $D_{G_{t}(f)}(v_{s})=\sum_{r=1}^{t}\mathbb{1}\left\\{v_{r}\to
v_{s}\text{ in
}\mathcal{T}_{t}\right\\}+\mathbb{1}\left\\{\ell(v_{r})=v_{s},U_{r}\geq
f(r)\right\\}.$
The contribution of the first term in the RHS of the above identity remains
stable under $f$ and $h$, however, being $f(r)\leq h(r)$ for all $r\leq t$, we
automatically have
$\mathbb{1}\left\\{\ell(v_{r})=v_{s},U_{r}\geq
f(r)\right\\}\geq\mathbb{1}\left\\{\ell(v_{r})=v_{s},U_{r}\geq h(r)\right\\},$
which implies that $D_{G_{t}(f)}(v_{s})\geq D_{G_{t}(h)}(v_{s})$.
To finish the proof of this part, notice that, if $v_{s}$ is truly a vertex in
$G_{t}(h)$ but is glued to its $\ell$-label under $f$, this operation just
increases the maximum degree in the sense that the degree of $\ell(v_{s})$
inherits all the contributions of $v_{s}$. Thus, if $v_{\textit{max}}$ is a
vertex that achieves $D_{\textit{max}}(G_{t}(h))$, then if $v_{\textit{max}}$
is also a vertex of $G_{t}(f)$ by the above discussion, its degree is at least
$D_{\textit{max}}(G_{t}(h))$. On the other hand, if it is identified to
$\ell(v_{\textit{max}})$, then the degree of $\ell(v_{\textit{max}})$ is also
at least the maximum degree of $G_{t}(h)$.
Proof of part (b): Observe that if $u$ and $w$ are vertices whose $U$-label is
less than $f$ and such that $u\leftrightarrow w$ in $G_{t}(f)$, then $u$ and
$w$ are possibly not connected by a single edge in $G_{t}(h)$, in other words,
if $\mathrm{dist}_{G_{t}(f)}(u,w)=1$ then $\mathrm{dist}_{G_{t}(h)}(u,w)\geq
1$. Therefore, if $v_{0}=u,v_{1},\cdots,v_{k}=w$ is a minimal path in
$G_{t}(f)$, by the previous observation, this path induces a path in
$G_{t}(h)$ whose length is equal to or greater than $k$.
On the other hand, if $u=v_{0},v_{1},\cdots,v_{k}=w$ is a path realizing the
minimal distance between $u$ and $w$, in $G_{t}(h)$ we have that this path
induces another path in $G_{t}(f)$ of same length or less. To see this, just
notice that, if $U_{j}>f(j)$ and $\ell(v_{j})$ points to a vertex outside the
path for some $j\in\\{2,3,\cdots,k-1\\}$, then, after identifying $v_{j}$ to
its $\ell$-label, this operation does not increase the path length. This is
enough to conclude the proof.
## 3\. Large Cliques: proof of Theorem 1
In this section we keep exploring the machinery developed in previous
sections. Here, we transpose the existence of large cliques using the edge-
step function, i.e., if $h$ is such that $G_{t}(h)$ has a clique of order $K$
and $f$ another function satisfying $f(s)\leq h(s)$ for all $s\in(1,\infty)$,
then the clique existence propagates to $G_{t}(f)$ too but with a possibly
$f$-dependent order.
Our strategy to prove Theorem 1 will be to apply the results obtained for the
function identically equal to $p$, with $p\in(0,1)$ (which we denote by
$\mathrm{cons}_{p}$) in [1] and propagate them to smaller functions. For the
sake of the reader’s convenience, we state and comment the aforementioned
results here. Given $m\in\mathbb{Z}_{+}$, we order the vertices of
$G_{t}(\mathrm{cons}_{p})$ from oldest to earliest and then divide them into
blocks of size $m$. We then denote by $d_{t,m}(j)$ the sum of all degrees of
the vertices from the $j$-th block. The following theorem has a very involved
notation, but in essence it provides with high probability an explicit
polynomial lower bound for $d_{t,m}(j)$.
###### Theorem (Theorem 2 from [1] ).
Given $p\in(0,1]$, let $\xi\in(0,2(2-p)^{-1}-1)$ and fix $m\in\mathbb{Z}_{+}$
sufficiently large. Define
$\zeta_{m}:=\frac{(1-p)}{2(2-p)m^{\xi}}.$
and let
$1<R<m(1-p/2)(1-\zeta_{m}).$
There exists a positive constant $c=c(m,R,p)$ such that, for
$\beta\in(0,(1-p/2)(1-\zeta_{m}))\textit{ and }j\geq m^{\frac{2}{1-p}}+1,$
we have
(3.1) $\mathbb{P}\left(d_{t,m}(j)<t^{\beta}\right)\leq
c\frac{j^{R}}{t^{R-\beta
R(1-p/2)^{-1}(1-\zeta_{m})^{-1}}}+\frac{m}{[(j-1)m]^{99}}.$
Using the above bound one can then prove
###### Theorem (Theorem $1$ from [1]).
For any $\varepsilon>0$ and every $p\in(0,1)$, it follows that
(3.2) ${\mathbb{P}}\left(\exists K_{n}\subset
G_{t}(\mathrm{cons}_{p}),\textit{ such that
}n=t^{(1-\varepsilon)\frac{(1-p)}{2-p}}\right)=1-o(1).$
We can now use the doubly-labeled tree together with the above results in
order to prove Theorem 1. It will be useful to recall our special notation to
the expected number of vertices,
(3.3) $F(t):={\mathbb{E}}V(G_{t}(f))=1+\sum_{s=2}^{t}f(s).$
###### Proof of Theorem 1.
Given $\delta>0$ sufficiently small, let $\varepsilon,p=\delta/6$ so that
$\alpha=\alpha(p,\varepsilon):=\frac{1-p}{2-p}(1-\varepsilon)>\frac{1}{2}\left(1-\frac{\delta}{2}\right).$
In the proof of Theorem $1$ of [1] one uses Theorem $2$ of [1] to show that
there exist a fixed integer $m=m(\varepsilon,p)>0$ and a small number
$\varepsilon^{\prime}\in(0,\alpha)$ with the following property: if one
divides the set of vertices born between times $t^{\varepsilon^{\prime}}$ and
$t^{\alpha}$ into disjoint subsets of $m$ vertices born _consecutively_ , then
with high probability (at least $1$ minus a polynomial function of $t$) one
can choose a vertex from each of these subsets in such a way that the subgraph
induced by the set of chosen vertices is a complete subgraph of
$G_{t}(\mathrm{cons}_{p})$.
Having the above in mind and letting $m$ be the auxiliary $m$ that appears in
proof of Theorem 1 of [1], define the following event
(3.4)
$A_{p,\varepsilon^{\prime},m,k}:=\left\\{\begin{array}[]{c}\exists\\{t_{1},\cdots,t_{k}\\}\subset[t]\text{
such that
}t_{j}\in(t^{\varepsilon^{\prime}}+(j-1)m,t^{\varepsilon^{\prime}}+jm]\\\
\text{and }v_{t_{j}}\leftrightarrow v_{t_{i}}\textit{ in
}G_{t}(\mathrm{cons}_{p})\text{ for all }i,j\in[k]\end{array}\right\\}.$
In the context of the doubly labeled tree process, the event
$A_{p,\varepsilon^{\prime},m,k}$ says that we may find $k$ vertices of
$\mathcal{T}_{t}$ with the property that the $j$-th vertex $v_{t_{j}}$ was
added by the doubly-labeled tree process sometime in the interval
$(t^{\varepsilon^{\prime}}+(j-1)m,t^{\varepsilon^{\prime}}+jm]$ and its
$U_{t_{j}}$-label is less than $p$. Moreover, when we apply
$\mathrm{cons}_{p}$ on $\mathcal{T}_{t}$ in the event
$A_{p,\varepsilon^{\prime},m,k}$, all these $k$ vertices form a complete graph
in the resulting graph $G_{t}(\mathrm{cons}_{p})$. Theorem $1$ of [1] states
that setting $k=t^{\alpha}-t^{\varepsilon^{\prime}}$, the event
$A_{p,\varepsilon^{\prime},m,k}$ occurs with probability at least
$1-t^{-\eta}$, for some positive small $\eta$ depending on $p$, which, in our
case, is a function of $\delta$. Thus, for $t$ large enough, we may simply use
that
(3.5) ${\mathbb{P}}\left(A_{p,\varepsilon^{\prime},m,t^{\alpha}}\right)\geq
1-\frac{1}{\log t},$
this bound will be useful for the proof of Corollary 1.4. This is why we are
exchanging a polynomial decay by a $\log$ one, to get rid off the dependency
on $\delta$ and consequently simplify latter arguments.
Now, let $t$ be large enough so that $f(t^{\varepsilon^{\prime}})<p$. This is
possible since $f$ decreases to zero. Also notice that if $u$ and $v$ are
vertices of $G_{t}(f)$ added after time $t^{\varepsilon^{\prime}}$ and are
connected in $G_{t}(\mathrm{cons}_{p})$, then they are connected in $G_{t}(f)$
as well. Moreover, since $f$ is non-increasing and the $U$-labels are assigned
independently and according to a uniform distribution on $[0,1]$, we have for
all $j\leq k$
(3.6) ${\mathbb{P}}\left(U_{t_{j}}\leq
f(t_{j})\;\middle|\;A_{p,\varepsilon^{\prime},m,k}\right)\geq
p^{-1}f(t^{\varepsilon^{\prime}}+jm).$
Setting $k=(t^{\alpha}-t^{\varepsilon^{\prime}})/m$ and using that $f$ is non-
increasing, we have that
(3.7) $m\sum_{j=0}^{k}f(t^{\varepsilon^{\prime}}+jm)\geq
F(t^{\alpha})-F(t^{\varepsilon^{\prime}}+m)\geq F(t^{\alpha})/2$
for large enough $t$.
By the independence of the $U$-labels, it follows that conditioned on
$A_{p,\varepsilon^{\prime},m,k}$, the random variable that counts how many
vertices of the clique in $G_{t}(\mathrm{cons}_{p})$ remain vertices of
$G_{t}(f)$ as well is a sum of $k$ independent random variables taking values
on $\\{0,1\\}$ and whose expected value is greater than $F(t^{\alpha})/2mp$.
Finally, by Chernoff bounds, this random variable is at least half its
expected value with probability at least $1-\exp\\{-F(t^{\alpha})/16mp\\}$.
This and (3.5) gives us that there exists a constant $C_{\delta}$ such that
(3.8) ${\mathbb{P}}\left(\nexists K_{n}\subset G_{t}(f),\textit{ such that
}n\geq C_{\delta}F(t^{\frac{1-\delta}{2}})\right)\leq\frac{1}{\log
t}+e^{-F(t^{\alpha})/16mp},$
which proves the Theorem.
## 4\. Technical estimates for the degree
In this section we develop technical estimates related to the degree of a
given vertex. We begin by stating one of the most fundamental identities in
the study of preferential attachment models: the conditional distribution of
the increment of the degree of a given vertex. Given $v\in V(G_{t}(f))$, we
have
(4.1) $\begin{split}\mathbb{P}\left(\Delta
D_{t}(v)=0\middle|\mathcal{F}_{t}\right)&=f(t+1)\left(1-\frac{D_{t}(v)}{2t}\right)+(1-f(t+1))\left(1-\frac{D_{t}(v)}{2t}\right)^{2}\\\
\mathbb{P}\left(\Delta
D_{t}(v)=1\middle|\mathcal{F}_{t}\right)&=f(t+1)\frac{D_{t}(v)}{2t}+2(1-f(t+1))\frac{D_{t}(v)}{2t}\left(1-\frac{D_{t}(v)}{2t}\right),\\\
\mathbb{P}\left(\Delta
D_{t}(v)=2\middle|\mathcal{F}_{t}\right)&=(1-f(t+1))\frac{D_{t}(v)^{2}}{4t^{2}}.\end{split}$
To see why the above identities hold true, observe for example that in order
for $\Delta D_{t}(v)=0$, either a vertex step was taken, and the vertex did
not connect to $v$, or an edge step was taken and neither of the endpoints of
the new edge connected to $v$. The other equations follow from analogous
reasonings. As a direct consequence, we obtain
(4.2) $\begin{split}{\mathbb{E}}\left[\Delta
D_{t}(v)\middle|\mathcal{F}_{t}\right]&=1\cdot
f(t+1)\cdot\frac{D_{t}(v)}{2t}+1\cdot
2(1-f(t+1))\frac{D_{t}(v)}{2t}\left(1-\frac{D_{t}(v)}{2t}\right)\\\
&\quad+2\cdot(1-f(t+1))\frac{D^{2}_{t}(v)}{4t^{2}}\\\
&=\left(1-\frac{f(t+1)}{2}\right)\frac{D_{t}(v)}{t}.\end{split}$
Using the above equation repeatedly one obtains, conditioned on the event
where the vertex $v$ is born at time $t_{0}$,
(4.3)
$\begin{split}{\mathbb{E}}\left[D_{t}(v)\right]&={\mathbb{E}}\left[{\mathbb{E}}\left[D_{t}(v)\middle|\mathcal{F}_{t-1}\right]\right]\\\
&=\left(1+\frac{1}{t-1}-\frac{f(t)}{2(t-1)}\right){\mathbb{E}}\left[D_{t-1}(v)\right]\\\
&=\prod_{s=t_{0}}^{t-1}\left(1+\frac{1}{s}-\frac{f(s+1)}{2s}\right).\end{split}$
We state a lower bound for the degree, whose proof comes from a direct
application of Theorem $2$ of [1] and Proposition 2.5, which assures that the
maximum degree is a decreasing graph observable.
###### Lemma 1.
Let $f$ be an edge-step function such that $f(t)\rightarrow 0$ as $t$ goes to
infinity. Then, for any fixed $\varepsilon>0$ we have
${\mathbb{P}}\left(D_{\textit{max}}\left(G_{t}(f)\right)<t^{1-\varepsilon}\right)\leq
t^{-2}.$
###### Proof.
By Theorem $2$ of [1], w.h.p, $D_{\text{max}}(G_{t}(p))$ is at least
$t^{(1-\varepsilon)(1-p/2)}$, for fixed $\varepsilon>0$ and $p\in(0,1)$. Thus,
taking $p_{0}$ small enough, we have that $D_{\text{max}}(G_{t}(p_{0}))$ is at
least $t^{1-\varepsilon}$, w.h.p.
Then, using Proposition 2.5, we have that for large enough $t$,
$D_{\text{max}}(G_{t}(f))\geq D_{\text{max}}(G_{t}(p_{0}))$ which proves the
Lemma.
Our objective now is to obtain a polynomial lower bound for the degree of
older vertices, which will be important in the proof of the upper bound for
the diameter in Theorem 3. We begin with an upper bound for the expectation of
the multiplicative inverse of the degree. Recall the definition of the process
$(Z_{t})_{t\geq 1}$, consisting of independent Bernoulli variables that
dictate whether a vertex-step or an edge-step is performed at time $t$.
###### Lemma 2.
Given any edge-step function $f$, consider the process $\\{G_{t}(f)\\}_{t\geq
1}$. Denote by $v_{i}$ the vertex born at time $i\in\mathbb{N}$. We have
(4.4) ${\mathbb{E}}\left[(D_{t}(v_{i}))^{-1}Z_{i}\right]\leq
f(i)\left(\frac{t-1}{i}\right)^{-\frac{1}{6}}.$
And consequently
(4.5)
${\mathbb{P}}\left(D_{t}(v_{i})\leq\left(\frac{t-1}{i}\right)^{\frac{1}{12}}\middle|Z_{i}=1\right)\leq\left(\frac{t-1}{i}\right)^{-\frac{1}{12}}.$
###### Proof.
If $Z_{i}=1$, then for every $s\geq 1$ we have that $\Delta D_{s}(v_{i})\geq
0$, that $D_{s+1}(v_{i})\leq D_{s}(v_{i})+2\leq 3D_{s}(v_{i})$, and that
$D_{s}(v_{i})$ is $\mathcal{F}_{s}$ measurable. Together with (4.2), these
facts imply. on the event $\\{Z_{i}=1\\}$,
(4.6)
$\begin{split}{\mathbb{E}}\left[\frac{1}{D_{s+1}(v_{i})}-\frac{1}{D_{s}(v_{i})}\middle|\mathcal{F}_{s}\right]&={\mathbb{E}}\left[-\frac{\Delta
D_{s}(v_{i})}{D_{s+1}(v_{i})D_{s}(v_{i})}\middle|\mathcal{F}_{s}\right]\\\
&\leq-\frac{1}{3(D_{s}(v_{i}))^{2}}\left(1-\frac{f(s+1)}{2}\right)\frac{D_{s}(v_{i})}{s}\\\
&\leq-\frac{1}{6s}\frac{1}{D_{s}(v_{i})},\end{split}$
since $f(k)\leq 1$ for very $k\in\mathbb{N}$. Therefore,
(4.7)
${\mathbb{E}}\left[Z_{i}\cdot(D_{s+1}(v_{i}))^{-1}\middle|\mathcal{F}_{s}\right]\leq
Z_{i}(1-(6s)^{-1})(D_{s}(v_{i}))^{-1}.$
Iterating the above argument from $i$ until $t$, we obtain
(4.8) ${\mathbb{E}}\left[Z_{i}\cdot(D_{t}(v_{i}))^{-1}\right]\leq
f(i)\prod_{s=i}^{t-1}\left(1-\frac{1}{6s}\right)\leq
f(i)\exp\left\\{-\frac{1}{6}\sum_{s=i}^{t-1}\frac{1}{s}\right\\}\leq
f(i)\left(\frac{t-1}{i}\right)^{-\frac{1}{6}},$
proving (4.4). Equation (4.5) is then obtained by an elementary application of
the Markov inequality:
(4.9)
$\begin{split}{\mathbb{P}}\left(D_{t}(v_{i})\leq\left(\frac{t-1}{i}\right)^{\frac{1}{12}}\middle|Z_{i}=1\right)&={\mathbb{P}}\left((D_{t}(v_{i}))^{-1}\geq\left(\frac{t-1}{i}\right)^{-\frac{1}{12}}\middle|Z_{i}=1\right)\\\
&\leq\left(\frac{t-1}{i}\right)^{\frac{1}{12}}{\mathbb{E}}\left[(D_{t}(v_{i}))^{-1}\middle|Z_{i}=1\right]\\\
&\leq\left(\frac{t-1}{i}\right)^{-\frac{1}{12}}.\end{split}$
We now provide an elementary consequence of the above result, which uses the
union bound in order to show that, with high probability, every vertex born
before time $t^{\frac{1}{12}}$ has degree at least $t^{\frac{1}{15}}$ by time
$t$.
###### Lemma 3.
Using the same notation as in Lemma 2 we have, for every edge-step function
$f$ and for sufficiently large $t\in\mathbb{N}$,
(4.10) ${\mathbb{P}}\left(\exists i\in\mathbb{N},1\leq i\leq
t^{\frac{1}{12}},\text{ such that }Z_{i}=1\text{ and }D_{t}(v_{i})\leq
t^{\frac{1}{15}}\right)\leq Ct^{-\frac{1}{144}}.$
###### Proof.
By the union bound and equation (4.5), we have that the probability in the
left hand side of (4.10) is smaller than or equal to
(4.11)
$\begin{split}\sum_{i=1}^{t^{\frac{1}{12}}}{\mathbb{P}}\left(Z_{i}=1,D_{t}(v_{i})\leq
t^{\frac{1}{15}}\right)&\leq\sum_{i=1}^{t^{\frac{1}{12}}}f(i){\mathbb{P}}\left(D_{t}(v_{i})\leq\left(\frac{t-1}{i}\right)^{\frac{1}{12}}\middle|Z_{i}=1\right)\\\
&\leq\sum_{i=1}^{t^{\frac{1}{12}}}\left(\frac{t-1}{i}\right)^{-\frac{1}{12}}\\\
&\leq Ct^{-\frac{1}{12}+\frac{1}{12}\left(1-\frac{1}{12}\right)}\\\ &\leq
Ct^{-\frac{1}{144}},\end{split}$
finishing the proof of the Lemma.
## 5\. General lower bound for the diameter: proof of Theorem 2
The proof of Theorem 2 follows a second moment argument. The idea is to count
the number of “long” (the specific size depending on $f$ and $t$) _isolated
paths_ in $G_{t}(f)$. We begin by showing that the expected number of isolated
paths goes to infinity with $t$ in Lemma 4. We then show in Lemma 5 that the
presence of a specific isolated path is almost independent from the presence
of some other given isolated path whenever said paths are disjoint. This
“almost independence” makes the second moment of the number of such paths very
close to the first moment squared. The proof is then completed via the Payley-
Zygmund inequality.
We start by defining precisely what we mean by an isolated path.
###### Definition 2 (Isolated path).
Let $l$ be a positive integer. Let $\vec{t}=(t_{1},..,t_{l})$ be a vector of
distinct positive integers. We say that this vector corresponds to an isolated
path $\\{v_{t_{1}},\dots,v_{t_{l}}\\}$ in $G_{t}(f)$ if and only if:
* •
$t_{l}\leq t$;
* •
$t_{i}<t_{j}$ whenever $1\leq i<j\leq l$;
* •
during each time $t_{i}$, $i=1,\dots,l$, a vertex-step is performed;
* •
for every integer $k\leq l$, the subgraph induced by the vertices
$\\{v_{t_{i}}\\}_{1\leq i\leq k}$ is connected in $G_{t_{k}}(f)$;
* •
for $i=1,\dots,l-1$, the degree of $v_{t_{i}}$ in $G_{t}(f)$ is $2$. The
degree of $v_{t_{l}}$ in $G_{t}(f)$ is $1$.
In other words, an isolated path $\\{v_{t_{i}}\\}_{1\leq i\leq l}$ is a path
where each vertex $v_{t_{i}}$, for $i=2,\dots,l$, is born at time $t_{i}$ and
makes its first connection to its predecessor $v_{t_{i-1}}$. Other than that,
no other vertex or edge gets attached to $\\{v_{t_{i}}\\}_{1\leq i\leq l}$. We
will denote $\\{v_{t_{i}}\\}_{1\leq i\leq l}$ by $v_{\vec{t}}$.
Given $\xi\in(0,1)$, denote by $\mathcal{S}_{l,\xi}(t)$ the set of all
isolated paths in $G_{t}(f)$ of size $l$ whose vertices were created between
times $\xi t$ and $t$. Our first goal is to obtain lower bounds for
${\mathbb{E}}\left[|\mathcal{S}_{l,\xi}(t)|\right]$:
###### Lemma 4.
Let $f$ be a non-increasing edge-step function, then, for any $0<\xi<1$ and
any integer $l$, the following lower bound holds:
(5.1)
${\mathbb{E}}\left[|\mathcal{S}_{l,\xi}(t)|\right]\geq\binom{(1-\xi)t}{l}\frac{f(t)^{l}}{(2t)^{l-1}}\left(1-\frac{2l}{\xi
t}\right)^{t}.$
Furthermore, for
(5.2) $l\leq\frac{1}{3}\left(\frac{\log t}{\log\log t}\wedge\frac{\log
t}{-\log f(t)}\right),$
we have that, for sufficiently large $t$,
(5.3) ${\mathbb{E}}\left[|\mathcal{S}_{l,\xi}(t)|\right]\geq t^{\frac{1}{4}}.$
###### Proof.
The random variable $|\mathcal{S}_{l,\xi}(t)|$ can be written as
(5.4)
$|\mathcal{S}_{l,\xi}(t)|=\sum_{t_{1}<t_{2}<\dots<t_{l}}\mathbb{1}\left\\{v_{\vec{t}}\in\mathcal{S}_{l,\xi}(t)\right\\}\quad\implies\quad{\mathbb{E}}\left[|\mathcal{S}_{l,\xi}(t)|\right]=\sum_{t_{1}<t_{2}<\dots<t_{l}}\mathbb{P}\left(v_{\vec{t}}\in\mathcal{S}_{l,\xi}(t)\right).$
So it will be important to obtain a proper lower bound for
$\mathbb{P}\left(v_{\vec{t}}\in\mathcal{S}_{l,\xi}(t)\right)$. Given a time
vector of an isolated path $\vec{t}=(t_{1},\dots,t_{l})$ such that $\xi t\leq
t_{1}<t_{2}<\dots<t_{l}\leq t$, it follows that
(5.5)
$\mathbb{P}\left(v_{\vec{t}}\in\mathcal{S}_{l,\xi}(t)\right)\geq\frac{f(t)^{l}}{(2t)^{l-1}}\left(1-\frac{2l}{\xi
t}\right)^{t},$
since in order for $v_{\vec{t}}$ to be in $\mathcal{S}_{l,\xi}(t)$, we need to
assure that $l$ vertices are born exactly at times $t_{1},\dots,t_{l}$ (which
happens with probability greater than $f(t)^{l}$, by the monotonicity of $f$),
that $v_{t_{i}}$ connects to $v_{t_{i-1}}$ for every $i=2,\dots,l$ (which
happens with probability greater than $(2t)^{-(l-1)}$), and that no other
vertex or edge connects to $v_{\vec{t}}$ until time $t$ (which happens with
probability greater than $\left(1-\frac{2l}{\xi t}\right)^{t}$).
Finally, by counting the number of possible ways to choose
$t_{1}<t_{2}<\dots<t_{l}$ so that $t_{i}\in[\xi t,t]$ for $1\leq i\leq l$, we
obtain 5.1.
We now assume $l$ to be such that 5.2 holds. Stirling’s formula gives us
$\begin{split}\log\left(\binom{(1-\xi)t}{l}\right)&\geq
c+(1-\xi)t\log((1-\xi)t)-(1-\xi)t+\frac{\log((1-\xi)t)}{2}\\\ &\quad-l\log
l+l-\frac{\log l}{2}\\\
&\quad-((1-\xi)t-l)\log((1-\xi)t-l)+(1-\xi)t-l-\frac{\log((1-\xi)t-l)}{2},\end{split}$
Since $l\ll t$, we obtain
(5.6) $\log\left(\binom{(1-\xi)t}{l}\right)\geq l\log t-l\log l-cl.$
Equation (5.5) then implies, again for sufficiently large $t$,
(5.7)
$\log\left(\mathbb{P}\left(v_{\vec{t}}\in\mathcal{S}_{l,\xi}(t)\right)\right)\geq
l\log f(t)-(l-1)\log t-cl.$
Combining the above inequality with (5.1), (5.2), and (5.6) gives us that, for
large enough $t$,
$\begin{split}{\mathbb{E}}\left[|\mathcal{S}_{l,\xi}(t)|\right]\geq\exp\left\\{l\log
t-l\log l+l\log f(t)-(l-1)\log t-cl\right\\}\geq t^{\frac{1}{4}},\end{split}$
since
$l\log l\leq\frac{1}{3}\frac{\log t}{\log\log t}\log\log
t\left(1-\frac{\log\log\log t+\log 3}{\log\log t}\right)\leq\frac{\log t}{3},$
which finishes the proof of the lemma.
Some new notation will be useful throughout the proof of Theorem 2:
###### Definition 3 (Degree of an isolated path).
Given an isolated path $v_{\vec{t}}=\\{v_{t_{j}}\\}_{1\leq j\leq l}$, we
denote by $D_{r}(v_{\vec{t}})$ the sum of the degrees of each of its vertices
at time $r$, i.e. :
$D_{r}(v_{\vec{t}})=\sum_{t_{i}\in\vec{t}}D_{r}(v_{t_{i}}),$
where we assumed that $D_{r}(v_{t_{i}})=0$ if $t_{i}<r$.
Note that if $r>t_{l}$ and $v_{\vec{t}}$ has size $l$, then
$D_{r}(v_{\vec{t}})=2l-1$. Furthermore, by the same reasoning as in (4.1),
(5.8) $\begin{split}{\mathbb{P}}\left(\Delta
D_{s}(v_{\vec{t}})=0~{}\middle|\mathcal{F}_{s}\right)&=f(s+1)\left(1-\frac{D_{s}(v_{\vec{t}})}{2s}\right)+(1-f(s+1))\left(1-\frac{D_{s}(v_{\vec{t}})}{2s}\right)^{2}\\\
&=1-\left(1-\frac{f(s+1)}{2}-(1-f(s+1))\frac{D_{s}(v_{\vec{t}})}{4s}\right)\frac{D_{s}(v_{\vec{t}})}{s}.\end{split}$
Paley-Zigmund’s inequality (see e.g. section $5.5$ of [22]) assures us that,
for any $0\leq\theta\leq 1$,
(5.9)
$\mathbb{P}\left(|\mathcal{S}_{l,\xi}(t)|>\theta{\mathbb{E}}\left[|\mathcal{S}_{l,\xi}(t)|\right]\right)\geq(1-\theta)^{2}\frac{\left({\mathbb{E}}\left[|\mathcal{S}_{l,\xi}(t)|\right]\right)^{2}}{{\mathbb{E}}\left[|\mathcal{S}_{l,\xi}(t)|^{2}\right]}.$
If we are able to guarantee that
1. (i)
${\mathbb{E}}\left[|\mathcal{S}_{l,\xi}(t)|\right]\rightarrow\infty$;
2. (ii)
$\left({\mathbb{E}}\left[|\mathcal{S}_{l,\xi}(t)|\right]\right)^{2}=(1-o(1)){\mathbb{E}}\left[|\mathcal{S}_{l,\xi}(t)|^{2}\right]$;
then by choosing
$\theta=\theta(t)={\mathbb{E}}\left[|\mathcal{S}_{l,\xi}(t)|\right]^{-1/2}$,
we see that
$\mathbb{P}\left(|\mathcal{S}_{l,\xi}(t)|>{\mathbb{E}}\left[|\mathcal{S}_{l,\xi}(t)|\right]^{1/2}\right)\geq\left(1-{\mathbb{E}}\left[|\mathcal{S}_{l,\xi}(t)|\right]^{-1/2}\right)^{2}(1-o(1))=1-o(1),$
thus guaranteeing with probability $1-o(1)$ the existence of many isolated
paths of length $l$, finishing the proof of the theorem.
By Lemma 4, we know that item (i) is true. Therefore from now on we will focus
on proving item (ii).
In order for the isolated path $v_{\vec{t}}$ to appear the following must
happen:
* •
a vertex $v_{t_{1}}$ must be be created at time $t_{1}$, which happens with
probability $f(t_{1})$;
* •
between times $t_{1}+1$ and $t_{2}-1$ there can be no new connection to
$v_{t_{1}}$, which, by (5.8), happens with probability
$\prod_{r_{1}=t_{1}+1}^{t_{2}-1}\left(1-\left(1-\frac{f(r_{1})}{2}-(1-f(r_{1}))\frac{1}{4(r_{1}-1)}\right)\frac{1}{r_{1}-1}\right);$
* •
In general, at time $t_{k}$ a vertex $v_{t_{k}}$ is created and makes its
first connection to $v_{t_{k-1}}$, no new connection is then made to
$\\{v_{t_{j}}\\}_{1\leq j\leq k}$ between times $t_{k}+1$ and $t_{k+1}-1$ for
every $k=2,\dots,l-1$, all this happens with probability equal to
$\displaystyle
f(t_{k})\frac{1}{2(t_{k}-1)}\prod_{r_{k}=t_{k}+1}^{t_{k+1}-1}\left(1-\left(1-\frac{f(r_{k})}{2}-(1-f(r_{k}))\frac{D_{r_{k}}(v_{\vec{t}})}{4(r_{k}-1)}\right)\frac{D_{r_{k}}(v_{\vec{t}})}{r_{k}-1}\right)$
$\displaystyle=f(t_{k})\frac{1}{2(t_{k}-1)}\prod_{r_{k}=t_{k}+1}^{t_{k+1}-1}\left(1-\left(1-\frac{f(r_{k})}{2}-(1-f(r_{k}))\frac{2k-1}{4(r_{k}-1)}\right)\frac{2k-1}{r_{k}-1}\right);$
* •
finally, a vertex $v_{t_{l}}$ is born at time $t_{l}$, connects to
$v_{t_{l-1}}$ and no new connection is made to $\\{v_{t_{j}}\\}_{1\leq j\leq
l}$ between times $t_{l}+1$ and $t$.
This implies
(5.10)
$\begin{split}\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle{\mathbb{P}}\left(v_{\vec{t}}\in\mathcal{S}_{l,\xi}(t)\right)$\mbox{}\hfil\\\
&=f(t_{1})\prod_{r_{1}=t_{1}+1}^{t_{2}-1}\left(1-\left(1-\frac{f(r_{1})}{2}-(1-f(r_{1}))\frac{1}{4(r_{1}-1)}\right)\frac{1}{r_{1}-1}\right)\\\
&\quad\times\dots\times
f(t_{k})\frac{1}{2(t_{k}-1)}\prod_{r_{k}=t_{k}+1}^{t_{k+1}-1}\left(1-\left(1-\frac{f(r_{k})}{2}-(1-f(r_{k}))\frac{2k-1}{4(r_{k}-1)}\right)\frac{2k-1}{r_{k}-1}\right)\\\
&\quad\times\dots\times
f(t_{l})\frac{1}{2(t_{l}-1)}\prod_{r_{l}=t_{l}+1}^{t}\left(1-\left(1-\frac{f(r_{l})}{2}-(1-f(r_{l}))\frac{2l-1}{4(r_{l}-1)}\right)\frac{2l-1}{r_{l}-1}\right).\end{split}$
Given two time vectors $\vec{r}$ and $\vec{t}$, we note that
${\mathbb{P}}\left(v_{\vec{t}},v_{\vec{r}}\in\mathcal{S}_{l,\xi}(t)\right)$ is
only nonzero if $\vec{r}$ and $\vec{t}$ have either disjoint or identical sets
of entries. Our focus now is on proving the following lemma:
###### Lemma 5.
Let $l$ be such that (5.2) is satisfied. For two isolated paths with disjoint
time vectors $\vec{t}$ and $\vec{r}$, we have
(5.11)
$\frac{{\mathbb{P}}\left(v_{\vec{t}}\in\mathcal{S}_{l,\xi}(t)\right){\mathbb{P}}\left(v_{\vec{r}}\in\mathcal{S}_{l,\xi}(t)\right)}{{\mathbb{P}}\left(v_{\vec{t}},v_{\vec{r}}\in\mathcal{S}_{l,\xi}(t)\right)}=1+o(1).$
###### Proof.
To prove the above result we will make a comparison between the two
probabilities terms
${\mathbb{P}}\left(v_{\vec{t}},v_{\vec{r}}\in\mathcal{S}_{l,\xi}(t)\right)$
and
${\mathbb{P}}\left(v_{\vec{t}}\in\mathcal{S}_{l,\xi}(t)\right){\mathbb{P}}\left(v_{\vec{r}}\in\mathcal{S}_{l,\xi}(t)\right)$.
We can write both these terms as products in the manner of (5.10). We can then
compare the terms from these products associated to each time $s\in[\xi t,t]$.
There are two cases we must study.
Case 1: $s\in\vec{t}$ but $s\notin\vec{r}$ ($s\notin\vec{t}$ but
$s\in\vec{r}$.).
The product term related to time $s$ in
${\mathbb{P}}\left(v_{\vec{t}},v_{\vec{r}}\in\mathcal{S}_{l,\xi}(t)\right)$ is
(5.12) $\frac{f(s)}{2(s-1)},$
since a new vertex is created and then makes its first connection specifically
to the latest vertex of $\vec{t}$. On the other hand, the term related to time
$s$ in
${\mathbb{P}}\left(v_{\vec{t}}\in\mathcal{S}_{l,\xi}(t)\right){\mathbb{P}}\left(v_{\vec{r}}\in\mathcal{S}_{l,\xi}(t)\right)$
is
$\frac{f(s)}{2(s-1)}\left(1-\left(1-\frac{f(s)}{2}-(1-f(s))\frac{D_{s-1}(v_{\vec{r}})}{4(s-1)}\right)\frac{D_{s-1}(v_{\vec{r}})}{s-1}\right),$
since the term related to $s$ in the product form of
${\mathbb{P}}\left(v_{\vec{t}}\in\mathcal{S}_{l,\xi}(t)\right)$ continues to
be equal to (5.12), but the related term in
${\mathbb{P}}\left(v_{\vec{r}}\in\mathcal{S}_{l,\xi}(t)\right)$ is
(5.13)
$\left(1-\left(1-\frac{f(s)}{2}-(1-f(s))\frac{D_{s-1}(v_{\vec{r}})}{4(s-1)}\right)\frac{D_{s-1}(v_{\vec{r}})}{s-1}\right).$
The above expression is the term that will appear regarding the time $s$ in
the fraction in the left hand side of (5.11). This case occurs $2l$ times
since the isolated paths are disjoint. Thus, recalling that $s\in[\xi t,t]$,
$l\leq 3^{-1}\log(t)/\log(\log(t))$, and that the degree of each isolated path
is at most $2l-1$, we obtain that there exist constants $c_{1},c_{2}>0$ such
that we can bound the product of all the terms of the form (5.13) from above
by
$\left(1-\frac{c_{1}}{t}\right)^{2l},$
and from below by
$\left(1-\frac{c_{2}l}{t}\right)^{2l}.$
Observe that both products go to $1$ as $t$ goes to infinity.
Case 2: $s\notin\vec{t}$ and $s\notin\vec{r}$.
In ${\mathbb{P}}\left(v_{\vec{r}}\in\mathcal{S}_{l,\xi}(t)\right)$ as well as
in ${\mathbb{P}}\left(v_{\vec{t}}\in\mathcal{S}_{l,\xi}(t)\right)$ we see
terms of the form (5.13), since we must avoid the isolated paths in both
events. But in the term related to
${\mathbb{P}}\left(v_{\vec{t}},v_{\vec{r}}\in\mathcal{S}_{l,\xi}(t)\right)$ we
actually observe
$\left(1-\left(1-\frac{f(s)}{2}-(1-f(s))\frac{(D_{s-1}(v_{\vec{t}})+D_{s-1}(v_{\vec{r}}))}{4(s-1)}\right)\frac{(D_{s-1}(v_{\vec{t}})+D_{s-1}(v_{\vec{r}}))}{s-1}\right),$
since we must guarantee that neither isolated path receives a connection. We
note however that
$\displaystyle\left(1-\left(1-\frac{f(s)}{2}-(1-f(s))\frac{D_{s-1}(v_{\vec{r}})}{4(s-1)}\right)\frac{D_{s-1}(v_{\vec{r}})}{s-1}\right)$
$\displaystyle\phantom{**********************}\times\left(1-\left(1-\frac{f(s)}{2}-(1-f(s))\frac{D_{s-1}(v_{\vec{t}})}{4(s-1)}\right)\frac{D_{s-1}(v_{\vec{t}})}{s-1}\right)\quad$
$\displaystyle=\left(1-\left(1-\frac{f(s)}{2}-(1-f(s))\frac{(D_{s-1}(v_{\vec{t}})+D_{s-1}(v_{\vec{r}}))}{4(s-1)}\right)\frac{(D_{s-1}(v_{\vec{t}})+D_{s-1}(v_{\vec{r}}))}{s-1}\right)$
$\displaystyle\phantom{**********************}\times\left(1+O\left(\frac{l^{2}}{t^{2}}\right)\right),$
since $D_{s-1}(v_{\vec{t}}),D_{s-1}(v_{\vec{r}})\leq 2l-1$ and $s\geq\xi t$.
In the fraction in the left hand side of (5.11), we will then have $\Theta(t)$
terms of the form
$\left(1+O\left(\frac{l^{2}}{t^{2}}\right)\right),$
But again, as in Case $1$, their product goes to 1 as $t\to\infty$ since
$l^{2}=o(t)$. This finishes the proof of the Lemma.
We can finally finish the proof of the theorem.
###### Proof of Theorem 2.
Let again $l$ be such that (5.2) is satisfied, and consider $\xi\in(0,1)$.
Since it is impossible for two non disjoint and non equal isolated paths to
exist at the same time, we have that
(5.14)
$\begin{split}{\mathbb{E}}\left[|\mathcal{S}_{l,\xi}(t)|^{2}\right]&={\mathbb{E}}\left[\left(\sum_{\vec{t}}\mathbb{1}\left\\{v_{\vec{t}}\in\mathcal{S}_{l,\xi}(t)\right\\}\right)\left(\sum_{\vec{r}}\mathbb{1}\left\\{v_{\vec{r}}\in\mathcal{S}_{l,\xi}(t)\right\\}\right)\right]\\\
&=\sum_{\begin{subarray}{c}\vec{t},\vec{r}\\\
\text{disjoint}\end{subarray}}{\mathbb{P}}\left(v_{\vec{t}},v_{\vec{r}}\in\mathcal{S}_{l,\xi}(t)\right)+{\mathbb{E}}\left[|\mathcal{S}_{l,\xi}(t)|\right],\end{split}$
and that
$\left({\mathbb{E}}\left[|\mathcal{S}_{l,\xi}(t)|\right]\right)^{2}=\sum_{\begin{subarray}{c}\vec{t},\vec{r}\\\
\text{disjoint}\end{subarray}}{\mathbb{P}}\left(v_{\vec{t}}\in\mathcal{S}_{l,\xi}(t)\right){\mathbb{P}}\left(v_{\vec{r}}\in\mathcal{S}_{l,\xi}(t)\right)+\sum_{\begin{subarray}{c}\vec{t},\vec{r}\\\
\vec{r}\cap\vec{t}\neq\emptyset\end{subarray}}{\mathbb{P}}\left(v_{\vec{t}}\in\mathcal{S}_{l,\xi}(t)\right){\mathbb{P}}\left(v_{\vec{r}}\in\mathcal{S}_{l,\xi}(t)\right).$
Therefore, by lemmas 4 and 5, we obtain
(5.15)
$\begin{split}\frac{{\mathbb{E}}\left[|\mathcal{S}_{l,\xi}(t)|^{2}\right]}{\left({\mathbb{E}}\left[|\mathcal{S}_{l,\xi}(t)|\right]\right)^{2}}&=\frac{\sum_{\begin{subarray}{c}\vec{t},\vec{r}\\\
\text{disjoint}\end{subarray}}{\mathbb{P}}\left(v_{\vec{t}},v_{\vec{r}}\in\mathcal{S}_{l}(t)\right)}{\left({\mathbb{E}}\left[|\mathcal{S}_{l,\xi}(t)|\right]\right)^{2}}+\frac{{\mathbb{E}}\left[|\mathcal{S}_{l,\xi}(t)|\right]}{\left({\mathbb{E}}\left[|\mathcal{S}_{l,\xi}(t)|\right]\right)^{2}}\\\
&\leq\frac{\sum_{\begin{subarray}{c}\vec{t},\vec{r}\\\
\text{disjoint}\end{subarray}}{\mathbb{P}}\left(v_{\vec{t}},v_{\vec{r}}\in\mathcal{S}_{l,\xi}(t)\right)}{\sum_{\begin{subarray}{c}\vec{t},\vec{r}\\\
\text{disjoint}\end{subarray}}{\mathbb{P}}\left(v_{\vec{t}}\in\mathcal{S}_{l,\xi}(t)\right){\mathbb{P}}\left(v_{\vec{r}}\in\mathcal{S}_{l,\xi}(t)\right)}+\frac{1}{{\mathbb{E}}\left[|\mathcal{S}_{l,\xi}(t)|\right]}\\\
&\leq\frac{\sum_{\begin{subarray}{c}\vec{t},\vec{r}\\\
\text{disjoint}\end{subarray}}(1+o(1)){\mathbb{P}}\left(v_{\vec{t}}\in\mathcal{S}_{l,\xi}(t)\right){\mathbb{P}}\left(v_{\vec{r}}\in\mathcal{S}_{l,\xi}(t)\right)}{\sum_{\begin{subarray}{c}\vec{t},\vec{r}\\\
\text{disjoint}\end{subarray}}{\mathbb{P}}\left(v_{\vec{t}}\in\mathcal{S}_{l,\xi}(t)\right){\mathbb{P}}\left(v_{\vec{r}}\in\mathcal{S}_{l,\xi}(t)\right)}+\frac{1}{{\mathbb{E}}\left[|\mathcal{S}_{l,\xi}(t)|\right]}\\\
&\leq 1+o(1),\end{split}$
which proves the desired result.
## 6\. General upper bound for the diameter: proof of Theorem 3
In this section we provide a proof for Theorem 3. The main idea is to use
Lemmas 1 and 3 to show that, with high probability, all vertices born up to
time $t^{\frac{1}{12}}$ are in a connected component with diameter $2$. We
then use a first moment estimate (Lemma 7 below) to show that the lengths of
the paths formed by newer vertices have the desired upper bound.
Given $k\in\mathbb{N}$ and $k$ times $s_{1},\dots,s_{k}\in\mathbb{N}$ such
that $s_{1}<\dots<s_{k}$, we say that the vector of times
$\vec{s}=(s_{1},\dots,s_{k})$ is a _vertex path_ if in the process
$\\{G_{t}(f)\\}_{t\geq 1}$, at each time $s_{j}$, for $j=2,\dots,k$, a vertex
is born and makes its first (vertex-step) connection to the vertex born at
time $s_{j-1}$. We denote by
$\\{s_{1}\leftarrow s_{2}\leftarrow\dots\leftarrow s_{k}\\}$
the event where $\vec{s}$ is a vertex path. We will show:
###### Lemma 6.
Using the notation above, we have, for each vector
$\vec{s}=(s_{1},\dots,s_{k})$ and step function $f$,
(6.1) $\displaystyle{\mathbb{P}}\left(s_{1}\leftarrow
s_{2}\leftarrow\dots\leftarrow s_{k}\right)\leq
f(s_{1})\frac{s_{k}-1}{s_{1}+1}\prod_{m=2}^{k}\frac{f(s_{m})}{2(s_{m}-1)}.$
###### Proof.
Consider the events $\\{s_{1}\leftarrow s_{2}\leftarrow\dots\leftarrow
s_{k-1}\\}$ and $\\{s_{k-1}\leftarrow s_{k}\\}$, defined analogously as the
event in the above equation, but for the vectors $(s_{1},\dots,s_{k-1})$ and
$(s_{k-1},s_{k})$ respectively. We have
(6.2) $\begin{split}{\mathbb{P}}\left(s_{1}\leftarrow
s_{2}\leftarrow\dots\leftarrow
s_{k}\right)&={\mathbb{E}}\left[\mathbb{1}\\{s_{1}\leftarrow
s_{2}\leftarrow\dots\leftarrow s_{k-1}\\}{\mathbb{P}}\left(s_{k-1}\leftarrow
s_{k}\middle|\mathcal{F}_{s_{k}-1}\right)\right]\\\
&={\mathbb{E}}\left[\mathbb{1}\\{s_{1}\leftarrow
s_{2}\leftarrow\dots\leftarrow
s_{k-1}\\}f(s_{k})\cdot\frac{D_{s_{k}-1}(v_{s_{k-1}})}{2(s_{k}-1)}\right].\end{split}$
But, crucially, conditioned on the event where a vertex is born at time
$s_{k-1}$, the degree of said vertex at time $s_{k}-1$ depends only on the
connections made after time $s_{k-1}$, and is therefore independent of the
indicator function above. We then obtain, by (4.3),
(6.3)
$\begin{split}\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle{\mathbb{P}}\left(s_{1}\leftarrow
s_{2}\leftarrow\dots\leftarrow s_{k}\right)$\mbox{}\hfil\quad\\\
&={\mathbb{E}}\left[\mathbb{1}\\{s_{1}\leftarrow
s_{2}\leftarrow\dots\leftarrow
s_{k-1}\\}f(s_{k})\cdot\frac{D_{s_{k}-1}(v_{s_{k-1}})}{2(s_{k}-1)}\middle|Z_{s_{k-1}}=1\right]f(s_{k-1})\\\
&=f(s_{k-1})f(s_{k}){\mathbb{P}}\left(s_{1}\leftarrow
s_{2}\leftarrow\dots\leftarrow
s_{k-1}\middle|Z_{s_{k-1}}=1\right){\mathbb{E}}\left[\frac{D_{s_{k}-1}(v_{s_{k-1}})}{2(s_{k}-1)}\middle|Z_{s_{k-1}}=1\right]\\\
&={\mathbb{P}}\left(s_{1}\leftarrow s_{2}\leftarrow\dots\leftarrow
s_{k-1}\right)\frac{f(s_{k})}{2(s_{k}-1)}\prod_{m=s_{k-1}}^{s_{k}-2}\left(1+\frac{1}{m}-\frac{f(m+1)}{2m}\right)\\\
&\leq{\mathbb{P}}\left(s_{1}\leftarrow s_{2}\leftarrow\dots\leftarrow
s_{k-1}\right)\frac{f(s_{k})}{2(s_{k}-1)}\exp\left\\{\sum_{m=s_{k-1}}^{s_{k}-2}\left(\frac{1}{m}-\frac{f(m+1)}{2m}\right)\right\\},\end{split}$
by elementary properties of the exponential function. Iterating the above
argument and recalling that the vertex $s_{1}$ is born with probability
$f(s_{1})$, we obtain
(6.4) $\begin{split}{\mathbb{P}}\left(s_{1}\leftarrow
s_{2}\leftarrow\dots\leftarrow s_{k}\right)&\leq
f(s_{1})\exp\left\\{\sum_{m=s_{1}}^{s_{k}-2}\left(\frac{1}{m}-\frac{f(m+1)}{2m}\right)\right\\}\prod_{m=2}^{k}\frac{f(s_{m})}{2(s_{m}-1)}\\\
&\leq
f(s_{1})\frac{s_{k}-1}{s_{1}+1}\prod_{m=2}^{k}\frac{f(s_{m})}{2(s_{m}-1)},\end{split}$
finishing the proof of the lemma.
Given $k,t_{0},t\in\mathbb{N}$, denote by $\mathcal{V}_{k,t_{0}}(t)$ the set
of all vertex-paths of length $k$ whose vertices were born between times
$t_{0}$ and $t$. Our goal now is to bound from above the expectation of
$|\mathcal{V}_{k,t_{0}}(t)|$, this will be accomplished in the next lemma.
###### Lemma 7.
Using the notation above defined, we have, for $k,t_{0},t\in\mathbb{N}$,
(6.5) ${\mathbb{E}}\left[|\mathcal{V}_{k,t_{0}}(t)|\right]\leq
C_{1}\exp\left\\{2\log
t-(k-2)\left(\log(k-2)+C_{2}+\log\left(\sum_{j=t_{0}}^{t}\frac{f(j)}{j-1}\right)\right)\right\\}.$
###### Proof.
We will use Lemma 6 and the an application of the union bound. First, fix
$s_{1},s_{k}\in\mathbb{N}$ such that $t_{0}\leq s_{1}<s_{k}\leq t$. We have,
by Stirling’s approximation formula and the positivity of the terms involved,
(6.6) $\begin{split}\sum_{\begin{subarray}{c}s_{2},s_{3},\dots,s_{k-1}\\\
s_{1}<s_{2}<\dots<s_{k-1}<s_{k}\end{subarray}}\prod_{m=2}^{k-1}\frac{f(s_{m})}{s_{m}-1}&\leq\frac{1}{(k-2)!}\left(\sum_{j=t_{0}}^{t}\frac{f(j)}{j-1}\right)^{k-2}\\\
&\leq
C\exp\left\\{-(k-2)\left(\log(k-2)-1-\log\left(\sum_{j=t_{0}}^{t}\frac{f(j)}{j-1}\right)\right)\right\\}.\end{split}$
We can then show, by the above equation, Lemma 6, and the union bound,
(6.7)
$\begin{split}{\mathbb{E}}\left[|\mathcal{V}_{k,t_{0}}(t)|\right]&\leq\sum_{\begin{subarray}{c}s_{1},\dots,s_{k}\\\
t_{0}\leq s_{1}<\dots<s_{k}\leq
t\end{subarray}}{\mathbb{P}}\left(s_{1}\leftarrow
s_{2}\leftarrow\dots\leftarrow s_{k}\right)\\\
&\leq\sum_{\begin{subarray}{c}s_{1},\dots,s_{k}\\\ t_{0}\leq
s_{1}<\dots<s_{k}\leq
t\end{subarray}}f(s_{1})\frac{s_{k}-1}{s_{1}+1}\prod_{m=2}^{k}\frac{f(s_{m})}{2(s_{m}-1)}\\\
&=\frac{1}{2^{k-1}}\sum_{\begin{subarray}{c}s_{1},s_{k}\\\ t_{0}\leq
s_{1}<s_{k}\leq
t\end{subarray}}\frac{f(s_{1})f(s_{k})}{s_{1}+1}\sum_{\begin{subarray}{c}s_{2},\dots,s_{k-1}\\\
s_{1}<s_{2}<\dots<s_{k-1}<s_{k}\end{subarray}}\prod_{m=2}^{k-1}\frac{f(s_{m})}{s_{m}-1}\\\
&\leq
C\exp\left\\{-(k-2)\left(\log(k-2)+C_{2}-\log\left(\sum_{j=t_{0}}^{t}\frac{f(j)}{j-1}\right)\right)\right\\}\sum_{\begin{subarray}{c}s_{1},s_{k}\\\
t_{0}\leq s_{1}<s_{k}\leq t\end{subarray}}\frac{f(s_{1})f(s_{k})}{s_{1}+1}\\\
&\leq C\exp\left\\{2\log
t-(k-2)\left(\log(k-2)+C_{2}-\log\left(\sum_{j=t_{0}}^{t}\frac{f(j)}{j-1}\right)\right)\right\\},\end{split}$
thus concluding the proof of the result.
We can finally finish the proof of Theorem 3.
###### Proof of Theorem 3.
We prove part (a) first.
Proof of part (a): This part follows immediately from the fact that the B-A
random tree has diameter bounded by $\log t$ a.a.s (see Theorem $7.1$ of [27])
combined with Proposition 2.5 which states that the diameter is an increasing
graph observable.
Proof of part (b): Recall that, in this part of the theorem, $f$ is under
condition (S), which holds if $\sum_{s=2}^{\infty}f(s)/s$ is finite.
For $t_{0},t\in\mathbb{N}$ and $\delta\in(0,1)$, let $A_{\delta}(t_{0},t)$ be
the event where every vertex born before time $t_{0}$ has degree at least
$t^{\delta}$ in $G_{t}(f)$. For $\varepsilon\in(0,1)$, let
$B_{\varepsilon}(t)$ be the event where there exists a vertex $v$ in
$V(G_{t}(f))$ such that $D_{t}(v)\geq t^{1-\varepsilon}$. Then, by Lemmas 1
and 3, we have
(6.8) ${\mathbb{P}}\left(A_{\frac{1}{15}}(t^{\frac{1}{12}},t)\right)\geq
1-Ct^{-144^{-1}},\quad\quad{\mathbb{P}}\left(B_{\frac{1}{30}}(t)\right)\geq
1-t^{-2}.$
We have, on the event where two vertices $u_{1},u_{2}\in V(G_{t}(f))$ are such
that $D_{t}(u_{1})\geq t^{15^{-1}}$ and $D_{t}(u_{2})\geq t^{1-30^{-1}}$,
(6.9) $\begin{split}{\mathbb{P}}\left(u_{1}\nleftrightarrow u_{2}\text{ in
}G_{2t}(f)\right)&\leq\prod_{s=t+1}^{2t}\left(1-\frac{f(s)}{2(s-1)}\right)\left(1-\frac{t^{15^{-1}}t^{1-30^{-1}}}{4(s-1)^{2}}\right)\\\
&\leq\exp\left\\{-\frac{t^{15^{-1}}t^{1-30^{-1}}}{16t^{2}}\cdot t\right\\}\\\
&=\exp\left\\{-\frac{t^{-30^{-1}}}{16}\right\\}.\end{split}$
Recall the notation $v_{i}$ symbolizing the vertex born at time
$i\in\mathbb{N}$. Together with (6.8) and the union bound, the above equation
implies
$\begin{split}\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle{\mathbb{P}}\left(\exists
i,j\in\mathbb{N},\text{ such that }1\leq i<j\leq
t^{12^{-1}},Z_{i}=Z_{j}=1,\text{ and }\mathrm{dist}(v_{i},v_{j})>2\text{ in
}G_{2t}(f)\right)$\mbox{}\hfil\phantom{***}\\\ &\leq
Ct^{-144^{-1}}+t^{-2}+\sum_{1\leq i<j\leq
t^{\frac{1}{12}}}{\mathbb{P}}\left(A_{\frac{1}{15}}(t^{\frac{1}{12}},t),B_{\frac{1}{30}}(t),\mathrm{dist}(v_{i},v_{j})>2\text{
in }G_{2t}(f)\right)\\\ &\leq
Ct^{-144^{-1}}+t^{-2}+t^{-6^{-1}}\exp\left\\{-\frac{t^{-30^{-1}}}{16}\right\\}\\\
&\leq Ct^{-144^{-1}}.\end{split}$
This implies the existence of a constant $C_{1}>0$ such that the probability
of there existing two vertices born before time $C_{1}t^{12^{-1}}$ such that
the distance between said vertices is larger than $2$ in $G_{t}(f)$ is
polynomially small in $t$. We now turn our attention to vertices born after
$t^{\frac{1}{13}}$. We will use Lemma 7 in order to bound the probability of
there existing long vertex-paths formed by vertices born after
$t^{\frac{1}{13}}$, the notion of a “long” path being $f$-dependent. Let
$V_{t}(t^{\frac{1}{13}})$ denote the set of all vertices of $V(G_{t}(f))$ born
before time $t^{\frac{1}{13}}$, let ${\bf
d}_{\mathrm{max}}(t^{\frac{1}{13}},t)$ be the length of a maximal vertex-path
of vertices born between times $t^{\frac{1}{13}}$ and $t$. Let $u_{1},u_{2}\in
V(G_{t}(f))$. Since $G_{t}(f)$ is connected,
(6.10)
$\begin{split}\mathrm{dist}(u_{1},u_{2})&\leq\mathrm{dist}(u_{1},V_{t}(t^{\frac{1}{13}}))+\mathrm{diam}(V_{t}(t^{\frac{1}{13}}))+\mathrm{dist}(u_{2},V_{t}(t^{\frac{1}{13}}))\\\
&\leq 2{\bf
d}_{\mathrm{max}}(t^{\frac{1}{13}},t)+\mathrm{diam}(V_{t}(t^{\frac{1}{13}})).\end{split}$
But we know that $\mathrm{diam}(V_{t}(t^{\frac{1}{13}}))\leq 2$ with high
probability. Bounding ${\bf d}_{\mathrm{max}}(t^{\frac{1}{13}},t)$ then gives
us an a.a.s. upper bound for the diameter of $G_{t}(f)$.
Now, given $t\in\mathbb{N}$, if
(6.11) $k\geq 3\left(\frac{\log
t}{-\log\left(\sum_{s=t^{\frac{1}{13}}}^{t}\frac{f(s)}{s-1}\right)}\wedge\frac{\log
t}{\log\log t}\right),$
then, by Lemma 7, and since
$\log\left(\sum_{j=t^{\frac{1}{13}}}^{t}\frac{f(j)}{j-1}\right)$ is eventually
negative for large $t$ (recall again that $f$ satisfies (S)), we have,
(6.12) $\begin{split}{\mathbb{P}}\left({\bf
d}_{\mathrm{max}}>k\right)&\leq{\mathbb{E}}\left[|\mathcal{V}_{k,t^{1/13}}(t)|\right]\\\
&\leq C_{1}\exp\left\\{2\log
t-(k-2)\left(\log(k-2)+C_{2}-\log\left(\sum_{j=t^{\frac{1}{13}}}^{t}\frac{f(j)}{j-1}\right)\right)\right\\}\\\
&\leq Ct^{-\frac{1}{2}}.\end{split}$
The above upper bound and (6.10) proves part (b).
Proof of part (c): Recall that in this part, we are under condition (Lκ),
which holds if, for $\kappa\in(0,1)$ and every $t\in\mathbb{N}$ sufficiently
large, one has
(6.13) $\sum_{s=t^{1/13}}^{t}\frac{f(s)}{s}<(\log t)^{\kappa}.$
Then, let $k$ be so that
$k\geq\frac{3}{1-\kappa}\frac{\log t}{\log\log t}.$
We then obtain, again by Lemma 7, for sufficiently large $t$,
(6.14)
$\begin{split}\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle{\mathbb{P}}\left({\bf
d}_{\mathrm{max}}>k\right)$\mbox{}\hfil\quad\quad\\\
&\leq{\mathbb{E}}\left[|\mathcal{V}_{k,t^{1/13}}(t)|\right]\\\ &\leq
C_{1}\exp\left\\{2\log
t-(k-2)\left(\log(k-2)+C_{2}-\log\left(\sum_{j=t^{\frac{1}{13}}}^{t}\frac{f(j)}{j-1}\right)\right)\right\\}\\\
&\leq C_{1}\exp\left\\{2\log t-\frac{3}{1-\kappa}\frac{\log t}{\log\log
t}\left((1-\kappa)\log\log t-C-\log\log\log t\right)\right\\}\\\ &\leq
Ct^{-\frac{1}{2}}.\end{split}$
Finally, the above upper bound together with (6.10) finishes the proof of the
Theorem.
## 7\. The family of regularly varying functions
In this section we explore our results when more information on $f$ is
provided. In particular, we assume that $f$ satisfies condition (RVγ) for
$\gamma\in[0,1)$. Recall that this condition holds whenever
$\exists\gamma,\text{ such that }\forall
a>0,\;\lim_{t\to\infty}\frac{f(at)}{f(t)}=\frac{1}{a^{\gamma}}.$
Also recall that when $f$ satisfies the above condition for $\gamma>0$ we say
it is a regular varying function with index of regular variation $-\gamma$.
The case $\gamma=0$ is said to be slowly varying.
To prove the results under the assumption of regular variation our arguments
rely on the theorems from Karamata’s theory. In particular, the Representation
Theorem (Theorem 1.4.1 of [8]) and Karamata’s theorem (Proposition 1.5.8 of
[8]). The former states that if $f$ is a regularly varying function with index
$-\gamma$, then there exists a slowly varying function $\ell$ such that $f$ is
of the form
(7.1) $f(t)=\frac{\ell(t)}{t^{\gamma}}.$
Whereas, the latter states that if $\ell$ is a slowly varying function, then,
for any $a<1$
(7.2)
$\int_{1}^{t}\frac{\ell(x)}{x^{a}}\mathrm{d}x\sim\frac{\ell(t)t^{1-a}}{1-a},$
and for $a>1$, we have
(7.3)
$\int_{t}^{\infty}\frac{\ell(x)}{x^{a}}\mathrm{d}x\sim\frac{\ell(t)}{(a-1)t^{a-1}}.$
We begin by proving Corollary 1.4, which gives lower and upper bounds for the
clique number $\omega(G_{t}(f))$.
###### Proof of Corollary 1.4.
We prove part (a) first.
Proof of part (a): The lower bound follows from (7.1), when $f$ is regularly
varying, (7.2) and Theorem 1. Just observe that, if $f$ is regularly varying,
then $F(t)=\Theta({t^{1-\gamma}})$, whereas if it is slowly, we have
$F(t)=\Theta(f(t)t)$. In the latter case, we appeal to another result from
Karamata’s Theory (Corollary A.6 of [2]) which assures that for all
$\varepsilon>0$,
(7.4) $\frac{f(t)}{t^{\varepsilon}}\stackrel{{\scriptstyle
t\to\infty}}{{\longrightarrow}}0.$
Then, by Theorem 1, $G_{t}(f)$ has, with probability $1-o(1)$, a clique of
order $C_{\varepsilon}F(t^{\frac{1}{2}(1-\frac{\varepsilon}{2})})$, which
gives the desired lower bound for $\omega(G_{t}(f))$. The upper bound comes
from the deterministic bound that says that a graph with $t$ (simple) edges
has at most $t^{\frac{3}{2}}$ triangles (see e.g. Theorem $4$ of [24]), and
the fact that a complete subgraph with $k$ vertices has $6^{-1}k(k-1)(k-2)$
triangles.
Proof of part (b): First observe that for any small $\varepsilon>0$ by (7.2),
$F(t^{\frac{1}{2}(1-\varepsilon)})\gg\log\log t$ for large enough $t$. Thus,
by (3.8), we have that
(7.5) ${\mathbb{P}}\left(\omega(G_{t}(f))\geq
C_{\varepsilon}F(t^{\frac{1-\varepsilon}{2}})\right)\geq 1-\frac{2}{\log t}.$
Now, define the following sequence of deterministic times
(7.6) $t_{k}:=(1+\varepsilon)^{k^{2}}.$
By the first Borel-Cantelli Lemma and (7.5) we have that,
(7.7)
$\liminf_{k\to\infty}\frac{\log\omega(G_{t_{k}}(f))}{\log(C_{\varepsilon}F(t_{k}^{\frac{1-\varepsilon}{2}}))}\geq
1,\;\text{a.s.}$
Using that $f$ is slowly varying, Karamata’s Theorem (7.2), and (7.4), we also
have that
(7.8)
$\lim_{t\to\infty}\frac{\log(C_{\varepsilon}F(t^{\frac{1-\varepsilon}{2}}))}{\log
F(t^{\frac{1}{2}})}=1-\frac{\varepsilon}{2}.$
For the same reason, it also follows that
(7.9) $\lim_{t\to\infty}\frac{\log
F(t^{\frac{1}{2}})}{\log(7t^{\frac{1}{2}})}=1.$
Then (7.7) yields
(7.10) $\liminf_{k\to\infty}\frac{\log\omega(G_{t_{k}}(f))}{\log
F(t_{k}^{\frac{1}{2}})}\geq 1-\varepsilon,\;\text{a.s.}.$
And using the upper bound given in part (a) and (7.9) we also obtain
(7.11) $\limsup_{k\to\infty}\frac{\log\omega(G_{t_{k}}(f))}{\log
F(t_{k}^{\frac{1}{2}})}\leq 1,\;\text{a.s.}$
Now, note that $\omega(G_{t}(f))\leq\omega(G_{t+1}(f))$, since $G_{t}(f)$ is
contained in $G_{t+1}(f)$. Thus, for any time $t\in(t_{k},t_{k+1})$ it follows
(7.12) $\begin{split}\frac{\log(\omega(G_{t_{k}}(f)))}{\log
F(t_{k+1}^{\frac{1}{2}})}\leq\frac{\log(\omega(G_{t}(f)))}{\log
F(t^{\frac{1}{2}})}\leq\frac{\log(\omega(G_{t_{k+1}}(f)))}{\log
F(t_{k}^{\frac{1}{2}})}.\end{split}$
Using (7.10), (7.11) and that
$\lim_{k\to\infty}\frac{\log F(t_{k+1}^{\frac{1}{2}})}{\log
F(t_{k}^{\frac{1}{2}})}=1$
we finally obtain that
(7.13)
$1-\varepsilon\leq\liminf_{t\to\infty}\frac{\log(\omega(G_{t}(f)))}{\log
F(t^{\frac{1}{2}})}\leq\limsup_{t\to\infty}\frac{\log(\omega(G_{t}(f)))}{\log
F(t^{\frac{1}{2}})}\leq 1,\;\text{a.s.}.$
Since $\varepsilon$ was arbitrarily chosen, we conclude the proof.
The next step is to prove the constant order of the diameter of the graphs
generated by regularly varying functions and how it depends on the index of
regular variation on infinity. This result is stated on Theorem 4 and we
provide a proof for it below.
###### Proof of Theorem 4.
We begin proving the lower bound.
Lower bound: By the Representation Theorem (7.1) we have that there exists a
slowly varying function $\ell$ such that $f(t)=\ell(t)/t^{\gamma}$. Moreover,
(7.4) implies that
(7.14) $\frac{\log\ell(t)}{\log t}\to 0.$
Thus, we have that
(7.15) $-\frac{\log t}{\log f(t)}=\frac{\log
t}{\gamma\log(t)+\log\ell(t)}=\frac{1-o(1)}{\gamma}.$
Applying Theorem 2 gives us a lower bound of order $\gamma^{-1}$.
Upper bound: By the Representation Theorem and (7.4), we have that $f$ also
satisfies condition (S). Just notice that
(7.16)
$\sum_{s=1}^{\infty}\frac{f(s)}{s}=\sum_{s=1}^{\infty}\frac{\ell(s)}{s^{1+\gamma}}<\infty.$
And by Karamata’s Theorem (Equation (7.3) in particular) we have that
(7.17)
$\sum_{s=t^{\frac{1}{13}}}^{t}\frac{\ell(s)}{s^{1+\gamma}}\sim\frac{\ell(t)}{t^{\gamma\frac{\gamma}{13}}}\implies-\frac{\log
t}{\log\left(\sum_{s=t^{\frac{1}{13}}}^{t}\frac{\ell(s)}{s^{1+\gamma}}\right)}\sim\frac{\log
t}{\frac{\gamma}{13}\log t+\log\ell(t)}=\frac{13(1-o(1))}{\gamma}$
And finally, applying Theorem 3 we prove the result.
## 8\. Final comments
We end this paper with a brief discussion on the affine version of our model
and how dropping some regularity conditions on $f$ may produce a sequence of
graphs $\\{G_{t}\\}_{t\in\mathbb{N}}$ that has a subsequence which is
essentially of complete graphs and another one which is close to the BA-model.
### Affine version
At the introduction, we have discussed the affine version of the PA-rule,
which we recall below.
${\mathbb{P}}\left(v_{t+1}\rightarrow
u\middle|G_{t}\right)=\frac{\text{degree}(u)+\delta}{\sum_{w\in
G_{t}}(\text{degree}(w)+\delta)}.$
In [2], the authors showed that the effect of the affine term $\delta$
vanishes in the long run when one is dealing with the empirical degree
distribution. I.e., their results show that $\delta$ has no effect on the
degree sequence of the graphs, what is not observed in the affine version of
the BA-model, for which the exponent of the power-law distribution depends on
$\delta$, see [13]. However, regarding the diameter, we believe $\delta$ may
have an increasing/decreasing effect on the diameter’s order. One also may
find interesting to consider $\delta=\delta(t)$ and investigate which one
takes over: is it the edge-step function or the affine term?
### Dropping regularity conditions
In this part we illustrate that dropping some assumptions on $f$ may produce a
somewhat pathological sequence of graphs. For instance, if we drop the
assumption of $f$ being non-increasing, we may obtain a sequence of graphs
whose diameter sequence oscillates between $1$ and $\log t$. More generally,
the sequence of graphs oscillates between graphs similar to the BA-random
tree, $\\{\mathrm{BA}_{t}\\}_{t\in\mathbb{N}}$, and graphs close to complete
graphs.
Let $(t_{k})_{k\in\mathbb{N}}$ be the following sequence: $t_{0}=1$ and
$t_{k+1}:=\exp\\{t_{k}\\}$, for $k>1$. Now, let $h$ be the edge-step function
defined as follows
(8.1) $h(t)=\begin{cases}1&\quad\text{ if }t\in[t_{2k},t_{2k+1}],\\\
0&\quad\text{ if }t\in(t_{2k+1},t_{2k+2}).\end{cases}$
The idea behind such $h$ is that between times $[t_{2k},t_{2k+1}]$ the process
behaves essentially as the traditional BA-model, whereas at interval
$(t_{2k+1},t_{2k+2})$ the process “messes things up” connecting almost all
vertices by only adding new edges. Moreover, in both regimes the process has
time enough to “forget about what was built in the past”.
Using Lemma 3 and reasoning as in (6.9), one may prove that
$\mathrm{diam}_{G_{t_{2k+1}^{13}}(h)}G_{t_{2k+1}}(h)\leq 2,\text{ a.a.s.}$
However, the process does not add any new vertex in the interval
$(t_{2k+1},t_{2k+2})$. Therefore, $\mathrm{diam}G_{t_{2k+2}}(h)\leq 2$, a.a.s.
On the other hand, if we sample $G_{t_{2k+3}}(h)$ and $\mathrm{BA}_{t_{2k+3}}$
from the doubly-labeled tree $\mathcal{T}_{t_{2k+3}}$ it follows that the
quantity
$\max_{u>t_{2k+2}}\mathrm{dist}(u,\mathcal{T}_{t_{2k+2}})$
remains the same under $h$ and under $f\equiv 1$, since paths using vertices
added after $t_{2k+2}$ belong to the graphs generated by both functions.
Finally, noting that $\mathcal{T}_{t_{2k+2}}$ has diameter at most $\log
t_{2k+2}=t_{2k+1}$ w.h.p, which means it is a very small graph when compared
to $\mathcal{T}_{t_{2k+3}}$, it is not hard to see that the diameter of
$G_{2k+3}(h)$ has the same order the diameter from $\mathrm{BA}_{t_{2k+3}}$.
Observe that the function $h$ may be constructed considering it equal to any
other edge-step function $f$ instead of the constant case $1$. Roughly
speaking, when we sample both processes from the doubly-labeled random tree,
up to a very small subgraph, the graph $G_{t_{2k+1}}(h)$ is similar to
$G_{t_{2k+1}}(f)$, whereas, $G_{t_{2k}}(h)$ is a graph of diameter at most $2$
and whose vertices have degree at least a subpolynomial of $t_{2k}$.
The conclusion is that if we drop some monoticity assumption on $f$, we may
obtain a sequence of graphs having at least two subsequences that are
completely different as graphs.
Acknowledgements We are thankful to Roberto Imbuzeiro Oliveira for the many
inspiring and pleasant conversations. C.A. was partially supported by Fundação
de Amparo à Pesquisa do Estado de São Paulo (FAPESP), grants 2013/24928-2 and
2015/18930-0, and by the Deutsche Forschungsgemeinschaft (DFG). R.R. was
partially supported by Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq) and by the project Stochastic Models of Disordered and
Complex Systems. The Stochastic Models of Disordered and Complex Systems is a
Millennium Nucleus (NC120062) supported by the Millenium Scientific Initiative
of the Ministry of Science and Technology (Chile). R.S. has been partially
supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico
(CNPq) and by FAPEMIG (Programa Pesquisador Mineiro), grant PPM 00600/16.
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|
# Learning Dense Visual Correspondences
in Simulation to Smooth and Fold Real Fabrics
Aditya Ganapathi1, Priya Sundaresan1, Brijen Thananjeyan1, Ashwin
Balakrishna1,
Daniel Seita1, Jennifer Grannen1, Minho Hwang1, Ryan Hoque1,
Joseph E. Gonzalez1, Nawid Jamali2, Katsu Yamane2, Soshi Iba2, Ken Goldberg1
1AUTOLab at the University of California, Berkeley, USA2Honda Research
Institute, USACorrespondence to Aditya Ganapathi<EMAIL_ADDRESS>
(112001)
###### Abstract
Robotic fabric manipulation is challenging due to the infinite dimensional
configuration space, self-occlusion, and complex dynamics of fabrics. There
has been significant prior work on learning policies for specific deformable
manipulation tasks, but comparatively less focus on algorithms which can
efficiently learn many different tasks. In this paper, we learn visual
correspondences for deformable fabrics across different configurations in
simulation and show that this representation can be used to design policies
for a variety of tasks. Given a single demonstration of a new task from an
initial fabric configuration, the learned correspondences can be used to
compute geometrically equivalent actions in a new fabric configuration. This
makes it possible to robustly imitate a broad set of multi-step fabric
smoothing and folding tasks on multiple physical robotic systems. The
resulting policies achieve $80.3\%$ average task success rate across 10 fabric
manipulation tasks on two different robotic systems, the da Vinci surgical
robot and the ABB YuMi. Results also suggest robustness to fabrics of various
colors, sizes, and shapes. See https://tinyurl.com/fabric-descriptors for
supplementary material and videos.
## I Introduction
Robot fabric manipulation has applications in folding laundry [46, 17, 4, 25],
bed making [37], surgery [43, 42, 38, 8], and manufacturing [27, 45]. However,
while robots are able to learn general purpose policies to manipulate a
variety of rigid objects with increasing reliability [22, 6, 28, 14, 20],
learning such policies for manipulating deformable objects remains an open
problem due to difficulties in sensing and control. While there is significant
prior work on geometric [35, 46, 1, 23] and learning based approaches [47, 36,
37] for fabric manipulation, these approaches often involve designing or
learning task-specific manipulation policies, making it difficult to
efficiently reuse information for different tasks.
In this work, we leverage recent advances in dense keypoint learning [6] to
learn visual point-pair correspondences across fabric in different
configurations. Then, given a single offline demonstration of a fabric
manipulation task from a given configuration, we utilize the learned
correspondences to compute geometrically equivalent actions to complete the
task on a similar fabric in a different configuration. For example, a human
might provide a sequence of actions that would fold a T-shirt when it is
placed neck up in a smoothed configuration. However, when a robot is operating
at test time, it will likely encounter a different T-shirt whose color, size
and pose may differ from the T-shirt used for the demonstration. We find that
learning visual correspondences that are invariant across these fabric
attributes provides a powerful representation for defining controllers that
can generalize to the above variations.
Figure 1: We use learned visual correspondences across different fabric
configurations to perform a variety of fabric manipulation tasks on the ABB
YuMi (top) and the da Vinci Research Kit (bottom). Given a single
demonstration of smoothing or folding, the robot uses the learned
correspondences to compute geometrically equivalent actions for fabric of
different color and in different initial configurations. This enables robust
one-shot imitation learning of tasks that involve smoothing then folding.
We extend work by [41], which leverages dense object descriptors [6] to learn
visual correspondences for rope using synthetic depth data in simulation.
These correspondences are then used to learn new rope manipulation tasks such
as rearrangement or knot tying given a single task demonstration. We find that
similar visual correspondence learning methods are also effective for learning
correspondences between different fabric configurations using task-agnostic
RGB data collected entirely in simulation and can be used to perform fabric
manipulation tasks. Precisely, given a user demonstration of the task from a
given initial fabric configuration, we leverage the learned visual
correspondences to perform the same task from different initial configurations
by computing geometrically equivalent actions using the correspondences. This
approach has a number of appealing properties. First, visual correspondences
can be learned purely in simulation without task-specific data and widely
applied to a variety of real fabric manipulation tasks with no further
training. Second, training in simulation enables sufficient data variety
through domain randomization, making it possible to learn correspondences that
generalize to fabrics with different colors, shapes, and configurations.
Third, since perception and control are decoupled, the same perception module
can be used on different robots with no additional training.
We contribute (1) a framework for learning dense visual correspondences of
fabric in simulation using dense object descriptors from [6, 41] and applying
them to manipulation tasks on real fabrics with unseen colors, scales, and
textures, (2) a data generation pipeline for collecting images of fabrics and
clothing in Blender [3] and a testbed to experiment with different
manipulation policies on these fabrics in simulation and (3) physical
experiments on both the da Vinci Research Kit (dVRK) [15] and the ABB YuMi
suggesting that the learned descriptors transfer effectively on two different
robotic systems. We experimentally validate the method on 10 different tasks
involving 5 T-shirts and 5 square fabrics of varying dimensions and colors and
achieve an average task success rate of $80.3\%$.
## II Related Work
Fabric manipulation is an active area of robotics research [2, 33, 18, 11].
Over the past decade, the research has primarily been focused on three
different categories: perception-based manipulation, learning-based algorithms
in the real world, and learning-based algorithms in simulation which are then
transferred to real robots.
Traditional Vision-Based Algorithms for Fabric Manipulation: Much of the prior
work on perception-based deformable object manipulation relies on traditional
image processing and vision techniques to estimate the state of the fabric.
This state estimation is then used to define geometric controllers which bring
the fabric into some desired configuration. However, due to limitations in
these traditional vision algorithms, most prior work makes specific
assumptions on the fabric’s initial configurations or requires more complex
robotic manipulators to bring the fabric into a desired starting
configuration. For example, Miller _et al_. [25] demonstrate a robust folding
pipeline for clothing by fitting a polygonal contour to the fabric and
designing a geometric controller on top of it, but assume that the initial
state of the fabric is flat. Sun _et al_. [39, 40] perform effective fabric
smoothing by estimating the wrinkles in the fabric, but condition on a near-
flat starting fabric. Other work relies on “vertically smoothing” fabrics
using gravity [26, 16, 17, 4, 23] to standardize the initial configuration and
to expose fabric corners before attempting the task, which is difficult for
large fabrics or single-armed robots.
Learning-Based Algorithms in the Real World: More recent approaches have
transitioned to end to end learning of fabric manipulation directly on a real
system, but these approaches have struggled to generalize to a variety of
fabrics and tasks due to the high volume of training data required. For
example, [5] use model-based reinforcement learning to learn fabric
manipulation policies which generalize to many tasks, but require several days
of continuous data collection on a real physical system and perform relatively
low precision tasks. Jia _et al_. [12, 13] show impressive collaborative
human-robot cloth folding under the assumption that fabric has already been
grasped and is in a particular starting configuration, and [35] demonstrate
deformable object manipulation while requiring task-specific kinesthetic
demonstrations. In follow-up work, [19] consider many of the same tasks as in
this paper and demonstrate that policies can be learned to fold fabric using
reinforcement learning with only one hour of experience on a real robot. In
contrast, we learn entirely in simulation and decouple perception from
control, making it easier to generalize to different fabric colors and shapes
and flexibly deploy the learned policies on different robots.
Sim-to-Real Learning-Based Algorithms: Due to the recent success of sim-to-
real transfer [32, 44], many recent papers leverage simulation to collect
large amounts of training data which is used to learn fabric manipulation
policies. Seita _et al_. [37, 36] and [47] followed-up on the smoothing task
from [39] by generalizing to a wider range of initial fabric states using
imitation learning (DAgger [31]), and reinforcement learning (Soft Actor-
Critic [9]) respectively, but still tailor policies specifically for
smoothing. Similarly, [24] learn fabric folding policies by using deep
reinforcement learning augmented with task-specific demonstrations. These
works use simulation to optimize fabric manipulation policies for specific
tasks. In follow-up and concurrent work, [10] and [48] use simulation to train
fabric manipulation policies using model-based reinforcement learning for
multiple tasks. In contrast, we leverage simulation to learn visual
representations of fabric to capture its geometric structure without task-
specific data or a model of the environment and then use this representation
to design intuitive controllers for several tasks from different starting
configurations.
Dense Object Descriptors: We learn visual representations for fabric by using
dense object descriptors [6, 34], which were shown to enable task oriented
manipulation of various rigid and slightly deformable objects [6]. This
approach uses a deep neural network to learn a representation which encourages
corresponding pixels in images of an object in different configurations to
have similar representations in embedding space. Such descriptors can be used
to design geometrically structured manipulation policies for grasping [6],
assembly [49], or for learning from demonstrations [7]. [41] extend this idea
to manipulation of ropes, and demonstrate that deformation-invariant dense
object descriptors can be learned for rope using synthetic depth data in
simulation and then transferred to a real physical system. [41] then use the
learned descriptors to imitate offline demonstrations of various rope
manipulation tasks. In this work, we apply the techniques from [41] to learn
descriptors which capture geometric correspondence across different fabric
configurations from synthetic RGB images and use them for 2D fabric
manipulation.
## III Problem Definition
### III-A Assumptions
We assume a deformable object is place on a planar workspace in an initial
configuration $\xi_{1}$ observed by an overhead camera corresponding RGB image
$I_{1}:=I_{1}(\xi_{1})\in\mathbb{R}^{W\times H\times 3}$. As in prior work
[36, 47], we assume that fabric manipulation tasks can be completed by a
sequence of actions where each includes grasping at a _pick point_ , pulling
to a _place point_ without changing the orientation of the end-effector, and
releasing at the place point. We additionally assume access to a single
demonstration of each task in the form of a sequence of pick and place actions
from some arbitrary initial fabric configuration $\xi_{1}$. These
demonstrations can be collected offline, such as through a GUI where a user
clicks on an image of fabric to indicate pick and place point pixels. However,
the fabric used to create the instruction does not have to be of the same
color, the same size or in the same initial configuration as the fabric the
robot sees at test time. The only requirement is that the fabric be of a
similar geometry. For example, T-shirts can be compared to other instances of
T-shirts, but not to pants or long-sleeved shirts.
### III-B Task Definition
Define the action at step $j$ as
$\mathbf{a}_{j}=((x_{g},y_{g})_{j},(x_{p},y_{p})_{j})$ (III.1)
where $(x_{g},y_{g})_{j}$ and $(x_{p},y_{p})_{j}$ are the pixel coordinates of
a grasp point on the fabric and place point respectively in image $I_{j}$ at
time $j$. The robot grasps the world coordinate associated with the grasp
point and then moves to the world coordinate associated with the place point
without changing the end effector orientation. This causes the fabric located
at $(x_{g},y_{g})_{j}$ in the image to be placed on top of the world
coordinate associated with $(x_{p},y_{p})_{j}$ with the same surface normals
as before. In future work, we will investigate how to execute more complex
actions that result in reversed surface normals, which requires a rotation
motion during the action. We are given a sequence of actions
$\left(\mathbf{a}_{j}\right)_{j=1}^{n}$ executed on a fabric starting in
configuration $\xi_{1}$ and corresponding observations
$\left(I_{j}\right)_{j=1}^{n}$. Then at test-time, a similar object is dropped
onto the surface in a previously unseen configuration and the goal is to
generate a corresponding sequence of actions for a fabric in some previously
unseen configuration. Specifically, the robot generates a new sequence of
actions:
$\Big{(}\mathbf{a}_{j}^{\prime}\Big{)}_{j=1}^{n}=\Big{(}d_{I_{j}\rightarrow
I_{j}^{\prime}}(x_{g},y_{g})_{j},\;\;d_{I_{j}\rightarrow
I_{j}^{\prime}}(x_{p},y_{p})_{j}\Big{)}_{j=1}^{n}$ (III.2)
for $j\in\\{1,\ldots,n\\}$ where $d_{I_{j}\rightarrow
I_{j}^{\prime}}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ is a function which
estimates the corresponding point $(x^{\prime},y^{\prime})_{j}$ in
$I_{j}^{\prime}$ given a point $(x,y)_{j}$ in $I_{j}$. This function is
difficult to compute directly from images in general, and even more so
difficult to compute for images of highly deformable objects due to their
infinite degrees of freedom and tendency to self-occlude. Thus, we leverage
dense object descriptors [6] to approximate $d_{I_{j}\rightarrow
I_{j}^{\prime}}$ for any $I_{j}$ and $I_{j}^{\prime}$, as described in
Sections V and VI.
## IV Simulator
We use Blender 2.8, an open-source simulation and rendering engine [3]
released in mid-2019, to both create large synthetic RGB training datasets and
model the fabric dynamics for simulated experiments using its in-built fabric
solver based on [30, 29]. We simulate T-shirts and square fabrics, each of
which we model as a polygonal mesh made up of 729 vertices, a square number we
experimentally tuned to trade-off between fine-grained deformations and
reasonable simulation speed. These meshes can be easily constructed in Blender
by starting with a planar grid of vertices and then removing vertices and
edges to create desired shapes. See Figure 2 for an illustration. Each vertex
on the mesh has a global coordinate which we can query directly through
Blender’s API, allowing for easily available ground truth information about
various locations on the mesh and their pixel counterparts. See Figure 2 in
the supplement for examples of these meshes. We can also simulate finer-
grained manipulation of the mesh including grasps, pulls, and folds. To
implement the action space defined in Section III, we first deproject the
pixel corresponding to the pick point and map it to the vertex whose global
coordinates are closest in $\mathbb{R}^{3}$ to the pixel’s deprojected
coordinates. We then directly manipulate this vertex by pinning it and
translating it over a sequence of 30 frames. This generalized form of
manipulation allows us to easily execute experiments in simulation. See the
supplement for further details on how we pin vertices, the overall technique
and experiments in simulation.
## V Dense Shape Descriptor Training
### V-A Dense Object Descriptor Training Procedure
We consider an environment with a deformable fabric on a flat tabletop and
learn policies that perform smoothing and folding tasks. The policies we train
use point-pair correspondences that are generated between overhead images of
the fabric in different configurations. We generate deformation-invariant
correspondences by training dense object descriptors [41, 6] on synthetically
generated images of the fabric in different configurations.
Figure 2: Fabric Meshes: Examples of the meshes generated in Blender for both
square cloth (left) and t-shirts (right). The ground-truth vertices are
highlighted in the second and fourth columns.
In [6], an input image $I$ is mapped to a descriptor volume
$Z=f_{\theta}(I)\in\mathbb{R}^{W\times H\times D}$, where each pixel $(i,j)$
has a corresponding descriptor vector $Z_{i,j}\in\mathbb{R}^{D}$. Descriptors
are generated by a Siamese network $f_{\theta}$ and are guided closer together
for corresponding pixels in images and pushed apart by at least some margin
$M$ for non-corresponding pairs by minimizing a pixel-wise contrastive loss
function during training [6]. Corresponding pairs of pixels represent the same
point on an object. In this work, we also train a Siamese network to output
descriptors that are close for corresponding pairs of pixels and separated for
non-corresponding pixel pairs for overhead images of fabric in different
configurations. Since ground-truth pixel correspondences are difficult to
obtain in images across deformations of a real fabric, we train the network on
synthetic RGB data from Blender (see Section IV), where perfect information
about the pixel correspondences is available through the global coordinates of
the fabric mesh’s vertices. Note that during training, the sampled image
inputs to the Siamese network are of the same fabric type to ensure valid
correspondences. That is, two different images of T-shirts can be passed into
the network, but not a T-shirt and square fabric. Figure 3 demonstrates the
pipeline for predicting descriptors for correspondence generation. The learned
descriptors can then be used to approximate the correspondence function
$d_{I\rightarrow I^{\prime}}$ described in Section III:
$\displaystyle\left((i^{\prime\prime}_{l},j^{\prime\prime}_{l})\right)_{l=1}^{k}$
$\displaystyle=\operatorname*{arg\,min}_{(i^{\prime}_{1},j^{\prime}_{1})...(i^{\prime}_{k},j^{\prime}_{k})}\sum_{l=1}^{k}\lVert
f_{\theta}(I)_{i,j}-f_{\theta}(I^{\prime})_{i^{\prime}_{l},j^{\prime}_{l}}\rVert_{2}$
$\displaystyle\text{s.t.
}(i^{\prime}_{n},j^{\prime}_{n})\neq(i^{\prime}_{m},j^{\prime}_{m})\;\forall
m,n\in[k]$ $\displaystyle d_{I\rightarrow I^{\prime}}(i,j)$
$\displaystyle=\operatorname*{arg\,min}_{(i^{\prime},j^{\prime})}\sum_{l=1}^{k}\lVert(i^{\prime},j^{\prime})-(i^{\prime\prime}_{l},j^{\prime\prime}_{l})\rVert_{2}$
The first equation computes the top $k$ pixel matches based on their distance
in descriptor space and the second equation computes the geometric median of
these matches in pixel space. In the physical setup, we experimentally found
$k=20$ to give us the most robust predictions.
Figure 3: Learning Visual Correspondences: pipeline for training dense object
nets for robot fabric manipulation. Left: we train a dense correspondence
network on pairs of simulated fabric images to learn pixel-wise
correspondences using a pixel-wise contrastive loss. Right: we use the learned
descriptors for policy optimization. We can use correspondence to map a
reference action to a new fabric configuration. For example, we show an image
of a wrinkled fabric in “State 2,” and we can use descriptors to figure out
the action needed to smooth the fabric from “State 2” to “State 1.”
### V-B Dataset Generation and Domain Randomization
To enable generalization of the learned descriptors to a range of fabric
manipulation tasks, we generate a diverse dataset of initial fabric
configurations. The first step simulates dropping the fabric onto the planar
workspace while executing similar pinning actions to those described in
Section IV on an arbitrary subset of vertices, causing some vertices to fall
due to gravity while others stay fixed. We then release the pinned vertices 30
frames later so that they collapse on top of the fabric. This allows us to
create realistic deformations in the mesh. We then export RGB images which
serve as inputs to the Siamese network, pixel-wise annotations which gives us
correspondences, and segmentation masks which allow us to sample matches on
the fabric.
Simulating soft-body animations is in general a computationally time-consuming
process which makes it difficult to render large datasets in short periods of
time. We take steps toward mitigating this issue by rendering 10 images per
drop, allowing us to collect 10x as much data in the same time period. In
simulation, we found that the test time pixel match error was unaffected when
including these unsettled images of the fabric in the dataset. We additionally
make use of domain randomization [32, 44] by rendering images of the scene
while randomizing parameters including mesh size, lighting, camera pose,
texture, color and specularity (see supplement for further details). We also
restrict the rotation about the z-axis to be between $(-\pi/4,\pi/4)$ radians
to reduce ambiguity during descriptor training due to the natural symmetry of
fabrics such as squares. To randomize the image background, we sample an image
from MSCOCO [21] and “paste” the rendered fabric mask on top. For our
experiments, we generated one (domain-randomized) dataset, including both
T-shirts and square fabric, and train a single model which we used for the
experiments in Section VII. For reference, generating a single dataset of
7,500 images, half T-shirts and half square cloth, with 729 annotations per
image takes approximately 2 hours on a 2.6GHz 6-core Intel Core i7 MacBook
Pro.
## VI Descriptor-Parameterized Controller
As discussed in Section III-B, the robot receives a demonstration of the task
consisting of actions $\left(\mathbf{a}_{j}\right)_{j=1}^{n}$ and observations
$\left(I_{j}\right)_{j=1}^{n}$. At execution time, the robot starts with the
fabric in a different configuration, and the fabric itself may have a
different texture or color. At time $j\in[n]$, the robot observes
$I^{\prime}_{j}$ then executes
$\pi_{j}(I_{j}^{\prime})=\left(d_{I_{j}\rightarrow
I_{j}^{\prime}}(x_{g},y_{g})_{j},\;\;d_{I_{j}\rightarrow
I_{j}^{\prime}}(x_{p},y_{p})_{j}\right)$ where $d_{I_{j}\rightarrow
I_{j}^{\prime}}$ is defined in Section V-A. $\pi_{j}$ uses the geometric
structure learned by the descriptor network to identify semantically relevant
pixels in $I_{j}^{\prime}$ to generate actions that manipulate these
keypoints. Because the descriptor network learns about task-agnostic fabric
geometry, we train one network for a variety of tasks and use it across
different fabric configurations from the ones supplied in demonstrations.
Additionally, we do not require separate networks for each fabric type we wish
to train.
For example, one step of a task could involve grasping the top-right corner of
the fabric and taking an action to place it in alignment with the bottom-left
corner, thereby folding the fabric. The robot could receive an offline
demonstration of this task on an initially flat fabric, but then be asked to
perform the same task on a crumpled, rotated fabric. To do this, the robot
must be able to identify the corresponding points in the new fabric
configuration (top-right and bottom-left corners) and define a new action to
align them. $\pi_{j}$ computes correspondences for the grasp and place points
across the demonstration frame and the new observation to generate a
corresponding action for the new configuration.
### VI-A Fabric Smoothing
In the square fabric smoothing task, the robot starts with a crumpled fabric
and spreads it into a smooth configuration on a planar workspace as in [36].
To complete this task, we use the approach from [36] and iterate over fabric
corners, pulling each one to their target locations on an underlying plane.
However, while [36] design a policy to do this using ground-truth knowledge of
the fabric in simulation, we alternatively locate corners on the crumpled
fabric using a learned descriptor network and a source image of a flat fabric
where the corners are labeled. For the T-shirt smoothing task, we apply a
similar method but instead iterate over the corners of the sleeves and the
base of the T-shirt.
### VI-B Fabric Folding
The fabric folding task involves executing a sequence of folds on a fairly
smooth starting configuration. For each folding task, we use a single offline
demonstration containing up to 4 pick and place actions, each defined by a
pick and drop pixel location and collected by a human via a GUI. The
descriptor-parameterized controller is then executed in an open-loop manner.
Figure 4: Fabric Specifications: Images and dimensions of the square fabrics
and shirts we use in experiments.
## VII Experiments
We experimentally evaluate (1) the quality of the learned descriptors and
their sensitivity to training parameters and (2) the performance of the
descriptor-based policies from Section VI on two physical robotic systems, the
da Vinci Research Kit (dVRK) [15] and the ABB YuMi. Results suggest that the
learned descriptors and the resulting policies are robust to changes in fabric
configuration and color.
### VII-A Tasks
We consider 10 fabric manipulation tasks executed on a set of 5 T-shirts and 5
square fabrics in the real world:
1. 1.
Single Fold (SF): A single fold where one corner is pulled to its opposing
corner.
2. 2.
Double Inward Fold (DIF): Two opposing corners are folded to the center of the
fabric.
3. 3.
Double Triangle Fold (DTF): Two sets of opposing corners are aligned with each
other.
4. 4.
Double Straight Fold (DSF): The square cloth is folded in half twice, first
along the horizontal bisector and then along the vertical bisector.
5. 5.
Four Corners Inward Fold (FCIF): All four corners are sequentially folded to
the center of the cloth.
6. 6.
T-Shirt Sleeves Fold (TSF): The two sleeves of a t-shirt are folded to the
center of the shirt.
7. 7.
T-Shirt Sleeve to Sleeve Fold (TSTSF): The left sleeve of a T-shirt is folded
to the right sleeve of the T-shirt.
8. 8.
Smoothing (S): Fabric is flattened from a crumpled state.
9. 9.
Smoothing + Double Triangle Fold (SDTF): Fabric is smoothed then the DTF is
executed.
10. 10.
Smoothing + Sleeve to Sleeve Fold (SSTSF): T-shirt is smoothed then TSTSF is
executed.
All fabrics are varied either in dimension or color according to Figure 4.
Additionally, we execute a subset of these tasks in simulation. A single
visual demonstration consisting of up to 4 actions is provided to generate a
policy which the robot then tries to emulate in the same number of actions.
Figure 5: Policy Rollouts: We visualize policy execution on the YuMi for
tasks 2, 3, 4, 5, 6 and 7 as described in Section VII-A. The first four
columns show the folding instructions on some initial fabric and the last four
columns show the corresponding folds executed on novel starting configurations
for a different fabric. Figure 6: Full Folding Sequence: The first and
second row is a time-lapse of a sequence of 6 actions taken by the YuMi and
dVRK respectively, and with actions overlaid by red arrows, to successively
smooth a wrinkled fabric and then fold it according to task 3 in Section
VII-A. The third row is a time-lapse of a sequence of 5 actions taken by the
YuMi to complete task 10 in Section VII-A. Here, robot actions are overlaid
with blue arrows.
### VII-B Experimental Setup
We execute fabric folding and smoothing experiments on the dVRK [15] and ABB
YuMi robot. The dVRK is equipped with the Zivid OnePlus RGBD sensor that
outputs $1900\times 1200$ pixel images at 13 FPS at depth resolution $0.5$ mm.
The workspace of the dVRK is only $5"\times 5"$, so we use only square fabric
of the same dimension while varying the color according to Figure 4.
Manipulating small pieces of fabric into folds is challenging due to the
elasticity of the fabric, so we add weight to the fabric by dampening it with
water. Additionally, we place a layer of 1 inch foam rubber below the fabric
to avoid damaging the gripper. The YuMi has a $36"\times 24"$ workspace, and
since only one arm is utilized resulting in a more limited range of motion, we
only manipulate at most $12"\times 12"$ pieces of fabric which we do not
dampen. In this setup we use a 1080p Logitech webcam to collect overhead color
images. For the YuMi, we use both T-shirts and square fabric of varying
dimension and color but go no lower than $9"\times 9"$ fabrics due to its
larger gripper. Finally, for both robots, we use a standard pixel to world
calibration procedure to get the transformation from pixel coordinates to
planar workspace coordinates.
For both robots, we follow the same experimental protocol. We manually place
the fabric in configurations similar to those shown in Figure 2 and deform
them by pulling at multiple locations on the fabric. To obtain image input for
the descriptor networks, we crop and resize the overhead image to be
$485\times 485$ such that the fabric is completely contained within the image.
Although lighting conditions, camera pose and workspace dimensions are
significantly different between the two robotic systems, no manual changes are
made to the physical setup. We find that the learned descriptors are
sufficiently robust to handle this environmental variability.
We evaluate the smoothing task by computing the coverage of the cropped
workspace before and after execution. For the folding tasks, as in [19], we
consider an outcome a success if the final state is visually consistent with
the goal image. Conventional quantitative metrics such as intersection of
union between the final state and a target image provide limited diagnostic
information when starting configurations are significantly different as in the
presented experiments.
### VII-C Results
We evaluate the smoothing and folding policies on both the YuMi and dVRK on
square fabrics and T-shirts. Table II shows the success rates of our method on
all proposed tasks in addition to a breakdown of the failure cases detailed in
Table III. We observe that the descriptor-parameterized controller is able to
successfully complete almost all folding tasks at least $75\%$ of the time,
and the smoothing policies are able to increase coverage of the cloth to over
$83\%$ (Table I). The execution of the smoothing policy followed by the double
triangle folding policy results in successful task completion $6/10$ and
$8/10$ times on the YuMi and dVRK respectively. We find that the most frequent
failure mode is an unsuccessful grasp of the fabric which is compounded for
tasks that require more actions. Though this is independent of the quality of
the learned descriptors, it highlights the need for more robust methods to
grasp highly deformable objects.
Task | Robot | Avg. Start Coverage | Avg. End Coverage
---|---|---|---
S | YuMi | $71.4\pm 6.2$ | $83.2\pm 8.1$
S | dVRK | $68.4\pm 4.4$ | $86.4\pm 5.2$
Table I: Physical Fabric Smoothing Experiments: We test the smoothing policies
designed in Section VI on the YuMi and the dVRK robots. Both robots are able
to increase coverage during the smoothing task by $11-22$ percent on average.
Task | Robot | # Actions | Success | Error A | Error B | Error C
---|---|---|---|---|---|---
SF | YuMi | 1 | 18/20 | 2 | 0 | 0
SF | dVRK | 1 | 20/20 | 0 | 0 | 0
DIF | YuMi | 2 | 16/20 | 3 | 0 | 1
DIF | dVRK | 2 | 20/20 | 0 | 0 | 0
DTF | YuMi | 2 | 14/20 | 3 | 2 | 1
DTF | dVRK | 2 | 18/20 | 0 | 2 | 0
TSF | YuMi | 2 | 15/20 | 3 | 0 | 2
SDTF | YuMi | 6 | 6/10 | 2 | 1 | 1
SDTF | dVRK | 6 | 8/10 | 0 | 2 | 0
DSF | YuMi | 3 | 15/20 | 1 | 1 | 3
DSF | dVRK | 3 | 17/20 | 1 | 0 | 2
FCIF | YuMi | 4 | 13/20 | 5 | 1 | 1
FCIF | dVRK | 4 | 18/20 | 0 | 1 | 1
TSTSF | YuMi | 1 | 17/20 | 2 | 0 | 1
SSTSF | YuMi | 5 | 6/10 | 2 | 0 | 2
Table II: Physical Fabric Folding Experiments: We test the folding policies
from Section VI on the YuMi and the dVRK. We observe both robots are able to
perform almost all folding tasks at least $75$ percent of the time. The YuMi
is able to perform the smoothing then folding task $6/10$ times and the dVRK
is able to do so $8/10$ times.
Error | Description
---|---
A | Gripper picks up more than one layer of fabric or fabric slips out of gripper due to inaccurate depth of grasp
B | Pick or drop correspondence error greater than 30 pixels (10% of cloth width) or pick correspondence not on fabric mask
C | Unintended physics: resulting fold does not hold due to variable stiffness of the fabric, friction of the fabric, or friction of the underlying plane
Table III: Failure Mode Categorization
## VIII Discussion and Future Work
We present an approach for multi-task fabric manipulation by using visual
correspondences learned entirely in simulation. Experiments suggest that the
learned correspondences are robust to different fabric colors, shapes,
textures, and sizes and make it possible to efficiently learn 10 different
fabric smoothing and folding tasks on two different physical robotic systems
with no training in the real world. In future work, we plan to explore
hierarchical fabric manipulation policies, where visual correspondences can be
used to define coarse action plans while a closed loop controller can be
learned to realize these plans. We will also explore more complex fabric
manipulation tasks, such as wrapping rigid objects, in which reasoning about
fabric dynamics is critical or tasks involving manipulating multiple fabrics
simultaneously.
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## IX Appendix
The appendix is organized as follows:
* •
Appendix A contains additional details on the fabric simulator
* •
Appendix B contains additional details on the experiments conducted in
simulation
* •
Appendix C contains results for simulation experiments
* •
Appendix D shows visualizations of the learned descriptor mappings.
* •
Appendix E contains images from additional physical trials executed on the
dVRK and YuMi.
* •
Appendix F conducts a detailed study on the effect of various hyperparameters
on the quality of the learned visual correspondences.
### IX-A Fabric Simulator Details
We use Blender 2.8 to both create dynamic cloth simulations and to render
images of the fabric in different configurations. As can be seen in Figure 2,
we are able to retrieve the world coordinates of each vertex via Blender’s API
which we then use to find ground truth pixel correspondences through an
inverse camera to world transformation. This allows us to create dense pixel-
vertex annotations along the surface of the fabric which we feed to into the
descriptor training procedure. Figure 9 is a visualization of the learned
descriptors and Figure 7 contains examples of the domain randomized training
data we generate through Blender.
#### IX-A1 Fabric Model
To generate the square cloth in Blender, we first import a default square mesh
and subdivide it three times to create a grid of $27\times 27$ grid of
vertices. We found that this number of square vertices resulted in a visually
realistic animation in comparison to our real fabrics. We additionally add
0.02 meter thickness to the cloth to increase its weight which creates more
realistic collision physics. In order to apply Blender’s in-built cloth
physics to the mesh, we simply make use of the cloth physics modifier through
which we are able modify the parameters shown in Table IV. Internally, Blender
simulates fabric physics for polygonal meshes with gravitational forces,
damping, stiffness, and by interconnecting the mesh vertices with four types
of virtual springs: tension springs, compression springs, shear springs, and
angular bending springs. Each vertex also exerts repulsive forces within a
self-contained virtual sphere on vertices both within fabric and in
surrounding objects, to simulate self-collisions and collisions with other
objects. We visually tune the simulator by replaying a fabric folding action
while varying parameters, most notably the friction coefficients and spring
elasticity constants. From observing videos of the folding actions, we settle
on the parameter values specified in Table IV. A visualization of these steps
can be seen in the top row of Figure 8. To generate the t-shirt mesh, we
similarly import a default square mesh and subdivide it three times, but also
delete all vertices that do not lie in a predefined t-shirt cutout of the
square mesh which results in the bottom right image of Figure 2.
#### IX-A2 Manipulation with Hook Objects
We utilize hook objects to take actions in the Blender simulator. A hook
object attaches to a mesh vertex and exerts a proportional sphere of influence
over the selected vertex and those in its vicinity, pulling the fabric in the
direction of movement. We simulate a grasp, drag, and drop of the fabric by
assigning a hook object to a fabric vertex, moving this hook over a series of
frames to the deprojected pixel drop location, and removing the hook object
assignment to release the cloth.
Figure 7: Examples of domain-randomized images of the starting fabric states
encountered in the dataset generation phase described in Section V-B. The
first two columns show examples of images with a square fabric, and the last
two columns show similar examples but with a t-shirt. Figure 8: The top row
illustrates the the process of creating cloth in Blender from a default square
mesh. The second row is an example of a starting configuration generated by
dropping the cloth from a fixed height and pinning a single arbitrary vertex.
The pinned vertex is labeled by the red circle. The third row illustrates
frames from a folding action in the simulator and the last row shows the
corresponding rendered images of the settled cloth before and after the
action. Table IV: Blender Cloth Simulation Parameters
Parameter | Explanation | Value
---|---|---
Quality Steps | quality of cloth stability and collision response | 5.0
Speed Multiplier | how fast simulation progresses | 1.0
Cloth Mass (kg) | – | 0.3
Air Viscosity | air damping | 1.0
Tension Springs | tension damping/stretching | 5.0
Compression Springs | compression damping/stretching | 5.0
Shear Springs | damping of shear behavior | 5.0
Bending Springs | damping of bending behavior | 0.5
Friction | friction with self-contact | 5
Self-Collision Distance (m) | per-vertex spherical radius for repulsive forces | 0.015
Figure 9: Visualization of the 3-dimensional descriptors learned via the
training procedure described in Section V by mapping each pixel’s descriptor
vector to an RGB vector. Thus, similar colors across the images of columns two
and four represent corresponding points on the square cloth. Figure 10:
Additional rollouts of the smoothing task from randomly chosen starting
configurations. The learned descriptors are used to locate the corners of the
fabric and successively pull them to a reference location in an image of the
flat cloth. Figure 11: Additional rollouts of the smoothing task from
randomly chosen starting configurations. The learned descriptors are used to
locate the corners of the fabric and successively pull them to a reference
location in an image of the flat cloth. Figure 12: Additional rollouts of
the folding tasks described in Section VII-A from arbitrary starting
configurations. The left column, center column and right column contain
results for tasks 4, 3 and 2 respectively.
#### IX-A3 Starting Configurations and Actions
To generate varied starting configurations, we simulate dropping the fabric
from 0.2 meters above the workspace while pinning it an arbitrary subset of
the 729 vertices. After 30 frames in the animation, the pinned vertices are
released and are allowed to settle for another 30 frames. This creates natural
deformation in the cloth and introduces a wide range of starting
configurations to the training dataset. A sequence of these steps is shown in
the second row of Figure 8. When running simulated experiments, taking pick
and place actions requires manipulating the cloth via hook objects as defined
in Section IV. A sequence of frames throughout the course of an action using a
hook object as well as the corresponding rendered frames are shown in the last
two rows of Figure 8.
### IX-B Simulation Experiment Details
In simulation, we conduct 50 trials of the first 4 folding tasks described in
VII-A on a domain randomized test set generated as described in V-B. We
consider an outcome a success if the final state is visually consistent with
the target image. We additionally declared a failure when the planned pick and
drop pixels were more than 50 pixels away from their correct ground truth
locations which we had access to in Blender. Note that this is neither a
sufficient nor necessary condition for a successful fold, but nevertheless
serves as a decent heuristic. While we considered more quantitative metrics
such as structural similarity between the target image and the final state and
summed distance between corresponding vertices on the mesh, these metrics are
insufficient when the test time starting configuration is significantly
different from the demonstration configuration.
### IX-C Simulation Experiment Results
We evaluate the folding policies designed in Section VI in the simulated
fabric environment. The policies successfully complete the tasks $84$ to $96$
percent of the time (Table V).
Task | Success Rate
---|---
Single Fold | 46/50
Double Inward Fold | 48/50
Double Triangle Fold | 42/50
T-Shirt Sleeves Fold | 44/50
Table V: Simulated Fabric Folding Experiments: We observe that the system is
able to successfully complete the tasks $84$ to $96$ percent of the time in
simulation. Success is determined by visual inspection of the cloth after the
sequence of actions is executed. Example simulation rollouts are shown in
Figure 14.
### IX-D Descriptor Mapping Visualizations
We present the descriptor volumes produced by a model trained to output
3-dimensional descriptors. We coarsely visualize the volumes by presenting
them as RGB images (Figure 9), and observe that corresponding pixels of the
cloth map to similar colors in the descriptor volumes across configurations.
### IX-E Physical Trial Trajectories
In this section, we present additional trials of the physical experiments
conducted using the descriptor-based policies for smoothing (Figure 10, Figure
11) and folding (Figure 12).
### IX-F Descriptor Quality Ablations
Figure 13: Ablation studies: We study the sensitivity of the learned dense
object descriptors as described in Sections V and IX-F to training parameters.
Starting from top left, and proceeding clockwise, we test the effect of
testing on RGB vs depth images, on the descriptor dimension (either 3, 9, or
16), on the number of ground truth annotations, and whether domain
randomization is used. All results are evaluated using pixel match error on a
held-out set of image pairs. Figure 14: Simulation Policy Visualization:
Visualization of the policy executed in simulation (with Blender) using
learned descriptors for folding tasks 2 and 4 described in VII-A. The first
two columns show the corresponding folding instructions from a web interface
(pick-and-place actions shown with red arrows) for tasks 2 and 4. The third
column shows images of the previously unseen initial configurations of fabrics
before the actions, while the last two columns show the result of executing
descriptor-parameterized actions. Results suggest that the learned descriptors
can be used to successfully perform a variety of folding tasks from varying
initial configurations.
To investigate the quality of the learned descriptors with training process
described in Section V, we perform four sets of ablation studies. We evaluate
the quality of learned descriptors in a manner similar to [41] by evaluating
the $\ell_{2}$ pixel distance of the pixel match error on a set of 100 pairs
of held-out validation set images, where for each we sample 100 pixel pairs.
We study the effect of training descriptors on (1) RGB or depth images, (2)
using descriptor dimension 3, 9, or 16, (3) using 200, 450, or 700 ground-
truth annotated images, and (4) whether domain randomization is used or not.
Results suggest that the learned descriptors are best with RGB data, with
descriptor dimension between 3 and 16 and with domain randomization, though
the performance is generally insensitive to the parameter choices, suggesting
a robust training procedure. Based on these results, we use RGB images with
domain randomization, and with descriptor dimension 3 for all simulated
experiments for both the t-shirt and square fabric. We use RGB, domain-
randomized, 9-dimensional descriptors for real fabric experiments. See Figure
13 for plots.
|
# Path integral control under McKean-Vlasov dynamics
Timothy Bennett111Corresponding author<EMAIL_ADDRESS>222Department of Mathematics and Statistics, University of South Alabama,
Mobile, AL, 36688, United States.
###### Abstract
We investigate the complexities of the McKean–Vlasov optimal control problem,
exploring its various formulations such as the strong and weak formulations,
as well as both Markovian and non-Markovian setups within financial markets.
Furthermore, we examine scenarios where the law governing the control process
impacts the dynamics of options. By conceptualizing controls as probability
measures on a fitting canonical space with filtrations, we unlock the
potential to devise classical measurable selection methods, conditioning
strategies, and concatenation arguments within this innovative framework.
These tools enable us to establish the dynamic programming principle under a
wide range of conditions.
## Introduction:
The Black-Scholes model stands as the quintessential example of financial
models, embodying key principles that have influenced subsequent developments.
Its widespread adoption reflects the common adjustments and trade-offs
necessary to translate theoretical concepts into practical tools for financial
analysis and decision-making. Let us dive deep into the core principles of
this model. The main aim of creating the Black-Scholes model is to evaluate
and manage options, operating on the premise that the volatility of the
underlying asset remains constant (Fournié et al.,, 1997). Building on this
assumption, it provides an analytical formula for calculating the value of a
European option, which relies on the volatility parameter of the system.
However, the model’s main drawback arises from the fact that in actuality,
volatility fluctuates, and the market does not necessarily conform to the
values predicted by this model (Pramanik,, 2020).
However, practitioners in financial markets have widely embraced the Black-
Scholes model, albeit with some modifications that deviate from the original
theoretical expectations. These adjustments include anticipating negotiations
for transactions in terms of implicit volatility rather than option prices,
implementing theoretical delta hedging as recommended by the model, and
recognizing that the implicit volatility tends to be higher for options
further from the money and closer to maturity (Pramanik, 2021c, ; Pramanik and
Polansky,, 2021, 2023). Consequently, the Black-Scholes model retains
significant practical utility, although its role differs from the primary
hypotheses outlined in the original theory. While one might assume that the
compromises and adaptations made in utilizing the Black-Scholes model are
unique to this particular context, we argue that they offer broader insights
applicable across various fields beyond finance.
So far in our initial discussion, we overlooked the inclusion of control
theory. Control theory encompasses two primary forms: open loop and closed
loop control. In open loop control, the decision of the policymaker remains
uninfluenced by the state variable’s information. Conversely, closed loop
control entails the policymaker’s decision being contingent upon all available
state variable information (Pramanik, 2024a, ). Integrating control variables
is crucial in understanding the dynamics of financial markets. For instance,
consider an entrepreneur aiming to maximize long-term profits, assuming
control over advertising expenditure. If they increase spending on commercials
presently, they may afford a celebrity endorsement, thereby boosting sales
probability in subsequent periods (Pramanik and Polansky,, 2020). Conversely,
heightened sales in the current period provide more funds for future
commercial investment. This exemplifies closed loop control within financial
markets. Furthermore, all of theses financial models are not always linear
stochastic differential equations (SDEs) in practice. Therefore, the
importance of non-linear SDEs come into place. One such SDE is McKean-Vlasov
(McKean Jr,, 1966) system. The next section gives an overview of it.
## Overview of McKean-Vlasov SDEs:
McKean-Vlasov type SDEs are commonly known as nonlinear SDEs, with _non-
linear_ highlighting the potential dependence of coefficients on the
solutions’ marginal distributions rather than the state variable itself. The
distinctive aspect of these coefficients, even when deterministic, leads to
non-local feedback effects, precluding the conventional Markov property
(Carmona and Delarue,, 2015). Incorporating the marginal distribution into the
system’s state could restore the Markov property but at the expense of
significantly increasing the complexity by introducing a probability measure,
rendering the system infinite-dimensional. While analyzing the infinitesimal
generator can utilize tools developed for infinite-dimensional differential
operators, conventional differential calculus, even in infinite dimensions,
struggles to ensure the second component of the state process matches the
statistical distribution of the first component (Carmona and Delarue,, 2015).
Nevertheless, pathwise analysis complexity remains manageable and can be
effectively addressed using standard stochastic calculus tools. This points
towards devising a wholly probabilistic approach to address optimal control in
these nonlinear systems.
The optimal control of systems governed by McKean-Vlasov SDEs appears to be a
relatively novel challenge, curiously overlooked in stochastic control
literature. Typically, solving a McKean-Vlasov SDE involves employing a fixed-
point argument: Initially, a set of potential marginal distribution candidates
is established, and then the resulting standard SDE is solved, with the fixed-
point argument entailing the requirement that the solution’s marginal
distributions match the initially chosen ones. Introducing stochastic control
introduces an additional layer of optimization atop this fixed-point process.
This formulation shares many similarities with the mean-field game (MFG)
problem, as initially proposed by Lasry and Lions, (2007). In Carmona et al.,
(2013), the similarities and disparities between these two problems are
identified and discussed. It is notably emphasized that optimizing first and
subsequently seeking a fixed point yields the solution to a mean-field game
problem, while first determining the fixed point and then optimizing leads to
the resolution of the optimal control for McKean-Vlasov SDEs (Polansky and
Pramanik,, 2021; Pramanik, 2024b, ). The solutions to both problems describe
equilibrium states of large populations whose interactions and objectives
follow mean-field patterns. The nuances between these equilibrium concepts are
subtle and hinge on the optimization component’s formulation within the
equilibrium model. Linear quadratic models are presented in Carmona et al.,
(2013) to offer illustrative examples of these distinctions.
As McKean-Vlasov SDEs exhibit non-Markovian dynamics, it is logical to tackle
optimization through a tailored version of the Pontryagin stochastic maximum
principle, rather than forcing fit with the Hamilton-Jacobi-Bellman (HJB)
framework. The stochastic Pontryagin method hinges on introducing and
manipulating adjoint processes, which are solutions to adjoint backward
stochastic differential equations (BSDEs). These equations involve partial
derivatives of the Hamiltonian function concerning the state variable (Carmona
and Delarue,, 2015). However, in the context of McKean-Vlasov SDEs, the
marginal distributions of solutions serve as full-fledged variables of the
Hamiltonian function, necessitating differentiation in the pursuit of critical
points. This discrepancy likely contributes to the impasse in existing
literature. Until now, only dynamics contingent on moments of the marginal
distributions have been explored, where differentiability with respect to the
measure can be achieved through conventional calculus chain rules (Carmona and
Delarue,, 2015). The right notation of differentiability was introduced in
Lasry and Lions, 2006a and Lasry and Lions, 2006b .
## Pontryagin principle and Path integral control:
The stochastic Pontryagin principle stands as a potent tool, yet its
illuminating insights often necessitate restrictive assumptions on the models.
For instance, findings in Carmona and Delarue, (2015) lean on a set of
technical prerequisites that confine the scope of models to those
characterized by coefficients primarily linear in state, control, and measure
variables, alongside costs that exhibit convexity concerning the state and
control variables. While these conditions may seem limiting, they are typical
in applications of the stochastic Pontryagin principle to control problems.
Notably, assuming the convexity of the control space is done for simplicity’s
sake. More extensive spaces could be accommodated, albeit at the expense of
employing spike variation techniques and introducing an additional adjoint
equation. However, Carmona and Delarue, (2015) opted against pursuing this
broader generality, as it would entail added complexity in notation and
technicalities, potentially obscuring the core focus of their research,
especially without specific application-driven motivation (Pramanik, 2022a, ;
Pramanik, 2022b, ; Pramanik, 2023b, ; Pramanik, 2023c, ).
The essential aspect of the stochastic Pontryagin principle involves seeking
within the control set for a potential minimizer of the Hamiltonian (often
referred to as satisfying the Isaacs condition), while the complementary
aspect suggests incorporating the minimizer’s formula into both the forward
equation governing dynamics and the adjoint backward equation describing
adjoint processes (Hua et al.,, 2019; Pramanik,, 2016; Pramanik, 2021b, ). The
inclusion of the minimizer in these equations establishes a significant
interconnection between the forward and backward equations, effectively
reducing the control problem solution to that of a forward-backward stochastic
differential equation (FBSDE). Applying this approach to the current scenario
entails investigating FBSDEs where the marginal distributions of solutions
emerge in the coefficients of the equations. Carmona and Delarue, (2015) term
these equations “mean-field FBSDEs” or “FBSDEs of McKean-Vlasov type,”
asserting that their study had not been undertaken prior to their research.
They noted that while a general existence result was proposed in Carmona and
Delarue, (2013), one of the assumptions therein prevents its application to
the linear quadratic model.
A specific variant of the stochastic Pontryagin principle, such as the
Feynman-type path integral method introduced by Pramanik, (2020), can be
utilized and further developed by Pramanik and Polansky, 2024a to obtain an
analytical solution for the system in question (Pramanik and Polansky, 2024b,
). When the state variable is high-dimensional and the system dynamics are
nonlinear, as seen in Merton-Garman-Hamiltonian SDEs, numerical construction
of an HJB equation becomes extremely difficult. The Feynman-type path integral
approach effectively overcomes this challenge of dimensionality and provides a
localized analytical solution. To employ this approach, we first establish a
stochastic Lagrangian for each continuous time point within the interval
$s\in[0,t]$, where $t>0$. Subsequently, we divide this entire time span into
$n$ equal-length subintervals and define a Riemann measure corresponding to
the state variable for each subinterval. After constructing a Euclidean action
function, we obtain a Schroedinger-type equation via Wick rotation. By
enforcing the first-order conditions with respect to both the state and
control variables, we determine the solution of the system. This methodology
holds potential application in cancer research (Dasgupta et al.,, 2023;
Hertweck et al.,, 2023; Kakkat et al.,, 2023; Khan et al.,, 2023; Vikramdeo et
al.,, 2023; Khan et al.,, 2024).
## Further impact:
We gain insight into the challenge of optimal control for mean-field
stochastic differential equations, also known as McKean–Vlasov stochastic
differential equations in academic literature. This problem presents a
stochastic control scenario where the state process is driven by SDEs with
coefficients influenced by the present time, the trajectories of the state
process, and its distribution (or conditional distribution in cases involving
shared noise). Likewise, the reward functionals can be influenced by the
distribution of the state process (Pramanik, 2023a, ).
A common limitation of the aforementioned results is their general requirement
for some Markovian property of the system or its distribution, along with
strong regularity assumptions on the coefficients and reward functions under
consideration. This might come as a surprise to those familiar with the
classical stochastic Pontryagin principle (Pramanik, 2021a, ). In stochastic
control problems, it is feasible to utilize measurable selection arguments to
derive the stochastic Pontryagin principle without necessitating more than
mild measurability assumptions (Pramanik and Maity,, 2024). Typically, two
crucial elements are needed to establish the dynamic programming principle:
first, ensuring the stability of controls concerning conditioning and
concatenation, and second, the measurability of the associated value function.
Measurable selection arguments offer a framework to validate the conditioning,
concatenation, and measurability requirements of the associated value function
without relying on strong assumptions. Through the concept of relaxed control,
interpreting a control as a probability measure on a canonical space, and
leveraging martingale problem theory, researchers have demonstrated a
stochastic Pontryagin principle using straightforward and transparent
reasoning (Djete et al.,, 2022). In the future, we aim to explore scenarios
where the drift and diffusion components, as well as the reward functions in
option markets, can depend on the joint conditional distribution of the state
process path and the closed-loop control.
## References
* Carmona and Delarue, (2013) Carmona, R. and Delarue, F. (2013). Mean field forward-backward stochastic differential equations. Electronic Communications in Probability, 18:1–15.
* Carmona and Delarue, (2015) Carmona, R. and Delarue, F. (2015). Forward–backward stochastic differential equations and controlled mckean–vlasov dynamics. The Annals of Probability, pages 2647–2700.
* Carmona et al., (2013) Carmona, R., Delarue, F., and Lachapelle, A. (2013). Control of mckean–vlasov dynamics versus mean field games. Mathematics and Financial Economics, 7:131–166.
* Dasgupta et al., (2023) Dasgupta, S., Acharya, S., Khan, M. A., Pramanik, P., Marbut, S. M., Yunus, F., Galeas, J. N., Singh, S., Singh, A. P., and Dasgupta, S. (2023). Frequent loss of cacna1c, a calcium voltage-gated channel subunit is associated with lung adenocarcinoma progression and poor prognosis. Cancer Research, 83(7_Supplement):3318–3318.
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* Fournié et al., (1997) Fournié, E., Lasry, J.-M., and Lions, P.-L. (1997). Some nonlinear methods for studying far-from-the-money contingent claims. Publ. Newton Inst, pages 115–45.
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* Lasry and Lions, (2007) Lasry, J.-M. and Lions, P.-L. (2007). Mean field games. Japanese journal of mathematics, 2(1):229–260.
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* Polansky and Pramanik, (2021) Polansky, A. M. and Pramanik, P. (2021). A motif building process for simulating random networks. Computational Statistics & Data Analysis, 162:107263.
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* Pramanik and Maity, (2024) Pramanik, P. and Maity, A. K. (2024). Bayes factor of zero inflated models under jeffereys prior. arXiv preprint arXiv:2401.03649.
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* Pramanik and Polansky, (2023) Pramanik, P. and Polansky, A. M. (2023). Scoring a goal optimally in a soccer game under liouville-like quantum gravity action. In Operations Research Forum, volume 4, page 66. Springer.
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* Vikramdeo et al., (2023) Vikramdeo, K. S., Anand, S., Sudan, S. K., Pramanik, P., Singh, S., Godwin, A. K., Singh, A. P., and Dasgupta, S. (2023). Profiling mitochondrial dna mutations in tumors and circulating extracellular vesicles of triple-negative breast cancer patients for potential biomarker development. FASEB BioAdvances, 5(10):412.
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# Chiral valley phonons and flat phonon bands in moiré materials
Indrajit Maity, Arash A. Mostofi Johannes Lischner<EMAIL_ADDRESS>Departments of Materials and Physics and the Thomas Young Centre for Theory
and Simulation of Materials, Imperial College London, South Kensington Campus,
London SW7 2AZ, UK
###### Abstract
We investigate the chirality of phonon modes in twisted bilayer
$\mathrm{WSe_{2}}$ and demonstrate distinct chiral behavior of the
$K/K^{\prime}$ valley phonons for twist angles close to $0^{\circ}$ and close
to $60^{\circ}$. In particular, multiple chiral non-degenerate $K/K^{\prime}$
valley phonons are found for twist angles near $60^{\circ}$ whereas no non-
degenerate chiral modes are found for twist angles close to $0^{\circ}$.
Moreover, we discover two sets of emergent chiral valley modes that originate
from an inversion symmetry breaking at the moiré scale and find similar modes
in moiré patterns of strain-engineered bilayers $\mathrm{WSe_{2}}$ and
$\mathrm{MoSe_{2}/WSe_{2}}$ heterostructures. At the energy gap between
acoustic and optical modes, the formation of flat phonon bands for a broad
range of twist angles is observed in twisted bilayer $\mathrm{WSe_{2}}$. Our
findings are relevant for understanding electron-phonon and exciton-phonon
scattering in moiré materials and also for the design of phononic analogues of
flat band electrons.
Introduction. Chirality - the characteristic of an object that can be
distinguished from its mirror image - plays a fundamental role in physics,
chemistry, and biology. For example, different enantiomers of chiral molecules
absorb different amounts of right-handed and left-handed polarized light,
which enables their spectroscopic characterization through measurement of the
circular dichroism [1]. In condensed matter physics, the chirality of
electrons gives rise to many exotic effects, such as Klein tunneling [2] in
graphene, and the chiral magnetic effect in three-dimensional semi-metals [3,
4].
Recently, the study of chirality in phonons has attracted significant
interest. For example, Zhang and Niu [5] predicted the existence of chiral
phonons at the $K$ and $K^{\prime}$ valleys of monolayer $\mathrm{WSe_{2}}$,
which was subsequently verified by experiments that measured the circular
dichroism of phonon-assisted intervalley transitions of holes [6]. The
interaction of such chiral valley phonons with other quasiparticles is
relevant for understanding and controlling many electronic and optical
phenomena [7, 8, 9, 10, 11, 12, 13]. For instance, Li et al. demonstrated that
the coupling between a chiral valley phonon and an intervalley exciton in
monolayer $\mathrm{WSe_{2}}$ can lead to a long exciton lifetime maintaining
the valley polarization, which is important for valley-excitonics [12]. Chen
et al. observed the entanglement of chiral phonon modes in monolayer
$\mathrm{WSe_{2}}$ and single photons emitted from an embedded quantum dot in
the material, which promises to reveal new avenues for phonon-driven
entanglement of quantum dots [14].
Besides monolayer $\mathrm{WSe_{2}}$, chiral phonons have been found in other
materials [15, 16, 17, 18, 19]. A particularly promising platform for
observing and manipulating chiral phonons are twisted bilayers of two-
dimensional (2D) materials. Since the discovery of flat electronic bands in
twisted bilayer graphene [20], such moiré materials have emerged as rich
systems to investigate the properties of correlated electrons, excitons and
phonons [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37,
38, 39, 40, 41, 42]. For example, at small twist angles phonon properties are
significantly modified as a consequence of large atomic reconstructions [43,
38, 44] which result in the localization of incipient phason modes [45] and
optical phonon modes [45, 46, 47, 48, 49, 50, 51]. To date, however, the
chirality of phonons in twisted bilayer systems has not been studied.
In this Letter, we study the chirality of phonon modes in moiré materials.
Focusing on twisted bilayer $\mathrm{WSe_{2}}$, we first demonstrate
qualitatively different phonon chiralities for twist angles close to
$0^{\circ}$ and close to $60^{\circ}$ and explain these differences by
analyzing the folding of the monolayer $K$ and $K^{\prime}$-valleys into the
Brillouin zone of the twisted bilayer. For twist angles near $60^{\circ}$, we
find two sets of emergent chiral phonons modes which originate from symmetry
breaking at the moiré scale. Finally, we discover flat chiral phonon bands in
the energy gap between acoustic and optical modes without the requirement of a
magic angle. Such flat phonon bands have recently been reported in
metamaterials [52, 53, 54, 55], but not yet in moiré systems.
Methods. For twist angles near $0\degree$ ($\theta$) or $60\degree$
($60^{\circ}-\theta$), the size of the moiré unit cell becomes very large
making standard first-principles approaches for calculating phonon properties
unfeasible. We therefore use simpler models for interatomic interactions.
Specifically, intralayer interactions in $\mathrm{WSe_{2}}$ and
$\mathrm{MoSe_{2}}$ are described using a Stillinger-Weber potential, which
has been demonstrated to give accurate results for monolayers [56, 57]. For
interlayer interactions, we use the Kolmogorov-Crespi potential that can
correctly reproduce the interlayer binding energy landscape obtained using
first-principles calculations [58]. The relaxed atomic positions of our
twisted bilayer systems are determined using the implementation of these
potentials within the LAMMPS code [59]. Phonon frequencies and polarization
vectors are obtained by diagonalizing the dynamical matrix using a modified
version of the PHONOPY code [60] (see Supplementary Information (SI), Sec. I,
for additional details). For monolayer graphene, we use the REBO potential
[61].
Following Zhang and Niu [5], we define the chirality of a phonon with crystal
momentum $\mathbf{q}$ and band index $n$ along the $z$-direction (assuming
that the 2D material lies in the $x$-$y$ plane) as
$\begin{split}S^{z}_{n{\bf q}}=&\bra{\epsilon_{n{\bf
q}}}\widehat{S}^{z}\ket{\epsilon_{n{\bf q}}}\\\ =&\bra{\epsilon_{n{\bf
q}}}\sum_{j=1}^{N}\Big{[}\ket{R_{j}}\bra{R_{j}}-\ket{L_{j}}\bra{L_{j}}\Big{]}\ket{\epsilon_{n{\bf
q}}},\end{split}$ (1)
where $\ket{\epsilon_{n{\bf q}}}$ is the normalized polarization vector,
$\widehat{S}^{z}$ denotes the phonon circular polarization operator and $N$ is
the number of atoms in the moiré unit cell. $\ket{R_{j}}$ and $\ket{L_{j}}$
denote the right and left circularly polarized basis vectors for atom $j$. For
example, for the first two atoms these basis vectors are given by
$|R_{1}\rangle=1/\sqrt{2}\times{(1,i,0,0,\ldots)^{\mathrm{T}}}$,
$|L_{1}\rangle=1/\sqrt{2}\times{(1,-i,0,0,\ldots)^{\mathrm{T}}}$,
$|R_{2}\rangle=1/\sqrt{2}\times{(0,0,1,i,0,0,\ldots)^{\mathrm{T}}}$, and
$|L_{2}\rangle=1/\sqrt{2}\times{(0,0,1,-i,0,0,\ldots)^{\mathrm{T}}}$. The
phonon mode is linearly polarized when $S_{n\bf{q}}^{z}=0$, circularly
polarized when $S_{n\bf{q}}^{z}=\pm 1$ and elliptically polarized when
$0<|S_{n\bf{q}}^{z}|<1$. We refer to both the circularly and elliptically
polarized modes as chiral phonons. Note that for systems with time-reversal
symmetry, we have $S^{z}_{n{\bf q}}=-S^{z}_{n-{\bf q}}$, whereas systems with
inversion-symmetry obey $S^{z}_{n{\bf q}}=S^{z}_{n-{\bf q}}$. Therefore, the
existence of non-degenerate chiral phonon modes requires the breaking of one
of these symmetries [5, 6].
As a test, we first study the chirality of phonons in monolayer graphene
(which has both time-reversal and inversion symmetry) and monolayer
$\mathrm{WSe_{2}}$ (which is not inversion symmetric). The results are shown
in Sec. II of the SI [62]. As expected, all graphene phonons are achiral. For
$\mathrm{WSe_{2}}$, we find multiple non-degenerate chiral phonon modes near
the $K$ and $K^{\prime}$ valleys, consistent with previous first-principles
calculations [6], and the signs of the chiralities in the two valleys are
opposite.
Figure 1: (a),(b): Chiral phonons in twisted bilayer $\mathrm{WSe_{2}}$ at
twist angles of $21.78^{\circ}$ and $38.22^{\circ}$, respectively. Dark solid
lines represent phonon bands and blue and red color indicates the associated
chirality. (c),(d): Red circles denote the unfolded $K_{\mathrm{M}}$ point of
the moiré BZ into the monolayer BZ for $21.78^{\circ}$ and $38.22^{\circ}$,
respectively. The green and blue solid lines represent the unrotated and
rotated monolayer BZs, whereas the black solid line represents the moiré BZ.
Chiral valley phonons in twisted bilayer ${WSe_{2}}$. The naturally occuring
bilayer $\mathrm{WSe_{2}}$ has 2H stacking and is inversion symmetric [63].
The untwisted bilayer, therefore, does not exhibit chiral valley phonons.
However, the introduction of a twist between the layers results in the
emergence of chiral phonons, as demonstrated in Figs. 1(a) and (b), which show
the phonon chirality $S^{z}$ of each phonon mode at twist angles of
$21.78^{\circ}$ and $38.22^{\circ}$, respectively. At $21.78^{\circ}$, all
phonons near the $K_{\mathrm{M}}$ and $K^{\prime}_{\mathrm{M}}$ valleys are
achiral (with the subscript $\mathrm{M}$ denoting that these k-points belong
to the moiré Brillouin zone), but there are some chiral phonons between
$\Gamma_{\mathrm{M}}$ and $M_{\mathrm{M}}$. In contrast, we find several
chiral phonon modes in the $K_{\mathrm{M}}$/$K^{\prime}_{\mathrm{M}}$ valleys
for the $38.22^{\circ}$ twist angle (but none between $\Gamma_{\mathrm{M}}$
and $M_{\mathrm{M}}$). Qualitatively similar results are found for other twist
angles near $0\degree$ and $60\degree$, respectively.
The different behaviour of the $K_{\mathrm{M}}$/$K^{\prime}_{\mathrm{M}}$
valley phonons for twist angles near $0^{\circ}$ and near $60^{\circ}$ can be
understood by analyzing the folding of the monolayer phonon modes into the
smaller moiré Brillouin zone. For the untwisted system (i.e.,
$\theta=0^{\circ}$), the $K$ point of the top layer (denoted $K_{\mathrm{t}}$)
coincides with the $K$ point of the bottom layer (denoted $K_{\mathrm{b}}$)
and similarly $K^{\prime}_{\mathrm{t}}$ and $K^{\prime}_{\mathrm{b}}$
coincide. When instead a small, but finite twist angle is considered, the BZs
of the two layers are rotated relative to each other and a smaller moiré BZ is
created, see Fig. 1(c). To understand which crystal momenta $\mathbf{q}_{i}$
of the monolayer BZs fold onto a specific point $\mathbf{Q}_{\mathrm{M}}$ in
the moiré BZ, we “unfold” $\mathbf{Q}_{\mathrm{M}}$ by adding all possible
reciprocal lattice vectors ${\bf G}_{i}$ corresponding to the moiré crystal
according to [64]
${\bf q}_{i}={\bf Q}_{\mathrm{M}}+{\bf G}_{i}.$ (2)
Figure 1(c) shows the set of points in the monolayer BZs that result from
unfolding $K_{\mathrm{M}}$ for a twist angle of $21.78^{\circ}$. It can be
seen that both $K^{\prime}_{\mathrm{t}}$ and $K_{\mathrm{b}}$ fold onto
$K_{\mathrm{M}}$. Since the chirality of the phonons at
$K^{\prime}_{\mathrm{t}}$ and $K_{\mathrm{b}}$ have equal magnitudes, but
opposite signs, the phonons of the twisted system have a vanishing chirality.
Instead, for the system with $60^{\circ}$ twist angle, $K_{\mathrm{t}}$
coincides with $K^{\prime}_{\mathrm{b}}$ and similarly
$K^{\prime}_{\mathrm{t}}$ and $K_{\mathrm{b}}$ coincide. Fig. 1(d) shows that
for a twist angle near $60^{\circ}$, $K_{\mathrm{M}}$ unfolds onto
$K^{\prime}_{\mathrm{t}}$ and $K^{\prime}_{\mathrm{b}}$ which have phonons
with the same chirality sign. As a consequence, the phonons of the twisted
bilayer exhibit a non-vanishing chirality at the $K_{\mathrm{M}}$ valley (and
also for the $K^{\prime}_{\mathrm{M}}$ valley).
Figure 2: (a) Chiral $K$-valley phonon modes of monolayer $\mathrm{WSe_{2}}$
(first column) and twisted bilayer $\mathrm{WSe_{2}}$ at different twist
angles (columns labelled $\beta$). For comparison, results are also shown for
twisted bilayers without interlayer interactions (columns labelled $\alpha$).
The colour of the symbols indicates the chirality of the phonon modes. The
emergent chiral modes are highlighted by curly brackets. (b)-(e) Absolute
value of the in-plane components of the normalized polarization vector for
some of the chiral modes for a twist angle of $58^{\circ}$. In (b) and (c),
the polarization vector of W atoms in the top layer is used, In (d) and (e),
the polarization vector of Se atoms in the top layer is shown. Different
stacking locations and associated chiralities within the moiré unit cell,
which is denoted by dashed lines, are indicated.
Figure 2(a) shows the chiralities of phonons in the $K_{\mathrm{M}}$-valley as
a function of twist angle near $60\degree$ (columns labelled $\beta$) and
compares them to the $\mathrm{WSe_{2}}$ monolayer result (the first column)
and also to the results of _decoupled_ twisted bilayers at the same twist
angles (columns labelled $\alpha$), in which no interlayer interactions
between the layers are considered. Note that no atomic relaxations occur in
decoupled twisted bilauers. For a twist angle of $38.22\degree$, the phonon
chiralities of the twisted bilayer are very similar to those of the monolayer
and also to those of the decoupled system. As the twist angle approaches
$60\degree$, two novel sets of chiral valley phonons emerge [indicated by the
curly brackets in Fig. 2(a)] in the twisted system which are absent in the
monolayer and in the decoupled bilayer: one set with frequencies near 10
$\mathrm{cm^{-1}}$ and the other with frequencies near 160 $\mathrm{cm^{-1}}$.
The frequencies of the first set soften as the twist angle approaches
$60\degree$.
To understand the origin of these emergent chiral valley phonons, we plot the
magnitude of the polarization vector in the moiré unit cell for 58∘, see Figs.
2(b)-(e). Figs. 2(b)-(c) show results for the two emergent valley phonons with
the largest chiralities from the low-energy set (near
$10~{}\mathrm{cm^{-1}}$). In particular, the mode with
$6.2~{}\mathrm{cm^{-1}}$ has a chirality of $-0.56$ and is localized in the
$\mathrm{B^{Se/Se}}$ stacking regions while the one with
$9.6~{}\mathrm{cm^{-1}}$ has a chirality of 0.35 and is localized in the
$\mathrm{B^{W/W}}$ stacking regions. Analyzing the contributions to the total
chirality of these modes, we find that both layers and both W and Se atoms
contribute. Figs. 2(d)-(e) show the polarization vectors of the two valley
phonons belonging to the second emergent set (with frequencies near
$160~{}\mathrm{cm^{-1}}$) that have the largest chiralities. The mode with
frequency $159.2~{}\mathrm{cm^{-1}}$ is localized in the $\mathrm{B^{Se/Se}}$
region while the one with frequency $159.4~{}\mathrm{cm^{-1}}$ is
approximately localized around the $\mathrm{B^{W/W}}$ region. Interestingly,
only the Se atoms contribute to the total chiralities of these modes.
The emergence of these novel chiral valley phonons can be explained by an
inversion symmetry breaking on the moiré scale that occurs as the twist angle
approaches $60\degree$. Near this twist angle, the moiré unit cell contains
three high-symmetry stackings: 2H, $\mathrm{B^{W/W}}$, and
$\mathrm{B^{Se/Se}}$. Among these stackings 2H is most stable and for twist
angles larger than $\sim~{}55\degree$ atomic relaxations result in a
significant increase in the regions with 2H stacking as well as the formation
of domain walls [43] (see SI, Sec. III). It can be shown [43] that this system
can be mapped onto an equivalent system of two particles (with one particle
representing $\mathrm{B^{W/W}}$ regions and the other $\mathrm{B^{Se/Se}}$
regions) that occupy different sublattices of a triangular lattice and are
connected by springs. Since the energies of $\mathrm{B^{W/W}}$ and
$\mathrm{B^{Se/Se}}$ regions are different, the two particles have different
masses and therefore the effective system lacks inversion symmetry. This in
turn results in the emergence of novel chiral phonon modes. In contrast, the
twisted bilayer near $0\degree$ features three high-symmetry stackings: AA,
$\mathrm{B^{Se/W}}$, and $\mathrm{B^{W/Se}}$. Among these stackings
$\mathrm{B^{Se/W}}$ and $\mathrm{B^{W/Se}}$ are the most stable stackings.
This system can be mapped onto a set of particles (representing $\mathrm{AA}$
regions) which occupy the sites of a triangular lattice and are connected by
springs. This effective system exhibits inversion symmetry and therefore no
emergent chiral phonon modes are observed.
Our analysis shows that the polarization vectors of chiral phonons near
$0^{\circ}$ and $60^{\circ}$ are qualitatively different from the those of the
monolayer phonons. This suggests that commonly used approaches for calculating
phonon properties in twisted bilayers that are based on a zone folding
procedure of monolayer results are unreliable [65, 66].
Figure 3: (a) The phonon band structure of twisted bilayer $\mathrm{WSe_{2}}$
at $58^{\circ}$ in the vicinity of the energy gap between acoustic and optical
modes. (b) Chirality $S^{z}$ (b) and band width (c) of flat acoustic
($\mathrm{A1^{\prime}}$) and optical ($\mathrm{O1^{\prime}}$) phonon bands as
function of twist angle near $60^{\circ}$. (d) and (e): Polarization vectors
of $\mathrm{A1^{\prime}}$ and $\mathrm{O1^{\prime}}$ modes at a twist angle of
$58^{\circ}$. Different stackings are indicated by colored symbols and the
moiré unit cell is outlined using dashed gray lines. In (d), the displacements
of W atoms in the top layer is shown, while in (e) that of Se atoms in the top
layer was used.
Phononic band flattening in twisted bilayer $\mathrm{WSe_{2}}$. Figure 3(a)
demonstrates the emergence of flat phonon bands in the vicinity of the energy
gap between acoustic and optical modes for a $58\degree$ twisted bilayer. This
phononic band gap is inherited from the monolayer which has an energy gap of
17.9 $\mathrm{cm^{-1}}$ between the acoustic and optical phonon modes with the
highest-energy acoustic mode being located at $K$ and the lowest-energy
optical mode at the $\Gamma$ point of the BZ. The highest-energy acoustic mode
at $K$ is non-degenerate and chiral with $S^{z}=-0.5$, while the lowest-energy
optical modes at $\Gamma$ are degenerate and achiral. Fig. 3(b) shows that the
chirality associated with the highest-energy acoustic mode at $K_{\mathrm{M}}$
(denoted as $\mathrm{A1^{\prime}}$) changes dramatically and becomes achiral
as the twist angle approaches $60^{\circ}$. This change in chirality is caused
by the localization of the phonon mode in regions with inversion-symmetric 2H
stacking (see discussion below). In contrast, the lowest-energy optical mode
at $K_{\mathrm{M}}$ (denoted as $\mathrm{O1^{\prime}}$) becomes chiral as the
twist angle approaches $60^{\circ}$. Fig. 3(c) shows the bandwidths of the A1′
and O1′ modes and demonstrates that these bands become extremely flat as the
twist angle approaches $60^{\circ}$ and the size of the moiré cell increases.
Similar to the electronic bandwidths in twisted transition metal
dichalcogenide bilayers, the flattening does not occur at a specific magic
angle, unlike the case of twisted bilayer graphene [20]. Interestingly, the
widths of the flat phonon bands are significantly smaller than the width of
the flat electron bands at the same twist angle. For example, at a twist angle
of 56.5∘ of the twisted bilayer of TMDs, the width of the electronic valence
band is 5 meV [27] and the width of the largest acoustic phonon band is 0.1
meV.
The flattening of the phonon bands is a consequence of both zone folding and
the localization of phonons in real space (see SI, Sec. IV for details). Figs.
3(d) and (e) show the polarization vectors (at $K_{\mathrm{M}}$) of the A1′
and O1′ modes for a twist angle of $58^{\circ}$. The A1′ mode is localized in
regions of 2H stacking, while O1′ is localized in $\mathrm{B^{Se/Se}}$ regions
as well as the domain walls. It is interesting to note that the localization
of the highest acoustic phonons is strikingly similar to that of the highest
electronic valence states in these systems [67]. Similar band flattening is
also observed for twist angles near $0^{\circ}$ (see SI[62], Sec. IV), where
the highest-energy acoustic mode is localized in the
$\mathrm{B^{W/Se}/B^{Se/W}}$ stacking region and the lowest-energy optical
mode is localized in regions of $\mathrm{AA}$ stacking and domain walls.
The distinct localization of the acoustic and optical modes can be explained
by analyzing the frequencies of these modes in the various untwisted bilayers
with high-symmetry stacking arrangements. For example, the A1′ mode originates
from the $K$ point of the monolayer BZ. For the untwisted bilayers, the phonon
frequencies at $K$ are 141.08 cm-1 for $\mathrm{B^{Se/Se}}$ stacking, 141.1
cm-1 for $\mathrm{B^{W/W}}$ stacking and 141.8 cm-1 for $\mathrm{2H}$
stacking. Therefore, the highest-energy acoustic mode localizes on the
$\mathrm{2H}$ regions as the moiré unit cell increases in size.
Chiral valley phonons in strain-engineered moiré materials and
heterostructures. Besides twisting, there are two approaches to create a moiré
pattern in bilayer systems: (i) by applying strain to only one of two
identical layers and (ii) by stacking two different 2D materials on top of
each other. We investigate the existence of chiral valley phonons in both
cases. In contrast to the twisted bilayer at small twist angles, we find that
the strain-engineered bilayer exhibits chiral phonons in the $K_{\mathrm{M}}$
valley (see Sec. V of the SI for a summary of these modes and more details
[62]). In addition, emergent chiral phonon modes with frequencies of
$\sim~{}10~{}\mathrm{cm^{-1}}$ and $\sim~{}156~{}\mathrm{cm^{-1}}$ are found.
Finally, we also observe emergent chiral phonon modes in a
$\mathrm{WSe_{2}/MoSe_{2}}$ heterostructure at a twist angle of $3.14^{\circ}$
(see SI [62], Sec. V).
Summary. In this paper, we have demonstrated the presence of chiral phonons in
twisted bilayer WSe2 as well as other moiré materials and studied their
properties. We have found that the phonon chirality depends on the twist angle
with systems near $0^{\circ}$ exhibiting qualitatively different chiralities
than systems near $60^{\circ}$. Very close to $60^{\circ}$, we observe
emergent chiral modes as well as flat phonon bands. The predicted chiral
properties of phonons in twisted bilayer materials can be measured with
helicity-resolved Raman spectroscopy [15, 68]. While the resolution of such
techniques is typically not high enough to access individual modes in systems
near 0∘ and 60∘, they can potentially measure the k-point and helicity-
resolved phonon density of states (see SI, Sec. VII). Future work should
investigate the scattering of chiral phonons with other quasiparticles, such
as electrons, excitons, and photons and the effects of such scattering
processes on the electronic and optical properties of moiré materials.
###### Acknowledgements.
Acknowledgments This project has received funding from the European Union’s
Horizon 2020 research and innovation programme under the Marie Skłodowska-
Curie grant agreement No 101028468. The authors acknowledge support from the
Thomas Young Centre under grant TYC-101, and discussions with Rup Chowdhury,
Nikita Tepliakov and Saurabh Srivastav.
Supplementary Information (SI) :
Chiral valley phonons and flat phonon bands in moiré materials
## I I: Simulation details
### I.1 Generation of structures
#### Moiré pattern due to twist:
The commensurate twist angles that give rise to the least-area moiré unit-cell
for twisted bilayer of $\mathrm{WSe_{2}}$ are given by,
$\cos(\theta_{i})=\frac{3i^{2}+3i+1/2}{3i^{2}+3i+1}$ (3)
for an integer $i$ [69]. One of the set of moiré lattice vectors are- ${\bf
A_{1}}=i{\bf a_{1}}+(i+1){\bf a_{2}}$ and ${\bf A_{2}}=-(i+1){\bf
a_{1}}+(2i+1){\bf a_{2}}$ with ${\bf a_{1}}=(1/2,\sqrt{3}/2)a_{0}$ and ${\bf
a_{2}}=(-1/2,\sqrt{3}/2)a_{0}$, where $a_{0}$ is the lattice constant of singe
layer $\mathrm{WSe_{2}}$. For twist angles ($\theta_{i}$) close to
$0^{\circ}$, we construct the moiré unit-cell starting from AA stacking using
TWISTER [70]. The high-symmetry stackings present in the moiré patterns are-
$\mathrm{AA}$ (with W and Se of the top layer are directly above W and Se of
bottom layer), $\mathrm{B^{W/Se}}$ (Bernal stacking with W of the top layer
directly above Se of bottom layer), and $\mathrm{B^{Se/W}}$ (Bernal stacking
with Se of the top layer directly above W of bottom layer). These stackings
are sometimes referred to as AA, AB, and BA, as well. For twist angles
($60^{\circ}-\theta_{i}$) close to $60^{\circ}$, we construct the moiré unit-
cell starting from 2H stacking. The high-symmetry stackings present in the
moiré patterns are- $\mathrm{2H}$ (with W and Se of the top layer are directly
above Se and W of bottom layer), $\mathrm{B^{Se/Se}}$ (Bernal stacking with Se
of the top layer directly above Se of bottom layer), and $\mathrm{B^{W/W}}$
(Bernal stacking with W of the top layer directly above W of bottom layer).
These stackings are sometimes referred to as AA′, A′B, and AB′, respectively.
Twist angles | Number of atoms | Moiré length (in Å)
---|---|---
21.78∘ | 42 | 8.8
9.4∘ | 222 | 20.4
7.3∘ | 366 | 26.1
6∘ | 546 | 31.9
4.41∘ | 1626 | 54.6
3.14∘ | 1986 | 60.4
2∘ | 4902 | 94.9
38.22∘ | 42 | 8.8
50.6∘ | 222 | 20.4
52.7∘ | 366 | 26.1
54∘ | 546 | 31.9
55.59∘ | 1626 | 54.6
56.86∘ | 1986 | 60.4
58∘ | 4902 | 94.9
58.31∘ | 6846 | 113.3
Table 1: Moiré patterns of twisted bilayer $\mathrm{WSe_{2}}$ studied in this
work.
#### Moiré pattern due to strain
We use the unit-cell lattice vectors of single layer of $\mathrm{WSe_{2}}$ as
${\bf a_{1}}=a_{0}(1,0,0)$, ${\bf a_{2}}=a_{0}(1/2,\sqrt{3}/2,0)$, and ${\bf
a_{3}}=a_{0}(0,0,30)$, where $a_{0}$ is the lattice constant. We can create
the moiré patterns in two ways: by straining only one of the layer of
$\mathrm{WSe_{2}}$, and by straining both the layers. We use the first
approach in this work. We apply a tensile strain to the bottom layer. We
discuss the details of this approach below. The periodicity of the strained
bilayer is computed in the following manner:
$\begin{bmatrix}m&0\\\ 0&m\end{bmatrix}\begin{pmatrix}{\bf a_{1}}\\\ {\bf
a_{2}}\end{pmatrix}=\begin{bmatrix}n&0\\\ 0&n\end{bmatrix}\begin{pmatrix}{\bf
a_{1}^{b}}\\\ {\bf a_{2}^{b}}\end{pmatrix}$ (4)
In the case of 3.3 % biaxial strain, we use $m=30$, and $n=29$ starting from
AA stacking configuration. The moiré lattice constant of the strained bilayer
is 98.9 Å and contains 5223 atoms.
#### Moiré heterostructure
We replace the top layer of bilayer of $\mathrm{WSe_{2}}$ by
$\mathrm{MoSe_{2}}$ and construct moiré lattice by rotating the layer. Since
the experimental lattice constants of these two materials are similar, we
simply replace one of the $\mathrm{WSe_{2}}$ layer by $\mathrm{MoSe_{2}}$ in
the twisted bilayer $\mathrm{WSe_{2}}$ and create the moiré heterostructure.
### I.2 Structural relaxations
The moiré patterns are relaxed using the LAMMPS package with the Stillinger-
Weber [56], and Kolmogorov-Crespi [58] potentials to capture the intralayer
and interlayer interactions of the twisted bilayer of $\mathrm{WSe_{2}}$,
respectively. The Kolmogorov-Crespi parameters used in this work can correctly
reproduce the interlayer binding energy landscape, obtained using density
functional theory. The atomic relaxations produced using these parameters are
in excellent agreement with relaxations performed using density functional
theory. The intralayer Stillinger-Weber potential quite accurately captures
the full phonon dispersion, in particular the acoustic modes. The accurate
potentials used in our simulations are expected to capture the phonon
renormalization of the moiré materials. At first, we relax the simulation box
to ensure the in-plane pressure is as small as possible. Next, we relax the
atoms within a fixed simulation box with the force tolerance of $10^{-5}$ eV/Å
for any atom along any direction.
### I.3 Force constants calculations
For monolayer $\mathrm{WSe_{2}}$, we use a $6\times 6\times 1$ supercell to
compute the force constants. For large twist angles of twisted bilayer
$\mathrm{WSe_{2}}$, we use a $2\times 2\times 1$ supercell to compute the
force constants. The moiré patterns contain thousands of atoms for small twist
angles. Therefore, we use $1\times 1\times 1$ supercell to compute the force-
constants.
## II II : Chiral phonons in 2D materials
### II.1 Absence of chiral valley-phonons in graphene
Figure 4: Chirality of phonons in single layer of graphene. The solid lines represent the phonon frequencies, and hexagons (absent in the figure as $S^{z}=0$) represent the computed $S_{z}$ values after summing over all the basis atoms. There are no non-degenerate chiral modes in single layer of graphene due to inversion symmetry. $\omega$ | $S^{z}$ | $S^{z}_{C1}$ | $S^{z}_{C{2}}$
---|---|---|---
(in $\mathrm{cm^{-1}}$) | | |
505.2 | 0.0 | 0.0 | 0.0
505.2 | 0.0 | 0.0 | 0.0
821.1 | 0.0 | 0.5 | -0.5
1201.5 | 0.0 | -0.5 | 0.5
1201.5 | 0.0 | -0.5 | 0.5
1724.5 | 0.0 | 0.5 | -0.5
Table 2: Phonon modes, total chirality, and chirality of the each types of
atoms in the case of single-layer of graphene. Within the unit-cell, two atoms
can have opposite chirality, but the total chirality is 0.
$\omega=505.2$ cm-1 $\omega=505.2$ cm-1 $\omega=821.1$ cm-1 $\omega=1201.5$
cm-1 $\omega=1201.5$ cm-1 $\omega=1724.5$ cm-1 Eigenvectors associated with
phonon modes in grapheneThe Blue circles denote the Carbon atoms
### II.2 Presence of chiral non-degenerate valley phonons in monolayer
$\mathrm{WSe_{2}}$
Figure 5: Chirality of phonons in monolayer of $\mathrm{WSe_{2}}$. The solid lines represent the phonon frequencies, and hexagons represent the computed $S_{z}$ values after summing over all the basis atoms. There are multiple non-degenerate chiral modes in monolayer of $\mathrm{WSe_{2}}$ due to inversion symmetry breaking. $\omega$ | $S^{z}$ | $S^{z}_{W1}$ | $S^{z}_{Se{2}}$ | $S^{z}_{Se{3}}$
---|---|---|---|---
(in $\mathrm{cm^{-1}}$) | | | |
118.9 | -0.38 | 0.0 | -0.19 | -0.19
121.2 | 0.76 | 0.76 | 0.0 | 0.0
141.4 | -0.49 | -0.75 | 0.13 | 0.13
200.66 | 0.0 | 0.0 | 0.0 | 0.0
212.06 | -1.0 | 0.0 | -0.5 | -0.5
216.52 | 1.0 | 0.0 | 0.5 | 0.5
240.28 | 0.49 | -0.25 | 0.37 | 0.37
249.2 | -0.62 | 0.0 | -0.31 | -0.31
255.38 | 0.24 | 0.24 | 0.0 | 0.0
Table 3: Chirality associated with all the valley phonon modes (at $K$) in
single layer of $\mathrm{WSe_{2}}$.
$\omega=118.9$ cm-1 $\omega=121.2$ cm-1 $\omega=141.4$ cm-1 $\omega=200.6$
cm-1 $\omega=212$ cm-1 $\omega=216.5$ cm-1 $\omega=240.3$ cm-1 $\omega=249.2$
cm-1 $\omega=255.4$ cm-1 Eigenvectors associated with phonon modes in single
layer $\mathrm{WSe_{2}}$The Blue and red circles denote the Se and W atoms,
respectively.
## III III : Interlayer separation for twist angles close to $60^{\circ}$ and
$0^{\circ}$
Figure 6: (a),(b): Significant structural relaxations and inversion symmetry
breaking at the moiré scale illustrated using interlayer separation for twist
angles of $58^{\circ}$ and $2^{\circ}$, respectively. The associated color bar
denotes the magnitude of interlayer separation and is in Å. The moiré unit
cell and stackings are marked.
## IV IV : Flat phonon bands
Figure 7: (a), (b) Evidence of band-flattening for a twist angle of 3.14∘ and $56.86^{\circ}$ twisted bilayer $\mathrm{WSe_{2}}$ near the acoustic-optic phonon band gap. The blue lines denote the calculations with twisted bilayers and the red dashed lines denote calculations with decoupled twisted bilayers. Importantly, the band flattens and degeneracies in the band-structure of the decoupled systems are significantly reduced. (c),(d) ((e),(f)) Absolute value of the polarization vectors at $\Gamma_{\mathrm{M}}$ of $\mathrm{A1}$ and $\mathrm{O1}$ modes at a twist angle of $3.14^{\circ}$ ($56.86^{\circ}$). Different stackings are indicated by colored symbols and the moiré unit cell is outlined using dashed gray lines. In (c),(e) the displacements of W atoms in the top layer is shown, while in (d),(f) that of Se atoms in the top layer was used. (g) The band-flattening is indicated by the reduction of band-width with twist angles. | Bandwidth in cm-1
---|---
| Twisted bilayer | Decoupled twisted bilayer
Twist angle | A1 | O1 | A1 | O1
55.59∘ | 1.029 | 0.185 | 1.321 | 0.327
56.86∘ | 0.427 | 0.069 | 0.672 | 0.110
Table 4: Comparison of bandwidths in the twisted bilayers that include all the
relaxation effects and in the decoupled twisted bilayers due to zone-folding
for $55.59^{\circ}$ and $56.86^{\circ}$ twisted bilayers. Note that no atomic
relaxations occur in decoupled twisted bilayers.
## V V : Chiral valley phonons in strain engineered moiré and
$\mathrm{MoSe_{2}/WSe_{2}}$ heterostructure
Figure 8: (a),(b)-Chiralities at K point for all the phonon modes in 3.3$\%$
biaxially strained moiré of $\mathrm{WSe_{2}}$ and the
$\mathrm{MoSe_{2}/WSe_{2}}$ heterostructure. The left column of (a) denote the
non-interacting and the right column of (a) interacting bilayer calculations.
Similarly, we show the results for (b). The emergent chiral modes are
highlighted by curly brackets.
We apply a bi-axial tensile strain to the bottom layer of an AA stacked
bilayer $\mathrm{WSe_{2}}$. The resulting strained structure contains three
unique high-symmetry stackings: $\mathrm{A\widetilde{A}}$,
$\mathrm{B^{W/\widetilde{Se}}}$ and $\mathrm{B^{Se/\widetilde{W}}}$, where the
tilde indicates a sublattice of the strained layer. These stackings are
distinct from those of the twisted bilayer near $0^{\circ}$ as they break
layer symmetry.
## VI VI: Achiral phonon modes at $K/K^{\prime}$ for twist angles close to
$0^{\circ}$
Figure 9: Chiral phonons in twisted bilayer $\mathrm{WSe_{2}}$ at twist angle
of (a) $4.41^{\circ}$, (b) $55.59^{\circ}$, (c) $3.14^{\circ}$ and (d)
$56.86^{\circ}$). Black solid circles represent phonon bands (computed at a
few discrete points) and blue and red color indicates the associated chirality
(computed at the same set of discrete points). Moreover, the size of the
colored circles are proportional to absolute value of the chirality. As this
figure is somewhat difficult to interpret, we have also directly inspected the
computed chirality values at $\mathrm{K_{M}}$ and find that there are non-
degenerate chiral modes for twist angles close to 60∘ but none for twist
abgles close to 0∘.
## VII VII: Chirality resolved and total density of states
Figure 10: (a),(b),(c): Total and chirality resolved density of states (DOS).
We have used $36\times 36\times 1$, $12\times 12\times 1$, and $3\times
3\times 1$ grids to compute the phonon DOS for monolayer $\mathrm{WSe_{2}}$,
$38.22^{\circ}$ twisted bilayer $\mathrm{WSe_{2}}$, and $55.59^{\circ}$
twisted bilayer $\mathrm{WSe_{2}}$, respectively. The delta function in DOS
calculation has been replaced by a Gaussian with a standard deviation of 4.8
cm-1. Figure 11: (a),(b),(c): $k$-point resolved density of states (DOS). We
have used $36\times 36\times 1$, $12\times 12\times 1$, and $3\times 3\times
1$ k-point grids to compute the phonon DOS for monolayer $\mathrm{WSe_{2}}$,
$38.22^{\circ}$ twisted bilayer $\mathrm{WSe_{2}}$, and $55.59^{\circ}$
twisted bilayer $\mathrm{WSe_{2}}$, respectively. A Gaussian broadening with a
standard deviation of 4.8 cm-1 has been used.
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|
# SYGMA: System for Generalizable Modular Question Answering Over Knowledge
Bases
Sumit Neelam, Udit Sharma, Hima Karanam, Shajith Ikbal, Pavan Kapanipathi
Ibrahim Abdelaziz, Nandana Mihindukulasooriya, Young-Suk Lee, Santosh
Srivastava
Cezar Pendus, Saswati Dana, Dinesh Garg, Achille Fokoue, G P Shrivatsa Bhargav
Dinesh Khandelwal, Srinivas Ravishankar, Sairam Gurajada, Maria Chang,
Rosario Uceda-Sosa, Salim Roukos, Alexander Gray, Guilherme Lima
Ryan Riegel, Francois Luus, L Venkata Subramaniam
IBM Research
###### Abstract
Knowledge Base Question Answering (KBQA) tasks that involve complex reasoning
are emerging as an important research direction. However, most KBQA systems
struggle with generalizability, particularly on two dimensions: (a) across
multiple reasoning types where both datasets and systems have primarily
focused on multi-hop reasoning, and (b) across multiple knowledge bases, where
KBQA approaches are specifically tuned to a single knowledge base. In this
paper, we present SYGMA, a modular approach facilitating generalizability
across multiple knowledge bases and multiple reasoning types. Specifically,
SYGMA contains three high level modules: 1) KB-agnostic question understanding
module that is common across KBs 2) Rules to support additional reasoning
types and 3) KB-specific question mapping and answering module to address the
KB-specific aspects of the answer extraction. We demonstrate effectiveness of
our system by evaluating on datasets belonging to two distinct knowledge
bases, DBpedia and Wikidata. In addition, to demonstrate extensibility to
additional reasoning types we evaluate on multi-hop reasoning datasets and a
new Temporal KBQA benchmark dataset on Wikidata, named TempQA-
WD111https://github.com/IBM/tempqa-wd, introduced in this paper. We show that
our generalizable approach has better competetive performance on multiple
datasets on DBpedia and Wikidata that requires both multi-hop and temporal
reasoning.
## 1 Introduction
The goal of Knowledge Base Question Answering (KBQA) systems is to answer
natural language questions by retrieving and reasoning over facts in a
Knowledge Base (KB). KBQA has gained significant popularity both due to its
practical real-world applications and challenging research problems associated
with it (Fu et al. 2020). However, most existing datasets and research in this
area are primarily focused on single/multi-hop reasoning on a single knowledge
base (Trivedi et al. 2017; Usbeck et al. 2017; Yih et al. 2016). Consequently,
this has encouraged research in methodologies that are tuned to a restricted
set of reasoning types on a single knowledge base, in turn lacking
generalizability (Kapanipathi et al. 2021; Zou et al. 2014a; Sorokin and
Gurevych 2018a). In this work, we propose a modular approach that is
generalizable across knowledge bases and reasoning types.
Figure 1: A description of how information about Nero’s predecessor is
represented in DBpedia vs. Wikidata
Open Knowledge Bases such as DBpedia (Auer et al. 2007), Wikidata (Vrandečić
and Krötzsch 2014), and Freebase (Bollacker et al. 2008) form the basis of
many KBQA datasets. Each of these knowledge Bases, however, have different
representations. For instance: information associated to the question “Who was
roman emperor before Nero?" from DBpedia and Wikidata are shown in Figure 1.
We can see that Wikidata represents properties of facts such as replaces,
temporal and spatial with reification.222Information about facts are
explicitly mapped by making facts as primitive nodes. On the other hand,
DBpedia manages to represent it as a simple fact with the relationship to
existing entity nodes. Handling such different representations is an
unexplored challenge and requires to be addressed for a generalizable KBQA
approach.The second aspect where KBQA approaches fail to generalize is in
handling complex reasoning types such as temporal and spatial reasoning. Table
1 shows examples of questions that require different types of reasoning to
answer them. Most research have focused on multi-hop reasoning questions
(Bordes et al. 2015a; Dubey et al. 2019; Berant et al. 2013), lacking both
approaches and datasets that are adaptable to complex reasoning types.
Particularly a single approach that can answer questions with different
reasoning types.
In this paper, we present a modular approach called SYGMA (System for
Generalizable and Modular question Answering over knowledge bases), that is
built on a framework adaptable to different KB representations and reasoning
types. SYGMA introduces intermediate representations based on lambda calculus
to adapt to new reasoning types and representations. SYGMA , following the
recent advances in KBQA such as NSQA (Kapanipathi et al. 2021) and ReTraCk
(Chen et al. 2021), is a modular approach with independent Abstract Meaning
Representation, Entity Linking, and Relation Linking modules. In contrast to
NSQA which is tuned specifically to DBpedia-based KBQA datasets, SYGMA is
generalizable to multiple knowledge bases and multiple reasoning types. In
order to evaluate our generalizable approach, we consider datasets that have:
(a) DBpedia and Wikidata as knowledge bases, and (b) multi-hop and temporal as
the reasoning types.
In pursuit of this goal, we also address the lack of temporal reasoning
datasets on knowledge bases that we are considering in this work, specifically
Wikidata. We annotate and create TempQA-WD; a Wikidata-based dataset focused
on temporal reasoning. TempQA-WD is a parallel dataset to TempQuestions
dataset (Jia et al. 2018) which is a Freebase based temporal reasoning dataset
derived from three other datasets Free917 (Cai and Yates 2013), WebQuestions
(Berant et al. 2013) and ComplexQuestions (Bao et al. 2016). While the
Freebase dataset consists of question answer pairs along with temporal signal
classifications, our dataset is annotated with intermediate SPARQL queries
enabling evaluation of different modules in a KBQA pipeline.
| Category
---
Example
| Single-Hop
---
| _Who directed Titanic Movie?_
---
SPARQL: select distinct ?a where {
wd:Q44578 wdt:P57 ?a}
| Multi-hop
---
| _Which movie is directed by James Cameron_
---
_starring Leonardo DiCaprio?_
SPARQL: select distinct ?a where
{?a wdt:P57 wd:Q42574.
?a wdt:P161 wd:Q38111. }
| Temporal
---
| _Who was the US President_
---
_during cold war?_ SPARQL: in Figure 3
Table 1: Examples of Single-hop, Multi-hop and Temporal reasoning questions on
Wikidata.
In summary, the main contributions of this paper are:
* •
A modular, generalizable approach, called SYGMA, for KBQA with lambda calculus
based intermediate representations that enable adaptation to: (a) multiple
knowledge bases, specifically DBpedia and Wikidata, and (b) multiple reasoning
types such as multi-hop and temporal reasoning.
* •
A benchmark dataset, called TempQA-WD, for building temporal-aware KBQA
systems on Wikidata with parallel annotations on Freebase.
* •
Experimental results show SYGMA achieves state-of-the-art numbers on LC-QuAD
1.0, WebQSP-WD, and comparable numbers on QALD-9 dataset. We also present
baseline numbers for Simple WebQuestions and new dataset introduced in this
paper; TempQA-WD.
## 2 Related Work
Datasets | Knowledge Base | Emphasis | SPARQL | Intermediate | Natural questions
---|---|---|---|---|---
| | | | Evaluation |
QALD-9 | DBpedia | Multi-hop | ✓ | ✓ | ✓
LC-QuAD 1.0 | DBpedia | Multi-hop | ✓ | ✓ | ✗
LC-QuAD 2 | DBpedia, Wikidata | Multi-hop | ✓ | ✓ | ✗
Simple Questions | Freebase, DBpedia, Wikidata | Single-hop | ✓ | ✓ | ✓
Web Questions | Freebase | Multi-hop | ✗ | ✗ | ✓
WebQSP | Freebase, Wikidata | Multi-hop | ✓ | ✓ | ✓
Complex Web Questions | Freebase | Multi-hop | ✗ | ✗ | ✓
TempQuestions | Freebase | Temporal | ✗ | ✗ | ✓
TempQA-WD | Freebase, Wikidata | Temporal | ✓ | ✓ | ✓
Table 2: This table compares most of the KBQA datasets based on features
relevant to the work presented in this paper.
KBQA Systems: KBQA approaches can be primarily categorized into two groups:
(a) Question-to-entities: where techniques output the answer entities from the
knowledge graph ignoring the SPARQL query (Saxena, Tripathi, and Talukdar
2020; Sun et al. 2018; Vakulenko et al. 2019), and (b) semantic parsing based:
where approaches output intermediate SPARQL queries (or logic forms) that can
retrieve the answer from the knowledge base (Singh et al. 2018; Kapanipathi et
al. 2021; Zou et al. 2014b). First, in question-to-entities category,
(Vakulenko et al. 2019) leverages a message passing technique on a two hop
graph from the entities mentioned in the question to retrieve answers from the
knowledge graph. (Saxena, Tripathi, and Talukdar 2020) encodes both text and
entities, text based on language models and entities based on knowledge graph
embeddings (Trouillon et al. 2016) and shows that text can help KBQA in an
incomplete setting. In contrast to question-to-entities approach, the semantic
parsing based approaches improves interpretability and facilitates evaluation
of different modules as shown by Frankestein (Singh et al. 2018), and NSQA
(Kapanipathi et al. 2021). Both, Frankestein (Singh et al. 2018), and NSQA
(Kapanipathi et al. 2021) follow a pipeline based approach with the
differentiating factor being the use of Abstract Meaning Representation as a
starting point by NSQA. We build on top of these representational efforts and
introduce $\lambda$-expressions with the required additional functions for
adapting to new knowledge graphs and new reasoning types.
KBQA Datasets: Over the years, many question answering datasets have been
developed for KBQA, such as Free917 (Cai and Yates 2013), SimpleQuestions
(Bordes et al. 2015b), WebQuestions (Berant et al. 2013), QALD-9 (Usbeck et
al. 2017), LC-QuAD 1.0 (Trivedi et al. 2017), and LC-QuAD 2.0 (Dubey et al.
2019). In Table 2, we compare each of these datasets across the following
features: (a) underlying KB, including subsequent extensions, e.g. Wikidata
(Diefenbach et al. 2017) and DBpedia (Azmy et al. 2018) based versions of
SimpleQuestions, as well as the Wikidata subset of WebQSP (Sorokin and
Gurevych 2018a); (b) reasoning types that are emphasized in the dataset; (c)
availability of SPARQL queries, entities, and relationships for intermediate
evaluation; and (d) the use of question templates, which can often generate
noisy, unnatural questions. As Table 2 shows, our dataset is distinguished
from prior work in its emphasis on temporal reasoning, its application to both
Freebase and Wikidata, and its annotation of intermediate representations and
SPARQL queries. The most relevant KBQA dataset to our work is TempQuestions
(Jia et al. 2018), upon which we base TempQA-WD, as described in dataset
section. CRONQUESTIONS (Saxena, Chakrabarti, and Talukdar 2021) is another
temporal QA dataset proposed recently over a small subset of Wikidata. It has
template based questions targeted at temporal KG embeddings and QA.
## 3 SYGMA: System Description
Motivated by Kapanipathi et al. (2021), SYGMA is designed as a neuro-symbolic
system, avoiding the need for end-to-end training. Figure 2 shows the overall
KBQA pipeline with independent modules and their intermediate representations.
The approach is designed with the goal of generalization across multiple KBs
and reasoning types. Supporting generalization is accomplished in two stages
as shown in Figure 2, driven by an example in Figure 3. The first is a KB-
agnostic question understanding stage, which takes in a natural language
question as input and outputs an intermediate representation that can be
common across different KBs. The second is question mapping and reasoning
stage where, first, the necessary heuristics to introduce templates for new
reasoning types are applied. Next, the KB-agnostic representation is mapped to
the vocabulary of the KG to output KB-specific representation that has a
deterministic mapping to SPARQL. We use Wikidata and DBpedia to show
generalization across KBs, and use multi-hop and temporal reasoning to
evaluate across reasoning types. Before we get into the details of these
modules and intermediate representations, we give a brief description of the
Lambda calculus and the temporal functions used by SYGMA to generate its
intermediate logic representation.
Figure 2: Architecture of SYGMA that shows the pipeline with independent
modules and intermediate representations. The intermediate representations are
supported by appropriate heuristics to facilitate generalizability. Figure 3:
An illustration of the outputs at the intermediate stages of the pipeline in
SYGMA
Lambda Calculus $\lambda$-calculus, by definition, is considered the smallest
universal programming language that expresses any computable function. In
particular, we have adopted Typed $\lambda$-Calculus presented in (Zettlemoyer
and Collins 2012) which support addition of new higher order functions
necessary for handling various reasoning types. We use constants and logical
connectives like AND, OR, NEGATION and functions like argmin, argmax, count,
etc., presented in this work. Apart from this, we also added new temporal
functions to demonstrate the system’s adaptability to support new reasoning
types. For example, consider the following question and its corresponding
logical form:
Question: | When was Barack Obama born?
---|---
Logical Form: | | $\lambda$t. born(b,“Barack Obama") $\wedge$
---
interval(t, b)
Here, b is instance variable for event born(b, “Barack Obama") and interval(t,
b) finds time for the event denoted by b. Variable t is unknown which is
marked as $\lambda$ variable.
Temporal Functions: We introduce interval, overlap, before, after, teenager,
year; where interval gets time interval associated with event and overlap,
before, after are used to compare temporal events. teenager gets teenager age
interval for a person, and year return year of a date.333Although there are
many other possible temporal functions for specific English words, such as
adult and tween, we defined teenager because it was featured in the datasets
explored here.
### 3.1 KB-Agnostic Question Understanding
The modules in this stage aim at deriving logical expressions of the natural
language question that is common for all the knowledge bases. In particular,
we use the formalism of $\lambda$-calculus for logical representation, i.e.,
to map questions to their corresponding $\lambda$-expressions representing the
semantics.
#### AMR
In order to have a generic parse (logical expression) that is common across
KBs, SYGMA performs initial language understanding using Abstract Meaning
Representation (AMR) (Banarescu et al. 2013), a semantic representation
language based on PropBank. AMR encodes the meaning of a sentence into a
rooted directed acyclic graph where nodes represent concepts and edges
represent relations. We adopt an action pointer transformer architecture of
Zhou et al. (2021) for transition-based AMR parsing and self-training
technique of Lee et al. (2020) for domain adaptation to KBQA.
AMR provides generic representation that can be used for multi-hop reasoning
(Kapanipathi et al. 2021). However for temporal expressions with :time
relation, we have to augment the AMR annotations with implicit predicates that
cannot be captured by ellipsis and/or re-entrancies. An example of AMR
annotation for the question Who was US president during cold war? is given in
Figure 3. The representation encodes time edge with cold war event as sub-
event of time. Also in case of before/after constraints it explicitly captures
the constraint as part of the time edge. If there are no explicit constraints
like before/after or ordinal in the time edge, we treat them as overlap
constraints.
#### KB-Agnostic Lambda Expression
AMR is a fairly granular representation. However, for a KBQA system,
specifically on KGs such as DBpedia, Wikidata, and Freebase, such granularity
can add noise. Therefore to construct KB-agnostic $\lambda$-expressions of the
questions from their corresponding AMRs and their identified entity and
relation mentions, we use the transformational heuristics described in Table
3. The table shows the transformation from AMR frame (high level) to the
corresponding Lambda expression. The rule type describes where the rule is in
general applied to all reasoning (base) vs temporal. We handle these as
conjunction of multiple triples in the KB with some projection variable and/or
numerical operation to get the final answer. We use AMR unknown/imperative
constructs to identify the projection variables and AMR polarity/interrogative
frames for queries which fall into boolean answers or ASK questions in SPARQL.
The next set of rules present in Table 3 are few sample template rules used to
capture temporal events and constraints to help in temporal reasoning. Figure
3 gives an example question that falls into the temporal overlap rule in the
table and how the AMR constructs are used to split the events is highlighted
with nodes being bold in the table. Complete set of rules used can be found in
the supplementary material. Note that we covered temporal reasoning as an
additional form of reasoning in the current system. However, more reasoning
templates like spatial reasoning with additional operators like coordinates(),
south() …etc can be added. Table 3 shows an example template for spatial
questions such as "which states are to the south of California?".
| Rule
---
Type
AMR A = (v/frame …) | Lambda Expression L = $\psi$(v)
| Base
---
| (v/frame :arg0(v0/frame0) … :argn(vn/framen))
---
| frame(v, v0, … vn) $\land$ $\psi$(v0) $\land$ … $\psi$(vn)
---
| Base
---
| (v/frame :arg1(a/amr-unknown) … :argn(vn/framen))
---
| $\lambda$a. $\psi$(v)
---
| Numerical
---
| (v/frame :arg0(v0/frame0 :quant(a/amr-unknown)) …
---
:argn(vn/framen))
| count($\lambda$v0. $\psi$(v))
---
| Temporal
---
| (v/frame :arg0(v0/frame0) … :argn(vn/framen)
---
:time(a/amr-unknown))
| $\lambda$ev. $\psi$(v) $\land$ interval(ev, v)
---
| Temporal
---
| (v/frame :arg0(a/amr-unknown) … :argn(vn/framen)
---
:time(b/before :op1(n/nested-frame)))
| argmax($\lambda$a. $\psi$(v) , $\lambda$a. $\lambda$ev. $\psi$(n) $\land$
interval(ev, v)
---
$\land$ interval(en, n) $\land$ before(ev, en), 0, 1)
| Temporal
---
| (v/frame :arg0(a/amr-unknown) … :argn(vn/framen)
---
:time(n/nested-frame))
| $\lambda$a. $\psi$(v) $\land$ $\psi$(n) $\land$ interval(ev, v) $\land$
interval(en, n)
---
$\land$ overlap(ev, en)
| Spatial
---
| (b/be-located-at-91 :arg0(a/amr-unknown),
---
:mod(s/south) :op1(n/nested-frame))
| $\lambda$a. $\psi$(b)$\land$ $\psi$(n) $\land$ coordinate(cb, b) $\land$
coordinate(cn, n)
---
$\land$ south(cb, cn)
Table 3: AMR to Lambda Translation
### 3.2 Question Mapping and Reasoning
This is the second stage which comprises of modules that transform the KB-
agnostic $\lambda$-expression of the questions into KB-specific
$\lambda$-expression. These modules are entity linking and relationship
linking that are specific to the underlying KB along with any KB specific
rules to handle special forms of transformation or reasoning. The modules are
interchangeable to different knowledge graphs. Currently, we demonstrate the
adaptability of these modules to DBpedia and Wikidata.
#### KB-Specific Lambda Expression:
Structurally, it is similar to $\lambda$-expression with all entities and
relations mapped to KB entities and relations respectively.
_Entity Linking:_ The goal of Entity Linking is to map entity mentions as
captured in the $\lambda$-expression of the question to their corresponding
KB-specific entities. We use a recently proposed zero-shot entity linking
approach called BLINK (Wu et al. 2020). For a question where entity mentions
are already identified, bi-encoder piece of the BLINK is used to predict top-K
entities. For this prediction, we use an entity dictionary of 5.9M English
Wikipedia entities, mapped to their corresponding Wikidata and DBpedia
entities.
_Relation Linking:_ SYGMA’s Relation linking component takes in the question
text and an AMR graph as input and returns a ranked list of KG relationships.
For instance, given a question such as "Who was the US President during cold
war?" (see Fig 3) and the corresponding AMR, the goal of the relation linking
component is to find the corresponding Wikidata KB relations position held
(P39), start time (P580), end time (P582). For this task, we use state-of-the-
art AMR-based relation linking approach with models built for both DBpedia and
Wikidata (Naseem et al. 2021).
SPARQL Query: This module maps KB-specific $\lambda$-expressions onto SPARQL
queries through a deterministic approach. Each KB-specific
$\lambda$-expression construct is mapped to an equivalent SPARQL construct, as
rules given in Table 4. Each of lambda expressions contains one or more terms
such that each term $T_{i}$ is comprised of one or more predicates connected
via $\land$ or $\lor$. Translation of different predicates is present in Table
4. Some of the KB-spe cific aspects like handling reification in Wikidata are
shown in the Table. To get the time interval in case of reified events, Start
time(wdt:P580), end time(wdt:P582), or point in time(wdt:P585) connected to
intermediate statement node are used.
Type | Expression/Predicate E | SPARQL S = $\phi$(E)
---|---|---
| $\lambda$ abstraction
---
| $\lambda$x.T
---
| SELECT DISTINCT ?x WHERE { $\phi$(T) }
---
| Count expression
---
| count($\lambda$x.T)
---
| SELECT (COUNT(?x) AS ?c) WHERE { $\phi$(T) }
---
| Argmax expression
---
| argmax($\lambda$x.T1, $\lambda$x.$\lambda$y. T2, O, L)
---
| SELECT DISTINCT ?x WHERE { $\phi$(T1) $\phi$(T2)
---
} ORDER BY DESC(?yStart) LIMIT L OFFSET O
| KB Predicate
---
| IRIp(r, s$|$IRIs, o$|$IRIo)
---
| ?s$|$IRIs IRIp ?o$|$IRIo.
---
| Interval predicate
---
for reified facts
| wdt:PID(e, s$|$wd:QID1, o$|$wd:QID2)
---
$\land$ interval(ei, e)
| ?s$|$wd:QID1 p:PID ?e. ?e ps:PID ?o$|$wd:QID2.
---
?e pq:P580 ?eiStart. ?e pq:P582 ?eiEnd.
| Overlap predicate
---
| overlap(e1i e2i)
---
| FILTER(?e1iStart<=?e2iEnd && ?e2iStart<=?e1iEnd)
---
| Before predicate
---
| before(e1i, e2i)
---
| FILTER(?e1iEnd<=?e2iStart)
---
| After predicate
---
| after(e1i, e2i)
---
| FILTER(?e1iStart>=?e2iEnd)
---
Table 4: Translation of KB Specific Lambda Expression
## 4 TempQA-WD Dataset
In order to evaluate our approach on temporal reasoning, specifically on
DBpedia and Wikidata as KGs, we require a dataset to be based on one of these
two KGs. TempQuestions (Jia et al. 2018) is the first KBQA dataset intended to
focus specifically on temporal reasoning. However, TempQuestions is based on
Freebase.We adapt TempQuestions to Wikidata444Creating a parallel dataset on
two KGs (DBpedia and Wikidata) is non trivial and labor intensive. Therefore,
we opted to create a dataset based on Wikidata alone since it is the most up-
to-date KG. to create a temporal QA dataset that has three desirable
properties. First, we create a generalizable benchmark that has parallel
answer annotations on two KBs. Second, we take advantage of Wikidata’s
evolving, up-to-date knowledge. Lastly, we enhance TempQuestions with SPARQL
queries which was missing in original dataset. We also add entity, relation,
intermediate lambda expression annotations for a subset of the dataset that
are used by the SYGMA.
There has been two previous attempts at transferring Freebase-QA questions to
Wikidata; namely WebQSP-WD (Sorokin and Gurevych 2018a) and SimpleQuestions-
WD(SWQ-WD) (Diefenbach, Singh, and Maret 2017). SWQ-WD contains only single
triple questions whereas WebQSP-WD have only the final question answers that
map directly to corresponding entities in Wikidata. However, as stated
(Sorokin and Gurevych 2018a), one challenge is that not all Freebase answers
can be directly mapped to entities in Wikidata. For example, the Freebase
answer annotation for the question “When did Moscow burn?” is “1812 Fire of
Moscow”, despite the year being entangled with the event itself. In contrast,
Wikidata explicitly represents the year of this event, with an entity for
“Fire in Moscow” and an associated year of “1812”. Thus, a direct mapping
between the two answers is not possible, as it would amount to a false
equivalence between “1812 Fire of Moscow” and “1812”.
In order to address such issues, we enlisted a team of annotators to manually
create and verify SPARQL queries, ensuring not only that the SPARQL
formulation was correct, but that the answers accurately reflected the
required answer type (as in the “Fire in Moscow” example above) and the
evolving knowledge in Wikidata. Having SPARQL queries also facilitates
intermediate evaluation of the approaches that use semantic parsing to
directly generate the query or the query graph, increasing interpretability
and performance in some cases (Sorokin and Gurevych 2018a).
Dataset Details
Dataset | Size | Answer | Add’l Details
---|---|---|---
TempQuestions(Freebase) | 1271 | Freebase | A
TempQA-WD (Wikidata) | 839 | Wikidata | A + B
A Subset of TempQA-WD | 175 | Wikidata | A + B + C
Table 5: Benchmark dataset details. TempQuestions denote original dataset with
Freebase answers (Jia et al. 2018). Second row is the subset adapted to
Wikidata and the third row is the devset taken out of it. Set-A denote
TempQuestions with- {temporal signal, question type, data source}. Set-B
-{Wikidata SPARQL , answer, category}. Set-C-{AMR, $\lambda$-expression,
entities, relations, KB-specific $\lambda$-expression}
Table 5 gives details of our new benchmark dataset. We took all the questions
from TempQuestions dataset (of size $1271$) and chose a subset for which we
could find Wikidata answers. This subset has 839 questions that constitute our
new dataset, TempQA-WD. We annotated this set with their corresponding
Wikidata SPARQL queries and Wikidata answers. We also retained the Freebase
answers from the TempQuestions dataset effectively creating parallel answers
from two KBs. Additionally, we added a question complexity label to each
question, according to the reasoning complexity required to answer the
question. Details of categorization are present in supplementary material.
Within this dataset, we chose a subset of $175$ questions for detailed
annotations as described in Set-C in the table description leaving $664$
question as test set.
## 5 Evaluation
KB $\rightarrow$ | DBpedia | Wikidata
---|---|---
DataSet $\rightarrow$ | LC-QuAD 1.0 | QALD-9 | WebQSP-WD | SWQ-WD | TempQA-WD
Reasoning $\rightarrow$ | multi-hop | multi-hop | multi-hop | single-hop | Temporal
System $\downarrow$ | P | R | F1 | P | R | F1 | P | R | F1 | P | R | F1 | P | R | F1
WDAqua | 0.22 | 0.38 | 0.28 | 0.26 | 0.26 | 0.25 | - | - | - | - | - | - | - | - | -
gAnswer | - | - | - | 0.29 | 0.32 | 0.29 | - | - | - | - | - | - | - | - | -
QAMP | 0.25 | 0.50 | 0.33 | - | - | - | - | - | - | - | - | - | - | - | -
NSQA | 0.44 | 0.45 | 0.44 | 0.31 | 0.32 | 0.31 | - | - | - | - | - | - | - | - | -
GGNN | - | - | - | - | - | - | 0.27 | 0.32 | 0.26 | - | - | - | - | - | -
SYGMA | 0.47 | 0.48 | 0.47 | 0.29 | 0.30 | 0.29 | 0.32 | 0.36 | 0.31 | 0.42 | 0.55 | 0.44 | 0.32 | 0.34 | 0.32
Table 6: SYGMA’s Performance across knowledge bases and reasoning types.
Baseline numbers are taken from Kapanipathi et al. (2021) and Sorokin and
Gurevych (2018b). P-Precision, R-Recall
#### Implementation Details:
We implemented our pipeline using a Flow Compiler (Chakravarti et al. 2019)
stitching individual modules exposed as gRPC services. We defined the ANTLR
grammar to define $\lambda$-expressions. KB-Specific $\lambda$-expression to
SPARQL module is implemented in Java using Apache Jena SPARQL modules, to
create SPARQL objects and generate the final SPARQL query that is run on
target KB end point. The rest of the modules are implemented in Python and are
exposed as gRPC services.
#### Datasets:
We considered two different KBs namely DBpedia and Wikidata to evaluate our
system. We did not consider Freebase as its no longer actively maintained and
is not up-to-date. Along with the new benchmark dataset TempQA-WD introduced
in this paper, we also evaluate our baseline system on two other benchmark
datasets adopted to Wikidata, to evaluate its effectiveness beyond temporal
questions. They are SWQ-WD (Diefenbach et al. 2017), which consists of 14894
train and 5622 test set questions, and WebQSP-WD (Sorokin and Gurevych 2018b),
which consists of 2880 train and 1033 test set questions. For DBpedia, we used
QALD-9 (Usbeck et al. 2017), that has 408 training and 150 test questions, and
LC-QuAD 1.0 (Trivedi et al. 2017) data set, that has 4k training set and 1k
test set. Each of these datasets operate on their own version of the DBpedia
and SPARQLs provided and we used the same instance of DBpedia for evaluation.
The baseline system is tuned with dev sets of 175 from TempQA-WD and 200 from
SWQ-WD train and 200 from LC-QuAD 1.0 and evaluated on all the the five test
sets.
#### Baselines
We Compare SYGMA with various KBQA baseline systems as given in Table 6. To
our knowledge there is no other system that works across both DBpedia and
Wikidata. We took the numbers reported from NSQA (Kapanipathi et al. 2021)
work, which is current state-of-the-art on both LC-QuAD 1.0 and QALD-9
datasets. We also compare our work with GGNN (Sorokin and Gurevych 2018a)
which is the only known benchmark for WebQSP-WD dataset. We also compare our
work with WDAqua (Diefenbach, Singh, and Maret 2017), gAnswer (Zou et al.
2014b) and QAMP (Vakulenko et al. 2019) systems.
### 5.1 Results & Discussion
Table 6 shows performance of SYGMA on two different KBs with five different
datasets described in Section 5. We also show the type of reasoning required
for each dataset along with the precision/recall and F1 measure. We use GERBIL
(Usbeck et al. 2019) to compute performance metrics from the pairs of gold
answers and system generated answers from the pipeline. To our knowledge, ours
is the first system reporting KBQA numbers across two different KBs and
varying reasoning types. We get state of the art numbers beating NSQA
(Kapanipathi et al. 2021) on LC-QuAD 1.0 and also achieve comparable numbers
for QALD. For WebQSP-WD dataset we get state of the art numbers nearly $20\%$
gain over GGNN (Sorokin and Gurevych 2018a). The accuracy numbers for Wikidata
datasets show that there is ample scope for improvement for different modules.
This gets evident when we look at the module performances on a small set of
development sets across datasets. For WebQSP-WD we did not have any ground
truth for evaluating entity linking and relation linking.
AMR: Table 7 show the performance of the AMR parser on the 5 development sets
in Smatch (standard measure used to measure AMRs) and Exact Match ($\%$ of
questions that match fully with ground truth AMR).
Dataset | KB | Smatch | Exact Match
---|---|---|---
LC-QuAD 1.0 | DBpedia | 87.6 | 30.0
QALD-9 | DBpedia | 89.3 | 41.8
WebQSP-WD | Wikidata | 88.0 | 43.8
SWQ-WD | Wikidata | 83.0 | 37.8
TempQA-WD | Wikidata | 89.6 | 39.8
Table 7: AMR parser performances on the development sets
Entity Linking: Table 8 below captures the performance of entity linking
module on different datasets. The question level accuracy is computed by
considering all the entities present in each question. However, mention level
accuracy is computed by considering single mention at a time. We can see that
entity linking performance for Wikidata datasets is low compared to DBpedia.
One of the reasons for is all these datasets are adopted from Freebase dataset
and the way entities represented in these two KBs is different. Also, very
little research is done on entity linking for Wikidata and KBQA in general.
Dataset | KB | Accuracy (%)
---|---|---
Mention Level | Question Level
LC-QuAD 1.0 | DBpedia | $86.81\;(91.94)$ | $84.00\;(90.50)$
QALD-9 | DBpedia | $89.52\;(94.28)$ | $89.79\;(93.87)$
TempQA-WD | Wikidata | $74.01\;(82.47)$ | $57.14\;(69.71)$
SWQ | Wikidata | $72.27\;(83.18)$ | $70.14\;(81.59)$
Table 8: Entity linking performance on development sets when gold mentions are
provided. The numbers inside parentheses denote the Hits@5 scores.
Relation Linking: Table 9 captures the performance of the relation linking
module. It demonstrates that relation linking is still a challenging tasks
specially with multi-hop reasoning and temporal reasoning, where the query
graph is disconnected across events, there is room for improvement.
Dataset | KB | Precision | Recall | F1
---|---|---|---|---
LC-QuAD 1.0 | DBpedia | 0.52 | 0.50 | 0.50
QALD-9 | DBpedia | 0.55 | 0.53 | 0.53
TempQA-WD | Wikidata | 0.43 | 0.43 | 0.42
SWQ | Wikidata | 0.67 | 0.68 | 0.67
Table 9: Relation linking performance on development sets.
#### Ablation Study:
To gain more insights on the performance, we also did an ablation study of
SYGMA using TempQA-WD dev set for Wikidata and 100 question dev set from LC-
QuAD 1.0 that we manually annotated with ground truth for all the modules.
This shows the impact of individual module on overall performance is
evaluated. Table 10 shows the results. For example GT-$AMR$ refers to the case
where ground truth AMR is fed directly into $\lambda$-module. The table shows
large jump in accuracy (in both the datasets) when fed with ground truth
entities (GT-EL) and ground truth relations (GT-RL). This points to the need
for improved entity linking and relation linking on both datasets across KBs.
| TempQA-WD | LC-QuAD 1.0
---|---|---
| P | R | F1 | P | R | F1
NO GT | 0.47 | 0.50 | 0.47 | 0.50 | 0.52 | 0.50
GT-AMR | 0.50 | 0.51 | 0.50 | 0.51 | 0.53 | 0.51
GT-$\lambda$ | 0.52 | 0.53 | 0.52 | 0.52 | 0.53 | 0.52
GT-EL | 0.60 | 0.62 | 0.60 | 0.63 | 0.66 | 0.64
GT-RL | 0.92 | 0.93 | 0.92 | 0.99 | 0.98 | 0.98
GT-KB-$\lambda$ | 0.93 | 0.93 | 0.93 | 1.0 | 0.99 | 0.99
GT-SPARQL | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0
Table 10: Ablation Study on TempQA-WD and LC-QuAD 1.0 dev sets. P-Precision,
R-Recall
## 6 Conclusion
In this paper, we present SYGMA system that is generalizable across knowledge
bases and reasoning types. We introduce KB-agnostic question understanding
component that is common across KBs with AMR based intermediate
$\lambda$-Calculus representation. Question Mapping and reasoning module on
the specific KB is customized per KB. we also presented rule modules that can
aid in adding new reasoning type. We introduced a new benchmark dataset
TempQA-WD for temporal KBQA on Wikidata. Experimental results show that SYGMA
indeed achieves its generalization goals with state of the art results on LC-
QuAD 1.0 and WebQSP-WD.
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MIDLMedical Imaging with Deep Learning 2022 Short Paper – MIDL 2022 submission
– Under Review Haoyue ZhangHaoyue Zhang and Jennifer S Polson contributed
equally.1,2<EMAIL_ADDRESS>
Jennifer S Polson1,2<EMAIL_ADDRESS>
Eric J<EMAIL_ADDRESS>
Kambiz Nael3<EMAIL_ADDRESS>
William Speier1,3<EMAIL_ADDRESS>
Corey W. Arnold1,2,3,4<EMAIL_ADDRESS>
1 Computational Diagnostics Lab, University of California, Los Angeles, CA
90024 USA
2 Department of Bioengineering, University of California, Los Angeles, CA
90024 USA
3 Department of Radiology, University of California, Los Angeles, CA 90024,
USA
4 Department of Pathology, University of California, Los Angeles, CA 90024,
USA
# Predicting Thrombectomy Recanalization from CT Imaging Using Deep Learning
Models
###### Abstract
For acute ischemic stroke (AIS) patients with large vessel occlusions,
clinicians must decide if the benefit of mechanical thrombectomy (MTB)
outweighs the risks and potential complications following an invasive
procedure. Pre-treatment computed tomography (CT) and angiography (CTA) are
widely used to characterize occlusions in the brain vasculature. If a patient
is deemed eligible, a modified treatment in cerebral ischemia (mTICI) score
will be used to grade how well blood flow is reestablished throughout and
following the MTB procedure. An estimation of the likelihood of successful
recanalization can support treatment decision-making. In this study, we
proposed a fully automated prediction of a patient’s recanalization score
using pre-treatment CT and CTA imaging. We designed a spatial cross attention
network (SCANet) that utilizes vision transformers to localize to pertinent
slices and brain regions. Our top model achieved an average cross-validated
ROC-AUC of 77.33 $\pm$ 3.9%. This is a promising result that supports future
applications of deep learning on CT and CTA for the identification of eligible
AIS patients for MTB.
###### keywords:
Acute Ischemic Stroke, Computed Tomography, Deep Learning, Vision Transformers
††editors: Under Review for MIDL 2022
## 1 Introduction
Stroke is the fifth leading cause of death and the leading cause of long-term
disability; of the 795,000 new and recurrent strokes each year, acute ischemic
stroke (AIS) accounts for 87% of cases [Tsao et al.(2022)]. Mechanical
thrombectomy (MTB) is the leading treatment for patients with clots in large
blood vessels. In this procedure, a blood clot is surgically removed from an
artery to achieve recanalization, i.e., restored blood flow. As a standard
measurement for recanalization achieved, a modified treatment in cerebral
ischemia (mTICI) score [Tomsick(2007)] is assigned to patients post-treatment.
This post-treatment score is clinically significant, as it has been shown that
favorable scores, i.e., mTICI 2c or greater, are associated with better
clinical outcomes in the long term [Ángel Chamorro et al.(2017)]. Unfavorable
scores (mTICI less than 2c) indicate that the treatment did not effectively
clear the blood vessel. Imaging has been identified as one modality to
illustrate patient physiology that could influence the likelihood of a
successful MTB procedure. Predicting final mTICI score prior to a procedure
can provide doctors and with more more information when considering treatment
options. Deep learning has been shown to leverage the amount of detail in
images to improve prediction accuracy [LeCun et al.(2015)LeCun, Bengio, and
Hinton]. Current literature presents models that perform semi-automated
prediction of mTICI score based on pre-treatment CT imaging, with inconsistent
performance [Hilbert et al.(2019), Siddiqui et al.(2021)]. We propose a fully
automated model that uses both CT and CTA images to predict mTICI score post-
treatment, incorporating attention modules into a deep learning network to
effectively localize to informative stroke regions without requiring manual
segmentation.
## 2 Data and Methods
The cohort used for this study comprises patients treated from 2012-2019. A
patient was included in the cohort if they had CT and CTA imaging, underwent
thrombectomy for stroke, and were assigned an mTICI score post-MTB. Of the 254
eligible patients, 69 patients were excluded due to missing either CT or CTA
series, and 8 were excluded to due unclear stroke location, leaving 177
patients total. The dataset matched demographic distributions seen in other
stroke studies, and the target labels were approximately balanced. Patient
images were processed using a previously published pipeline adapted for CT,
which included brain extraction and registration to a CT template in MNI space
[Zhang et al.(2021b)].
Figure 1: Overview of the proposed deep learning architecture. The figure
details the entire architecture, neighborhood branch modules, spatial
attention transformer module (SAT), and cross attention transformer module
(CAT).
Utilizing a ResNet backbone, CT and CTA images were used as a slice-wise input
(input size 26x224x224) for a global 2D convolutional block, chosen because of
the large slice thickness found in stroke protocols [He et al.(2016)]. The
outputs from neighboring slices were then fed into ResNet34-based branches
that shared weights across slice neighborhoods. The model leveraged two
versions of transformer attention modules. A spatial attention transformer
(SAT) utilized multi-head attention on each slice to focus on salient regions
[Dosovitskiy et al.(2021)]. Within each neighborhood branch, a cross attention
transformer (CAT) identified important slices. Finally, the branch outputs
were subjected to a weighted softmax layer to ultimately generate binary
predictions. The architecture and modules are summarized in Figure 1. The
model was trained for 200 epochs with early stopping, using the Adam optimizer
with weight decay, a batch size of 12, and a learning rate of 0.0001.
Model | ROC-AUC | Accuracy | Precision | Sensitivity | Specificity
---|---|---|---|---|---
Siddiqui et al. | 0.717 | – | – | – | –
Hilber et al. | 0.65 $\pm$ 0.10 | – | – | – | –
Radiomics | 0.6859 $\pm$ 0.043 | 0.6877 $\pm$ 0.039 | 0.6417 $\pm$ 0.068 | 0.7425 $\pm$ 0.123 | 0.6421 $\pm$ 0.126
ResNet34 | 0.5840 $\pm$ 0.036 | 0.5656 $\pm$ 0.046 | 0.5410 $\pm$ 0.067 | 0.8500 $\pm$ 0.300 | 0.3253 $\pm$ 0.296
SCANet | 0.7732 $\pm$ 0.039 | 0.7523 $\pm$ 0.042 | 0.7424 $\pm$ 0.145 | 0.8250 $\pm$ 0.174 | 0.6905 $\pm$ 0.215
Table 1: Performance of our current model benchmarked against results from
literature as well as previously published models applied to this cohort
[Hilbert et al.(2019), Siddiqui et al.(2021)].
## 3 Results and Discussion
The results of our experiments are summarized in Table 1. The average ROC-AUC
achieved by SCANet was 0.7732 $\pm$ 0.039. This is a significant improvement
over the previously published fully automatic deep learning model[Siddiqui et
al.(2021)]. Our method also demonstrates higher and more robust performance
metrics than the state-of-the-art model requiring manual clot segmentation
[Hilbert et al.(2019)]. In addition to the literature benchmarks, SCANet
performs better than a radiomics-based model and standard deep learning
architecture when trained on the same cohort [Zhang et al.(2021a)].
Clinicians decide to perform MTB based on likelihood of successful
recanalization, but it is unknown what factors underlie MTB responses.
Clinical images such as CT and CTA contain valuable information to predict
procedure outcome, and deep learning models have the capability to learn
representations from highly dimensional imaging data. This study sought to
predict final MTB recanalization in a fully automatic manner, leveraging
recent advances in vision transformers to localize to the stroke region. We
showed that our proposed model outperforms prior fully- and semi-automated
machine and deep learning models. The primary limitation of our study is the
small sample size, which precludes more robust validation. A few future
directions include experimenting on a larger dataset across several
institutions, optimizing the preprocessing pipeline to more effectively
preserve high resolution CTA, and correlation of the immediate treatment
response with long-term outcomes. These steps can produce a model that more
accurately predicts MTB recanalization, in turn helping doctors and patients
in the treatment decision process.
## References
* [Dosovitskiy et al.(2021)] Alexey Dosovitskiy et al. An image is worth 16x16 words: Transformers for image recognition at scale. _ICLR_ , 2021.
* [He et al.(2016)] Kaiming He et al. Deep residual learning for image recognition. In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , pages 770–778, 2016.
* [Hilbert et al.(2019)] Adam Hilbert et al. Data-efficient deep learning of radiological image data for outcome prediction after endovascular treatment of patients with acute ischemic stroke. _Computers in Biology and Medicine_ , 115:103516, 2019.
* [LeCun et al.(2015)LeCun, Bengio, and Hinton] Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. Deep learning. _Nature_ , 521(7553):436–444, 2015.
* [Siddiqui et al.(2021)] Fazeel M Siddiqui et al. Quantitative assessment of hyperdense sign measured by hounsfield units is associated with unsuccessful mechanical thrombectomy. _Clinical Neuroradiology_ , 31(4):1111–1119, 2021\.
* [Tomsick(2007)] Thomas A. Tomsick. Timi, tibi, tici: I came, i saw, i got confused. _American Journal Of Neuroradiology_ , 28(2):382–384, 2007.
* [Tsao et al.(2022)] Connie W. Tsao et al. Heart disease and stroke statistics-2022 update: A report from the american heart association. _Circulation_ , 145(8):e391–e426, 2022.
* [Zhang et al.(2021a)] Haoyue Zhang et al. A machine learning approach to predict acute ischemic stroke thrombectomy reperfusion using discriminative mr image features. In _2021 IEEE EMBS International Conference on Biomedical and Health Informatics (BHI)_ , pages 1–4, 2021a.
* [Zhang et al.(2021b)] Haoyue Zhang et al. Intra-domain task-adaptive transfer learning to determine acute ischemic stroke onset time. _Computerized Medical Imaging and Graphics_ , 90:101926, 2021b.
* [Ángel Chamorro et al.(2017)] Ángel Chamorro et al. Complete reperfusion is required for maximal benefits of mechanical thrombectomy in stroke patients. _Scientific Reports_ , 7(1):11636, 2017.
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We consider a combined system of regular delivery trucks and crowdsourced drones, available via a sharing economy platform, to provide a technology-assisted crowd-based last-mile delivery experience. We develop analytical models and methods for a system in which package delivery is performed by a big truck carrying many packages to a neighborhood or a town in a metropolitan area and then the packages are assigned to crowdsourced drone operators to deliver them to their final destinations. We develop several optimization models for various cases of the problem and use a combination of heuristic algorithms to solve this NP-hard problem. Finally, we present computational results for the models and algorithms, and conduct an exhaustive sensitivity analysis to check the influence of different parameters and assumptions on the final results. We also provide extensive managerial insights on the benefits of our proposed model if implemented in practice.
§ INTRODUCTION
The number of deliveries and the revenue obtained from delivery operations have been growing continuously and rapidly during the last two decades, thanks to the exponential growth of e-commerce. However, the efficiency of delivery operations remains a big challenge.
The last mile of delivery process has consistently been one of the most expensive (nearly or even more than 50% of the total cost), least efficient, and most polluting part of the entire parcel delivery supply chain <cit.>; the fact that Amazon Flex has been paying $18-$25 for Uber-like package delivery services <cit.>, while they have not increased their hourly wages to $15 up until just recently <cit.>, speaks to the expensiveness of the last-mile delivery operations.
The expensiveness of last-mile delivery is due to several factors including the facts that it is a labor-intense operation, it is a scattered operation serving different individual customers at dispersed places, which often results in underutilized carrier capacity and the least economy of scale in the whole delivery operation, and that such deliveries are usually very time-consuming because of road congestion, accessibility of the destinations, and most importantly unattended deliveries. The rapidly increasing importance of same day and same hour delivery in our lives will make this operation even more inefficient. Such deliveries are mostly used for low-value high-frequency products such as grocery items for which the shipping cost could quickly become disproportionate in the eyes of consumers.
The advancement of technology can revolutionize the conventional delivery practices and boost the efficiency. Among these advancements are the recent efforts to adopt autonomous vehicles, unmanned aerial vehicles (UAVs), automated guided vehicles (AGVs), and other droids in package delivery operations. The integration of autonomous and semi-autonomous technologies into the last-mile delivery operations in a centralized or decentralized manner have the potential to remove or mitigate the long-lasting factors such as pooling and routing inefficiencies that have been contributing to the expensiveness of last-mile delivery. Utilizing drones, for example, can lead to significant savings in the delivery operation and have been explored for a few years <cit.>. According to <cit.>, Amazon's shipping costs in the period between 2016 and 2019 was around $6,400M, $8,025M, $10,025M, $12,375M, and Prime Air operations could save about $1,044M, $2,677M, $5,147M, $8,549M from these costs.
In an earlier work <cit.>, we have shown that for a centralized delivery system to be competitive with the decentralized household shopping model, a very large portion of the population have to adopt the centralized system and shows inefficient pooling as the primary cause of inefficient last-mile delivery. In this paper, we analyze the impact of decentralization, in particular crowdsourcing of the last part of the last-mile delivery operations when integrated with new technologies, on the efficiency of pooling and clustering customers.
To reduce the cost of the last mile delivery operation and reducing the delivery time (customer's waiting time), we analyze the traditional last mile delivery operation when conducted in coordination with crowdsourced vehicles, bikes, or drones, as illustrated in Figure <ref>. These crowdsourced delivery operators can work in the context of sharing economy and form one side of a two-sided market platform. In fact, this provides a combination of centralized and decentralized systems.
Sharing economy platforms, crowdsourced agents, and last-mile delivery.
In this paper, we combine the crowdsourced autonomous delivery vessels with regular delivery trucks, vans, cars, and bikes
to provide a technology-assisted last-mile delivery experience and a better and smoother transition to a fully autonomous parcel delivery ecosystem. We develop analytical models and methods for one of these intermediary systems, in which package delivery is performed by a big truck carrying a large number of packages to a neighborhood or a town in a metropolitan area and then assign the packages to crowdsourced delivery agents who operate drones[The technology around civil applications of drones is developing rapidly, that can be observed by a quick look into the websites of major companies in this field. Many companies have developed different types of drones for video recording or delivering products. DJI is a famous manufacturer of drones for customers to record videos or take photos. Phantom 4 Pro V2.0 is one of its recent products which costs $1599 and can travel 30 minutes with a full battery. Its speed is around 45 mph and weighs 1375 g. Amazon, Google, Domino, and DHL are all developing their own drones for delivering packages. Amazon's Prime Air can carry products with a weight of 2.26 kg and fly for 30 minutes with 50 mph speed, while it can be controlled 10 miles away from the base. One could compare this to the Google Wing that has a capacity of 1.5 kg and can fly at 75 mph within 8.7 miles from the base. DHL's commercial drones are the biggest ones with 2200 mm diagonal size and can fly at 43.5 mph within 7.5 miles from the base carrying packages up to 3 kg.] to deliver them to their final destinations. Other applications such as post disaster response operations in which trucks cannot access all customers and have to rely on drones in the neighborhood can also be considered for this problem. We introduced this problem in our preliminary work <cit.>. In this paper, we expand that and study and analyze this problem in a more comprehensive way. To the best of our knowledge, besides our earlier extended abstract, this is the first work studying this problem. A combination of heuristic algorithms is used to solve this NP-hard problem and an exhaustive sensitivity analysis is done to check the influence of different parameters and assumptions such as speed ratio of drones and trucks, the number of drones in the service region, and the distribution of the customers. The simulation results show significant savings in the total delivery cost under reasonable assumptions.
§.§ Related Work
Relevant work in the literature are mainly in two categories: drone-aided delivery and crowdsourced delivery, with the latter being more relevant to our work. The literature of drone-aided delivery is so young but has been expanding rapidly. It can be divided into four main categories; travelling salesman problem (TSP) with drones, vehicle routing problem (VRP) with drones, Done(-only) delivery problem, and carrier-vehicle with drones <cit.>. Murray and Chu <cit.> introduced routing problems for two combined systems of truck and drone, namely flying sidekick travelling salesman problem (FSTSP) and parallel drone scheduling TSP. The first problem and its variants have been called with several names including HorseFly Problem, TSPD, Truck-and-Drone TSP, etc. Since this first paper in 2015 the literature on such problems have been growing rapidly (see <cit.> for example).
Although these problems resemble some similarities with the problem we study in this paper, but they are essentially solving a different problem. The main difference is that all of these problems are defined under a centralized delivery system, while our problem combines a centralized truck delivery system and decentralized drone delivery system, and to the best of our knowledge is the first to do so. We will show that the drone(-only) delivery problem is the centralized delivery version of a special case of our problem when the range and speed of drones are large enough, while the pure TSP model is the centralized alternative for the case where the flying range and speed of drones are rather small. For a review on the literature of centralized drone-aided delivery problems please see <cit.>.
Crowdsourced delivery, in essence, can be categorized under pickup and delivery problem (PDP), where the goal is to transport parcels from origins to destinations while minimizing costs. The PDP has been studied for almost half a century; see <cit.> for overview on this body of literature. Unlike the traditional PDP setting, our problem has on-demand features, which is also somewhat related to the literature of dynamic PDP <cit.>, and the delivery operation is partially conducted on a sharing economy platform. Sharing economy indicates a system in which people share access to goods and services as opposed to ownership and it has been extensively studied <cit.>. However, many aspects of the application of sharing economy systems, especially those with (semi-)autonomous technologies, in logistics and delivery services has received less attention <cit.>. The paper <cit.> introduced the idea of “crowdphysics” – crowdsourcing tasks that require people to collaborate and synchronize both in time and physical space and discussed the concept using a crowd-powered delivery service. The paper <cit.> introduced a typology of the early startup businesses.
A first framework for crowd logistics was proposed in <cit.>. The paper <cit.> developed a more comprehensive framework for crowd-based operations in city-level logistics. The authors discussed different kinds of crowd-based logistics including crowdsourced delivery, cargo-hitching (i.e., integration of freight and passenger transport), receiving packages through pickup locations, providing storage by the crowd, and returning (reverse flow) logistics. There have been a number of theoretical and experimental research related to each of these topics, out of which we review those relevant to crowdsourced delivery.
In 2013 Wal-Mart started to consider the idea the idea of encouraging individuals/shoppers in a store who are willing to deliver packages for online customers on their way back home <cit.>. Later, Amazon started considering the crowd as couriers <cit.>. Many researchers explored the crowdsourced delivery service with crowdsourced drivers coming to the package center to pickup the packages and deliver them to their destination. Among the relevant theoretical research, the paper <cit.> studied this idea and formulated it as a vehicle routing problem with occasional drivers (VRPOD). The paper <cit.> studied this problem more in-depth under both static and dynamic settings, in which the latter incorporates probabilistic information about future online orders and in-store customer arrivals. The paper <cit.> proposed a stochastic optimization model for the same problem.
A similar problem with stochastic supply of crowd vehicle and time windows was analyzed in <cit.>. The authors presented a two-stage stochastic model, an exact branch and price algorithm, a column generation heuristic.
The paper <cit.> also considered a group of ad-hoc drivers for performing parcel delivery tasks and proposed a route planning system that automatically makes the matching between ad-hoc drivers and these tasks. The platform also operates a fleet of dedicated vehicles for delivery tasks that cannot be completed by the ad-hoc drivers. The authors present a rolling horizon framework and an exact solution approach based on a matching formulation to solve the problem. They also compared their results with the traditional delivery system and concluded that the use of crowdsourced drivers can significantly reduce the costs.
On the experimental side, the paper <cit.> tried to leverage social networks to find available crowd couriers. They compared three delivery systems with respect to their greenhouse gas emission: the traditional delivery system, a pickup location delivery system (PLS) in which parcels are delivered to a pickup center and customers pick them up on their way to home, work, etc, and finally a socially networked PLS (SPLS) where the last-mile delivery from the pickup locations are done by agents socially connected to the customers. They showed that the PLS system is worse than the traditional delivery system but SPLS significantly reduces the emission.
The paper <cit.> also studied the idea of creating a social network-supported last mile delivery and developed a simulation model for it, in which they exploit the social networks of the online customer to find potential couriers in their networks.
A trial with crowd participants for book delivery in a Finish library was conducted in <cit.> that showed crowdsourced delivery reduces vehicle miles traveled.
The paper <cit.> used a survey to analyze potential driver behavior in choosing to change status from pure commuters to traveler-shippers. Meanwhile, the paper <cit.> also created a survey to study the determinants of crowd-shipping acceptance among drivers. The paper <cit.> developed an agent-based simulation model for the crowdsourced last-mile delivery service with the existence of central pickup location/warehouse and identified the important factors influencing its performance. They set the packages to have three types of accepted delivery time windows and three kinds of crowdsourced agents, including driving, cycling, or walking, delivering one package each time.
In contrast to an entirely crowdsourced fleet, the performance of a hybrid fleet of privately owned delivery vehicles and crowdsourced assets was simulated in <cit.>.
Relatively fewer papers in the literature have considered a cooperative delivery system with cooperation between traditional delivery trucks and crowdsourced carriers or between the crowdsources themselves. The paper <cit.> evaluated the use of shared mobility for last mile delivery services in coordination with delivery trucks. They tried to minimize the combined transportation and outsourcing cost of the trucks and shared mobility. They also considered minimizing greenhouse gas emissions as one of their objectives.
They used an analytical model and found that crowdsourced shared mobility is not as economically scalable as the conventional truck-only system with respect to the operating costs, due to the fact that the payment to crowdsourced drivers is route specific and accounts for the competition with the ride-share service market. However, they state that a transition towards this model can create other economic and operational benefits such as reducing the truck fleet size, avoiding high-demand areas and peak hours, and adjusting vehicle loading capacities.
Cooperation is usually conducted through intermediary depots. A crowd-delivery problem with a number of pop stations as intermediary between trucks and crowdsource workers was modeled as a network min-cost flow problem in <cit.> in which workers are assigned to their closest pop station.
The paper <cit.> considered cyclists and pedestrians as crowdsources who are close to customers and can perform the last-leg parcel delivery and the first-leg parcel pickup with relaying the parcels with a truck carrier at some relay points selected from a pre-defined set of such points. They formulated the problem as a mixed integer non-linear program to simultaneously select crowdsources and relay points and find the truck routes and schedule. The crowdsource routes and their feasibility are determined by the crowdsources themselves before making a bid (that includes the job they are willing to do and the price) and the platform only selects the the best offered bids.
Pedestrians were also considered in <cit.> as crowd delivery partners assigned to their closest relay point determined by a circle packing scheme.
A similar problem was discussed in <cit.> and was formulated as a vehicle routing problem (VRP) type model, in which customers can be served by either an operator's truck or a crowdsourcing partner that received the package at a relay point.
The paper <cit.> also considered a VRPOD with time windows and intermediate depots. The authors also considered the scenario in which the crowdsourced drivers are forbidden at the central depot and can only visit the transshipment centers. Similar transshipment points with occasional drivers were considered in a pickup and delivery problem in <cit.>. The same problem was considered in <cit.> assuming the existence of pickup locations where the packages could be dropped off and the customers can walk and pick up their packages there. A multi-objective version was analyzed in <cit.>. Another collaborative system between delivery truck and crowdsourced drivers was considered in a recent paper <cit.> for which the authors present a heuristic algorithm that solves the involved routing and assignment problems separately.
The idea of relaying packages have also been used in a setting that crowd agents could cooperate with each other. The paper <cit.> analyzed a multi-driver multi-parcel matching problem (MDMPMP) based on the multi-hop ride sharing problem, in which parcels may be transported by a single or by multiple drivers, being relayed between drivers on the path from parcels origin to its destination. Crowd drivers have preset routes from their origins to their destinations from which can deviate slightly to pick up or drop off a package.
There is a lack of a study on the design of a cooperative delivery system with a truck and autonomous or semi-autonomous crowdsourced carriers. In this paper, we fill this gap.
§.§ Regulations
Before we get into the details of the problem and our solution approach, it is necessary to discuss the applicability of the studied problem. One important area that needs to be considered in drone-related operations is their regulations. In 2016, the U.S. government announced that commercial use of drones needs to be registered by the Federal Aviation Administration(FAA) and Department of Transportation(DOT) and need to follow the new regulations <cit.>. The person who operates the drones needs either to be at least 16 years old and get a remote pilot certificate with a small unmanned aircraft systems (UAS) rating or to be directly supervised by someone who already has the certificate <cit.>.
The regulations state that all drones including their carried materials or products must weigh less than 55 lbs which is 25 kg, the limit speed of drones is 100 mph, and the maximum altitude is 400 feet above ground level. The drones need to be within the visual line-of-sight of the remote pilot during the flight and need to be close enough to receive the command from the pilot. Meanwhile, all the drones must only be operated in daylight which is from 30 minutes before official sunrise to 30 minutes after official sunset. They need to yield the way to other aircraft and the minimum weather visibility is 3 miles from a control station. One person can only operate one drone at a time and cannot manipulate a moving vehicle unless it is a sparsely populated area. Moreover, drones cannot carry hazardous materials <cit.>.
Other regulations are mainly related to the safety issues that drones need to be checked before their launching and the object being carried by the drone is securely attached. On the other hand, most of the restrictions can be waived if the applicant demonstrates that the operation can safely be conducted under the terms of a certificate of waiver <cit.>.
§ PROBLEM STATEMENT
Consider a region $R$ as a residential area in which one truck has to go through all the neighborhoods to deliver some packages. There are also private drone operators in the area that could deliver packages from the truck to their final destination (households). When the truck stops at a neighborhood corner, the crowd-based drones, after receiving an order from the courier, will fly from their base to that corner to pick up the packages, deliver them to the customers and go back to their base, i.e., operator's house, for recharging its battery. The objective is to design a coordinated system between the truck and these crowdsourced drones in a way that minimizes the total time spent on fulfilling the demand of all customers in that area.
Figure <ref> shows an illustration of this cooperative delivery between a truck and crowd-sourced drones.
In this problem we assume that:
* Each drone can only carry one item at a time.
* The charging time for drones at its home base is 0 (i.e., changing quickly to a new battery).
* Each drone base location launches only one drone.
* The speed of all drones is set to be fixed and equal to each other.
* There is no weight limit for a drone to carry the package. So all packages of a customer will be delivered in one trip.
* Only one truck is used and its capacity is sufficient for the entire demand in the region.
* If there is no drone nearby, the truck will serve all the customers in that neighborhood.
* Each drone can be only used at one truck's stop (cluster center).
* The truck will wait at the pickup center to track the drones until the driver receives the confirmation that the packages are delivered to the customers by the drone and send necessary instructions to drone operators in case of delivery failures.
* The compensation mechanism for drone delivery tasks is assumed to be set by the platform separately.
A schematic representation of the proposed crowdsourced drone delivery.
Depending on whether there are one or multiple pickup centers, or whether drones need to go back to their home bases after each delivery or not, we define four problems as followings:
Problem I–(One Center with Recharging): Problem with one pickup center with the assumption that after each delivery, drones should stop at their home bases (for recharging/replacing battery) before revisiting the pickup center for another delivery.
Problem II–(One Center with Revisiting): Problem with one pickup center with the assumption that after each delivery, drones could revisit the pickup center for another delivery before going back to their home bases.
Problem III–(Multiple Centers with Recharging): Problem with multiple pickup centers with the assumption that after each delivery, drones should stop at their home base (for recharging/replacing battery) before revisiting the pickup centers for another delivery.
Problem IV–(Multiple Centers with Revisiting): Problem with multiple pickup centers with the assumption that after each delivery, drones could revisit the pickup centers for another delivery before going back to their home base.
The four considered problems.
Recharging Revisiting
One Center Problem I Problem II
Multiple Centers Problem III Problem IV
<Ref> summarizes the conditions of these four problems. In the followings we will model these four problems. In the rest of this section, we delve into each of these problems and model them mathematically. The following notations are commonly used in all of the presented models: we denote the set of customer nodes with $C$ and set of drone nodes with $D$ and let $v_D$ be the speed of drones, $v_T$ be the speed of trucks, $L$ be the longest distance (range) a drone can travel, and $M$ to be a sifficiently large number. Note that setting $L=0$ reduces problems III and IV to TSP.
§.§ Problem I–(One Center with Recharging)
In this problem we have one pickup center and assume that drones should go back to their home base for recharging after each delivery. There is no truck route and truck (or a store, warehouse, or distribution center) operates as a depot for a fleet of crowdsourced drones to pick up the package and deliver them to the customers. The problem with one center is important to study because it sets a a basis for the multiple centers problems and also it helps to understand the dynamics inside a cluster, introduced later, in a better way. The insight behind our algorithm is partly driven by this sub-problem. Before we present our model for this problem we define the sets, parameters, and variables in Table <ref>.
Sets, parameters, and variables used in the one center model.
Parameters/Variables Description
$d_{ij}$ Route length going from node $i\in D$ to the center node (0) and then to node $j\in C$ and back to node $i$
$x_{ij}$ Binary decision variable. It is $1$ when a drone travels from node $i\in D$ to the center node 0 and then to node $j\in C$ and back to node $i$. It is $0$ otherwise.
$q_{i}$ Total travel time of the drone with base at node $i$
$Q$ Maximum time spent by all drones (makespan)
A mixed integer linear programming (MILP) model for this problem can be written as follows:
\begin{eqnarray}
\minimize \qquad Q & & \quad \qquad \st \nonumber \\
\sum_{i\in D} x_{ij} & = & 1\,, \qquad \forall j\in C \label{eq:Problem1SatisfyAll} \\
d_{ij}x_{ij} & \leq & L\,, \qquad \forall i\in D, \; \forall j\in C \label{eq:Problem1DroneRange} \\
\frac{1}{v_D} \sum_{j \in C} d_{ij}x_{ij} & \leq & q_i \,, \qquad \forall i \in D \label{eq:Problem1DroneTime} \\
0 \;\; \leq \;\; q_i & \leq & Q\,, \qquad \forall i\in D \label{eq:Problem1TotalTime} \\
x_{ij} & \in & \{0, 1\}\,, \quad \forall i\in D, j\in C \nonumber
\end{eqnarray}
The objective is to minimize the maximum time of drone routes (makespan). Constraint (<ref>) makes sure all customers are visited once. Constraint (<ref>) ensures that the distance traveled by each drone does not exceed the maximum travel distance (range) of drones. Constraints (<ref>) finds the time spent by each drone and Constraint (<ref>) calculates the maximum time among all drones. An alternative and less efficient but more flexible (adjustable) model for this problem is provided in <ref> of the Online Supplement of this paper.
§.§ Problem II–(One Center with Revisiting)
In this problem we consider one pickup center (no truck route) and after each delivery, drones could revisit the pickup center for another delivery before going back to their home bases.
We first note that a feasible drone route in this case is a route that starts from a drone node $i\in D$, satisfies the demand of one or multiple customer nodes after picking their packages up at the center node, and comes back to the starting drone node while not violating the drone range $L$.
Let $\Pi_i$ denote the set of feasible drone routes starting from drone node $i\in D$ and let $\Pi_i(j) \subset \Pi_i$ be the set of routes that contain the customer node $j\in C$. We use $k$ to index these routes. For any route $\pi_{ik} \subset \Pi_i$ let $d_{\pi_{ik}}$ denote its total length. We define our decision variable as
x_π_ik = {
1 if route π_ik ∈Π_i is selected;
0 . .
Then we can form an MILP formulation:
\begin{eqnarray}
\minimize \qquad Q \qquad & & \qquad \qquad \st \nonumber \\
& & \nonumber \\
\sum_{i} \sum_{\pi_{ik} \in \Pi_i(j)} x_{\pi_{ik}} & = & 1\,, \quad \qquad \forall j\in C \label{eq:Problem2SatisfyAll} \\
d_{\pi_{ik}} x_{\pi_{ik}} & \leq & L \;\; \quad \qquad \forall i\in D,\; \pi_{ik} \in \Pi_i \label{eq:Problem2DroneRange} \\
\frac{1}{v_D} \sum_{\pi_{ik} \in \Pi_i} d_{\pi_{ik}} x_{\pi_{ik}} & \leq & q_i\,, \quad \qquad \forall i \in D \label{eq:Problem2DroneTime} \\
0 \;\; \leq \;\; q_i & \leq & Q\,, \quad \qquad \forall i\in D \label{eq:Problem2TotalTime} \\
x_{\pi_{ik}} & \in & \{0, 1\}\,, \;\;\quad \forall i\in D, \; \pi_{ik} \in \Pi_i \nonumber %\\
\end{eqnarray}
Constraint (<ref>) ensures all customers are visited once.
Constraints (<ref>) and (<ref>) find the maximum time among all drone routes.
An alternative and less efficient but more flexible (adjustable) model for this problem is provided in <ref> of the Online Supplement of this paper.
§.§ Problem III–(Multiple Centers with Recharging)
In this problem we can have multiple pickup centers and drones should go back to their home base for recharging after each delivery. All the customer and drone nodes are in a
region $R$. We first assume that truck nodes or centers can be placed anywhere in $R$ (we will change this later). This implies that all deliveries are done by the drones and the truck carries package through different neighborhoods and stops at one center in each neighborhood where it passes the packages to the drones.
Although the pickup centers are not necessarily the same as customers locations but if distance between pickup centers and some customers are less than or equal to a threshold, we assume that those customers are served by the truck.
The sets, parameters, and variables used in our model are defined in Table <ref>. Note that some of the variables are infinite dimensional.
Sets, parameters, and variables used in the general (multi-center) model.
Parameters/Variables Description
$d_{ij}$ Distance between node $i\in D$ and $j\in C$
$\bm{y}_i$ Vector of location of drone $i \in D$
$\bm{z}_j$ Vector of location of customer $j \in C$
$\tau$ Distance threshold for serving the customer by the truck (if the distance between truck's stop and a customer is less than or equal to this threshold)
$N_T$ Total number of truck centers
$\bm{x}_m$ Vector of location of center $m$ with $\bm{x}_m \in R$ and $m = 1, \dots, N_T$
$D \times R \times C \xrightarrow{} \{0,1\}$. It is $1$ when drone travels from node $i\in D$ to center $\bm{x}_m$ then to node $j\in C$ and back to node $i$. It is $0$ otherwise.
$\zeta_{\bm{x}_mj}$ $R \times C \xrightarrow{} \{0,1\}$. It is $1$ when customer $j\in C$ is served by the truck that has stopped at location $\bm{x}_m$ and 0 otherwise
$q_{i\bm{x}_m}$ Total distance travelled by drone node $i$ assigned to center $\bm{x}_m$
$Q_{\bm{x}_m}$ Maximum distance among all $q_{i\bm{x}_m}$
$R \times R \xrightarrow{} \{0,1\}$. It is $1$ if the path from center $\bm{x}_m$ to $\bm{x}_n$ is used by the truck
$u_{\bm{x}_m}$ A dummy variable
We can model this problem in its most general sense as an infinite dimensional optimization problem that follows:
\begin{eqnarray}
\minimize \; \frac{1}{v_T} \iint_{\bm{x}_m\,, \bm{x}_n \in R} \|\bm{x}_m-\bm{x}_n\| \gamma_{(\bm{x}_m\bm{x}_n)} \, dA & + & \iint_{\bm{x}_m \in R} Q_{\bm{x}_m} \, dA \qquad \qquad \st \nonumber \\
\sum_{i\in D}\iint_{\bm{x}_m \in R} \xi_{i\bm{x}_mj}\, dA & = & 1\,, \qquad \qquad \forall j\in C \label{eq:Problem3CentersAnywhere-AllCustomersServed} \\
\|\bm{x}_m - \bm{z}_j\| & \leq & \tau + M (1- \zeta_{\bm{x}_mj}) \,, \quad \forall j \in C\,, \; \forall \bm{x}_m \in R \label{eq:Problem3CentersAnywhere-TruckServing} \\
(\|\bm{x}_m - \bm{y}_i\| + \|\bm{x}_m - \bm{z}_j\| + d_{ij}) \, \xi_{i\bm{x}_mj} & \leq &L\,, \qquad \qquad \forall i \in D\,, \; \forall j \in C\,, \; \forall \bm{x}_m \in R \label{eq:Problem3CentersAnywhere-DroneRange} \\
\frac{1}{v_D} \sum_{j\in C} (\|\bm{x}_m - \bm{y}_i\| + \|\bm{x}_m - \bm{z}_j\| + d_{ij}) \, \xi_{i\bm{x}_mj} \, (1-\zeta_{\bm{x}_mj}) & \leq & q_{i\bm{x}_m} \,, \qquad \forall i \in D\,, \; \forall \bm{x}_m \in R \label{eq:Problem3CentersAnywhere-CenterDroneTime} \\
q_{i\bm{x}_m} - \frac{1}{v_D} \max_{j} d_{ji} \xi_{i\bm{x}_mj} & \leq & Q_{\bm{x}_m}\,, \qquad \forall i\in D\,, \; \forall \bm{x}_m \in R \label{eq:Problem3CentersAnywhere-CenterMakespan} \\
\iint_{\bm{x}_m \in R,\bm{x}_m\neq \bm{x}_n}{\gamma_{(\bm{x}_m\bm{x}_n)}} dA & = & 1\,, \qquad \qquad \forall \bm{x}_n \in R \label{eq:Problem3CentersAnywhere-TSPassignment1} \\
\iint_{\bm{x}_n \in R,\bm{x}_n\neq \bm{x}_m}\gamma_{(\bm{x}_m\bm{x}_n)} dA & = & 1\,, \qquad \qquad \forall \bm{x}_m \in R \label{eq:Problem3CentersAnywhere-TSPassignment2} \\
\iint_{\bm{x}_m\,, \bm{x}_n \in R} \gamma_{\bm{x}_m\bm{x}_n} dA & = & N_T\,, \label{eq:Problem3CentersAnywhere-CentersNumber} \\
u_{\bm{x}_m}-u_{\bm{x}_n} + N_T \; \gamma_{\bm{x}_m\bm{x}_n} & \leq & N_{T}-1 \,, \qquad \forall \; 2\leq u_{\bm{x}_m} \neq u_{\bm{x}_n} \leq N_T \label{eq:Problem3CentersAnywhere-TSPsubtour} \\
d_{i\bm{x}_mj}, \; d_{\bm{x}_m\bm{x}_n}, \; q_{i\bm{x}_m}, \; Q_{\bm{x}_m} & \geq & 0\,, \qquad \qquad \; \forall i\in D\,, \; \forall j\in C\,, \; \forall \bm{x}_m \in R \nonumber \\
\zeta_{\bm{x}_mj}, \; \xi_{i\bm{x}_mj}, \; \gamma_{\bm{x}_m\bm{x}_n} & \in & \{0,1\} \,, \qquad \forall j\in C\,, \; \forall \bm{x}_m\,, \bm{x}_n \in R \nonumber \\
N_T, \; u_{\bm{x}_m} & \in & \N \,, \qquad \qquad \forall \bm{x}_m \in R \nonumber %\\
\end{eqnarray}
The objective function minimizes the sum of the time that truck takes to travel between the pickup centers on its route and the total time that it takes to serve clusters of customers with drones at these stopping points.
Constraint (<ref>) ensures all customer are visited once.
Constraint (<ref>) controls whether a customer is served by the truck.
Constraint (<ref>) enforces the drone battery limit. Constraints (<ref>) and (<ref>) calculates the time cost in each center, excluding the time spent on the last edge of last route of each drone.
Constraint (<ref>)-(<ref>) are travelling salesman problem (TSP) constraints for the truck as well as to calculate number of pickup centers.
This is obviously a complex problem to solve and to make it easier we need some relaxations. To do this, we assume that the truck only stops at a customer location and while stopping there that location will serve as a pickup center for drones as well. Luckily, this simple and very reasonable relaxation, makes the problem much easier to model and solve. This assumption is confirmed to be valid and beneficial by findings of <cit.> that showed the benefits of having dedicated drivers in parallel to crowd drivers in the context of dynamic ride-sharing, where the dedicated drivers, in addition to satisfying part of the demand, could also serve riders for whom no ad-hoc drivers are available.
The sets, parameters, and variables used in this new model are defined in Table <ref>.
Sets, parameters, and variables used in the general (multi-center) model.
Parameters/Variables Description
$d_{pp'}$ Distance between node $p\in C$ and $p'\in C$
$d^p_{ij}$ Route length going from node $i\in D$ to the center node $p\in C$ and then to customer node $j\in C$ and back to node $i$
$\delta_{ji}$ Distance of customer node $j\in C$ to the drone node $i\in D$
$y_p$ Binary decision variable that is 1 if node $p\in C$ is served by the truck and 0 otherwise
$x^p_{ij}$ Binary decision variable that is $1$ when a drone travels from node $i\in D$ to the center node $p\in C$ and then to customer node $j\in C$ and back to node $i$ and is $0$ otherwise
$z_{ip}$ Binary decision variable that is $1$ when drone $i\in D$ is used for delivery at center node $p\in C$ and is $0$ otherwise
$q_{ip}$ Total travel time of the drone with base at node $i\in D$ that is assigned to center $p\in C$
$Q_p$ Maximum time spent by all drones assigned to center $p\in C$
$\Delta_{ip}$ Maximum length of the last tour segment among all tours formed by drone $i \in D$ through center $p \in C$
$\eta^{p}_{ij}$ Binary decision variable equal to 1 if $d_{ji}x^p_{ij} = \Delta_{ip}$ and 0 otherwise
$\gamma_{pp'}$ Binary decision variable equal to 1 if the path from $p$ to $p'$ has been used by the truck
An MILP model for this problem can be written as following:
\begin{eqnarray}
\minimize \;\;\; \frac{1}{v_T}\sum_{p\in C}\sum_{p'\in C} d_{pp'}\gamma_{pp'} & + & \sum_{p} Q_p \qquad \qquad \qquad \st \nonumber \\
\sum_{p\in C} y_p & \geq & 1 \; , \label{eq:Problem3CentersOnCustomersMinOneTruckNode} \\
\sum_{i\in D}\sum_{p\in C} x^p_{ij} & = & 1-y_j \; , \quad \qquad \qquad \forall j\in C \label{eq:Problem3CentersOnCustomersAssignment} \\
\sum_{i\in D} z_{ip} & \leq & M y_p \; , \quad \qquad \qquad \forall p\in C \label{eq:Problem3CentersOnCustomersTruckThenDrone} \\
\sum_{p\in C} z_{ip} & \leq & 1 \; , \quad \qquad \qquad \qquad \forall i\in D \label{eq:Problem3CentersOnCustomersEachDroneOneCenter} \\
\sum_{j \in C} x^p_{ij} & \leq & M z_{ip} \; , \quad \qquad \qquad \forall i\in D, p\in C \label{eq:Problem3CentersOnCustomersIfDroneAtCenter} \\
d^p_{ij} x^p_{ij} & \leq & L \; , \quad \qquad \qquad \qquad \forall i\in D, j,p\in C \label{eq:Problem3CentersOnCustomersDroneRange} \\
\frac{1}{v_D}\sum_{j\in C} d^p_{ij}x^p_{ij} - M(1-z_{ip}) & \leq & q_{ip} \; , \qquad \qquad \qquad \forall i\in D, p\in C \label{eq:Problem3CentersOnCustomersCenterDroneTime} \\
q_{ip} - (1/v_D) \max_{j} \delta_{ji} x^p_{ij} & \leq & Q_p \; , \qquad \qquad \qquad \forall i\in D, p\in C \label{eq:Problem3CentersOnCustomersCenterMakespan} \\
2\gamma_{pp'} & \leq & y_{p} + y_{p'} \; , \qquad \qquad \forall p\neq p' \in C
\label{eq:Problem3CentersOnCustomersTSPtruckNodesMakePath} \\
\sum_{p\in C} \gamma_{pp'} & = & y_{p'} \; , \qquad \qquad \qquad \forall p'\in C \label{eq:Problem3CentersOnCustomersTSPassignment1} \\
\sum_{p'\in C} \gamma_{pp'} & = & y_p \; , \qquad \qquad \qquad \forall p \in C \label{eq:Problem3CentersOnCustomersTSPassignment2} \\
\sum_{p\in S} \; \sum_{p'\in S,\, p'\neq p} \gamma_{pp'} & \leq & |S| -1 \; , \qquad \qquad \forall S \subset C\,, \; 2 \leq |S| \leq |C|-2 \label{eq:Problem3CentersOnCustomersTSPsubtour} \\
M \gamma_{pp} + \sum_{j\in C} y_{j} -1 & \leq & M \; , \qquad \qquad \qquad \forall p \in C \label{eq:Problem3CentersOnCustomersTSPorStar} \\
x^p_{ij}, \; \gamma_{pp'}, \; y_p & \in & \{0,1\} \;, \qquad \qquad \forall i\in D, j,p,p' \in C \nonumber \\
q_{ip}, \; z_{ip}, \; Q_p & \geq & 0 \;, \qquad \qquad \qquad \forall i\in D, p\in C \nonumber %\\
\end{eqnarray}
The objective function minimizes the sum of the time that truck takes to travel between the pickup centers on its route and the total time that it takes to serve customers with crowdsourced drones at these stopping points.
Constraint (<ref>) ensures that at least one customer node is set to be truck node.
Constraint (<ref>) ensures all customers are visited once, either by the truck or by one of the drones. Constraint (<ref>) ensures that a node can serve as pickup center for drones if it is visited by the truck.
Constraint (<ref>) enforces the assumption that each drone can only be used at one truck center.
Constraint (<ref>) ensures that customers are served by a drone through a center node if that drone is used at that center.
Constraint (<ref>) ensures that a drone cannot fly more than its range (battery limit).
Constraints (<ref>) and (<ref>) are used to find the time spent at each stop of the truck to serve a group of customers.
Note that truck does not have to wait at a pickup center after all packages in that neighborhood are delivered so the time spent on the last segment of the last route for each drone is deducted. This implies that we may prefer to assign the drone to a farther customer.
Constraint (<ref>) is to make sure that the truck tour is formed through truck nodes.
Constraints (<ref>)–(<ref>) are TSP constraints for the truck. Note that we treat the set of constraints in (<ref>) as lazy constraints and add them to our model in the implementation in a lazy fashion only when they are needed.
Finally, constraint (<ref>) allows the model to switch from having a TSP tour to a one center star shape (hub-and-spoke) model when the range and speed of drones are large enough.
To linearize the
$\max_{j} \delta_{ji}x^p_{ij}$ term in (<ref>), we replace this term with $\Delta_{ip}$ and constraint (<ref>) with
\begin{eqnarray}
\hspace{0.6in} q_{ip} - \frac{1}{v_D} \Delta_{ip} & \leq & Q_p \; , \qquad \qquad \qquad \forall i\in D, p\in C \label{eq:Problem3CentersOnCustomersCenterMakespanLinearized}
\end{eqnarray}
and add the following four constraints
\begin{eqnarray}
\hspace{1.7in} \delta_{ji}x^p_{ij} & \leq & \Delta_{ip} \;, \quad \qquad \qquad \qquad \forall i\in D, p\in C, j\in C \label{eq:Problem3CentersOnCustomersCenterMakespanLinearized1} \\
\Delta_{ip} & \leq & \delta_{ji}x^p_{ij} + M(1-\eta^p_{ij}) \;, \quad \forall i\in D, p\in C, j\in C \label{eq:Problem3CentersOnCustomersCenterMakespanLinearized2} \\
\sum_{j\in C} \eta^p_{ij} & \geq & z_{ip} \;, \qquad \qquad \qquad \qquad \forall i\in D, p\in C \label{eq:Problem3CentersOnCustomersCenterMakespanLinearized3} \\
\eta^p_{ij} & \leq & x^p_{ij} \qquad \qquad \qquad \qquad \forall i\in D, p\in C, j\in C \label{eq:Problem3CentersOnCustomersCenterMakespanLinearized4} \\
\eta^p_{ij} & \in & \{0,1\} \quad \qquad \qquad \qquad \forall i\in D, p\in C, j\in C \nonumber
\end{eqnarray}
to form an MILP model.
We solved this model for small instances but for larger problems we rely on a heuristic algorithm.
§.§ Problem IV–(Multiple Centers with Revisiting)
In this problem we consider multiple pickup centers and assume that after each delivery, drones could revisit the pickup center for another delivery before going back to their home bases. Similar to the latter part of the previous section, we assume the pickup centers (truck nodes) are a subset of customer nodes.
Also, similar to Problem II, we define a feasible drone route as a route that starts from a drone node $i\in D$, satisfies the demand of one or multiple customer nodes after picking their packages up at a center node, and comes back to the starting drone node while not violating the drone range $L$. We assume that the revisits happen to the same center and the drone could start serving through another center after going back to its base. Relaxing this assumption is doable but comes at huge computational costs. We consider no restriction on the number of revisits for each drone.
We pre-calculate all such routes.
The sets, parameters, and variables used in our model are defined in Table <ref>.
Sets, parameters, and variables used in the general (multi-center) model.
Sets/Parameters/Variables Description
$\Pi^p_{i}$ Set of all feasible routes that start from drone node $i\in D$ and go through pickup center $p \in C$
$\Pi^p_{i}(j)$ Set of all feasible routes that start from drone node $i\in D$, go through pickup center $p \in C$, and traveled through customer node $j \in C$
$d_{pp'}$ Distance between node $p\in C$ and $p'\in C$
$d_{\pi^p_{ik}}$ Length of route $\pi^p_{ik} \in \Pi^p_i$ for drone node $i \in D$
$\delta_{\pi^p_{ik}}$ Length of the last line segment of route $\pi^p_{ik}$
$y_p$ Binary decision variable that is 1 if node $p\in C$ is served by the truck and 0 otherwise
$x_{\pi^p_{ik}}$ Binary decision variable that is $1$ when a drone travels along the route $\pi^p_{ik}\in \Pi^p_{i}$ and is $0$ otherwise
$z_{ip}$ Binary decision variable that is $1$ when drone $i\in D$ is used for delivery at center node $p\in C$ and is $0$ otherwise
$q_{ip}$ Total travel time of the drone with base at node $i\in D$ that is assigned to center $p\in C$
$Q_p$ Maximum time spent by all drones assigned to center $p\in C$
$\Delta_{ip}$ Maximum length of the last tour segment among all routes $\pi^p_{ik}\in \Pi^p_{i}$
$\eta_{\pi^p_{ik}}$ Binary decision variable equal to 1 if $\delta_{\pi^p_{ik}}x_{\pi^p_{ik}} = \Delta_{ip}$ and 0 otherwise
$\gamma_{pp'}$ Binary decision variable equal to 1 if the path from $p$ to $p'$ has been used by the truck
An MILP model for this problem can be written as following:
\begin{eqnarray}
\minimize \;\;\; \frac{1}{v_T}\sum_{p\in C}\sum_{p'\in C} d_{pp'}\gamma_{pp'} & + & \sum_{p} Q_p \qquad \qquad \st \nonumber \\
\sum_{p\in C} y_p & \geq & 1 \; , \label{eq:Problem4CentersOnCustomersMinOneTruckNode} \\
\sum_{p\in C}\sum_{i\in D} \sum_{\pi^p_{ik}\in \Pi^p_i(j)}x_{\pi^p_{ik}} & = & 1-y_j \; , \quad \qquad \forall j\in C \label{eq:Problem4CentersOnCustomersAssignment} \\
\sum_{i\in D} z_{ip} & \leq & M y_p \; , \qquad \qquad \forall p\in C \label{eq:Problem4CentersOnCustomersTruckThenDrone} \\
\sum_{p\in C} z_{ip} & \leq & 1 \; , \quad \qquad \qquad \qquad \forall i\in D \label{eq:Problem4CentersOnCustomersEachDroneOneCenter} \\
\sum_{\pi^p_{ik}\in \Pi^p_i} x_{\pi^p_{ik}} & \leq & M z_{ip} \; , \qquad \qquad \forall i\in D, p\in C \label{eq:Problem4CentersOnCustomersIfDroneAtCenter} \\
\frac{1}{v_D} \sum_{\pi^p_{ik}\in \Pi^p_i} d_{\pi^p_{ik}} x_{\pi^p_{ik}} - M(1-z_{ip}) & \leq & q_{ip} \; , \quad \qquad \qquad \forall i\in D, \; p\in C \label{eq:Problem4CentersOnCustomersCenterDroneTime} \\
q_{ip} - (1/v_D) \max_{\pi^p_{ik}\in \Pi^p_i} \delta_{\pi^p_{ik}}x_{\pi^p_{ik}} & \leq & Q_p \; , \quad \qquad \qquad \forall i\in D, \; p\in C \label{eq:Problem4CentersOnCustomersCenterMakespan} \\
2\gamma_{pp'} & \leq & y_{p} + y_{p'} \; , \qquad \qquad \forall p\neq p'\in C \label{eq:Problem4CentersOnCustomersTSPtruckNodesMakePath} \\
\sum_{p\in C} \gamma_{pp'} & = & y_{p'} \; , \qquad \qquad \qquad \forall p'\in C \label{eq:Problem4CentersOnCustomersTSPassignment1} \\
\sum_{p'\in C} \gamma_{pp'} & = & y_p \; , \qquad \qquad \qquad \forall p \in C \label{eq:Problem4CentersOnCustomersTSPassignment2} \\
\sum_{p\in S} \; \sum_{p'\in S,\, p'\neq p} \gamma_{pp'} & \leq & |S| -1 \; , \quad \qquad \forall S \subset C\,, \; 2 \leq |S| \leq |C|-2 \label{eq:Problem4CentersOnCustomersTSPsubtour} \\
M \gamma_{pp} + \sum_{j\in C} y_{j} -1 & \leq & M \; , \qquad \qquad \qquad \forall p \in C \label{eq:Problem4CentersOnCustomersTSPorStar} \\
x_{\pi^p_{ik}}, \; \gamma_{pp'},\; y_p & \in & \{0,1\} \; , \; \qquad \qquad \forall i\in D,\; p,p'\in C,\; \pi^p_{ik} \in \Pi^p_i \nonumber \\
q_{ip}, \; Q_p & \geq & 0 \;, \qquad \qquad \qquad \forall i\in D,\; p\in C \nonumber %\\
\end{eqnarray}
The objective function minimizes the sum of the time that spent on the truck route and the drone routes.
Constraint (<ref>) ensures that at least one customer node is set to be truck node (pickup center).
Constraint (<ref>) ensures either the truck or one of the drones visits each of the customers.
Constraint (<ref>) ensures that a customer nodes can only be used as pickup center for drones if that node is visited by the truck.
Constraint (<ref>) enforces the assumption that each drone can only be used at one truck center.
Constraint (<ref>) enforces drone routes to be formed at pickup centers.
Constraints (<ref>) and (<ref>) are used to find the time spent at each stop of the truck to serve a group of customers. Note that at each center the truck may not wait until the last drone route is completed, which may, at times, prioritize a farther customer over a closer customer for a drone.
Constraint (<ref>) is to make sure that the truck tour is formed through truck nodes.
Constraints (<ref>)–(<ref>) are TSP constraints for the truck. Note that we treat the set of constraints in (<ref>) as lazy constraints and add them to our model in the implementation in a lazy fashion only when they are needed.
Finally, constraint (<ref>) allows the model to switch from having a TSP tour to a one center star shape (hub-and-spoke) model when the range and speed of drones are large enough.
To linearize the $\max_{\pi^p_{ik}\in \Pi^p_i} \delta_{\pi^p_{ik}}x_{\pi^p_{ik}}$ term in (<ref>), we replace this term with $\Delta_{ip}$ and constraint (<ref>) with
\begin{eqnarray}
\hspace{0.1in} q_{ip} - \frac{1}{v_D} \Delta_{ip} & \leq & Q_p \; , \qquad \qquad \forall i\in D, p\in C \label{eq:Problem4CentersOnCustomersCenterMakespanLinearized}
\end{eqnarray}
and add the following four constraints
\begin{eqnarray}
\hspace{1.7in} \delta_{\pi^p_{ik}}x_{\pi^p_{ik}} & \leq & \Delta_{ip} \;, \qquad \qquad \qquad \forall i\in D, p\in C, \, \pi^p_{ik}\in \Pi^p_i \label{eq:Problem4CentersOnCustomersCenterMakespanLinearized1} \\
\Delta_{ip} & \leq & \delta_{\pi^p_{ik}}x_{\pi^p_{ik}} + M(1-\eta_{\pi^p_{ik}}) \;, \;\; \forall i\in D, p\in C, \pi^p_{ik}\in \Pi^p_i \label{eq:Problem4CentersOnCustomersCenterMakespanLinearized2} \\
\sum_{\pi^p_{ik}\in \Pi^p_i} \eta_{\pi^p_{ik}} & \geq & z_{ip} \;, \label{eq:Problem4CentersOnCustomersCenterMakespanLinearized3} \\
\eta_{\pi^p_{ik}} & \leq & x_{\pi^p_{ik}} \;, \;\; \qquad \qquad \forall i\in D, p\in C, \pi^p_{ik}\in \Pi^p_i \label{eq:Problem4CentersOnCustomersCenterMakespanLinearized4} \\
\eta_{\pi^p_{ik}} & \in & \{0,1\} \;, \qquad \qquad \forall i\in D, p\in C, \, \pi^p_{ik}\in \Pi^p_i \nonumber
\end{eqnarray}
to form an MILP model. We also solved this model for small instances but for larger problems we rely on our heuristic algorithm.
§ ALGORITHM
It is clear that the problem is NP-hard and we have to take a sub-optimal approach to solve the problem.
Our algorithm consists of several sub-routines as explained in the following. The high-level steps of our algorithm are as follows:
Step 1: Running a binary search and the $k$-means clustering algorithm to find the minimum feasible $k$ to cluster all customers into $k$ clusters with radius $L/4$ and finding the cluster centers.
Step 2: Partitioning the region with the Voronoi tessellation generated by the $k$ cluster center points found in Step 1 and assigning customers and drone bases to their nearest centers.
Step 3: Running a Tabu Search algorithm to solve the sub-problem in each cluster where parcels are assigned to drones to be delivered to customers in a way that minimizes total delivery time.
Step 4: Solving the truck tour by the Lin-Kernighan-Helsgaun (LKH) algorithm to go through all cluster centers as well as all customer nodes that do not have any drone node in that cluster.
We first cluster all customers into $k$ groups, where each group will have a center that will serve as the truck's stoping point, by solving a $k$-means clustering problem to ensure all customers are within radius $L/4$ of their closest centers. A similar idea for finding truck's stopping points for launching drones in the centralized system of combined truck and drone has been previously shown to be efficient <cit.>.
To solve the $k$-means clustering problem we use Lloyd's algorithm <cit.>, which is described in <ref> of the Online Supplement of this paper. This minimizes the total squared Euclidean distances between customer nodes and their closest cluster centers. The choice of radius $L/4$ is to make sure that with one full battery the drones in each cluster can finish the delivery and return to their base. The number $k$ is the smallest integer that makes this feasible and will be found using a binary search. The feasibility here means that all customers are located in at least one circle that is centered at a cluster center with radius $L/4$. The clustering step is shown in <ref>. The final geographic partitioning, i.e., identifying the boundaries of each cluster, is done using a Voronoi partitioning scheme <cit.> with the cluster centers as the generators. This determines the assignment of customer nodes and drone nodes to their closest cluster centers.
These nodes would be assigned to the nearest center only if their distance to the center is less than $L/4$.
Clusters with no drone will be served by the truck. The partitioning step is illustrated in <ref>.
In the next step, in each cluster (Voronoi cell), a Tabu Search algorithm <cit.> will be used to solve the one center problem. The steps of a generic Tabu Search algorithm are explained in in <ref> of the Online Supplement of this paper. In each cluster, several drones will fly from their origin (base), go to the truck center, deliver the packages to the customer and go back to their base. We represent a solution of the Tabu Search with a list of all nodes. As an example consider the following solution
8 1 5 7 1 2 9 1 3 10 1 4 9 1 6
for a cluster (one center problem) with 5 customer nodes and 4 drone nodes. In this solution, $1$ represents the truck node, $2$ to $6$ represents the customer nodes and $7$ to $10$ represents the drone nodes. Therefore, a solution representation like this indicates that a drone from location 8 goes to the cluster center (truck node) at location 1 and then goes to customer node 5, while drones from locations 7 and 10 also go to the cluster center and deliver the packages to customer nodes 2 and 4, respectively. Drone from location 9 first serves the customer node 3 and comes back to its home base to change the battery, then it goes to serve the customer node 6. In this solution, four drones will deliver packages for five customers, and the solution for one cluster is shown in <ref>, where the black square is the cluster center, blue stars are the customer-owned drones' location and small black circles are the customers' location. It is clear that each such list represents a unique solution of the problem but each solution can be represented with multiple such lists.
The Tabu Search algorithm is applied to this solution to find a new solution. During each iteration, an action will be applied to the current solution. An action is either to randomly switch the position of customer nodes (switching 3 and 5 in the following example)
8 1 3 7 1 2 9 1 5 10 1 4 9 1 6
or randomly change a drone node among all the drone nodes in that cluster (changing 7 to 9 in the following example).
8 1 5 9 1 2 9 1 3 10 1 4 9 1 6
This gives us an action list that has a size of $|C|(|C|-1)/2+|D|$.
In the final step of the algorithm, we solve a travelling salesman problem (TSP) to find the truck tour among all cluster centers and all customer nodes that do not have any drone node nearby. All customers in any cluster that has no drones in it, will be served by the truck. Lin-Kernighan-Helsgaun (LKH) algorithm <cit.> is used for this purpose. The LKH algorithm is a local search heuristic to solve TSP with swapping operations inside the tours to make a new tour. It is an iterative algorithm with two set of links, with each link connecting two nodes, and in each step it will decide pairs of links that need to be switched by deleting links in the first set from the tour and adding the links from the second set to the tour (a generalization of the well-known 2-opt and 3-opt exchanges). <ref> shows the TSP tour in our example. It can be seen that there are some cluster centers that are not on the truck route. This is because in those clusters all of the customers are served by the truck and thus there is no reason for the truck to go to the cluster center and wait there for the drones anymore.
$k$-means clustering
Voronoi Partition
One cluster result
Truck route
Illustration of the main steps of the Algorithm for a problem with 40 customers and $k=9$ clusters. The $k$-means clustering step is shown in (fig:ClusteringStep) and the Voronoi partitioning step is illustrated in (fig:VoronoiStep). (fig:OneClusterTS) shows the routes for the drones in the upper left cluster and (fig:TruckRouteTSP) shows the truck route going through cluster centers and customer nodes that are not served by the drones.
§ IMPROVEMENTS OF THE ALGORITHM
Next, we will show that some further improvements can be applied to the proposed model with sharing drones. In this paper, we analyze four such improvements. In the first three, we try to find the best location and the best number of truck centers by moving these centers to one of the customer nodes and merging these centers, if we could still cover the all customers using the new centers. In the last improvement, we try to utilize the battery of drones, that is if the battery remaining in the drones can deliver another package, instead of going back to the base, we let them to go back to the cluster center to pick up the next package immediately after delivering the previous one.
§.§ Moving the Cluster Centers to Their Nearest Feasible Customer Nodes
Both of our MILP models developed for Problems III and IV assume pickup centers (truck nodes) as a subset of customer nodes. We will also show in <ref> that in our algorithm moving the cluster centers to some well-selected customer nodes, provided that the new pickup center still could serve all of its assigned customers via the available assigned drones to that center, improves the performance of the algorithm. Therefore, to make the comparison between the optimization models and the algorithm valid, we incorporate this modification into our algorithm.
In the first modification we move the cluster center to the nearest customer node to that center provided that the center could still cover all of its assigned customers using its assigned drones as illustrated in <ref>. While we can save one drone trip to the customer that is now served by the truck, the length of the truck route or the routes of the other drones could either decrease or increase. We will analyze this trade-off in Section <ref>. <ref> summarizes the steps of the algorithm with this modification. One could take an alternative approach by incorporating the $k$-medoids algorithm that automatically selects one of the central customers in each cluster as the cluster center <cit.>.
Moving cluster centers
§.§ Moving the Cluster Centers to Nearest Feasible Customer Node to Area Center
The modification in Algorithm 1 may provide a slight improvement. However, we can do better by moving the cluster centers strategically having the goal of making the truck route shorter. The intuition behind this is due to the general understanding that the drones are faster than trucks especially in urban delivery operations. Also, in the centralized alternative (TSPD) it is known that for having any significant gain by combining a truck and a drone in delivery operations we must have the speed of the drone at least twice as much as the speed of the truck <cit.> and it is most often assumed to have drones faster than the truck. Thus, since drone has speed advantage, we expect to gain in delivery time by decreasing the length of truck route and increasing the length of drone routes. We will analyze this trade-off in Section <ref>. Therefore, we modify the algorithm as follows.
We first find the centroid of the cluster centers (truck nodes). Let's call it $\bm{g}$. Then, for each cluster center we find one of its feasible assigned customer nodes closest to $\bm{g}$ among all other assigned customers to this center and closer to $\bm{g}$ than the cluster center itself. We choose this customer node, if existing, as the new cluster center (truck node). The steps of the algorithm with this modification are summarized in <ref>.
§.§ Merging Centers
We can still do a significant improvement to Algorithm <ref> by trying to merge any two centers if all customers assigned to one could be served via the other by using drones assigned to either of these two centers. This comes from our assumption that the drone is faster than the truck and making the truck route shorter by reducing the number of centers could benefit us in the total delivery time. We first sort centers found in Algorithm <ref> with respect to their distance from $\bm{g}$ in a descending order. We then start from the the first center in the list, which is the farthest from $\bm{g}$, and check the possibility of its merge into its closest center. If it is feasible, we assign this center, which itself is a customer node, and all customer nodes assigned to this center, to its closest center and we serve them using combined set of drones assigned to these two centers. The feasibility here is defined as whether drones in this combined set could serve all customers assigned to these two centers via the latter center while staying within their flying range $L$. This also limits the impact of the very restricting constraint of considering a range of $L/4$ in the initial clustering step. Once we have checked the possibility of a merge for all $k$ centers and if at least one merge is happened, we repeat this step with the remaining number of centers. This process continues until no more merge is possible. The pseudocode of the algorithm with this modification is presented in <ref>.
$\noun{RechargingRouteFinderMC1}\:(R,C,D,L)$ Generates the truck route and all the drone routes after finding the best location for the truck centers in the continuous space and then moving those centers to the nearest customer's place.
Input region $R$, set of customer nodes $C$, set of drone nodes $D$, and drone range $L$.
Truck route $T_0$ and routes of drones $T_j\,, \; j\in D$ used for serving the customers.
Let $T_0$ denote the truck route and $T_j$ denote the route for drone $j\in D$
Find a feasible $k$ and cluster all customer nodes in $C$ into $k$ clusters within distance $L/4$ using a binary search and $k$-means clustering algorithms
Let $\bm{m}_1,...,\bm{m}_k$ be the cluster centers and generate their Voronoi tessellation
Let $V_1,...,V_k$ be the $k$ Voronoi cells and use them to assign all customer and drone nodes to their corresponding centers if their distance to the center is less than $L/4$
Let $\bm{z}_{\ell} = \argmin_{\bm{z}_j \in V_{\ell}} \|\bm{z}_j - \bm{m}_{\ell}\|\,, \; \forall j\in C$
Let $\bm{m}_{\ell}^{\prime} = \bm{z}_{\ell}$
Let $C_{\ell}=\{\bm{z}_j \in V_{\ell} \,, \; j\in C\}$
$\bm{z}_j \in C_{\ell} \; \&\& \; \bm{z}_j \neq \bm{m}_{\ell}^{\prime}$
$\bm{z}_j$ can be served via $\bm{m}_{\ell}^{\prime}$ using drones in $V_{\ell}$
$|C_{\ell}\backslash \{\bm{m}_{\ell}^{\prime}\}| = 1$ &&
$2*\|\bm{z}_j - \bm{m}_{\ell}^{\prime}\| / v_{T} \leq (\|\bm{y}_i - \bm{m}_{\ell}^{\prime}\| + \|\bm{m}_{\ell}^{\prime} - \bm{z}_j\| + \|\bm{z}_j - \bm{y}_i\|)/v_{D} \,, \; \bm{y}_i \in V_{\ell}\,, \; i\in D$
Let $\bm{z}_j$ be a truck node, i.e., to be served by the truck
Assign $\bm{z}_j$ to $\bm{m}_{\ell}^{\prime}$, i.e., to be served by a drone
Let $\bm{z}_j$ be a truck node, i.e., to be served by the truck
Update $k$ to be the updated number of truck nodes and reindex them as $\bm{m}_1^{\prime},...,\bm{m}_k^{\prime}$
Run the Tabu Search algorithm to find the drone routes in all Voronoi cells, i.e., $T_j\,, \; j \in D$
Set $T_0 = \{\bm{m}_1^{\prime}, \bm{m}_2^{\prime}\}$
$k\geq 3$
Run the Lin-Kernighan-Helsgaun algorithm to find the truck route $T_0$ through $\bm{m}_1^{\prime},...,\bm{m}_k^{\prime}$
$T_0$ and $T_j\,, \; j\in D$
$\noun{RechargingRouteFinderMC2}\:(R,C,D,L)$ Generates the truck route and all the drone routes. It first finds the best location for the truck centers and then moves those centers to the nearest customer's place. Finally, it finds the centroid of these centers and moves these centers to one of their assigned customers that is closest to this centroid and keeps the delivery feasible.
Input region $R$, set of customer nodes $C$, set of drone nodes $D$, and drone range $L$.
Truck route $T_0$ and routes of drones $T_j\,, \; j\in D$ used for serving the customers.
Let $T_0$ denote the truck route and $T_j$ denote the route for drone $j\in D$
Find a feasible $k$ and cluster all customer nodes in $C$ into $k$ clusters within distance $L/4$ using a binary search and $k$-means clustering algorithms
Let $\bm{m}_1,...,\bm{m}_k$ be the cluster centers and generate their Voronoi tessellation
Let $V_1,...,V_k$ be the $k$ Voronoi cells and use them to assign all customer and drone nodes to their corresponding centers if their distance to the center is less than $L/4$
Let $\bm{z}_{\ell} = \argmin_{\bm{z}_j \in V_{\ell}} \|\bm{z}_j - \bm{m}_{\ell}\|\,, \; \forall j\in C$
Let $\bm{m}_{\ell}^{\prime} = \bm{z}_{\ell}$
Let $C_{\ell}=\{\bm{z}_j \in V_{\ell} \,, \; j\in C\}$
$\bm{z}_j \in C_{\ell} \; \&\& \; \bm{z}_j \neq \bm{m}_{\ell}^{\prime}$
$\bm{z}_j$ can be served via $\bm{m}_{\ell}^{\prime}$ using drones in $V_{\ell}$
$|C_{\ell}\backslash \{\bm{m}_{\ell}^{\prime}\}| = 1$ &&
$2*\|\bm{z}_j - \bm{m}_{\ell}^{\prime}\| / v_{T} \leq (\|\bm{y}_i - \bm{m}_{\ell}^{\prime}\| + \|\bm{m}_{\ell}^{\prime} - \bm{z}_j\| + \|\bm{z}_j - \bm{y}_i\|)/v_{D} \,, \; \bm{y}_i \in V_{\ell}\,, \; i\in D$
Let $\bm{z}_j$ be a truck node, i.e., to be served by the truck
Assign $\bm{z}_j$ to $\bm{m}_{\ell}^{\prime}$, i.e., to be served by a drone
Let $\bm{z}_j$ be a truck node, i.e., to be served by the truck
Update $k$ to be the updated number of truck nodes and reindex them as $\bm{m}_1^{\prime},...,\bm{m}_k^{\prime}$
Let $\bm{g}$ be the centroid of $\bm{m}_1^{\prime},...,\bm{m}_k^{\prime}$
Set $d_j = \|\bm{z}_j - \bm{g}\|\,, \; \forall j\in C$ and $d_{\bm{m}_{\ell}^{\prime}} = \|\bm{m}_{\ell}^{\prime} - \bm{g}\|$
Let $A_{\ell}$ be an array of customer nodes in $V_{\ell}$ sorted in ascending order of $d_j\,, \; \forall j\in C$
Find the customer node $\bm{z}_{\ell}$ with the smallest index in $A_{\ell}$ and $d_{\bm{z}_{\ell}} \leq d_{\bm{m}_{\ell}^{\prime}}$ that has at least one drone within distance $L/4$ and if chosen as cluster center we can still serve all other customers in $V_{\ell}$ using drones in $V_{\ell}$ and let $\bm{m}_{\ell}^{\prime\prime} = \bm{z}_{\ell}$. If no such point exists let $\bm{m}_{\ell}^{\prime\prime} = \bm{m}_{\ell}^{\prime}$
Run the Tabu Search algorithm to find the drone routes in all Voronoi cells, i.e., $T_j\,, \; j \in D$
Set $T_0 = \{\bm{m}_1^{\prime\prime}, \bm{m}_2^{\prime\prime}\}$
$k\geq 3$
Run the Lin-Kernighan-Helsgaun algorithm to find the truck route $T_0$ through $\bm{m}_1^{\prime\prime},...,\bm{m}_k^{\prime\prime}$
$T_0$ and $T_j\,, \; j\in D$
$\noun{RechargingRouteFinderMC3}\:(R,C,D,L)$ – Generates the truck route and all the drone routes. It first finds the best location for the truck centers and then moves those centers to the nearest customer's place. It then finds the centroid of these centers and moves these centers to one of their assigned customers that is closest to this centroid and keeps the delivery feasible. Finally, it tries to merge these centers to shorten the truck tour.
Input region $R$, set of customer nodes $C$, set of drone nodes $D$, and drone range $L$.
Truck route $T_0$ and routes of drones $T_j\,, \; j\in D$ used for serving the customers.
Let $T_0=\emptyset$ denote the truck route and $T_j=\emptyset$ denote the route for drone $j\in D$
Find a feasible $k$ and cluster all customer nodes in $C$ into $k$ clusters within distance $L/4$ using a binary search and $k$-means clustering
Let $\bm{m}_1,...,\bm{m}_k$ be the cluster centers and generate their Voronoi tessellation
Let $V_1,...,V_k$ be the $k$ Voronoi cells and use them to assign all customer and drone nodes to their corresponding centers within distance $L/4$
Let $\bm{z}_{\ell} = \argmin_{\bm{z}_j \in V_{\ell}} \|\bm{z}_j - \bm{m}_{\ell}\|\,, \; \forall j\in C$
Let $\bm{m}_{\ell}^{\prime} = \bm{z}_{\ell}$
Let $C_{\ell}=\{\bm{z}_j \in V_{\ell} \,, \; j\in C\}$
$\bm{z}_j \in C_{\ell} \; \&\& \; \bm{z}_j \neq \bm{m}_{\ell}^{\prime}$
$\bm{z}_j$ can be served via $\bm{m}_{\ell}^{\prime}$ using drones in $V_{\ell}$
$|C_{\ell}\backslash \{\bm{m}_{\ell}^{\prime}\}| = 1$ &&
$2*\|\bm{z}_j - \bm{m}_{\ell}^{\prime}\| / v_{T} \leq (\|\bm{y}_i - \bm{m}_{\ell}^{\prime}\| + \|\bm{m}_{\ell}^{\prime} - \bm{z}_j\| + \|\bm{z}_j - \bm{y}_i\|)/v_{D} \,, \; \bm{y}_i \in V_{\ell}\,, \; i\in D$
Let $\bm{z}_j$ be a truck node, i.e., to be served by the truck
Assign $\bm{z}_j$ to $\bm{m}_{\ell}^{\prime}$, i.e., to be served by a drone
Let $\bm{z}_j$ be a truck node, i.e., to be served by the truck
Update $k$ to be the updated number of truck nodes and reindex them as $\bm{m}_1^{\prime},...,\bm{m}_k^{\prime}$
Let $\bm{g}$ be the centroid of $\bm{m}_1^{\prime},...,\bm{m}_k^{\prime}$
Set $d_j = \|\bm{z}_j - \bm{g}\|\,, \; \forall j\in C$ and $d_{\bm{m}_{\ell}^{\prime}} = \|\bm{m}_{\ell}^{\prime} - \bm{g}\|$
Let $A_{\ell}$ be an array of customer nodes in $V_{\ell}$ sorted in ascending order of $d_j\,, \; \forall j\in C$
Find the customer node $\bm{z}_{\ell}$ with the smallest index in $A_{\ell}$ and $d_{\bm{z}_{\ell}} \leq d_{\bm{m}_{\ell}^{\prime}}$ that has at least one drone within distance $L/4$ and if chosen as cluster center we can still serve all other customers in $V_{\ell}$ using drones in $V_{\ell}$ and let $\bm{m}_{\ell}^{\prime\prime} = \bm{z}_{\ell}$. If no such point exists let $\bm{m}_{\ell}^{\prime\prime} = \bm{m}_{\ell}^{\prime}$
Let $\Omega = [\bm{m}_1^{\prime\prime},...,\bm{m}_{k}^{\prime\prime}]$, $\Omega^{\prime} = \emptyset$, and $count=k$
$count > 0$
Set $count=0$
$k = 1$
Sort centers $\bm{m}_{\ell}^{\prime\prime} \in \Omega, \ell=1,...,k$ in a descending order of their distances to $\bm{g}$ and reindex them accordingly
Reallocate customer and drone nodes to their nearest centers. For customer nodes pick the closest center that could serve it
Let $\bm{\mu} = \argmin_{\bm{m}_{\ell}^{\prime\prime} \in \Omega \backslash \{\Omega^{\prime}\cup \{\bm{m}_{r}^{\prime\prime}\}\}} \|\bm{m}_{\ell}^{\prime\prime} - \bm{m}_{r}^{\prime\prime}\|\,, \; \ell=1,...,k$ and let $s$ be its index in $\Omega$, i.e., $\bm{\mu} = \bm{m}_{s}^{\prime\prime}$. Break a tie with closeness to $\bm{g}$
Let $feasibility = true$
$\bm{z}_j \in V_r \cup V_s\,, \; j\in C$
Find $\bm{y}_i \in V_{r} \cup V_{s}\,, \; i\in D$ such that $\|\bm{y}_i - \bm{\mu}\| + \|\bm{\mu} - \bm{z}_j\| + \|\bm{z}_j - \bm{y}_i\| \leq L$. If no such drone exists, let $feasibility = false$
$feasibility = false$
$feasibility = true$
$k = 2$
Merge the center with fewer customers (say $\bm{m}_{2}^{\prime\prime}$) into the center with more customers (say $\bm{m}_{1}^{\prime\prime}$) and reallocate its assigned customers to that center
Set $\Omega^{\prime} = \Omega^{\prime} \cup \{\bm{m}_{2}^{\prime\prime}\}$
Merge the center $\bm{m}_{r}^{\prime\prime}$ and $V_r$ into $\bm{m}_{s}^{\prime\prime}$ and $V_s$ and allocate its assigned (customer and drone) nodes to $\bm{m}_{s}^{\prime\prime}$
Set $\Omega^{\prime} = \Omega^{\prime} \cup \{\bm{m}_{r}^{\prime\prime}\}$
Set $count = count +1$
Set $\Omega = \Omega \backslash \Omega^{\prime}$, $k = length(\Omega)$, and $\Omega^{\prime} = \emptyset$
Run the Tabu Search algorithm to find the drone routes in all Voronoi cells, i.e., $T_j\,, \; j \in D$
Set $T_0 = \{\bm{m}_1^{\prime\prime}, \bm{m}_2^{\prime\prime}\}$
$k\geq 3$
Run the Lin-Kernighan-Helsgaun algorithm to find the truck route $T_0$ through $\bm{m}_1^{\prime\prime},...,\bm{m}_k^{\prime\prime}$
$T_0$ and $T_j\,, \; j\in D$
§.§ Utilization of Battery
In Problem IV, we assumed that drones can revisit the pickup center (truck's stopping place) as long as they do not violate their range (battery duration) limit. Therefore, we need to modify the algorithm to utilize the battery of the drones and allow drones to revisit their assigned pickup center before returning to their base. We assume no limit on the number of times a drone could go back to a pickup center to deliver another package as long as it remains within the range. By implementing this modification, obviously, we may expect some savings since if one drone can deliver another package without going back to the base, it can save time on that trip. However, this adds to the waiting time of the truck and makes the truck route costlier. This trade-off is analyzed in Section <ref>. We apply this modification to all three Algorithms <ref>, <ref>, and <ref>.
We skip presenting the pseudocode here for brevity (the only difference is allowing revisiting in range conditions and in the TS step). The modification in the tabu search step is applied by changing the solution representation. Our original example of a solution in a cluster for a problem with 5 customer nodes and 4 drone nodes can change as following
8 1 5 7 1 2 9 1 3 1 6 10 1 4
In this solution, drone from location 9 first serves the customer node 3 and then revisits the truck node $1$ (without going to its base first) to serve the customer node 6.
§ COMPUTATIONAL RESULTS
As problems III and IV are the generalized versions of Problems I and II, we only do our computational results for Problems III and IV. We will see that for certain values of input parameters we actually get solutions for the special cases of Problems I and II. We tested the MILP model of Problems III and IV on a set of two synthetic problem instances with 16 and 12 ($P1$) and 16 and 16 ($P2$) customer and drone nodes, respectively, distributed uniformly at random in a unit square. The speed of drones is set to be 2 times the speed of the truck in both models. For the drone flying range $L$ we took the maximum length of a drone in a unit box, i.e., $2+\sqrt{2} \simeq 3.4$, and with increments of 0.2 we generated 17 instance for each problem. In the tabu search step of the algorithms we have set |Tabu List| = 0.7 * |Action List| with a termination condition of 1000 iterations. We have also set the precision to be $\epsilon=10^{-5}$ for all solutions.
We implemented our models in Julia 1.9.2 using Gurobi Optimizer 10.0.2 on
an Intel Core i7-5600U @2.6GHz, 8GB DDR3 RAM computer. We ran our algorithms for the same problem instances on the same computer using MATLAB R2022b.
We also ran our algorithms on larger problem instances for comparison with the results of traditional centralized delivery system (pure TSP model) and the coordinated truck and drone delivery (TSP-D model).
§.§ Results for Multiple Centers with Recharging
Table <ref> compares the computational results of the our optimization model for Problem III and the proposed algorithms on the 34 instances of $P1$ and $P2$. As it is evident from the table, our algorithms are much faster than the optimization model. The average running time of the final version of our algorithm (Algorithm <ref>) is 8.94 seconds versus 885.13 seconds of the optimization model for $P1$ and 9.56 seconds versus 172.23 seconds for $P2$. The average (maximum) optimality gap of the solutions provided by Algorithm <ref> for $P1$ and $P2$ is 11.92% (57.66%) and 16.61% (102.71%), respectively.
Figure <ref> illustrates optimal solutions obtained by our optimization model and optimal/suboptimal solutions obtained by Algorithm <ref> for $P1$ and $P2$ for select instances of $L$.
Comparison between the results of the optimization model for Problem III (Multiple Centers with Recharging) and the algorithms on two synthetic examples with 16 and 12 ($P1$) and 16 and 16 ($P2$) customers nodes and drone nodes, respectively, distributed randomly in a unit box and for different values of $L$. The column “Time (s)” shows the computational time in seconds. The column “Gap” presents the optimality gap of the solutions found by the algorithms. The instances for which the optimal solution is found by our algorithms are shown in bold (0.00% gap).
11cP1 (16-12) – Recharging
2-12 2[4]*L 2cOpt. Model 3cAlgorithm 1 3cAlgorithm 2 3cAlgorithm 3
2-12 Time (s) Obj. Value Time (s) Obj. Value Gap (%) Time (s) Obj. Value Gap (%) Time (s) Obj. Value Gap (%)
1c|0.2 2.80 3.3325 0.36 3.3325 0.00% 0.36 3.3325 0.00% 0.42 3.3325 0.00%
1c|0.4 21.93 3.2561 0.63 3.2837 0.85% 0.61 3.2837 0.85% 0.62 3.3385 2.53%
1c|0.6 384.27 3.0865 0.70 3.2659 5.81% 0.67 3.2868 6.49% 1.12 3.7672 22.05%
1c|0.8 7772.69 2.4655 0.89 3.2056 30.02% 0.81 3.2881 33.36% 1.65 2.7818 12.83%
1c|1 483.70 1.5588 0.96 3.6451 133.85% 0.92 3.6421 133.66% 1.62 2.4575 57.66%
1c|1.2 5445.16 1.4339 1.14 2.8359 97.78% 1.14 2.8080 95.84% 3.56 1.7979 25.39%
1c|1.4 11.45 0.5176 1.19 3.1534 509.28% 1.35 2.9181 463.82% 13.04 0.6887 33.07%
1c|1.6 97.43 0.4828 1.05 3.1534 553.13% 1.17 2.9218 505.17% 12.88 0.5029 4.16%
1c|1.8 43.86 0.4686 3.12 1.7029 263.40% 3.18 1.7107 265.07% 13.28 0.5849 24.82%
1c|2 100.87 0.4686 3.03 1.7029 263.40% 3.06 1.4503 209.51% 12.84 0.4772 1.84%
1c|2.2 92.24 0.4686 2.96 1.7029 263.40% 2.95 1.4503 209.51% 12.83 0.4869 3.90%
1c|2.4 97.30 0.4686 12.63 0.4737 1.09% 12.82 0.4869 3.90% 13.11 0.4832 3.12%
1c|2.6 100.90 0.4686 12.64 0.4694 0.18% 12.62 0.4755 1.47% 14.06 0.4734 1.03%
1c|2.8 96.93 0.4686 12.66 0.4836 3.21% 12.85 0.4832 3.12% 12.69 0.4775 1.91%
1c|3 98.45 0.4686 12.91 0.4832 3.12% 12.96 0.5029 7.32% 12.86 0.4828 3.03%
1c|3.2 98.74 0.4686 12.78 0.4836 3.21% 12.71 0.4734 1.03% 12.86 0.4846 3.41%
1c|3.4 98.49 0.4686 12.78 0.4755 1.47% 13.15 0.4832 3.12% 12.59 0.4775 1.91%
mygray Average 885.13 - 5.44 - 125.48% 5.49 - 114.31% 8.94 - 11.92%
11cP2 (16-16) – Recharging
2-12 2[4]*L 2cOpt. Model 3cAlgorithm 1 3cAlgorithm 2 3cAlgorithm 3
2-12 Time (s) Obj. Value Time (s) Obj. Value Gap (%) Time (s) Obj. Value Gap (%) Time (s) Obj. Value Gap (%)
1c|0.2 2.63 3.5817 0.55 3.5817 0.00% 0.44 3.5817 0.00% 0.47 3.5817 0.00%
1c|0.4 82.08 3.5817 0.46 3.6717 2.51% 0.40 3.6717 2.51% 0.49 3.7663 5.15%
1c|0.6 19.84 3.2325 0.35 3.4684 7.30% 0.33 3.7302 15.40% 0.55 3.7508 16.03%
1c|0.8 90.52 2.7967 0.90 3.5360 26.43% 0.66 3.5360 26.43% 1.11 3.3767 20.74%
1c|1 209.00 2.2349 1.25 3.6115 61.59% 1.22 3.6115 61.59% 1.22 3.6115 61.59%
1c|1.2 631.79 1.7369 1.28 3.3403 92.31% 1.31 3.3403 92.31% 3.16 2.5199 45.08%
1c|1.4 1645.00 1.1662 1.17 3.3403 186.43% 1.20 3.3403 186.43% 3.26 2.3639 102.71%
1c|1.6 69.15 0.4673 1.38 3.2021 585.27% 1.36 2.9233 525.62% 15.25 0.4673 0.00%
1c|1.8 16.60 0.4438 1.19 3.5377 697.22% 1.25 3.3313 650.72% 15.41 0.4438 0.00%
1c|2 6.51 0.4438 1.13 3.0457 586.34% 1.18 3.0333 583.56% 15.05 0.5803 30.76%
1c|2.2 37.22 0.4438 3.78 1.6636 274.90% 3.44 1.6636 274.90% 15.66 0.4438 0.00%
1c|2.4 21.44 0.4438 3.56 1.6636 274.90% 3.40 1.6636 274.90% 15.02 0.4438 0.00%
1c|2.6 13.89 0.4438 3.41 1.6636 274.90% 3.43 1.6636 274.90% 14.99 0.4438 0.00%
1c|2.8 11.76 0.4438 14.77 0.4438 0.00% 15.72 0.4438 0.00% 15.50 0.4450 0.28%
1c|3 23.71 0.4438 15.00 0.4438 0.00% 14.89 0.4450 0.28% 15.15 0.4438 0.00%
1c|3.2 23.79 0.4438 14.71 0.4598 3.61% 14.75 0.4438 0.00% 15.26 0.4438 0.00%
1c|3.4 22.98 0.4438 14.87 0.4438 0.00% 14.88 0.4438 0.00% 14.94 0.4438 0.00%
mygray Average 172.23 - 4.69 - 180.81% 4.70 - 174.68% 9.56 - 16.61%
$P1,\;L = 0.2$
$P1,\;L = 0.6$
$P1,\;L = 0.8$
$P1,\;L = 0.2$
$P1,\;L = 0.6$
$P1,\;L = 0.8$
$P1,\;L = 1.0$
$P1,\;L = 1.4$
$P1,\;L = 1.8$
$P1,\;L = 1.0$
$P1,\;L = 1.4$
$P1,\;L = 1.8$
$P2,\;L = 0.2$
$P2,\;L = 0.6$
$P2,\;L = 0.8$
$P2,\;L = 0.2$
$P2,\;L = 0.6$
$P2,\;L = 0.8$
$P2,\;L = 1.0$
$P2,\;L = 1.4$
$P2,\;L = 1.8$
$P2,\;L = 1.0$
$P2,\;L = 1.4$
$P2,\;L = 1.8$
Examples of optimal solutions of Problem III (multiple centers with recharging) obtained by the optimization model for problems $P1$ with 16 customer nodes and 12 drone nodes and $P2$ with 16 customer nodes and 16 drone nodes for select values of drone range $L$ are provided on the left three columns. Their corresponding (optimal/suboptimal) solutions obtained by Algorithm <ref> is shown on the three columns on the right.
It can be seen that drones are not necessarily assigned to their closest center or closest customer. This is due to the fact that the truck is not required to wait at a center while its assigned drones are traversing the last edge of their route (after delivery) to go back to their base.
For small drone range such as $L = 0.2$ (with a 2-to-1 relative drone speed assumption) we are better off to not use the drones and for very large drone ranges such as $L\geq 1.8$ the problem reduces to a one center (drone-only) delivery problem.
§.§ Results for Multiple Centers with Revisiting
We also tested the MILP model of Problem IV and our algorithms
on the same problem instances with the same relative drone speed of 2.
<ref> summarizes the results of this experiment. We observe that the average computational time of the optimization model, when compared to that of Problem III, is significantly increased. However, this is mostly due to one to two values of $L$ and for the rest the run time is closer to, although still higher than, what we observed for Problem III. This increase is reasonable because the solution space of Problem IV is substantially larger than the solution space of Problem III. The revisiting model achieved better solutions than the recharging model in 13 instances of $P1$ and one instance of $P2$.
The average running time of the final version of our algorithm (Algorithm <ref>) is 6.11 seconds versus 3397.84 seconds of the optimization model for $P1$ and 6.66 seconds versus 1202.10 seconds for $P2$. The average (maximum) optimality gap of the solutions provided by Algorithm <ref> for $P1$ and $P2$ is 28.01% (279.14%) and 17.82% (83.38%), respectively.
<ref> illustrates solutions of the considered instances of Problem IV solved by our optimization model for select values of $L$ as well as the corresponding solutions obtained by Algorithm <ref>.
Comparison between the results of the optimization model for Problem IV (Multiple Centers with Revisiting) and the algorithms on two synthetic examples with 16 and 12 and 16 and 16 customers nodes and drone nodes, respectively, distributed randomly in a unit box and for different values of $L$. The column “Time (s)” shows the computational time in seconds. The column “Gap” presents the optimality gap of the solutions found by the algorithms. The instances for which the optimal solution is found by our algorithms are shown in bold (0.00% gap). The underlined optimal objective values show the instances that the revisiting model achieved better solution than the recharging model.
11cP1 (16-12) – Revisiting
2-12 2[4]*L 2cOpt. Model 3cAlgorithm 1 3cAlgorithm 2 3cAlgorithm 3
2-12 Time (s) Obj. Value Time (s) Obj. Value Gap (%) Time (s) Obj. Value Gap (%) Time (s) Obj. Value Gap (%)
1c|0.2 2.99 3.3325 0.40 3.3325 0.00% 0.37 3.3325 0.00% 0.48 3.3325 0.00%
1c|0.4 18.87 3.2561 0.63 3.6008 10.59% 0.51 3.5956 10.42% 0.61 3.3176 1.89%
1c|0.6 954.06 3.0865 0.48 3.4071 10.39% 0.46 3.4071 10.39% 0.75 3.6581 18.52%
1c|0.8 12324.75 2.4651 0.52 3.4377 39.45% 0.61 3.4765 41.03% 1.25 2.8226 14.50%
1c|1 1758.83 1.5250 0.68 3.2693 114.38% 0.83 3.3081 116.92% 1.09 2.2191 45.51%
1c|1.2 40669.77 1.3796 0.95 3.2953 138.86% 0.87 2.7749 101.14% 2.39 1.9162 38.89%
1c|1.4 34.30 0.5029 0.99 2.7771 452.25% 0.91 2.7862 454.06% 2.03 1.9066 279.14%
1c|1.6 56.83 0.4742 1.11 2.9175 515.21% 1.09 2.6562 460.12% 9.84 0.5181 9.25%
1c|1.8 87.52 0.4563 2.23 1.6961 271.72% 2.21 1.6961 271.72% 9.48 0.6040 32.37%
1c|2 120.90 0.4563 2.35 1.6742 266.91% 2.40 1.4503 217.86% 9.44 0.4690 2.79%
1c|2.2 142.73 0.4563 2.42 1.6742 266.91% 2.31 1.4447 216.61% 9.55 0.4775 4.66%
1c|2.4 178.49 0.4563 9.54 0.4709 3.21% 9.74 0.4727 3.60% 9.61 0.4775 4.66%
1c|2.6 228.82 0.4563 9.47 0.4805 5.31% 9.32 0.4856 6.43% 9.55 0.4846 6.20%
1c|2.8 226.00 0.4563 9.60 0.4832 5.90% 9.59 0.4686 2.70% 9.53 0.4694 2.88%
1c|3 272.74 0.4563 9.39 0.4828 5.81% 9.49 0.4828 5.81% 9.38 0.4710 3.23%
1c|3.2 335.45 0.4563 9.61 0.4828 5.81% 9.40 0.4767 4.47% 9.42 0.4828 5.81%
1c|3.4 350.22 0.4563 9.39 0.4828 5.81% 9.45 0.4742 3.93% 9.51 0.4832 5.90%
mygray Average 3397.84 - 4.10 - 124.62% 4.09 - 113.36% 6.11 - 28.01%
11cP2 (16-16) – Revisiting
2-12 2[4]*L 2cOpt. Model 3cAlgorithm 1 3cAlgorithm 2 3cAlgorithm 3
2-12 Time (s) Obj. Value Time (s) Obj. Value Gap (%) Time (s) Obj. Value Gap (%) Time (s) Obj. Value Gap (%)
1c|0.2 3.44 3.5817 0.31 3.5817 0.00% 0.33 3.5817 0.00% 0.33 3.5817 0.00%
1c|0.4 55.07 3.5817 0.28 3.5817 0.00% 0.30 3.5817 0.00% 0.38 3.7663 5.15%
1c|0.6 16.14 3.2325 0.38 3.7280 15.33% 0.34 3.7280 15.33% 0.43 3.7508 16.03%
1c|0.8 97.30 2.7967 0.58 3.5360 26.43% 0.65 3.5360 26.43% 0.85 3.3767 20.74%
1c|1 475.36 2.2349 0.86 3.5589 59.24% 1.10 3.5571 59.16% 0.93 3.5588 59.23%
1c|1.2 7490.24 1.7369 1.15 3.1890 83.60% 1.07 3.1868 83.47% 1.06 3.1851 83.38%
1c|1.4 10233.86 1.1598 1.13 3.1689 173.23% 1.09 3.1851 174.64% 2.16 2.1160 82.45%
1c|1.6 24.63 0.4673 1.09 3.1689 578.16% 1.04 2.9051 521.71% 10.62 0.5055 8.18%
1c|1.8 57.27 0.4438 0.95 3.0112 578.58% 1.06 2.9436 563.34% 10.80 0.4774 7.59%
1c|2 54.01 0.4438 0.98 3.4090 668.23% 0.96 3.1674 613.78% 10.66 0.4438 0.00%
1c|2.2 81.60 0.4438 5.38 1.8542 317.85% 5.40 1.8129 308.55% 10.80 0.4675 5.36%
1c|2.4 124.37 0.4438 2.16 2.1089 375.25% 2.12 2.0368 358.99% 10.83 0.4631 4.37%
1c|2.6 167.88 0.4438 2.13 2.1235 378.54% 2.06 1.7725 299.44% 10.55 0.4640 4.56%
1c|2.8 280.61 0.4438 10.69 0.4438 0.00% 10.65 0.4438 0.00% 10.78 0.4701 5.94%
1c|3 295.28 0.4438 10.61 0.4461 0.53% 10.65 0.4438 0.00% 10.56 0.4438 0.00%
1c|3.2 460.98 0.4438 10.63 0.4673 5.30% 10.78 0.4626 4.25% 10.71 0.4438 0.00%
1c|3.4 517.70 0.4438 10.79 0.4438 0.00% 10.77 0.4673 5.30% 10.77 0.4438 0.00%
mygray Average 1202.10 - 3.54 - 191.78% 3.55 - 178.49% 6.66 - 17.82%
$P1,\;L = 0.2$
$P1,\;L = 0.6$
$P1,\;L = 0.8$
$P1,\;L = 0.2$
$P1,\;L = 0.6$
$P1,\;L = 0.8$
$P1,\;L = 1.0$
$P1,\;L = 1.4$
$P1,\;L = 1.8$
$P1,\;L = 1.0$
$P1,\;L = 1.4$
$P1,\;L = 1.8$
$P2,\;L = 0.2$
$P2,\;L = 0.6$
$P2,\;L = 0.8$
$P2,\;L = 0.2$
$P2,\;L = 0.6$
$P2,\;L = 0.8$
$P2,\;L = 1.0$
$P2,\;L = 1.4$
$P2,\;L = 1.8$
$P2,\;L = 1.0$
$P2,\;L = 1.4$
$P2,\;L = 1.8$
Examples of solutions for Problem IV (multiple centers with revisiting) on $P1$ and $P2$ instances for select values of maximum drone range $L$ obtained by the optimization model (optimal solutions) on the three columns on the left and by the revisiting version of Algorithm <ref> (optimal/suboptimal solutions) on the three columns on the right.
Similar to the results of Problem III, small drone range makes the problem closer to a pure TSP (truck-only) model and for reasonably large $L$ the problem gets closer and closer to the one center (drone-only) delivery problem.
It can be seen that for some large enough values of $L$ in both model and algorithm solutions, e.g., in (fig:model-P1-L1.8) and (fig:MC3-P2-L1.4), some of the drones were able to revisit the pickup center after delivering a package at a customer node. Such incidents are distinguishable by hub-and-spoke type (non-triangular) drone routes.
§.§ Similarities Between the Results of the Two Problems
It is apparent in Tables <ref> and <ref> that the objective function value for the optimization model and the algorithms decreases in $L$ as expected.
We also see in Tables <ref> and <ref> that starting from $L=1.8$ the optimal objective function value remains the same. This is because at this level, the range of the drone is long enough to serve all customers with drones and thus the truck would not be used. This means for large enough $L$ and large enough relative speed of drone to truck, the problem essentially reduces to a one center problem with recharging (Problem I). This is also visible in Figures <ref> and <ref>.
Furthermore, it can be seen in the results from both Problems III and IV that the running time of the optimization model and the optimality gap of our algorithm are both very sensitive to the values of $L$. It is much easier to solve the problem when $L$ is either small or large. Optimality gap for the middle values of $L$ (between 1.0 to 1.4 in these instances) is also larger.
If the drone range is very low, which means drones can only fulfill the demand of nearby customer nodes, the problem is a pure or nearly TSP problem which is much faster to solve since its feasible region is smaller. In contrast, if the drone maximum allowed distance is in a reasonable range (between 1.0 to 1.4 in these instances), drones can be a big help in delivering the packages while still leaving a good portion of the customers to be served by the truck. Finding both drone routes and the truck route takes considerable time in this case, although the lion share of total time is due to the drone routes. Therefore, the feasibility region is much bigger than a TSP which increases the run time and makes finding optimal or high quality solutions harder. On the other end of the range, if the drone range is large enough (e.g., larger than 1.8), we can do all deliveries with drones without using the truck. In this case, the solving time is reasonable but still a bit higher than pure TSP. This is because larger values of $L$ increase the number of feasible drone routes. Moreover, we still have to determine which node(s) will serve as the truck node(s), and what's the best truck route (if any). This explains why there is a little bit of increase in solving time at the right tail of the drone range compared to the left tail. Although the solution space might be even bigger than that of the mid-level range of $L$ (e.g., between 1.0 to 1.4), it is much faster than that case because good upper bounds are found quickly (with for example fewer (or even one) centers) which helps skipping a large number of feasible solutions that use bigger number of centers.
For the same reasons we can see a similar behavior for the optimality gap
and when $L$ is either small or large, the gap between the algorithm and the optimal solution is much smaller than the results of the mid-level cases of drone range.
§.§ Comparison with Traditional Delivery System
We also compared our approach with the traditional centralized truck-only delivery system in which a single truck would deliver all packages, i.e., a pure TSP model. We ran both recharging and revisiting versions of Algorithm <ref>, which provides a decentralized alternative, as well as the LKH algorithm for the pure TSP model on several problem instances with 60 customer nodes and 40 drone nodes distributed uniformly at random in a unit square. Again, we set the relative speed of drone to be 2. For drone range we chose $L=0.8$ to ensure having both truck and drone routes.
Figure <ref> shows solutions obtained by our algorithm presented side-by-side with the corresponding TSP solution for each instance. It is easy to see the advantages of our proposed decentralized system over the conventional centralized delivery system.
On these examples, the average delivery time when we combine a truck with crowdsourced drones was 4.6280, while the average delivery time for the same problem if the truck serves all the customers was 6.2035. This crude comparison suggests an almost 25.32% improvement in the efficiency of the last-mile delivery, in our randomly generated examples, if we combine truck delivery and drone delivery in the context of sharing economy platforms. Certainly, this would highly depend on the chosen input parameters that we investigate further in Sections <ref> and <ref>.
Total Time = 6.1880
Total Time = 5.0042
Total Time = 5.8312
Total Time = 4.5983
Total Time = 6.5363
Total Time = 4.9806
Total Time = 6.2583
Total Time = 3.9290
Sample solutions for a multiple center with recharging problem with 60 customer nodes and 40 drone nodes for $L=0.8$ obtained by our Algorithm <ref>, representing a decentralized delivery system, for different random examples in a unit box with their objective function values are illustrated side by side to their corresponding TSP solutions (centralized truck-only delivery system) for the same instances obtained by the LKH algorithm.
§ SENSITIVITY ANALYSIS
In the rest of the paper, we perform an extensive sensitivity analysis with respect to several factors to study their impact on the quality of the solution and savings of the shared delivery system compared to the traditional truck-only delivery.
These factors include speed of drones, number of available drones, a measure combining speed and number of drones, and customer distribution. Moreover, we analyze the impact of improvements presented in <ref> for different values of the input parameters.
Finally, a comparison is made between three models to measure the impact of shared delivery model on carbon footprint. These three models are the traditional truck-only delivery (TSP) model, delivery with a truck and a drone where the truck carries a drone and both deliver packages in a coordinated way (we call this model TSPD), and our truck and shared drone delivery model (TSP-SD). Almost all simulations are done on an instance of the recharging problem with 60 customer nodes and several drone nodes (the number varies based on the input parameters) distributed uniformly at random or according to a distribution (depending on the analysis) in a unit square assuming $L=0.8$. In most cases, the revisiting model delivers similar results but wherever we expect a significant difference we run the simulation on the revisiting problem as well.
To analyze the sensitivity of the proposed models and algorithms that combines a truck with crowdsourced drones in the last-mile delivery operation to input parameters, we consider the impact of several factors including relative speed of drones, relative number of crowdsourced drones, and the customer and drone distribution on the solutions obtained by our algorithm. We expect similar behavior in the optimal solutions. Here we define two parameters as: $\rho_1 = \frac{V_d}{V_t}=\frac{\text{speed of drone}}{\text{speed of truck}}$ and $\rho_2 = \frac{m}{n}=\frac{\text{number of drones}}{\text{number of customers}}$. In terms of customer distributions, a multivariate normal distribution is assumed and its variance will be adjusted. We compare the results with respect to the savings in time by using the crowdsourced drones that is
\[
\frac{\text{Time cost of TSP (truck only) - Time cost of TSP-SD (crowdsourced drones combined with a truck)}}{\text{Time cost of TSP (truck only)}} \,,
\]
i.e., we compare our solutions with those obtained by the LKH algorithm.
When comparing Algorithm <ref> to the results of LKH for pure TSP, one should keep in mind that any savings suggested by our algorithm in despite the fact that for our algorithm with $L=0.8$, we usually have a considerable optimality gap, while the optimality gap of LKH for pure TSP is almost zero. This difference could be seen as additional savings if our problem is solved to optimality.
§.§ Relative Speed of Drones
To analyze the sensitivity with respect to $\rho_1$, we adjust this parameter in the range $0.75\leq \rho_1 \leq 3$ that means the drone speed can vary from 0.75 times the truck speed to three times the truck speed, while we fix the number of drones to be half or twice of the number of customers, i.e., $\rho_2\in\{0.5,2\}$.
The choice of the range for $\rho_1$ is driven by the fact that it is known in the literature of the centralized alternative (TSPD) that for having any significant gain by combining a truck and a drone in delivery operations we must have $\rho_1 \geq 2$ <cit.> and it is most often assumed to have $\rho_1 \geq 1$.
$\rho_2 = 0.5$
$\rho_2 = 2$
Sensitivity analysis of the results of Algorithm <ref> with respect to the relative speed of drones for two cases of drones supply.
From <ref> we can find that, when there is considerable shortage of drones relative to the number of customers, only when the drone speed is less than 0.75 times the speed of the truck ($\rho_1 \leq 1.15$), the TSP model will have less time cost than the model with shared drones. Also, from <ref> we can observe that
when relative drone supply increases to 2 the chart shifts to left, so to speak, and we see a positive and significant savings for the shared model for all values of $\rho_1$ in the considered range ($0.75 \leq \rho_1 \leq 3$).
In both situations, the TSP with sharing drones model has a consistently better performance than the TSP model when drones are reasonably faster than the speed of the truck ($\rho_1 \geq 1.15$). Intuitively, the combined model gains help from the drones by working (potentially) in parallel to each other and to the truck, and on the other hand loses some time by idling the truck at the cluster centers to wait for the drones to fly back and forth and deliver packages. The overall, tradeoff between these two opposite forces shows that the combined model has a significant advantage over the pure TSP model and the proposed model of combined truck and shared drones performs significantly better. This advantage becomes bigger by increasing drones speed relative to the speed of truck ($\rho_1$) with a diminishing return. Both these properties are quite consistent with the results of <cit.> that shows the gain for a TSPD model over TSP is proportional to $\sqrt{\rho_1}$. Finally, we also see that increasing the supply of drones ($\rho_2$) increases the savings. This phenomena is similar to growing the supply side in other sharing economy (two-sided market) applications.
§.§ Relative Number of Drones
We analyze the sensitivity with respect to $\rho_2$, by varying this parameter in the range $0.3 \leq \rho_2 \leq 2$, while we fix the speed of drones to either be one or two times that of the truck, i.e., $\rho_1\in\{1,2\}$. The choice of the range for $\rho_2$ is motivated by the well-known importance of having a balance between supply and demand in two-sided markets, i.e., $\rho_2=1$.
$\rho_1 = 1$
$\rho_1 = 2$
Sensitivity analysis of the results of Algorithm <ref> with respect to the relative number of drones.
From <ref>, we can find that if the speed of the drones is equal to the speed of the truck, for the TSP-SD model it is sufficient to have $\rho_2 \geq 0.6$ to consistently outperform the pure TSP model. Increasing the relative speed of drones to 2 (<ref>) both increases the magnitude of savings and the enlarges the range of $\rho_2$ that deliver positive savings ($\rho_2 \geq 0.45$). The same increasing and concave pattern can be observed in $\rho_2$ as well. Comparing <ref> to <ref> we see that the sensitivity of the performance of TSP-SD to relative speed of drones is much more than its sensitivity to relative number of drones. It is reasonable to see that the number of drones is not as important as the speed of drones. This is because, if drones are fast enough, we only need a few drones in that area to make the performance of the model with drones better than the normal TSP model, while having huge number of slow moving drones may still force us to prefer truck over the drone on many of the deliveries.
§.§ Combining
In this section, we try to combine the analysis of $\rho_1$ and $\rho_2$ together to see how much we can improve the delivery performance by including the crowd-owned drones when we consider both parameters at the same time, i.e., comparing TSP-SD with TSP. We let the parameters to vary in the ranges $0.75\leq \rho_1 \leq 3$ and $0.3 \leq \rho_2 \leq 2$.
The sensitivity analysis of the results of Algorithm <ref> with respect to the relative speed of drones combined with the relative number of drones. The positive savings in (fig:SA3CompleteRange) are shown in (fig:SA3PositiveRange).
From <ref>, we can see that if both $\rho_1$ and $\rho_2$ are small, the normal TSP will have much better performance than the model with sharing drones (TSP-SD), which we can also observe from the two previous sections, especially in <ref>. On the other hand, from <ref>, which shows the range with positive savings, we can see that the performance of the TSP-SD model improves as $\rho_1$ and $\rho_2$ increase. It reaches its maximum percentage of savings around $63.47\%$ when both $\rho_1$ and $\rho_2$ are around our set upper bounds of 3 and 2, respectively, implying that the improvement could increase further by relaxing this upper end of the range. Meanwhile, the growth rate becomes smaller as $\rho_1$ and $\rho_2$ grow, which is apparent by the concavity of the curve and suggests a diminishing return phenomena. We expect even bigger savings for the revisiting model but the pattern stays the same. For the other centralized alternative (TSPD) it is shown that for $\rho_1 \leq 2$ the gain in delivery time by using a drone is insignificant <cit.>. However, our shared drone delivery system shows considerable gain for a wider range of $\rho_1$ that makes our model more applicable across a variety of available drone technologies and regulatory environments.
§.§ Population Distribution
We consider two cases with different population distributions. The first case has one population center of 60 customers and 40 drone service providers distributed inside a unit box according to a multivariate Gaussian distribution with mean $\mu=(0.5,0.5)$ and covariance matrices $\Sigma=\usebox{\smlmats}$, where $\sigma^2 \in [0.005,0.1]$ with increments of 0.005.
The second case is a clustered data with four population centers of 60 customers and 40 drone providers (across the clusters) distributed inside a unit box according to an even mixture of four truncated multivariate Gaussian distributions with means $\mu_{1},\mu_{2},\mu_{3},\mu_{4}=(0.25.0.25),(0.25,0.75),(0.75,0.75),(0.25,0.75)$ and covariance matrices $\Sigma_{1}=\Sigma_{2}=\Sigma_{3}=\Sigma_{4}=\usebox{\smlmats}$, where $\sigma^2 \in [0.005,0.1]$ with increments of 0.005. In both cases we assume $L=0.8$ and $\rho_1 = 2$ and compare the results of Algorithm <ref> with the results of LKH for pure TSP for different values of $\sigma^2$. The <ref> shows examples of generated instances with the solutions found by the recharging version of our Algorithm <ref>. Complete results for these instances are presented in <ref>.
One Population Center with $\sigma^2$ = 0.1
One Population Center with $\sigma^2$ = 0.005
Four Population Centers with $\sigma^2$ = 0.1
Four Population Centers with $\sigma^2$ = 0.005
Two cases of population distribution with two values of $\sigma^2$ and the solutions of Algorithm <ref> for them.
We see that having a clustered population may make a TSP tour a necessity in our solution regardless of how large drone range is relative of the variance. However, we can expect shorter TSP tour when variance is smaller.
One Population Center
Four Population Centers
Comparison of TSP and TSP-SD models for two cases of population distribution for varying values of variance.
We can see from <ref> that the proposed combined model of truck and sharing economy drones (TSP-SD) consistently and significantly performs better than the pure TSP (truck-only) model in both cases and across the range of the variance. Its best performance usually happens when customers are concentered in one place, i.e., very small variance, whether we have one center or several centers of population. This is consistent with real-life applications as in the metropolitan areas population distribution is usually nonuniform and residences usually live near each other in a neighborhood or town while having several neighborhoods or towns in the metropolitan area with some distances from each other.
When the variance is low, we can see savings of up to $80\%$ in the delivery time, when compared to TSP, by using drones in the once center case (<ref>). Although this savings becomes smaller when the distribution of the customers gets closer to a uniforms distribution, we can still expect at least $30\%$ savings in the delivery time.
When there are four centers of population in a large city, we can still see between $17\%$ and $45\%$ savings in the delivery time (<ref>). Unlike the one population center case that shows a negative correlation between the savings and $\sigma^2$, in four population centers case we see that the savings stays more or less flat that is due to the clustered nature of the population.
These observations on the sensitivity of our proposed model to the customer distribution is consistent with findings of <cit.> that suggests that the benefits of the crowdsourced delivery depends on the spatial characteristics of the network (of pickup/delivery/transfer locations and roads).
§.§ Impact of Moving Cluster Centers and Merging Centers
Here, we compare the performance of our original heuristic, presented in <ref> in which the cluster centers (truck nodes) could be anywhere in the continuous space, with the final improved <ref>, and
Algorithm <ref> discussed in <ref> in which the cluster centers (truck nodes) are located at some customer nodes.
As mentioned in Sections <ref> and <ref>, moving a cluster center to customer nodes needs to satisfy one condition that is the new
center needs to be able to still cover all of its assigned customers using its assigned drones. When merging the centers, as discussed in <ref>, this feasibility is also satisfied in another way by reallocation of customers. <ref> compares an example solution for before and after moving the cluster centers customer nodes (i.e., the original heuristic vs. <ref>).
Without Moving Centers
After Moving Centers
A comparison of solution of our original heuristic, presented in <ref>, in (fig:SA_OHMC3_OH) and the solution obtained by <ref> after moving cluster centers to customer nodes and merging centers in (fig:SA_OHMC3_MC3). In this instance we have 60 customer nodes, 30 drone nodes ($\rho_2=0.5$), distributed uniformly at random in a unit box with $\rho_1=3$ and $L=0.8$.
Sensitivity analysis of the impact of moving cluster centers to customer nodes via <ref> and <ref> and further merging them together in <ref> on the performance when compared to the original heuristic presented in <ref>. (fig:SA_MCrho2Half) shows the results for $\rho_2=0.5$ and (fig:SA_MCrho2Two) presents the results for $\rho_2=2$. The range of relative speed in both cases is $0.75 \leq \rho_1 \leq 3$. The sensitivity of the impact of this improvement with respect to drone range in illustrated in fig:SA_MC_L which shows the maximum improvement for middle range $L$.
To evaluate the overall impact of this change, we use assume $0.75 \leq \rho_1 \leq 3$ and $\rho_2 \in \{0.5,2\}$ for our simulation. The results of comparison between <ref> and our original heuristic of <ref> is presented in <ref>. In Figures <ref> and <ref> We see about 45% and 38% savings on average for two values of $\rho_2$, respectively, which suggest a slight decrease in savings by increasing $\rho_2$. This is because when we have an abundance of drone supply the original heuristic could also perform well making the gap between the two algorithms narrower. However, this average decrease is quite small. The savings are more or less flat across different values of $\rho$ as well. This suggests the robustness of the implemented improvements with respect to $\rho_1$ and $\rho_2$, providing more than roughly 40% savings in the delivery time when compared to the algorithm before improvement. However, the design of <ref> hints more sensitivity with respect to drone range $L$ for the purpose of this comparison. This analysis for increments of 0.4 in $L$ is shown in <ref>. We see the biggest improvement (up to 68%) for the middle range $L$ where we needed it the most as already seen in Tables <ref> and <ref>. For very large $L$ both problems can result to one truck node (serving all customers with drones). If the location of this center is the same in the solutions obtained by these two algorithms, the only remaining factor making a difference in the solution is the drone routes obtained by the tabu search step. This explains the small gap for $L=2.8$.
§.§ Revisiting versus Recharging
Comparing the revisiting and recharging modes using the results of the algorithm is tricky. When using the proposed algorithm we could see revisiting model as a battery utilization modification to the algorithm that allows the drones to revisit the truck node after a delivery to pick up another package if their battery limits allow that. However, one should keep in mind that achieving any improvement by this modification highly depends on the problem instance and the input parameters. When comparing the results of the optimization models we expect to see further improvement by allowing revisiting for the drones that was identified in <ref> in several instances. However, comparing the results of <ref> in Tables <ref> and <ref> we can see a mixed situation and in some instances the recharging model is providing a better solution. This is due to the additional complexity of the revisiting model that makes it much harder for the algorithm to find better solutions for certain instances of the problem, resulting in a larger optimality gap and potentially weaker solution than the recharging model. In other words, we are dealing with a larger feasibility set potentially containing a better optimal solution but depending on the complexity of the problem instance we may end up with a better or worse solution than the output of the recharging model.
<ref> shows an example of implementing this modification that results to weaker solutions for the revisiting model. When revisiting is desired we suggest to run both recharging and revisiting versions of the algorithm and pick the best. For smaller size problems we can obviously solve the problem with the optimization model of Problem IV to expect better results. Further adjustment of the parameters of the algorithm to better handle the complexity of the revisiting model for such unfriendly instances may also help to achieve positive savings compared to the recharging version of <ref>.
Savings analysis when $\rho_2=0.5$
Savings analysis when $\rho_2=2$
Illustration of the impact of battery utilization modification that shows savings, for various values of $\rho_1$ with $\rho_2=\{0.5,2\}$, by comparison of three algorithms including recharging and revisiting versions of <ref> and the original heuristic of <ref>.
§ THE IMPACT ON CARBON FOOTPRINT AND COMPARISON WITH TSPD
It is clear that drone delivery systems such as our proposed combined system and the coordinated but centralized system of a truck and a drone in which the drone is controlled from the truck (the TSPD model[We follow the name used by <cit.>. This is also called horsefly problem, truck-and-drone TSP, etc.]) have less carbon footprint than a truck-only (pure TSP) delivery system since drones are much more energy-efficient than trucks.
In order to evaluate the energy efficiency of our proposed shared drone delivery model, we compare it with the TSPD model. First, note that if in both models we assume the drones are capable of carrying more than one package, the shared delivery model would have less carbon footprint since it becomes very similar to the comparison of the generalized traveling salesman problem (GTSP) and the TSP done in the earlier paper <cit.> that showed the household-level transportation has its own economies of scale and most likely outperforms a centralized delivery system.
Now assume the drones can only carry one package per trip (which is the case in most current delivery drone technologies) and have to return to their home base (for replacing battery or recharging) after each delivery. The comparison would obviously depend on the balance between the supply of drones and demand in deliveries. The <ref> shows a comparison between the two systems in two situations when there is only one drone in each cluster ($\rho_2=0.18$ for this instance with 60 customers), i.e., extreme shortage of drone supply, and <ref> illustrates the results of this comparison when we increase drone supply to more reasonable levels in sharing economy environments, which as we will discuss next makes this comparison more fair as well. We use the ratio $\frac{\text{Time cost of TSPD - Time cost of TSP-SD}}{\text{Time cost of TSPD}}$ to calculate the percentage of savings.
For the TSPD model we assumed that drone can pickup a package at one stopping point and returns to the truck after delivering it at either the same point or at another stopping point which makes the model more flexible and more efficient. We also assume no restriction of the drone range. Both of these assumptions give an advantage to this model over our TSP-SD. We solved this problem via the algorithms developed in our working papers <cit.> and picked the best result.
Figures <ref> and <ref> show the solution of the TSPD model and the sharing model TSP-SD, respectively, for a problem with 60 customers distributed uniformly at random in a unit box. We have set $\rho_1=3$ and $L=0.8$ for both models. Using the binary search and $k$-means step of <ref> to find a feasible $k$ we found that $k=11$ cluster centers are needed. The number of clusters is further reduced by <ref> to 9 via merging centers as visible in <ref>. Assuming one drone in each of the original 11 clusters gives $\rho_2 = 0.18$.
It is clear from <ref> that with this restrictive assumption the TSPD model will have much better performance than our TSP-SD model. This is understandable as in the shared model, the truck needs to wait at the cluster centers for one drone to serve all the customers which looks very inefficient when drone supply is limited. The drone-traveling time is huge in this circumstance and the truck idles when it could serve customers. In contrast, in the TSPD model the truck can serve another customer while the drone is delivering the package as well. It is obvious that the TSPD model is faster than the shared model if there is only one drone in each cluster since there is no such huge waiting time for the truck. TSPD can be seen as a model that has one drone for each customer in the neighborhood. To see this more clearly, imagine a very large relative drone speed, i.e., letting $\rho_1 \rightarrow \infty$. In such extreme case the truck will stop at some point in the region and all the customers will be served by the drone, forming a star-shape combined route. This situation is equivalent to a situation where each customer node sends its own drone to the truck to pick up its package and bring it back to customer's home, i.e., $\rho_2=1$. The sharing economy model assumes reasonable drone supply (such as a perfect balance of supply-demand with $\rho_2=1$) with limited relative drone speed $\rho_1$. Obviously, this situation justifies using the truck for serving some of the customers and leaves the question of finding the optimal trade-off. Therefore, to have a fair comparison we have to increase the relative number of drones in the sharing model. Note that the assumption of unlimited drone range ($L \rightarrow \infty$) for the TSPD model also gives a significant advantage to that model. This is one of the reasons behind the fact that TSPD is showing less sensitivity to $\rho_1$ in its considered range as can be observed in <ref> and Figures <ref> – fig:TSPDvsSharing-Savings-rhoTwoTwo. Of course, this will change when $\rho_1$ increases significantly beyond this range.
Solution of the TSPD model
Solution of the TSP-SD model with one drone in each cluster
Comparison of TSPD model and TSP-SD model with one drone in each cluster ($\rho_2=0.18$)
Comparison of the TSPD model the shared delivery model (TSP-SD) assuming only one drone in each cluster of the shared model.
To investigate the impact of the balance between the supply of drones and the demand in deliveries, we did a sensitivity analysis in terms of the number of drones. In <ref>, it is clear that when the relative number of drones increases to half of the customers, the average savings of the sharing model can be as much as about 6%, an increase by 137 percentage points compared to the case with one drone in each cluster. We see in <ref> that from $\rho_1=1.35$ we start to see positive savings in this case.
Furthermore, when there is the same number of drones and customers in the neighborhood ($\rho_2 = 1$), the sharing model will have better performance compared to the TSPD model on all values of $\rho_1$, and can achieve about $30\%$ savings on average. If we further increase the drone supply to $\rho_2 = 2$ there is more than $46\%$ savings on average in the delivery time. This is consistent with other sharing economy application and shows that to use shared drones in a delivery system, we need to have enough supply (of drones) relative to the demand in deliveries in order to have a better performance than the centralized alternative of coordinated truck and drone delivery system. Clearly, the gap between the TSPD and our TSP-SD model when we allow drones to revisit the pickup centers becomes wider, at least when we compare the optimal solutions. Assuming a larger $L$ will also increase this gap significantly.
Comparison of the TSPD model the shared delivery model (TSP-SD) for different values of relative number and speed of drones.
§ CONCLUSION
In this paper we have developed a shared last-mile delivery model in which a truck carries packages to a neighborhood and then outsources the last piece of the trip to private drone operators whose service can be utilized on a sharing economy platform. We have developed several optimization models for different versions of the problem and proposed efficient algorithms to solve them. Our computational analysis show that the shared delivery model (decentralized model) is much more efficient than the traditional truck-only delivery model (centralized model) in almost all possible scenarios and offers significant savings in the delivery time. This is aligned with the results from <cit.>. The comparison between the shared delivery model and the coordinated delivery system, in which a truck carries and controls a drone during the delivery operation, depends on other factors such as number of available drones in the platform, their capacity and speed. Our analysis in this case also shows a considerable overall advantage for the decentralization. For future work, one may look into considering different factors such as time windows for delivery to customers, time windows for drone availability, ability of drones to carry multiple packages at the same time, weight capacity for the drones, and combination of the system with crowdsourced drivers. Another avenue for extending this work is to incorporate pickup stations into the problem. Finally, it would be interesting to consider stochastic models: one example is to assume that some of the drones with some probability could fail to deliver and have to bring the package back to the truck and that customer must be then served by the truck.
§ ACKNOWLEDGMENTS
The authors gratefully acknowledge support from a Tier-2 grant from Northeastern University (NU) as well as an NSF Planning Grant for establishing an engineering research center (ERC). Authors would also like to thank NU's SHARE Group, led by Ozlem Ergun, for their support and for creating an environment in which this research could be accomplished.
*Online Supplement to “Last Mile Delivery with Drones and Sharing Economy”
§ ALTERNATIVE MODEL FOR PROBLEM I
For the alternative formulation of Problem I, we change some of the settings of the model as detailed in Table <ref>.
Sets, parameters, and variables used in the one center model.
Sets/Parameters/Variables Description
$C$ Customer Nodes
$D$ Drone Nodes
$T$ Truck nodes, i.e., only one node here. We call it node $0$.
$d_{ij}$ Distance between node $i$ and $j$ for $i,j\in C\cup D \cup T$
$v_D$ Speed of drones
$L$ Longest distance a drone can travel without charging battery
$x_{ij}$ $\in \N$. The number of trips from node $i$ to node $j$.
$q_{i}$ Total travel time of the drone with base at node $i$
$Q$ Maximum time spent by all drones (makespan)
Then, an alternative mixed integer linear programming (MILP) model for this problem can be written as follows:
\begin{eqnarray}
\minimize \qquad Q & & \quad \qquad \st \nonumber \\
& & \nonumber \\
\sum_{i\in D} x_{ji} & = & 1 \,, \qquad \forall j \in C \label{eq:Problem1AlternativeDroneSatisfyAll} \\
x_{0i} + \sum_{j\in C} x_{ij} & = & 0 \,, \qquad \forall i \in D \label{eq:Problem1AlternativeNoSelfNoDirect} \\
x_{0j} & = & 1 \,, \qquad \forall j \in C \label{eq:Problem1AlternativeCenterSatisfyAll} \\
x_{i0} - \sum_{j\in C}x_{ji} & = & 0 \,, \qquad \forall i \in D \label{eq:Problem1AlternativeDroneNodeFlowBalance} \\
\sum_{i \in D}x_{i0} - \sum_{j\in C}x_{0j} & = & 0\,, \qquad \label{eq:Problem1AlternativeCenterFlowBalance} \\
(d_{i0}+d_{0j}+ d_{ji})x_{ji} & \leq & L\,, \qquad \forall i\in D, \; \forall j\in C \label{eq:Problem1AlternativeDroneRange} \\
\frac{1}{v_D}\sum_{j\in C}(d_{i0}+d_{0j}+ d_{ji})x_{ji} & \leq & q_i \,, \qquad \forall i\in D \label{eq:Problem1AlternativeDroneTime} \\
q_i & \leq & Q \,, \qquad \forall i\in D \label{eq:Problem1AlternativeTotalTime} \\
x_{0,j}\, \; x_{ji} & \in & \{0,1\} \,, \qquad \forall i,j\in C\cup D \nonumber \\
x_{i0} & \in & \N \,, \qquad \forall i \in D \nonumber \\
q_i\,, \; Q & \geq & 0 \,, \qquad \forall i\in D \nonumber
\end{eqnarray}
Constraint (<ref>) ensures all customers are visited once by a drone.
Constraints (<ref>) and (<ref>) ensure all customers are serviced via the center (truck node).
Constraint (<ref>) and (<ref>) enforce the balance of outbound and inbound flows at each drone node and at the pickup center.
Constraint (<ref>) is the drone range limit.
Constraints (<ref>) sums the total time (distance) of all routes starting from each drone node. Constraint (<ref>) finds the maximum time (distance) among all drones.
§ ALTERNATIVE MODEL FOR PROBLEM II
For the alternative formulation of Problem I, we change some of the settings of the model as detailed in Table <ref>. We know that each drone can make multiple trips and in each trip it can serve multiple customers. This is obviously much more complicated than the model for Problem I. To do this in an easier way, we hypothetically each trip initiated from a drone node is made with a new drone. Therefore, we index different trips as different drones. Since each drone node would need at most $|C|$ trips to serve all customers we define set $K$ as the set of all hypothetical drones in which the first $|C|$ members are associated with drone node 1, the second $|C|$ members are associated with the second drone node, and so on and so forth. Therefore, we have $|K|=|C|\times|D|$.
Sets, parameters, and variables used in the one center model.
Sets/Parameters/Variables Description
$C$ Customer Nodes
$D$ Drone Nodes
$K$ Set of artificial drones representing trips for drones in $D$
$T$ Truck nodes, i.e., only one node here. We call it node $0$.
$d_{ij}$ distance between node $i$ and $j$ for $i,j\in C\cup D \cup T$
$v_D$ Speed of drones
$L$ Longest distance a drone can travel without charging battery
$x_{ijk}$ Binary variable. It is $1$ when drone $k$ travels from node $i$ to node $j$ and $0$ otherwise.
$Z_{ijk}$ Binary variable. It is $1$ if both $x_{i0k},i\in D$ and $x_{0jk},j\in C$ are 1 and 0 otherwise.
$Y_{ijk}$ Binary variable. It is $1$ if both $x_{i0k},i\in D$ and $x_{j0k},j\in C$ are 1 and 0 otherwise.
$q_{i}$ Total travel time of the drone with base at node $i$
$Q$ Maximum time spent by all drones (makespan)
Then, an alternative mixed integer linear programming (MILP) model for this problem can be written as follows:
\begin{eqnarray}
\minimize \qquad Q & & \quad \qquad \st \nonumber \\
& & \nonumber \\
\sum_{i\in D\cup \{0\}}\, \sum_{k\in K} x_{jik} & = & 1\,, \qquad \forall j\in C \label{eq:Problem2AlternativeDroneSatisfyAll} \\
\sum_{k\in K} x_{0ik} + \sum_{j\in C} \sum_{k\in K} x_{ijk} & = & 0 \,, \qquad \forall i \in D \label{eq:Problem2AlternativeNoSelfNoDirect} \\
\sum_{k\in K} x_{0jk} & = & 1 \,, \qquad \forall j \in C \label{eq:Problem2AlternativeCenterSatisfyAll-1} \\
\sum_{i\in D\cup C}x_{i0k} - \sum_{j \in C} x_{0jk} & = & 0\,, \qquad \forall k\in K \label{eq:Problem2AlternativeCenterSatisfyAll-2} \\
\sum_{k\in K} x_{i0k} - \sum_{j \in C} \sum_{k\in K} x_{jik} & = & 0\,, \qquad \forall i\in D \label{eq:Problem2AlternativeDroneNodeFlowBalance} \\
\sum_{i \in D\cup C} \sum_{k\in K} x_{i0k} - \sum_{j\in D\cup C} \sum_{k\in K} x_{0jk} & = & 0\,, \qquad \label{eq:Problem2AlternativeCenterFlowBalance} \\
\sum_{i \in D} d_{i0}x_{i0k} + \sum_{j\in C}(d_{0j}x_{0jk} + d_{j0}x_{j0k}) + \sum_{j\in C} \sum_{i \in D} d_{ji} x_{jik} & \leq & L\,, \qquad \forall k \in K \label{eq:Problem2AlternativeDroneRange} \\ |
# Can mobility induce orders in active XY spins on a substrate?
Astik Haldar<EMAIL_ADDRESS>Theory Division, Saha Institute of
Nuclear Physics, HBNI, 1/AF Bidhannagar, Calcutta 700064, West Bengal, India
Apurba Sarkar<EMAIL_ADDRESS>School of Mathematical &
Computational Sciences, Indian Association for the Cultivation of Science,
Kolkata-700032, West Bengal, India Swarnajit Chatterjee
<EMAIL_ADDRESS>Center for Biophysics & Department for
Theoretical Physics, Saarland University, 66123 Saarbrücken, Germany Abhik
Basu<EMAIL_ADDRESS><EMAIL_ADDRESS>Theory Division, Saha
Institute of Nuclear Physics, HBNI, 1/AF Bidhannagar, Calcutta 700064, West
Bengal, India
###### Abstract
We elucidate how the interplay between diffusive mobility and number
conservation can make nearly phase-ordered active XY spins on a substrate
stable. For wide-ranging model parameters, it has stable uniform phases with
orientational order logarithmically stronger or weaker than in equilibrium,
together with miniscule (i.e., hyperuniform) or giant number fluctuations,
respectively, forming a new universality class. For other parameters, it has
no stable uniformly ordered phase. Our theory sheds light on wide-ranging
systems, e.g., active superfluids on substrates, synchronization of
oscillators, active carpets of cilia and bacterial flagella, and sandblasting.
Nonequilibrium systems can be strikingly different from their equilibrium
analogs: they can display both ordered states or instabilities not found in
their equilibrium analogs. For instance, a two-dimensional (2D) collection of
self-propelled particles can be in orientationally long-range ordered states
in the presence of finite noises Marchetti _et al._ (2013); Toner and Tu
(1995), which is prohibited in equilibrium systems with continuous symmetries
Mermin and Wagner (1966). Driven systems can show novel instabilities as well,
e.g., driven 2D isotropic Bose systems, e.g., driven exciton polariton fluids
on a substrate Carusotto and Ciuti (2013), cannot exhibit off-diagonal
algebraic correlations (i.e., 2D superfluidity) on a substrate, but show only
short-range order (SRO) for any weak noises Altman _et al._ (2015), unlike
quasi-long-range order (QLRO) at low enough temperature ($T$) in its
equilibrium counterpart Chaikin and Lubensky (1995).
In this Letter, we study diffusively moving nearly phase-ordered active XY
spins on a 2D solid substrate, as illustrated in Fig. 1. The “activity” of the
model stems from the propensity of the spins to rotate in response to the
local spin concentration and the magnitude of the local phase difference.
Absence of self-propulsion of the spins makes this active model distinct from
the celebrated “moving XY” Toner-Tu model Toner and Tu (1995) or the
microscopic Vicsek model Vicsek _et al._ (1995) for flocks. We develop the
hydrodynamic theory for this system, supplemented by a lattice-gas type
microscopic agent-based model constructed by us. Surprisingly, this model
encompasses both the striking features of nonequilibrium systems vis-a-vis
their equilibrium counterparts, viz., novel order and instabilities in
different parameter ranges, and belongs to a heretofore unstudied universality
class.
Figure 1: Schematic diagram of the model. Each lattice site contains an
arbitrary number of XY spins, with the phase at a site being the average
phases of all the spins there, marked by bluish thick arrows. The phase
changes by local “active rotations” in response to the local spin
concentration and magnitude of the nearest-neighbor phase differences, and by
simple relaxation (thick yellow arrow). The density changes via currents
having diffusive and local phase difference-dependent parts, denoted by the
red broken horizontal arrow.
Interestingly, this theory also describes the 2D Kardar-Parisi-Zhang (KPZ)
equation Kardar _et al._ (1986); Barabási _et al._ (1995) for surface growth
or erosion (“sandblasting”) with a conserved species on it.
Our most striking result is that, in contrast to either non-number conserving
or immobile XY spins on a substrate that admit only SRO Altman _et al._
(2015), the mobility and number conservation together in this system can lead
to stable uniformly ordered phases; see supplemental movie MOV1 sup . These
ordered phases are distinguished by varieties of stable orientational order
for a wide range of model parameters. Although like QLRO, the variance of the
orientational fluctuations grows with the system size $L$, it does so either
logarithmically slower than QLRO (i.e. stronger order than QLRO or “SQLRO”),
or faster than QLRO (i.e. weaker order than QLRO or “WQLRO”), depending upon
the model parameters. In a surprising correspondence with phase fluctuations,
the density fluctuations are either generically miniscule, or giant,
respectively. Intriguingly, fluctuations, which are underdamped in these
ordered states, are characterized by continuously varying scaling exponents.
For other choices of the parameters, there are no stable uniform phase-ordered
states. Without mobility or number conservation, our model predicts short-
range phase-ordering; see supplemental movie MOV2 sup . In the equilibrium
limit, the model shows QLRO phase order and normal number fluctuations.
Our model provides a broad perspective concerning the impact of number
fluctuations on the orientational order in wide classes of 2D driven systems
with rotational invariance at low noises that exhibit QLRO in equilibrium. For
instance, can an isotropic driven number-conserving mobile ordered 2D
condensate exist, and if so, with what type of order? Likewise, generic
conditions for synchronization in phase-locked states of mobile oscillators,
having the same $U(1)$ internal symmetry as XY spins, is a question of
paramount significance Pikovsky _et al._ (2003); Dörfler and Bullo (2014);
Strogatz (2018); Uriu _et al._ (2010); Uriu and Morelli (2014); Zhou _et
al._ (2016); Levis _et al._ (2017); Frasca _et al._ (2008); Peruani _et
al._ (2010); Levis _et al._ (2017); Banerjee and Basu (2017). Other
experimentally realizable systems motivating our theory include “active
carpets” of cilia or bacterial flagella, modeled as active rotors grafted on
the carpet and can order Uchida and Golestanian (2010), and take part in
nutrient transport Mathijssen _et al._ (2018) or mucous in respiratory tract
D. R. Brumley and Goldstein (2012), monolayers of spinning colloids Bruot and
Cicuta (2016); Zhang _et al._ (2021), and in vitro magnetic cilia carpets et
al (2020). Lastly, possible suppression of the thresholdless instability of 2D
KPZ surfaces is an overarching theoretical issue, given the paradigmatic
status of the KPZ equation in nonequilibrium physics Barabási _et al._
(1995).
Below we outline our hydrodynamic theory that quantitatively explains
indispensable role of the conserved density fluctuations to sustain order
here, as dramatically manifested in movies MOV1 vis-à-vis MOV2 sup . Details
can be found in the associated long paper (ALP) Haldar _et al._ (2021).
Due to friction from the substrate, there is no momentum conservation, so the
only conserved variable on the surface is the spin number density $c({\bf
x},t)$. In addition, for nearly phase-ordered spins, the broken symmetry phase
fluctuations $\theta({\bf x},t)$ about an arbitrary reference state are slow
variables with relaxation rates diverging in the long wavelength limit, but
the amplitude fluctuations relax fast. Therefore, $\theta({\bf x},t)$, and
$c({\bf x},t)$ are the only hydrodynamic variables. For a driven system, we
must write down the equations of motion by appealing to the general symmetries
of the underlying microscopic dynamics (i.e., translation and rotation) and
the phase-ordered state (here, arbitrariness of the reference state) and
conservation laws. Additional equilibrium requirements like detailed balance
do not apply to our nonequilibrium system. Retaining up to the lowest order
symmetry-permitted nonlinear terms in $\theta$ and spatial gradients, the
dynamical equation of $\theta({\bf x},t)$ has the form
$\partial_{t}\theta=\kappa\nabla^{2}\theta+\frac{\lambda}{2}({\bm{\nabla}}\theta)^{2}+\Omega(c)+f_{\theta},$
(1)
where $\Omega(c)$ is a general function of $c$. In case of fluctuating
surfaces, $\theta({\bf x},t)$ is the local height with respect to an arbitrary
base plane Barabási _et al._ (1995). Number density $c$ follows a
conservation law with a current ${\bf J}_{c}$ given by
${\bf
J}_{c}=-D{\bm{\nabla}}c-\lambda_{0}\tilde{\Omega}(c){\bm{\nabla}}\theta.$ (2)
Again we have truncated up to the lowest order in $\theta$ and spatial
gradients; $\tilde{\Omega}(c)$ is yet another function of $c$. While we have
constructed (1) and (2) by writing down all the rotation-invariant leading
order terms, each term actually carries a simple physical interpretation. The
linear $\kappa\nabla^{2}\theta$ term in (1), and $D{\bm{\nabla}}c$ term in (2)
are just the equilibrium spin relaxation and particle diffusion, respectively;
$\kappa>0$ and $D>0$, respectively, are the spin stiffness and diffusivity.
The $\lambda$\- and $\Omega(c)$-terms in (1) represent, respectively, active
rotations of the spins in response to local gradients of $\theta$, and local
spin concentration. The $\lambda_{0}$-term in (2) models particle currents in
response to spatially nonuniform phases. Gaussian noises $f_{\theta}$ and
$f_{c}$ are white and conserved noises, respectively, with zero mean, and
variances
$\displaystyle\langle f_{\theta}({\bf
x},t)f_{\theta}(0,0)\rangle=2D_{\theta}\delta^{2}({\bf x})\delta(t),$ (3)
$\displaystyle\langle f_{c}({\bf
x},t)f_{c}(0,0)\rangle=2D_{c}(-\nabla^{2})\delta^{2}({\bf x})\delta(t),$ (4)
consistent with $\theta$ and $c$ being, respectively, a non-conserved and a
conserved variable. Writing $c({\bf x},t)=c_{0}+\delta c({\bf
x},t),\,\langle\delta c\rangle=0$, we obtain
$\displaystyle\frac{\partial\theta}{\partial
t}=\kappa\nabla^{2}\theta+\Omega_{1}\delta
c+\frac{\lambda}{2}({\bm{\nabla}}\theta)^{2}+\Omega_{2}(\delta
c)^{2}+f_{\theta},$ (5) $\displaystyle\frac{\partial\delta c}{\partial
t}=\lambda_{0}\tilde{\Omega}_{0}\nabla^{2}\theta+D\nabla^{2}\delta
c+\lambda_{0}\tilde{\Omega}_{1}{\bm{\nabla}}\cdot(\delta
c{\bm{\nabla}}\theta)+f_{c},$ (6)
where we have retained the most relevant nonlinear terms in fields and
gradients. Parameters $\Omega_{1}\equiv\partial\Omega/\partial
c|_{c=c_{0}},\,\Omega_{2}\equiv\partial^{2}\Omega/\partial^{2}c|_{c=c_{0}},\,\tilde{\Omega}_{0}\equiv\tilde{\Omega}|_{c=c_{0}},\,\tilde{\Omega}_{1}\equiv\partial\tilde{\Omega}/\partial
c|_{c=c_{0}}$. Couplings $\lambda,\Omega_{2},\lambda_{0}\tilde{\Omega}_{1}$
have arbitrary signs. For active carpets with grafted rotors, $c$ represents
the concentration of nutrients or other chemical substance, moved around by
the beating of cilia or bacterial flagella Uchida and Golestanian (2010);
Mathijssen _et al._ (2018). Purely for symmetry reasons, Eqs. (5) and (6)
describe the hydrodynamics of immobile active XY spins interacting with an
incompressible binary fluid in its well-mixed phase on a substrate, where $c$
now is the binary fluid order parameter, and also chiral active hexatics on a
substrate Maitra _et al._ (2020). One-dimensional versions of (5) and (6) are
same as the hydrodynamic equations of a one-dimensional sedimenting crystal
Lahiri and Ramaswamy (1997).
It is useful to study Eqs. (5) and (6) in their linear limit first, by setting
$\lambda=\Omega_{2}=\tilde{\Omega}_{1}=0$. When
$\Omega_{1}\tilde{\Omega}_{0}\lambda_{0}>0$, uniform states are linearly
stable. Small fluctuations around these linearly stable states travel non-
dispersively, i.e., with a wavespeed independent of wavevector $k$. The
correlators in the linear theory are exactly calculated conveniently in
Fourier space. For instance, the phase and density autocorrelation functions
show QLRO and “normal number fluctuations (NNF)”, indistinguishable from the
2D equilibrium XY model and a non-critical equilibrium system with short-range
interactions. In particular, equal-time phase correlator
$C^{0}_{\theta\theta}(k)\equiv\langle|\theta({\bf
k},t)|^{2}\rangle_{0}\approx\frac{\overline{D}}{2\Gamma k^{2}},$ (7)
in the long wavevlength limit, where $\overline{D}\equiv D_{\theta}+D_{c}$ and
$\Gamma\equiv(\kappa+D)/2$; “0” refers to a linear theory result. This in turn
gives for the variance
$\Delta_{\theta}^{0}\equiv\langle\theta^{2}({\bf
x},t)\rangle_{0}=\frac{\overline{D}}{4\pi\Gamma}\ln\left(\frac{L}{a_{0}}\right)$
(8)
in 2D; $a_{0}$ is a small-scale cutoff. Equation (8) corresponds to a
logarithmically rough Edward-Wilkinson (EW) Barabási _et al._ (1995) surface
at 2D. Furthermore
$C_{\theta\theta}^{0}(r)\equiv\langle\left[\theta({\bf x+r},t)-\theta({\bf
x},t)\right]^{2}\rangle_{0}\approx\frac{\overline{D}}{2\pi\Gamma}\ln(r/a_{0})$
(9)
for large $r$ Chaikin and Lubensky (1995). Equations (8) and (9) imply QLRO.
The equal-time density correlator
$C_{cc}^{0}(k)\equiv\langle|\delta c({\bf
k},t)|^{2}\rangle_{0}\approx\frac{\overline{D}}{2\Gamma}$ (10)
is independent of $k$ in the long wavelength limit. In real space,
$C^{0}_{cc}(r)$ vanishes for $r\gg\zeta$, a microscopic length characterizing
the short-range interactions. This further means that the standard deviation
of the number fluctuations of $N$ spins contained in a fixed open area,
$\sqrt{\langle N^{2}\rangle-\langle N\rangle^{2}}\equiv\sigma(N_{0})$ scales
with the mean $\langle N\rangle\equiv N_{0}$ as $\sqrt{N_{0}}$, giving NNF as
expected in a non-critical equilibrium system with short-range interactions.
If $\Omega_{1}\tilde{\Omega}_{0}\lambda_{0}<0$, the uniform states are
linearly unstable with a growth rate proportional to $k$.
The Fluctuation-Dissipation-Theorem (FDT) Chaikin and Lubensky (1995) is
broken in the linearized theory. This manifests in the non-vanishing cross-
correlator $C_{\times}({\bf k})\equiv\langle\theta({\bf-k},t)\delta c({\bf
k},t)\rangle$, which is a model parameter-dependent constant in the limit of
small $k$.
It now behooves us to find whether nonlinear effects are relevant (in the RG
or renormalization group sense), and the scaling properties of any ordered
states that are robust against finite noises. To study this, we perform one-
loop perturbative RG analysis on Eqs. (5) and (6) at 2D, similar to the RG
calculations on the KPZ equation Forster _et al._ (1977); Barabási _et al._
(1995), or the coupled Burgers-like equation for Magnetohydrodynamics Basu
_et al._ (1999); Basu and Frey (2004, 2009). It turns out, as discussed below,
that the nonlinearities either introduce logarithmic modulations to the
scaling of the linearly stable states, or to destroy those states altogether.
As usual, the RG is done by tracing over the short wavelength Fourier modes of
the fields Forster _et al._ (1977); Barabási _et al._ (1995); Hohenberg and
Halperin (1977), by expanding in the dimensionless coupling constant
$g\sim\lambda^{2}\overline{D}/\Gamma^{3}$; see ALP for technical details. It
predicts that not all but some of the linearly stable states are robust
against noises. This is controlled by
$\mu_{1}\equiv\Omega_{2}/\lambda,\,\mu_{2}\equiv\lambda_{0}\tilde{\Omega}_{1}/\lambda$,
that are marginal (in the RG sense) at the one-loop order. We find that for
wide ranges of $\mu_{1},\,\mu_{2}$, renormalized scale-dependent
$g(k)=1/[\Delta_{1}(\mu_{1},\mu_{2})\ln(\Lambda/k)]$ flows to zero very
slowly, where $\Delta_{1}>0$ as $k\rightarrow 0$ at 2D, under successive
applications of the RG procedure, for stable ordered phases;
$\Lambda=2\pi/a_{0}$ is an upper wavevector cutoff. This gives, as obtained
from the RG flow equations, that renormalized, scale-dependent
$\Gamma(k)=\Gamma[\ln(\Lambda/k)]^{\eta_{2}}$ and
$\overline{D}(k)=\overline{D}[\ln(\Lambda/k)]^{\eta_{1}}$, both diverging
logarithmically when $k\rightarrow 0$;
$\eta_{1}(\mu_{1},\mu_{2}),\,\eta_{2}(\mu_{1},\mu_{2})>0$ are constants
related to $\Delta_{1}(>0)$. These log-divergences due to the “slow” vanishing
of the coupling constant are reminiscent of the logarithmic anomalous
elasticity in three-dimensional equilibrium smectics Grinstein and Pelcovits
(1981, 1982). The resulting renormalized theory, owing to $g(k)\rightarrow 0$
as $k\rightarrow 0$, is effectively linear, albeit with renormalized
parameters. Straightforward calculations of the renormalized correlation
functions show that the stable ordered states have scaling properties
essentially same as in the linear theory, modulated only by logarithmic
corrections, which either strengthen or weaken the linear theory order. For
instance, the renormalized phase correlator $C_{\theta\theta}^{R}(k)$ reads
$C^{R}_{\theta\theta}(k)\approx\frac{\overline{D}}{2\Gamma
k^{2}[\ln(\Lambda/k)]^{\eta}},$ (11)
for $k\rightarrow 0$; $\eta\equiv\eta_{2}-\eta_{1}$. Here and below, $R$
refers to renormalized quantities. Exponent $\eta$ varies continuously with
$\mu_{1},\,\mu_{2}$ and can be positive or negative; detailed calculations
show $\eta<1/3$ always Haldar _et al._ (2021). For $\eta>(<)0$,
$C^{R}_{\theta\theta}(k)\ll(\gg)C^{0}_{\theta\theta}(k)$ as $k\rightarrow 0$,
demonstrating strong suppression (enhancement) of fluctuations in the long
wavelength limit. Next, the renormalized variance
$\Delta_{\theta}^{R}\equiv\langle\theta^{2}({\bf x},t)\rangle_{R}$ now
acquires a novel $L$-dependence:
$\Delta_{\theta}^{R}\approx\frac{\overline{D}}{4\pi\Gamma}\left[\ln\left(\frac{L}{a_{0}}\right)\right]^{1-\eta},$
(12)
This means $\Delta_{\theta}^{R}$ can grow, respectively, logarithmically
slower or faster with $\ln L$ than QLRO, representing order stronger or weaker
than QLRO, named, respectively, SQLRO, or WQLRO; for $\eta=0$, QLRO is
retrieved. Equation (12) also implies a surface logarithmically smoother or
rougher than the 2D EW surface. Likewise, the renormalized correlator
$C_{\theta\theta}^{R}(r)$, related to the inverse Fourier transform of
$C_{\theta\theta}^{R}(k)$, scales as
$C_{\theta\theta}^{R}(r)\equiv\langle[\theta({\bf x+r},t)-\theta({\bf
x},t)]^{2}\rangle_{R}\approx\frac{\overline{D}}{2\pi\Gamma}[\ln(r/a_{0})]^{1-\eta},$
(13)
for large $r$. Related to $C^{R}_{\theta\theta}(r)$, the renormalized spin
correlation function $C_{ZZ}^{R}(r)$ for large $r$ is
$C^{R}_{ZZ}(r)\equiv\langle\cos[\theta({\bf x+r},t)-\theta({\bf
x},t)]\rangle_{R}\approx(r/a_{0})^{-\tilde{\gamma}(r)}$ (14)
where the $r$-dependent exponent $\tilde{\gamma}(r)$ has a complex form:
$\tilde{\gamma}(r)\equiv\frac{\overline{D}}{4\pi\Gamma}[\ln(r/a_{0})]^{-\eta}$
(15)
for large $r$. For $\eta>0(<0)$, clearly $C_{ZZ}(r)$ decays much slower
(faster) for large $r$, giving SQLRO (WQLRO).
Next in contrast to its linear theory analog, renormalized density correlator
$C_{cc}^{R}(k)$ picks up a weak $k$-dependence:
$C_{cc}^{R}(k)\approx\frac{\overline{D}}{2\Gamma}\left[\ln\left(\frac{\Lambda}{k}\right)\right]^{-\eta}$
(16)
in the hydrodynamic limit $k\rightarrow 0$. Evidently, for $\eta>0$, the
density fluctuations are strongly suppressed in the limit $k\rightarrow 0$,
implying miniscule number fluctuations (MNF) or hyperuniformity Torquato and
Stillinger (2003) vis-a-vis for $\eta=0$, same as the equilibrium result. In
contrast, if $\eta<0$ the density fluctuations are hugely enhanced when
$k\rightarrow 0$. This is giant number fluctuations (GNF), often encountered
in orientationally ordered active fluids Marchetti _et al._ (2013), and also
in equilibrium superfluids Chaikin and Lubensky (1995). Lastly, the equal-time
renormalized density autocorrelator $C_{cc}^{R}(r)$, the inverse Fourier
transform of $C_{cc}^{R}(k)$, is
$C^{R}_{cc}(r)\approx\frac{\overline{D}}{4\pi\Gamma}\frac{-\eta}{r^{2}[\ln(r/a_{0})]^{(1+\eta)}},$
(17)
for large $r$, $r/a_{0}\gg 1$. Thus for $\eta>(<)0$, $C^{R}_{cc}(r)$ falls off
relatively faster (slower) for MNF (GNF). By using (17) we re-express MNF
(GNF) for $\eta>(<)0$:
$\sigma(N_{0})\propto\sqrt{N_{0}/(\ln N_{0})^{\eta}}<(>)\sqrt{N_{0}},$ (18)
in the renormalized theory. For $\eta=0$, (15) and (18) reduce to the well-
known results for the 2D equilibrium XY model and an equilibrium system with
NNF having short-range interactions away from any critical point.
We thus show SQLRO (WQLRO) phase order is necessarily accompanied by MNF
(GNF).
Like $C_{cc}^{R}(k)$, renormalized cross-correlation function
$C_{\times}^{R}(k)$ also picks up a weak $k$-dependence:
$C_{\times}^{R}(k)\propto\left[\ln(\frac{\Lambda}{k})\right]^{(1-3\eta)/2}\times{\cal
O}(1)$ for small $k$. Thus FDT remains broken in the renormalized theory.
The model shows breakdown of conventional dynamic scaling in the ordered
phases: the form of $\Gamma(k)$ in the renormalized theory shows that the
diffusive scaling of time $t$ with $r$ is modulated by logarithmic corrections
as
$t\propto r^{2}/[\ln(r/a_{0})]^{(1-\eta)/2}$ (19)
for large $r$. Thus the fluctuations relax logarithmically faster than
ordinary diffusion or in the linear theory. Also, they relax faster with WQLRO
($\eta<0$) than SQLRO ($\eta>0$).
Equations (11)-(19) collectively define the new universality class,
characterized by the SQLRO-MNF and WQLRO-GNF correspondence and continuously
varying scaling exponents.
For other choices of $\mu_{1},\,\mu_{2}$, uniform ordered states get unstable
due to nonlinear effects. In the equilibrium limit, the nonlinearities in (5)
and (6) vanish, and QLRO phase-order together with density NNF follows. A
phase diagram of the model in the $\mu_{1}-\mu_{2}$ plane over limited ranges
of $\mu_{1},\,\mu_{2}$ is shown in Fig. 2(a).
Numerical results from our agent-based model indeed confirm the existence of
stable order with SQLRO/MNF and WQLRO/GNF as generic, holding beyond any
perturbative calculations, and that topological defects in the active XY model
(not included in the hydrodynamic theory) do not proliferate and destroy order
at low-enough non-zero noises. It consists of $N_{\text{tot}}$ XY spins on a
2D square lattice of size $L\times L$, with periodic boundary conditions and a
mean spin number density $c_{0}\equiv N_{\text{tot}}/L^{2}$. The microscopic
update rules for $\theta$ depend on $c$ and the lattice analogs of the phase
difference-dependent active rotations of the spins along with equilibrium
relaxation processes; $c$ changes via hopping to the nearest neighbor sites,
provided either the density or the phase in the randomly chosen neighboring
target site is different. Stochasticity is introduced in the update of
$\theta$ by adding a (small) white noise $\xi$ to it Vicsek _et al._ (1995),
and to $c$ through hopping of particles in randomly selected sites to randomly
chosen nearest neighbors; see ALP for details. Our numerical studies reveal
stable ordered states at finite noise $\xi$; see movie MOV1 sup . Furthermore,
in these ordered states, depending upon certain parameters $g_{1},\,g_{2}$,
which are the substitutes of $\mu_{2},\,\mu_{1}$ here (see ALP for precise
correspondence) and a few other parameters that define the relative
probabilities of the various microscopic dynamical processes Haldar _et al._
(2021), numerical analogs of $\Delta_{\theta}^{R}(L)$ can grow slower and
faster than $\ln(L/a_{0})$, implying SQLRO and WQLRO respectively. In
addition, $\sigma(N_{0})$ as a function of $\ln\,N_{0}$ in a given open area
grows logarithmically slower and faster than $\sqrt{N_{0}}$, showing MNF and
GNF, correspondingly with SQLRO and WQLRO, respectively. Generic dependence of
$\eta$ on $g_{1},\,g_{2}$ are also found; see ALP Haldar _et al._ (2021). Two
representative plots from the simulations of the agent-based model in Fig.
2(b) and (c) show the $\ln L$-dependence of $\langle\theta^{2}\rangle$ with
SQLRO and WQLRO, and correspondingly, $\ln\,N_{0}$-dependence of
$\sigma(N_{0})$ displaying MNF and GNF, respectively. These establish the
SQLRO (WQLRO) and MNF (GNF) correspondence in the agent-based model, in
agreement with the hydrodynamic theory 111The quantitative accuracy of these
plots is restricted due to the limited range of $L$ used in the simulations..
Figure 2: (a) Schematic phase diagram in the $\mu_{1}-\mu_{2}$ plane in the
linearly stable hydrodynamic theory. SQLRO/MNF, WQLRO/GNF and disordered
regions are marked in different colors. QLRO/NNF is the (blue) line
demarcating SQLRO/MNF and WQLRO/GNF regions. Plots from agent-based
simulations: (b) $\langle\theta^{2}\rangle/\ln L$ versus $\ln L$ showing SQLRO
and WQLRO, and (c) $\sigma(N_{0})/\sqrt{N_{0}}$ versus $\ln\,N_{0}$ showing
MNF and GNF ($L=128$) with $c_{0}=5,\,\xi=0.1$ and (SQLRO/MNF):
$g_{1}=1.0,\,g_{2}=0.02$, and (WQLRO/GNF): $g_{1}=2.0,\,g_{2}=0.03$,
respectively. The red broken horizontal lines in (b) and (c) correspond to the
equilibrium results. The SQLRO-MNF and WQLRO-GNF correspondences are clearly
established in the agent-based model (see text).
The simulations also detect two kinds of disordered states, corresponding to
the two different routes to disorder (linear instability and nonlinear
effects) in the hydrodynamic theory; see ALP Haldar _et al._ (2021) for
details.
For immobile spins with uniform density, $\delta c=0$, or if there is no
number conservation, e.g., if the spins are created and destroyed, $\delta c$
relaxes fast, and is a non-hydrodynamic variable. Then Eq. (5) reduces to the
KPZ equation with only SRO in 2D Kardar _et al._ (1986); Barabási _et al._
(1995); Altman _et al._ (2015), in agreement with our numerical studies in
its immobile limit (movie MOV2 sup ). Heuristically, fluctuations of the
conserved density $c$ create an effective long-range interactions between
phases at distant locations, which for appropriately chosen parameters can
suppress the instability of the 2D KPZ equation, ultimately sustaining order.
We have, therefore, presented the hydrodynamic theory of phase-ordering in
diffusively mobile active XY model on a substrate, supplemented by numerical
studies of an equivalent agent-based model. Existence of stable SQLRO and
WQLRO phase orders, correspondingly with MNF and GNF, respectively, uncovering
a novel correspondence, are predicted in wide ranges of the parameter space.
This is our central conclusion. Absence of ordered states elsewhere in the
parameter space is also revealed.
Acknowledgement:- AH and AB thank T. Banerjee and N. Sarkar for critical
reading and helpful suggestions. AS thanks University Grants Commission (UGC),
India for a research fellowship. SC is financially supported by the German
Research Foundation (DFG) within the Collaborative Research Center SFB 1027
and Indian Association for the Cultivation of Science, Kolkata. AS and SC also
acknowledges Prof. R. Paul (IACS, Kolkata) for some invaluable suggestions and
discussions and are also thankful to him for providing the computational
resources. AB thanks J. Toner for valuable discussions in the early stages of
this work, and comments concerning derivations of Eqs. (17) and (18), and P.
K. Mohanty, A. Maitra and D. Levis for helpful comments, and the SERB, DST
(India) for partial financial support through the MATRICS scheme [file no.:
MTR/2020/000406]. AB has designed the problem, AH has contributed to the
analytical part of the work, AS and SC have contributed equally to the
numerical part of the work. All four authors jointly wrote the manuscript.
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* Note (1) The quantitative accuracy of these plots is restricted due to the limited range of $L$ used in the simulations.
|
# A Zeroth-Order Momentum Method for Risk-Averse
Online Convex Games
Zifan Wang, Yi Shen, Zachary I. Bell, Scott Nivison, Michael M. Zavlanos, and
Karl H. Johansson *This work is supported in part by AFOSR under award
#FA9550-19-1-0169 and by NSF under award CNS-1932011.Zifan Wang and Karl H.
Johansson are with Division of Decision and Control Systems, School of
Electrical Enginnering and Computer Science, KTH Royal Institute of
Technology, and also with Digital Futures, SE-10044 Stockholm, Sweden. Email:
<EMAIL_ADDRESS>Shen and Michael M. Zavlanos are with the Department
of Mechanical Engineering and Materials Science, Duke University, Durham, NC,
USA. Email: {yi.shen478<EMAIL_ADDRESS>I. Bell and Scott
Nivison are with the Air Force Research Laboratory, Eglin AFB, FL, USA. Email:
{scott.nivison<EMAIL_ADDRESS>
###### Abstract
We consider risk-averse learning in repeated unknown games where the goal of
the agents is to minimize their individual risk of incurring significantly
high cost. Specifically, the agents use the conditional value at risk (CVaR)
as a risk measure and rely on bandit feedback in the form of the cost values
of the selected actions at every episode to estimate their CVaR values and
update their actions. A major challenge in using bandit feedback to estimate
CVaR is that the agents can only access their own cost values, which, however,
depend on the actions of all agents. To address this challenge, we propose a
new risk-averse learning algorithm with momentum that utilizes the full
historical information on the cost values. We show that this algorithm
achieves sub-linear regret and matches the best known algorithms in the
literature. We provide numerical experiments for a Cournot game that show that
our method outperforms existing methods.
## I Introduction
In online convex games [1, 2], agents interact with each other in the same
environment in order to minimize their individual cost functions. Many
applications, including traffic routing [3] and online marketing [4], among
many others [5], can be modeled as online convex games. The individual cost
functions depend on the joint decisions of all agents and are typically
unknown but can be accessed via querying. Using this information, the agents
can sequentially select the best actions that minimize their expected
accumulated costs. This can be done using online learning algorithms whose
performance is typically measured using the notions of regret [5] that
quantify the gap between the agents’ online decisions and the best decisions
in hindsight. Many such online learning algorithms have been recently
proposed, and have been shown to achieve sub-linear regret, which indicates
that optimal decisions are eventually selected; see e.g., [1, 2, 6]. In this
paper, here we focus on online convex games for high-stakes applications,
where decisions that minimize the expected cost functions are not necessarily
desirable. For example, in portfolio management, a selection of assets with
the highest expected return is not necessarily desirable since it may have
high volatility that can lead to catastrophic losses. To capture unexpected
and catastrophic outcomes, risk-averse criteria have been widely proposed in
place of the expectation, such as the Sharpe Ratio [7] and Conditional Value
at Risk (CVaR) [8].
In this paper, we consider online convex games with risk-averse agents that
employ CVaR as risk measure. We assume that only cost function evaluations at
selected decisions are accessible. Despite the practical utility of CVaR as a
measure of risk in high-stakes applications, the theoretical analysis of risk-
averse learning methods that employ the CVaR is limited, mainly since the CVaR
values must be estimated from the distributions of the unknown cost functions.
To avoid this problem, [9] proposes a variational definition of CVaR that
allows to reformulate the computation of CVaR values into an optimization
problem by introducing additional variables. Building on this reformulation,
many risk-averse learning methods have been proposed, including [10, 11, 12],
and their convergence rates have been analysed. Specifically, [10] provides
performance guarantees for stochastic gradient descent-type algorithms in
risk-averse statistical learning problems; [11] addresses risk-averse online
convex optimization problems with bandit feedback; and [12] proposes an
adaptive sampling strategy for CVaR learning, and transforms the supervised
learning problem into a zero-sum game between the sampler and the learner.
However, all the above works focus on single-agent learning problems.
To the best of our knowledge, multi-agent risk-averse learning problems have
not been analyzed in the literature, with only a few exceptions [13], [14].
The work in [13] solves multi-armed bandit problems with finite actions, which
are different from the convex games with continuous actions considered here.
Most closely related to this paper is our recent work in [14] proposing a new
risk-averse learning algorithm for online convex games, where the agents
utilize the cost evaluations at each time step to update their actions and
achieve no-regret learning with high probability. To improve the learning rate
of this algorithm, [14] also proposes two variance reduction methods on the
CVaR estimates and gradient estimates, respectively. In this paper, we build
on this work to significantly improve the performance of the algorithm using a
notion of momentum that combines these variance reduction techniques.
Specifically, the proposed new algorithm reuses samples of the cost functions
from the full history of data while appropriately decreasing the weights of
out-of-date samples. The idea is that, using more samples, the unknown
distribution of the cost functions needed to estimate the CVaR values can be
better estimated. To do so, at each time step, the proposed algorithm
estimates the distribution of the cost functions using both the immediate
empirical distribution estimate and the distribution estimate from the
previous time step, which summarizes all past cost values. Then, to construct
the gradient estimates, more weight is allocated to more recent information,
similar to momentum gradient descent methods [15]. To reduce the variance of
the gradient estimate, we employ residual feedback [16], which uses previous
gradient estimates to update the current one.
Although [14] showed that reusing samples from the last iteration or residual
feedback can individually improve the performance of the algorithm, it also
identified that the combination of sample reuse and residual feedback is an
unexplored and theoretically nontrivial problem. In this work, we combine
sample reuse and residual feedback by proposing a novel zeroth-order momentum
method that reuses all historical samples, compared to samples from only the
last iteration. An additional important contribution of this work is that the
proposed zeroth-order momentum method subsumes Algorithm 3 in [14] as a
special case by appropriately choosing the momentum parameter. We show that
our proposed method theoretically matches the best known result in [14] but
empirically outperforms other existing methods.
The rest of the paper is organized as follows. In Section II, we formally
define the problem and provide some preliminary results. In Section III, we
summarize results in [14] that are needed to develop our proposed method. In
Section IV, we present our proposed method and analyze its regret. In Section
V, we numerically verify our method using a Cournot game example. Finally, in
Section VI, we conclude the paper.
## II Problem definition
We consider a repeated game $\mathcal{G}$ with $N$ risk-averse agents. Each
agent selects an action $x_{i}$ from the convex set
$\mathcal{X}_{i}\subseteq\mathbb{R}^{d_{i}}$, and then receives a stochastic
cost
$J_{i}(x_{i},x_{-i},\xi_{i}):\mathcal{X}\times\Xi_{i}\rightarrow\mathbb{R}$,
where $x_{-i}$ represents all agents’ actions except for agent $i$, and
$\mathcal{X}=\Pi_{i=1}^{N}\mathcal{X}_{i}$ is the joint action space. We
sometimes instead write $J_{i}(x,\xi_{i})$ for ease of notation, where
$x=(x_{i},x_{-i})$ is the concatenated vector of all agents’ actions. The cost
$J_{i}(x_{i},x_{-i},\xi_{i})$ is stochastic, as $\xi_{i}\in\Xi_{i}$ is a
stochastic variable. We assume that the diameter of the convex set
$\mathcal{X}_{i}$ is bounded by $D_{x}$ for all $i=1,\ldots,N$. In what
follows, we make the following assumptions on the class of games we consider
in this paper.
###### Assumption 1.
The function $J_{i}(x_{i},x_{-i},\xi_{i})$ is convex in $x_{i}$ for every
$\xi_{i}\in\Xi_{i}$ and bounded by $U$, i.e., $|J_{i}(x,\xi_{i})|\leq U$, for
all $i=1,\ldots,N$.
###### Assumption 2.
$J_{i}(x,\xi_{i})$ is $L_{0}$-Lipschitz continuous in $x$ for every
$\xi_{i}\in\Xi_{i}$, for all $i=1,\ldots,N$.
Assumptions 1 and 2 are common assumptions in the analysis of online learning
in games [17].
The goal of the risk-averse agents is to minimize the risk of incurring
significantly high costs, for possibly different risk levels. In this work, we
utilize CVaR as the risk measure. For a given risk level $\alpha_{i}\in[0,1]$,
CVaR is defined as the average of the $\alpha_{i}$ percent worst-case cost.
Specifically, we denote by $F_{i}^{x}(y)=\mathbb{P}\\{J_{i}(x,\xi_{i})\leq
y\\}$ as the cumulative distribution function (CDF) of the random cost
$J_{i}(x,\xi_{i})$ for agent $i$; and by $J^{\alpha_{i}}$ the cost value at
the $1-\alpha_{i}$ quantile of the distribution, also called the Value at Risk
(VaR). Then, for a given risk level $\alpha_{i}\in[0,1]$, CVaR of the cost
function $J_{i}(x,\xi_{i})$ of agent $i$ is defined as
$\displaystyle C_{i}(x):$
$\displaystyle={\rm{CVaR}}_{\alpha_{i}}[J_{i}(x,\xi_{i})]$ $\displaystyle:$
$\displaystyle=\mathbb{E}_{F}[J_{i}(x,\xi_{i})|J_{i}(x,\xi_{i})\geq
J^{\alpha_{i}}].$
Notice that the CVaR value is determined by the distribution function
$F_{i}^{x}(y)$ for a given $\alpha_{i}$, so we sometimes write CVaR as a
function of the distribution function, i.e.,
${\rm{CVaR}}_{\alpha_{i}}[F_{i}^{x}]:={\rm{CVaR}}_{\alpha_{i}}[J_{i}(x,\xi_{i})]$.
In addition, we assume that the agents cannot observe other agents’ actions,
but the only information they can observe is the cost evaluations of the
jointly selected actions at each time step.
Given Assumptions 1 and 2, the following lemma lists properties of CVaR, that
will be important in the analysis that follows. The proof can be found in
[11].
###### Lemma 1.
Given Assumptions 1 and 2, we have that $C_{i}(x_{i},x_{-i})$ is convex in
$x_{i}$ and $L_{0}$-Lipschitz continuous in $x$, for all $i=1,\ldots,N$.
The following additional assumption on the variation of the CDFs is needed for
the analysis that follows.
###### Assumption 3.
Let $F^{w}_{i}(y)=\mathbb{P}\\{J_{i}(w,\xi_{i})\leq y\\}$ and
$F^{v}_{i}(y)=\mathbb{P}\\{J_{i}(v,\xi_{i})\leq y\\}$. There exist a constant
$C_{0}>0$ such that
$\displaystyle\mathop{\rm{sup}}_{y}|F^{w}_{i}(y)-F^{v}_{i}(y)|\leq
C_{0}\left\|w-v\right\|.$
Assumption 3 states that the difference between two CDFs can be bounded by the
difference between the corresponding two action profiles. It is less
restrictive than Assumption 3 in [14] since it does not depend on the
algorithm but only on the cost functions.
A common measure of the ability of the agents to learn online optimal
decisions that minimize their individual risk-averse objective functions
$C_{i}(x)$ is the algorithm regret, which is defined as the difference between
the actual rewards and best rewards that the agent could have achieved by
playing the single best action in hindsight. Suppose the action sequences of
agent $i$ and the other agents in the team are $\\{\hat{x}_{i,t}\\}_{t=1}^{T}$
and $\\{\hat{x}_{-i,t}\\}_{t=1}^{T}$, respectively. Then, we define the regret
of agent $i$ as
$\displaystyle{\rm{R}}_{C_{i}}(T)=\sum_{t=1}^{T}C_{i}(\hat{x}_{i,t},\hat{x}_{-i,t})-\mathop{\rm{min}}_{\tilde{x}_{i}\in\mathcal{X}_{i}}\sum_{t=1}^{T}C_{i}(\tilde{x}_{i},\hat{x}_{-i,t}).$
An algorithm is said to be no-regret if its regret grows sub-linearly with the
number of episodes $T$. In this paper, we propose a no-regret learning
algorithm, so that $\lim_{T\rightarrow\infty}\frac{{\rm{R}}_{C_{i}}(T)}{T}=0$,
$i=1,\ldots,N$.
## III Preliminary Results
In this section, we summarize some results from [14] that lay the foundation
for the subsequent analysis. Note that the CVaR values depend on the
distributions of the cost functions, which are generally unknown. To estimate
the distribution of the cost functions, and subsequently the CVaR values with
only a few samples, a sampling strategy is proposed in [14] that uses a
decreasing number of samples with the number of iterations. Specifically, at
episode $t$, the sampling strategy used by the agents is designed as follows
$\displaystyle n_{t}=\lceil bU^{2}(T-t+1)^{a}\rceil,$ (1)
where $\lceil\cdot\rceil$ is the regularized ceiling function, $T$ is the
number of episodes, $U$ is the bound of $J_{i}$ as in Assumption 1, and
$a,b\in(0,1)$ are parameters to be selected later.
Using the cost evaluations at each episode, zeroth-order methods can be
employed to estimate the CVaR gradients and then update the agents’ actions.
Specifically, at episode $t$, the agents calculate the number of samples
$n_{t}$ according to (1). Then, each agent perturbs the current action
$x_{i,t}$ by an amount $\delta u_{i,t}$, where $u_{i,t}\in\mathbb{S}^{d_{i}}$
is a random perturbation direction sampled from the unit sphere
$\mathbb{S}^{d_{i}}\subset\mathbb{R}^{d_{i}}$ and $\delta$ is the size of this
perturbation. Next the agents play their perturbed actions
$\hat{x}_{i,t}=x_{i,t}+\delta u_{i,t}$ for $n_{t}$ times, and obtain $n_{t}$
cost evaluations which are utilized to update their actions. To facilitate the
theoretical analysis, we define the $\delta$-smoothed function
$C_{i}^{\delta}(x)=\mathbb{E}_{w_{i}\sim\mathbb{B}_{i},u_{-i}\sim\mathbb{S}_{-i}}[C_{i}(x_{i}+\delta
w_{i},x_{-i}+\delta u_{-i})]$, where $\mathbb{S}_{-i}=\Pi_{j\neq
i}\mathbb{S}_{j}$, and $\mathbb{B}_{i}$, $\mathbb{S}_{i}$ denote the unit ball
and unit sphere in $\mathbb{R}^{d_{i}}$, respectively. The size of the
perturbation $\delta$ here is related to a smoothing parameter that controls
how well $C_{i}^{\delta}(x)$ approximates $C_{i}(x)$. As shown in [14], the
function $C_{i}^{\delta}(x)$ satisfies the following properties.
###### Lemma 2.
Let Assumptions 1 and 2 hold. Then we have that
1. 1.
$C_{i}^{\delta}(x_{i},x_{-i})$ is convex in $x_{i}$,
2. 2.
$C_{i}^{\delta}(x)$ is $L_{0}$-Lipschitz continuous in $x$,
3. 3.
$|C_{i}^{\delta}(x)$-$C_{i}(x)|\leq\delta L_{0}\sqrt{N}$,
4. 4.
$\mathbb{E}[\frac{d_{i}}{\delta}C_{i}(\hat{x}_{t})u_{i,t}]=\nabla_{i}C_{i}^{\delta}(x_{t})$.
The last property in Lemma 2 shows that the term
$\frac{d_{i}}{\delta}C_{i}(\hat{x}_{t})u_{i,t}$ is an unbiased estimate of the
gradient of the smoothed function $C_{i}^{\delta}(x)$. Next we present a lemma
that helps bound the distance between two CVaR values by the distance between
the two corresponding CDFs. The proof can be found in [14].
###### Lemma 3.
Let $F$ and $G$ be two CDFs of two random variables and suppose the random
variables are bounded by $U$. Then we have that
$\displaystyle|{\rm{CVaR}}_{\alpha}[F]-{\rm{CVaR}}_{\alpha}[G]|\leq\frac{U}{\alpha}\mathop{\rm{sup}}_{y}|F(y)-G(y)|.$
## IV A zeroth-order momentum method
In this section, we present a new one-point zeroth-order momentum method that
combines sample reuse as in Algorithm 2 in [14] and residual feedback as in
Algorithm 3 in [14], but unlike Algorithm 2 in [14] that reuses samples only
from the last iteration, it uses all past samples to update the agents’
actions. The new algorithm is given as Algorithm 1.
Algorithm 1 Risk-averse learning with momentum
0: Initial value $x_{0}$, step size $\eta$, parameters $a$, $b$, $\delta$,
$T$, risk level $\alpha_{i}$, $i=1,\cdots,N$.
1: for ${\rm{episode}}\;t=1,\ldots,T$ do
2: Select $n_{t}=\lceil bU^{2}(T-t+1)^{a}\rceil$
3: Each agent samples $u_{i,t}\in\mathbb{S}^{d_{i}}$, $i=1,\ldots,N$
4: Each agent play $\hat{x}_{i,t}=x_{i,t}+\delta u_{i,t}$, $i=1,\ldots,N$
5: for $j=1,\ldots,n_{t}$ do
6: Let all agents play $\hat{x}_{i,t}$
7: Obtain $J_{i}(\hat{x}_{i,t},\hat{x}_{-i,t},\xi_{i}^{j})$
8: end for
9: for agent $i=1,\ldots,N$ do
10: Build EDF $\bar{F}_{i,t}(y)$
11: Estimate CVaR: ${\rm{CVaR}}_{\alpha_{i}}[\bar{F}_{i,t}]$
12: Construct gradient
estimate$\bar{g}_{i,t}=\frac{d_{i}}{\delta}\left({\rm{CVaR}}_{\alpha_{i}}[\bar{F}_{i,t}]-{\rm{CVaR}}_{\alpha_{i}}[\bar{F}_{i,t-1}]\right)u_{i,t}$
13: Update $x$:
$x_{i,t+1}\leftarrow\mathcal{P}_{\mathcal{X}_{i}^{\delta}}(x_{i,t}-\eta\bar{g}_{i,t})$
14: end for
15: end for
Using the sampling strategy in (1), each agent plays the perturbed action
$\hat{x}_{i,t}$ for $n_{t}$ times and obtains $n_{t}$ samples at episode $t$.
For agent $i$, we denote the CDF of the random cost
$J_{i}(\hat{x}_{t},\xi_{i})$ that is returned by the perturbed action
$\hat{x}_{t}$ as
$F_{i,t}^{\hat{x}_{t}}(y)=\mathbb{P}\\{J_{i}(\hat{x}_{t},\xi_{i})\leq y\\}$.
Since $\hat{x}_{t}$ depends on $t$, we write $F_{i,t}$ for
$F_{i,t}^{\hat{x}_{t}}$ for ease of notation. Using bandit feedback in the
form of finitely many cost evaluations, the agents cannot obtain the accurate
CDF $F_{i,t}$. Instead, they construct an empirical distribution function
(EDF) $\hat{F}_{i,t}$ of the cost $J_{i}(\hat{x}_{t},\xi_{i})$ using $n_{t}$
cost evaluations by
$\displaystyle\hat{F}_{i,t}(y)=\frac{1}{n_{t}}\sum_{j=1}^{n_{t}}\mathbf{1}\\{J_{i}(\hat{x}_{t},\xi_{i}^{j})\leq
y\\}.$ (2)
To improve estimate of the CDF, we utilize past samples for all episodes
$t\geq 2$, and construct a modified distribution estimate $\bar{F}_{i,t}$ by
adding a momentum term:
$\displaystyle\bar{F}_{i,t}(y)=\beta\bar{F}_{i,t-1}(y)+(1-\beta)\hat{F}_{i,t}(y),$
(3)
where $\beta\in[0,1)$ is the momentum parameter. For $t=1$, we set
$\bar{F}_{i,t}(y)=\hat{F}_{i,t}(y)$. The agents utilize the distribution
estimate $\bar{F}_{i,t}$ to calculate the CVaR estimates and further construct
the gradient estimate using residual feedback as
$\displaystyle\bar{g}_{i,t}=\frac{d_{i}}{\delta}\left({\rm{CVaR}}_{\alpha_{i}}[\bar{F}_{i,t}]-{\rm{CVaR}}_{\alpha_{i}}[\bar{F}_{i,t-1}]\right)u_{i,t},$
(4)
where $\delta$ is the size of the perturbation on the action $x_{i,t}$ defined
above. Depending on the size of the perturbation, the played action
$\hat{x}_{i,t}$ may be infeasible, i.e.,
$\hat{x}_{i,t}\not\in\mathcal{X}_{i}$. To handle this issue, we define the
projection set
$\mathcal{X}_{i}^{\delta}=\\{x_{i}\in\mathcal{X}_{i}|{\rm{dist}}(x_{i},\partial\mathcal{X}_{i})\geq\delta\\}$
that we use in the projected gradient-descent update
$\displaystyle
x_{i,t+1}=\mathcal{P}_{\mathcal{X}_{i}^{\delta}}(x_{i,t}-\eta\bar{g}_{i,t}).$
(5)
Note that the agents are not able to obtain accurate values of
$C_{i}(\hat{x}_{t})$ using finite samples. In fact, if we use the distribution
estimate $\bar{F}_{i,t}$ in (3) to calculate the CVaR values, there will be a
CVaR estimation error, which is defined as
$\displaystyle\bar{\varepsilon}_{i,t}:={\rm{CVaR}}_{\alpha_{i}}[\bar{F}_{i,t}]-{\rm{CVaR}}_{\alpha_{i}}[F_{i,t}].$
As a result, the gradient estimate $\bar{g}_{i,t}$ in (4) is biased since
$\mathbb{E}[\bar{g}_{i,t}]=\mathbb{E}\left[\frac{d_{i}}{\delta}(C_{i}(\hat{x}_{t})+\bar{\varepsilon}_{i,t})u_{i,t}\right]=\nabla_{i}C_{i}^{\delta}(x_{t})+\mathbb{E}\left[\frac{d_{i}}{\delta}\bar{\varepsilon}_{i,t}u_{i,t}\right]$,
with the bias captured by the last term. In what follows, we present a lemma
that bounds the sum of the CVaR estimation errors.
###### Lemma 4.
Suppose that Assumption 3 holds. Then, the sum of the CVaR estimation errors
satisfies
$\displaystyle\sum_{t=1}^{T}\left\|\bar{\varepsilon}_{i,t}\right\|\leq\frac{U}{\alpha_{i}}\Big{(}\frac{\beta}{1-\beta}e_{i,1}+\frac{\beta\Sigma_{1}}{1-\beta}T+\sum_{t=1}^{T}r_{t}\Big{)}$
(6)
with probability at least $1-\gamma$, where
$e_{i,t}:=\mathop{\rm{sup}}_{y}|\bar{F}_{i,t}(y)-F_{i,t}(y)|$,
$r_{t}:=\sqrt{\frac{{\rm{ln}}(2T/\gamma)}{2bU^{2}(T-t+1)^{a}}}$, and
$\Sigma_{1}:=C_{0}\eta\frac{d_{i}}{\delta}U\sqrt{N}+2C_{0}\delta$.
###### Proof.
To bound the CVaR estimation error, it suffices to bound the CDF differences
using Lemma 3. Recall the definition of $\bar{F}_{i,t}$ in (3). Then, we have
that
$\displaystyle\mathop{\rm{sup}}_{y}|\bar{F}_{i,t}(y)-F_{i,t}(y)|$
$\displaystyle=$
$\displaystyle\mathop{\rm{sup}}_{y}|\beta\bar{F}_{i,t-1}(y)+(1-\beta)\hat{F}_{i,t}(y)-F_{i,t}(y)|$
$\displaystyle=$
$\displaystyle\mathop{\rm{sup}}_{y}|\beta(\bar{F}_{i,t-1}(y)-{F}_{i,t-1}(y))+\beta({F}_{i,t-1}(y)-{F}_{i,t}(y))$
$\displaystyle+(1-\beta)(\hat{F}_{i,t}(y)-{F}_{i,t}(y))|$ $\displaystyle\leq$
$\displaystyle\beta\mathop{\rm{sup}}_{y}|\bar{F}_{i,t-1}(y)-{F}_{i,t-1}(y)|+\beta\mathop{\rm{sup}}_{y}|{F}_{i,t-1}(y)-{F}_{i,t}(y)|$
$\displaystyle+(1-\beta)\mathop{\rm{sup}}_{y}|\hat{F}_{i,t}(y)-{F}_{i,t}(y)|.$
(7)
We now bound the latter two terms in the right-hand-side of (IV) separately.
Using Assumption 3 and Lemma 2, we have that
$\displaystyle\mathop{\rm{sup}}_{y}|F_{i,t}(y)-F_{i,t-1}(y)|$
$\displaystyle\leq$ $\displaystyle
C_{0}\left\|\hat{x}_{t}-\hat{x}_{t-1}\right\|\leq
C_{0}\left\|x_{t}-x_{t-1}\right\|+2C_{0}\delta$ $\displaystyle\leq$
$\displaystyle C_{0}\eta\left\|\bar{g}_{t}\right\|+2C_{0}\delta\leq
C_{0}\eta\frac{d_{i}}{\delta}U\sqrt{N}+2C_{0}\delta:=\Sigma_{1},$ (8)
where the last inequality holds since
$\left\|\bar{g}_{t}\right\|=\sqrt{\sum_{i=1}^{N}\left\|\bar{g}_{i,t}\right\|^{2}}\leq\frac{\sqrt{N}d_{i}U}{\delta}$.
Then, applying the Dvoretzky–Kiefer–Wolfowitz (DKW) inequality, we obtain that
$\displaystyle\mathbb{P}\left\\{\mathop{\rm{sup}}_{y}|F_{i,t}(y)-\hat{F}_{i,t}(y)|\geq\sqrt{\frac{{\rm{ln}}(2/\bar{\gamma})}{2n_{t}}}\right\\}\leq\bar{\gamma}.$
(9)
Define the events in (9) as $A_{t}$ and denote $\gamma=\bar{\gamma}T$. Then,
the following inequality holds
$\displaystyle\mathop{\rm{sup}}_{y}|F_{i,t}(y)-\hat{F}_{i,t}(y)|\leq\sqrt{\frac{{\rm{ln}}(2T/\gamma)}{2n_{t}}},\forall
t=1,\ldots,T,$ (10)
with probability at least $1-\gamma$ since
$1-\mathbb{P}\\{\bigcup_{t=1}^{T}A_{t}\\}\geq
1-\sum_{t=1}^{T}\mathbb{P}\\{A_{t}\\}\geq 1-T\frac{\gamma}{T}\geq 1-\gamma$.
Substituting the bounds in (IV) and (10) into (IV), and using the definition
of $e_{i,t}$, we get that
$\displaystyle e_{i,t}\leq\beta e_{i,t-1}+\beta\Sigma_{1}+(1-\beta)r_{t}.$
(11)
Note that (11) holds for all $t\in\\{2,\cdots,T\\}$. By iteratively using this
inequality, we have that
$\displaystyle
e_{i,t}\leq\frac{\beta}{1-\beta}\Sigma_{1}+(1-\beta)\sum_{k=0}^{k-2}\beta^{k}r_{t-k}+\beta^{t-1}e_{1,i}.$
(12)
Taking the sum over $t=1,\ldots,T$ of both sides of (12), we have that
$\displaystyle\sum_{t=1}^{T}e_{i,t}\leq\frac{\beta}{1-\beta}e_{i,1}+\frac{\beta\Sigma_{1}}{1-\beta}T+\sum_{t=1}^{T}r_{t},$
(13)
with probability at least $1-\gamma$. Finally, using Lemma 3, we can obtain
the desired result. ∎
Lemma 4 bounds the CVaR estimation error caused by using past samples. The
next lemma quantifies the error due to residual feedback in the gradient
estimate (4). Specifically, it bounds the second moment of the gradient
estimate.
###### Lemma 5.
Let Assumptions 1, 2 and 3 hold. Suppose that the action is updated as in (5).
Then, the second moment of the gradient estimate satisfies
$\displaystyle\sum_{t=1}^{T}\left\|\bar{g}_{t}\right\|^{2}\leq\frac{1}{1-\sigma}\left\|\bar{g}_{1}\right\|^{2}+\frac{1}{1-\sigma}\Sigma_{2}T,$
(14)
where $\bar{g}_{t}:=(\bar{g}_{i,t},\bar{g}_{-i,t})$ is the concatenated vector
of all agents’ gradient estimates,
$\sigma=\frac{4d_{i}^{2}L_{0}^{2}N\eta^{2}}{\delta^{2}}$,
$\Sigma_{2}=16d_{i}^{2}N^{2}L_{0}^{2}+\frac{48C_{0}^{2}d_{i}^{4}U^{4}N\eta^{2}\beta^{2}}{\delta^{4}(1-\beta)^{2}}(\sum_{i}\frac{1}{\alpha_{i}^{2}})+\frac{192C_{0}^{2}d_{i}^{2}U^{2}\beta^{2}}{(1-\beta)^{2}}(\sum_{i}\frac{1}{\alpha_{i}^{2}})+\frac{12\ln(2T/\gamma)d_{i}^{2}}{b\delta^{2}}(\sum_{i}\frac{1}{\alpha_{i}^{2}})+\frac{24d_{i}^{2}U^{2}\beta^{2}e_{m}^{2}}{\delta^{2}}(\sum_{i}\frac{1}{\alpha_{i}^{2}})$,
and $e_{m}:=\mathop{\max}_{i}\\{e_{i,1}\\}$.
###### Proof.
Using the fact that $r_{t}\leq\sqrt{\frac{{\rm{ln}}(2T/\gamma)}{2bU^{2}}}:=r$
and (12), we have that
$\displaystyle\left\|e_{i,t}\right\|^{2}$
$\displaystyle\leq\left\|\frac{\beta}{1-\beta}\Sigma_{1}+r+\beta
e_{m}\right\|^{2}$ $\displaystyle\leq
3\left\|\frac{\beta}{1-\beta}\Sigma_{1}\right\|^{2}+3\left\|r\right\|^{2}+3\left\|\beta
e_{m}\right\|^{2},$ (15)
where the last inequality holds since $\left\|a+b+c\right\|^{2}\leq
3\left\|a\right\|^{2}+3\left\|b\right\|^{2}+3\left\|c\right\|^{2}$. Then,
applying Lemma 3 to (IV), we obtain that
$\displaystyle\left\|\bar{\varepsilon}_{i,t}\right\|^{2}\leq\frac{U^{2}}{\alpha_{i}^{2}}\left(\frac{3\beta^{2}\Sigma_{1}^{2}}{(1-\beta)^{2}}+3\left\|r\right\|^{2}+3\beta^{2}\left\|e_{m}\right\|^{2}\right).$
(16)
By the definition of $\bar{g}_{i,t}$ in (4), we have that
$\displaystyle\left\|\bar{g}_{i,t}\right\|^{2}=\frac{d_{i}^{2}}{\delta^{2}}\left(({\rm{CVaR}}_{\alpha_{i}}[\bar{F}_{i,t}]-{\rm{CVaR}}_{\alpha_{i}}[\bar{F}_{i,t-1}])u_{i,t}\right)^{2}$
$\displaystyle\leq\frac{d_{i}^{2}}{\delta^{2}}\left({\rm{CVaR}}_{\alpha_{i}}[\bar{F}_{i,t}]-{\rm{CVaR}}_{\alpha_{i}}[\bar{F}_{i,t-1}]\right)^{2}\left\|u_{i,t}\right\|^{2}$
$\displaystyle\leq\frac{d_{i}^{2}}{\delta^{2}}\Big{(}2({\rm{CVaR}}_{\alpha_{i}}[F_{i,t}]-{\rm{CVaR}}_{\alpha_{i}}[F_{i,t-1}])^{2}$
$\displaystyle\quad+2(\bar{\varepsilon}_{i,t}-\bar{\varepsilon}_{i,t-1})^{2}\Big{)}\left\|u_{i,t}\right\|^{2}$
$\displaystyle\leq\frac{d_{i}^{2}}{\delta^{2}}\Big{(}2L_{0}^{2}\left\|\hat{x}_{t}-\hat{x}_{t-1}\right\|^{2}+4|\bar{\varepsilon}_{i,t}|^{2}+4|\bar{\varepsilon}_{i,t-1}|^{2}\Big{)},$
(17)
where the last inequality is due to the fact that the CVaR function is
Lipschitz continuous as shown in Lemma 1. We now bound the term
$\left\|\hat{x}_{t}-\hat{x}_{t-1}\right\|^{2}$ in (IV). Recalling the update
rule
$x_{i,t}=\mathcal{P}_{\mathcal{X}_{i}^{\delta}}(x_{i,t-1}-\eta_{i}\bar{g}_{i,t-1})$,
we have that
$\displaystyle\left\|\hat{x}_{t}-\hat{x}_{t-1}\right\|^{2}=\left\|x_{t}-x_{t-1}+\delta
u_{t}-\delta u_{t-1}\right\|^{2}$ $\displaystyle\leq$ $\displaystyle
2\left\|x_{t}-x_{t-1}\right\|^{2}+2\delta^{2}(2\left\|u_{t}\right\|^{2}+2\left\|u_{t-1}\right\|^{2})$
$\displaystyle\leq$ $\displaystyle
2\left\|x_{t}-x_{t-1}\right\|^{2}+8N\delta^{2}\leq
2\eta^{2}\left\|\bar{g}_{t-1}\right\|^{2}+8N\delta^{2}.$
Substituting the above inequality and the inequality in (16) into the right
hand side of (IV) and summing both sides over all agents $i=1,\ldots,N$, we
have that
$\displaystyle\left\|\bar{g}_{t}\right\|^{2}=\sum_{i=1}^{N}\left\|\bar{g}_{i,t}\right\|^{2}$
$\displaystyle\leq\frac{d_{i}^{2}N}{\delta^{2}}\Big{(}4L_{0}^{2}\eta^{2}\left\|\bar{g}_{t-1}\right\|^{2}+16L_{0}^{2}N\delta^{2}\Big{)}$
$\displaystyle\quad+\frac{8d_{i}^{2}}{\delta^{2}}\sum_{i=1}^{N}\frac{U^{2}}{\alpha_{i}^{2}}\left(\frac{3\beta^{2}\Sigma_{1}^{2}}{(1-\beta)^{2}}+3\left\|r\right\|^{2}+3\beta^{2}\left\|e_{m}\right\|^{2}\right)$
$\displaystyle\leq\sigma\left\|\bar{g}_{t-1}\right\|^{2}+16d_{i}^{2}N^{2}L_{0}^{2}+\frac{12\ln(2T/\gamma)d_{i}^{2}}{b\delta^{2}}(\sum_{i}\frac{1}{\alpha_{i}^{2}})$
$\displaystyle\quad+\frac{24d_{i}^{2}U^{2}\beta^{2}}{\delta^{2}(1-\beta)^{2}}(C_{0}\eta\frac{d_{i}}{\delta}U\sqrt{N}+2C_{0}\delta)^{2}(\sum_{i}\frac{1}{\alpha_{i}^{2}})$
$\displaystyle\quad+\frac{24d_{i}^{2}U^{2}\beta^{2}e_{m}^{2}}{\delta^{2}}(\sum_{i}\frac{1}{\alpha_{i}^{2}})$
$\displaystyle\leq\sigma\left\|\bar{g}_{t-1}\right\|^{2}+\Sigma_{2},$ (18)
where the last inequality is due to the fact that
$(C_{0}\eta\frac{d_{i}}{\delta}U\sqrt{N}+2C_{0}\delta)^{2}\leq
2C_{0}^{2}\eta^{2}\frac{d_{i}^{2}}{\delta^{2}}U^{2}N+8C_{0}^{2}\delta^{2}$.
Summing (IV) over all episode $t=1,\ldots,T$ completes the proof. ∎
Before proving the main theorem, we present a regret decomposition lemma that
links the regret to the errors bounds on the CVaR estimates and gradient
estimates as provided in Lemmas 4 and 5.
###### Lemma 6.
Let Assumptions 1, 2 and 3 hold. Then, the regret of Algorithm 1 satisfies
$\displaystyle{\rm{R}}_{C_{i}}(T)\leq$
$\displaystyle\frac{D_{x}^{2}}{2\eta}+\frac{\eta}{2}\mathbb{E}\big{[}\sum_{t=1}^{T}\left\|\bar{g}_{i,t}\right\|^{2}\big{]}+(4\sqrt{N}+\Omega)L_{0}\delta
T$
$\displaystyle+\frac{d_{i}D_{x}}{\delta}\sum_{t=1}^{T}|\bar{\varepsilon}_{i,t}|,$
(19)
where $\Omega>0$ is a constant that represents the error from projection
$\mathcal{P}$.
###### Proof.
The proof can be adapted from Lemma 5 in [14] and is omitted due to the space
limitations. ∎
###### Theorem 1.
Let Assumptions 1, 2 and 3 hold and select
$\eta=\frac{D_{x}}{d_{i}L_{0}N}T^{-\frac{3a}{4}}$,
$\delta=\frac{D_{x}}{N^{\frac{1}{6}}}T^{-\frac{a}{4}}$,
$\beta=\frac{1}{U^{2}T^{\frac{a}{4}}}$. Suppose that $n_{t}$ is chosen as in
(1) with $a\in(0,1)$, and the EDF and the gradient estimate are defined as in
(2) and (4), respectively. Then, when $T\geq(8N^{\frac{2}{3}})^{\frac{1}{a}}$,
Algorithm 1 achieves regret
${\rm{R}}_{C_{i}}(T)=\mathcal{O}(D_{x}d_{i}L_{0}NS(\alpha)\ln(T/\gamma)T^{1-\frac{a}{4}}),$
with probability at least $1-\gamma$ and where
$S(\alpha):=\sum_{i=1}^{N}\frac{1}{\alpha_{i}^{2}}$.
###### Proof.
Substituting the bounds in Lemmas 4 and 5 into Lemma 6, we have that
$\displaystyle{\rm{R}}_{C_{i}}(T)\leq$
$\displaystyle\frac{D_{x}^{2}}{2\eta}+\frac{\eta}{2}\left(\frac{1}{1-\sigma}\left\|\bar{g}_{1}\right\|^{2}+\frac{1}{1-\sigma}\Sigma_{2}T\right)$
$\displaystyle+\frac{d_{i}D_{x}U}{\delta\alpha_{i}}\Big{(}\frac{\beta}{1-\beta}e_{i,1}+\frac{\beta\Sigma_{1}}{1-\beta}T+\sum_{t=1}^{T}r_{t}\Big{)}$
$\displaystyle+(4\sqrt{N}+\Omega)L_{0}\delta T.$ (20)
Substituting $\delta$, $\eta$ and $\beta$ into the above inequality, we can
obtain the desired result. The detailed proof is straightforward and is
omitted due to the space limitations. ∎
Note that Algorithm 1 in fact achieves the regret
${\rm{R}}_{C_{i}}(T)=\tilde{\mathcal{O}}(T^{1-\frac{a}{4}})$, in which we use
the notation $\tilde{\mathcal{O}}$ to hide constant factors and poly-
logarithmic factors of $T$. This result matches the best result in [14].
Compared to [14], the analysis of Algorithm 1 that combines historical
information and residual feedback is nontrivial and requires an appropriate
selection of the momentum parameter to show convergence. Note that Algorithm 3
in [14] is a special case of Algorithm 1 for the momentum parameter $\beta=0$.
## V Numerical Experiments
In this section, we illustrate the proposed algorithm on a Cournot game.
Specifically, we consider two risk-averse agents $i=1,2$ that compete with
each other. Each agent determines the production level $x_{i}$ and receives an
individual cost feedback which is given by
$J_{i}=1-(2-\sum_{j}x_{j})x_{i}+0.2x_{i}+\xi_{i}x_{i}$, where $\xi_{i}\sim
U(0,1)$ is a uniform random variable. Here we utilize the term $\xi_{i}x_{i}$
to represent the uncertainty occurred in the market, which is proportional to
the production level $x_{i}$. The agents have their own risk levels
$\alpha_{i}$ and they aim to minimize the risk of incurring high costs, i.e.,
the CVaR of their cost functions.
Note that the regret for agent $i$ depends on the sequence of the other
agent’s actions $\\{x_{-i,t}\\}_{t=1}^{T}$, and this sequence depends on the
algorithm. Therefore, it is not appropriate to compare different algorithms in
terms of the regret. Instead, we use the empirical performance to evaluate the
algorithms. Specifically, we compute the CVaR values at each episode, and
observe how fast the agents can minimize the CVaR values during the learning
process.
To compute the distribution function $\bar{F}_{i,t}$, we divide the interval
$[0,U]$ into $n$ bins of equal width, and we approximate the expectation by
the sum of finite terms. We compare our zeroth-order momentum method with the
two algorithms in [14], which we term as the algorithm with sample reuse and
the algorithm with residual feedback, respectively. The number of samples
$n_{t}$ at each episode for Algorithm 1 is shown in Figure 1. We set the
momentum parameter $\beta=0.5$. Each algorithm is run for 20 trials and the
parameters of these algorithms are separately optimally tuned. The CVaR values
of these algorithms are presented in Figure 2. We observe that all the
algorithms finally converge to the same CVaR values, but the zeroth-order
momentum method proposed here converges faster. This is because our method
uses all past samples to get a better estimate of the CVaR values and thus
allows for a larger step size. Moreover, we also observe that the standard
deviation of both our proposed algorithm and the algorithm with residual
feedback is sufficiently small, which verifies the variance reduction effect
of using residual feedback.
Figure 1: The number of samples of Algorithm 1.
Figure 2: Comparative results among our proposed zeroth-order momentum method,
and the three algorithms in [14]. Shaded areas represent $\pm$ one standard
deviation over 20 runs.
Figure 3: CVaR values achieved by Algorithm 1 with different $\beta$ values.
Shaded areas represent $\pm$ one standard deviation over 20 runs.
Moreover, we explore the effect of different momentum parameters $\beta$. In
Figure 3, we present the achieved CVaR values of our method for various
$\beta$ values and all other parameters unchanged. Recalling the definition of
the distribution estimate in (3), large values of $\beta$ place more weight on
outdated cost values and thus reduce the convergence speed.
## VI Conclusion
In this work, we proposed a zeroth-order momentum method for online convex
games with risk-averse agents. The use of momentum that employs past samples
allowed to improve the ability of the algorithm to estimate the CVaR values.
Zeroth-order estimation of the CVaR gradients using residual feedback allowed
us to reduce the variance of the CVaR gradient estimates. We showed that the
proposed algorithm theoretically achieves no-regret learning with high
probability, matching the best known result in literature. Moreover, we
provided numerical simulations that demonstrated the superior performance of
our method in practice. While sample reuse and residual feedback had been both
separately shown to improve the performance of online learning, here we showed
that their combination yields yet better performance.
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|
# Measuring the magnetic origins of solar flares, CMEs and Space Weather
Philip Judge High Altitude Observatory, National Center for Atmospheric
Research, Boulder CO 80307-3000, USA Matthias Rempel High Altitude
Observatory, National Center for Atmospheric Research, Boulder CO 80307-3000,
USA Rana Ezzeddine Department of Astronomy, University of Florida, 211
Bryant Space Sciences Center, Gainesville, Florida, 32611, USA Lucia Kleint
Université de Genève, Centre Universitaire d’Informatique, 7, route de Drize,
1227 Carouge, Switzerland Ricky Egeland High Altitude Observatory, National
Center for Atmospheric Research, Boulder CO 80307-3000, USA Svetlana V.
Berdyugina Leibniz-Institut fuer Sonnenphysik (KIS), Schöneckstrasse 6,
79104, Freiburg, Germany Thomas Berger Space Weather Technology, Research
and Education Center, 3775 Discovery Drive N433 Boulder, CO 80309 Paul Bryans
High Altitude Observatory, National Center for Atmospheric Research, Boulder
CO 80307-3000, USA Joan Burkepile High Altitude Observatory, National Center
for Atmospheric Research, Boulder CO 80307-3000, USA Rebecca Centeno High
Altitude Observatory, National Center for Atmospheric Research, Boulder CO
80307-3000, USA Giuliana de Toma High Altitude Observatory, National Center
for Atmospheric Research, Boulder CO 80307-3000, USA Mausumi Dikpati High
Altitude Observatory, National Center for Atmospheric Research, Boulder CO
80307-3000, USA Yuhong Fan High Altitude Observatory, National Center for
Atmospheric Research, Boulder CO 80307-3000, USA Holly Gilbert High Altitude
Observatory, National Center for Atmospheric Research, Boulder CO 80307-3000,
USA Daniela A. Lacatus High Altitude Observatory, National Center for
Atmospheric Research, Boulder CO 80307-3000, USA
###### Abstract
We take a broad look at the problem of identifying the magnetic solar causes
of space weather. With the lackluster performance of extrapolations based upon
magnetic field measurements in the photosphere, we identify a region in the
near UV part of the spectrum as optimal for studying the development of
magnetic free energy over active regions. Using data from SORCE, Hubble Space
Telescope, and SKYLAB, along with 1D computations of the near-UV (NUV)
spectrum and numerical experiments based on the MURaM radiation-MHD and
HanleRT radiative transfer codes, we address multiple challenges. These
challenges are best met through a combination of near UV lines of bright Mg
II, and lines of Fe II and Fe I (mostly within the $4s-4p$ transition array)
which form in the chromosphere up to $2\times 10^{4}$ K. Both Hanle and Zeeman
effects can in principle be used to derive vector magnetic fields. However,
for any given spectral line the $\tau=1$ surfaces are generally geometrically
corrugated owing to fine structure such as fibrils and spicules. By using
multiple spectral lines spanning different optical depths, magnetic fields
across nearly-horizontal surfaces can be inferred in regions of low plasma
$\beta$, from which free energies, magnetic topology and other quantities can
be derived.
Based upon the recently-reported successful suborbital space measurements of
magnetic fields with the CLASP2 instrument, we argue that a modest space-borne
telescope will be able to make significant advances in the attempts to predict
solar eruptions. Difficulties associated with blended lines are shown to be
minor in an Appendix.
Sun: atmosphere - Sun: magnetic fields
## 1 Introduction
Commenting on the origin of solar flares, in 1960 Gold & Hoyle wrote
> The requirements of the theory can therefore be stated quite definitely.
> Magnetic field configurations must be found that are capable of storing
> energy densities hundreds of times greater than occur in any other form, and
> that are stable most of the time. A situation that occurs only a small
> fraction of the time must be able to lead to instability in which this
> energy can rapidly be dissipated into heat and mass motion…
These requirements have remained unchanged in the intervening six decades. We
now have a “standard model” of flares (Carmichael, 1964; Sturrock, 1966;
Hirayama, 1974; Kopp & Pneuman, 1976), involving a loss of equilibrium of
arcades of closed magnetic flux systems, particle acceleration and,
frequently, ejection of plasmoids. The instability can only arise when the
free magnetic energy stored in the coronal magnetic fields exceeds certain
thresholds. This is, however, a necessary but not sufficient condition for
flaring and plasmoid ejection, because other global and local properties of
the magnetic field, notably the topology, also determine the stabilility of
coronal MHD systems (Low, 1994).
Enormous interest in solar eruptions is driven by society’s need to understand
origins of “space weather” (e.g. Eddy, 2009) as well as its intrinsic
scientific challenges. However, much like weather prediction before the advent
of modern instrumentation and computers, predicting the properties of
eruptions and flares has remained a difficult challenge primarily due to the
lack of quantitative data available for analysis of the pre-flare state and
flare-triggering mechanism(s). In particular, the only high-resolution, high
cadence, data available to date has been the magnetic field state in the
photosphere, which has been used to define “boundary conditions” to the
overlying coronal magnetic field, where most of the free energy is stored
prior to eruption. Herein lies the basic problem: magnetic fields measured in
the photosphere are not ideally suited to the task. We elaborate upon this
below.
The situation described by Gold & Hoyle (1960) draws analogy with trying to
predict the timing and location of lightning strikes (Judge, 2020). By
measuring updrafts and/or observing cloud buildup one can identify the causes
of the buildup of electrostatic energy, but lightning occurs in a predictable
way only in a statistical sense. In-situ measurements of electric fields are
required even then to make probabilistic forecasts of lightning strikes to the
ground, without reference to intra- or inter- cloud discharges. In this sense
the weaknesses found in previous prediction methods is perhaps unsurprising.
The difficulty of the task has been summarized in the write-up of an inter-
agency “All Clear Workshop” (Barnes et al., 2016) in which the authors
conclude with the sobering statement that
> For M-class flares and above, the set of methods tends toward a weakly
> positive skill score (as measured with several distinct metrics), with no
> participating method proving substantially better than climatological
> forecasts.
Steady improvements in the performance of flare forecasts has been documented
in more recent years (e.g. Leka et al., 2019). However, these authors make
explicit the need to augment existing photospheric magnetic data in order to
proceed, with a heterogeneous mix of theory and observation:
> quantitative “modern” forecasts incorporate …physical understanding as they
> often characterize coronal magnetic energy storage by proxy, with the
> parameterizations of photospheric magnetograms.
In other words, measurements of the photospheric magnetic field can be related
only indirectly and through certain ad-hoc parameterizations, to the magnetic
free energy of the overlying corona, which is the ultimate origin of
instabilities leading to flares and CMEs.
With the availability of a decade of high-resolution photospheric magnetic
field data from NASA’s Solar Dynamics Observatory (SDO) mission, recent
efforts have focused on statistical pattern recognition methods for flare
prediction that generally fall under the “machine learning (ML)” sub-field of
artificial intelligence research. For example, a study by Bobra & Couvidat
(2015) used photospheric vector magnetic data of 2071 active regions from the
Helioseismic and Magnetic Imager (HMI) instrument on SDO. They predicted
strong flares using a Support Vector Machine (SVM) algorithm applied to a set
of 25 parameters derived from the vector magnetic field measured in SHARPs
(Space-weather HMI Active Region Patches), which are cut-outs around sunspot
active regions. They achieved a high True Skill Statistic (TSS) of $\approx$
0.8, but at the cost of getting a significant number of false positives, a
common characteristic of all ML flare prediction algorithms optimized for TSS
in the strongly unbalanced training sets which naturally contain many more “no
flare” examples than “flare” examples. Later studies (Park et al., 2018; Liu
et al., 2019; Chen et al., 2019; Li et al., 2020) applied recurrent
architectures such as Long Short-Term Memory (LSTM) systems and deep learning
systems including Convolution Neural Network (CNN) and autoencoder
architectures to explore feature sets that extend beyond the magnetic field-
derived features of earlier ML flare prediction systems. However these studies
do not demonstrate significant increases in skill scores relative to earlier
approaches.
These studies have used only photospheric measurements of magnetic fields.
Investigations have also been carried out that include information from higher
atmospheric layers, e.g., by incorporating SDO Atmospheric Imaging Array (AIA)
data into existing ML flare prediction systems (Nishizuka et al., 2018; Jonas
et al., 2018), or by using extrapolated coronal magnetic field models (Korsós
et al., 2020). Chromospheric UV spectral lines from IRIS have also been
scrutinized for additional flare prediction utility (Panos & Kleint, 2020).
These studies all show some degree of improvement in flare prediction metrics
when including chromospheric or derived coronal information. More recently,
Deshmukh et al. (2020) showed that combining Topological Data Analysis (TDA)
of the radial magnetic field structure in active regions with the SHARPs
vector magnetic field metadata results in similar or slightly higher skill
scores compared to using selected SHARPs metadata alone. This result implies
that inclusion of high-resolution imaging of active region and flows may be an
important complement to spectroscopic measurements when deducing atmospheric
conditions that evolve to a flaring state.
Physically, however, solar flare prediction based upon photospheric data is
ultimately confronted with basic difficulties, beyond the well-known
disambiguation problem associated with native symmetries in the Zeeman effect.
The photosphere is a dense radiating layer of high density ($10^{-7}$ g cm-3)
plasma. Outside of sunspot umbrae the photospheric magnetic field is unable to
suppress convective motions which therefore control the magnetic fields
threading the fluid, a case where the plasma-$\beta=$ gas pressure / magnetic
pressure, is generally $>1$. In traversing the chromosphere, emerging magnetic
fields drop in strength algebraically but the plasma densities and energy
densities drop exponentially (by 7 and 5 orders of magnitude, respectively),
so that the tenuous plasma at the coronal base is in a low-$\beta$ state. One
immediate practical problem is that tiny dynamical changes which may be
unobservable in the dense photosphere can have enormous consequences for the
tenuous plasma and magnetic fields above: the problem is observationally ill-
posed. As a result of these different photospheric and coronal plasma regimes,
the time scales for significant photospheric magnetic field evolution are
typically measured in minutes to hours whereas the dynamic evolution and
reconnection of the coronal magnetic field configurations leading to flares,
after a quasi-static buildup phase, is measured in seconds. This fundamental
mismatch in dynamical time scales evidently cannot be overcome through any
type of analysis of photospheric magnetic field data and leads to the
conclusion that quantitative analysis of the magnetic field and flows in the
chromosphere and corona are necessary to catalyze significant breakthroughs.
Figure 1: Contribution functions are shown as a function of temperature from
the model C of Fontenla et al. (1993). The emergent intensities are over-
plotted as solid lines for a ray from close to disk center
($\cos\theta=0.887$, $\theta$ the angular distance from disk center as
measured from Sun center). The Ca II lines at 393 and 854 nm span from 3.8 to
4.0 in log$T$ (the latter forming in deeper cooler layers than the former),
and the 1083 nm line of He I forms within the last pressure scale height of
the chromosphere. Notice the extension of contributions for the Mg II
resonance lines beyond all others, to $\log T_{e}\geq 4.3$, and the multiple
lines of Fe II between 259 and 261 nm whose cores form close to 3.9-4.0 in
log$T$, and weaker lines in the 275 nm region with stronger contributions from
lower temperatures. A modest atomic model for Fe II was adopted to examine the
line cores formed in non LTE in the upper chromosphere. The calculations used
the RH computer program Uitenbroek (2001). Far more complete LTE calculations
are made in the Appendix where spectral line blends are discussed.
In order to address this and other questions, our purpose here is to explore
the entire solar spectrum to attempt to identify more direct ways to identify
the free energy and topology of magnetic fields above active regions. In this
way we hope to put the prediction of solar magnetic eruptions, the source of
all major space weather events, on a firm observational basis. The specific
problem we address is reviewed in section 2. It will involve spectro-
polarimetry.
Before proceeding, we point out that our goals may on first reading appear
similar to efforts measuring only intensities of EUV and UV emission lines
(e.g. De Pontieu et al., 2020), an endeavor with a history of 6+ decades yet
able to study only the effects and not causes of magnetic energy storage and
release. Instead our work is closer to an important study by Trujillo Bueno et
al. (2017), but the latter focuses more on the physical processes behind the
magnetic imprints across the solar UV spectrum, including the near-UV spectral
region of the Mg II $h$ & $k$ lines. We focus instead on finding an optimal
set of observations which target the problem of the build-up and release of
magnetic energy. We argue that spectro-polarimetry of the near-UV solar
spectrum is indeed a profitable avenue for research.
## 2 Observational signatures of energy build-up and release
### 2.1 Measuring magnetic fields in and above chromospheric plasma
Two approaches are in principle possible: in-situ measurements and remote
sensing. Clearly in-situ methods are far beyond the capabilities of known
technology. The state-of-the-art mission Parker Solar Probe (Fox et al., 2016)
has a perihelion distance of just below ten solar radii, far above active
regions where most of the free magnetic energy exists. Therefore we limit our
discussion to measuring magnetic fields remotely, an area which has a firm
physical basis (Landi degl’Innocenti & Landolfi, 2004).
Briefly, two types of remote-sensing measurements have been made: direct
measurements of light emitted from within coronal plasma, and measurements
close to the corona’s lower boundary, some 9 pressure scale heights above the
photosphere. Coronal measurements include (see, for example, reviews by Judge
et al., 2001; Casini et al., 2017; Kleint & Gandorfer, 2017, for more complete
references):
* •
Measurements of line-of-sight (LOS) field strengths using differences between
ordinary and extraordinary rays of magneto-ionic theory at GHz frequencies
Gelfreikh (1994).
* •
Measurements of field strengths across surfaces determined typically by a
harmonic of gyro-resonant motions of electrons at frequencies of several GHz,
frequencies uncontaminated by bremsstrahlung opacity only when the field
strength exceeds $\approx$200 G (White, 2004).
* •
Serendipitous level-crossings in atomic ions produces changes in EUV spectra
through level mixing by the Zeeman effect (Li et al., 2017), yielding magnetic
field strengths.
* •
A long history of measurements of magnetic dipole lines during eclipses or
with coronagraphs hold great promise for measuring vector magnetic fields in
the corona. The infrared, coronagraphic DKIST facility offers significant new
opportunities for advancements using such lines (e.g. Judge et al., 2013).
* •
Occultations of background radio sources by the corona have revealed
integrated properties of coronal magnetic fields, weighted by electron density
through the Faraday effect (Bird & Edenhofer, 1990), a technique also applied
to signals from spacecraft away from the Sun-Earth line (Bird, 1982).
Of the second type, we can list:
* •
Several strong lines measured using spectro-polarimetric measurements of the
Zeeman effect from the ground can yield LOS components of chromospheric
magnetic fields. Very high photometric precision is however required for
vector fields, beyond most current capabilities (Uitenbroek, 2010), owing to
the second-order $\epsilon^{2}$ signature of the Zeeman effect in the small
ratio $\epsilon$ of magnetic splitting to the line Doppler width (defined
below). These include Na I D lines, Ca II H, K and infrared triplet lines.
Synthetic Ca II line profiles are shown in the non-LTE calculations using the
RH non-LTE program including partial redistribution (Uitenbroek, 2001) in
Figure 1.
* •
The He I 1083 nm triplet has a proven record of magnetic field determinations
in a variety of cool structures such as spicules (Centeno et al., 2010),
filaments (Kuckein et al., 2012; Díaz Baso et al., 2019) and cool loops
(Solanki et al., 2003), in the vicinity of the base of the corona before,
during and after flares (Judge et al., 2014) even extending far into the
corona (Judge et al., 2012; Schad et al., 2016). Both the Zeeman and Hanle
effects contribute to the formation of this line, owing to the large
contribution of anisotropic radiation incident on the helium atoms to the
statistical equilibrium in the triplet states (e.g. Trujillo Bueno & Asensio
Ramos, 2007).
* •
Recently, sub-orbital rockets have carried sensitive spectropolarimeters to
observe the resonance lines of hydrogen (Kano et al., 2017) and singly-ionized
magnesium (Trujillo Bueno et al., 2018; Ishikawa et al., 2021). Both Zeeman
and Hanle effects have revealed information on magnetic fields near the very
top of the solar chromosphere.
* •
Measurements at 30 GHz and above made with the ALMA interferometer facility
can in principle yield line-of-sight magnetic field strengths using
Gelfreikh’s (1994) theory of continuum polarization (Loukitcheva, 2014, 2020),
but we have been unable to find reports of solar polarization measurements
with ALMA. It can observe from 0.4 to 8.8 mm wavelengths (750 and 34 Ghz
respectively), thereby sampling heights from 600 km to 2000 km in the
chromosphere. Transition region plasma under non-flaring conditions is tenuous
and thin. It contributes very little to the far brighter chromospheric
emission at the high frequencies measured by ALMA.
Next we assess the ability of these techniques to address the specific problem
of interest.
Figure 2: Images are shown to highlight the corrugated surfaces at which the
centers of UV lines are formed (close to $\tau=1$) in MURaM calculations of
the upper solar chromosphere. The calculations are from a numerical experiment
where a magnetic flux system is emerging from beneath the solar atmosphere.
The emergent fields carry with them plasma, which is revealed by the extended
heights of the $\tau=1$ surfaces, which can exceed 10 Mm. Typically the line
shifts in the lower transition region are smaller than the line widths (Athay
& Dere, 1991). Therefore the $\tau=1$ surfaces were computed without taking
into account Doppler shifts, at the centers of lines of Mg II $k$, and several
lines of Fe II. The $x-y$ images of height of formation are plotted in order
of decreasing opacity from Mg II to Fe II 2769.75 Å. In the bottom right
panel, slices of these $\tau=1$ surfaces are shown, taken along the black line
in the five other panels. But the black lines in this final panel are magnetic
lines of force in the $y-z$ plane. Evidently, a combination of these and other
lines of iron with a wide range of opacities spans the ranges of excursions of
the surfaces away from a horizontal plane.
### 2.2 Establishing methods to measure free magnetic energy and topology
While information on magnetism above the photosphere is contained in all the
above strategies, few offer ways to assess the necessary quantities concerning
magnetic free energy as magnetic fields emerge, build and change topology as
it evolves and is suddenly released. If we consider measurements within the
corona, we would have to probe the 3D magnetic structure. However, the corona
is optically thin (with the exception of gyro-resonance emission which yields
just the magnetic field strength within resonant surfaces, whose geometry is
not known from one line of sight), which means that recovery of the 3D coronal
field would require measurements from at least two different lines of sight
(e.g. Kramar et al., 2006). Until suitable instruments are flown on spacecraft
away from the Sun-Earth line, the required stereoscopic views will be
unavailable. Therefore we must seek alternatives.
When chromospheric, transition region and coronal plasmas are connected by
magnetic lines of force (Judge, 2021), the plasma pressure near the top of the
chromosphere is close to that of the overlying corona. Accurate measurements
of magnetic fields using spectral lines formed in plasma up to a few times
$10^{4}$ K can therefore be used to inform us on the evolving magnetic state
of the overlying corona (Trujillo Bueno et al., 2017). Selected spectral line
profiles formed across the upper chromosphere are shown in Figure 1, intended
to illustrate near UV lines in comparison to familiar lines commonly observed
from the ground. The solar spectrum between 250 and 281 nm is dominated by
line transitions of Mg II and many lines belonging to the $4s-4p$ transition
array in neutral and singly ionized iron. In the figure, contributions to the
intensities of Mg II and typical Fe II lines are shown as a function of
electron temperature. This abscissa was chosen instead of height and continuum
optical depth because in the particular 1-dimensional model adopted, the
temperature gradient in the transition region is very large, making lines
appear to originate from the same heights.
The regime of plasma $\beta=8\pi p/B^{2}<1$ in upper chromospheric, transition
region and coronal lines is the same, if the photospheric magnetic field has
expanded to fill the volume. In active region plasma this appears to be the
case (Ishikawa et al., 2021). When measurements are made close to the force-
free state ($\beta\ll 1$), then theory can be reliably invoked in at least two
ways: application of the magnetic virial theorem (Chandrasekhar, 1961), and
extrapolation of fields with observational boundary conditions compatible with
the magnetostatic conditions (Wiegelmann & Sakurai, 2021). Any waves and/or
tangential discontinuities measured will also be of importance to assessing
the free magnetic energy and its evolution. Fleishman et al. (2017) have
confirmed these theoretical concepts quantitatively using model simulations.
Skeptics of our statements might rightfully recall that the chromosphere is
finely structured (see, e.g., the remarkable structure in narrow-band images
of Cauzzi et al., 2008; Wang et al., 2016; Robustini et al., 2018). It is also
known, for example from limb observations, that transition region plasma at
intermediate temperatures has contributions from structures far from a level
surface which might otherwise be amenable to mathematical techniques to
estimate free energies and extrapolated magnetic fields. A non-level,
“corrugated” surface of unit optical depth in a given spectral line can lead
to spurious inferences of magnetic structure even within low-$\beta$ plasmas
(see, e.g., section 4.3 of the paper by Trujillo Bueno et al., 2017, and our
Figure 2 below). The question then arises, is the interface between
chromosphere and corona just too finely structured to make measurements of
magnetic fields there of little use?
Hints from two pieces of work suggest that this is not the case. Firstly, the
CLASP2 measurements reported by Ishikawa et al. (2021) have shown that, at an
angular resolution of 1” which is far smaller than the scales of large coronal
structures leading to flares, the fields measured in the core of the Mg II $h$
& $k$ lines across active regions are unipolar and spatially smooth over large
areas. Any mixed polarity fields undetected by CLASP2 must lie below 1” scales
and will therefore not penetrate more than a few hundred km above the
photosphere. But the fields measured by CLASP2 may well contain unseen
tangential discontinuities arising from small-scale angular displacements
across flux surfaces (Judge et al., 2011). Such features would constitute an
additional source of free magnetic energy whose consequences might then be
explored using the very techniques advocated for below. Secondly, Judge (2021)
has shown that the attention to the dynamics and fine structure of transition
region plasma has been over-emphasized, statistically. Most of the
chromosphere-corona transition region on the Sun is indeed in a thermal
interface connected by magnetic lines of force, and not isolated, as was
claimed by Hansteen et al. (2014) and references therein, all of them based on
circumstantial evidence accumulated using spectral line intensities alone.
One example is shown in Figure 2. This figure is based on a MURaM flux
emergence simulation in a 98 Mm wide domain, reaching from -8 to 75 Mm in the
vertical direction. The snapshot corresponds to a highly energized corona, in
which a strong bipolar field has emerged into a previously quiet region. Ten
minutes after this, a flare and a mass ejection occurred. The simulation is
based on the Rempel (2017) version of MURaM, which uses a simplified
description of the chromosphere. The $\tau=1$ surfaces were computed using
parameterized li¡ne opacities at line center and integrating downwards from
the corona. The opacities were derived using a function, which for the Mg II
$k$-line for example, are of the form
$\kappa_{\nu=\nu_{0}}=\frac{\pi
e^{2}f_{ij}}{m_{e}c}\frac{n_{\mathrm{i}}}{n_{\mathrm{Mg^{+}}}}\frac{n_{\mathrm{Mg}^{+}}}{n_{\mathrm{Mg}}}\frac{n_{\mathrm{Mg}}}{n_{\mathrm{H}}}n_{\mathrm{H}}\phi_{\nu_{0}},$
(1)
in which the lower level population $n_{\mathrm{i}}$ is expanded as a product
of ionization fraction $\frac{n_{\mathrm{Mg}^{+}}}{n_{\mathrm{Mg}}}$ (a
function of electron temperature $T_{e}$), the abundance
$\frac{n_{\mathrm{Mg}}}{n_{\mathrm{H}}}$, and the hydrogen density
$n_{\mathrm{H}}$ from the model. The first factor is the frequency-integrated
cross section of the $k$ line from fundamental constants and the absorption
oscillator strength $f_{ij}$ between levels $i$ and $j$, and $\phi_{\nu}$ the
line profile at frequency $\nu$, normalized so that
$\int_{0}^{\infty}\phi_{\nu}d\nu=1$. The profile is dominated for these ions
by an assumed microturbulence of 10 km s-1. The term
$\frac{n_{\mathrm{Mg}^{+}}}{n_{\mathrm{Mg}}}$ is computed assuming that
neutral Mg is negligible, but allowing for electron impact ionization to Mg2+
and higher ions and radiative recombination using data from Allen (1973).
This figure illustrates approximate line formation heights, sufficient for the
present paper. The excursions in height corresponding to the emerging flux
system vary from a maximum near 10 Mm for Mg II $k$, to a few Mm for a line of
Fe II at 276.9 nm. As can be seen, these corrugations can be probed with
multiple spectral features with different opacities. Thus measurements at the
top of the chromosphere, properly sampling the potential excursions of the
isotherms and isobars in the transition between chromosphere and corona,
appear to be a promising approach to the problem. At the very least, such new
magnetic measurements could be compared in detail with advanced simulations of
the Sun’s atmosphere (Gudiksen et al., 2011; Rempel, 2017). Below we will
suggest one approach to this problem based upon machine learning techniques.
Figure 3: A low-resolution UV irradiance spectrum from the Whole Heliospheric
Interval in 2008 is shown (from
https://lasp.colorado.edu/lisird/data/whi_ref_spectra). The near UV region has
a photon flux density far higher than the majority of vacuum UV lines below
200 nm. The exception is H L$\alpha$ which roughly 10x smaller than the flux
density at 260 nm and longer wavelengths.
### 2.3 The near UV solar spectrum and the chromosphere-corona interface
A standard, low resolution solar UV spectrum is shown in Figure 3. After an
extensive search, we have identified a spectral range of 256-281 nm as a
promising part of the solar spectrum for measurement of vector magnetic fields
at the base of the corona. Our search, driven by the need to probe in detail
magnetic structure in height above the solar surface and/or depth along the
LOS of prominences, led to the choice of important transitions in the 4s-4p
transition array of Fe II. While there are strong Fe II 4s-4p transitions in
the region near 234 nm, the solar fluxes are weaker than in the 256-281 nm
range. Our wavelength selection is significantly broader than the 279.2-280.7
nm spectral region of CLASP2, which includes the Mg II $h$ & $k$ line cores
and magnetically-sensitive wings (Auer et al., 1980; Belluzzi & Trujillo
Bueno, 2012; Alsina Ballester et al., 2016; del Pino Alemán et al., 2016) the
3p-3d lines (del Pino Alemán et al., 2020), and photospheric lines of Mn I
(Ishikawa et al., 2021). Both Hanle and Zeeman effects can be brought to bear
on the problem depending on the strength of the magnetic fields found across
the solar atmosphere. The Hanle effect can be important to remove the 180o
ambiguity present in Zeeman effect measurements. The rationale for this
spectral region is based upon requirements demanded by the desire to measure
vector magnetic fields within force-free ($\approx$ low $\beta$) plasma at the
coronal base, over an area covering a typical active region. An angular
resolution sufficient to resolve the thermodynamic fine structure is also a
constraint. Such features are most clearly seen only in images of features
within the narrow spectral lines chromospheric images. This constraint poses
difficulties for current observations at radio and sub-mm wavelengths, and in
hard X-rays. Soft X-rays ($\leq 1$ keV) have no known sensitivity to magnetic
fields of solar magnitude, and along with EUV photons cannot escape from or
penetrate deep into the chromosphere. We are then left to consider vacuum UV
to infrared wavelengths.
We further eliminate wavelengths above the atmospheric cutoff at 310 nm
because of the lack of spectral lines that can span many decades in optical
depths above the chromosphere. The Balmer and higher line series of hydrogen
are poor choices for Zeeman diagnostic work, the lightness of hydrogen makes
lines broad and patterns overlapping. For Ba-$\alpha$, seven anomalous Zeeman
patterns will lie within half a wave number for typical solar field strengths
even in the photosphere, thereby making the observation of anomalous Zeeman
patterns extremely difficult. The Hanle effect has been explored by Štěpán &
Trujillo Bueno (2010) in their comparison of 1D radiative transfer
calculations with limb observations of the quiet Sun by Gandorfer (2000). The
value of H$\alpha$ as a diagnostic of magnetic field remains a subject for
further research.
The strongest line in the chromospheric visible spectrum is Ca II $K$ has just
$1/20$ the opacity of the Mg II $k$ line (see Figures 1 and 5). Also shown in
Figure 1 is the well-known line of He I at 1083 nm which is present in active
region plasma. However, the line, belonging to the triplet system, populated
largely by recombination following EUV photoionization, tends to form where
coronal ionizing photons (and perhaps even hot particles) cannot penetrate the
chromosphere. Our non-LTE calculations (Figure 1) reveal that significant
opacity in the line is narrowly confined between surfaces of column mass well
below the height of the NUV Mg II lines, because of the depth of penetration
of vacuum UV radiation from the overlying corona in the continua of H and He.
When included along with the Ca II resonance and infrared triplet lines, the
optical- near UV regions accessible from the ground extend neither high
enough, nor do they sample opacity with the fineness of UV lines (Figures 1, 2
and 5. Lines of Mn I are sufficiently close to the strong Mg II lines for the
wings of the latter to contribute to opacity, but this was not accounted for
in the plot. Therefore heights of formation of Mn I lines are lower limits. ).
The solar irradiance spectrum at vacuum UV wavelengths (below 200 nm, and
above 91 nm below which the Lyman continuum absorbs line radiation), including
H L$\alpha$ and many chromospheric and transition region lines, is shown in
Figure 3. All are substantially weaker than the Mg II lines. The polarimetric
measurements necessary to infer magnetic fields require large photon fluxes.
The VUV region is a factor of at least 10-100 times dimmer than the NUV. Also,
the Zeeman signatures vary in proportion to the wavelengths observed. Further,
most of the transition region lines are optically thin, so that there is no
single “depth of formation” of a given line. The exceptions are the Lyman
lines of hydrogen and 58.4 nm resonance line of He I. But using only this
region of the spectrum would also fail to sample surfaces of multiple optical
depths, and thus fail to address the need to span the anticipated corrugated
isothermal and isobaric surfaces (Figure 2), particularly in active regions,
the sources of most solar eruptions.
Figure 4: A smoothed UV spectrum of $\alpha$ Cen A, obtained by the Hubble
Space Telescope, is displayed as a proxy for the mean solar intensity
spectrum. The large number of lines of Fe I and Fe II are indicated and
annotated with their multiplet terms. The Mg II spectrum is also annotated
including the $3s-3p$ $h$ and $k$ lines, and the $3p-3d$ lines. Notice that
the permitted Fe II lines (shown with solid bars with downward ticks) consist
of the sextet-D resonance lines $(4s-4p)$ near 2600 Å, and quartet transitions
at other wavelengths that occur between metastable (even parity) quartet lower
levels and odd-parity quartets of the same transition array ($4s-4p)$.
In contrast, the NUV region (from say 200 to 310 nm, the atmospheric cutoff)
is comparatively brighter, containing lines (Mg II) whose cores form within
lower transition region. Compared with lines accessible from the ground, lines
have smaller Zeeman signatures, but have increased linear polarization
generated by the higher radiation anisotropies at UV wavelengths (much
stronger limb-darkening). Thus, the Hanle signals will statistically be
stronger than at visible wavelengths. Further, the region contains a plethora
of lines of neutral and singly ionized iron within the $4s-4p$ transition
array, including resonance lines near 260 nm (see Figure 4). Their complex
spectra span and sample multiple depths across the chromosphere, extending
into lower transition region plasma (see in particular the contours of unit
optical depth surfaces for several lines in the lower right panel of Figure 2,
and the plot of relative optical depths in Figure 5).
The richness of the Fe II spectrum itself has been known for decades in
astronomy (e.g. Viotti et al., 1988). Figure 4 identifies lines and multiplets
of Fe plotted above a smoothed Hubble Space Telescope spectrum of $\alpha$ Cen
A, which is a good proxy for disk-averaged intensity spectrum of the Sun.
(There is no disk-averaged solar spectrum of comparable quality). All of the
strong transitions of iron lie near or below the Mg II lines near 280 nm. NUV
spectra obtained at the solar limb during the SKYLAB era revealed a
spectacular array of Fe II lines in emission against the darkness of space
(Doschek et al., 1977). Particularly strong in the limb spectra (Figure 10)
are multiplets in the sextet system near 259-263 nm, and quartet transitions
in the 273 nm region. These Fe II lines were readily detected up to 8” above
the limb in the SKYLAB data, the limb itself is already far higher than the
photosphere. Such emission is very likely associated with the kind of
structures modeled in Figure 2. A downside of the richness of the spectra of
iron is the possibility of blended lines. The important issues of blended
spectra lines at NUV wavelengths and effects of missing opacity are addressed
in the Appendix. They are shown to be a minor issue, affecting only a few
lines in the regions between 256 and 281 nm, owing to the overwhelming
strength of the Fe II lines formed high enough in the chromosphere to aid in
the diagnosis of magnetism there. Between 281 and 310 nm, there are just 10
“wide” lines, i.e. strong enough to exhibit wing absorption profiles, compared
with 38 listed by Moore et al. (1982) between 256 and 281 nm. None of the
lines above 281 nm form high enough in the Sun’s atmosphere to contribute
significantly beyond measurements of the chromospheric magnetic field at
shorter wavelengths.
Figure 5: Properties of spectral lines calculated using line center opacities
as a function of wavelength are shown. Circles show lines of singly-charged
ions, squares show lines of neutral species. Symbol sizes reflect critical
Hanle field strengths. Relative optical depths of the lines are shown on the
ordinate. The Mn I lines should include wing opacity from the Mg II line, but
this was omitted here. The Mn I optical depths are thus a lower limit. The “?”
for the He I 1083 nm line refers to the unknown relative optical depth of this
line because of peculiarities in the formation of this multiplet in the Sun
(e.g. Judge et al., 2015, who discuss this multiplet in the context of a well-
observed flare).
Relative optical depths at the centers of the lines in this region, and “Hanle
field strengths” are shown in Figure 5, including lines of He, Mg, Ca, Mn and
Fe. These Hanle field strengths (see equations 2 and 3 below) are those near
which the emergent polarized spectra are strongly sensitive to magnetic field
strength and direction, (Landi degl’Innocenti & Landolfi, 2004, see below).
### 2.4 Spectral signatures of magnetic fields
The Zeeman effect is well-known in laboratory and astrophysics as a
characteristic splitting in intensity ($I$) spectra. But in the Sun and stars,
the Stokes parameters $I,Q,U$ and $V$ are used to measure magnetic fields in
commonly encountered situations where the widths of the lines exceed the
Zeeman splitting (this is assumed in the following discussion). As noted
above, Zeeman signals depend upon the quantity
$\varepsilon=\omega_{B}/\Delta\omega_{0}$, where $\omega_{B}$ is the Larmor
frequency of the gyrating ion:
$\hbar\omega_{B}=\mu_{0}B.$ (2)
Here $\mu_{0}$ is the Bohr magneton, and $B$ the magnetic field strength in G.
$\Delta\omega_{0}$ is the line width (Doppler width for chromospheric and
coronal lines). The amplitude of circular polarization measured by Stokes $V$
varies linearly with $\epsilon$, the linear polarization profiles $Q$ and $U$
have amplitudes varying as $\varepsilon^{2}$. While $B$ is measured in Gauss,
it is important to remember that unless contributions from non-magnetic
regions to $I,Q,U$ and $V$ can be determined, the polarized components can be
used to determine only the average flux density per unit area in Mx cm-2, not
field strength.
Table 1: Prominent lines and blends in the solar chromospheric spectrum
$\lambda$ nm | Landé g | log $\omega_{L}$ | log $\sum_{i}A_{ji}$ | $B_{HANLE}$ | log $\tau_{0}$ | Ion | blend
---|---|---|---|---|---|---|---
lab. (air) | factor | rad sec-1 | sec-1 | G | | | severity∗ Ion $\lambda$
256.254 | 1.21 | 7.03 | 8.49 | 28.8 | -2.54 | Fe II |
256.348 | 1.10 | 6.99 | 8.49 | 31.9 | -2.87 | Fe II | 1 Fe I .341
256.691 | 0.83 | 6.86 | 8.49 | 42.2 | -3.30 | Fe II |
257.437 | 1.30 | 7.06 | 8.51 | 28.4 | -4.33 | Fe II | 1 Co I .351
257.792 | 1.33 | 7.07 | 8.49 | 26.4 | -3.32 | Fe II | 1 Mg I .888
258.258 | 1.47 | 7.11 | 8.49 | 23.9 | -3.19 | Fe II |
258.588 | 1.50 | 7.12 | 8.46 | 21.7 | -2.03 | Fe II | 2 Fe I .588
259.055 | 1.50 | 7.12 | 8.48 | 22.8 | -4.47 | Fe II | 1 Co I .059
259.154 | 1.49 | 7.12 | 8.49 | 23.5 | -3.24 | Fe II |
259.373 | 2.17 | 7.28 | 8.49 | 16.2 | -4.02 | Fe II | 1 Mn I .372
259.837 | 1.50 | 7.12 | 8.46 | 21.7 | -1.93 | Fe II |
259.940 | 1.56 | 7.14 | 8.46 | 21.1 | -1.39 | Fe II | 1 Fe I .957
260.709 | 1.50 | 7.12 | 8.66 | 34.8 | -1.99 | Fe II |
261.107 | 1.90 | 7.22 | 8.49 | 18.4 | -4.23 | Fe II | 1 Cr II? .104
261.187 | 1.59 | 7.15 | 8.46 | 20.5 | -1.79 | Fe II |
261.382 | 1.50 | 7.12 | 8.46 | 21.7 | -2.20 | Fe II |
261.762 | 1.66 | 7.16 | 8.46 | 19.6 | -2.36 | Fe II |
262.041 | 1.87 | 7.22 | 8.66 | 27.9 | -3.75 | Fe II |
262.167 | 3.34 | 7.47 | 8.46 | 9.8 | -2.76 | Fe II |
262.567 | 1.50 | 7.12 | 8.46 | 21.9 | -2.08 | Fe II |
262.829 | 1.50 | 7.12 | 8.66 | 34.8 | -2.21 | Fe II |
263.105 | 1.50 | 7.12 | 8.46 | 21.7 | -2.02 | Fe II | 1 Fe II .045†
263.132 | 1.50 | 7.12 | 8.46 | 21.7 | -1.97 | Fe II |
264.112 | 1.87 | 7.22 | 8.51 | 19.8 | -4.62 | Fe II | 1 Ti I .089
268.351 | 1.84 | 7.21 | 8.51 | 20.1 | -4.99 | Fe II | 2 Cr II .345
269.283 | 1.93 | 7.23 | 8.47 | 17.2 | -4.40 | Fe II | 1 Fe I .265, Fe II .260
270.938 | 2.10 | 7.27 | 8.47 | 15.8 | -4.30 | Fe II | 1 Cr II .931
271.441 | 1.50 | 7.12 | 8.60 | 30.3 | -3.09 | Fe II |
271.670 | 1.33 | 7.07 | 8.46 | 24.8 | -3.11 | Fe II | 1 Mn II .680
271.903 | 1.25 | 7.04 | 8.25 | 16.3 | -4.45 | Fe I | 2 Fe I .903
272.090 | 1.17 | 7.01 | 8.28 | 18.4 | -4.76 | Fe I |
272.358 | 1.00 | 6.94 | 8.27 | 21.1 | -5.22 | Fe I |
272.488 | 1.20 | 7.02 | 8.47 | 27.7 | -3.05 | Fe II | 1 Fe I .495
272.754 | 1.50 | 7.12 | 8.60 | 30.4 | -3.05 | Fe II |
273.073 | 0.80 | 6.85 | 8.47 | 41.6 | -3.19 | Fe II |
273.245 | 1.45 | 7.10 | 8.46 | 22.8 | -4.92 | Fe II | 1 Fe II .233
273.697 | 1.50 | 7.12 | 8.60 | 30.3 | -3.22 | Fe II | 1 Fe I .696
273.731 | 2.00 | 7.25 | 8.27 | 10.5 | -5.12 | Fe I |
273.955 | 1.43 | 7.10 | 8.61 | 32.1 | -2.31 | Fe II |
274.241 | 1.67 | 7.17 | 8.28 | 12.9 | -5.01 | Fe I | 1 Ti I .230, V II .243
274.320 | 0.50 | 6.64 | 8.47 | 66.6 | -2.81 | Fe II |
274.407 | 2.50 | 7.34 | 8.27 | 8.4 | -5.51 | Fe I |
274.648 | 0.90 | 6.90 | 8.47 | 37.0 | -2.59 | Fe II |
274.698 | 1.37 | 7.08 | 8.60 | 33.2 | -2.66 | Fe II | 1 Fe I .698
274.918 | 1.20 | 7.02 | 8.60 | 38.0 | -3.02 | Fe II |
274.932 | 1.07 | 6.97 | 8.46 | 30.9 | -2.38 | Fe II |
274.949 | 0.00 | 4.31 | 8.60 | … | -3.24 | Fe II |
275.014 | 1.58 | 7.14 | 8.25 | 12.8 | -5.04 | Fe I |
275.574 | 1.17 | 7.01 | 8.46 | 28.1 | -2.17 | Fe II |
275.633 | 2.00 | 7.25 | 8.28 | 10.7 | -5.52 | Fe I | 2 Fe I .633, Cr II .630
276.181 | 1.50 | 7.12 | 8.60 | 30.4 | -3.25 | Fe II | 1 Fe I .178, Cr I .174
276.893 | 1.50 | 7.12 | 8.60 | 30.3 | -3.08 | Fe II |
277.273 | 1.50 | 7.12 | 8.61 | 30.6 | -3.14 | Fe II | 1 Fe I .283
277.534 | 1.34 | 7.07 | 8.46 | 24.2 | -6.06 | Fe II | 1 ?
279.565 | 1.17 | 7.01 | 8.43 | 25.3 | 0.00 | Mg II |
279.482 | 1.98 | 7.24 | 8.57 | 21.3 | -5.65 | Mn I |
279.827 | 1.70 | 7.17 | 8.57 | 24.8 | -5.79 | Mn I |
280.108 | 0.84 | 6.87 | 8.57 | 50.3 | -5.96 | Mn I |
280.270 | 1.33 | 7.07 | 8.43 | 21.9 | -0.30 | Mg II |
285.213 | 1.00 | 6.94 | 8.69 | 55.8 | -2.20 | Mg I |
393.366 | 1.17 | 7.01 | 8.17 | 14.3 | -1.55 | Ca II |
854.200 | 1.10 | 6.99 | 7.00 | 1.0 | -2.50 | Ca II |
Landé g-factors are computed using LS coupling. The final column lists the
severity of the blend (as assessed by Moore et al., 1982), and the wavelength
of the blended line (decimal nm). ∗The severity is: 0=unblended; 1=moderate
blend; 2=severe blend. The data in Figure 5 are taken directly from this
table. †The strong line of Fe II multiplet 1 is blended with a line of
multiplet 171.
The Hanle effect has a quite different and complementary behavior independent
of the Doppler width $\Delta\omega_{0}$. It requires a pre-existing state of
polarization induced by symmetry breaking effects such as anisotropic and/or
polarized incident radiation. When the radiation is anisotropic, such as
radiation emerging into the upper chromosphere from the photosphere below, the
atom develops atomic polarization (a particular imbalance of populations of
magnetic substates called the atomic “alignment”). In the presence of a
magnetic field, these populations and associated atomic coherences
(essentially representing the entanglement of neighboring atomic states) are
altered by the magnetic field in the regime where
$g_{ji}\omega_{B}\mathcal{T}_{j}\sim 1,$ (3)
where $g_{ji}$ is the effective Landé $g-$factor for the transition from upper
level $j$ to lower $i$, and $\mathcal{T}_{j}=1/\sum_{i}A_{ji}$ is the inverse
lifetime of level $j$, the sum taken over all lower levels $i$. Figure 5 shows
spectral lines of Fe and Mg, their wavelengths, optical depths, with symbol
sizes indicating the magnetic field for which equality holds in equation (3),
using data compiled in the NIST spectroscopy database. These Hanle field
strengths vary between roughly 1 G and 70 G for the different lines, field
strengths that are anticipated in quiet and active regions of the solar
chromosphere. The data shown in Figure 5 are listed in Table 1, the final
column of which which also addresses the potentially serious problem of
spectral line blends. This problem is shown in the Appendix to be easily dealt
with.
In a classical picture, when $g_{ji}\omega_{B}\mathcal{T}_{j}\approx 1$, the
ion on average gyrates through 1 radian before it emits photons sharing the
common upper level. When $\omega_{B}\mathcal{T}_{j}\gg 1$, the “strong-field
limit” of the Hanle effect, the radiating ions exhibit many gyrations before
emitting a photon and all “memory” of the particular orientation of the
radiating ions is statistically averaged out. This condition is the same as
that found for coronal forbidden lines where $\mathcal{T}_{j}$ is typically
between 1 and 0.01 seconds, not $10^{-8}$ seconds for the permitted lines
studied here (Casini et al., 2017). The only signature of the magnetic field
is in its direction projected on to the plane-of-the-sky, modulo 90o. If
$\omega_{B}\mathcal{T}_{i}\ll 1$ (the weak field limit) the magnetic field
enters as a small perturbation in expressions for the emergent radiation and
the magnetic effects cannot easily be observed. But near the Hanle regime
(equation 3) the magnetic field is encoded as a rotation and de-polarization
of the emergent linearly polarized radiation. The symmetry broken by the mis-
alignment between the directed radiation vector and magnetic field vector
resolves ambiguities associated with Zeeman measurements. This effect may be
useful for identifying tangential discontinuities (changes of magnetic field
direction not strength, across a flux surface).
In practice, both the Zeeman and Hanle effects can be used in a complementary
fashion to find signatures of magnetism in the Sun’s atmosphere using such
spectra (Trujillo Bueno et al., 2018; Ishikawa et al., 2021).
Finally, we have investigated whether molecular transitions are significant in
this spectral region, using the line list complied by R. Kurucz111
http://kurucz.harvard.edu/linelists.html and the synthesis codes by Berdyugina
et al. (2003). In particular, we have found that OH transitions from the
electronic A–X system are abundant in this spectral region, with stronger
lines toward the red. They form below most chromospheric lines owing to
chromospheric stratification and the rapid drop of molecular number density
with gas pressure, but remarkably, can also form above deeper photospheric
lines. A few moderately strong OH features may be visible at wavelengths
between strong atomic lines in the quiet sun spectrum. They become especially
important in models of sunspot umbrae. Numerous molecular CO lines in this
spectral region are quite weak, but they are so many that the continuum level
is effectively reduced by a few percent, with stronger lines toward the blue.
If observed, both OH and CO lines are potentially interesting for sensing
magnetic fields using both the Zeeman and Hanle effects (Berdyugina &
Solanki, 2002; Berdyugina et al., 2003; Berdyugina & Fluri, 2004), as well as
temperature and pressure near the interface between the photosphere and
chromosphere. (The reader can refer to the paper by Landi Degl’Innocenti, 2007
for elementary theory of molecules in magnetic fields applied to solar
conditions).
We conclude that, based on known atomic and molecular data, there are many
lines of Fe II whose cores are not strongly blended. The case of blended lines
from sunspot umbrae requires further work outside of the scope of this paper.
### 2.5 Diagnostic Sensitivity of Mg II k
Figure 6: Observing geometry for uniform field computations with HanleRT used
in the panels of Figure 7. Note the convention for negative LOS $\theta$.
Figure 7: _Top:_ “Heat Map” of the Mg II k core polarization amplitude for a
uniform magnetic field inclined at $\theta_{B}=-30^{\circ}$ from vertical.
Gray regions are where the amplitude falls below 0.1% of the intensity, here
taken as a typical instrumental SNR. _Bottom:_ Derivative of polarization
amplitude data from the top panel with respect to magnetic field to produce a
“heat map” of the response. The color scale gives the relative change in
amplitude of the spectral feature in response to a 20% change in $|B|$.
Regions of the map brighter than the 0.1% value on the color scale
(highlighted by black box) indicate an observing geometry that could infer
$|B|$ to within 20% assuming an instrumental polarization SNR of 0.1% of the
intensity. See text for further details.
In Figures 1 and 2 we saw how the Mg II $k$ line has contributions from plasma
that is highest in the Sun’s atmosphere. The $h$ line is a factor of two less
opaque, but it has a higher Landé g-factor than $k$ and so is more sensitive
to the Zeeman effect (Ishikawa et al., 2021). However, its upper $J=1/2$ level
is unpolarizable and so it can have no Hanle effect in the line core.
Both the Zeeman and Hanle effects can and should be used to probe
chromospheric magnetism. The Zeeman effect produces the largest polarimetric
signals when fields are relatively strong (average flux densities of a few
hundred Mx cm-2 were measured using the weak field limit applied to the Mg
lines by Ishikawa et al., 2021). However, for weaker fields, the Hanle effect
becomes the dominant source of polarized light variations in response to the
solar magnetic fields. Following early theoretical work by Belluzzi & Trujillo
Bueno (2012), the scattering polarization, Hanle and Zeeman effects across the
$h$ and $k$ lines was investigated in semi-empirical 1D models (Alsina
Ballester et al., 2016; del Pino Alemán et al., 2016, 2020). The Hanle field
of the $k$ line is 23 G (see Figure 5). These authors demonstrated that, for
field strengths of the order of 10 G, measurable variations of the order of
$\sim$1% in the fractional linear polarization of Mg II $k$ are produced by
the Hanle effect. Here, we extend this work with a larger grid of LOS angles
and magnetic field strengths (see Figure 6 for the geometry) and show
polarization amplitude and magnetic response “heat maps” in Figure 7. As in
del Pino Alemán et al. (2020) we used HanleRT to compute the intensity and
polarization from a Mg II 3-term atom including PRD effects in atmosphere C of
Fontenla et al. (1993). We computed profiles for a grid of uniform magnetic
field vectors inclined at $\theta_{B}=-30^{\circ}$ from vertical, with fixed
azimuth, and with $|B|$ ranging from 1 to 200 G at 1 G intervals. These values
of $|B|$ are about 20 and 10 times smaller and larger, respectively, than the
Hanle field. We computed the emergent Stokes profiles for 13 LOS viewing
angles $\theta$ ranging from $+75^{\circ}{}$ to $-75^{\circ}{}$ at 15∘
intervals, as well as $\mu=\cos(\theta)=0.1$, which corresponds to
$\theta\sim\pm 84.5$∘ or a limb distance of about 5′′. We examine the relative
variation in the amplitude of the fractional polarization (hereafter simply
“response“), $|B|\cdot dA_{i}/dB$, where $i$ indicates the polarization $Q$,
$U$, or $V$, and $A_{i}$ is the amplitude, $A_{i}=({\rm max}(i)-{\rm
min}(i))/I\cdot 100\%$, within the Mg II k core region. The top panel of 7
shows the amplitude $A_{i}$, while the bottom panel shows the response with
respect to magnetic field over the full range of $|B|$ and LOS $\theta$. The
units of the response are $\%I/\%B$, and we have scaled the data such that the
denominator represents a 20% change in $|B|$. Under this normalization, a
response of 1 corresponds to a 1% amplitude change for a 20% change in $|B|$.
In order to assess the practical application of these calculations, we
consider noise at the level of $\sim 0.1\%$ of intensity ($10^{-3}I$). Those
regions in the response heat map brighter than that value on the color scale
are where the instrument would be able to infer components of the magnetic
field to within 20%. We conclude that we would be able to reasonably estimate
this vector orientation for magnetic field strengths of $\sim$5 to 50 G at
observation angles from about $\theta=\pm 45^{\circ}$ to the limb due to the
Hanle effect in $Q$ and/or $U$, and stronger line-of-sight fields at nearly
any observation angle from the Zeeman effect in Stokes $V$. The largest
sensitivity in $Q$ and $U$ occurs for observations near the limb and field
strengths below the critical Hanle field. (Dark vertical strips in the
response of Stokes $U$ are due to local maxima in the amplitude curve near the
critical Hanle field for Mg II $k$; compare to the top panel of Figure 7). The
$V$ amplitude is constant and zero at $\theta=60^{\circ}$ as this is the angle
where the magnetic field vector is orthogonal to the line-of-sight. Measurable
linear polarization is produced at that geometry, however its variation with
magnetic field is too low to be used as a diagnostic. “Heat map” diagrams for
other field inclination angles are qualitatively similar, with the regions of
maximum response due to the Hanle and Zeeman effects shifting in relation to
the field & LOS geometry.
We conclude that diagnosing magnetic field from the Hanle effect in Mg II $k$
is feasible for a large range of geometries and with achievable signal-to-
noise ratios. The range of field strengths to which the Hanle effect is
sensitive complement those higher field strengths ($|B|\geq 50$G) to which the
Zeeman effect is sensitive, for the case of the Mg II $k$ line.
Additional work on the Hanle effect in the lines of Fe II is underway and will
be reported elsewhere. Importantly, there is at least one line (at 2744.01 Å)
with a Hanle field above 60 G, and several where the critical field exceeds
that of the Mg II $k$ line.
### 2.6 Addressing the “corrugation problem”
Figure 2 shows how state-of-the-art numerical models lead to contours in
optical depth which can be far from horizontal. The free energy in these MHD
simulations generated by convection and emerging flux leads to highly dynamic
evolution in the lower density regions of most interest to us at the top of
the chromosphere. The geometric distortions of the $\tau=1$ surfaces in these
tenuous regions naturally reflect the low inertia of the plasma, subjected to
Lorentz, pressure gradient and gravitational forces, evolving in response to
the forcing from below. But, as noted earlier, the extreme corrugations of
order 10 Mm are not negligible compared with the scale of the emerging flux
system itself. The question then arises, how can we infer magnetic field at a
constant geometric height, if we are to apply mathematical techniques to
estimate the free energy and topology within the corona? Progress can be
anticipated without such techniques, by direct comparison of numerical
prediction and measurements. But this question must be addressed if we are to
measure the development of free energy in the overlying corona in time, in
general.
Let us assume that the calculations are a reasonable, “first order”
approximation to the kind of geometric distortions of the $\tau=1$ surfaces of
sets of lines. The advantage of observing the 259-281 nm region is that we
know the order of the opacities of each line in a sequence: the Mg II lines
are the most opaque, next come the sextet system transitions of Fe II and then
the quartet systems, each with its spread of oscillator strengths. Thus, given
a calculation, we can (as in Figure 2) find the $\tau=1$ surfaces. We could
compute Stokes profiles and associate a magnetic field from such synthetic
data, knowing the heights of each line’s formation.
But this is not the situation we face with the remotely-sensed Stokes data, we
have no knowledge other than the Stokes profiles of many lines, on a two
dimensional geometric image of the Sun at each wavelength. Fortunately, this
problem fits well into the picture of machine learning algorithms (e.g. Bobra
& Mason, 2021). Synthetic data can be used to train an algorithm to find level
geometric surfaces (Asensio Ramos & Díaz Baso, 2019) of direct use for
mathematics to derive quantities of interest. The large coverage of optical
depths of the 256-281 nm region and the variety of Hanle field strengths
strongly suggest that, if the models are sufficiently complete to represent
the atmospheric dynamics and temperature structure, then such an algorithm
should have a non-zero probability of yielding the desired results. The
combined use of Hanle and Zeeman effects enhances this probability, at least
in principle. We are exploring ways in which this program of research might be
implemented.
## 3 Discussion
In this paper we have examined the general problem of identifying the sources
of magnetic energy within coronal plasma, which leads to flaring, plasmoid and
CME ejections. By a process of elimination, we arrived at a wavelength range
between 2560 and 2810 Å, which appears to be the most promising range in which
to measure conditions at the coronal base. Our conclusions build upon the
important study of Trujillo Bueno et al. (2017), not only by seeking answers
to questions concerning the magnetic free energy and topology, aimed at
understanding the origins of phenomena responsible for Space Weather. Our
conclusions also differ by proposing a potentially ideal, relatively narrow
spectral region between 256 and 281 nm, expanding significantly the
279.2-280.7 region selected by the CLASP2 mission, to serve as a basis for
future, more targeted research.
To achieve measurements in this region clearly will require a space-based
platform for spectropolarimetry. These conclusions are based upon several
criteria: The spectrum must be bright enough; spectral features must extend as
high as possible; the spectral features must contain information on magnetic
fields through both the Zeeman and Hanle effects; the spectra must be
demonstrably unblended; lines with multiple opacities are needed to span
across the corrugated surfaces that are present at the coronal base.
Thus we suggest that a novel spectropolarimeter, building upon a heritage from
SKYLAB, SMM (UVSP), SoHO (SUMER), IRIS, CLASP and CLASP2, be considered for
flight to address the problems of interest to science and society. Such a
mission would complement the unique capabilties of the Daniel K. Inouye Solar
Telescope which can observe important chromospheric lines (include all those
shown in the top row of Figure 1) from the ground, at a sub-arcsecond spatial
resolution and/or high cadence.
Finally, it is important to note that the CLASP2 measurements of Ishikawa et
al. (2021) achieved signal-to-noise ratios of over 1000 in the Mg II line
cores. This demonstrates that a modest, D=25 cm telescope operating at NUV
wavelengths is sufficient for delivering the weak signals required for
diagnosis of magnetic fields, over active regions at least, where the magnetic
flux density exceeds 100 Mx cm-2.
Thanks to Roberto Casini and Alfred de Wijn for the many fruitful discussions
regarding the Hanle effect and requirements to achieve new results using
diverse spectral lines. NCAR is funded by the National Science Foundation. PGJ
is grateful to the NSF for funding his research at a national center without
any specific goal. The authors thank an anonymous referee for their helpful
comments.
## Line blends and missing opacity
The NUV region is well-known as a crowded region of the solar spectrum, and
also a region where some opacity appears to be “missing”. The effects of this
missing opacity are examined below using comparisons of detailed simulations
and observations. It can be significant at certain wavelengths, but does it
not affect the cores of strong chromospheric lines. (See also discussions by
Fontenla et al., 2011; Peterson et al., 2017). Line blending is however a
concern.
We have addressed blending using data from the Hubble Space Telescope, from
the SO82B spectrograph on SKYLAB, and using sophisticated radiative transfer
models. The models are based on Local Thermodynamic Equilibrium (LTE)
radiative transfer models computed using TURBOSPECTRUM (Plez, 2012), and
adopting a custom calculated Solar 1D, spherical MARCS model atmosphere
(Gustafsson et al., 2008). A complete linelist in the corresponding spectral
region was adopted in the computations from the VALD
database222http://vald.astro.uu.se/ (Ryabchikova et al., 2011). The linelist
includes up-to-date atomic data, including line-broadening parameters from
collisions with neutral hydrogen atoms, calculated ab-initio following Barklem
et al. (1998) for all neutral atoms including dominant Mg II lines and iron
group elements.
A comparison of high resolution Hubble spectra of $\alpha$ Cen A with the
models is informative. A typical example in an interesting region of the NUV
spectrum is shown in Figure 8. The blue line represents the flux spectrum
computed for the Sun, the red line the spectrum of $\alpha$ Cen A. The flux
densities shown are computed at the respective stellar surfaces, with no ad-
hoc adjustments of either scale.
Figure 8: Flux spectra computed for a Solar atmospheric model (blue) using 1D,
LTE radiative transfer calculations (see Section Line blends and missing
opacity) are shown along with spectra of the Sun-like star $\alpha$ Cen A
(red) obtained with the STIS instrument on the HST. The data are a composite
of several exposures from the E230H grating, assembled and processed by Ayres
(2014). The STIS data were shifted according to the redshift of 22.4 km s-1,
and converted to flux at the stellar surface using the angular diameter of
9.09 milli-arcsec. The spectra are compared with no adjustment of flux or
wavelength scale. The qualitative agreement is remarkable. The units are Å
(wavelength) and erg cm-2 s-1 Å-1 (flux, $F=\pi\overline{I}$, where
$\overline{I}$ is the mean disk intensity).
The comparison is remarkable, particularly in the cores of some of the strong
lines, most of which belong to Fe II. Wavelengths are evident (near 2623 Å for
example) where opacity appears to be missing (blue line above red line), but
in the Fe II lines identified in the figure there is no evidence for missing
opacity, if anything, the observed line cores in red lie above the blue line,
indicating emission from the star’s chromosphere.
The model calculations also permit us to identify possible blends (Figure 9).
This figure shows two computations, black is the full model calculation (also
shown in blue in Figure 8), and in this figure blue is a calculation using
opacity only from the Fe II lines. When significant differences between black
and blue lines exist, these are wavelengths where opacities are present in
lines other than Fe II. Conversely, when they agree, the Fe II spectrum
dominates and these are essentially unblended wavelengths. As noted in the
main text, the wavelength region above 281 and below 310 nm does not contain
strong lines formed in the middle-high chromosphere, therefore these are not
shown here. Above 281 nm the lack of such lines makes this a less desirable
region because blending becomes a bigger problem.
Figure 9: Two calculations are shown for the solar NUV spectrum: one (black)
is made with the full set of lines, and the other (red) is made where lines of
Fe II are the only sources of opacity. Strong lines from the quartet system of
Fe II are annotated, along with lines of Fe I whose opacity is much smaller.
The cores of the strong lines of Fe II are dominated by Fe II itself, the
region above $\approx$ 265 nm is, aside from narrow lines, dominated by wings
of the Mg II $h$ and $k$ lines.
Lastly we can, to some extent, check for blends independently of the model
computations. Rutten & Stencel (1980) were able to find weak lines in the
wings of the Ca II $H$ and $K$ lines looking to data obtained near the solar
limb. In Figure 10 we show (with an arbitrary flux calibration), spectra of
$\alpha$ Cen A with spectra from the SO82B spectrograph on SKYLAB. The latter
are taken from the scans across a quiet region of solar limb obtained on
August 27 1973, the slit being tangential to the limb and stepped by 2”
between pointings. These are the photographic data analyzed by Doschek et al.
(1977). The “intensities” plotted here are photographic densities. But these
densities suffice for us to find significant blends, which will potentially be
revealed by changes in the spectra as the spectrograph slit was repointed from
2” inside the limb to 6” outside, in 2” steps.
Figure 10: The left panel shows the spectral region close to the sextet
resonance lines of Fe II is shown from $\alpha$ Cen A (top), and 5 spectra
representative of the spectra as a function of position relative to the solar
limb. The right panel shows the 2765-2780 Å region. Blue lines mark
wavelengths of Fe II transitions. The three other lines seen in emission in
the right panel are from Cr II at 2766.52 Å, and questionable identifications
of Fe II 2767.50, a transition between high lying states, and of Mg I at
2779.8 Å (Doschek et al., 1977).
Different behaviors are seen depending on the line of interest. In the
wavelength range from 2595 to 2615 Å (left panel, showing resonance lines of
the sextet system) there is no evidence of blended features. But the right
panel of Figure 10 shows a spectral region containing lines of the quartet
system of Fe II. In contrast to the resonance sextet transitions, here we find
that essentially no Fe II lines are strong enough to be unblended. However, a
plot of similar transitions (not shown) between 2750 and 2765 Å has stronger
lines at 2756 and 2762 Å which are not strongly blended. These conclusions
have been verified using NRL report 8653 (Moore et al., 1982), from which the
last column of Table 1 has been constructed. There are only four lines which
are classified as severely blended of all the quartets, sextets of Fe II and
quintets of Fe I.
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# A construction of deformations to general algebras
Dave Bowman, Dora Puljić, Agata Smoktunowicz
###### Abstract
One of the questions investigated in deformation theory is to determine to
which algebras can a given associative algebra be deformed. In this paper we
investigate a different but related question, namely: for a given associative
finite-dimensional $\mathbb{C}$-algebra $A$, find algebras $N$ which can be
deformed to $A$. We develop a simple method which produces associative and
flat deformations to investigate this question. As an application of this
method we answer a question of Michael Wemyss about deformations of
contraction algebras.
## Introduction
Michael Wemyss and Will Donovan developed a method for characterising
commutative rings which appear in algebraic geometry, by using methods from
noncommutative ring theory. In [DW16] they introduced contraction algebras
which provide important insight into resolutions of noncommutative
singularities and invariants of flops. Contraction algebras can be described
in a purely algebraic way, by using generators and relations. They appear in
many questions which can be investigated by noncommutative ring theorists who
are not familiar with advanced methods of algebraic geometry. Some of these
questions are related to nilpotent rings. In 2022, Gavin Brown and Michael
Wemyss described all finite dimensional contraction algebras which are local
[BW22]. Notice that the Jacobson radical of a finite dimensional algebra is
nilpotent, and hence contraction algebras which are local are very close to
being nilpotent. Some other questions involve characterisation of contraction
algebras which are not local.
Deformations of contraction algebras give insight into invariants of flops. In
this context Michael Wemyss asked questions about deformations of contraction
algebras which are local. Using geometric methods he was able to conjecture
which rings can be obtained as deformations of these contraction algebras.
Since contraction algebras are noncommutative, the usual methods using
derivations as in [CGG89] cannot be applied. Another approach can be to use
Hochschild cohomology and the Gerstenhaber bracket, however this often leads
to complicated calculations as seen in [SW13].
In this paper, we develop a method to calculate deformations of noncommutative
local algebras. In particular we answer a question of Michael Wemyss regarding
deformations of contraction algebras. For information about contraction
algebras we refer reader to [Wem23, DW16].
In [HT18, Conjecture 4.3] Hua and Toda conjectured that there exists an
algebraic flat deformation of the contraction algebra. As an application we
confirm this conjecture for one of the contraction algebras constructed in
[BW22]. Note that our method can be applied to other contraction algebras
described in [BW22]. Observe that Theorem $4.2$ of [HT18] gives important
information about deformations of contraction algebras which were obtained by
using geometric methods. More information related to the existence of
geometric deformations is given on pages 7-9 in [Tod15].
In [SW13] deformations of graded Hecke algebras were constructed using novel
methods. It is known that deformations of graded Hecke algebras give important
information about the Hecke algebras. Our method could be also used as another
method of choice for constructing deformations of graded Hecke algebras. Some
other methods of constructing deformations of associative algebras were
described in [Wit19].
## 1 Preliminaries
###### Definition 1.1.
Let $S$ be a subset of a $k$-algebra $A$ for a commutative unital ring $k$.
Then $S$ is a generating set if all elements of $A$ can be written as a
$k$-linear sum of products of elements of $S$ using the operations in $A$.
###### Definition 1.2 ([Wit19]).
A formal deformation $(A_{t},\ast)$ of a $k$-algebra $A$ is an associative
$k$-bilinear multiplication $\ast$ on the $k[t]$-module $A[t]$, such that in
the specification at $t=0$, the multiplication corresponds to that on $A$;
this multiplication is required to be determined by a multiplication on
elements of $A$ and extended to $A[t]$ by the Cauchy product rule.
Any multiplication $\ast$ as in Definition 1.2 is determined by products of
pairs of elements of $A$: for $a,b\in A$, write
$a\ast b=\sum_{i\geq 0}\mu_{i}(a\otimes b)t^{i},$
where $\mu_{0}$ denotes the usual product in $A$ and $\mu_{i}$ for $i\geq 1$
are $k$-linear functions from $A\otimes A$ to $A$.
###### Definition 1.3 ([FO19]).
Let $A$ be a $k$-algebra and for $t\in\mathbb{C}$ let $\\{A_{t}\\}$ be the
family of algebras arising from a deformation $*$ of $A$. Then $*$ is a flat
deformation if each $A_{t}$ has the same dimension.
###### Notation 1.4.
We will denote by $\mathbb{C}[x,y]$ the polynomial ring over $\mathbb{C}$ in
two non-commuting variables, $x$ and $y$.
###### Notation 1.5.
Consider monomials of the ring $\mathbb{C}[x,y]$. We can order the monomials
using shortlex ordering by specification $1<x<y$. We denote the monomials of
$\mathbb{C}[x,y]$ by $p_{1},p_{1},p_{2}\dots$ where $p_{i}<p_{j}$ for $i<j$.
###### Notation 1.6.
We will denote by $\mathbb{C}[x,y][t]$ the polynomial ring in variable $t$
with coefficients from the polynomial ring in two non-commuting variables $x$
and $y$, with coefficients from $\mathbb{C}$.
###### Notation 1.7.
For $a_{i},b_{i}\in A$ we define $f:\mathbb{C}[x,y][t]\rightarrow A[t]$ to be
a homomorphism of $\mathbb{C}$-algebras such that
$f:x\mapsto\sum_{i=1}^{n}a_{i}t^{i},\quad
f:y\mapsto\sum_{i=1}^{n}b_{i}t^{i},\quad f:t\mapsto t.$
We will denote $\ker f$ by $I$.
###### Notation 1.8.
For an element $p_{i}$ in $\mathbb{C}[x,y]$ we will denote by
$\overline{p_{i}}$ the same element $p_{i}$, but with all instances of $x$
replaced by $\sum_{i=1}^{n}a_{i}t^{i}$, and all instances of $y$ replaced by
$\sum_{i=1}^{n}b_{i}t^{i}$. We will denote by $\overline{\overline{p_{i}}}$
the same element as $p_{i}$, but with all instances of $x$ replaced by
$\sum_{i=1}^{n}a_{i}$, and all instances of $y$ replaced by
$\sum_{i=1}^{n}b_{i}$.
###### Notation 1.9.
Let $R$ be a ring and $I$ be an ideal of $R$, then elements of the factor ring
$R/I$ will be denoted as $r+I$, where $r\in R$. Notice that $r+I=s+I$ for
$r,s\in R$ if and only if $r-s\in I$.
## 2 Algorithm
In this section we shall describe the simplest case of our proposed method.
This requires the data of a $\mathbb{C}$-algebra $A$ and a two element
generating set $\\{a,b\\}$. We shall state the steps of this algorithm and
then provide notes regarding important components. Then some examples will be
displayed.
###### Method 1.
We fix a finite dimensional $\mathbb{C}$-algebra $A$ with two generators
$a,b$. Then:
1. 1.
We consider $A[t]$, the polynomial algebra over $A$. We will use $t$ as the
parameter of our deformation. We define:
$\displaystyle x$ $\displaystyle:=ta,$ $\displaystyle y$ $\displaystyle:=tb.$
2. 2.
We calculate:
$\displaystyle x^{2}$ $\displaystyle=t^{2}a^{2},$ $\displaystyle xy$
$\displaystyle=t^{2}ab,$ $\displaystyle yx$ $\displaystyle=t^{2}ba,$
$\displaystyle y^{2}$ $\displaystyle=t^{2}b^{2},$
and continue to calculate larger products of $x,y$. We shall then proceed with
a Diamond Lemma111See section 1 of [Ber78] like decomposition of large
products of $a,b$ in terms of smaller products of $a,b$. This will cause all
elements of sufficiently large length to have a power of $t$ as a factor. In
doing so we will obtain relations on $x,y$ and products thereof. We terminate
this process when we have enough relations to decompose any large product into
only multiples of $x,y$ and powers of $t$. Our relations will be given by
polynomials $p_{1},\dots,p_{m}\in\mathbb{C}[x,y][t]$ for some
$m\in\mathbb{N}.$ This terminates finitely because $A$ finite dimensional.
3. 3.
We present the algebra:
$\mathcal{N}:={\mathbb{C}[x,y][t]}/{\left<p_{1},\dots,p_{m}\right>}.$
We note that sufficiently large products of $x,y$ will obtain a factor of $t$.
Now we evaluate $t$ at various values in $T=[0,\infty)$. We denote by $N_{s}$
the algebra that arises from $\mathcal{N}$ by evaluating $t$ at $s$ and in
particular we write $N=N_{0}$. By step (2) $N$ is local, in Section 5 we
consider a more general method where $N$ is not necessarily local. The algebra
$\mathcal{N}$ and the family of algebras $\\{N_{t}\\}$ are the output of this
method.
By Theorem 3.8 the family $\\{N_{t}\\}$ is a deformation of $N$ and by
Proposition 3.9 it holds that $A\in\\{N_{t}\\}$.
Changing the chosen generators $a,b$ can result in different algebras $N_{0}$
that have $A$ as an associative deformation.
### 2.1 Notes on the method
We give some notes before the examples:
* •
In step (1) we are trying to capture the behaviour of the elements in $A$, but
in a way that is controlled by $t$. This enables us to make use of a
presentation while still having access to our deformation parameter $t$.
* •
We note that since $a,b$ generate $A$ some linear combination of products of
$a,b$ will be the identity. In step (2) it is important we find a relation on
$x,y,t$ and $1\in A$.
* •
In step (3) we discard $A$ entirely. However, because of the relations
$p_{i}$, $x$ and $y$ “remember” that they came from $A$ and that they form a
generating set. This step results in something subtle: the monomials $x,y\in
N$ are no longer dependent on $t$ (and thus do not vanish when we set $t=0$)
but their products maintain their dependence on $t$. We have loosened our grip
on $x,y$ just enough for the deformation to take place.
* •
We note that our deformation will be associative since the multiplication of
$N_{t}$ will inherit from the associative multiplication $*$ of the algebra
$\mathcal{N}$.
* •
This algorithm is easily generalised to algebras that require a larger
generating set. Suppose a finite dimensional $\mathbb{C}$-algebra $A$ has $n$
generators $\\{a_{1},\dots,a_{n}\\}$. Then in step (1) we would define
$x_{i}:=ta_{i}$ for $1\leq i\leq n$ and proceed as described in Method 1. The
resulting family of algebras will also be a flat and associative deformation
of $A$.
### 2.2 Examples
Here we shall work through some examples. We shall start with a simple example
where $A=\mathbb{C}\oplus\mathbb{C}$ and then give a more complicated one
where $A=M_{2}(\mathbb{C})$, the algebra of $2\times 2$ matrices over
$\mathbb{C}$. Practically it can be useful to fix a basis for the algebra $A$
as this helps find the decompositions in step (2).
###### Example 2.1.
We consider $A=\mathbb{C}\oplus\mathbb{C}$ and fix $a=(i,0),\;b=(0,1)$.
Clearly $\mathbb{C}$-linear combinations of $a$ and $b$ span $A$ so our basis
is just $\\{a,b\\}$. We write $\mathbbm{1}:=(1,1)$. Now we consider $A[t]$ and
define:
$\displaystyle x:=ta=(ti,0),$ $\displaystyle y:=tb=(0,t).$
We calculate:
$\displaystyle x^{2}=t^{2}a^{2}=(-t^{2},0)=tix,$ $\displaystyle
xy=t^{2}ab=(0,0)=0,$ $\displaystyle yx=t^{2}ab=(0,0)=0,$ $\displaystyle
y^{2}=t^{2}b^{2}=(0,t^{2})=ty,$ $\displaystyle t\mathbbm{1}=-ix+y.$
Thus we have relations:
$\displaystyle p_{1}$ $\displaystyle=x^{2}-tix=0,$ $\displaystyle p_{2}$
$\displaystyle=xy=0,$ $\displaystyle p_{3}$ $\displaystyle=yx=0,$
$\displaystyle p_{4}$ $\displaystyle=y^{2}-ty=0,$ $\displaystyle p_{5}$
$\displaystyle=t\mathbbm{1}-(-ix+y).$
Since $xy=yx=0$ and squares of $x,y$ can be reduced, we have enough relations
to decompose any large product into only multiples of $x,y$ and powers of $t$.
Thus we present:
$N:={\mathbb{C}[x,y][t]}/{\left<p_{1},p_{2},p_{3},p_{4},p_{5}\right>}.$
We observe:
$N_{0}={\mathbb{C}[x,y]}/{\left<x^{2},xy,yx,y^{2},-ix+y\right>},$
and
$N_{1}={\mathbb{C}[x,y]}/{\left<x^{2}-ix,xy,yx,y^{2}-y,1+ix-y\right>.}$
We see that $x,y$ behave exactly as $a,b\in A$ and by Proposition 3.9. It is
easily checked that the isomorphism is given by:
$\displaystyle\phi:N_{1}$ $\displaystyle\to A$ $\displaystyle 1$
$\displaystyle\mapsto\mathbbm{1}$ $\displaystyle x$
$\displaystyle\mapsto(i,0)$ $\displaystyle y$ $\displaystyle\mapsto(0,1).$
Thus we have produced an algebra $N_{0}$ that has
$A=\mathbb{C}\oplus\mathbb{C}$ as a deformation.
###### Example 2.2.
We consider $A=M_{2}(\mathbb{C})$ and fix:
$a=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},\quad b=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}.$
We denote by $\mathbbm{1}$ the multiplicative unit in $A$. Next we compute:
$\displaystyle\begin{pmatrix}1&0\\\
0&0\end{pmatrix}=\frac{1}{2}(a+b^{2})=e_{1},$
$\displaystyle\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}=\frac{1}{2}(ab+b)=e_{2},$
$\displaystyle\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}=\frac{1}{2}(b-ab)=e_{3},$
$\displaystyle\begin{pmatrix}0&0\\\
0&1\end{pmatrix}=\frac{1}{2}(b^{2}-a)=e_{4}$
and thus $a,b$ generate $A$ as an algebra. Now we consider $A[t]$ and define:
$\displaystyle x:=ta,$ $\displaystyle y:=tb.$
We compute:
$\displaystyle x^{2}=t^{2}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}=t^{2}\mathbbm{1},$ $\displaystyle
xy=t^{2}\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}=t^{2}(e_{2}-e_{3}),$
$\displaystyle yx=t^{2}\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}=t^{2}(e_{3}-e_{2}),$ $\displaystyle
y^{2}=t^{2}\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}=t^{2}\mathbbm{1}.$
We have obtained the relations:
$\displaystyle p_{1}=x^{2}-t^{2}\mathbbm{1},$ $\displaystyle
p_{2}=x^{2}-y^{2}=0,$ $\displaystyle p_{3}=xy+yx=0,$ $\displaystyle
p_{4}=y^{2}-t^{2}\mathbbm{1}.$
Now we consider larger products to produce more relations:
$\displaystyle x^{3}=t^{3}\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}=t^{2}x,$
$\displaystyle y^{3}=t^{3}\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}=t^{2}y.$
Hence we have also obtained:
$\displaystyle p_{5}=x^{3}-t^{2}x=0,$ $\displaystyle p_{6}=y^{3}-t^{2}y=0.$
These allow us to derive the following:
$\displaystyle x^{2}y=y^{3}=yx^{2}=y^{3}=t^{2}y,$ $\displaystyle
y^{2}x=x^{3}=xy^{2}=x^{3}=t^{2}x,$
which are enough to reduce an arbitrary product of $x,y$. Thus we present:
$N:={\mathbb{C}[x,y][t]}/{\left<p_{1},p_{2},p_{3},p_{4},p_{5},p_{6}\right>}.$
We observe:
$N_{0}={\mathbb{C}[x,y]}/{\left<x^{2},y^{2},xy+yx\right>},$
and
$N_{1}={\mathbb{C}[x,y]}/{\left<x^{2}-1,y^{2}-1,xy+yx,x^{3}-x,y^{3}-y\right>}.$
By Proposition 3.9 we have $N_{1}\cong A$. It is easily checked that the
isomorphism is given by:
$\displaystyle\phi:N_{1}$ $\displaystyle\to A$ $\displaystyle 1$
$\displaystyle\mapsto\mathbbm{1}$ $\displaystyle x$
$\displaystyle\mapsto\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$ $\displaystyle
y$ $\displaystyle\mapsto\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}.$
Thus we have produced an algebra $N_{0}$ that has $A=M_{2}(\mathbb{C})$ as a
deformation.
## 3 Deformations to $A$
We now consider the general case in which $A$ is an associative finite
dimensional $\mathbb{C}$-algebra generated by some elements
$\sum_{i=1}^{n}a_{i},\sum_{i=1}^{n}b_{i}$ for some $a_{i},b_{i}\in A$.
### 3.1 Specification at $t=0$
In this section we define a multiplication on $\mathbb{C}[x,y][t]$ which gives
the algebra $N$ at specification $t=0$.
###### Proposition 3.1.
Let $\mathbb{C}[t]$ denote the polynomial ring in variable $t$. Further,
assume that for every $r\in A$ there exists $i=i(r)$ such that
$rt^{i}\in\text{im}f$. There exist elements
$q_{1},q_{2},\dots,q_{n}\in{\mathbb{C}}[x,y][t]$ such that for every $k$ we
have
$f(p_{k})\in\sum_{i=1}^{n}\mathbb{C}[t]f(q_{i}).$ (1)
Moreover $n=\dim A.$
###### Proof.
We let $\\{e_{1},\dots,e_{n}\\}$ be a basis of $A$. Let $k_{1}$ be the
smallest possible integer such that for some integer $j_{1}$ we have
$e_{j_{1}}t^{k_{1}}+\sum_{\begin{subarray}{c}i=1\\\ i\neq
j_{1}\end{subarray}}^{n}\alpha_{i}(t)e_{i}\in\text{im}f$
for some $\alpha_{i}(t)\in\mathbb{C}[t]$. Let $k_{2}$ be the smallest possible
integer such that for some integer $j_{2}\neq j_{1}$ we have
$e_{j_{2}}t^{k_{2}}+\sum_{\begin{subarray}{c}i=1\\\ i\neq
j_{1},j_{2}\end{subarray}}^{n}\alpha^{\prime}_{i}(t)e_{i}\in\text{im}f$
for some $\alpha^{\prime}_{i}(t)\in\mathbb{C}[t]$. Continuing in this way, we
define a basis
$\\{q_{1},\dots,q_{n}\\}$ such that
$f(q_{i})=e_{j_{m}}t^{k_{m}}+\sum_{\begin{subarray}{c}i=1\\\ i\neq
j_{1},\dots,j_{m}\end{subarray}}^{n}\alpha_{i}(t)e_{i}.$
Now notice that $f(q_{1}),f(q_{2}),\dots,f(q_{n})$ are linearly independent
over $\mathbb{C}$ as $e_{1},e_{2},\dots,e_{n}$ are linearly independent over
$\mathbb{C}$.
We now prove that $f(q_{1})+t\text{im}f,\dots,f(q_{n})+t\text{im}f$ span the
$\mathbb{C}$-algebra $\frac{\text{im}f}{t\text{im}f}$ over $\mathbb{C}$, and
hence they are a basis for $\frac{\text{im}f}{t\text{im}f}$. Let
$z_{1}\in\text{im}f$ and write
$z_{1}=\sum_{i=1}^{n}r_{i}(t)e_{j_{i}}t^{s_{i}}$
for some $r_{i}\in\mathbb{C}[t]-t\mathbb{C}[t]$ and some
$j_{i},s_{i}\in\mathbb{N}$.
Notice that for the smallest $i$ such that $r_{i}(t)\neq 0$, $s_{i}\geq k_{i}$
for $k_{i}$ as above. Let
$z_{2}=z_{1}-f(q_{1})t^{s_{1}-k_{1}}r_{1}(t).$
Note that we have
$z_{2}\in\sum_{i=2}^{n}\mathbb{C}[t]e_{j_{i}}$
so we can write
$z_{2}=\sum_{i=2}^{n}r^{2}_{i}(t)e_{j_{i}}t^{s^{2}_{i}}$
for some $r^{2}_{i}(t)\in\mathbb{C}[t]-t\mathbb{C}[t]$ and
$s^{2}_{i}\in\mathbb{N}$. Notice that $q_{2}\in\text{im}f$. Now let
$z_{3}=z_{2}-f(q_{2})t^{s^{2}_{2}-k_{2}}r^{2}_{2}(t).$
Continuing in this way we eventually arrive at $q_{n+1}=0.$ Hence by
summing the following equations
$\displaystyle 0$
$\displaystyle=z_{n}-f(q_{n})t^{s^{n}_{n}-k_{n}}r^{n}_{n}(t),$
$\displaystyle\vdots$ $\displaystyle z_{3}$
$\displaystyle=z_{2}-f(q_{2})t^{s^{2}_{2}-k_{2}}r^{2}_{2}(t)$
and
$z_{2}=z_{1}-f(q_{1})t^{s_{1}-k_{1}}r_{1}(t)$
we arrive at
$z_{1}=\sum_{l=1}^{n}f(q_{l})t^{s^{\prime}_{l}-k_{l}}r^{l}_{l}(t)\in\sum_{l=1}^{n}\mathbb{C}[t]f(q_{l})$
for some $s^{\prime}_{l}\in\mathbb{N}$ as required. ∎
Remark. Notice that for $k\in\mathbb{N}$ and $\alpha_{j}\in\mathbb{C}[t]$
$f(p_{k})-\sum_{j=1}^{n}\alpha_{j}f(q{{}_{j}})=0$
if and only if
$p_{k}-\sum_{j=1}^{n}\alpha_{j}q_{j}\in I.$
###### Proposition 3.2.
For any $k,m\in\\{1,2,\dots,n\\}$ there exist $\zeta_{i,k}\in\mathbb{C}$,
$\xi_{i,k}(t)\in{\mathbb{C}}[t]$ such that
$p_{k}-\sum_{i=1}^{n}(\zeta_{i,k}q_{i}+t\xi_{i,k}(t)q_{i})\in I.$
In particular, there exist $\zeta_{i,k,m}\in\mathbb{C}$,
$\xi_{i,k,m}(t)\in{\mathbb{C}}[t]$ such that
$q_{k}q_{m}-\sum_{i=1}^{n}(\zeta_{i,k,m}q_{i}+t\xi_{i,k,m}(t)q_{i})\in I.$
###### Proof.
For each element $p_{k}\in{\mathbb{C}}[x,y][t]$ we have
$p_{k}\in\sum_{i=1}^{n}{\mathbb{C}}[t]q_{i}+I.$
This follows by taking $f^{-1}$ of equation (1). Hence, for each
$k\in\mathbb{N}$ there exist $\sigma_{i,k}\in\mathbb{C}[t]$ such that
$p_{k}-\sum_{i=1}^{n}\sigma_{i,k}q_{i}\in I,$
where $I=\ker f$. Notice that we can write
$\sigma_{i,k}(t)=\zeta_{i,k}+t\xi_{i,k}(t)$ for some
$\zeta_{i,k}\in\mathbb{C}$, $\xi_{i,k}(t)\in{\mathbb{C}}[t]$, as to separate
the terms dependent on $t$. The above then becomes
$p_{k}-\sum_{i=1}^{n}(\zeta_{i,k}q_{i}+t\xi_{i,k}(t)q_{i})\in I.$
∎
Reasoning similarly as in the proof of Proposition 3.1 we get the following
corollary.
###### Corollary 3.3.
Suppose $\sum_{i=1}^{n}\alpha_{i}q_{i}\in I$ for some
$\alpha_{i}\in{\mathbb{C}}[t]$. Then $f(\sum_{i}^{n}\alpha_{i}q_{i})=0$ and
consequently $\alpha_{i}=0$ for every $i\in\\{1,2,\dots,n\\}$.
Moreover, all elements of $I$ are $\mathbb{C}[t]$-linear combinations of
elements
$p_{k}-\sum_{i=1}^{n}(\zeta_{i,k}q_{i}+t\xi_{i,k}(t)q_{i})$
for some $k\in\mathbb{N},\zeta_{i,k}\in\mathbb{C}$,
$\xi_{i,k}(t)\in{\mathbb{C}}[t]$.
###### Notation 3.4.
We denote by $J$ be the set consisting of $\mathbb{C}$-linear combinations of
elements
$p_{k}-\sum_{i=1}^{n}\zeta_{i,k}q_{i}\in{\mathbb{C}}[x,y],$ (2)
where $\zeta_{i,k}$ are as in Proposition 3.2. Let $\langle J\rangle$ be the
ideal of ${\mathbb{C}}[x,y]$ generated by elements from $J$. Further, let $N$
be the quotient algebra $\mathbb{C}[x,y]/\langle J\rangle$.
###### Corollary 3.5.
We have $e\in J$ if and only if $e+e^{\prime}\in I$ for some $e^{\prime}\in
t{\mathbb{C}}[x,y][t]$. In particular $J\subseteq I+t{\mathbb{C}}[x,y][t]$,
and hence $\langle J\rangle\subseteq I+t{\mathbb{C}}[x,y][t]$.
###### Proposition 3.6.
The dimension of $N$ equals $n$. Moreover, elements $q_{k}+J\in N$ are a basis
of $N$ as a $\mathbb{C}$-vector space.
###### Proof.
Notice that expression (2) implies that
${\mathbb{C}}[x,y]\subseteq\sum_{k=1}^{n}\mathbb{C}q_{k}+\langle J\rangle,$
and hence $\\{q_{k}+\langle J\rangle\mid k\in\\{1,2,\ldots,n\\}\\}$ span $N$
as a $\mathbb{C}$-vector space. Therefore, the dimension of $N$ does not
exceed $n$.
We now show that the elements $q_{k}+\langle J\rangle\in N$ for
$k\in\\{1,2,\ldots,n\\}$ are linearly independent over $\mathbb{C}$. Suppose
on the contrary that we have
$\sum_{i=1}^{n}\xi_{i}q_{i}\in\langle J\rangle$
for some $\xi_{i}\in\mathbb{C}$. By Corollary 3.5 and Proposition 3.1 we have
$\langle J\rangle\subseteq I+t{\mathbb{C}}[x,y][t]\subseteq
I+t\sum_{i=1}^{n}{\mathbb{C}}[t]q_{i}.$
It follows that there is $e^{\prime}\in t\sum_{i=1}^{n}{\mathbb{C}}[t]q_{i}$
such that $\sum_{i=1}^{n}\xi_{i}q_{i}-e^{\prime}\in I$, contradicting
Corollary 3.3. ∎
Observe now that the following holds:
###### Proposition 3.7.
We can present $N$ as an $\mathbb{C}$-algebra with generators
$d_{1},\dots,d_{n}$, which span $N$ as a $\mathbb{C}$-vector space, subject to
relations
$d_{k}d_{m}=\sum_{i=1}^{n}\zeta_{i,k,m}d_{i}$
where $\zeta_{i,k,m}$ are as in Proposition 3.2.
###### Proof.
We let ${\mathbb{C}}[q_{1},\ldots,q_{n}]$ be the subalgebra of
${\mathbb{C}}[x,y]$ generated by elements
$q_{1},\ldots,q_{n}\in{\mathbb{C}}[x,y]$. Similarly, let
${\mathbb{C}}[d_{1},\ldots,d_{n}]$ be the free $\mathbb{C}$-algebra generated
by elements $d_{1},\ldots,d_{n}$. Let
$\zeta:{\mathbb{C}}[d_{1},\ldots,d_{n}]\rightarrow{\mathbb{C}}[x,y]/\langle
J\rangle$ be defined as $\zeta(d_{i})=q_{i}+\langle J\rangle$.
We now show that $x+\langle J\rangle,y+\langle J\rangle\in\text{im}\zeta$.
Notice that since $f(x)\in\text{im}f$ we have
$f(x)\in\sum_{i=0}^{\infty}{\mathbb{C}}[t]f(q_{i}).$
Hence we have $f(x)=f(x^{\prime})$ for some
$x^{\prime}\in\sum_{i=0}^{\infty}{\mathbb{C}}[t]f(q_{i}).$ Then
$x-x^{\prime}\in I$, and so $x-x^{\prime\prime}\in J$ for some
$x^{\prime\prime}\in\sum_{i=0}^{\infty}{\mathbb{C}}q_{i}.$ Then
$x^{\prime\prime}+\langle J\rangle\in\text{im}\zeta$, and so $x+\langle
J\rangle\in\text{im}\zeta$. An analogous argument shows $y+\langle
J\rangle\in\text{im}\zeta$.
Let
$J^{\prime}=\ker(h)\subseteq{\mathbb{C}}[d_{1},\ldots,d_{n}].$
Then by the First Isomorphism Theorem for rings
${\mathbb{C}}[d_{1},\ldots,d_{n}]/J^{\prime}$ is isomorphic to
${\mathbb{C}}[x,y]/\langle J\rangle=N$.
Let $J^{\prime\prime}$ be the ideal of ${\mathbb{C}}[d_{1},\ldots,d_{n}]$
generated by elements
$d_{k}d_{m}-\sum_{i=1}^{n}\zeta_{i,k,m}d_{i}$
where $\zeta_{i,k,m}$ are as in Proposition 3.2. Observe that
$\zeta\left(d_{k}d_{m}-\sum_{i=1}^{n}\zeta_{i,k,m}d_{i}\right)=q_{k}q_{m}-\sum_{i=1}^{n}\zeta_{i,k,m}q_{i}+\langle
J\rangle=0+\langle J\rangle.$
Therefore $J^{\prime\prime}\subseteq J^{\prime}$. It follows that the
dimension of ${\mathbb{C}}[d_{1},\dots,d_{n}]/J^{\prime}$ does not exceed the
dimension of ${\mathbb{C}}[d_{1},\dots,d_{n}]/J^{\prime\prime}$.
We now show that $J^{\prime}=J^{\prime\prime}$. Notice that
${\mathbb{C}}[d_{1},\ldots,d_{n}]/J^{\prime\prime}$ is at most
$n$-dimensional, since every element can be presented as a linear combination
of elements $d_{i}+J^{\prime\prime}$ for $i=1,2,\ldots,n$. On the other hand
${\mathbb{C}}[d_{1},\ldots,d_{n}]/J^{\prime}$ is isomorphic to
${\mathbb{C}}[x,y]/\langle J\rangle$, and hence is $n$-dimensional (by
Proposition 3.6). Hence by comparing dimensions we get that the dimension of
${\mathbb{C}}[d_{1},\dots,d_{n}]/J^{\prime\prime}$ does not exceed the
dimension of ${\mathbb{C}}[d_{1},\dots,d_{n}]/J^{\prime}$. Previously we
showed that ${\mathbb{C}}[d_{1},\dots,d_{n}]/J^{\prime}$ does not exceed the
dimension of ${\mathbb{C}}[d_{1},\dots,d_{n}]/J^{\prime\prime}$. It follows
that $J^{\prime}=J^{\prime\prime}$, as required. ∎
We now define a multiplication on $N$ which gives a formal deformation to the
algebra $A$.
###### Theorem 3.8.
Let $d_{1},\dots,d_{n}$ be free generators of the $\mathbb{C}$-algebra
$\mathbb{C}[d_{1},\dots,d_{n}]$ and suppose $\zeta_{i,k,m}\in\mathbb{C}$,
$\xi_{i,k,m}(t)\in{\mathbb{C}}[t]$ are as in Proposition 3.2. Then the
multiplication rule
$d_{k}\ast_{t}d_{m}=\sum_{i=1}^{n}(\zeta_{i,k,m}d_{i}+t\xi_{i,k,m}(t)d_{i}).$
gives a formal deformation such that $\ast_{0}$ gives the multiplication on an
algebra isomorphic to $N$.
###### Proof.
Recall relations from Proposition 3.2 that for any $k,m\in\\{1,2,\dots,n\\}$
there exist $\zeta_{i,k,m}\in\mathbb{C}$, $\xi_{i,k,m}(t)\in{\mathbb{C}}[t]$
such that
$q_{k}q_{m}-\sum_{i=1}^{n}(\zeta_{i,k,m}q_{i}+t\xi_{i,k,m}(t)q_{i})\in I.$ (3)
We introduce notation $[q_{i}]:=q_{i}+I$ for elements of
${\mathbb{C}}[x,y]/I$. Hence we obtain the following relations corresponding
to (3) which give the multiplicative table on ${\mathbb{C}}[x,y]/I$. We have
$[q_{k}][q_{m}]-\sum_{i=1}^{n}(\zeta_{i,k,m}[q_{i}]+t\xi_{i,k,m}(t)[q_{i}]).$
Notice that these relations give a multiplication
$d_{k}\ast_{t}d_{m}=\sum_{i=1}^{n}(\zeta_{i,k,m}d_{i}+t\xi_{i,k,m}(t)d_{i})$
(4)
for $d_{i}\in\mathbb{C}[x,y].$ Now recall that $N$ is isomorphic to
${\mathbb{C}}[d_{1},\ldots,d_{n}]/J^{\prime\prime}$ where $J^{\prime\prime}$
is the ideal of ${\mathbb{C}}[d_{1},\ldots,d_{n}]$ generated by relations
$d_{k}d_{m}-\sum_{i=1}^{n}\zeta_{i,k,m}d_{i}.$
Therefore by setting $t=0$ in (4) we obtain the multiplication rule for
${\mathbb{C}}[d_{1},\ldots,d_{n}]/J^{\prime\prime}$. ∎
### 3.2 Specification at $t=1$
In this section we define a multiplication on $\mathbb{C}[x,y][t]$ which gives
a formal deformation of $N$, such that at specification $t=1$ we get an
algebra isomorphic to $A$.
###### Proposition 3.9.
Let $\xi:\mathbb{C}[x,y][t]\rightarrow A$ be a homomorphism of
$\mathbb{C}$-algebras such that
$\xi(x)=\sum_{i=0}^{n}a_{i},\quad\xi(y)=\sum_{i=0}^{n}b_{i},\quad\xi(t)=1.$
Further, assume that for every $r\in A$ there exists $i=i(r)$ such that
$rt^{i}\in\text{im}f$. Then $\ker\xi=\langle I,t-1\rangle.$
###### Proof.
We let $e=\sum_{i}\alpha_{i}p_{i}t^{\beta{i}}\in\ker\xi$ where $p_{i}$ are
monomials in $\mathbb{C}[x,y]$, $\alpha_{i}\in\mathbb{C}$ and
$\beta_{i}\in\mathbb{N}$. We will show there exists $\gamma_{i}\in\mathbb{N}$
such that $\hat{e}:=\sum_{i}\alpha_{i}p_{i}t^{\gamma_{i}}\in I$, so that $e\in
I+\langle t-1\rangle$ as $e-\hat{e}\in\langle t-1\rangle.$ This follows since
$e-\hat{e}=\sum_{i}\alpha_{i}p_{i}t^{\beta_{i}}-\sum_{i}\alpha_{i}p_{i}t^{\gamma_{i}}=\sum_{i}\alpha_{i}p_{i}(t^{\beta_{i}}-t^{\gamma_{i}})=\sum_{i}\alpha_{i}p_{i}t^{m}(t^{n}-1)\in\langle
t-1\rangle$
for some $m,n\in\mathbb{N}$.
Recall Notation 1.8 and note that
$\xi(e)=\sum_{i}\alpha_{i}\overline{\overline{p_{i}}}=0$. Notice that by
assumption there exist $j_{i}\in\mathbb{N}$ such that
$\overline{\overline{p_{i}}}t^{j_{i}}\in\text{im}f$. We let
$f(c_{i})=\overline{\overline{p_{i}}}t^{j_{i}}$ for some
$c_{i}\in\mathbb{C}[x,y][t]$. Hence for large enough $k\in\mathbb{N}$ we have
$f\left(\sum_{i}\alpha_{i}c_{i}t^{k-j_{i}}\right)=\sum_{i}\alpha_{i}\overline{\overline{p_{i}}}t^{k}=0.$
∎
###### Proposition 3.10.
Let notation be as in Proposition 3.9, and suppose that $\sum_{i=0}^{n}a_{i}$
and $\sum_{i=0}^{n}b_{i}$ generate $A$ as a $\mathbb{C}$-algebra. We have
$\frac{{\mathbb{C}}[x,y][t]/I}{\langle t-1+I\rangle}\cong A.$
###### Proof.
Note that the ideal $\langle t-1+I\rangle$ of ${\mathbb{C}}[x,y][t]/I$ equals
the set
$\frac{{\mathbb{C}}[x,y][t](t-1)+I}{I}=\frac{\langle
I,t-1\rangle}{I}=\\{g(t-1)+I\mid g\in{\mathbb{C}}[x,y][t]\\}.$
By the Third Isomorphism Theorem we have
$\frac{{\mathbb{C}}[x,y][t]/I}{\langle
I,t-1\rangle/I}\cong\frac{{\mathbb{C}}[x,y][t]}{\langle I,t-1\rangle}.$
By Proposition 3.9 we have $\langle I,t-1\rangle=\ker\xi$ and as $\xi$ is
onto,
$\frac{{\mathbb{C}}[x,y][t]}{\ker\xi}\cong A$
by the First Isomorphism Theorem. ∎
We now define a multiplication on ${\mathbb{C}}[x,y][t]/\langle t-1,I\rangle$
which gives a deformation to $A$ at $t=1$.
###### Theorem 3.11.
Let notation be as in Proposition 3.9, and suppose that $\sum_{i=0}^{n}a_{i}$
and $\sum_{i=0}^{n}b_{i}$ generate $A$ as a $\mathbb{C}$-algebra. Let
$d_{1},\dots,d_{n}$ be free generators of the $\mathbb{C}$-algebra
$\mathbb{C}[d_{1},\dots,d_{n}]$ and suppose $\zeta_{i,k,m}\in\mathbb{C}$,
$\xi_{i,k,m}(t)\in{\mathbb{C}}[t]$ are as in Proposition 3.2. Then the
multiplication rule
$d_{k}\ast_{t}d_{m}=\sum_{i=1}^{n}(\zeta_{i,k,m}d_{i}+t\xi_{i,k,m}(t)d_{i}).$
gives a formal deformation such that $\ast_{1}$ gives the multiplication on an
algebra generated by $d_{1},\dots,d_{n}$ isomorphic to
${\mathbb{C}}[x,y][t]/\langle I,t-1\rangle$.
###### Proof.
We show that the algebra ${\mathbb{C}}[x,y][t]/\langle I,t-1\rangle$ is
isomorphic to the algebra ${\mathbb{C}}[d_{1},d_{2},\ldots d_{n}]/I^{\prime}$
where $I^{\prime}$ is the ideal of ${\mathbb{C}}[d_{1},d_{2},\ldots d_{n}]$
generated by elements
$d_{k}\ast_{1}d_{m}-\sum_{i=1}^{n}(\zeta_{i,k,m}d_{i}+\xi_{i,k,m}(1)d_{i})$
for $\zeta_{i,k,m}\in\mathbb{C}$, $\xi_{i,k,m}(t)\in{\mathbb{C}}[t]$ as in
Proposition 3.2.
We let
$\delta:{\mathbb{C}}[d_{1},\ldots,d_{n}]\rightarrow{\mathbb{C}}[x,y][t]/\langle
I,t-1\rangle$ be such that
$\delta(d_{i})=q_{i}+\langle I,t-1\rangle,$
for $i\in\\{1,2,\ldots,n\\}$. Observe that $I^{\prime}\subseteq\ker(\delta)$
since
$\displaystyle\delta\left(d_{k}*d_{m}-\sum_{i=1}^{n}(\zeta_{i,k,m}d_{i}+\xi_{i,k,m}(1)d_{i})\right)$
$\displaystyle=q_{k}*q_{m}-\sum_{i=1}^{n}(\zeta_{i,k,m}q_{i}+\xi_{i,k,m}(1)q_{i})$
$\displaystyle\quad+\langle I,t-1\rangle$ $\displaystyle=\langle I,t-1\rangle$
since $q_{k}\ast
q_{m}-\sum_{i=1}^{n}(\zeta_{i,k,m}q_{i}+\xi_{i,k,m}(1)q_{i})\in I+\langle
t-1\rangle$.
Therefore the dimension of ${\mathbb{C}}[d_{1},\ldots,d_{n}]/\ker(\delta)$
does not exceed the dimension of
${\mathbb{C}}[d_{1},\ldots,d_{n}]/I^{\prime}$. Notice that
${\mathbb{C}}[d_{1},\ldots,d_{n}]/I^{\prime}$ is spanned by elements $d_{i}+I$
as a vector space, and hence has dimension at most $n$. On the other hand, by
the First Isomorphism Theorem for rings:
${\mathbb{C}}[d_{1},\ldots,d_{n}]/ker(\delta)\cong\text{im}(\delta)={\mathbb{C}}[x,y][t]/\langle
I,t-1\rangle,$ (5)
which in turn is isomorphic to $A$ by Proposition 3.9, and hence has dimension
$n$. It follows that $I^{\prime}=\ker(\delta)$ and hence
${\mathbb{C}}[d_{1},\ldots,d_{n}]/ker(\delta)$ is isomorphic to
${\mathbb{C}}[d_{1},\ldots,d_{n}]/I^{\prime}$. Where the equality in Equation
5 is true by the proof of Proposition 3.7.
∎
###### Remark 3.12.
Deformations at $t\neq 0$ The results from this section, hold with analogous
proofs for a specialisation at other $t\neq 0$. Let $z\in\mathbb{C}$ and
$z\neq 0$, the the deformation at $t=z$ is also isomorphic to $A$, provided
that $\sum_{i=0}^{n}a_{i}z^{i}$ and $\sum_{i=0}^{n}b_{i}z^{i}$ generate $A$ as
a $\mathbb{C}$-algebra.
###### Remark 3.13.
Since the dimension of all the algebras in $\\{N_{t}\\}_{t\in[0,1]}$ remains
$n=\dim(A)$ the deformation arising from $*$ is flat.
## 4 Application
In this section we prove a conjecture of M. Wemyss using the method in section
3. The statement is that of the following theorem.
###### Theorem 4.1.
Let
$N=\frac{\mathbb{C}[x,y]}{\langle xy+yx,x^{3}+y^{2},y^{3}\rangle}.$
Then $N$ deforms into $A=M_{2}(\mathbb{C})\oplus\mathbb{C}^{\oplus 5}$.
We prove this theorem through a series of lemmas. Firstly, we fix notation in
addition to the notation of Theorem 4.1.
###### Notation 4.2.
Let $i_{1},i_{2}$ be roots of the polynomial $x^{2}-x+1$.
###### Notation 4.3.
Let $e\in\mathbb{C}$ be such that for $c=\frac{i_{1}-i_{2}}{\sqrt{1+e^{2}}}$
the polynomial
$g(z)=(2z-1)^{2}z^{3}+c^{2}$
has $5$ distinct roots $\alpha_{1},\dots,\alpha_{5}\in\mathbb{C}$. Further, we
assume that $e\neq 0$, $c^{2}\neq 3$ and $\alpha_{i}\neq\frac{1}{2}.$ Notice
that $g(z)$ and $z^{3}+1$ have no common roots, so $\alpha_{i}\neq-1$ and
$\alpha_{j}\notin\\{i_{1},i_{2}\\}$.
###### Notation 4.4.
Let $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5})$ and
$\beta=(\beta_{1},\beta_{2},\beta_{3},\beta_{4},\beta_{5})$ for
$\beta_{i}\in\mathbb{C}$ such that
$(\beta_{1},\beta_{2},\beta_{3},\beta_{4},\beta_{5})(2\alpha_{1}-1,2\alpha_{2}-1,2\alpha_{3}-1,2\alpha_{4}-1,2\alpha_{5}-1)=c(1,1,1,1,1).$
Notice that such $\beta_{i}$ exist since $\alpha_{i}\neq\frac{1}{2}$.
###### Lemma 4.5.
Let $\xi:\mathbb{C}[x,y][t]\rightarrow A$ be a homomorphism of
$\mathbb{C}$-algebras such that
$\displaystyle\xi(x)$ $\displaystyle=\begin{pmatrix}i_{1}&0\\\
0&i_{2}\end{pmatrix}+\alpha,\quad\xi(y)=\begin{pmatrix}1&e\\\
e&-1\end{pmatrix}+\beta,\quad\xi(t)=1.$
Then $\xi(x),\xi(y)$ and $1$ generate $A$ as an algebra.
###### Proof.
Observe that $\xi(x^{j}(x^{2}-x+1))=\alpha^{j}(\alpha^{2}-\alpha+1)$ (where
$\alpha^{j}(\alpha^{2}-\alpha+1)=(\alpha_{1}^{j}(\alpha_{1}^{2}-\alpha_{1}+1),\dots,\alpha_{5}^{j}(\alpha_{5}^{2}-\alpha_{5}+1))$).
Now let $W$ be the matrix with entries
$W_{ij}=\alpha_{i}^{j}(\alpha_{i}^{2}-\alpha_{i}+1)$. Let $V$ be the
Vandermonde matrix with entries $V_{ij}=\alpha_{i}^{j}$. Let $D$ be a diagonal
matrix with entries $D_{ij}=\delta_{ij}(\alpha_{i}^{2}-\alpha_{i}+1)$. It
follows that $W=DV.$ Now since $V$ is nonsingular, $W$ is nonsingular. Now
this implies that any $5$-dimensional vector in $A$ is in $\text{im}\xi$. Then
$\xi(x)-\alpha\in\text{im}\xi$ and hence
$\begin{pmatrix}i_{1}&0\\\ 0&i_{2}\end{pmatrix}\in\text{im}\xi.$
Similarly, $\xi(y)-\beta\in\text{im}\xi$ and hence
$\begin{pmatrix}1&e\\\ e&-1\end{pmatrix}\in\text{im}\xi.$
Now recall that $i_{1},i_{2}$ are roots of the polynomial $z^{3}+1$, so we
have
$\begin{pmatrix}i_{1}&0\\\ 0&i_{2}\end{pmatrix}^{3}=\begin{pmatrix}-1&0\\\
0&-1\end{pmatrix}.$
Notice that every diagonal matrix is a linear combination of
$\begin{pmatrix}i_{1}&0\\\ 0&i_{2}\end{pmatrix}$ and $\begin{pmatrix}-1&0\\\
0&-1\end{pmatrix}$. Now, it can be checked that these matrices are sufficient
to generate all matrices in $M_{2}(\mathbb{C})$.
∎
###### Notation 4.6.
Let $f:\mathbb{C}[x,y][t]\rightarrow A[t]$ be a homomorphism of
$\mathbb{C}$-algebras such that
$\displaystyle f(x)$ $\displaystyle=t^{2}\begin{pmatrix}i_{1}&0\\\
0&i_{2}\end{pmatrix}+t^{2}\alpha,\quad
f(y)=\frac{1}{\sqrt{1+e^{2}}}t^{3}\begin{pmatrix}1&e\\\
e&-1\end{pmatrix}+t^{3}\beta,\quad f(t)=t\mathbbm{1},$
where $\mathbbm{1}$ is the identity element in $A$. We write
$\tilde{1}:=(1,1,1,1,1)$. As usual we denote by $1$ the identity element in
$\mathbb{C}$ and thus in $\mathbb{C}[x,y][t]$.
We now prove Theorem 4.1.
###### Proof.
Referring to Notation 4.4 and 4.6 we calculate:
$\displaystyle f(x^{2})$
$\displaystyle=t^{4}\Big{(}\begin{pmatrix}i_{1}^{2}&0\\\
0&i_{2}^{2}\end{pmatrix}+\alpha^{2}\Big{)},$ $\displaystyle f(y^{2})$
$\displaystyle=t^{6}\Big{(}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}+\beta^{2}\Big{)},$ $\displaystyle f(xy)$
$\displaystyle=\frac{t^{5}}{\sqrt{1+e^{2}}}\Big{(}\begin{pmatrix}i_{1}&i_{1}e\\\
i_{2}e&-i_{2}\end{pmatrix}+\alpha\beta\Big{)},$ $\displaystyle f(yx)$
$\displaystyle=\frac{t^{5}}{\sqrt{1+e^{2}}}\Big{(}\begin{pmatrix}i_{1}&i_{2}e\\\
i_{1}e&-i_{2}\end{pmatrix}+\alpha\beta\Big{)}.$
Note that $f(x^{2}-t^{2}x+t^{4})=t^{4}(\alpha^{2}-\alpha+\tilde{1})$ and hence
$\displaystyle f(x(x^{2}-t^{2}x+t^{4}))$
$\displaystyle=t^{6}\alpha(\alpha^{2}-\alpha+\tilde{1}),$ $\displaystyle
f(x^{2}(x^{2}-t^{2}x+t^{4}))$
$\displaystyle=t^{8}\alpha^{2}(\alpha^{2}-\alpha+\tilde{1}),$ $\displaystyle
f(y(x^{2}-t^{2}x+t^{4}))$
$\displaystyle=t^{7}\beta(\alpha^{2}-\alpha+\tilde{1}).$
Observe that
$\displaystyle f(xy+yx)$
$\displaystyle=\frac{t^{5}}{\sqrt{1+e^{2}}}\begin{pmatrix}2i_{1}&e\\\
e&-2i_{2}\end{pmatrix}+t^{5}2\alpha\beta,$ $\displaystyle f(t^{2}y)$
$\displaystyle=\frac{t^{5}}{\sqrt{1+e^{2}}}\begin{pmatrix}1&e\\\
e&-1\end{pmatrix}+t^{5}\beta,$ $\displaystyle
f(t^{5}\frac{i_{2}-i_{1}}{\sqrt{1+e^{2}}})$
$\displaystyle=\frac{t^{5}}{\sqrt{1+e^{2}}}\begin{pmatrix}i_{2}-i_{1}&0\\\
0&i_{2}-i_{1}\end{pmatrix}+\frac{t^{5}(i_{2}-i_{1})}{\sqrt{1+e^{2}}}\tilde{1}.$
Note that $2i_{1}-1+(i_{2}-i_{1})=0$ and $-2i_{2}+1+(i_{2}-i_{1})=0$, so
$f(xy+yx-t^{2}y+\frac{t^{5}(i_{2}-i_{1})}{\sqrt{1+e^{2}}})=2\alpha\beta-\beta+\frac{i_{2}-i_{1}}{\sqrt{1+e^{2}}}\tilde{1}.$
Now notice that
$2\alpha\beta-\beta+\frac{i_{2}-i_{1}}{\sqrt{1+e^{2}}}\tilde{1}=0$ by
assumptions of Notation 4.4. It follows that
$xy+yx-t^{2}y+\frac{t^{5}(i_{2}-i_{1})}{\sqrt{1+e^{2}}}\in I$ and so
$xy+yx\in\langle J\rangle$ is a relation in $N$ as defined in Notation 3.4.
Notice that by Notation 4.4 we have $\beta^{2}(2\alpha-1)^{2}=c^{2}\tilde{1}$.
Now it follows by Notation 4.3 that
$\alpha^{3}+\beta^{2}=0,$ (6)
and thus $x^{3}+y^{2}\in\langle J\rangle$ is a relation in $N$.
We now proceed to show that $y^{3}\in\langle J\rangle$. Notice that:
$f(y^{3})=\frac{t^{9}}{\sqrt{1+e^{2}}}\begin{pmatrix}1&e\\\
e&-1\end{pmatrix}+t^{9}\beta^{3}.$
Hence $y^{3}-t^{6}y=t^{9}(\beta^{3}-\beta).$ Therefore it is sufficient to
show that $y^{3}-t^{6}y+q\in I$ for some $q\in t\mathbb{C}[x,y][t]$. We
denote:
$\displaystyle r_{1}$ $\displaystyle:=t^{5}(x^{2}-t^{2}x+t^{4})$
$\displaystyle r_{2}$ $\displaystyle:=t^{3}x(x^{2}-t^{2}x+t^{4})$
$\displaystyle r_{3}$ $\displaystyle:=tx^{2}(x^{2}-t^{2}x+t^{4})$
$\displaystyle r_{4}$ $\displaystyle:=t^{2}y(x^{2}-t^{2}x+t^{4})$
Notice that the elements
$\displaystyle f(r_{1})$
$\displaystyle=t^{9}(\alpha^{2}-\alpha+\tilde{1})=:t^{9}e_{1},$ $\displaystyle
f(r_{2})$
$\displaystyle=t^{9}\alpha(\alpha^{2}-\alpha+\tilde{1})=:t^{9}e_{2},$
$\displaystyle f(r_{3})$
$\displaystyle=t^{9}\alpha^{2}(\alpha^{2}-\alpha+\tilde{1})=:t^{9}e_{3},$
$\displaystyle f(r_{4})$
$\displaystyle=t^{9}\beta(\alpha^{2}-\alpha+\tilde{1})=:t^{9}e_{4}$
are in $t\mathbb{C}[x,y][t]$. Note that it suffices to show that
$y^{3}-t^{6}y+q\in I$ for some
$q\in\mathbb{C}r_{1}+\mathbb{C}r_{2}+\mathbb{C}r_{3}+\mathbb{C}r_{4}$.
Therefore it suffices to show that
$\beta^{3}-\beta\in\mathbb{C}e_{1}+\mathbb{C}e_{2}+\mathbb{C}e_{3}+\mathbb{C}e_{4}.$
(7)
Now relation (7) is equivalent to
$(2\alpha-1)\beta(\beta^{2}-1)\in\mathbb{C}(2\alpha-1)e_{1}+\mathbb{C}(2\alpha-1)e_{2}+\mathbb{C}(2\alpha-1)e_{3}+\mathbb{C}(2\alpha-1)e_{4},$
which in turn is equivalent to
$-c(\alpha^{3}+1)\in\mathbb{C}(2\alpha-1)e_{1}+\mathbb{C}(2\alpha-1)e_{2}+\mathbb{C}(2\alpha-1)e_{3}+\mathbb{C}(2\alpha-1)e_{4}.$
(8)
Now, notice that $\alpha^{3}+1=(\alpha+1)(\alpha^{2}+\alpha+1)$ and that
$\alpha+1$ has non-zero entries, so multiplying relation (8) by $\alpha+1$ we
get an equivalent relation
$\displaystyle(\alpha^{3}+1)(\alpha+1)\in\mathbb{C}$
$\displaystyle(2\alpha-1)(\alpha^{3}+1)+\mathbb{C}(2\alpha-1)(\alpha^{3}+1)\alpha$
$\displaystyle+\mathbb{C}(2\alpha-1)(\alpha^{3}+1)\alpha^{2}+\mathbb{C}(\alpha^{3}+1).$
Notice that the right hand side can be rewritten as
$\mathbb{C}(\alpha^{3}+1)+\mathbb{C}(\alpha^{3}+1)\alpha+\mathbb{C}(\alpha^{3}+1)\alpha^{2}+\mathbb{C}(\alpha^{3}+1)\alpha^{3}$.
∎
## 5 A more general method
In this section we consider a more general method that results in algebras $N$
that are not necessarily local and have more generators. Furthermore, we
consider $A$ to have finitely many generators.
Let $A$ be a finitely generated $\mathbb{C}$-algebra.
1. 1.
We consider a free $\mathbb{C}$-algebra $F$ with $j$ generators
$x_{1},\dots,x_{j}$ for some $j\in\mathbb{N}$. We define:
$\displaystyle f:F[t]$ $\displaystyle\to A[t]$ $\displaystyle x_{i}$
$\displaystyle\mapsto\sum\limits_{l=0}^{m}a_{i,l}t^{l}$
for some $m\in\mathbb{N}$ and some $a_{i,l}\in A$. We also assume that for
each $z\in\mathbb{C}$, $z\neq 0$ elements $\sum\limits_{l=0}^{m}a_{i,l}z^{l}$
for $i=1,2,\ldots j$ generate $A$. In other words, images of the generators
generate A when $t$ is set to non-zero numbers.
2. 2.
We define
$\mathcal{N}:=\frac{\mathbb{C}[x_{1},\dots,x_{j}][t]}{I},$
where $I:=\ker(f)$.
3. 3.
We assume that for every $r\in A$ there exists $i=i(r)$ such that
$rt^{i}\in\text{im}f$.
4. 4.
Our algebra $N=N_{0}$ is isomorphic to
$\frac{\mathcal{N}}{\langle t+I\rangle}$
and deforms flatly and associatively to $A$. Moreover $N$ is isomorphic to
$\frac{\mathbb{C}[x_{1},\dots,x_{j}][t]}{\langle I,t\rangle},$ where $\langle
I,t\rangle$ is the ideal of $\mathbb{C}[x_{1},\dots,x_{j}][t]$ generated by
elements of $I$ and by $t$.
If $j=2$ then this method corresponds to the method described in section 3
provided that $a_{i,0}=0$ for all $i$.
Proof. When reasoning as in the previous chapters we will use at all places
$\mathbb{C}[x_{1},\ldots,x_{j}]$ instead of $\mathbb{C}[x,y]$, and we will use
$\mathbb{C}[x_{1},\ldots,x_{j}][t]$ instead of $\mathbb{C}[x,y][t]$. Let $A$
have dimension $n$ over $\mathbb{C}$. Recall that we
$\mathcal{N}:=\frac{\mathbb{C}[x_{1},\dots,x_{j}][t]}{I},$
where $I:=\ker(f)$.
* •
Reasoning analogously as in Proposition 3.1 we obtain that there are
$q_{1},\ldots,q_{n}\in F$ such that
$f(F[t])\subseteq\sum_{i=1}^{n}{\mathbb{C}}f(q_{i}),$
and hence, since $I=\ker f$, we get
$F(t)\subseteq\sum_{i=1}^{n}{\mathbb{C}}q_{i}+I.$
In particular, there are $\zeta_{i,k,m}\in\mathbb{C}$,
$\xi_{i,k,m}(t)\in{\mathbb{C}}[t]$ such that
$q_{k}\cdot q_{m}-(\sum_{i=1}^{n}\zeta_{i,k,m}q_{i}+t\xi_{i,k,m}q_{i})\in I.$
* •
Reasoning similarly as in Theorem 3.8 we obtain that if $d_{1},\ldots,d_{n}$
are free generators then
$d_{k}*d_{m}-(\sum_{i=1}^{n}\zeta_{i,k,m}d_{i}+t\xi_{i,k,m}d_{i}),$
gives a formal deformation such that $*_{0}$ gives a multiplication on $N$
where $N$ is the $\mathbb{C}$-algebra with generators $d_{1},\ldots,d_{n}$
subject to relations
$d_{k}*_{0}d_{m}=\sum_{i=1}^{n}\zeta_{i,k,m}d_{i}.$
Observe that $*$ defines a multiplication on the algebra
$\mathbb{C}[d_{1},\ldots,d_{n}][t]/I^{\prime}$, where $I^{\prime}$ is the
ideal of $\mathbb{C}[d_{1},\ldots,d_{n}][t]$ generated by elements
$d_{k}*d_{m}-(\sum_{i=1}^{n}\zeta_{i,k,m}d_{i}+t\xi_{i,k,m}d_{i}).$
Notice that $d_{1},\ldots,d_{n}$ are free generators of the free algebra
$\mathbb{C}[d_{1},\ldots,d_{n}]$. Notice that the algebra
$\mathbb{C}[d_{1},\ldots,d_{n}][t]/I^{\prime}$ is isomorphic to algebra
$\mathcal{N}$. To see this consider the map
$\sigma\mathbb{C}[d_{1},\ldots,d_{n}][t]\rightarrow{\mathcal{N}}$ given by
$d_{i}\rightarrow q_{i}+I.$
Notice that $I^{\prime}\subseteq\ker\sigma$ since
$(q_{k}+I)*(q_{m}+I)=\sum_{i=1}^{n}\zeta_{i,k,m}(q_{i}+I)+t\xi_{i,k,m}(q_{i}+I).$
Recall that $\ker\sigma=I^{\prime}$ because if $a\in ker\sigma$ then by using
relations from $I^{\prime}$ we can present $a$ as $a^{\prime}+i$ where
$i^{\prime}\in I$ and where $a^{\prime}=\sum_{i=1}^{n}\alpha_{i}(t)d_{i}$ for
some $\alpha_{i}(t)\in{\mathbb{C}}[t]$. Notice that then
$\sigma(i^{\prime})=0$ and hence
$\sigma(a)=\sigma(a^{\prime})=\sigma(\sum_{i=1}^{n}\alpha_{i}(t)d_{i})=\sum_{i=1}^{n}\alpha_{i}(t)q_{i}.$
Reasoning as in Corollary $3.3$ we get that this implies that all
$\alpha_{i}=0$, hence $a^{\prime}=0$, hence $a=i^{\prime}$. Therefore,
$\ker\sigma\subseteq I$ and consequently $\ker\sigma=I^{\prime}$.
Therefore, by the First Isomorphism Theorem for algebras
${\mathbb{C}}[d_{1},\ldots,d_{n}][t]/I^{\prime}$ is isomorphic to $Im(\sigma)$
which is equal to ${\mathcal{N}}$. To see that $\sigma$ is surjective, notice
that $q_{1}+I,\ldots,q_{n}+I$ span $\mathcal{N}$ as a linear space (as
mentioned before this can be proved similarly as Proposition 3.1).
* •
Reasoning as in Proposition 3.6 we obtain that $N$ is $n$ dimensional
$\mathbb{C}$-algebra.
* •
Reasoning similarly as in Proposition 3.7 we obtain that the algebra $N$ is
isomorphic to the algebra $\frac{\mathbb{C}[x_{1},\dots,x_{j}][t]}{\langle
I,t\rangle}$ where ${\langle I,t\rangle}$ is the ideal of
$\mathbb{C}[x_{1},\dots,x_{j}][t]$ generated by elements from $I$ and by $t$.
By the Third Isomorphism Theorem $N$ is isomorphic to
$\frac{\mathcal{N}}{\langle t+I\rangle}.$
This can be seen by reasoning similarly as in Proposition 3.10 but taking $t$
instead of $t-1$ in the proof.
* •
We will now consider deformations $*_{1}$ at $t=1$. Consider first the
homomorphism of $\mathbb{C}$-algebras $\xi:F[t]\rightarrow A$ given by
$\xi(x_{i})=\sum_{l=0}^{m}a_{i,l}$ for $i=1,2,\ldots,j$.
Reasoning similarly as in Proposition 3.9 we obtain that $\ker\xi=\langle
I,t-1\rangle$. Reasoning analogously as in the last two lines of Proposition
3.10 we obtain that $F[t]/\ker\xi$ is isomorphic to $A$.
* •
Reasoning as in Theorem 3.11 we see that the algebra
${\mathbb{C}}[d_{1},\ldots,d_{n}][t]/I$ deforms at $t=1$ to the algebra
$F[t]/{\langle I,t-1\rangle}=F[t]/{\ker\xi}$ which is isomorphic to $A$.
* •
We will now consider deformations $*_{z}$ at $t=z$ where $z\in\mathbb{C},z\neq
0,$ $z\neq 1$. Consider first the homomorphism of $\mathbb{C}$-algebras
$\xi:F[t]\rightarrow A$ given by $\xi(x_{i})=\sum_{l=0}^{m}a_{i,l}z^{l}$ for
$i=1,2,\ldots,j$.
Reasoning similarly as in Proposition 3.9 we obtain that $\ker\xi=\langle
I,t-z\rangle$ (this can be done by using $t^{\prime}=t/z$ at the place of $t$
in the proof of Proposition 3.9, and in the first line declaring that
$p_{i}\in F$ instead of $p_{i}\in{\mathbb{C}}[x,y]$).
Next, reasoning analogously as in the last two lines of Proposition 3.10 we
obtain that $F[t]/\ker\xi$ is isomorphic to $A$ (this follows since elements
$\xi(x_{i})$ generate $A$).
Next, reasoning as in Theorem 3.11 we see that the algebra
${\mathbb{C}}[d_{1},\ldots,d_{n}][t]/I$ deforms at $t=z$ to the algebra
$F[t]/{\langle I,t-z\rangle}=F[t]/{\ker\xi}$ which is is isomorphic to $A$. ∎
## 6 Future Work
In this section we pose some questions about Method 1 and suggest some
generalisations.
###### Question 6.1.
Let $A$ be a $\mathbb{C}$-algebra and suppose $G_{1}=\\{a,b\\}$ and
$G_{2}=\\{a^{\prime},b^{\prime}\\}$ are two distinct generating sets of $A$.
Under what conditions does Method 1 using $A,G_{1}$ and $A,G_{2}$ result in
the same algebra $N_{0}$ that has $A$ as a deformation?
###### Question 6.2.
Let $A$ be a $\mathbb{C}$-algebra. Under what conditions does Method 1 only
produce one algebra $N$ that has $A$ as a deformation.
###### Question 6.3.
Let $A$ be a $\mathbb{C}$-algebra. Does their exist a finitely terminating
algorithm that will produce all the algebras $N$ arising from Method 1 that
have $A$ as a deformation?
###### Question 6.4.
Let $A$ be a $\mathbb{C}$-algebra. Do ring theoretic properties of the
generators $a,b$ (for example being idempotent, being irreducible idempotent,
prime) determine any properties of the algebra $N$ arising from Method 1 using
$A,\\{a,b\\}$?
###### Question 6.5.
Let $A$ be a $\mathbb{C}$-algebra. Do ring theoretic properties of $A$ (for
example being semi-simple, local) determine any properties of the algebra $N$
arising from Method 1 using $A$?
###### Question 6.6.
Can Method 1 be generalised to be used for algebras over a general commutative
ring?
#### Acknowledgements
We are grateful to Michael Wemyss for suggesting his interesting questions
which inspired this paper, and for useful comments about contraction algebras
and their applications in geometry. The third author acknowledges support from
the EPSRC programme grant EP/R034826/1 and from the EPSRC research grant
EP/V008129/1.
## References
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* [BW22] Gavin Brown and Michael Wemyss. Local normal forms of noncommutative functions, 2022.
* [CGG89] Vincent Coll, Murray Gerstenhaber, and Anthony Giaquinto. An explicit deformation formula with noncommuting derivations. Israel Math. Conf. Proc., 1:396–403, 01 1989.
* [DW16] Will Donovan and Michael Wemyss. Noncommutative deformations and flops. Duke Mathematical Journal, 165(8), June 2016. arXiv:1309.0698 [math].
* [FO19] Boris Feigin and Alexandre Odesski. Flat deformations of algebras and functional equations. Journal of Combinatorial Algebra, 3(3):215–236, September 2019\.
* [HT18] Zheng Hua and Yukinobu Toda. Contraction algebra and invariants of singularities. Int. Math. Res. Not. IMRN, (10):3173–3198, 2018.
* [SW13] Anne V. Shepler and Sarah Witherspoon. PBW deformations of skew group algebras in positive characteristic, December 2013. arXiv:1312.3616 [math].
* [Tod15] Yukinobu Toda. Non-commutative width and Gopakumar-Vafa invariants. Manuscripta Math., 148(3-4):521–533, 2015.
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* [Wit19] Sarah J. Witherspoon. Hochschild cohomology for algebras, volume 204 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, [2019] ©2019.
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# Extracting non-local inter-polaron interactions from collisional dynamics
Junichi Takahashi Department of Electronic and Physical Systems, Waseda
University, Tokyo 169-8555, Japan Hiroyuki Tajima Department of Mathematics
and Physics, Kochi University, Kochi 780-8520, Japan Quantum Hadron Physics
Laboratory, RIKEN Nishina Center, Wako, Saitama 351-0198, Japan Eiji Nakano
Department of Mathematics and Physics, Kochi University, Kochi 780-8520, Japan
Kei Iida Department of Mathematics and Physics, Kochi University, Kochi
780-8520, Japan
###### Abstract
This study develops a novel experimental method of deducing the profile of
interaction induced between impurities in a trapped gas of ultracold
Fermi/Bose atoms, which are often referred to as Fermi/Bose polarons. In this
method, we consider a two-body Fermi/Bose polaron collision experiment in
which impurities and atoms interact only weakly. Numerical simulations of the
quantum dynamics reveal the possibility to obtain information regarding the
non-local induced interaction between two polarons from a measured profile of
the polaron wave packet at several snapshots. This is because the potential of
the induced interaction is well balanced by the quantum potential whenever the
WKB approximation for the relevant Schrödinger equation is applicable.
Introduction— To describe interparticle interactions is indispensable in
various fields of physics. At extremely high energy, fundamental interactions
are mediated by gauge bosons Book_Weinberg1995 , while in atomic nuclei, the
nuclear force at a distance occurs through exchange of mesons Taketani1951 .
In conventional superconductors, attractive electron–electron interactions are
induced by phonons Bardeen1957 . In high-$T_{\rm c}$ superconductors,
furthermore, background spin and/or charge fluctuations are expected to play a
role Scalapino1995 .
Ultracold atomic systems can be used as a platform for studying medium-induced
interactions. Particularly, such systems can exhibit high tunability and
pureness, thereby serving as excellent quantum simulators for studying quantum
many-body theories Bloch2008 ; Giorgini2008 ; Dalfovo1999 . For example, cold
atomic Fermi systems with a long scattering length close to the unitary limit
are good simulators with regard to low density neutron matter Horikoshi2019 .
They also possess the unique feature of having the real-time dynamics and
momentum distribution measured Pethick-Smith . Moreover, atomic Fermi/Bose
polarons, which are quasiparticles that consist of impurities (minority atoms)
immersed in a degenerate Fermi gas Schirotzek2009 ; Frohlich2011 ;
Kohstall2012 or a Bose-Einstein condensate Jorgensen2016 ; Hu2016 ;
Rentrop2016 , have been realized. Since the impurities have the mass and the
interaction with each other modified from those in vacuum by the impurity-
medium interaction, a great deal of research has been conducted both from
theoretical and experimental sides. Theoretically, the Ruderman-Kittel-Kasuya-
Yoshida (RKKY) type Ruderman1 ; Kasuya1 ; Yoshida1 and Yukawa/Efimov type
interactions have been proposed as typical examples of non-local induced
interactions between heavy Fermi and Bose polarons Nishida2009 ; Suchet2017 ;
Heiselberg2000 ; Yu2012 ; Enss2020 ; Naidon2018 , respectively. In connection
with such induced interactions, moreover, bi-polaron formation for Bose
polarons Naidon2018 ; Camacho-Guardian2018 ; Dehkharghani2018 ; Volosniev2015
; Mistakidis2019 and the energy shift and broadening of Fermi polarons
Giraud2012 ; Mistakidis2019c ; Tajima2018 have been actively discussed.
Recently, the presence of non-local interactions between Fermi polarons has
been experimentally confirmed DeSalvo2019 ; Edri2020 . Here, the formation of
solitons through the corruption of a Bose gas was found to be different from
the behavior predicted when only local interactions occur in a boson-fermion
mixture, which is an evidence for the existence of a non-local interaction
mediated by fermions. The non-local nature of such interactions, however,
remains to be investigated experimentally. Given that similar induced
interactions occur in various fields of physics, therefore, real cold atom
experiments in this direction would serve as a cornerstone.
We now address the question “What is enough information to find induced
interactions between particles in a medium?” For concreteness, we consider the
case in which two polarons are approximately described by a two-body time-
dependent Schrödinger equation (TDSE) with a mediated interaction instead of
dealing with the medium explicitly. In this case, the simplest answer is a
two-body polaron wavefunction. That is, the interaction can be obtained from
the wavefunction by transforming the TDSE as
$U(r)=\Psi^{-1}\left[i\hbar\frac{\partial}{\partial t}-h_{1}({\bf
x}_{1})-h_{2}({\bf x}_{2})\right]\Psi$, where $\Psi$ is the wavefunction,
$h_{i}$ denotes the one-particle Hamiltonian, $r=|{\bf x}_{1}-{\bf x}_{2}|$,
and $U$ denotes the interaction between two polarons. This means that one can
in principle constitute the inter-polaron interaction $U$ as function of the
relative coordinate if the $\Psi$ keeps nonvanishing in the range of
$U(r)\not=0$ throughout such experiments as the collisional ones Sommer2011 ;
Joseph2011 ; Valtolina2017 ; Reynold2020 . In reality, however, it is
difficult to simultaneously obtain the phase and amplitude of the wavefunction
by experiments. In fact, at a given snapshot, only the amplitude of the
wavefunction can be measured via the square root of the probability density of
the wavefunction. For trapped cold atomic systems, one can use, e.g., a wave
packet of a non-interacting Bose gas as an impurity and then identify the
number density of the Bose gas with the probability density of the impurity.
In this study, we propose a method of deducing the non-local interaction
between two such impurities in a medium from the density of the Bose gas at
several snapshots. It is noteworthy that the idea of deriving bulk
thermodynamic quantities from the density of a single-species cold atomic gas
has been already proposed Ho2010 . Despite similarity in strategy, there is a
crucial difference from our proposal. Authors in Ref. Ho2010 use the density
distribution to obtain the thermodynamic properties of the corresponding
homogeneous system within local density approximation (LDA), while information
about the non-local interaction between atoms in a trap, which is of interest
here, is out of the reach of LDA.
Formulation— For future possible experimental realizations, we focus on a two-
polaron collision in a harmonic trap filled with a Fermi/Bose atomic gas. Each
impurity, which is a Bose condensate wave packet, is initially localized by a
separate confining harmonic potential, released at $t=0$, and allowed to
collide with another impurity. Note that experimental setups that involve a
scenario highly similar to that considered in this study have been proposed
Mistakidis2019b ; Magierski2019 ; Kwasniok2020 ; Tajima2020 , while a part of
them have been realized in experiment by using optical tweezers Reynold2020 .
We assume that at $t=0$, each impurity has already been immersed in the
majority gas long enough to form a polaron and that at $t\geq 0$ a medium-
induced interaction occurs between the polarons. Furthermore, we consider the
polarons to be distinguishable particles that do not interact directly with
each other and to remain robust during the time evolution owing to coupling
between the impurity and the majority gas that is too weak for the entire
system to be relaxed. In a regime of strong coupling, on the other hand, a
hydrodynamic description looks more relevant. In fact, a hydrodynamic analysis
of collisions of two polaronic clouds in a similar setup has been performed
Tajima2020 ; in this case, induced inter-polaron interactions have been
incorporated even inside a cloud but only within the LDA.
In the weak coupling regime of interest here, the dynamics of the two polarons
essentially obeys the TDSE: $i\hbar\partial_{t}\Psi=[h({\bf x}_{1})+h({\bf
x}_{2})+U_{\rm med}(r)]\Psi$, where $h({\bf x}_{i})=-\hbar^{2}\nabla^{2}_{{\bf
x}_{i}}/2m+V_{\rm HO}({\bf x}_{i})$ with the trap potential $V_{\rm HO}({\bf
x}_{i})=m\omega^{2}{\bf x}_{i}^{2}/2$ and $U_{\rm med}(r)$ denotes an
interaction mediated by the majority gas. Here, for simplicity, we assume that
the trap frequency $\omega$ and the bare mass $m$ of an impurity atom are the
same between the polarons and that the dynamics is 1D. We next rewrite the
TDSE in the center-of-mass frame by using the relative coordinate
$x=x_{1}-x_{2}$ as $i\hbar\frac{\partial\varphi}{\partial
t}=\left[-\frac{\hbar^{2}}{2m_{\rm r}}\frac{\partial^{2}}{\partial
x^{2}}+V_{\rm r,HO}(x)+U_{\rm med}(|x|)\right]\varphi$ where $m_{\rm r}=m/2$
is the reduced mass of the impurities, $V_{\rm r,HO}(x)=\frac{1}{2}m_{\rm
r}\omega^{2}x^{2}$, and $\varphi(x,t)$ is the relative wavefunction of the
polarons. We then substitute the wavefunction
$\psi=\sqrt{\rho(x,t)}\exp[iS(x,t)/\hbar]$ into the TDSE, thereby obtaining
the coupled equations that are the equation of continuity and the quantum
Hamilton–Jacobi equation Book_Tannor2007
$\displaystyle C(S;x,t)+Q(\rho;x,t)+U_{\rm med}(|x|)=0,$ (1)
with $C(S;x,t)=\frac{\partial S}{\partial t}+\frac{1}{2m_{\rm
r}}\left(\frac{\partial S}{\partial x}\right)^{2}+V_{\rm r,HO}$ and
$Q(\rho:x,t)=-\frac{\hbar^{2}}{2m_{\rm
r}\sqrt{\rho}}\frac{\partial^{2}\sqrt{\rho}}{\partial x^{2}}$. Here, $C$ is
the potential given by the phase fluctuations, while $Q$ is the potential
given by the density fluctuaions, which is often referred to as the quantum
potential. Equation (1) shows that the interaction $U_{\rm med}$ is determined
by the balance of those potentials.
Analysis and Results — We proceed to solve the TDSE for a particular
interaction and obtain density profiles of a polaron to calculate the quantum
potential. Since each polaron is initially trapped in the respective confining
harmonic potential, we set the initial two-polaron wavefunction as a pair of
the Gaussian wave packets via
$\Psi(x_{1},x_{2},t=0)=\phi_{+}(x_{1})\phi_{-}(x_{2})$, with
$\phi_{\pm}(x)=\frac{1}{(2\pi\eta^{2})^{\frac{1}{4}}}\exp\left[-\frac{(x\pm
x_{0})^{2}}{4\eta^{2}}\right],$ where the two parameters $x_{0}$ and $\eta$
controls the initial position and the width of the respective wave packet. For
such nodeless wave packets, the WKB approximation is expected to be good. We
solve the TDSE under the corresponding initial condition
$\varphi(x,t=0)=\frac{1}{(4\pi\eta^{2})^{\frac{1}{4}}}\exp\left[-\frac{(x-2x_{0})^{2}}{8\eta^{2}}\right],$
by using the second-order split-step Fourier method. We adopt (i) RKKY, (ii)
Yukawa, and (iii) Efimov type interactions as $U_{\rm med}$. We note that the
interactions (i)–(iii), which are derived for a 3D homogeneous medium, are
used for simplicity and that the concrete form of the interactions is not
important for the present purpose of finding a way of capturing the mediated
interactions through experiments unless the range of the interactions is
significantly long.
(i) The RKKY type interaction is as follows:
$\displaystyle U_{\rm med}(|x|)=\frac{\hbar^{2}Ma^{2}}{2\pi M_{\rm
r}^{2}}\frac{2k_{{\rm F}}|x|\cos(2k_{\rm F}|x|)-\sin(2k_{\rm
F}|x|)}{|x|^{4}},$ (2)
where $a$ is the $s$-wave scattering length between a majority atom of mass
$M$ and an impurity, taken to be independent of which impurity the majority
atom interacts with, $k_{\rm F}$ is the Fermi wavenumber of the majority gas,
and $M_{\rm r}=mM/(m+M)$. Figure 1 presents the probability densities for the
relative two-polaron wave packet before and after the collision of a pair of
the polaron wave packets that evolve from various initial conditions under the
influence of $U_{\rm med}$. We set the number of atoms in the majority gas as
a typical one, $N=10^{5}$, while assuming $(k_{\rm F}a)^{2}=2$ and $M=m$. It
is interesting to note that before the collision, the width of the two-polaron
relative wave packet increases (decreases) with $t$ in the case of Fig. 1(d)
(Fig. 1(a)), while remaining almost unchanged with $t$ in the cases of Figs.
1(b) and 1(c). This behavior stems from the fact that the Gaussian wave packet
for a free particle with mass $m_{\rm r}$ has the width increased with time
like $\sqrt{2}\eta\sqrt{1+(\hbar t/4m_{\rm r}\eta^{2})^{2}}$, while, as two
polarons approach each other, the confining potential acts to reduce the width
of the wave packet like $\hbar/\sqrt{2m_{\rm r}(E-V_{\rm r,HO})}$ with the
relative energy $E$. Incidentally, the collision time is of order $t_{c}$
(quarter of the dipole oscillation period) in the cases of Figs. 1(a)–(c),
while being well before $t_{c}$ in the case of Fig. 1(d).
Figure 1: Probability density of the two-polaron relative wave packet (time
vs. relative position) for (a) $x_{0}/R_{\rm F}=0.5,\,\eta/R_{\rm F}=0.1$, (b)
$x_{0}/R_{\rm F}=0.5,\,\eta/R_{\rm F}=0.05$, (c) $x_{0}/R_{\rm
F}=0.25,\,\eta/R_{\rm F}=0.05$, and (d) $x_{0}/R_{\rm F}=0.25,\,\eta/R_{\rm
F}=0.01$. Here, $R_{\rm F}$ and $E_{\rm F}$ denote the Thomas–Fermi radius and
Fermi energy, respectively. The Fermi energy is given by $E_{\rm
F}=\frac{1}{2}m\omega^{2}R_{\rm F}^{2}=(6N)^{\frac{1}{3}}\hbar\omega$. The
time $t_{c}$ at which the center of the relative wave packet reaches $x=0$ can
be estimated from $t_{c}\omega_{\rm
F}\simeq\frac{1}{4}\frac{2\pi}{\omega}\omega_{\rm F}\simeq 125,$ where
$\omega_{\rm F}=E_{\rm F}/\hbar$. Figure 2: Relative density profile
$\rho(x,t)$ at three values of $t\omega_{\rm F}$ of order or greater than the
collision time. The distinction between (a)–(d) corresponds to the difference
in initial parameters as in Fig. 1. Figure 3: Same as Fig. 2 for minus the
quantum potential. For comparison, the RKKY interaction employed to solve the
TDSE is also plotted in dotted line. For detailed movies, see Animation .
Figure 2 depicts the relative density $\rho(x,t)$ at three different times,
most of which are close to the collision time extracted from Fig. 1. Numerical
data in the regions of $x<0$ and $x>0$ represent the behavior of the
transmitted component and of the incident and reflected components of the
relative wave packet, respectively. Once this kind of profiles are obtaind by
experiments, one can use them as input data in deriving the quantum potential.
We proceed to exhibit, in Fig. 3, minus the quantum potential $-Q(\rho;x,t)$
that can be derived from pseudo data for $\rho(x,t)$ shown in Fig. 2. One can
observe from Fig. 3 that the quantum potential in the transmitted wave regime
($x<0$) well reproduces the oscillating pattern of the RKKY interaction for
any initial condition, while the reproducibility of the RKKY interaction
itself depends on the initial condition. Comparing Figs. 3 (a)–(c), in which
cases the width of the transmitted relative wave packet just after the
collision is different, we find it advantageous for the transmitted wave
packet to be sufficiently spread. Remarkably, Fig. 3 (d) shows an even better
reproducibility of the interaction for the initial condition that allows the
two respective wave packets to spread and merge well before the centers of
these packets come together at the origin at a timescale $t_{c}$ of the dipole
oscillation. This supports the tendency that the wider the transmitted wave
packet, the higher the reproducibility of the interaction from the quantum
potential. This tendency in turn ensures $C(S;x,t)\simeq 0$ and $U_{\rm
med}\simeq-Q(\rho;x,t)$ in the region of $x<0$ where the WKB approximation is
a good approximation because of the sufficiently spread wave packet and gentle
spatial dependence of the confining potential.
In the region of $x>0$ in each panel of Figs. 1 and 2, on the other hand, a
density oscillation pattern emerges after the collision. This pattern stems
naturally from interference between the incident and reflected components of
the relative wave packet, while the presence of the reflected component arises
mainly from a repulsive part of $U_{\rm med}$. As can be seen from Fig. 3,
however, the periodicity of this density oscillation is totally different from
that of $U_{\rm med}$. In other words, such quantum interference leads
inevitably to $C\not\simeq 0$. Then, the WKB approximation is no longer valid,
and hence the mediated interaction cannot be reproduced by $-Q$.
Let us turn to the (ii) Yukawa and (iii) Efimov type interactions:
$\displaystyle U_{\rm
med}(|x|)=\begin{cases}U_{0}\frac{\exp(-\kappa_{0}|x|)}{\kappa_{0}|x|}&({\rm
Yukawa}\,{\rm type}),\\\ U_{1}\frac{\kappa_{1}^{-2}}{|x|^{2}}&({\rm
Efimov}\,{\rm type}).\end{cases}$ (3)
We again confine ourselves to sufficiently spread relative wave packets for
the WKB approximation to hold at least in the region of $x<0$. Figure 4 shows
minus the relative quantum potentials obtained by numerically simulating
collision dynamics of two Bose polarons that interact with (ii) or (iii). We
can observe that the quantum potential tends to well reproduce the respective
mediated interaction in the regime ($x<0$), just like the RKKY case shown in
Fig. 3. Comparing Figs. 4 (a) and (b), in which cases the initial position of
the relative wave packet is different under the same Yukawa interaction, one
can see that $U_{\rm med}$ is more reproducible in the former than in the
latter. On the other hand, comparing Figs. 4 (c) and (d), which are the Efimov
counterparts to Figs. 4 (a) and (b), one can realize that $U_{\rm med}$ is
more reproducible in the latter than in the former. These results imply that
tuning of the initial energy would be desirable for better reproduction of
$U_{\rm med}$ in a manner that depends on the range of the mediated
interaction. We remark in passing that in contrast to the RKKY case, almost no
density oscillation emerges in the region of $x>0$ for the Yukawa and Efimov
type interactions, whose purely attractive nature produces almost no reflected
wave packet. Then, the WKB approximation is expected to be valid even for
$x>0$. A significant deviation of $-Q$ from $U_{\rm med}$ that can be observed
in this region for the Efimov type interaction, therefore, suggests that the
long-range nature distorts the incident component of the relative wave packet
even at a semiclassical level.
Figure 4: Minus the quantum potential at three different times as plotted for
the Yukawa (upper panels) and Efimov (lower panels) type interactions with
$U_{0}/\hbar\omega=U_{1}/\hbar\omega=-0.1$ and $\kappa_{0}R_{\rm
HO}=\kappa_{1}R_{\rm HO}=20$, where $R_{\rm HO}=\sqrt{\hbar/m_{r}\omega}$. In
panels (a) and (c) [(b) and (d)], the initial parameters are set to
$x_{0}/R_{\rm HO}=0.25$ and $\eta/R_{\rm HO}=0.05$ ($x_{0}/R_{\rm HO}=1$ and
$\eta/R_{\rm HO}=0.05$). The dotted line in each panel denotes the mediated
interaction involved. For detailed movies, see Animation .
Summary— This study has developed a novel practical method of deducing the
induced interactions between two impurities from a measured profile of the
polaron wave packet at several snapshots of a two-polaron collisional dynamics
in the case in which impurities and medium atoms interact only weakly. The key
to success in this method is the validity of the WKB approximation, which is
satisfied for sufficiently spread polaron wave packets. We have successfully
demonstrated by solving the TDSE the possibility of reproducing such
interactions as the RKKY, Yukawa, and Efimov type from the measured profile of
the wave packets via the quantum potential.
In real possible experiments, one would be prepared to confine a degenerate
Fermi gas or Bose condensate with a sufficiently large trap and treat two non-
interacting Bose gases with different internal states as impurities by putting
each of them in a sufficiently narrow trap that is located at a symmetrical
position with respect to the large trap of a majority gas and then by
releasing the two Bose gases from the trap at the same time. We expect that
observation of transmitted wave packets after the collision would be possible
owing to the different internal states, which would help to distinguish
between two impurities. We emphasize that our strategy to deduce the mediated
interaction from the probability density of the two-polaron wavefunction is
different from the method of estimating the potential from a phase shift as
used in an inverse scattering method InverseScattering .
In general, the full profile of the medium-induced non-local interaction that
occurs between particles is theoretically unknown. In this regard, we believe
that the method proposed here could help to easily and quantitatively obtain
information regarding mediated interactions from experiments. It is
nevertheless foreseeable that in actual experiments the interaction cannot be
described by the type of mediated interactions used in this study due to
various effects such as the finite volume one. If the experiment is carried
out under appropriate initial conditions, however, the resultant deviation
between the empirically deduced interactions and the full interaction could be
duly reduced. Also, our proposed method might open an opportunity to study
nuclear interactions that are in principle microscopically known from QCD
Ishii2007 ; Ishii2012 ; Iritani2019 , from a different viewpoint that utilizes
a cold atomic system as a quantum simulator. As a next step, we will attempt
to test our method for 3D collision systems and investigate the applicability
to general interactions.
Acknowledgments— We would like to thank K. Nishimura, T. Hata, K. Ochi, and Y.
Yamanaka for useful discussions. This work was supported in part by Grants-in-
Aid for Scientific Research from JSPS (Nos. 17K05445, 18K03501, 18H05406,
18H01211, and 19K14619).
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Supplemental online material for: “Extracting non-local inter-polaron
interactions from collisional dynamics”
In the supplimental material, we show the list of movies.
List of movies:
* 1.
File: `RKKY_x0_050_eta_010.mp4`
Dynamics of the relative density and minus the quantum potential that
corresponds to Figs. 2(a) and 3(a).
Youtube: `https://youtu.be/By_ZALuI6A8`
* 2.
File: `RKKY_x0_050_eta_005.mp4`
Dynamics of the relative density and minus the quantum potential that
corresponds to Figs. 2(b) and 3(b).
Youtube: `https://youtu.be/dTKHjs05SfQ`
* 3.
File: `RKKY_x0_025_eta_005.mp4`
Dynamics of the relative density and minus the quantum potential that
corresponds to Figs. 2(c) and 3(c).
Youtube: `https://youtu.be/9iXiWWM9H6o`
* 4.
File: `RKKY_x0_025_eta_001.mp4`
Dynamics of the relative density and minus the quantum potential that
corresponds to Figs. 2(d) and 3(d).
Youtube: `https://youtu.be/Q74kZCrYWrY`
* 5.
File: `Yukawa_x0_025_eta_005.mp4`
Dynamics of the relative density and minus the quantum potential that
corresponds to Fig. 4(a).
Youtube: `https://youtu.be/nTRTT_8gGME`
* 6.
File: `Yukawa_x0_100_eta_005.mp4`
Dynamics of the relative density and minus the quantum potential that
corresponds to Fig. 4(b).
Youtube: `https://youtu.be/tJkNVh91SrY`
* 7.
File: `Efimov_x0_025_eta_005.mp4`
Dynamics of the relative density and minus the quantum potential that
corresponds to Fig. 4(c).
Youtube: `https://youtu.be/fK-G-StErAM`
* 8.
File: `Efimov_x0_100_eta_005.mp4`
Dynamics of the relative density and minus the quantum potential that
corresponds to Fig. 4(d).
Youtube: `https://youtu.be/2RqUnCu-Shg`
|
# An Exploration of Mars Colonization with Agent-Based Modeling
Edgar Arguello 111Authors are listed in alphabetical order. Sam Carter
Cristina Grieg Michael Hammer Chris Prather Clark Petri Anamaria Berea222
Department of Computational Social Sciences, George Mason University
## 1 Introduction
Establishing a human settlement on Mars is an incredibly complex engineering
problem. The inhospitable nature of the Martian environment requires any
habitat to be largely self-sustaining. Beyond mining a few basic minerals and
water [4], the colonizers will be dependent on earth resupply and
replenishment of necessities via technological means, i.e., splitting martian
water into oxygen for breathing and hydrogen for fuel [6].
Our goal is to better understand the interactions of future Martian colonists
through an Agent-Based Modeling (ABM) approach. Accounting for engineering and
technological limitations, we draw on research regarding high performing teams
in isolated and high stress environments (ex: submarines, Arctic exploration,
war) to include psychological components within the ABM. Interactions between
agents with different psychological profiles are modeled at the individual
level while global events such as accidents or delays in earth resupply affect
the colony as a whole.
## 2 Space Settlements Literature Review
### 2.1 Use of Agent-Based Models to Simulate Small-Scale Communities
Given the many facets to the problem, a complex systems approach is
appropriate to extend our insights and avoid uninformed predictions later
[10]. ABMs may prove useful to help understand the ecological and demographic
conditions needed for sustainable resource-management in a constrained human
habitat [10]. It has been demonstrated that integrating Geographic Information
System (GIS) overlays and multi-agent simulation provides a means to explore
group behaviors which may be affected by an agent’s spatial knowledge of their
terrain [16]. ABMs provide useful “laboratories” for competing hypotheses to
identify rules of behavior that lead to group dynamics [16], which we would
then like to apply to our simulation of a potential Mars settlement.
### 2.2 Team and Interpersonal Psychology
The modern era of teams working in extreme conditions commenced with the
advent of nuclear submarines in the 1950s, the space race of the 1960s, and
expanded Arctic exploration of the 1900s [14]. These pursuits specifically
buoyed interest in how teams perform in extreme environments, what makes them
successful, and how individuals cope with the challenges presented by such
conditions. The technical sophistication necessary for persistent survival in
submarine missions or space exploration, coupled with their isolation and
inherent risks, position those endeavors as proxies for initializing a martian
colony ABM.
In a study of Antarctic exploration teams, researchers found that
psychological discomfort that occurred was limited to the individual or
between individuals levels and not the group as a whole [8]. Furthermore, the
researchers identified a steady linear increase in group tension over the
course of the mission. The interplay of individual interactions as well as a
broad and continual drain on psychological well-being, albeit minor, are
concepts which underpin much of our psychological modeling. It should be noted
that specific personality profiles as determined by psychological examination
were predictive of interpersonal cohesion in some instances [8], though these
considerations are an area for future research.
Attempts to connect quantitative biomarkers such as cortisol to psychological
examination performance while operating in extreme environments have shown
inconclusive results [18]. However, the same study did identify overarching
coping mechanisms of a submarine crew. While these mechanisms showed no
correlation to the personality profiles, they were beneficial. Traits such as
strong achievement motivation and interpersonal orientation were associated
with superior coping during submarine missions [18]. In another study of
submarine service, optimism, humor accompanied by cynicism, and a strong
perception of the submarine service were the primary coping mechanisms
exhibited by the crew [9]. Additionally, having some semblance of occasional
privacy with their own bed was a highly valuable coping mechanism, a
consideration for future research with more granular spatial modeling.
Research conducted on Soldiers and Marines during their deployment to Iraq
demonstrated that the intensity of combat situations is the primary
determinant of behavioral health challenges [2]. Other deployment-issues, such
as their length and frequency, also impacted the psychological health of the
studied Soldiers and Marines, albeit to a lesser extent than direct combat.
The present literature demonstrates that team and individual psychological
success in extreme environments can be broadly attributed to coping capacity,
which we define as the ability of people, organizations and systems, using
available skills and resources, to manage adverse conditions, risk or
disasters. Within the model, coping capacity is aggregated from a series of
sub-categories and quantitatively incorporated on a Cronbach alpha scale for
each individual agent. Cronbach’s alpha scoring is drawn from the
psychological changes in a 100-Day remote field groups study, where
researchers examined two Australian teams traversing across the Lambert
Glacier Basin, paying close attention to personality characteristics,
environmental factors, and interpersonal factors as predictors of Group
Tensions, Personal Morale, Emotional State, Cognitive Readiness, and the
Team’s Work-Life [8].
The agents within the model are categorized into four separate personality
categories: Agreeable, Social, Reactive, and Neurotic. Their coping ability
and resilience are linked to personality profiles typified by strong
instrumentality and achievement motivation combined with interpersonal
sensitivity. The four categories are drawn from the Positive Instrumental-
Expressive, the Hostile, Competitive Interpersonal orientation, and the low
achievement motivation combined with passive-aggressive characteristics linked
to Personality Characteristics Inventory (PCI) profiles which NASA has used
for screening, monitoring, and evaluating subjects in isolated and confined
extreme (ICE) environments [18]. The categorical measurements are as follows:
* •
Agreeables - Individuals with the lowest degree of competitiveness, low
aggressiveness, and not fixated of stringent routine. Cronbach Alpha score
starting at 0.97
* •
Socials - Individuals with a medium degree of competitiveness, extroverted,
require social interaction, but are not fixated on stringent routines.
Cronbach Alpha score starting at 0.94
* •
Reactives - Individuals with a medium degree of competitiveness, competitive
interpersonal orientation, and fixated on stringent routines. Cronbach Alpha
score starting at 0.89
* •
Neurotics - Individuals with a high degree of competitiveness, highly
aggressive interpersonal characteristics, and challenged ability to adapt to
boredom or a change in routine. Cronbach Alpha score starting at 0.84
Additionally, each agent is granted skills associated with their civilian and
military occupational specialties consistent with NASA’s Human Factors and
Behavioral Performance Element research, which analyzed the abilities that are
generalizable across circumstances and crew roles and those that will be
required by all crew members during a 30-month expedition to Mars [1].
The skills are broken up into two categories; Management (Skill 1) and
Engineer (Skill 2), which is extrapolated from NASA’s more comprehensive chart
of Leader, Pilot, Physician, Biologist, Geologist, Computer, Electrician,
Mechanic, and Combined [1]. The psychological structure of the model is
explored in more detail in section 3.
### 2.3 Martian Environment and Local Economy
Our challenge is to examine the unique institutional conditions required to
manage a settlement at such distance from Earth. In particular, we sought to
address the following questions: What conditions are needed to maintain a
stable colony on Mars? What combination of personality types would do best in
this hostile environment? How many resources are needed for two years between
resupplies and assuming occasional accidents? To tackle these questions, we
examined research relating to similar questions in four contexts: existing
ABMs, economic and related literature, ISS and Antarctica outpost data, and
agricultural research.
Collectively, the research suggests important design principles for Martian
economic modeling. Economic modeling of a remote settlement involves modeling
the external dependencies (food, etc.) and the internal labor and trade
economy of the settlement itself. Food is the single largest external
dependency. For example, food shipments arrive at the McMurtry and other
Antarctica settlements only once annually, so this provides us with a starting
point for calculating the estimated bi-annual shipments to Mars. Consequently,
the frequency and size of external food shipment, as well as the degree to
which a Mars settlement would attempt to grow food, are essential variables to
calibrate to model a sustainable Mars settlement.
Second, waste disposal and recycling (including carbon dioxide from humans and
other types of emissions from plants) are costs that must be included. Our
current focus is on supporting a stable population on Mars, so waste disposal,
recycling, economic gain from mining minerals, and other second order
considerations were left for future work.
Lastly, the degree of task differentiation in a settlement will directly
correlate to the extent that labor and food shipments generate an internal
bartering economy. Task differentiation is itself largely driven by the size
of the settlement, which is also critical to other parts of the economic
model. Developing a Martian economy is also left for future work.
### 2.4 External Martian Economy
To examine how a Martian economy might function outside of a colony, we looked
at other harsh environments with possible mineral deposits; the oceans and
Antarctica. Various countries have exploited mineral deposits in the oceans
before there were laws in place to govern mining or for the protection of
marine vegetation and wildlife. There is a call now to bring together the
fragmented efforts and explicitly address the protection of ocean
ecosystems.[12]
Exploration of Antarctica has gone differently given the lack of obvious
mineral deposits and the harsher environment. After the initial exploration
period during colonial times, there was a scientific period when all activity
on Antarctica was frozen and all signers of the Antarctic Treaty agreed to
collaborative, scientific exploration and experiments were allowed.[17] With
the possibility of mineral discovery, efforts are now underway to expand on
the original treaty to address additional uses.[3]
### 2.5 Energy Sources in Space
Energy sources are another key consideration for colonization. Solar power is
limited by daylight hours, seasonal variability, and dust accumulation on
solar panels.[11] We chose to model our colony on the nuclear power generator
employed by NASA and the U.S. Department of Energy for the Perseverance rover
in 2011. The Perseverance uses a Multi-Mission Radioisotope Thermoelectric
Generator (MMRTG) power system to provide a steady supply of electricity. The
MMRTG is expected to operate for at least 14 years, which is more than seven
times longer than the prime mission.[11]
Nuclear fission can be utilized to provide surface power on the moon and Mars,
and small, light fission reactors could provide up to 10 kilowatts of
electrical power for at least 10 years, according to a NASA website. This
would provide enough electricity to power several households. Together NASA
and DOE plan to design and test one of these low-enriched uranium systems on
the moon by the late 2020s.[5] For our Mars settlement we will assume a safe,
stable nuclear power generator.
### 2.6 Radiation
International space scientists have written on the threat particle radiation
poses to future human colonists of Mars. After four years on Mars, colonists
would be exposed to a level of radiation above what is assessed safe for
humans, according to the article.[19]Additionally the mission should depart
for Mars when solar activity is at its peak (the solar maximum), because the
increased solar activity deflects the most dangerous particles from distant
galaxies.
## 3 ABM description and ODD
### 3.1 KEY ASSUMPTIONS
Due to the complexity of the model, a variety of base assumptions were made:
1) Assume the Mars colony has already been constructed, that food, air and
water are being produced locally and will not require a start-up time to
create. 2) Assume a nuclear generator, similar to the Perseverance power
source, has been installed and the habitat has a steady electricity source for
at least 7 years.
### 3.2 PROBLEM FORMULATION
The main goal of the model is to simulate a theoretical habitat environment on
Mars for a colony carrying out the mining of minerals to be sent back to
Earth. There is an emphasis on the mental state of each Martian agent, since
that directly impacts the success of the mission.
### 3.3 MODEL DESCRIPTION
1\. Purpose
The agent-based model MARS-COLONY is intended to provide insights into the
possible scenarios that might develop from varying initial population sizes,
personality type and skill levels of colonists sent to run a mining colony for
minerals. The goal is to aid decision-makers with planning by highlighting
potential pit falls. To make the model as realistic as possible, research is
used on the studied coping capacities of common personality types, and random
accidents are introduced both transporting food from Earth, as well habitat
disasters on Mars that also lower food supplies. Accidents not only threaten
food supplies, but considerably increase stress levels of colony members.
The model was developed using NetLogo 6.2.
2\. Entities, State Variables, Scales There are two collectives and two agent
types in the model. The first agent type is the Martian, which represents a
single human member of the settlement. The entire group of Martian agents form
the collective “Martians”. The second agent type is the stressor, which can be
triggered by a settlement habitat accident or an earth shipping disaster. When
triggered, the stressor interacts with the Martians in the settlement to
reduce their coping capacity and health. A stressor can dissipate over time,
or can be removed by the Martian agents if they succeed a skills check to
recover from the habitat accident. Table 1 lists the attributes and starting
attribute values for Martian agents, and Table 2 lists the attribute and
starting values for stressor agents.
Attribute Name | Description | Starting Value (or Range)
---|---|---
coping-capacity | psychological coping capacity of agents | 0.84 – 0.98
resilience | coping category | “neurotic”, “reactive”, “social”, or “agreeable”
skill-1 | value of skill 1 | 0 - 100
skill-2 | value of skill 2 | 0 - 100
health | agent health value | 100
food | food available for the given agent | 1
air | air available for the given agent | 1
water | water available for the given agent | 1
waste | waste produced by the given agent | 1
partner | the martian with which the agent is partnered, or ”nobody” if not partnered | nobody
taskmate-food | taskmate for food production | nobody
taskmate-water | taskmate for water production | nobody
taskmate-air | taskmate for air production | nobody
taskmate-accident | taskmate for habitat accident recovery | nobody
taskmate-waste | taskmate for waste removal | nobody
Table 1: Agent Attributes: Martian Attribute Name | Description | Starting Value (or Range)
---|---|---
identity | two types of stressor identity externalities | “Shipping” or “Habitat”
Table 2: Entity Attributes: Stressor Figure 1: Martian Attributes
The settlement acts as additional entity, capable of storing air, food, water,
and waste, and having a set production capacity for these resources. Martian
agents interact with the settlement by working together to produce resources,
store resources, and consume settlement resources. The settlement is a 50x50
grid, with each grid cell having an individual resource production capacity,
but collective and unlimited storage. The geospatial location of each Martian
agent is randomly assigned during model initialization, and agents move
randomly around the grid. Table 3 lists the attributes and starting values for
the Martian settlement.
Attribute Name | Description | Starting Value (or Range)
---|---|---
p-food | the rate at which food, on average, can be grown per time step in each patch of the settlement | 0.5
p-air | air recycled per patch of the settlement | 5.88
p-water | water recycled per patch of the settlement | 28
p-waste | non-recyclable waste (i.e., fecal matter, plant emissions) processed per patch of the settlement | 0
settlement-air | total air available for settlement | 5.88 * number-of-martians * 156
settlement-water | total water available for settlement | 28 * number-of-martians * 156
settlement-food | total food available for settlement | 10.5 * number-of-martians * 156
settlement-waste | total waste produced by settlement | 0
settlement-minerals | total minerals available for settlement | 0
Table 3: Entity Attributes: Settlement
The environment is the final entity, which determines the probability of a
habitat accident, earth shipment frequency and success probability, time
scale, technological efficiency, and production skill requirements. Table 4
lists the environmental attributes and starting values.
Attribute Name | Description | Starting Value (or Range)
---|---|---
ticks | time step for model | 1 tick = 1 week
minerals-shipment | amount of minerals received in each shipment from Earth | 100
food-shipment | amount of food received in each shipment from Earth | 10.5 * number-of-martians * 78
shipment-frequency | frequency of shipments from Earth | 78 ticks
technology | technological efficiency | 0.5
energy | energy (currently unused) | 1.0
food-s1 | amount of skill 1 required to produce food | 0 - 100
food-s2 | amount of skill 2 required to produce food | 0 - 100
water-s1 | amount of skill 1 required to produce water | 0 - 100
water-s2 | amount of skill 2 required to produce water | 0 - 100
air-s1 | amount of skill 1 required to produce air | 0 - 100
air-s2 | amount of skill 2 required to produce air | 0 - 100
waste-s1 | amount of skill 1 required to produce waste | 0 - 100
waste-s2 | amount of skill 2 required to produce waste | 0 - 100
accident-s1 | amount of skill 1 required to recover from a habitat accident | 0 - 100
accident-s2 | amount of skill 2 required to recover from a habitat accident | 0 - 100
Table 4: Entity Attributes: Environment
3\. Process overview and scheduling The only user input parameter currently in
the model is the initial number of Martians, which ranges from 4 – 152, in
increments of 4. When the model is initialized, the total number of Martians
are divided equally between four possible psychological categories: neurotic,
reactive, social, and agreeable. Martians, Stressors, the Settlement, and the
Environment are initialized with the values listed in Tables 1 – 4.
At each tick, Martians are able to move, sleep, team up with other Martians,
produce resources, consume resources, and engage socially. To move, the
Martians face a random direction and move forward one grid cell. Next, they
receive additional health from sleeping, with total health capped at 100.
Martians are then able to partner with other Martians to perform a skills
check for production. If the skills check is successful, they produce food,
water, and air, each with a unique skills check. Waste is produced as a by-
product of successful production. After resources are produced, they are then
consumed. Each Martian has a weekly requirement for food, water, and air. If
an individual Martian produces enough resources to meet this requirement, they
consume the defined amount and any remainder is contributed to the settlement
resource pool. If the Martian did not produce enough resources to meet the
weekly requirement, they consume the required resource amount from the
settlement resource pool. If the settlement resource pool does not have enough
resources to meet a Martian’s weekly requirement, the Martian’s health level
is reduced. Once the Martian’s health reaches zero, the Martian dies and is
removed from the model. Additionally, there is a small random chance of death
for each Martian to represent unforeseen mortality.
The settlement restores its production capacity for each grid cell and may
consume resources at each tick. Martian production is limited by the total
capacity of each settlement grid cell, and reduces the amount of producible
resources on that cell during the time step. At the next time step, the
resources are renewed to maintain a set weekly production capacity. The
settlement may also consume minerals, which are received from earth shipment,
to improve technological efficiency, which then improves Martian production
capability.
At each time step, the environment has a chance to add new Martians,
experience an earth shipping disaster, receive an earth shipment, and to
experience a habitat accident. Every 78 weeks, there is a chance that 4 new
Martians may be added to the colony from thr resupply shuttle, with a random
psychological category assigned upon addition. Also at 78 weeks, an earth
shipment may be received or there is a small random chance that an earth
shipping disaster may occur. If the earth shipment is received, the settlement
receives food and minerals, and improves technological efficiency. If an earth
shipping disaster occurs, the shipment is not received, and an earth shipping
disaster stressor is generated. There is also a small random chance of a
habitat accident occurring at each time step. If a habitat accident does
occur, it randomly affects either the food, water, air, or mineral settlement
resource supplies, and reduces the selected resource by half. It also
generates a habitat accident stressor.
If a stressor is generated, it has a chance to interact with Martians or to
dissipate over time. Habitat accident stressors persist until they dissipate
or are removed by a successful Martian skill check. While active, they
continue to reduce the settlement resource supplies. After 4 time steps, the
habitat accident stressor will dissipate and no longer affect the settlement
resource supplies.
4\. Design Concept
_Basic Principles_ The model was developed to explore the psychological,
social, technological, economic, and logistical factors that would influence
the long-term viability of a human Martian settlement. Martian settlers
interact with each other while producing and consuming resources, which
affects their overall health. They are divided into four personality types
that determine the degree and direction of this effect. Each Martian also
possesses a skillset that determines their success in producing resources for
themselves and the settlement. The settlement is supplied by regular shipments
from earth, which are susceptible to shipping disasters, and the settlement
can experience a habitat accident, both of which negatively influence
settlement supplies and overall Martian health.
_Emergence_ The primary observed emergent phenomenon occurs in the decline of
the Martian population. While the members of the settlement have an equal
probability of being affected by lack of settlement resources, habitat
accidents, or earth shipping disasters, Martians with the “neurotic”
psychology die at a much higher rate than those of other psychologies, and
once their population reaches a low enough level, the settlement population
stabilizes. Martians with the neurotic psychology and a high coping capacity
benefit the least from interaction with other Martians, and are penalized the
most if they have a low coping capacity. Our results suggest that this effect
is a driver of the Martian population decline, and once minimized or removed,
can produce a stable settlement.
_Adaptation_ The Martian agents do not have explicit adaptation, but do have
implicit adaptation in the use of their assigned skills. During each time
step, the model seeks to partner two Martian agents whose skills sum to the
required production threshold for resources or for habitat accident recovery
in the case of a habitat accident. This represents the idea that Martian
settlers would work together based on their actual skills to accomplish
various tasks needed to provide for the settlement, and would adapt as needed
for these tasks.
_Objectives_ The overall objective of the model is to produce a long-term
stable Martian population. Each Martian’s objective is to partner with other
Martians to produce resources and recover from habitat accidents. If Martians
are unable to produce sufficient resources, the population will begin to
decline, making future production more difficult due to a decline in the
available skills required for production.
_Sensing_ Martian agents move around the grid space and are able to sense
other agents within 3 grid spaces of themselves. If there are one or more
Martian agents within this range, a random agent is selected as a partner, and
the agents interact to increase or decrease overall health and coping
capacity, depending on current coping capacity and each agent’s psychological
category.
_Interaction_ In addition to interacting with agents in close spatial
proximity, as described in the section above, Martian agents also interact to
produce resources, recover from habitat accidents, and remove waste from the
settlement. Pairs of agents are selected by the model based on their skill
levels to achieve a set productivity threshold, which results in successful
achievement of these activities. If a pair cannot be made to reach the
threshold, the unpaired agents are not successful in production, recovery, or
waste removal. Stressor agents interact with the settlement to reduce resource
supplies or prevent an earth shipment from resupplying food and minerals.
_Stochasticity_ Stochasticity appears in the assignment of agent skill levels
and coping capacity, Martian agent movement, Martian death, Martian agent
addition, earth shipping disaster, habitat accidents, and Martian agent
teaming for production and interaction. When Martian agents are generated,
levels for their skills are assigned from a random uniform distribution for
the first skill, and the second skill then sums with the first to equal one
hundred. Martian agents move randomly around the grid space, and face a small
random probability of death at each time step. There are also random
probabilities of Martian agents being added to the model, the occurrence of an
earth shipping disaster when a shipment is due to arrive, or a habitat
accident occurring, with the type of resource affected by the habitat accident
also randomly selected. When agents pair up for resource production, habitat
accident recovery, or waste removal, the pairing is random given the pair
reaches the production threshold requirement. If agents are within three grid
cells of each other, they can randomly be assigned to interact to affect
health and coping capacity.
_Collectives_ Martian agents form a settlement collective, drawing on a pool
of shared resources and producing an overall amount of settlement waste. If an
individual agent is unable to produce enough resources for their own needs in
a given time step, they may draw on the collective settlement resources. If
they produce excess resources, they are contributed to the settlement resource
pool.
_Observation_ Outputs collected from the model for observation include coping
capacity statistics, number of Martian agents, total settlement resources,
shipments received, shipping disasters and habitat accidents experienced,
population coping and resilience, overall Martian health, and time elapsed.
5\. Initialization
The setup procedure imports a Martian landscape of geographic files. The
global variables are then checked for total water, food and air for the
habitat. Also total energy and technology levels needed to run the habitat and
mine minerals. Additional parameters are assigned for the frequency of
shipment and habitat accidents. Every 78 ticks there is also a slight chance
of a birth or death in the community. The individual Martian attributes are
then calculated by dividing resources equally, and randomly assigning
personality types and skill.
6\. Input Data Geographic Information System (GIS) files of the Mars landscape
have been loaded for awareness, but no input data is used in this model. Mars
exploration is still in the early stages and applicable data is not yet
available.
### 3.4 SETTING PARAMETERS
One of the challenges of agent-based modeling is determining how to set the
initial parameters for the model. Empirical data are often a source of initial
parameters, even in far-flung scenarios such as past civilizations or extra-
planetary settlements. Our model draws upon a number of sources to set our
parameters.
Global Variables. The Earth shipment and shipment frequency variables are
discussed above. This discussion also excludes shipping and accident counter
variables, which are discussed under results. The other global variables
include the settlement’s total resource stockpiles (settlement-air,
settlement-food, and settlement-water). The values of the parameters are
primarily derived from a review of the resource storage and production
capabilities of the International Space Station (ISS). NASA states that a
single ISS resident consumes 5.88kg of air, 28L of water, and 10.5kg of food
in a given week.[7][13][15] Our initial settlement-food, settlement-air, and
settlement-water assume a ramp-up, pre-settlement series of deliveries that
stockpile supplies equal to two earth shipment cycles (156 weeks/time steps).
However, there are proven technologies from the ISS to allow for the
sustainable production of air and water. Conversely, there is as yet no method
for the sustainable production of food in low gravity. As a result, we assume
that earth shipments will deliver food but not water or air. Earth shipments
deliver sufficient food for one additional earth shipment cycle (10.5kg of
food per settler per week).
Our global variables also define several other rates necessary to the function
of the settlement. The technology parameter sets a rate of aggregate
technological efficiency. This term serves as a modifier to the amount of
production generated by labor in the settlement. Technology is initially set
at 0.5
For this early version of the model, energy serves as a similar term that
specifies the amount of resource consumption required for the settlement to
function. This placeholder value is set at 1.0.
Lastly, the global variables set the skill levels necessary for settlement
production functions. An assumed score of 100 is needed across two sets of
skills (-s1 and -s2) to successfully accomplish each of four tasks (food,
water, air production, and accident recovery). Each pair of scores related to
resource production is set with the same values. For example, food-s1 is set
as a random integer between 0 and 100. Food-s2 is set as the difference
between 100 and food-s1 (100 - food-s1). Accident recovery sets both both
accident-s1 and accident-s2 as a random integer from 0 to 100. This represents
the ex ante uncertainty about what types of skills may be needed for a given
emergency. It is important to remember that these values set the required
skill score that must be met for successful production; they do not guarantee
that settlers will have these skills.
Settler Variables. Settlers are assigned two skills. Skill 1 is set as a
random number from 0 to 100. Skill 2 is set as 100 - Skill 1, such that each
settler has a total skill level of 100. Settlers have a partner variable (all
settlers begin unpartnered) and indicator variables that store their task
assignment.
Each settler is created with one of four resilience types: nuerotic, reactive,
social, and agreeable. Each of these resilience types are assigned a coping
score based on our research in team function and group theory. These coping
scores are 0.84, 0.89, 0.94, and 0.98. Coping scores determine how settlers
continue to produce and consume resources after adverse events such as a
habitat accident.
Settlement (Patch) Variables. Patch variables in this model represent the
modest estimated amount of food production that could be generated in a garden
or farm (0.5 kg per patch per time step), the WRPS + WTS air production system
(5.88kg per patch per time step), and OPS water recycling capacity (28L per
patch per time step). These production levels are activated in the production
functions only with the requisite levels of settler skill levels, as discussed
above.
Exogenous Variables. This model contains only two exogenous variables: the
starting number of settlers and the number of time steps in a given model run.
### 3.5 IMPLEMENTATION VERIFICATION
Our code shows cyclical variability in the population sizes and is able to
stabilize. The code appears to run correctly, the colonists have plenty of
resources, and we are able to focus on the interaction between the different
personality types. However we are unable to validate the values since there
has not been an actual Mars colony we can compare data to so model data serves
largely to highlight problems for consideration.
## 4 Scenarios for a Stable Mars Settlement
### 4.1 A Stable Population Size
Our goal was to see which initial population sizes lead to a stable colony
size. We did 5 runs of our model, for 28 Earth years, varying the initial
population size from 10 to 170 by steps of 10. Given that there are 4 critical
tasks that are needed continuously (air, water, food production and waste
removal) in addition to handling disasters, and two skills needed for each
task, we chose a population size of 10 as the minimum needed for a ”stable”
colony size. The population is allowed to dip below 10 as long as it bounces
back within 1.5 years, or the amount of time between Earth resupply shuttles.
A plot of our runs below shows that all initial population sizes above 50 were
able to maintain a population above 10 for all time steps. We will therefore
only consider initial populations below 50 as we try to determine a minimum
initial population which is able to maintain, or bounce back quickly to, a
stable colony size equal to or greater than 10 for all 28 years.
Figure 2: Variation in number of agents at each time Step (one week) in the
simulation.
Since our 4 personality types are equally distributed at setup, for our focus
on smaller initial populations we started at 10 (the minimum for a stable
colony), then increased to 50 with a step size of 4. Each time tick in our
model represents a week, so 1.5 years is equivalent to 84 ticks.
For an initial population of 10, the plot below shows the population sizes
over 28 years with prolonged time periods when the population drops below 10
Figure 3: Population sustainability with a start of population = 10.
For an initial population size of 22, the plot below show the population sizes
over 28 years remains above 10, meeting our definition of a stable colony.
Figure 4: Population sustainability with a start of population = 22.
A summary of which small initial population sizes resulted in stable colonies
over 28 years, in 5 runs, is given in the table below. For the purpose of this
model, an initial population of 22 was the minimum required to maintain a
viable colony size over the long run.
Initial Population Size | Does Population Size Bounce Back Above 10 or Not
---|---
10 | No Bounce Back
14 | No Bounce Back
18 | No Bounce Back
22 | Successful Bounce Back
26 | No Bounce Back
30 | Successful Bounce Back
34 | Successful Bounce Back
38 | No Bounce Back
42 | Successful Bounce Back
46 | Successful Bounce Back
50 | Successful Bounce Back
Table 5: Small Initial Populations Which Do and Do Not Lead to Stable Colonies
### 4.2 An Agreeable Personality Type Does Best
In all runs, the Agreeable personality type was the only one to survive the
full duration of model runs. This is likely because it has the highest coping
capability, and after long periods of time every agent has been exposed to a
series of stressor interactions, as well as space and habitat accidents. The
picture of the MARS-COLONY NetLogo model below includes a visualization of
total personality types over time, and the yellow Agreeable personality type
is most resilient, the Neurotic personality type was the most likely to fail,
the Reactive and Social personality types alternated in between. While this
model assigns equal numbers of each personality type, future work could try
adjusting the proportion of each to possibly lead to a lower required minimum
initial population. For example, a crew of all Agreeable personalities may be
more successful.
Figure 5: Screenshot of the simulation in NetLogo.
### 4.3 Maintaining Sufficient Resources Despite Accidents
Our data analysis shows that our food supplies are more than sufficiently
resupplied with earth shipments, for the majority of time steps, even in
scenarios where there are shipping disasters. Since the focus of this study
was to examine the interplay of the 4 personality types we did not assume any
production of air, water or food on the colony. Instead we provided multiple
years of initial supplies, with resupplies on incoming shuttles. Similarly we
were not concerned with the amount of waste generated by the colonists. Future
work can focus on this area of the model and explore the trade off between
resources needed from Earth, and amounts that could be generated on Mars. This
is a vital component to establishing a stable colony which adds to the
complexity of the model.
## 5 Conclusion
In summary, the MARS-COLONY Agent-Based Model offers a preliminary look at
necessary conditions to establish a stable mining colony on Mars. The main
focus is on the personality types of colonists selected and how they perform
throughout their time on Mars, using their skills to mine minerals and react
to random resupply shuttle accidents or habitat disasters. The stress caused
by accidents, as well as from interacting with other colonists, takes a toll
and Agreeable personality types were assessed to be the most enduring for the
long term, whereas Neurotics showed least adaptation capacity. This study
assumed a steady available nuclear energy supply and sufficient initial air,
water and food with sufficient resupply in order to focus on personality and
mental health of colonists, however those are recommended areas of future
work.
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|
# Game of arrivals at a two queue network with heterogeneous customer routes
Agniv Bandyopadhyay
School of Technology and Computer Science
Tata Institute of Fundamental Research
Mumbai, India
&Sandeep Juneja
School of Technology and Computer Science
Tata Institute of Fundamental Research
Mumbai, India
###### Abstract
We consider a queuing network that opens at a specified time, where customers
are non-atomic and belong to different classes. Each class has its own route,
and as is typical in the literature, the costs are a linear function of
waiting and service completion time. We restrict ourselves to a two class, two
queue network: this simplification is well motivated as the diversity in
solution structure as a function of problem parameters is substantial even in
this simple setting (e.g., a specific routing structure involves eight
different regimes), suggesting a combinatorial blow up as the number of
queues, routes and customer classes increase. We identify the unique Nash
equilibrium customer arrival profile when the customer linear cost preferences
are different. This profile is a function of problem parameters including the
size of each class, service rates at each queue, and customer cost
preferences. When customer cost preferences match, under certain parametric
settings, the equilibrium arrival profiles may not be unique and may lie in a
convex set. We further make a surprising observation that in some parametric
settings, customers in one class may arrive in disjoint intervals. Further,
the two classes may arrive in contiguous intervals or in overlapping
intervals, and at varying rates within an interval, depending upon the problem
parameters.
_K_ eywords strategic arrivals $\cdot$ queuing network games $\cdot$
population games.
## 1 Introduction
Queueing games or games where strategic customers, served by a queue or a
queuing network, decide on actions such as which queue to join, whether to
join, when to join, what level of priority to select and so on, are well
studied in the literature (see [6, 7, 9] for surveys). In this paper we focus
on the queuing arrival game where customers decide on when to join a queuing
network. This is in contrast to much of the literature that considers arrival
games to a single queue (see, e.g., [2, 5, 8, 9, 12, 13, 15]). Applications of
such arrival games to multiple queues are many: Customers arriving to a health
facility where they queue up to meet a doctor and then may queue to get tests
done and/or procure medicines; in banks where different customers may need to
go to different and multiple counters; in cafeteria where customers queue up
for different and multiple food items, and so on. In consonance with much of
the existing literature, we model customers as non-atomic ‘infinitesimal’
fluid particles with costs that are linear in waiting time and in time to
service, customers are served in a first-come-first-serve manner and service
facility opens at time zero (see [11, 12, 15]). In [13], uniqueness of
equilibrium solution was proven in a single queue setting with stochastic
service rates and large no of users. Moreover, [13] showed that, the
equilibrium solution with a large no of users is well approximated by the
corresponding fluid system and thus lending support to the fluid analysis.
This fluid setting has resulted in unique and elegant, easily calculable
customer equilibrium profiles for single queues as well as certain symmetric
queuing networks where customers have homogeneous travel routes (see [11,
12]).
To keep the discussion simple we focus on a two queue, two class customer
setting where each class of customers has a distinct route, and customers in
each queue are served in a first-come-first-served manner. While this set-up
may be practically interesting, our main contributions are theoretical: Our
key aim is to test whether the elegance and simplicity of customer equilibrium
behaviour to a single queue extends to more general queuing networks in
presence of heterogeneous routing.
Even in the simple two queue setting, we see that, unlike for the single
queue, here the solution structure and order of arrivals in equilibrium is a
function of all the problem parameters, i.e., linear coefficients of the cost
function, the queue service rates and the population size of each class of
customers. For one set of customer travel routes we observe that depending
upon problem parameters, there exist eight distinct solution structures. This
suggests that as the number of queues increase there may be a rapid blow-up in
the solution structures. This may make the problem of identifying and learning
the correct structure computationally prohibitive. In this paper, we do not
address the issue of customers learning the equilibrium profile by repeatedly
playing the game. Limiting behaviour of players repeatedly updating their
action in a game using a simple rule, often called a ‘no-regret’ policy, are
studied in [1] and we refer the reader to [3] for a comprehensive exposition
of this literature.
Our other broad contributions/observations are: 1) We find that similar to the
single queue setting, the equilibrium profile of arriving customers is unique
for a wide set of parameters. However, interestingly, when customer cost
preferences across classes are identical up to a constant, there may be
multiple equilibrium arrival profiles, all lying in a convex set that we
identify. Although there are many arrival profiles in equilibrium in this
case, they all have identical social cost.
2) In [12], the equilibrium profile is determined for the case when multiple
classes of customers with linear costs are arriving at a single queue again.
They find that different classes of customers arrive in non-overlapping and
contiguous intervals. In our two queue setting we find that depending upon the
problem parameters, in equilibrium, arrivals may come in non-overlapping and
contiguous intervals, in overlapping intervals, or under certain parametric
settings, a class of customers may even arrive in disjoint intervals.
Moreover, we show that whether the classes will arrive over overlapping sets
or not is independent of the population sizes and decided entirely by the
queue service rates and customer preferences.
Related literature: The arrival games to queues were first considered by [5].
The concert queueing game is the fluid setting was introduced in [12]. The
arrival game in a fluid network of bottleneck queues including tandem,
Trellis, and general feed-forward networks was considered in [11], where they
characterized the equilibrium arrival profile in each of these topologies.
Transportation modelling community has extensively studied arrival games.
Vickrey in [16], introduced the morning commute problem. Unlike the concert
queuing game, in these transportation problems, the service facility has no
predetermined opening time. Instead, the customers have a preferred time to
complete service and a cost for arriving too early or too late (see [10]).
This led to a huge literature on arrival games to a bottleneck queue, the
impact of tolls, etc. (see [14] for an extensive list of references). Much of
the transportation literature considers single queue settings. Lindsey, in an
influential work [15], establishes the existence of equilibrium arrival
profile for multiple classes of customers with general non-linear cost
functions arriving at a bottleneck queue with a constant service rate, through
intricate fixed point arguments. Our work differs from transportation
literature in that we consider a two queue network with heterogeneous arrival
routes, linear costs, and in this setting we are able characterize the
equilibrium user arrival profiles in a closed form, and for a large class of
parameters, show that these profiles are unique.
Outline of the paper: In Section 2 we provide the background to the arrival
queueing game and overview the two-class, two queue, two-route networks that
we consider. We emphasize on two heterogeneous routes networks 1) where the
departures of the two classes are through different queues (Heterogeneous
Departure System or HDS) and 2) where the arrivals enter at different queues
(Heterogeneous Arrival System or HAS). In Section 3, we identify the
equilibrium arrival profile for all possible parameters for arriving customers
for HDS. In particular, we see that these parameters can be partitioned into
four distinct regions each having a separate solution structure, when the two
customer classes have unequal preferences. In Section 4 we similarly analyze
HAS. Here we discover that the parameter space can be partitioned into eight
regions based on the solution structure, when the two customer classes have
unequal preferences. Moreover, for both HDS and HAS, when the groups have
identical preference, we identify a parametric regime where unique equilibrium
exists, as well as a parametric regime where the equilibrium is non-unique and
the set of equilibrium profiles is convex. We end with a brief conclusion in
Section 5. In the main body, we have confined our discussion to the main proof
ideas behind our results and have kept the detailed proofs in the appendix.
## 2 Preliminaries
### 2.1 Fluid Model
We consider a fluid model having two classes of customers or users. The size
of each class $i=1,2$ is given by a positive quantity $\Lambda_{i}>0$. In
every class $i=1,2$ individual users are infinitesimal and the set of all
users in class $i$ is given by the points in the interval $[0,\Lambda_{i}]$.
We define functions $F_{i}:\mathbb{R}\to[0,\Lambda_{i}]$ for $i=1,2$ such
that, $F_{i}(t)$ denotes the amount of users of class $i$ that arrive by time
$t$. We call $F_{i}$ the arrival profile of class $i$ users. Therefore, each
$F_{i}$ is non-decreasing and satisfies $F_{i}(-\infty)=0$ and
$F_{i}(+\infty)=\Lambda_{i}$. We consider $F_{i}$ that are right-continuous
and can be expressed as a sum of a non-decreasing absolutely continuous
function and a non-decreasing discrete function. We call the pair
$\mathbf{F}=\\{F_{1},F_{2}\\}$ as the joint arrival profile of the two
classes.
We consider a network comprising of two queues, both starting service at time
$t=0$. Let $\mu_{1}$ and $\mu_{2}$, respectively, denote the deterministic
fixed service rates at the two queues after they start service. We consider
four routes of the two arriving classes to the two queues. These are displayed
in Table 1.
Instance I Instance II Instance III (Heterogeneous Departure System or HDS)
Instance IV (Heterogeneous Arrival System or HAS)
Table 1: Various instances with two groups traveling through two queues
connected in tandem
Instance I is equivalent to two groups of users arriving at a two-layer tandem
network to travel by the same path. By Theorem 5 of [11], the instance is
equivalent to the case where the two groups are arriving at a single queue of
capacity $\min\\{\mu_{1},\mu_{2}\\}$. Instance II is equivalent to the case
where the two queues independently serve the two groups and therefore is
equivalent to two independent instances of a single queue with a single class
customer arrivals. Hence, the first two instances are reducible to instances
with just one queue studied in [12]. In this paper we study the arrival
equilibrium behaviour in the other two instances III and IV. We refer to them
as Heterogeneous Departure (HDS) and Heterogeneous Arrival Systems (HAS),
respectively.
### 2.2 Waiting and Departure Times
To specify the waiting and departure times in a system, first consider a
single queue setting where $A(t)$ denotes the total mass of users of all
classes that have arrived at the queue by time $t$. Let $\mu$ denote the
service rate. Then at time $t$, the length of the waiting queue developed in
that queue will be (see Theorem 6.5 in [4]):
$\displaystyle Q(t)$
$\displaystyle=A(t)-\mu\cdot\max\\{t,0\\}+\sup_{s\in[0,t]}\max\left\\{\mu
s-A(s),0\right\\}.$ (1)
We assume that if there is a jump in the arrival profile $A$ at time $t$, the
arrivals are positioned in the queue at uniformly random order. As a result, a
user arriving at that queue at time $t$ will suffer an expected waiting time
of
$\displaystyle W(t)$
$\displaystyle=\frac{Q(t+)+Q(t-)}{2\mu}+\max\\{0,-t\\},~{}\text{and departs at
time}~{}~{}\tau(t)=W(t)+t,$ (2)
where $Q(t+)$ and $Q(t-)$ respectively denote the right and left limits of $Q$
at time $t$. Note that if the queue length process $Q(\cdot)$ is continuous
(which is the case if $A(\cdot)$ is absolutely continuous), waiting time as a
function of time $t$ will be $W(t)=\frac{Q(t)}{\mu}+\max\\{0,-t\\}$.
If the arrival profile $A(\cdot)$ is absolutely continuous, by (1) and (2),
the departure time as a function of time $t$ will be:
$\tau(t)=\frac{A(t)}{\mu}+\sup_{s\in[0,t]}\max\left\\{0,s-\frac{A(s)}{\mu}\right\\}$.
Whenever $Q(t)>0$, the term $\sup_{s\in[0,u]}\max\\{\mu s-A(s),0\\}$ is
independent of the choice of $u\in[t-\delta,t+\delta]$ for $\delta>0$ and
sufficiently small, and when $t<0$, $\tau_{1}(t)=\frac{A(t)}{\mu}$. If
$A(\cdot)$ is absolutely continuous, its derivative $A^{\prime}(\cdot)$ will
exist a.e. As a result,
$\displaystyle\tau^{\prime}(t)$
$\displaystyle=\frac{A^{\prime}(t)}{\mu}~{}~{}\text{a.e. in the closure of the
set of times}~{}\\{s~{}|~{}s<0~{}\text{or}~{}Q(s)>0\\}.$ (3)
The above observation will be useful in our analysis of HDS and HAS in the
later sections.
###### Definition 2.1.
We say the queue is _engaged_ at time $t$ if $t$ lies in the closure of the
set $\\{s~{}|~{}Q(s)>0\\}$.
By (3), after the queue starts serving, users depart at rate $\mu$ whenever
the queue is engaged. We introduce the following notation:
* •
$A_{i}(t)$ be the total mass of customers of both the groups who have arrived
at queue $i=1,2$ till time $t$ (Note that while $F_{i}$ denotes arrival
profile corresponding to user class $i$, $A_{i}$ denotes overall arrival
profile to queue $i$.).
* •
$Q_{i}(t)$ and $W_{i}(t)$ be the length of the waiting queue, and the waiting
time that a customer arriving in queue $i$ at time $t$ will observe. Let
$\tau_{i}(t)$ denote the time that customer will depart the system.
* •
$W_{\mathbf{F}}^{(j)}(t)$ and $\tau_{\mathbf{F}}^{(j)}(t)$ be the waiting and
departure times from the network suffered by a class $j$ user arriving at time
$t$ for $j\in\\{1,2\\}$. Explicit dependence on $\mathbf{F}$ in this notation
is useful to our analysis later.
For both the queues $i=1,2$ upon defining the arrival profile $A_{i}(\cdot)$,
using (1) and (2), $Q_{i}(\cdot)$, $W_{i}(\cdot)$ and $\tau_{i}(\cdot)$ are
well-defined. Now we specify the waiting and departure times of both the
queues in HDS and HAS as functions of time, under the assumption that the
joint arrival profile $\mathbf{F}=\\{F_{1},F_{2}\\}$ is absolutely continuous
(we later argue by Lemma 2.1 that considering absolutely continuous joint
arrival profiles are sufficient for identifying equilibrium behavior).
1. HDS
: Arrival profiles at individual queues are $A_{1}(t)=F_{1}(t)+F_{2}(t)$ and
$A_{2}(t)=F_{2}(\tau_{1}^{-1}(t))$, where
$\tau_{1}^{-1}(t)=\sup\\{s~{}|~{}\tau_{1}(s)\leq t\\}$. Both $A_{1}(\cdot)$
and $A_{2}(\cdot)$ are absolutely continuous. With this,
$W_{\mathbf{F}}^{(1)}(t)=W_{1}(t)$, $\tau_{\mathbf{F}}^{(1)}(t)=\tau_{1}(t)$,
$W_{\mathbf{F}}^{(2)}(t)=W_{1}(t)+W_{2}(\tau_{1}(t))$, and
$\tau_{\mathbf{F}}^{(2)}(t)=\tau_{1}(t)+W_{2}(\tau_{1}(t))$.
2. HAS
: Arrival profile at individual queues are $A_{1}(t)=F_{1}(t)$ and
$A_{2}(t)=F_{1}(\tau_{1}^{-1}(t))+F_{2}(t)$ where
$\tau_{1}^{-1}(t)=\sup\\{s~{}|~{}\tau_{1}(s)\leq t\\}$. Both $A_{1}(\cdot)$
and $A_{2}(\cdot)$ are absolutely continuous. With this,
$W_{\mathbf{F}}^{(1)}(t)=W_{1}(t)+W_{2}(\tau_{1}(t))$,
$\tau_{\mathbf{F}}^{(1)}(t)=\tau_{1}(t)+W_{2}(\tau_{1}(t))$,
$W_{\mathbf{F}}^{(2)}(t)=W_{2}(t)$, and
$\tau_{\mathbf{F}}^{(2)}(t)=\tau_{2}(t)$.
### 2.3 Solution Concept
We assume that every user in group $i$ ($i\in\\{1,2\\}$) has a cost function
linear in her waiting and departure times from the network given by:
$C_{\mathbf{F}}^{(i)}(t)=\alpha_{i}\cdot
W_{\mathbf{F}}^{(i)}(t)+\beta_{i}\cdot\tau_{\mathbf{F}}^{(i)}(t)$, where
$\alpha_{i}$ and $\beta_{i}$ are positive constants quantifying the cost
suffered by a class $i$ user for unit waiting time and delay in departure.
###### Definition 2.2 (Support of arrival profile).
Given an arrival profile $t\mapsto B(t)$ such that $B(+\infty)<\infty$, the
support of $B$, denoted by $\mathcal{S}(B)$, is defined as the smallest closed
set having a $B$-measure equal to $B(+\infty)$.
###### Definition 2.3 (Equilibrium Arrival Profile (EAP)).
The joint arrival profile
$\mathbf{F^{\star}}=\\{F^{\star}_{1},F^{\star}_{2}\\}$ is an Equilibrium
Arrival Profile of this game if for both the groups $i\in\\{1,2\\}$:
$t\in\mathcal{S}(F_{i}^{\star})$ and $\tilde{t}\in\mathbb{R}$ implies
$C_{\mathbf{F^{\star}}}^{(i)}(t)\leq C_{\mathbf{F^{\star}}}^{(i)}(\tilde{t})$.
In particular, the arrival profile is iso-cost along its support.
Note that, the EAP doesn’t change upon normalizing the cost function of both
the classes $i=1,2$ by multiplying $1/(\alpha_{i}+\beta_{i})$. For simplicity,
and without loss of generality, we assume both the classes $i=1,2$ have their
normalized cost function, which are:
$C_{\mathbf{F}}^{(i)}(t)=\gamma_{i}W_{\mathbf{F}}^{(i)}(t)+(1-\gamma_{i})\tau_{\mathbf{F}}^{(i)}(t)$
where $\gamma_{i}=\frac{\alpha_{i}}{\alpha_{i}+\beta_{i}}$ quantifies the
preference of every class $i$ user. A value of $\gamma_{i}$ close to $1$
indicates users in group $i$ prefer late departure compared to waiting a long
time in the network and $\gamma_{i}$ close to $0$ implies the opposite. So, we
use $\gamma_{i}$ to quantify the cost preference of every group $i$ user.
###### Remark 1.
_EAP captures the aggregate equilibrium behavior of the group. We can
equivalently define Nash equilibrium at individual level where under it no
individual has unilateral incentive to deviate. As is well known and discussed
in more detail in[12], the two concepts are equivalent._
###### Lemma 2.1.
In every EAP, $\mathbf{F}=\\{F_{1},F_{2}\\}$ of the HDS and HAS, the arrival
profiles $F_{1}$ and $F_{2}$ are absolutely continuous.
Proof of the above lemma is similar to proof of statement (ii) of Lemma 1 in
[12]. On assuming contradiction, if any of the arrival profiles have a jump,
any user arriving in that jump will be strictly better off arriving slightly
early and as a result the arrival profile cannot be an EAP.
We argued before that Instances I and II in Table 1 are reducible to instances
where one or more groups of users having distinct preferences are arriving to
a single queue. [12] show that when two classes of customers having cost
preferences $\gamma_{1}$ and $\gamma_{2}$ arrive at a single queue with
service rate $\mu$, the EAP has a simple structure. The class with smaller
$\gamma_{i}$ comes first at arrival rate
$\mu\cdot\min\\{\gamma_{1},\gamma_{2}\\}$ over an interval, while the next
class arrives at a contiguous but non-overlapping interval, at rate
$\mu\cdot\max\\{\gamma_{1},\gamma_{2}\\}$. Fig 1 illustrates this EAP and the
resulting queue length with the assumption $\gamma_{1}<\gamma_{2}$ and is
useful to contrast with the various EAP structures that we find for HDS and
HAS in Sections 3 and 4 below. The queue length process is constructed
assuming that in the EAP, class 2 users start arriving from a positive time,
which is equivalent to saying, masses of the two classes satisfy
$\Lambda_{1}>\left(\frac{1}{\gamma_{2}}-1\right)\Lambda_{2}$.
Figure 1: EAP structure (left) and resulting queue length process (right) when
two classes of users with cost preferences $\gamma_{1}$ and $\gamma_{2}$
(assuming $\gamma_{1}<\gamma_{2}$) are arriving at a queue of capacity $\mu$.
The support boundaries are
$T_{2,f}=\frac{\Lambda_{1}+\Lambda_{2}}{\mu},~{}T_{2,a}=\frac{\Lambda_{1}}{\mu}-\left(\frac{1}{\gamma_{2}}-1\right)\frac{\Lambda_{2}}{\mu}$
and
$T_{1,a}=-\left(\frac{1}{\gamma_{1}}-1\right)\frac{\Lambda_{1}}{\mu}-\left(\frac{1}{\gamma_{2}}-1\right)\frac{\Lambda_{2}}{\mu}$.
The queue length process is illustrated only for the situation $T_{2,a}>0$, or
equivalently $\Lambda_{1}>\left(\frac{1}{\gamma_{2}}-1\right)\Lambda_{2}$. Red
and blue, respectively, represents class 1 and 2 populations. The black dashed
line represents the total waiting mass of the two classes in the plot for
queue length.
## 3 Heterogeneous Departure Systems (HDS)
In this section, we consider the situation where the two classes arrive at the
first queue and depart from different queues, as illustrated in Table 1. If
$\mu_{1}\leq\mu_{2}$, class 2 users arrive at queue 2 at a maximum rate of
$\mu_{1}$ and as a result, queue 2 remains empty and the cost of class 2 is
unaffected by the second queue. Thus, if $\mu_{1}\leq\mu_{2}$, the instance
becomes equivalent to both the groups arriving at a queue of capacity
$\mu_{1}$. The problem is identical to the two-class, single queue case
studied in [12]. Therefore, in subsequent discussion, we restrict ourselves to
HDS with $\mu_{1}>\mu_{2}$. We further consider the case
$\gamma_{1}\neq\gamma_{2}$ separately from $\gamma_{1}=\gamma_{2}$ since the
latter displays different behaviour.
### 3.1 Unequal Preferences: $\gamma_{1}\neq\gamma_{2}$
Structural properties of EAP. We identify several structural properties that
every EAP of HDS satisfies. We will exploit these properties later to narrow
our search of an EAP. Many of these properties are true even when the two
groups have equal preference, i.e., $\gamma_{1}=\gamma_{2}$, and we use them
later in Section 3.2.
As mentioned earlier, when $\mu_{1}\leq\mu_{2}$, the second queue is not
relevant to equilibrium behaviour, and the two classes arrive in disjoint,
contiguous intervals in the order of increasing $\gamma$’s. Lemma 3.1 shows
that the EAP has a similar arrival pattern up to a threshold for
$\mu_{1}>\mu_{2}$.
###### Lemma 3.1 (Threshold Behavior).
If $\gamma_{1}\neq\gamma_{2}$, in the EAP, the two classes arrive over
contiguous intervals with disjoint interiors if and only if
$\mu_{1}\leq\mu_{2}\cdot\max\left\\{1,\frac{\gamma_{2}}{\gamma_{1}}\right\\}$.
Below we sketch the proof of sufficiency of the condition stated in Lemma 3.1.
Proving the other direction requires exploiting the behavior of the two queues
in EAP and also the structure of $\mathcal{S}(F_{1})$ and
$\mathcal{S}(F_{2})$. So, we prove that after stating Lemma 3.5.
Proof sketch of sufficiency in Lemma 3.1: The detailed proof of sufficiency of
the condition
$\mu_{1}\leq\mu_{2}\cdot\max\left\\{1,\frac{\gamma_{2}}{\gamma_{1}}\right\\}$
is in Lemma A.3 and is similar to the proof sketch we will present here, but
instead uses some other supplementary lemmas and tools. First we show via
contradiction that if
$\mu_{1}\leq\mu_{2}\cdot\max\left\\{1,\frac{\gamma_{2}}{\gamma_{1}}\right\\}$,
$\mathcal{S}(F_{1})$ and $(\mathcal{S}(F_{2}))^{o}$ cannot overlap. We later
argue via Lemma 3.2 (stated later) that in every EAP
$\mathbf{F}=\\{F_{1},F_{2}\\}$, $\mathcal{S}(F_{1})$ and $\mathcal{S}(F_{2})$
are intervals. As a result, by the previous two statements, sufficiency of the
stated condition will follow.
If
$\mu_{1}\leq\mu_{2}\cdot\max\left\\{1,\frac{\gamma_{2}}{\gamma_{1}}\right\\}$,
and $\mathcal{S}(F_{1}),(\mathcal{S}(F_{2}))^{o}$ overlap, we can find
$t_{1},t_{2}\in\mathcal{S}(F_{1})$ such that
$[t_{1},t_{2}]\subseteq\mathcal{S}(F_{2})$. Note that queue 1 must be engaged
in $(t_{1},t_{2})$, otherwise the class 1 user arriving at $t_{2}$ is strictly
better off arriving at the time when queue 1 is empty in $(t_{1},t_{2})$.
Hence using 3,
$(C_{\mathbf{F}}^{(1)})^{\prime}(t)=\tau_{1}^{\prime}(t)-\gamma_{1}=\frac{F_{1}^{\prime}(t)+F_{2}^{\prime}(t)}{\mu_{1}}-\gamma_{1}$
in $[t_{1},t_{2}]$. Now $t_{1},t_{2}\in\mathcal{S}(F_{1})$ implies
$C_{\mathbf{F}}^{(1)}(t_{2})=C_{\mathbf{F}}^{(1)}(t_{1})$, which by
integrating
$(C_{\mathbf{F}}^{(1)})^{\prime}(t)=\frac{F_{1}^{\prime}(t)+F_{2}^{\prime}(t)}{\mu_{1}}-\gamma_{1}$
in $[t_{1},t_{2}]$ gives
$F_{1}(t_{2})+F_{2}(t_{2})-(F_{1}(t_{1})+F_{2}(t_{1}))=\mu_{1}\gamma_{1}(t_{2}-t_{1}).$
Therefore $\mu_{1}\gamma_{1}(t_{2}-t_{1})$ is the total mass of the two groups
that have arrived in $[t_{1},t_{2}]$.
Since $[t_{1},t_{2}]\subseteq\mathcal{S}(F_{2})$, we must have
$(C_{\mathbf{F}}^{(2)})^{\prime}(t)=0$ and hence
$(\tau_{\mathbf{F}}^{(2)})^{\prime}(t)=\gamma_{2}$ in $[t_{1},t_{2}]$. As
$\tau_{\mathbf{F}}^{(2)}(t)=\tau_{2}(\tau_{1}(t))$, we have
$(\tau_{\mathbf{F}}^{(2)})^{\prime}(t)=\tau_{2}^{\prime}(\tau_{1}(t))\tau_{1}^{\prime}(t)$
in $[t_{1},t_{2}]$, assuming the derivatives exist. Later we argue in Lemma
3.3 that, queue 2 must remain engaged at $\tau_{1}(t)$ for every
$t\in[t_{1},t_{2}]$. Therefore, using (3),
$\tau_{2}^{\prime}(\tau_{1}(t))=\frac{A_{2}^{\prime}(\tau_{1}(t))}{\mu_{2}}$,
and hence
$(\tau_{\mathbf{F}}^{(2)})^{\prime}(t)=\frac{A_{2}^{\prime}(\tau_{1}(t))\tau_{1}^{\prime}(t)}{\mu_{2}}$
in $[t_{1},t_{2}]$. Since $A_{2}(\tau_{1}(t))=F_{2}(t)$, the previous
statement implies
$(\tau_{\mathbf{F}}^{(2)})^{\prime}(t)=\frac{F_{2}^{\prime}(t)}{\mu_{2}}$ in
$[t_{1},t_{2}]$. Combining this with the observation that
$(\tau_{\mathbf{F}}^{(2)})^{\prime}(t)=\gamma_{2}$ a.e. in $[t_{1},t_{2}]$, we
have $F_{2}^{\prime}(t)=\mu_{2}\gamma_{2}$ a.e. in $[t_{1},t_{2}]$. Therefore,
$F_{2}(t_{2})-F_{2}(t_{1})=\mu_{2}\gamma_{2}(t_{2}-t_{1}).$
Since a positive mass of both the classes arrive in $[t_{1},t_{2}]$, we must
have
$F_{1}(t_{2})+F_{2}(t_{2})-(F_{1}(t_{1})+F_{2}(t_{1}))>F_{2}(t_{2})-F_{2}(t_{1})$,
which implies $\mu_{1}\gamma_{1}>\mu_{2}\gamma_{2}$ by the previous arguments.
This contradicts our assumption that
$\mu_{1}\leq\mu_{2}\max\left\\{1,\frac{\gamma_{2}}{\gamma_{1}}\right\\}$. ∎
Lemma 3.2 and 3.4 imply EAPs can only be piece-wise linear arrival profiles
with intervals as support. Proof of Lemma 3.2 (in A.1) is done via
contradiction by arguing if there is a gap, we can identify a user who can
improve her cost by arriving at a different time.
###### Lemma 3.2 (Structure of supports).
If $\mu_{1}>\mu_{2}$ and $\mathbf{F}=\\{F_{1},F_{2}\\}$ is an EAP, then
$\mathcal{S}(F_{1})$ and $\mathcal{S}(F_{1})\cup\mathcal{S}(F_{2})$ must be
intervals. Additionally, if $\gamma_{1}\neq\gamma_{2}$, $\mathcal{S}(F_{2})$
must also be an interval.
For a joint arrival profile $\mathbf{F}=\\{F_{1},F_{2}\\}$, we define the
quantities $T_{i,a}\overset{def.}{=}\inf\mathcal{S}(F_{i})$ and
$T_{i,f}\overset{def.}{=}\sup\mathcal{S}(F_{i})$ for the two classes
$i\in\\{1,2\\}$. For every EAP $\mathbf{F}$,
$\mathcal{S}(F_{1}),~{}\mathcal{S}(F_{2})$ must be compact, as cost of the two
classes over their support must be finite. As a result, the support boundaries
$T_{1,a},T_{1,f},T_{2,a},T_{2,f}$ must be finite. By Lemma 3.2, we can further
say, $\mathcal{S}(F_{1})=[T_{1,a},T_{1,f}]$ and
$\mathcal{S}(F_{2})=[T_{2,a},T_{2,f}]$, such that union of the two intervals
is also an interval.
Lemma 3.3 is about the behavior of queue 2 in equilibrium.
###### Lemma 3.3.
If $\mu_{1}>\mu_{2}$ and $\gamma_{1}\neq\gamma_{2}$, then under EAP, for every
$t\in(T_{2,a},T_{2,f})$, $\tau_{1}(t)$ belongs to the closure of the set
$\\{s~{}|~{}Q_{2}(s)>0\\}$.
Proof sketch of Lemma 3.3: We consider two separate cases:
1. 1.
If $t\in(T_{2,a},T_{2,f})/(T_{1,a},T_{1,f})$, two possibilities are there. If
queue 1 is engaged at $t$, class 2 users arrive from queue 1 to 2 at rate
$\mu_{1}>\mu_{2}$ at $\tau_{1}(t)$, making queue 2 engaged. Otherwise, if
queue 1 is empty at $t$, in EAP, the network cannot be empty at $t$ with a
positive mass of class 2 users arriving in $(t,T_{2,f})$. As a result, queue 2
must be engaged at $t=\tau_{1}(t)$.
2. 2.
If $t\in(T_{1,a},T_{1,f})\cap(T_{2,a},T_{2,f})$, we assume a contradiction,
i.e., queue 2 is empty in some neighbourhood of $\tau_{1}(t)$. As a result, by
continuity of $\tau_{1}(\cdot)$, only queue 1 will be serving in some
neighbourhood of $t$. Therefore
$\tau_{\mathbf{F}}^{(1)}(\cdot)=\tau_{\mathbf{F}}^{(2)}(\cdot)=\tau_{1}(\cdot)$
in that neighbourhood. Also, for both the classes $i=1,2$ to be iso-cost in
that neighbourhood, we need
$(C_{\mathbf{F}}^{(i)})^{\prime}(s)=\tau_{1}^{\prime}(s)-\gamma_{i}=0$. This
gives us $\tau_{1}^{\prime}(s)=\gamma_{1}=\gamma_{2}$, contradicting
$\gamma_{1}\neq\gamma_{2}$.
∎
Lemma 3.4 states conditions on the arrival rates necessary for the two classes
to have constant cost over their support in any EAP.
###### Lemma 3.4 (Rates of arrival).
If $\mu_{1}>\mu_{2}$, $\gamma_{1}\neq\gamma_{2}$, and
$\mathbf{F}=\\{F_{1},F_{2}\\}$ is an EAP, the following properties must be
true almost everywhere:
$\displaystyle F_{1}^{\prime}(t)$
$\displaystyle=\begin{cases}\mu_{1}\gamma_{1}~{}~{}&\text{if}~{}t\in\mathcal{S}(F_{1})/\mathcal{S}(F_{2}),\\\
\mu_{1}\gamma_{1}-\mu_{2}\gamma_{2}~{}~{}&\text{if}~{}t\in\mathcal{S}(F_{1})\cap\mathcal{S}(F_{2}),\end{cases}~{}~{}\text{and,}~{}F_{2}^{\prime}(t)=\mu_{2}\gamma_{2}~{}~{}\text{if}~{}t\in\mathcal{S}(F_{2}),$
where from Lemma 3.1, $\mathcal{S}(F_{1})\cap\mathcal{S}(F_{2})$ has zero
measure if $\mu_{1}\gamma_{1}\leq\mu_{2}\gamma_{2}$.
In the proof of Lemma 3.4 (in A.1), we first observe that both the classes
$i=1,2$ have $C_{\mathbf{F}}^{(i)}(t)=\tau_{\mathbf{F}}^{(i)}(t)-\gamma_{i}t$
constant in $[T_{i,a},T_{1,f}]$, causing
$(\tau_{\mathbf{F}}^{(i)})^{\prime}(t)=\gamma_{i}$. The rest of the proof
relies on relating $(\tau_{\mathbf{F}}^{(i)})^{\prime}(t)$ with the arrival
rates $F_{1}^{\prime}(t)$ and $F_{2}^{\prime}(t)$ of the two classes. Towards
that, we use (3) and leverage the facts that queue 1 has positive waiting time
in $(T_{1,a},T_{1,f})$, (otherwise, if $Q_{1}(t)=0$ at some
$t\in(T_{1,a},T_{1,f})$, every class 1 user arriving in $(t,T_{1,f}]$ is
strictly better off arriving at time $t$) and queue 2 stays engaged in
$[\tau_{1}(T_{2,a}),\tau_{1}(T_{2,f})]$ (by Lemma 3.3).
Lemma 3.5 is about the state of the two queues at support boundaries $T_{1,f}$
and $T_{2,f}$. Proof of Lemma 3.5 is done via a contradiction by showing that
if the specified properties do not hold, a user can reduce her cost by
arriving at a different time.
###### Lemma 3.5.
If $\mu_{1}>\mu_{2}$ and $\gamma_{1}\neq\gamma_{2}$, then under EAP, queue
length at the second queue is zero at $\tau_{1}(T_{2,f})$. If, in addition, we
have $\mu_{1}\gamma_{1}>\mu_{2}\gamma_{2}$, queue length at the first queue
equals zero at $T_{1,f}$.
Proof of necessity of the condition in Lemma 3.1: We prove via contradiction
that
$\mu_{1}\leq\mu_{2}\cdot\max\left\\{1,\frac{\gamma_{2}}{\gamma_{1}}\right\\}$
is necessary for the intervals $[T_{1,a},T_{1,f}]$ and $[T_{2,a},T_{2,f}]$ to
overlap. This forms the other direction of Lemma 3.1. On assuming a
contradiction, we must have
$\mu_{1}>\mu_{2}\cdot\max\left\\{1,\frac{\gamma_{2}}{\gamma_{1}}\right\\}$ and
interiors of $[T_{1,a},T_{1,f}]$, $[T_{2,a},T_{2,f}]$ are disjoint. Now by
Lemma 3.2, union of $[T_{1,a},T_{1,f}]$ and $[T_{2,a},T_{2,f}]$ must be an
interval. This leaves us two possibilities:
1. 1.
If $T_{1,f}=T_{2,a}$, by Lemma 3.5, queue 1 will be empty at $T_{1,f}$, making
the whole network empty at $T_{1,f}$. As a result, every class 2 user will be
strictly better off arriving at $T_{1,f}$.
2. 2.
If $T_{2,f}=T_{1,a}$, queue 1 must have a positive waiting time in
$(T_{2,a},T_{2,f}]$ and as a result, class 2 users arrive at queue 2 at rate
$\mu_{1}>\mu_{2}$ in $[0,\tau_{1}(T_{2,f})]$, causing
$Q_{2}(\tau_{1}(T_{2,f}))=(\mu_{1}-\mu_{2})\cdot\tau_{1}(T_{2,f})>0$,
contradicting Lemma 3.5. ∎
Specification of the EAP. Theorem 3.1 specifies the unique EAP of this regime
and we mention below the support boundaries of the unique EAP, which we will
refer to later in Theorem 3.1.
1. 1.
If $\gamma_{1}\leq\frac{\mu_{2}}{\mu_{1}}\gamma_{2}$, then
$\displaystyle T_{1,a}$
$\displaystyle=-\left(\frac{1}{\gamma_{1}}-1\right)\frac{\Lambda_{1}}{\mu_{1}}-\left(\frac{1}{\gamma_{2}}-1\right)\frac{\Lambda_{2}}{\mu_{2}},~{}T_{1,f}=T_{2,a}=\frac{\Lambda_{1}}{\mu_{1}}-\left(\frac{1}{\gamma_{2}}-1\right)\frac{\Lambda_{2}}{\mu_{2}},~{}\text{and}~{}T_{2,f}=\frac{\Lambda_{1}}{\mu_{1}}+\frac{\Lambda_{2}}{\mu_{2}}.$
(4)
2. 2.
If $\gamma_{1}>\frac{\mu_{2}}{\mu_{1}}\gamma_{2}$ and
$\Lambda_{1}\geq\min\left\\{\frac{1-\gamma_{2}}{1-\gamma_{1}}\left(\frac{\mu_{1}\gamma_{1}}{\mu_{2}\gamma_{2}}-1\right),\frac{\mu_{1}}{\mu_{2}}-1\right\\}\cdot\Lambda_{2}$,
then
1. a)
if $\gamma_{1}<\gamma_{2}$, then
$\displaystyle\vspace{-0.05in}T_{1,a}$
$\displaystyle=-\frac{1-\gamma_{1}}{\mu_{1}\gamma_{1}}\left[\Lambda_{1}+\frac{1-\gamma_{2}}{1-\gamma_{1}}\Lambda_{2}\right],~{}T_{1,f}=\frac{1}{\mu_{1}}\left[\Lambda_{1}+\frac{1-\gamma_{2}}{1-\gamma_{1}}\Lambda_{2}\right],$
$\displaystyle T_{2,a}$
$\displaystyle=\frac{1}{\mu_{1}}\left[\Lambda_{1}-\frac{\mu_{1}-\mu_{2}\gamma_{2}}{\mu_{2}\gamma_{2}}\frac{1-\gamma_{2}}{1-\gamma_{1}}\Lambda_{2}\right],~{}\text{and}~{}T_{2,f}=\frac{1}{\mu_{1}}\left[\Lambda_{1}+\frac{\mu_{1}(\gamma_{2}-\gamma_{1})+\mu_{2}\gamma_{2}(1-\gamma_{2})}{\mu_{2}\gamma_{2}(1-\gamma_{1})}\Lambda_{2}\right].$
(5)
2. b)
if $\gamma_{1}>\gamma_{2}$, then
$\displaystyle\vspace{-0.05in}T_{1,a}$
$\displaystyle=\frac{\gamma_{1}-\gamma_{2}}{\gamma_{1}}\frac{\Lambda_{2}}{\mu_{1}\gamma_{1}-\mu_{2}\gamma_{2}}-\frac{1-\gamma_{1}}{\gamma_{1}}\frac{\Lambda_{1}+\Lambda_{2}}{\mu_{1}},~{}T_{1,f}=\frac{\Lambda_{1}+\Lambda_{2}}{\mu_{1}},$
$\displaystyle T_{2,a}$
$\displaystyle=-\frac{1-\gamma_{1}}{\gamma_{1}}\frac{\Lambda_{1}}{\mu_{1}}-\left(\frac{\gamma_{1}}{\gamma_{2}}+(1-\gamma_{1})\frac{\mu_{2}}{\mu_{1}}-1\right)\frac{\Lambda_{2}}{\mu_{2}\gamma_{1}},~{}\text{and}~{}T_{2,f}=-\frac{1-\gamma_{1}}{\gamma_{1}}\frac{\Lambda_{1}}{\mu_{1}}+\left(1-(1-\gamma_{1})\frac{\mu_{2}}{\mu_{1}}\right)\frac{\Lambda_{2}}{\mu_{2}\gamma_{1}}.$
(6)
3. 3.
If $\gamma_{1}>\frac{\mu_{2}}{\mu_{1}}\gamma_{2}$ and
$\Lambda_{1}<\min\left\\{\frac{1-\gamma_{2}}{1-\gamma_{1}}\left(\frac{\mu_{1}\gamma_{1}}{\mu_{2}\gamma_{2}}-1\right),\frac{\mu_{1}}{\mu_{2}}-1\right\\}\cdot\Lambda_{2}$,
then
$\displaystyle~{}T_{1,a}$
$\displaystyle=\frac{1-\gamma_{2}}{\mu_{1}-\mu_{2}\gamma_{2}}\left[\Lambda_{2}-\frac{(1-\gamma_{1})\mu_{1}}{(1-\gamma_{2})(\mu_{1}\gamma_{1}-\mu_{2}\gamma_{2})}\Lambda_{1}\right],~{}T_{1,f}=\frac{\Lambda_{1}+(1-\gamma_{2})\Lambda_{2}}{\mu_{1}-\mu_{2}\gamma_{2}}$
$\displaystyle T_{2,a}$
$\displaystyle=-\frac{1-\gamma_{2}}{\gamma_{2}}\frac{\Lambda_{2}}{\mu_{2}},~{}\text{and}~{}T_{2,f}=\frac{\Lambda_{2}}{\mu_{2}}.$
(7)
###### Theorem 3.1.
If $\mu_{1}>\mu_{2}$ and $\gamma_{1}\neq\gamma_{2}$, HDS has a unique EAP. The
arrival rates in the EAP are given below along with the support boundaries:
1. 1.
If $\gamma_{1}\leq\frac{\mu_{2}}{\mu_{1}}\gamma_{2}$,
$F_{1}^{\prime}(t)=\mu_{1}\gamma_{1}$ for $t\in[T_{1,a},T_{1,f}]$ and
$F_{2}^{\prime}(t)=\mu_{2}\gamma_{2}$ for $t\in[T_{2,a},T_{2,f}]$ where
$T_{1,a},T_{1,f},T_{2,a}$ and $T_{2,f}$ are given in (4).
2. 2.
If $\frac{\mu_{2}}{\mu_{1}}\gamma_{2}<\gamma_{1}<\gamma_{2}$,
1. 2a.
when
$\Lambda_{1}\geq\frac{1-\gamma_{2}}{1-\gamma_{1}}\left(\frac{\mu_{1}\gamma_{1}}{\mu_{2}\gamma_{2}}-1\right)\Lambda_{2}$,
$\displaystyle F_{1}^{\prime}(t)$
$\displaystyle=\begin{cases}\mu_{1}\gamma_{1}~{}~{}&\text{if}~{}t\in[T_{1,a},T_{2,a}],\\\
\mu_{1}\gamma_{1}-\mu_{2}\gamma_{2}~{}~{}&\text{if}~{}t\in[T_{2,a},T_{1,f}],\end{cases}~{}~{}\text{and}~{}F_{2}^{\prime}(t)=\mu_{2}\gamma_{2}~{}~{}\text{if}~{}t\in[T_{2,a},T_{2,f}]$
where $T_{1,a},T_{1,f},T_{2,a}$ and $T_{2,f}$ are given in ( ‣ 2.).
2. 2b.
when
$\Lambda_{1}<\frac{1-\gamma_{2}}{1-\gamma_{1}}\left(\frac{\mu_{1}\gamma_{1}}{\mu_{2}\gamma_{2}}-1\right)\Lambda_{2}$,
$F_{1}^{\prime}(t)=\mu_{1}\gamma_{1}-\mu_{2}\gamma_{2}$ for
$t\in[T_{1,a},T_{1,f}]$, and $F_{2}^{\prime}(t)=\mu_{2}\gamma_{2}$ for
$t\in[T_{2,a},T_{2,f}]$, where $T_{1,a},T_{1,f},T_{2,a}$ and $T_{2,f}$ are
given in (3.).
3. 3.
If $\gamma_{2}<\gamma_{1}$,
1. 3a.
when $\Lambda_{1}\geq\left(\frac{\mu_{1}}{\mu_{2}}-1\right)\Lambda_{2}$,
$\displaystyle F_{1}^{\prime}(t)$
$\displaystyle=\begin{cases}\mu_{1}\gamma_{1}-\mu_{2}\gamma_{2}~{}~{}&\text{if}~{}t\in[T_{1,a},T_{2,f}],\\\
\mu_{1}\gamma_{1}~{}~{}&\text{if}~{}t\in[T_{2,f},T_{1,f}],\end{cases}~{}~{}\text{and}~{}F_{2}^{\prime}(t)=\mu_{2}\gamma_{2}~{}~{}\text{if}~{}t\in[T_{2,a},T_{2,f}]$
where $T_{1,a},T_{1,f},T_{2,a}$ and $T_{2,f}$ are given in ( ‣ 2.).
2. 3b.
when $\Lambda_{1}<\left(\frac{\mu_{1}}{\mu_{2}}-1\right)\Lambda_{2}$, the EAP
has a closed form same as case 2b.
###### Remark 2.
_Figures 2, 3, 4, and 5 show the illustrative EAPs and resulting queue length
processes, respectively, for cases 1, 2a, 2b and 3a of Theorem 3.1. EAP
structure in case 3b of Theorem 3.1 is similar to case 2b and is illustrated
in Figure 4. The structure of the queue length processes in certain regimes
depend on the support boundaries and may vary accordingly. In the figures
referred above, we have illustrated only one possible structure of the queue
length process. Moreover, we have mentioned the conditions to be satisfied by
the support boundaries for the attainment of that structure in the caption. In
all these figures, red and blue, respectively, represent class 1 and 2
populations, and the black dashed line represents the total mass of the two
populations waiting in queue 1._
###### Remark 3.
_The illustrative EAPs referred to in the previous remark also displays the
threshold behavior stated in Lemma 3.1. Since
$\mu_{1}\leq\mu_{2}\cdot\max\left\\{1,\frac{\gamma_{2}}{\gamma_{1}}\right\\}$
in case 1 of Theorem 3.1, we can see the support intervals of the two classes
to have disjoint interiors in Figure 2. On the other hand, since cases 2 and 3
of Theorem 3.1 have
$\mu_{1}>\mu_{2}\cdot\left\\{1,\frac{\gamma_{2}}{\gamma_{1}}\right\\}$, two
classes arrive over overlapping intervals, as can be seen in Figure 3, 4, and
5._
The following lemma specifies the arrival order of the two classes in EAP and
is necessary for proving Theorem 3.1.
###### Lemma 3.6.
If $\mu_{1}>\mu_{2}$ and $\gamma_{1}\neq\gamma_{2}$, the support boundaries of
EAP of HDS satisfy:
1. 1.
$T_{1,f}=T_{2,a}$ if $\gamma_{1}\leq\frac{\mu_{2}}{\mu_{1}}\gamma_{2}$,
2. 2.
$T_{1,f}\leq T_{2,f}$ if
$\frac{\mu_{2}}{\mu_{1}}\gamma_{2}<\gamma_{1}<\gamma_{2}$, and
3. 3.
$T_{1,a}>T_{2,a}$ if $\gamma_{1}>\gamma_{2}$.
Proof of Lemma 3.6 (in A.1) is done via contradiction by showing that if the
stated property doesn’t hold, a user can improve her cost by arriving at a
different time.
###### Remark 4.
_The arrival orders anticipated by Lemma 3.6 can be observed in the
illustrative EAPs referred earlier. By Lemma 3.6, in case 2
($\gamma_{2}>\gamma_{1}>\frac{\mu_{2}}{\mu_{1}}\gamma_{2}$) of Theorem 3.1,
class 2 population finishes arrival after class 1, as can be observed in
Figure 3 and 4. Similarly for case 3 ($\gamma_{1}>\gamma_{2}$) of Theorem 3.1,
class 1 population starts arrival after class 2, as can be observed in Figure
5 and 4. The class arriving first in case 2 and finishing late in case 3 is
the one having a population size significantly larger among the two classes.
In case 2 (or case 3): 1) when $\Lambda_{1}>c\Lambda_{2}$, class 1 starts
arriving before class 2 (or finishes arriving after class 2), 2) when
$\Lambda_{1}=c\Lambda_{2}$, both classes start arriving from the same time (or
finishes arriving at the same time), 3) when $\Lambda_{1}<c\Lambda_{2}$, class
1 starts arriving after class 2 starts (or finishes arriving before class 2),
where
$c=\min\left\\{\frac{1-\gamma_{2}}{1-\gamma_{1}}\left(\frac{\mu_{1}\gamma_{1}}{\mu_{2}\gamma_{2}}-1\right),\frac{\mu_{1}}{\mu_{2}}-1\right\\}$
for both case 2 and 3. _
Key steps in the proof of Theorem 3.1 (details in A.1.1): By Lemma 3.2 we only
consider candidates $\mathbf{F}=\\{F_{1},F_{2}\\}$ which are absolutely
continuous with supports of the form $\mathcal{S}(F_{1})=[T_{1,a},T_{1,f}]$,
$\mathcal{S}(F_{2})=[T_{2,a},T_{2,f}]$, such that their union is an interval
and the arrival rates $F_{1}^{\prime}(\cdot),~{}F_{2}^{\prime}(\cdot)$ satisfy
the property in Lemma 3.4. We eliminate candidates not satisfying structural
properties desired of an EAP and will be eventually left with only one
candidate, which is the only candidate that can qualify as an EAP. We then
prove that the only remaining candidate is indeed an EAP. Thus, existence and
uniqueness of EAP follows.
We follow this agenda for the three cases: 1)
$\gamma_{1}\leq\frac{\mu_{2}}{\mu_{1}}\gamma_{2}$, 2)
$\frac{\mu_{2}}{\mu_{1}}\gamma_{2}<\gamma_{1}<\gamma_{2}$, and 3)
$\gamma_{2}<\gamma_{1}$. The final candidate we get in these cases have their
arrival rates and support boundaries same as the joint arrival profiles
mentioned in Theorem 3.1 under the respective cases.
Case 1 $\mathbf{\gamma_{1}\leq\frac{\mu_{2}}{\mu_{1}}\gamma_{2}}$: By Lemma
3.1, every EAP must have $[T_{1,a},T_{1,f}]$ and $[T_{2,a},T_{2,f}]$ disjoint.
Lemma 3.7 helps us substantially narrow our search for an EAP.
###### Lemma 3.7.
Under case 1 $\gamma_{1}\leq\frac{\mu_{2}}{\mu_{1}}\gamma_{2}$, every EAP has
(4) as support boundaries $T_{1,a},T_{1,f},T_{2,a},T_{2,f}$.
Proof Sketch: By Lemma 3.6, we must have $T_{1,f}=T_{2,a}$. By Lemma 3.4,
$F_{1}^{\prime}(t)=\mu_{1}\gamma_{1}$ in $[T_{1,a},T_{1,f}]$ and
$F_{2}^{\prime}(t)=\mu_{2}\gamma_{2}$ in $[T_{1,f},T_{2,f}]$. Therefore,
$T_{1,a},T_{1,f},T_{2,f}$ must satisfy:
$\displaystyle T_{1,f}$
$\displaystyle=T_{1,a}+\frac{\Lambda_{1}}{\mu_{1}\gamma_{1}}~{}\text{and}~{}T_{2,f}=T_{1,f}+\frac{\Lambda_{2}}{\mu_{2}\gamma_{2}}.$
(8)
Now, queue 2 must be empty at $\tau_{1}(T_{2,f})$ (by Lemma 3.5) and must stay
engaged in the image of $[T_{1,f},T_{2,f}]$ under $\tau_{1}(\cdot)$ (by Lemma
3.3), which is $[\tau_{1}(T_{1,f}),\tau_{1}(T_{2,f})]$. Since, the first and
last class 2 users arrive at queue 2, respectively at times
$\tau_{1}(T_{1,f})$ and $\tau_{1}(T_{2,f})$, the previous statement implies,
$\displaystyle\mu_{2}\cdot(\tau_{1}(T_{2,f})-\tau_{1}(T_{1,f}))$
$\displaystyle=\Lambda_{2}$ (9)
For queue 2 to be empty at $\tau_{1}(T_{2,f})$, queue 1 must also be empty at
$T_{2,f}$. Otherwise if queue 1 has a positive waiting time at $T_{2,f}$,
class 2 users will be arriving at queue 2 from queue 1 at rate
$\mu_{1}>\mu_{2}$ in $[\tau_{1}(T_{2,f}-\delta),\tau_{1}(T_{2,f})]$, where
$\delta>0$ picked sufficiently small such that queue 1 has positive waiting
time in $[T_{2,f}-\delta,T_{2,f}]$. By (3),
$\tau_{1}^{\prime}(t)=\frac{F_{2}^{\prime}(t)}{\mu_{1}}=\frac{\mu_{2}\gamma_{2}}{\mu_{1}}>0$
in $[T_{2,f}-\delta,T_{2,f}]$, giving us
$\tau_{1}(T_{2,f})>\tau_{1}(T_{2,f}-\delta)$. As a result,
$Q_{2}(\tau_{1}(T_{2,f}))\geq(\mu_{1}-\mu_{2})\cdot(\tau_{1}(T_{2,f})-\tau_{1}(T_{2,f}-\delta))>0$,
which contradicts Lemma 3.5. With queue 1 empty at $T_{2,f}$, we have
$\tau_{1}(T_{2,f})=T_{2,f}$.
Note that queue 1 must have a positive waiting time in $(T_{1,a},T_{1,f}]$,
otherwise, every class 2 user arriving after $T_{1,f}$ is strictly better off
arriving at the time queue 1 is empty. Applying (3), we have
$\tau_{1}(T_{1,f})-\tau_{1}(T_{1,a})=\frac{F_{1}(T_{1,f})-F_{1}(T_{1,a})}{\mu_{1}}=\frac{\Lambda_{1}}{\mu_{1}}$.
Since the network cannot be empty at time zero, we have $T_{1,a}<0$. This
gives us $\tau_{1}(T_{1,a})=0$ and hence,
$\tau_{1}(T_{1,f})=\frac{\Lambda_{1}}{\mu_{1}}$. Putting
$\tau_{1}(T_{2,f})=T_{2,f}$ and
$\tau_{1}(T_{1,f})=\frac{\Lambda_{1}}{\mu_{1}}$ in (9), we get
$T_{2,f}=\frac{\Lambda_{1}}{\mu_{1}}+\frac{\Lambda_{2}}{\mu_{2}}$. Using this
in (8) and by the fact $T_{2,a}=T_{1,f}$, we get the values of
$T_{1,a},T_{1,f},T_{2,a},T_{2,f}$ mentioned in (4). ∎
The only candidate having arrival rates satisfying Lemma 3.4 and support
boundaries from Lemma 3.7 is the joint arrival profile under case 1 of Theorem
3.1. Proving that this unique remaining candidate is an EAP requires analyzing
the behavior of the two queues induced by this candidate.s Details of it are
in A.1.
Figure 2: Illustrative EAP (left) and resulting queue length process (right)
of HDS with $\mu_{1}>\mu_{2}$ under case 1:
$\gamma_{1}\leq\frac{\mu_{2}}{\mu_{1}}\cdot\gamma_{2}$ and the condition
$T_{1,f}>0$ in (4). $T\in(T_{1,f},T_{2,f})$ is the time at which queue 1
empties after time zero. In queue 1, since class 1 users stop arriving after
$T_{1,f}$, class 1 waiting mass decreases faster in
$[T_{1,f},\tau_{1}(T_{1,f})]$ than in $[0,T_{1,f}]$.
Case 2 $\frac{\mu_{2}}{\mu_{1}}\cdot\gamma_{2}<\gamma_{1}<\gamma_{2}$: By
Lemma 3.1, $[T_{1,a},T_{1,f}]$ and $[T_{2,a},T_{2,f}]$ must overlap, and by
Lemma 3.6, class 2 shouldn’t finish arriving before class 1, i.e.,
$T_{1,f}\leq T_{2,f}$. Therefore, every EAP must have its support boundaries
$T_{1,a},T_{1,f},T_{2,a},T_{2,f}$ ordered by one of the following two ways:
1. Type I:
$T_{1,a}\leq T_{2,a}<T_{1,f}\leq T_{2,f}$ with $T_{1,a}<0$, and,
2. Type II:
$T_{2,a}<T_{1,a}<T_{1,f}\leq T_{2,f}$ with $T_{2,a}<0$.
Note that, we need $\min\\{T_{1,a},T_{2,a}\\}<0$, otherwise, any user will be
strictly better off arriving at time zero when the network is empty. The
following lemma now specifies the necessary and sufficient conditions for
existence of an EAP under these two types. Existence of unique EAP in case 2
of Theorem 3.1 follows trivially from this lemma.
###### Lemma 3.8.
If $\gamma_{2}>\gamma_{1}>\frac{\mu_{2}}{\mu_{1}}\gamma_{2}$, the following
statements are true:
* •
There exists an EAP under Type I if and only if
$\Lambda_{1}\geq\frac{1-\gamma_{2}}{1-\gamma_{1}}\left(\frac{\mu_{1}\gamma_{1}}{\mu_{2}\gamma_{2}}-1\right)\Lambda_{2}$
and if it exists, it will be unique with a closed form same as the joint
arrival profile under case 2a of Theorem 3.1.
* •
There exists an EAP under Type II if and only if
$\Lambda_{1}<\frac{1-\gamma_{2}}{1-\gamma_{1}}\left(\frac{\mu_{1}\gamma_{1}}{\mu_{2}\gamma_{2}}-1\right)\Lambda_{2}$
and if it exists, it will be unique with a closed form same as the joint
arrival profile under case 2b of Theorem 3.1.
Proof Sketch of Lemma 3.8: Identifying the support boundaries: For every EAP
under Type I, we identify the following system of equations to be satisfied by
its support boundaries $T_{1,a},T_{1,f},T_{2,a},T_{2,f}$:
1) By Lemma 3.4, $F_{2}^{\prime}(t)=\mu_{2}\gamma_{2}$ in $[T_{2,a},T_{2,f}]$,
giving us: $T_{2,f}=T_{2,a}+\frac{\Lambda_{2}}{\mu_{2}\gamma_{2}}$.
2) Since $T_{1,a}<0$, queue 1 starts serving at time zero and must have
positive waiting time in $(0,T_{1,f})$. By Lemma 3.5, queue 1 must be empty at
time $T_{1,f}$. Therefore, $F_{1}(T_{1,f})+F_{2}(T_{1,f})=\mu_{1}T_{1,f}$. Now
$F_{1}(T_{1,f})=\Lambda_{1}$ and
$F_{2}(T_{1,f})=\mu_{2}\gamma_{2}(T_{1,f}-T_{2,a})$ (by Lemma 3.4). Plugging
this in, we get:
$\Lambda_{1}+\mu_{2}\gamma_{2}(T_{1,f}-T_{2,a})=\mu_{1}T_{1,f}$.
3) By definition of EAP,
$C_{\mathbf{F}}^{(1)}(T_{1,a})=C_{\mathbf{F}}^{(1)}(T_{1,f})$. Queue 1 empties
at $T_{1,f}$ (by Lemma 3.5), giving us
$C_{\mathbf{F}}^{(1)}(T_{1,f})=(1-\gamma_{1})T_{1,f}$. The first class 1 user
arriving at time $T_{1,a}$ gets served by queue 1 at time zero and waits for
$-T_{1,a}$ time, giving us $C_{\mathbf{F}}^{(1)}(T_{1,a})=-\gamma_{1}T_{1,a}$.
Equating these two costs, we get:
$T_{1,a}=-\left(\frac{1}{\gamma_{1}}-1\right)T_{1,f}$.
4) By Lemma 3.3, queue 2 stays engaged in
$[\tau_{1}(T_{2,a}),\tau_{1}(T_{2,f})]$ and by Lemma 3.5, queue 2 empties at
$\tau_{1}(T_{2,f})$. In this way, queue 2 serves the class 2 population in
$[\tau_{1}(T_{2,a}),\tau_{1}(T_{2,f})]$, giving us
$\mu_{2}\cdot(\tau_{1}(T_{2,f})-\tau_{1}(T_{2,a}))=\Lambda_{2}$. Queue 1
empties at $T_{1,f}$ (by Lemma 3.5) and after $T_{1,f}$, since class 2 users
arrive at rate $\mu_{2}\gamma_{2}<\mu_{1}\gamma_{1}<\mu_{1}$, queue 1 remains
empty at $T_{2,f}$, giving us $\tau_{1}(T_{2,f})=T_{2,f}$. Since queue 1 has
positive waiting time in $(T_{1,a},T_{2,a}]$, we have
$\tau_{1}(T_{2,a})-\tau_{1}(T_{1,a})=\frac{F_{1}(T_{2,a})-F_{1}(T_{1,a})}{\mu_{1}}=\gamma_{1}(T_{2,a}-T_{1,a})$
(by (3) and Lemma 3.4), which upon putting $\tau_{1}(T_{1,a})=0$ gives us
$\tau_{1}(T_{2,a})=\gamma_{1}(T_{2,a}-T_{1,a})$. Placing the expressions
obtained for $\tau_{1}(T_{2,a})$ and $\tau_{1}(T_{2,f})$ into
$\mu_{2}(\tau_{1}(T_{2,f})-\tau_{1}(T_{2,a}))=\Lambda_{2}$, we get,
$\mu_{2}(T_{2,f}-\gamma_{1}(T_{2,a}-T_{1,a}))=\Lambda_{2}$.
Getting the necessary condition: Solution of the identified system of
equations is ( ‣ 2.) and therefore, every EAP under Type I must have ( ‣ 2.)
as support boundaries. Now ( ‣ 2.) must be ordered by $T_{1,a}\leq
T_{2,a}<T_{1,f}\leq T_{2,f}$ to represent an EAP of Type I. Note that ( ‣ 2.)
satisfies
$T_{2,a}-T_{1,a}=\frac{1}{\mu_{1}\gamma_{1}}\left[\Lambda_{1}-\frac{1-\gamma_{2}}{1-\gamma_{1}}\left(\frac{\mu_{1}\gamma_{1}}{\mu_{2}\gamma_{2}}-1\right)\Lambda_{2}\right]$.
As a result, for $T_{2,a}\geq T_{1,a}$, we need
$\Lambda_{1}\geq\frac{1-\gamma_{2}}{1-\gamma_{1}}\left(\frac{\mu_{1}\gamma_{1}}{\mu_{2}\gamma_{2}}-1\right)\Lambda_{2}$
and therefore this is a necessary condition for existence of an EAP under Type
I.
Proving sufficiency of the obtained necessary condition: Now, if
$\Lambda_{1}\geq\frac{1-\gamma_{2}}{1-\gamma_{1}}\left(\frac{\mu_{1}\gamma_{1}}{\mu_{2}\gamma_{2}}-1\right)\Lambda_{2}$,
it is easy to verify that ( ‣ 2.) satisfies $T_{1,a}\leq T_{2,a}<T_{1,f}\leq
T_{2,f}$. Therefore, with the necessary condition satisfied and upon plugging
in arrival rates using Lemma 3.4, we get a candidate having arrival rates and
support boundaries same as the joint arrival profile mentioned under case 2a
of Theorem 3.1. This candidate satisfies
$F_{1}(T_{1,f})=\Lambda_{1},~{}F_{2}(T_{2,f})=\Lambda_{2}$ and will be the
only candidate under Type I to qualify as an EAP. Proving that this candidate
is an EAP requires analyzing the behavior of the two queues. Details of this
argument is in A.1. Therefore, it follows that the obtained necessary
condition is also sufficient for existence of an EAP under Type I, and once it
is satisfied, there is a unique EAP under Type I which has a closed form same
as the one mentioned under case 2a of Theorem 3.1. Thus the first statement of
the lemma follows.
The second statement is proved via an argument similar to the one used for
proving the first statement via identifying a system of equations satisfied by
the support boundaries and getting the necessary condition by imposing
$T_{2,a}<T_{1,a}$ on the solution to that system. Details of the argument in
A.1.1. ∎
Case 3: $\gamma_{1}>\gamma_{2}$: By Lemma 3.1, $[T_{1,a},T_{1,f}]$ and
$[T_{2,a},T_{2,f}]$ must overlap, and by Lemma 3.6, $T_{1,a}>T_{2,a}$.
Therefore, the support boundaries of any EAP must be ordered by the following
two orderings:
1. Type I:
$T_{2,a}<T_{1,a}<T_{2,f}\leq T_{1,f}$ , and
2. Type II:
$T_{2,a}<T_{1,a}<T_{1,f}<T_{2,f}$.
For both the above types, every EAP must have $T_{2,a}<0$. The following lemma
specifies the necessary and sufficient condition for existence of an EAP under
the two types mentioned above. Existence of unique EAP under case 3 of Theorem
3.1 follows trivially from this lemma.
###### Lemma 3.9.
If $\gamma_{1}>\gamma_{2}$, the following statements are true:
1. 1.
There exists an EAP under Type I if and only if
$\Lambda_{1}\geq\left(\frac{\mu_{1}}{\mu_{2}}-1\right)\Lambda_{2}$ and if it
exists, it has a closed form same as the one mentioned under case 3a of
Theorem 3.1.
2. 2.
There exists an EAP under Type II if and only if
$\Lambda_{1}<\left(\frac{\mu_{1}}{\mu_{2}}-1\right)\Lambda_{2}$ and if it
exists, it has a closed form same as the one mentioned under case 3b of
Theorem 3.1.
Proof of the above lemma follows an argument similar to the proof of Lemma 3.8
by identifying a system of linear equations to be satisfied by the support
boundaries of any EAP under the two types. Details are in A.1.1. ∎
The proof of the main results in Sections 3.2, 4.1 and 4.2 will follow a
similar sequence of arguments as used in the proof of Theorem 3.1, through
elimination of candidates not satisfying structural properties of every EAP in
that regime.
Figure 3: Illustrative EAP (left) and resulting queue length process (right)
of HDS with $\mu_{1}>\mu_{2}$ under case 2a of Theorem 3.1 with the assumption
$T_{2,a}<0$ in ( ‣ 2.). Queue 1 divides its capacity between the two classes
from $\tau_{1}(T_{2,a})$ proportionally to their arrival rate. As a result,
queue 1 serves class 1 users at a slower rate in $[\tau_{1}(T_{2,a}),T_{1,f}]$
than in $[0,\tau_{1}(T_{2,a})]$, causing class 1 mass in queue 1 to decrease
at a slower rate in the former interval than in the latter one. Figure 4:
Illustrative EAP (left) of HDS with $\mu_{1}>\mu_{2}$ under case 2b and 3b of
Theorem 3.1. The resulting queue length process (right) is for case 2b with
the assumption $T_{2,a}<0$ in (3.). Queue 1 divides its capacity between the
two classes from $\tau_{1}(T_{1,a})$ proportionally to their service rates. As
a result, queue 1 serves class 2 users at a lesser rate in
$[\tau_{1}(T_{1,a}),T_{1,f}]$ than in $[0,\tau_{1}(T_{1,a})]$, causing class 2
waiting mass to decrease at a lesser rate in the former interval than in the
latter one. In queue 2, waiting mass increases in
$[\tau_{1}(T_{1,a}),T_{1,f}]$ because class 2 arrival rate to queue 2 is
$\mu_{2}\gamma_{1}/\gamma_{2}>\mu_{2}$. However the waiting mass increases in
$[\tau_{1}(T_{1,a}),T_{1,f}]$ at a rate slower than in
$[0,\tau_{1}(T_{1,a})]$, since the arrival rate in the formal interval is
$\mu_{2}\gamma_{2}/\gamma_{1}<\mu_{1}=$ arrival rate in the latter interval.
For case 3b of Theorem 3.1, with the assumption $T_{1,a}<0$ on (3.), queue 1
length process has the same structure. Queue 2 length process of case 3b is
also same as case 2b, except for one region: queue 2 length decreases in
$[\tau_{1}(T_{1,a}),T_{1,f}]$, since class 2 arrival rate to queue 2 is
$\mu_{2}\gamma_{2}/\gamma_{1}<\mu_{2}$ in case 3b. Figure 5: Illustrative EAP
(left) and resulting queue length process (right) of HDS with
$\mu_{1}>\mu_{2}$ under case 3a of Theorem 3.1 with the assumption $T_{1,a}<0$
and
$\tau_{1}(T_{1,a})=\frac{\mu_{2}\gamma_{2}}{\mu_{1}}(T_{1,a}-T_{2,a})<T_{2,f}$
in ( ‣ 2.). In $[\tau_{1}(T_{1,a}),T_{1,f}]$, queue 1 divides its capacity
between the two classes proportionally to their arrival rates. As a result,
class 2 mass in queue 1 decreases at a lesser rate in
$[\tau_{1}(T_{1,a}),T_{2,f}]$ than in $[0,\tau_{1}(T_{1,a})]$. After
$T_{2,f}$, as class 2 users stop arriving, the class 2 mass in queue 1
decreases to zero at a higher rate in $[T_{2,f},\tau_{1}(T_{2,f})]$ than in
$[\tau_{1}(T_{1,a}),T_{2,f}]$. Class 1 mass in queue 1 initially decreases in
$[\tau_{1}(T_{1,a}),T_{2,f}]$. Since class 1 users increase their arrival rate
after $T_{2,f}$, slope of the red line increases and the class 1 mass in queue
1 can even increase in $[T_{2,f},\tau_{1}(T_{2,f})]$ if
$\mu_{1}\gamma_{1}>\mu_{1}-\mu_{2}\gamma_{2}/\gamma_{1}$. After
$\tau_{1}(T_{2,f})$, queue 1 allocates the entire capacity to class 1 and the
class 1 mass in queue 1 decreases to zero in $[\tau_{1}(T_{2,f}),T_{1,f}]$.
### 3.2 Equal Preferences: $\gamma_{1}=\gamma_{2}=\gamma$
Structural properties of any EAP. We first identify the structural properties
of any EAP in this regime and later we exploit them to refine our search of
EAP. By Lemma 3.2, we can restrict our search to joint arrival profiles having
supports satisfying $\mathcal{S}(F_{1})=[T_{1,a},T_{1,f}]$ and
$\mathcal{S}(F_{1})\cup\mathcal{S}(F_{2})=[T_{a},T_{f}]$ for some
$T_{f}>T_{a}$ and $T_{1,f}>T_{1,a}$. Lemma 3.10 states the arrival rates of
the two classes in EAP and is proved (in A.2) via an argument similar to the
one used for proving Lemma 3.4. Lemma 3.11 is proved via contradiction, by
showing that, if the stated condition doesn’t hold, users of one of the
classes can improve by arriving at a different time.
###### Lemma 3.10 (Rate of arrival).
If $\mu_{1}>\mu_{2}$ and $\gamma_{1}=\gamma_{2}=\gamma$, the following
properties must be true almost everywhere for every EAP
$\mathbf{F}=\\{F_{1},F_{2}\\}$,
$\displaystyle F_{1}^{\prime}(t)+F_{2}^{\prime}(t)$
$\displaystyle=\mu_{1}\gamma~{}~{}~{}\text{if $t\in\mathcal{S}(F_{1})$, and,}$
$\displaystyle F_{2}^{\prime}(t)$
$\displaystyle\leq\mu_{2}\gamma~{}~{}~{}\text{if $t\in\mathcal{S}(F_{2})$,
with equality when $Q_{2}(\tau_{1}(t))>0$}.$
###### Lemma 3.11.
If $\mu_{1}>\mu_{2}$ and $\gamma_{1}=\gamma_{2}=\gamma$, in the EAP,
1. 1.
If $\mathcal{S}(F_{2})$ has a gap $(t,t+\delta]$ at $t\in\mathcal{S}(F_{2})$
for some $\delta>0$ sufficiently small, queue 2 must be empty at
$\tau_{1}(t)$.
2. 2.
Queue 1 must be empty time at $T_{1,f}$.
Specification of the EAP.
###### Theorem 3.2.
If $\mu_{1}>\mu_{2}$ and $\gamma_{1}=\gamma_{2}=\gamma$, EAP of HDS exists
and:
1. 1.
If $\Lambda_{1}<\left(\frac{\mu_{1}}{\mu_{2}}-1\right)\Lambda_{2}$, the EAP is
unique and its arrival rates and support boundaries are:
$\displaystyle F_{1}^{\prime}(t)$
$\displaystyle=(\mu_{1}-\mu_{2})\gamma~{}\text{for}~{}t\in[T_{1,a},T_{1,f}]~{}\text{and},~{}F_{2}^{\prime}(t)=\mu_{2}\gamma~{}\text{for}~{}t\in[T_{a},T_{f}],$
$\displaystyle\text{where},~{}T_{1,a}$
$\displaystyle=\frac{1-\gamma}{\mu_{1}-\mu_{2}\gamma}\left[\Lambda_{2}-\frac{\mu_{1}}{(\mu_{1}-\mu_{2})\gamma}\Lambda_{1}\right],~{}T_{1,f}=\frac{\Lambda_{1}+(1-\gamma)\Lambda_{2}}{\mu_{1}-\mu_{2}\gamma},~{}T_{a}=-\frac{1-\gamma}{\gamma}\frac{\Lambda_{2}}{\mu_{2}},~{}\text{and}~{}T_{f}=\frac{\Lambda_{2}}{\mu_{2}}.$
2. 2.
If $\Lambda_{1}\geq\left(\frac{\mu_{1}}{\mu_{2}}-1\right)\Lambda_{2}$, the set
of EAPs is the convex set of joint arrival profiles
$\mathbf{F}=\\{F_{1},F_{2}\\}$ satisfying: 1)
$\mathcal{S}(F_{1})=[T_{a},T_{f}]$,
$\mathcal{S}(F_{2})\subseteq[T_{a},T_{f}]$, for $i=1,2$
$F_{i}(T_{f})=\Lambda_{i}$, 2)
$F_{1}^{\prime}(t)+F_{2}^{\prime}(t)=\mu_{1}\gamma$, and 3)
$F_{2}^{\prime}(t)\leq\mu_{2}\gamma$ for every $t\in[T_{a},T_{f}]$, where
$T_{a}=-\left(\frac{1}{\gamma}-1\right)\frac{\Lambda_{1}+\Lambda_{2}}{\mu_{1}}$
and $T_{f}=\frac{\Lambda_{1}+\Lambda_{2}}{\mu_{1}}$.
We only provide the key steps of the proof of Theorem 3.2 below. The details
are given in A.2.1. Following two supporting lemmas are necessary for proving
Theorem 3.2.
###### Lemma 3.12.
In every EAP, the cost of the class 2 users remains constant over
$[T_{a},T_{f}]$.
###### Lemma 3.13.
In every EAP, if $t\in\mathcal{S}(F_{1})\cap\mathcal{S}(F_{2})$, then class 2
users will arrive at queue 2 at a maximum rate of $\mu_{2}$ at time
$\tau_{1}(t)$.
Proofs of Lemma 3.12 and Lemma 3.13 are in A.2. Lemma 3.12 is proved via
exploiting Lemma 3.11 and the fact that the two classes have equal cost
preferences. Proof of Lemma 3.13 is based on the observation that
$A_{2}(\tau_{1}(t))=F_{2}(t)$. As a result, at time $\tau_{1}(t)$, class 2
users arrive at queue 2 at rate
$A_{2}^{\prime}(\tau_{1}(t))=\frac{F_{2}^{\prime}(t)}{\tau_{1}^{\prime}(t)}$,
assuming that the derivatives exist. Following this, we use Lemma 3.10 and (3)
to upper bound the rate of arrival.
Key steps of proof of Theorem 3.2: By Lemma 3.2, we consider candidates
$\mathbf{F}=\\{F_{1},F_{2}\\}$ which are absolutely continuous with
$\mathcal{S}(F_{1})=[T_{1,a},T_{1,f}],~{}\mathcal{S}(F_{1})\cup\mathcal{S}(F_{2})=[T_{a},T_{f}]$
and have arrival rates $F_{1}^{\prime}(\cdot),~{}F_{2}^{\prime}(\cdot)$ given
by Lemma 3.10. We eliminate candidates which do not satisfy the structural
properties of any EAP and narrow our search of an EAP to a smaller set of
candidates. We follow this agenda separately over two subsets of candidates
with: 1) $T_{f}>T_{1,f}$ and 2) $T_{f}=T_{1,f}$. Lemma 3.14 and 3.15 provides
the necessary and sufficient condition for existence of EAPs of the mentioned
two types. The statement of Theorem 3.2 follows from these two lemmas.
###### Lemma 3.14.
If $\gamma_{1}=\gamma_{2}=\gamma$, there exists an EAP with $T_{f}>T_{1,f}$ if
and only if $\Lambda_{1}<\left(\frac{\mu_{1}}{\mu_{2}}-1\right)\Lambda_{2}$,
and if it exists, it will be unique with a closed form same as the joint
arrival profile mentioned under case 1 of Theorem 3.2.
###### Lemma 3.15.
If $\gamma_{1}=\gamma_{2}=\gamma$, there exists an EAP with $T_{f}=T_{1,f}$ if
and only if
$\Lambda_{1}\geq\left(\frac{\mu_{1}}{\mu_{2}}-1\right)\Lambda_{2}$, and if it
exists, the set of all EAPs with $T_{1,f}=T_{f}$ is the set of joint arrival
profiles mentioned under case 2 of Theorem 3.2.
Below we sketch the proof of Lemma 3.14 and 3.15. The remaining details are
presented in A.2.1.
Proof sketch of Lemma 3.14: Identifying the support boundaries: First we
identify the support boundaries $T_{a},T_{f}$ of any EAP with $T_{f}>T_{1,f}$:
1. 1)
Exploiting Lemma 3.13 and 3.11, we argue that queue 2 has a positive waiting
time in $(0,T_{f})$ and empties at $T_{f}$. As a result, queue 2 serves the
whole class 2 population in $(0,T_{f})$, giving us
$T_{f}=\frac{\Lambda_{2}}{\mu_{2}}$.
2. 2)
Using Lemma 3.12 we have,
$C_{\mathbf{F}}^{(2)}(T_{a})=C_{\mathbf{F}}^{(2)}(T_{f})$. The class 2 user
arriving at $T_{a}$ gets served at time zero causing
$C_{\mathbf{F}}^{(2)}(T_{a})=-\gamma T_{a}$. Since the network is empty at
$T_{f}$, $C_{\mathbf{F}}^{(2)}(T_{f})=(1-\gamma)T_{f}$. Therefore,
$C_{\mathbf{F}}^{(2)}(T_{a})=C_{\mathbf{F}}^{(2)}(T_{f})$ implies
$T_{a}=-\left(\frac{1}{\gamma}-1\right)T_{f}=-\left(\frac{1}{\gamma}-1\right)\frac{\Lambda_{2}}{\mu_{2}}$.
3. 3)
The class 2 population of mass $\Lambda_{2}$ has to arrive in the interval
$\left[T_{a},T_{f}\right]$ of length $\frac{\Lambda_{2}}{\mu_{2}\gamma_{2}}$
at maximum rate of $\mu_{2}\gamma_{2}$ (by Lemma 3.10). This leaves us with
only one possible arrival rate for class 2 users:
$F_{2}^{\prime}(t)=\mu_{2}\gamma_{2}$ in $[T_{a},T_{f}]$. As a result, by
Lemma 3.10, we must have $F_{1}^{\prime}(t)=(\mu_{1}-\mu_{2})\gamma$ in
$[T_{1,a},T_{1,f}]$, giving us
$T_{1,f}=T_{1,a}+\frac{\Lambda_{1}}{(\mu_{1}-\mu_{2})\gamma}$.
4. 4)
Since $T_{a}<0$, queue 1 starts serving from time zero and has positive
waiting time in $[0,T_{1,f})$. By Lemma 3.11, queue 1 empties at $T_{1,f}$.
Therefore, $\mu_{1}T_{1,f}=F_{1}(T_{1,f})+F_{2}(T_{1,f})=\Lambda_{1}+\newline
\mu_{2}\gamma\cdot(T_{1,f}-T_{a})$.
Getting the necessary condition: Solving the above system of equations, we get
$T_{a}=-\left(\frac{1}{\gamma}-1\right)\frac{\Lambda_{2}}{\mu_{2}}$,
$T_{f}=\frac{\Lambda_{2}}{\mu_{2}}$,
$T_{1,a}=\frac{1-\gamma}{\mu_{1}-\mu_{2}\gamma}\left[\Lambda_{2}-\frac{\mu_{1}}{(\mu_{1}-\mu_{2})\gamma}\Lambda_{1}\right]$,
$T_{1,f}=\frac{\Lambda_{1}+(1-\gamma)\Lambda_{2}}{\mu_{1}-\mu_{2}\gamma}$ and
as a result, every EAP with $T_{f}>T_{1,f}$ has $T_{a},T_{f},T_{1,a},T_{1,f}$
same as the ones we identified. For $T_{f}>T_{1,f}$, we need
$\Lambda_{1}<\left(\frac{\mu_{1}}{\mu_{2}}-1\right)\Lambda_{2}$ and therefore,
this is a necessary condition for existence of an EAP with $T_{f}>T_{1,f}$.
Identifying the unique EAP: If the necessary condition
$\Lambda_{1}<\left(\frac{\mu_{1}}{\mu_{2}}-1\right)\Lambda_{2}$ is satisfied,
it is easy to verify that, the support boundaries obtained above follow the
ordering $T_{a}<T_{1,a}<T_{1,f}<T_{f}$. With this ordering satisfied, the only
candidate with $T_{f}>T_{1,f}$, which qualifies to be an EAP is:
$F_{1}^{\prime}(t)=(\mu_{1}-\mu_{2})\gamma$ in $[T_{1,a},T_{1,f}]$ and
$F_{2}^{\prime}(t)=\mu_{2}\gamma$ in $[T_{a},T_{f}]$, with
$T_{a},T_{f},T_{1,a},T_{1,f}$ same as we identified before. Moreover, the
candidate obtained is same as the joint arrival profile under case 1 of
Theorem 3.2.
Proving sufficiency of the obtained necessary condition: Another interesting
observation is, if
$\Lambda_{1}<\left(\frac{\mu_{1}}{\mu_{2}}-1\right)\Lambda_{2}$, the obtained
candidate with $T_{f}>T_{1,f}$ is same as the EAPs in cases 2b and 3b of
Theorem 3.1, respectively, upon taking limits
$\gamma_{1}=\gamma,~{}\gamma_{2}\to\gamma+$ and
$\gamma_{2}=\gamma,~{}\gamma_{1}\to\gamma+$. Proving that this candidate is an
EAP follows by the argument used for proving the unique Type II candidate in
Lemma 3.8 is an EAP (in A.1.1), except replacing $T_{2,a},T_{2,f}$ with
$T_{a},T_{f}$ and taking $\gamma_{1}=\gamma_{2}=\gamma$. Therefore, the
statement of Lemma 3.14 stands proved. ∎
Proof sketch of Lemma 3.15: Identifying $\mathbf{T_{a}}$: Since
$[0,T_{f}]\subseteq\mathcal{S}(F_{1})$, queue 1 must have a positive waiting
time in $[0,T_{f})$. By Lemma 3.11, queue 1 empties at time $T_{f}$. Therefore
we get $\mu_{1}T_{f}=F_{1}(T_{f})+F_{2}(T_{f})=\Lambda_{1}+\Lambda_{2}$
implying $T_{f}=\frac{\Lambda_{1}+\Lambda_{2}}{\mu_{1}}$.
Identifying $\mathbf{T_{f}}$: By Lemma 3.12, we have
$C_{\mathbf{F}}^{(2)}(T_{a})=C_{\mathbf{F}}^{(2)}(T_{f})$. Using the argument
used in the proof sketch of Lemma 3.14 for finding $T_{a}$, we get
$T_{a}=-\left(\frac{1}{\gamma}-1\right)T_{f}=-\left(\frac{1}{\gamma}-1\right)\frac{\Lambda_{1}+\Lambda_{2}}{\mu_{1}}$.
Getting the necessary condition: By Lemma 3.10, all class 2 users arrive in
$[T_{a},T_{f}]$ at a maximum rate of $\mu_{2}\gamma$. Therefore for existence
of an EAP with $T_{f}=T_{1,f}$, we need the necessary condition
$\mu_{2}\gamma\cdot(T_{f}-T_{a})\geq\Lambda_{2}$, which after some
manipulation gives us
$\Lambda_{1}\geq\left(\frac{\mu_{1}}{\mu_{2}}-1\right)\Lambda_{2}$.
Identifying the set of EAPs: By Lemma 3.10, $\mu_{1}\gamma$ is the maximum
rate at which the entire population of mass $\Lambda_{1}+\Lambda_{2}$ can
arrive within the time interval $[T_{a},T_{f}]$ of length
$\frac{\Lambda_{1}+\Lambda_{2}}{\mu_{1}\gamma}$. Therefore, we must have
$F_{1}^{\prime}(t)+F_{2}^{\prime}(t)=\mu_{1}\gamma$ in $[T_{a},T_{f}]$. Also
by Lemma 3.10, we have $F_{1}^{\prime}(t)+F_{2}^{\prime}(t)=\mu_{1}\gamma$
only in $\mathcal{S}(F_{1})=[T_{1,a},T_{1,f}]\subseteq[T_{a},T_{f}]$, which
implies $T_{1,a}=T_{a}$. By Lemma 3.13, class 2 users arrive at queue 2 from
queue 1 in $[0,T_{f}]$ at a maximum rate of $\mu_{2}$. Hence queue 2 stays
empty and as a result, by Lemma 3.10, $F_{2}^{\prime}(t)\leq\mu_{2}\gamma$ for
every $t\in[T_{a},T_{f}]$. Therefore, our reduced set of candidates with
$T_{f}=T_{1,f}$ will be the set of all joint arrival profiles
$\mathbf{F}=\\{F_{1},F_{2}\\}$ satisfying: 1)
$\mathcal{S}(F_{1})=[T_{a},T_{f}]$ and
$\mathcal{S}(F_{2})\subseteq[T_{a},T_{f}]$, for $i=1,2$
$F_{i}(T_{f})=\Lambda_{i}$, 2)
$F_{1}^{\prime}(t)+F_{2}^{\prime}(t)=\mu_{1}\gamma$, and 3)
$F_{2}^{\prime}(t)\leq\mu_{2}\gamma$ for every $t\in[T_{a},T_{f}]$ where
$T_{a}=-\frac{1-\gamma}{\gamma}\frac{\Lambda_{1}+\Lambda_{2}}{\mu_{1}}$ and
$T_{f}=\frac{\Lambda_{1}+\Lambda_{2}}{\mu_{1}}$, which is exactly the set
mentioned under the case 2 of Theorem 3.2.
Proving sufficiency of the obtained necessary condition: It is easy to verify
that, if the necessary condition
$\Lambda_{1}\geq\left(\frac{\mu_{1}}{\mu_{2}}-1\right)\Lambda_{2}$ holds, the
set of candidates with $T_{f}=T_{1,f}$ mentioned earlier is non-empty. Two
elements of the set are the limits of the EAPs in cases 2a and 3a of Theorem
3.1, respectively, when $\gamma_{1}=\gamma,~{}\gamma_{2}\to\gamma+$ and
$\gamma_{2}=\gamma,~{}\gamma_{1}\to\gamma+$. After this, we prove that, if
$\Lambda_{1}\geq\left(\frac{\mu_{1}}{\mu_{2}}-1\right)\cdot\Lambda_{2}$, every
joint arrival profile $\mathbf{F}=\\{F_{1},F_{2}\\}$ in the obtained set of
candidates is an EAP. This step requires exploiting Lemma 3.12 and the fact
that queue 2 stays empty when users are arriving by such a candidate and the
details of it are in A.2.1. Hence, the statement of Lemma 3.15 stands proved.
∎
## 4 Heterogeneous Arrival System (HAS)
In this section, we consider the two groups arrive at different queues and
depart from the same queue as illustrated in Table 1. Namely, the $i$-th group
arrive at $i$-th queue and both the groups depart from the 2nd queue. We
consider the case $\gamma_{1}\neq\gamma_{2}$ and $\gamma_{1}=\gamma_{2}$
separately, since the later displays different behaviors.
### 4.1 Unequal Preferences $\gamma_{1}\neq\gamma_{2}$
As for HDS, here too we narrow down our search for EAP by exploiting its
structural properties. The detailed account of these structural properties as
well as detailed proofs of results in this section are given in B.1. The main
results in this section are proved by arguments similar to those used for
proving Theorem 3.1 for HDS, except that we now exploit a different set of
structural properties and the no. of different regimes is significantly larger
and diverse for HAS. In this section, we state the key results and after
Theorem 4.2 in Remark 7, we discuss two interesting cases: 1) where the EAP
corresponds to arrivals in a stream coming in disjoint intervals and 2) where
all the arrivals for class 2 arrive before time zero.
The following lemma identifies an important threshold property of an EAP.
Proof of Lemma 4.1 is in B.1 and follows a sequence of arguments similar to
the one used for proving Lemma 3.1.
###### Lemma 4.1 (Threshold Behavior).
If $\gamma_{1}\neq\gamma_{2}$, in EAP, queue 2 serves the two classes over
disjoint sets of time if and only if $\mu_{1}\geq\mu_{2}\gamma_{2}$.
Proof sketch: Proof of Lemma 4.1 is similar to that of Lemma 3.1. First we
argue via a contradiction that if $\mu_{1}\geq\mu_{2}\gamma_{2}$, then queue 2
cannot serve the two classes together in an EAP. If queue 2 is serving the two
classes together, we argue that, queue 1 must stay engaged in a neighbourhood
of that time for class 1 to be iso cost (by Lemma B.1 in B.1). As a result,
class 1 users arrive from queue 1 to 2 at rate $\mu_{1}$. For class 2 have
constant cost in that neighbourhood, we argue that, users of the two classes
arrive at queue 2 at a combined rate of $\mu_{2}\gamma_{2}$. Since queue 2
serves a positive mass of both the classes together, the combined arrival rate
must be strictly larger than the arrival rate of class 1 to queue 2, implying
$\mu_{2}\gamma_{2}>\mu_{1}$ and contradicting $\mu_{1}\geq\mu_{2}\gamma_{2}$.
The other direction is proved via contradiction by exploiting the structures
of $\mathcal{S}(F_{1}),~{}\mathcal{S}(F_{2})$ in an EAP and showing that, if
$\mu_{1}<\mu_{2}\gamma_{2}$ and queue 2 is serving the two classes over
disjoint sets of times, we can find a user who can decrease her cost by
arriving at a different time. ∎
Specification of the EAP. The two regimes $\mu_{1}<\mu_{2}\gamma_{2}$ and
$\mu_{1}\geq\mu_{2}\gamma_{2}$ exhibit substantially different EAP structures,
owing to the threshold behavior given by Lemma 4.1. For conciseness, we
specify the unique EAP in these two regimes separately in Theorem 4.1 (for
$\mu_{1}<\mu_{2}\gamma_{2}$) and 4.2 (for $\mu_{1}\geq\mu_{2}\gamma_{2}$).
Proofs of these theorems (in B.1) have a structure similar to the proof of
Theorem 3.1.
For the two classes $i=1,2$, we define the quantities
$T_{i,a}=\inf\mathcal{S}(F_{i})$ and $T_{i,f}=\sup\mathcal{S}(F_{i})$. In an
EAP, both the sets $\mathcal{S}(F_{1}),~{}\mathcal{S}(F_{2})$ are compact and
therefore $T_{1,a},T_{1,f},T_{2,a},T_{2,f}$ are all finite.
Regime I $\mu_{1}<\mu_{2}\gamma_{2}$. We mention below the support boundaries
of the unique EAP, which we will refer later in Theorem 4.1:
1. 1.
If $\mu_{1}<\mu_{2}\gamma_{2}$ and
$\Lambda_{1}\geq\max\left\\{\frac{1-\gamma_{2}}{1-\gamma_{1}},1\right\\}\cdot\frac{\mu_{1}}{\mu_{2}-\mu_{1}}\Lambda_{2}$,
then
$\displaystyle T_{1,a}$
$\displaystyle=-\frac{1-\gamma_{1}}{\gamma_{1}}\frac{\Lambda_{1}}{\mu_{1}}+\frac{1-\gamma_{2}}{\gamma_{1}}\frac{\Lambda_{2}}{\mu_{2}-\mu_{1}},~{}T_{1,f}=\frac{\Lambda_{1}}{\mu_{1}},~{}T_{2,a}=-\frac{1-\gamma_{2}}{\gamma_{2}}\frac{\Lambda_{2}}{\mu_{2}-\mu_{1}},~{}\text{and}~{}T_{2,f}=\frac{\Lambda_{2}}{\mu_{2}-\mu_{1}}.$
(10)
2. 2.
If $\mu_{1}<\mu_{2}\gamma_{2}$ and
$\Lambda_{1}<\max\left\\{\frac{1-\gamma_{2}}{1-\gamma_{1}},1\right\\}\cdot\frac{\mu_{1}}{\mu_{2}-\mu_{1}}\Lambda_{2}$,
then
1. a)
if $\gamma_{1}>\gamma_{2}$, then
$\displaystyle T_{1,a}$
$\displaystyle=\frac{1}{\mu_{2}}\left(\Lambda_{2}-\frac{1-\gamma_{1}}{1-\gamma_{2}}\frac{\mu_{2}-\mu_{1}}{\mu_{1}}\Lambda_{1}\right),~{}T_{1,f}=\frac{1}{\mu_{2}}\left(\Lambda_{2}+\frac{(\gamma_{1}-\gamma_{2})\mu_{2}+(1-\gamma_{1})\mu_{1}}{(1-\gamma_{2})\mu_{1}}\Lambda_{1}\right),$
$\displaystyle T_{2,a}$
$\displaystyle=-\frac{1}{\mu_{2}\gamma_{2}}\left((1-\gamma_{1})\Lambda_{1}+(1-\gamma_{2})\Lambda_{2}\right),~{}\text{and}~{}T_{2,f}=\frac{1}{\mu_{2}}\left(\Lambda_{2}+\frac{1-\gamma_{1}}{1-\gamma_{2}}\Lambda_{1}\right).$
(11)
2. b)
if $\gamma_{1}<\gamma_{2}$, then
$\displaystyle T_{1,a}$
$\displaystyle=-\left(\frac{\gamma_{2}}{\gamma_{1}}-1\right)\frac{\Lambda_{1}}{\mu_{1}},~{}T_{1,f}=\frac{\Lambda_{1}}{\mu_{1}},~{}T_{2,a}=-\frac{1-\gamma_{2}}{\gamma_{2}}\frac{\Lambda_{1}+\Lambda_{2}}{\mu_{2}},~{}\text{and}~{}T_{2,f}=\frac{\Lambda_{1}+\Lambda_{2}}{\mu_{2}}.$
(12)
###### Theorem 4.1.
If $\gamma_{1}\neq\gamma_{2}$ and $\mu_{1}<\mu_{2}\gamma_{2}$, HAS has a
unique EAP which has the following arrival rates and support boundaries:
1. 1.
If $\gamma_{1}>\gamma_{2}$:
1. 1a.
when
$\Lambda_{1}\geq\frac{1-\gamma_{2}}{1-\gamma_{1}}\frac{\mu_{1}}{\mu_{2}-\mu_{1}}\Lambda_{2}$:
$\displaystyle F_{1}^{\prime}(t)$
$\displaystyle=\begin{cases}\frac{\mu_{1}\gamma_{1}}{\gamma_{2}}~{}~{}&\text{for}~{}t\in[T_{1,a},\tau_{1}^{-1}(T_{2,f})],\\\
\mu_{1}\gamma_{1}~{}~{}&\text{for}~{}t\in[\tau_{1}^{-1}(T_{2,f}),T_{1,f}],\end{cases}~{}\text{and}~{}F_{2}^{\prime}(t)=\begin{cases}\mu_{2}\gamma_{2}~{}~{}&\text{for}~{}t\in[T_{2,a},0],\\\
\mu_{2}\gamma_{2}-\mu_{1}~{}~{}&\text{for}~{}t\in[0,T_{2,f}],\\\ \end{cases}$
where $T_{1,a},T_{1,f},T_{2,a}$ and $T_{2,f}$ are given by (10).
2. 1b.
when
$\Lambda_{1}<\frac{1-\gamma_{2}}{1-\gamma_{1}}\frac{\mu_{1}}{\mu_{2}-\mu_{1}}\Lambda_{2}$:
$\displaystyle F_{1}^{\prime}(t)$
$\displaystyle=\begin{cases}\frac{\mu_{1}\gamma_{1}}{\gamma_{2}}~{}~{}&\text{for}~{}t\in[T_{1,a},\tau_{1}^{-1}(T_{2,f})],{}\\\
\mu_{1}\gamma_{1}~{}~{}&\text{for}~{}t\in[\tau_{1}^{-1}(T_{2,f}),T_{1,f}],\\\
\end{cases}~{}\text{and}~{}F_{2}^{\prime}(t)=\begin{cases}\mu_{2}\gamma_{2}~{}~{}&\text{for}~{}t\in[T_{2,a},T_{1,a}],\\\
\mu_{2}\gamma_{2}-\mu_{1}~{}~{}&\text{for}~{}t\in[T_{1,a},T_{2,f}],\\\
\end{cases}$
where $T_{1,a},T_{1,f},T_{2,a}$ and $T_{2,f}$ are given by ( ‣ 2.).
2. 2.
If $\gamma_{1}<\gamma_{2}$:
1. 2a.
when $\Lambda_{1}\geq\frac{\mu_{1}}{\mu_{2}-\mu_{1}}\Lambda_{2}$, the EAP has
closed form same as case 1a.
2. 2b.
when $\Lambda_{1}<\frac{\mu_{1}}{\mu_{2}-\mu_{1}}\Lambda_{2}$:
$\displaystyle F_{1}^{\prime}(t)$
$\displaystyle=\frac{\mu_{1}\gamma_{1}}{\gamma_{2}}~{}~{}\text{for}~{}t\in[T_{1,a},T_{1,f}],~{}\text{and}~{}F_{2}^{\prime}(t)=\begin{cases}\mu_{2}\gamma_{2}~{}~{}&\text{for}~{}t\in[T_{2,a},0]\cup[T_{1,f},T_{2,f}],\\\
\mu_{2}\gamma_{2}-\mu_{1}~{}~{}&\text{for}~{}t\in[0,T_{1,f}],\end{cases}$
where $T_{1,a},T_{1,f},T_{2,a}$ and $T_{2,f}$ are given by (12).
###### Remark 5.
_Illustrative EAPs and resulting queue length processes of cases 1a, 1b, 2a
and 2b of Theorem 4.1 are illustrated, respectively, in Figures 6, 7, 8 and 9.
Structure of the queue length processes in some regimes might vary depending
on the support boundaries and related quantities. We have illustrated only one
possible structure of the queue length process in those regimes and mentioned
the conditions on the support boundaries for the attainment of that structure
in the caption. Red and blue respectively denote class 1 and 2 populations. In
the plot on right-top, the black dashed line represents the total waiting mass
of the two classes in queue 2. The same is true for the illustrative EAPs and
resulting queue length processes referred to in Theorem 4.2._
Figure 6: Illustrative EAP (left) and queue length process (right) for HAS
with $\mu_{1}<\mu_{2}\gamma_{2}$ under case 1a of Theorem 4.1 and the
assumption $T=T_{1,a}+\frac{\gamma_{2}}{\gamma_{1}}T_{2,f}>0$ on (10).
$T\overset{def.}{=}\tau_{1}^{-1}(T_{2,f})$. Figure 7: Illustrative EAP (left)
and queue length process (right) for HAS with $\mu_{1}<\mu_{2}\gamma_{2}$
under case 1b of Theorem 4.1. $T\overset{def.}{=}\tau_{1}^{-1}(T_{2,f})$.
Figure 8: Illustrative EAP (left) and queue length process (right) for HAS
with $\mu_{1}<\mu_{2}\gamma_{2}$ under case 2a of Theorem 4.2 and the
assumption $T=T_{1,a}+\frac{\gamma_{2}}{\gamma_{1}}T_{2,f}>0$ on (10).
$T\overset{def.}{=}\tau_{1}^{-1}(T_{2,f})$. Figure 9: Illustrative EAP for
HAS with $\mu_{1}<\mu_{2}\gamma_{2}$ under case 2b of Theorem 4.2 and the
assumption $\tau_{2}(0)=-\gamma_{2}T_{2,a}<T_{1,f}$ on (12). Mass of class 2
waiting in queue 2 can increase in $[T_{1,f},\tau_{2}(T_{1,f})]$ if
$\mu_{2}\gamma_{2}>\mu_{2}-\mu_{1}/\gamma_{2}$.
Regime II $\mu_{1}\geq\mu_{2}\gamma_{2}$. The quantity
$T\overset{def}{=}\inf\\{t>0~{}|~{}Q_{2}(t)=0\\}$, representing the time at
which queue 2 empties for the first time, is necessary to describe the unique
EAP in Theorem 4.2.
We mention below the support boundaries and $T$ of the unique EAP, which we
will refer later in the statement of Theorem 4.2:
1. 1.
If $\mu_{2}\gamma_{1}>\mu_{1}\geq\mu_{2}\gamma_{2}$ and
$\Lambda_{1}\geq\frac{\mu_{1}}{(1-\gamma_{1})\mu_{2}}\Lambda_{2}$; or
$\mu_{1}\geq\mu_{2}\cdot\max\\{\gamma_{1},\gamma_{2}\\}$ and
$\left(\frac{\mu_{2}}{\mu_{1}}-1\right)\Lambda_{1}>\Lambda_{2}$, then
$\displaystyle T_{1,a}$
$\displaystyle=\frac{\Lambda_{2}}{\mu_{2}\gamma_{1}}-\left(\frac{1}{\gamma_{1}}-1\right)\frac{\Lambda_{1}}{\mu_{1}},~{}T_{1,f}=\frac{\Lambda_{1}}{\mu_{1}},~{}T=\frac{\Lambda_{2}}{\mu_{2}-\mu_{1}},~{}T_{2,a}=-\frac{\Lambda_{2}}{\mu_{2}\gamma_{2}},~{}\text{and}~{}T_{2,f}=0.$
(13)
2. 2.
If $\mu_{2}\gamma_{1}>\mu_{1}\geq\mu_{2}\gamma_{2}$ and
$\Lambda_{1}<\frac{\mu_{1}}{(1-\gamma_{1})\mu_{2}}\Lambda_{2}$, then
$\displaystyle T_{1,a}=T_{2,f}$
$\displaystyle=\frac{1}{\mu_{2}}\left(\Lambda_{2}-\frac{(1-\gamma_{1})\mu_{2}}{\mu_{1}}\Lambda_{1}\right),~{}T_{1,f}=\gamma_{1}\frac{\Lambda_{1}}{\mu_{1}}+\frac{\Lambda_{2}}{\mu_{2}},$
$\displaystyle T$
$\displaystyle=\frac{\Lambda_{2}}{\mu_{2}}+\frac{(1-\gamma_{1})\Lambda_{1}}{\mu_{2}-\mu_{1}},~{}\text{and}~{}T_{2,a}=-\left(\frac{1}{\gamma_{2}}-1\right)\frac{\Lambda_{2}}{\mu_{2}}-(1-\gamma_{1})\frac{\Lambda_{1}}{\mu_{1}}.$
(14)
3. 3.
If $\mu_{1}\geq\mu_{2}\cdot\max\\{\gamma_{1},\gamma_{2}\\}$ and
$\left(\frac{\mu_{2}}{\mu_{1}}-1\right)\Lambda_{1}\leq\Lambda_{2}\leq\left(\frac{1}{\max\\{\gamma_{1},\gamma_{2}\\}}-1\right)\Lambda_{1}$:
$\displaystyle T_{1,a}$
$\displaystyle=\frac{1}{\mu_{2}}\left(\Lambda_{2}-\left(\frac{1}{\gamma_{1}}-1\right)\Lambda_{1}\right),~{}T=T_{1,f}=\frac{\Lambda_{1}+\Lambda_{2}}{\mu_{2}},~{}T_{2,a}=-\frac{\Lambda_{2}}{\mu_{2}\gamma_{2}},~{}\text{and}~{}T_{2,f}=0.$
(15)
4. 4.
If $\mu_{1}\geq\mu_{2}\gamma_{1}>\mu_{2}\gamma_{2}$ and
$\Lambda_{2}>\left(\frac{1}{\gamma_{1}}-1\right)\Lambda_{1}$:
$\displaystyle T_{1,a}=T_{2,f}$
$\displaystyle=-\left(\frac{1}{\gamma_{1}}-1\right)\frac{\Lambda_{1}}{\mu_{2}}+\frac{\Lambda_{2}}{\mu_{2}},~{}T_{1,f}=T=\frac{\Lambda_{1}+\Lambda_{2}}{\mu_{2}},~{}\text{and}$
$\displaystyle T_{2,a}$
$\displaystyle=-\left(\frac{1}{\gamma_{1}}-1\right)\frac{\Lambda_{1}}{\mu_{2}}-\left(\frac{1}{\gamma_{2}}-1\right)\frac{\Lambda_{2}}{\mu_{2}}.$
(16)
5. 5.
If $\mu_{1}\geq\mu_{2}\gamma_{2}>\mu_{2}\gamma_{1}$ and
$\Lambda_{2}>\left(\frac{1}{\gamma_{2}}-1\right)\Lambda_{1}$:
$\displaystyle T_{1,a}$
$\displaystyle=-\left(\frac{1}{\gamma_{1}}-\frac{1}{\gamma_{2}}\right)\frac{\Lambda_{1}}{\mu_{2}},~{}T_{1,f}=\frac{\Lambda_{1}}{\mu_{2}\gamma_{2}},~{}T=T_{2,f}=\frac{\Lambda_{1}+\Lambda_{2}}{\mu_{2}},~{}\text{and}~{}T_{2,a}=-\frac{1-\gamma_{2}}{\gamma_{2}}\frac{\Lambda_{1}+\Lambda_{2}}{\mu_{2}}.$
(17)
###### Theorem 4.2.
If $\mu_{1}\geq\mu_{2}\gamma_{2}$ and $\gamma_{1}\neq\gamma_{2}$, HAS has a
unique EAP. Moreover, the EAP has the following arrival rates and support
boundaries:
1. 1.
If $\mu_{2}\gamma_{1}>\mu_{1}\geq\mu_{2}\gamma_{2}$:
1. 1a.
when $\Lambda_{1}\geq\frac{\mu_{1}}{(1-\gamma_{1})\mu_{2}}\Lambda_{2}$:
$\displaystyle F_{1}^{\prime}(t)$
$\displaystyle=\begin{cases}\mu_{2}\gamma_{1}~{}~{}&\text{for}~{}t\in[T_{1,a},\tau_{1}^{-1}(T)],\\\
\mu_{1}\gamma_{1}~{}~{}&\text{for}~{}t\in[\tau_{1}^{-1}(T),T_{1,f}],\end{cases}~{}\text{and}~{}F_{2}^{\prime}(t)=\mu_{2}\gamma_{2}~{}~{}\text{for}~{}t\in\left[T_{2,a},T_{2,f}\right]$
where $T_{1,a},T_{1,f},T_{2,a},T_{2,f}$ and $T$ are in (13).
2. 1b.
when $\Lambda_{1}<\frac{\mu_{1}}{(1-\gamma_{1})\mu_{2}}\Lambda_{2}$,
$\displaystyle F_{1}^{\prime}(t)$
$\displaystyle=\begin{cases}\mu_{2}\gamma_{1}~{}~{}&\text{for}~{}t\in[T_{1,a},\tau_{1}^{-1}(T)],\\\
\mu_{1}\gamma_{1}~{}~{}&\text{for}~{}t\in[\tau_{1}^{-1}(T),T_{1,f}],\end{cases}~{}\text{and}~{}F_{2}^{\prime}(t)=\mu_{2}\gamma_{2}~{}~{}~{}\text{for}~{}t\in[T_{2,a},T_{2,f}]$
where $T_{1,a},T_{1,f},T_{2,a},T_{2,f}$ and $T$ are in (2.).
2. 2.
If $\mu_{1}\geq\mu_{2}\gamma_{1}>\mu_{2}\gamma_{2}$:
1. 2a.
when $\left(\frac{\mu_{2}}{\mu_{1}}-1\right)\Lambda_{1}>\Lambda_{2}$, the EAP
has a closed form same as case 1a.
2. 2b.
when
$\left(\frac{1}{\gamma_{1}}-1\right)\Lambda_{1}\geq\Lambda_{2}\geq\left(\frac{\mu_{2}}{\mu_{1}}-1\right)\Lambda_{1}$,
$\displaystyle F_{1}^{\prime}(t)$
$\displaystyle=\mu_{2}\gamma_{1}~{}~{}\text{for}~{}t\in[T_{1,a},T_{1,f}]~{}\text{and}~{}F_{2}^{\prime}(t)=\mu_{2}\gamma_{2}~{}~{}\text{for}~{}t\in\left[T_{2,a},T_{2,f}\right]$
where $T_{1,a},T_{1,f},T_{2,a},T_{2,f}$ and $T$ are in (15).
3. 2c.
when $\Lambda_{2}>\left(\frac{1}{\gamma_{1}}-1\right)\Lambda_{1}$,
$\displaystyle F_{1}^{\prime}(t)$
$\displaystyle=\mu_{2}\gamma_{1}~{}~{}~{}\text{for}~{}t\in[T_{1,a},T_{1,f}]~{}\text{and}~{}F_{2}^{\prime}(t)=\mu_{2}\gamma_{2}~{}~{}~{}\text{for}~{}t\in[T_{2,a},T_{2,f}]$
where $T_{1,a},T_{1,f},T_{2,a},T_{2,f}$ and $T$ are in (4.).
3. 3.
If $\mu_{1}\geq\mu_{2}\gamma_{2}>\mu_{2}\gamma_{1}$:
1. 3a.
when $\left(\frac{\mu_{2}}{\mu_{1}}-1\right)\Lambda_{1}>\Lambda_{2}$, the EAP
has a closed form same as case 1a.
2. 3b.
when
$\left(\frac{1}{\gamma_{2}}-1\right)\Lambda_{1}\geq\Lambda_{2}\geq\left(\frac{\mu_{2}}{\mu_{1}}-1\right)\Lambda_{1}$,
the EAP has a closed form same as case 2b.
3. 3c.
when $\Lambda_{2}>\left(\frac{1}{\gamma_{2}}-1\right)\Lambda_{1}$,
$\displaystyle F_{1}^{\prime}(t)$
$\displaystyle=\mu_{2}\gamma_{1}~{}~{}\text{for}~{}t\in[T_{1,a},T_{1,f}]~{}\text{and}~{}F_{2}^{\prime}(t)=\mu_{2}\gamma_{2}~{}~{}\text{for}~{}t\in[T_{2,a},0]\cup[T_{1,f},T_{2,f}]$
where $T_{1,a},T_{1,f},T_{2,a},T_{2,f}$ and $T$ are in (17).
###### Remark 6.
_Illustrative EAPs and resulting queue length processes of cases 1a, 1b, 2a,
2b, 2c, 3a, 3b, and 3c of Theorem 4.2 are illustrated, respectively, in
Figures 10, 11, 12, 13, 14, 15, 16 and 17._
###### Remark 7.
_We show by Lemma B.6 and B.12 in B.1 that in every EAP of HAS under case 3
($\mu_{1}\geq\mu_{2}\gamma_{2}>\mu_{2}\gamma_{1}$) of Theorem 4.2, class 1
users arrive from queue 1 to queue 2 in $[0,T_{1,f}]$ at rate $\mu_{1}$. As a
result, by Lemma 4.1, class 2 users cannot arrive in $[0,T_{1,f}]$. However,
in every EAP, class 2 users start arriving before time zero (otherwise queue 2
stays empty at time zero and every class 2 user can be strictly better off
arriving at time zero). Therefore there are two possible ways in which class 2
users can arrive in any EAP: 1) the whole class 2 population arrives before
time zero, or, 2) a fraction of the class 2 population arrives after $T_{1,f}$
and the rest arrives before time zero. While proving the existence of a unique
EAP in case 3 of Theorem 4.2, we argue that, if $\Lambda_{2}$ is under an
identified threshold of $\left(\frac{1}{\gamma_{2}}-1\right)\Lambda_{1}$, the
whole class 2 population arrives before time zero and waits in queue 2 at time
zero, as can be seen for cases 3a and 3b, respectively, in Figure 15 and 16.
After $\Lambda_{2}$ crosses that threshold, a fraction of the class 2
population of mass
$\gamma_{2}\left[\Lambda_{2}-\left(\frac{1}{\gamma_{2}}-1\right)\Lambda_{1}\right]$
arrives after $T_{1,f}$ and the rest arrives before time zero, as can be seen
for case 3c in Figure 17._
Figure 10: Illustrative EAP (left) and resulting queue length process (right)
of HAS with $\mu_{1}\geq\mu_{2}\gamma_{2}$ under case 1a of Theorem 4.2 with
the assumption $T^{0}>0$, where $T^{0}\overset{def.}{=}\tau_{1}^{-1}(T)$.
Figure 11: Illustrative EAP (left) and resulting queue length process (right)
of HAS with $\mu_{2}\gamma_{1}>\mu_{1}\geq\mu_{2}\gamma_{2}$ under case 1b of
Theorem 4.2. $T^{0}\overset{def.}{=}\tau_{1}^{-1}(T)$.
.
Figure 12: Illustrative EAP (left) and resulting queue length process (right)
of HAS with $\mu_{1}\geq\mu_{2}\gamma_{1}>\mu_{2}\gamma_{2}$ under case 2a of
Theorem 4.2 with the assumption $T^{0}>0$ where
$T^{0}\overset{def.}{=}\tau_{1}^{-1}(T)$. Figure 13: Illustrative EAP (left)
and queue length process (right) of HAS with
$\mu_{1}\geq\mu_{2}\gamma_{1}>\mu_{2}\gamma_{2}$ under case 2b of Theorem 4.2
with assumption $\tau_{2}(0)<T_{1,idle}$, where $T_{1,idle}$ is the time at
which queue 1 empties after the network starts at time zero. In the
illustration, we assumed $\mu_{1}<\mu_{2}$, causing mass of class 1 users
waiting in queue 2 to decrease in $[\tau_{2}(0),T_{1,idle}]$, but it can be
non-decreasing if $\mu_{1}\geq\mu_{2}$. For the same assumption, the overall
mass of the two classes is decreasing in $[0,T_{1,idle}]$ and can be non-
decreasing if $\mu_{1}\geq\mu_{2}$. Figure 14: Illustrative EAP (left) and
resulting queue length process (right) of HAS with
$\mu_{1}\geq\mu_{2}\gamma_{1}>\mu_{2}\gamma_{2}$ under case 2c of Theorem 4.2.
Figure 15: Illustrative EAP of HAS (left) and queue length process (right)
with $\mu_{1}\geq\mu_{2}\gamma_{2}>\mu_{2}\gamma_{1}$ under case 3a of Theorem
4.2 with the assumption $T^{0}=\tau_{1}^{-1}(T)>0$. Figure 16: Illustrative
EAP (left) and resulting queue length process (right) of HAS with
$\mu_{1}\geq\mu_{2}\gamma_{2}>\mu_{2}\gamma_{1}$ under case 3b of Theorem 4.2
with the assumption $\tau_{2}(0)<T_{1,idle}$, where $T_{1,idle}$ is the time
at which queue 1 empties after the network starts at time zero. Figure 17:
Illustrative EAP (left) and queue length process (right) of HAS with
$\mu_{1}\geq\mu_{2}\gamma_{2}>\mu_{2}\gamma_{1}$ under case 3c of Theorem 4.2
with the assumption $T_{1,f}>\tau_{2}(0)>T_{1,idle}$, where $T_{1,idle}$ is
the time at which queue 1 empties after time zero.
### 4.2 Equal Preferences $\gamma_{1}=\gamma_{2}=\gamma$
We state the main result of this section separately for two regimes as Theorem
4.3 (for $\mu_{1}<\mu_{2}\gamma$), and 4.4 (for $\mu_{1}\geq\mu_{2}\gamma$),
since they have different EAP structures. Proofs of these two theorems (in
B.2) are similar in structure to that of Theorem 3.2.
$T_{1,a},T_{1,f},T_{2,a},T_{2,f}$ are as in the unequal preference case. Let
$T_{a}=\inf\mathcal{S}(F_{1})\cup\mathcal{S}(F_{2})$ and
$T_{f}=\sup\mathcal{S}(F_{1})\cup\mathcal{S}(F_{2})$.
###### Theorem 4.3.
If $\mu_{1}<\mu_{2}\gamma$,
1. 1.
when $\Lambda_{1}>\frac{\mu_{1}}{\mu_{2}-\mu_{1}}\Lambda_{2}$, the EAP is
unique and has the following arrival rates and support boundaries:
$\displaystyle F_{1}^{\prime}(t)$
$\displaystyle=\begin{cases}\mu_{1}~{}~{}&\text{if}~{}~{}t\in[T_{1,a},\tau_{1}^{-1}(T_{2,f})],\\\
\mu_{1}\gamma~{}~{}&\text{if}~{}~{}t\in[\tau_{1}^{-1}(T_{2,f}),T_{1,f}].\end{cases},~{}\text{and}~{}F_{2}^{\prime}(t)=\begin{cases}\mu_{2}\gamma~{}~{}&\text{if}~{}t\in[T_{2,a},0],\\\
\mu_{2}\gamma-\mu_{1}~{}~{}&\text{if}~{}t\in[0,T_{2,f}],\end{cases}$
$\displaystyle\text{where}~{}T_{1,a}$
$\displaystyle=-\frac{1-\gamma}{\gamma}\left(\frac{\Lambda_{1}}{\mu_{1}}-\frac{\Lambda_{2}}{\mu_{2}-\mu_{1}}\right),~{}T_{1,f}=\frac{\Lambda_{1}}{\mu_{1}},~{}T_{2,a}=-\frac{1-\gamma}{\gamma}\frac{\Lambda_{2}}{\mu_{2}-\mu_{1}},~{}\text{and}~{}T_{2,f}=\frac{\Lambda_{2}}{\mu_{2}-\mu_{1}}.$
2. 2.
when $\Lambda_{1}\leq\frac{\mu_{1}}{\mu_{2}-\mu_{1}}\Lambda_{2}$, set of EAPs
is given by the convex set of joint customer arrival profiles
$\\{F_{1},F_{2}\\}$ satisfying: 1)
$F_{1}^{\prime}(t)=0,~{}F_{2}^{\prime}(t)=\mu_{2}\gamma$ for all
$t\in[T_{a},0]$, 2)
$F_{1}^{\prime}(t)\leq\mu_{1},~{}F_{1}^{\prime}(t)+F_{2}^{\prime}(t)=\mu_{2}\gamma$
for all $t\in[0,T_{f}]$, and 3)
$F_{1}(T_{f})=\Lambda_{1},~{}F_{2}(T_{f})=\Lambda_{2}$, where
$T_{a}=-\left(\frac{1}{\gamma}-1\right)\frac{\Lambda_{1}+\Lambda_{2}}{\mu_{2}}$
and $T_{f}=\frac{\Lambda_{1}+\Lambda_{2}}{\mu_{2}}$.
###### Remark 8.
_If $\Lambda_{1}>\frac{\mu_{1}}{\mu_{2}-\mu_{1}}\Lambda_{2}$, the EAPs in case
1a and 2a of Theorem 4.1, respectively, upon taking limits
$\gamma_{1}\to\gamma+,~{}\gamma_{2}=\gamma$ and
$\gamma_{1}=\gamma,~{}\gamma_{2}\to\gamma+$ converges to the EAP under case 1
of Theorem 4.3. On the other hand, if
$\Lambda_{1}\leq\frac{\mu_{1}}{\mu_{2}-\mu_{1}}\Lambda_{2}$, the EAPs in case
1b and 2b of Theorem 4.1, respectively, upon taking limits
$\gamma_{1}\to\gamma+,~{}\gamma_{2}=\gamma$ and
$\gamma_{1}=\gamma,~{}\gamma_{2}\to\gamma+$ might converge to different
limits, but both the limits will be contained in the convex set of EAPs under
case 2 of 4.1._
The quantity $T\overset{def.}{=}\inf\left\\{t>0~{}|~{}Q_{2}(t)=0\right\\}$
will be necessary to describe the structure of the EAP in the regime
$\mu_{1}\leq\mu_{2}\gamma$.
###### Theorem 4.4.
If $\mu_{1}\geq\mu_{2}\gamma$,
1. 1.
when $\left(\frac{\mu_{2}}{\mu_{1}}-1\right)\Lambda_{1}>\Lambda_{2}$, the EAP
is unique and has the following arrival rates and support boundaries:
$\displaystyle F_{1}^{\prime}(t)$
$\displaystyle=\begin{cases}\mu_{2}\gamma~{}~{}&\text{if}~{}t\in[T_{1,a},\tau_{1}^{-1}(T)],\\\
\mu_{1}\gamma~{}~{}&\text{if}~{}t\in[\tau_{1}^{-1}(T),T_{1,f}],\end{cases},~{}\text{and}~{}F_{2}^{\prime}(t)=\mu_{2}\gamma~{}~{}\text{if}~{}t\in\left[-\frac{\Lambda_{2}}{\mu_{2}\gamma},0\right],$
$\displaystyle\text{where}~{}T_{1,a}$
$\displaystyle=\frac{\Lambda_{2}}{\mu_{2}\gamma}-\left(\frac{1}{\gamma}-1\right)\frac{\Lambda_{1}}{\mu_{1}},~{}T=\frac{\Lambda_{2}}{\mu_{2}-\mu_{1}},~{}\text{and}~{}T_{1,f}=\frac{\Lambda_{1}}{\mu_{1}}.$
2. 2.
when
$\left(\frac{1}{\gamma}-1\right)\Lambda_{1}\geq\Lambda_{2}\geq\left(\frac{\mu_{2}}{\mu_{1}}-1\right)\Lambda_{1}$,
the EAP is unique and has the following arrival rates and support boundaries:
$\displaystyle F_{1}^{\prime}(t)$
$\displaystyle=\mu_{2}\gamma~{}~{}\text{if}~{}t\in[T_{1,a},T_{1,f}],~{}\text{and}~{}F_{2}^{\prime}(t)=\mu_{2}\gamma~{}~{}\text{if}~{}t\in\left[-\frac{\Lambda_{2}}{\mu_{2}\gamma},0\right],$
$\displaystyle\text{where}~{}T_{1,a}$
$\displaystyle=\frac{1}{\mu_{2}}\left(\Lambda_{2}-\left(\frac{1}{\gamma}-1\right)\Lambda_{1}\right),~{}\text{and}~{}T=T_{1,f}=\frac{\Lambda_{1}+\Lambda_{2}}{\mu_{2}}.$
3. 3.
If $\Lambda_{2}>\left(\frac{1}{\gamma}-1\right)\Lambda_{1}$, set of EAPs is
given by the convex set of joint customer arrival profiles $\\{F_{1},F_{2}\\}$
satisfying: 1) $F_{1}^{\prime}(t)=0$ and $F_{2}^{\prime}(t)=\mu_{2}\gamma$ for
all $t\in[T_{a},0]$, 2) $F_{1}^{\prime}(t)+F_{2}^{\prime}(t)=\mu_{2}\gamma$
for all $t\in[0,T_{f}]$, and 3)
$F_{1}(T_{f})=\Lambda_{1},~{}F_{2}(T_{f})=\Lambda_{2}$, where
$T_{a}=-\left(\frac{1}{\gamma}-1\right)\frac{\Lambda_{1}+\Lambda_{2}}{\mu_{2}}$
and $T_{f}=\frac{\Lambda_{1}+\Lambda_{2}}{\mu_{2}}$.
###### Remark 9.
_If $\left(\frac{\mu_{2}}{\mu_{1}}-1\right)\Lambda_{1}>\Lambda_{2}$, the EAPs
in case 2a and 3a of Theorem 4.2, respectively upon taking limits
$\gamma_{1}\to\gamma+,\gamma_{2}=\gamma$ and
$\gamma_{1}=\gamma,\gamma_{2}\to\gamma+$, converges to the EAP under case 1 of
Theorem 4.4. If
$\left(\frac{1}{\gamma}-1\right)\Lambda_{1}\geq\Lambda_{2}\geq\left(\frac{\mu_{2}}{\mu_{1}}-1\right)\Lambda_{1}$,
upon taking the same limits, the EAPs in case 2b and 3b of Theorem 4.2
converges to the EAP under case 2 of Theorem 4.4. If
$\Lambda_{2}>\left(\frac{1}{\gamma}-1\right)\Lambda_{1}$, upon taking the same
limits, the EAPs in case 2c and 3c of Theorem 4.2 converges to different
limits and both of them are contained in the convex set of EAPs under case 3
of Theorem 4.4._
## 5 Conclusion
Arrival games to a single queue have been well studied in the literature, but
the important area of arrival games to a general queuing network, has limited
literature. As we discovered, this may be because the problem complexity may
increase substantially with the network size. We studied a simple two queue,
two class network with a linear cost structure, where we analyzed in detail
two heterogeneous routing configurations: (1) where the customer classes
arrived at the same queue but departed in different queues, and (2) where the
customer classes arrived at different queues but departed from the same queue.
We discovered non-trivial customer behaviour not apparent in a single queue or
in networks studied thus far in the literature. In a specific setting we found
that the parameter space had to be partitioned into eight distinct regions,
where each region had its own closed form parametric representation of the
arrival equilibrium profile. We found that although for most set of
parameters, the equilibrium profile is unique, there exists settings where the
collection of equilibrium profiles is not unique but a convex set. While in a
single queue, multi-class customers arrive in contiguous, non-overlapping
intervals, in our two queue setting there are regions where, in equilibrium,
different class arrival times may overlap. Further, there exist regions, where
a single class customer may arrive in disjoint intervals. The broad message of
the paper is mixed and motivates further research - it suggests that even for
more complex networks one may expect a unique equilibrium profile for most
parameter settings. It also suggests that the number of different solution
structures may blow up with the network size, so that learning the structure
in any instance may be a difficult task.
## Acknowledgement
Our work is supported by Department of Atomic Energy, Government of India,
under project no. RTI4001.
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## Appendix
For the ease of analysis, we define the set
$E_{j}\overset{def.}{=}\\{t~{}|~{}t<0~{}\text{or}~{}Q_{j}(t)>0\\}$ to be the
set of all times when queue $j$ has a positive waiting time for $j=1,2$. For
any set $S\subseteq\mathbb{R}$, we use the notations $\overline{S}$ and
$S^{o}$ to respectively denote the closure and interior of $S$.
By Lemma 2.1, we consider only absolutely continuous arrival profiles $F_{1}$
and $F_{2}$ and as a result, $\mathcal{S}(F_{1})$ and $\mathcal{S}(F_{2})$
cannot have isolated points.
## Appendix A Proofs of Lemmas in Section 3
As argued in Section 3, we assume $\mu_{1}>\mu_{2}$ for the following
discussion.
### A.1 Proofs of Lemmas in Section 3.1 (HDS with $\gamma_{1}\neq\gamma_{2}$)
Several of the lemmas proved below will extend to the case where
$\gamma_{1}=\gamma_{2}$. So, whenever some lemma is applicable only for the
case of unequal preferences, we mention it explicitly in the lemma statement.
Lemma A.1 specifies the state of queue 2 when users of both classes are
arriving together in queue 1 and is a special case of Lemma 3.3. This lemma
helps us prove Lemma 3.1 and together with Lemma A.2, it will help us
calculate the arrival rate of the two classes in equilibrium.
###### Lemma A.1.
If $\gamma_{1}\neq\gamma_{2}$ and $\mu_{1}>\mu_{2}$, in the EAP,
$t\in(\mathcal{S}(F_{1}))^{o}\cap(\mathcal{S}(F_{2}))^{o}$ implies
$\tau_{1}(t)\in\overline{E_{2}}$.
###### Proof.
Assuming contradiction, i.e., $\tau_{1}(t)\notin\overline{E_{2}}$, there is a
neighbourhood $[\tau_{1}(t)-\delta,\tau_{1}(t)+\delta]$ of $\tau_{1}(t)$,
where queue 2 is empty. Since the arrival profiles $F_{1},~{}F_{2}$ are
absolutely continuous, $\tau_{1}(\cdot)$ will be continuous. As a result, we
have
$[t-\epsilon,t+\epsilon]\subseteq(\mathcal{S}(F_{1}))^{o}\cap(\mathcal{S}(F_{2}))^{o}$
such that, $Q_{2}(\tau_{1}(s))=0$ for all $s\in[t-\epsilon,t+\epsilon]$. Hence
every class 2 user arriving in $[t-\epsilon,t+\epsilon]$ waits only at queue
1. For both the classes $i=1,2$ departure time
$\tau_{\mathbf{F}}^{(i)}(s)=\tau_{1}(s)$ in $[t-\epsilon,t+\epsilon]$ and
hence cost $C_{\mathbf{F}}^{(i)}(s)=\tau_{1}(s)-\gamma_{i}s$. Since,
$[t-\epsilon,t+\epsilon]\subseteq(\mathcal{S}(F_{1}))^{o}\cap(\mathcal{S}(F_{2}))^{o}$,
by definition of EAP, for $i=1,2$ $C_{\mathbf{F}}^{(i)}(\cdot)$ is constant in
$[t-\epsilon,t+\epsilon]$. Therefore,
$(C_{\mathbf{F}}^{(i)})^{\prime}(s)=\tau_{1}^{\prime}(s)-\gamma_{i}=0$ in
$[t-\epsilon,t+\epsilon]$, giving us
$\tau_{1}^{\prime}(s)=\gamma_{1}=\gamma_{2}$, contradicting the assumption
$\gamma_{1}\neq\gamma_{2}$. ∎
Lemma A.2 helps us relate the arrival rate of class 2 users and will be
referred to ubiquitously while proving the structural properties.
###### Lemma A.2.
For every absolutely continuous joint user arrival profile
$\\{F_{1},F_{2}\\}$, $\tau_{\mathbf{F}}^{(2)}(\cdot)$ is differentiable a.e.
in the set of times
$\tau_{1}^{-1}(\overline{E_{2}})\overset{def.}{=}\\{t~{}|~{}\tau_{1}(t)\in\overline{E_{2}}\\}$
with derivative
$(\tau_{\mathbf{F}}^{(2)})^{\prime}(t)=\frac{F_{2}^{\prime}(t)}{\mu_{2}}$ a.e.
As a consequence,
$(C_{\mathbf{F}}^{(2)})^{\prime}(t)=\frac{F_{2}^{\prime}(t)}{\mu_{2}}-\gamma_{2}$
a.e. in $\tau_{1}^{-1}(\overline{E_{2}})$.
###### Proof.
We divide the proof into two separate cases:
1. 1.
In $\overline{E_{1}}^{c}$, we will have $\tau_{1}(t)=t$ and as a result
$\overline{E_{1}}^{c}\cap\tau_{1}^{-1}(\overline{E_{2}})=\overline{E_{1}}^{c}\cap\overline{E_{2}}$.
Therefore, using (3)
$(\tau_{\mathbf{F}}^{(2)})^{\prime}(t)=\tau_{2}^{\prime}(t)=\frac{F_{2}^{\prime}(t)}{\mu_{2}}$
a.e. in $\overline{E_{1}}^{c}\cap\tau_{1}^{-1}(\overline{E_{2}})$.
2. 2.
In $\overline{E_{1}}$, by (3), $\tau_{1}^{\prime}(\cdot)$ exists a.e. Again
using (3), since $\tau_{1}(t)\in\overline{E_{2}}$, we have
$\tau_{2}^{\prime}(\tau_{1}(t))=\frac{A_{2}^{\prime}(\tau_{1}(t))}{\mu_{2}}$.
Therefore,
$(\tau_{\mathbf{F}}^{(2)})^{\prime}(t)=\frac{A_{2}^{\prime}(\tau_{1}(t))\tau_{1}^{\prime}(t)}{\mu_{2}}$
a.e. in $\overline{E_{1}}\cap\tau_{1}^{-1}(\overline{E_{2}})$. Since
$A_{2}(\tau_{1}(t))=F_{2}(t)$, by chain rule,
$(\tau_{\mathbf{F}}^{(2)})^{\prime}(t)=\frac{F_{2}^{\prime}(t)}{\mu_{2}}$ a.e.
in $\overline{E_{1}}\cap\tau_{1}^{-1}(\overline{E_{2}})$.
Since $C_{\mathbf{F}}^{(2)}(t)=\tau_{\mathbf{F}}^{(2)}(t)-\gamma_{2}t$, the
second statement of the lemma follows trivially. ∎
Lemma B.5 implies that
$\mu_{1}\leq\mu_{2}\cdot\max\left\\{1,\frac{\gamma_{2}}{\gamma_{1}}\right\\}$
is sufficient for $\mathcal{S}(F_{1})$ and $(\mathcal{S}(F_{2}))^{o}$ to be
disjoint, and is helpful for proving sufficiency of the condition in Lemma
3.1.
###### Lemma A.3.
If $\mu_{1}\gamma_{1}\leq\mu_{2}\gamma_{2}$, in every EAP,
$\mathcal{S}(F_{1})$ and $(\mathcal{S}(F_{2}))^{o}$ are disjoint.
###### Proof.
Proof of Lemma A.3 follows the argument mentioned after Lemma 3.1 for proving
sufficiency of
$\mu_{1}\leq\mu_{2}\cdot\max\left\\{1,\frac{\gamma_{2}}{\gamma_{1}}\right\\}$
for $\mathcal{S}(F_{1}),\mathbf{S}(F_{2})$ to be non-overlapping, except while
proving
$(C_{\mathbf{F}}^{(2)})^{\prime}(t)=\frac{F_{2}^{\prime}(t)}{\mu_{2}}-\gamma_{2}$
in $[t_{1},t_{2}]$, we use Lemma B.1 to argue queue 2 stays engaged at
$\tau_{1}(t)$ and then use Lemma A.2 to have
$(\tau_{\mathbf{F}}^{(2)})^{\prime}(t)=\frac{F_{2}^{\prime}(t)}{\mu_{2}}$ in
$[t_{1},t_{2}]$.
∎
As we stated in the main body, $T_{i,a}=\inf\mathcal{S}(F_{i})$ and
$T_{i,f}=\sup\mathcal{S}(F_{i})$ for the two classes $i=1,2$. Note that
$T_{1,a},T_{1,f},T_{2,a},T_{2,f}$ are all finite if
$\mathbf{F}=\\{F_{1},F_{2}\\}$ is an EAP. Lemma A.4 and A.5 are respectively
the first and second statements of Lemma 3.5.
###### Lemma A.4.
(First statement of Lemma 3.5) In the EAP, if
$\mathcal{S}(F_{2})=[T_{2,a},T_{2,f}]$, then queue 2 will have zero waiting
time at $\tau_{1}(T_{2,f})$.
###### Proof.
Assume contradiction, i.e., $\tau_{1}(T_{2,f})\in E_{2}$. For $\delta>0$
chosen sufficiently small, image of $[T_{2,f},T_{2,f}+\delta]$ under
$\tau_{1}(\cdot)$ is contained in $E_{2}$. Therefore by Lemma A.2,
$(\tau_{\mathbf{F}}^{(2)})^{\prime}(t)=\frac{F_{2}^{\prime}(t)}{\mu_{2}}$ in
$[T_{2,f},T_{2,f}+\delta]$. Since $F_{2}^{\prime}(t)=0$ in
$(T_{2,f},T_{2,f}+\delta]$, we get
$\tau_{\mathbf{F}}^{(2)}(T_{2,f}+\delta)=\tau_{\mathbf{F}}^{(2)}(T_{2,f})$. As
a result, the class 2 user arriving at $T_{2,f}$ will be strictly better off
arriving at $T_{2,f}+\delta$, contradicting that $\\{F_{1},F_{2}\\}$ is an
EAP. ∎
We now prove below Lemma 3.2 about the structure of the supports of the two
classes in EAP.
Proof of Lemma 3.2: (1) On assuming contradiction, there exists
$t_{1},t_{2}\in\mathcal{S}(F_{1})\cup\mathcal{S}(F_{2})$ such that
$t_{2}>t_{1}$ and $F_{1}(t_{1})+F_{2}(t_{1})=F_{1}(t_{2})+F_{2}(t_{2})$. Note
that $t_{2}<\max\\{T_{1,f},T_{2,f}\\}$. Otherwise, $\max\\{T_{1,f},T_{2,f}\\}$
will be an isolated point of $\mathcal{S}(F_{1})\cup\mathcal{S}(F_{2})$,
contradicting absolute continuity of $F_{1}$ and $F_{2}$.
With $t_{2}<\max\\{T_{1,f},T_{2,f}\\}$, a positive mass of users must arrive
after $t_{2}$. Therefore, none of the classes can have zero waiting time in
$[t_{1},t_{2}]$, implying $[t_{1},t_{2}]\subseteq E_{1}\cup E_{2}$. There can
be two situations now:
* •
If $t_{1}\in E_{1}$: For $\delta>0$ picked sufficiently small,
$[t_{1},t_{1}+\delta]\subseteq E_{1}$. Since $A_{1}^{\prime}(s)=0$ for every
$s\in[t_{1},t_{1}+\delta]$, by (3), $\tau_{1}(t_{1})=\tau_{1}(t_{1}+\delta)$.
As a result, for both classes $i=1,2$
$\tau_{\mathbf{F}}^{(i)}(t_{1})=\tau_{\mathbf{F}}^{(i)}(t_{1}+\delta)$.
Therefore, the user arriving at $t_{1}$ will be strictly better off arriving
at $t_{1}+\delta$.
* •
If $t_{1}\notin E_{1}$: Since no users arrive in $[t_{1},t_{2}]$, queue 1 has
zero waiting time in $[t_{1},t_{2}]$ (by (3)) implying,
$[t_{1},t_{2}]\subseteq E_{2}$. Therefore some positive mass of class 2 users
must have arrived before $t_{1}$. Let $\tilde{t}=\sup\\{t\leq
t_{1}~{}|~{}t\in\mathcal{S}(F_{2})\\}=$ the last time before $t_{1}$ some
class 2 user has arrived. Since no class 2 users arrives in
$[\tilde{t},t_{2}]$, $Q_{2}(\cdot)$ cannot increase in
$[\tau_{1}(\tilde{t}),\tau_{1}(t_{2})]$. Therefore, $t_{2}=\tau_{1}(t_{2})\in
E_{2}$ implies $[\tau_{1}(\tilde{t}),\tau_{1}(t_{2})]\subseteq E_{2}$. With
this observation, since $F_{2}^{\prime}(s)=0$ in $(\tilde{t},t_{2})$, we
conclude $\tau_{\mathbf{F}}^{(2)}(\tilde{t})=\tau_{\mathbf{F}}^{(2)}(t_{2})$
(by Lemma A.2). Therefore, the class 2 user arriving at $\tilde{t}$ will be
strictly better off arriving at $t_{2}$ instead.
In both cases, $\mathbf{F}=\\{F_{1},F_{2}\\}$ will not be an EAP,
contradicting our assumption.
(2) On assuming contradiction, there exists $t_{1},t_{2}\in\mathcal{S}(F_{2})$
such that $F_{2}(t_{1})=F_{2}(t_{2})$. We must have $t_{2}<T_{2,f}$,
otherwise, $T_{2,f}$ will be an isolated point in $\mathcal{S}(F_{2})$. Since
$\mathcal{S}(F_{1})\cup\mathcal{S}(F_{2})$ is an interval, we must have
$[t_{1},t_{2}]\subseteq\mathcal{S}(F_{1})$. Two situations might arise:
* •
If $\tau_{1}(t_{1})\in E_{2}$: By continuity of $\tau_{1}(\cdot)$, we can find
$\delta\in(0,t_{2}-t_{1})$ sufficiently small such that image of
$[t_{1},t_{1}+\delta]$ under $\tau_{1}(\cdot)$ is contained in $E_{2}$. Since
$F_{2}^{\prime}(t)=0$ in $[t_{1},t_{2}]$, by Lemma A.2, the previous statement
implies
$\tau_{\mathbf{F}}^{(2)}(t_{1})=\tau_{\mathbf{F}}^{(2)}(t_{1}+\delta)$.
Therefore the class 2 user arriving at $t_{1}$ will be strictly better off
arriving at $t_{1}+\delta$.
* •
If $\tau_{1}(t_{1})\notin E_{2}$: With no class 2 user arriving at queue 2 in
$[\tau_{1}(t_{1}),\tau_{1}(t_{2})]$, we have
$[\tau_{1}(t_{1}),\tau_{1}(t_{2})]\subseteq E_{2}^{c}$. Therefore for
$t\in[t_{1},t_{2}]$,
$\tau_{\mathbf{F}}^{(1)}(t)=\tau_{\mathbf{F}}^{(2)}(t)=\tau_{1}(t)$. Now
$[t_{1},t_{2}]\subseteq\mathcal{S}(F_{1})$ implies
$C_{\mathbf{F}}^{(1)}(\cdot)$ is constant in $[t_{1},t_{2}]$. Hence
$(C_{\mathbf{F}}^{(1)})^{\prime}(t)=\tau_{1}^{\prime}(t)-\gamma_{1}=0$ in
$(t_{1},t_{2})$, implying $\tau_{1}^{\prime}(t)=\gamma_{1}$. As a result, in
$(t_{1},t_{2})$,
$(C_{\mathbf{F}}^{(2)})^{\prime}(t)=\tau_{1}^{\prime}(t)-\gamma_{2}=\gamma_{1}-\gamma_{2}\neq
0$. Now if $\gamma_{1}>\gamma_{2}$, class 2 user arriving at time $t_{2}$ will
be better off arriving at time $t_{1}$ and vice-versa when
$\gamma_{1}<\gamma_{2}$.
$\\{F_{1},F_{2}\\}$ cannot be an EAP in both the situations.
(3) We consider two separate situations for proving $\mathcal{S}(F_{1})$ is an
interval:
* •
If $\mu_{1}\gamma_{1}\neq\mu_{2}\gamma_{2}$: On assuming contradiction, there
exists $t_{1},t_{2}\in\mathcal{S}(F_{1})$ such that,
$F_{1}(t_{1})=F_{1}(t_{2})$. We must have $t_{2}<T_{1,f}$, otherwise,
$T_{1,f}$ will be an isolated point in $\mathcal{S}(F_{1})$. Since
$\mathcal{S}(F_{1})\cup\mathcal{S}(F_{2})$ is an interval, we must have
$[t_{1},t_{2}]\subseteq\mathcal{S}(F_{2})$. Also, we must have
$[t_{1},t_{2}]\subseteq E_{1}$, since a positive mass of class 1 users arrive
after $t_{2}$. This implies, class 2 users will be arriving at queue 2 at a
rate $\mu_{1}>\mu_{2}$ in $[\tau_{1}(t_{1}),\tau_{1}(t_{2})]$ and therefore
$[\tau_{1}(t_{1}),\tau_{1}(t_{2})]\subseteq\overline{E_{2}}$. Using Lemma A.2
and the definition of EAP, in $[t_{1},t_{2}]$,
$(C_{\mathbf{F}}^{(2)})^{\prime}(t)=(\tau_{\mathbf{F}}^{(2)})^{\prime}(t)-\gamma_{2}=\frac{F_{2}^{\prime}(t)}{\mu_{2}}-\gamma_{2}=0$
a.e. implying $F_{2}^{\prime}(t)=\mu_{2}\gamma_{2}$ a.e. Since
$[t_{1},t_{2}]\subseteq E_{1}$, by (3),
$\tau_{1}^{\prime}(t)=\frac{F_{2}^{\prime}(t)}{\mu_{1}}=\frac{\mu_{2}\gamma_{2}}{\mu_{1}}$
a.e. in $[t_{1},t_{2}]$. As a result,
$(C_{\mathbf{F}}^{(1)})^{\prime}(t)=\tau_{1}^{\prime}(t)-\gamma_{1}=\frac{\mu_{2}\gamma_{2}}{\mu_{1}}-\gamma_{1}$
a.e. in $[t_{1},t_{2}]$. Now if $\mu_{1}\gamma_{1}>\mu_{2}\gamma_{2}$, class 1
user arriving at time $t_{1}$ will be strictly better off arriving at time
$t_{2}$ and vice-versa if $\mu_{1}\gamma_{1}<\mu_{2}\gamma_{2}$. Therefore
$\\{F_{1},F_{2}\\}$ will not be an EAP.
* •
If $\mu_{1}\gamma_{1}=\mu_{2}\gamma_{2}$: Since $\mu_{1}>\mu_{2}$,
$\mu_{1}\gamma_{1}=\mu_{2}\gamma_{2}$ implies $\gamma_{1}<\gamma_{2}$. By 2nd
statement of the lemma, $\mathcal{S}(F_{2})$ must be the interval
$[T_{2,a},T_{2,f}]$. By Lemma A.3, $\mathcal{S}(F_{1})$ and
$(T_{2,a},T_{2,f})$ are disjoint and their union is an interval. As a result,
$\mathcal{S}(F_{1})$ is union of two intervals, one ending at $T_{2,a}$ and
the other one starting from $T_{2,f}$. We now show that $\mathcal{S}(F_{1})$
has no element larger than $T_{2,f}$. This implies that $\mathcal{S}(F_{1})$
is an interval ending at $T_{2,a}$. For this, we again assume a contradiction
and suppose that there exists $T>T_{2,f}$ such that
$[T_{2,f},T]\subseteq\mathcal{S}(F_{1})$. By Lemma A.4, queue 2 must be empty
at $\tau_{1}(T_{2,f})$. Therefore for $t\in[T_{2,f},T]$,
$\tau_{\mathbf{F}}^{(1)}(t)=\tau_{\mathbf{F}}^{(2)}(t)=\tau_{1}(t)$. Since
$[T_{2,f},T]\subseteq\mathcal{S}(F_{1})$, we will have
$(C_{\mathbf{F}}^{(1)})^{\prime}(t)=\tau_{1}^{\prime}(t)-\gamma_{1}=0$
implying $\tau_{1}^{\prime}(t)=\gamma_{1}$ in $[T_{2,f},T]$. Using this, in
$[T_{2,f},T]$,
$(C_{\mathbf{F}}^{(2)})^{\prime}(t)=\tau_{1}^{\prime}(t)-\gamma_{2}=\gamma_{1}-\gamma_{2}<0$.
Hence, the class 2 user arriving at $T_{2,f}$ is strictly better off arriving
at $T$, implying that $\\{F_{1},F_{2}\\}$ is not an EAP. ∎
By Lemma 3.2, for every EAP $\\{F_{1},F_{2}\\}$, we must have
$\mathcal{S}(F_{1})=[T_{1,a},T_{1,f}]$, $\mathcal{S}(F_{2})=[T_{2,a},T_{2,f}]$
and their union must be an interval.
Proof of Lemma 3.3: If $t\in(T_{1,a},T_{1,f})\cap(T_{2,a},T_{2,f})$, the
statement follows from Lemma A.1. Otherwise, we can obtain $\delta>0$ such
that either $[t,t+\delta]$ or
$[t-\delta,t]\subseteq(T_{2,a},T_{2,f})/(T_{1,a},T_{1,f})$. Now if $t\in
E_{1}$, we can choose $\delta>0$ small enough such that
$[t-\delta,t+\delta]\subseteq E_{1}$ As a result,
$\tau_{1}(t-\delta)<\tau_{1}(t)<\tau_{1}(t+\delta)$ (by (3)) and class 2 users
arrive in queue 2 at a rate $\mu_{1}>\mu_{2}$ in
$[\tau_{1}(t),\tau_{1}(t+\delta)]$ or $[\tau_{1}(t-\delta),\tau_{1}(t)]$
implying that $\tau_{1}(t)\in\overline{E_{2}}$. Otherwise, if $t\in
E_{1}^{c}$, since a positive mass of class 2 users arrive in $(t,T_{2,f}]$, we
must have $t=\tau_{1}(t)\in E_{2}$. Therefore, the lemma stands proved. ∎
Proof of Lemma 3.4: By Lemma 3.3, if $t\in\mathcal{S}(F_{2})$,
$\tau_{1}(t)\in\overline{E_{2}}$, which by Lemma A.2 implies
$(C_{\mathbf{F}}^{(2)})^{\prime}(t)=\frac{F_{2}^{\prime}(t)}{\mu_{2}}-\gamma_{2}$
a.e. in $\mathcal{S}(F_{2})$. Since $C_{\mathbf{F}}^{(2)}(\cdot)$ is constant
in $\mathcal{S}(F_{2})$, we must have $F_{2}^{\prime}(t)=\mu_{2}\gamma_{2}$
a.e. in $\mathcal{S}(F_{2})$.
Note that $t\in\mathcal{S}(F_{1})=[T_{1,a},T_{1,f}]$ implies
$t\in\overline{E_{1}}$. Therefore, by (3),
$(C_{\mathbf{F}}^{(1)})^{\prime}(t)=(\tau_{\mathbf{F}}^{(1)})^{\prime}(t)-\gamma_{1}=\frac{F_{1}^{\prime}(t)+F_{2}^{\prime}(t)}{\mu_{1}}-\gamma_{1}$
a.e. in $\mathcal{S}(F_{1})$. Since $C_{\mathbf{F}}^{(1)}(\cdot)$ is constant
in $\mathcal{S}(F_{1})$, we must have
$F_{1}^{\prime}(t)+F_{2}^{\prime}(t)=\mu_{1}\gamma_{1}$ a.e. in
$\mathcal{S}(F_{1})$. As $F_{2}^{\prime}(t)=\mu_{2}\gamma_{2}$ a.e. in
$\mathcal{S}(F_{2})$, we must have,
$\displaystyle F_{1}^{\prime}(t)$
$\displaystyle=\begin{cases}\mu_{1}\gamma_{1}~{}~{}&\text{if}~{}t\in\mathcal{S}(F_{1})/\mathcal{S}(F_{2}),~{}\text{and},\\\
\mu_{1}\gamma_{1}-\mu_{2}\gamma_{2}~{}~{}&\text{if}~{}t\in\mathcal{S}(F_{1})\cap\mathcal{S}(F_{2}).\end{cases}$
Note that, the above arrival rates are positive a.e. because, by Lemma A.3 and
3.2, if $\mu_{1}\gamma_{1}\leq\mu_{2}\gamma_{2}$,
$\mathcal{S}(F_{1})\cap\mathcal{S}(F_{2})$ has zero Lebesgue measure. ∎
###### Lemma A.5.
(Second statement of Lemma 3.5) In the EAP, if
$\mu_{1}\gamma_{1}>\mu_{2}\gamma_{2}$ and
$\mathcal{S}(F_{1})=[T_{1,a},T_{1,f}]$, then queue 1 will have zero waiting
time at $T_{1,f}$.
###### Proof.
Assume a contradiction, i.e., $T_{1,f}\in E_{1}$ with
$\mu_{1}\gamma_{1}>\mu_{2}\gamma_{2}$. Two situations are possible:
* •
If $(T_{1,f},\infty)\cap\mathcal{S}(F_{2})=\emptyset$, by (3),
$\tau_{\mathbf{F}}^{(1)}(T_{1,f})=\tau_{\mathbf{F}}^{(1)}(T_{idle})$, where
$T_{idle}>T_{1,f}$ is the time queue 1 empties after $T_{1,f}$. Hence, the
class 1 user arriving at $T_{1,f}$ is strictly better off arriving at
$T_{idle}$.
* •
If a positive mass of class 2 users arrive after $T_{1,f}$, by Lemma 3.2 and
3.4, they will arrive over an interval $[T_{1,f},T]$ at rate
$\mu_{2}\gamma_{2}$, where $T>T_{1,f}$. Therefore, picking
$\delta\in(0,T-T_{1,f})$ sufficiently small, we can have
$[T_{1,f},T_{1,f}+\delta]\subseteq E_{1}$. By (3), for
$t\in[T_{1,f},T_{1,f}+\delta]$,
$(C_{\mathbf{F}}^{(1)})^{\prime}(t)=\tau_{1}^{\prime}(t)-\gamma_{1}=\frac{F_{2}^{\prime}(t)}{\mu_{1}}-\gamma_{1}=\frac{\mu_{2}\gamma_{2}}{\mu_{1}}-\gamma_{1}<0$
a.e. Hence the class 1 user arriving at $T_{1,f}$ will be strictly better off
arriving at $T_{1,f}+\delta$.
For both the situations above, we get $\\{F_{1},F_{2}\\}$ will not be an EAP.
∎
Proof of Lemma 3.5: Lemma 3.5 follows from Lemma A.4 and A.5. ∎
Proof of Lemma 3.1: By Lemma A.3 and 3.2, we get
$\mu_{1}\leq\mu_{2}\cdot\max\\{1,\frac{\gamma_{2}}{\gamma_{1}}\\}$ is
sufficient for $[T_{1,a},T_{1,f}]$ and $[T_{2,a},T_{2,f}]$ to have disjoint
interiors. Proving that the condition is necessary follows from the argument
after Lemma 3.5 in Section 3.1. Hence Lemma 3.1 stands proved. ∎
Proof of Lemma 3.6: (1) If $T_{1,f}=T_{2,a}$, queue 1 is empty at $T_{1,f}$
(by Lemma 3.5), making the network empty at $T_{1,f}$. As a result, every
class 2 user will be strictly better off arriving at $T_{1,f}$ and
$\mathbf{F}$ will not be an EAP. Therefore, the only other possibility is
$T_{2,f}=T_{1,a}$.
(2) On assuming a contradiction, if $T_{2,f}<T_{1,f}$, by Lemma 3.1, we must
have $T_{2,f}\in(T_{1,a},T_{1,f})$. Since group 1 users will arrive in
$[T_{2,f},T_{1,f}]$, we must have
$[T_{2,f},T_{1,f}]\subseteq\overline{E_{1}}$. Using Lemma A.4, queue 2 has a
zero waiting time for $t\geq\tau_{1}(T_{2,f})$ and as a consequence
$\tau_{\mathbf{F}}^{(2)}(t)=\tau_{1}(t)$ for $t\geq T_{2,f}$. Therefore, for
$t\in[T_{2,f},T_{1,f}]$,
$\displaystyle(C^{(2)}_{\mathbf{F}})^{\prime}(t)$
$\displaystyle=(\tau_{F}^{(2)})^{\prime}(t)-\gamma_{2}=\tau_{1}^{\prime}(t)-\gamma_{2}$
$\displaystyle=\frac{F_{1}^{\prime}(t)}{\mu_{1}}-\gamma_{2}~{}~{}$ (using (3)
and the fact that $[T_{2,f},T_{1,f}]\in\overline{E_{1}}$)
$\displaystyle=\gamma_{1}-\gamma_{2}<0.~{}~{}$ (using Lemma 3.4).
Thus, the group 2 user arriving at $T_{2,f}$ will be strictly better off by
arriving at $T_{1,f}$. Hence $\mathbf{F}$ will not be an EAP.
(3) Let $T=\min\left\\{T_{1,f},T_{2,f}\right\\}$. By Lemma 3.1,
$\max\\{T_{1,a},T_{2,a}\\}<T$. If we assume contradiction to the statement, we
must have $T_{1,a}\leq T_{2,a}<T$. As a result, class 2 users are arriving in
$[T_{2,a},T]$ alongside class 1 users. Using Lemma 3.4, for every
$t\in[T_{2,a},T]$,
$\displaystyle F_{2}^{\prime}(t)$
$\displaystyle=\mu_{2}\gamma_{2},~{}\text{and},$ $\displaystyle
F_{1}^{\prime}(t)$ $\displaystyle=\mu_{1}\gamma_{1}-\mu_{2}\gamma_{2}.$
Moreover, for $t\in[T_{2,a},T]$, we have $A_{2}(\tau_{1}(t))=F_{2}(t)$. Again
$[T_{2,a},T]\subseteq\mathcal{S}(F_{1})$ implies
$[T_{2,a},T]\subseteq\overline{E_{1}}$ and therefore by (3)
$\tau_{1}^{\prime}(t)=\frac{F_{1}^{\prime}(t)+F_{2}^{\prime}(t)}{\mu_{1}}=\gamma_{1}$
a.e. in $[T_{2,a},T]$. Hence,
$A_{2}^{\prime}(\tau_{1}(t))=\frac{F_{2}^{\prime}(t)}{\tau_{1}^{\prime}(t)}=\mu_{2}\frac{\gamma_{2}}{\gamma_{1}}<\mu_{2}$
a.e. in $[T_{2,a},T]$. This implies that class 2 users are arriving at queue 2
at a rate $<\mu_{2}$ from the time when queue 2 starts serving them, i.e.,
$\tau_{1}(T_{2,a})$. Therefore $Q_{2}(t)=0$ for all
$t\in[\tau_{1}(T_{2,a}),\tau_{1}(T)]$, which contradicts Lemma A.1. ∎
#### A.1.1 Proof of Theorem 3.1
Below we mention those parts of the proof of Theorem 3.1 which were not
covered in the proof sketch mentioned in Section 3.1. Of the skipped portion
is the proof of existence and uniqueness of EAP for case 3,
$\gamma_{1}>\gamma_{2}$, owing to its similarity in arguments with case 2
$\gamma_{2}>\gamma_{1}>\frac{\mu_{2}}{\mu_{1}}\cdot\gamma_{2}$. Here, we cover
those portions of the proof for each of the three cases in which the theorem
statement is divided.
Case 1 $\gamma_{1}\leq\frac{\mu_{2}}{\mu_{1}}\gamma_{2}$: For convenience of
the reader, we mention the unique candidate we obtained by Lemma 3.4 and 3.7:
$\displaystyle F_{1}^{\prime}(t)$
$\displaystyle=\mu_{1}\gamma_{1}~{}\text{if}~{}t\in[T_{1,a},T_{1,f}]~{}\text{and}~{}F_{2}^{\prime}(t)=\mu_{2}\gamma_{2}~{}\text{if}~{}t\in[T_{1,f},T_{2,f}].$
(18)
We first verify that, when the two classes arrive by the unique remaining
candidate profile, for both the classes $i=1,2$ cost derivative satisfies 1)
$(C_{\mathbf{F}}^{(i)})^{\prime}(t)\leq 0$ in $(-\infty,T_{i,a})$, 2)
$(C_{\mathbf{F}}^{(i)})^{\prime}(t)=0$ in $[T_{i,a},T_{i,f}]$, and 3)
$(C_{\mathbf{F}}^{(i)})^{\prime}(t)\geq 0$ in $(T_{i,f},\infty)$. The
preceding statement implies both the classes have their cost constant on the
support interval and higher outside of it, implying the candidate is an EAP.
Following our agenda, we go by the following sequence of arguments.
* •
In $(-\infty,T_{1,a}]$: $\tau_{\mathbf{F}}^{(i)}(t)=0$ for both classes
$i\in\\{1,2\\}$ in $(-\infty,T_{1,a}]$. As a result,
$(C_{\mathbf{F}}^{(i)})^{\prime}(t)=-\gamma_{i}<0$ in $(-\infty,T_{1,a}]$.
* •
In $[T_{1,a},T_{1,f}]$: Queue 1 stays engaged in $[T_{1,a},T_{1,f}]$, since
$A_{1}(t)=F_{1}(t)>\mu_{1}\cdot\max\\{t,0\\}$ in $[T_{1,a},T_{1,f}]$. Hence by
(3), for both the classes $i\in\\{1,2\\}$,
$(C_{\mathbf{F}}^{(i)})^{\prime}(t)=\frac{F_{1}^{\prime}(t)}{\mu_{1}}-\gamma_{i}=\gamma_{1}-\gamma_{i}$.
As a result, for $i=1$, $(C_{\mathbf{F}}^{(1)})^{\prime}(t)=0$ and for $i=2$,
$(C_{\mathbf{F}}^{(2)})^{\prime}(t)=\gamma_{1}-\gamma_{2}<0$.
* •
In $[T_{1,f},T_{2,f}]$: Since $\mu_{1}\cdot
T_{2,f}=\Lambda_{1}+\frac{\mu_{1}}{\mu_{2}}\Lambda_{2}>\Lambda_{1}+\Lambda_{2}=A_{1}(T_{2,f})$,
queue 1 empties at some time $T\in(T_{1,f},T_{2,f}]$. As class 2 users arrive
at a constant rate of $\mu_{2}\gamma_{2}$ in $[T_{1,f},T_{2,f}]$, queue 1
stays engaged in $[T_{1,f},T]$ and empty in $[T,T_{2,f}]$. Therefore, in
$[T_{1,f},T]$, by (3),
$(C_{\mathbf{F}}^{(1)})^{\prime}(t)=\frac{F_{2}^{\prime}(t)}{\mu_{1}}-\gamma_{1}=\frac{\mu_{2}\gamma_{2}-\mu_{1}\gamma_{1}}{\mu_{1}}\geq
0$ and in $(T,T_{2,f}]$, since queue 1 is empty,
$(C_{\mathbf{F}}^{(1)})^{\prime}(t)=1-\gamma_{1}\geq 0$. Now we argue that
queue 2 remains engaged in $[\tau_{1}(T_{1,f}),T_{2,f}]$ by considering two
parts separately:
* –
In $[\tau_{1}(T_{1,f}),T]$, class 2 users arrive at queue 2 from queue 1 at
rate $\mu_{1}>\mu_{2}$ in that period, making queue 2 engaged.
* –
In $[T,T_{2,f}]$, since $A_{2}^{\prime}(t)=\mu_{2}\gamma_{2}<\mu_{2}$,
$A_{2}(t)-\mu_{2}\cdot(t-\tau_{1}(T_{1,f}))$ is a decreasing function.
Combining this with the fourth equation
$A_{2}(T_{2,f})=\mu_{2}\cdot(T_{2,f}-\tau_{1}(T_{1,f}))$ in the identified
linear system, we get $A_{2}(t)>\mu_{2}\cdot(t-\tau_{1}(T_{1,f}))$ in
$[T,T_{2,f})$, implying queue 2 stays engaged.
Therefore, by (3), for $t\in[T_{1,f},T_{2,f}]$,
$\tau_{\mathbf{F}}^{(2)}(t)=\tau_{2}(\tau_{1}(t))=\tau_{1}(T_{1,f})+\frac{A_{2}(\tau_{1}(t))}{\mu_{2}}=\frac{\Lambda_{1}}{\mu_{1}}+\frac{F_{2}(t)}{\mu_{2}}$.
Hence,
$(C_{\mathbf{F}}^{(2)})^{\prime}(t)=(\tau_{\mathbf{F}}^{(2)})^{\prime}(t)-\gamma_{2}=\frac{F_{2}^{\prime}(t)}{\mu_{2}}-\gamma_{2}=0$
in $[T_{1,f},T_{2,f}]$. Upon summarizing, for $t\in[T_{1,f},T_{2,f}]$, we get
$(C_{\mathbf{F}}^{(1)})^{\prime}(t)\geq 0$ and
$(C_{\mathbf{F}}^{(2)})^{\prime}(t)=0$.
* •
In $[T_{2,f},\infty)$: By the last step, both the queues stay empty after
$T_{2,f}$. As a result, for both the classes $i\in\\{1,2\\}$,
$(C_{\mathbf{F}}^{(i)})^{\prime}(t)=1-\gamma_{i}>0$ in $[T_{2,f},\infty)$.
Case 2 $\frac{\mu_{2}}{\mu_{1}}\gamma_{2}<\gamma_{1}<\gamma_{2}$: Skipped
parts of the proof of Lemma 3.8: Identifying the unique candidate for Type II:
For every EAP under Type II, we obtain the following system of equations to be
satisfied by the support boundaries $T_{1,a},T_{1,f},T_{2,a},T_{2,f}$:
1. 1.
By Lemma 3.4, $F_{1}^{\prime}(t)=\mu_{1}\gamma_{1}-\mu_{2}\gamma_{2}$ in
$[T_{1,a},T_{1,f}]$, giving us:
$T_{1,f}=T_{1,a}+\frac{\Lambda_{1}}{\mu_{1}\gamma_{1}-\mu_{2}\gamma_{2}}$.
2. 2.
By Lemma 3.4, $F_{2}^{\prime}(t)=\mu_{2}\gamma_{2}$ in $[T_{2,a},T_{2,f}]$,
giving us: $T_{2,f}=T_{2,a}+\frac{\Lambda_{2}}{\mu_{2}\gamma_{2}}$.
3. 3.
Applying the argument used for getting the third equation of Type I, we must
have $F_{1}(T_{1,f})+F_{2}(T_{1,f})=\mu_{1}T_{1,f}$. Plugging in
$F_{1}(T_{1,f})=\Lambda_{1}$ and
$F_{2}(T_{1,f})=\mu_{2}\gamma_{2}(T_{1,f}-T_{2,a})$ (by Lemma 3.4), we get:
$\Lambda_{1}+\mu_{2}\gamma_{2}(T_{1,f}-T_{2,a})=\mu_{1}T_{1,f}$.
4. 4.
By an argument similar to the one used for getting the fourth equation of Type
I, we have $\mu_{2}\cdot(\tau_{1}(T_{2,f})-\tau_{1}(T_{2,a}))=\Lambda_{2}$.
Since queue 1 is empty at $T_{1,f}$ (by Lemma 3.5) and class 2 users arrive at
rate $\mu_{2}\gamma_{2}<\mu_{1}$ (by Lemma 3.4), queue 1 stays empty at
$T_{2,f}$, causing $\tau_{1}(T_{2,f})=T_{2,f}$. Queue 1 serves the first class
2 user at time zero, causing $\tau_{1}(T_{2,a})=0$. Plugging these in, we get:
$T_{2,f}=\frac{\Lambda_{2}}{\mu_{2}}$. ∎
The solution to the above system of equations is in (3.). The support
boundaries in (3.) must satisfy $T_{2,a}<T_{1,a}<T_{1,f}\leq T_{2,f}$ to
represent an EAP under Type II. Imposing $T_{1,a}>T_{2,a}$ on (3.), we get the
condition
$\Lambda_{1}<\frac{1-\gamma_{2}}{1-\gamma_{1}}\left(\frac{\mu_{1}\gamma_{1}}{\mu_{2}\gamma_{2}}-1\right)\Lambda_{2}$.
Moreover, if
$\Lambda_{1}<\frac{1-\gamma_{2}}{1-\gamma_{1}}\left(\frac{\mu_{1}\gamma_{1}}{\mu_{2}\gamma_{2}}-1\right)\Lambda_{2}$,
(3.) satisfies $T_{2,a}<T_{1,a}<T_{1,f}\leq T_{2,f}$ and upon plugging in
arrival rates from Lemma 3.4, we get a candidate having arrival rates and
support boundaries same as the joint arrival profile mentioned under case 2b
of Theorem 3.1. This candidate satisfies
$F_{1}(T_{1,f})=\Lambda_{1},~{}F_{2}(T_{2,f})=\Lambda_{2}$ and will be the
only Type II candidate to qualify as an EAP. Upon proving that this identified
Type II candidate is an EAP, the second statement of Lemma 3.8 will follow.
Proving that the unique candidate obtained is an EAP: For convenience of the
reader we mention the candidates we were left with for both the types:
1. 1.
If
$\Lambda_{1}\geq\frac{1-\gamma_{2}}{1-\gamma_{1}}\left(\frac{\mu_{1}\gamma_{1}}{\mu_{2}\gamma_{2}}-1\right)\Lambda_{2}$,
the unique Type I candidate is:
$\displaystyle F_{1}^{\prime}(t)$
$\displaystyle=\begin{cases}\mu_{1}\gamma_{1}~{}~{}&\text{if}~{}t\in[T_{1,a},T_{2,a}),\\\
\mu_{1}\gamma_{1}-\mu_{2}\gamma_{2}~{}~{}&\text{if}~{}t\in[T_{2,a},T_{1,f}],\end{cases}~{}\text{and}~{}F_{2}^{\prime}(t)=\mu_{2}\gamma_{2}~{}~{}\text{if}~{}t\in[T_{2,a},T_{2,f}]$
(19)
where $T_{1,a},T_{1,f},T_{2,a},T_{2,f}$ are in ( ‣ 2.).
2. 2.
If
$\Lambda_{1}<\frac{1-\gamma_{2}}{1-\gamma_{1}}\left(\frac{\mu_{1}\gamma_{1}}{\mu_{2}\gamma_{2}}-1\right)\Lambda_{2}$,
the unique Type II candidate is:
$\displaystyle F_{1}^{\prime}(t)$
$\displaystyle=\mu_{1}\gamma_{1}-\mu_{2}\gamma_{2}~{}~{}\text{if}~{}t\in[T_{2,a},T_{1,f}],~{}\text{and}~{}F_{2}^{\prime}(t)=\mu_{2}\gamma_{2}~{}~{}\text{if}~{}t\in[T_{2,a},T_{2,f}]$
(20)
where $T_{1,a},T_{1,f},T_{2,a},T_{2,f}$ are in (3.).
By the following sequence of arguments, for both the types, we argue that if
users of the two classes arrive by the unique remaining candidate, both
classes $i=1,2$ have their cost derivative satisfying:
$(C_{\mathbf{F}}^{(i)})^{\prime}(t)\leq 0$ in $(-\infty,T_{i,a})$,
$(C_{\mathbf{F}}^{(i)})^{\prime}(t)=0$ in $[T_{i,a},T_{i,f}]$, and
$(C_{\mathbf{F}}^{(i)})^{\prime}(t)\geq 0$ in $(T_{i,f},\infty)$. This will
imply, for both the classes, cost is constant in the support interval and
higher outside and hence the candidate is an EAP.
1. 1.
In $(-\infty,\min\\{T_{1,a},T_{2,a}\\}]$: Note that for
$t\in(-\infty,\min\\{T_{1,a},T_{2,a}\\}]$, $\tau_{\mathbf{F}}^{(i)}(t)=0$ for
both the classes $i\in\\{1,2\\}$. As a result,
$(C_{\mathbf{F}}^{(i)})^{\prime}(t)=-\gamma_{i}<0$ for $i\in\\{1,2\\}$.
2. 2.
In $[\min\\{T_{1,a},T_{2,a}\\},T_{1,f}]$: Obtained candidates of both the
types satisfy $A_{1}^{\prime}(t)=F_{1}^{\prime}(t)+F_{2}^{\prime}(t)<\mu_{1}$
in $[\min\\{T_{1,a},T_{2,a}\\},T_{1,f}]$, making $A_{1}(t)-\mu_{1}t$
decreasing in $[0,T_{1,f}]$. Since both the types satisfy
$A_{1}(T_{1,f})=\mu_{1}T_{1,f}$, the preceding statement imply
$A_{1}(t)>\mu_{1}t$ in $[0,T_{1,f})$ and therefore, queue 1 is engaged in
$[\min\\{T_{1,a},T_{2,a}\\},T_{1,f}]$ and empty after $T_{1,f}$. Using (3) and
the arrival rates from (19) and (20), we get
$(C_{\mathbf{F}}^{(1)})^{\prime}(t)=\tau_{1}^{\prime}(t)-\gamma_{1}=\frac{F_{1}^{\prime}(t)+F_{2}^{\prime}(t)}{\mu_{1}}-\gamma_{1}=0$
for $t\in[\min\\{T_{1,a},T_{2,a}\\},T_{1,f}]$. For analyzing the cost of the
second class, we consider the two types separately:
* •
In Type I ($T_{1,a}\leq T_{2,a}<T_{1,f}$): In $[T_{1,a},T_{2,a}]$,
$(C_{\mathbf{F}}^{(2)})^{\prime}(t)=\tau_{1}^{\prime}(t)-\gamma_{2}=\frac{F_{1}^{\prime}(t)+F_{2}^{\prime}(t)}{\mu_{1}}-\gamma_{2}=\gamma_{1}-\gamma_{2}<0$.
Upon observing $A_{2}(\tau_{1}(t))=F_{2}(t)$, we can write using chain rule
that
$A_{2}^{\prime}(\tau_{1}(t))=\frac{(A_{2}\circ\tau_{1})^{\prime}(t)}{\tau_{1}^{\prime}(t)}=\frac{F_{2}^{\prime}(t)}{\tau_{1}^{\prime}(t)}$,
whenever the derivatives exists. Now for $t\in[T_{2,a},T_{1,f}]$, using (3),
rate of arrival of class 2 users in queue 2 at $\tau_{1}(t)$ is
$A_{2}^{\prime}(\tau_{1}(t))=\frac{F_{2}^{\prime}(t)}{\tau_{1}^{\prime}(t)}=\mu_{1}\cdot\frac{F_{2}^{\prime}(t)}{F_{1}^{\prime}(t)+F_{2}^{\prime}(t)}=\frac{\mu_{2}\gamma_{2}}{\gamma_{1}}>\mu_{2}$.
As a result, queue 2 remains engaged in $[\tau_{1}(T_{2,a}),T_{1,f}]$. With
this, the time of service of class 2 users arriving in $[T_{2,a},T_{1,f}]$ is
$\tau_{\mathbf{F}}^{(2)}(t)=\tau_{2}(\tau_{1}(t))=\tau_{1}(T_{2,a})+\frac{A_{2}(\tau_{1}(t))}{\mu_{2}}=\tau_{1}(T_{2,a})+\frac{F_{2}(t)}{\mu_{2}}$.
Using this and the arrival rates from (19),
$(C_{\mathbf{F}}^{(2)})^{\prime}(t)=(\tau_{\mathbf{F}}^{(2)})^{\prime}(t)-\gamma_{2}=\frac{F_{2}^{\prime}(t)}{\mu_{2}}-\gamma_{2}=0$
in $[T_{2,a},T_{1,f}]$.
* •
In Type II ($T_{2,a}<T_{1,a}<T_{1,f}$): In $[T_{2,a},T_{1,a}]$, rate of
arrival of class 2 users at queue 2 at time $\tau_{1}(t)$ is
$\mu_{1}>\mu_{2}$, and $\tau_{1}(T_{2,a})=0$. As a result, queue 2 has a
positive waiting queue at $\tau_{1}(T_{1,a})$. Following a calculation similar
to the one used in the previous step for Type I, rate of arrival of class 2
users at queue 2 at time $\tau_{1}(t)$ is
$\frac{\mu_{2}\gamma_{2}}{\gamma_{1}}\geq\mu_{2}$ when
$t\in[T_{1,a},T_{1,f}]$. The last two statements imply queue 2 has positive
waiting time in $(0,T_{1,f}]$. Using this, time of service of class 2 users
arriving in $[T_{2,a},T_{1,f}]$ is
$\tau_{\mathbf{F}}^{(2)}(t)=\tau_{2}(\tau_{1}(t))=\tau_{1}(T_{2,a})+\frac{A_{2}(\tau_{1}(t))}{\mu_{2}}=\frac{F_{2}(t)}{\mu_{2}}$
and as a result,
$(C_{\mathbf{F}}^{(2)})^{\prime}(t)=\frac{F_{2}^{\prime}(t)}{\mu_{2}}-\gamma_{2}=0$.
3. 3.
In $[T_{1,f},T_{2,f}]$: Queue 1 remains empty in this region. On the other
hand, queue 2 remains engaged in $[T_{1,f},T_{2,f}]$, since |
# TIT/HEP-689 May 2022 Analytic continuation for giant gravitons
Yosuke Imamura111E-mail<EMAIL_ADDRESS>
Department of Physics, Tokyo Institute of Technology,
Tokyo 152-8551, Japan
We investigate contributions of giant gravitons to the superconformal index.
We concentrate on coincident giant gravitons wrapped around a single cycle,
and each contribution is obtained by a certain variable change for fugacities
from the index of the worldvolume theory on the giant gravitons. Because we
treat the index as a series of fugacities and the variable change relates
different convergence regions, we need an analytic continuation before summing
up such contributions. We propose a systematic prescription for the
continuation. Although our argument is based on some unproved assumptions, it
passes non-trivial numerical checks for some examples. With the prescription
we can calculate the indices of the M5-brane theories from those of the
M2-brane theories, and vice versa.
###### Contents
1. 1 Introduction
2. 2 Plethystic exponential and plethystic logarithm
3. 3 Variable changes and analytic continuation
4. 4 Numerical tests
1. 4.1 Double expansion of the Schur index for $T=D3$
2. 4.2 Simple expansion of the Schur index for $T=D3$
3. 4.3 Superconformal index for $T=D3$
4. 4.4 M5 from M2
5. 4.5 M2 from M5
5. 5 Conclusions and Discussion
6. A Full index of $6$-dim $A_{N-1}$ SCFT
## 1 Introduction
For the last few decades the AdS/CFT correspondence [1, 2, 3] has been playing
an important role in the progress of string theory and quantum field theories.
It provides novel approaches for investigation of different physical
quantities in various situations. It is very powerful for the analysis of
large $N$ gauge theories, and even in the finite $N$ region, where the Planck
length is not negligible, it is possible to calculate supersymmetry protected
quantities like $R$-charges of gauge invariant operators by using string
theory in the AdS background.
Generating functions of such quantities have been calculated on the gravity
side of the AdS/CFT corresponsence. For example, the superconformal index of
the ${\cal N}=4$ $U(N)$ SYM in the large $N$ limit was reproduced as the index
of the bulk supergravity modes [4]. Extended branes are important when $N$ is
finite. It is known that not only branes wrapped around topologically non-
trivial cycles [5, 6] but also branes wrapped around trivial cycles can be
stable and BPS. Such branes carry the same quantum numbers with point-like
gravitons, and are called giant gravitons [7, 8, 9, 10]. The BPS partition
function for scalar fields in 4d ${\cal N}=4$ SYM was reproduced by geometric
quantization of giant gravitons in [11], and the same quantity was also
obtained by using the dual giants in [12]. Similar analysis was done in [13]
for the 3d and 6d theories realized on M2 and M5-branes.
The contribution of giant gravitons with different wrapping numbers to the
superconformal index was suggested by a characteristic form of the analytic
result in [14] for the unrefined Schur index of ${\cal N}=4$ SYM. (See also
[15] for generalization to ${\cal N}=2$ quiver gauge theories.) It has been
confirmed by studying fluctuation modes of giant gravitons in [16, 17, 18] for
the ${\cal N}=4$ SYM, and for more general theories in [19, 20, 21, 22, 23].
In all cases the index $I_{N}^{(T)}$ of a theory $T$ with finite $N$ whose
holographic dual is string/M theory in $AdS\times M_{T}$ is given by a
multiple expansion of the form
$\displaystyle\frac{I_{N}^{(T)}}{I_{\infty}^{(T)}}=\sum_{m_{1}=0}^{\infty}\cdots\sum_{m_{d}=0}^{\infty}x_{1}^{m_{1}N}\cdots
x_{d}^{m_{d}N}F^{(T)}_{m_{1},\ldots,m_{d}},$ (1)
where $m_{1},\ldots,m_{d}$ are wrapping numbers of giant gravitons around
appropriately chosen $d$ supersymmetric cycles in $M_{T}$. The number of
cycles $d$ depends on the theory $T$. For each set of the wrapping numbers the
function $F^{(T)}_{m_{1},\ldots,m_{d}}$ is the index of the field theory
realized on the corresponding system of giant gravitons and
$F_{0,\ldots,0}^{(T)}=1$. A typical structure of the theory is shown in Figure
1.
Figure 1: A typical structure of the theory realized on a system of giant
gravitons. Each vertex represents the theory $T_{i}$ with rank $m_{i}$
realized on $m_{i}$ coincident giant gravitons wrapped around cycle $i$. Edges
connecting vertices are degrees of freedom arising on intersections of cycles.
$d=3$ case is shown.
It is the direct product of theories $T_{i}$ realized on cycles $i=1,\ldots,d$
coupling to the degrees of freedom arising on their intersections.
$F^{(T)}_{m_{1},\ldots,m_{k}}$ are $N$-independent, and the $N$-dependence
appears only through the prefactor $x_{1}^{m_{1}N}\cdots x_{d}^{m_{d}N}$.
Similar expansion of the index is also proposed in [24] based on a
complementary analysis on the gauge theory side, and such expansions were
named “the giant graviton expansions”. See also [25]. In their analysis the
correspondents for giant gravitons and fluctuations on them are baryon
operators and their modifications, respectively. Although there are some
technical subtleties about the relation between these expansions obtained on
the two sides of the AdS/CFT correspondence they are expected to be
essentially the same.
In this work we focus on the contribution of giant gravitons wrapped around a
single cycle $i$. Let $F^{(T)}_{i,m_{i}}=F^{(T)}_{0,\ldots,m_{i},\ldots,0}$ be
such contributions. It is essentially the index $I_{m_{i}}^{(T_{i})}$ of the
theory $T_{i}$ realized on the worldvolume of $m_{i}$ coincident giant
gravitons. More precisely, $F_{i,m}^{(T)}$ and $I_{m}^{(T_{i})}$ are related
by a simple variable change of the fugacities as we will explain shortly. A
purpose of this work is to study this relation in detail.
For concreteness let us consider the Schur index [26] of the ${\cal N}=4$
$U(N)$ SYM. Let $J_{1}$ and $J_{2}$ be the angular momenta and $R_{x}$,
$R_{y}$, and $R_{z}$ be the $R$ charges. The Schur index is defined by
$\displaystyle I_{N}^{(D3)}(x,y)=\mathop{\rm
tr}\nolimits[(-1)^{F}q^{J_{1}}x^{R_{x}}y^{R_{y}}],\quad q=xy,$ (2)
where we denote the ${\cal N}=4$ $U(N)$ SYM by $T=D3$. The constraint $q=xy$
is necessary to preserve the supersymmetry. We eliminate $q$ by the constraint
and treat the index as a function of $x$ and $y$.
The giant graviton expansion for the Schur index is given in [17] as the
double expansion
$\displaystyle\frac{I_{N}^{(D3)}}{I^{(D3)}_{\infty}}=\sum_{m_{x}=0}^{\infty}\sum_{m_{y}=0}^{\infty}x^{m_{x}N}y^{m_{y}N}F^{(D3)}_{m_{x},m_{y}}.$
(3)
The two non-negative integers $m_{x}$ and $m_{y}$ are wrapping numbers over
the cycles $i=x$ and $i=y$, respectively, which are respectively defined to be
the $R_{x}$ and $R_{y}$ fixed roci in $M_{D3}=S^{5}$.222We use $x$ and $y$ for
the labels of cycles. These should not be confused with the fugacities. A
special property of the Schur index is that the functions
$F_{m_{x},m_{y}}^{(D3)}$ are factorized [17]:
$\displaystyle
F_{m_{x},m_{y}}^{(D3)}=(xy)^{m_{x}m_{y}}F^{(D3)}_{x,m_{x}}F^{(D3)}_{y,m_{y}}.$
(4)
Therefore, we only need to determine $F_{i,m}^{(D3)}$ to write down the
expansion (3). $F_{i,m}^{(D3)}$ is the index of the ${\cal N}=4$ $U(m)$ SYM
realized on $m$ coincident giant gravitons wrapped around the cycle $i(=x,y)$.
The theories $T$, $T_{x}$, and $T_{y}$ are accidentally the same in this case,
and the corresponding indices $I_{N}^{(D3)}$, $F_{x,N}^{(D3)}$, and
$F_{y,N}^{(D3)}$ are also essentially the same functions. Careful analysis of
the action of superconformal algebra on the $AdS$ boundary and that on giant
gravitons wrapped around the cycle $x$ shows that they are related by the
involution [16]
$\displaystyle(H,J_{1},J_{2},R_{x},R_{y},R_{z})\rightarrow(H-2R_{x},R_{y},R_{z},-R_{x},J_{1},J_{2}).$
(5)
A similar relation also holds for cycle $y$. This means that
$I_{N}^{(D3)}(x,y)$ and $F_{i,N}^{(D3)}(x,y)$ are related by the variable
changes [16]
$\displaystyle F_{x,N}^{(D3)}(x,y)=I_{N}^{(D3)}(x^{-1},xy),\quad
F_{y,N}^{(D3)}(x,y)=I_{N}^{(D3)}(xy,y^{-1}).$ (6)
See also [24] for a derivation of (6) on the gauge theory side.
Despite the simplicity of (6), it is not straightforward to calculate
$F_{i,N}^{(D3)}$ from $I_{N}^{(D3)}$, and vice versa, because we usually do
not have analytic form of the indices, and only series expansions are
available. The variable changes relate different convergence regions of the
functions, and it is a non-trivial problem to generate one from another. In
this paper, we propose a simple prescription to obtain the functions
$F_{i,N}^{(T)}$ from $I_{N}^{(T_{i})}$.
To understand the relation, we need to carefully analyze the dependence of the
series expansion on the choice of expansion variables. Different choices of
expansion variables give different series expansions. Indeed, the expansion
found in [24] is the simple expansion over a single non-negative integer $m$:
$\displaystyle\frac{I_{N}^{(T)}}{I_{\infty}^{(T)}}=\sum_{m=0}^{\infty}x^{mN}F_{i,m}^{(T)}.$
(7)
This looks different from the multiple expansion (1). Once we have understood
the relation between $I_{N}^{(T_{i})}$ and $F_{i,N}^{(T)}$, we can give a
partial explanation to this difference.
In the process of confirming the relation (7) the analytic continuation has
been necessarily done in [24] for functions appearing in (7) based on a pole
structure commonly found in the functions. Our prescription is useful to
clarify the pole structure and enable us to study the relations like (6) in a
systematic way.
The paper is organized as follows.
In the next section we prepare tools used in the following sections. In
particular, we define “a domain” specifying the expansion variables. We also
define the plethystic exponential and the plethystic logarithm with emphasis
on the dependence on domains. In section 3 we explain the necessity of the
analytic continuation, and propose a simple prescription to realize it. We
numerically test the proposed method in Section 4 for the maximally
supersymmetric theories realized on D3, M5, and M2-branes. The last section is
devoted to conclusions and discussion.
## 2 Plethystic exponential and plethystic logarithm
The calculation of the index $I$ of a free field theory usually starts from
the analysis of single-particle states. The corresponding index is called the
letter index. It is obtained by summing up the contributions from all one-
particle states. If there are two fugacities $x$ and $y$ it is given by333We
mainly consider the case with two variables. Generalization is
straightforward.
$\displaystyle i=\sum_{(n_{x},n_{y})\in R}c_{n_{x},n_{y}}x^{n_{x}}y^{n_{y}},$
(8)
where $R$ is a certain region on the $2$-dimensional charge lattice specifying
a set of monomials appearing in the expansion. We call such a region “a
domain.” We usually adopt $R$ such that it covers all one-particle states. We
call such a domain “a physical domain”. For the Schur index of $T=D3$ we can
take
$\displaystyle R_{0}=\\{(n_{x},n_{y})|n_{x},n_{y}\geq 0\\}\backslash(0,0)$ (9)
as a physical domain. Later we will discuss other choices of $R$. We call the
series associated with a domain $R$ the $R$-series. For a free field theory
the series sums up into a simple rational function. For example, the letter
index of the ${\cal N}=4$ $U(1)$ vector multiplet is [4]
$\displaystyle i^{(D3)}_{1}=1-\frac{(1-x)(1-y)}{1-xy},$ (10)
and that of the type IIB supergravity multiplet in $AdS_{5}\times S^{5}$,
which is dual to the large $N$ SYM, is [4]
$\displaystyle
i^{(D3)}_{\infty}=\frac{1}{1-x}+\frac{1}{1-y}-\frac{1}{1-xy}-1.$ (11)
By expanding these functions into terms appearing in the domain $R_{0}$ we can
obtain the set of coefficients in (8). In the following we use the notation
$i|_{R}$ when we want to emphasize that $i$ is given as an $R$-series. We
graphically express a series by plotting the coefficients in the two-
dimensional lattice as shown in Figure 2.
Figure 2: (a) The letter index $i_{1}^{(D3)}$ of the ${\cal N}=4$ vector
multiplet. (b) The letter index $i_{\infty}^{(D3)}$ of the type IIB
supergravity multiplet in $AdS_{5}\times S^{5}$.
Once we have obtained the letter index (8), the index for multi-particle
states is uniquely determined by considering the combinatorics of the letters.
The solution to this problem is
$\displaystyle I=\mathop{\rm Pexp}\nolimits i,$ (12)
where $\mathop{\rm Pexp}\nolimits$ is the plethystic exponential defined by
$\displaystyle\mathop{\rm Pexp}\nolimits
f(x,y)=\exp\left(\sum_{m=1}^{\infty}\frac{1}{m}f(x^{m},y^{m})\right).$ (13)
For the definition (13) to make sense the condition $|x^{n_{x}}y^{n_{y}}|<1$
must hold for all terms appearing in the sum (8). This means all terms in $f$
must be contained in the domain $R_{(x,y)}$ defined by
$\displaystyle R_{(x,y)}:=\\{(n_{x},n_{y})|n_{x}\log|x|+n_{y}\log|y|<0\\}.$
(14)
For example, if we take $x$ and $y$ satisfying $|x|=|y|<1$, (14) gives the
upper half plane with the horizontal boundary excluded, and all points of
$i_{1}^{(D3)}$ and $i_{\infty}^{(D3)}$ shown in Figure 2 are contained in the
domain. (Namely, we can use $R_{(x,y)}$ as a physical domain.) If $i$ is
expanded with such a domain $R$, $I$ given by (12) can also be expanded in a
similar form;
$\displaystyle I=1+\sum_{(n_{x},n_{y})\in
R}c^{\prime}_{n_{x},n_{y}}x^{n_{x}}y^{n_{y}}$ (15)
with the same domain $R$. We can show that the map from
$\\{c_{n_{x},n_{y}}\\}$ to $\\{c^{\prime}_{n_{x},n_{y}}\\}$ is one to one if
* •
$R$ is a sector with the center angle $\theta\leq\pi$.
* •
$R\cap(-R)=\phi$.
The latter is the condition for the boundary of $R$. We do not include the
origin, and if $\theta=\pi$ we can include at most one of two boundary rays.
We always require $R$ to satisfy these conditions, and then we can define the
inverse operation to $\mathop{\rm Pexp}\nolimits$:
$\displaystyle i=\mathop{\rm Plog}\nolimits I.$ (16)
This is called the plethystic logarithm. It is important that we can introduce
the group structure for $\mathop{\rm Pexp}\nolimits i$ under the
multiplication, which corresponds to the addition for $i$.
Even for an interacting theory we can calculate the index by using
localization method if it is a Lagrangian theory. For the ${\cal N}=4$ SYM
with $U(N)$ gauge group the index is given by
$\displaystyle
I^{(D3)}_{N}|_{R_{(x,y)}}=\frac{1}{N!}\int\prod_{a=1}^{N}\frac{dz_{a}}{2\pi
iz_{a}}\prod_{a\neq b}\left(1-\frac{z_{a}}{z_{b}}\right)\mathop{\rm
Pexp}\nolimits\left(i_{1}^{(D3)}|_{R_{(x,y)}}\sum_{a,b}\frac{z_{a}}{z_{b}}\right),$
(17)
where $z_{a}$ ($a=1,\ldots,N$) are gauge fugacities. As is explicitly shown we
suppose $i_{1}^{(D3)}$ is given as an $R_{(x,y)}$-series. Usually $R_{(x,y)}$
is assumed to be a physical domain. This means that $|x^{n_{x}}y^{n_{y}}|<1$
is satisfied for all terms in $i_{1}^{(D3)}|_{R_{(x,y)}}$. If we take the unit
circle $|z_{a}|=1$ for the integration contours
$|x^{n_{x}}y^{n_{y}}\frac{z_{a}}{z_{b}}|<1$ is also satisfied, and
$\mathop{\rm Pexp}\nolimits$ in (17) makes sense. The contour integrals
extract the contribution of gauge invariant states and give $I_{N}$ as an
$R_{(x,y)}$-series.
## 3 Variable changes and analytic continuation
In the following we frequently use the variable changes in (6), and it is
convenient to introduce $\sigma_{x}$ and $\sigma_{y}$ to represent these
variable changes as follows.
$\displaystyle\sigma_{x}(x,y,q)=(x^{-1},q,y),\quad\sigma_{y}(x,y,q)=(q,y^{-1},x).$
(18)
If we use the triangular lattice to express the series expansion as in Figure
2, $\sigma_{x}$ and $\sigma_{y}$ are the reflections through the lines
perpendicular to $n_{x}$ and $n_{y}$ axes, respectively. With the maps
$\sigma_{i}$ ($i=x,y$) the relations in (6) are expressed in the simple form
$\displaystyle F_{i,N}^{(D3)}=\sigma_{i}I_{N}^{(D3)}\quad(i=x,y).$ (19)
We want to confirm that $F_{i,N}^{(D3)}$ given by (19) correctly reproduce
$I_{N}^{(D3)}$ via (3). Let $R$ be a physical domain used for the expansion of
$I_{N}^{(D3)}$. To confirm the relation (3) holds we need
$F_{i,N}^{(D3)}|_{R}$. However, the variable change $\sigma_{i}$ transforms
$I_{N}^{(D3)}|_{R}$ into $F_{i,N}^{(D3)}|_{\sigma_{i}R}$, where $\sigma_{i}R$
is the image of $R$ under the map $\sigma_{i}$. Therefore, to compare the left
hand side and the right hand side of (3), we need to resum or analytically
continue $F_{i,N}^{(D3)}|_{\sigma_{i}R}$ to $F_{i,N}^{(D3)}|_{R}$.
In general, if we have $i|_{R}$ for a domain $R$, it can also be regarded as
$i|_{R^{\prime}}$ for another domain $R^{\prime}$ which contains $R$ as a
subset. Therefore, it is convenient to define “a maximal domain” which cannot
be enlarged any more. $R$ is a maximal domain if
$\displaystyle R\cup(-R)\cup O=L,$ (20)
where $O$ is the origin and $L$ is the whole plane. A maximal domain $R$ is a
sector with center angle $\pi$, and only one of the boundary rays is included
in $R$.
Let us suppose that $f|_{R^{\prime}}$ for a maximal domain $R^{\prime}$ is
given and we want to obtain $(\mathop{\rm Pexp}\nolimits f)|_{R}$ for another
maximal domain $R\neq R^{\prime}$. We divide $R^{\prime}$ into the following
two parts.
$\displaystyle R_{1}=R^{\prime}\backslash R,\quad R_{2}=R^{\prime}\cap R.$
(21)
By definition, $R^{\prime}=R_{1}\cup R_{2}$ and $R=(-R_{1})\cup R_{2}$.
Correspondingly, we divide $f$ into two parts $f_{1}$ and $f_{2}$ so that
$f|_{R^{\prime}}=f_{1}|_{R_{1}}+f_{2}|_{R_{2}}$. The plethystic exponential of
$f|_{R^{\prime}}$ is factorized into two factors:
$\displaystyle\mathop{\rm Pexp}\nolimits f|_{R^{\prime}}=(\mathop{\rm
Pexp}\nolimits f_{1}|_{R_{1}})(\mathop{\rm Pexp}\nolimits f_{2}|_{R_{2}}).$
(22)
The factor $\mathop{\rm Pexp}\nolimits f_{2}|_{R_{2}}$ can be regarded as an
$R$-series. What we need to do to obtain $(\mathop{\rm Pexp}\nolimits f)|_{R}$
is to rewrite the other factor $\mathop{\rm Pexp}\nolimits f_{1}|_{R_{1}}$ as
a $(-R_{1})$-series. This can be done by using analytic continuation as
follows. Let us suppose that $f_{1}$ is given by
$\displaystyle f_{1}(x,y)=\sum_{(n_{x},n_{y})\in
R_{1}}c_{n_{x},n_{y}}x^{n_{x}}y^{n_{y}}.$ (23)
For values of $x$ and $y$ such that $R_{1}\subset R_{(x,y)}$ we can rewrite
the plethystic exponential in the product form
$\displaystyle\mathop{\rm Pexp}\nolimits
f_{1}|_{R_{1}}=\prod_{(n_{x},n_{y})\in
R_{1}}\frac{1}{(1-x^{n_{x}}y^{n_{y}})^{c_{n_{x},n_{y}}}}.$ (24)
Once we have obtained this expression, we can analytically continue this
function to values of $x$ and $y$ such that $-R_{1}\subset R_{(x,y)}$. Then,
we can rewrite (24) as
$\displaystyle\prod_{(n_{x},n_{y})\in
R_{1}}\frac{(-x^{n_{x}}y^{n_{y}})^{-c_{n_{x},n_{y}}}}{(1-x^{-n_{x}}y^{-n_{y}})^{c_{n_{x},n_{y}}}}=P(x,y)\mathop{\rm
Pexp}\nolimits f_{1}(x^{-1},y^{-1})|_{-R_{1}},$ (25)
where $P(x,y)$ is the monomial function
$\displaystyle P(x,y)=\prod_{(n_{x},n_{y})\in
R_{1}}(-x^{n_{x}}y^{n_{y}})^{-c_{n_{x},n_{y}}}.$ (26)
It is important that if $f_{1}$ has infinitely many terms (26) may not be
well-defined, and then our prescription does not work.
By combining (25) with the factor $\mathop{\rm Pexp}\nolimits f_{2}$ we obtain
the analytically continued plethystic exponential associated with the domain
$R$:
$\displaystyle\mathop{\rm Pexp}\nolimits_{R}f\equiv P\mathop{\rm
Pexp}\nolimits f^{\rm mod}|_{R}$ (27)
where $f^{\rm mod}$ is defined by
$\displaystyle f^{\rm mod}(x,y)=f_{1}(x^{-1},y^{-1})+f_{2}(x,y).$ (28)
Each point $(n_{x},n_{y})\in R_{1}$ in $f$ is moved to the opposite point
$(-n_{x},-n_{y})\in-R_{1}$ in the modified function $f^{\rm mod}$, and all
terms of $f^{\rm mod}$ are contained in $R$.
In fact, essentially the same prescription has been used in previous works.
Because giant gravitons are wrapped around topologically trivial cycles, there
are unwrapping modes with negative excitation energies. Such modes are handled
in [16, 17, 18] by applying the prescription explained above to the integrand
of the gauge fugacity integrals. The variable change of the index after the
gauge fugacity integrals has also been considered in [24]. Although detailed
explanation is not given, the pole structure caused by the factor (24) was
pointed out, and essentially the same method seems to be used.
The prescription explained above can be summarized in the following equations.
$\displaystyle F_{i,N}^{(T)}|_{R}=\mathop{\rm
Pexp}\nolimits_{R}\sigma_{i}i_{N}^{(T_{i})},\quad i_{N}^{(T_{i})}=\mathop{\rm
Plog}\nolimits I_{N}^{(T_{i})}.$ (29)
We first calculate $i_{N}^{(T)}$ as the plethystic logarithm of
$I_{N}^{(T_{i})}$, and then we calculate the analytically continued plethystic
exponential.
Unfortunately, it will turn out that this prescription works only in limited
cases. If $\sigma_{i}i_{N}^{(T_{i})}$ has infinitely many points outside the
domain $R$, then the factor (26) becomes an infinite product which in general
does not converge. Even so, we will find that (29) is quite effective for
simple expansion (7).
## 4 Numerical tests
In this section we numerically calculate $F_{i,N}^{(T)}$ from
$I_{N}^{(T_{i})}$ by (29) in some examples and confirm the consistency with
known results. The main results in 4.2 and 4.3 have been already given in
[24].
### 4.1 Double expansion of the Schur index for $T=D3$
We first consider the double giant graviton expansion of the Schur index of
$T=D3$ studied in [17]. Let us define ${\mathfrak{q}}$ and $u$ by
$\displaystyle x={\mathfrak{q}}u,\quad y={\mathfrak{q}}u^{-1},\quad
q={\mathfrak{q}}^{2}.$ (30)
In [17] functions $F_{i,N}^{(D3)}$ are treated as ${\mathfrak{q}}$-series and
the coefficients are given as rational functions of $u$.444The fugacity
${\mathfrak{q}}$ is denoted by $q$ in [17]. Here we do the further $u^{-1}$
expansion ($u$ expansion around $u=\infty$) after the
${\mathfrak{q}}$-expansion. This corresponds to the maximal physical domain
(Figure 3 (a))
$\displaystyle R_{qu}$ $\displaystyle=\lim_{{\mathfrak{q}}\rightarrow
0,|u|>1}R_{(x,y)}$
$\displaystyle=\\{(n_{x},n_{y})|n_{x}+n_{y}>0\\}\cup\\{(n_{x},n_{y})|n_{x}+n_{y}=0,n_{y}>0\\}.$
(31)
Figure 3: The maximal domains $R_{qu}$ and $R_{yx}$
Let us first consider $F_{x,1}^{(D3)}=\mathop{\rm
Pexp}\nolimits(\sigma_{x}i_{1}^{(D3)})$ and $F_{y,1}^{(D3)}=\mathop{\rm
Pexp}\nolimits(\sigma_{y}i_{1}^{(D3)})$. We find $\sigma_{x}i_{1}^{(D3)}$ has
one term $x^{-1}={\mathfrak{q}}^{-1}u^{-1}$ that is not contained in $R_{qu}$,
and we have to move it to the opposite point to define the modified letter
index. For $\sigma_{y}i_{1}^{(D3)}$ we need to move two points corresponding
to $xy^{-1}=u^{2}$ and $y^{-1}={\mathfrak{q}}^{-1}u$. See Figure 4.
Figure 4: The modification of letter indices $\sigma_{x}i_{1}^{(D3)}$ and
$\sigma_{y}i_{1}^{(D3)}$
Following the prescription explained in the previous section, we can calculate
$F_{x,1}^{(D3)}|_{R_{qu}}$ and $F_{y,1}^{(D3)}|_{R_{qu}}$. The results are
shown in Figure 5.
Figure 5: (a) $F_{x,1}^{(D3)}|_{R_{qu}}$ (b) $F_{y,1}^{(D3)}|_{R_{qu}}$
These agree with the results in [17]:555$F_{x,N}(x,y)$ is denoted by
$F_{N}(q,u)$ in [17] and $F_{y,N}(x,y)$ corresponds to $F_{N}(q,u^{-1})$.
$F_{N}$ are given in [17] as ${\mathfrak{q}}$-series. We need further $u^{-1}$
expansion to obtain (32), (33), (34), and (35).
$\displaystyle F_{x,1}^{(D3)}$
$\displaystyle=(-u-u^{-1}-u^{-3}-\cdots){\mathfrak{q}}$
$\displaystyle+(-u^{2}+1){\mathfrak{q}}^{2}+(-u^{3}+u^{-3}){\mathfrak{q}}^{3}+(-u^{4}+1-u^{-2}+u^{-6}){\mathfrak{q}}^{4}+\cdots,$
$\displaystyle F^{(D3)}_{y,1}$
$\displaystyle=(u^{-3}+u^{-5}+u^{-7}+\cdots){\mathfrak{q}}$
$\displaystyle+(1-u^{-2}){\mathfrak{q}}^{2}+(u^{3}-u^{-3}){\mathfrak{q}}^{3}+(u^{6}-u^{2}+1-u^{-4}){\mathfrak{q}}^{4}+\cdots.$
(32)
This is not surprising because essentially the same prescription is used in
[17] for the single giant graviton sector.
Next, let us apply (29) to $F_{i,N}^{(D3)}$ with $N\geq 2$. We first calculate
$I_{N}^{(D3)}$ by (17), and calculate their plethystic logarithms
$i_{N}^{(D3)}=\mathop{\rm Plog}\nolimits I_{N}^{(D3)}$. $i_{N}^{(D3)}$ have
information not only about letters generating the operator spectra but also
about non-trivial constraints among letters called “syzygies” [27]. The
results for $N=2,3,4$ are shown in Figure 6.
Figure 6: $i_{N}^{(D3)}$ for $N=2,3,4$ are shown. The distributions are left-
right symmetric, and we show only the left half ($n_{x}\geq n_{y}$) of each
plot. The dotted lines show a numerical cut-off $n_{x}+n_{y}\leq 20$.
The asymptotic behavior of the distribution of dots is important for the
following analysis. We find in the plots in Figure 6 that except for finite
number of dots near the origin almost all dots are contained in the sectors
bounded by $E_{1}:n_{y}=Nn_{x}$ and $E_{2}:n_{x}=Nn_{y}$.
For $N=2$ the image of $E_{2}$ under $\sigma_{x}$, $\sigma_{x}E_{2}$,
coincides with the right half of the horizontal line, which is the boundary of
$R_{qu}$ (Figure 7).
Figure 7: The modification of $\sigma_{x}i_{2}^{(D3)}$ for $R_{qu}$. The
dotted line shows a numerical cut-off.
There are three points outside $R_{qu}$ corresponding to terms
$x^{-1}={\mathfrak{q}}^{-1}u$, $x^{-2}={\mathfrak{q}}^{-2}u^{2}$, and
$x^{-2}y={\mathfrak{q}}^{-1}u^{3}$. These give the factor
$P=-{\mathfrak{q}}^{4}u^{-6}=-x^{5}y^{-1}$, and the analytically continued
plethystic exponential is shown in Figure 8.
Figure 8: $F_{x,2}^{(D3)}|_{R_{qu}}$ is shown. The dotted line shows a
numerical cut-off.
Again, we find agreement with the corresponding result in [17]:
$\displaystyle F_{x,2}^{(D3)}$
$\displaystyle=(-u^{6}+u^{4}+2+u^{-2}+3u^{-4}+2u^{-6}+\cdots){\mathfrak{q}}^{4}$
$\displaystyle+(-u^{9}+2u^{5}){\mathfrak{q}}^{5}+(-u^{12}+2u^{6}+2){\mathfrak{q}}^{6}+\cdots.$
(33)
Next, let us consider $F_{y,2}^{(D3)}$. This time, $\sigma_{y}E_{1}$ is the
semi-infinite horizontal line, and unlike the previous case the line is not
contained in $R_{qu}$. This means that there are infinitely many points
outside $R_{qu}$, and the factor (26) becomes infinite product. Therefore,
(29) does not work. In fact, this is also expected from $F_{y,2}^{(D3)}$ in
[17]:
$\displaystyle F_{y,2}^{(D3)}({\mathfrak{q}},u)$
$\displaystyle=(2u^{-10}+u^{-12}+3u^{-14}+\cdots){\mathfrak{q}}^{4}+\cdots.$
(34)
As we see, the leading term has coefficient $2$, and it is impossible to
obtain such a series by the plethystic exponential.
For $N=3$ and $4$ the situation is worse. Both $\sigma_{x}$ and $\sigma_{y}$
maps one of $E_{1}$ and $E_{2}$ below the boundary of $R_{qu}$, and there are
infinitely many points outside $R_{qu}$, which make the factor (26) ill-
defined. We can also see that the leading term coefficients of the functions
in [17] are not $\pm 1$.
$\displaystyle F_{x,3}^{(D3)}$
$\displaystyle=(-2u^{15}+u^{13}+\cdots){\mathfrak{q}}^{9}+\cdots,$
$\displaystyle F_{y,3}^{(D3)}$
$\displaystyle=(5u^{-21}+2u^{-23}+\cdots){\mathfrak{q}}^{9}+\cdots,$
$\displaystyle F_{x,4}^{(D3)}$
$\displaystyle=(-5u^{28}+2u^{26}+\cdots){\mathfrak{q}}^{16}+\cdots,$
$\displaystyle F_{y,4}^{(D3)}$
$\displaystyle=(14u^{-36}+5u^{-38}+\cdots){\mathfrak{q}}^{16}+\cdots.$ (35)
These expansions cannot be reproduced by the relation (29).
### 4.2 Simple expansion of the Schur index for $T=D3$
Let us consider the simple giant graviton expansion investigated in [24]. The
index is treated in [24] as a series obtained by the successive expansions in
which $y$ expansion is carried out first and then $x$ expansion is done for
the coefficients of the first expansion. The associated maximal physical
domain is (Figure 3 (b))
$\displaystyle R_{yx}=\lim_{y\rightarrow
0,|x|<1}R_{(x,y)}=\\{(n_{x},n_{y})|n_{y}>0\\}\cup\\{(n_{x},0)|n_{x}>0\\}.$
(36)
Let us start with the analysis of $F_{x,1}^{(D3)}|_{R_{yx}}$ and
$F_{y,1}^{(D3)}|_{R_{yx}}$.
Figure 9: (a) The modification of $\sigma_{x}i_{1}$. (b) $\sigma_{y}i_{1}$
For $\sigma_{x}i_{1}^{(D3)}$, there is one dot outside $R_{yx}$ (Figure 9
(a)). This is the same as in Figure 4 (a), and hence we obtain the same result
for $F_{x,1}^{(D3)}$ as the previous subsection (Figure 5 (a)). However, for
$\sigma_{y}i_{1}^{(D3)}$, there are infinitely many points outside $R_{yx}$
corresponding to terms $y^{-1}x^{k}$ ($k=0,1,2,\ldots$) (Figure 9 (b)), and
(26) becomes
$\displaystyle P=\prod_{k=0}^{\infty}(-yx^{-k}).$ (37)
Although this is an infinite product, we can give it significance. $P$
includes the factor $y^{+\infty}$, and the contribution decouples. We can
simply treat $F_{y,1}^{(D3)}|_{R_{yx}}$ as $0$. Namely, as far as single-
wrapping giant graviton contributions are concerned, the difference between
the multiple expansion (1) and the simple expansion (7) comes from the
different choice of the domains.
Let us proceed to $N\geq 2$ contributions. For $\sigma_{y}i_{N}^{(D3)}$, we
find infinitely many points outside $R_{qu}$, just like the $N=1$ case.
Because both positive and negative coefficients appear, it is not clear
whether they decouple like $F_{y,1}^{(D3)}$. Here, based on the analysis in
[24], we simply assume their decoupling and focus only on $F_{x,N}^{(D3)}$.
In $i_{N}^{(D3)}$ shown in Figure 6 we find $N$ dots on the positive part of
the $n_{x}$ axis corresponding to the $1/2$ BPS operators $\mathop{\rm
tr}\nolimits X^{k}$ ($k=1,2,\ldots,N$). By $\sigma_{x}$ they are mapped to the
$N$ points on the negative part of the $n_{x}$ axis. Because they are not
contained in $R_{yx}$ we need to move them back to the opposite points on the
positive part of the $n_{x}$ axis (Figure 10).
Figure 10: The modification of $i_{N}^{(D3)}$ for $R_{yx}$ for $N=2,3,4$. (a)
$\sigma_{x}i_{2}$. (b) $\sigma_{x}i_{3}$. (c) $\sigma_{x}i_{4}$. The dotted
lines show a numerical cut-off
The factorization corresponding to (22) is
$\displaystyle
I_{N}^{(D3)}(x,y)=\frac{1}{\prod_{k=1}^{N}(1-x^{k})}I^{\prime(D3)}_{N}(x,y),$
(38)
The fractional factor coming from the $N$ points on the $n_{x}$ axis and
produces poles on the $x$-plane at $k$-roots of unity with $1\leq k\leq N$. As
is pointed out in [24] these are only poles we need to take care of in the
analytic continuation. The function $I^{\prime(D3)}_{N}$ corresponds to all
other points with $n_{y}\geq 1$ in the range $0\leq n_{x}\leq(N+1)n_{y}$. The
coefficient of $y$-expansion $I^{\prime(D3)}_{N}$ at order $n_{y}$ is a
Laurant polynomial of $x$ consisting of terms in this range. Therefore, we can
safely perform the variable change $\sigma_{x}$ for the function
$I^{\prime(D3)}_{N}$. After the variable change we obtain
$\displaystyle
F_{x,N}^{(D3)}=\frac{(-1)^{N}x^{\frac{N(N+1)}{2}}}{\prod_{k=1}^{N}(1-x^{k})}\sigma_{x}I^{\prime(D3)}_{N}$
(39)
where $\sigma_{x}I^{\prime(D3)}_{N}$ has terms in the range $-Nn_{y}\leq
n_{x}\leq n_{y}$ for each order of its $y$-expansion. Including the fractional
factor with the numerator $(-1)^{N}x^{\frac{N(N+1)}{2}}$, we obtain
$F_{x,N}^{(D3)}|_{R_{yx}}$ consisting of terms in the region (Figure 11)
$\displaystyle\frac{N(N+1)}{2}-Nn_{y}\leq n_{x}.$ (40)
Figure 11: (a) $F_{x,2}^{(D3)}|_{R_{yx}}$. (b) $F_{x,3}^{(D3)}|_{R_{yx}}$.
(c) $F_{x,4}^{(D3)}|_{R_{yx}}$. The dotted lines show a numerical cut-off.
As is confirmed in [24], the simple expansion (7) correctly reproduces the
index $I_{N}^{(D3)}$ for different values of $N$. See Figure 12.
Figure 12: The result of numerical check for $N=3$. (a) The ratio
$I_{3}^{(D3)}/I_{\infty}^{(D3)}$. (b)
$(I_{3}^{(D3)}/I_{\infty}^{(D3)})-\sum_{m=0}^{4}x^{mN}F_{x,m}^{(D3)}$. The
dotted lines show numerical cut-offs. The red lines labeled by $m$ show
thresholds for respective wrapping numbers. Up to expected errors due to the
$m=5$ contribution indicated by the line labeled by $m=5$, all terms are
correctly canceled.
### 4.3 Superconformal index for $T=D3$
In the analysis of the Schur index we could focus on $F_{i,N}^{(D3)}|_{R}$
thanks to the factorization (4). This is not the case for the superconformal
index without the Schur limit taken. The superconformal index of the ${\cal
N}=4$ SYM is defined by [4]
$\displaystyle I^{(D3)}=\mathop{\rm
tr}\nolimits[(-1)^{F}q^{J_{1}}p^{J_{2}}x^{R_{x}}y^{R_{y}}z^{R_{z}}],\quad
qp=xyz.$ (41)
The giant graviton expansion investigated in [18] is a triple expansion with
functions $F_{m_{x},m_{y},m_{z}}^{(D3)}$ in the summand, and the involution
(5) gives the following relation between $F_{x,N}^{(D3)}=F_{N,0,0}^{(D3)}$ and
$I_{N}^{(D3)}$ [16, 24]:
$\displaystyle F_{x,N}^{(D3)}(p,q,x,y,z)=I_{N}^{(D3)}(y,z,x^{-1},p,q).$ (42)
Let $\sigma_{x}$ denote this variable change. $F_{0,N,0}^{(D3)}$ and
$F_{0,0,N}^{(D3)}$ are also related to $I_{N}^{(D3a)}$ via similar variable
changes, which we denote by $\sigma_{y}$ and $\sigma_{z}$, respectively.
Due to the absence of the factorization, we need to calculate functions
$F_{m_{x},m_{y},m_{z}}^{(D3)}$ directly by using localization formula similar
to (17). The letter index appearing in the integrand of the localization
formula contains $\sigma_{x}i_{1}^{(D3)}$, $\sigma_{y}i_{1}^{(D3)}$, and
$\sigma_{z}i_{1}^{(D3)}$ at the same time, where $i_{1}^{(D3)}$ is given by
[4]
$\displaystyle i_{1}^{(D3)}=1-\frac{(1-x)(1-y)(1-z)}{(1-q)(1-p)}.$ (43)
Even if we use a maximal domain we cannot cover three functions
$\sigma_{i}i_{1}^{(D3)}$ with a single domain, and we need to use deformed
contours for the gauge fugacity integrals [18, 25].
Despite the absence of the factorization, the analysis in [24] suggests that
only $F_{x,m}^{(D3)}$ contribute to the index if we use an appropriate domain,
and then our relation (29) is still useful to obtain $F_{x,N}^{(D3)}$ from
$I_{N}^{(D3)}$. Corresponding to four independent variables we need to use a
four-dimensional lattice to express full expansion of the index. To make two-
dimensional plot possible we project the four-dimensional lattice onto the
two-dimensional lattice corresponding to the unrefinement $y=z$ and $p=q$.
Then, the index becomes function of $x$ and $y$ just like the Schur
index.666The projection is used only when we show functions as two-dimensional
plots, and we do not take any unrefinement in the calculation. We regard the
index as a function of $x$, $\sqrt{yz}$, $y/z$, and $q/p$, and treat
$\sqrt{yz}$ as if $y$ in the previous subsection. Although we neglect $y/z$
and $q/p$ in the graphical expression, we keep all variables in the
calculation. The letter indices $i_{1}^{(D3)}$ and $\sigma_{i}i_{1}^{(D3)}$
($i=x,y,z$) after the projection are shown in Figure 13.
Figure 13: (a) The letter index $i_{1}^{(D3)}$ (b) The modification of
$\sigma_{x}i_{1}^{(D3)}$ for $R_{yx}$. (c) $\sigma_{y}i_{2}^{(D3)}$ and
$\sigma_{z}i_{2}^{(D3)}$ are identical after the projection to the two-
dimensional lattice.
We see that if we use the domain $R_{yx}$ $F_{y,1}^{(D3)}$ and
$F_{z,1}^{(D3)}$ decouple just like the Schur index, and $F_{x,1}^{(D3)}$ is
the only contribution with $m_{x}+m_{y}+m_{z}=1$. Although we cannot prove the
decoupling of all $F_{m_{x},m_{y},m_{z}}^{(D3)}$ with $m_{y}+m_{z}\geq 1$, the
analysis in [24] shows that we can reproduce the finite $N$ index by the
simple expansion containing only $F_{x,m}^{(D3)}$. See Figure 14 for the
result of a numerical test.
Figure 14: (a) The ratio of the superconformal indices
$I_{N}^{(D3)}/I_{\infty}^{(D3)}$ for $N=3$. (b)
$(I_{N}^{(D3)}/I_{\infty}^{(D3)})-\sum_{m=0}^{4}x^{mN}F_{x,N}^{(D3)}$ for
$N=3$. The dotted lines show numerical cut-offs. The red lines labeled by $m$
show thresholds for respective wrapping numbers. Only small number of dots,
which are consistent with the higher order contributions with $m\geq 5$, are
left.
### 4.4 M5 from M2
Let us apply our method to the theory realized on $N$ coincident M5-branes,
which we denote by $T=M5$. It is the $6$-dim ${\cal N}=(2,0)$ theory of
$A_{N-1}$ type together with a free tensor multiplet.
The superconformal index is defined by [28]
$\displaystyle I_{N}^{(M5)}=\mathop{\rm
tr}\nolimits[(-1)^{F}\check{q}_{1}^{\check{J}_{12}}\check{q}_{2}^{\check{J}_{34}}\check{q}_{3}^{\check{J}_{56}}\check{x}_{1}^{\check{R}_{12}}\check{x}_{2}^{\check{R}_{34}}]\quad\check{q}_{1}\check{q}_{2}\check{q}_{3}=\check{x}_{1}\check{x}_{2},$
(44)
where $\check{J}_{12}$, $\check{J}_{34}$, and $\check{J}_{56}$ are Cartan
generators of $SO(6)_{\rm spin}$ and $\check{R}_{12}$ and $\check{R}_{34}$ are
Cartan generators of $SO(5)_{R}$.
The index is given as a double giant graviton expansion associated with two
two-cycles in [21], and the single giant graviton sector has been studied. The
cycles are defined as the fixed loci of $\check{R}_{12}$ and $\check{R}_{34}$
in $M_{M5}=S^{4}$. We call $\check{R}_{12}$ ($\check{R}_{34}$) fixed locus
“12-cycle” (“34-cycle”). M2-branes wrapped around these cycles contribute to
the index.
The superconformal index of the $3$d ${\cal N}=8$ SCFT realized on coincident
M2-branes is defined by [28]
$\displaystyle I_{N}^{(M2)}=\mathop{\rm
tr}\nolimits[(-1)^{F}\hat{q}^{\hat{J}_{12}}\hat{x}_{1}^{\hat{R}_{12}}\hat{x}_{2}^{\hat{R}_{34}}\hat{x}_{3}^{\hat{R}_{56}}\hat{x}_{4}^{\hat{R}_{78}}]\quad\hat{q}=\hat{x}_{1}\hat{x}_{2}\hat{x}_{3}\hat{x}_{4},$
(45)
where $\hat{J}_{12}$ is the spin and $\hat{R}_{12}$, $\hat{R}_{34}$,
$\hat{R}_{56}$, and $\hat{R}_{78}$ are $SO(8)_{R}$ Cartan generators. We can
calculate this index for an arbitrary $N$ by applying the localization method
[29] to the ABJM theory with Chern-Simons level $k=1$ [30].
The Cartan generators of the $3$d ${\cal N}=8$ superconformal algebra acting
on M2-branes wrapped around $12$-cycle and those of the $6$d ${\cal N}=(2,0)$
superconformal algebra acting on the $AdS_{7}$ boundary are related by [21]
$\displaystyle(\hat{H},\hat{J}_{12},\hat{R}_{12},\hat{R}_{34},\hat{R}_{56},\hat{R}_{78})=(\tfrac{1}{2}\check{H}-\tfrac{3}{2}\check{R}_{12},\check{R}_{34},\check{J}_{12},\check{J}_{34},\check{J}_{56},-\check{R}_{12}).$
(46)
Correspondingly, the fugacities in (44) and those in (45) are related by
$\displaystyle\sigma_{12}(\hat{q},\hat{x}_{1},\hat{x}_{2},\hat{x}_{3},\hat{x}_{4})=(\check{x}_{2},\check{q}_{1},\check{q}_{2},\check{q}_{3},\check{x}_{1}^{-1}).$
(47)
The variable change associated with $34$-cycle, $\sigma_{34}$, is given by
swapping $\check{x}_{1}$ and $\check{x}_{2}$ after $\sigma_{12}$. the M2-giant
contributions $F_{12,N}^{(M5)}$ are obtained from the index $I_{N}^{(M2)}$ by
the relation
$\displaystyle F_{12,N}^{(M5)}=\sigma_{12}I_{N}^{(M2)}.$ (48)
Again, there are four independent variables and the lattice is four-
dimensional. We want to define a projection to two-dimensional lattice to show
an expansion as a two-dimensional plot. We introduce variables $x$, $y$, and
$u_{i}$ ($i=1,2,3$) by
$\displaystyle\check{q}_{i}=yu_{i}\quad(u_{1}u_{2}u_{3}=1),\quad\check{x}_{1}=x,\quad\check{x}_{2}=x^{-1}y^{3}.$
(49)
We focus on $x$ and $y$ to show series in figures.
We also rewrite the fugacities for M2-branes as follows:
$\displaystyle\hat{q}=xy^{3},\quad\hat{x}_{i}=yu_{i}\quad(u_{1}u_{2}u_{3}=1),\quad\hat{x}_{4}=x.$
(50)
Then, the variable change (47) becomes
$\displaystyle\sigma_{12}(x,y,u_{i})=(x^{-1},y,u_{i}).$ (51)
Namely, (48) becomes
$\displaystyle F_{12,N}^{(M5)}(x,y,u_{i})=I_{N}^{(M2)}(x^{-1},y,u_{i}).$ (52)
As in the previous examples, let us first look at the letter index for a
single giant graviton. The theory on a single M2-brane is the free theory of
an ${\cal N}=8$ scalar multiplet with the letter index [28]
$\displaystyle
i_{1}^{(M2)}=\frac{\hat{x}_{1}+\hat{x}_{2}+\hat{x}_{3}+\hat{x}_{4}-\hat{q}(\hat{x}_{1}^{-1}+\hat{x}_{2}^{-1}+\hat{x}_{3}^{-1}+\hat{x}_{4}^{-1})}{1-\hat{q}}.$
(53)
We show $i_{1}^{(M2)}$, $\sigma_{12}i_{1}^{(M2)}$, and
$\sigma_{34}i_{1}^{(M2)}$ projected on the two-dimensional plane in Figure 15.
Figure 15: (a) The letter index $i_{1}^{(M2)}$. (b) The modification of
$\sigma_{1}i_{1}^{(M2)}$ for $R_{yx}$. (c) Infinitely many dots of
$\sigma_{2}i_{1}^{(M2)}$ are out of $R_{yx}$.
If we take the domain $R_{yx}$ defined in (36), $\sigma_{34}i_{1}^{(M2)}$ has
infinitely many points outside $R_{yx}$. They give the factor
$\displaystyle
P=\prod_{k=1}^{\infty}(-y^{3}x^{-k})\prod_{k=1}^{\infty}(-yx^{-k})^{-3}.$ (54)
If we take the product for each $k$ first, we obtain
$\prod_{k=1}^{\infty}x^{2k}=x^{\infty}$ and the contribution decouples. Just
like the previous examples, we simply assume that $F_{m_{12},m_{34}}^{(M5)}$
with $m_{34}\geq 1$ decouple, and let us consider the simple expansion
associated with the $12$-cycle
$\displaystyle\frac{I_{N}^{(M5)}}{I_{\infty}^{(M5)}}=\sum_{m=0}^{\infty}x^{mN}F_{12,m}^{(M5)}.$
(55)
Let us numerically confirm (55) holds for small $N$. $I_{N}^{(M5)}$ with
$N=-1,0,1$ and $\infty$ are given by
$\displaystyle I_{-1}^{(M5)}=0,\quad I_{0}^{(M5)}=1,\quad
I_{1}^{(M5)}=\mathop{\rm Pexp}\nolimits i_{1}^{(M5)},\quad
I_{\infty}^{(M5)}=\mathop{\rm Pexp}\nolimits i_{\infty}^{(M5)},$ (56)
where $i_{1}^{(M5)}$ and $i_{\infty}^{(M5)}$ are the letter indices of the
tensor multiplet and the supergravity multiplet in $AdS_{7}\times S^{4}$. They
are given by [28]
$\displaystyle
i_{1}^{(M5)}=\frac{\check{x}_{1}+\check{x}_{2}-\check{q}_{1}\check{q}_{2}-\check{q}_{2}\check{q}_{3}-\check{q}_{3}\check{q}_{1}+\check{x}_{1}\check{x}_{2}}{(1-\check{q}_{1})(1-\check{q}_{2})(1-\check{q}_{3})},$
(57)
and
$\displaystyle
i_{\infty}^{(M5)}=\frac{\check{x}_{1}+\check{x}_{2}-\check{q}_{1}\check{q}_{2}-\check{q}_{2}\check{q}_{3}-\check{q}_{3}\check{q}_{1}+\check{x}_{1}\check{x}_{2}(\check{q}_{1}+\check{q}_{2}+\check{q}_{3}-\check{x}_{1}-\check{x}_{2})}{(1-\check{x}_{1})(1-\check{x}_{2})(1-\check{q}_{1})(1-\check{q}_{2})(1-\check{q}_{3})}.$
(58)
By using these we can calculate the left hand side of (55). In the following
we calculate the right hand side of (55).
We first calculate the index $I_{N}^{(M2)}$ for different $N$ with the
physical domain $R_{yx}$. This is done by applying localization method [29] to
the ABJM theory [30]. The plethystic logarithms $i_{N}^{(M2)}$ for small $N$
are shown in Figure 16.
Figure 16: $i_{N}^{(M2)}=\mathop{\rm Plog}\nolimits I_{N}^{(M2)}$ for
$N=1,2,3,4$ are shown. As shown by the dotted lines the $y$ expansions are cut
off at $n_{y}=3$.
They have similar structure to $i_{N}^{(D3)}$. In particular, for each $N$
there are $N$ dots on the positive part of the vertical axis corresponding to
$N$ $1/2$ BPS operators. After the variable change (51), these dots have to be
moved to the opposite points to obtain the modified letter index. The
expansion of the analytically continued plethystic exponential
$F_{12,N}^{(M5)}|_{R_{yx}}=\mathop{\rm
Pexp}\nolimits_{R_{yx}}\sigma_{12}i_{N}^{(M2)}$ for $N=1,2,3,4$ are shown in
Figure 17.
Figure 17: M2-giant contributions $F_{12,m}^{(M5)}$ with wrapping number
$m=1,2,3,4$ are shown. As shown by the dotted lines the $y$ expansions are cut
off at $n_{y}=3$.
For $N\geq 2$ the dots are in the region above the line
$n_{x}=\frac{m(m+1)}{2}+mN-mn_{y}$. ($N=1$ case is exceptional and the line is
$n_{x}=1-\frac{2}{3}n_{y}$.)
With the functions $F_{12,m}^{(M5)}$ obtained above we can confirm that (55)
holds for $N=-1,0,1$ up to expected errors. See Figure 18.
Figure 18: The results of consistency check for $N=-1,0,1$.
$I_{N}^{(M5)}/I_{\infty}^{(M5)}$ for $N=0$ and $N=1$ are shown in (b) and (d),
respectively. Trivial one, $I_{-1}^{(M5)}/I_{\infty}^{(M5)}=0$ is not shown.
$(I_{N}^{(M5)}/I_{\infty}^{(M5)})-\sum_{m=0}^{4}x^{mN}F_{12,m}^{(M5)}$ for
$N=-1$, $0$, and $1$ are shown in (a), (c), and (e),respectively. As shown by
the dotted lines the $y$ expansions are cut off at $n_{y}=3$. In (a), (c), and
(e), all terms below the $m=5$ lines are correctly canceled, and only expected
errors due to $m\geq 5$ contributions are left.
This strongly suggests that (55) correctly gives $I_{N}^{(M5)}$ for an
arbitrary $N$. Although we only show the two-dimensional plots, we can
calculate the full superconformal index. See Appendix A.
We can also compare our results with the analytic result for the Schur-like
limit [31, 32] obtained from the analysis of five-dimensional SYM. It is known
that by setting $\check{q}_{1}=\check{x}_{1}$, the index $I_{N}^{(M5)}$
reduces to the following function of the single variable $\check{x}_{1}$:
$\displaystyle I_{N}^{(M5)}=\mathop{\rm
Pexp}\nolimits\frac{\check{x}_{1}+\check{x}_{1}^{2}+\cdots+\check{x}_{1}^{N}}{1-\check{x}_{1}}.$
(59)
If we take the same limit in our results shown in Appendix A we obtain
$\displaystyle I_{2}^{(M5)}$
$\displaystyle=1+\check{x}_{1}+3\check{x}_{1}^{2}+5\check{x}_{1}^{3}+10\check{x}_{1}^{4}+\underline{15\check{x}_{1}^{5}}+{\cal
O}(\check{x}_{1}^{6}),$ $\displaystyle I_{3}^{(M5)}$
$\displaystyle=1+\check{x}_{1}+3\check{x}_{1}^{2}+6\check{x}_{1}^{3}+12\check{x}_{1}^{4}+\underline{20\check{x}_{1}^{5}}+{\cal
O}(\check{x}_{1}^{6}),$ $\displaystyle I_{4}^{(M5)}$
$\displaystyle=1+\check{x}_{1}+3\check{x}_{1}^{2}+6\check{x}_{1}^{3}+13\check{x}_{1}^{4}+\underline{22\check{x}_{1}^{5}}+{\cal
O}(\check{x}_{1}^{6}),$ $\displaystyle I_{5}^{(M5)}$
$\displaystyle=1+\check{x}_{1}+3\check{x}_{1}^{2}+6\check{x}_{1}^{3}+13\check{x}_{1}^{4}+\underline{23\check{x}_{1}^{5}}+{\cal
O}(\check{x}_{1}^{6}).$ (60)
We only showed terms independent of $\check{q}_{2}$ and $\check{q}_{3}$. The
underlines indicate the terms are incorrect. These are consistent with (59)
because in our numerical results the $y$ expansion is cut off at $n_{y}=3$ and
the expected order of errors is ${\cal O}(\check{x}_{1}^{4})$ or higher. The
$\check{q}_{2}$ or $\check{q}_{3}$-dependent terms which we did not show in
(60) are also consistent with the expected errors.
Note that the Schur-like limit is ill-defined for each M2-giant contribution
because the limit corresponds to $\hat{x}_{1}=\hat{x}_{4}^{-1}$ via (47). The
terms $\hat{x}_{1}$ and $\hat{x}_{1}^{-1}$ appear at the same time in the
letter index $i_{1}^{(M2)}$ in the limit, and there is no domain covering both
of them. We need to take the limit after summing up the contributions of
M2-giants.
### 4.5 M2 from M5
It is possible to interchange the roles of M2 and M5 in the previous
subsection. Namely, we can calculate the finite $N$ index of M2 theory by
summing up M5-giant contributions by the relation
$\displaystyle\frac{I_{N}^{(M2)}|_{R_{yx}}}{I_{\infty}^{(M2)}|_{R_{yx}}}=\sum_{m=0}^{\infty}x^{mN}(\sigma_{12}^{-1}I_{m}^{(M5)})|_{R_{yx}},$
(61)
where the large $N$ index is given by $I_{\infty}^{(M2)}=\mathop{\rm
Pexp}\nolimits i_{\infty}^{(M2)}$ with the letter index [28]
$\displaystyle
i_{\infty}^{(M2)}=\frac{(1-\hat{q}\hat{x}_{1}^{-1})(1-\hat{q}\hat{x}_{2}^{-1})(1-\hat{q}\hat{x}_{3}^{-1})(1-\hat{q}\hat{x}_{4}^{-1})}{(1-\hat{x}_{1})(1-\hat{x}_{2})(1-\hat{x}_{3})(1-\hat{x}_{4})(1-\hat{q})^{2}}-\frac{1-\hat{q}+\hat{q}^{2}}{(1-\hat{q})^{2}}.$
(62)
The calculation to test (61) is parallel to the previous example, and we only
give a brief explanation.
In [21] four five-cycles in $M_{M5}=S^{7}$ were taken into account. As the
previous examples we can show for a single M5 giant that if we take $R_{yx}$
only one of the four cycles, $\hat{R}_{78}$ fixed locus, gives non-trivial
contribution. Hence, we focus on $F_{78,N}^{(M2)}$ associated with the cycle.
By using the results of previous subsection for $I_{N}^{(M5)}$ we can obtain
the plethystic logarithms $i_{N}^{(M5)}=\mathop{\rm Plog}\nolimits
I_{N}^{(M5)}$ for small $N$ shown in Figure 19.
Figure 19: $i_{N}^{(M5)}$ for $N=1,2,3,4,5$ are shown. As shown by the dotted
lines the $y$ expansions are cut off at $n_{y}=3$.
As the previous examples we find $N$ dots associated with the $1/2$ BPS
operators on the positive part of the vertical axis for each $N$. The
functions $F_{78,N}^{(M2)}=\mathop{\rm
Pexp}\nolimits_{R_{yx}}\sigma_{12}^{-1}i_{N}^{(M5)}$ are shown in Figure 20.
Figure 20: M5 giants contributions $F_{78,N}^{(M2)}$ for $N=1,2,3,4,5$ are
shown. As shown by the dotted lines the $y$ expansions are cut off at
$n_{y}=3$.
The results of a numerical check of (61) for $N=-1,0,1$ are shown in Figure
21.
Figure 21: $(I_{N}^{(M2)}/I_{\infty}^{(M2)})$ for $N=0,1$ are shown in (b)
and (d), respectively. The trivial one $I_{-1}^{(M2)}/I_{\infty}^{(M2)}=0$ is
not shown.
$(I_{N}^{(M2)}/I_{\infty}^{(M2)})-\sum_{m=0}^{5}x^{mN}F_{78,m}^{(M2)}$ for
$N=-1,0,1$ are shown in (a), (c), and (e), respectively. As shown by the
dotted lines the $y$ expansions are cut off at $n_{y}=3$. In (a), (c), and (e)
all terms below $m=6$ lines are correctly canceled, and only expected errors
due to $m\geq 6$ contributions are left.
## 5 Conclusions and Discussion
We investigated the giant graviton expansions. In particular, we concentrated
on coincident giant gravitons wrapped around a single cycle. Because the
expansion domain for giant gravitons and that of the boundary theory are
different, we need to perform an analytic continuation to relate them. We
proposed a simple prescription (29) to realize it.
We explicitly showed for the Schur index of $T=D3$ that different choices of
the expansion domains give different functions for single giant graviton
contributions $F_{i,1}^{(D3)}$ ($i=x,y$). If we use $R_{qu}$ in (31) both
$F_{x,1}^{(D3)}$ and $F_{y,1}^{(D3)}$ contribute to the Schur index, while if
we use $R_{yx}$ in (36), $F_{y,1}^{(D3)}$ does not contribute. Although we
have not proved $F_{y,N\geq 2}^{(D3)}$ also vanish, this partially explains
why there are two different expansions: the simple expansion found in [24] and
the multiple expansion in [17]. It is surprising that they give the same
result even though some contributions are lost in the simple expansion. This
may be related to the fact that the set of functions $F_{m_{x},m_{y}}$ are
strongly constrained. For example, we can formally substitute negative $N$ to
expansion (1) or (7), and find that the result is vanishing. This gives very
strong constraints on the functions. This implies that the functions share
common information, and only small subset appearing in the simple expansion
may be sufficient to give the complete answer. It would be interesting to
investigate the structure of the constrained set of the functions.
An important application of our method is the calculation of the full
superconformal index of the 6d ${\cal N}=(2,0)$ $A_{N-1}$ SCFT. It is in
principle possible to calculate $I_{N}^{(M5)}$ for an arbitrary $N$ up to an
arbitrary order starting from the indices $I_{m}^{(M2)}$ of the M2-brane
theories with different $m$. Some results are shown in Appendix A.
An important merit of our method is that the theory on giant gravitons does
not have to be Lagrangian theories unlike the method adopted in [17, 18, 25],
which uses deformed contours in the gauge fugacity integrals to realize the
analytic continuation. We only need the final expression of the index of the
giant graviton theory. It enables us to apply the method to M5-giants, as was
demonstrated in 4.5. Another example with non-Lagrangian giant gravitons is
$AdS_{5}\times S^{5}$ with a $7$-brane insertion [33, 34]. As is demonstrated
in [23], the giant graviton expansion works well for this system at least for
the leading contribution. There are higher order contributions coming from
giant gravitons coincident with the $7$-brane, on which a non-Lagrangian
theory is realized. We expect our method is useful for the analysis of such
contributions.
We studied two expansion domains $R_{yx}$ and $R_{qu}$ for the Schur index of
$T=D3$. An advantage of $R_{qu}$ over $R_{yx}$ is that for each value of
$m=m_{x}+m_{y}$ the symmetry between $x$ and $y$ is manifest. Unfortunately,
it turned out that our method has limited applicability for the calculation
with the domain $R_{qu}$ due to the ill-defined factor $P$. It would be very
nice if we can somehow regularize the product in $P$ when it is infinite and
make it possible to apply the method to an arbitrary contribution
$F_{i,m}^{(T)}$.
In general, the functions $F_{m_{1},\ldots,m_{d}}^{(T)}$ do not factorize into
$F_{i,m}^{(T)}$. In the case of D3-giants it is possible to write
$F_{m_{1},\ldots,m_{d}}^{(T)}$ in the matrix integral form. Even so, it is not
clear how we should choose the integration contours. Because we cannot take a
physical domain of expansion, using unit circles for integration contours is
not justified. Although some rules for contours have been proposed [18, 25],
it is desirable to find more efficient method of calculation applicable to
non-Lagrangian giant gravitons.
Another important problem is to clarify to what extent the giant graviton
expansion works. In this work we focused only on the maximally supersymmetric
theories: 4d ${\cal N}=4$ SYM on D3-branes, 3d ${\cal N}=8$ SCFT on M2-branes,
and 6d ${\cal N}=(2,0)$ SCFT on M5-branes. The analysis on the gauge theory
side [24, 25] found the structure of the giant graviton expansions in variety
of theories. It would be interesting to study the applicability of our method
to more general class of theories which have holographic dual description.
In addition to the giant graviton expansions (1) and (7), another similar
expansion was proposed in [35]. It may be interesting to study the relation
among them.
## Acknowledgments
The author would like to thank S. Murayama and D. Yokoyama for valuable
discussions. The work of Y. I. was partially supported by Grand-in-Aid for
Scientific Research (C) (No.21K03569), Ministry of Education, Science and
Culture, Japan. This work used computational resources TSUBAME3.0
supercomputer provided by Tokyo Institute of Technology.
## Appendix A Full index of $6$-dim $A_{N-1}$ SCFT
Let $\chi_{[a,b]}$ be the $SU(3)$ character of the representation with Dynkin
label $[a,b]$ defined such that $\chi_{[1,0]}=u_{1}+u_{2}+u_{3}$. The
superconformal indices $I_{N}^{(M5)}$ for $N=2,3,4,5$ are given as follows.
$\displaystyle I^{(M5)}_{2}$
$\displaystyle=(1+\check{x}_{1}+2\check{x}_{1}^{2}+2\check{x}_{1}^{3}+3\check{x}_{1}^{4}+3\check{x}_{1}^{5}+4\check{x}_{1}^{6}+4\check{x}_{1}^{7}+5\check{x}_{1}^{8}+5\check{x}_{1}^{9}+6\check{x}_{1}^{10}+\cdots)$
$\displaystyle+y\chi_{[1,0]}(\check{x}_{1}+2\check{x}_{1}^{2}+3\check{x}_{1}^{3}+4\check{x}_{1}^{4}+5\check{x}_{1}^{5}+6\check{x}_{1}^{6}+7\check{x}_{1}^{7}+8\check{x}_{1}^{8}+9\check{x}_{1}^{9}+10\check{x}_{1}^{10}+\cdots)$
$\displaystyle+y^{2}[\chi_{[0,1]}(-1-2\check{x}_{1}-3\check{x}_{1}^{2}-3\check{x}_{1}^{3}-4\check{x}_{1}^{4}-4\check{x}_{1}^{5}-5\check{x}_{1}^{6}-5\check{x}_{1}^{7}-6\check{x}_{1}^{8}-6\check{x}_{1}^{9}-7\check{x}_{1}^{10}+\cdots)$
$\displaystyle\quad+\chi_{[2,0]}(\check{x}_{1}+3\check{x}_{1}^{2}+5\check{x}_{1}^{3}+8\check{x}_{1}^{4}+10\check{x}_{1}^{5}+13\check{x}_{1}^{6}+15\check{x}_{1}^{7}+18\check{x}_{1}^{8}+20\check{x}_{1}^{9}+23\check{x}_{1}^{10}+\cdots)]$
$\displaystyle+y^{3}[(\check{x}_{1}^{-1}+2+\check{x}_{1}-\check{x}_{1}^{3}-2\check{x}_{1}^{4}-3\check{x}_{1}^{5}-4\check{x}_{1}^{6}-5\check{x}_{1}^{7}-6\check{x}_{1}^{8}-7\check{x}_{1}^{9}-8\check{x}_{1}^{10}+\cdots)$
$\displaystyle\quad+\chi_{[1,1]}(-1-3\check{x}_{1}-5\check{x}_{1}^{2}-6\check{x}_{1}^{3}-6\check{x}_{1}^{4}-6\check{x}_{1}^{5}-6\check{x}_{1}^{6}-6\check{x}_{1}^{7}-6\check{x}_{1}^{8}-6\check{x}_{1}^{9}-6\check{x}_{1}^{10}+\cdots)$
$\displaystyle\quad+\chi_{[3,0]}(\check{x}_{1}+3\check{x}_{1}^{2}+7\check{x}_{1}^{3}+11\check{x}_{1}^{4}+16\check{x}_{1}^{5}+21\check{x}_{1}^{6}+26\check{x}_{1}^{7}+31\check{x}_{1}^{8}+36\check{x}_{1}^{9}+41\check{x}_{1}^{10}+\cdots)]$
$\displaystyle+\cdots$ (63) $\displaystyle I^{(M5)}_{3}$
$\displaystyle=(1+\check{x}_{1}+2\check{x}_{1}^{2}+3\check{x}_{1}^{3}+4\check{x}_{1}^{4}+5\check{x}_{1}^{5}+7\check{x}_{1}^{6}+8\check{x}_{1}^{7}+10\check{x}_{1}^{8}+12\check{x}_{1}^{9}+14\check{x}_{1}^{10}+\cdots)$
$\displaystyle+y\chi_{[1,0]}(\check{x}_{1}+2\check{x}_{1}^{2}+4\check{x}_{1}^{3}+6\check{x}_{1}^{4}+9\check{x}_{1}^{5}+12\check{x}_{1}^{6}+16\check{x}_{1}^{7}+20\check{x}_{1}^{8}+25\check{x}_{1}^{9}+30\check{x}_{1}^{10}+\cdots)$
$\displaystyle+y^{2}[\chi_{[0,1]}(-1-2\check{x}_{1}-4\check{x}_{1}^{2}-5\check{x}_{1}^{3}-7\check{x}_{1}^{4}-8\check{x}_{1}^{5}-10\check{x}_{1}^{6}-11\check{x}_{1}^{7}-13\check{x}_{1}^{8}-14\check{x}_{1}^{9}-16\check{x}_{1}^{10}+\cdots)$
$\displaystyle\quad+\chi_{[2,0]}(\check{x}_{1}+3\check{x}_{1}^{2}+6\check{x}_{1}^{3}+11\check{x}_{1}^{4}+17\check{x}_{1}^{5}+25\check{x}_{1}^{6}+34\check{x}_{1}^{7}+45\check{x}_{1}^{8}+57\check{x}_{1}^{9}+71\check{x}_{1}^{10}+\cdots)]$
$\displaystyle+y^{3}[(\check{x}_{1}^{-1}+2+2\check{x}_{1}+\check{x}_{1}^{2}-\check{x}_{1}^{3}-4\check{x}_{1}^{4}-8\check{x}_{1}^{5}-12\check{x}_{1}^{6}-18\check{x}_{1}^{7}-24\check{x}_{1}^{8}-31\check{x}_{1}^{9}-39\check{x}_{1}^{10}+\cdots)$
$\displaystyle\quad+\chi_{[1,1]}(-1-3\check{x}_{1}-6\check{x}_{1}^{2}-10\check{x}_{1}^{3}-13\check{x}_{1}^{4}-16\check{x}_{1}^{5}-18\check{x}_{1}^{6}-19\check{x}_{1}^{7}-19\check{x}_{1}^{8}-19\check{x}_{1}^{9}-17\check{x}_{1}^{10}+\cdots)$
$\displaystyle\quad+\chi_{[3,0]}(\check{x}_{1}+3\check{x}_{1}^{2}+8\check{x}_{1}^{3}+15\check{x}_{1}^{4}+26\check{x}_{1}^{5}+40\check{x}_{1}^{6}+58\check{x}_{1}^{7}+79\check{x}_{1}^{8}+105\check{x}_{1}^{9}+133\check{x}_{1}^{10}+\cdots)]$
$\displaystyle+\cdots$ (64) $\displaystyle I^{(M5)}_{4}$
$\displaystyle=(1+\check{x}_{1}+2\check{x}_{1}^{2}+3\check{x}_{1}^{3}+5\check{x}_{1}^{4}+6\check{x}_{1}^{5}+9\check{x}_{1}^{6}+11\check{x}_{1}^{7}+15\check{x}_{1}^{8}+18\check{x}_{1}^{9}+23\check{x}_{1}^{10}+\cdots)$
$\displaystyle+y\chi_{[1,0]}(\check{x}_{1}+2\check{x}_{1}^{2}+4\check{x}_{1}^{3}+7\check{x}_{1}^{4}+11\check{x}_{1}^{5}+16\check{x}_{1}^{6}+23\check{x}_{1}^{7}+31\check{x}_{1}^{8}+41\check{x}_{1}^{9}+53\check{x}_{1}^{10}+\cdots)$
$\displaystyle+y^{2}[\chi_{[0,1]}(-1-2\check{x}_{1}-4\check{x}_{1}^{2}-6\check{x}_{1}^{3}-9\check{x}_{1}^{4}-11\check{x}_{1}^{5}-15\check{x}_{1}^{6}-17\check{x}_{1}^{7}-21\check{x}_{1}^{8}-23\check{x}_{1}^{9}-27\check{x}_{1}^{10}+\cdots)$
$\displaystyle\quad+\chi_{[2,0]}(\check{x}_{1}+3\check{x}_{1}^{2}+6\check{x}_{1}^{3}+12\check{x}_{1}^{4}+20\check{x}_{1}^{5}+32\check{x}_{1}^{6}+47\check{x}_{1}^{7}+68\check{x}_{1}^{8}+92\check{x}_{1}^{9}+124\check{x}_{1}^{10}+\cdots)]$
$\displaystyle+y^{3}[(\check{x}_{1}^{-1}+2+2\check{x}_{1}+2\check{x}_{1}^{2}-4\check{x}_{1}^{4}-10\check{x}_{1}^{5}-18\check{x}_{1}^{6}-30\check{x}_{1}^{7}-44\check{x}_{1}^{8}-62\check{x}_{1}^{9}-84\check{x}_{1}^{10}+\cdots)$
$\displaystyle\quad+\chi_{[1,1]}(-1-3\check{x}_{1}-6\check{x}_{1}^{2}-11\check{x}_{1}^{3}-17\check{x}_{1}^{4}-23\check{x}_{1}^{5}-30\check{x}_{1}^{6}-36\check{x}_{1}^{7}-40\check{x}_{1}^{8}-43\check{x}_{1}^{9}-43\check{x}_{1}^{10}+\cdots)$
$\displaystyle\quad+\chi_{[3,0]}(\check{x}_{1}+3\check{x}_{1}^{2}+8\check{x}_{1}^{3}+16\check{x}_{1}^{4}+30\check{x}_{1}^{5}+50\check{x}_{1}^{6}+79\check{x}_{1}^{7}+117\check{x}_{1}^{8}+168\check{x}_{1}^{9}+231\check{x}_{1}^{10}+\cdots)]$
$\displaystyle+\cdots$ (65) $\displaystyle I^{(M5)}_{5}$
$\displaystyle=(1+\check{x}_{1}+2\check{x}_{1}^{2}+3\check{x}_{1}^{3}+5\check{x}_{1}^{4}+7\check{x}_{1}^{5}+10\check{x}_{1}^{6}+13\check{x}_{1}^{7}+18\check{x}_{1}^{8}+23\check{x}_{1}^{9}+30\check{x}_{1}^{10}+\cdots)$
$\displaystyle+y\chi_{[1,0]}(\check{x}_{1}+2\check{x}_{1}^{2}+4\check{x}_{1}^{3}+7\check{x}_{1}^{4}+12\check{x}_{1}^{5}+18\check{x}_{1}^{6}+27\check{x}_{1}^{7}+38\check{x}_{1}^{8}+53\check{x}_{1}^{9}+71\check{x}_{1}^{10}+\cdots)$
$\displaystyle+y^{2}[\chi_{[0,1]}(-1-2\check{x}_{1}-4\check{x}_{1}^{2}-6\check{x}_{1}^{3}-10\check{x}_{1}^{4}-13\check{x}_{1}^{5}-18\check{x}_{1}^{6}-22\check{x}_{1}^{7}-28\check{x}_{1}^{8}-32\check{x}_{1}^{9}-38\check{x}_{1}^{10}+\cdots)$
$\displaystyle\quad+\chi_{[2,0]}(\check{x}_{1}+3\check{x}_{1}^{2}+6\check{x}_{1}^{3}+12\check{x}_{1}^{4}+21\check{x}_{1}^{5}+35\check{x}_{1}^{6}+54\check{x}_{1}^{7}+81\check{x}_{1}^{8}+116\check{x}_{1}^{9}+163\check{x}_{1}^{10}+\cdots)]$
$\displaystyle+y^{3}[(\check{x}_{1}^{-1}+2+2\check{x}_{1}+2\check{x}_{1}^{2}+\check{x}_{1}^{3}-3\check{x}_{1}^{4}-10\check{x}_{1}^{5}-20\check{x}_{1}^{6}-36\check{x}_{1}^{7}-57\check{x}_{1}^{8}-86\check{x}_{1}^{9}-123\check{x}_{1}^{10}+\cdots)$
$\displaystyle\quad+\chi_{[1,1]}(-1-3\check{x}_{1}-6\check{x}_{1}^{2}-11\check{x}_{1}^{3}-18\check{x}_{1}^{4}-27\check{x}_{1}^{5}-37\check{x}_{1}^{6}-48\check{x}_{1}^{7}-59\check{x}_{1}^{8}-69\check{x}_{1}^{9}-75\check{x}_{1}^{10}+\cdots)$
$\displaystyle\quad+\chi_{[3,0]}(\check{x}_{1}+3\check{x}_{1}^{2}+8\check{x}_{1}^{3}+16\check{x}_{1}^{4}+31\check{x}_{1}^{5}+54\check{x}_{1}^{6}+89\check{x}_{1}^{7}+138\check{x}_{1}^{8}+208\check{x}_{1}^{9}+300\check{x}_{1}^{10}+\cdots)]$
$\displaystyle+\cdots$ (66)
We can also calculate the index of $A_{N-1}$ SCFT $I_{N}^{(A)}$ by removing
the contribution of the free tensor multiplet by
$\displaystyle I_{N}^{(A)}=\frac{I_{N}^{(M5)}}{I_{1}^{(M5)}}.$ (67)
The numerical results for $N=2,3,4,5$ are as follows:
$\displaystyle I^{(A)}_{2}$
$\displaystyle=(1+\check{x}_{1}^{2}+\check{x}_{1}^{4}+\check{x}_{1}^{6}+\check{x}_{1}^{8}+\check{x}_{1}^{10}+\cdots)$
$\displaystyle+y\chi_{[1,0]}(\check{x}_{1}^{2}+\check{x}_{1}^{4}+\check{x}_{1}^{6}+\check{x}_{1}^{8}+\check{x}_{1}^{10}+\cdots)$
$\displaystyle+y^{2}[\chi_{[0,1]}(-\check{x}_{1}-\check{x}_{1}^{3}-\check{x}_{1}^{5}-\check{x}_{1}^{7}-\check{x}_{1}^{9}+\cdots)$
$\displaystyle\quad+\chi_{[2,0]}(\check{x}_{1}^{2}+2\check{x}_{1}^{4}+2\check{x}_{1}^{6}+2\check{x}_{1}^{8}+2\check{x}_{1}^{10}+\cdots)]$
$\displaystyle+y^{3}[(1-\check{x}_{1}+\check{x}_{1}^{2}-x^{3}+\check{x}_{1}^{4}-\check{x}_{1}^{5}+\check{x}_{1}^{6}-\check{x}_{1}^{7}+\check{x}_{1}^{8}-\check{x}_{1}^{9}+\check{x}_{1}^{10}+\cdots)$
$\displaystyle\quad+\chi_{[1,1]}(-\check{x}_{1}-2\check{x}_{1}^{3}+\check{x}_{1}^{4}-2\check{x}_{1}^{5}+\check{x}_{1}^{6}-2\check{x}_{1}^{7}+\check{x}_{1}^{8}-2\check{x}_{1}^{9}+\check{x}_{1}^{10}+\cdots)$
$\displaystyle\quad+\chi_{[3,0]}(\check{x}_{1}^{2}+2\check{x}_{1}^{4}+3\check{x}_{1}^{6}+3\check{x}_{1}^{8}+3\check{x}_{1}^{10}+\cdots)]$
$\displaystyle+\cdots$ (68) $\displaystyle I^{(A)}_{3}$
$\displaystyle=(1+\check{x}_{1}^{2}+\check{x}_{1}^{3}+\check{x}_{1}^{4}+\check{x}_{1}^{5}+2\check{x}_{1}^{6}+\check{x}_{1}^{7}+2\check{x}_{1}^{8}+2\check{x}_{1}^{9}+2\check{x}_{1}^{10}+\cdots)$
$\displaystyle+y\chi_{[1,0]}(\check{x}_{1}^{2}+\check{x}_{1}^{3}+\check{x}_{1}^{4}+2\check{x}_{1}^{5}+2\check{x}_{1}^{6}+2\check{x}_{1}^{7}+3\check{x}_{1}^{8}+3\check{x}_{1}^{9}+3\check{x}_{1}^{10}+\cdots)$
$\displaystyle+y^{2}[\chi_{[0,1]}(-\check{x}_{1}-\check{x}_{1}^{2}-\check{x}_{1}^{3}-2\check{x}_{1}^{4}-\check{x}_{1}^{5}-2\check{x}_{1}^{6}-2\check{x}_{1}^{7}-2\check{x}_{1}^{8}-2\check{x}_{1}^{9}-3\check{x}_{1}^{10}+\cdots)$
$\displaystyle\quad+\chi_{[2,0]}(\check{x}_{1}^{2}+\check{x}_{1}^{3}+2\check{x}_{1}^{4}+3\check{x}_{1}^{5}+4\check{x}_{1}^{6}+4\check{x}_{1}^{7}+6\check{x}_{1}^{8}+6\check{x}_{1}^{9}+7\check{x}_{1}^{10}+\cdots)]$
$\displaystyle+y^{3}[(1-\check{x}_{1}^{4}-\check{x}_{1}^{5}-\check{x}_{1}^{6}-2\check{x}_{1}^{7}-2\check{x}_{1}^{8}-2\check{x}_{1}^{9}-3\check{x}_{1}^{10}+\cdots)$
$\displaystyle\quad+\chi_{[1,1]}(-\check{x}_{1}-\check{x}_{1}^{2}-2\check{x}_{1}^{3}-3\check{x}_{1}^{4}-2\check{x}_{1}^{5}-3\check{x}_{1}^{6}-3\check{x}_{1}^{7}-2\check{x}_{1}^{8}-3\check{x}_{1}^{9}-3\check{x}_{1}^{10}+\cdots)$
$\displaystyle\quad+\chi_{[3,0]}(\check{x}_{1}^{2}+\check{x}_{1}^{3}+2\check{x}_{1}^{4}+4\check{x}_{1}^{5}+5\check{x}_{1}^{6}+6\check{x}_{1}^{7}+9\check{x}_{1}^{8}+10\check{x}_{1}^{9}+11\check{x}_{1}^{10}+\cdots)]$
$\displaystyle+\cdots$ (69) $\displaystyle I^{(A)}_{4}$
$\displaystyle=(1+\check{x}_{1}^{2}+\check{x}_{1}^{3}+2\check{x}_{1}^{4}+\check{x}_{1}^{5}+3\check{x}_{1}^{6}+2\check{x}_{1}^{7}+4\check{x}_{1}^{8}+3\check{x}_{1}^{9}+5\check{x}_{1}^{10}+\cdots)$
$\displaystyle+y\chi_{[1,0]}(\check{x}_{1}^{2}+\check{x}_{1}^{3}+2\check{x}_{1}^{4}+2\check{x}_{1}^{5}+4\check{x}_{1}^{6}+4\check{x}_{1}^{7}+6\check{x}_{1}^{8}+6\check{x}_{1}^{9}+9\check{x}_{1}^{10}+\cdots)$
$\displaystyle+y^{2}[\chi_{[0,1]}(-\check{x}_{1}-\check{x}_{1}^{2}-2\check{x}_{1}^{3}-2\check{x}_{1}^{4}-3\check{x}_{1}^{5}-3\check{x}_{1}^{6}-4\check{x}_{1}^{7}-4\check{x}_{1}^{8}-5\check{x}_{1}^{9}-5\check{x}_{1}^{10}+\cdots)$
$\displaystyle\quad+\chi_{[2,0]}(\check{x}_{1}^{2}+\check{x}_{1}^{3}+3\check{x}_{1}^{4}+3\check{x}_{1}^{5}+7\check{x}_{1}^{6}+7\check{x}_{1}^{7}+12\check{x}_{1}^{8}+12\check{x}_{1}^{9}+19\check{x}_{1}^{10}+\cdots)]$
$\displaystyle+y^{3}[(1+\check{x}_{1}^{2}-\check{x}_{1}^{3}-3\check{x}_{1}^{5}-2\check{x}_{1}^{6}-6\check{x}_{1}^{7}-5\check{x}_{1}^{8}-9\check{x}_{1}^{9}-9\check{x}_{1}^{10}+\cdots)$
$\displaystyle\quad+\chi_{[1,1]}(-\check{x}_{1}-\check{x}_{1}^{2}-3\check{x}_{1}^{3}-3\check{x}_{1}^{4}-6\check{x}_{1}^{5}-5\check{x}_{1}^{6}-8\check{x}_{1}^{7}-6\check{x}_{1}^{8}-10\check{x}_{1}^{9}-6\check{x}_{1}^{10}+\cdots)$
$\displaystyle\quad+\chi_{[3,0]}(\check{x}_{1}^{2}+\check{x}_{1}^{3}+3\check{x}_{1}^{4}+4\check{x}_{1}^{5}+9\check{x}_{1}^{6}+10\check{x}_{1}^{7}+17\check{x}_{1}^{8}+20\check{x}_{1}^{9}+30\check{x}_{1}^{10}+\cdots)]$
$\displaystyle+\cdots$ (70) $\displaystyle I^{(A)}_{5}$
$\displaystyle=(1+\check{x}_{1}^{2}+\check{x}_{1}^{3}+2\check{x}_{1}^{4}+2\check{x}_{1}^{5}+3\check{x}_{1}^{6}+3\check{x}_{1}^{7}+5\check{x}_{1}^{8}+5\check{x}_{1}^{9}+7\check{x}_{1}^{10}+\cdots)$
$\displaystyle+y\chi_{[1,0]}(\check{x}_{1}^{2}+\check{x}_{1}^{3}+2\check{x}_{1}^{4}+3\check{x}_{1}^{5}+4\check{x}_{1}^{6}+6\check{x}_{1}^{7}+8\check{x}_{1}^{8}+10\check{x}_{1}^{9}+13\check{x}_{1}^{10}+\cdots)$
$\displaystyle+y^{2}[\chi_{[0,1]}(-\check{x}_{1}-\check{x}_{1}^{2}-2\check{x}_{1}^{3}-3\check{x}_{1}^{4}-3\check{x}_{1}^{5}-5\check{x}_{1}^{6}-5\check{x}_{1}^{7}-7\check{x}_{1}^{8}-7\check{x}_{1}^{9}-9\check{x}_{1}^{10}+\cdots)$
$\displaystyle\quad+\chi_{[2,0]}(\check{x}_{1}^{2}+\check{x}_{1}^{3}+3\check{x}_{1}^{4}+4\check{x}_{1}^{5}+7\check{x}_{1}^{6}+10\check{x}_{1}^{7}+15\check{x}_{1}^{8}+19\check{x}_{1}^{9}+27\check{x}_{1}^{10}+\cdots)]$
$\displaystyle+y^{3}[(1+\check{x}_{1}^{2}-\check{x}_{1}^{4}-2\check{x}_{1}^{5}-4\check{x}_{1}^{6}-7\check{x}_{1}^{7}-10\check{x}_{1}^{8}-14\check{x}_{1}^{9}-19\check{x}_{1}^{10}+\cdots)$
$\displaystyle\quad+\chi_{[1,1]}(-\check{x}_{1}-\check{x}_{1}^{2}-3\check{x}_{1}^{3}-4\check{x}_{1}^{4}-6\check{x}_{1}^{5}-9\check{x}_{1}^{6}-10\check{x}_{1}^{7}-13\check{x}_{1}^{8}-15\check{x}_{1}^{9}-16\check{x}_{1}^{10}+\cdots)$
$\displaystyle\quad+\chi_{[3,0]}(\check{x}_{1}^{2}+\check{x}_{1}^{3}+3\check{x}_{1}^{4}+5\check{x}_{1}^{5}+9\check{x}_{1}^{6}+14\check{x}_{1}^{7}+21\check{x}_{1}^{8}+30\check{x}_{1}^{9}+42\check{x}_{1}^{10}+\cdots)]$
$\displaystyle+\cdots$ (71)
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