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+MNRAS 000, 1–14 (0000)
+Preprint 10 January 2023
+Compiled using MNRAS LATEX style file v3.0
+MulGuisin, a Topological Clustering Algorithm, and Its
+Performance as a Cosmic Structure Finder
+Young Ju1,2, Inkyu Park1,2⋆, Cristiano G. Sabiu1,2 and Sungwook E. Hong,3,4
+1Department of Physics, University of Seoul, 163 Seoulsiripdae-ro, Dongdaemun-gu, Seoul 02504, Republic of Korea
+2Natural Science Research Institute, University of Seoul, 163 Seoulsiripdae-ro, Dongdaemun-gu, Seoul 02504, Republic of Korea
+3Korea Astronomy and Space Science Institute, 776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea
+4Astronomy Campus, University of Science and Technology, 776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea
+10 January 2023
+ABSTRACT
+We introduce a new clustering algorithm, MulGuisin (MGS), that can find galaxy clusters using topological informa-
+tion from the galaxy distribution. This algorithm was first introduced in an LHC experiment as a Jet Finder software,
+which looks for particles that clump together in close proximity. The algorithm preferentially considers particles with
+high energies and merges them only when they are closer than a certain distance to create a jet. MGS shares some
+similarities with the minimum spanning tree (MST) since it provides both clustering and graph-based topology in-
+formation. Also, similar to the density-based spatial clustering of applications with noise (DBSCAN), MGS uses the
+ranking or the local density of each particle to construct clustering. In this paper, we compare the performances of
+clustering algorithms using some controlled data and some realistic simulation data as well as the SDSS observation
+data, and we demonstrate that our new algorithm find clusters most efficiently and it defines galaxy clusters in a way
+that most closely resembles human vision.
+Key words: large-scale structure of Universe, galaxies: clusters: general, methods: statistical, software: data analysis
+1 INTRODUCTION
+In the standard ΛCDM cosmology paradigm, structures in
+the universe grow in a hierarchical manner (e.g., White &
+Rees 1978; Fall & Efstathiou 1980; Blumenthal et al. 1984).
+It means that smaller structures of matter start forming ear-
+lier, and the more massive structures form by the merging and
+accretion of smaller structures at later epoch of the universe.
+Therefore, understanding cosmic structures in the universe
+at various scales is crucial for understanding the nature of
+our universe. For example, numerous statistics of large-scale
+structures, such as topological analyses (e.g., Gott et al. 1986;
+Park & Gott 1991; Park & Kim 2010; Appleby et al. 2017,
+2018), Alcock-Paczynski tests (e.g., Alcock & Paczynski 1979;
+Ballinger et al. 1996; Li et al. 2014; Park et al. 2019), and
+small-scale redshift-space distortion (RSD; e.g., Sheth 1996;
+DeRose et al. 2019; Tonegawa et al. 2020), have been used to
+constraint cosmological parameters such as the matter den-
+sity parameter (Ωm) and the equation-of-state parameter of
+dark energy (ωde).
+While numerous statistics of cosmic structures have been
+used to understand the evolution and structure formation
+⋆ E-mail: icpark@uos.ac.kr
+of our universe, the exact definition of cosmic structures re-
+mains unclear. This is mainly because the matter distribution
+on large scale is continuous, and therefore, there exists no
+specific discrete boundary for each structure. Also, the mem-
+bership for a certain structure might change if one considers
+other properties than just position, such as dynamics, mass,
+and so on (e.g., Serra & Diaferio 2013; Gifford et al. 2013).
+Due to this ambiguity, numerous clustering algorithms have
+been proposed and used in the astronomical community. For
+example, Knebe et al. (2011) compared the various properties
+of dark matter (DM) halos found by 17 different halo-finding
+algorithms run on the same cosmological N-body simulation.
+Galaxy clustering algorithms have been used as essential
+tools for identifying galaxy clusters or super-clusters, as well
+as for investigating the large structure of the universe, includ-
+ing the filament structures. The most commonly used galaxy
+clustering algorithms in astronomy research are the friends-
+of-friends (FoF; Davis et al. 1985) and the minimum span-
+ning tree (MST; Borůvka 1926). These algorithms were in-
+troduced in the 1980s and have been widely used as standard
+galaxy clustering algorithms. In recent years, with the rapid
+development of machine learning (ML) technology, clustering
+algorithms such as DBSCAN (Density-based Spatial Cluster-
+ing of Applications with Noise; Ester et al. 1996) have also
+been applied to galaxy clustering. These clustering softwares,
+© 0000 The Authors
+arXiv:2301.03278v1  [astro-ph.IM]  9 Jan 2023
+
+2
+Y. Ju et al.
+including the ML based one, show comparable performance
+in galaxy clustering and produce consistent clustering results.
+However, the results of clustering do not always represent the
+clusters that the human eye can find. There are cases where a
+distribution clearly contains a cluster but it is not recognized
+as such by clustering algorithms, and there are cases where
+it is clearly divided into two clusters, visually, but appears as
+a single lump to the software.
+As such, we have explored the possibility of developing an
+algorithm that creates galaxy clusters in a way that more
+closely resembles how the human eye and brain identify pat-
+terns. One approach we considered was to adapt jet-finding
+software used in high-energy particle physics research, with a
+particular focus on the MulGuisin (MGS) algorithm as a po-
+tentially suitable software for galaxy clustering. MulGuisin
+(ᄆ
+ᅮ
+ᆯᄀ
+ᅱᄉ
+ᅵ
+ᆫ) is a Korean word for a ghost that lives in water
+and is a figure that often appears in old Korean stories. The
+MGS algorithm started with the idea that the ghosts hiding
+in the water could be found in the order of height by simply
+draining the water from the lake.
+Initially, we copied the MGS software from the A Toroidal
+LHC Apparatus (ATLAS) Jet-Finding library released in the
+early 1990s and developed it into a 3D galaxy clustering al-
+gorithm. We then made several sample galaxy distributions
+to check the performance of MGS and compared the results
+to those produced by other standard clustering algorithms
+such as FoF, MST, and DBSCAN. As a result, it was found
+that MGS had characteristics that other algorithms could not
+show, and it was also found that the cluster results created by
+MGS were most similar to the cluster results that the human
+eye found.
+Since the clusters created by MGS show different shapes
+when compared with clusters formed by other algorithms,
+and the number of clusters and the size distribution of clus-
+ters are quite different from those of classical algorithms, us-
+ing MGS makes a big difference in searching halos and super-
+clusters, and can yield a different interpretation for the large
+scale structures of the universe. Therefore, we anticipate that
+this new algorithm will be used as a new methodology in
+galaxy cluster research and furthermore used to create new
+interpretations in cosmology studies.
+The structure of this paper is as follows. In Section 2, we
+introduce the MulGuisin clustering algorithm, as well as FoF,
+MST, and DBSCAN as benchmark clustering algorithms for
+comparison. In Section 3, we describe both controlled random
+data and realistic galaxy distribution data that we will use
+for the performance test. We apply the above four clustering
+algorithms to the data and compare their performances in
+Section 4, and we summarize our results in Section 5.
+2 METHODS
+A halo or galaxy cluster is a group of galaxies held together by
+gravity. Finding such cluster structures in galaxy distribution
+data is very important for astronomical research because it
+provides a tool to study super-clusters, filaments, and even
+bigger the large scale structure of the universe.
+A cluster can be defined as a concentration of points or
+cells in a localized volume. The task of cluster identification
+has been extensively studied in the field of computational
+science, and a wide range of clustering algorithms have been
+developed for this purpose. Because different algorithms have
+different strengths and weaknesses, it is important for re-
+searchers to carefully select the algorithm that best suits their
+specific research purpose.
+In this section, we first introduce our MulGuisin clustering
+algorithm and introduce two clustering tools that are widely
+used in the field of astronomy and a newly developed clus-
+tering program through machine learning.
+2.1 MulGuisin galaxy clustering algorithm
+The MulGuisin (MGS) clustering algorithm was first intro-
+duced as a jet finder for the Large Hadron Collider (LHC)
+physics in the ATLAS Collaboration (Bosman et al. 1998).
+The algorithm is neither a variant of the conventional cone
+algorithm nor a variant of the kT algorithm that is used in
+various collider experiments as the standard tools for finding
+jets. Although it has shown some improvements in jet recon-
+struction performance, such as optimized jet orientation and
+jet energy resolution, but has not been used as a standard
+jet-finding tool for LHC experiments.
+Fig. 1 shows how the MGS algorithm works. The MGS
+algorithm first finds the most massive point from the input
+data and names it a cluster seed. Then it finds the second
+massive point and decides whether the point should belong
+to the first cluster or stand alone as the seed of a new cluster.
+This decision is made by checking how close the test point
+is to any neighboring clusters, for which we introduce a pa-
+rameter called linking length (ℓMST). That is, if the distance
+between the test point and the closest point in the cluster
+is less than this parameter, the test point is attached to the
+cluster, otherwise, it becomes the seed of a new cluster. The
+algorithm then finds the next massive point and repeats the
+above process until there are no more points left to test. At
+this stage, all points are converted into clusters. Of course,
+some points do not belong to any cluster and remain.
+Fig. 2 is an illustration to explain how the MGS algorithm
+creates galaxy clusters. In the figure, the points are sorted in
+order according to their mass and number. And according to
+this order, they become new cluster seeds or stick to exist-
+ing clusters. After going through the process, a cluster forms
+a tree-like structure that is sequentially connected accord-
+ing to the order of mass. The points in a cluster then form
+branches and nodes, and from the characteristic structure of
+such tree shape, one may able to study the topology of the
+galaxy cluster.
+2.2 Benchmark Algorithms
+In order to compare the performance of the MGS algorithm
+with those of other standard clustering algorithms, we select
+three benchmark algorithms, mostly based on their popular-
+ity in the astronomical community, mathematical clarity, and
+versatility. They are the friends-of-friends (FoF), minimum
+spanning tree (MST), and the density-based spatial cluster-
+ing of applications with noise (DBSCAN). Here we briefly
+introduce each package and describe how they make clus-
+ters.1
+1 Note that running our MGS algorithm from scratch may take
+a longer time than the above benchmark algorithms, especially
+MNRAS 000, 1–14 (0000)
+
+MulGuisin Clustering Algorithm
+3
+Figure 1. Schematic flow chart to describe how MulGuisin (MGS) algorithm works
+Figure 2. Diagram showing how the MulGuisin (MGS) algorithm works to identify clusters. Each gray circle represents a galaxy, and the
+size of the circle denotes its local density, with the number specifying the galaxies ranking in descending order of density. In this specific
+example, 21 galaxies are grouped into 3 clusters and 2 isolated galaxies.
+MNRAS 000, 1–14 (0000)
+
+galaxies
+No
+Yes
+clusters
+No
+Yes1st cluster
+8
+5
+10
+numberofchildrene3
+12
+1st seed
+19
+linking distance
+isolatedgalaxy
+3rd cluster
+Oth generation
+3rd seed
+2nd cluster
+2ndseed
+3
+1st generation
+13
+9
+16
+14
+2nd generation
+b
+15
+4th generation
+21
+isolatedgalaxy4
+Y. Ju et al.
+2.2.1 Friends-of-Friends (FoF)
+The friends-of-friends (FoF) algorithm is a commonly used
+technique for identifying clusters in astrophysical data
+(Huchra & Geller 1982; Tago et al. 2008; Duarte & Mamon
+2014; Tempel et al. 2016). This algorithm has a single free
+parameter, the linking length (ℓFoF), which determines the
+distance threshold for linking two data points. Points that are
+within this distance of each other are considered to be con-
+nected, and all connected points are grouped together into a
+single cluster.
+One limitation of the FoF algorithm is that it can be diffi-
+cult to choose an appropriate linking length. Different values
+of this parameter can result in clusters of different shapes
+or numbers, making it challenging to determine the optimal
+value (Tago et al. 2008).2 In this study, we use the Halotools
+implementation of the FoF algorithm (Hearin et al. 2017) to
+identify clusters in our datasets by applying various ℓFoF.
+Unless otherwise noted, we assume all FoF groups containing
+two or more members as clusters.
+2.2.2 Minimum Spanning Tree (MST)
+Galaxy data can be represented as a graph, with each galaxy
+represented as a node and the distance between two galaxies
+represented as an edge. The minimum spanning tree (MST)
+algorithm is a method for constructing a unique network from
+this data by connecting all nodes with minimum edges. Unlike
+other clustering algorithms, the MST does not require the use
+of a free parameter such as a linking length to construct the
+entire network. However, the MST connects all nodes and
+may not produce clusters with shapes that accurately reflect
+those of the original clusters.
+Nevertheless, MST has been used in cosmology to study
+the large-scale structure of the universe (Barrow et al. 1985;
+Krzewina & Saslaw 1996; Naidoo et al. 2020). In this study,
+we use the MiSTree package (Naidoo 2019) to construct MSTs
+from our galaxy data. Then, we find clusters from the single
+MST tree by cutting nodes longer than the linking length
+(ℓMST). Similar to the FoF case, we apply various values of
+ℓMST and assume all tree segments containing two or more
+members as clusters.
+2.2.3 Density-based Spatial Clustering of Applications with
+Noise (DBSCAN)
+The use of machine learning (ML) techniques is widespread
+in astronomy, as they enable the identification of patterns in
+data using algorithms. ML algorithms can be classified based
+on the type of data they are applied to, and one type, called
+when the number of data points is large. We found that most of
+the MGS calculation time, for a large number of data points, is
+taken in constructing the Voronoi tesselation and calculating the
+local density for each point. If we separate MGS into a density
+calculation and a tree building part, we found that the tree building
+takes a similar time to the benchmark algorithms.
+2 Note that the appropriate choice linking length for identifying
+DM halos from the DM particles in the N-body simulations is well
+known (ℓFoF ≃ 0.2⟨dparticle⟩) (More et al. 2011). However, the
+optimal choice of linking length in general clustering problems is
+not well known.
+unsupervised ML, is used with unlabeled data. Clustering al-
+gorithms, a subcategory of unsupervised ML algorithms, are
+used to group together data points with similar properties.
+One popular clustering algorithm is DBSCAN (density-based
+spatial clustering of applications with noise), which has been
+applied in a variety of contexts (Ester et al. 1996; Sander
+et al. 2017).
+DBSCAN is a density-based clustering algorithm that
+groups together data points based on their local density. In
+this algorithm, each cluster is identified by defining its core,
+which consists of high-density points within a certain dis-
+tance. The definition of core requires two free parameters,
+min_samples and eps, which determine the minimum num-
+ber of neighbors a point must have within a given radius in
+order to be considered as the core. Then, other points that are
+directly reachable from some core points within eps are also
+considered part of the cluster, while other points are labeled
+as noise.
+In this study, we use the scikit-learn package (Pedregosa
+et al. 2011) to implement the DBSCAN algorithm and iden-
+tify clusters in our data by applying various eps (or, the “link-
+ing length” in DBSCAN (ℓDBSCAN)). Unless otherwise noted,
+we assume min_samples = 3.
+2.3 A Simple 2D Toy Model Test
+To see how the shape of the clusters generated by the MGS
+algorithm differs from the results of other clustering algo-
+rithms, we created simple simulation data and compared the
+results. We first assume that there are 5 clusters in 2D space,
+and consider the case where each cluster contains 50 galaxies
+equally. The width of the galaxy distribution of each cluster
+was fixed to 10. The coordinates of the two-dimensional space
+span from 0 to 100 on both the X- and Y-axes, and the posi-
+tion of each cluster is set to have three different distributions,
+from far away from each other to all close together, as shown
+in Fig. 3 column (a) in rows (1), (2) and (3).
+As shown in Fig. 3, both the MGS and MST algorithms
+correctly find 5 clusters when the distances among the clus-
+ters are sufficiently far apart. However, when the clusters get
+closer together, MST can’t differentiate between the clus-
+ters and starts recognizing them as one big cluster. Even
+for the cases where clusters are attached to each other as
+shown in Fig. 3 (3), MGS still recognizes four among five
+true clusters like the human eyes can distinguish each clus-
+ter, whereas MST recognizes 4 adjacent clusters as one huge
+cluster. These differences can create serious differences in re-
+sults when studying the number and mass distributions of
+clusters.
+Note that, although we leave its details as future works,
+the tree structures made by the MGS algorithm have a non-
+negligible number of long nodes connecting two distant points
+in the cluster, while the MST algorithm connects only rea-
+sonably nearby points. This is because the MGS algorithm
+connects data points based on their local density, not only the
+distance between the points. Therefore, if two highly dense
+points are within the linking length, then they would be con-
+nected in the MGS but may not be in the MST.
+In the next section, we will compare the performance of
+MGS and other algorithms with more realistic 3D data.
+MNRAS 000, 1–14 (0000)
+
+MulGuisin Clustering Algorithm
+5
+0
+20
+40
+60
+80
+100
+(1)
+LL = 10
+0
+20
+40
+60
+80
+100
+(2)
+0
+20
+40
+60
+80
+100
+(a)
+0
+20
+40
+60
+80
+100
+(3)
+0
+20
+40
+60
+80
+100
+(b)
+0
+20
+40
+60
+80
+100
+(c)
+Figure 3. A simple 2D toy model test of the MGS algorithm by comparing it with the MST algorithm. (a) Input distributions of 5
+clusters with different degrees of separation from each other ((1)–(3)). Background color denotes the galaxy number distribution we used
+for generating the galaxies. (b) Clusters found by MGS and their tree structures. (c) Clusters found by MST and their tree structures.
+3 DATA
+Our final goal is to apply the MGS algorithm described in
+Section 2 to the galaxy clusters or other large-scale struc-
+tures of the universe. However, since some inconsistencies ex-
+ist between various clustering algorithms for finding clusters
+or other large-scale structures (e.g., see Knebe et al. 2011,
+and references therein), we cannot compare the MGS clus-
+ters found in the realistic data with their “truth”.
+Therefore, we apply two types of data sets in this section
+to compare the performance between MGS and other bench-
+mark algorithms. The first sets, called the “controlled ran-
+dom data” (D1–D3), are those that we design all properties
+of clusters, including their positions and member galaxy dis-
+tributions. Since we already know the true information of
+each cluster, we can test which algorithms predict the true
+clusters better in which conditions. The next sets, called the
+“realistic data” (D4), are the observational and simulation
+data sets of galaxies around z ≃ 0, and we focus on com-
+paring the properties of predicted clusters in each algorithm.
+Table 1 summarizes the data sets we use in this work.
+3.1 Controlled Random Data (D1–D3)
+3.1.1 Different Spatial Dispersion (D1)
+We use controlled, simulated data to evaluate the perfor-
+mance of the MGS algorithm in comparison to other clus-
+tering algorithms. These data are generated randomly and
+allow us to control the shape and distribution of clusters to
+test the algorithms under different conditions. The first set
+of data consists of 100 galaxies per cluster and 50 clusters
+within a 3-dimensional cubic volume of space with a side
+length 200 h−1Mpc. The cluster center positions are chosen
+randomly, and the galaxies in each cluster are distributed ac-
+cording to a Gaussian distribution with a variable standard
+deviation (σ) that controls the spatial dispersion. The D1-LD
+data has a low spatial dispersion (σ = 1 h−1Mpc), leading to
+well-separated clusters, while the D1-HD data has a higher
+MNRAS 000, 1–14 (0000)
+
+6
+Y. Ju et al.
+Data set
+Description
+D1-LD
+50 randomly positioned clusters, each of which contains 100 galaxies randomly spread by the 3D Gaussian
+distribution with standard deviation σ = 1 h−1Mpc. The total number of galaxies is 5,000.
+D1-HD
+Same as D1-LD, but with the greater standard deviation σ = 10 h−1Mpc.
+D2-NA
+Same as D1-HD, but the number of galaxies in each cluster follows an exponential random distribution.
+The total number of galaxies is 7,041.
+D2-LA
+Same as D2-NA, but adding uniformly randomly distributed noisy galaxies to the entire box to increase
+the total galaxy number density 1.5 times of D2-NA.The total number of galaxies is 12,041.
+D2-HA
+Same as D2-LA, but adding more noisy galaxies so that the total galaxy number is twice D2-NA. The
+total number of galaxies is 17,041.
+D3-HOD
+500 randomly positioned clusters with a mass distribution similar to the Press-Schechter mass function.
+The number of galaxies for each cluster follows HODa for massive halos (M ⩾ 1013 h−1M⊙). The
+galaxies are spread by the NFW profile with the concentration parameter to 10. The total number of
+galaxies is 50,257.
+D4-SDSS
+Volume-limited sample of the KIAS-VAGCb with absolute r-band magnitude Mr − 5 log h < −20.
+D4-HR4
+Four lightcone data of mock galaxy catalogs from the Horizon Run 4 simulationc with a similar condition
+to D4-SDSS.
+a Kravtsov et al. (2004). b Choi et al. (2010a). c Kim et al. (2015); Hong et al. (2016).
+Table 1. Name and description of galaxy data sets that we use in this analysis. The box size of all controlled data (D1–D3) is
+(200 h−1Mpc)3.
+X
+100
+75
+50
+25
+0
+25
+50
+75
+100
+Y
+100
+75
+50
+25
+0
+25
+50
+75
+100
+Z
+0
+25
+50
+75
+100
+125
+150
+175
+200
+X
+100
+50
+0
+50
+100
+Y
+100
+50
+0
+50
+100
+Z
+0
+50
+100
+150
+200
+D1-LD
+D1-HD
+X
+100
+75
+50
+25
+0
+25
+50
+75
+100
+Y
+100
+75
+50
+25
+0
+25
+50
+75
+100
+Z
+0
+25
+50
+75
+100
+125
+150
+175
+200
+X
+100
+50
+0
+50
+100
+Y
+100
+50
+0
+50
+100
+Z
+0
+50
+100
+150
+200
+D1-LD
+D1-HD
+X
+0
+50
+100
+150
+200
+Y
+0
+50
+100
+150
+200
+Z
+0
+50
+100
+150
+200
+D3-HOD
+X
+100
+50
+0
+50
+100
+Y
+100
+50
+0
+50
+100
+Z
+0
+50
+100
+150
+200
+X
+100
+50
+0
+50
+100
+Y
+100
+50
+0
+50
+100
+Z
+0
+50
+100
+150
+200
+X
+100
+50
+0
+50
+100
+Y
+100
+50
+0
+50
+100
+Z
+0
+50
+100
+150
+200
+D2-NA
+D2-LA
+D2-HA
+X
+100
+50
+0
+50
+100
+Y
+100
+50
+0
+50
+100
+Z
+0
+50
+100
+150
+200
+X
+100
+50
+0
+50
+100
+Y
+100
+50
+0
+50
+100
+Z
+0
+50
+100
+150
+200
+X
+100
+50
+0
+50
+100
+Y
+100
+50
+0
+50
+100
+Z
+0
+50
+100
+150
+200
+D2-NA
+D2-LA
+D2-HA
+X
+100
+50
+0
+50
+100
+Y
+100
+50
+0
+50
+100
+Z
+0
+50
+100
+150
+200
+X
+100
+50
+0
+50
+100
+Y
+100
+50
+0
+50
+100
+Z
+0
+50
+100
+150
+200
+X
+100
+50
+0
+50
+100
+Y
+100
+50
+0
+50
+100
+Z
+0
+50
+100
+150
+200
+D2-NA
+D2-LA
+D2-HA
+Figure 4. Three-dimensional galaxy distributions of controlled data sets used in this paper. Top: D1-LD(left), D1-HD(middle), and D3-
+HOD(right). Bottom: D2-NA(left), D2-LA(middle), and D2-HA(right), with noisy additional galaxies shown as yellow dots. See Table 1
+for details.
+MNRAS 000, 1–14 (0000)
+
+MulGuisin Clustering Algorithm
+7
+spatial dispersion (σ = 10 h−1Mpc), resulting in clusters that
+are closer together. Upper left and middle panels of Fig. 4
+show the distribution of galaxies in the D1-LD and D1-HD
+data sets.
+3.1.2 Additional Noisy Galaxies (D2)
+We generate additional controlled data sets that are similar
+to D1 but with slightly different characteristics. We again
+place 50 cluster centers at the same positions as D1, but this
+time we use an exponential distribution to generate a variable
+number of galaxies for each cluster:
+P(Ngal) =
+�
+�
+�
+1
+∆N exp
+�
+−Ngal − N0
+∆N
+�
+if Ngal > N0
+0
+otherwise
+.
+(1)
+Here, we set N0 and ∆N as 50 and 100, respectively, so
+that the minimum number of galaxies per cluster and the
+total number of galaxies roughly match with D1. The galax-
+ies are spatially distributed according to a Gaussian function
+centred on the cluster’s centre with a standard deviation of
+σ = 10 h−1Mpc, as was done for D1-HD, and we call this
+new controlled data D2-NA. The lower left panel of Fig. 4
+shows the distribution of galaxies in the D2-NA data. The
+total number of galaxies in this data set is 7,041.
+In addition to D2-NA, we introduce two more data sets
+that are created by adding unclustered galaxies, which are
+sampled uniformly in the entire box. We add these ‘noisy’
+galaxies so as to test how the algorithms are affected by the
+background density. The lower middle panel of Fig. 4 shows
+the D2-LA data, where we add 5,000 galaxies (yellow dots)
+to increase the galaxy number density by 1.5 times compared
+to D2-NA. On the other hand, the lower right panel shows
+the D2-HA data, where we add 10,000 galaxies to make the
+total galaxy number density twice that of D2-NA.
+3.1.3 HOD-based Mock Galaxies (D3-HOD)
+We generate a third set of controlled, simulated data to cre-
+ate a more complex environment for testing the performance
+of the MGS algorithm. We use an analytic formula to model
+the distribution of galaxies in this data set. First, we create
+500 cluster center positions by sampling uniform random dis-
+tribution within a (200 h−1Mpc)3 box. Then, we obtain the
+normalized version of Press-Schechter halo mass function at
+z = 0(Press & Schechter 1974), with a concordance ΛCDM
+cosmology to the Planck 2015 data (Planck Collaboration
+et al. 2016), for massive halos Mhalo > 1013 h−1M⊙ using the
+Colossus package (Diemer 2018). We then obtain masses for
+each of the 500 clusters by randomly sampling for the mass
+function.3
+We then use this information to generate a distribution
+of the number of galaxies using a halo occupation distribu-
+tion (HOD) model. The mean halo occupation is typically as-
+sumed to follow a power law at massivehalo masses (Berlind
+3 Note that neither the positions nor the mass distribution of clus-
+ters in D3-HOD follows the estimation from the standard cosmol-
+ogy. However, here we focus only on providing complex environ-
+ments, and therefore, such differences do not affect our motivation.
+See Section 3.2 for realistic data sets instead.
+& Weinberg 2002; Kravtsov et al. 2004):
+Navg(Mcluster) =
+�
+�
+�
+�Mcluster
+M1
+�α
+if Mhalo > Mmin
+0
+otherwise
+,
+(2)
+where α, Mmin, and M1 correspond to the power-law in-
+dex, cutoff halo mass where halo cannot contain galaxies,
+and the mass scale containing a single galaxy at the given
+condition of galaxy sample. Here, we use α = 0.87 and
+Mmin = 1013 h−1M⊙ by following Kravtsov et al. (2004). We
+set M1 = 1011 h−1M⊙ so that the minimum number of galax-
+ies for each cluster is set as 50. Also, for simplicity, we calcu-
+late the actual number of galaxies at each cluster by applying
+the ceiling to Navg.
+Next, we use the Colossus package to create an Navarro-
+Frenk-White (NFW) profile (Navarro et al. 1996)
+ρ(x) = Mcluster
+4πR3
+vir
+��
+ln(1 + cs) −
+cs
+1 + cs
+�
+x(x + c−1
+s )2�−1
+,
+(3)
+where x ≡ r/Rvir. The concentration parameter for the NFW
+profile cs is fixed as 10, and the virial radii Rvir is determined
+by the cluster mass accordingly. We then randomly distribute
+the galaxies according to this profile, and the resulting data
+set consists of 50,257 galaxies. The upper right panel of Fig. 4
+shows the distribution of galaxies in D3-HOD.
+3.2 Realistic Data: SDSS & Horizon Run 4 (D4)
+In the previous subsection, we described a set of controlled
+data catalogues for which we can carefully control the prop-
+erties of the clusters. Such data are useful for testing the
+performance of MGS over other benchmark algorithms by
+comparing the properties of identified clusters with the in-
+put truth. However, the true distribution of galaxies in the
+universe differs from these controlled random data in the fol-
+lowing ways. First, unlike those in the controlled random data
+with low noise levels, the boundaries of clusters in the uni-
+verse are often not clearly defined (e.g., Serra & Diaferio 2013;
+Gifford et al. 2013). Also, the spatial distribution of galax-
+ies in each cluster may not follow spherical symmetry (e.g.,
+Limousin et al. 2013, for a good review). Furthermore, the
+redshift-space distortion elongates spherical clusters in real
+space, which may require that we separate linking lengths
+between the radial and tangential directions (Farrens et al.
+2011; Tempel et al. 2016).
+Therefore, it is necessary to adopt a realistic galaxy dis-
+tribution for a fair performance test of the MGS algorithm.
+However, unlike for the case of the controlled random data
+where we know the answer, we may only study the dif-
+ference between the cluster properties from the MGS and
+other benchmark algorithms. Here we use observational data
+and four corresponding sets of mock simulation data — the
+volume-limited KIAS-Value Added Galaxy Catalog (KIAS-
+VAGC) of the Sloan Digital Sky Survey (SDSS) Main Galaxy
+Sample with r-band absolute magnitude Mr − 5 log h < −20
+(Choi et al. 2010a) and the lightcone mock galaxy samples
+from the Horizon Run 4 simulation (Kim et al. 2015; Hong
+et al. 2016).
+MNRAS 000, 1–14 (0000)
+
+8
+Y. Ju et al.
+3.2.1 Volume-limited KIAS-VAGC (D4-SDSS)
+The KIAS Value-Added Galaxy Catalog (KIAS-VAGC; Choi
+et al. 2010a) is an upgraded version of the New York Uni-
+versity Value-Added Galaxy Catalog (NYU-VAGC; Blanton
+et al. 2005), which is part of the Sloan Digital Sky Survey
+(SDSS) Data Release 7 (Abazajian et al. 2009), by adding
+some missing redshifts to improve spectroscopic complete-
+ness. This catalog has been widely used in numerous studies,
+including cosmic voids statistics (Pan et al. 2012; Hoyle et al.
+2012), largest structures of universe (Park et al. 2012), frac-
+tion of barred galaxies (Lee et al. 2012a), and the properties
+of active galactic nuclei (AGN; Lee et al. 2012b; Hwang et al.
+2012; Bae & Woo 2014).
+Most of the KIAS-VAGC galaxies were observed with the
+apparent r-band magnitude limit r = 17.6. It means that, in
+terms of absolute magnitude, the catalog contains less bright
+galaxies at lower redshifts, while only very bright galaxies
+could be seen at higher redshifts. Therefore, for a fair com-
+parison between galaxies over a wide redshift range, we ap-
+ply a “volume-limited” selection by selecting galaxies brighter
+than a certain absolute r-band magnitude (Choi et al. 2010b).
+Here, we use Mr − 5 log h < −20. By combining with the
+given apparent r-band magnitude limit, such absolute mag-
+nitude cutoff naturally provides the upper redshift bound of
+our volume-limited sample (z < 0.107; left panel of Fig. 5).
+We also apply the lower redshift bound z > 0.02, by consider-
+ing the incompleteness of the galaxy sample below the given
+redshift.
+In addition to the volume-limited selection in the redshift-
+magnitude plane, we also apply a sky selection for simplifica-
+tion. Specifically, we select galaxies within the SDSS Survey
+coordinate −33.5◦ < η < 36.5◦ and −48◦ < λ < 51◦, in or-
+der to maximize the sky area with a simple geometry, and
+to avoid issues arising from a complicated boundary (right
+panel of Fig. 5).
+3.2.2 Horizon Run 4 (D4-HR4)
+The Horizon Run 4 simulation (HR4; Kim et al. 2015) is an
+extremely large cosmological N-body simulation that uses
+6, 3003 DM particles within a periodic cube with a comoving
+volume V = (3.15 h−1cGpc)3. It assumes a vanilla ΛCDM
+cosmological model in concordance with the Wilkinson Mi-
+crowave Anisotropy Probe (WMAP) 5th-year result (Dunkley
+et al. 2009). Among 2,001 timesteps between z = 100 to 0,
+75 coarse timesteps with mean time difference ∆t = 0.18 Gyr
+are chosen between z = 12 to 0 to build a merging tree of
+FoF halos. The FoF linking length is 0.2 times the particle
+mean separation, and we identify halos only whose mass is
+greater than M min
+halo = 2.7 × 1011 h−1M⊙.
+The mock galaxies are then produced by so-called the
+most bound halo particle (MBP)-galaxy abundance match-
+ing method (Hong et al. 2016). We find MBPs for all halos in
+the merging tree and adopt their positions and peculiar ve-
+locities as those of corresponding mock galaxies. The “mass”
+of mock galaxies, which is used as a proxy of stellar mass or
+luminosity, is defined as the mass of their hosting halos. For
+satellite halos, we identify their MBPs at the timestep just
+before the infall event and trace them until they are totally
+absorbed toward their central halo by tidal disruption. For
+estimating the tidal disruption timescale tmerge), we adopt a
+modified model of Jiang et al. (2008),
+tmerge
+tdyn
+= (0.94ϵ0.60 + 0.60)/0.86
+ln[1 + (Mhost/Msat)]
+�Mhost
+Msat
+�α
+,
+(4)
+where ϵ, Mhost, Msat, tdyn are the circularity of the satellite’s
+orbit, the mass of central and satellite halos, and the orbital
+period of virialized objects, respectively. We adopt α = 1.5
+for a better match of the galaxy two-point correlation func-
+tion (2pCF) at scales less than 1 h−1Mpc at a given spatial
+resolution of the HR4 (Zehavi et al. 2011; Park et al. 2019).
+Then the mass of survived satellite galaxies is defined as the
+mass of their hosting halos just before the infall.
+After producing snapshot mock galaxy catalogs for coarse
+timesteps, we then produce lightcone mock galaxy catalogs
+up to z = 1.5. The all-sky lightcone DM particle data of the
+HR4 were created during the simulation by stacking the co-
+moving shells at the corresponding redshifts. Then, we com-
+pare the IDs of the galaxy MBPs at each coarse timestep
+snapshots and those of DM particles at the lightcone data
+with the coarse comoving shells. If the MBP ID of a given
+mock galaxy matches that of a particle in the lightcone data,
+we assign a galaxy in the lightcone data. Here, we adopt
+the position and peculiar velocity from the particle at the
+lightcone data, while the galaxy “mass” comes from the mock
+galaxy at the nearest snapshot.
+After creating the all-sky lightcone mock galaxy catalog,
+we cut it in a similar way to the volume-limited KIAS-VAGC
+sample. First, we apply the redshift space distortion (RSD)
+for each mock galaxy for a fair comparison with observation,
+by using real-space positions and peculiar velocities. Then,
+we apply the same redshift range 0.02 < z < 0.107 and set
+the lower bound of galaxy “mass,” so that the galaxy number
+density of the HR4 lightcone data is identical to that of KIAS-
+VAGC. After that, we create four non-overlapping subsets
+from it with the same angular geometry as our SDSS Survey
+coordinate selection.
+During the analysis, we found that the fiber collision in the
+fiber-fed spectroscopic observations affects various clustering
+statistics (Zehavi et al. 2002; Guo et al. 2012; Reid et al. 2014;
+Tonegawa et al. 2020). Therefore, for a fair comparison, our
+HR4 mock galaxy catalogs also need to follow the same fiber
+collision condition as the KIAS-VAGC. To do so, we select
+pairs of mock galaxies whose angular distance is less than
+55 arcseconds and keep only one from each pair by random
+selection. Because SDSS observations were partially overlap-
+ping, some close-pairs have both redshifts. In order to reflect
+this, we only fiber-collide 60% of the close pairs.
+4 RESULTS
+We test MGS and the other algorithms using the 3 controlled
+data and observation data. We run 4 algorithms with various
+linking-length and find out the number of clusters. The same
+process is repeated by changing linking-length and we check
+the tendency of the number of clusters.
+We use the three controlled data sets and observation data
+to evaluate the performance of the MGS algorithm and com-
+pare it to other clustering algorithms. We run each of the
+four algorithms with different values of the linking length
+and count the number of clusters identified by each algo-
+rithm. We repeat this process for a range of linking lengths
+MNRAS 000, 1–14 (0000)
+
+MulGuisin Clustering Algorithm
+9
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+Redshift
+23
+22
+21
+20
+19
+18
+17
+r
+5logh
+150
+100
+50
+0
+50
+100
+150
+ [degree]
+60
+40
+20
+0
+20
+40
+60
+ [degree]
+Figure 5. Selection of the volume-limited sample of the KIAS Value-Added Galaxy Catalog (KIAS-VAGS) used in this study (red boxes).
+Left: Volume-limited selection in the redshift vs. absolute r-band magnitude plane with Mr − 5 log h < −20. Right: Sky selection in SDSS
+Survey coordinates (η, λ).
+30
+35
+40
+45
+50
+Number of Clusters
+D1-LD
+MGS
+MST
+FoF
+DBSCAN
+2
+4
+6
+8
+10
+Linking-length
+0
+10
+20
+30
+40
+50
+Number of Clusters
+D1-HD
+Figure 6. The number of clusters as a function of linking length
+for D1-LD (top panel) and D1-HD (bottom). Each of the 4 clus-
+tering algorithms are indicated using different colors and symbols.
+Note that MST and DBSCAN show considerable overlap. The Hor-
+izontal dash shows the original number of clusters, which is 50.
+and analyze the trends in the number of clusters identified
+by each algorithm. This allows us to assess the sensitivity of
+the algorithms to the choice of linking length and to compare
+their performance in identifying clusters in the different data
+sets.
+4.1 Results with Controlled Data
+Fig. 6 shows the number of clusters identified by each algo-
+rithm as a function of the linking length for the controlled
+data set 1. The top panel shows the results for the D1-
+LD data, which consists of well-separated clusters. The al-
+gorithms are expected to identify 50 clusters in this data set.
+All four algorithms perform well in identifying the clusters,
+but the MGS algorithm stands out for its ability to accu-
+rately identify the correct number of clusters. In particular,
+for large linking lengths, the FoF and DBSCAN algorithms
+identify fewer than 50 clusters, because they connect neigh-
+boring clusters and merge them into a single cluster.
+The bottom panel of Fig. 6 shows the results for the D1-
+HD data, which has a higher level of spatial dispersion and
+some clusters that are close to each other. For small link-
+ing lengths, the algorithms identify fewer than 50 clusters
+because the linking length is not sufficient to connect the
+galaxies in these clusters. As the linking length increases, the
+behavior of the algorithms becomes more distinct. The MGS
+algorithm continues to accurately identify the correct number
+of clusters, while the other algorithms identify fewer clusters
+due to the merging of originally separate clusters. The MGS
+algorithm is able to track the structure of the clusters and
+identify their boundaries, leading to more accurate results in
+this type of data.
+Fig. 7 shows the results for controlled data set 2, which
+includes the D2-NA data with no additional galaxies and the
+D2-LA and D2-HA data with additional galaxies. The top
+panel shows the number of clusters identified by each algo-
+rithm for the D2-NA data. When this data was generated,
+the minimum number of galaxies per cluster was set to 50.
+In the region of small linking lengths, all algorithms iden-
+tify fewer than 50 clusters because the linking length is too
+small to connect the galaxies in the clusters. As a result,
+the clusters identified by the algorithms have fewer than 50
+member galaxies, and are therefore not considered as true
+clusters. For larger linking lengths, particularly those larger
+than 5, the difference between the MGS algorithm and the
+other algorithms becomes more pronounced. The MGS al-
+gorithm continues to accurately identify the correct number
+of clusters, while the other algorithms identify fewer clusters
+due to the merging of originally separate clusters.
+The behavior of the algorithms with additional galaxies is
+even more distinct. The middle panel of Fig. 7 shows the re-
+sults for the D2-LA data, where the other three algorithms
+identify only a single cluster for very large linking lengths.
+As the linking length increases, the algorithms merge several
+clusters into a single giant cluster, resulting in a significantly
+lower number of clusters than the original data. This rapid
+MNRAS 000, 1–14 (0000)
+
+10
+Y. Ju et al.
+10
+20
+30
+40
+50
+60
+Number of Clusters
+D2-NA
+d
+= 10.43
+Nmin = 50
+MGS
+MST
+FoF
+DBSCAN
+10
+20
+30
+40
+50
+60
+Number of Clusters
+D2-LA
+d
+= 8.73
+Nmin = 56
+2
+4
+6
+8
+10
+12
+14
+Linking-length
+0
+10
+20
+30
+40
+50
+60
+Number of Clusters
+D2-HA
+d
+= 7.77
+Nmin = 59
+Figure 7. Same as Fig. 6, but with D2-NA(top), D2-LA(middle),
+and D2-HA(bottom). The vertical dashed line is mean-separation
+of data (⟨d⟩). Since each data set has a different overall number
+density, we assign different minimum number of member galaxies
+(Nmin) to define clusters.
+increment of a single giant cluster is called “percolation,” and
+it is known to occur at linking length similar to the mean-
+separation (ℓ ≃ ⟨d⟩) for the ideal random Poisson graph (Dall
+& Christensen 2002). Fig. 7 clearly shows that such percola-
+tion occurs at ℓ ≃ ⟨d⟩ for all three benchmark algorithms.
+Note that, however, the percolation occurs at the low-
+est linking length in FoF, while both MST and DBSCAN
+share a similar value of linking length at percolation. This
+is because FoF does not have an additional consideration
+for limiting the cluster boundary that exists in the other
+two algorithms (minimize the number of edges in MST, and
+core definition in DBSCAN). Fig. 8 shows the 3D distribu-
+tions of clustering results from various algorithms at linking
+length ℓ = 11 h−1Mpc, which is longer than the mean sep-
+aration ⟨d⟩ = 8.73 h−1Mpc. As expected, three benchmark
+algorithms show percolation (blue color), while our MGS al-
+gorithm successfully reconstructs most of the true clusters.
+Note that only one giant cluster is found in the FoF algo-
+rithm, while both MST and DBSCAN have two additional
+small clusters (green and orange colors).
+The behavior of the MGS algorithm for the D2-HA data
+is slightly different. In the region of small linking lengths,
+the MGS algorithm accurately identifies the correct number
+of clusters. However, for larger linking lengths, particularly
+ℓ > 13 h−1Mpc, the MGS algorithm identifies additional clus-
+ters that were not present in the original data. These “fake”
+clusters are not true clusters and are not representative of the
+underlying structure of the data. This behavior highlights the
+ability of the MGS algorithm to identify clusters in data with
+a complex distribution of galaxies but also underscores the
+importance of choosing an appropriate linking length to avoid
+identifying false clusters.
+Fig. 9 shows the number of clusters for controlled 3 data
+with a more complex environment than D1–D2. At ℓ ≳
+⟨d⟩/2 ≈ 3 h−1Mpc, the number of clusters using FoF and
+DBSCAN decreases as the linking length increases, resulting
+in the percolation at ℓ ≳ ⟨d⟩.The MST shows a flat curve
+when the linking length is larger than ∼ 8 h−1Mpc. This is
+because MST connects all galaxies with minimal edge first,
+and then we cut off the links with linking length. Therefore,
+if there were no links longer than 8 h−1Mpc in the original
+tree, then cutting the links with any longer linking length
+than 8 h−1Mpc would not change the result. So, the number
+of clusters using MST shows a constant value.
+In contrast, the number of clusters identified by the MGS
+algorithm slowly decreases as the linking length increases.
+This is because the clusters in this data set are close to each
+other and are easily merged by the algorithm for large linking
+lengths. However, the MGS algorithm is able to identify clus-
+ters based on density, which allows it to retain the structure
+of the clusters even for large linking lengths. This is the main
+advantage of the MGS algorithm compared to the other three
+algorithms, which are not able to accurately identify clusters
+in complex data sets.
+4.2 Results with Observational and Cosmological
+Simulation Data
+Fig. 10 shows the results of the four algorithms applied to
+both KIAS-VAGC observational data and four sets of HR4
+lightcone data. We track the number of detected clusters
+changing with both linking lengths and with the minimum
+number of member galaxies from 2 to 5.
+For all four clustering algorithms, the HR4 simulation re-
+sults match well with the observations within cosmic vari-
+ance, especially for n ⩾ 5 at ℓ ≳ ⟨dparticle⟩ = 0.5 h−1Mpc.
+On the other hand, HR4 tends to underestimate the num-
+ber of clusters for a smaller minimum number of member
+galaxies and/or smaller linking length ℓ ≲ 0.5 h−1Mpc. This
+may mean that, despite the agreement with the observation in
+terms of 2pCF below 1 h−1Mpc-scale, some disagreements ex-
+ist between HR4 and observation in terms of the higher-order
+statistics in smaller scales than the particle mean separation
+scale.
+One notable feature of the MGS algorithm is that it does
+not create a single giant cluster for large linking lengths. In-
+stead, the algorithm identifies a number of smaller clusters,
+even for large linking lengths. This is in contrast to the other
+three algorithms, which all create a single giant cluster for
+large linking lengths. This difference highlights the ability of
+the MGS algorithm to accurately identify clusters in data
+with a complex distribution of galaxies.
+Fig. 11 shows the number of member galaxies for the 1st
+MNRAS 000, 1–14 (0000)
+
+MulGuisin Clustering Algorithm
+11
+X
+100
+50
+0
+50
+100
+Y
+100
+50
+0
+50
+100
+Z
+0
+50
+100
+150
+200
+X
+100
+50
+0
+50
+100
+Y
+100
+50
+0
+50
+100
+Z
+0
+50
+100
+150
+200
+X
+0
+50
+100
+150
+200
+Y
+0
+50
+100
+150
+200
+Z
+0
+50
+100
+150
+200
+X
+100
+50
+0
+50
+100
+Y
+100
+50
+0
+50
+100
+Z
+0
+50
+100
+150
+200
+MGS
+MST
+FoF
+DBSCAN
+Figure 8. 3D distribution of clustering results from MGS and other benchmark algorithms in D2-LA with linking length ℓ = 11 h−1Mpc,
+which is longer than the mean-separation ⟨d⟩ = 8.73 h−1Mpc. Color indicates the cluster membership. MGS finds 49 clusters among 50
+true clusters, while other algorithms connect most of galaxies and finally make a giant cluster (blue color).
+to 4th largest clusters identified by each algorithm in D4-
+SDSS and D4-HR4. Similar to Fig. 10, both results from the
+simulation and observation data match well with each other.
+The top left panel of the figure shows the shape of the largest
+cluster for each algorithm. As the linking length increases
+over certain value, the largest cluster identified by the FoF,
+MST, and DBSCAN algorithms contains all of the galaxies in
+the data, while the MGS algorithm identifies a cluster with
+only a portion of the galaxies. This indicates that the MGS
+algorithm is able to identify multiple clusters even for large
+linking lengths, while the other algorithms merge all of the
+galaxies into a single giant cluster.
+The main difference between the MGS algorithm and the
+other three algorithms becomes particularly clear when ex-
+amining the number of member galaxies in the 2nd to 4th
+largest clusters (upper right and bottom panels of Fig. 11). As
+the linking length increases, the number of member galaxies
+in these clusters identified by the FoF, MST, and DBSCAN
+algorithms decreases to zero. This is because the first largest
+cluster identified by these algorithms took all the galaxies
+in the data, leaving no galaxies to be considered for further
+clustering. In contrast, the MGS algorithm is able to identify
+multiple clusters even with large linking lengths as the largest
+cluster does not monopolize all galaxies. This demonstrates
+the ability of the MGS algorithm to accurately identify clus-
+ters in data with a complex distribution of galaxies.
+Fig. 12 shows the 30 largest clusters found by the MGS
+algorithm in the D4-SDSS data with a linking length ℓMGS =
+10 h−1Mpc. While such a large choice of linking length makes
+a single giant cluster in all other three benchmark algorithms
+(see Fig. 11), none of the 30 clusters suffer percolation. All
+30 largest clusters are well-separated and have some even dis-
+tribution of galaxies in the XY-plane (that is, the tangential
+plane). On the other hand, most of the clusters have some-
+what elongated features in the line-of-sight direction, which
+clearly shows the Finger-of-God effect due to the RSD. There-
+fore, although this needs further inspection, we consider that
+the 30 largest clusters found by the MGS algorithm could be
+MNRAS 000, 1–14 (0000)
+
+12
+Y. Ju et al.
+0
+2
+4
+6
+8
+10
+12
+14
+Linking-Length
+0
+100
+200
+300
+400
+500
+Number of cluster
+D3-HOD
+MGS
+MST
+FoF
+DBSCAN
+Figure 9. Same as Figs. 6–7, but with D3-HOD.
+100
+101
+102
+103
+104
+Number of clusters
+MGS
+n
+2
+n
+3
+n
+4
+n
+5
+MST
+100
+101
+Linking length(h
+1Mpc)
+100
+101
+102
+103
+104
+Number of clusters
+FoF
+100
+101
+Linking length(h
+1Mpc)
+DBSCAN
+Figure 10. Same as Figs. 6, 7 & 9, but with D4-SDSS (thick
+lines) and D4-HR4. For D4-HR4, the average values and the ranges
+between minimums and the maximums of 4 data samples are drawn
+as thin lines and error bars. Results from each clustering algorithm
+are shown on different panels, while the color indicates the different
+choices of the minimum number of member galaxies to identify
+clusters.
+the actual large structures similar to galaxy (super)clusters
+in real space.
+5 CONCLUSIONS
+The MulGuisin (MGS) algorithm is a powerful technique for
+identifying clusters in data from astrophysical simulations
+and observations. It consistently produces results closer to
+those inferred from human visual inspection. In comparison
+to other clustering algorithms, such as the friends-of-friends
+(FoF) algorithm, the minimum spanning tree (MST) algo-
+100
+101
+102
+103
+104
+105
+Number of galaxies
+1st cluster
+MGS
+MST
+FoF
+DBSCAN
+2nd cluster
+10
+1
+100
+101
+Linking length (h
+1Mpc)
+100
+101
+102
+103
+104
+105
+Number of galaxies
+3rd cluster
+10
+1
+100
+101
+Linking length (h
+1Mpc)
+4th cluster
+Figure 11. The number of member galaxies for the 1st, 2nd, 3rd
+and 4th largest clusters in D4-SDSS (thick lines) and in D4-HR4
+as a function of linking length. For D4-HR4, the average values
+and the ranges between minimums and the maximums of 4 data
+samples are drawn as thin lines and error bars. Color indicates the
+different clustering algorithms.
+rithm, and the DBSCAN algorithm, the MGS algorithm has
+several advantages. The MGS algorithm is able to take into
+consideration the local density and is able to accurately iden-
+tify clusters even in complex data sets with a large number
+of galaxies. In contrast, the FoF, MST, and DBSCAN algo-
+rithms often merge clusters into a single giant cluster for large
+linking lengths, losing the ability to accurately identify indi-
+vidual clusters. This characteristic of the MGS algorithm is
+particularly important for analyzing data from astrophysical
+simulations and observations.
+In this proof of concept work we have shown that the jet-
+finding algorithm MGS can be applied to mock Galaxy data
+resulting in reliable cluster identification. However the iden-
+tification of clusters in real observation is a difficult issue due
+to survey incompleteness, selection effects, redshift-space dis-
+tortions, etc. In future work we will test MGS in the presence
+of realistic observational systematic effects.
+MGS also provides auxiliary topological information such
+as the number and length of connections for each galaxy. In
+future work we will explore the use of this enhanced enhanced
+information in testing or constraining cosmological models.
+ACKNOWLEDGEMENTS
+The authors thank Changbom Park, Dongsu Bak, and Ena
+Choi for helpful discussions. This research was supported by
+Basic Science Research Program through the National Re-
+search Foundation of Korea(NRF) funded by the Ministry
+of Education(grant number) S.E.H. was supported by the
+project ᄋ
+ᅮᄌ
+ᅮᄀ
+ᅥᄃ
+ᅢᄀ
+ᅮᄌ
+ᅩᄅ
+ᅳ
+ᆯ ᄋ
+ᅵᄋ
+ᅭ
+ᆼᄒ
+ᅡ
+ᆫ ᄋ
+ᅡ
+ᆷᄒ
+ᅳ
+ᆨᄋ
+ᅮᄌ
+ᅮ ᄋ
+ᅧ
+ᆫᄀ
+ᅮ (“Under-
+MNRAS 000, 1–14 (0000)
+
+MulGuisin Clustering Algorithm
+13
+200
+100
+0
+100
+200
+X (h
+1Mpc)
+200
+100
+0
+100
+200
+Y (h
+1Mpc)
+0
+100
+200
+300
+400
+500
+Z (h
+1Mpc)
+200
+100
+0
+100
+200
+Y (h
+1Mpc)
+200
+100
+0
+100
+200
+X (h
+1Mpc)
+0
+100
+200
+300
+400
+500
+Z (h
+1Mpc)
+X (h
+1Mpc)
+200
+100
+0
+100
+200
+Y (h
+1Mpc)
+200
+100
+0
+100
+200
+Z (h
+1Mpc)
+0
+100
+200
+300
+400
+500
+Figure 12. Top 30 largest clusters (colors) found by the MGS in the D4-SDSS galaxies (gray dots). The observer is located at the
+origin. The linking length is ℓMGS = 10 h−1Mpc, where all D4-SDSS galaxies fall into a single giant cluster in all other three benchmark
+algorithms (see Fig. 11). Note that, even in such a large linking length, none of the 30 largest clusters suffers percolation.
+standing Dark Universe Using Large Scale Structure of the
+Universe”), funded by the Ministry of Science. C.G.S is sup-
+port via the Basic Science Research Program from the Na-
+tional Research Foundation of South Korea (NRF) funded
+by the Ministry of Education (2018R1A6A1A06024977 and
+2020R1I1A1A01073494).
+This work was supported by the Supercomputing Cen-
+ter/Korea Institute of Science and Technology Information,
+with supercomputing resources including technical support
+(KSC-2013-G2-003), and the simulation data were trans-
+ferred through a high-speed network provided by KRE-
+ONET/GLORIAD.
+Funding for the SDSS and SDSS-II has been provided by
+the Alfred P. Sloan Foundation, the Participating Institu-
+tions, the National Science Foundation, the US Department
+of Energy, the National Aeronautics and Space Administra-
+tion, the Japanese Monbukagakusho, the Max Planck Society,
+and the Higher Education Funding Council for England. The
+SDSS website is http://www.sdss.org/.
+The SDSS is managed by the Astrophysical Research Con-
+sortium for the Participating Institutions. The Participating
+Institutions are the American Museum of Natural History,
+Astrophysical Institute Potsdam, University of Basel, Uni-
+versity of Cambridge, Case Western Reserve University, Uni-
+versity of Chicago, Drexel University, Fermilab, the Institute
+for Advanced Study, the Japan Participation Group, Johns
+Hopkins University, the Joint Institute for Nuclear Astro-
+physics, the Kavli Institute for Particle Astrophysics and Cos-
+mology, the Korean Scientist Group, the Chinese Academy of
+Sciences (LAMOST), Los Alamos National Laboratory, Max
+Planck Institute for Astronomy (MPIA), the Max Planck In-
+stitute for Astrophysics (MPA), New Mexico State Univer-
+sity, Ohio State University, University of Pittsburgh, Uni-
+versity of Portsmouth, Princeton University, the US Naval
+Observatory, and the University of Washington.
+MNRAS 000, 1–14 (0000)
+
+14
+Y. Ju et al.
+DATA AVAILABILITY
+The up-to-date MulGuisin algorithm can be downloaded at
+https://github.com/youngju20/Mulguisin.
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+MNRAS 000, 1–14 (0000)
+

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+Publications of the Astronomical Society of Australia (), 1–11
+doi:
+ARTICLE
+Milliarcsecond Structures of Variable Peaked-Spectrum Sources
+K. Ross,1 C. Reynolds,2 N. Seymour,1 J. R. Callingham,3,4 N. Hurley-Walker,1 and H. Bignall5,2
+1International Centre for Radio Astronomy Research, Curtin University, Bentley, WA 6102, Australia
+2 CSIRO, Space and Astronomy, P.O. Box 1130, Bentley, WA 6102, Australia
+3Leiden Observatory, Leiden University, PO Box 9513, Leiden, 2300 RA, The Netherlands
+4ASTRON, Netherlands Institute for Radio Astronomy, Oude Hoogeveensedijk 4, Dwingeloo, 7991 PD, The Netherlands
+5Manly Astrophysics, 15/41-42 East Esplanade, Manly, NSW 2095, Australia
+Author for correspondence: K. Ross, Email: kathryn.ross@icrar.org.
+(Received 03 Aug 2022; revised 24 Nov 2022; accepted 31 Dec 2022; first published online XX)
+Abstract
+Spectral variability offers a new technique to identify small scale structures from scintillation, as well as determining the absorption mechanism
+for peaked-spectrum (PS) radio sources. In this paper, we present very long baseline interferometry (VLBI) imaging using the Long Baseline
+Array (LBA) of two PS sources, MRC 0225–065 and PMN J0322–4820, identified as spectrally variable from observations with the Murchison
+Widefield Array (MWA). We compare expected milliarcsecond structures based on the detected spectral variability with direct LBA imaging.
+We find MRC 0225–065 is resolved into three components, a bright core and two fainter lobes, roughly 430 pc projected separation. A
+comprehensive analysis of the magnetic field, host galaxy properties, and spectral analysis implies that MRC 0225–065 is a young radio
+source with recent jet activity over the last 102–103 years. We find PMN J0322–4820 is unresolved on milliarcsecond scales. We conclude
+PMN J0322–4820 is a blazar with flaring activity detected in 2014 with the MWA. We use spectral variability to predict morphology and find
+these predictions consistent with the structures revealed by our LBA images.
+1.
+Introduction
+Peaked-spectrum (PS) sources, are a subset of active galac-
+tic nuclei (AGN) that are identified by a peak in their radio
+spectral energy distribution (O’Dea & Saikia, 2021), and are
+also often associated with compact morphologies (≲ 20 kpc;
+Phillips & Mutel, 1982; Tzioumis et al., 2010). PS sources
+provide an interesting population of AGN as the evolutionary
+pathway from PS source to extended (≳ 30 kpc) AGN is still
+unclear. Two contending theories hypothesise the nature and
+evolutionary pathway of PS sources: the youth scenario, where
+the age of the PS source is ≤ 105 years and has not yet had
+ample time to grow to the large-scale AGN (O’Dea & Baum,
+1997; Owsianik & Conway, 1998; Tinti & de Zotti, 2006);
+and the frustration scenario, when the PS source is confined
+by a dense cloud of the interstellar medium (ISM) of the host
+galaxy environment (van Breugel et al., 1984; Wilkinson et al.,
+1984; O’Dea et al., 1991). Furthermore, recent identifications
+of embedded PS cores within remnant ageing lobes has been
+attributed to restarted and episodic AGN activity (Hernández-
+García et al., 2019), i.e. a cyclical evolution rather than linear
+evolution.
+Compact symmetric objects (CSOs) are a subset of PS
+sources with similar morphologies to large scale AGN, namely
+a central region (often quite faint, if detected) with emission
+either side associated with hot spots and/or lobes. Unlike
+typical AGN, CSOs show emission only on very compact scales,
+typically ≤1 kpc, and thus require high-resolution imaging to
+detect (Phillips & Mutel, 1982; Gugliucci et al., 2005). CSOs
+are generally considered young AGN (< 104 yr; O’Dea &
+Baum, 1997; Owsianik & Conway, 1998; Tinti & de Zotti,
+2006), which may evolve into typical, radio-loud AGN.
+Previous attempts to discriminate between youth and frus-
+tration scenarios have relied on spectral modelling and high-
+resolution imaging (e.g. Marr et al., 2014; Keim et al., 2019)
+using very long baseline interferometry (VLBI). The cause of
+absorption at low-frequencies, producing the spectral peak,
+has typically been attributed to synchrotron-self absorption
+(SSA) and/or free-free absorption (FFA) for the youth and
+frustration scenarios respectively (Tingay & de Kool, 2003;
+Callingham et al., 2015). Unfortunately, without sufficient
+sampling below the spectral turnover, the cause of absorption
+is often ambiguous (Callingham et al., 2017). In rare cases, a
+SSA model can be ruled out if the optically thin spectral index
+is sufficiently steep (α ≥ 2.5)a.
+Many PS sources have been identified as a CSOs (e.g.
+0108+388, 0710+439 and 2352+495; Readhead et al., 1996).
+As CSOs are typically considered to be young AGN, identify-
+ing PS sources that are also CSOs could help to differentiate
+between the youth and frustrations scenarios. However, identi-
+fying CSOs requires high resolution (mas) observations using
+VLBI. Likewise, PS sources sometimes display extremely asym-
+metrical mas structures, likely due to an inhomogeneous sur-
+rounding environment influencing their growth (Orienti et al.,
+2006; Keim et al., 2019), compared with a fairly symmetrical
+morphology associated with CSOs with minor asymmetries
+likely coming from orientation effects (Orienti & Dallacasa,
+2008). VLBI can also be used to measure proper motion of
+aWe assume a power-law relation where Sν = S0να, thus the sign of α,
+being negative or positive, also indicates either the optically thin or thick
+spectral index respectively.
+arXiv:2301.00977v1  [astro-ph.GA]  3 Jan 2023
+
+2
+K. Ross et al.
+hot-spots in lobes to estimate kinematic ages of ≤ 3×103years
+(Polatidis & Conway, 2003; Gugliucci et al., 2005), consistent
+with the theory that CSOs are young AGN. Indeed, Gugliucci
+et al. (2005) find a majority of the CSOs with age estimates
+were ≤ 500 yrs, suggesting CSOs may be short lived and few
+would continue to grow to the scale of typical AGN, thereby
+explaining the large fraction of CSO and PS sources thought
+to be young relative to the number of large-scale radio galaxies
+(O’Dea & Saikia, 2021). VLBI of PS sources can thus help to
+identify populations of CSOs and elucidate the youth scenario
+and AGN evolution.
+Spectral variability at radio frequencies offers a new tech-
+nique for identifying young or frustrated candidates. Many
+variability surveys have identified PS sources that lost their PS
+classification over time (Tinti et al., 2005; Torniainen et al.,
+2005; Ross et al., 2021, hereafter R21), or showed a signifi-
+cant change in spectral shape likely due to a variable opacity
+from the inhomogeneous surrounding ISM (Tingay et al.,
+2015; Ross et al., 2022, hereafter R22). Thus the population of
+known PS sources, which is already biased from sparse spectral
+coverage from a range of instruments and times, is likely con-
+taminated by temporary PS sources. This is particularly true
+at higher frequencies (∼GHz), which is sensitive to emission
+from the core/jets. PS sources with a peak at lower frequen-
+cies (∼MHz) appear to be less contaminated by sources only
+showing a temporary peak (Callingham et al., 2017, R21).
+Spectral variability offers the a new technique to find and
+exclude contaminating “temporary” PS sources, as well as iden-
+tify CSO candidates with a decreased risk of contaminating
+sources. Variability of PS sources has been used to infer the
+presence of compact (µas – mas) features based on scintillation
+(Fanti et al., 1979; Chhetri et al., 2018, R21). Such compact
+features are common for CSOs, but VLBI is required for con-
+firmation of a CSO classification. Spectral variability has also
+found PS sources that show changing spectral shape, inconsis-
+tent with scintillation, which suggests that some PS sources
+are frustrated or contaminating blazars (R22).
+This paper aims to investigate the milliarcsecond scale struc-
+tures of variable PS sources using VLBI to test predictions based
+on spectral variability. In particular, we investigate PS sources
+that have shown a consistent spectral shape with a variable
+overall flux density, consistent with scintillation, suggesting a
+compact feature on milliarcsecond scales (R21, R22), and use
+VLBI to test a CSO classification. We also investigate variable
+PS sources that R21 found as changing spectral shape. They
+concluded the short timescale (∼1 year), and variable spectral
+shape is inconsistent with interstellar scintillation and present
+it as a blazar caught flaring.
+In Section 2, we describe the three variable PS sources of
+this study, in Section 3 we describe the observational strategy
+and data reduction. Section 4 outlines the results of the LBA
+imaging. We discuss the host galaxy properties including
+their linear size compared to turnover in Section 5.1, the mid-
+infrared (MIR) and optical emission in Section 5.2 and the radio
+properties in Section 5.3. In Section 6 we present the likely
+absorption mechanisms and source classification of our targets.
+We adopt the standard Λ-cold dark matter cosmological model,
+with ΩM = 0.286, ΩΛ = 0.714, and the Hubble constant
+H0 = 69.6 km s–1 Mpc–1 (Wright, 2006; Hinshaw et al., 2013)
+2.
+Target Selection
+Targets were selected for LBA imaging with the goal of com-
+paring direct imaging of milliarcsecond structures with pre-
+dicted morphologies based on their variability. Three targets
+were selected based on the variability detected by R21 and
+R22. MRC 0225–065 (GLEAM J022744-062106) was initially
+identified as variable in R21 but further monitoring over a
+year found no evidence of variability (R22). As such, it was
+predicted MRC 0225–065 would have resolved structures on
+milliarcsecond scales with a compact feature ≲ 25 mas, re-
+sulting in variability from refractive interstellar scintillation
+(RISS) on a longer timescale with a dampened modulation
+index due to the extended structure. Conversely, PMN J0322–
+4820 (GLEAM J032237–482010) was selected due to the vari-
+able spectral shape identified in R21. To explain the variable
+spectral shape, R21 concluded PMN J0322–4820 was likely a
+blazar caught flaring in 2014. As such, it was predicted to show
+a compact morphology even on milliarcsecond scales. Finally,
+MRC 2236-454 (GLEAM J223933–451414) was identified by
+R21 as the only PS source in their sample that showed sig-
+nificant variability but maintained a constant peak frequency
+below 231 MHz. A low peak frequency is typically associated
+with PS sources that are of the order of tens of kilo-parsecs
+across, but the RISS detected by R22 suggested MRC 2236-454
+is dominated by a compact feature, and showed variability due
+to a surrounding inhomogeneous environment. As such, it was
+predicted MRC 2236-454 may be resolved on milliarcsecond
+scales and show an asymmetrical morphology, often associ-
+ated with frustrated sources in an inhomogeneous surrounding
+environment (Orienti et al., 2006).
+3.
+LBA Observations and Data Reduction
+3.1
+Observations
+LBA observations were taken on November 23, 2020 and
+February 17, 2021 as part of project V600. The November
+observation was centered at 2.4 GHz and the February obser-
+vation was centered at 8.3 GHz and both utilised 128 MHz of
+bandwidth in dual polarizations. Stations used in each obser-
+vation and their diameter is listed in Table 1. Both observa-
+tions cycled through phase calibrator scans and target scans
+of lengths 2 min and 5 min, respectively. However, the spatial
+separation of each target and their respective phase calibrator
+meant each target had a different number of scans. A summary
+of the targets, phase calibrators and number of scans each is
+presented in Table 2.
+Parkes at 2.4 GHz, and Katherine at both frequencies, ob-
+served using their native linear feeds. These were converted
+to a circular polarization basis post-correlation using the Pol-
+Convert software (Martí-Vidal et al., 2016)
+3.2
+Data Processing and Calibration
+After correlation, data calibration and processing were done
+using the NRAO’s Astronomical Imaging Processing System
+
+Publications of the Astronomical Society of Australia
+3
+Table 1. LBA stations included in observations
+Name
+Code
+Diameter (m)
+Nov20
+Feb21
+ATCA, phased up
+At
+5×22
+Y
+Y
+Mopra
+Mp
+22
+Y
+Y
+Parkes
+Pa
+64
+Y
+Y
+Hobart
+Ho
+26
+Y
+Y
+Ceduna
+Cd
+30
+Y
+Y
+Yarragadee
+Yg
+12
+Y
+Y
+Warkworth
+Ww
+12
+Y
+Y
+Hartebeesthoek
+Hh
+26
+Y
+Y
+Katherine
+Ke
+12
+Y
+Y
+Tidbinbilla
+Td
+34
+Y
+N
+Table 2. Targets, associated calibrators and number of LBA scans for each
+target source.
+Source Name
+Expected S5GHz (mJy)
+Number of scans
+MRC 0225–065
+0.238
+27
+PKS J0217+0144 (C)
+0.666
+27
+PMN J0322–4820
+0.112
+40
+PMN J0335-4837 (C)
+0.112
+40
+MRC 2236–454
+0.420
+48
+QSO B2227–445 (C)
+0.386
+48
+(AIPS) (Wells, 1985). The calibration and flagging followed
+the general procedure outlined in the AIPS cookbookb and
+was implemented in a semi-automated script with the Parsel-
+Tongue interface (Kettenis et al., 2006). Initial flagging of edge
+channels and RFI was done using UVFLG. Auto-correlations
+were scaled to unity across the band using ACCOR before
+removing gross residual instrumental delays using FRING on
+a short scan of a bright calibrator. Complex bandpass cor-
+rections were derived using BPASS. The system temperature
+and gain calibration were applied using APCAL. Delay, rate
+and phase calibrations were determined from fringe fitting
+using FRING from each target’s respective phase calibrator.
+A phase referenced image was created for all targets except
+for MRC 0225–065, as a first pass detection of the targets to
+determine if a phase shift was needed. Lastly, UVFIX was used
+to apply a phase shift to the data for any sources that were
+∼arcsecond away from the phase centre used in correlation.
+MRC 0225–065 had accurate VLBI coordinates and thus did
+not require a phase shift. The calibrated and phase shifted data
+were exported to be imaged using CASA.
+3.3
+Imaging and Self-Calibration
+Initial Stokes-I images were made with a quasi-natural weight-
+ing with robust parameter set to +1 (Briggs, 1995) using the
+tclean function in CASA (McMullin et al., 2007). Clean boxes
+were used but were tightly restricted for the models used for
+self-calibration to avoid inducing artificial structure from the
+bThe AIPS cookbook can be found here http://www.aips.nrao.edu/cook.
+html
+complex point-spread-function. For each image, phase only
+self calibration was performed and applied using the gaincal
+and applycal functions respectively. Due to the sparse (u, v)-
+coverage and low signal-to-noise (SNR), calibration solutions
+were inspected and applied without flagging solutions that
+had insufficient SNR. The slow rate of improvement necessi-
+tated several (∼9) rounds of self-calibration. The SNR of the
+main component and the root-mean-squared (rms) noise of
+the image were inspected after each self calibration iteration
+to ensure each round improved the overall image quality. For
+each source the initial model assumed for the self-calibration
+was an unresolved point source to avoid inducing any morpho-
+logical features. Any resolved components were included in
+subsequent rounds of imaging clean components and kept in
+the model for self-calibration if this reduced the rms noise of
+the image. The initial solution interval for the self calibration
+was set to the scan length and decreased in further rounds of
+self calibration. Phase only self calibration rounds were contin-
+ued until the rms noise of the image increased. A final round of
+both phase and amplitude self calibration was then performed
+(provided it reduced the rms of the final image) with the so-
+lution interval set to the scan length. For MRC 0225–065, an
+amplitude self-calibration was applied to both frequencies, but
+no amplitude self-calibration was applied to the 2.4 GHz image
+of PMN J0322–4820.
+4.
+Results
+Images of MRC 0225–065 at both 2.4 and 8.3 GHz are pre-
+sented in Figure 1, and an image of PMN J0322–4820 at
+2.4 GHz, presented in Figure 3. Unfortunately, due to large
+phase errors from a pointing offset, we were unable to re-
+cover images for MRC 2236–454 at either frequency, or for
+PMN J0322–4820 at 8.3 GHz, this was because the source po-
+sitions were beyond the observed correlated field of view for
+recovery in each case. For MRC 2236–454, the pointing offset
+was over 11 arcseconds for both the 2.4 GHz and 8.3 GHz ob-
+servations, thus the phase errors from this pointing offset was
+beyond recovery. PMN J0322–4820 also had a pointing offset
+of ≈ 11.5 arcseconds, however, given it was bright (∼ 0.2 Jy),
+there was sufficient sensitivity using a subset of antennas (flag-
+ging the Hartebeesthoek antenna), and a phase shift combined
+with self calibration to recover and image at 2.4 GHz. How-
+ever, this method was not possible at 8.3 GHz due to the smaller
+field-of-view and decreased sensitivity. Henceforth, we will
+only discuss the results for MRC 0225–065 and PMN J0322–
+4820.
+Table 3. Properties for each LBA image: synthesised beam size and rms
+background noise.
+Source, ν (GHz)
+rms (mJy/beam)
+θbeam,maj
+θbeam,min
+PA
+MRC 0225–065, 2.4
+2.7
+9.5
+3.2
+7.0
+MRC 0225–065, 8.3
+1.0
+4.4
+2.7
+83
+PMN J0322–4820, 2.4
+1.0
+30
+17
+-54
+
+4
+K. Ross et al.
+4.1
+MRC B0225–065
+MRC 0225–065 was resolved into three components morphol-
+ogy at both 2.4 GHz and 8.3 GHz, as shown in Figure 1. The
+final image was made with a robust parameter of -1 at 2.4 GHz
+and -0.5 at 8.3 GHz (Briggs, 1995). MRC 0225–065 is resolved
+into 3 regions: a bright, unresolved central component, with
+an upper limit of source size of 2.5 × 4 mas assuming the beam
+size at 8.3 GHz (labelled C in Figure 1), a fainter 16 × 11 mas
+Western region (L1) and even fainter 14 × 10 mas Eastern
+component (L2). The sizes of L1 and L2 are measured using
+the contours in the 2.4 GHz image. The triple morphology is
+roughly symmetrical with the distance between the C to L1
+and L2 being ∼ 40 mas each. Since it appears the components
+of MRC 0225–065 may be resolved, we measured their flux
+density over an irregular polygonc for each component.
+We recovered all the flux density predictions from the
+spectral fit to the R22 ATCA observations at 2.4 GHz, but
+found that ∼ 35% of the flux density was lost at 8.3 GHz.
+The flux densities for each component and their spectral index
+are presented in Table 4. The irregular polygon was shaped
+based on contour levels to ensure only real flux was included in
+the final measurement. However, the missing flux density at
+8.3 GHz may be due to extended structure being resolved out.
+Consequently, the estimates for the spectral index presented
+in Table 4 should be considered lower limits.
+Table 4. Flux densities and two component spectral index for each compo-
+nent of MRC 0225–065 found in the LBA images. The uncertainties for the
+fluxdensitiesaremeasuredcalculatedusingthemeasureduncertaintyfrom
+polygon flux and the rms noise of the image. The uncertainty for α is calcu-
+lated using standard propagation of errors. The model prediction is calcu-
+lated from the best spectral fit, a double SSA spectral model with an expo-
+nential break.
+Component
+S2.4GHz (mJy)
+S8.3GHz (mJy)
+α
+C
+270±10
+78±7
+-0.95±0.08
+L1
+121±8
+30±5
+-1.1±0.2
+L2
+56±7
+18±4
+-0.9±0.2
+Integrated LBA
+447±14
+126±10
+-0.97±0.07
+Model Prediction
+400
+195
+N/A
+The symmetrical triple morphology suggests MRC 0225–
+065 is a CSO candidate with a core (C) and two lobes (L1
+and L2).
+The spectral index of the central component is
+αC = –0.95 ± 0.08, which is far steeper than expected for
+a typical AGN “core", generally expected to have a α ≥ –0.5
+(Orienti et al., 2006; Hardcastle & Looney, 2008). However,
+components have previously been identified as cores with spec-
+tral indices as steep as –0.7 (Orienti et al., 2006). We present
+the SED for MRC 0225–065 in Figure 2 including the MWA
+flux densities from R22 as well as the flux densities and power-
+law spectral model for each LBA component. The entire SED
+is fit, using the most recent MWA epoch (2020-09), with a
+double SSA model with an exponential break, which assumes
+two synchrotron emitting regions that are self-absorbed and
+cusing https://github.com/nhurleywalker/polygon-flux, (Hurley-Walker
+et al., 2019)
+ageing producing the exponential break, νb, separate from the
+peak frequency. The break frequency is the frequency where
+the spectrum begins to steepen as the electrons are ageing
+and experiencing energy losses (Turner et al., 2018). We fit
+the spectral model using the UltraNest packaged (Buchner,
+2021), which uses a nested sampling Monte Carlo algorithm.
+From the double SSA spectral model, we find the peak frequen-
+cies for the two SSA components to be νp,1 =400±100 MHz
+and νp,2=112±90 MHz, and find νb =14.3±2.7 GHz.
+MRC 0225–065 has a spectroscopic redshift of 0.445 (Al-
+bareti et al., 2017); thus, 1 mas corresponds to a linear scale of
+5.25 pc. Using this redshift, we find the projected linear size
+of MRC 0225–065 (from L1 to L2) to be ∼430 pc, the linear
+distance from the core to either lobe to be ∼210 pc and place
+an upper limit on the size of component C to be ≤26 pc.
+4.2
+PMN J0322–4820
+Due to difficulties in the phase calibration, we were only able to
+produce a high quality image of J0322–483 at 2.4 GHz, shown
+in Figure 3. We do not resolve PMN J0322–4820 and it is
+confined to the size of the beam: 56 × 40 mas. The final image
+was made using a robust parameter of +0.5, and by flagging the
+Hartebeesthoek antenna, thus the beam size for PMN 0322–
+4820 compared to MRC 0225–065 for the same frequency is
+much larger. Details of the image properties are presented in
+Table 3. Compared to the spectral model fit to the ATCA and
+2014 MWA observations, 18% of the flux density was missing.
+We used a reported photometric redshift for PMN J0322–4820
+of 0.16 (Bilicki et al., 2014), thus 1 mas corresponds to a linear
+size of 2.650 pc. We place an upper limit on the source size of
+148 pc.
+5.
+Discussion
+In this section, we will present a comprehensive analysis of both
+MRC 0225–065 and PMN J0322–4820 to produce a unified
+perspective of these two sources with the aim of concluding
+whether they are young or frustrated PS sources. In Sec-
+tion 5.1, we present our two sources in the linear size and
+turnover relation, in Section 5.2, we discuss the host galaxy
+properties according to mid-infrared, optical observations and
+radio properties.
+5.1
+Linear Size and Turnover Relation
+PS sources follow an inverse relation between their linear size
+and intrinsic turnover frequency, often referred to as the linear
+size turnover relation, first presented by O’Dea (1998). This
+relation is directly predicted from the youth scenario (O’Dea,
+1998) where the peak frequency is due to SSA and thus the
+linear size is directly related to the peak frequency (Keller-
+mann & Pauliny-Toth, 1981). While modifications to models
+in the frustration scenario can reproduce this relation (Bick-
+nell et al., 2018), it is generally understood that PS sources
+that fall below the linear size-turnover relation are likely com-
+pact beyond what is expected for a young source and a thus
+dhttps://johannesbuchner.github.io/UltraNest/
+
+Publications of the Astronomical Society of Australia
+5
+40
+20
+0
+-20
+-40
+40
+20
+0
+-20
+-40
+Relative R.A. (mas)
+Relative Dec (mas)
+C
+L1
+L2
+MRC 0225-065 at 2.4GHz
+52.5pc
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+Intensity (Jy/beam)
+40
+20
+0
+-20
+-40
+40
+20
+0
+-20
+-40
+Relative R.A. (mas)
+Relative Dec (mas)
+C
+L1
+L2
+MRC 0225-065 at 8.3GHz
+52.5pc
+0.00
+0.01
+0.02
+0.03
+0.04
+Intensity (Jy/beam)
+Figure 1. LBA images of MRC 0225–065 at 2.4 GHz (lef) and 8.3 GHz (right). Beam sizes are shown with a white ellipse in the bottom lef corner of each image
+and dimensions are specified in Table 3. Contours are placed at (-3, 3, 4, 5, 6, 7, 10, 20, 50, 100, 200, 400, 800, 1600) times the rms noise of the image, also
+specified in Table 3. Pixel brightness is plotted in a linear scale following the colour-bars to the right of each image. The resolved regions are labelled C, L1,
+L2 and properties of each region are outlined in Table 4. Relative R.A and Dec are calculated from the position of the core (C) component with coordinates:
+J2000 02h27m44.5s -06d21m06.7s.
+0.1
+0.2
+0.5
+1.0
+2.0
+5.0
+10.0
+Frequency (GHz)
+0.10
+0.20
+0.30
+0.40
+0.50
+0.60
+0.70
+0.80
+Flux Density (Jy)
+MRC 0225-065
+2013
+2014
+2020-04
+2020-05
+2020-07
+2020-09
+ATCA 2020
+LBA int
+C
+L1
+L2
+40
+20
+0
+-20
+-40
+40
+20
+0
+-20
+-40
+Relative R.A. (mas)
+Relative Dec (mas)
+C
+L1
+L2
+MRC 0225-065 Spectral Index Map
+52.5pc
+−1.8
+−1.6
+−1.4
+−1.2
+−1.0
+−0.8
+−0.6
+−0.4
+−0.2
+α
+Figure 2. Spectral energy distribution (SED) for MRC 0225–065 (lef) and spectral index map (right). The spectral index map was created using by convolving
+both the 8.3 GHz image and 2.4 GHz image to the same resolution. Data included in the SED are from R21 and R22 monitoring (circles) and coloured according
+toepoch. LBAfluxdensitiesareplottedassquareswiththeintegratedfluxdensityofLBAplottedasblacksquares. ThespectralfittoeachLBApointisapower-
+law with spectral index presented in Table 4. The grey spectral model to the entire SED is a double SSA model with an exponential break. Supplementary
+data included: TIFR GMRT 150 MHz Sky Survey Alternative Data Release 1 (TGSS-ADR1; Intema, H. T. et al., 2017) (grey cross), Molonglo Reference Catalogue
+(MRC; Large et al., 1981, 1991) (grey +), Rapid ASKAP Continuum Survey (RACS; McConnell et al., 2020; Hale et al., 2021) (grey ‘Y’), NRAO VLA Sky Survey (NVSS;
+Condon et al., 1998), Australia Telescope 20 GHz (AT20G; Murphy et al., 2010) (grey right arrow).
+
+6
+K. Ross et al.
+200
+100
+0
+-100
+-200
+200
+100
+0
+-100
+-200
+Relative R.A. (mas)
+Relative Dec (mas)
+C
+L1
+L2
+PMN J0322-4820 at 2.4GHz
+132.5pc
+0.00
+0.02
+0.04
+0.06
+0.08
+Intensity (Jy/beam)
+0.1
+0.2
+0.5
+1.0
+2.0
+5.0
+10.0
+Frequency (GHz)
+0.10
+1.00
+0.50
+Flux Density (Jy)
+PMN J0322-4820
+2013
+2014
+ATCA 2020
+LBA
+Figure 3. LBA image for PMN J0322–4820 at 2.4 GHz (lef) and associated SED (right). The beam size is shown with a white ellipse in the bottom lef corner
+and dimensions are specified in Table 3. Contours are placed at (-3, 3, 4, 5, 6, 7, 10, 20, 50, 100, 200, 400, 800, 1600) times the rms noise of the image, also
+specified in Table 3. Pixel brightness is plotted in a linear scale following the colour-bars to the right of the image. Relative R.A and Dec are calculated from the
+central coordinate: J2000 03h22m38.0s -48d20m16.2s. Data included in SED is from R21 and R22 (circles) and coloured according to epoch. LBA flux density
+is plotted as a blue square. The grey spectral model to the entire SED is a single SSA model with an exponential break. Supplementary data included is: TIFR
+GMRT 150 MHz Sky Survey Alternative Data Release 1 (TGSS-ADR1; Intema, H. T. et al., 2017) (grey cross), Sydney University Molonglo Sky Survey (SUMSS;
+Mauch et al., 2003) (grey star), Rapid ASKAP Continuum Survey (RACS; McConnell et al., 2020; Hale et al., 2021) (grey ‘Y’).
+assumed to be frustrated. We plot both MRC 0225–065 and
+PMN J0322–4820 on the linear size-turnover relation in Fig-
+ure 4, along with other known PS sources, details of which
+are discussed by Keim et al. (2019). It is evident from Figure 4,
+that MRC 0225–065 is entirely consistent with the relation
+whereas PMN J0322–4820 sits somewhat below the relation,
+particularly since the linear size is an upper limit. This would
+suggest MRC 0225–065 is consistent with the youth scenario
+whereas PMN J0322–4820 may be frustrated. However, it is
+worth nothing, R21 identified PMN J0322–4820 as a variable
+PS source with a changing spectral shape, and thus concluded
+it was likely a blazar. Furthermore, R21 found the peak fre-
+quency changed from ∼320 MHz in 2013 to ∼145 MHz in
+2014. As the peak frequency is variable and PMN J0322–4820
+is known to exhibit a changing spectral shape, its position on
+the linear size-turnover relation will also vary, shown by the
+error bar in Figure 4 corresponding to the range of the peak
+frequency from 2013 to 2014. Most likely, PMN J0322–4820
+is only a temporary PS source and thus should not be included
+in this relation nor when considering the PS population at
+large.
+5.2
+Host Galaxy Properties
+5.2.1
+WISE Colours
+MIR colour selection techniques using the Wide-Field Infrared
+Survey Explorer (Wright et al., 2010, WISE) are widely used to
+efficiently distinguish between AGN and star-forming galax-
+ies.
+WISE is a MIR all sky survey covering four photometric
+bands: 3.4, 4.6, 12, and 22 µm referred to as W1, W2, W3, and
+W4 respectively. The MIR wavelengths are sensitive to the
+emission from hot dust in the torus of the AGN, allowing for
+the identification of AGN where X-ray and optical emission
+10
+2
+10
+1
+100
+101
+102
+Linear Size (kpc)
+102
+103
+104
+Rest-Frame Peak Frequency (MHz)
+J0227-0621
+J0322-482
+Figure 4. Rest frame peak frequency versus linear size. Sources in black are
+describedinKeimetal.(2019). Thedashedlineisthefittotherelationfound
+by Orienti & Dallacasa (2014). Arrows indicate maximum linear sizes for un-
+resolved sources. MRC 0225–065 (pink circle) and PMN J0322–4820 (purple
+circle) are plotted with linear sizes calculated from LBA images. The error
+bars for MRC 0225–065 represent the range for peak frequencies calculated
+in R21.
+
+Publications of the Astronomical Society of Australia
+7
+may be blocked by intervening gas and dust. This also makes
+AGN stand out from star-bursting galaxies or stars due to their
+extremely red MIR emission (Lonsdale et al., 2015). Obscured
+AGN with red MIR emission have been identified by their MIR
+colours, often by their place in a colour-colour diagram (Jarrett
+et al., 2011; Lonsdale et al., 2015). The bulk of sources centred
+around W1 – W2 = 1.2 and W2 – W3 = 3 correspond to the
+region typically associated with quasars and AGN. MRC 0225–
+065 is found in the region typically associated with emission
+from star formation or stellar emission; i.e. there is no evidence
+of hot AGN dust, however, there is evidence for moderate star
+formation. As we know MRC 0225–065 is an AGN, it is likely
+the emission at MIR is a combination of these two processes.
+PMN J0322–4820 is well within the elliptical regime, thus
+has low emission from star formation and no evidence of hot
+AGN dust. Blazars are typically found to dominate the top
+right region of the WISE colour-colour plot as the MIR emis-
+sion is dominated by the emission of the blazar over the galaxy
+(and associated stellar emission). A compact morphology and
+variable spectral shape suggest PMN J0322–4820 is a blazar.
+However, the WISE colours of PMN J0322–4820 suggest that
+the host galaxy is an elliptical with predominantly red optical
+emission. Therefore, the emission from the potential radio
+blazar is not dominant in the MIR. While it is more common
+to find blazars in the top right region of the WISE colour-
+colour plot, the MIR colours, which suggest the host galaxy
+for PMN J0322–4820 is an elliptical, are still consistent with a
+blazar classification (Yang et al., 2015; D’Abrusco et al., 2019).
+5.2.2
+Optical Spectra
+MRC 0225–065 has an optical spectrum from the 13th data
+release of the Sloan Digital Sky Survey (Albareti et al., 2017,
+SDSS). From the fitted spectrum, Albareti et al. (2017) report
+a spectroscopic redshift for MRC 0225–065 of z = 0.445 and
+classify it as a broad-line, starburst quasar. The spectrum
+additionally has low-ionisation nuclear emission-line region
+(LINER) properties, evident from the strong NII, SiII and OI
+lines. A LINER has a high energy radiation field. There is still
+debate about whether this is AGN emission or star formation,
+but likely the combination of the broad lines, strong OIII
+emission and radio-loudness of MRC 0225–065 is evidence
+of AGN. From the broad Hα, we can calculate the velocity
+dispersion according to:
+d(velocity) = cd(λ)
+λ0
+,
+(1)
+where c is the speed of light, d(λ) is the wavelength dispersion
+from the spectral fit, and λ0 is the rest-frame wavelength of
+Hα. Using the reported fit to the broad Hα from SDSS where
+λobserved = 9486 Å, we use the equivalent width, EW= 30±4 Å,
+and find the velocity dispersion to be 900±100 km/s. This large
+velocity dispersion may be from an extreme star formation
+wind but it is also indicative of the broad-line regions from an
+AGN, which is more consistent given our radio observations
+identify MRC 0225–065 as an AGN. The broad Hα, and large
+velocity dispersion, is consistent with an AGN that is quite
+obscured, as reported by Albareti et al. (2017) who classify
+it as a broad-line quasar. Perhaps of more interest are the
+starburst properties of MRC 0225–065, namely OII and OIII
+emission lines, identified by Albareti et al. (2017). Both OII and
+OIII are forbidden lines with different origins: OII is mostly
+due to star formation and thus is often used as an indicator
+for star formation in galaxies; OIII is due to an AGN and
+can be used as a proxy for the AGN bolometric luminosity.
+This is also consistent with the WISE colours discussed in
+Section 5.2.1, which find MRC 0225–065 consistent with a
+galaxy with emission coming from both the AGN and star
+formation. Combining the radio, MIR and optical properties
+of MRC 0225–065, it is likely this galaxy has moderate star
+formation with an obscured AGN.
+5.3
+Radio Properties of MRC B0225–065
+Combining the spectral information and high resolution re-
+solved structure of MRC 0225–065, we are able to determine
+several intrinsic properties that can help differentiate between
+SSA and FFA models. In this section, we estimate the magnetic
+field strength and spectral ages to assess whether MRC 0225–
+065 is consistent with the youth scenario. We do not consider
+PMN J0322–4820 in this section due to its unresolved mor-
+phology (even on mas scales) and since the radio variability
+suggests it is a blazar with an added beaming effect producing
+Doppler boosting and thus many of the assumptions required
+for these calculations no longer hold.
+5.3.1
+Magnetic Field
+As a means of evaluating the validity of SSA compared to
+an FFA, we can calculate the magnetic field estimates based
+on a pure SSA model and on equipartition. Equipartition
+assumes there is equal energy between the radiating particles
+and the magnetic field. The comparison between magnetic
+field estimates based on an SSA model and equipartition has
+been used as evidence both for the SSA model (when the
+estimates are in agreement; Orienti & Dallacasa, 2008) and
+against (when there is a clear disparity; Keim et al., 2019). In
+this section, we will first estimate the magnetic field assuming
+a purely SSA model, then assuming equipartition and compare
+these to determine whether SSA is a reasonable model for
+MRC 0225–065.
+We can estimate the magnetic field strength, in Gauss,
+based on a purely SSA spectral model, BSSA, according to:
+BSSA ≈
+(νpeak/f (αthin))5θsrc,min2θsrc,max2
+Speak
+2(1 + z)
+,
+(2)
+where νpeak is the observed peak frequency in GHz, Speak is the
+flux density in Jy at the peak frequency for the source at redshift
+z with angular minor and major component axis, θsrc,min and
+θsrc,max, in mas (Kellermann & Pauliny-Toth, 1981). We note,
+f (αthin) is as defined by Kellermann & Pauliny-Toth (1981),
+where it is loosely related to αthin. We take f (αthin) = 8 based
+on values from Marscher (1983); Orienti & Dallacasa (2008).
+
+8
+K. Ross et al.
+Now, assuming equipartition, we calculate the magnetic
+field strength, in Gauss, according to (Miley, 1980), as Bequi
+by assuming the component has cylindrical symmetry such
+that the width of the source on the sky is equivalent to the line
+of sight path-length.
+For both calculations, we calculate BSSA and Bequi for the
+compact core region rather than the total source, to ensure we
+are comparing a homogeneous region (Orienti & Dallacasa,
+2008; Keim et al., 2019). For MRC 0225–065, using Equa-
+tion 2, we estimate the magnetic field strength for a purely
+SSA model to be BSSA ≈6±7 mG for the core region where
+θsrc = 2.5 × 4 mas. To estimate Bequi, we assume a filling fac-
+tor η = 1 and set k = 1e and find Bequi ≈6±2 mG. As BSSA is
+within the uncertainties of Bequi, it suggests the core region of
+MRC 0225–065 is in equipartition and consistent with a pure
+SSA model. While this does not exclude the FFA model, it
+does provide supportive evidence for the SSA model. Further-
+more, it may not be a valid assumption that MRC 0225–065
+is in equipartition, thus the equation from Miley (1980) for
+Bequi would not be a reasonable estimate of the magnetic field
+strength.
+We can also use the estimated magnetic field to calculate
+the age of the electron population as a proxy for the age of the
+jets/lobes. Calculating the spectral age of the electron popula-
+tion requires an accurate estimate of the break frequency, νb.
+We can thus calculate the spectral age, τspec, according to:
+τspec =
+aB1/2
+B2 + BiC2
+�
+νb(1 + z)
+�–1/2
+where
+BiC = 0.318(1 + z)2
+a =
+�243πme5c2
+4µ02e7
+�1/2
+(3)
+where BiC is the magnitude of the microwave background
+magnetic field in nT, B is the magnetic field of the source
+in nT, νb is the break frequency in GHz, and the constants
+me, c, µ0, and e are the mass of an electron, speed of light,
+magnetic permeability of free space, and charge of an electron,
+respectively.
+It is possible the core is actually an unresolved double of
+more recent AGN activity than the outer lobes, producing
+the steep (α ≲ –1, see Table 4) spectral index. We assume a
+constant expansion speed, v, and use the linear sizes to estimate
+the dynamical age, τdyn, of the core and outer lobes. Using
+the magnetic field calculated for the core region assuming
+equipartition, i.e. setting B = Bequi = 6 ± 2 mG, and deter-
+mining a break frequency, we can estimate the spectral age
+of the core. Using a break frequency of νb = 14.3 ± 2.7 GHz,
+calculated from the double SSA spectral model fit, we estimate
+the spectral age of the core to be τspec ≈ 700 ± 100 years.
+ek = 1 is equivalent to the minimum energy condition, however values for
+k have ranged from 1 to 100, where k = 100 produces an order of magnitude
+difference in Bequi (Pacholczyk & Roberts, 1971; Miley, 1980)
+We then calculate an upper limit on the expected expansion
+velocity of v ≤ 0.13 c (using simple speed = distance/time argu-
+ments) for the core using the upper limit for the linear source
+size of θsrc ≤ 26 pc, as outlined in Section 4.1. An expansion
+velocity of v = 0.13 c is well within previous measurements of
+the expansion speeds for compact AGN that have been found
+to range from 0.1 c up to 0.7 c (Polatidis & Conway, 2003;
+An & Baan, 2012; Orienti & Dallacasa, 2020). The range of
+expansion velocities would correspond to a range in dynamical
+ages for the core of 100 ≲ τdyn ≲ 900 years. If we assume the
+expansion velocity of the core of “inner lobes" is roughly equal
+to that of the outer lobes from a previous epoch of activity, we
+can place an upper limit on the dynamical ages of the outer
+lobes. We calculate the distance between the core and L1 as
+∼ 210 pc, which corresponds to a dynamical age of 5000 years
+for an expansion velocity of 0.13 c. For the range of dynamical
+ages for typical PS sources, we expect the age of the outer lobes
+to be 1000 ≲ τdyn ≲ 7000 years. Previous estimates for the
+ages of PS sources using similar assumptions have estimated
+ages from ∼ 101 to ∼ 105 years (Orienti et al., 2010), which
+is entirely consistent with our age estimates for both the inner
+core and outer lobes.
+As the ages, expansion velocities, and magnetic fields that
+we calculate are all consistent with the SSA model and a youth
+scenario, it appears MRC 0225–065 is more consistent with
+a young CSO rather than a frustrated compact AGN. How-
+ever, there are several caveats and assumptions made in these
+calculations. Thus, while these results are consistent with
+the evolutionary scenario of MRC 0225–065 being the youth
+model, it is not sufficient for excluding the frustration scenario
+entirely.
+6.
+AUnifiedPerspectiveofMRCB0225–065andPMNJ0322–
+4820
+Combining all the information we have obtained about MRC 0225–
+065, we begin to create a unified perspective that suggests
+MRC 0225–065 is a CSO with a peaked spectrum best ex-
+plained by SSA and recent jet activity over the last 102–103 years.
+A summary of the evidence in support of this conclusion are
+as follows:
+• Variability: R21 identified spectral variability of MRC 0225–
+065 with a constant spectral shape, consistent with vari-
+ability due to RISS. Further spectral variability monitor-
+ing by R22 detected no further variability, suggesting a
+resolved structure but consistent PS source classification.
+This observation suggests it is unlikely MRC 0225–065 is
+a contaminating blazar or source with only a temporary
+PS source classification, such as frustrated sources with an
+inhomogeneous surrounding medium.
+• Radio morphology: Previously, it has been suggested
+frustrated PS sources are more likely to show an asymmet-
+rical morphology due to the asymmetrical environment
+confining the growth of the lobes. Inversely, this suggests
+young PS sources that are not frustrated may be more
+likely to show a symmetrical morphology like that of a
+
+Publications of the Astronomical Society of Australia
+9
+CSO. MRC 0225–065 has a very symmetrical morphology
+according to our LBA images, suggesting it may not be
+interacting with its surrounding environment.
+• Linear size and turnover relation: We find MRC 0225–
+065 is entirely consistent with the linear size turnover rela-
+tion, a natural product of the youth scenario. Although, it
+can be reproduced in certain frustration models.
+• Host galaxy: Using the MIR colours reported in by WISE
+and the optical spectrum from SDSS, we identify the MRC 0225–
+065 as having an obscured AGN with moderate star forma-
+tion. Since the AGN does not dominate the entire MIR and
+optical emission, and there is still star formation present, it
+is possible the AGN has only recently been switched on
+and thus has not yet quenched all star formation in the
+galaxy, which is not surprising given the compact size of
+MRC 0225–065.
+• Magnetic field: Estimating the magnetic field using a
+purely SSA model and comparing it to the magnetic field
+calculated assuming equipartition are entirely consistent,
+suggesting the SSA model is a reasonable model for MRC 0225–
+065
+• Spectral ages: Using spectral modelling of the break fre-
+quency, we estimate the age of the radio emission (from
+the core and lobes) to be roughly 700 years, consistent with
+estimates of the age of PS sources in the youth scenario.
+• Dynamical ages: Using the linear size from our LBA im-
+ages and previous measurements of expansion velocity we
+estimate MRC 0225–065 has two major epochs of activity,
+one between 1000 to 7000 years ago and another more
+recently from 100 to 900 years ago. This is also consistent
+with previous estimates of the ages for young PS sources.
+Furthermore, due to the missing flux density at 8.3 GHz,
+this estimate should be considered an upper limit as the
+spectral indices for each component may be artificially
+steepened by the missing flux density.
+We therefore conclude, MRC 0225–065 is likely a young AGN
+and with the peak occurring due to SSA.
+Likewise, combining all information of PMN J0322–4820,
+we can also begin to create a unified picture that PMN J0322–
+4820 is a blazar. A summary of the evidence for this conclusion
+are:
+• Spectral variability: R21 identified PMN J0322–4820 as
+a variable source in and classified it as showing a changing
+spectral shape. The dramatic change in spectral shape in the
+megahertz regime on a timescale of ∼ 1 year is inconsistent
+with evolutionary models for PS sources and predicted
+variability due to RISS. The changing spectral shape is
+most easily explained by the dynamical nature of blazars.
+• Radio morphology: The high resolution image of PMN J0322–
+4820 using the LBA found it was still compact on mas scales.
+This is also entirely consistent with a blazar morphology,
+which appears compact due to orientation effects.
+• Linear size and turnover relation: PMN J0322–4820 sits
+well below the linear size and turnover relation typically
+associated with PS sources. This could either be because
+it is a frustrated source and is thus more compact than
+expected for it’s predicted age. However, more likely, is
+that the temporary peak detected with the MWA in 2014
+was a result of the variability of a blazar with effects like
+Doppler boosting influencing measurements and thus the
+spectral peak is unrelated to the source age or absorption
+mechanisms.
+• WISE MIR Colours: PMN J0322–4820 has WISE colours
+typically associated with elliptical galaxies and/or LERGs/BL
+Lac blazars.
+We therefore identify PMN J0322–4820 as a new blazar where
+the jets are oriented along the line-of-sight. However, PMN J0322–
+4820 was not in the ROMA-bzcat catalogue of γ-ray emitting
+blazars. This is potentially due to the steep spectrum at fre-
+quencies over 1 GHz where PMN J0322–4820 is too faint to be
+detected by traditional blazar searches. We suggest further ob-
+servations using higher frequency observations in the X-ray or
+γ regimes to search for any high frequency counterpart (Mas-
+saro et al., 2009, 2015). We conclude PMN J0322–4820 should
+not be included in any future population studies of PS sources
+as it is a contaminating blazar and not a genuine PS source.
+Furthermore, this highlights the possibility of a population
+of blazars with steep spectra at high frequencies (ν ≥ 1 GHz)
+that aren’t detected in traditional blazar searches and thus may
+be contaminating populations of PS sources. Low-frequency
+spectral variability thus presents as a new method for identify-
+ing blazar candidates.
+7.
+Conclusion
+We have sought to compare detections of spectral variabil-
+ity for two PS sources with small scale (∼mas) morphology
+and structures. The images produced using observations with
+the LBA have identified one resolved and one unresolved PS
+source. We have also combined our observations with archival
+observations of the host galaxies of our sources to provide
+evidence for either the youth or frustration scenario.
+We find PMN J0322–4820 is unresolved with the LBA at
+2.4 GHz, and pace an upper limit of the source size to be 148 pc,
+using a photometric redshift of 0.16. In R21, PMN J0322–4820
+was found to show a changing spectral shape and was presented
+as a blazar candidate. Comparing our compact morphology
+with the spectral variability of R21, we find PMN J0322–4820
+is consistent with a blazar classification, and suggest high fre-
+quency (X-ray or Gamma) to confirm.
+We resolve MRC 0225–065 into three components at both
+2.4 GHz and 8.3 GHz: a bright central region containing
+∼50% of the total flux density, and two fainter regions roughly
+equal distance from the central region. In R21 and R22,
+MRC 0225–065 was found to show low levels of variability
+with a constant spectral shape, and presented as showing vari-
+ability due to ISS from a compact morphology with resolved
+structure on mas scales. We find the projected linear size to
+be 430 pc, using a spectroscopic redshift of 0.445. Using spec-
+tral modelling, we calculate the magnetic field assuming a
+purely SSA model, and find it is in agreement with the mag-
+netic field calculated assuming equipartition. We therefore
+conclude MRC 0225–065 is a young CSO, with a PS classifi-
+
+10
+K. Ross et al.
+cation due to SSA. We found the core to have a spectral age of
+τspec = 700 ± 100 years, which is consistent with previous age
+estimates of young CSO sources of 101 – 105 years (Orienti
+et al., 2010; Orienti & Dallacasa, 2020). Furthermore, we use
+the spectral age of the core and the upper limit of core size to
+calculate and expected expansion velocity (assuming the simple
+relation speed = distance/time), and place an upper limit on
+the expansion velocity of the lobes to be v = 0.13c, well within
+previous measurements of expansion velocities for PS sources
+of 0.1c ≲ v ≲ 0.7c (Orienti & Dallacasa, 2020). Lastly, we
+use this to estimate the dynamical age of the outer lobes and
+estimate their age to be τdyn ≈ 5000 years, again, well within
+previous estimates of ages for young PS sources.
+Our findings highlight the advantage of spectral variability
+in identifying different milliarcsecond structures in PS sources
+traditionally acquired using VLBI. Furthermore, we have con-
+firmed the use of identifying contaminating sources displaying
+only a temporary spectral peak and present spectral variability
+as a new method for identifying steep spectrum blazars. We
+also suggest future observations of MRC 0225–065 to search
+for direct observations of expansion to better constraining the
+expansion velocity and age. We recommend observations of
+MRC 0225–065 with the VLBA for improved sensitivity and
+more u, v-coverage on short baselines to recover more flux
+density from extended structures. Likewise, with improved ac-
+curacy of the position for MRC 2236-454, we suggest another
+VLBI observation.
+Acknowledgement
+We thank the referees for their comments that improved the
+overall quality of this work. KR acknowledges a Doctoral
+Scholarship and an Australian Government Research Training
+Programme scholarship administered through Curtin Univer-
+sity of Western Australia. JRC thanks the Nederlandse Organ-
+isatie voor Wetenschappelijk Onderzoek (NWO) for support
+via the Talent Programme Veni grant. NHW is supported
+by an Australian Research Council Future Fellowship (project
+number FT190100231) funded by the Australian Government.
+The Long Baseline Array is part of the Australia Telescope
+National Facility https://ror.org/05qajvd42 which is funded by
+the Australian Government for operation as a National Facility
+managed by CSIRO. This work was supported by resources
+provided by the Pawsey Supercomputing Centre with funding
+from the Australian Government and the Government of West-
+ern Australia. LBA data was correlated at the Pawsey Super-
+computer Centre using the DiFX software (Deller et al., 2011).
+This scientific work uses data obtained from Inyarrimanha
+Ilgari Bundara/the Murchison Radio-astronomy Observatory.
+We acknowledge the Wajarri Yamaji People as the Traditional
+Owners and native title holders of the Observatory site. The
+Australian SKA Pathfinder is part of the Australia Telescope
+National Facility https://ror.org/05qajvd42 which is managed
+by CSIRO. Operation of ASKAP is funded by the Australian
+Government with support from the National Collaborative
+Research Infrastructure Strategy. ASKAP uses the resources of
+the Pawsey Supercomputing Centre. Establishment of ASKAP,
+the Murchison Radio-astronomy Observatory and the Pawsey
+Supercomputing Centre are initiatives of the Australian Gov-
+ernment, with support from the Government of Western Aus-
+tralia and the Science and Industry Endowment Fund. This
+paper includes archived data obtained through the CSIRO
+ASKAP Science Data Archive, CASDA (https://data.csiro.au).
+This research made use of NASA’s Astrophysics Data System,
+the VizieR catalog access tool, CDS, Strasbourg, France. We
+also make use of the IPYTHON package (Pérez & Granger,
+2007); SciPy (Virtanen et al., 2020); MATPLOTLIB, a PYTHON
+library for publication quality graphics (Hunter, 2007); AS-
+TROPY, a community-developed core PYTHON package for
+astronomy (Astropy Collaboration et al., 2013; Price-Whelan
+et al., 2018); PANDAS, a data analysis and manipulation PYTHON
+module (pandas development team, 2020; Wes McKinney,
+2010); and NUMPY (van der Walt et al., 2011). We also made
+extensive use of the visualisation and analysis packages DS9f
+and Topcat (Taylor, 2005). This work was compiled in the
+useful online LATEX editor Overleaf.
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+Generalized PTR: User-Friendly Recipes for Data-Adaptive
+Algorithms with Differential Privacy
+Rachel Redberg, Yuqing Zhu, Yu-Xiang Wang
+University of California, Santa Barbara
+{rredberg, yuqingzhu, yuxiangw}@ucsb.edu
+January 3, 2023
+Abstract
+The “Propose-Test-Release” (PTR) framework [Dwork and Lei, 2009] is a classic recipe for
+designing differentially private (DP) algorithms that are data-adaptive, i.e. those that add less
+noise when the input dataset is “nice”. We extend PTR to a more general setting by privately
+testing data-dependent privacy losses rather than local sensitivity, hence making it applicable
+beyond the standard noise-adding mechanisms, e.g. to queries with unbounded or undefined
+sensitivity. We demonstrate the versatility of generalized PTR using private linear regression
+as a case study. Additionally, we apply our algorithm to solve an open problem from “Private
+Aggregation of Teacher Ensembles (PATE)” [Papernot et al., 2017, 2018] — privately releasing
+the entire model with a delicate data-dependent analysis.
+1
+Introduction
+The guarantees of differential privacy (DP) [Dwork et al., 2006] are based on worst-case outcomes
+across all possible datasets. A common paradigm is therefore to add noise scaled by the global
+sensitivity of a query f, i.e. the maximum change in f between any pair of neighboring datasets.
+A given dataset X might have a local sensitivity that is much smaller than the global sensitivity, in
+which case we can hope to add a smaller amount of noise (calibrated to the local rather than the
+global sensitivity) while achieving the same privacy guarantee. However, this must not be undertaken
+naïvely – the local sensitivity is a dataset-dependent function and so calibrating noise to the local
+sensitivity could leak information about the dataset [Nissim et al., 2007].
+The “Propose-Test-Release” (PTR) framework [Dwork and Lei, 2009] resolves this issue by introducing
+a test to privately check whether a proposed bound on the local sensitivity is valid. Only if the
+test “passes” is the output released with noise calibrated to the proposed bound on the local
+sensitivity.
+PTR is a powerful and flexible tool for designing data-adaptive DP algorithms, but it has several
+limitations. First, it applies only to noise-adding mechanisms which calibrate noise according to the
+sensitivity of a query. Second, the test in “Propose-Test-Release” is computationally expensive for all
+but a few simple queries such as privately releasing the median or mode. Third, while some existing
+works [Decarolis et al., 2020, Kasiviswanathan et al., 2013, Liu et al., 2021] follow the approach of
+testing “nice” properties of a dataset before exploiting these properties in a private release to PTR 1,
+1We refer to these as PTR-like methods.
+1
+arXiv:2301.00301v1  [cs.LG]  31 Dec 2022
+
+there has not been a systematic recipe for discovering which properties should be tested.
+In this paper, we propose a generalization of PTR which addresses these limitations. The centerpiece
+of our framework is a differentially private test on the data-dependent privacy loss. This test does
+not directly consider the local sensitivity of a query and is therefore not limited to additive noise
+mechanisms. Moreover, in many cases, the test can be efficiently implemented by privately releasing
+a high-probability upper bound, thus avoiding the need to search an exponentially large space of
+datasets. Furthermore, the derivation of the test itself often spells out exactly what properties of the
+input dataset need to be checked, which streamlines the design of data-adaptive DP algorithms.
+Our contributions are summarized as follows:
+1. We propose a generalization of PTR which can handle algorithms beyond noise-adding
+mechanisms. Generalized PTR allows us to plug in any data-dependent DP analysis to
+construct a high-probability DP test that adapts to favorable properties of the input dataset –
+without painstakingly designing each test from scratch.
+2. We demonstrate that many existing examples of PTR and PTR-like algorithms can be unified
+under the generalized PTR framework, sometimes resulting in a tighter analysis (see an
+example of report-noisy-max in Sec A.1).
+3. We show that one can publish a DP model through privately upper-bounding a one-dimensional
+statistic — no matter how complex the output space of the mechanism is. We apply this result
+to solve an open problem from PATE [Papernot et al., 2017, 2018].
+4. Our results broaden the applicability of private hyper-parameter tuning [Liu and Talwar, 2019,
+Papernot and Steinke, 2021] in enabling joint-parameter selection of DP-specific parameters
+(e.g., noise level) and native parameters of the algorithm (e.g., learning rate, regularization
+weight), which may jointly affect the data-dependent DP losses.
+2
+Related Work
+Data-dependent DP algorithms. Privately calibrating noise to the local sensitivity is a well-
+studied problem. One approach is to add noise calibrated to the smooth sensitivity [Nissim et al.,
+2007], an upper bound on the local sensitivity which changes slowly between neighboring datasets.
+An alternative to this – and the focus of our work – is Propose-Test-Release (PTR) [Dwork and
+Lei, 2009], which works by calculating the distance Dβ(X) to the nearest dataset to X whose local
+sensitivity violates a proposed bound β. The PTR algorithm then adds noise to Dβ(X) before
+testing whether this privately computed distance is sufficiently large.
+PTR spin-offs abound. Notable examples include stability-based methods [Thakurta and Smith,
+2013] (stable local sensitivity of 0 near the input data) and privately releasing upper bounds of local
+sensitivity [Kasiviswanathan et al., 2013, Liu et al., 2021, Decarolis et al., 2020]. We refer readers to
+Chapter 3 of Vadhan [2017] for a concise summary of these classical results. Recent work [Wang
+et al., 2022] has provided Rényi DP bounds for PTR and demonstrated its applications to robust
+DP-SGD. Our work (see Section 5.2) also considers applications of PTR in data-adaptive private
+deep learning: Instead of testing the local sensitivity of each gradient step as in Wang et al. [2022],
+our PTR-based PATE algorithm tests the data-dependent privacy loss as a whole.
+Liu et al. [2021] proposed a new variant called High-dimensional Propose-Test-Release (HPTR). HPTR
+provides a systematic way of solving DP statistical estimation problems by using the exponential
+2
+
+mechanism (EM) with carefully constructed scores based on certain one-dimensional robust statistics,
+which have stable local sensitivity bounds. HPTR focuses on designing data-adaptive DP mechanisms
+from scratch; our method, in contrast, converts existing randomized algorithms (including EM and
+even some that do not satisfy DP) into those with formal DP guarantees. Interestingly, our proposed
+method also depends on a one-dimensional statistic of direct interest: the data-dependent privacy
+loss.
+Data-dependent DP losses. The flip side of data-dependent DP algorithms is the study of
+data-dependent DP losses [Papernot et al., 2018, Soria-Comas et al., 2017, Wang, 2017], which fix
+the randomized algorithm but parameterize the resulting privacy loss by the specific input dataset.
+For example: In the simple mechanism that adds Laplace noise with parameter b, data-dependent
+DP losses are ϵ(X) = ∆LS(X)/b. The data-dependent DP losses are often much smaller than the DP
+loss, but they themselves depend on the data and thus may reveal sensitive information; algorithms
+satisfying a data-dependent privacy guarantee are not formally DP with guarantees any smaller
+than that of the worst-case. Existing work has considered privately publishing these data-dependent
+privacy losses [Papernot et al., 2018, Redberg and Wang, 2021], but notice that privately publishing
+these losses does not improve the DP parameter of the given algorithm. Part of our contribution is
+to resolve this conundrum by showing that a simple post-processing step of the privately released
+upper bound of ϵ(Data) gives a formal DP algorithm.
+Private hyper-parameter tuning. Our work has a nice connection with private hyper-parameter
+tuning. Prior work [Liu and Talwar, 2019, Papernot and Steinke, 2021] requires each candidate
+configuration to be released with the same DP (or Rényi DP) parameter set. Another hidden
+assumption is that the parameters must not be privacy-correlated (i.e., parameter choice will not
+change the privacy guarantee). Otherwise we need to use the largest DP bound across all candidates.
+For example, Liu and Talwar [2019] show that if each mechanism (instantiated with one group of
+hyper-parameters) is (ϵ, 0)-DP, then running a random number of mechanisms and reporting the best
+option satisfies (3ϵ, 0)-DP. Our work directly generalizes the above results by (1) considering a wide
+range of hyper-parameters, either privacy-correlated or not; and (2) requiring only that individual
+candidates to have a testable data-dependent DP.
+3
+Preliminaries
+Datasets X, X′ ∈ X are neighbors if they differ by no more than one datapoint – i.e., X ≃ X′ if
+d(X, X′) ≤ 1. We will define d(·) to be the number of coordinates that differ between two datasets
+of the same size n: d(X, Y ) = #{i ∈ [n] : Xi ̸= Yi}.
+We use || · || to denote the radius of the smallest Euclidean ball that contains the input set, e.g.
+||X|| = supx∈X ||x||.
+The parameter φ denotes the privacy parameters associated with a mechanism (e.g. noise level,
+regularization). Mφ is a mechanism parameterized by φ. For mechanisms with continuous output
+space, we will take Pr[M(X) = y] to be the probability density function of M(X) at y.
+Definition 3.1 (Differential privacy [Dwork et al., 2006]). Fix ϵ, δ ≥ 0. A randomized algorithm
+M : X → S satisfies (ϵ, δ)-DP if for all neighboring datasets X ≃ X′ and for all measurable sets
+S ⊂ S,
+Pr
+�
+M(X) ∈ S
+�
+≤ eϵPr
+�
+M(X′) ∈ S
+�
++ δ.
+Suppose we wish to privately release the output of a real-valued function f : X → R. We can do so
+3
+
+by calculating the global sensitivity ∆GS, calibrating the noise scale to the global sensitivity and
+then adding sampled noise to the output.
+Definition 3.2 (Local / Global sensitivity). The local ℓ∗-sensitivity of a function f is defined as
+∆LS(X) = max
+X≃X′ ||f(X) − f(X′)||∗ and the global sensitivity of f is ∆GS = supX ∆LS(X).
+3.1
+Propose-Test-Release
+Calibrating the noise level to the local sensitivity ∆LS(X) of a function would allow us to add less
+noise and therefore achieve higher utility for releasing private queries. However, the local sensitivity
+is a data-dependent function and naïvely calibrating the noise level to ∆LS(X) will not satisfy
+DP.
+PTR resolves this issue in a three-step procedure: propose a bound on the local sensitivity, privately
+test that the bound is valid (with high probability), and if so calibrate noise according to the bound
+and release the output.
+PTR privately computes the distance Dβ(X) between the input dataset X and the nearest dataset
+X′′ whose local sensitivity exceeds the proposed bound β:
+Dβ(X) = min
+X′′ {d(X, X′′) : ∆LS(X′′) > β}.
+Algorithm 1 Propose-Test-Release [Dwork and Lei, 2009]
+1: Input: Dataset X; privacy parameters ϵ, δ; proposed bound β on ∆LS(X); query function
+f : X → R.
+2: if Dβ(X) + Lap
+� 1
+ϵ
+�
+≤ log(1/δ)
+ϵ
+then output ⊥,
+3: else release f(X) + Lap
+�
+β
+ϵ
+�
+.
+Theorem 3.3. Algorithm 1 satisfies (2ϵ, δ)-DP. [Dwork and Lei, 2009]
+Rather than proposing an arbitrary threshold β, one can also privately release an upper bound of
+the local sensitivity and calibrate noise according to this upper bound. This was used for node DP
+in graph statistics [Kasiviswanathan et al., 2013], and for fitting topic models using spectral methods
+[Decarolis et al., 2020].
+4
+Generalized PTR
+This section introduces the generalized PTR framework. We first formalize the notion of data-
+dependent differential privacy that conditions on an input dataset X.
+Definition 4.1 (Data-dependent privacy). Suppose we have δ > 0 and a function ϵ : X → R. We
+say that mechanism M satisfies (ϵ(X), δ) data-dependent DP2 for dataset X if for all possible output
+sets S and neighboring datasets X′,
+Pr
+�
+M(X) ∈ S
+�
+≤ eϵ(X)Pr
+�
+M(X′) ∈ S
+�
++ δ,
+Pr
+�
+M(X′) ∈ S
+�
+≤ eϵ(X)Pr
+�
+M(X) ∈ S
+�
++ δ.
+2We will sometimes write that M(X) satisfies ϵ(X) data-dependent DP with respect to δ.
+4
+
+In generalized PTR, we propose a value φ for the randomized algorithm M, which could be a noise
+scale or regularization parameter – or a set including both. For example, φ = (λ, γ) in Example 4.4.
+We then say that Mφ is the mechanism M parameterized by φ, and ϵφ(X) its data-dependent
+DP.
+The following example illustrates how to derive the data-dependent DP for a familiar friend – the
+Laplace mechanism.
+Example 4.2. ( Data-dependent DP of Laplace Mechanism.) Given a function f : X → R, we will
+define
+Mφ(X) = f(X) + Lap (φ) .
+We then have
+log Pr[Mφ(X) = y]
+Pr[Mφ(X′) = y] ≤ |f(X) − f(X′)|
+φ
+.
+Maximizing the above calculation over all possible outputs y and using Definition 4.1,
+ϵφ(X) =
+max
+X′:X′≃X
+|f(X) − f(X′)|
+φ
+= ∆LS(X)
+φ
+.
+The data-dependent DP ϵφ(X) is a function of both the dataset X and the parameter φ. Maximizing
+ϵφ(X) over X recovers the standard DP guarantee of running M with parameter φ.
+Algorithm 2 Generalized Propose-Test-Release
+1: Input: Dataset X; mechanism Mφ : X → R and its privacy budget ϵ, δ; (ˆϵ, ˆδ)-DP test T ; false
+positive rate ≤ δ′; data-dependent DP function ϵφ(·) w.r.t. δ.
+2: if not T (X) then output ⊥,
+3: else release θ = Mφ(X).
+Theorem 4.3 (Privacy guarantee of generalized PTR). Consider a proposal φ and a data-dependent
+DP function ϵφ(X) w.r.t. δ. Suppose that we have an (ˆϵ, ˆδ)-DP test T : X → {0, 1} such that when
+ϵφ(X) > ϵ,
+T (X) =
+�
+0 with probability 1 − δ′,
+1 with probability δ′.
+Then Algorithm 2 satisfies (ϵ + ˆϵ, δ + ˆδ + δ′)-DP.
+Proof sketch. There are three main cases to consider:
+1. We decide not to run Mφ.
+2. We decide to run Mφ and ϵφ(X) > ϵ;
+3. We decide to run Mφ and ϵφ(X) ≤ ϵ.
+5
+
+In the first case, the decision to output ⊥ is post-processing of an (ˆϵ, ˆδ)-DP mechanism and inherits
+its privacy guarantees. The second case occurs when the (ˆϵ, ˆδ)-DP test "fails" (produces a false
+positive) and occurs with probability at most δ′. The third case is a composition of an (ˆϵ, ˆδ)-DP
+algorithm and an (ϵ, δ)-DP algorithm.
+Generalized PTR is a strict generalization of Propose-Test-Release. For some function f, define Mφ
+and T as follows:
+Mφ(X) = f(X) + Lap(φ);
+T (X) =
+�
+0
+if Dβ(X) + Lap
+� 1
+ϵ
+�
+> log(1/δ)
+ϵ
+,
+1
+otherwise.
+Notice that our choice of parameterization is φ = β
+ϵ , where φ is the scale of the Laplace noise. In
+other words, we know from Example 4.2 that ϵφ(X) > ϵ exactly when ∆LS(X) > β.
+For noise-adding mechanisms such as the Laplace mechanism, the sensitivity is proportional to the
+privacy loss (in both the global and local sense, i.e. ∆GS ∝ ϵ and ∆LS ∝ ϵ(X)). Therefore for these
+mechanisms the only difference between privately testing the local sensitivity (Algorithm 1) and
+privately testing the data-dependent DP (Theorem 4.3) is a change of parameterization.
+4.1
+Limitations of local sensitivity
+Why do we want to generalize PTR beyond noise-adding mechanisms? Compared to classic PTR, the
+generalized PTR framework allows us to be more flexible in both the type of test conducted and also
+the type of mechanism whose output we wish to release. For many mechanisms, the local sensitivity
+either does not exist or is only defined for specific data-dependent quantities (e.g., the sensitivity of
+the score function in the exponential mechanism) rather than the mechanism’s output.
+The following example illustrates this issue.
+Example 4.4 (Private posterior sampling). Let M : X × Y → Θ be a private posterior sampling
+mechanism [Minami et al., 2016, Wang et al., 2015, Gopi et al., 2022] for approximately minimizing
+FX(θ).
+M samples θ ∼ P(θ) ∝ e−γ(FX(θ)+0.5λ||θ||2) with parameters γ, λ. Note that γ, λ cannot be appro-
+priately chosen for this mechanism to satisfy DP without going through a sensitivity calculation of
+arg min FX(θ). In fact, the global and local sensitivity of the minimizer is unbounded even in linear
+regression problems, i.e when FX(θ) = 1
+2||y − Xθ||2.
+Output perturbation algorithms do work for the above problem when we regularize, but they
+are known to be suboptimal in theory and in practice [Chaudhuri et al., 2011]. In Section 5.1
+we demonstrate how to apply generalized PTR to achieve a data-adaptive posterior sampling
+mechanism.
+Even in the cases of noise-adding mechanisms where PTR seems to be applicable, it does not lead to
+a tight privacy guarantee. Specifically, by an example of privacy amplification by post-processing
+(Example A.1 in the appendix), we demonstrate that the local sensitivity does not capture all
+sufficient statistics for data-dependent privacy analysis and thus is loose.
+6
+
+4.2
+Which φ to propose
+The main limitation of generalized PTR is that one needs to “propose” a good guess of parameter φ.
+Take the example of φ being the noise level in a noise-adding mechanism. Choosing too small a φ
+will result in a useless output ⊥, while choosing too large a φ will add more noise than necessary.
+Finding this ’Goldilocks’ φ might require trying out many different possibilities – each of which will
+consume privacy budget.
+This section introduces a method to jointly tune privacy parameters (e.g., noise scale) along with
+parameters related only to the utility of an algorithm (e.g., learning rate or batch size in stochastic
+gradient descent) – while avoiding the ⊥ output.
+Algorithm 3 takes a list of parameters as input, runs generalized PTR with each of the parameters,
+and returns the output with the best utility. We show that the privacy guarantee with respect to ϵ
+is independent of the number of φ that we try.
+Formally, let φ1, ..., φk be a set of hyper-parameters and ˜θi ∈ {⊥, Range(M)} denotes the output of
+running generalized PTR on a private dataset X with φi. Let Xval be a public validation set and
+q(˜θi) be the score of evaluating ˜θi with Xval (e.g., validation accuracy). The goal is to select a pair
+(˜θi, φi) such that DP model ˜θi maximizes the validation score.
+The generalized PTR framework with privacy calibration is described in Algorithm 3. The privacy
+guarantee of Algorithm 3 is an application of Liu and Talwar [2019].
+Algorithm 3 PTR with hyper-parameter selection
+1: Input: Privacy budget per PTR algorithm (ϵ∗, δ∗), cut-off T, parameters φ1:k, flipping probability
+τ and validation score function q(·).
+2: Initialize the set S = ∅.
+3: Draw G from a geometric distribution Dτ and let ˆT = min(T, G).
+4: for i = 1 ,..., ˆT do
+5:
+pick a random φi from φ1:k.
+6:
+evaluate φi: (˜θi, q(˜θi)) ← Algorithm 2(φi, (ϵ∗, δ∗)).
+7:
+S ← S ∪ {˜θi, q(˜θi)}.
+8: end for
+9: Output the highest scored candidate from S.
+Theorem 4.5 ( Theorem 3.4 Liu and Talwar [2019] ). Fix any τ ∈ [0, 1], δ2 > 0 and let T = 1
+τ log 1
+δ2 .
+If each oracle access to Algorithm 2 is (ϵ∗, δ∗)-DP, then Algorithm 3 is (3ϵ∗ +3
+√
+2δ∗,
+√
+2δ∗T +δ2)-DP.
+The theorem implies that one can try a random number of φ while paying a constant ϵ. In practice,
+we can roughly set τ =
+1
+10k so that the algorithm is likely to test all k parameters. We emphasize
+that the privacy and the utility guarantee (stated in the appendix) is not our contribution. But the
+idea of applying generalized PTR to enforce a uniform DP guarantee over all choices of parameters
+with a data-dependent analysis is new, and in our opinion, significantly broadens the applicability to
+generic hyper-parameter tuning machinery from Liu and Talwar [2019].
+4.3
+Construction of the DP test
+Classic PTR uses the Laplace mechanism to construct a differentially private upper bound of Dβ(X),
+the distance from input dataset X to the closest dataset whose local sensitivity exceeds the proposed
+7
+
+bound β. The tail bound of the Laplace distribution then ensures that if Dβ(X) = 0 (i.e. if
+∆LS(X) > β), then the output will be released with only a small probability δ.
+The following theorem shows that we could instead use a differentially private upper bound of the
+data-dependent DP ϵφ(X) in order to test whether to run the mechanism Mφ.
+Theorem 4.6 (Generalized PTR with private upper bound). Suppose we have a differentially private
+upper bound of ϵφ(X) w.r.t. δ such that with probability at least 1 − δ′, ϵP
+φ (X) > ϵφ(X). Further
+suppose we have an (ˆϵ, ˆδ)-DP test T such that
+T(X) =
+�
+1
+if ϵP
+φ (X) < ϵ,
+0
+otherwise.
+Then Algorithm 2 is (ϵ + ˆϵ, δ + ˆδ + δ′)-DP.
+In Section 5.2, we demonstrate that one can upper bound the data-dependent DP through a
+modification of the smooth sensitivity framework applied on ϵφ(X). Moreover, in Section 5.1 we
+provide a direct application of Theorem 4.6 with private linear regression by making use of the
+per-instance DP technique [Wang, 2017].
+The applications in Section 5 are illustrative of two distinct approaches to constructing the DP test
+for generalized PTR:
+1. Private sufficient statistics release (used in the private linear regression example of Section 5.1)
+specifies the data-dependent DP as a function of the dataset and privately releases each
+data-dependent component.
+2. The second approach (used in the PATE example of Section 5.2) uses the smooth sensitivity
+framework to privately release the data-dependent DP as a whole, and then construct a
+high-confidence test using the Gaussian mechanism.
+These two approaches cover most of the scenarios arising in data-adaptive analysis. For example,
+in the appendix we demonstrate the merits of generalized PTR in handling data-adaptive private
+generalized linear models (GLMs) using private sufficient statistics release. Moreover, sufficient
+statistics release together with our private hyper-parameter tuning (Algorithm 3) can be used to
+construct data-adaptive extensions of DP-PCA and Sparse-DP-ERM (see details in the future work
+section).
+5
+Applications
+In this section, we put into action our approaches to construct the DP test and provide applications
+in private linear regression and PATE.
+5.1
+Private Linear Regression
+Theorem 5.1 ([Wang, 2017]). For input data X ∈ X and Y ∈ Y, define the following:
+• λmin(X) denotes the smallest eigenvalue of XT X;
+• ||θ∗
+λ|| is the magnitude of the solution θ∗
+λ = (XT X + λI)−1XT Y ;
+• and L(X, y) := ||X||(||X||||θ∗
+λ|| + ||Y||) is the local Lipschitz constant, denoted L in brief.
+8
+
+10
+1
+100
+10
+2
+6 × 10
+3
+2 × 10
+2
+3 × 10
+2
+4 × 10
+2
+MSE
+UCI Bike dataset (n = 17379, d = 17)
+AdaOPS
+non-private
+OutPert
+OPS
+OPS with PTR
+(a) Bike dataset
+10
+1
+100
+2 × 10
+2
+3 × 10
+2
+4 × 10
+2
+6 × 10
+2
+MSE
+UCI elevators dataset (n = 8752, d = 18)
+AdaOPS
+non-private
+OutPert
+OPS
+OPS with PTR
+(b) Elevators dataset
+Figure 1: Differentially private linear regression algorithms on UCI datasets. y-axis reports the MSE
+error with confidence intervals. ϵ is evaluated with δ = 1e − 6.
+For brevity, denote λ∗ = λ + λmin(X). The algorithm used in Example 4.4 with parameter φ = (λ, γ)
+obeys (ϵφ(Z), δ) data-dependent DP for each dataset Z = (X, Y ) with ϵφ(Z) equal to
+�
+γL2 log(2/δ)
+λ∗
++
+γL2
+2(λ∗ + ||X||2) + 1 + log(2/δ)||X||2
+2(λ∗)
+.
+Notice that the data-dependent DP is a function of (λmin, L, ||θ∗
+λ||, λ, γ), where (λmin, L, ||θ∗
+λ||) are
+data-dependent quantities. One can apply the generalized PTR framework as in the following
+example.
+Example 5.2 (OPS with PTR). We demonstrate here how to apply generalized PTR to the one-
+posterior sample (OPS) algorithm, a differentially private mechanism which outputs one sample from
+the posterior distribution of a Bayesian model with bounded log-likelihood.
+• Propose φ = (λ, γ).
+• Based on (λ, γ), differentially privately release λmin, ||θ∗
+λ||, L with privacy budget (ϵ, δ/2).
+• Condition on a high probability event (with probability at least 1 − δ/2) of λmin, ||θ∗
+λ||, L, test if
+ϵP
+φ (X) is smaller than the predefined privacy budget (ˆϵ, ˆδ), where ϵP
+φ (X) denotes the sanitized
+data-dependent DP.
+• Based on the outcome of the test, decide whether to release θ ∝ e− γ
+2 ||Y −Xθ||2+λ||θ||2.
+Theorem 5.3. The algorithm outlined in Example 5.2 satisfies (ϵ + ˆϵ, δ + ˆδ)-DP.
+The main idea of the above algorithm boils down to privately releasing all data-dependent quantities
+in data-dependent DP, constructing high-probability confidence intervals of these quantities, and
+then deciding whether to run the mechanism M with the proposed parameters. We defer the details
+of the privacy calibration of data-dependent quantities to the appendix.
+One may ask why we cannot directly tune privacy parameters (λ, γ) based on the sanitized data-
+dependent DP. This is because, in many scenarios, data-dependent quantities depend on the choice of
+privacy parameters, e.g., ||θ∗
+λ|| is a complicated function of λ. Thus, the optimization on λ becomes
+9
+
+a circular problem — to solve λ, we need to sanitize ||θ∗
+λ||, which needs to choose a λ to begin with.
+Alternatively, generalized PTR provides a clear and flexible framework to test the validity of privacy
+parameters adapted to the dataset.
+Remark 5.4. The above “circular” issue is even more serious for generalized linear models (GLMs)
+beyond linear regression. The data-dependent DP there involves a local strong-convexity parameter,
+a complex function of the regularizer λ and we only have zeroth-order access to. In the appendix,
+we demonstrate how to apply generalized PTR to provide a generic solution to a family of private
+GLMs where the link function satisfies a self-concordance assumption.
+We next apply Algorithm 3 for Example 5.2 with UCI regression datasets. Standard z-scoring is
+applied and each data point is normalize with a Euclidean norm of 1. We consider (60%, 10%, 30%)
+splits for training, validation and testing test.
+Baselines
+• Output Perturbation (Outpert) [Chaudhuri et al., 2011]: θ = (XT X + λI)−1XT y. Release
+ˆθ = θ + b with an appropriate λ, where b is a Gaussian random vector.
+• Posterior sampling (OPS). Sample ˆθ ∼ P(θ) ∝ e−γ(F(θ)+0.5λ||θ||2) with parameters γ, λ.
+• Adaptive posterior sampling (AdaOPS) [Wang, 2018]. Run OPS with (λ, γ) chosen adaptively
+according to the dataset.
+Outpert and OPS serve as two non-adaptive baselines. In particular, we consider OPS-Balanced [Wang,
+2018], which chooses λ to minimize a data-independent upper bound of empirical risk and dominates
+other OPS variants. AdaOPS is one state-of-the-art algorithm for adaptive private regression, which
+automatically chooses λ by minimizing an upper bound of the data-dependent empirical risk.
+We implement OPS-PTR as follows: propose a list of λ through grid search (we choose k = 30 and λ
+ranges from [2.5, 2.510] on a logarithmic scale); instantiate Algorithm 3 with τ = 0.1k, T = 1
+τ log(1/δ2)
+and δ2 = 1/2δ; calibrate γ to meet the privacy requirement for each λ. sample ˆθ using (λ, γ) and
+return the one with the best validation accuracy. Notice that we use a “no ⊥” variant of Algorithm 2
+as the calibration of γ is clear given a fixed λ and privacy budget (see more details in the appendix).
+We can propose various combinations of (λ, γ) for more general applications.
+Figure 1 demonstrates how the MSE error of the linear regression algorithms varies with the privacy
+budget ϵ. OutPert suffers from the large global sensitivity of output θ. OPS performs well but does
+not benefit from the data-dependent quantities. AdaOPS is able to adaptively choose (λ, γ) based
+on the dataset, but suffers from the estimation error of the data-dependent empirical risk. On the
+other hand, OPS-PTR selects a (λ, γ) pair that minimizes the empirical error on the validation set
+directly, and the privacy parameter γ adapts to the dataset thus achieving the best result.
+5.2
+PATE
+In this section, we apply the generalized PTR framework to solve an open problem from the Private
+Aggregation of Teacher Ensembles (PATE) [Papernot et al., 2017, 2018] — privately publishing the
+entire model through privately releasing data-dependent DP losses. Our algorithm makes use of the
+smooth sensitivity framework [Nissim et al., 2007] and the Gaussian mechanism to construct a high-
+probability test of the data-dependent DP. The one-dimensional statistical nature of data-dependent
+DP enables efficient computations under the smooth sensitivity framework. Thus, this approach is
+generally applicable for other private data-adaptive analysis beyond PATE.
+10
+
+PATE is a knowledge transfer framework for model-agnostic private learning. In this framework, an
+ensemble of teacher models is trained on the disjoint private data and uses the teachers’ aggregated
+consensus answers to supervise the training of a “student” model agnostic to the underlying machine-
+learning algorithms. By publishing only the aggregated answers and by the careful analysis of the
+“consensus”, PATE has become a practical technique in recent private model training.
+The tight privacy guarantee of PATE heavily relies on a delicate data-dependent DP analysis, for
+which the authors of PATE use the smooth sensitivity framework to privately publish the data-
+dependent privacy cost. However, it remains an open problem to show that the released model is DP
+under data-dependent analysis. Our generalized PTR resolves this gap by carefully testing a private
+upper bound of the data-dependent privacy cost. Our algorithm is fully described in Algorithm 4,
+where the modi��cation over the original PATE framework is highlighted in blue.
+Algorithm 4 takes the input of privacy budget (ϵ′, ˆϵ, δ), unlabeled public data x1:T and K teachers’
+predictions on these data. The parameter ϵ denotes the privacy cost of publishing the data-dependent
+DP and ϵ′ is the predefined privacy budget for testing. nj(xi) denotes the the number of teachers
+that agree on label j for xi and C denotes the number of classes. The goal is to privately release a
+list of plurality outcomes — argmaxj∈[C]nj(xi) for i ∈ [T] — and use these outcomes to supervise
+the training of a “student” model in the public domain. The parameter σ1 denotes the noise scale
+for the vote count.
+In their privacy analysis, Papernot et al. [2018] compute the data-dependent RDPσ1(α, X) of labeling
+the entire group of student queries. RDPσ1(α, X) can be orders of magnitude smaller than its data-
+independent version if there is a strong agreement among teachers. Note that RDPσ1(α, X) is a
+function of the RDP order α and the dataset X, analogous to our Definition 4.1 but subject to
+RDP [Mironov, 2017].
+Theorem 5.5 ([Papernot et al., 2018]). If the top three vote counts of xi are n1 > n2 > n3 and
+n1 − n2, n2 − n3 ≫ σ1, then the data-dependent RDP of releasing argmaxj{nj + N(0, σ2
+1)} satisfies
+(α, exp{−2α/σ2
+1}/α)-RDP and the data-independent RDP (using the Gaussian mechanism) satisfies
+(α, α
+σ2
+1 )-RDP.
+Algorithm 4 PATE with generalized PTR
+1: Input: Unlabeled public data x1:T , aggregated teachers prediction n(·), privacy parameter
+ˆϵ, ϵ′, δ, noisy parameter σ1.
+2: Set α = 2 log(2/δ)
+ˆϵ
++ 1, σs = σ2 =
+�
+3α+2
+ˆϵ
+, δ2 = δ/2, smoothness parameter β = 0.2
+α .
+3: Compute noisy labels: yip ← argmaxj∈[C]{nj(xi) + N(0, σ2
+1)} for all i ∈ [1 : T].
+4: RDPσ1(α, X) ← data-dependent RDP at the α-th order.
+5: SSβ(X) ← the smooth sensitivity of RDPupper
+σ1
+(α, X).
+6: Privately release µ := log(SSβ(X)) + β · N(0, σ2
+2) +
+�
+2 log(2/δ2) · σ2 · β
+7: RDPupper
+σ1
+(α) ← an upper bound of data-dependent RDP through Lemma 5.6.
+8: ϵσ1 ← DP guarantee converted from RDPupper
+σ1
+(α).
+9: If ϵ′ ≥ ϵσ1 return a student model trained using (x1:T ; yp
+1:T ).
+10: Else return ⊥.
+However, RDPσ1(α, X) is data-dependent and thus cannot be revealed. The authors therefore
+privately publish the data-dependent RDP using the smooth sensitivity framework [Nissim et al., 2007].
+The smooth sensitivity calculates a smooth upper bound on the local sensitivity of RDPσ1(α, X),
+11
+
+15
+20
+25
+30
+35
+40
+45
+50
+Noise scale 
+1
+1
+2
+3
+4
+5
+ Gaussian mechanism
+PATE-PTR ( +
+1)
+data-dependent DP (non-private)
+(a) High consensus and strong data-dependent DP
+15
+20
+25
+30
+35
+40
+45
+50
+Noise scale 
+1
+1
+2
+3
+4
+5
+ Gaussian mechanism
+PATE-PTR ( +
+1)
+data-dependent DP (non-private)
+(b) Low consensus and low data-dependent DP
+Figure 2: Privacy and utility tradeoffs with PATE. When σ1 is aligned, three algorithms provide the
+same utility. y-axis plots the privacy cost of labeling T = 200 public data with δ = 10−5. The left
+figure considers the high-consensus case, where the data-adaptive analysis is preferred.
+denoted as SSβ(X), such that SSβ(X) ≤ eβSSβ(X′) for any neighboring dataset X and X′. By
+adding Gaussian noise scaled by the smooth sensitivity (i.e., release ϵσ1(α, X) + SSβ(X) · N(0, σ2
+s)),
+the privacy cost is safely published.
+Unlike most noise-adding mechanisms, the standard deviation σs cannot be published since SSβ(X)
+is a data-dependent quantity. Moreover, this approach fails to provide a valid privacy guarantee
+of the noisy labels obtained through the PATE algorithm, as the published privacy cost could be
+smaller than the real privacy cost. Our solution in Algorithm 4 looks like the following:
+• Privately release an upper bound of the smooth sensitivity SSβ(X) with eµ.
+• Conditioned on a high-probability event of eµ, publish the data-dependent RDP with RDPupper
+σ1
+(α).
+• Convert RDPupper
+σ1
+(α) back to the standard DP guarantee using RDP to DP conversion at δ/2.
+• Test if the converted DP is above the predefined budget ϵ′.
+The following lemma states that RDPupper
+σ1
+(α) is a valid upper bound of the data-dependent
+RDP.
+Lemma 5.6 (Private upper bound of data-dependent RDP). We are given a RDP function
+RDP(α, X) and a β-smooth sensitivity bound SS(·) of RDP(α, X). Let µ (defined in Algorithm 4)
+denote the private release of log(SSβ(X)). Let the (β, σs, σ2)-GNSS mechanism be
+RDPupper(α):=RDP(α,X)+SSβ(X)·N(0,σ2
+s)+σs
+�
+2 log( 2
+δ2 )eµ
+Then, the release of RDPupper(X) satisfies (α, 3α+2
+2σ2s )-RDP for all 1 < α <
+1
+2β; w.p. at least 1 − δ2,
+RDPupper(α) is an upper bound of RDP(α, X).
+The proof (deferred to the appendix) makes use of the facts that: (1) the log of SSβ(X) has a
+bounded global sensitivity β through the definition of smooth sensitivity; (2) releasing RDPσ1(α, X)+
+SSβ(X) · N(0, σ2
+s) is (α, α+1
+σ2s )-RDP (Theorem 23 from Papernot et al. [2018]).
+Now, we are ready to state the privacy guarantee of Algorithm 4.
+12
+
+Theorem 5.7. Algorithm 4 satisfies (ϵ′ + ˆϵ, δ)-DP.
+In the proof, the choice of α ensures that the cost of the δ/2 contribution (used in the RDP-to-DP
+conversion) is roughly ˆϵ/2. Then the release of RDPupper
+σ1
+(α) with σs =
+�
+2+3α
+ˆϵ
+accounts for another
+cost of (ϵ/2, δ/2)-DP.
+Empirical results. We next empirically evaluate Algorithm 4 (PATE-PTR) on the MNIST dataset.
+Following the experimental setup from Papernot et al. [2018], we consider the training set to be the
+private domain, and the testing set is used as the public domain. We first partition the training set
+into 400 disjoint sets and 400 teacher models, each trained individually. Then we select T = 200
+unlabeled data from the public domain, with the goal of privately labeling them. To illustrate the
+behaviors of algorithms under various data distributions, we consider two settings of unlabeled
+data, high-consensus and low-consensus. In the low-consensus setting, we choose T unlabeled data
+such that there is no high agreement among teachers, so the advantage of data-adaptive analysis is
+diminished. We provide further details on the distribution of these two settings in the appendix.
+Baselines.
+We consider the Gaussian mechanism as a data-independent baseline, where the
+privacy guarantee is valid but does not take advantage of the properties of the dataset. The data-
+dependent DP ( Papernot et al. [2018]) serves as a non-private baseline, which requires further
+sanitation. Note that these two baselines provide different privacy analyses of the same algorithm
+(see Theorem 5.5).
+Figure 2 plots privacy-utility tradeoffs between the three approaches by varying the noise scale σ1.
+The purple region denotes a set of privacy budget choices (ˆϵ + ϵ′ used in Algorithm 4) such that the
+utility of the three algorithms is aligned under the same σ1. In more detail, the purple region is
+lower-bounded by ˆϵ+ϵσ1. We first fix σs = σ2 = 15 such that ˆϵ is fixed. Then we empirically calculate
+the average of ϵσ1 (the private upper bound of the data-dependent DP) over 10 trials. Running
+Algorithm 4 with any choice of ˆϵ + ϵ′ chosen from the purple region implies ϵ′ > ϵσ1. Therefore,
+PATE-PTR will output the same noisy labels (with high probability) as the two baselines.
+Observation As σ1 increases, the privacy loss of the Gaussian mechanism decreases, while the
+data-dependent DP curve does not change much. This is because the data-dependent DP of each
+query is a complex function of both the noise scale and the data and does not monotonically
+decrease when σ1 increases (see more details in the appendix). However, the data-dependent DP still
+dominates the Gaussian mechanism for a wide range of σ1. Moreover, PATE-PTR nicely interpolates
+between the data-independent DP guarantee and the non-private data-adaptive DP guarantee. In the
+low-consensus case, the gap between the data-dependent DP and the DP guarantee of the Gaussian
+mechanism unsurprisingly decreases. Meanwhile, PATE-PTR (the purple region) performs well
+when the noise scale is small but deteriorates when the data-independent approach proves more
+advantageous. This example demonstrates that using PTR as a post-processing step to convert
+the data-dependent DP to standard DP is effective when the data-adaptive approach dominates
+others.
+6
+Limitations and Future Work
+One weakness of generalized PTR is that it requires a case-specific privacy analysis. Have we simply
+exchanged the problem of designing a data-adaptive DP algorithm with the problem of analyzing
+the data-dependent privacy loss? We argue that this limitation is inherited from classic PTR. In
+situations where classic PTR is not applicable, we’ve outlined several approaches to constructing the
+13
+
+DP test for our framework (see Sections 4.3 and 5.2).
+Furthermore, the data-dependent privacy loss is often more straightforward to compute than local
+sensitivity, and often exists in intermediate steps of classic DP analysis already. Most DP analysis
+involves providing a high-probability tail bound of the privacy loss random variable. If we stop
+before taking the max over the input dataset, then we get a data-dependent DP loss right away (as
+in Example 4.2).
+There are several exciting directions for applying generalized PTR to more problems. Sufficient
+statistics release and our private hyperparameter tuning (Algorithm 3) can be used to construct
+data-adaptive extensions of DP-PCA [Dwork et al., 2014] and Sparse-DP-ERM [Kifer et al., 2012].
+For DP-PCA we could use our Algorithm 3 to tune the variance of the noise added to the spectral
+gap; for Sparse-DP-ERM we would test the restricted strong convexity parameter (RSC), i.e. not
+adding additional regularization if the RSC is already large.
+7
+Conclusion
+Generalized PTR extends the classic “Propose-Test-Release” framework to a more general setting by
+testing the data-dependent privacy loss of an input dataset, rather than its local sensitivity. In this
+paper we’ve provided several examples – private linear regression with hyperparameter selection and
+PATE – to illustrate how generalized PTR can enhance DP algorithm design via a data-adaptive
+approach.
+Acknowledgments
+The work was partially supported by NSF Award # 2048091 and the Google Research Scholar Award.
+Yuqing was supported by the Google PhD Fellowship.
+14
+
+Contents
+1
+Introduction
+1
+2
+Related Work
+2
+3
+Preliminaries
+3
+3.1
+Propose-Test-Release . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+4
+Generalized PTR
+4
+4.1
+Limitations of local sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+4.2
+Which φ to propose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+4.3
+Construction of the DP test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+5
+Applications
+8
+5.1
+Private Linear Regression
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+5.2
+PATE
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+6
+Limitations and Future Work
+13
+7
+Conclusion
+14
+A Omitted examples in the main body
+15
+A.1 Limits of the classic PTR in private binary voting . . . . . . . . . . . . . . . . . . . .
+15
+A.2 Self-concordant generalized linear model (GLM) . . . . . . . . . . . . . . . . . . . . .
+18
+A.3 Differentially privately release λmin
+�
+∇2F(θ)
+�
+. . . . . . . . . . . . . . . . . . . . . .
+21
+A.4 Other applications of generalized PTR . . . . . . . . . . . . . . . . . . . . . . . . . .
+22
+B Omitted proofs in Section 4
+23
+C Experimental details
+23
+C.1 Experimental details in private linear regression . . . . . . . . . . . . . . . . . . . . .
+23
+C.2 Details of PATE case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+24
+D Omitted proofs in private GLM
+26
+D.1 Per-instance DP of GLM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+26
+A
+Omitted examples in the main body
+In this appendix, we provide more examples to demonstrate the merits of generalized PTR. We
+focus on a simple example of post-processed Laplace mechanism in Section A.1 and then an example
+on differentially private learning of generalized linear models in Section 4. In both cases, we observe
+that generalized PTR provides data-adaptive algorithms with formal DP guarantees, that are simple,
+effective and not previously proposed in the literature (to the best of our knowledge).
+A.1
+Limits of the classic PTR in private binary voting
+The following example demonstrates that classic PTR does not capture sufficient data-dependent
+quantities even when the local sensitivity exists and can be efficiently tested.
+15
+
+Example A.1. Consider a binary class voting problem: n users vote for a binary class {0, 1} and
+the goal is to output the class that is supported by the majority. Let ni denote the number of people
+who vote for the class i. We consider the report-noisy-max mechanism:
+M(X) : argmaxi∈[0,1]ni(X) + Lap(b),
+where b = 1/ϵ denotes the scale of Laplace noise.
+In the example, we will (1) demonstrate the merit of data-dependent DP; and (2) empirically compare
+classic PTR with generalized PTR.
+We first explicitly state the data-dependent DP.
+Theorem A.2. The data-dependent DP of the above example is
+ϵ(X) := max
+X′ {| log p
+p′ |, | log 1 − p
+1 − p′ |},
+where p := Pr[n0(X) + Lap(1/ϵ) > n1(X) + Lap(1/ϵ)] and p′ := Pr[n0(X′) + Lap(1/ϵ) > n1(X′) +
+Lap(1/ϵ)]. There are four possible neighboring datasets X′ : n0(X′) = max(n0(X) ± 1, 0), n1(X′) =
+n1(X) or n0(X′) = n0(X), n1(X′) = max(n1(X) ± 1, 0).
+In Figure 3(a), we empirically compare the above data-dependent DP with the Laplace mechanism
+by varying the gap between the two vote counts |n0(X) − n1(X)|. The noise scale is fixed to ϵ = 10.
+The data-dependent DP substantially improves over the standard DP if the gap is large. However,
+the data-dependent DP is a function of the dataset. We next demonstrate how to apply generalized
+PTR to exploit the data-dependent DP.
+Notice that the probability n0(X) + Lap(1/ϵ) > n1(X) + Lap(1/ϵ) is equal to the probability that a
+random variable Z := X − Y exceeds ϵ(n1(X) − n0(X)), where X, Y are two independent Lap(1)
+distributions. We can compute the pdf of Z through the convolution of two Laplace distributions,
+which implies fX−Y (z) = 1 + |z|
+4e|z| . Let t denote the difference between n1(X) and n0(X), i.e.,
+t = n1(X) − n0(X). Then we have
+p = Pr[Z > ϵ · t] =
+2 + ϵ · t
+4 exp(ϵ · t)
+Similarly, p′ =
+2 + ϵ · (t + ℓ)
+4 exp(ϵ · (t + ℓ)), where ℓ ∈ [−1, 1] denotes adding or removing one data point to
+construct the neighboring dataset X′. Therefore, we can upper bound log(p/p′) by
+log p
+p′ =
+2 + ϵ · t
+4 exp(ϵ · t) · 4 exp(ϵ(t + ℓ))
+2 + ϵ · (t + ℓ)
+≤ ϵ · log
+�
+2 + ϵt
+2 + ϵ(t + 1)
+��
+= ϵ log
+�
+1 −
+ϵ
+2 + ϵ(t + 1)
+�
+Then we can apply generalized PTR by privately lower-bounding t.
+On the other hand, the local sensitivity ∆LS(X) of this noise-adding mechanism is 0 if t > 1.
+Specifically, if the gap is larger than one, adding or removing one user will not change the result. To
+16
+
+0
+5
+10
+15
+20
+25
+30
+35
+40
+The gap t=|n0(X)
+n1(X)|
+0
+2
+4
+6
+8
+10
+ data-dependent DP
+Laplace mechanism
+(a) data-dependent DP vs Laplace mechanism
+10
+28
+10
+23
+10
+18
+10
+13
+10
+8
+10
+3
+102
+Error
+10
+2
+10
+1
+ 
+ Gen-PTR(
+p + )
+classic PTR
+Laplace mechanism
+(b) Privacy-utility tradeoff between three approaches.
+Figure 3: In Figure 3(a), we compare the privacy guarantee by varying the gap. In Figure 3(b) We
+fix t = n0(X) − n1(X) = 100 and compare privacy cost when the accuracy is aligned. Gen-PTR with
+any choice of privacy budget (˜ϵ + ϵ′) chosen from the purple region would achieve the same utility as
+Laplace mechanism but with a smaller privacy cost. The curve of Gen-PTR is always below than
+that of the classic PTR, which implies that Gen-PTR can result a tighter privacy analysis when the
+utility is aligned.
+apply classic PTR, we let γ(X) denote the distance to the nearest dataset X
+′′ such that ∆LS > 0
+and test if γ(X) + Lap(1/ϵ) > log(1/δ)
+ϵ
+. Notice in this example that γ(X) = max(t − 1, 0) can be
+computed efficiently. We provide the detailed implementation of these approaches.
+1. Gen PTR: lower bound t with tp = t − log(1/δ)
+˜ϵ
++ Lap(1/˜ϵ). Calculate an upper bound of
+data-dependent DP ϵp using Theorem A.2 with tp. The algorithm then tests if ϵp is within an
+predefined privacy budget ϵ′. If the test passes, the algorithm returns argmaxi∈[0,1]ni(X) +
+Lap(1/ϵ) satisfies (˜ϵ + ϵ′, δ)-DP.
+2. classic PTR: lower bound t with tp = t − log(1/δ)
+˜ϵ
++ Lap(1/˜ϵ). If tp > 1, classic PTR outputs
+the ground-truth result else returns a random class. This algorithm satisfies (˜ϵ, δ)-DP.
+3. Laplace mechanism. M(X) : argmaxi∈[0,1]ni(X) + Lap(1/ϵ). M is (ϵ, δ)-DP.
+We argue that though the Gen-PTR and the classic PTR are similar in privately lower-bounding
+the data-dependent quantity t, the latter does not capture sufficient information for data-adaptive
+analysis. That is to say, only testing the local sensitivity restricts us from learning helpful information
+to amplify the privacy guarantee if the test fails. In contrast, our generalized PTR, where privacy
+parameters and the local sensitivity parameterize the data-dependent DP, can handle those failure
+cases nicely.
+To confirm this conjecture, Figure 3(b) plots a privacy-utility trade-off curve between these three
+approaches. We consider a voting example with n0(X) = n1(X) + 100 and t = 100, chosen such
+that the data-adaptive analysis is favorable.
+In Figure 3(b), we vary the noise scale b = 1/ϵ between [0, 0.5]. For each choice of b, we plot the
+privacy guarantee of three algorithms when the error rate is aligned. For Gen-PTR, we set ˜ϵ = 1
+2b
+and empirically calculate ϵp over 100000 trials.
+17
+
+In the plot, when ϵ ≪ log(1/δ)
+t
+, the classic PTR is even worse than the Laplace mechanism. This is
+because the classic PTR is likely to return ⊥ while the Laplace mechanism returns argmaxi∈[0,1]ni(X)+
+Lap(1/ϵ), which contains more useful information. Compared to the Laplace mechanism, Gen-PTR
+requires an extra privacy allocation ˜ϵ to release the gap t. However, it still achieves an overall smaller
+privacy cost when the error rate ≤ 10−5 (the purple region). Meanwhile, Gen-PTR dominates the
+classic PTR (i.e., the dashed black curve is always below the blue curve). Note that the classic PTR
+and the Gen-PTR utilize the gap information differently: the classic PTR outputs ⊥ if the gap is
+not sufficiently large, while the Gen-PTR encodes the gap into the data-dependent DP function
+and tests the data-dependent DP in the end. This empirical result suggests that testing the local
+sensitivity can be loosely compared to testing the data-dependent DP. Thus, Gen-PTR could provide
+a better privacy-utility trade-off.
+A.2
+Self-concordant generalized linear model (GLM)
+In this section, we demonstrate the effectiveness and flexibility of generalized PTR in handling a
+family of GLMs where the link function satisfies a self-concordance assumption. This section is
+organized as follows:
+• Introduce a family of GLMs with the self-concordance property.
+• Introduce a general output perturbation algorithm for private GLMs.
+• Analyze the data-dependent DP of GLMs with the self-concordance property.
+• Provide an example of applying our generalized PTR framework to logistic regression.
+Consider the empirical risk minimization problem of the generalized linear model
+θ∗ = argminθ
+�
+i=1n
+li(θ) + r(θ),
+where l : R × R → R belongs to a family of convex GLMs: li(θ) = l(y, xT
+i θ). Let r : Rd → R be a
+regularization function.
+We now define the self-concordance property.
+Definition A.3 (Generalized self-concordance [Bach, 2010]). A convex and three-times differentiable
+function f : Θ → R is R-generalized-self-concordant on an open nonempty convex set Θ∗ ⊂ Θ with
+respect to norm ∥ · ∥ if for all u ∈ Θ∗ and all v ∈ Rd,
+∇3f(u)[v, v, v] ≤ 2R∥v∥(∇2f(u)[v, v]).
+The closer R is to 0, the “nicer” — more self-concordant — the function is. A consequence of (gener-
+alized) self-concordance is the spectral (multiplicative) stability of Hessian to small perturbations of
+parameters.
+Lemma A.4 (Stability of Hessian[Nesterov and Nemirovskii, 1994, Theorem 2.1.1], [Bach, 2010,
+Proposition 1]). Let Hθ := ∇2Fs(θ). If Fs is R-self-concordant at θ, then for any v such that
+R∥v∥Hθ < 1, we have that
+(1 − R∥v∥Hθ)2∇2Fs(θ) ≺ ∇2Fs(θ + v)
+≺
+1
+(1 − R∥v∥Hθ)2 ∇2Fs(θ).
+18
+
+If instead we assume Fs is R-generalized-self-concordant at θ with respect to norm ∥ · ∥, then
+e−R∥v∥∇2Fs(θ) ≺ ∇2Fs(θ + v) ≺ eR∥v∥∇2Fs(θ)
+The two bounds are almost identical when R∥v∥ and R∥v∥θ are close to 0. In particular, for x ≤ 1/2,
+we have that e−2x ≤ 1 − x ≤ e−x.
+In particular, the loss function of binary logistic regression is 1-generalized self-concordant.
+Example A.5 (Binary logistic regression). Assume ∥x∥2 ≤ 1 for all x ∈ X and y ∈ {−1, 1}. Then
+binary logistic regression with datasets in X × Y has a log-likelihood of F(θ) = �n
+i=1 log(1 + e−yixT
+i θ).
+The univariate function l := log(1 + exp(·)) satisfies
+|l′′′| =
+����
+exp (·)(1 − exp (·))
+(1 + exp (·))3
+���� ≤
+exp (·)
+(1 + exp (·))2 := l′′.
+We next apply the modified output perturbation algorithm to privately release θ∗. The algorithm is
+simply:
+1. Solve
+θ∗ = argminθ
+n
+�
+i=1
+li(θ) + r(θ).
+2. Release
+ˆθ = θ∗ + Z,
+where γ > 0 is a tuning parameter and Z ∼ N(0, γ−1(�n
+i=1 ∇2li(θ) + ∇2r(θ))−1).
+The data-dependent DP of the above procedure is stated as follows.
+Theorem A.6 (Data-dependent DP of GLM). Denote the smooth part of the loss function Fs =
+�n
+i=1 l(yi, < xi, · >) + rs(·). Assume the following:
+1. The GLM loss function l is convex, three-times continuously differentiable and R-generalized-
+self-concordant w.r.t. ∥ · ∥2,
+2. Fs is locally α-strongly convex w.r.t. ∥ · ∥2,
+3. and in addition, denote L := supθ∈[θ∗,˜θ∗] |l′(y, xT θ)|, β := supθ∈[θ∗,˜θ∗] |l′′(y, xT θ)|. That is, ℓ(·)
+is L-Lipschitz and β-smooth.
+We then have the data-dependent DP
+ϵ(Z) ≤ R(L + β)
+α
+(1 + log(2/δ)) + γL2
+α
++
+�
+γL2
+α
+log(2/δ).
+The proof follows by taking an upper bound of the per-instance DP loss (Theorem D.1) ϵ(Z, z) over
+z = (x, y) ∈ (X, Y).
+Notice that the Hessians can be arbitrarily singular and α could be 0, which leads to an infinite
+privacy loss without additional assumptions. Thus, we will impose an additional regularization of
+form λ
+2||θ||2, which ensures that for any dataset FS is λ-strongly convex.
+This is not yet DP because it is still about a fixed dataset. We also need a pre-specified privacy
+budget (ϵ, δ). We next demonstrate how to apply the generalized PTR to provide a general solution
+to the above GLM, using logistic regression as an example.
+19
+
+Remark A.7 (Logistic regression). For logistic regression, we know L ≤ 1, β ≤ 1/4 and if ∥x∥2 ≤ 1,
+it is 1-generalized self-concordant. For any dataset Z = (X, y), the data-dependent DP ϵ(X) w.r.t.
+δ can be simplified to:
+1.25
+α (1 + log(2/δ)) + γ
+α +
+�γ
+α log(2/δ)
+Now, the data-dependent DP is a function of α and γ, where α denotes the local strong convexity at
+θ∗
+λ and γ controls the noise scale. We next show how to select these two parameters adapted to the
+dataset.
+Example A.8. We demonstrate here how we apply generalized PTR to output perturbation of the
+logistic regression problem.
+1. Take an exponential grid of parameters {λ} and propose each λ.
+2. Solve for θ∗
+λ = argminθF(θ) + λ∥θ∥2/2
+3. Calculate the smallest eigenvalue λmin(∇2F(θ∗
+λ)) (e.g., using power method).
+4. Differentially privately release λmin with λp
+min := max{λmin+
+√
+log(4/δ)
+ϵ/2
+·∆GS·Z−
+√
+2 log(4/δ)·log(1/δ)∆GS
+ϵ/2
+, 0},
+where ∆GS denote the global sensitivity of λmin using Theorem A.11.
+5. Let ϵp(·) be instantiated with ϵ(X) w.r.t. δ from Remark A.7, where α = λp
+min + λ. Then,
+conditioned on a high probability event, ϵp(·) (a function of γ) is a valid DP bound that holds
+for all datasets and all parameters γ.
+6. Calculate the maximum γ such that ϵp
+δ/2(γ) ≤ ϵ/2.
+7. Release ˆθ ∼ N(θ∗
+λ, γ−1∇2Fs(θ∗
+λ)−1).
+8. Evaluate the utility on the validation set and return the (λ, γ) pair that leads to the highest
+utility.
+Theorem A.9. For each proposed λ, the algorithm that releases ˆθ ∼ N(θ∗
+λ, γ−1∇2Fs(θ∗
+λ)−1) is
+(ϵ, 2δ)-DP.
+Proof. The proof follows the recipe of generalized PTR with private upper bound (Example 4.6). First,
+the release of λmin(∇2F(θ∗
+λ)) is (ϵ/2, δ/2)-DP. Then, with probability at least 1 − δ, ϵp
+δ(·) > ϵδ(X)
+holds for all X and γ. Finally, γ is chosen such that the valid upper bound is (ϵ/2, δ/2)-DP.
+For the hyper-parameter tuning on λ (Steps 1 and 8), we can use Algorithm 3 to evaluate each λ.
+Unlike Example 5.2, the λmin(∇2F(θ∗
+λ)) is a complicated data-dependent function of λ. Thus, we
+cannot privately release the data-dependent quantity λmin(∇2F(θ∗
+λ)) without an input λ. The PTR
+approach allows us to test a number of different λ and hence get a more favorable privacy-utility
+trade-off.
+An interesting perspective of this algorithm for logistic regression is that increasing the regularization
+α is effectively increasing the number of data points within the soft “margin”3 of separation, hence a
+larger contribution to the Hessian from the loss function.
+3If we think of logistic regression as a smoothed version of SVM, then increasing α leads to more support vectors.
+The “margin” is “softer” in logistic regression, but qualitatively the same.
+20
+
+Remark A.10. The PTR solution for GLMs follows a similar recipe: propose a regularization
+strength λ; construct a lower bound of the strong convexity α at the optimal solution θ∗
+λ; and test
+the validity of data-dependent DP using Theorem D.1.
+Before moving on to other applications of generalized PTR, we will show how to differentially
+privately release λmin according to the requirements of the logistic regression example.
+A.3
+Differentially privately release λmin (∇2F(θ))
+To privately release λmin∇2F(θ), we first need to compute its global sensitivity. Once we have that
+then we can release it differentially privately using either the Laplace mechanism or the Gaussian
+mechanism.
+Theorem A.11 (Global sensitivity of the minimum eigenvalue at the optimal solution). Let
+F(θ) = �n
+i=1 fi(θ) + r(θ) and ˜F(θ) = F(θ) + f(θ) where f1, ..., fn are loss functions corresponding
+to a particular datapoint x. Let θ∗ = argminθF(θ) and ˜θ∗ = argminθ ˜F(θ). Assume f is L-Lipschitz
+and β-smooth, r(θ) is λ-strongly convex, and F and ˜F are R-self-concordant. If in addition, λ ≥ RL,
+then we have
+sup
+X,x
+(λmin(∇2F(θ∗
+λ)) − λmin(∇2 ˜F( ˜θ∗
+λ))) ≤ 2RL + β.
+Proof.
+λmin(∇2F(θ∗
+λ)) − λmin(∇2 ˜F( ˜θ∗
+λ))
+= (λmin(∇2F(θ∗
+λ)) − λmin(∇2 ˜F(θ∗
+λ)))
++ (λmin(∇2 ˜F(θ∗
+λ)) − λmin(∇2 ˜F( ˜θ∗
+λ))).
+(1)
+We first bound the part on the left. By applying Weyl’s lemma λ(X + E) − λ(X) ≤ ||E||2, we have
+sup
+x ||∇2F(θ∗
+λ) − ∇2
+˜
+F(θ∗
+λ)||2 = ||∇2f(θ∗
+λ)||2 ≤ β
+(2)
+In order to bound the part on the right, we apply the semidefinite ordering using self-concordance,
+which gives
+e−R∥ ˜
+θ∗
+λ−θ∗
+λ∥∇2 ˜F( ˜θ∗
+λ) ≺ ∇2 ˜F(θ∗
+λ) ≺ eR∥ ˜
+θ∗
+λ−θ∗
+λ∥∇2 ˜F( ˜θ∗
+λ).
+By the Courant-Fischer Theorem and the monotonicity theorem, we also have that for the smallest
+eigenvalue
+e−R∥ ˜
+θ∗
+λ−θ∗
+λ∥λmin
+�
+∇2 ˜F( ˜θ∗
+λ)
+�
+≤ λmin
+�
+∇2 ˜F(θ∗
+λ)
+�
+≤ eR∥ ˜
+θ∗
+λ−θ∗
+λ∥λmin
+�
+∇2 ˜F( ˜θ∗
+λ)
+�
+.
+(3)
+Moreover by Proposition D.2, we have that
+∥ ˜θ∗
+λ − θ∗
+λ∥2 ≤
+∥∇f( ˜θ∗λ)∥
+λmin
+�
+∇2 ˜F( ˜θ∗
+λ)
+� ≤
+L
+λmin
+�
+∇2 ˜F( ˜θ∗
+λ)
+�.
+If λmin
+�
+∇2 ˜F( ˜θ∗
+λ)
+�
+≥ RL, then use that ex − 1 ≤ 2x for x ≤ 1. Substituting the above bound to (3)
+then to (1) together with (2), we get a data-independent global sensitivity bound of
+λmin(∇2F(θ∗
+λ)) − λmin(∇2 ˜F( ˜θ∗
+λ)) ≤ 2RL + β
+21
+
+as stated.
+Proposition A.12. Let ∥ · ∥ be a norm and ∥ · ∥∗ be its dual norm. Let F(θ), f(θ) and ˜F(θ) =
+F(θ) + f(θ) be proper convex functions and θ∗ and
+˜
+theta
+∗ be their minimizers, i.e., 0 ∈ ∂F(θ∗) and
+0 ∈ ∂ ˜F(
+˜
+theta
+∗). If in addition, F, ˜F is α, ˜α-strongly convex with respect to ∥ · ∥ within the restricted
+domain θ ∈ {tθ∗ + (1 − t)˜θ∗ | t ∈ [0, 1]}. Then there exists g ∈ ∂f(θ∗) and ˜g ∈ ∂f(˜θ∗) such that
+∥θ∗ − ˜θ∗∥ ≤ min
+� 1
+α∥˜g∥∗, 1
+˜α∥g∥∗
+�
+.
+Proof. Apply the first order condition to F restricted to the line segment between ˜θ∗ and θ∗, we get
+F(˜θ∗) ≥ F(θ∗) + ⟨∂F(θ∗), ˜θ∗ − θ∗⟩ + α
+2 ∥˜θ∗ − θ∗∥2
+(4)
+F(θ∗) ≥ F(˜θ∗) + ⟨∂F(˜θ∗), θ∗ − ˜θ∗⟩ + α
+2 ∥˜θ∗ − θ∗∥2
+(5)
+Note by the convexity of F and f, ∂ ˜F = ∂F + ∂f, where + is the Minkowski Sum. Therefore,
+0 ∈ ∂ ˜F(˜θ∗) implies that there exists ˜g such that ˜g ∈ ∂f(˜θ∗) and −˜g ∈ ∂F(˜θ∗). Take −˜g ∈ ∂F(˜θ∗) in
+Equation 10 and 0 ∈ ∂F(θ∗) in Equation 9 and add the two inequalities, we obtain
+0 ≥ ⟨−˜g, θ∗ − ˜θ∗⟩ + α∥˜θ∗ − θ∗∥2
+≥ −∥˜g∥∗∥θ∗ − ˜θ∗∥ + α∥˜θ∗ − θ∗∥2.
+For ∥˜θ∗ − θ∗∥ = 0 the claim is trivially true; otherwise, we can divide both sides of the above
+inequality by ∥˜θ∗ − θ∗∥ and get ∥θ∗ − ˜θ∗∥ ≤ 1
+α∥˜g∥∗.
+It remains to show that ∥θ∗ − ˜θ∗∥ ≤ 1
+˜α∥g∥∗. This can be obtained by exactly the same arguments
+above but applying strong convexity to ˜F instead. Note that we can actually get something slightly
+stronger than the statement because the inequality holds for all g ∈ ∂f(θ∗).
+A.4
+Other applications of generalized PTR
+Besides one-posterior sampling for GLMs, there are plenty of examples that our generalized-PTR
+could be applied, e.g., DP-PCA [Dwork et al., 2014] and Sparse-DP-ERM [Kifer et al., 2012] (when
+the designed matrix is well-behaved).
+[Dwork et al., 2014] provides a PTR style privacy-preserving principle component analysis (PCA).
+The key observation of [Dwork et al., 2014] is that the local sensitivity is quite “small” if there is a
+large eigengap between the k-th and the k + 1-th eigenvalues. Therefore, their approach (Algorithm
+2) chooses to privately release a lower bound of the k-th eigengap (k is fixed as an input) and use
+that to construct a high-confidence upper bound of the local sensitivity.
+For noise-adding mechanisms, the local sensitivity is proportional to the data-dependent loss and
+generalized PTR is applicable. We can formulate the data-dependent DP of DP-PCA as follows:
+Theorem A.13.
+For a given matrix A ∈ Rm×n, assume each row of A has a bounded ℓ2 norm
+being 1. Let Vk denotes the top k eigenvectors of AT A and dk denotes the gap between the k-th
+and the k + 1-th eigenvalue. Then releasing VkV T
+k + E, where E ∈ Rn×n is a symmetric matrix
+with the upper triangle is i.i.d samples from N(0, σ2) satisfies (ϵ(A), δ) data-dependent DP and
+ϵ(A) =
+2√
+log(1.25/δ)
+σ(dk−2)
+.
+22
+
+The proof is based on the local sensitivity result from [Dwork et al., 2014] and the noise calibration
+of Gaussian mechanism.
+We can combine Theorem A.13 with our Algorithm 3 to instantiate the generalized PTR framework.
+The improvement over Dwork et al. [2014] will be to allow joint tuning of the parameter k and the
+noise variance (added to the spectral gap dk).
+B
+Omitted proofs in Section 4
+The utility of Algorithm 3 depends on how many rounds that Algorithm 2 is invoked. We next
+provide the utility guarantee of Algorithm 3, which follows a simplification of the result in the
+Section A.2 of Papernot and Steinke [2021].
+Theorem B.1. Suppose applying Algorithm 2 with each φi has an equal probability to achieve the
+highest validation score. Let ˆT denotes the number of invocation of Algorithm 2, where ˆT follows a
+truncated geometric distribution. Then the expected quantile of the highest score candidate is given
+by E ˆT
+�
+1 −
+1
+ˆT+1
+�
+.
+In practice, we can roughly set τ =
+1
+10k so that the algorithm is likely to test all k parameters.
+Proof. Suppose each oracle access to Q(X) has a probability 1/k of achiving the best validation
+accuracy. Let β denote the probability that A (shorthand for Algorithm 3) outputs the best choice
+of φi.
+β = 1 − Pr[A(X)is not best]
+= 1 − E ˆT
+�
+Pr[Q(X)is not best]
+ˆT
+�
+= 1 − E ˆT
+�
+(1 − 1
+k)
+ˆT
+�
+.
+Let f(x) = E[x ˆT ]. Applying a first-order approximation on f(1 − 1
+k), we have f(1 − 1
+k) ≈ f(1) −
+f′(1) · 1
+k = 1 − E[ ˆT]/k. Then, if k is large and we choose τ = 0.1/k, A can roughly return the best
+φi.
+C
+Experimental details
+C.1
+Experimental details in private linear regression
+We start with the privacy calibration of the OPS-PTR algorithm.
+Algorithm 5 provides the detailed privacy calibration of the private linear regression problem.
+Theorem C.1. Algorithm 5 is (ϵ, 2δ)-DP.
+Proof. There are three data-dependent quantities in Theorem 5.1: λmin, ||θ∗
+λ|| and L. First, notice
+that λmin has a global sensitivity of ||X||2 by Weyl’s lemma. Under the assumption ||X||2 ≤ 1, we
+privately release λmin using (ϵ/4, δ/3) in Step 3. Notice that with probability at least 1 − δ/2, ˜λmin
+is a lower bound of λmin.
+23
+
+Algorithm 5 OPS-PTR: One-Posterior Sample with propose-test-release (no-“perp” version)
+1: Input: Data X, y. Private budget : ϵ, δ, proposed regularizer λ.
+2: Calculate the minimum eigenvalue λmin(XT X).
+3: Sample Z ∼ N(0, 1) and privately release ˜λmin = max
+�
+λmin +
+√
+log(6/δ)
+ϵ/4
+Z −
+√
+2 log(6/δ)·log(2/δ)
+ϵ/4
+, 0
+�
+4: Calculate ˆθ = (XT X + λI)−1XT y.
+5: Sample Z ∼ N(0, 1) and privately release ∆ = log(||Y|| + ||X||||ˆθ||) + log(1+||X||2/(λ+˜λmin))
+ϵ/(4√
+6/δ)
+Z +
+log(1+||X||2/(λ+˜λmin))
+ϵ/(4√
+2 log(6/δ) log(2/δ)).
+6: Set the local Lipschitz ˜L := ||X||e∆.
+7: Calibrate γ with Theorem 5.1(δ/3, ϵ/2.)
+8: Output ˜θ ∼ p(θ|X, y) ∝ e− γ
+2 ||y−Xθ||2+λ||θ||2
+Then, we apply Lemma C.2 from
+Wang [2018] to privately release log(||Y|| + ||X||||ˆθ||) using
+(ϵ/4, δ/3). Note that both the local Lipschitz constant L and the norm ||θ∗
+λ|| are functions of
+log(||Y|| + ||X||||ˆθ||). Thus, we can construct a private upper bound of these by post-processing of
+∆.
+Then, with probability at least 1 − δ (by a union bound over ˜λmin and ∆), instantiating Theorem 5.1
+with ˜λmin and ˜L provides a valid upper bound of the data-dependent DP. We then tune the parameter
+γ using the remaining privacy budget (ϵ/2, δ/3).
+Lemma C.2 (Lemma 12 [Wang, 2018]). Let θ∗
+λ be the ridge regression estimate with parameter
+λ and the smallest eigenvalue of XT X be λmin, then the function log(||Y + ||X||||θ∗
+λ||) has a local
+sensitivity of log(1 +
+||X||2
+λmin+λ ).
+C.2
+Details of PATE case study
+Definition C.3 (Renyi DP [Mironov, 2017]). We say a randomized algorithm M is (α, ϵM(α))-RDP
+with order α ≥ 1 if for neighboring datasets X, X′
+Dα(M(X)||M(X′)) :=
+1
+α − 1 log Eo∼M(X′)
+�� Pr[M(X) = o]
+Pr[M(X′) = o]
+�α�
+≤ ϵM(α).
+At the limit of α → ∞, RDP reduces to (ϵ, 0)-DP. We now define the data-dependent Renyi DP
+that conditioned on an input dataset X.
+Definition C.4 (Data-dependent Renyi DP [Papernot et al., 2018]). We say a randomized algorithm
+M is (α, ϵM(α, X))-RDP with order α ≥ 1 for dataset X if for neighboring datasets X′
+Dα(M(X)||M(X′)) :=
+1
+α − 1 log Eo∼M(X′)
+�� Pr[M(X) = o]
+Pr[M(X′) = o]
+�α�
+≤ ϵM(α, X).
+RDP features two useful properties.
+24
+
+Lemma C.5 (Adaptive composition). ϵ(M1,M2) = ϵM1(·) + ϵM2(·).
+Lemma C.6 (From RDP to DP). If a randomized algorithm M satisfies (α, ϵ(α))-RDP, then M
+also satisfies (ϵ(α) + log(1/δ)
+α−1 , δ)-DP for any δ ∈ (0, 1).
+Definition C.7 (Smooth Sensitivity). Given the smoothness parameter β, a β-smooth sensitivity
+of f(X) is defined as
+SSβ(X) := max
+d≥0 e−βd ·
+max
+˜
+X′:dist(X, ˜
+X′)≤d
+∆LS( ˜X′)
+Lemma C.8 (Private upper bound of data-dependent RDP, Restatement of Theorem 5.6). ] Given
+a RDP function RDP(α, X) and a β-smooth sensitivity bound SS(·) of RDP(α, X). Let µ (defined
+in Algorithm 4) denote the private release of log(SSβ(X)). Let (β, σs, σ2)-GNSS mechanism be
+RDPupper(α):=RDP(α,X)+SSβ(X)·N(0,σ2
+s)+σs
+�
+2 log( 2
+δ2 )eµ
+Then, the release of RDPupper(X) satisfies (α, 3α+2
+2σ2s )-RDP for all 1 < α <
+1
+2β; w.p. at least 1 − δ2,
+RDPupper(α) is an upper bound of RDP(α, X).
+Proof sketch. We first show that releasing the smooth sensitivity SSβ with eµ satisfies (α,
+α
+2σ2
+2 )-RDP.
+Notice that the log of SSβ(X) has a bounded global sensitivity β (Definition C.7 implies that
+| log SSβ(X) − log SSβ(X′)| ≤ β for any neighboring dataset X, X′). By Gaussian mechanism,
+scaling noise with βσ2 to log SSβ(X) is (α,
+α
+2σ2
+2 )-RDP. Therefore, the release of RDP(α, X) is
+(α, ϵs(α) +
+α
+2σ2
+2 )-RDP. Since the release of f(X) + SSβ(X) · N(0, σ2
+s) is (α, α+1
+σ2s )-RDP (Theorem 23
+from Papernot et al. [2018]) for α <
+1
+2β, we have ϵs(α) +
+α
+2σ2
+2 = 3α+2
+2σ2s .
+We next prove the second statement. First, notice that with probability at least 1−δ2/2, eµ ≥ SSβ(X)
+using the standard Gaussian tail bound. Let E denote the event that eµ ≥ SSβ(X).
+Pr
+�
+RDPupper(α) ≤ RDP(α, X)
+�
+= Pr
+�
+RDPupper(α) ≤ RDP(α, X)|E
+�
++ Pr
+�
+RDPupper(α) ≤ RDP(α, X)|Ec
+�
+≤ Pr
+�
+RDPupper(α) ≤ RDP(α, X)|E
+�
++ δ2/2
+= Pr
+�
+N(0, σ2
+s) · SSβ(X) ≥ σs ·
+�
+2 log(2/δ2)eµ|E
+�
+�
+��
+�
+denoted by(∗)
++δ2/2
+Condition on the event E, eµ is a valid upper bound of SSβ(X), which implies
+(∗) ≤ Pr[N(0, σ2
+s) · SSβ(X) ≥ σs ·
+�
+2 log(2/δ2)SSβ(X)|E] ≤ δ2/2
+Therefore, with probability at least 1 − δ2, RDPupper(α) ≥ RDP(α, X).
+Theorem C.9 (Restatement of Theorem 5.7). Algorithm 4 satisfies (ϵ′ + ˆϵ, δ)-DP.
+25
+
+Proof. The privacy analysis consists of two components — the privacy cost of releasing an upper
+bound of data-dependent RDP (ϵupper(α) := ϵs(α)+
+α
+2σ2
+2 and the valid upper bound ϵp
+σ1(α). First, set
+α = 2 log(2/δ)
+ϵ
++ 1 and use RDP to DP conversion with δ/2 ensures that the cost of δ/2 contribution
+to be roughly ϵ/2 (i.e., log(2/δ)
+α−1
+= ϵ/2). Second, choosing σs =
+�
+2+3α
+ϵ
+gives us another ϵ/2.
+Experimental details K = 400 teacher models are trained individually on the disjoint set using
+AlexNet model. We set σ2 = σs = 15.0. Our data-dependent RDP calculation and the smooth-
+sensitivity calculation follow Papernot et al. [2018]. Specifically, we use the following theorem
+(Theorem 6 from Papernot et al. [2018]) to compute the data-dependent RDP of each unlabeled
+data x from the public domain.
+Theorem C.10 (data-dependent RDP Papernot et al. [2018]). Let ˜q ≥ Pr[M(X) ̸= Argmaxj∈[C]nj(x)],
+i.e., an upper bound of the probability that the noisy label does not match the majority label. Assume
+α ≤ µ1 and ˜q ≤ e(µ2−1)ϵ2/
+�
+µ1
+µ1−1 ·
+µ2
+µ2−1
+�µ2
+, then we have:
+ϵM(α, X) ≤
+1
+α − 1 log
+�
+(1 − ˜q) · A(˜q, µ2, ϵ2)α−1 + ˜q · B(˜q, µ1, ϵ1)α−1
+�
+where A(˜q, µ2, ϵ2) := (1 − ˜q)/
+�
+1 − (˜qeϵ2)
+µ2−1
+µ2
+�
+, B(˜q, µ1, ϵ1) = eϵ1/˜q
+1
+µ1−1 , µ2 = σ1 ·
+�
+log(1/˜q), µ1 =
+µ2 + 1, ϵ1 = µ1/σ2
+1 and ϵ2 = µ2/σ2
+2.
+In the experiments, the non-private data-dependent DP baseline is also based on the above theorem.
+Notice that the data-dependent RDP of each query is a function of ˜q, where ˜q denotes an upper
+bound of the probability where the plurality output does not match the noisy output.
+˜q is a
+complex function of both the noisy scale and data and is not monotonically decreasing when σ1 is
+increasing.
+Simulation of two distributions. The motivation of the experimental design is to compare
+three approaches under different data distributions. Notice that there are K = 400 teachers, which
+implies the number of the vote count for each class will be bounded by 400. In the simulation of
+high-consensus distribution, we choose T = 200 unlabeled public data such that the majority vote
+count will be larger than 150 (i.e., maxj∈[C] nj(x) > 150). For the low-consensus distribution, we
+choose to select T unlabeled data such that the majority vote count will be smaller than 150.
+D
+Omitted proofs in private GLM
+D.1
+Per-instance DP of GLM
+Theorem D.1 (Per-instance differential privacy guarantee). Consider two adjacent data sets Z and
+Z′ = [Z, (x, y)], and denote the smooth part of the loss function Fs = �n
+i=1 l(yi, ⟨xi, ·⟩) + rs(·) (thus
+˜Fs = Fs + l(y, ⟨x, ·⟩). Let the local neighborhood be the line segment between θ∗ and ˜θ∗. Assume
+1. the GLM loss function l be convex, three-time continuous differentiable and R-generalized-self-
+concordant w.r.t. ∥ · ∥2,
+2. Fs is locally α-strongly convex w.r.t. ∥ · ∥2,
+3. and in addition, denote L := supθ∈[θ∗,˜θ∗] |l′(y, xT θ)|, β := supθ∈[θ∗,˜θ∗] |l′′(y, xT θ)|.
+26
+
+Then the algorithm obeys (ϵ, δ)-pDP for Z and z = (x, y) with any 0 < δ < 2/e and
+ϵ ≤ ϵ0(1 + log(2/δ)) + e
+RL∥x∥2
+α
+�γL2∥x∥2
+H−1
+2
++
+�
+γL2∥x∥2
+H−1 log(2/δ)
+�
+where ϵ0 ≤ e
+RL∥x∥2
+α
+− 1 + 2β∥x∥2
+H−1
+1
++ 2β∥x∥2
+˜H−1
+2 . If we instead assume that l is R-self concordant.
+Then the same results hold, but with all e
+RL∥x∥2
+α
+replaced with (1 − RL∥x∥H−1)2.
+Under the stronger three-times continuous differentiable assumption, by mean value theorem, there
+exists ξ on the line-segment between θ∗ and ˜θ∗ such that
+H =
+�� 1
+t=0
+∇2Fs((1 − t)θ∗ + t˜θ∗)dt
+�
+= ∇2Fs(ξ).
+The two distributions of interests are N(θ∗, [γ∇2Fs(θ∗)]−1) and N(˜θ∗, [γ∇2Fs(˜θ∗)+∇2l(y, xT ˜θ∗)]−1).
+Denote [∇2Fs(θ∗)]−1 =: Σ and [∇2Fs(˜θ∗)+∇2l(y, xT ˜θ∗)]−1 =: ˜Σ. Both the means and the covariance
+matrices are different, so we cannot use multivariate Gaussian mechanism naively. Instead we will
+take the tail bound interpretation of (ϵ, δ)-DP and make use of the per-instance DP framework as
+internal steps of the proof.
+First, we can write down the privacy loss random variable in analytic form
+log |Σ|−1/2e− γ
+2 ∥θ−θ∗∥2
+Σ−1
+|˜Σ|−1/2e− γ
+2 ∥θ−˜θ∗∥2
+˜Σ−1
+= 1
+2 log
+�|Σ−1|
+|˜Σ−1|
+�
+�
+��
+�
+(∗)
++ γ
+2
+�
+∥θ − θ∗∥2
+Σ−1 − ∥θ − ˜θ∗∥2
+˜Σ−1
+�
+�
+��
+�
+(∗∗)
+The general idea of the proof is to simplify the expression above and upper bounding the two terms
+separately using self-concordance and matrix inversion lemma, and ultimately show that the privacy
+loss random variable is dominated by another random variable having an appropriately scaled shifted
+χ-distribution, therefore admits a Gaussian-like tail bound.
+To ensure the presentation is readable, we define a few short hands. We will use H and ˜H to denote
+the Hessian of Fs and Fs + f respectively and subscript 1 2 indicates whether the Hessian evaluated
+at at θ∗ or ˜θ∗. H without any subscript or superscript represents the Hessian of Fs evaluated at ξ as
+previously used.
+(∗) = 1
+2 log |H1|
+|H|
+|H|
+|H2|
+|H2|
+| ˜H2|
+≤ 1
+2
+�
+log |H1|
+|H| + log |H|
+|H2| + log |H2|
+| ˜H2|
+�
+By the R-generalized self-concordance of Fs, we can apply Lemma D.3,
+−∥θ∗ − ξ∥2R ≤ log |H1|
+|H| ≤ R∥θ∗ − ξ∥2,
+−R∥ξ − ˜θ∗∥2 ≤ log |H|
+|H2| ≤ R∥ξ − ˜θ∗∥2.
+The generalized linear model ensures that the Hessian of f is rank-1:
+∇2f(˜θ∗) = l′′(y, xT ˜θ∗)xxT
+and we can apply Lemma ?? in both ways (taking A = H2 and A = ˜H2) and obtain
+|H2|
+| ˜H2|
+=
+1
+1 + l′′(y, xT ˜θ∗)xT H−1
+2 x
+= 1 − l′′(y, xT ˜θ∗)xT ˜H2x
+27
+
+Note that l′′(y, xT ˜θ∗)xT ˜H−1
+2 x is the in-sample leverage-score and l′′(y, xT ˜θ∗)xT H−1
+2 x is the out-
+of-sample leverage-score of the locally linearized problem at ˜θ∗. We denote them by µ2 and µ′
+2
+respectively (similarly, for the consistency of notations, we denote the in-sample and out of sample
+leverage score at θ∗ by µ1 and µ′
+1 ).
+Combine the above arguments we get
+(∗) ≤R∥θ∗ − ξ∥2 + R∥ξ − ˜θ∗∥2 + log(1 − µ2) ≤ R∥θ∗ − ˜θ∗∥2 + log(1 − µ2)
+(6)
+(∗) ≥ − R∥θ∗ − ˜θ∗∥2 − log(1 − µ2).
+(7)
+We now move on to deal with the second part, where we would like to express everything in terms of
+∥θ − θ∗∥H1, which we know from the algorithm is χ-distributed.
+(∗∗) = γ
+2
+�
+∥θ − θ∗∥2
+H1 − ∥θ − θ∗∥2
+H2 + ∥θ − θ∗∥2
+H2 − ∥θ − ˜θ∗∥2
+H2 + ∥θ − ˜θ∗∥2
+H2 − ∥θ − ˜θ∗∥2
+˜H2
+�
+By the generalized self-concordance at θ∗
+e−R∥θ∗−˜θ∗∥2∥ · ∥2
+H1 ≤ ∥ · ∥2
+H2 ≤ eR∥θ∗−˜θ∗∥2∥ · ∥2
+H1
+This allows us to convert from ∥ · ∥H2 to ∥ · ∥H1, and as a consequence:
+��∥θ − θ∗∥2
+H1 − ∥θ − θ∗∥2
+H2
+�� ≤ [eR∥θ∗−˜θ∗∥2 − 1]∥θ − θ∗∥2
+H1.
+Also,
+∥θ − θ∗∥2
+H2 − ∥θ − ˜θ∗∥2
+H2 =
+�
+˜θ∗ − θ∗, 2θ − 2θ∗ + θ∗ − ˜θ∗�
+H2 = 2⟨θ − θ∗, ˜θ∗ − θ∗⟩H2 − ∥θ∗ − ˜θ∗∥2
+H2
+Therefore
+���∥θ − θ∗∥2
+H2 − ∥θ − ˜θ∗∥2
+H2
+��� ≤ 2∥θ − θ∗∥H2∥θ∗ − ˜θ∗∥H2 + ∥θ∗ − ˜θ∗∥2
+H2
+≤ 2eR∥˜θ∗−θ∗∥2∥θ − θ∗∥H1∥θ∗ − ˜θ∗∥H + eR∥˜θ∗−θ∗∥2∥θ∗ − ˜θ∗∥2
+H.
+Then lastly we have
+0 ≥ ∥θ − ˜θ∗∥2
+H2 − ∥θ − ˜θ∗∥2
+˜H2 = −l′′(y, xT ˜θ∗)
+�
+⟨x, θ − θ∗⟩ + ⟨x, θ∗ − ˜θ∗⟩
+�2
+≥ −2β∥x∥2
+H−1
+1 ∥θ − θ∗∥2
+H1 − 2β∥x∥2
+H−1∥θ∗ − ˜θ∗∥2
+H
+���∥θ − ˜θ∗∥2
+H2 − ∥θ − ˜θ∗∥2
+˜H2
+��� ≤ 2β∥x∥2
+H−1
+1 ∥θ − θ∗∥2
+H1 + 2β∥x∥2
+H−1∥θ∗ − ˜θ∗∥2
+H
+Combine the above derivations, we get
+|(∗∗)| ≤ γ
+2
+�
+a∥θ − θ∗∥2
+H1 + b∥θ − θ∗∥H1 + c
+�
+(8)
+where
+a :=
+�
+eR∥θ∗−˜θ∗∥2 − 1 + 2β∥x∥2
+H−1
+1
+�
+b :=2eR∥θ∗−˜θ∗∥2∥θ∗ − ˜θ∗∥H
+c :=(eR∥θ∗−˜θ∗∥2 + 2β∥x∥2
+H−1)∥θ∗ − ˜θ∗∥2
+H
+28
+
+Lastly, by (6) and (8),
+����log p(θ|Z)
+p(θ|Z′)
+���� ≤ R∥θ∗ − ˜θ∗∥2 + log(1 − µ2) + γ
+2[aW 2 + bW + c].
+where according to the algorithm W := ∥θ − θ∗∥H1 follows a half-normal distribution with σ =
+γ−1/2.
+By standard Gaussian tail bound, we have for all δ < 2/e.
+P(|W| ≤ γ−1/2�
+log(2/δ)) ≤ δ.
+This implies that a high probability upper bound of the absolute value of the privacy loss random
+variable log p(θ|Z)
+p(θ|Z′) under p(θ|Z). By the tail bound to privacy conversion lemma (Lemma ??), we
+get that for any set S ⊂ Θ P(θ ∈ S|Z) ≤ eϵP(θ ∈ S|Z′) + δ for any 0 < δ < 2/e and
+ϵ = R∥θ∗ − ˜θ∗∥2 + log(1 − µ2) + γc
+2 + a
+2 log(2/δ) + γ1/2b
+2
+�
+log(2/δ).
+Denote v := θ∗ − ˜θ∗, by strong convexity
+∥v∥2 ≤ ∥∇l(y, xT θ)[˜θ∗]∥2/α = |l′|∥x∥2/α ≤ L∥x∥2/α
+and
+∥v∥H ≤ ∥∇l(y, xT θ)[˜θ∗]∥H−1 = |l′|∥x∥H−1 ≤ L∥x∥H−1.
+Also use the fact that | log(1 − µ2)| ≤ 2µ2 for µ2 < 0.5 and µ2 ≤ β∥x∥2
+˜H−1
+2 , we can then combine
+similar terms and have a more compact representation.
+ϵ ≤ ϵ0(1 + log(2/δ)) + e
+RL∥x∥2
+α
+�γL2∥x∥2
+H−1
+2
++
+�
+γL2∥x∥2
+H−1 log(2/δ)
+�
+where
+ϵ0 ≤ e
+RL∥x∥2
+α
+− 1 + 2β∥x∥2
+H−1
+1
++ 2β∥x∥2
+˜H−1
+2
+is the part of the privacy loss that does not get smaller as γ decreases.
+Proposition D.2. Let ∥ · ∥ be a norm and ∥ · ∥∗ be its dual norm. Let F(θ), f(θ) and ˜F(θ) =
+F(θ) + f(θ) be proper convex functions and θ∗ and
+˜
+theta
+∗ be their minimizers, i.e., 0 ∈ ∂F(θ∗) and
+0 ∈ ∂ ˜F(
+˜
+theta
+∗). If in addition, F, ˜F is α, ˜α-strongly convex with respect to ∥ · ∥ within the restricted
+domain θ ∈ {tθ∗ + (1 − t)˜θ∗ | t ∈ [0, 1]}. Then there exists g ∈ ∂f(θ∗) and ˜g ∈ ∂f(˜θ∗) such that
+∥θ∗ − ˜θ∗∥ ≤ min
+� 1
+α∥˜g∥∗, 1
+˜α∥g∥∗
+�
+.
+Proof. Apply the first order condition to F restricted to the line segment between ˜θ∗ and θ∗, there
+are we get
+F(˜θ∗) ≥ F(θ∗) + ⟨∂F(θ∗), ˜θ∗ − θ∗⟩ + α
+2 ∥˜θ∗ − θ∗∥2
+(9)
+F(θ∗) ≥ F(˜θ∗) + ⟨∂F(˜θ∗), θ∗ − ˜θ∗⟩ + α
+2 ∥˜θ∗ − θ∗∥2
+(10)
+29
+
+Note by the convexity of F and f, ∂ ˜F = ∂F + ∂f, where + is the Minkowski Sum. Therefore,
+0 ∈ ∂ ˜F(˜θ∗) implies that there exists ˜g such that ˜g ∈ ∂f(˜θ∗) and −˜g ∈ ∂F(˜θ∗). Take −˜g ∈ ∂F(˜θ∗) in
+Equation 10 and 0 ∈ ∂F(θ∗) in Equation 9 and add the two inequalities, we obtain
+0 ≥ ⟨−˜g, θ∗ − ˜θ∗⟩ + α∥˜θ∗ − θ∗∥2 ≥ −∥˜g∥∗∥θ∗ − ˜θ∗∥ + α∥˜θ∗ − θ∗∥2.
+For ∥˜θ∗ − θ∗∥ = 0 the claim is trivially true, otherwise, we can divide the both sides of the above
+inequality by ∥˜θ∗ − θ∗∥ and get ∥θ∗ − ˜θ∗∥ ≤ 1
+α∥˜g∥∗.
+It remains to show that ∥θ∗ − ˜θ∗∥ ≤ 1
+˜α∥g∥∗. This can be obtained by exactly the same arguments
+above but applying strong convexity to ˜F instead. Note that we can actually get something slightly
+stronger than the statement because the inequality holds for all g ∈ ∂f(θ∗).
+A consequence of (generalized) self-concordance is the spectral (multiplicative) stability of Hessian
+to small perturbations of parameters.
+Lemma D.3 (Stability of Hessian[Nesterov and Nemirovskii, 1994, Theorem 2.1.1], [Bach, 2010,
+Proposition 1]). Let Hθ := ∇2Fs(θ). If Fs is R-self-concordant at θ. Then for any v such that
+R∥v∥Hθ < 1, we have that
+(1 − R∥v∥Hθ)2∇2Fs(θ) ≺ ∇2Fs(θ + v) ≺
+1
+(1 − R∥v∥Hθ)2 ∇2Fs(θ).
+If instead we assume Fs is R-generalized-self-concordant at θ with respect to norm ∥ · ∥, then
+e−R∥v∥∇2Fs(θ) ≺ ∇2Fs(θ + v) ≺ eR∥v∥∇2Fs(θ)
+The two bounds are almost identical when R∥v∥ and R∥v∥θ are close to 0, in particular, for x ≤ 1/2,
+e−2x ≤ 1 − x ≤ e−x.
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+32
+

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+EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE
+MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG
+Abstract. We generate anti-self-polar polytopes via a numerical implementation of the
+gradient flow induced by the diameter functional on the space of all finite subsets of the
+sphere, and prove related results on the critical points of the diameter functional as well as
+results about the combinatorics of such polytopes. We also discuss potential connections to
+Borsuk’s conjecture.
+Contents
+1.
+Introduction
+1
+2.
+Pointwise extremal sets
+3
+2.1.
+The pyramid construction
+4
+2.2.
+Construction of k-stacks
+5
+3.
+Minimal sets on S2 with diameter below the first accumulation critical value
+7
+3.1.
+Configuration space
+8
+3.2.
+Finiteness results
+8
+3.3.
+A labeling strategy for the points in Bk
+9
+4.
+Anti-self-polar polytopes
+10
+4.1.
+ASP polytopes
+11
+4.2.
+Borsuk’s conjecture
+12
+4.3.
+Proof of Lovasz’s theorem
+13
+4.4.
+4-dimensional polytopes
+15
+5.
+Implementation of the diameter gradient flow
+16
+6.
+Computational results
+17
+6.1.
+Pointwise extremal configurations on S2
+18
+6.2.
+Pointwise extremal configurations on S3
+20
+Appendix A.
+Semi-algebraic sets
+21
+References
+21
+1. Introduction
+Let (X, dX) be a metric space. The Kuratowski embedding x �→ dX(x, ·) is an embedding
+of X into L∞(X), the space of all bounded real-valued functions on X with the uniform
+norm. When X is the unit sphere with its geodesic distance, the homotopy types of the
+r-neighborhoods Br(X, L∞(X)) in the Kuratowski embedding of X were studied by Katz
+in [Kat91]. The values at which the homotopy type changes are closely related to the critical
+configurations of the diameter functional diam of X which maps a finite subset A of X to
+diam(A) := maxa,a′∈A dX(a, a′). When X is the unit circle, such critical values turn out to
+be exactly one-half of the diameter values of odd regular polygons inscribed in S1. Note that
+1
+arXiv:2301.13076v1  [math.CO]  30 Jan 2023
+
+2
+MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG
+the vertex sets of odd regular polygons are exactly the configurations that are local minima
+of the diameter functional on the space of all finite subsets of S1 equipped with Hausdorff
+distance. In [Kat89], Katz studied the diameter-extremal configurations on S2 and S3. The
+latter provide candidates for testing Borsuk’s conjecture in R4 (see below).
+Recently, Lim, M´emoli, and Okutan [LMO22, Theorem 5] proved that the homotopy types
+of neighborhoods of the Kuratowski embedding of X are naturally homotopy equivalent to
+the so-called Vietoris–Rips complexes of X, a central object in the field of applied algebraic
+topology. Therefore, the study of diameter-extremal configurations is also of interest for
+understanding the properties of the Vietoris-Rips complex of spheres [AA17, AAF18].
+In this paper, we extend the investigation of diameter-extremal configurations on spheres
+started in [Kat89].
+In the S1 case, the critical values of the diameter functional form a
+convergent sequence with the only accumulation point being π. It is natural to wonder to
+what extent a similar behavior is true on S2. We consider two canonical families of diameter-
+extremal configurations on S2 which we call pyramids Ak that contains 2k + 2 points (see
+Section 2.1) and stacked-triangles Bk that contains 3k + 1 points (see Section 2.2). Both
+families contain infinitely many members with diameters monotonically approaching 2π
+3 . We
+prove in Theorem 3.10 that 2π
+3 is in fact the first accumulation point of the set of critical
+values of the diameter functional. In Proposition 3.17, we prove that the two families Ak and
+Bk do not exhaust all the possible configurations with similar diameter bounds, and in fact
+there are infinitely many additional diameter-extremal configurations. Diameter-extremal
+configuration with 3k points can be found by performing diameter gradient flow on a certain
+subset of Bk. When k is odd, by a parity argument, the resulting configuration cannot be
+an instance of Ak or Bk.
+We next devise and implement a computational algorithm (see Algorithm 1) that attempts
+to produce diameter extremal configurations. We use this algorithm to find new configu-
+rations not in Ak or Bk. Furthermore, we found configurations not isometric to the ones
+produced in the course of proving Proposition 3.17. See Table 1 for a complete list of all the
+configurations we found in this way with up to 10 points. The list contains 10 previously
+unknown configurations where 8 of those exhibit Z2 symmetry and the remaining two are
+asymmetric; see Figures 7 and 8.
+The convex hulls of certain diameter-extremal configurations give rise to anti-self-polar
+polytopes (ASP), for example, the regular tetrahedron and any Ak or Bk. ASPs are polytopes
+P characterized by the property that the polar of P equals −P (see Definition 4.3). ASPs
+have been studied by Lov´asz in the context of answering a question by Erd¨os and Graham
+[Lov83] and were also considered in [Kat89, Section 5] in the context of Borsuk’s conjecture.
+Borsuk’s conjecture (see Section 4.2) for a finite point set X in Rn is equivalent to the
+property that the chromatic number of the diameter graph (see Definition 4.7) of X is
+bounded above by n + 1. We continue to explore the suggestion in [Kat89] to use diameter-
+extremal configurations on S3 to test Borsuk’s conjecture in R4 (a case that is still open).
+As shown by Lovasz [Lov83], the chromatic number of the diameter graph associated to
+any ASP in Rn is at least n + 1. An ASP for which the inequality is strict would disprove
+Borsuk’s conjecture.
+It was conjectured in [Kat89] that the number of edges in the diameter graph of an ASP
+4-polytope with v vertices is at least 3v − 5. We use Kalai’s inequality from [Kalai94, Sec-
+tion 4.3] to prove such a bound in Theorem 4.21 below. We then formulate conjectures
+
+EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE
+3
+about the number of edges in the diameter graph for more general subsets on S3, see Conjec-
+tures 4.22 and 4.23. A calculation based on these two conjectures suggests that the maximum
+possible chromatic number of the diameter graph of a finite subset X ⊆ R4 is 6 instead 5,
+the number predicted by Borsuk’s conjecture.
+We perform experiments attempting to identify diameter-extremal configurations on the
+three-dimensional sphere. The interest in these experiments is twofold. On the one hand,
+it is naturally interesting to obtain an understanding of critical configurations beyond the
+case of S1 and S2. On the other hand, whereas Borsuk’s conjecture is known to be true
+in dimensions 2 and 3 but false in dimensions 64 and higher, its status for dimension 4 is
+unknown. Hence, by the above, it is tempting to seek a diameter-extremal configuration X of
+S3 whose convex hull is an ASP such that its diameter graph has chromatic number at least
+6. We discovered 65 new configurations on S3 not obtained by the pyramid construction
+(see 2.1) on a previously known configuration on S2; see Theorem 6.1. However, all the
+diameter graphs of these configurations have a chromatic number precisely 5.
+Acknowledgements. This work was partially supported by BSF #2020124, NSF CCF
+#1740761, and NSF IIS #1901360.
+2. Pointwise extremal sets
+Let Sn ⊆ Rn+1 be the unit sphere with its geodesic distance. For a subset Y ⊆ Sn, its
+diameter diam(Y ) is computed with respect to the geodesic distance on the sphere.
+Definition 2.1 (Taut sets in Sn). A finite subset Y ⊂ Sn is taut if one of the following
+equivalent conditions is satisfied:
+(1) the convex hull of Y contains the origin;
+(2) there are non-negative real numbers {ay}y∈Y , not all zero, satisfying
+�
+y∈Y
+ay y = 0,
+where y denotes the position vector of the point y ∈ Rn+1.
+Jung’s theorem immediately gives the following result.
+Proposition 2.2. If Y ⊂ Sn is taut, then diam(Y ) ≥ arccos
+� −1
+n+1
+�
+.
+The following observation will be useful in the sequel.
+Corollary 2.3. Let Y ⊂ Sn be a taut set such that |Y | = n+2 and diam(Y ) < arccos
+�
+− 1
+n
+�
+.
+Then the dimension of the vector space spanned by Y is equal to n + 1.
+In particular, if {a1, . . . , an+2} is any set of non-negative coefficients such that
+n+2
+�
+i=1
+aiyi = 0,
+then all ai must be positive.
+Proof. Suppose the vector space spanned by all points in Y is of dimension at most n. Then,
+the set {y1, y2, . . . , yn+2} must lie on some great sphere Sn−1 ⊆ Sn and it must be taut in
+Sn−1. Then, by Proposition 2.2, the set Y must have diameter at least arccos
+�
+− 1
+n
+�
+which
+contradicts the assumptions on Y . This concludes the first part of the proof.
+For the second part, without loss of generality, we assume that a1 = 0, then the set of vectors
+
+4
+MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG
+{y2, y3 . . . , yn+2} is linearly dependent and hence dim(span{y2, y3, . . . , yn+2}) < n + 1. The
+contradiction with the first part establishes the result.
+□
+Let Y be a subset of a metric space (X, dX). For any two points y, y′ ∈ Y , we say that y
+and y′ are comaximal in Y if dX(y, y′) = diam(Y ). In such a case, y is called a comaximal
+point with y′. We use the notation comaxY (y) to denote the set of all points in Y which are
+comaximal with y.
+For two points x, x′ ∈ Sn with distance less than π, there is a unique arclength-parametrized
+geodesic γx,x′ connecting x to x′ such that γx,x′(0) = x. Consider the unit tangent vector
+˙γx,x′(0) in the tangent space TxSn.
+We recall the notion of pointwise extremal subsets in Sn as in [Kat89].
+Definition 2.4 ([Kat89]). Let Y ⊆ Sn be a finite subset with no antipodal pairs. We say
+that y ∈ Y is held (in place) by Y if the set of vectors ˙γy,y′(0) as y′ runs over comaxY (y) is
+a taut set. We say that Y is pointwise extremal if every point y ∈ Y is held by Y .
+When n = 1, it is not difficult to see that, for all integers k ≥ 1, the vertex set of an
+inscribed regular (2k + 1)-gon is pointwise extremal. The following proposition shows the
+converse.
+Proposition 2.5. Let Y ⊆ S1 be a pointwise extremal set containing no pair of antipodal
+points. Then Y is the vertex set of an odd regular polygon inscribed in S1.
+Proof. Let y ∈ Y and let D = diam(Y ). Let RD be the clockwise rotation on S1 by angle D.
+As y is held by Y ⊆ S1, the set Y must contain both points in S1 at distance D from y. In
+particular, the set Y is invariant under the rotation RD. As Y is a finite subset, the quotient
+D
+2π must be rational. Let m
+n be the representation of
+D
+2π in lowest terms. Then the orbit of
+y under the rotation RD forms the vertex set of an inscribed regular n-gon Y ′ ⊆ S1. As Y
+does not contain any antipodal pairs, n is necessarily odd. Therefore, Y contains the vertex
+set of a odd regular n-gon Y ′ of the same diameter as Y . Then Y must coincide with Y ′ as
+adding any additional point to the set Y ′ would strictly increase the diameter.
+□
+2.1. The pyramid construction. In this section, we describe a class of pointwise extremal
+subsets of Sn called pyramids in [Kat89]. For any pointwise extremal subset Y ⊂ Sn−1, the
+pyramid construction provides a corresponding pointwise extremal subset in Sn that consists
+of a rescaled copy of Y together with one extra point. Let Sn ⊆ Rn+1 be the unit sphere.
+Let Z = (0, . . . , 0, 1) denote the “north pole”. Let xn+1 be the last coordinate of Rn+1. Then
+for each plane {xn+1 = a}, a ∈ R that meets Sn at more than one point, the intersection is
+a rescaled copy of Sn−1 which we call a horizontal section. Each horizontal section contains
+a suitable rescaled copy of Y which is isometrically embedded into it.
+Definition 2.6. The pyramid over Y is the subset of Sn consisting of the north pole Z
+together with a rescaled copy Y ′ of Y inside some horizontal section such that the diameter
+of Y ′ equals the distance from Z to the horizontal section. Denote by Pyr(Y ) the pyramid
+over a pointwise extremal subset Y .
+Let x, y ∈ Y be points with dSn−1(x, y) = diam(Y ). Let x′, y′ ∈ Pyr(Y ) be points corre-
+sponding to x, y. Then the triple Z, x′, y′ is the vertex set of a spherical equilateral triangle,
+with spherical angle ∢ x′Zy′ = diam(Y ). Applying the spherical theorem of cosines to the
+geodesic triangle △ x′Zy′, we obtain the following relationship:
+diam(Pyr(Y )) = arcsec
+�
+sec
+�
+diam(Y )
+�
+− 1
+�
+.
+
+EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE
+5
+Example 2.7 (The Ak family in S2). Let k ≥ 1. We apply the pyramid construction to the
+regular (2k + 1)-gon on S1 to obtain a pointwise extremal configuration Ak ⊆ S2, consisting
+of the north pole of S2 together with a suitably rescaled copy of the regular (2k + 1)-gon, so
+that diam(Ak) = arcsec
+�
+sec
+� 2kπ
+2k+1
+�
+− 1
+�
+; see Figure 1. In particular, the diameter diam(Ak)
+tends to 2π
+3 as k goes to infinity.
+Figure 1. The configuration A2 consists of the north pole and the vertices
+of a regular pentagon.
+2.2. Construction of k-stacks. Following [Kat89], let a β-digon be the convex region on
+S2 bounded by two meridians (great semicircles joining the north and south poles), with
+angle β between the two meridians.
+Given a β-digon, we now introduce a procedure that will be used to produce a certain
+type of pointwise extremal set Y ⊆ Sn called a k-stack. The digon procedure is a “walking
+process” on the digon that takes as input an odd integer 2k + 1 ≥ 3 and outputs a suitable
+step length d1 > β.
+We start walking with equal steps from the north pole on alternating sides of the digon,
+with step length d1 calibrated so as to get exactly to the south pole after 2k + 1 steps; see
+Figure 2.
+Let Z ∈ Sn be the north pole. A regular n-simplex inscribed in the equator Sn−1 ⊂ Sn
+defines n + 1 meridians passing through the vertices of the simplex.
+Let ℓ ∈ (0, π). The set of points on Sn which are at distance ℓ away from the north pole Z
+is a rescaled (n−1)-sphere Sn−1
+ℓ
+, namely a horizontal section of Sn. The intersection between
+Sn−1
+ℓ
+and the set of n + 1 meridians is the vertex set of an inscribed n-simplex in Sn−1
+ℓ
+.
+Let k ≥ 1. A k-stacked configuration Y (see Figure 2) consists of the north pole Z together
+with the union of the vertex sets of k stacked n-simplices each obtained as the intersection of
+a horizontal (n − 1)-sphere Sn−1
+ℓi
+with the n + 1 meridians. The distances ℓ1, . . . , ℓk between
+the horizontal sections and the north pole are determined by the digon procedure as follows.
+Let d1 be the step length that comes from the digon procedure with input 2k + 1. Consider
+the sequence of numbers {dj}2k+1
+j=0
+where dj is the distance to the north pole of the point
+obtained after walking j steps in via the digon procedure. Then, the sequence of numbers
+{ℓi}1≤i≤k is defined in terms of {di}2k+1
+i=0
+by setting
+ℓi = d2i
+for
+1 ≤ i ≤ k.
+Note that d2k = diam(Y ) and d1 = π − d2k.
+Given an odd integer 2k + 1 ≥ 3, the following system of equations summarizes the
+computation of di for 1 ≤ i ≤ 2k + 1.
+
+6
+MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG
+Figure 2. Each value di in Equation (2.1) is the distance between the point
+pi shown in this figure and the north pole. For 1 ≤ i ≤ 2k + 1, the distance
+between pi and pi+1 is d1. The two conditions in Equation (2.1) are obtained by
+requiring p0 to be the north pole and p2k+1 to be the south pole. The conditions
+in the second line of Equation (2.1) are obtained by applying the theorem of
+cosines for the geodesic spherical triangles with vertices {Z, pi, pi+1}, for each
+1 ≤ i ≤ 2k + 1.
+The third line Equations (2.1) is obtained by symmetry
+considerations.
+Let βn = arccos( 1
+n). The values {di}0≤i≤2k+1 are determined by n and k via the following
+equations (see Figure 2):
+(2.1)
+�
+�
+�
+�
+�
+d0 = 0, d2k+1 = π
+cos(di) cos(di+1) + sin(di) sin(di+1) cos(βn) = cos(d1), 1 ≤ i ≤ 2k
+di + d2k+1−i = π, 0 ≤ i ≤ 2k + 1.
+Remark 2.8. Let d1 be the output of the digon procedure with input 2k + 1 on a digon of
+angle β. If we perform the “walking process” on a digon of angle π − β with complementary
+step length π − d1, we will eventually get close to the south pole (but will not reach it) and
+then will start walking back to the north pole and reach it after 2k + 1 steps. If we add an
+edge between the points that we traveled during the “walking process”, we obtain the diameter
+graph (see Definition 4.7) of a regular 2k + 1-gon.
+Example 2.9 (The Bk family in S2). When n = 2, for each k, we denote the stacks that
+result from the digon procedure by Bk, which consists of the vertices of k stacked triangles
+(2-simplices) together with the north pole. Note that B1 coincides with the configuration
+A1 from Example 2.7. By construction, diam(Bk) = π − d1 < π − arccos( 1
+2) =
+2π
+3 and
+limk→∞ diam(Bk) = 2π
+3 .
+
+EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE
+7
+Figure 3. The configuration B2 that consists of the north pole and vertices
+of two stacked triangles. The green dash lines are meridians; the red dot is the
+north pole, and points of the same color are of the same distance to the north
+pole.
+Example 2.10 (The Tk family in S3). Let n = 3. For each k, we denote the stacks that
+result from the digon procedure by Tk, which consists of the vertices of k stacked tetrahedra
+together with the north pole.
+3. Minimal sets on S2 with diameter below the first accumulation critical
+value
+Let d > 0 and let D(S2, d) be the set of all finite subsets Y ⊂ S2 with diam(Y ) < d.
+As each finite subset on S2 is closed, the Hausdorff distance dH is a metric on D(S2, d).
+Definition 3.1 (Diameter-extremal sets in D(S2, d) [Kat89]). A subset Y ∈ D(S2, d) is called
+diameter-extremal for the diameter functional if there is a little-o function such that
+diam(Y ) ≤ diam(Y ′) + o( dH(Y, Y ′))
+for all Y ′ ⊂ S2. In other words, we have
+lim
+dH(Y ′,Y )→0
+diam(Y ′) − diam(Y )
+dH(Y ′, Y )
+≥ 0.
+Remark 3.2. An n-point set Y is diameter-extremal if and only if at the corresponding
+point in the configuration space (S2)×n, the gradients of the distances between pairs of points
+at maximal distance form a taut set (see further in Section 3.1).
+Lemma 3.3 ([Kat89, Corollary 3.4]). A diameter-extremal set Y ∈ D(S2, 2π
+3 ) is necessarily
+pointwise extremal.
+Definition 3.4 (Minimal set in D(S2, d) [Kat89]). A subset Y ∈ D(S2, d) is called a minimal
+set if there is some δ > 0 such that diam(Y ) ≤ diam(Y ′) for all finite subsets Y ′ with
+dH(Y, Y ′) ≤ δ.
+Clearly, every minimal set is diameter-extremal. In fact, there is a converse.
+Theorem 3.5 ( [Kat89, Theorem 2]). Every diameter-extremal set in D(S2, 2π
+3 ) is a minimal
+set on S2.
+By a mountain-pass argument, one obtains the following consequence.
+Lemma 3.6 ([Kat89, Corollary 2]). There is exactly one (up to congruence) minimal set in
+each connected component of D(S2, 2π
+3 ).
+
+8
+MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG
+3.1. Configuration space. We will now estimate the number of such connected compo-
+nents. We use the notation
+k�
+diam≤d
+S2 to denote the set of all tuples (y1, . . . , yk) in �k S2 such
+that the diameter of its associated set {y1, . . . , yk} is less than or equal to d. Note that, for
+any ϵ > 0, we have a natural continuous map
+k
+�
+diam≤d
+S2 −→ D(S2, d + ϵ).
+By realizing
+k�
+diam≤d
+S2 as a closed semi-algebraic set, we obtain the following upper bound on
+the number of connected components in
+k�
+diam≤d
+S2.
+Lemma 3.7. Let k ≥ 0. We set sk = 2k+ k(k+1)
+2
+. Then, for every d > 0, the number b0(k, d)
+of connected components of
+k�
+diam≤d
+S2 satisfies
+b0(k, d) ≤ 2sk(4sk − 1)3k−1.
+Proof. We will first describe the set
+k�
+diam≤d
+S2 as a closed basic semi-algebraic set in R3k. Let
+xi,j, where 1 ≤ i ≤ k and 1 ≤ j ≤ 3, denote the standard coordinates in R3k. Then the set
+k�
+diam≤d
+S2 is characterized by the following conditions:
+�
+x2
+i,1 + x2
+i,2 + x2
+i,3 = 1
+for all 1 ≤ i ≤ k,
+(xi,1 − xi′,1)2 + (xi,2 − xi′,2)2 + (xi,3 − xi′,3)2 ≤ d2
+for all 1 ≤ i < i′ ≤ k.
+Therefore the set
+k�
+diam≤d
+S2 is a basic semi-algebraic set given by sk = 2k + k(k+1)
+2
+non-strict
+inequalities. Then Theorem A.5 implies that b0(k, d) ≤ 1
+2(2sk + 2)(2sk + 1)3k−1.
+□
+3.2. Finiteness results. Lemma 4.1 and Lemma 4.3 in [Kat89] imply the following result.
+Lemma 3.8 ([Kat89]). Let 0 < d <
+2π
+3 .
+Let Y
+∈ D(S2, d) be a pointwise extremal
+set.
+Then for any pair of distinct points y, y′ in Y , the distance dS2(y, y′) is at least
+arccos
+�
+2 cos2(d)
+cos2(d/2) − 1
+�
+.
+By a packing argument on the sphere, we obtain the following result.
+Corollary 3.9. For each ϵ > 0, there is a positive integer N(ϵ) such that every pointwise
+extremal subset Y of diameter less than 2π
+3 − ϵ contains fewer than N(ϵ) points.
+Theorem 3.10. For each 0 < ϵ < 2π
+3 , there are only finitely many diameter-extremal sets
+in D(S2, 2π
+3 − ϵ).
+In particular,
+2π
+3 is the first accumulation point of the critical values of the diameter
+functional of S2.
+
+EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE
+9
+Proof. Let dϵ =
+2π
+3 − ϵ. By Theorem 3.5, it suffices to show that there are only finitely
+many minimal sets in D(S2, dϵ). By Corollary 3.9, there is some N such that every pointwise
+extremal set in D(S2, dϵ) contains no more than N points.
+Therefore the image of the
+continuous map φ
+φ :
+N
+�
+diam≤dϵ
+S2 −→ D(S2, 2π
+3 )
+contains all pointwise extremal configurations with diameter less than or equal to dϵ. By
+Lemma 3.3, the image of φ (in particular) contains all minimal sets with diameter not
+exceeding dϵ.
+Let Cϵ be the number of connected components which contain a minimal set with diameter
+no more than dϵ. By Lemma 3.6, the number of minimal sets in D(S2, dϵ) is at most Cϵ. As
+the image of φ contains all minimal sets with diameter no more than dϵ, the number Cϵ is
+bounded by the rank of the map
+φ∗ : H0
+�
+N
+�
+diam≤dϵ
+S2
+�
+−→ H0
+�
+D(S2, 2π
+3 )
+�
+.
+The claim now follows by invoking the upper bound on the dimension of H0
+�
+N�
+diam≤dϵ
+S2
+�
+from Lemma 3.7.
+□
+3.3. A labeling strategy for the points in Bk. Recall that Bk ⊆ S2 consists of the north
+pole and the vertices of k stacked triangles, and that the vertices of the stacked triangles are
+distributed along three meridians.
+We label the north pole as Z, then label the vertices of the i-th triangle (counting from the
+north pole) by Pi, Qi, Ri in such a way that all the Pi, 1 ≤ i ≤ k are on a common longitude
+and similarly for all Qi, 1 ≤ i ≤ k and Ri, 1 ≤ i ≤ k.
+Definition 3.11. The subset dEBk is obtained by removing the points with indexes in E
+from Bk.
+Definition 3.12. A set Y ⊆ S2 is separable if for each pair of points x, y ∈ Y there are two
+other points z, w ∈ Y such that the 4-tuple {x, y, z, w} is taut.
+Lemma 3.13 ([Kat89, Lemma 4.1]). A pointwise extremal subset Y ⊂ S2 with diam(Y ) < 2π
+3
+is necessarily separable.
+The proof of the above lemma in [Kat89] gives the following stronger result.
+Lemma 3.14. Let Y ⊂ S2 be a subset with diam(Y ) < 2π
+3 . Suppose x ∈ Y is held by Y .
+Then for any other point y ∈ Y , there exist z, w ∈ Y such that the four-point set {x, y, z, w}
+is taut.
+We will now analyze variations of subsets which are continuous with respect to the Haus-
+dorff distance.
+Lemma 3.15. Let {Yt, t ∈ [0, 1]} be a continuous family of subsets of S2 with at most 4
+points. Suppose the following two conditions hold:
+• the set Y0 is taut,
+• Yt ∈ D(S2, 2π
+3 ) for every t ∈ [0, 1].
+
+10
+MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG
+Then Yt is taut for each t ∈ [0, 1].
+Proof. As the set Y0 is taut and diam(Y ) < 2π
+3 , Corollary 2.3 implies that the convex hull
+H0 of Y0 is a tetrahedron and that the origin 0 is in interior of H0. For each t ∈ [0, 1] let Ht
+be the convex hull of the set Yt. To show that each set Yt is taut, it suffices to show that the
+origin 0 stays in the interior of Ht for all t ∈ [0, 1].
+Suppose the contrary. Let t0 be the supremum of t such that 0 is in the interior of Ht
+for all smaller values of t. Either Ht0 is nondegenerate and then 0 must belong to one of its
+(triangular) faces, or it is degenerate, i.e., lies in a plane through the origin. In either case,
+we obtain a taut subset of the circle given by the intersection of the plane with the sphere,
+and can apply Jung’s theorem.
+Namely, by Proposition 2.2 we obtain diam(Yt0) ≥ 2π
+3 , contradicting the hypothesis Yt0 ∈
+D(S2, 2π
+3 ) and proving the lemma.
+□
+Corollary 3.16. Let Yt, t ∈ [0, 1] be a path in D(S2, 2π
+3 ). If a certain 4-tuple in Y0 is taut,
+then it continues to be taut for all t ∈ [0, 1].
+Proposition 3.17. There exist infinitely many (up to congruence) pointwise extremal sets
+in D(S2, 2π
+3 ) that are not contained in the family Ak or Bk.
+Proof. Since each connected component contains a (unique) minimal set, it suffices to show
+that for each k, the configuration dPkBk is separable.
+By Lemma 3.14, we can separate most pairs of points from dPkBk except for a pair of
+points from the triple of points at maximal distance from Pk, namely the points Z, Q1, and
+R1. Let us check that such pairs don’t coalesce, either. This is immediate from the fact that
+if we remove all layers except the first and the k-th, the remaining configuration is in the
+connected component in D(S2, 2π
+3 ) of the 7-point minimal set B2. Thus, by Corollary 3.16,
+it suffices to check that if we remove P2 from B2, no remaining points coalesce. This can be
+checked directly, and also follows from the fact that the diameter flow applied to the 6-point
+configuration dP2B2 produces the 6-point minimal set A2 (see Section 5).
+□
+4. Anti-self-polar polytopes
+In this paper, we adopt the following restricted definition of a polytope: a (convex) polytope
+will be the convex hull of any finite set of points in Rn.
+Definition 4.1. The affine hull aff(S) of a set S ⊆ Rn is
+aff(S) =
+� k
+�
+i=1
+αixi
+����� k > 0, xi ∈ S, αi ∈ R,
+k
+�
+i=1
+αi = 1
+�
+.
+We now give the formal definition of a face of a polytope following [Zie12].
+Definition 4.2 ([Zie12, Definition 2.1]). Let P ⊆ Rd be a convex polytope. A linear inequality
+⟨c, x⟩ ≤ c0 is valid for P if it is satisfied for all points x ∈ P. A face of P is any set of the
+form
+F = P ∩
+�
+x ∈ Rd : ⟨c, x⟩ = c0
+�
+where ⟨c, x⟩ ≤ c0 is a valid inequality for P.
+
+EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE
+11
+The dimension of a polytope P is defined to be the dimension of its affine hull aff(P)
+(regarded as an affine space). A 3-dimensional polytope is a polyhedron. The codimension-
+one faces of a polytope P are called facets; the codimension-two faces are called ridges. If
+each face of P is a simplex, then P is called a simplicial polytope. We will use fi(P) to
+denote the number of i-faces of the polytope P. When there is no risk of confusion, we will
+denote fi(P) by just fi. For a n-dimensional polytope, the vector (f0, f1, . . . , fn−1) is called
+the f-vector of P.
+4.1. ASP polytopes. In [Lov83], Lov´asz introduced the following type of polytopes which
+we will refer to as anti-self-polar (ASP) polytopes.1
+Our terminology will be justified in
+Remark 4.4.
+Definition 4.3 (Anti-self-polar polytopes). Let P ⊆ Rn be a n-dimensional polytope. We
+say that P is anti-self-polar (ASP) if the following three conditions hold:
+(1) P is inscribed in the unit sphere Sn−1 ⊆ Rn.
+(2) P is circumscribed around a sphere centered at the origin with radius s for some
+0 < s < 1.
+(3) There is a bijection σ between vertices and facets of P such that if v is any vertex
+then the facet σ(v) is orthogonal to the vector v.
+Remark 4.4. Let P ⊂ Rn be a polytope containing the origin 0. Let Sn−1
+r
+(0) be the sphere
+centered at 0 ∈ Rn with radius r > 0. The polar body of P with respect to the sphere Sn−1
+r
+(0)
+is defined to be the set
+polarr(P) = {x ∈ Rn| ⟨x, y⟩ ≤ r2 for all y ∈ P}.
+As shown in [Hor21], the condition for an ASP polytope in Rn can be restated using the
+terminology of polar bodies. In terms of our definition of polarity, if P is an ASP polytope,
+then there exists some r such that the following relation holds; see [Hor21, Lemma 1].
+polarr(P) = −P.
+The polar body description shows that for each 0 ≤ i ≤ n − 1, the bijection σ in condition
+(3) can be extended to a bijection between the set of i-dimensional faces and the set of
+(n − i − 1)-dimensional faces; see [Hor21, Lemma 2].
+Proposition 4.5 ([Kat89, Remark after Theorem 1]). Let Y ⊂ S2 be a pointwise extremal
+subset with diam(Y ) < 2π
+3 . Then the convex hull of Y is an ASP polyhedron.
+Remark 4.6. The result above no longer holds if the restriction on the diameter is removed.
+A counterexample is given by an 8-point configuration Y ⊆ S2 consisting of the vertices of
+an antiprism over a square (see Figure 4). If the diameter of Y is exactly attained by the
+diagonals of the two squares and by the pairs that consist of a vertex of one square and one
+of the two farthest vertices of the other square, then Y is pointwise extremal. However, the
+convex hull of Y is not ASP. Indeed, note that the top square is a facet of the convex hull
+of Y . If the convex hull of Y were ASP, then there would be a vertex y0 ∈ Y such that the
+distance from y0 to each vertex of the top square would equal diam(Y ). But, our construction
+of Y does not satisfy this.
+1Lov´asz [Lov83] and Horv`ath[Hor21] use the terminology “strongly self-dual polytopes”.
+
+12
+MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG
+Figure 4. The antiprism on a square.
+Definition 4.7. Let Y ⊆ Rn be a finite subset. The diameter graph G(Y ) of Y is defined
+to be the graph with vertex set V (G) = Y and two vertices y, y′ in G are connected if and
+only if y and y′ are comaximal in Y .
+Given a polytope P, we will refer to the diameter graph of the vertex set of P simply as
+the diameter graph of P. We denote the diameter graph of P by G(P).
+Definition 4.8. The chromatic number χ(G) of a graph G is the smallest number of colors
+needed to color the vertices so that no two adjacent vertices share the same color.
+The following property of the diameter graph G(P) of an ASP polytope P follows from
+[Lov83, Lemma 2 and Lemma 3]. Recall that σ denotes the bijection between the vertex set
+and the set of facets of P. In [Lov83, Lemma 1], it is shown that for any two vertices v, v′
+of P, the condition v ∈ σ(v′) is equivalent to v′ ∈ σ(v).
+Proposition 4.9. Let P be an ASP polytope. Two vertices v, v′ in G(P) are connected by
+an edge in G(P) if and only if v ∈ σ(v′), when viewed as vertices in P.
+Theorem 4.10 ([Lov83, Theorem 2]). The diameter graph G(P) of an n-dimensional ASP
+polytope P ⊆ Rn satisfies χ(G(P)) ≥ n + 1.
+The proof of the theorem is discussed in Section 4.3. The chromatic number of a diameter
+graph G(Y ) of a subset Y ⊂ Rn is closely related to the following conjecture of Borsuk.
+4.2. Borsuk’s conjecture.
+Conjecture 4.11 (Borsuk’s conjecture). Let Y be a bounded subset of Rn. Then there is a
+partition of Y into n + 1 sets each of which has a smaller diameter than Y .
+For finite subsets, Borsuk’s conjecture has the following equivalent form in terms of diam-
+eter graphs:
+For every finite bounded subset Y ⊆ Rn, the chromatic number of the diam-
+eter graph G(Y ) of Y is no greater than n + 1.
+To see the above equivalence, a partition {Y1, . . . , Yk} of Y is equivalent to a coloring of
+Y by requiring that two points are of the same color if and only if they both belong to
+
+EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE
+13
+some Yi, 1 ≤ i ≤ k. Therefore, since Y is a finite set, the condition that the diameter of each
+subset Yi is less than the diameter of Y is equivalent to requiring that the coloring associated
+to the partition {Y1, . . . , Yk} has the property that no two adjacent vertices in the diameter
+graph G(Y ) share the same color.
+Borsuk’s conjecture holds when n = 2 (Borsuk [Bor33]) and n = 3 (Perkal [Per47]). The
+general conjecture was disproved by Khan and Kalai [KK93]. The lowest dimensional coun-
+terexample currently known was constructed by Jenrich and Brouwer (and based on a con-
+struction by Bondarenko) in dimension 64 [JB14]. For additional information on the histori-
+cal developments on the construction of counterexamples to Borsuk’s conjecture, see [Rai13,
+Section 2].
+Remark 4.12. Let Y ⊂ Sn−1 be a finite subset. Given a regular geodesic n + 1-simplex
+∆geodesic
+n+1
+, Sn−1 can be partitioned into n + 1 connected parts {X1, X2, . . . , Xn+1} where each
+Xi contains the interior of one of the faces of ∆geodesic
+n+1
+. Therefore, by coloring points of
+Y according to which partition set Xi the point belongs to, we obtain a proper coloring of
+the diameter graph of Y provided that the diameter diam(Y ) diameter of Y is greater than
+ηn−1, the diameter of a face of ∆geodesic
+n+1
+. The above coloring strategy was first described in
+[Lov83, Section 0]. Though notice that [Lov83] made a mistake in computing the exact value
+of ηn−1 [Rai12, Rai13]. The correct values of ηn−1 first appeared in [San46] and reproduced
+in the context of ASP polytopes in [Hor21].
+By Theorem 4.10 and the fact that Borsuk’s conjecture is true for n = 3, the chromatic
+number χ(G(P)) of an ASP polyhedron P ⊆ R3 equals 4. In Figures 7 and 8, we display
+4-colorings of the diameter graphs of all the ASP polyhedra in Tables 5 and 6.
+Remark 4.13. Borsuk’s conjecture is still open for 4 ≤ n ≤ 63. Theorem 4.10 suggests
+that ASP polytopes are a natural source of potential counterexamples to Borsuk’s conjecture.
+Additionally, by Proposition 4.5, pointwise extremal configurations are closely related to ASP
+polytopes. In Section 6.2, we present some pointwise extremal subsets on S3 obtained through
+computer experiments. However, the pointwise extremal subsets that we have found so far
+all have chromatic number 5; cf. Section 6.2.
+4.3. Proof of Lovasz’s theorem. Theorem 4.10 was proved in [Lov83] by analyzing the
+neighborhood complex of the diameter graph of ASP polytopes.
+Definition 4.14 (Neighborhood complex). Let G be a finite graph.
+The neighborhood
+complex N(G) is the simplicial complex with vertex set V (G) such that a subset A ⊆ V (G)
+forms a simplex if and only if the points of A have a neighbor in common.
+In [Lov78], Lov´asz shows the following lower bound of the chromatic number of a graph
+with respect to the connectivity of its neighborhood complex. Recall a topological space X
+is k-connected if its homotopy groups are trivial up to degree k.
+Theorem 4.15 ([Lov78]). Let G be a graph and suppose that N(G) is k-connected (k ≥ 0).
+Then χ(G) ≥ k + 3.
+Lemma 4.16 ([Lov83, Lemma 4]). Let P be an ASP polytope and G(P) be its diameter
+graph. Then N(G(P)) is homotopy equivalent to the boundary of P.
+Proof of Theorem 4.10. By Lemma 4.16, N(G) is homotopy equivalent to the boundary of
+P.
+
+14
+MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG
+As P is a (convex) polytope, the boundary of P is homeomorphic to Sn−1. Hence N(G)
+is homotopy equivalent to Sn−1. Therefore, N(G) is (n − 2) connected. By Theorem 4.15,
+χ(G) ≥ n + 1.
+□
+Let d ≥ 2 and n ≥ 1 be integers and let e(d, n) be the maximum possible number of edges
+in the diameter graph of a subset of Rd with n points. When d = 2, it is shown in [HP34]
+that e(2, n) = n. This fact leads to one proof of Borsuk’s conjecture for finite subsets Y of
+R2. When d = 3, it was conjectured by V´azsonyi that e(3, n) = 2n − 2; see [Erd46]. The
+V´azsonyi’s conjecture was proved independently by Gr¨unbaum [Gr¨u56], Heppes [Hep56] and
+Straszewicz [Str57]. As mentioned in Heppes [Hep56], V´azsonyi’s conjecture implies that
+Borsuk’s conjecture is true for finite subsets in R3. We have already seen in Theorem 4.10
+that the diameter graph of an ASP polytope has high chromatic number, suggesting a
+possible approach to seeking higher-dimensional counterexamples.
+We now introduce a set of enumerative invariants fij(P) of a polytope P which will be
+used below. Informally, for i < j, fij(P) counts the number of pairs “i-face contained in a
+j-face” in the polytope P. Precisely,
+fij(P) := ♯{(φi, φj) | φi is a i-face of P, φj is a j-face of P, and φi ⊆ φj.}
+When there is no risk of confusion, we will simply use fij to denote fij(P). Thus f01 is the
+number of pairs “vertex contained in an edge”, namely just twice the number f1 of edges
+in P.
+Lemma 4.17. Let P be an anti-self-polar polytope of dimension d + 1. Let e(G(P)) be the
+number of edges in the graph G(P). Then f0d(P) = 2e(G(P)).
+Proof. Let V be the set of vertices of P and let W be the set of faces in P. Recall that σ
+denotes the bijection between V and the set of facets of P. By Proposition 4.9, we have
+2e(G(P)) =
+�
+v∈V
+f0(σ(v)) =
+�
+φd⊂W
+f0(φd) = f0d.
+The second equality above follows from the definition of σ.
+□
+Proposition 4.18. Every ASP polyhedron P ⊆ R3 satisfies e(G(P)) = 2f0 − 2.
+Proof. By Lemma 4.17, we have 2e(G(P)) = f02. Furthermore by duality we have f01 = f12.
+This enables us to give a possibly generalizable proof as follows.
+Note that, each face has as many vertices as edges and therefore f02 = f12. By duality,
+f12 = f01 which is twice the number of edges, namely 2f1. Thus the number of maximal
+distances is the same as the number of edges.
+Meanwhile by the formula for the Euler
+characteristic, for an anti-self-polar polyhedron we have f1 = 2f0 − 2. Altogether, we have
+f02 = f12 = f01 = 2f1 = 2(2f0 − 2).
+Thus the number of maximal distances is also 2f0 − 2.
+□
+In fact, it is shown in [Kat89] that every pointwise extremal set in S2 with diameter less
+than 2π
+3 exhibits the maximum number of possible edges.
+Theorem 4.19 ([Kat89, Theorem 1]). Suppose Y ⊂ S2 is a pointwise extremal set with
+N = |Y | and diam(Y ) < 2π
+3 . Then the number of edges in the diameter graph G(Y ) equals
+2N − 2.
+
+EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE
+15
+As noted in [Kat89, page 118], the example of the antiprism on a square constructed
+in Remark 4.13 shows that the above result is no longer true if we remove the diameter
+constraint: the diameter graph of the antiprism on a square has 8 vertices but only 12 edges.
+4.4. 4-dimensional polytopes. Consider the V´azsonyi’s problem in R4, that is, for a fixed
+n, determine the maximal possible number of edges e(4, n) amongst the diameter graphs of
+all possible n point sets in R4.
+Example 4.20. Let m be a positive integer and let Y := A ∪ B ⊂ S3 be a subset consisting
+of 2m points constructed as follows. The set A consists of m points on an arc of length less
+than π
+2 on a great circle whereas the (disjoint) set B consists of m points also on an arc of
+length less than π
+2 on an orthogonal great circle.
+Then each pair of points a ∈ A, b ∈ B is comaximal in Y . Thus e(4, n) is at least quadratic
+in n. It is shown in [Erd67] that e(4, n) exactly has quadratic growth rate in n.
+For an anti-self-polar polytope P ⊆ R4, we prove the following lower bound on the number
+of edges in the diameter graph G(P), originally conjectured in [Kat89, Section 5].
+Theorem 4.21. Let P ⊆ R4 be a 4-dimensional anti-self-polar polytope. Then the number
+of edges e(G(P)) in the diameter graph G(P) is at least 3f0(P) − 5.
+Proof. By Lemma 4.17, the assertion is equivalent to the bound f03(P) ≥ 6f0(P) − 10. For
+each facet φ of P, let aj
+φ be the number of j-gons occurring as faces of φ, and let aj denote
+the total number of j-gons occurring as faces of P. Kalai [Kalai94, Section 4.3] proved that
+every 4-dimensional polytope satisfies g2 ≥ 0 or equivalently
+a4 + 2a5 + · · · ≥ 4f0(P) − f1(P) − 10.
+Let φ run through all the facets of P. By Euler’s formula, we have
+f03(P) =
+�
+φ
+f0(φ)
+=
+�
+φ
+2 + f1(φ) − f2(φ)
+=
+�
+φ
+2 + 1
+2(3a3
+φ + 4a4
+φ + 5a5
+φ + · · · ) − f2(φ)
+=
+�
+φ
+2 + 1
+2f2(φ) + 1
+2(a4
+φ + 2a5
+φ + · · · )
+= 2f3(P) + f2(P) + 1
+2
+�
+φ
+(a4
+φ + 2a5
+φ + · · · )
+= 2f3(P) + f2(P) + (a4 + 2a5 + · · · )
+≥ 2f3(P) + f2(P) + 4f0(P) − f1(P) − 10
+= (2f3(P) + 4f0(P)) + (f2(P) − f1(P)) − 10
+= 6f0(P) − 10
+by duality, as required.
+□
+The above results suggest formulating the following conjectures.
+
+16
+MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG
+Conjecture 4.22. Every ASP polytope P ⊆ R4 satisfies e(G(P)) = 3f0(P) − 5.
+In Section 6.2, we report 65 configurations that we generate through numerical experi-
+ments. Each of those configurations confirms the above conjecture.
+Conjecture 4.23. Every subset X ⊆ S3 with diam(X) > π
+2 satisfies e(G(X)) ≤ 3|X| − 5.
+Assuming these conjectures and by an argument similar to the case of the S2 discussed
+on page 14, one can show that the chromatic number of the diameter graph of any set X
+in S3 with its diameter greater than π
+2 would be at most 6. Indeed, Conjecture 4.23 implies
+that one can always choose a point x0 ∈ X comaximal with at most 5 other points, by the
+pigeonhole principle. Thus, if X − {x0} can be colored with 6 colors, then X can be so
+colored, also, by using the color not used up by any of its 5 (or fewer) comaximal points, and
+we conclude by induction. The fact that this calculation produces the number 6 instead of
+5 would provide weak evidence toward the possibility that the Borsuk number of R4 might
+be the former rather than the latter.
+5. Implementation of the diameter gradient flow
+This section describes the implementation of the diameter gradient flow on spheres. Given
+a finite subset Y of Sn, we first test whether every point in Y is held. If there is a point y
+that is not held by Y , we then move y in the direction that points toward the center of the
+minimum bounding sphere of the tangent vectors determined by points in comaxY (y). We
+continue this process until every point in Y is held. In other words the point y is updated to
+a point yt =
+y0+tv0
+||y0+tv0|| where t > 0 is a parameter value determined through the Armijo rule
+[Arm66], and v0 is the unit tangent vector at y that points toward the center of the minimum
+
+EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE
+17
+bounding sphere of the set {˙γy,y′ | y′ ∈ comaxyY }. The pseudocode of the algorithm is shown
+below.
+Algorithm 1: DiameterGradientFlow
+Input: An initial finite subset Y on unit sphere Sn.
+Parameters: β, η ∈ (0, 1) for determing Armijo Rule stepsa.
+Output: The extremal configurations obtained under the diameter gradient flow
+with initial condition Y .
+1 Function IsHeld(y, Y ):
+2
+E ← comaxY (y)
+3
+Ty(E) ← {˙γy,y′ for y′ ∈ comaxY (y)}
+4
+if 0 in the convex hull of Ty(E) then
+5
+return True
+6
+else
+7
+return False
+8
+end if
+9
+10 Function Main(Y , β, η):
+11
+/* Initialize convergence tag
+*/
+12
+tag = False
+13
+while tag == False do
+14
+for y0 ∈ Y do
+15
+if IsHeld(y0, Y ) then
+16
+tag == True
+17
+else
+18
+E ← comaxY (y)
+19
+Ty0(E) ← {˙γy,y′} for y′ ∈ E}
+20
+v0 ← center of the minimum bounding sphere of Ty(E).
+21
+/* Determine the step size tk > 0 using Armijo Rule
+*/
+22
+tk = maxl∈N0 βl
+s.t.
+diam
+�
+Y \{y0} ∪
+�
+y0+tkv0
+||y0+tkv0||
+��
+≤ diam(Y ) − βlη
+23
+Y ← Y \{y0} ∪
+�
+y0+tkv0
+||y0+tkv0||
+�
+24
+tag == False
+25
+break
+26
+end if
+27
+end for
+28
+end while
+29 return
+asee [Arm66]
+6. Computational results
+In this section, we describe our computational results regarding pointwise extremal config-
+urations on S2 and S3 using the Algorithm 1. In most of our experiments, we set parameters
+β = 0.5, η = 0.001 and use the Python package MINIBALL([Dev21]) for finding the optimal
+direction for decreasing the diameter by moving a single point.
+
+18
+MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG
+6.1. Pointwise extremal configurations on S2. In this section, we present the compu-
+tational results from running the diameter gradient flow Algorithm 1 with initial sets, which
+are obtained by removing up to six points from Bk (Example 2.9) with k ≤ 5.
+In total, we obtain 54 configurations. We present in Table 1 the configurations with up to
+10 points2 that we found upon convergence of the gradient flow.
+Shape
+v
+f
+r
+t
+Diameter
+Symmetry Group
+Initial Set
+A1(= B1)
+4
+3
+4
+4
+1.91064
+S3
+dZB2
+A2
+6
+5
+1
+5
+2.03446
+D5
+dP2B2
+B2
+7
+4
+3
+4
+2.07654
+S3
+dZB3
+C1
+8
+5
+1
+4
+2.08707
+Z2
+d{Q1,P3}B3
+A3
+8
+7
+1
+7
+2.06459
+D7
+d{P1,P3}B3
+C2
+9
+5
+1
+3
+2.09335
+Z2
+d{P1,R1,Q3,Q4}B4
+C3
+9
+5
+1
+4
+2.09079
+Z2
+dP3B3
+C4
+9
+6
+1
+5
+2.09016
+Z2
+dP1B3
+B3
+10
+4
+6
+4
+2.09303
+S3
+dZB4
+D1
+10
+5
+1
+3
+2.09409
+{e}
+d{P1,Q1,P2,Q4,R4,R5}B5
+C5
+10
+5
+1
+4
+2.09317
+Z2
+d{P1,R3,Q4}B4
+C6
+10
+5
+2
+4
+2.09356
+Z2
+d{P1,Q1}B4
+C7
+10
+5
+3
+4
+2.09240
+Z2
+d{P1,R3,P4}B4
+D2
+10
+6
+1
+4
+2.09360
+{e}
+d{P1,R3,R4}B4
+C8
+10
+7
+1
+6
+2.09174
+Z2
+d{P1,P3,Q4}B4
+A4
+10
+9
+1
+9
+2.07654
+D9
+d{P1,P3,P4}B4
+Table 1.
+Pointwise extremal configurations on S2 with up to v = 10 vertices,
+sorted first by v, then by f (maximal number of edges in a face), then by r
+(number of faces with a maximal number of edges), then by t (number of
+triangles in the configuration’s diameter graph). For each of the 10 pointwise
+extremal configurations that we found, in the last column we list one initial
+set which leads to that configuration under the diameter gradient flow (a given
+pointwise extremal configuration may be reached from different initial sets).
+2An interactive visualization of the table can be found through the link:
+https://ndag.github.io/
+anti-self-dual-polyhedra/
+
+EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE
+19
+(a) C1
+(b) C2
+(c) C3
+(d) C4
+(e) C5
+(f) C6
+(g) C7
+(h) C8
+Figure 5. Eight Z2 symmetric pointwise extremal configurations with at
+most 10 points.
+(a) D1
+(b) D2
+Figure 6. Two asymmetric pointwise extremal configurations with 10 points.
+
+20
+MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG
+(a) C1
+(b) C2
+(c) C3
+(d) C4
+(e) C5
+(f) C6
+(g) C7
+(h) C8
+Figure 7. Diameter graphs of Z2 symmetric pointwise extremal configura-
+tions with less than 10 with a minimal coloring. Note that all diameter graphs
+above can be colored with four colors.
+(a) D1
+(b) D2
+Figure 8. The diameter graph of the two asymmetric pointwise extremal
+configurations D1 and D2.
+6.2. Pointwise extremal configurations on S3. In this section we present some compu-
+tational results on pointwise extremal configurations on S3. Recall that Tk ⊆ S3 denotes the
+k-stack; cf. Example 2.10. The Tk consists of the north pole and the vertices of k stacked
+3-simplices, for a total of 4k + 1 points.
+We use similar indexing for the points in Tk, that is, the north pole is denoted Z, then
+we label the verticees of i-th tetrahedron (counting from the north pole) by Pi, Qi, Ri, Si
+in such a way that all the Pi, 1 ≤ i ≤ k are on a common longitude and similarly for all
+Qi, 1 ≤ i ≤ k, Ri, 1 ≤ i ≤ k, and Si, 1 ≤ i ≤ k.
+Theorem 6.1. Applying the diameter gradient flow to the initial sets of the diameter gradient
+flow be the subsets of T1, T2, T3, T4 with at most four points removed, one obtains at least 65
+distinct pointwise-extremal configurations which are not pyramids. 3
+3A comprehensive table containing statistics for the 65 configurations, similar to Table 1, can be accessed
+through the following link: https://ndag.github.io/anti-self-dual-polyhedra/table.html
+
+EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE
+21
+Through exact calculation via the Python package NetworkX([HSS]), we find that the
+diameter graph of each of these 65 configurations has chromatic number equal to 5 and also
+satisfies e = 3v − 5 where e and v are the number of edges and the number of vertices in the
+diameter graph, respectively.
+Appendix A. Semi-algebraic sets
+Let k ≥ 1. Let R[x1, . . . , xk] be the k-dimensional ring of polynomials with real coefficients.
+We now introduce the notion of semi-algebraic subset following [BCR13].
+Definition A.1 ([BCR13, Definition 2.1.4]). Let {ri}s
+i=1 be a set of positive integers. A
+semi-algebraic subset of Rn is a subset of the form
+s�
+i=1
+ri
+�
+j=1
+{x ∈ Rn | fi,j ∗i,j 0} ,
+where fi,j ∈ R [X1, . . . , Xn] and the operation ∗i,j is either < or =, for i = 1, , . . . , s and
+j = 1, . . . , ri.
+Definition A.2. A collection A of subsets of a set X is called an algebra of sets if A
+contains the empty set and is closed under finite union, finite intersection and under taking
+complements.
+Remark A.3. Semi-algebraic subsets of Rn form the smallest algebra of sets that contains
+all sets of the form
+{x ∈ Rn | f(x) > 0} , where f ∈ R [X1, . . . , Xn] .
+Definition A.4 ([BCR13, Definition 2.7.1]). A basic open semi-algebraic subset of Rn is a
+set of the form
+{x ∈ Rn | f1(x) > 0, . . . , fs(x) > 0}
+where f1, . . . , fs ∈ R [X1, . . . , Xn]. A basic closed semi-algebraic subset of Rn is a set of the
+form
+{x ∈ Rn | f1(x) ≥ 0, . . . , fs(x) ≥ 0}
+where f1, . . . , fs ∈ R [X1, . . . , Xn]
+By applying Morse theory, Milnor [Mil64] obtained the following bound on the number of
+Betti numbers of a closed basic semi-algebraic set.
+Theorem A.5 ([Mil64, Theorem 3]). If X ⊂ Rn is a basic closed semi-algebraic subset
+defined by p polynomial inequalities f1 ≥ 0, . . . , fp ≥ 0 of degree ≤ d, then the sum of the
+Betti numbers of X is at most 1
+2(dp + 2)(dp + 1)n−1.
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+Phys. Sci., 440, Kluwer, Dordrecht, 1994.
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+Sci. Math.(Szeged) 45 (1983), no. 1-4, 317–323.
+[Mil64] John Milnor, On the Betti numbers of real varieties, Proceedings of the American Mathematical
+Society 15 (1964), no. 2, 275–280.
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+
+EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE
+23
+[Wal70] David W Walkup, The lower bound conjecture for 3-and 4-manifolds, Acta Mathematica 125 (1970),
+75–107.
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+Bar Ilan University.
+Email address: katzmik@math.biu.ac.il
+The Ohio State University.
+Email address: facundo.memoli@gmail.com
+University of Utah.
+Email address: qswang@math.utah.edu
+

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+Astronomy & Astrophysics manuscript no. kink_cutoff
+©ESO 2023
+January 10, 2023
+Cut-off of transverse waves through the solar transition region
+Gabriel Pelouze1, 2, Tom Van Doorsselaere2 , Konstantinos Karampelas2, 3 , Julia M. Riedl2 , and Timothy Duckenfield2
+1 Université Paris-Saclay, CNRS, Institut d’astrophysique spatiale, 91405, Orsay, France
+2 Centre for mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven, Celestijnenlaan 200B, 3001 Leuven,
+Belgium.
+e-mail: tom.vandoorsselaere@kuleuven.be
+3 Department of Mathematics, Physics and Electrical Engineering, Northumbria University, Newcastle upon Tyne, NE1 8ST, UK
+Received 23 September 2022 / Accepted 3 January 2023
+ABSTRACT
+Context. Transverse oscillations are ubiquitously observed in the solar corona, both in coronal loops and open magnetic flux tubes.
+Numerical simulations suggest that their dissipation could heat coronal loops, counterbalancing radiative losses. These models rely
+on a continuous driver at the footpoint of the loops. However, analytical works predict that transverse waves are subject to a cut-off in
+the transition region. It is thus unclear whether they can reach the corona, and indeed heat coronal loops.
+Aims. Our aims are to determine how the cut-off of kink waves affects their propagation into the corona, and to characterize the
+variation of the cut-off frequency with altitude.
+Methods. Using 3D magnetohydrodynamic simulations, we modelled the propagation of kink waves in a magnetic flux tube, embed-
+ded in a realistic atmosphere with thermal conduction, that starts in the chromosphere and extends into the corona. We drove kink
+waves at four different frequencies, and determined whether they experienced a cut-off. We then calculated the altitude at which the
+waves were cut-off, and compared it to the prediction of several analytical models.
+Results. We show that kink waves indeed experience a cut-off in the transition region, and we identified the analytical model that
+gives the best predictions. In addition, we show that waves with periods shorter than approximately 500 s can still reach the corona by
+tunnelling through the transition region, with little to no attenuation of their amplitude. This means that such waves can still propagate
+from the footpoints of loop, and result in heating in the corona.
+Key words. Sun: atmosphere – Sun: oscillations – magnetohydrodynamics (MHD) – waves – methods: numerical
+1. Introduction
+Recent advances in observations and modelling have shown
+that magnetohydrodynamic (MHD) waves could significantly
+contribute to the heating of the solar corona (see review by
+Van Doorsselaere et al. 2020). In particular, transverse waves are
+ubiquitously observed, and they come in several kinds. The type
+that was first discovered are the transverse waves that are impul-
+sively excited after a flare (Nakariakov et al. 1999). However,
+these transverse waves are only sporadically excited and do not
+play an important role in the energy budget of the solar corona
+(Terradas & Arregui 2018). Later on, it was discovered that the
+corona is filled by small-amplitude transverse waves (Tomczyk
+et al. 2007; Tomczyk & McIntosh 2009; McIntosh et al. 2011;
+Tian et al. 2012). These were observed in coronal loops as prop-
+agating (Tiwari et al. 2019) or standing waves (Anfinogentov
+et al. 2015). These low-amplitude transverse waves were also
+observed as propagating waves in open-field regions (Thurgood
+et al. 2014; Morton et al. 2015). These low-amplitude waves
+show little-to-no decay (Morton et al. 2021) and are thus named
+“decayless”.
+Because the flare-excited standing waves are rapidly decay-
+ing (Goddard et al. 2016; Nechaeva et al. 2019) due to reso-
+nant absorption (Goossens et al. 2002) and non-linear Kelvin-
+Helmholtz instability (KHI) damping (Terradas et al. 2008; An-
+tolin et al. 2014; Van Doorsselaere et al. 2021; Arregui 2021),
+it is generally thought that the decayless waves must be con-
+tinuously supplied with energy to counteract its strong damp-
+ing. Several mechanisms for excitation have been proposed: slip-
+stick driving with steady flows (Nakariakov et al. 2016; Karam-
+pelas & Van Doorsselaere 2020), vortex shedding (Nakariakov
+et al. 2009; Karampelas & Van Doorsselaere 2021) or footpoint
+driving (Nisticò et al. 2013; Karampelas et al. 2017) through p-
+modes (Morton et al. 2019) or convective shuffling. The latter
+option of footpoint driving has had some success in generating
+standing mode decayless waves (Afanasyev et al. 2020), which
+counterbalance the non-linear damping through the KHI (Guo
+et al. 2019) and lead to heating of loops (Shi et al. 2021).
+However, for the driving of decayless waves through their
+footpoints, it is not well understood how the transverse waves
+propagate through the complicated structure of the chromo-
+sphere and transition region. The simulations of transverse-wave
+induced KHI heating (e.g. Karampelas et al. 2019) only take into
+account the coronal part of the loop, that is imposing a driver at
+the top of the transition region. To properly model the whole loop
+evolution due to the wave heating, it is essential to also model the
+wave driver in the photosphere, and accurately capture its influ-
+ence on the coronal loop dynamics.
+In plane-parallel atmospheres, the propagation of fast and
+slow waves has been well studied. It was found that these modes
+couple efficiently to Alfvén waves through resonant absorption
+(Hansen & Cally 2009; Cally & Andries 2010; Khomenko &
+Cally 2012). Currently, investigations are ongoing to what hap-
+pens if the cross-field structuring is included into the wave prop-
+agation model (Cally & Khomenko 2019; Riedl et al. 2019,
+2021). Another crucial ingredient is the wave’s behaviour in
+Article number, page 1 of 8
+arXiv:2301.03100v1  [astro-ph.SR]  8 Jan 2023
+
+A&A proofs: manuscript no. kink_cutoff
+strong (i.e. non-WKB) stratification. It is well-known that slow
+waves experience a cut-off while propagating through a stratified
+medium (Bel & Leroy 1977). This has been verified observation-
+ally (Jess et al. 2013) and numerically (Felipe et al. 2018). Still,
+up to now, it is unknown if a similar cut-off exists for transverse
+waves in structured media. For the driving of the observed decay-
+less waves in the corona, this is a crucial property to understand.
+Several analytical works predict that transverse waves are
+cut-off in the transition below a given frequency. The first for-
+mula was derived by Spruit (1981):
+ω2
+Sp81 = g
+8H
+1
+2β + 1,
+(1)
+where g is the gravity projected along the loop, H the pressure
+scale height, and β the ratio between the gas and magnetic pres-
+sures. For a typical isothermal atmosphere, this corresponds to
+a cut-off period of 700 s (Spruit 1981). However, Lopin et al.
+(2014) showed that this classical cut-off is suppressed when the
+radial component of the magnetic field is taken into account.
+Lopin & Nagorny (2017) later showed that transverse waves can
+still be cut-off, provided a non-isothermal atmosphere. They pre-
+dict the following cut-off frequency:
+ω2
+LN17 =
+c2
+k0
+4H0H(z)
+�
+δ2
+B
+dH(z)
+dz
++ H2(z)
+z2
+�
+,
+(2)
+where z is the altitude, ck0 is the kink speed at the base of atmo-
+sphere (z = z0), H is the pressure scale height, H0 = H(z0), and
+δ2
+B =
+�
+B2
+0i − B2
+0e
+�
+/
+�
+B2
+0i + B2
+0e
+�
+is the relative difference between
+the magnetic field inside (B0,i) and outside (B0,e) the flux tube, at
+z = z0. Finally, an alternative formula was derived by Snow et al.
+(2017):
+ω2
+Sn17 = v2
+A(z)
+4z2 ,
+(3)
+where z is the altitude, and vA is the Alfvén speed.
+In this article, we modelled the propagation of kink waves
+in an open magnetic flux tube, embedded in a non-isothermal
+atmosphere. The atmosphere extends from the chromosphere to
+the corona, and includes gravitational stratification and thermal
+conduction (Sect. 2). We drove kink waves at different periods,
+and determined whether they experienced a cut-off (Sect. 3).
+We compare these results to the three analytical formulas given
+above in Sect. 4, and summarize our conclusions in Sect. 5.
+2. Numerical model: magnetic flux tube through the
+transition region
+We modelled a vertical magnetic flux tube of radius R = 1 Mm
+embedded in a stratified atmosphere, starting in the chromo-
+sphere (altitude z = 0 Mm) and extending through the transi-
+tion region (z ≈ 4 Mm) into the corona. Kink waves were ex-
+cited in the flux tube by applying a monoperiodic driver at the
+bottom of the domain (z = 0 Mm). In the upper half of the do-
+main (z > 50 Mm), we implemented a “velocity rewrite layer”
+to absorb the kink waves. The driver and the velocity rewrite
+layer are described in Sect. 2.1. A sketch of the domain is shown
+on Fig. 1. We solved the 3D MHD evolution of this tube using
+the PLUTO code (Mignone et al. 2007), version 4.3. This code
+solves the conservative MHD equations (mass continuity, mo-
+mentum conservation, energy conservation, and induction equa-
+tion). We used the corner transport upwind finite volume scheme,
+x [Mm]
+z [Mm]
+Driver
+Transition 
+region
+Velocity 
+rewrite 
+layer
+Corona
+Magnetic tube
+Kink
+wave
+2 Mm
+Chromosphere
+0
+−8
+8
+0
+100
+50
+0
+3
+−3
+y [Mm]
+Fig. 1. Sketch of the simulation domain, showing the magnetic flux
+tube, the location of the kink wave driver (bottom boundary), chromo-
+sphere, transition region, corona, and velocity rewrite layer.
+where characteristic tracing is used for the time stepping, and a
+linear spatial reconstruction with a monotonized central differ-
+ence limiter is performed. The magnetic field divergence was
+kept small using the extended divergence cleaning method (gen-
+eralized Lagrange multiplier, or GLM), and flux was computed
+with the linearized Roe Riemann solver. We did not include ex-
+plicit viscosity, resistivity, or cooling. However, numerical dis-
+sipation results in higher effective viscosity and resistivity than
+what is expected for the solar corona, as discussed by Karam-
+pelas et al. (2019). We included a modified thermal conduction,
+as described below.
+The transition region between the chromosphere and the
+corona is characterized by a very sharp temperature gradient.
+Resolving such gradient requires a very high resolution along
+the tube (∼ 1 km in the transition region). In order to keep com-
+putational costs reasonable, we artificially broadened the tran-
+sition region (thus reducing the temperature gradient). To that
+end, we modified the thermal conductivity using the method de-
+veloped by Linker et al. (2001); Lionello et al. (2009); Miki´c
+et al. (2013). Below the cut-off temperature Tc = 2.5 · 105 K,
+the parallel thermal conductivity was set to κ∥ = C0T 5/2
+c
+with
+C0 = 9 · 10−12 Wm−1K−7/2. Above Tc, κ∥ = C0T 5/2. This al-
+lowed us to use a resolution of 98 km along the tube. This grid
+allows to fully resolve the broadened transition region, which
+has a minimum temperature scale length of 1.6 Mm (see John-
+ston & Bradshaw 2019). The dimensions of the domain were
+(Lx, Ly, Lz) = (16, 6, 100) Mm. We used a uniform grid of
+400 × 150 × 1024 cells, with a size of 40 km in the x and y direc-
+tions, and 98 km in the z direction. Furthermore, we verified that
+the results did not change significantly when using a resolution
+of 40 km in the z direction. To that end, we ran a separate sim-
+ulation and verified that the resulting cut-off altitude and com-
+parison to the analytical formulas (see Sect. 4) were not strongly
+modified. We note that such resolution is too costly in terms of
+compute time to be used for all simulations in this work.
+The strong stratification in the transition region makes it
+challenging to obtain a relaxed initial state for the model. We
+first initialized the domain with a field-aligned hydrostatic equi-
+librium (Sect. 2.2). We then let the simulation relax in 2D for
+47 ks (Sect. 2.3). Finally, we filled the 3D domain with this re-
+Article number, page 2 of 8
+
+G. Pelouze et al.: Cut-off of transverse waves
+0
+20
+40
+60
+80
+100
+Altitude [Mm]
+0.9995
+0.9996
+0.9997
+0.9998
+0.9999
+1.0000
+Velocity rewrite coefficient αv
+αv(t≤15.7 ks)
+αv(t=18.8 ks)
+αv(t=21.9 ks)
+αv(t=25.1 ks)
+αv(t=28.2 ks)
+αv(t≥31.3 ks)=αv,3D
+Fig. 2. Velocity-rewrite coefficient αv, applied to the velocity above
+50 Mm so that upper-propagating waves are not reflected back into the
+domain. αv is shown for different times of the 2D relaxation run. The
+last profile (t ≥ 31.3 ks) is also applied in the 3D driven simulations.
+laxed state through cylindrical symmetry, where we drove kink
+waves of different periods for a duration up to 2.7 ks (Sect. 2.4).
+2.1. Boundary conditions and driver
+We first describe the boundary conditions used for the relaxation
+(2D) and kink wave (3D) simulations.
+Bottom boundary
+At the bottom boundary (base of the chro-
+mosphere, z = 0), the density and pressure were extrapolated
+using the hydrostatic equilibrium equation. The magnetic field
+was extrapolated using the zero normal-gradient condition de-
+scribed by Karampelas et al. (2019, section 2.4). For vz, we ei-
+ther imposed a reflective boundary condition (2D relaxation, see
+Sect. 2.3), or imposed vz = 0 (in 3D, see Sect. 2.4). We verified
+that both boundary conditions give the same results in 3D sim-
+ulations. The parallel velocity components vx and vy were set to
+obey either a zero-gradient boundary condition (2D relaxation),
+or to follow a driver that excites kink waves (in 3D). We used
+a monoperiodic, dipole-like, driver developed by Pascoe et al.
+(2010) and updated by Karampelas et al. (2017). Inside the tube,
+the driver imposes:
+�
+vx(x, y, t), vy(x, y, t)
+�
+= {v(t), 0} ,
+(4)
+where v(t) = v0 cos (2πt/P0), with v0 the driver amplitude, set
+to 2 km s−1. The driver period, P0, was set to different values
+in order to test the cut-off of kink waves. Outside the tube, the
+driver imposes:
+�
+vx(x, y, t), vy(x, y, t)
+�
+= v(t)R2
+�
+(x − x0(t))2 − y2, 2 (x − x0(t)) y
+�
+�
+(x − x0(t))2 + y2�2
+,
+(5)
+where x0(t) = v0P0/(2π) · sin (2πt/P0) is the centre of the tube’s
+footpoint at time t. This driver generates a kink wave polarized
+in the x direction.
+Upper boundary At the upper boundary (top of the corona, z =
+100 Mm), the magnetic field was kept symmetric. All other vari-
+ables obeyed a reflective boundary condition. In order to absorb
+the upwards waves excited by the driver, we artificially modified
+the velocity in the upper half of the domain (z > 50 Mm). At
+each time step, after solving the MHD equations, we decreased
+each component of the velocity vi by multiplying it by a quantity
+αv ≲ 1:
+v′
+i = αv(t, z)vi.
+(6)
+In the driven 3D simulations αv was kept constant in time, and
+varied linearly along the loop, from 1 at z = zv = 50 Mm, to
+αv,min = 0.9995 at z = L = 100 Mm:
+αv,3D(z) =
+�������
+1
+if z ≤ zv,
+1 − �1 − αv,min
+� � z−zv
+L−zv
+�
+else.
+(7)
+In the 2D relaxation run, the first third of the simulation (t1/3 =
+15.7 ks) was run without modifying the velocity (i.e. αv = 1).
+During the second third, αv was linearly ramped down in time to
+match the profile αv,3D(z) described above. Finally, the last third
+of the simulation was run with the constant αv,3D(z):
+αv,2D(z, t) =
+�����������
+1
+if t ≤ t1/3,
+1 − �1 − αv,3D(z)� � t−t1/3
+t1/3
+�
+if t1/3 < t ≤ 2t1/3,
+αv,3D(z)
+else.
+(8)
+The evolution of αv is shown in Fig. 2. This “velocity rewrite
+layer” can successfully absorb the kink waves that are excited
+by the driver at the bottom of the chromosphere. As a result,
+these waves are not reflected at the upper boundary, and do not
+propagate downwards back into the domain. We stress that the
+solution obtained inside the velocity rewrite layer (i.e. above z =
+50 Mm) is not physical, and that this layer should be considered
+as a part of the upper boundary.
+Side boundaries At the side boundaries (x and y axes), all vari-
+ables obeyed a zero-gradient boundary condition. In the 2D re-
+laxation run, we only simulated half of the tube radius (x > 0).
+For these simulations, we imposed a reflective boundary condi-
+tion on all variables at the centre of the tube (x = 0).
+2.2. Initial conditions: field-aligned hydrostatic equilibrium
+The simulation was initialized with a uniform vertical magnetic
+field of magnitude B0 = 42 G. Along the tube, we imposed
+the following temperature profile, derived from Aschwanden &
+Schrijver (2002):
+T(x, y, z) =
+���������
+Tch
+if z ≤ ∆ch,
+Tch + (Tcor(x, y) − Tch)
+�
+1 −
+� L−z
+L−∆ch
+�2�0.3
+else,
+(9)
+where z is the altitude, L is the height of the computational
+domain, ∆ch = 4 Mm is thickness of the chromosphere, and
+Tch = 20 000 K is the temperature in the chromosphere. We de-
+fined the transverse temperature profile at the top of the domain,
+Tcor(x, y), as:
+Tcor(x, y) = Tcor,ext + (Tcor,int − Tcor,ext)ζ(x, y),
+(10)
+where Tcor,int = 1.2 MK is the temperature inside the tube, and
+Tcor,ext = 3.6 MK is the temperature outside the tube. The shape
+of the profile was set by ζ(x, y):
+ζ(x, y) = 1
+2
+�
+1 − tanh
+�� �
+x2 + y2/R − 1
+�
+b
+��
+,
+(11)
+Article number, page 3 of 8
+
+A&A proofs: manuscript no. kink_cutoff
+0.1
+1
+10
+100
+Altitude [Mm]
+10−2
+10−1
+100
+Temperature [MK]
+Tint
+Text
+10−13
+10−12
+10−11
+10−10
+10−9
+10−8
+Density [kg m⁻³]
+ρint
+ρext
+39
+40
+41
+42
+43
+44
+Magnetic field [G]
+Bint
+Bext
+(a) Field-aligned hydrostatic equilibrium
+0.1
+1
+10
+100
+Altitude [Mm]
+10−2
+10−1
+100
+Temperature [MK]
+Tint
+Text
+10−13
+10−12
+10−11
+10−10
+10−9
+10−8
+Density [kg m⁻³]
+ρint
+ρext
+9
+10
+11
+12
+13
+14
+Magnetic field [G]
+Bint
+Bext
+(b) 2D magnetohydrodynamic relaxation
+Fig. 3. Temperature (black), density (red), and magnetic field magnitude (blue) profiles inside (r = 0 Mm; solid lines) and outside (r = 8 Mm;
+dashed lines) the flux tube. (a) After solving the field-aligned hydrostatic equilibrium. (b) After the 2D magnetohydrodynamic relaxation.
+where R = 1 Mm is the tube radius, and b = 5 is a dimensionless
+number setting the width of the inhomogeneous layer between
+the interior and exterior of the tube (l ≈ 6R/b). ζ(x, y) is close to
+1 inside the tube, and to 0 outside.
+We also set the density at the bottom of the chromosphere
+(z = 0) to:
+ρch(x, y, z = 0) = ρch,ext + (ρch,int − ρch,ext)ζ(x, y),
+(12)
+where ρch,int = 3.51 · 10−8 kg m−3 is the density inside the tube,
+and ρch,ext = 1.17 · 10−8 kg m−3 is the density outside. We then
+integrated the field-aligned hydrostatic equilibrium equation nu-
+merically using a Crank-Nicholson scheme. The profiles of the
+imposed temperature and of the density resulting from the inte-
+gration are shown in Fig. 3 (a). The temperature contrast (interior
+temperature divided by exterior temperature) is 1 in the chromo-
+sphere, and decreases to 1/3 in the corona. The density contrast
+is 3 in the chromosphere, increases to around 7 in the transition
+region, and decreases again to about 4 in the upper corona. The
+pressure contrast is 3 in the chromosphere, and slowly decreases
+to reach 1.2 in the upper corona.
+However, this initial state is not in magnetohydrostatic
+(MHS) equilibrium, because the pressure varies across the flux
+tube, while the magnetic field does not. To fix this, we let the
+tube relax by running a 2D magnetohydrodynamic simulation
+(Sect. 2.3). We then used this relaxed state to initialize the 3D
+simulation of kink waves (Sect. 2.4).
+2.3. Flux tube relaxation (2D)
+In order to obtain a flux tube in MHS equilibrium, we first
+run a 2D simulation, initialized with the initial state described
+in Sect. 2.2. The MHD equations were solved in a longitudi-
+nal plane at y = 0 (see Fig. 1), with x ∈ [0, 8.56] Mm, and
+z ∈ [0, 100] Mm. We used a uniform grid of 64 × 2048 cells with
+a size of 134 km×49 km. The resolution along z is higher than in
+the 3D runs in order to resolve the sharper gradients in the tran-
+sition region (see Fig. 3). We verified that a resolution of 40 km
+in the x direction yielded the same results, by running a separate
+2D simulation followed by a 3D driven simulation (P0 = 200 s),
+and verifying that the cut-off altitude and comparison to the an-
+alytical formulas (Sect. 4) were not significantly modified.
+We let the system evolve for 47 ks, during which the velocity
+rewrite parameter αv varied as described in Eq. (8). As a result
+of the relaxation, periodic longitudinal flows with a velocity of
+about 15 km s−1 develop along the tube. They are damped during
+the later stages of the simulation, as the velocity rewrite layer is
+gradually introduced. At the end of the relaxation run, residual
+velocities are lower than 0.5 km s−1 everywhere in the domain.
+The resulting temperature, density, and magnetic field profiles
+are shown on Fig. 3 (b). Compared to the initial state (Fig. 3 a),
+the transition region is significantly broadened, with a thickness
+of about 7 Mm. This is the direct result of the modified thermal
+conductivity used in this setup, and allows for a coarser resolu-
+tion along the loop in the 3D simulations. In addition, the tem-
+perature and density decrease, both inside and outside the tube.
+Overall, the density contrast (ρint/ρext) decreases: it reaches 1
+in the chromosphere, 1.2 in the transition region, and 1.8 in the
+corona. The temperature contrast also changes to about 1.3 in
+the transition, and about 0.8 in the corona. Finally, the magnetic
+field amplitude contrast remains very close to 1 everywhere in
+the domain (0.97 in the chromosphere and 1 in the corona), with
+a magnitude of about 11 G everywhere in the domain. Compared
+to the initial uniform magnetic field, the magnitude is divided by
+about four, while the contrast remains close to 1. The final tem-
+perature and density profile significantly differ from the initial
+conditions of 2D relaxation run. However, this is not an issue, as
+the goal of this study is to investigate how the analytical formulas
+we consider (Spruit 1981; Lopin & Nagorny 2017; Snow et al.
+2017) predict the cut-off frequency for a given temperature and
+density profile. By using the relaxed profiles as an input to these
+analytical formulas, we obtained predictions for the relaxed sys-
+tem.
+This relaxed 2D simulation was then mapped onto the 3D
+domain through cylindrical symmetry. We used a rotation about
+the line x = 0 (i.e. the centre of the loop), and a trilinear interpo-
+lation to project onto the 3D Cartesian grid.
+2.4. Kink waves propagation (3D)
+In order to simulate the propagation of kink waves from the chro-
+mosphere to the corona, we drove the 3D simulations with the
+monoperiodic, dipole-like, driver described in Eqs. (4) and (5).
+We ran four simulations, with different driver periods P0: 200 s,
+Article number, page 4 of 8
+
+G. Pelouze et al.: Cut-off of transverse waves
+0
+200
+400
+Time [s]
+0
+10
+20
+30
+40
+50
+Altitude [Mm]
+(a) P0 =200 s
+−15
+−10
+−5
+0
+5
+10
+15
+Velocity [km s⁻¹]
+0
+250
+500
+750
+1000
+Time [s]
+(b) P0 =335 s
+−6
+−4
+−2
+0
+2
+4
+6
+Velocity [km s⁻¹]
+0
+500
+1000
+1500
+2000
+Time [s]
+(c) P0 =700 s
+−3
+−2
+−1
+0
+1
+2
+3
+Velocity [km s⁻¹]
+0
+1000
+2000
+Time [s]
+(d) P0 =2000 s
+−2
+−1
+0
+1
+2
+Velocity [km s⁻¹]
+Fig. 4. Kink waves transverse velocity (vx) at the loop centre (x = y = 0), as a function of altitude and time. The velocity is shown for four 3D
+simulations with different driver periods P0, after an initial settling time of 2P0 (for P0 = 200 s, 335 s and 700 s), or 0.42P0 (for P0 = 2000 s). The
+dashed black lines represent a propagation at the kink speed (see Eq. (13)), and are independent of the driver period.
+335 s, 700 s, and 2000 s. The propagating kink waves generated
+by the driver are absorbed by the velocity rewrite layer at the top
+of the domain, and are thus not reflected downwards. The first
+three simulations were run for a duration of 5P0. The last simula-
+tion was run for 1.75P0. At the beginning of the simulations, the
+system goes through an initial transitory phase before the propa-
+gating kink wave is fully established (i.e. its amplitude does not
+change with time). We waited for 2P0 (0.42P0 for P0 = 2000 s)
+for the kink wave to enter a stable sinusoidal regime. After this
+duration, we saved high-cadence snapshots at the centre of the
+loop (line x = y = 0). For all further analysis, we used the snap-
+shots saved after the transitory phase. The transverse velocity vx
+at the loop centre is shown in Fig. 4. As can be seen on this
+figure, the amplitude of the kink wave decreases as the period
+increases. For the two longer driver periods (700 and 2000 s),
+the amplitude of the kink wave is small enough for some pertur-
+bations to become visible. They travel at the Alfvén speed, and
+appear to be triggered by the flows remaining after the relaxation
+(see Sect. 2.3). These perturbations have amplitudes smaller than
+0.2 km s−1, and should thus have no effect on the wave.
+3. Results: cut-off and tunnelling of transverse
+waves
+In order to determine whether the kink waves driven in the 3D
+simulations are experiencing a cut-off, we looked at the evolution
+of the velocity amplitude (Sect. 3.1), as well as the phase speed
+(Sect. 3.2) as a function of altitude. The analysis of these profiles
+allows us to establish that the transverse waves are subject to a
+low-frequency cut-off in the transition region.
+3.1. Wave amplitude increases with frequency
+In order to compute the velocity amplitude of the kink wave, we
+fitted the function Ax(z) sin (ω(z)t + φ(z)) to the transverse ve-
+locity vx(z, t), at each altitude (z). Ax(z) is the velocity amplitude,
+ω(z) is the kink wave frequency, and φ(z) is the phase. The fre-
+quency varies by less than 1 % with altitude, confirming theoret-
+ical understanding. The velocity amplitude is shown in Fig. 5.
+In all simulations, the wave amplitude increases with altitude,
+because of the density decreases with altitude and energy con-
+servation. Across simulations, the amplitude at a given altitude
+increases with the frequency of the wave. This means that kink
+waves with higher frequencies propagate better from the chro-
+0.1
+1
+10
+Altitude [Mm]
+2
+4
+6
+8
+10
+12
+14
+16
+Velocity amplitude [km s⁻¹]
+P0 =200 s
+P0 =335 s
+P0 =700 s
+P0 =2000 s
+0.1
+1
+10
+2.0
+2.2
+2.4
+2.6
+0.05
+50
+Fig. 5. Velocity amplitude of kink waves, as a function of altitude. The
+velocity is shown for four different driver periods (P0). The inset has the
+same axes as the main figure, with a zoom-in on the vertical axis.
+mosphere to the corona. This would be consistent with the low-
+frequency cut-off predicted by analytical models (see Sect. 1).
+3.2. Evanescent waves in the transition region
+To determine the altitude at which the waves are cut-off, we
+compared their phase speed vp(z) to the kink speed of the
+flux tube ck(z). The inverse phase speed is equivalent to the
+phase difference ∆φ(z) between two altitudes separated by ∆z:
+1/vp(z) = ∆φ(z)/(ω∆z). The phase difference has been success-
+fully used to determine the cut-off frequency of acoustic and
+slow-magnetosonic waves in observations (Centeno et al. 2006;
+Felipe et al. 2010; Krishna Prasad et al. 2017; Felipe et al. 2018),
+and in simulations (Felipe & Sangeetha 2020). In these articles,
+the authors determine the phase speed for a wide range of fre-
+quencies, but at a limited number of altitude positions. In the
+present study however, we could only examine four frequencies,
+because of the high computational cost of a simulation. However,
+we computed the phase difference at all altitudes of the simula-
+tion domain. This allows us to determine the altitude at which
+the wave is cut-off.
+Article number, page 5 of 8
+
+A&A proofs: manuscript no. kink_cutoff
+The phase speed at a given altitude z was computed from the
+transverse velocity in the cells above and below, that is vx(t, z +
+∆z/2) and vx(t, z − ∆z/2), where ∆z = 98 km is the cell size.
+We apodized these velocity time series with a Hann window,
+and computed the cross-correlation C(τ, z) = vx(t, z + ∆z/2) ⋆
+vx(t, z−∆z/2). We then determined the time delay ∆τ(z), by find-
+ing the maximum of C(τ, z). To that end, we fitted the function
+A + B cos (ω(τ − ∆τ)/δ) to C(τ, z), with τ ∈ [−P0/4, +P0/4]. Fi-
+nally, the phase difference was given by ∆φ(z) = ω∆τ(z), and the
+inverse phase speed by 1/vp(z) = ∆τ(z)/∆z. The inverse phase
+speed is shown on Fig. 6, alongside the inverse kink speed for
+the simulated flux tube. The kink speed ck is calculated using:
+c2
+k(z) = ρi(z)v2
+A i(z) + ρe(z)v2
+A e(z)
+ρi(z) + ρe(z)
+,
+(13)
+where ρ(z) is the density, vA(z) = B(z)/
+�
+µ0ρ(z) is the Alfvén
+speed, B(z) is the magnetic field amplitude, and µ0 is the mag-
+netic permittivity of vacuum. The indices i and e correspond,
+respectively, to internal and external quantities relatively to the
+flux tube, and are taken at x = 0 and x = 8 Mm.
+In simulations with short driver periods, the inverse phase
+speed is somewhat smaller than the inverse kink speed in the
+chromosphere and transition region (vp/ck ≈ 2 for P0 = 200 s,
+and 5 for P0 = 335 s), and equals the inverse kink speed in the
+corona. On the other hand, in simulations with longer periods,
+the inverse phase speeds are much lower than the inverse kink
+speed below a given altitude. For P0 = 700 s, 1/vp is about 250
+times smaller than 1/ck below z = 1 Mm. For P0 = 2000 s, a
+similar drop occurs below z = 20 Mm.
+For a propagating kink wave, the inverse phase speed is ex-
+pected to be equal to the inverse kink speed. Conversely, stand-
+ing and evanescent (i.e. cut-off) waves have inverse phase speeds
+smaller than the inverse kink speed. Thus, the decreased inverse
+phase speed for higher periods indicates that the waves are cut-
+off in at least some regions.
+To distinguish between the standing and evanescent cases,
+we have also looked at the wave amplitude (Fig. 5). In the
+absence of vertical stratification, the amplitude of evanescent
+waves decreases with altitude. However, in a stratified atmo-
+sphere (our case), the amplitude increases with altitude because
+of the density decrease, even for evanescent waves. On Fig. 5, the
+amplitude of waves with longer periods (for which 1/vp ≪ 1/ck)
+increases less with altitude compared to waves with shorter pe-
+riods (for which 1/vp ≲ 1/ck). We thus conclude that the waves
+with longer periods are evanescent in parts of the low atmo-
+sphere, where their inverse phase speed is much lower than the
+inverse kink speed. This means that these long-period waves are
+cut-off in the transition region.
+3.3. Wave tunnelling at higher frequencies
+Waves with shorter periods (P0 = 200 and 335 s) also show signs
+of cut-off at low altitudes. Below z = 3 Mm, the inverse phase
+speed 1/vp is lower than the inverse kink speed 1/ck (Fig. 6),
+and the amplitude increase with altitude is smaller for P0 = 335 s
+than for P0 = 200 s (Fig. 5). However, this cut-off is significantly
+weaker than in the long-period case. This is explained by the fact
+that the cut-off region (where 1/vp < 1/ck) is narrower for short
+periods (∼ 1 Mm) than for long periods (∼ 10 Mm). As a result,
+short-period waves can tunnel through the cut-off region, and
+propagate into the corona. Furthermore, the weak attenuation in
+the cut-off region (1/vp ≲ 1/ck) results further reduces the effect
+of the cut-off.
+0.1
+1
+10
+Altitude [Mm]
+10−1
+100
+101
+102
+1/v [s Mm⁻¹]
+1/ck
+1/vp (P0 =200 s)
+1/vp (P0 =335 s)
+1/vp (P0 =700 s)
+1/vp (P0 =2000 s)
+50
+Fig. 6. Inverse phase speed of the kink wave (1/vp), and inverse kink
+speed of the flux tube (1/ck), as a function of altitude. The phase speed
+is given for four different driver periods (P0).
+0.1
+1
+10
+Altitude [Mm]
+10−3
+10−2
+10−1
+ωc [s⁻¹]
+Models
+Sp81
+Sn17
+LN17 (z0 = 24 km)
+LN17 (z0 = 659 km)
+LN17 (z0 = 1343 km)
+LN17 (z0 = 1978 km)
+Simulations
+tr =0.2
+tr =0.3
+tr =0.4
+tr =0.5
+50
+Fig. 7. Kink wave cut-off frequency as a function of altitude, from an-
+alytical models (left column of the legend), and from our numerical
+simulations (right column of the legend). We show the analytical pre-
+dictions of Spruit (1981, SP81), Snow et al. (2017, Sn17), and of Lopin
+& Nagorny (2017, LN17) (coloured lines). For the last model, we com-
+puted the cut-off frequency for different values of z0, the “base of the
+atmosphere”. We show the cut-off altitude (zc) for the four simulations
+that we ran with different driver frequencies (black markers). The cut-
+off altitudes are computed with different thresholds tr, indicated on the
+legend and described in the text.
+4. Discussion: comparison to analytical formulas
+In order to compare our simulations to the analytical models,
+we quantified the cut-off frequency as a function of altitude. We
+define zc, the altitude at which ck/vp goes above a given threshold
+tr. This corresponds to the altitude where the wave leaves the cut-
+off regime and enters the propagating regime. That is, the cut-
+off altitude. We computed zc for four values of tr between 0.2
+and 0.5. Considering the four simulations with different driver
+frequencies ω, we obtained the cut-off altitude as a function of
+the frequency, zc(ω). We compare this to the cut-off frequency as
+a function of altitude, ωc(z), predicted by the analytical models
+presented in Sect. 1.
+Article number, page 6 of 8
+
+G. Pelouze et al.: Cut-off of transverse waves
+On Fig. 7, we show the cut-off frequency and altitude com-
+puted in our simulations, for different values of tr (black points).
+On the same figure, we show the predictions of the analytical
+formulas of Spruit (1981, Eq. (1)), Lopin & Nagorny (2017,
+Eq. (2)), and Snow et al. (2017, Eq. (3)) (coloured lines), com-
+puted for the temperature and density profiles used in our simu-
+lations. We implement the formula of Lopin & Nagorny (2017)
+for different values of z0, defined by the authors as “the base of
+the atmosphere”, with no further details. Because this quantity
+is not accurately defined, we used four values of z0 in the range
+of 24 km (bottom cell of our simulation domain), to 1978 km.
+This loosely defined parameter broadens the range for the cut-
+off frequencies predicted by this formula. While the match is
+rather loose, the cut-off altitude zc(ω) measured in our simula-
+tions matches the overall variation the cut-off frequency ωc(z)
+predicted by the Lopin & Nagorny (2017) formula. In particular,
+the shape of the profiles are in good agreement. On the contrary,
+the Snow et al. (2017) model correctly predicts the cut-off fre-
+quency only in the lower transition region, but fails to do so in
+the upper transition region and corona. In particular, their model
+predicts a slower decrease of the cut-off frequency above 20 Mm,
+while the simulations and the Lopin & Nagorny (2017) show a
+continued decrease. Finally, the Spruit (1981) predictions are off
+by almost an order of magnitude at all altitudes. Thus, the for-
+mula of Lopin & Nagorny (2017) best predicts the cut-off fre-
+quency of transverse waves at different altitudes.
+While the broadened transition region in our simulations
+could affect the altitude-dependence of the cut-off frequency, this
+should have little impact on the validation of the analytical for-
+mulas. Indeed, these formulas include the atmospheric stratifi-
+cation through altitude-dependent profiles of either the pressure
+scale height or the Alfvén speed (see Sect. 1). Because they make
+no hypothesis on these profiles, they should be valid regardless
+of the atmosphere considered. As such, the agreement with the
+simulations should not depend on the broadening of the transi-
+tion region, provided the appropriate profile is fed into the for-
+mulas. After validating the Lopin & Nagorny (2017) formula by
+comparing it to our simulations, it should be applicable to other
+stratification profiles.
+We note that while analytical formulas can predict the kink
+cut-off frequency, this is not sufficient to know whether a kink
+wave with a given frequency will propagate into the corona. To
+that end, the thickness of the cut-off region and the strength of
+the attenuation have to be taken into account. As shown by our
+simulations, kink waves with higher frequencies (≥ 3 mHz) can
+propagate into the corona by tunnelling through a region where
+they are cut-off (Sect. 3.3). Furthermore, these waves only expe-
+rience a weak attenuation, because their frequency is close to the
+cut-off frequency. In fact, the cut-off frequency does not consti-
+tute a clear-cut boundary between oscillatory and non-oscillatory
+solutions. This was also reported for sound waves by Felipe &
+Sangeetha (2020). Although the question of whether a solution
+is oscillating is well-defined mathematically, this is not straight-
+forward to translate into a single cut-off frequency (Schmitz &
+Fleck 1998). For this reason, there exist several canonical def-
+initions for cut-off frequencies, set within the continuous vari-
+ation between the oscillating and non-oscillating regimes (see
+e.g. Schmitz & Fleck 1998 for sound waves in the solar atmo-
+sphere). As a result, cut-off frequencies are bound to be mere
+indications, rather than strong constraints, on the physical be-
+haviour of a wave (Chae & Litvinenko 2018).
+5. Conclusions
+Transverse waves are a candidate mechanism for heating the so-
+lar corona. However, several analytical models predicted that
+they are cut-off in the transition region. In order to assess
+whether transverse waves can indeed heat the corona, it is thus
+crucial to determine whether they can propagate through the
+transition region. To that end, we have simulated the propagation
+of transverse kink waves in an open magnetic flux tube, embed-
+ded in an atmosphere extending from the chromosphere to the
+corona. We found that transverse waves are indeed cut-off in the
+lower solar atmosphere. However, only waves with low frequen-
+cies (ν ≲ 2 mHz) are significantly affected. At higher frequen-
+cies, the cut-off occurs in a very thin layer (∼ 1 Mm), and results
+in a weak attenuation. In this case, waves can tunnel through
+the cut-off layer, experiencing little to no amplitude attenuation.
+This means that transverse waves with high frequencies are able
+to transport energy from the chromosphere to the corona, where
+it can be dissipated and result in heating.
+Furthermore, we compared our simulations to several ana-
+lytical models that predict the cut-off frequency of transverse
+waves. We conclude that the formula proposed by Lopin &
+Nagorny (2017) gives the best prediction. While our simulations
+use a broadened transition, we expect it to have little impact on
+the validation of analytical formulas. As such, the formula by
+Lopin & Nagorny (2017) should be able to predict the cut-off
+frequency for any atmospheric stratification profile. We note that
+while the cut-off frequency is a good first indicator of whether a
+wave can propagate into the corona, it cannot alone predict the
+whole behaviour of the wave. In particular, waves with frequen-
+cies just below the cut-off frequency (that should thus be cut-off)
+can still reach the corona, thanks to a combination of tunnelling,
+and weak attenuation.
+Acknowledgements. This project has received funding from the European Re-
+search Council (ERC) under the European Union’s Horizon 2020 research and
+innovation program (grant agreement No. 724326). GP was supported by a
+CNES postdoctoral allocation. TVD was supported by the European Research
+Council (ERC) under the European Union’s Horizon 2020 research and inno-
+vation programme (grant agreement No 724326) and the C1 grant TRACEs-
+pace of Internal Funds KU Leuven. K.K. recognises support from a postdoctoral
+mandate from KU Leuven Internal Funds (PDM/2019), from a UK Science and
+Technology Facilities Council (STFC) grant ST/T000384/1, and from a FWO
+(Fonds voor Wetenschappelijk Onderzoek – Vlaanderen) postdoctoral fellowship
+(1273221N). The results received support from the FWO senior research project
+with number G088021N. Software: Astropy (Astropy Collaboration et al. 2013,
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+

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+1 
+ 
+Determination of the Zak phase of one-dimensional photonic 
+systems via far-field diffraction 
+ 
+C. Liu*, H.R. Wang*, and H.C. Onga) 
+ 
+Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong, People’s 
+Republic of China 
+ 
+Bloch waves in 1D periodic systems carry Zak phase, which plays a key role in determining 
+the band topology.  In general, for systems that possess inversion symmetry, the Zak phase of 
+an isolated band is quantized as 0 or  and is associated with the spatial field symmetries at the 
+Brillouin zone center and boundary.  The phase is  if the field symmetries are different but is 
+0 when they are the same.  Since the radiation losses from leaky systems are strongly associated 
+with the Bloch waves, one may probe the far-field continuum to determine the Zak phases.  
+Here, we formulate the diffractions from photonic systems at the zone center and boundary and 
+find their spectral profiles reveal the Bloch wave symmetries and thereby the corresponding 
+Zak phase.  The field symmetries also generalize the occurrence of bound states in the 
+continuum at high symmetry points.  For verification, we have studied the Zak phases of one-
+dimensional TM plasmonic and TE photonic crystals by electrodynamic simulations and 
+measuring the optical properties of plasmonic crystals using Fourier space diffraction 
+spectroscopy and common path interferometry.  In addition, a topological protected interface 
+state is demonstrated when two 0 and  systems are joined together.  The results prove our 
+method provides a simple way for characterizing the band topology of non-Hermitian systems 
+via far-fields.       
+ 
+* These authors equally contributed to this work 
+a) hcong@phy.cuhk.edu.hk 
+ 
+
+2 
+ 
+I. 
+INTRODUCTION 
+Topological physics has attracted a widespread of interest not only in condensed matter 
+physics [1-3] but also in other branches such as ultracold atom [4,5], electromagnetism [6-8], 
+mechanics [9], acoustics [10,11], and oceanography [12]. Much attention in this field is focused 
+on realizing the so-called topologically protected states, which support robust wave 
+propagation against perturbation and disorder [1-12]. To produce such states, two systems that 
+are topologically trivial and nontrivial are brought together to facilitate the occurrence of 
+topological phase transition at the interface. As most of the matters are topologically trivial, 
+the identification and the growth of different classes of topological systems are currently under 
+intensive investigation [13,14].  Likewise, developing methods to characterize the topological 
+properties of the systems is equally important.     
+In analogy to the Su-Schrieffer-Heeger (SSH) model, the band topology of one-
+dimensional (1D) periodic systems is determined by Zak phase, , which is a geometric phase 
+[15,16].  For a particular th isolated band,  emerges when the Bloch wave travels in 
+momentum space adiabatically along the band across the first Brillouin zone from k = -/P to 
+/P, where P is the period of the system. [16].  If systems possess inversion symmetry,  is 
+quantized as either 0 or  [16].   defines the topological invariant of two band systems.  For 
+systems that support higher order bands, the topology of the band gap of interest is the 
+summation of all  below that gap, giving rise to a  summation that is either even or odd 
+multiple of  for indicating whether the system is topologically trivial or nontrivial [17,18].  A 
+zero-dimensional interface state is then formed between two odd and even  systems.    
+One notable feature that comes with  is the distinctive spatial wave symmetries at the 
+Brillouin zone center and boundary of the band [16-18].  The field symmetries, typically even 
+
+3 
+ 
+and odd with respect to the unit cell center, are the same for  = 0 system but different when 
+ =  [18].  The association between n and the field symmetry can be understood from the 
+standpoint of Wannier function, which sums the Bloch waves carrying all k along a band [19].  
+Consider the Bloch waves at the zone center and boundary that have the same field symmetry, 
+the Wannier function has either the 
+(
+)
+( )
+W
+x
+W
+x
+−
+=
+ or 
+(
+)
+( )
+W
+x
+W
+x
+−
+= −
+ spatial dependence, 
+leading to  = 
+( )
+2
+2
+x W
+x
+dx
+P
+
+
+−
+ = 0 [16].  On the other hand, for the waves that exhibit 
+different spatial symmetries at two high symmetry points, the Wannier function now shows 
+(
+)
+( )
+W
+x
+P
+W
+x
+− +
+=
+ or 
+(
+)
+( )
+W
+x
+P
+W
+x
+− +
+= −
+ dependence, which gives  =  [16].  
+Therefore, instead of tracing the Bloch waves one by one along the band to determine n, one 
+can simply examine the field symmetries.  However, how to measure the Bloch wave symmetry 
+remains challenging.  
+To date, there have been only a few studies focusing on measuring the geometric phase, 
+either Zak or Berry phase.  Demler and Bloch are among the first to combine Bloch oscillation 
+and interferometry in a dimerized cold atom system to mobilize the Bloch wave across the 
+Brillouin zone and subsequently measure  [20,21].  They prove  =  evolves when the 
+intercell interaction is stronger than that of intracell.  Cardano et al have demonstrated the use 
+of mean displacement method to determine  in a chiral Floquet system [22].  Such method is 
+then extended to other more generalized SSH systems where the next nearest neighbor 
+interaction is strong enough to break the chiral symmetry [23].  While most of them trace the 
+Bloch waves, Gorlach et al adopt an alternative approach by probing the spectral positions of 
+the dipolar (bright) and quadrupolar (dark) characteristics of far-field radiations, which scale 
+with the topological invariant of the system as deduced by using temporal coupled mode theory 
+
+4 
+ 
+(CMT) [24].  When the bright and dark radiation bands are at longer and shorter wavelengths, 
+the system is trivial, but becomes nontrivial upon switching places.  However, their method is 
+limited to the lowest band gap at the zone center.  Recently, Chan and his coworkers have 
+formulated that the sign of the reflection phase for wavelengths within the th band gap can 
+resolves the trivial and nontrivial  [17,18].  The determination of  via measuring the 
+reflection phase of the band gap is then demonstrated in several photonic and acoustic systems 
+[25-28].     
+Here, we further extend the CMT to formulate the diffractions arising from 1D leaky 
+optical systems and show the mirror symmetric diffraction orders taken at the zone center and 
+boundary directly reveal the near-field symmetries and thereby the corresponding .  It is found 
+the odd and even near-field symmetries dictate the far-field interferences, shaping the overall 
+radiation profiles including the bound states in the continuum (BICs) [29-35] and Fano 
+resonances [36].  We find destructive interference always occurs between the diffraction orders 
+of the first band gap at the zone center, resulting in a symmetry-protected quasi-BIC [34].  To 
+verify the CMT, we first conduct finite-difference time-domain (FDTD) simulations on 1D Au 
+plasmonic and SiO2/Au photonic crystals which respectively support TM- and TE-polarized 
+surface waves and the results agree very well with the theory.  We then fabricate plasmonic 
+crystals (PmCs) with different geometries and measure their polarization- and angle-resolved 
+diffraction and phase profiles by Fourier space spectroscopy and common path interferometry 
+to study .  Changing the groove width of PmCs leads to band inversion and thus effectively 
+varies the band topology.  Finally, a topological protected interface state is demonstrated by 
+joining two topological trivial and nontrivial PmCs together.   
+II. 
+TEMPORAL COUPLED MODE THEORY 
+
+5 
+ 
+At high symmetry points in 1D Brillouin zone, two degenerate but counter propagating 
+Bloch modes interact with each other to yield two coupled modes separated by an energy gap 
+[37,38].  Such interaction can be described within the framework of CMT [37-40].  As shown 
+in Fig. 1(a), for an optically thick system that possesses inversion symmetry, the dynamics of 
+two mode amplitudes, a1 and a2, taken under TM or TE polarization can be written as: 
+ 
+1
+1
+2
+2
+o
+c
+T
+c
+o
+a
+a
+d
+i
+K
+s
+a
+a
+dt
+
+
+
+
++
+
+
+
+
+
+
+=
++
+
+
+
+
+
+
+
+
+
+
+
+
+,  
+ 
+ 
+(1) 
+where 
+o
+  and 
+c
+  are the complex frequency and coupling constant, which are expressed as 
+(
+) 2
+o
+o
+a
+r
+i
+
+
+=
++
+ +
+ and 
+c
+i
+
+
+
+=
++
+, where o is the resonant angular frequency, a and 
+r are the absorption and radiative decay rates, and  and  are the real and imaginary parts of 
+the coupling constant.  For a given polarization, the discrete incoming power amplitude vector 
+is  
+0
+T
+N ,
+,
+N ,
+s
+s
+s
+s
++
+−
++
++
++
+= 
+
+
+ , where the subscript N is an integer  0.  
+0,
+s + is denoted 
+as the surface normal power and 
+N ,
+s
++  are two mirror symmetric powers defined obliquely with 
+respect to the surface normal.  
+1
+0 1
+1
+2
+0 2
+2
+N ,
+,
+N ,
+T
+N ,
+,
+N ,
+K
+
+
+
+
+
+
+−
+−
+
+
+= 
+
+
+
+, where 
+1
+N,
+
+ and 
+2
+N,
+
+ are the 
+complex in-coupling constants for inputting energy from the continuum to a1 and a2.  N 
+depends on the number of available ports, which is governed by the diffraction equation as 
+(
+)
+m
+m
+P sin
+sin
+
+
+
+=
+−
+, where m is the diffraction order,  is the incident polar angle, and m 
+is the diffraction angle [41].  For example, as shown in Fig. 1(b), for the lowest band gap at the 
+zone center,  = 0o, such that only one m = 0th propagating order exists in free space.  For the 
+second band gap at the zone boundary where 
+2P sin
+
+
+=
+, two m = 0th and 1st orders are 
+present at 
+m
+
+
+=  .  In general, zone center supports an odd number of ports including 
+0
+  
+whereas an even number of ports is found at zone boundary where 
+0
+  is always zero [41]. 
+
+6 
+ 
+To see how the field symmetry is revealed, we solve the eigenvalues and eigenvectors of 
+the homogeneous part of Eq. (1) by diagonalization.  The complex frequencies of the coupled 
+modes as:
+(
+)
+(
+)
+(
+)
+2
+o
+a
+r
+i
+
+
+
+
+ =
+
++
+ +
+
+, indicating their spectral positions and decay 
+rates depend on  and .  For the real part, we see the spectral positions of the coupled modes 
+are determined by the magnitude and sign of  and they are separated by an energy gap = 2.  
+On the other hand, for the imaginary part, one mode has larger decay rate whereas another one 
+has lower, featuring the bright (dipolar) and dark (quadrupolar) modes [42].  In particular, if 
+2
+0
+r
+
+ −
+=
+, one coupled mode exhibits zero radiation damping, resulting in a quasi-BIC [34].  
+The unit eigenvectors are 
+1
+2
+1
+2
+1
+2
+a
+a
+a
+a
+a
+a
++
+−
++
+
+
+
+
+=
+
+
+
+
+−
+
+
+
+
+, which are orthogonal and carry odd and even 
+symmetries with respect to the unit cell center.  As a result, for an isolated energy band,  = 0 
+if both the eigenvectors at the zone center and boundary are either a+  or a−  but =  if they are 
+different.   
+We study the spatial field symmetries of a  for TM and TE polarized waves.  Leaky 
+evanescent waves are considered here as an example.  For TM modes such as Bloch-like 
+surface plasmon polaritons (SPPs) propagating in the x-direction, the magnetic fields of a+  are 
+( )(
+)
+1
+2
+x
+x
+z
+ik x
+ik x
+k z
+k
+ˆ
+H
+H
+Ae
+u
+x
+e
+e
+y
+−
+−
++
+=
+−
+, where A is a constant, kx and kz are the propagation 
+constants in the x- and z-directions, and 
+( )
+ku
+x  is the periodic function [43].  
+( )
+ku
+x  is assumed 
+to be an even function for simplicity as its symmetry does not affect the Zak phase results.  The 
+corresponding 
+electric 
+fields 
+are 
+(
+)
+( )
+(
+)
+(
+)
+(
+)
+1
+2
+2
+zk z
+k
+z
+x
+x
+x
+H
+H
+A
+ˆ
+ˆ
+E
+e
+u
+x
+k sin k x x
+k cos k x z
+i
+−
+
++
+−
+=
+=
++
+− 
+
+, revealing the in-plane x- 
+and out-of-plane z-components are odd and even in the x-direction, or 
+( )
+(
+)
+x
+x
+E
+x
+E
+x
+= −
+−
+ and 
+
+7 
+ 
+( )
+(
+)
+z
+z
+E
+x
+E
+x
+=
+−
+.  Likewise, for a− , we have even 
+( )
+(
+)
+x
+x
+E
+x
+E
+x
+=
+−
+ and odd 
+( )
+(
+)
+z
+z
+E
+x
+E
+x
+= −
+−
+.  Conversely, for TE modes such as waveguide modes, the in-plane electric 
+fields of a+  and a−  are 
+( )
+(
+)
+2
+zk z
+k
+x
+ˆ
+iAe
+u
+x sin k x y
+−
+ and 
+( )
+(
+)
+2
+zk z
+x
+ˆ
+Ae
+u x cos k x y
+−
+, giving rise to 
+odd 
+( )
+(
+)
+y
+y
+E
+x
+E
+x
+= −
+−
+ and even 
+( )
+(
+)
+y
+y
+E
+x
+E
+x
+=
+−
+, respectively.  Therefore, for the in-plane 
+components, the TM and TE polarized a+  and a−  are odd and even in the x-direction.   
+Once the field symmetries of a  are known, their spectral positions will then be deduced 
+via far-field.  By using conservation of energy and time reversal symmetry, the outgoing ports 
+are expressed as  
+ 
+1
+2
+a
+s
+C s
+K a
+−
++
+
+
+=
++
+
+
+
+
+, where  
+0
+T
+N ,
+,
+N ,
+s
+s
+s
+s
+−
+−
+−
+−
+−
+= 
+
+
+  and C is the 
+nonresonant scattering matrix [38].  We find the transformation matrix to be 
+1
+1
+1
+1
+1
+2
+T
+T
+
+
+=
+
+
+−
+
+
+ 
+so that the outgoing fields can now be rewritten as: 
+ 
+ 
+ 
+1
+2
+1
+2
+0 1
+0 2
+0 1
+0 2
+1
+2
+1
+2
+1
+1
+2
+2
+N ,
+N ,
+N ,
+N ,
+T
+,
+,
+,
+,
+N ,
+N ,
+N ,
+N ,
+a
+s
+C s
+T K
+C s
+a
+a
+a
+
+
+
+
+
+
+
+
+
+
+
+
+−
+−
+−
+−
++
+−
++
++
++
+−
+−
++
+−
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
++
+−
+=
++
+=
++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
++
+−
+
+
+
+
+. 
+(2) 
+Eq. (2) can be further simplified by using the relationships between 
+n,i
+ −
+ and 
+n,i
+
+, where i = 1 
+or 2 and n  N is the diffraction order.  As provided in the Supplementary Information [44], 
+given the fact that both far- and near-fields should follow the same spatial symmetry, the 
+radiation patterns of TM a  arising from the interferences between the decay ports should 
+preserve the same 
+( )
+(
+)
+F
+F
+x
+x
+E
+x
+E
+x
+= −
+−
+ and 
+( )
+(
+)
+F
+F
+x
+x
+E
+x
+E
+x
+=
+−
+ dependences, where the 
+superscript 
+F 
+denotes 
+the 
+far-fields, 
+leading 
+to 
+(
+)
+1
+2
+1
+2
+n,
+n,
+n,
+n,
+
+
+
+
+−
+−
++
+= −
++
+ and 
+1
+2
+1
+2
+n,
+n,
+n,
+n,
+
+
+
+
+−
+−
+−
+=
+−
+ for a+  and a− .  Likewise, for TE a , 
+( )
+(
+)
+F
+F
+y
+y
+E
+x
+E
+x
+= −
+−
+ and 
+
+8 
+ 
+( )
+(
+)
+F
+F
+y
+y
+E
+x
+E
+x
+=
+−
+ also give 
+(
+)
+1
+2
+1
+2
+n,
+n,
+n,
+n,
+
+
+
+
+−
+−
++
+= −
++
+ and 
+1
+2
+1
+2
+n,
+n,
+n,
+n,
+
+
+
+
+−
+−
+−
+=
+−
+.  More 
+importantly, both polarizations indicate 
+,1
+,2
+n
+n
+
+−
+= −
+ and 
+,1
+,2
+n
+n
+
+
+−
+= −
+, which agree with the 
+fact that the system should fulfill the inversion symmetry requirement.  However, 
+1
+n,
+ −
+ (
+2
+n,
+ −
+) 
+is not necessarily equal to 
+1
+n,
+
+(
+2
+n,
+
+).  In addition, for a+ , 
+(
+)
+0 1
+0 2
+0 1
+0 2
+,
+,
+,
+,
+
+
+
+
++
+= −
++
+ implies the 
+normal diffraction order is always missing, resulting in an even number of decay ports at both 
+the zone center and boundary.  Therefore, at the zone center for TM and TE polarizations, Eq. 
+(2) can be reduced as: 
+ 
+(
+)
+0
+0
+1
+1
+0
+2
+2
+2
+N
+N
+N ,
+N
+N
+,
+N
+N
+N ,
+N
+N
+s
+s
+C s
+a
+a
+s
+
+
+
+
+
+
+
+
+
+−
+−
+−
+−
+−
++
++
+−
+−
+−
+−
+−
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+=
++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+−
+−
++
+
+
+
+
+
+
+,  
+ 
+(3) 
+where the 1,2 subscripts are now dropped.  On the other hand, at the zone boundary, the 
+outgoing fields carry the same analytical form as Eq. (3) except 
+0,
+s − = 0 since 
+0
+0
+ =
+. 
+Eq. (3) reveals additional information about the occurrence of quasi-BIC at high 
+symmetry points.  In general, quasi-BIC occurs when all the decay ports are zero.  Therefore, 
+at the zone center, unless 
+0
+  = 0, quasi-BIC can only be observed from a+ .  Particularly, for 
+the lowest zone center band gap where only the N = 0 port is present, an a+  quasi-BIC is always 
+present, making it symmetry protected [34].  However, for higher order band gaps, while the 
+normal N = 0 port is still zero, other N > 0 ports are not necessary.  Quasi-BIC can still be 
+found if 
+n
+n
+
+
+− =
+.  In other words, if all the mirror symmetric decay ports of the uncoupled 
+mode are identical and in-phase, destructive interferences occur everywhere across all 
+diffraction orders, resulting in quasi-BIC.  Such special condition can only be met for certain 
+tailored system geometry.  If 
+n
+n
+
+
+− 
+, a+  appears as bright or dark mode depending on the 
+
+9 
+ 
+sign of .  On the other hand, at the zone boundary where 
+0,
+s −is always zero, a+  or a−  can be 
+quasi-BIC if 
+n
+n
+
+−
+=
+ or 
+n
+n
+
+−
+= −
+ is fulfilled.      
+We then explicitly formulate the diffraction orders.  By considering only one single 
+incidence port q such that  
+0
+0
+T
+q,
+s
+s
++
++
+
+
+= 
+ , the coupled mode amplitudes are 
+(
+)
+(
+)
+,
+1
+2
+q
+q
+qs
+a
+i
+−
++
++
++
+−
+=
+−
+
+
+
+
+ and 
+(
+)
+(
+)
+,
+1
+2
+q
+q
+qs
+a
+i
+
+
+
+
+−
++
+−
+−
++
+=
+−
+.  Two mirror symmetric n  N diffraction 
+orders thus are:  
+(
+)(
+)
+(
+)
+(
+)(
+)
+(
+)
+(
+)(
+)
+(
+)
+(
+)(
+)
+(
+)
+,
+,
+,
+,
+1
+1
+,
+2
+2
+1
+1
+,
+2
+2
+n
+n
+q
+n
+n
+n
+n
+q
+q
+n
+n
+q
+q
+n
+n
+q
+q
+n
+n
+q
+q
+q
+s
+c
+s
+s
+s
+i
+i
+i
+c
+i
+−
+−
+−
++
++
+−
++
+−
+−
+−
+−
++
+−
+−
+−
+−
+−
++
+−
+−
+−
++
++
++
+−
+−
+−
+−
+=
+−
++
+−
+=
++
++
++
+−
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+ 
+(4) 
+where 
+n
+c  are the complex nonresonant scattering coefficients.  We see from Eq. (4) that the 
+radiations from a+  and  a−  have odd and even symmetries [37].  While a−  gives two in phase 
+diffraction orders, those from a+  are  out of phase.  Therefore, by fitting the magnitude and 
+phase, 
+2
+,
+,
+n
+q
+s
+s
+
+−
++  and 
+(
+)
+,
+,
+arg
+n
+q
+s
+s
+
+−
++ , spectra of any pair of oblique mirror diffraction orders 
+at the zone center and boundary with Eq. (4) to determine their relative phase, the spectral 
+positions 
+(
+)
+Re
+
+
+ can be deduced to find out whether a+  or a−  is associated with the energy 
+band of interest.         
+III. 
+FINITE-DIFFERENCE TIME DOMAIN SIMULATION 
+We verify the CMT model by FDTD simulations.  Two types of optical systems are 
+considered, and they are 1D Au plasmonic and SiO2/Au photonic crystals.  While the plasmonic 
+crystals (PmCs) support TM-polarized Bloch-like SPPs [45], the photonic crystals (PhCs) 
+excite TE waveguide modes [46].  We will present the results of PmCs here and those of the 
+
+10 
+ 
+PhCs are provided in the Supplementary Information [44].  For the PmCs, the unit cell is shown 
+in Fig. 2(a), with the period P and groove height H are set at 900 nm and 50 nm, respectively, 
+and the groove width W is varied from 100 and 700 nm with a step size of 150 nm.  The 
+corresponding TM-polarized k- and wavelength-resolved total reflectivity, which sums all the 
+diffraction orders, mappings are calculated along the -X direction in Fig 2(b) – (f), showing 
+the presence of the dispersive ±1 and -2 Bloch-like SPP bands, which follow the phase 
+matching equation given as 
+2
+2
+1
+1
+2
+Au
+SP
+Au
+n
+k
+P
+
+
+
+
+
+
+
+ =
++
+
+
+
+
++ 
+
+
+
+, where 
+Au
+
+ is the dielectric constant 
+of Au and nSP is the SPP band, as illustrated by the dash lines in Fig 2(b) [37,45].  More 
+importantly, one sees ±1 SPPs cross at k = 0 m-1 and +1 and -2 SPPs cross at k = /P m-1, 
+yielding two band gaps at  = 925 and 650 nm for the zone center and boundary.  In agreement 
+with the CMT model, the coupled modes exhibit dark and bright radiation characteristics.   
+We attempt to determine the Zak phase of the +1 SPP band.  At the zone center for all 
+PmCs, the dark mode is quasi-BIC and located at the +1 band for W = 100 – 400 nm but flips 
+to the -1 band when W increases further.  The corresponding reflectivity spectra are plotted in 
+Fig. 3(a) for illustration, clearly showing only one single reflectivity dip as the bright mode.  
+As a result, we conclude a+  locates at the +1 band for W = 100 – 400 nm but flips to the -1 
+band for wider W.  On the other hand, at the zone boundary, we can no longer differentiate the 
+spectral positions of a  simply by examining the total reflectivity spectra because two dark 
+and bright modes are present.  Since only a pair of mirror symmetric m = 0th and 1st, or n = 1, 
+diffraction orders is available, Fig. 3(b) & (c) show the simulated 
+2
+1,
+1,
+s
+s
+ −
++
+ and 
+(
+)
+1,
+1,
+arg s
+s
+ −
++  spectra and we fit them by by Eq. (4) to determine the relative phases between 
+the diffraction pairs of two modes.  The best fits are displayed as the solid lines.  The 
+corresponding 
+(
+)
+Re   of all PmCs are summarized in Table 1, in which the highlights are the 
+
+11 
+ 
+coupled modes sitting on the +1 band at the zone center (high energy mode) and boundary (low 
+energy mode).  If the highlights at two regions are either a+  or a− , the Zak phase is 0, but  
+when they are different.  As a result, by comparing the modes at the zone center and boundary 
+of the +1 band,  =  for W = 100, 250, 550 nm but  = 0 for 400 and 700 nm.   
+To confirm our findings, we have simulated the near-field intensity profiles at the zone 
+center and boundary of the +1 band by FDTD in Fig. 4(a) & (b) for different W.  At the zone 
+center, we see the profiles are even with respect to the groove center for W = 100 – 400 nm but 
+change to odd afterwards [18,47].  On the other hand, the profiles at the zone boundary are odd 
+for W = 100, 250, and 700 nm but are even for 400 and 550 nm.  As a result, the field 
+symmetries indicate  =  for W = 100, 250 and 550 nm but 0 for 400 and 700 nm, in consistent 
+with the far-field simulations.  In addition, we have calculated the near-field patterns across the 
+first Brillouin zone for all PmCs in the Supplementary Information [44] and then employ the 
+Wilson loop method to directly determine  given as 
+( )
+P
+P
+X
+k dk
+
+
+−
+, where 
+( )
+X
+k  is the Berry 
+connection given as 
+( ) ( )
+( ) ( )
+*
+,k
+k
+unit cell
+*
+k
+,k
+unit cell
+u
+( x )
+i
+u
+x
+x
+dx
+k
+u
+x
+x u
+( x )dx
+
+
+
+
+
+
+  [47,48].  The evolutions of the individal 
+phase difference, which is 
+( )
+X
+k
+k
+ , of  the +1 band as a fucntion of k with k = 0.04π/P are 
+plotted in Fig. 4(c).  The integrated areas yield the  phases that once again support our results. 
+IV. 
+EXPERIMENTAL VERIFICATION 
+A series of 1D periodic Au rectangular groove PmCs has been fabricated by focused ion 
+beam (FIB) and their scanning electron microscopy (SEM) images are shown in the insets of 
+Fig. 5(a) – (e), showing they have P = 900 nm, H = 50 nm, and W varying from 100 to 700 nm 
+[47].  After the sample preparation, the PmCs are then transferred to a homebuilt Fourier space 
+
+12 
+ 
+optical microscope described in the Supplementary Information for angle- and wavelength-
+resolved diffraction measurements [44].  Briefly, a supercontinuum generation laser is 
+illuminated on the sample at a well-defined incident angle  via the microscope objective lens 
+and the signals from the sample are collected by the same objective lens in which the diffraction 
+orders are projected onto the momentum space [49,50].  By using an aperture to filter out the 
+desired diffraction order, a spectrometer-based CCD detector and a common path 
+interferometer are used for measuring the magnitude and phase spectra [51,52].   
+By varying  sequentially and at the same time measuring the total reflection spectra, we 
+contour plot the TM-polarized reflectivity mappings in Fig. 5(a) – (e) for different W along the 
+-X direction.  They show ±1 and -2 SPP bands are present, and the bands are consistent with 
+the phase-matching equation as illustrated by the dash lines.  From the mappings, we see at 
+normal incidence, or the zone center, BIC-like mode is always observed near the band gap.  
+The +1 band has a+  for W = 100 – 400 nm but a−  for wider W.  On the other hand, at the zone 
+boundary where +1 and -2 SPPs cross at  ~ 20.5o, we see the dark and bright modes are found 
+and their positions depend on W.  To estimate the spectral positions of a , we measure the 
+corresponding m = 0th and 1st, or n = ±1, reflectivity and TM-TE phase difference spectra in 
+Fig. 6(a) & (b) and fit them by Eq. (4) to determine 
+(
+)
+Re   in Table 1, which shows the +1 
+band is a−  for W = 100, 250 and 700 nm is a+  for 400 and 550 nm.  Therefore,  =  for W = 
+100, 250 and 550 nm but = 0 for 400 and 700 nm.   
+Finally, we demonstrate a topologically protected state is formed at the interface between 
+two topological trivial and nontrivial PmCs [47].  We construct a heterostructure by joining 
+two W = 100 and 400 nm PmCs together.  In prior to joining, we have examined by FDTD the 
+field symmetries at the zone center and boundary of two PmCs and determine the  of the 0, -
+1, and +1 SPP bands to be  ,   and  for W = 100 nm and  ,   and 0 for W = 400 nm.  
+
+13 
+ 
+Therefore, the sums of  give   and 0 for W = 100 and 400 nm PmCs, indicating the -2/+1 
+energy gaps at the zone boundary are topological trivial and nontrivial.  We then simulate the 
+heterostructure supercell as shown in Fig. 7(a) that consists of 14 unit cells of W = 100 and 400 
+nm PmCs on the right- and left-handed sides [47].  Fig. 7(b) shows the TM-polarized k- and 
+wavelength-resolved reflectivity mapping at the zone boundary along the -X direction, clearly 
+demonstrating a localized mode is located at k = 0.5/P or θ = 20.5o, and  ~ 640 nm in the 
+mid of the band gap.  We also have simulated the wavelength-dependent near-field mapping 
+of the heterostructure.  For different wavelengths, the near-field intensities at 20 nm above the 
+surface is simulated across the heterostructure and then contour plotted in Fig. 7(c), showing 
+the interface is located at x = 0 m and the trivial and nontrivial regions are at x > 0 m and < 
+0 m, respectively.  One sees two strong fields are visible at ~ 620 and 670 nm in the PmC 
+bulk regions away from the interface due to the excitations of the upper and lower coupled 
+modes.  However, the strongest field strength is observed at the interface, x = 0 µm, at 640 nm, 
+and it decays rapidly into the bulk regions, signifying the presence of a topologically protected 
+interface state [47].  We have prepared the heterostructure by FIB and its SEM image is shown 
+in Fig. 7(d) with W = 100 and 400 nm PmCs on the right- and left-hand sides.  The TM-
+polarized k- and wavelength-resolved reflectivity mapping of the sample is illustrated in Fig. 
+7(e), clearly showing an interface state is found at  = 20.5o and  ~ 625 nm in the +1/-2 band 
+gap at the zone boundary.    
+V. 
+CONCLUSION 
+In summary, we have formulated an analytical model based on temporal CMT to 
+determine the Zak phase of an isolated band in leaky photonic systems.  At the Brillouin zone 
+center and boundary, as the far- and near-fields of the systems share the same spatial symmetry, 
+the mirror symmetric diffractions are either in or  out of phase depending on the Bloch wave 
+
+14 
+ 
+symmetry.  Therefore, the near-field symmetries can be probed by studying the diffraction 
+profiles.  In addition, our model generalizes the occurrence of quasi-BIC at the high symmetry 
+points.  The interplay between the in-coupling constants of different ports plays a decisive role 
+in manifesting quasi-BICs.  For verification, we have studied 1D PmCs and PhCs that support 
+TM- and TE-polarized SPP and waveguide modes by FDTD and the results agree very well 
+with the theory.  We also have prepared 1D PmCs by FIB and examined their diffractions by 
+using Fourier space diffraction spectroscopy and common path interferometry for determining 
+the Zak phases.  In the end, a topological protected interface state is demonstrated by joining 
+two topological trivial and nontrivial PmCs together.      
+VI. 
+ACKNOWLEDGMENT   
+This research was supported by the Chinese University of Hong Kong through Area of 
+Excellence (AoE/P-02/12) and Innovative Technology Fund Guangdong-Hong Kong 
+Technology Cooperation Funding Scheme (GHP/077/20GD). 
+ 
+ 
+
+15 
+ 
+Reference 
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+1057 (2011). 
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+3. J.E. Moore, The birth of topological insulators, Nature 464, 194 (2010). 
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+Mod. Phys. 91, 015005 (2019). 
+5. D.W. Zhang, Y.Q. Zhu, Y.X. Zhao, H. Yan, S.L. Zhu, Topological quantum matter with 
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+Zi, Observation of polarization vortices in momentum space, Phys. Rev. Lett. 120, 186103 
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+dimensional periodic optical system, simulated near-field patterns of the +1 surface 
+plasmon polariton (SPP) band of 1D PmCs across the first Brillouin zone, FDTD 
+simulation results of 1D SiO2/Au photonic crystals (PhCs), and the Fourier space optical 
+
+19 
+ 
+microscope for angle- and wavelength resolved diffraction mapping and common path 
+interferometry 
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+arrays that support Bloch-like surface plasmon polaritons, Phys. Rev. Appl. 15, 024048 
+(2021). 
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+states in one-dimensional plasmonic crystals, Phys. Rev. B 106, 045401 (2022). 
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+numerical aperture microscope objective, Anal. Chem. 79, 2979 (2007). 
+50. F. Bleckmann, Z. Cherpakova, S. Linden, and A. Alberti, Spectral imaging of topological 
+edge states in plasmonic waveguide arrays, Phys. Rev. B 96, 045417 (2017). 
+51. Z.L. Cao, S.L. Wong, S.Y. Wu, H.P. Ho, and H.C. Ong, High performing phase-based 
+surface plasmon resonance sensing from metallic nanohole arrays, Appl. Phys. Lett. 104, 
+171116 (2014). 
+52. S.L. Wong and H.C. Ong, Phase difference mapping of two-dimensional metallic nanohole 
+arrays, Appl. Phys. Lett. 100, 233102 (2012). 
+ 
+ 
+
+20 
+ 
+ 
+Fig. 1.  (a) The schematic shows at the Brillouin zone center and boundary in 1D leaky optical 
+system, two Bloch-like modes a1,2 counter propagate in opposite directions with each supports 
+discrete in-coupling channels 
+1 2
+0 1 2
+1 2
+N, ,
+, ,
+N, ,
+
+
+−
+.  They interact with each other to form two 
+coupled a   at higher and lower energies separated by an energy band gap.  (b)  
+0 1 2
+0
+, ,
+
+
+ at 
+the zone center but 
+0 1 2
+0
+, ,
+
+=
+ at the zone boundary.  
+ 
+ 
+
+Second zone
+ boundary
+band gap
+Lowest zone
+center band gap
+Lowest zone
+boundary
+band gap21 
+ 
+ 
+Fig. 2. (a) The unit cell of 1D PmC for FDTD simulations.  The simulated TM-polarized k- 
+and wavelength-resolved total reflectivity mappings of PmCs with W = (b) 100, (c) 250, (d) 
+400, (e) 550, and (f) 700 nm taken along the -X direction. The white dash lines are calculated 
+by using the phase-matching equation, indicating ±1 and -2 Bloch-like SPPs are excited.  At 
+the zone center and boundary where k = 0 and 0.5, two energy band gaps are formed, featuring 
+two dark and bright modes are located above or below the gap.  Particularly, at k = 0, a quasi-
+BIC is observed at either above or below the gap.   
+ 
+
+(a)
+Air
+(b)
+-2 SPP
+p
+↑H
++1:
+SPP
+W
+Au
+、-1SPP
+C)
+(d)
+(f)
+e22 
+ 
+ 
+Fig. 3.  The TM-polarized total reflectivity spectra of PmCs taken at the zone center for 
+different W, exhibiting only one single reflectivity dip as the bright mode.  The red dash line 
+is the band gap center, indicating the quasi-BIC occurs at shorter wavelength for W = 100, 250 
+and 400 nm but longer wavelength for W = 550 and 700 nm.  At the zone boundary, two TM-
+polarized mirror symmetric n = -1 (black square) and 1 (red circle) (b) reflectivity and (c) phase 
+spectra for W = 100 (top) to 700 (bottom).  The green and blue solid lines are the best fits 
+determined by CMT.  
+
+XX
+X23 
+ 
+ 
+Fig. 4.  The FDTD simulated near-field patterns of the PmCs for different W taken at the 
+Brillouin zone (a) center and (b) boundary, showing their field symmetries are the same for W 
+= 400 and 500 nm but different for W = 100, 250, and 700 nm.  (c) The individual phase profiles 
+determined by the Wilson loop method.  The integration yields the Zak phase, indicating the 
+phase is 0 for W = 400 and 500 nm but  for W = 100, 250, and 700 nm. 
+
+(b)24 
+ 
+ 
+Fig. 5.  The measured TM-polarized k- and wavelength-resolved total reflectivity mappings of 
+PmCs with W = (a) 100, (b) 250, (c) 400, (d) 550, and (e) 700 nm taken along the -X direction. 
+The white dash lines are ±1 and -2 Bloch-like SPPs determined by the phase matching equation.  
+Two band gaps are formed at the zone center and boundary.  The insets are the corresponding 
+SEM images of the PmCs with the scale bare = 1 µm. 
+
+0..9
+600
+-2SPP
+0.8
+6/5
+0.7
+750
++1SPP
+0.6
+825
+0.5
+900
+1
+SPP
+(a)
+(b)
+(c)
+(d)
+(e
+975
+0..4
+80c0L0008060L000800L000s060L0008060L00025 
+ 
+ 
+Fig. 6.  At the zone boundary, two measured TM-polarized mirror symmetric n = -1 (black 
+square) and 1 (red circle) (b) reflectivity and (c) TM-TE phase difference spectra for W = 100 
+(top) to 700 (bottom).  The green and blue solid lines are the best fits determined by CMT.  
+ 
+ 
+ 
+
+26 
+ 
+ 
+Fig. 7.  (a) The schematic of the heterostructure by joining W = trivial 100 and nontrivial 400 
+nm PmCs.  The interface is marked by the dash line.  (b) The FDTD simulated TM-polarized 
+reflectivity mapping of the heterostructure taken at the zone boundary along the -X direction, 
+showing an interface state is found within the gap at  = 640 nm.  (c) The wavelength-
+dependent near-field intensity mapping simulated at 20 nm above the heterostructure.  The 
+interface is located at x = 0 m, showing strong field localization.  The strong fields at 620 and 
+670 nm arise from the PmC bulk regions. (d) The SEM image of the W = 100 and 400 nm with 
+the scale bar corresponding to 1 µm.  (e) The measured TM-polarized reflectivity mapping of 
+the heterostructure taken at the zone boundary along the -X direction, showing an interface 
+state is found within the gap at  = 625 nm.   
+
+W = 400nm
+W = 100nm
+nontrivial
+trivial
+0.9
+0.8
+0.7
+0.6
+0.52
+0.9
+0..8
+0..7
+0.6
+575
+575
+(b)
+e
+600
+600
+625
+625
+650
+650
+675
+675)
+700
+(25)
+/00
+0.4
+0.65)
+0.6
+0.25
+0.30
+0.35
+0.4.0
+0.415
+k (2/)
+k (2TN)
+700
+(c)
+0.9
+680
+0.8
+0..7
+660
+0.6
+0.5
+@)
+640
+0.4
+ABl
+interface state
+0.3
+620
+0.2
+0..1
+W = 400 nm
+W = 100 nm
+0
+600
+1000
+500
+0
+500
+1000
+x (nrm)27 
+ 
+ 
+ 
+ 
+ 
+100 nm 250 nm 400 nm 550 nm 700 nm 
+FDTD 
+Zone 
+center 
+(
+)
+Re +  (eV) 
+1.36 
+1.37 
+1.36 
+1.32 
+1.30 
+(
+)
+Re −  (eV) 
+1.32 
+1.31 
+1.33 
+1.36 
+1.37 
+Zone 
+boundary 
+(
+)
+Re +  (eV) 
+2.00 
+1.98 
+1.84 
+1.85 
+1.98 
+(
+)
+Re −  (eV) 
+1.82 
+1.89 
+1.99 
+1.96 
+1.85 
+Experiment 
+Zone 
+center 
+(
+)
+Re +  (eV) 
+1.36 
+1.36 
+1.36 
+1.33 
+1.32 
+(
+)
+Re −  (eV) 
+1.33 
+1.32 
+1.34 
+1.36 
+1.36 
+Zone 
+boundary 
+(
+)
+Re +  (eV) 
+2.02 
+2.01 
+1.95 
+1.96 
+2.02 
+(
+)
+Re −  (eV) 
+1.93 
+1.96 
+2.02 
+2.01 
+1.95 
+ 
+Table 1.  The FDTD and experimental 
+(
+)
+Re   at the Brillouin zone center and boundary for 
+the PmCs with different W.  The highlights are the coupled modes located on the +1 SPP band.  
+If the highlights at the zone center and boundary are both a+  or a− , the Zak phase is 0.  If not, 
+the Zak phase is .   
+ 
+ 
+
+28 
+ 
+Supplementary Information 
+Determination of the band topology of one-dimensional photonic 
+systems via far-field diffraction 
+ 
+C. Liu, H.R. Wang, and H.C. Ong 
+Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong, People’s 
+Republic of China 
+ 
+A. 
+Derivation of the connection between the far- and near-fields from one-dimensional 
+periodic optical system 
+ 
+ 
+Fig. S1.  The schematic of the 2N+1 diffraction orders arising from the coupled mode supported 
+on 1D periodic leaky system.  
+
+29 
+ 
+As shown in Fig. S1, for a one-dimensional optical leaky periodic system that possesses 
+inversion symmetry in the x-direction, at the Brillouin zone center and boundary, it supports 
+two Bloch-like coupled modes a  above and below the photonic band gap with each dissipates 
+a total of 2N + 1 mirror symmetric diffraction channels in free space, where N is the highest 
+diffraction order.  For TM- and TE-polarizations, both the near- and far-fields should carry the 
+same polarization and field symmetry in the x-y plane along the surface.  For example, for TM-
+polarization, in the far-field at zo above the system, the x-component of the electric field 
+( ,
+)
+F
+x
+o
+E
+x z
+ is expressed as the superposition of all diffraction orders: 
+1
+1
+1
+0
+1
+1
+1
+sin
+cos
+sin
+cos
+1
+1
+sin
+cos
+sin
+cos
+0
+1
+1
+cos
+cos
+cos
+cos
+N
+N
+N o
+N
+N
+N
+o
+o
+N
+N
+N
+o
+N
+N
+N o
+i
+ik
+x
+ik
+z
+i
+ik
+x
+ik
+z
+N
+N
+N
+N
+i
+ikz
+i
+ik
+x
+ik
+z
+i
+ik
+x
+ik
+z
+N
+N
+N
+N
+A e
+e
+e
+A
+e
+e
+e
+A e e
+A
+e
+e
+e
+A e
+e
+e
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+−
+−
+−
+−
++
+−
++
+−
++
+−
+−
+−
+−
+−
+−
++
+−
++
+−
+−
+−
+−
++
++
++
++
++
++
+,   (S1) 
+where An, n, and n are the diffraction amplitude, phase, and angle and the subscript n is the 
+diffraction order.  At the same time, for the near-field, the TM-polarized a+  is a standing wave 
+with 
+( )
+(
+)
+(
+)
+(
+)
+zk z
+k
+z
+x
+x
+x
+ˆ
+ˆ
+E
+e
+u
+x
+k sin k x x
+k cos k x z
+−
+
++
+, where kx and kz are the propagation 
+constants in the x- and z-directions and 
+( )
+ku
+x  is the periodic function.  Assume 
+( )
+ku
+x  is an 
+even function for simplicity, we see 
+( )
+x
+E
+x  is an odd function with 
+( )
+(
+)
+x
+x
+E
+x
+E
+x
+= −
+−
+ 
+dependence.  Therefore, Eq. (S1) should also exhibit 
+( )
+(
+)
+F
+F
+x
+x
+E
+x
+E
+x
+= −
+−
+ dependence, yielding 
+n
+n
+A
+A
+− =
+, 
+n
+n
+
+
+
+−
+=
++
+, and 
+0
+0
+A =
+ that indicate two mirror symmetric diffraction orders have 
+the same magnitude but are always  out of phase and the normal diffraction order is null.  As 
+a result, Eq. (S1) is rewritten as: 
+1
+1
+1
+1
+1
+1
+sin
+cos
+sin
+cos
+1
+1
+sin
+cos
+sin
+cos
+1
+1
+cos
+cos
+cos
+cos
+N
+N
+N o
+N
+N
+N
+o
+N
+N
+N
+o
+N
+N
+N o
+i
+ik
+x
+ik
+z
+i
+ik
+x
+ik
+z
+N
+N
+N
+N
+i
+ik
+x
+ik
+z
+i
+ik
+x
+ik
+z
+N
+N
+N
+N
+A e
+e
+e
+A
+e
+e
+e
+A
+e
+e
+e
+A e
+e
+e
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+−
+−
+−
+−
+−
+−
+−
+−
+−
+−
+−
+−
++
++
+−
+−
+. 
+(S2) 
+By matching Eq. (S2) with the outgoing power amplitudes of a+  from CMT, which are 
+,1
+,2
+0,1
+0,2
+,1
+,2
+1
+2
+N
+N
+N
+N
+a
+
+
+
+
+
+
+−
+−
++
++
+
+
+
+
+
+
+
+
++
+
+
+
+
+
+
++
+
+
+, we conclude 
+(
+)
+1
+2
+1
+2
+n,
+n,
+n,
+n,
+
+
+
+
+−
+−
++
+= −
++
+ and 
+0,1
+0,2
+0
+
+
++
+=
+.   Likewise, 
+for another coupled mode a−  where 
+( )
+(
+)
+(
+)
+(
+)
+zk z
+k
+z
+x
+x
+x
+ˆ
+ˆ
+E
+e
+u
+x
+k cos k x x
+k sin k x z
+−
+
++
+, we see
+
+30 
+ 
+( )
+(
+)
+x
+x
+E
+x
+E
+x
+=
+−
+ and have 
+n
+n
+A
+A
+− =
+, 
+n
+n
+
+−
+=
+ and 
+0
+0
+A 
+, indicating two mirror symmetric 
+orders are in phase and the normal diffraction order is present.  Therefore, Eq. (S1) for a−  is: 
+1
+1
+1
+0
+1
+1
+1
+sin
+cos
+sin
+cos
+1
+1
+sin
+cos
+sin
+cos
+0
+1
+1
+cos
+cos
+cos
+cos
+N
+N
+N o
+N
+N
+N
+o
+o
+N
+N
+N
+o
+N
+N
+N o
+i
+ik
+x
+ik
+z
+i
+ik
+x
+ik
+z
+N
+N
+N
+N
+i
+ikz
+i
+ik
+x
+ik
+z
+i
+ik
+x
+ik
+z
+N
+N
+N
+N
+A e
+e
+e
+A
+e
+e
+e
+A e e
+A
+e
+e
+e
+A e
+e
+e
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+−
+−
+−
+−
+−
+−
+−
+−
+−
+−
+−
+−
+−
++
++
++
++
++
++
+. (S3) 
+We then have 
+1
+2
+1
+2
+n,
+n,
+n,
+n,
+
+
+
+
+−
+−
+−
+=
+−
+ for the outgoing power amplitudes of a−  given as 
+,1
+,2
+0,1
+0,2
+,1
+,2
+1
+2
+N
+N
+N
+N
+a
+
+
+
+
+
+
+−
+−
+−
+−
+
+
+
+
+
+
+
+
+−
+
+
+
+
+
+
+−
+
+
+. 
+ 
+Finally, 
+two 
+conditions 
+(
+)
+1
+2
+1
+2
+n,
+n,
+n,
+n,
+
+
+
+
+−
+−
++
+= −
++
+ and 
+1
+2
+1
+2
+n,
+n,
+n,
+n,
+
+
+
+
+−
+−
+−
+=
+−
+ result in 
+,1
+,2
+n
+n
+
+−
+= −
+ and 
+,1
+,2
+n
+n
+
+
+−
+= −
+. 
+On the other hand, for TE-polarized Bloch-like coupled modes a , at zo in the free space above 
+the system, the y-component of the far-field electric field 
+( ,
+)
+F
+y
+o
+E
+x z
+ can be written as: 
+1
+1
+1
+0
+1
+1
+1
+sin
+cos
+sin
+cos
+1
+sin
+cos
+sin
+cos
+0
+1
+N
+N
+N o
+N
+N
+N
+o
+o
+N
+N
+N
+o
+N
+N
+N o
+i
+ik
+x
+ik
+z
+i
+ik
+x
+ik
+z
+N
+N
+i
+ikz
+i
+ik
+x
+ik
+z
+i
+ik
+x
+ik
+z
+N
+N
+A e
+e
+e
+A
+e
+e
+e
+A e e
+A
+e
+e
+e
+A e
+e
+e
+
+
+
+
+
+
+
+
+
+
+
+
+
+−
+−
+−
+−
++
+−
++
+−
++
+−
+−
+−
+−
+−
++
+−
+−
+−
++
++
++
++
++
++
+.   
+ 
+(S4) 
+The near-field of a+  where
+( )
+(
+)
+zk z
+k
+x
+ˆ
+E
+e
+u
+x sin k x y
+−
+
+, we have 
+( )
+(
+)
+y
+y
+E
+x
+E
+x
+= −
+−
+ such that 
+n
+n
+A
+A
+− =
+, 
+n
+n
+
+
+
+−
+=
++
+, and 
+0
+0
+A =
+, leading to 
+(
+)
+1
+2
+1
+2
+n,
+n,
+n,
+n,
+
+
+
+
+−
+−
++
+= −
++
+ and 
+0,1
+0,2
+0
+
+
++
+=
+.  
+Likewise, for a−  where 
+( )
+(
+)
+y
+y
+E
+x
+E
+x
+=
+−
+, we have 
+n
+n
+A
+A
+− =
+, 
+n
+n
+
+−
+=
+ and 
+0
+0
+A 
+, giving rise 
+to 
+1
+2
+1
+2
+n,
+n,
+n,
+n,
+
+
+
+
+−
+−
+−
+=
+−
+.  Therefore, two conditions give the same conclusion that 
+,1
+,2
+n
+n
+
+−
+= −
+ and 
+,1
+,2
+n
+n
+
+
+−
+= −
+ regardless of the polarization.  As a result, at the zone center for 
+TM- and TE-polarizations, the outgoing profile is:  
+ 
+ 
+(
+)
+0
+0
+1
+1
+0
+2
+2
+2
+N
+N
+N ,
+N
+N
+,
+N
+N
+N ,
+N
+N
+s
+s
+C s
+a
+a
+s
+
+
+
+
+
+
+
+
+
+−
+−
+−
+−
+−
++
++
+−
+−
+−
+−
+−
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+=
++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+−
+−
++
+
+
+
+
+
+
+,  
+ (S5) 
+
+31 
+ 
+where the 1,2 subscripts are now dropped.  We see quasi-BIC arises from a+  and it will occur 
+when 
+0
+n
+n
+
+
+− −
+=
+.  However, for the lowest band gap where only the normal diffraction order 
+is present, quasi-BIC always occur, making it symmetry protected.  On the other hand, at the 
+zone boundary where 
+0,
+s − is always 0, we have for TM- and TE-polarizations: 
+ 
+(
+)
+0
+1
+1
+0
+0
+2
+2
+N
+N
+N ,
+N
+N
+,
+N
+N
+N ,
+N
+N
+s
+s
+C s
+a
+a
+s
+
+
+
+
+
+
+
+
+−
+−
+−
+−
+−
++
++
+−
+−
+−
+−
+−
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+=
++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+−
+−
++
+
+
+
+
+
+
+.  
+(S6) 
+Quasi-BIC occurs depending on the interplay between 
+n
+−  and 
+n
+ .  a+  (a− ) is quasi-BIC if 
+0
+n
+n
+
+
+− −
+=
+ (
+0
+n
+n
+
+
+− +
+=
+) but dark and bright modes are present if 
+0
+n
+n
+
+
+− 
+
+.  
+B. 
+Simulated near-field patterns of the +1 surface plasmon polariton (SPP) band of 1D 
+PmCs across the first Brillouin zone  
+By using the dipole source excitation method, the complex near-field patterns along the +1 SPP 
+band of 1D Au PmCs with period = 900 nm, groove height = 50 nm and different groove widths 
+have been simulated.  The real and imaginary parts of the surface normal components, Re(Ez) 
+and Im(Ez), taken at 20 nm above the surface across the Brillouin zone from k = -/P to /P 
+m-1 are shown in Fig. S2 for groove width W = 100, 250, 400, 550 and 700 nm PmCs.  They 
+will then be used for determining the Zak phase by the Wilson loop method. 
+
+32 
+ 
+Fig. S2. The real and imaginary parts of the z-component of the near-field patterns of the PmCs 
+plotted as a function of k along the +1 SPP band in the first Brillouin zone for different W = 
+(a) & (b) 100, (c) & (d) 250, (e) & (f) 400, (g) & (h) 550, and (i) & (j) 700 nm.    
+ 
+ 
+ 
+
+(a
+(b)33 
+ 
+C. 
+FDTD results of 1D SiO2/Au photonic crystals (PhCs) 
+Fig. S3(a) shows the unit cell of the PhCs, which has 400 nm thick SiO2 coated on Au surface 
+with the period P and the groove height H being set at 900 nm and 200 nm whereas the groove 
+width W varied from 100 and 725 nm with a step size of 125 nm.  The corresponding TE-
+polarized k-resolved total reflectivity mappings are shown in Fig S3(b) – (f), showing the 
+dispersive ±1 and -2 photonic bands, which follow the phase matching equation given as 
+(
+)
+(
+)
+2
+2
+sin
+D
+D
+PhC
+n
+n
+m
+P
+
+ 
+=
++
+, where nD is the refractive index of SiO2 and mPhC is the 
+photonic band.  The calculations are superimposed in Fig 3(b).  We see mPhC = ±1 photonic 
+bands cross at k = 0 m-1 and mPhC = +1 and -2 bands cross at k = /P m-1, yielding two energy 
+band gaps at  = 930 – 1030 nm and 700 – 770 nm at the zone center and boundary.  At the 
+zone center, one symmetry protected quasi-BIC is always found, and it is located on the -1 
+band for W = 100 – 475 nm but flips to the +1 band when W increases further.  At the same 
+time, accidental quasi-BICs are also found along the +1 band at different k for all PhCs.        
+
+34 
+ 
+ 
+Fig. S3. (a) The FDTD unit cell of the PhC. The simulated TE-polarized k- and wavelength-
+resolved total reflectivity mappings of PhCs with W = (b) 100, (c) 225, (d) 350, (e) 475, (f) 
+600, and (g) 725 nm taken along the -X direction. The white dash lines are calculated by using 
+the phase-matching equation, indicating ±1 and -2 photonic band are present.  At the zone 
+center and boundary where k = 0 and 0.5, two energy band gaps are formed, featuring two dark 
+and bright modes are located above or below the gap.  Particularly, at k = 0, a symmetry 
+protected quasi-BIC is observed at either above or below the gap.  On the other hand, an 
+accidentally BIC is observed along the +1 band.   
+
+Air
+p
+SiO2
+H
+W
+Au
+(b)
+C
+-2 band
++1 band
+-1 band
+(d)
+(e)
+(f)
+935 
+ 
+We will focus on the modes located on the +1 band at the zone center and boundary and 
+determine their field symmetries as well as .  The reflectivity spectra of the PhCs taken under 
+normal incidence, i.e., at the zone center, are illustrated in Fig. S4(a), clearly showing only one 
+single reflectivity dip is present as the bright mode, verifying another coupled mode is quasi-
+BIC that does not produce any dip.  As quasi-BIC arises solely from a+  for the lowest band 
+gap, we deduce the coupled mode on the +1 band is symmetric a−  for W = 100 – 475 nm PhCs 
+but becomes asymmetric a+  for W = 600 and 725 nm PhCs.  On the other hand, the reflectivity 
+spectra taken at the zone boundary for all PhCs are shown in Fig. S4(b), showing two bright 
+and dark modes are present.    
+ 
+Fig. S4. The TE-polarized total reflectivity spectra of PhCs taken at the zone (a) center and (b) 
+boundary for different W.  At the zone center, only one single reflectivity dip is present as the 
+bright mode.  On the other hand, at the zone boundary, two bright and dark modes are present.   
+
+36 
+ 
+ 
+To determine the near-field symmetries of the PhCs at the zone boundary, the two mirror 
+symmetric diffraction and phase spectra are shown in Fig. S5 and they are fitted with 
+2
+1,
+1,
+s
+s
+ −
++
+ and 
+(
+)
+1,
+1,
+arg s
+s
+ −
++  from CMT.  The best fits are displayed as the solid lines and the 
+fitted results 
+(
+)
+Re   are tabulated in Table S1.  in which the highlights are the coupled modes 
+sitting on the +1 photonic band at the zone center (high energy mode) and boundary (low 
+energy mode).  If the highlights at two regions are either a+  or a− , the Zak phase is 0, but  
+when they are different.  As a result, we conclude the Zak phase of +1 band for W = 100, 225 
+and 600 nm is  but becomes 0 for W = 350, 475 and 725 nm.   
+ 
+ 
+ 
+ 
+ 
+100 nm 
+225 nm 
+350 nm 
+475 nm 
+600 nm 
+725 nm 
+Zone 
+center 
+(
+)
+Re +  (eV) 
+1.18 
+1.19 
+1.22 
+1.27 
+1.33 
+1.37 
+(
+)
+Re −  (eV) 
+1.21 
+1.26 
+1.28 
+1.29 
+1.29 
+1.31 
+Zone 
+boundary 
+(
+)
+Re +  (eV) 
+1.62 
+1.65 
+1.72 
+1.78 
+1.79 
+1.79 
+(
+)
+Re −  (eV) 
+1.67 
+1.69 
+1.69 
+1.71 
+1.77 
+1.84 
+ 
+Table S1.  The FDTD 
+(
+)
+Re   at the Brillouin zone center and boundary for the PhCs with 
+different W.  The highlights are the coupled modes located on the +1 photonic band.  If the 
+highlights at the zone center and boundary are both a+  or a− , the Zak phase is 0.  If not, the 
+Zak phase is .   
+   
+ 
+
+37 
+ 
+ 
+Fig. S5.  At the zone boundary, two TE-polarized mirror symmetric n = -1 (black square) and 
+1 (red circle) (a) reflectivity and (b) phase spectra of the PhCs for W = 100 (top) to 725 (bottom).  
+The green and blue solid lines are the best fits determined by CMT.  
+ 
+ 
+
+4
+XC38 
+ 
+To verify the Zak phases, we have simulated the real and imaginary parts of the surface normal 
+components, Re(Ez) and Im(Ez), taken at 20 nm above the surface across the Brillouin zone 
+from k = -/P to /P m-1 in Fig. S6 for all PhCs.  They will then be used for determining the 
+Zak phase by the Wilson loop method given as 
+( )
+P
+P
+X
+k dk
+
+
+−
+, where 
+( )
+X
+k   is 
+( ) ( )
+( ) ( )
+*
+,k
+k
+unit cell
+*
+k
+,k
+unit cell
+u
+( x )
+i
+u
+x
+x
+dx
+k
+u
+x
+x u
+( x )dx
+
+
+
+
+
+
+ .  The evolutions of the individal phase difference, which is 
+( )
+X
+k
+k
+ , of  the +1 band as a fucntion of k with k = 0.04π/P of all PhCs are plotted in Fig. 
+S7.  The integrated areas yield the Zak phases are  for W = 100, 225, and 600 nm and 0 for 
+W = 350, 475 and 725 nm, and they agree very well with earlier CMT results. 
+
+39 
+ 
+ 
+Fig. S6. The real and imaginary parts of the z-component of the near-field patterns of the PhCs 
+plotted as a function of k along the +1 photonic band in the first Brillouin zone for different W 
+= (a) & (b) 100, (c) & (d) 225, (e) & (f) 350, (g) & (h) 475, (i) & (j) 600, and (k) & (l) 725 nm.    
+
+(C)
+(e
+(g)
+(h)
+(i)
+(k)40 
+ 
+ 
+Fig. S7. The individual phase profiles of the PhCs with different W.  The integration yields the 
+Zak phase, indicating the phase  for W = 100, 225, and 600 nm and 0 for W = 350, 475 and 
+725 nm. 
+ 
+D. 
+Schematic of the Fourier space optical microscope for angle- and wavelength 
+resolved diffraction mapping and common path interferometry 
+Fig. S8 shows the schematic of the Fourier space optical microscope.  Briefly, a broadband 
+supercontinuum laser from a nonlinear photonic crystal fiber is collimated and then passed 
+through a set of linear polarizers, wave plates, and lenses before being focused onto the back 
+focal plane (BFP) of a 100X objective lens (OB) with numerical aperture = 0.9.  The light 
+exiting from the objective lens is then a collimated beam with well-defined linear polarization.  
+In addition, by displacing the focused spot across the BFP of the objective lens using a 
+motorized translation stage, the incident polar angle  of the collimated beam onto the sample 
+can be varied following sin = d/f, where d is the distance between the focused spot and the 
+optical axis of the BFP and f is the focal length of the objective lens.  In addition, the azimuth 
+angle  can be varied by a motorized rotation sample stage to align the incident plane to the -
+X direction of the PmC.  The diffractions from the PmC are then collected by the same objective 
+lens and are routed through a set of Fourier lens system so that the diffraction orders are 
+projected onto the momentum space.  By placing an aperture at the momentum space to filter 
+
+41 
+ 
+out the desired diffraction order, its intensity and phase spectra can be measured by a 
+spectrometer-based CCD detector and a common path interferometer [1]. 
+To perform common path interferometry, the 45o linearly polarized collimated beam with the 
+Jones vector given as 
+1
+1
+1
+2
+ 
+ 
+ 
+ is incident on the PmC.  The diffraction order from the PmC 
+after the aperture can be formulated as: 
+0
+0
+TM
+TE
+i
+TM
+PmC
+i
+TE
+r
+e
+J
+r
+e
+
+
+
+
+= 
+
+
+
+, where rTM,TE and TM,TE 
+are the magnitudes and phases for TM- and TE-polarizations.   The diffraction passes through 
+a quarter wave plate with the fast axis being placed at 45o with respect to the incident plane 
+and 
+a 
+motorized 
+rotatable 
+analyzer 
+with 
+angle 
+, 
+which 
+are 
+given 
+as 
+2
+( )
+2
+cos
+sin
+cos
+sin
+cos
+sin
+analyzer
+J
+
+
+
+
+
+
+
+
+
+= 
+
+
+
+ and 
+(45 )
+1
+1
+1
+1
+1
+2
+QWP
+i
+i
+J
+i
+i
+
+−
++
+
+
+=
+
+
++
+−
+
+
+.  The output vector is 
+( )
+(45 )
+1
+1
+1
+2
+analyzer
+QWP
+PmC
+J
+J
+J
+
+
+ 
+ 
+ 
+.  After some formulations, the intensities for different  = 0o, 
+±45o, 
+and 
+90o 
+can 
+be 
+written 
+as:
+(
+)
+2
+2
+2
+0
+1
+1
+( )
+2
+sin
+2
+4
+0
+i
+TM
+TE
+TM
+TE
+TM
+TE
+r
+e
+i r
+R
+r
+r
+r
+r
+
+
+
+
+
++
+=
+=
++
++
+
+
+
+
+, 
+(
+)
+2
+2
+45
+1
+( )
+2
+cos
+4
+TM
+TE
+TM
+TE
+R
+r
+r
+r
+r
+
+
++
+=
++
++
+, 
+(
+)
+2
+2
+45
+1
+( )
+2
+cos
+4
+TM
+TE
+TM
+TE
+R
+r
+r
+r
+r
+
+
+−
+=
++
+−
+, and 
+(
+)
+2
+2
+90
+1
+2
+sin
+4
+TM
+TE
+TM
+TE
+R
+r
+r
+r
+r
+
+=
++
+−
+, where 
+TM
+TE
+
+
+
+=
+−
+.  Therefore, the phase difference 
+between TM- and TE- polarized diffractions can be calculated by:
+0
+90
+45
+45
+( )
+( )
+tan ( )
+( )
+( )
+R
+R
+R
+R
+
+
+ 
+
+
++
+−
+−
+=
+−
+.   
+
+42 
+ 
+ 
+Fig. S8.  The schematic of the Fourier optical microscope. 
+ 
+Reference 
+53. Z.L. Cao, S.L. Wong, S.Y. Wu, H.P. Ho, and H.C. Ong, High performing phase-based 
+surface plasmon resonance sensing from metallic nanohole arrays, Appl. Phys. Lett. 104, 
+171116 (2014). 
+ 
+
+P: polarizer
+L1,L2,L3,L4: focusing lens
+OB: objective lens
+BS: beam splitter
+BFP: back focal plane

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@@ -0,0 +1,1122 @@
+Learning Coordination Policies over Heterogeneous Graphs for
+Human-Robot Teams via Recurrent Neural Schedule Propagation
+Batuhan Altundas1, Zheyuan Wang1, Joshua Bishop1 and Matthew Gombolay1
+Abstract— As human-robot collaboration increases in the
+workforce, it becomes essential for human-robot teams to
+coordinate efficiently and intuitively. Traditional approaches
+for human-robot scheduling either utilize exact methods that
+are intractable for large-scale problems and struggle to ac-
+count for stochastic, time varying human task performance,
+or application-specific heuristics that require expert domain
+knowledge to develop. We propose a deep learning-based
+framework, called HybridNet, combining a heterogeneous
+graph-based encoder with a recurrent schedule propagator for
+scheduling stochastic human-robot teams under upper- and
+lower-bound temporal constraints. The HybridNet’s encoder
+leverages Heterogeneous Graph Attention Networks to model
+the initial environment and team dynamics while accounting
+for the constraints. By formulating task scheduling as a se-
+quential decision-making process, the HybridNet’s recurrent
+neural schedule propagator leverages Long Short-Term Mem-
+ory (LSTM) models to propagate forward consequences of
+actions to carry out fast schedule generation, removing the
+need to interact with the environment between every task-
+agent pair selection. The resulting scheduling policy network
+provides a computationally lightweight yet highly expressive
+model that is end-to-end trainable via Reinforcement Learning
+algorithms. We develop a virtual task scheduling environment
+for mixed human-robot teams in a multi-round setting, capable
+of modeling the stochastic learning behaviors of human work-
+ers. Experimental results showed that HybridNet outperformed
+other human-robot scheduling solutions across problem sizes
+for both deterministic and stochastic human performance, with
+faster runtime compared to pure-GNN-based schedulers.
+I. INTRODUCTION
+With collaborative robots (cobots) becoming more avail-
+able in the industrial and manufacturing environments, robots
+and humans increasingly share the same work space to
+collaborate on tasks [1]. By removing the cage around tradi-
+tional robot platforms and integrating robots into dynamic,
+final assembly operations, manufacturers can see improve-
+ments in reducing a factory’s footprint and environmental
+costs as well as increased productivity [2]. In this paper,
+we focus on the problem of multi-agent task allocation and
+scheduling [3] with mixed human-robot teams over multiple
+iterations of the same task allocation problem. Our work
+accounts for and leverages stochastic, time-varying human
+task performance to quickly solve task allocation problems
+among team members to achieve a high-quality schedule
+with respect to the application-specific objective function
+*This work was supported in part by the Office of Naval Research
+under grant N00014-19-1-2076 and Naval Research Laboratory under grant
+N00173-21-1-G009.
+1Batuhan Altundas, Zheyuan Wang, Joshua Bishop and Matthew Gom-
+bolay are with the Institute for Robotics and Intelligent Machines, Georgia
+Institute of Technology, Atlanta, GA 30332, USA {baltundas3,
+pjohnwang, jbishop45, mgombolay3}@gatech.edu
+while satisfying the temporal constraints (i.e., upper and
+lower bound deadline, wait, and task duration constraints).
+Compared to task scheduling within multi-robot systems,
+the inclusion of human workers makes scheduling even more
+challenging because, while robots can be programmed to
+carry out certain tasks at a fixed rate, human workers typ-
+ically have latent, dynamic, and task-specific proficiencies.
+Effective collaboration in human-robot teams requires utiliz-
+ing the distinct abilities of each team member to achieve safe,
+effective, and fluent execution. For these problems, we must
+consider the ability of humans to learn and improve in task
+performance over time. To exploit this property, a scheduling
+algorithm must reason about a human’s latent performance
+characteristics in order to decide whether to assign the best
+worker to a task now versus giving more task experience
+to a person who is slower but has a greater potential for
+fluency at that particular task. However, it is non-trivial to
+infer human strengths and weaknesses while ensuring that
+the team satisfies requisite scheduling constraints, due to
+the uncertainty introduced by variability in task execution
+behavior across different individuals, as well as uncertainty
+on future task performance affected by human’s learning
+effects with practice [4]. Moreover, a lack of consideration
+for human preferences and perceived equality may, in the
+long run, put efficient behavior and fluent coordination at a
+contradiction [5].
+Recent advances in scheduling methods for human-robot
+teams have shown a significant improvement in the ability to
+dynamically coordinate large-scale teams in final assembly
+manufacturing [6], [7]. Prior approaches typically rely on
+an assumption of deterministic or static worker-task profi-
+ciencies to formulate the scheduling problem as a mixed-
+integer linear program (MILP), which is generally NP-hard
+[8]. Exact methods are hard to scale and often fail to consider
+the time-varying stochastic task proficiencies of human work-
+ers over multi-round schedule execution that could result
+in significant productivity gains. The heuristic approaches
+may be able to determine task assignments; however, such
+approaches required domain specific knowledge that takes
+years to gain. We desire a scalable algorithmic approach that
+can automatically learn to factor in the human behavior for
+fast and fluent human-robot teaming.
+Advancements in artificial intelligence have fostered the
+idea of leveraging deep neural networks (DNNs) to solve
+a plethora of problems in operations research [9]. DNNs
+can be trained to automatically explore the problem struc-
+ture and discover useful representations in high-dimensional
+data towards constructing high-quality solutions, without
+arXiv:2301.13279v1  [cs.AI]  30 Jan 2023
+
+Fig. 1.
+Overview of Multi-Round Environment with HybridNet Scheduler. Left: The Multi-Round Scheduling Environment is developed to simulate a
+human-robot scheduling problem over multiple iterative rounds of execution, accounting for changes in human task performance. Right: HybridNet consists
+of a heterogeneous graph-based encoder to extract high-level embeddings of the problem and a recurrent schedule propagator for fast schedule generation.
+hand-crafted feature engineering [10]. Particularly, promis-
+ing progress has been made in learning scalable solvers
+with graph neural networks via imitation learning (IL) or
+reinforcement learning (RL), outperforming state-of-the-art,
+approximate methods [11], [12], [13].
+To overcome the limitations of prior work, we propose
+a deep learning-based framework, called HybridNet, for
+scheduling stochastic human-robot teams under temporal
+constraints. Figure 1 shows the overall framework of our
+proposed method operating in a multi-round environment.
+HybridNet utilizes a heterogeneous graph-based encoder and
+a recurrent schedule propagator. The encoder extracts high
+level embeddings of the scheduling problem using a hetero-
+geneous graph representation of the problem extended from
+the simple temporal network (STN) [14]. By formulating
+task scheduling as a sequential decision-making process, the
+recurrent propagator uses Long Short Term Memory (LSTM)
+cells to carry out fast schedule generation. The resulted
+policy network provides a computationally lightweight yet
+highly expressive model that is end-to-end trainable via
+reinforcement learning algorithms.
+The primary contributions of our work are:
+• We propose a deep learning-based framework, Hybrid-
+Net, for human-robot coordination under temporal con-
+straints. HybridNet consist of a Heterogeneous Graph-
+based encoder and a Recurrent Schedule Propagator.
+The encoder extracts relevant information about the
+initial environment, while the Propagator generates the
+consequential models of each task-agent assignments
+based on the initial model. Inspired by the sensory
+encoding and recurrent processing of the brain, this
+approach allows for fast schedule generation, removing
+the need to interact with the environment between every
+task-agent pair selection.
+• We develop a virtual task scheduling environment for
+mixed human-robot teams in a multi-round setting,
+capable of modeling the stochastic learning behavior
+of human workers. We make our environment OpenAI
+gym-compatible and expect it to serve as a testbed to
+facilitate the development of human-robot scheduling
+algorithms. The implementation is publicly available.1
+• We present a novel policy model that jointly learns how
+to pick agents and tasks without interacting with the
+environment between intermediate scheduling decisions
+and only needs a single reward at the end of schedule.
+By factoring in the action space into an agent selec-
+tor and a task selector, we enable conditional policy
+learning with HybridNet. We account for the state and
+agent models when selecting the agents, and combine
+the information regarding the tasks, selected agent and
+the state for task assignment. As a result, HybridNet is
+end-to-end trainable via Policy Gradients algorithms.
+• We conducted extensive experiments to validate Hy-
+bridNet across a set of problem sizes. Results showed
+HybridNet consistently outperformed prior human-
+robot scheduling solutions under both deterministic and
+stochastic settings.
+II. RELATED WORK
+A. Multi-Agent Scheduling Problem
+Task assignment and scheduling of multi-agent systems is
+an optimization problem that has been studied for real world
+applications, both for Multi-Robot Task Allocation(MRTA)
+problem using traditional techniques [15] and deep learning
+based techniques [16] as well as for human-robot collab-
+oration [7]. Task Allocation can be formalised by Mixed
+Integer Linear Programming (MILP) to capture it’s con-
+straints. The exponential complexity of solving the MILP
+can be accelerated through constraint programming methods
+[7], [17], [18] or heuristic schedulers to leverage better
+scalability [19], [20]. Zhang et al. encoded task schedules
+as chromosomes for a genetic algorithm that optimized
+schedules for heterogeneous human-robot collaboration by
+repeatedly crossing over and mutating the solutions to find
+the optimal schedule. [21]
+1https://github.com/altundasbatu/HybridNet IROS2022
+
+Multi-Round Env
+HybridNet
+Schedule Propagator
+[wl|/1] 
+Encoder
+Problem Instance
+Input to
+Agent
+Learning Curve Models
+L'STM
+LSTM
+Sample
+Agent
+Agent Selector
+Embedding
+an
+Temporal Constraints
+Human-Robot Teams
+Agent
+Layer
+Layel
+HetGAT Layer
+HetGAT
+etGAT I
+State
+LSTM
+Agent Index
+Learning Curve
+State
+Repetition Tracker
+Estimator
+ Embeddings
+Task
+Sample
+(Task, Agent)
+ Round number
+Task Selector
+Embeddings
+a Task
+Picked
+Single assignment
+Evaluate
+Step
+Reward
+/Makespan
+Whole Schedule
+TrainingGombolay et al. present an algorithm to capture domain
+knowledge through scheduling policy requiring domain-
+expert demonstrations [22]. Wang et al. propose Schedu-
+leNet, a Heterogenous Graph Neural Networks-based model
+for task allocation under temporospatial constraints, trained
+through Imitation Learning using optimal schedule [23].
+ScheduleNet relies on interactive scheduling scheme, with
+constant update of an environment before reaching a com-
+plete schedule. These approaches require optimal schedules
+generated by other expert systemsto train and have high
+computational complexity that make their implementation
+costly.
+B. Modeling Human-Robot Teams
+As advancements in robot capability progress, they be-
+come safer and effective to use in conjunction with humans
+to complete specialized works. Liu et al. presents a model
+of human task completions, showing an increase in the task
+efficiency as a result of learning. This paper shows that
+prediction of human performance enhances the ability of
+the scheduling systems to explicitly reason about the agents’
+capabilities [4]. Prior work on behavioral teaming and the
+natural computational intractability of large-scale schedule
+optimization suggests that robots can offer a valuable service
+by designing and adapting schedules for human teammates.
+In our system, we leverage the findings of Liu et al. to
+account for humans learning over time, both in problem
+generation as part of the environment and a learning curve
+predictor as part of the scheduling policy. The human learn-
+ing curve follows an exponential function of generic form
+over the course of multiple iterations as shown in Equation
+1 [4]:
+y = c + ke−βi
+(1)
+where i is the number of iteration the human has previously
+executed a task and c, k, β parameters. We further account
+for the stochastic-nature of human learning in our environ-
+ment.
+C. Graph Neural Networks
+Graph Neural Networks (GNNs) are a class of deep neural
+networks that learn from unstructured data by representing
+objects as nodes and relations as edges and aggregating
+information from nearby nodes [24]. GNNs have been widely
+applied in graph-based problems such as node classification,
+link prediction and clustering, and they have shown to
+have an impressive performance [25]. The Heterogeneous
+Graph Attention Network presented in Wang et al. utilizes
+Deep Learning Algorithms to address the Scheduling Prob-
+lem, showing improved performance compared to non-Deep
+Learning Schedulers such as Earliest-Deadline First (EDF)
+[26] and Tercio [7] at the cost of increased computational
+complexity [23].
+D. LSTM Based Sequence Prediction
+The impact of the LSTM network has been notable
+in language modeling [27], speech-to-text transcription[28],
+machine translation [29], and other applications that involve
+predictive modeling [30], [31], [32]. The advantage of this
+lengthier path generated through the recurrent nature of the
+neural network is that it affords an opportunity to build a
+certain degree of intuition that can prove beneficial during
+all phases of the process [30], [33].
+III. HUMAN-ROBOT SCHEDULING PROBLEM
+A. Problem Overview
+In this paper, we focus on the problem of human-robot task
+allocation and scheduling with temporal constraints [15]. We
+describe the problem components using a 4-tuple ⟨a, τ, d, w⟩
+form. a represents all agents that belong to the human-robot
+team, τ represents all the tasks to be performed. Each task,
+τi, and agent, aj, have a task completion duration dur(τi, aj)
+and agents are capable of completing a sequence of tasks in
+order. d contains the set of deadline constraints, where di ∈ d
+specifies the tasks depending on τi [23]. w is the set of wait
+constraints where wij ∈ w denotes the wait time between
+tasks τi and τj. A Schedule, S, is a sequence of task-agent
+pairs ⟨τi, aj⟩ such that S contains all tasks in τ.
+B. Multi-Round Scheduling Environment
+The Multi-Round Scheduling Environment is developed
+to simulate a human-robot scheduling problem over multiple
+iterative rounds of execution, accounting for changes in
+the task performance of human workers based on previous
+round. Each round is a step in the OpenAI Gym-compatible
+environment, taking as input the complete set of task-agent
+pairs for the scheduling problem, simulating the sequential
+assignment of tasks to agents.
+Each round’s execution is considered finished when all
+the tasks are assigned to one of the agents or if the provided
+schedule is determined to be infeasible under the problem
+constraints. The environment checks the feasibility of the
+provided schedule given the constraints of the problem,
+and computes the total duration of task completion of the
+schedule if the schedule is feasible. If the schedule does not
+satisfy the constraints, it is determined to be infeasible and
+the list of tasks that could not been scheduled are returned.
+We formulate the Multi-Round Scheduling Environment as
+a Partially Observable Markov Decision Process (POMDP)
+using a six-tuple ⟨S, A, T, R, Ω, O, γ⟩ below:
+• States: The problem state S is a state of the Multi-
+Round Environment consistent of the state of the
+Agents.
+• Actions: Actions at round t within the Multi-Round
+Environment refers to a complete set of Task Alloca-
+tions made up of a list of task-agent pairs, denoted as
+At = [⟨τi1, aj1⟩, ⟨τi2, aj2⟩, ...] to be executed in order.
+• Transitions: T corresponds to executing the action in
+Multi-Round Scheduling Environment and proceed to
+next time step.
+• Rewards: Rt is based on the scheduling objective a user
+wants to optimize. In Section III-E we show how to
+compute Rt when optimizing makespan.
+• Observations: Ω is the estimated performance of all the
+task-agent pairs, plus the observable constraints.
+
+• Observation Function: O is handled by the Learning
+Curve Estimator explained in the Section III-D.
+• Discount factor, γ
+C. Agent Models
+The Multi-Round Environment stores the Agent informa-
+tion, allowing the environment to keep track of each agent
+and which tasks it has previously completed. The update of
+the Environment happens at the end of each round, allowing
+agents to modify themselves based on their internal models.
+to update the model based on the selected (task-agent) pairs
+for each round.
+1) Determinitic Robot Model: We generate the robot task
+completion times randomly through uniform distribution.
+2) Stochastic Human Model: We generate the human
+task completion times randomly based on Equation 1, such
+that the Environment can be setup to provide Deterministic
+and Stochastic performance for human learning. The task
+duration parameters of the human learning model, c, k, β, in
+Equation 1 are built from the randomly selected initial task
+completion time for round 0. For Stochastic performance,
+the standard deviations are used to sample from a Normal
+Distribution as presented in Liu et al. [4].
+D. Learning Curve Estimator
+The scheduler is given an estimate of the performance of
+the human agents for each task based on the information
+about the task duration of the previous executions of the
+task-agent pair through the Learning Curve Estimator as
+part of our OpenAI Gym-like Environment In our paper,
+we have implemented a black box model based on the
+insights presented in Liu et al.[4] to simulate a Stochastic
+Human Learning Estimator. As an Agent completes a task
+in multiple rounds, the Agent Model records the task comple-
+tion duration, allowing Learning Curve Estimator to predict
+the next task-agent duration more accurately. To represent
+the increase in accuracy from increase in information, we
+implemented a Learning Curve Estimator that generates an
+estimate of the human agent performance using the actual
+task performance as the mean of a Gaussian Distribution
+with noise that exponentially decreases with the number of
+repetitions of the same task for that agent in previous rounds.
+E. Reward Design
+The total reward, Rt, for the schedule generated by the
+multi-round scheduling environment is calculated based on
+feasible, A′, and infeasible, ˜A′, subsets of task allocations,
+such that At = A′
+t∪ ˜A′
+t. Specifically, the reward, Rt, is based
+on the expected reward for the feasible subset of task-agent
+assignments, Rt(A′
+t), and the reward from the assignment
+of the infeasible subset of task-agent assignments, Rt ˜A′
+t... ,
+based on the point estimate of the reward from assigning
+the incomplete task to the agent that will complete it in
+the longest possible duration, multiplied by an infeasible
+coefficient Ci as shown in equation 2:
+Rt =
+�
+i∈A′
+t
+R (τi, ai) + Cimaxaj
+�
+��
+i∈ ˜
+A′
+R (τi, aj)
+�
+�
+(2)
+The Total Schedule Reward, RS, favors schedules with
+more feasible task allocations and enables learning from
+infeasible explorations during training.
+IV. HYBRIDNET SCHEDULING POLICY
+As shown in Figure 1, our HybridNet framework consists
+of a heterogeneous graph-based encoder to learn high level
+embeddings of the scheduling problem, and a recurrent
+schedule propagator to generate the team schedule sequen-
+tially. This hybrid network architecture enables directly
+learning useful features from the problem structure, owing to
+the expressiveness of heterogeneous graph neural networks,
+and at the same time efficiently constructing the schedule
+with our LSTM-based propagator. As a result, HybridNet
+does not require interacting with the environment between
+every task-agent pair selection, which is necessary but com-
+putationally expensive in prior work [16], [23].
+We denote the policy learned by HybridNet as πθ(A|S),
+with θ representing the parameters of the neural network. At
+round t, an action takes the form of an ordered sequence
+of scheduling decisions, At = {d1, d2, ..., dn}, di = ⟨τi, aj⟩,
+where a latter decision, di, is conditioned on its former ones,
+d1:i−1. Then, the policy can be factorized as
+pθ(At|St) =
+n
+�
+i=1
+pθ(di|St, d1:i−1)
+(3)
+Using
+the
+Recurrent
+Schedule
+Propagator,
+HybridNet
+recursively
+computes
+the
+conditional
+probability,
+pθ(di|St, d1:i−1),
+for
+sampling
+a
+task-agent
+pair.
+At
+the end, the network collects all the decisions and sends to
+the environment for execution.
+A. Heterogeneous Graph Encoder
+We build our Encoder using the heterogeneous graph at-
+tention (HetGAT) layer proposed in [23] that has been shown
+effective in representation learning of multi-agent scheduling
+problems. At the start of each round for a given human-robot
+scheduling problem, the heterogeneous graph representation
+is built by extending from the simple temporal network
+(STN) that encodes the temporal constraints to include agent
+nodes and a state summary node. The metagraph of the
+resulted graph is shown in Figure 2, which summarizes
+all the node types and edge types. Then, a HetGAT layer
+computes the output node features by performing per-edge-
+type message passing followed by per-node-type feature
+reduction, while utilizing a feature-dependent and structure-
+free attention mechanism. We refer interested readers to [23]
+for full details of implementing a HetGAT layer.
+By stacking several HetGAT layers sequentially, we con-
+struct the Encoder that utilizes multi-layer structure to extract
+high-level embeddings of each node that will be send to
+the propagator for schedule generation. We follow the same
+
+Fig. 2.
+Metagraph of the heterogeneous graph built from the STN by
+adding agent and state summary nodes.
+hyper-parameters for HetGAT layers as provided in Wang et
+al. [23]
+B. Recurrent Schedule Propagator
+The HetGAT layers are computationally complex and
+require interactive scheduling to generate the initial model.
+By utilizing an LSTM based Recurrent Predictor, we prop-
+agate forward consequences of each task-agent assignment,
+recreating the encoded information about the environment
+without relying on the initial HetGAT Layer, significantly
+reducing the computational complexity of our scheduler.
+The Recurrent Schedule Propagator takes as input the
+Task, State and Agent embeddings generated by the Het-
+erogeneous Graph Encoder and sequentially generates task-
+agent pairs based on the encoded information. To predict the
+consecutive encoding of state and agents, we use an LSTM
+Model to recursively generate the Agent and State after
+each assignment of a task to an agent, without interacting
+with the Environment, outputting the sequential task-agent
+assignment for the complete set of tasks. The pseudo-code
+for scheduling generation with HybridNet is presented in
+Algorithm 1.
+As di = ⟨τi, aj⟩, we further factor pθ(di|St, d1:i−1) into
+an agent selector and a task selector. That is, πfactor(d|·) =
+πagent(aj|·) · πtask(τi|aj, ·). This factorization allows the
+policy to capture the underlying composite and conditional
+nature of the scheduling decisions, where the task to schedule
+is strongly dependent on the picked agent.
+The Agent Selector selects the new agent for the next deci-
+sion d based on the state and agent information. Specifically,
+the concatenated state-agent embeddings are processed by a
+feed-forward neural network, fa, to compute the likelihood
+of selecting each agent for the next task-agent pair, using
+Equation 4. A softmax operation is performed to convert
+the raw predictions into a probability distribution. After
+the selection of the agent, the agent embedding of the
+chosen agent is updated based on the selected task and state
+embeddings, as state change only happens for the assigned
+agent. This approach allows the agent selector to consider
+how busy each agent is, based on the inherent information
+Algorithm 1 Psuedocode for Schedule Generation
+Input: graph g, features f, unscheduled-Tasks u
+Output: schedule
+1: schedule = [ ], i = 1
+2: (ha1, ca1, ht1, ct1, hs1, cs1) ← Encoder(g, f)
+3: while |u| ̸= 0 do
+4:
+pai ← AgentSelector(hsi, hai)
+5:
+ai ← Sampling(pai)
+6:
+pti ← TaskSelector(hti, hsi, ai)
+7:
+ti ← Sampling(pti−1)
+8:
+schedule.append(⟨ti, ai⟩)
+9:
+unscheduledTasks.remove(ti)
+10:
+if |unscheduledTasks| == 0 then
+11:
+return schedule
+12:
+end if
+13:
+i ← i + 1
+14:
+hsi, csi ← LSTMs((hti−1[ti], hai−1[ai]),
+hai−1, cai−1)
+15:
+hai, cai ← LSTMa((hti−1[ti], hai−1[ai]),
+hai−1, cai−1)
+16: end while
+presented in the embeddings.
+πagent(aj|s) = softmaxi(fa([haj||hs]))
+(4)
+Next, the Schedule Propagator uses the Task Selector to
+assign the task for the selected agent based on the state, agent
+and unscheduled task embeddings. As shown in Equation 5,
+the Task Selector concatenates the state, selected agent and
+the unscheduled task embeddings and passes the combined
+information to a feedforward neural network, fτ, to calculate
+the likelihood of the task being assigned to the selected
+agent. After assigning to an agent for execution, the tasks
+are removed from the list of unscheduled tasks. Since the
+calculation of likelihood of each task is independent of each
+other up to the last softmax operation, the model is scalable
+and can be used for differentproblem sizes.
+πtask(τi|aj, s) = softmaxi(fτ([hτi||haj||hs]))
+(5)
+The key component of the Schedule Propagator is the use
+of LSTM. As shown in line 12 of Algorithm 1, after each
+task-agent pair selection, the state and agent embeddings are
+updated using the state LSTM and agent LSTM, respectively.
+The LSTM Cell stores the hidden and cell data from the
+previous step of the task allocation and predicts the next
+step based on the input using the Equation 6 [33].
+ft = σ(Wf[ht−1, xt] + bf)
+it = σ(Wi[ht, xt] + bi)
+˜ct = tanh(Wc[ht−1, xt] + bc)
+ct = ftct−1 + it˜ct
+ot = σ(Wo[ht−1, xt] + bo
+ht = ottanh(ct)
+(6)
+Where the Encoder produces initial hidden state, h1 and
+initial cell state c1 as an output in the form of [h1, c1].
+During testing, we utilize a batched sampling strategy for
+further performance gains. Specifically, we generate multiple
+schedules for the same task allocation problem every round.
+
+communicate
+Agent
+assignedTo
+State
+takeTime
+UseTime
+Task
+in
+temporalWe select the best performing schedule by computing the
+estimated makespan utilizing the Learning Curve Estimator
+and provide it to the Multi-Round Environment. More sam-
+pling improves solution quality at increased computation.
+C. Stochastic Policy Learning
+We train HybridNet in multi-round scheduling environ-
+ments using Policy Gradient methods that seek to directly
+optimize the model parameters based on rewards received
+from the environment [34]. Specifically, we compute the
+gradient of the model using the sum of the log likelihood
+of Agent and Task Selectors, as shown in Equation 7:
+∇θJ(θ) = Eπ(
+T
+�
+t
+Aπθ
+t (st, ⟨τi, ai⟩)
+∇θ(logπθ(τi|ai, st) + logπθ(ai|st))
+(7)
+In Equation 7, the advantage term, At is estimated by sub-
+tracting a “baseline” from the total future reward calculated
+in Equation 2. We calculate the “baseline” using the reward
+generated for the same task-allocation problem from multiple
+batches executed in multiple sequential rounds in the Multi-
+Round Environment. Each element of the batch solves the
+same scheduling problem and the environment is updated
+to account for the task-allocation of the previous round,
+updating the agent models. The gradients were calculated
+from Equation 7 to updated the model weights.
+Due to the combinatorial nature of the task scheduling
+problem, plus the stochasticity in human proficiency, learning
+a helpful value function as a baseline for computing the
+advantage term is non-trivial. Instead, we investigate two
+more accessible and efficient alternatives:
+• Step-based Baseline: During gradient estimation, the
+baseline value subtracted is set as the average return
+value across training episodes in the current batch.
+• Greedy Rollout Baseline: Greedy Rollout Baseline uses,
+πgreedy(A|S), a deterministic greedy version of the Hy-
+bridNet scheduler, to collect rewards in the environment.
+Its weights, θgreedy, are updated periodically by copying
+the weights from the current learner, πθ(A|S).
+V. EXPERIMENTAL RESULTS
+A. Data Generation
+We generate scheduling problems with deadline and wait
+constraints under different scales. For all scales, the deadline
+constraints are randomly generated for approximately 25% of
+the tasks from a range of [1, 5N] where N is the number of
+tasks. Approximately 25% of the tasks have wait constraints,
+and the duration of non-zero wait constraints is sampled from
+U([1, 10]). Task durations are clamped to 10 to 100.
+1) Small Scale: The small data set has 9 to 11 tasks with 2
+robots and 2 humans in a team. We generated 2000 Training
+Problems and 200 Test Problems.
+2) Medium Scale: The medium data set has 18 to 22
+tasks with 2 robots and 2 humans in a team. We generated
+2000 Training Problems and 200 Test Problems to inspect
+the scalability of our trained model.
+3) Large Scale: The large data set is defined as problems
+with 36 to 44 tasks chosen at random with 2 robots and 2
+humans in a team. We have generated 200 Test Problems
+to evaluated the HybridNet performance with zero training
+problems (i.e., zero-shot transfer to from the smaller scale
+datasets to the Large Scale dataset).
+To simulate the stochastic learning of human agents, for
+each Data Set noise is introduced to the Human Agent
+models by simulating the natural distribution of the c, k, β
+parameters of Equation 1. This allows for each Data Set to
+simulate Deterministic and Stochastic Human Performance.
+The stochastic model is clipped to fall within the specified
+range of task durations.
+B. Benchmarking
+We benchmark HybridNet against the following methods:
+• EDF: A ubiquitous heuristic algorithm, earliest deadline
+first (EDF), that selects from a list of available tasks the
+one with the earliest deadline, assigning it to the first
+available agent.
+• Genetic Algorithm: An Evolutionary Optimization Al-
+gorithm that uses Post-Processing on the Schedule
+Generated by EDF [21]. Genetic algorithm creates new
+schedules based on the initial schedule through iterative
+randomized mutations by swapping task allocations
+and task orders [4]. Each generation selects the top
+performing schedules, sorted on feasibility and total
+schedule completion time, and used as the baseline for
+creating new mutations. The Genetic Algorithm was run
+for 10 generation with 90 baseline schedules, 10 task
+allocation and 10 task order swapping mutations.
+Furthermore, we evaluate the functionality of the Re-
+current Schedule Propagator by comparing it against the
+following HybridNet variant:
+• HetGAT: We implement a HetGAT Scheduler based on
+the Encoder of HybridNet. After each task-agent pair
+assignment, instead of using the LSTM Cells to update
+the task, agent and state embeddings, it directly interacts
+with the environment to model the consequences of
+action with a new heterogeneous graph and re-computes
+those information from it.
+We evaluate HybridNet on three metrics: 1) Proportion
+of problems solved; 2) Adjusted makespan: determined by
+the average of the makespan of feasible schedules and the
+maximum possible makespan of the infeasible schedules;
+and 3) Runtime statistics. Runtime statistics for training and
+execution is compared for HybridNet and HetGAT Scheduler
+to model their computational complexity. Because HetGAT
+Scheduler relies on interactive scheduling through the envi-
+ronment after every task-agent pair allocation, we only train
+and evaluate it for Deterministic Human Performance.
+C. Model Details
+We implement HybridNet and HetGAT using PyTorch [35]
+and Deep Graph Library [36]. The HybridNet Encoder used
+in training/testing is constructed by stacking three multi-head
+
+TABLE I
+EVALUATION RESULTS: ADJUSTED MAKESPAN AND FEASIBILITY WITH DETERMINISTIC HUMAN TASK PROFICIENCY COMPARING BENCHMARKS
+WITH HYBRIDNET TRAINED ON SMALL AND MEDIUM SCALES, WITH SCHEDULES SAMPLED FROM SIZES 8 AND 16
+Training
+Methods
+Small
+Medium
+Large
+Makespan
+Feasibility (%)
+Makespan
+Feasibility (%)
+Makespan
+Feasibility (%)
+-
+EDF
+239.31
+73.00
+1109.85
+15.00
+2535.89
+1.00
+-
+Genetic Algorithm
+302.42 ± 0.77
+74.10 ± 0.30
+1180.07 ±2.54
+16.60 ± 0.70
+2542.79 ± 0.06
+1.00 ± 0.00
+Step-based
+HetGAT 8
+257.20 ± 0.18
+86.29 ± 0.08
+751.27 ± 1.29
+50.17 ± 0.14
+2123.96 ± 5.66
+17.12 ± 0.27
+HetGAT 16
+249.69 ± 0.30
+86.51 ± 0.09
+723.57 ± 0.94
+50.29 ± 0.11
+2081.65 ± 5.45
+17.15 ± 0.16
+Greedy
+HetGAT 8
+261.15 ± 0.09
+85.59 ± 0.10
+784.32 ± 0.52
+53.28 ± 0.17
+2017.25 ± 2.16
+23.98 ± 0.14
+HetGAT 16
+255.70 ± 0.23
+86.05 ± 0.15
+765.79 ± 0.96
+53.41 ± 0.08
+1983.73 ± 1.59
+23.84 ± 0.01
+Step-based
+HybridNet Small 8
+260.22 ± 0.15
+86.93 ± 0.10
+770.48 ± 1.07
+59.11 ± 0.35
+2005.80 ± 2.33
+30.65 ± 0.39
+HybridNet Small 16
+252.57 ± 0.49
+87.08 ± 0.10
+746.35 ± 0.52
+60.89 ± 0.36
+1953.65 ± 3.76
+33.24 ± 0.61
+Greedy
+HybridNet Small 8
+266.74 ± 0.31
+84.65 ± 0.32
+758.96 ± 2.27
+61.09 ± 0.43
+2049.32 ± 3.73
+28.74 ± 0.45
+HybridNet Small 16
+258.17 ± 0.45
+85.13 ± 0.20
+723.35 ± 1.70
+63.68 ± 0.49
+1973.15 ± 2.91
+32.46 ± 0.40
+Step-based
+HybridNet Medium 8
+-
+-
+722.85 ± 0.61
+64.69 ± 0.29
+2010.86 ± 1.97
+30.86 ± 0.45
+HybridNet Medium 16
+-
+-
+697.40 ± 2.04
+66.25 ± 0.51
+1944.72 ± 4.10
+33.88 ± 0.49
+Greedy
+HybridNet Medium 8
+-
+-
+692.01 ± 3.69
+68.33 ± 0.66
+2011.78 ± 5.08
+30.58 ± 0.87
+HybridNet Medium 16
+-
+-
+659.01 ± 0.89
+71.00 ± 0.45
+1936.97 ± 4.68
+34.66 ± 0.74
+TABLE II
+EVALUATION RESULTS: ADJUSTED MAKESPAN AND FEASIBILITY WITH STOCHASTIC HUMAN TASK PROFICIENCY
+Methods
+Small
+Medium
+Large
+Makespan
+Feasibility (%)
+Makespan
+Feasibility (%)
+Makespan
+Feasibility (%)
+EDF
+227.81± 6.17
+75.65 ± 1.21
+1071.02± 20.65
+17.30 ± 1.12
+2524.92± 8.95
+1.15 ± 0.23
+Genetic Algorithm
+283.79 ± 10.39
+77.45 ± 2.05
+1149.42 ± 12.14
+19.55 ± 1.31
+2541.20 ± 3.54
+1.05 ± 0.15
+HybridNet Small
+298.81 ± 0.96
+79.54 ± 0.52
+881.16 ± 2.89
+48.89 ± 1.09
+2141.80 ± 5.12
+23.51 ± 0.96
+HybridNet Medium
+-
+-
+859.99 ± 4.82
+51.94 ± 1.32
+2174.57 ± 8.53
+22.31 ± 0.94
+TABLE III
+EVALUATION RESULTS: RUNTIME PERFORMANCE ON SINGLE PROBLEM
+Methods
+HetGAT8
+HybridNet8
+HybridNet16
+Training Time (s)
+Small
+184.52 ± 18.00
+19.97 ± 0.91
+-
+Medium
+354.77 ± 38.31
+22.40 ± 6.52
+-
+Evaluation Time (s)
+Small
+22.91 ± 5.85
+10.94 ± 0.99
+18.95 ± 3.53
+Medium
+70.12 ± 8.67
+14.77 ± 1.42
+22.30 ± 7.55
+Large
+123.76 ± 32.32
+18.84 ± 7.38
+27.78 ± 16.52
+HetGAT layers (the first two use concatenation, and the last
+one uses averaging). The feature dimension of hidden layers
+= 64, and the number of heads = 8. The Recurrent Propagator
+utilizes a LSTMCell of size 32 followed by a fully-connected
+layer and a softmax layer. We set γ = 0.99, batch size =
+8 and used Adam optimizer [37] with a learning rate of
+2 × 10−3, and a weight decay of 5 × 10−4. We employed a
+learning rate decay of 0.5 every 4000 epochs. We evaluate the
+models using a batch size of 8 and 16. For the Multi-Round
+Environment, the infeasible reward coefficient Ci = 2.0 and
+total round number = 4. Both training and evaluation were
+conducted on a Quadro RTX 8000 GPU.
+D. Evaluation Results
+Table I shows the evaluation performance with Deter-
+ministic Human Proficiency in different scales. The Deter-
+ministic Human Proficiency means that during training and
+evaluation, human learning curve is known and execution
+is deterministic for every agent. In Table I, “Small” and
+“Medium” term after model name denotes the data scale the
+model was trained on and the number following it denotes
+the batch size for schedule sampling. The results show that
+HybridNet outperforms both EDF and Genetic Algorithm in
+adjusted makespan and percentage of feasibility. HybridNet
+trained on Small scale problems generalizes for both Medium
+and Large scale problems with similar or slightly worse
+performance than HybridNet trained on Medium. HybridNet
+and HetGAT performs similarly on all scales. This shows that
+HybridNet is capable of learning high performance policies
+by leveraging the Recurrent Schedule Propagator and without
+requiring interaction with the Environment.
+We provide the runtimes of training and evaluation for
+HetGAT and HybridNET in Table III. HybridNet is approx-
+imately 10 times faster in training compared to HetGAT
+Model and at least 2 times faster during evaluation for
+same batch size. EDF and Genetic Algorithm were evalu-
+ated through the CPU without GPU acceleration, making it
+infeasible to accurately compare the performance of the Deep
+Learning Models to the Traditional Models.
+We show that for HybridNet, step-based training has better
+performance over the greedy baseline, while for HetGAT
+model, greedy baseline training is better. We also observed
+that greedy baseline training reached convergence faster than
+step-based training (4500 epochs vs. 19000 epochs). Further
+investigation is worthwhile.
+Table II shows the evaluation performance with Stochas-
+tic Human Proficiency in different scales. The Stochastic
+Human Proficiency is presented as randomness in both the
+actual human execution within Multi-Round Environment
+and uncertainty within the Learning Curve Estimator used
+for schedule generation. The results show that HybridNet
+outperforms the EDF and Genetic Algorithm across different
+data scales. The largest performance gap was observed on
+large dataset (23.51% vs. 1.15%). Here, HetGAT model is
+not included as it requires interaction with the environment
+after every task-agent assignment to observe the outcome,
+which is not available until the whole schedule is generated
+and sent to the Stochastic Environment for execution to
+emulate real-world scenarios.
+VI. CONCLUSIONS
+We present a deep learning-based framework, called
+HybridNet, combining a heterogeneous graph-based en-
+coder with a recurrent schedule propagator, for scheduling
+
+stochastic human-robot teams under temporal constraints.
+The resulting policy network provides a computationally
+lightweight yet highly expressive model that is end-to-end
+trainable via reinforcement learning algorithms. We devel-
+oped a multi-round task scheduling environment for stochas-
+tic human-robot teams and conducted extensive experiments,
+showing that HybridNet outperforms other human-robot
+scheduling solutions across problem sizes. Future research
+includes integrating the learning-based human estimator into
+HybridNet, transfer learning across optimizing different ob-
+jective functions, and deploying the trained network in a real-
+world scenario.
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+

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+Fast spline detection in high density microscopy data
+Albert Alonso
+Julius B. Kirkegaard
+January 12, 2023
+Abstract
+Computer-aided analysis of biological microscopy data has seen a massive improvement with the utilization of
+general-purpose deep learning techniques. Yet, in microscopy studies of multi-organism systems, the problem of
+collision and overlap remains challenging. This is particularly true for systems composed of slender bodies such as
+crawling nematodes, swimming spermatozoa, or the beating of eukaryotic or prokaryotic flagella. Here, we develop
+a novel end-to-end deep learning approach to extract precise shape trajectories of generally motile and overlapping
+splines.
+Our method works in low resolution settings where feature keypoints are hard to define and detect.
+Detection is fast and we demonstrate the ability to track thousands of overlapping organisms simultaneously.
+While our approach is agnostic to area of application, we present it in the setting of and exemplify its usability on
+dense experiments of crawling Caenorhabditis elegans. The model training is achieved purely on synthetic data,
+utilizing a physics-based model for nematode motility, and we demonstrate the model’s ability to generalize from
+simulations to experimental videos.
+1
+Introduction
+Large-scale, high-throughput quantification of microscopy data have increasingly become possible with the aid of
+computer vision [1–6].
+In particular within the last decade, deep learning techniques [7–9] have improved and
+enabled accurate image analysis of microscopy data in a broad range of areas including cell counting [10, 11], cell
+segmentation [12–14], nucleus detection [6, 15], sub-cellular segmentation [16], drug discovery [17], cancer detection
+[18–20], and the identification of infectious diseases [21, 22]. Detection models serve as the fundamental operation
+in tracking procedures, and combined with suitable tracking algorithms, these can achieve morphologically resolved
+organism tracks that can accurately quantify organism motility [23], the application of which ranges from fundamental
+neuroscience [24–26] and the circuitry of simple organisms [27–30] to drug discovery [31–35].
+Multi-organism detection can be achieved at increasing levels of fidelity: at the crudest, only center-of-mass
+locations or bounding boxes are predicted [36] which does enable tracking of organisms but provide little morpho-
+logical information. In contrast, pixel-wise segmentation models [12] and pose estimation using keypoints [37] reveal
+accurate shape dynamics when employed on high-resolution data. However, these methods rely on high definition
+objects, as segmentation and prediction is highly sensible to noise. In particular for organisms that are long and
+slender, pixel-wise segmentation fails at low resolution as correct predictions require sub-pixel accuracy. Moreover,
+at high densities, these methods may fail due to their inability to properly handle overlap between organisms.
+Here, we consider the problem of studying slender organisms at low resolution and high density with the goal to
+enable both accurate identity tracking and quantification of shape dynamics. This problem has traditionally been
+approached by employing pixel-wise segmentation and subsequent skeletonization procedures [38–43], an approach
+that requires ad-hoc procedures to solve the problem of correctly identifying overlapping organisms [44], the com-
+binatorial complexity of which blows up at high densities. To this end we abandon pixel-wise output and instead
+construct a neural network architecture that predicts, potentially overlapping, splines directly [45–47]. Our method
+enables both accurate shape prediction and tracking in dense experiments of slender objects. This is applicable to a
+1
+arXiv:2301.04460v1  [cs.CV]  11 Jan 2023
+
+broad class of systems [Fig. 1], including tracking of nematode worms [48–50], spiral or elongated bacteria [51–54],
+spermatozoa [55, 56], the flagella of both eukaryotes [42, 43] and prokaryotes [57], and freely swimming flagella such
+those of microgametes [58].
+a
+b
+c
+d
+Figure 1: Microscopy images of different microorganisms whose slender structure and frequent overlaps makes them
+hard to detect using classical approaches. a. C. elegans motility experiment from the dataset of this paper. b.
+Motile, flexuous, thin, spiral-shaped B. pilosicoli bacteria. Still from Ref. [53]. c. Beating flagella of the green alga
+C. reinhardtii, provided by Kirsty Wan, University of Exeter. d. Swimming human spermatozoa. From dataset in
+Ref. [56].
+Our method relies on recent advances in deep learning [59–63] and extends these by few simple ideas: We found
+that humans are better at correctly resolving overlap between moving bodies when given access to videos rather
+than still micrographs. Thus, to allow the neural network to encode the identity of individual bodies as a function
+of their motion, the input to our neural network is taken to be short video clips rather than single frames. Our
+network outputs multiple independent predictions, and for each produces (1) the spline representing the centre-line
+of an object, (2) an estimated confidence score for the prediction, and (3) a latent vector, the space of which we
+induce a metric on that measures whether two predictions are trying to predict the same body. To train the network,
+each output quantity is associated with a specific loss term, where, importantly, the spline loss term is permutation-
+invariant in the labels. To resolve overlap, we do non-max suppression [36], but rather than measuring distances
+between spline predictions, we use the latent space output, which allows two predictions to be kept even though
+they are close in physical space. This enables correct predictions for data in which objects overlap very closely. Our
+method is further tailored to support the subsequent tracking process, which must link uniquely predictions from
+frame to frame. To that end, we not only predict the object location at a single timepoint, but also predict consecutive
+past and future splines. Using these time-resolved predictions in the linking process enables high-precision tracking
+even through dense regions.
+Our method is principally applicable to all microscopy datasets that involve slender bodies. In this paper, we
+focus on its applications for tracking dense experiments of crawling C. elegans worms, a popular model system
+2
+
+业in neuroscience [64], human diseases [65], drug discovery [32], motor control [66], memory [67], and ageing [68].
+Studies of C. elegans often rely on phenotypic assays that measure the motility of the nematode worms as function
+of some environmental condition or treatment [35, 69–81], the throughput of which can be massively increased if
+overlap between organisms can be tolerated. Likewise, resolving identities of organisms during overlap is crucial
+for studies of interactions between organisms [82]. Previous work on tracking C. elegans have generally employed
+classical computer vision approaches to accurately track single or a few high-definition worms [39, 83–86], or many
+low-resolution worms at non-overlapping densities [40, 87, 88], in some cases by utilizing a computational model of
+the worm motion for hypothesis tracking [39, 83]. Recently, deep learning techniques have been utilized to track
+C. elegans worms using e.g. bounding box predictions [89–91] and fully resolved centre-line splines in the case of
+isolated worms [92], allowing for detection also during periods of self-overlap.
+With this paper, we publish a dataset of videos of motile C. elegans worms imaged at a wide range of densities.
+The dataset includes ∼ 1,500 labelled splines that we use to evaluate, but not train, our detection model.
+We
+demonstrate that our model can be trained exclusively using synthetically generated data and yet generalizes well
+to real videos. Our method leverages the parallel capabilities of convolutional neural networks and is thus able to
+handle thousands of detections in a single pass, resulting in real-time detection at ∼ 90 Hz at 512 × 512 resolution
+on a single GPU. The code is open source and available at https://github.com/kirkegaardlab/deeptangle.
+2
+Results
+2.1
+Architecture
+Figure 2 illustrates the overall structure of our approach. Our model is based on single-stage detection models [36,
+59] that output many candidate predictions per target in a single forward pass and rely on a score system to prune
+until a single candidate is left for each target object. The performance of such single-stage models have been shown
+to enable accurate real-time bounding box detection [62]. The backbone of our neural network [Fig. 2a] consists
+of convolutional residual networks [60] with the small modification that we employ average pooling rather than
+max-pooling to avoid translational invariance in the spline predictions, which need to be accurate to a sub-pixel
+degree.
+We take the input to our model to be a stack consecutive frames in order to provide the model with a temporal
+context [Fig. 2c]. This has previously been shown to improve the detection of e.g. partially hidden objects [93]. In
+particular, in present case of motile slender objects where dynamic crossings and overlap between objects are very
+common, a temporal context can provide the necessary information to resolve the problem of correct identification.
+Furthermore, the temporal context allows the output of our model to include information on the motion of the
+splines, which we will further exploit for tracking purposes.
+The backbone of our neural network performs a 162-fold reduction in resolution when mapping the input images
+to feature space, from which the network outputs multiple anchored predictions. We choose the resulting number
+of candidates to be considerably larger than the number of objects in the frame, thus ensuring that all objects have
+suggestions. The anchored approach further means that the only restriction on input size is that its dimensions
+be divisible by 16, and, in particular, it allows training at a certain resolution H × W and subsequent inference at
+another H′ × W ′ without loss of accuracy.
+The output of our model is composed of spline predictions, confidence scores and latent vectors:
+Spline predictions
+We choose to represent the centre-line of the slender bodies of interest by arrays consisting of k
+equidistant points [Fig. 2d]. These coordinate arrays, which we refer to as splines, become high-precision descriptors
+even for complex shapes when k is chosen large. To reduce the complexity of predicting k points, we embed the
+spline representation with a principal component (PCA) transform A, the dimension κ of which can be much smaller
+than k [94]. The PCA components λ represent shape, and addition hereto, the network also predicts the offset x0 of
+3
+
+Neural Network
+I
+[
+H
+,
+W
+,
+T
+]
+fθ
+(λ, x0)
+[
+M
+,
+W
+T
+,
+m
+]
+z
+[
+M
+,
+W
+T
+,
+K
+,
+2
+]
+p
+[
+M
+,
+D
+]
+s
+[
+M
+]
+qϕ
+x = x0+Aλ
+a
+Latent Space
+Emergence
+of clusters
+1. Score Prunning
+rl
+Latent Space
+Best spline
+remaining
+2. Suppression
+Latent Space
+3. Repeat
+b
+Splines coordinates
+Synthetic
+I−
+I+
+I
+Real
+I−
+I+
+I
+L(lx, ls, lp)
+ˆz
+Neural
+Network
+Backpropagation
+
+
+z
+p
+s
+
+
+
+
+x−
+x
+x+
+
+
+Splines z
+Filtering
+x =
+
+
+(x0,0, y0,0)
+. . .
+(x0,k, y0,k)
+...
+...
+...
+(xn,0, yn,0)
+. . .
+(xn,k, yn,k)
+
+
+Predictions
+Spline points
+Visualization
+Visualization
+c
+• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ••x
+• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ••x
+.
+. ψi
+•
+(xi, yi)
+•
+•
+ds
+d
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+direct distance
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+•
+•
+flip distance
+e
+Figure 2:
+(a) Structure of the detection method. Trainable neural networks are colored in gray, and represent
+the convolutional neural network f(I; θ) and the latent space encoder q(λ, x0; φ). (b) Procedure to prune unfiltered
+predictions to final detections with the use of the encoded latent space vectors. (c) Method overview from the input
+clip I (we use a stack of 11 frames in this work) to the final matrix of splines x. The target frames [I−, I, I+] (center
+frames from the clip, orange) are explicitly shown for both the synthetic and real videos. Additionally, the training
+setup is represented using lighter color arrows; from synthetic data to loss backpropagation. After detection, direct
+visualization of the predicted splines x is possible. (d) Diagram with a spline descriptor composed of k equidistant
+points along the skeleton of the nematode. (e) Visual representation of the two distances used in Eq. (1), the
+minimum of which corresponds to correct head-tail alignment and is the one that will be used in the model.
+4
+
+CCthe spline, the internal calculation of which is done in a local coordinate system defined by the anchor points. Thus,
+instead of predicting 2k floating point values per spline, the network needs only output κ + 2.
+The temporal context of the input image stack permits output spline prediction also for the non-central images.
+In our approach, we predict a set of three splines z = [x−, x, x+] corresponding to the three central frames [I−, I, I+]
+of the input stack [Fig. 2c]. We consider the central spline x the main output, whereas the past x− and future x+
+splines are considered auxiliary predictions whose main purpose lie on their use during the latent space encoding as
+well as the tracking procedure.
+We define the similarity measure between two splines by the standard Euclidean distance. In the case of splines
+that look symmetric from either end, we exploit this symmetry and employ the flip-invariant distance defined by
+d2(x, x′) = min
+�
+k
+�
+i=1
+(xi − x′
+i)2,
+k
+�
+i=1
+(xi − x′
+k−i+1)2�
+,
+(1)
+as illustrated in Figure 2e.
+Likewise, we de��ne a distance between two collections of consecutive splines z, z′ by their weighted average
+d2
+s = �
+t ωt d2(zt, z′
+t), where the weights can be adjusted to give focus to central predictions, and for the present case
+we choose ω = 2ω− = 2ω+.
+The neural network is trained to minimize the distance d2
+s between predictions and labels. To do so, we let the
+independent predictors specialize for different shapes. This is achieved by using a permutation-invariant loss such
+that the total loss is computed as a sum over the labels only, each using the predictor that best match the labels.
+Thus many spline prediction will not contribute to the spline loss.
+Confidence scores
+Each independent prediction of the network includes a confidence score s, which is used to
+filter out bad candidates. In bounding box or mask detection, intersection over union (IoU) is commonly used to
+evaluate the accuracy of a prediction, however, this metric does not generalize well to spline predictions when there
+is overlap. Instead, we introduce a custom metric to define the goodness of a spline set z by comparing it to its label
+ˆz,
+ˆs = exp
+�
+−d2
+s(z,ˆz)/σ2
+s
+�
+.
+(2)
+Here, σs is a parameter that sets the scale over which the score varies. The metric is sensitive to perturbations on
+accurate predictions, i.e. predictions close to labels where ds → 0, but loses sensitivity the worse the predictions is.
+This is a useful feature as correct scoring for good predictions is crucial for choosing the best one, whereas low-scoring
+predictions are discarded in any case and their relative scoring therefore unimportant.
+The score prediction is trained using L2 loss. To avoid conflicting backwards error propagation between this task
+and that of spline prediction (as scoring bad predictions is easier), we stop the gradient flow in the computational
+graph on the last layer of the score-predicting part of fθ [Fig. 2a] such that it does not interfere with the accuracy
+of the predicted splines.
+Latent space for candidates suppression
+Finally, we need to ensure that there is only one prediction per
+object. Bounding box detectors let the user decide the fraction of overlap between prediction boxes of the same class
+that should be considered to be targeting the same object. As our method must work at high densities, this task is
+complicated by the fact that two predictions might be very close, even completely overlapping in the central frame,
+and yet represent different objects. The task of choosing a suitable cutoff distance is therefore difficult, and we make
+this a trainable task. We do so by embedding each prediction in a low-dimensional latent space in which comparison
+between predictions is cheap, thus allowing efficient and fast candidate suppression also at high densities.
+5
+
+Our method computes the latent vectors p for predictions using an auxiliary neural network, qφ which acts
+directly on the eigenvalues λ and offsets x0 rather than the more redundant spline coordinate points. We induce a
+Euclidean metric on the latent space with the interpretation that two predictions i, j are predicting the same object
+with probability
+P(i ↔ j) =
+�
+exp
+�
+−||pi − pj||2�
+if ||x0i − x0j|| ≤ σl,
+0
+otherwise.
+(3)
+Here, σl is a real-space visibility cutoff that prevents far predictions to interact in the encoded space, thus avoiding
+the need to scale the dimensionality of the latent space with the number of candidates or the input size. We note
+that when using the flip-invariant metric ds on splines, we explicitly construct the latent space encoder to likewise
+be flip-invariant (see Methods).
+To train the latent space, we make the assumption that during training predictors are ‘trying’ to predict the
+label closest to the prediction spline. Combined with the probability interpretation, this allows us to use binary
+cross entropy as a loss function for the probability defined in Eq. (3). To avoid wrong clustering between undefined
+close-by predictions, the loss contribution of each prediction is scaled by the product of their real scores ˆsiˆsj, this
+ensuring that the network focus its attention of good predictions that will not be filtered out. Finally, since the
+encoder should not to alter the performance of the spline suggestions, the loss on the latent space representations
+only updates the weights qφ of the encoder, but is trained concurrently with the main model.
+We employ non-max suppression to choose the best prediction of each object, but with distances measured in
+latent space, as illustrated in Fig. 2b. Concretely: Once all the predictions whose score is lower than a threshold
+τs have been discarded, multiple candidates are likely to still remain for each target object. The lack of low score
+predictions expose clusters in the latent space that correspond to single objects. We sort the remaining predictions
+by their score, automatically accepting the highest scored one. Once a prediction i is accepted, all predictions j
+that have a high probability P(i ↔ j) > τo of being the same object are removed. This is equivalent to setting an
+exclusion radius rl in the latent space as shown in Fig. 2b. We keep iterating on the remaining predictions, pruning
+the latent space until all candidates have been iterated. The final number of accepted predictions should equal the
+number of objects in the frame.
+Detection on dense C. elegans experiments
+To evaluate our approach, we study microscopy videos of crawling C. elegans worms. We are particularly interested
+in videos captured at much higher densities than those typically used in motility experiments. Thus we evaluate our
+model on wide-field videos captured under approximately uniform illumination [40], exemplified in Fig. 3a. In our
+dataset, the number of nematode worms vary ranging from from ∼ 400 with a small probability of overlap occurring
+( ≈ 0.05 average overlaps per worm) to extremely densely packed plates with up to ∼ 6,000 nematodes, where there
+is, on average, one overlap per worm. This means that in the dense plates, detection methods that stop tracking
+after contact between worms happens are rendered completely ineffective.
+Defining worm density ρ as the number of worms in a region per square millimeter, we find, as expected, a
+linear relation between the average amount of overlap per worm and the density [Fig.
+4a].
+Due to the spatial
+heterogeneity of the worm distribution inside the plate, higher densities can be observed when considering small
+regions. On 100 mm2 scales, the highest density in the dataset is ρ ∼ 2.5 mm−1, but this jumps to an extreme
+ρ ∼ 3.5 mm−1 when considering 10 mm2 regions, where humans begin to struggle to correctly identify worms. For
+quantitative evaluation of our model, ∼ 200 random regions of the videos were sampled and hand-labelled resulting
+in ∼ 1,500 labelled worm splines. A sample of frames are shown in Fig. 3b to provide a sense of the different densities
+encountered in the evaluation dataset, with the predictions of the model overlaid.
+6
+
+2736 x 2192
+a
+b
+Figure 3: Showcase of the capabilities of the method. (a) Detected splines predicted on an entire densely populated
+well plate with a single forward pass through the neural network. Inset shows a zoom-in section to demonstrate
+the accuracy of detection across the entire plate (except near borders, where the plate interferes). The total plate
+contains around 6,000 splines. (b) Close up evaluation of different experimental clips with different densities of
+worms.
+Simulation-based training.
+To train our network, we implement a physics-based synthetic dataset generator to
+exploit perfectly defined labels. This approach removes the need for a supervised dataset, and also allows labelled
+videos in situations where manual labeling may not be reliable, or where the subjectivity of the human labellers can
+result in inconsistent labels. Physics-based synthetic datasets have successfully been used to train systems on similar
+conditions, for instance where manual labelling may introduce unnecessary noise or bias to the model [16]. Our
+in-silico data generator has two main components: a physics-based model for the organism and a synthetic frame
+generator.
+In-silico worms are generated on demand every training step which removes the possibility of overfiting to the
+generated frames.
+In order to train the model to work effectively with a range of worm densities, we generate
+batches with different numbers of worms in a uniform manner, without bias towards low or high worm counts. This
+teaches the model to handle a variety of densities without overfitting to any specific case. And to make the model
+more robust, training also happens on densities whose manual annotation would be extremely challenging. The
+simulation and video synthesis are implemented in a GPU framework which enables fast end-to-end training without
+the performance penalization of data transferring between the accelerator and the host machine.
+We base the worm simulation on resistive force theory, as it has previously been shown to correctly predict the
+position of the skeleton for short spans of times [95]. Since the network only perceives the frames surrounding the
+target frames, we found the total duration of the clip to be short enough that a linear crawling model approximation
+fits our needs. The physics-based model should encapsulate all types of organism behavior. This can be achieved by
+7
+
+1 cmp= 0.19mm-2p= 0.79mm-2p= 0.91mm-2p= 1.57mm-2p= 1.97mm-2p = 3.28mm-2
+1 mm•
+•
+•
+•
+•
+•
+•
+•
+•
+•
+•
+•
+•
+•
+•
+•
+•
+•
+•
+•
+•
+b
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+Worm density in clip (mm−2)
+0.0
+0.5
+1.0
+1.5
+Average overlaps on a worm
+a
+c
+ρ = 2.3mm−2 δadtw = 0.6 px
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+Worm density in clip (mm−2)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Double prediction
+artefacts
+Double prediction
+artefacts
+Error distance δadtw dependance with density
+d
+0
+1
+2
+3
+4
+Worm density in clip (mm−2)
+0.996
+0.997
+0.998
+0.999
+1.000
+True Positive rate
+e
+0
+1
+2
+3
+4
+Worm density in clip (mm−2)
+0.00
+0.02
+0.04
+0.06
+0.08
+False Negative rate
+f
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Score threshold τs
+0.46
+0.48
+0.50
+0.52
+0.54
+0.56
+Average error distance δadtw
+τo = 0.1
+τo = 0.3
+τo = 0.5
+τo = 0.7
+g
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Score threshold τs
+0.95
+0.96
+0.97
+0.98
+0.99
+1.00
+Average TP rate
+h
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Score threshold τs
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Average FN rate
+i
+Figure 4:
+(a) Average number of overlaps counted on frames of pixel size 512 × 512 with different densities of
+worms (N = 90). (b) Illustration of the asymmetric dynamic time warping distance error corresponding to the
+average value of the orange euclidean distances between the prediction (green) and the labelled points (white). (c)
+Example frame with manually labelled points (white) and models predictions (colored). The metric is only evaluated
+in the lighter area of size 100 × 100. (d) Quantified accuracy of the detections by showing the distance to the
+manually labelled splines. Distributions for different densities are shown. The violin plots represent the 99 percentile
+of the data whereas outliers are plotted individually.
+(e–f) Rates for True Positive and False Negative on the
+manually annotated dataset. (g–i) Performance of the model with different combinations of score (τs) and overlap
+(τo) thresholds. N = 1,420.
+8
+
+oversampling the behavior, i.e. by making the simulations more diverse in the behavior than reality and thus hope
+to include all types of real behavior as well. Details on the worm simulation and video synthesis can be found in the
+methods section.
+Despite the potential for physics-based simulations to be used for synthetic training data, discrepancies with real
+data may lead to inaccuracies when applied to real microscopy images. This reality gap can be the result of an
+overly simplified motility model or physics model, or the result of imprecise video synthesis. The gap may be further
+increased by the fact that the model relies on the PCA transformation matrix A obtained on synthetic data, where the
+number of PCA components used have been chosen to accurately reproduce all synthetic patterns, but not necessarily
+to generalize to out-of-sample videos. Thus we find that our model is limited to accurate skeleton predictions only on
+shapes that resemble those produced by our simulations, and the goal of the simulations is therefore to reproduce a
+broad spectrum of possible motility patterns. Likewise, we find that our model is susceptible to the brightness of the
+videos, and accordingly we adjust the real videos to increase their resemblance to the training data (see Methods).
+Metrics
+Despite being trained exclusively on synthetic data, the model’s inference performance is very good on real
+clips. From visual inspection, no immediate discrepancies are observed between detections in low density clips and
+at high density [Fig. 3b]. Likewise, per design, the network accuracy is independent on the input clip dimensions,
+and the parallel structure of convolutions permits the use of large videos covering thousands of nematodes to be
+processed simultaneously in a single forward pass [Fig. 3a]. We note, however, that even though no quality impact
+on detections is observed when using large fields-of-view clips, there can be a dependency if non-uniform illumination
+is used as different sections of the frame may have different requirements for preprocessing.
+For a quantitative assessment of the method accuracy, we compare to the manually labelled dataset, an example
+of which alongside the model predictions can be seen in Fig. 4c. As the predictions are densely defined splines
+(here, ∼ 50 points), we introduce a custom metric to suitably evaluate the accuracy of the predictions using labels
+with lower fidelity. The metric used must be shift-invariant, as having points anywhere along the spline should yield
+zero error regardless of whether the label points precisely coincide with the prediction points or not. Likewise, label
+points should be monotonically assigned along the spline in order to avoid artificially reducing the error for strongly
+bent or self-coiling worms. Finally, it must be robust against the subjectivity of the labellers, as manual annotations
+might miss or avoid spots where visibility is low such as the end-points of the worms.
+To satisfy all these requirements, we introduce a metric based on the dynamical time warping (DTW) distance
+used to measure similarity between temporal curves. In our modified version, asymmetric DTW, summation only
+runs over label points. Thus, the metric δadtw is defined as follows: Let d(i, j) be the Euclidean distance between
+label point i and prediction line segment j, then
+δadtw = min
+α
+1
+N
+N
+�
+i=1
+d(i, α(i)),
+(4)
+where α : [1, N] → [1, M] is a monotonic (non-decreasing or non-increasing) assignments of the N label points to the
+M prediction line segments. A visual representation of the metric is shown in Figure 4b, and the O(NM) algorithm
+for its calculation is detailed in the Methods section.
+The results of evaluating the trained model on the labelled dataset are shown in Fig. 4. For reliable comparisons,
+we first solve the assignment algorithm for the label-prediction pairs. This means that in the case of two completely
+overlapped worms, two predictions need to be present to not count as a miss, and likewise, two predictions cannot be
+considered to target the same label. We find an average error of δadtw ≈ 0.54 px with no strong dependency between
+accuracy and density of worms [Fig. 4d], with the exception of a slight increase in error for extremely dense clips
+(∼ 3.5 mm−2). The average error corresponds to less than the width of a worm (≈ 2 px ≈ 50 µm), and part of this
+can be attributed to the fact that human accuracy is also near the half-pixel level [Fig. 4c]. Some outliers can be seen
+however, which can mostly be attributed to an artefact of the model, where the network mistakes a single long worm
+9
+
+for two overlapping shorter predictions. This effect seems particularly sensitive to incorrect intensity normalization
+of the videos.
+Let σϵ be a cutoff distance above which we no longer consider the predictions to be targeting the closest label. For
+all the figures in Figure 4, this cutoff is assumed to be σϵ = 3.0 px, and we observe no significant changes by tuning
+it within the range of sensible values. We define the True Positive (TP) rate as the fraction of predictions that both
+gets assigned a label and this label is within the the distance σϵ of the prediction. Figure 4e shows that the model
+rarely predicts a spline where there is nothing with a TP rate of 0.999. Nevertheless, there are some predictions that
+do not get assigned a label which can be attributed to the double-prediction artefacts just mentioned. The likelihood
+of this happening decreases with density, but the rate is so low that it is almost negligible. Similarly, we define the
+False Negative (FN) rate as the fraction of labels that are not assigned a prediction closer than σϵ. Fig. 4f shows
+that the model in general manages a low FN rate at around ∼ 0.015, but that this increases to a rate of ∼ 0.06
+at extreme densities such as ρ ≥ 3.0 mm−2, where clusters tend to be densely packed and manual labeling likewise
+becomes challenging.
+The filtering part of the model depends on the previously introduced thresholds τs and τo. The score threshold,
+0 < τs < 1, is used to prune low score predictions [Fig. 2b(1)], while the overlap threshold, 0 < τo < 1, is used to
+decide the probability of two independent predictions to be targeting the same object [Fig. 2b(2)]. Throughout this
+paper, we have set these to τs = τo = 0.5. However, due to their relevance in modifying the filtering process, we
+evaluate how different combinations of thresholds may alter the performance results. Figures 4g–i show the average
+performance obtained across all densities when filtering the predictions with variable thresholds. In spite of some
+dependency between worm density and TP/FN rates, we consider the average metric to be a good indicative of the
+performance on each case.
+Fig. 4g shows the effect of the thresholds on accuracy. No significant dependency on the thresholds is observed.
+This can be explained by the fact that accuracy is determined by the best predictors only, which are not discarded
+until a high τs is used, and ones those are removed, τo becomes irrelevant. Further, the fact that there is no notable
+difference between different values of τo indicates that the clusters are highly compact.
+In contrast, Fig. 4h shows that the TP rate has a stronger dependency on τo at low τs because low score predictions
+do not form compact clusters, and therefore a larger exclusion radius is required to discard them. Finally, Fig. 4i
+shows that misses only begin to occur once the best predictions are discarded, and a strong dependence on the τs is
+not observed before that point.
+2.2
+Tracking from consecutive detections
+Motility assays require not only accurate detections but also the ability to link these across frames to form time-
+resolved tracks of individual organisms. This is challenging at high densities where we have the breakdown of the
+assumption that the closest detected object to the previous frame corresponds to the same identity. In general,
+greedy approaches to particle tracking such as assigning directly the closest particle in consecutive frames frequently
+leads to failed tracks. Instead, the process of tracking can be efficiently formulated as a set of linear assignment
+problems [96]. Naturally, here we can expand upon particle tracking by using a metric that measures distances not
+between center-of-mass of the worms, but between the full splines as defined in Eq. (1). This works well for most
+predictions, but can fail for fast-moving worms or in dense clusters.
+A separate approach to tracking is Kalman filtering. This would require separate detection of entry and exit
+events of worms, as well as a probabilistic model for worm motility, which would most likely have to be highly non-
+linear. Kalman filtering is viable for the tracking of few organisms, but for present large-scale systems we require a
+more efficient approach. As previously mentioned, splines from adjacent frames are also predicted in order to embed
+temporal information into the latent vector. We propose a directed metric that leverages both past x− and future
+x+ spline predictions [Fig. 5a]. Thus to find a mapping σ from one frame to the next, we solve
+10
+
+xi
+xj
+x+
+i
+x+
+j
+xk
+xl
+x−
+k
+x−
+l
+d(x+
+i (t1), xk(t2)) + d(x+
+j (t1), xl(t2)) + d(xi(t1), x−
+k (t2)) + d(xj(t1), x−
+l (t2))
+Minimal assignment distance
+Forward distance df
+Backward distance db
+a
+No constrains
+Midpoint cutoff
+Final assignment
+t0
+t1
+t2
+0
+1
+2
+3
+4
+5
+0
+1
+2
+3
+4
+5
+0
+1
+2
+3
+4
+5
+t0
+t1
+t2
+0
+1
+2
+3
+4
+5
+0
+1
+2
+3
+4
+5
+0
+1
+2
+3
+4
+5
+t0
+t1
+t2
+0
+1
+2
+3
+4
+5
+0
+1
+2
+3
+4
+5
+0
+1
+2
+3
+4
+5
+b
+d
+1
+25
+49
+Position (k)
+0
+3
+6
+9
+12
+15
+18
+21
+24
+27
+Time (s)
+1
+25
+49
+Position (k)
+-π
+0
+π
+e
+f
+0.0
+0.5
+1.0
+1.5
+2.0
+Worm density in clip (mm−2)
+0.90
+0.95
+1.00
+Average track integrity
+Directed
+Normal
+c
+Only 135 tracks
+All tracks (∼ 6000)
+All tracks
+y
+x
+t
+h
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+Worm density in clip (mm−2)
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+mm/s
+σM ∼ 1/
+√
+N
+SE (σM) on CoM speed (vmm)
+g
+Figure 5:
+(a) Illustration of the directed distance used to assign consecutive detections the same identity. The
+simplified drawing shows two independent predictions at adjacent frames and showcases how the assignment scheme
+computes the identity by comparing future-present and past-present distances and choosing the assignment that
+minimizes their sum. (b) Diagram showcasing how using a location cutoff simplifies the assignment problem. Nodes
+represent independent detections at each frame whereas edge values are given by the directed distance measure. The
+assignment happens by minimizing the sum of edges at each timestep. (c) Comparison of using the straightforward
+spline distance and the proposed directed approach. The accuracy is evaluated by measuring the integrity of the
+tracks.
+In contrast to other metrics in this paper, this plot has been obtained using synthetic worms as long-
+term, accurate tracks are required to evaluate the tracking integrity (See Methods for details on Tracking integrity).
+(d) Qualitative example of 30 s trajectories of the center of mass of the nematodes in a dense experiment. The
+still background image represent the last frame of the video. To improve the visualization, a small subset of the
+trajectories are shown. In contrast, a corner of the frame is used to display all the trajectories to showcase the
+density of simultaneous tracks. (e) Two samples of the spline angle ψ of two randomly sampled nematodes from (d).
+(f) Undulations corresponding to 30 s of the detections relative to the center of mass coordinate of nine randomly
+sampled nematodes from (d). (g) Standard error value of the measurements of the center of mass speed as a function
+of density. (h) Showcase of the possible throughput of the method, by simultaneously tracking more than 6,000 tracks
+from a full dense plate. A small window on the tracks is shown to showcase their continuity.
+11
+
+CCσ = arg min
+σ
+��
+i
+d(xi(t), x−
+σi(t′)) + d(x+
+i (t), xσi(t′)
+�
+.
+(5)
+Identity assignment can be seen as a network flow global optimization where nodes represent detections and edges
+carry cost of assignment. To avoid having to perform all possible combinations of assignments, we include a physical
+distance threshold on the midpoint of the central spline. This threshold significantly simplifies the assignment scheme
+and improves the runtime of the filtering process [Fig. 5b].
+To quantify the performance of these methods, we define the tracking integrity ι as a scalar that indicates how
+consistent the assignment of a label to a prediction is along the tracked video. Perfect tracks have ι = 1, whereas
+labels that gets assigned two different identities for half of the duration of the video have ι = 1
+2, and so on (see
+Methods for a detailed definition). We evaluate this on synthetically generated videos of 10 seconds (200 frames)
+that have perfectly labelled tracks, the results of which are shown in Fig. 5c. On videos with densities up to 2.0
+mm−2, we achieve an average integrity of ι ≈ 0.97. This a ∼ 30 % improvement of the error over using direct spline
+assignment defined in Eq. (1). We observe that the integrity is almost perfect at low densities, but drops to ι ≈ 0.93
+at the highest densities.
+When applied to high density videos of C. elegans, the tracking method is able to keep track of individual worms
+as they pass through clusters of other worms [Fig. 5d]. In contrast to pixel-level classification of worms, our approach
+outputs splines directly, and thus subsequent analysis is straightforward. For instance, one may directly study the
+worm undulations [Fig. 5f] or extract the worm spline angle ψ = arctan(y(s, t) − y0(t), x(s, t) − x0(t)) to provide
+insight into the movement patterns and kinematics of the worm [Fig. 5e].
+One of the key advantages of our methods is its ability to collect a larger number of samples compared to
+traditional techniques, while still obtaining reliable results. As the standard error decreases with the number of
+samples, using our methods allows for metrics to be gathered with less uncertainty while still requiring the same
+experimental setup. For instance, Figure 5g) shows how the error of estimating the average speed of the center
+of mass of the nematodes decreases with density. This advantage can be extended to tracking large numbers of
+nematodes in crowded environments, such as extremely dense petri dishes where more than 6,000 concurrent tracks
+can be simultaneously computed [Fig. 5h]. Thus, with our method, we are able to collect a larger number of samples
+and obtain more precise and reliable results, even in challenging conditions.
+3
+Discussion
+We have introduced a novel deep learning approach for detecting and tracking slender bodies, such as crawling
+nematodes, in microscopy data. The presented convolutional neural network architecture is capable of accurately
+detecting a large number of overlapping organisms, a task that can be particularly challenging for standard methods
+such as bounding boxes and pixel-level classifiers due to the issue of occlusion and overlap. To address this, we have
+implemented a latent space encoding which allow us to filter by non-maximum suppression and effectively handle
+overlapping objects. Not only is our method capable of accurately detecting and tracking slender bodies, but it also
+demonstrates strong scalability, performing well across a range of input frame sizes and densities of bodies. This
+makes it an ideal tool for a variety of experimental settings where splines are useful descriptors, including studies of
+crawling nematodes, swimming spermatozoa and beating eukaryotic or prokaryotic flagella.
+Besides a suitable detector model, labeled training data is also needed. We have demonstrated that relying on a
+physics-based model to generate synthetic data is adequate to train our network to perform well on real data. This is
+a key achievement as it means that applications of our system for different experimental studies do not require large
+datasets to be procured, but rather the implementation of a suitable simulation. Our approach for synthetic data
+generation relies on over-sampling the behavior of the worms. This is naturally a trade-off as too extreme behavior
+can lead to datasets that are too hard for the neural network to replicate. For our model, we found that we slightly
+12
+
+undersampled certain worms shapes such as strong coiling, which the model therefore could struggle with identifying.
+Though we did not look into this here, an interesting avenue for future research would be to bootstrap synthetic
+motility models on small datasets of real organisms. In a similar fashion, the frame-generator procedure should
+oversample the textures, pixel intensities and noise of real videos. Here, it could be interesting to study whether style
+transfer [15] or diffusion models [97] could be used to further reduce the gap between training and inference data.
+For tracking, we introduced a directed metric that employs past and future spline predictions to link them across
+time. At very high densities this may still fail, in particular because the directed metric yields little advantage if
+predictions are missing in some frames. A potential way to improve on this could come from utilizing the latent space
+encoding as well. This would require temporal continuity in the latent space representation, which is achievable by
+modifying the associated loss function. This should enhance the integrity of tracking, as it could potentially be used
+to resolve issues such as switches by leveraging the separation of closely physical predictions with different temporal
+behaviour that characterises the latent encoding. We believe that these suggestions might be fruitful avenues for
+further research for improving deep learning models for dense detection of splines.
+In this paper, we have proposed a new approach for fast and precise detection and tracking of slender bodies in
+microscopy data. Its speed and accurate performance across a range of densities and sizes, combined with the ability
+to handle overlapping objects, makes it a valuable tool for a variety of experimental settings where precise tracking
+is essential for obtaining quantitative metrics.
+4
+Methods
+Convolutional neural network
+Most of the weights of the network are at the feature detection convolutional
+network whose backbone is made of four ResNet groups consisting of 2, 4, 4, 2 blocks with strides 1, 2, 1, 2, respectively.
+We modify the original ResNet architecture by replacing the initial max-pooling layer for an average-pool layer to
+avoid translational invariance. The final shape of the feature space is [H/16, W/16, C], with C being the number of
+candidates each cell proposes. We have set C = 8 for this project in order to fulfil the condition of M ≫ N even
+at high densities. All in all, there will always be C candidates per each cell regardless of input size, which leads to
+a large number of candidates to be sorted in the filtering process. The head of the convolutional neural network is
+composed of two fully connected layers of 512 and C · (3(m + 2) + 1) cells, respectively, with batch normalization
+in-between. Due to the orientation invariance of the loss function on the spline predictions, it is possible that the
+splines in the predicted set x−, x, x+ are not aligned. To remedy this, we aligned them by comparing with the
+eigenvalues of the flipped spline. In order to get the flipped eigenvalues λf, we use
+λf = A−1JAλ
+(6)
+where A is the PCA transformation matrix and J is the exchange matrix.
+Latent space encoder
+The encoder qφ is composed of two fully connected layers with batch normalization in-
+between. The input of the encoder is the vector of size 3(m + 2) characterizing the splines predictions and the
+output is D floating point values, corresponding to the coordinates of p in the D-dimensional latent space. We have
+found D = 8 to be a well-performing dimension in our experiments. Due to the orientation invariance of the splines
+predictions, we need to construct the encoder to cluster those splines regardless of orientations as well. To do so, the
+input values are expanded to include those of the flipped splines λ → (λ, λf) and both are fed to the same layer. To
+ensure symmetry, the output is then summed before passing through the last layer. In doing so, the encoder becomes
+independent of spline orientation.
+13
+
+Input clips pre-processing
+The images used to train the model have dark (small pixel intensity) background,
+as we employ zero-padded convolutional layers. This is relevant for real recordings, where a negative flip may be
+necessary to match the network requirements. During training, generated clips are normalized using a 1–99 percentile
+normalization. For real clips, we have found that accuracy is improved if we apply CLAHE (adaptive histogram
+equalization) before prediction. Likewise, a simple intensity correction factor µ may need to be applied to the videos
+in order to match the pixel profile of the simulated data. For our dataset, we use correction factors of µ ≈ 1.2, to
+get the best results. Note that we match real data to the synthetic as this avoids the need to retrain the network for
+different experimental setups.
+Loss functions
+Spline descriptors are trained as a regression problem. Thus, the loss contribution is given by
+the custom distance defined in Eq. (1). To enforce specialization on the predictors, and due to the number of
+predictions M being considerable larger than the number of bodies N, only the best predictors are accounted for in
+the loss. Nevertheless, there may be labels ˆx completely or partially outside the frame at tc, despite being inside at
+t0. To make sure not to punish bad predictions at the boundaries for not matching invisible splines, instead of using
+the number of simulated bodies N, the subset of bodies completely inside the frame Nv is used and the final loss
+expression is given by:
+lx = 1
+Nv
+Nv
+�
+i
+min
+m d2
+s(zm, ˆzi)
+(7)
+The score L2 loss is computed as the difference of the values predicted and the score the spline proposals should
+have. Thus, using Eq. (2), we train the predicted score of all predictions using:
+ls = 1
+M
+M
+�
+i
+�
+exp
+�
+− min
+n
+d2
+s(zi, ˆzn)
+σs
+�
+− s
+�2
+(8)
+Finally, the loss function for the latent space encoder is a modified cross entropy loss scaled by the product of scores.
+Denote Pi,j = P(i ↔ j) as defined in Eq.
+(3), then the encoder loss is defined as an average over all pairs of
+predictions ⟨i, j⟩ that are physically within the cutoff σl,
+lp = 1
+S ⟨ˆsiˆsj(tij log (Pi,j) + (1 − ti,j) log (1 − Pi,j))⟩⟨i,j⟩ ,
+(9)
+where S = � ˆsiˆsj, and ti,j indicates whether i and j are targeting the same label k, and is set by
+tij =
+�
+1
+if ki = kj
+0
+otherwise
+(10)
+with ki, kj being the closest labels to the predictions zi, zj respectively.
+Training details
+Training has been done from scratch, i.e. without the use of a pretrained backbone. During
+training, the frame size for the input clips used was 256×256, but due to the anchored approach this does not
+constrain inference to happen at the same resolution. Synthetic input is generated on demand and on device rather
+than using a fixed pre-generated dataset. Thus, the network never sees the same frame twice and there is no host-
+to-device data transfer. As mentioned in the main text, all networks are trained simultaneously, despite the weights
+of each one depending on different cost functions. The code has been written in Jax using Haiku and training has
+been carried on a cluster of 8 × NVIDIA A5000’s.
+14
+
+Inference
+Inference happens at any resolution whose dimensions are multiple of 16. The input frames need to be
+slightly pre-process as described in the previous sections. Candidate predictions are chosen using a score threshold,
+and non-maximum suppression in latent space is used for filtering. Due to the sequential nature of the filtering
+process, the implementation is written to use the CPU using numba.
+Worm simulation
+Worm trajectories are computed by employing a resisitve force theory crawling model used to
+predict rigid body motions of C. elegans from the undulations [95]. Thus, we ensure that from a given set of generated
+undulations, the produced motions will match those of real worms. From empirical observations, we propose a simple
+equation (Eq. (11)) to generate the undulation of the worms. We define the motions by the spline angle ψk(s) with
+s ∈ [0, 1] [Fig. 2d], and decompose this into a linear combination:
+ψ(s) = ψu(s, t) + ψs(s, t).
+(11)
+This logically separates the worm undulations into two types of motion: one corresponding to a sinusoidal motion
+ψs and one in which the whole body bends ψu. These we define by
+ψu(s, t) = A cos
+�2π
+T t + ρ1
+�
+cos (kusk + ρ2)
+(12)
+ψs(s, t) = ˜A cos
+�2π
+T t + kssk + ρ3
+�
+(13)
+where ˜A = 1
+2 (1 + | sin (2πt) |) A and the rest of parameters are sampled from random distributions. Although many
+improvements for the above equations can be suggested, we prefer to keep the model simple.
+Once the values of the parameters for ψ are generated all for the timesteps of the simulation, the positional
+coordinates are obtained using
+⃗x(s, t) = L
+� s
+0
+�cos (ψ(s′, t) + γ)
+sin (ψ(s′, t) + γ)
+�
+ds′
+(14)
+where γ is a random orientation and L is the length of the worm (also sampled). Once the skeleton is defined, the
+rigid body motions are predicted by solving [95]
+⃗F =
+� L
+0
+⃗f ds = 0,
+(15)
+⃗τ =
+� L
+0
+(⃗x − ⃗xCoM) × ⃗f ds = 0,
+(16)
+where the force ⃗f can be calculated from the spline velocity ⃗U = ∂t⃗x + V + Ω × (⃗x − ⃗xCoM) by
+⃗f = αt (ˆt · ⃗U) ˆt + αn (ˆn · ⃗U) ˆn.
+(17)
+Here, V and Ω are the center-of-mass velocity and rotational velocity (that we are solving for), and αt and αn = α αt
+is the tangential and normal drag coefficients, which is also sampled for (α > 1). We did not find a need for using a
+non-linear force theory. The simulation is run with Python 3.9 using the Jax library.
+15
+
+Video synthesis
+Given the labels for the splines positions, synthetic videos are generated to be used as input
+during training. In order to add width to each worm, we vary the local body radius r by a function of the form
+r(s) = ˜R |sin(arccos(as + b))|
+(18)
+The pixel values of those circles are calculated with anti-aliasing.
+Once the worms have been rendered, noise
+artefacts such as uneven background, blurring, Gaussian noise, etc. are added to replicate the observed conditions
+of real experiments. During training, standard augmentation techniques are applied as well. In the same manner as
+the motions simulation and the neural network training, frame generation is also written in Python using the Jax
+library in order to leverage GPU capabilities.
+Experimental dataset
+Videos of crawling C. elegans were filmed using the protocol described in Ref. [40].
+Manually annotated dataset
+The evaluation dataset is annotated using a custom tool that can be found at
+https://github.com/kirkegaardlab/deeptanglelabel. Around ∼ 1,500 splines have been annotated and this
+dataset (videos and labels) is included in the SI.
+Asymmetric dynamic time-warped error distance
+In order to evaluate the manually labelled dataset, we
+introduce an error distance that compares the similarity between two curves by calculating a distance between each
+point on one curve and the nearest segment on the other. The error distance used is a variation of the dynamic
+time warping distance, which is widely used for comparing time series data. We note that, just as is the case for the
+dynamic time warping distance, this is not a true distance in the mathematical sense.
+Algorithm 1: Algorithm for asymmetric dynamic time warping
+Data: Label curve defined by N points {pi}, and prediction curve defined by M line segments {sj}.
+Result: The asymmetric dynamically time-warped distance from label to prediction.
+Initialize matrices C, D with size [N, M].
+for i = 1 to N do
+for j = 1 to M do
+Di,j ← distance from point to segment(pi, sj)
+C1,1 ← D1,1
+for i = 2 to N do
+Ci,1 = Ci−1,1 + Di,1
+for j = 2 to M do
+C1,j = min (C1,j−1, D1,j−1)
+for i = 2 to N do
+for j = 2 to M do
+Ci,j = min (Ci,j−1, Ci−1,j + Di,j)
+return CN,M/N
+Tracking implementation
+Tracking is done by sequentially predicting individual frames. For better performance,
+batching of frames allows for parallel detections and can drastically reduce execution time. Nevertheless, due to the
+requirement of including surrounding frames for each detection, a considerable increase in memory usage is observed.
+Once a collection of spline detections is obtained, each prediction is adapted in order to make it work with the
+TrackPy Python library. Due to the peculiarity of our distance metric, we implement a custom neighbor strategy
+(see Code Availability) that avoids the assumption of a symmetric distance function.
+16
+
+It may happen that some detection artefacts appear during the sequential detection performed on tracking. We
+have implemented a quick check on the resulting tracks to make sure not to have stubs, and fix obvious branching
+of tracks due to these artefact. Slight increase in integrity is observed on dense clips.
+Tracking integrity
+Given a true label of a track of length N, we associate to this track at each time point i a
+prediction identity Ii. We may then define the integrity of the track as ι = 1/N 2 �N
+i=1
+�N
+j=1[Ii = Ij]. For instance,
+if a label is given identities I = [1, 1, 1, 5, 5, 5, 3, 3, 3] during the track, i.e. there have been two identity swaps, we find
+ι = 1
+3, which has the interpretation that the track was correct for a third of the time. This measure will in general
+scale like ι ∼ N −1, as longer tracks will have higher likelihood of identity swaps.
+Acknowledgments
+Video of C. elegans were provided by Celia Raimondi, Sunehera Sarwat and Michele Perni.
+This work was supported by the Novo Nordisk Foundation, Grant Agreement NNF20OC0062047.
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+Systematic design of a robust half-W1 photonic crystal waveguide
+for interfacing slow light and trapped cold atoms
+Adrien Bouscal,1 Malik Kemiche,2, 3 Sukanya Mahapatra,2 Nikos Fayard,4 J´er´emy
+Berroir,1 Tridib Ray,1 Jean-Jacques Greffet,4 Fabrice Raineri,2 Ariel Levenson,2
+Kamel Bencheikh,2 Christophe Sauvan,4 Alban Urvoy,1, ∗ and Julien Laurat1
+1Laboratoire Kastler Brossel, Sorbonne Universit´e, CNRS,
+ENS-Universit´e PSL, Coll`ege de France, 4 place Jussieu, 75005 Paris, France
+2Centre de Nanosciences et de Nanotechnologies, CNRS,
+Universit´e Paris-Saclay, 91120 Palaiseau, France
+3IMEP-LAHC, Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, Grenoble INP, 38000 Grenoble, France
+4Universit´e Paris-Saclay, Institut d’Optique Graduate School,
+CNRS, Laboratoire Charles Fabry, 91127 Palaiseau, France
+(Dated: January 13, 2023)
+Novel platforms interfacing trapped cold atoms and guided light in nanoscale waveguides are a
+promising route to achieve a regime of strong coupling between light and atoms in single pass,
+with applications to quantum non-linear optics and quantum simulation. A strong challenge for the
+experimental development of this emerging waveguide-QED field of research is to combine facilitated
+optical access for atom transport, atom trapping via guided modes and robustness to inherent
+nanofabrication imperfections. In this endeavor, here we propose to interface Rubidium atoms with
+a photonic crystal waveguide based on a large-index GaInP slab. With a specifically tailored half-
+W1 design, we show that a large coupling to the waveguide can be obtained and guided modes can
+be used to form two-color dipole traps for atoms at about 100 nm from the edge of the structure.
+This optimized device should greatly improve the level of experimental control and facilitate the
+atom integration.
+I.
+INTRODUCTION
+Interfacing cold neutral atoms and photons guided in
+nanoscale waveguides has raised a large interest over the
+recent years, with a wealth of emerging opportunities [1].
+Arrays of atoms can be trapped in the evanescent field
+of guided modes and the strong transverse confinement
+enables to increase the individual atom-photon coupling
+in single pass. Remarkable experimental advances have
+been obtained with optical nanofibers [2–5], exploiting
+collective effects and chiral properties to realize various
+all-fibered functionalities [6–11]. Beyond nanofibers, tai-
+lored dispersion relations that can be obtained in pho-
+tonic crystal waveguides (PCW) offer unique features.
+While the atom-photon coupling can be strongly en-
+hanced near a band edge, where guided modes can prop-
+agate slowly, atom-photon bound states can also appear
+for an atomic transition within a bandgap, with the ca-
+pability to implement tunable long-range atom-atom in-
+teractions. These features led to a variety of theoretical
+proposals for applications in quantum optics and many-
+body physics [12–14].
+Despite the promises of this new waveguide-QED
+paradigm, trapping atoms in the vicinity of such photonic
+crystal waveguides is still at its infancy. This combina-
+tion is a daunting challenge due to stringent requirements
+when considering real physical implementations. A first
+challenge is to keep the atoms as static as possible close to
+∗ Corresponding author: alban.urvoy@sorbonne-universite.fr
+the structure, so that they can interact with the evanes-
+cent mode. While tweezers can be used to maintain the
+atoms at a fixed distance [15–18], it is challenging to make
+an array of such atoms at distances on the 100 nm range.
+Dipole trapping by the evanescent field of guided modes
+is necessary but it has remained an important roadblock.
+Up to now, only a corrugated slot waveguide (so-called
+alligator waveguide) [19–23] has been implemented and
+first pioneering demonstrations obtained, albeit with a
+limited number of atoms and without stable trapping in
+the evanescent field. Some theoretical proposals on novel
+interesting structures supporting atom trapping in the
+evanescent field have emerged since, such as a slot [24] or
+a comb waveguide [25]. Structures must also provide a
+large optical access to bring atoms close to their surface.
+Eventually, in order to push experimental development,
+great care should be put in ensuring that the structure
+is robust against fabrication imperfections.
+In this paper, we design a novel platform for interfac-
+ing trapped cold atoms and a slow-mode photonic crystal
+waveguide.
+Building from the promises of W1 waveg-
+uides, made of a linear defect in a 2D photonic crystal,
+and initial work in Ref. [26], we propose a tailored plat-
+form for trapping arrays of Rubidium atoms in the prox-
+imity, as sketched in Fig. 1(a).
+Waveguides based on
+a 2D photonic crystal etched in a large refractive-index
+slab have well-known strengths and are widely used in the
+telecom range. Many techniques have been developed to
+shape their dispersion curve with astounding precision
+[27–31]. Strong coupling between a single emitter em-
+bedded in a W1 waveguide and the guided light has been
+demonstrated [32], and successfully exploited for quan-
+arXiv:2301.04675v1  [quant-ph]  11 Jan 2023
+
+2
+1D
+′
+t
+(a)
+L
+a
+r
+y1
+r3
+(b)
+680
+730
+780
+830
+880
+930
+980
+Wavelength [nm] 
+0.25
+0.30
+0.35
+0.40
+0.45
+0.50
+kx[in units of 2
+a ]
+320
+340
+360
+380
+400
+420
+440
+Frequency [THz]
+Radiative modes
+Bulk modes
+Bulk modes
+87Rb D2
+(c)
+o
+x
+y
+z
+FIG. 1. A half-W1 slow-mode photonic crystal waveguide coupled to cold atoms. (a) Sketch of the waveguide with an array of
+87Rb atoms trapped in the proximity, along the edge. Γ1D and Γ′ correspond to the decay rates in the guided mode and in the
+radiation continuum, respectively. The structure is etched in a GaInP membrane (refractive index n = 3.34) suspended in air,
+with a slab thickness t of 150 nm. (b) 2D scheme of the optimized waveguide. The initial unshifted and regularly distributed
+holes are shown as dotted lines. For the first three rows the position of the holes can be shifted along y and their radius tuned,
+amounting to 6 parameters (δyi,δri), i ∈ {1, 2, 3}. For the sake of clarity, only two parameters (δy1 and δr3) are displayed.
+(c) Bandstructure of the optimized structure calculated via FDTD simulation. The bulk modes propagate within the slab but
+are not guided on the edge of the PCW while the radiative modes are not guided at all. The 87Rb D2 line transition frequency
+is aligned with the linear part of a guided band, defined as the slow mode in the text.
+tum operations [33]. The proposed platform, sketched in
+Fig. 1(a), can be seen as half a W1 waveguide. As its
+W1 counterpart, it enables enables dispersion engineer-
+ing but in addition offers a 2π solid-angle optical access to
+the edge of the structure, allowing for simpler transport
+of atoms close to it [26]. We use a large refractive index
+GaInP slab that facilitates the design by offering more
+flexibility in the engineering of guided modes, and we
+show how to trap atoms in the proximity via additional
+guided modes. Our effort focuses at each step on making
+the design robust to imperfections and on assessing the
+experimental feasibility of the full platform.
+This paper is organized as follows. First, in section II
+we present the specific platform based on a half-W1
+waveguide realized in a GaInP slab with a high refractive
+index. We detail the optimization and the resulting ro-
+bustness to nanofabrication imperfections, and then pro-
+vide the achievable atom-photon coupling.
+Second, in
+section III we show that guided modes can be used to
+trap atoms in the proximity of the waveguide via a two-
+color evanescent dipole trap. Stable traps at about 115
+nm from the surface are obtained with low powers that
+are compatible with nanophotonic systems. A summary
+and outlook is provided in section IV.
+II.
+ENGINEERED HALF-W1 WAVEGUIDE FOR
+RUBIDIUM ATOMS
+In this section we introduce the specific half-W1 slow-
+mode waveguide designed in this work, based on GaInP.
+We identify the required geometrical parameters and
+then present the optimizations performed to increase the
+robustness to fabrication imperfections, leading thereby
+to linear bands.
+Finally, the expected coupling to the
+guided mode (i.e. Purcell factor) for atoms in the prox-
+imity from the surface is detailed.
+A.
+Description of the half-W1 GaInP waveguide
+A periodic modulation of the refractive index in a
+medium has deep consequences on light propagation.
+The wavevector can be constrained between −π/a and
+π/a, where a is the spatial period along the propaga-
+tion direction, and we observe the opening of photonic
+bandgaps. At the edge of the Brillouin zone (for k = π/a)
+the group velocity vanishes [34] and the Purcell factor di-
+verges (see Appendix A).
+Motivated by the proposal made in Ref. [26], we study
+a similar structure with a different material: GaInP. This
+material has been chosen for its advantageous optical
+and electronic properties. GaInP has a wide electronic
+bandgap below 1.85 eV [35], and as such is transparent
+for a wide range of wavelengths (from 670 nm up), mean-
+ing it could be used with several alkali. At 780 nm, its
+
+3
+775
+780
+785
+ [nm]
+0
+10
+20
+30
+40
+50
+group index ng
+(a)
+ = 9.2 nm 
+ navg =27.7
+775
+780
+785
+ [nm]
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+1D/
+0
+(b)
+F′ = 3, m′F = 3
+F = 2, mF = 2
+FIG. 2. Dispersion and atom-coupling properties for the half-
+W1 waveguide with the structure optimization specified in
+Table I. (a) Calculated group index ng for the slow mode. The
+dotted lines delimit the linear band region where the group
+index value is constant up to 15%.
+(b) Calculated Purcell
+factor Γ1D/Γ0 over the same range, for atoms trapped at 115
+nm from the structure. As it can be seen, ng is not the only
+parameter affecting this ratio, i.e., the field structure is also
+changing, but it is still critical as it diverges with ng just
+outside the plateau.
+refractive index is n = 3.34, reaching 3.55 at the elec-
+tronic band edge. This large index contrast with the air
+gives rise to band gaps that are wider and further away
+from the light line [34], allowing for more flexibility in
+the design of the trapping modes. Finally, this material
+has attracted some attention in the recent years as it is
+very convenient to operate in the telecom band due to
+its low two-photon absorption [36], and growth and fab-
+rication processes have therefore been developed and well
+mastered.
+As shown in figure 1(b), the holes etched in the GaInP
+slab do not go up to the edge, leaving a few hundreds
+of nanometers of unperturbed slab where the light can
+propagate. Being based on a 2D slab rather than a 1D
+structure, this geometry should be quite rigid and pre-
+vent detrimental effects from low frequency mechanical
+modes. The introduced symmetry breaking in the trans-
+verse direction allows for a more precise control on the
+dispersion properties of the waveguide since it offers ex-
+tra degrees of freedom [37], while significantly improving
+the optical access. This asymmetry has been harnessed
+in Ref. [38] to create many exotic dispersion bands as
+Dirac cones, multivalleys, or flat bands. Arrays of 87Rb
+will then be trapped on the edge of the waveguide thanks
+to a two-color dipole trap, at around 100 nm from the
+surface. For comparison, in tapered nanofiber platforms,
+atoms sit at more than 200 nm from the silica fiber.
+For a given thickness t, chosen here to be t = 150 nm,
+the first step to determine the geometrical parameters
+consists in finding the lattice period a and hold radius
+r of the 2D photonic crystal that allow for a bandgap
+at the Rubidium D2 transition. Indeed the width and
+position of the band gap is entirely determined by these
+values [34]. The band gap has to be wide enough to allow
+for at least two guided modes, one that crosses 780 nm,
+and a blue-detuned one for trapping, as described later.
+Guided bands appear when introducing the defect at the
+edge, and we can align the band of interest with respect
+to the D2 line by adjusting the width L.
+Given these constraints, the geometrical parameters of
+the waveguide are found to be: a = 212 nm, r = 63 nm
+and L = 337 nm for t = 150 nm.
+The corresponding
+band structure, computed with the 3D FDTD software
+Lumerical [39] is displayed in figure 1(c). Three guided
+bands can be found inside the band gap of the 2D pho-
+tonic crystal between 360 and 440 THz. The bulk modes
+are guided in the slab (kz imaginary) but can propagate
+in any direction in the plane, even inside the 2D array of
+holes (kx, ky real). Radiative modes have a real k vector
+in all directions and are therefore not guided.
+B.
+Imperfection-robust band engineering
+Nanofabrication inherently leads to imperfections,
+even if errors below 2 nm can be reached [40]. A spe-
+cific effort has been put in our design process to minimize
+the impact of such imperfections, thereby facilitating an
+experimental realization.
+As the Purcell factor diverges at the edge of the Bril-
+louin zone, one naive approach could be to align the D2
+line frequency to any band edge of the band structure.
+However, fabrication imperfections, to first order, lead
+to a shift of the energy of the band [41].
+The flatter
+the band, i.e., the smaller the group velocity, the more
+it is vulnerable to a shift in frequency [42]. If the D2
+line is aligned with the band edge, an infinitesimal shift
+to a lower energy will bring the atomic transition in the
+band gap of the 2D photonic crystal, impeding the prop-
+agation of the emitted light.
+In addition to a shift in
+frequency, the disorder introduced during the fabrication
+process can lead to strong localization of light inside the
+crystal [41, 43, 44].
+Following [26], two main criteria are to be considered
+when assessing the robustness of a structure: the group
+velocity has to be as independent of the frequency as
+possible at the band edge, i.e. ∂vg/∂ω|ωe ∼ 0, and the
+distance of the operation frequency to the band edge
+∆ω = |ω − ωe| has to be as large as possible.
+Designing slow modes with linear bands (i.e., an almost
+constant, large group index ng over the widest range of
+ω possible) allows us to fulfill these two criteria. First,
+a linear dispersion corresponds to a vanishing group ve-
+locity dispersion (GVD) and the atom-photon coupling
+Row
+Position δy (nm) Radius δr (nm)
+1
+42.7
+14.2
+2
+53.8
+-11.2
+3
+-3.7
+-10.8
+TABLE I. Calculated changes in row positions and holes radii
+via automatic differentiation optimization. All the rows after
+the third one are unperturbed.
+
+4
+-318
+-106 0 106
+318
+x [nm]
+-400
+-200
+0
+200
+400
+y [nm]
+(a)
+-200
+0
+200
+y [nm]
+-300
+-225
+-150
+-75
+0
+75
+150
+225
+300
+z [nm]
+(b)
+-212
+0
+212
+x [nm]
+-500
+-250
+0
+250
+500
+y [nm]
+(c)
+0
+0.2
+0.4
+0.6
+0.8
+1
+Intensity [norm.]
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+|C|
+FIG. 3. Slow mode structure at the 87Rb D2-line frequency. (a) Normalized intensity, in the (x, y)-plane at z = 0. (b) Same
+in the (y, z)-plane at x = −a/2, i.e., crossing the hole nearest to the slab edge. The mode is strongly expelled into the vacuum
+around the edge of the waveguide. (c) Polarization ellipticity z-component Cz in the XY plane at z = 0. The other components
+of the ellipticity vector are 0. |Cz| = 0 indicates a linear polarization, while we have |Cz|= 1 for a circularly polarized light.
+Close to the edge, the polarization has a large circular component due to the strong longitudinal component that appears when
+light is confined at the nanoscale. By taking z as the quantification axis, the polarization will be close to σ+ for atoms trapped
+in the proximity (91 to 99% at 115 nm from the surface).
+is proportional to the group index. Moreover, as shown
+in [28], it is possible to design a slow and linear band
+over a wide spectral range. As most fabrication imper-
+fections lead to a shift ∆ω of the guided bands, both
+these constraints aim at placing the relevant frequency
+at a position on the band where a small shift will affect
+the dispersion at the given frequency only slightly.
+It
+has been shown that linear bands can be achieved in at
+least two types of asymmetric PCWs [38]. Achieving such
+vanishing group velocity dispersion has been extensively
+studied in the context of W1 waveguides, by tuning the
+position of rows of holes [28, 45, 46], chirping the waveg-
+uide properties [47], or changing the size of the holes [29].
+Inspired by these previous optimization strategies, we
+set the radius of the first three rows of holes as well as
+their position along the y axis as optimization parame-
+ters, as depicted in figure 1(b). We then have 6 indepen-
+dent optimization parameters (δri, δyi), i ∈ {1, 2, 3}. As
+full 3D FDTD simulations are computationally intensive,
+we use the approximate method of Guided Mode Expan-
+sion (GME) [48] thanks to the legume [49] solver to faster
+compute the shape of the guided band. We optimize the
+shape of the slow-mode band by iteratively varying the
+parameters. At each iteration, a cost function enforcing
+the minimization of the group velocity dispersion (aver-
+aged over the wave vector interval) while setting a target
+ng value is evaluated and the (δri, δyi) varied thanks to
+automatic differentiation. After a few hundred iterations
+we obtain the optimal shifts for achieving this target ng
+value over the widest possible spectral range. Finally the
+optimized structure was simulated in full 3D FDTD to
+validate the results from the approximate GME method.
+In order for this optimization to give relevant results,
+ng has to be set to an experimentally realistic value, ide-
+ally below 60. Indeed, experiments have shown that it is
+extremely challenging to reach higher values for the group
+index without losses [50]. The most concluding optimiza-
+tion results are obtained for a target around ng = 30.
+The shifts in position and radius after optimization are
+given in table I and the corresponding band structure is
+presented in figure 1(c). Figure 2(a) shows that we engi-
+neered a band with a constant group index of 28 over a
+9 nm range, and hence reach similar performance than a
+previous optimization of a W1 waveguide [28]. This fea-
+ture offers a two-fold advantage. In addition to making it
+robust to shifts caused by fabrication imperfections, the
+optimization enables using the half-W1 waveguide in a
+large bandwidth regime (≥ 4 THz) with very little dis-
+persion.
+Finally, as seen in Appendix A, the Purcell factor is
+proportional to the group index. Keeping the group in-
+dex constant over a wide range enables to keep the Pur-
+cell factor constant in case of a shift, as shown in figure
+2(b). If it is necessary to have a constant group index it
+is not sufficient as the Purcell factor also depends on the
+shape of the mode of the electric field (equation A4). As
+we move along the guided band, the mode shape changes
+slightly, affecting the value of the Purcell factor.
+C.
+Strong coupling to the slow mode
+Given the optimized design, we now turn to the in-
+teraction between the slow mode and 87Rb atoms in the
+proximity. Taking into account the multilevel character
+of Rubidium, we defined a transition-dependent Purcell
+factor in equation (A4) given in Appendix A. The group
+velocity vg is evaluated from the simulated band struc-
+ture, while the other terms are computed from the field
+map of the guided mode. Indeed, the Purcell factor is
+
+5
+F′ = 3
+m′F =
+F = 2, mF = 2
+3
++
+2
+1
+-
+(a)
+0
+200
+400
+y [nm]
+0
+5
+10
+15
+1D/
+0
+-212
+0
+212
+x [nm]
+-400
+-200
+0
+200
+y [nm]
+(b)
+0
+200
+y [nm]
+-225
+-150
+-75
+0
+75
+150
+225
+z [nm]
+(c)
+10
+2
+10
+1
+100
+101
+1D/
+0
+0
+200
+0.0
+0.1
+FIG. 4. Excitation rates for 87Rb atoms in the waveguide proximity. (a) Allowed transitions on the D2 line for an atom in
+|F = 2, mF = +2⟩. Because of the large σ+ component (∼ 91% at the position of the atoms) and the values of the Clebsch-
+Gordan coefficients, the excitation probability to the |F′ = 3, mF′ = +3⟩ is 100 times higher than the σ− channel. The inset
+provides a zoom. (b) Purcell factor in the XY plane, at z = 0. (c). Purcell factor in the YZ plane, at x = −a/2. The red dots
+indicate the position of the atoms at 115 nm from the surface.
+proportional to the slow-mode intensity. Figure 3(a-b)
+show that in order to have the maximum coupling the
+atoms should be trapped close to the edge of the waveg-
+uide, aligning them to the holes of the first row.
+From figure 4(a) we see that for atoms in state
+|F = 2, mF = +2⟩ trapped at 115 nm from the edge, the
+Purcell factor reaches a value of 1.6. As shown in fig-
+ures 4(b) and 4(c), a small modulation in the x direction
+exists and the value of the Purcell factor decays rapidly
+when going further from the surface.
+To quantify the coupling of the atoms to the guided
+mode we also define the β factor, β = Γ1D/Γtot with
+Γtot = Γ1D + Γ′, and Γ′ the decay rate in all the other
+radiation modes than the guided slow mode. Because of
+the complex shape of the local density of states accessible
+to the atoms, the behaviour of Γ′ is hard to infer, but its
+modulation is expected to be minimal as seen in [25, 26].
+Hence we assume for the following Γ′ ≃ Γ0.
+At the position of the trap minimum, i.e at 115 nm
+from the surface, we find β = 0.62, very close to the
+averaged value ˜β = 0.57 for a thermal distribution (at a
+temperature of one tenth of the trap depth). This is at
+least 50 times better than the current systems involving
+nanofibers (β = 10−2) [9] and a significant improvement
+with respect to current PCW-based platforms (β = 0.45)
+[20].
+III.
+TRAPPING RUBIDIUM ATOMS NEAR A
+HALF-W1 WAVEGUIDE
+Simulations in the previous section were performed for
+atoms at 115 nm from the edge of the waveguide. Indeed,
+in the following we show a stable trapping scheme based
+on an evanescent two-color dipole trap formed by fast
+guided modes, allowing the atoms to be trapped between
+100 and 150 nm. This trap has been designed following
+the ideas implemented in optical nanofibers [2, 51], with
+blue- and red-detuned counter-propagating modes. Find-
+ing a stable trapping scheme that keeps the atoms close
+enough to the surface so that they can couple to the slow
+mode with a large Purcell factor is a critical requirement
+for experimental implementations.
+A.
+Two-color dipole trap structure
+In contrast with optical nanofibers, the guided modes
+are structured along the propagation direction due to the
+Bloch wave structure of the light field. The intensity of
+the modes, which is an important quantity when looking
+at dipole trapping, is periodic with period a, as shown in
+figure 3 for the slow mode. This feature constrains the
+position of the trapped atoms to the maxima of intensity
+of the red-detuned mode. It makes the search for a blue
+detuned mode more challenging as this one will also be
+structured, while a uniform one would work perfectly well
+to repel the atoms from the surface [3]. A blue-detuned
+beam with an intensity pattern completely out of phase
+with the red-detuned one is needed. Fortunately, modes
+separated by a band gap usually have intensity maxima
+shifted by a/2 [52]. We then use the highest guided band
+for the blue-detuned trap between 400 and 420 THz (fig-
+ure 1(c)).
+In order to have a full description of the potential
+seen by the atoms, we must add the Casimir-Polder
+(CP) interaction [53] between the atoms and the surface.
+Ground state fluctuations of the vacuum can polarize the
+atoms, even if they are not charged. When put in prox-
+imity to structures, the vacuum-induced dipole moments
+create mirror charges that act on the original dipole, lead-
+ing to an additional light shift.
+This CP potential is
+only significant at very close distances (≤ 150 nm) but
+is crucial as it reduces the local density of states and
+acts as an attractive potential close to the surface. For
+an atom in the proximity of an infinite dielectric plane
+
+6
+UCP = −C3/d3 [52], where d is the distance to the sur-
+face.
+As described in Appendix B, for a ground state
+87Rb atom close to a GaInP surface, we computed an
+approximate C3 = −9.25 × 10−49J.m3.
+B.
+Trapping potential simulation
+The
+trapping
+potentials
+were
+obtained
+via
+nanotrappy
+[54],
+a
+Python
+package
+developed
+by
+our group, to design, calculate and optimize dipole traps
+around nanoscale waveguides, making the search process
+faster and more systematic.
+Figure 5 shows a trapping potential in all 3 directions
+for an atom in the |F = 2, mF = +2⟩ hyperfine level. For
+this trap, a beam red-detuned from the D2 line of 87Rb at
+784.5 nm and a beam blue-detuned at 737 nm are used.
+A trap of depth 3 mK is obtained with a minimum at
+115 nm from the surface. The total powers are Pblue =
+1.65 mW and Pred = 0.1 mW, but a stable trap can be
+obtained on a wide range of powers. The main limitation
+can be the power handling of the structure which is still
+to be determined. In Ref. [36], power densities up to
+1 GW/cm2 were coupled to similar GaInP PCWs with
+group index 8.8.
+For our structure which has a cross
+section 10 times smaller and a group index 3 times bigger,
+this would be equivalent to coupling ≃ 100 mW into our
+waveguide. The proposed powers for the trap fall well
+below this bound.
+The trapping frequencies are large in the x and y direc-
+tions, with ωx = 2π×1.89 MHz and ωy = 2π×1.94 MHz.
+In the vertical direction however, an important anhar-
+monicity of the trap, discussed below, gives ωz = 2π ×
+133 kHz.
+A critical aspect is the trapping along the z direction.
+With small powers for the blue-detuned beam, a sym-
+metric double well around z = 0 appears. This can be
+detrimental for our platform as the atoms will be trapped
+at positions not corresponding to the maxima of the Pur-
+cell factor. Because of the fixed CP potential, rescaling
+the blue and red powers by the same factor does not lead
+to a simple reduction of the trap depth, it also goes with
+a shift in position, which might lead to reaching this two-
+well regime in the z direction. This explains the impor-
+tant depth of the simulated traps: the closer to the sur-
+face we want the trap minimum to be, the more we have
+to compensate for the CP potential with higher beam
+power, leading to higher trap depths. This phenomenon
+may come from the complex decay of the evanescent wave
+away from the surface pointed out in Ref. [55]. Indeed,
+this decay is usually multi-exponential, with decay rates
+not easily linked to the wavelength or the wavevector of
+the guided mode.
+Importantly, we also verified that we can achieve a sta-
+ble trap in the three directions for a wide range of wave-
+lengths, which is a valuable feature for finding the right
+trade-off between heating the atoms with off-resonant
+scattering and power handling of the waveguide. If we
+-212
+0
+212
+424
+x [nm]
+-500
+-250
+0
+250
+500
+y [nm]
+(a)
+-212
+0
+212
+424
+x [nm]
+-3.0
+-2.5
+-2.0
+-1.5
+-1.0
+-0.5
+0.0
+U [mK]
+(b)
+0
+100
+200
+300
+y [nm]
+-3.0
+-2.5
+-2.0
+-1.5
+-1.0
+-0.5
+0.0
+U [mK]
+(c)
+-200
+0
+200
+z [nm]
+-3.0
+-2.5
+-2.0
+-1.5
+-1.0
+-0.5
+0.0
+U [mK]
+(d)
+0
+20
+40
+60
+80
+100
+120
+140
+U [mK]
+FIG. 5. Calculated potential of the two-color dipole trap. (a)
+2D total trapping potential in the proximity of the waveguide,
+in the XY plane. The trapping potential is given along the
+three directions in (b), (c) and (d). A periodic stable trap with
+depth of about 3 mK is obtained with powers of 1.65 mW for
+each blue beam and 100 µW for each red one. The simulations
+are performed with the nanotrappy package.
+allow the blue power to go up to 3 mW, we can find a
+stable trap for blue wavelengths ranging from 724 nm to
+738 nm. Pushing the maximum allowed power to 5 mW
+we can find a trap for the full available blue-detuned air
+band, i.e. ∆λ = 21 nm. For the red-detuned laser, we
+have an available range from 780.5 nm up to 786 nm.
+Laser diodes are easily available on these wavelengths,
+reinforcing the feasibility of our platform.
+Finally, as briefly noted before, we used counter-
+propagating beams here instead of simple ones, albeit
+standing waves are not needed to get a periodic intensity
+modulation. The strong ellipticity of the guided modes
+(as shown in figure 3(c)), acts as a fictitious magnetic
+field on the atoms, splitting the Zeeman levels [56]. If
+we start from atoms evenly distributed in all the mF
+states, this effect would lead to a large inhomogeneous
+broadening up to a few GHz. It can be mitigated by us-
+ing counter-propagating trapping beams slightly detuned
+from each other, as used for blue detuned beams in some
+compensated nanofiber traps [3].
+Via nanotrappy, we
+estimated that adding a red-detuned laser at 280 GHz
+from the first one and a blue detuned at 250 GHz from
+the other reduces this broadening by 90%. Counterprop-
+agation creates a running wave at a velocity given by
+δω. This pattern propagates but at a speed so large the
+atoms only see the average of the potential.
+
+7
+IV.
+CONCLUSION
+Many experimental and technological challenges have
+yet to be overcome to enable further neutral-atom
+waveguide-QED protocols.
+As such, experimental ro-
+bustness of the targeted waveguide platforms is a critical
+requirement, as is evanescent trapping of atoms. In our
+work, we proposed and engineered a bona fide platform
+for trapping cold Rubidium atoms close to a half-W1
+photonic crystal waveguide based on high-index mate-
+rial GaInP. Atoms can be trapped between 100 and 150
+nm from the surface with compatible low-power incident
+light. At 115 nm, the slow mode couples to the atoms
+with a Purcell factor as high as 1.6 when the group index
+is taken around 30. This study has been carried out for
+conservative parameters and a strong focus on robustness
+against fabrication imperfections has been done by engi-
+neering the band structure for a large bandwidth, facili-
+tating first implementations. Future generations should
+support higher group index, albeit with narrower band-
+widths [28].
+This novel platform – tailor-designed for
+atom integration, robustness and large optical access –
+offers unique advantages for studying coherent and dissi-
+pative dynamics in the waveguide-QED framework.
+V.
+ACKNOWLEDGEMENTS
+This work was supported by the French National
+Research Agency (NanoStrong Project ANR-18-CE47-
+0008), by the R´egion Ile-de-France (DIM SIRTEQ), and
+by the European Union’s Horizon 2020 research and in-
+novation program under Grant Agreement No.899275
+(DAALI project). A.U. was supported by the European
+Union (Marie Curie Fellowship SinglePass 101030421).
+J.L. is a member of the Institut Universitaire de France.
+Appendix A: Theoretical framework: Reaching
+strong Purcell factor
+The coupling of the atoms to the guided mode of a
+waveguide can be characterized by the Purcell factor
+Γ1D/Γ0, which relates the decay rate of the atoms into
+the guided mode to the one into free space. For a mul-
+tilevel atom we can define the excitation rate ΓF,F ′,mF,q
+from the hyperfine level |F, mF⟩ to |F′, mF + q⟩ [57]:
+ΓF,F ′,mF,q = 2µ0| ⟨F||ˆd||F⟩|2
+¯h
+×
+ω2
+q|CmF ,q|2ˆeq · Im G(r, r; ωq) · ˆe∗
+q
+(A1)
+where the ˆeq, q ∈ {−1, 0, 1}, are the normalized dipole
+vectors over all the possible excitation channels (σ−, π,
+σ+ respectively) and ωq is the transition frequency be-
+tween the specified levels. G(r, r; ωq) is the value of the
+classical Green’s tensor at r for a dipole at the same po-
+sition.
+The CmF ,q are the Clebsch-Gordan coefficients
+given by:
+CmF ,q = (−1)F ′−1+mF √
+2F + 1
+�
+F ′
+1
+F
+m′
+F q −mF
+�
+. (A2)
+Note that only the Wigner 3j where m′
+F = mF + q are
+non zero.
+As we are looking at the decay into a defined guided
+mode, it is possible to write the Green’s tensor as a sum
+G(r, r; ωq) = G1D(r, r; ωq)+G′(r, r; ωq). From [58] (Eq.
+2.89) and [59] we have an analytical expression for the
+Green’s function of an effective 1D structured waveguide:
+G1D(r, r, ω) = i ac
+2ω
+� c
+vg
+�
+[E(r) ⊗ E∗(r)]
+�
+cell drϵ(r)|E(r)|2
+(A3)
+where E(r) is the electric field of the guided mode, a the
+period of the modulation and vg = ∂ω
+∂k the group velocity
+of the guided mode.
+By neglecting the Zeeman splitting of the mF levels
+(ωq constant), the excitation probability of a single atom
+in |F, mF⟩ into |F′, mF + q⟩ (through a single excitation
+channel) is hence given by:
+Γ1D,F,F ′,mF,q = 2πc
+ϵ0¯h
+| ⟨F||ˆd||F⟩|2
+λ0vg
+×
+|CmF ,q|2
+|ˆeq · E(r)|2
+�
+cell drϵ(r)|E(r)|2 .
+(A4)
+As such, we see that to reach high Purcell factors we
+must either decrease vg which can be achieved by dis-
+persion design, or maximize the normalized electric field
+amplitude at the position of the atom given by the second
+half of equation (A4).
+Since we are considering excitation probabilities (as in
+Figure 4(a)), we only consider the coupling between the
+atom and the injected mode, here the guided mode E(r)
+propagating along the positive x axis. If we were instead
+to consider decay rates we would have to modify slightly
+equation (A4) and sum the contributions of the coupling
+of the atom to both forward and backward propagating
+guided modes.
+Appendix B: Casimir-Polder interactions between
+GaInP and Rubidium atoms
+To the best of our knowledge, there were no previous
+computations of the C3 coefficient of the Casimir-Polder
+interactions between GaInP and Rubidium atoms. For
+a dielectric wall, we have to adapt the formula for C3 in
+the form [60]
+C3 ≈ ¯h
+4π
+� +∞
+0
+α(iξ)ϵ(iξ) − 1
+ϵ(iξ) + 1dξ.
+(B1)
+
+8
+We then have to evaluate α and ϵ over the imaginary axis.
+α is simply evaluated using the expression for the scalar
+polarizability of 87Rb in the ground state |F = 2⟩ with
+complex frequencies. As ϵ(ω) = ϵ′(ω) + iϵ′′(ω), we can
+get the dependence in ω from experimental data [35]. To
+evaluate it over the imaginary axis we use the Kramers-
+Kroenig relation that provides [61]
+ϵ(iξ) = 1 + 2
+π
+� +∞
+0
+ω[ϵ′(ω) − 1]
+ω2 + ξ2
+dω.
+(B2)
+Finally, for a ground state Rubidium atom close to a
+GaInP surface we get C3
+=
+9.25 × 10−49J.m3 =
+h × 1391 Hz.µm3.
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+

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@@ -0,0 +1,1645 @@
+Equilibrium Spacetime Correlations of the Toda Lattice on
+the Hydrodynamic Scale
+Guido Mazzuca∗, Tamara Grava†, Thomas Kriecherbauer‡, Kenneth T-R
+McLaughlin §, Christian B. Mendl¶, Herbert Spohn‖
+January 9, 2023
+Abstract
+We report on molecular dynamics simulations of spacetime correlations of the Toda lattice
+in thermal equilibrium. The correlations of stretch, momentum, and energy are computed
+numerically over a wide range of pressure and temperature. Our numerical results are com-
+pared with the predictions from linearized generalized hydrodynamics on the Euler scale. The
+system size is N = 3000, 4000 and time t = 600, at which ballistic scaling is well confirmed.
+With no adjustable parameters, the numerically obtained scaling functions agree with the
+theory within a precision of less than 3.5%.
+1
+Introduction
+A central goal of Statistical Mechanics is to explore the structure of equilibrium correlations for
+observables of physical interest. These could be static correlations, but more ambitiously also
+correlations in spacetime. An interesting, but very fine-tuned class of hamiltonians are integrable
+many-body systems, either classical or quantum. This choice restricts us to systems in one dimen-
+sion. Then, generically, static correlations have exponential decay whether the model is integrable
+or not. However, the dynamics of correlations is entirely different. In nonintegrable chains correla-
+tions propagate as a few narrow peaks at constant speed which then show characteristic sub-ballistic
+broadening. On the other hand for integrable models correlations still spread ballistically but now
+∗Department of Mathematics, The Royal Institute of Technology, Stockholm, Sweden.
+Email: mazzuca@kth.se
+†International School for Advanced Studies (SISSA), Trieste, Italy, School of Mathematics, University of Bristol,
+UK and INFN sezione di Trieste,
+Email: grava@sissa.it
+‡Department of Mathematics, Universit¨at Bayreuth, Germany
+Email: thomas.kriecherbauer@uni-bayreuth.de
+§Tulane University, New Orleans, United States
+Email: kmclaughlin@tulane.edu
+¶Technische Universit¨at M¨unchen Department of Informatics, Boltzmannstraße 3, 85748, Garching, Germany
+Email: christian.mendl@tum.de
+‖Technische Universit¨at M¨unchen Department of Mathematics and Physics, Boltzmannstraße 3 and James-
+Franck-Str. 1, 85748 Garching, Germany
+Email: spohn@ma.tum.de
+1
+arXiv:2301.02431v1  [cond-mat.stat-mech]  6 Jan 2023
+
+with a broad spectrum of velocities. Such behaviour was confirmed through a molecular dynamics
+(MD) simulation of the Ablowitz-Ladik model [32], an integrable discretization of the nonlinear
+Schr¨odinger equation. A further confirmation came from the simulation of the Toda chain [22]. On
+the theoretical side, the 2016 construction of generalized hydrodynamics (GHD) was an important
+breakthrough [3, 6]. This theory provides a powerful tool through which, at least in principle,
+the precise form of the spectrum of correlations can be predicted. With such a development MD
+simulations can also be viewed as probing the validity of GHD.
+From the side of condensed matter physics, integrable quantum models have received consider-
+able attention. Because of size limitations, the simulation of macroscopic profiles are preferred. But
+time correlations have also been studied through DMRG simulations [4,5,8,34]. In recent years,
+attention has been given to the spacetime spin-spin correlation of the XXZ model at half-filling
+and at the isotropic point [10,20,25]. The same quantity has also been investigated for a discrete
+classical chain with 3-spins of unit length and interactions such that the model is integrable [7]. A
+comparable situation occurs for the classical sinh-Gordon equation, which is integrable as a nonlin-
+ear continuum wave equation and possesses an integrable discretization, see [2] for MD simulations
+for equilibrium time correlations of the discrete model.
+In our contribution we study the correlations of the Toda chain in thermal equilibrium through
+MD simulations and compare with predictions from GHD. We will comment on the connection
+to [22] in the last section. To make our article reasonably self-contained we first discuss the Landau-
+Lifshitz theory for nonintegrable chains. This theory provides the connection between spacetime
+correlations and linearized hydrodynamics. For the Toda chain, the theory has to be extended so
+as to accommodate an infinite number of conserved fields. We report on MD simulations of the
+Toda chain and compare with linearized GHD.
+2
+Landau-Lifshitz theory
+The dynamics of the Toda chain is governed by the Hamiltonian
+H =
+�
+j∈Z
+� 1
+2p2
+j + exp(−(qj+1 − qj))
+�
+,
+(1)
+where (qj, pj) ∈ R2 are position and momentum of the j-th particle [43,44]. Introducing the j-th
+stretch (free volume) through rj = qj+1 − qj, the equations of motion read
+d
+dtrj = pj+1 − pj ,
+d
+dtpj = −e−rj + e−rj−1,
+j ∈ Z.
+(2)
+By tradition, one introduces coefficients for the range and strength of the interaction potential
+through (g/γ) exp(−γ(qj+1 − qj)). However, by a suitable change of spacetime scales, the form (2)
+can be regained, see the discussion in Section 5. The Toda hamiltonian has no free parameters.
+Since the equilibrium measure for (1) is of product form, static correlations are easily accessible.
+Time correlations are more challenging, see [36,37] for early attempts. A novel approach has been
+developed, now known as GHD. The guiding idea is to first identify the hydrodynamic equations
+for the Toda chain, which by necessity are a set of nonlinear coupled hyperbolic conservation laws.
+Given such an input one can construct the corresponding Landau-Lifshitz theory [13,24], as based
+on linearized GHD.
+Before entering into details, it will be useful to first recall the Landau-Lifshitz theory for a
+chain with a generic interaction potential, denoted by V (for the Toda lattice V (x) = e−x), see [38]
+2
+
+and references listed therein. Thus in (1) the interaction term reads V (qj+1 −qj) and the equations
+of motion become
+d
+dtrj = pj+1 − pj ,
+d
+dtpj = V ′(rj) − V ′(rj−1).
+To define spacetime correlations we first have to specify the random initial data modelling thermal
+equilibrium. By Galileian invariance one restricts to the case of zero average momentum. Then
+the Gibbs states are characterized by the inverse temperature β > 0 and a parameter P such
+that the physical pressure equals P/β. For simplicity, we will also refer to P as pressure. The
+allowed range of P depends on V . If V diverges faster than |x| for |x| → ∞, then P ∈ R. For the
+Toda lattice P > 0 because of the one-sided divergence of the exponential. In thermal equilibrium
+{(rj, pj), j ∈ Z} are a collection of i.i.d. random variables with single site probability density
+Z0(P, β)−1 exp
+�
+− β
+� 1
+2p2
+0 + V (r0)
+�
+− Pr0
+�
+.
+(3)
+Here Z0(P, β) is the normalizing partition function. Note that, with our convention, P and β appear
+linearly in the exponent. Expectations with respect to such i.i.d. random variables are denoted
+by ⟨·⟩P,β.
+We also shorten the notation for the covariance through ⟨X1X2⟩c
+P,β = ⟨X1X2⟩P,β −
+⟨X1⟩P,β⟨X2⟩P,β, where the particular random variables X1, X2 will be obvious from the context.
+For general V , the conserved fields are stretch, momentum, and energy with densities
+⃗Q(j) =
+�
+rj, pj, ej
+�
+,
+ej = 1
+2p2
+j + Vj,
+(4)
+using as shorthand Vj = V (rj). ⃗Q is a three-vector with components labeled by n = 0, 1, 2. The
+static space correlator is defined through
+Cm,n(j) = ⟨Qm(j)Qn(0)⟩c
+P,β
+(5)
+and the static susceptibility by summing over space,
+Cm,n =
+�
+j∈Z
+⟨Qm(j)Qn(0)⟩c
+P,β,
+m, n = 0, 1, 2. Since the underlying measure is product, only the j = 0 term is nonvanishing and
+C =
+�
+�
+�
+⟨r0r0⟩c
+P,β
+0
+⟨r0e0⟩c
+P,β
+0
+⟨p0p0⟩c
+P,β
+0
+⟨r0e0⟩c
+P,β
+0
+⟨e0e0⟩c
+P,β
+�
+�
+� ,
+the zero entries resulting from ⟨p0⟩P,β = 0, ⟨p3
+0⟩P,β = 0, and r0, p0 being independent random
+variables. Later on we will need the statistics of the conserved fields on the hydrodynamic scale.
+More precisely, for smooth test functions f, we consider the random field
+⃗ξϵ(f) = √ϵ
+�
+j∈Z
+f(ϵj)
+�⃗Q(j) − ⟨⃗Q(0)⟩P,β
+�
+.
+Then, by the central limit theorem for independent random variables,
+lim
+ϵ→0
+⃗ξϵ(f) =
+�
+R
+dxf(x)⃗u(x),
+3
+
+where the limit field ⃗u(x) is a Gaussian random field on R with mean zero, E(⃗u(x)) = 0, and
+covariance
+E(um(x)un(x′)) = Cm,nδ(x − x′),
+(6)
+in other words, ⃗u(x) is Gaussian white noise with correlated components.
+Microscopically, spacetime correlations are defined by evolving one of the observables to time
+t which yields
+Sm,n(j, t) = ⟨Qm(j, t)Qn(0, 0)⟩c
+P,β.
+(7)
+Note that the Gibbs measure is spacetime stationary and thus without loss of generality both
+arguments in Qn in (7) can be taken as (0, 0). To understand the structure of Sm,n one has to rely
+on approximations. For the long time ballistic regime a standard scheme is the Landau-Lifshitz
+theory, which views Qn(0, 0) as a small perturbation of the initial Gibbs measure at the origin.
+This perturbation will propagate and is then probed by the average of Qm at the spacetime point
+(j, t). For large (j, t) the microscopic dynamics is approximated by the Euler equations, but only
+in their linearized version since the perturbation is small. More concretely, the approximate theory
+will be a continuum field ⃗u(x, t) over R × R, which is governed by
+∂t⃗u(x, t) + A∂x⃗u(x, t) = 0 ,
+(8)
+with random initial conditions as specified in (6). The 3×3 matrix A is constant, i.e. independent
+of (x, t). To explain the structure of A requires some further efforts. We refer to [38] for more
+details and proofs of the key identities.
+From the equations of motion one infers that to each density Qn(j, t) there is a current density
+Jn(j, t) such that
+d
+dtQn(j, t) + Jn(j + 1, t) − Jn(j, t) = 0.
+Explicitly, the current densities are
+⃗J(j) = −(pj, V ′
+j−1, pjV ′
+j−1),
+(9)
+where we adopted the convention that omission of time argument t means time 0 fields. One then
+defines the static current-conserved field correlator
+Bm,n(j) = ⟨Jm(j)Qn(0)⟩c
+P,β,
+(10)
+and the corresponding susceptibility
+Bm,n =
+�
+j∈Z
+⟨Jm(j)Qn(0)⟩c
+P,β.
+Despite its asymmetric looking definition,
+Bm,n = Bn,m.
+(11)
+As a general property, Euler equations are built on thermally averaged currents. Linearizing
+them with respect to the average fields yields
+A = BC −1.
+4
+
+Here B appears when differentiating the average currents with respect to the chemical potentials
+and C −1 when switching from intensive to extensive variables. By construction C = C T and C > 0,
+in addition B = BT according to (11). Hence
+A = C 1/2C −1/2BC −1/2C −1/2,
+which ensures that A has real eigenvalues and a complete set of left-right eigenvectors. Anharmonic
+lattices are symmetric under time reversal, which implies the eigenvalues ⃗c = (−c, 0, c), with c > 0
+the isentropic speed of sound. We denote the right, resp. left eigenvectors of A by |ψα⟩ and ⟨ ˜ψα|,
+α = 0, 1, 2. With this input the solution to (8) with initial conditions (6) reads
+SLL
+m,n(x, t) = E
+�
+um(x, t)un(0, 0)
+�
+= (δ(x − At)C)m,n =
+2
+�
+α=0
+δ(x − cαt)(|ψα⟩⟨ ˜ψα|C)m,n
+with m, n = 0, 1, 2. There are three δ-peaks, the heat peak standing still and two sound peaks
+propagating in opposite directions with speed c. Specifying m, n, each peak has a signed weight
+which depends on C and the left-right eigenvectors of A.
+The Landau-Lifshitz theory asserts that the microscopic correlator
+Sm,n(j, t) ≃ SLL
+m,n(x, t)
+for j = ⌊xt⌋, ⌊·⌋ denoting integer part, with t sufficiently large.
+The reader might be disap-
+pointed by the conclusion. But with such basic information the fine-structure of the peaks can be
+investigated, in particular their specific sub-ballistic broadening and corresponding scaling func-
+tions [31,38,39].
+When turning to the Toda lattice, the conservation laws are now labeled by n = 0, 1, ... and
+thus A, B, C become infinite dimensional matrices. The corresponding Landau-Lifshitz theory has
+been worked out in [40]. As to be discussed in the following section, with appropriate adjustments
+Eq. (12) is still valid.
+3
+Toda lattice, linearized generalized hydrodynamics
+The conservation laws of the Toda lattice are obtained from a Lax matrix [11,26]. For this purpose,
+we first introduce the Flaschka variables
+aj = e−rj/2.
+Then the equations of motion become
+d
+dtaj = 1
+2aj(pj − pj+1),
+d
+dtpj = a2
+j−1 − a2
+j.
+(13)
+The Lax matrix, L, is defined by
+Lj,j = pj,
+Lj,j+1 = Lj+1,j = aj,
+j ∈ Z, and Li,j = 0 otherwise. Clearly L = LT. The conserved fields are labelled by nonnegative
+integers and have densities given by
+Q0(j) = rj,
+Qn(j) = (Ln)j,j ,
+(14)
+5
+
+with n ≥ 1. Note that Qn(j) is local in the sense that it depends only on the variables with indices
+in the interval [j − n, j + n]. An explicit expression for these quantities is given in [15]. For the
+current densities one obtains
+J0(j) = −pj,
+Jn(j) = (LnL
+↓)j,j,
+n = 1, 2, ... ,
+(15)
+where L↓ is the lower triangular part of L. Then under the Toda dynamics
+d
+dtQn(j, t) + Jn(j + 1, t) − Jn(j, t) = 0,
+which is the n-th conservation law in local form.
+The first items in the list are stretch and momentum for which our current definitions agree
+with those in (4), (9). However, for n = 2 one obtains (L2)0,0 = p2
+0 + a2
+−1 + a2
+0 and (L2L↓)0,0 =
+a2
+−1(p−1 + p0), which differs from (4), (9) on two accounts. First there is the trivial factor of 2.
+In our numerical plots we use the physical energy density ej. The second point is more subtle.
+Densities are not uniquely defined, since one can add a difference of some local function and its
+shift by one. When summing a particular choice for the density over some spatial interval, the
+result differs from another choice of the density by a boundary term only. Thus the bulk term will
+have a correction of order 1/(length of interval), which does not affect the hydrodynamic equations.
+For the currents the difference can be written as a total time derivative which is again a boundary
+term when integrating over some time interval. In this section we adopt the conventions (14) and
+(15), since the analysis heavily relies on the Lax matrix. Beyond n = 2, while the fields no longer
+have a name, they still have to be taken into account in a hydrodynamic theory.
+The infinite volume static field-field correlator is defined as in (5) and the current-field correlator
+as in (10). In particularly, B = BT. Of course, C, B are now matrices in the Hilbert space of
+sequences indexed by N0, i.e. the space ℓ2(N0). To distinguish 3 × 3 matrices from their infinite
+dimensional counterparts, for the latter we use standard italic symbols. The spacetime correlator
+of the Toda lattice is defined by
+Sm,n(j, t) = ⟨Qm(j, t)Qn(0, 0)⟩c
+P,β.
+(16)
+and we plan to construct its Landau-Litshitz approximation. In essence this amounts to an analysis
+of
+�
+eAtC
+�
+m,n,
+A = BC−1.
+(17)
+While we are mainly interested in the physical fields corresponding to the indices m, n = 0, 1, 2,
+for the operator in (17) an understanding of the infinite dimensional matrices is required.
+Starting from the basics, the free energy of the Toda lattice is given by
+Feq(P, β) = log
+�
+β/2π + P log β − log Γ(P).
+In particular, the average stretch, ν, is determined through
+ν(P, β) = ∂PFeq(P, β) = ⟨Q0(0)⟩P,β = log β − ψ(P),
+(18)
+with ψ the digamma function. Expectations of higher order fields can be written as moments of a
+probability measure denoted by νρp,
+κn = ⟨Qn(0)⟩P,β =
+�
+R
+dwνρp(w)wn,
+(19)
+6
+
+n ≥ 1. ρp is called particle density. To determine this density one first has to solve the thermody-
+namic Bethe equations (TBA). For this purpose we introduce the integral operator
+Tf(w) = 2
+�
+R
+dw′ log |w − w′|f(w′),
+w ∈ R, considered as an operator on L2(R, dw) and define the number density
+ρn(w) = e−ε(w),
+(20)
+with quasi-energies ε. The quasi-energies satisfy the TBA equation
+ε(w) = 1
+2βw2 − µ − (Te−ε)(w),
+(21)
+where the chemical potential µ has to be adjusted such that
+�
+R
+dwρn(w) = P.
+(22)
+Thereby the number density depends on the parameters P and β.
+The TBA equation is closely connected to the β-ensemble of random matrix theory. We rewrite
+(21) as
+− log ρn(w) = 1
+2αw2 − µ − αP(Tρn)(w).
+As α → ∞, the entropy term on the lefthand side can be neglected and one recognizes the defining
+equation for the Wigner semi-cirle law on the interval [−2
+√
+P, 2
+√
+P]. The Lax DOS is the P-
+derivative of ρn, which diverges as (w ± 2
+√
+P)−1/2 at the two borders. As α is lowered the borders
+become smeared to eventually cross over to a Gaussian.
+In practice, the TBA equation has to be solved numerically. But for thermal equilibrium an
+exact solution is available [1, 12, 35]. Denoting the solution of (21) for β = 1 and the constraint
+(22) by ρ∗
+n one has
+ρ∗
+n(w) =
+e−w2/2
+√
+2π| ˆfP(w)|2,
+ˆfP(w) =
+� ∞
+0
+dtfP(t)eiwt,
+fP(t) =
+√
+2π−1Γ(P)−1/2tP−1e− 1
+2 t2.
+(23)
+In our numerical simulations it is of advantage to use the exact solution.
+The TBA equation is a standard tool from GHD as one way to write the Euler-Lagrange
+equations for the variational principle associated with the generalized free energy. For the Toda
+lattice such a variational formula was obtained in [9,42]. Proofs using methods from the theory of
+large deviations and transfer operator method have also become available [16,27,29,30].
+Next we introduce the dressing transformation of some function f by
+f dr =
+�
+1 − Tρn
+�−1f
+with ρn regarded as a multiplication operator. Then number and particle density are related as
+ρn(w) =
+ρp(w)
+1 + Tρp(w)
+(24)
+with inverse
+ρp = (1 − ρnT)−1ρn = ρnςdr
+0 ,
+(25)
+7
+
+using the convention ςn(w) = wn.
+For the average currents similar identities are available.
+The central novel quantity is the
+effective velocity
+veff = ςdr
+1
+ςdr
+0
+,
+(26)
+see [3,6,41,45]. Then
+⟨J0(0)⟩P,β = −κ1,
+and, for n ≥ 1,
+⟨Jn(0)⟩P,β =
+�
+R
+dwρp(w)(veff(w) − κ1)wn.
+In thermal equilibrium we have κ1 = 0.
+Since in the following there will be many integrals over R, let us first introduce the abbreviation
+⟨f⟩ =
+�
+R
+dwf(w).
+With this notation the C matrix turns out to be of the form
+C0,0 = ν3⟨ρpςdr
+0 ςdr
+0 ⟩,
+C0,n = Cn,0 = −ν2⟨ρpςdr
+0 (ςn − κnς0)dr⟩,
+Cm,n = ν⟨ρp(ςm − κmς0)dr(ςn − κnς0)dr⟩,
+m, n ≥ 1. Note that the matrix C has the block structure
+C =
+�C0,0
+C0,n
+Cm,0
+Cm,n
+�
+,
+in the sense that Cm,n for m, n ≥ 1 follows a simple pattern. This structure will reappear for B
+and eAtC.
+The field-current correlator B can be computed in a similar fashion with the result
+B0,0 = ν2⟨ρp(veff − κ1)ςdr
+0 ςdr
+0 ⟩,
+B0,n = Bn,0 = −ν⟨ρp(veff − κ1)ςdr
+0 (ςn − κnς0)dr⟩,
+Bm,n = ⟨ρp(veff − κ1)(ςm − κmς0)dr(ςn − κnς0)dr⟩.
+As in (12), we want to determine the propagator of the Landau-Lifshitz theory, denoted by
+SLL
+m,n(x, t). In principle, all pieces have been assembled. However to figure out the exponential of A
+requires its diagonalization. Details can be found in [40] and we only mention that one constructs
+a linear similarity transformation, R, such that R−1AR is multiplication by
+ν−1(veff(w) − κ1)
+(30)
+in L2(R, dw). Here veff is the effective velocity defined in (26). Using the block convention as in
+(28), the spacetime correlator in the Landau-Lifshitz approximation is given by
+SLL(x, t) =
+�
+R
+dwδ
+�
+x − tν−1(veff(w) − κ1)
+�
+νρp(w)
+×
+�
+ν2ςdr
+0 (w)2
+νςdr
+0 (w)(ςn − κnς0)dr(w)
+νςdr
+0 (w)(ςm − κmς0)dr(w)
+(ςm − κmς0)dr(w)(ςn − κnς0)dr(w)
+�
+.
+8
+
+Note that SLL(x, 0) = δ(x)C. As a property of the Euler equations, the expression (31) possesses
+exact ballistic scaling,
+SLL
+m,n(x, t) = 1
+t SLL
+m,n(x/t, 1).
+(32)
+The correlator Sm,n(j, t) is computed in our MD simulations which will then be compared with
+SLL
+m,n(x, t).
+4
+Numerical simulations
+For a molecular dynamics simulation one has to first specify a finite ring [1, . . . , N] with suitable
+boundary conditions. For the dynamics of positions qj and momenta pj one imposes
+qN+1 = q1 + νN.
+(33)
+The parameter ν fixes the free volume per particle and can have either sign. In our simulation,
+we actually allow for a fluctuating free volume by choosing random initial conditions such that
+{r1, p1, . . . , rN, pN} are i.i.d. random variables with a single site distribution as specified in (3).
+Then the deterministic time evolution is governed by (13) with boundary conditions
+r0 = rN,
+pN+1 = p1.
+In fact, the boundary condition in (33) amounts to the micro-canonical constraint
+N
+�
+j=1
+rj = νN.
+If one sets ν = ⟨Q0(0)⟩P,β, then for large N, by the equivalence of ensembles, the two schemes
+for sampling the correlator Sm,n(j, t) should differ by the size of statistical fluctuations. For a
+few representative examples we checked that indeed the equivalence of ensembles holds for the
+particular observables under study.
+Returning to the choice of system size there is an important physical constraint. In all sim-
+ulations one observes a sharp right and left front, which travel with constant speed and beyond
+which spatial correlations are exponentially small. On a ring necessarily the two fronts will collide
+after some time. Such an encounter has a noticeable effect on the molecular dynamics which is not
+captured by the linearized GHD analysis. Therefore the simulation time is limited by the time of
+first collision. Indeed, we note in Figures 1-3 that both linearized GHD and MD clearly display
+maximal speeds of at most ∆j/∆t = 2 for the entire range of (P, β, m, n) displayed in these figures.
+Taking into account that the initial correlations are proportional to δ0j, we conclude that for a
+ring of size N = 3000 there will be no collision of the two fronts up to time t = 750 which is larger
+than t = 600 used in our simulations.
+Before displaying and discussing our results, we provide more details on numerically solving
+the TBA equations and on the actual scheme used for MD.
+4.1
+Details of the numerical implementation
+4.1.1
+Solving linearized GHD
+To numerically solve the linearized GHD equations, we use a numerical method similar to the one
+from [33]. First, Eq. (23) can be expressed in terms of the parabolic cylinder function Dν(z), which
+is readily available in Mathematica. This provides the solution to the TBA equations (21), (22).
+9
+
+Then, we use a simple finite element discretization of the w-dependent functions by hat func-
+tions, resulting in piecewise linear functions on a uniform grid. After precomputing the integral
+operator T in (20) for such hat functions, the dressing transformation (24) becomes a linear sys-
+tem of equations, which can be solved numerically. This procedure yields ςdr
+n , and subsequently
+ρp via (25) and veff via (26). The moments can be computed from κn =
+�
+R dwνρn(w)ςdr
+n (w), or
+(equivalently) Eq. (19).
+To evaluate the correlator in (31), we note that the delta-function in the integrand results in a
+parametrized curve, with the first coordinate (corresponding to x/t) equal to ˜veff from (30), and
+the second coordinate equal to the remaining terms in the integrand divided by the Jacobi factor
+| d
+dw ˜veff(w)| resulting from the delta-function.
+4.1.2
+Molecular dynamics simulations
+We approximate the expectation value that is contained in the MD-definition of the correlations
+Sm,n in equation (16) by the following numerical scheme, whose implementation program is written
+in Python, and can be found at [28]. First, we generate the random initial conditions distributed
+according to the Gibbs measure, as given by (3) for the i.i.d. random variables (rj, pj)1≤j≤N.
+Specifically, the variables pj are distributed according to a standard normal random variable, that
+we generate with Numpy v1.23’s native function random.default rng().normal [18], times 1/√β.
+It takes a brief calculation to see that rj can be chosen to be − ln(X/(2β)) where X is chi-square
+distributed with shape parameter 2P. We obtain the random variable X using Numpy v1.23’s
+native function random.default rng().chisquare. Having chosen the initial conditions in such
+a manner, we solve equation (2).
+For the evolution, we adapt the classical St¨ormer–Verlet algorithm [17] of order 2 to work with
+the variables (p, r). Specifically, we used a time step equal to δ = 0.05, and, given the solution
+(r(t), p(t)) at time t, we approximate the solution at time t + δ through the following scheme,
+pj
+�
+t + δ
+2
+�
+= pj(t) − δ
+2
+�
+e−rj(t) − erj−1(t)�
+,
+rj(t + δ) = rj(t) + δ
+�
+pj+1
+�
+t + δ
+2
+�
+− pj
+�
+t + δ
+2
+��
+,
+pj(t + δ) = pj
+�
+t + δ
+2
+�
+− δ
+2
+�
+e−rj(t+δ) − erj−1(t+δ)�
+,
+for all j = 1, . . . , N. In this part of the implementation, we extensively used the library Numba [23]
+to speed up the computations.
+Our approximation for the expectation Sm,n is then extracted from 3×106 trials with indepen-
+dent initial conditions. Here we take the empirical mean of all trials where for each trial we also
+take the mean of the N = 3000 sets of data that are generated by choosing each site on the ring
+for j = 0.
+To evaluate the quality of our numerical simulations, we have repeated the numerical experi-
+ments up to five times including variations for the length of the ring and evaluating the solutions at
+more intermediate time steps than displayed in the figures below. Furthermore, we have compared
+the results with the corresponding outcomes obtained by a MATLAB program that has been devel-
+oped independently from the Python program, and that follows a different numerical scheme. It
+uses MATLAB’s random number generators randn for initial momenta and rand combined with the
+10
+
+rejection method to produce initial stretches. The dynamics is then evaluated by the solver ode45,
+which exploits the Runge–Kutta method to numerically solve the Hamiltonian system associated
+with (1) on the ring. We found that the deviations between different experiments are comparable
+to the size of the amplitudes of the high frequency oscillations that are present in figures 4-5. These
+oscillations are due to the random fluctuations of the empirical means around their expectation
+values Sm,n. Agreement of different experiments up to the order of these oscillations therefore
+shows the consistency of the corresponding numerical results.
+We also want to mention that all the pictures that appeared in this paper are made using the
+library matplotlib [19].
+4.2
+Comparison of linearized GHD with MD at time t = 600
+We compare the GHD predictions with MD simulations for three different temperatures that
+correspond to β = 0.5 (Fig. 1), β = 1 (Fig. 2), and β = 2 (Fig. 3). For each β we choose three
+different values for the pressure parameter P in such a way that the corresponding mean stretches,
+given by (18), are positive (≈ 2.57) for low pressure, negative (≈ −0.42) for high pressure and
+approximately zero for medium pressure. We summarize their values in Table 1.
+pressure
+β = 0.5
+β = 1
+β = 2
+low
+P = 0.32, ⟨r⟩ ≈ +2.58
+P = 0.4, ⟨r⟩ ≈ +2.56
+P = 0.52, ⟨r⟩ ≈ +2.56
+medium
+P = 0.95, ⟨r⟩ ≈ −0.03
+P = 1.5, ⟨r⟩ ≈ −0.04
+P = 2.55, ⟨r⟩ ≈ −0.03
+high
+P = 1.21, ⟨r⟩ ≈ −0.42
+P = 2.0, ⟨r⟩ ≈ −0.42
+P = 3.53, ⟨r⟩ ≈ −0.42
+Table 1: Values for β and P and the corresponding mean stretches used in experiments
+In each of the nine cases we have evaluated the Landau-Lifshitz approximations SLL
+m,n(·, 1), see
+(31), of the correlators for all 0 ≤ n ≤ m ≤ 2 using the numerical scheme described in Section 4.1.1.
+Their graphs are displayed in Figures 1-3 as dashed lines. Note that the speeds of the sound peaks
+depend significantly on both pressure and temperature. Moreover, the predicted fine-structure of
+both the heat and the sound peaks are quite different for low pressure when compared to medium
+and high pressure.
+The colored lines in Figures 1-3 show our numerical results for the corresponding molecular
+dynamics. According to the predicted ballistic scaling (32) we plot tSm,n(j, t) as a function of
+j/t for t = 600. Here the values of Sm,n(j, t) are approximated using the numerics explained in
+Section 4.1.2.
+The agreement between linearized GHD and MD is striking, in particular since there are no
+adjustable parameters. In all of the 54 comparisons shown in Figures 1-3 the GHD predictions
+for the fine-structure of heat and sound peaks are in excellent agreement with the ones observed
+from molecular dynamics at time t = 600. As we show in more detail in the next subsection
+the largest deviations occur mostly near the sound peaks and do not exceed 3.5% of the peaks’
+maximal values.
+4.3
+Deviation of linearized GHD from MD at times t = 150 and t = 600
+The purpose of this subsection is twofold. On the one hand we have a look at the small differences
+between GHD predictions and molecular dynamics simulations that can hardly be detected in
+11
+
+1.0
+0.5
+0.0
+0.5
+1.0
+0
+2
+4
+6
+8
+S00, S11, S22,
+= 0.5, P = 0.32
+S00
+S11
+S22
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+0
+1
+2
+3
+4
+5
+6
+7
+S00, S11, S22,
+= 0.5, P = 0.95
+S00
+S11
+S22
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+0
+2
+4
+6
+8
+S00, S11, S22,
+= 0.5, P = 1.21
+S00
+S11
+S22
+1.0
+0.5
+0.0
+0.5
+1.0
+4
+2
+0
+2
+4
+S21, S20, S10,
+= 0.5, P = 0.32
+S21
+S20
+S10
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+4
+3
+2
+1
+0
+1
+2
+3
+4
+S21, S20, S10,
+= 0.5, P = 0.95
+S21
+S20
+S10
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+4
+3
+2
+1
+0
+1
+2
+3
+4
+S21, S20, S10,
+= 0.5, P = 1.21
+S21
+S20
+S10
+Figure 1: Toda correlation functions: GHD predictions y �→ SLL
+m,n(y, 1) vs. numerical simulations
+of the molecular dynamics y �→ tSm,n(yt, t) at t = 600 for β = 0.5 with low pressure (top), medium
+pressure (middle) and high pressure (bottom). Numerical simulations are colored according to
+the legend, the corresponding GHD predictions are displayed by dashed lines. Number of trials:
+3 × 106.
+12
+
+1.00
+0.75
+0.50
+0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+0
+2
+4
+6
+8
+S00, S11, S22,
+= 1.0, P = 0.40
+S00
+S11
+S22
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+S00, S11, S22,
+= 1.0, P = 1.50
+S00
+S11
+S22
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+0
+1
+2
+3
+4
+S00, S11, S22,
+= 1.0, P = 2.00
+S00
+S11
+S22
+1.00
+0.75
+0.50
+0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+3
+2
+1
+0
+1
+2
+3
+S21, S20, S10,
+= 1.0, P = 0.40
+S21
+S20
+S10
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2
+1
+0
+1
+2
+S21, S20, S10,
+= 1.0, P = 1.50
+S21
+S20
+S10
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+2
+1
+0
+1
+2
+S21, S20, S10,
+= 1.0, P = 2.00
+S21
+S20
+S10
+Figure 2: Toda correlation functions: GHD predictions y �→ SLL
+m,n(y, 1) vs. numerical simulations
+of the molecular dynamics y �→ tSm,n(yt, t) at t = 600 for β = 1.0 with low pressure (top), medium
+pressure (middle) and high pressure (bottom). Numerical simulations are colored according to
+the legend, the corresponding GHD predictions are displayed by dashed lines. Number of trials:
+3 × 106.
+13
+
+1.0
+0.5
+0.0
+0.5
+1.0
+0
+1
+2
+3
+4
+5
+6
+7
+S00, S11, S22,
+= 2.0, P = 0.52
+S00
+S11
+S22
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+S00, S11, S22,
+= 2.0, P = 2.55
+S00
+S11
+S22
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+S00, S11, S22,
+= 2.0, P = 3.53
+S00
+S11
+S22
+1.0
+0.5
+0.0
+0.5
+1.0
+2
+1
+0
+1
+2
+S21, S20, S10,
+= 2.0, P = 0.52
+S21
+S20
+S10
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+S21, S20, S10,
+= 2.0, P = 2.55
+S21
+S20
+S10
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+S21, S20, S10,
+= 2.0, P = 3.53
+S21
+S20
+S10
+Figure 3: Toda correlation functions: GHD predictions y �→ SLL
+m,n(y, 1) vs. numerical simulations
+of the molecular dynamics y �→ tSm,n(yt, t) at t = 600 for β = 2.0 with low pressure (top), medium
+pressure (middle) and high pressure (bottom). Numerical simulations are colored according to
+the legend, the corresponding GHD predictions are displayed by dashed lines. Number of trials:
+3 × 106.
+14
+
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+0.00
+0.25
+0.50
+0.75
+1.00
+t: 150
+t: 600
+GHD
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+0.02
+0.00
+0.02
+= 2.00, P = 2.55,  S1, 1
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+t: 150
+t: 600
+GHD
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+0.02
+0.00
+0.02
+0.04
+= 2.00, P = 2.55,  S1, 0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+0.0
+0.5
+1.0
+1.5
+t: 150
+t: 600
+GHD
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+0.02
+0.00
+0.02
+= 1.00, P = 1.50,  S1, 1
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+1
+0
+1
+t: 150
+t: 600
+GHD
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+0.02
+0.00
+0.02
+0.04
+= 1.00, P = 1.50,  S1, 0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+0.0
+0.5
+1.0
+1.5
+2.0
+t: 150
+t: 600
+GHD
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+0.05
+0.00
+0.05
+= 0.50, P = 0.95,  S1, 1
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+1
+0
+1
+t: 150
+t: 600
+GHD
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+0.050
+0.025
+0.000
+0.025
+0.050
+= 0.50, P = 0.95,  S1, 0
+Figure 4: Toda correlation functions S1,1 (left) and S1,0 (right) for medium pressure and increasing
+temperatures (top to bottom). For each value of β and P the top panels show GHD prediction
+vs. numerical simulations as in Figures 1-3 but with the the molecular dynamics evaluated at two
+times t = 150 and t = 600. The bottom panels display the differences between the GHD prediction
+and numerical simulations at time t = 150 (red) and at time t = 600 (green). Number of trials:
+3 × 106.
+15
+
+1.00
+0.75
+0.50
+0.25 0.00
+0.25
+0.50
+0.75
+1.00
+0
+2
+4
+6
+8
+t: 150
+t: 600
+GHD
+1.00
+0.75
+0.50
+0.25 0.00
+0.25
+0.50
+0.75
+1.00
+0.1
+0.0
+0.1
+0.2
+= 1.00, P = 0.40,  S0, 0
+1.00
+0.75
+0.50
+0.25 0.00
+0.25
+0.50
+0.75
+1.00
+3
+2
+1
+0
+1
+t: 150
+t: 600
+GHD
+1.00
+0.75
+0.50
+0.25 0.00
+0.25
+0.50
+0.75
+1.00
+0.1
+0.0
+0.1
+= 1.00, P = 0.40,  S2, 0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+0.00
+0.25
+0.50
+0.75
+1.00
+t: 150
+t: 600
+GHD
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+0.02
+0.00
+0.02
+= 1.00, P = 1.50,  S0, 0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+1.5
+1.0
+0.5
+0.0
+t: 150
+t: 600
+GHD
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+0.050
+0.025
+0.000
+0.025
+0.050
+= 1.00, P = 1.50,  S2, 0
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+t: 150
+t: 600
+GHD
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+0.02
+0.01
+0.00
+0.01
+0.02
+= 1.00, P = 2.00,  S0, 0
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+1.5
+1.0
+0.5
+0.0
+t: 150
+t: 600
+GHD
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+0.050
+0.025
+0.000
+0.025
+0.050
+= 1.00, P = 2.00,  S2, 0
+Figure 5: Toda correlation functions S0,0 (left) and S2,0 (right) for β = 1 and increasing pressure
+(top to bottom). For each value of β and P the top panels show GHD prediction vs. numerical
+simulations as in Figure 2 but with the the molecular dynamics evaluated at two times t = 150 and
+t = 600. The bottom panels display the differences between the GHD prediction and numerical
+simulations at time t = 150 (red) and at time t = 600 (green).Number of trials: 3 × 106.
+16
+
+Figures 1-3. On the other hand we indicate how these differences evolve in time by including time
+t = 150 for the molecular dynamics. Recall that the GHD predictions are time-invariant in the
+scaling y �→ tSm,n(yt, t) we have chosen, see (32).
+From the 54 comparisons that are displayed in Figures 1-3 we select 12 cases that are repre-
+sentative and show all the phenomena that we have observed. In Figure 4 we consider correlations
+S1,1 and S1,0 at medium pressure (cf. Table 1) for all three values of β. The small scale fluctuations
+displayed in the bottom panels are due to the approximation of expectation values by empirical
+averages. Their amplitudes become smaller if one increases the number of trials. Note that the
+difference in amplitudes of these fluctions between t = 150 and t = 600 is mostly due to the scaling
+y �→ tSm,n(yt, t) that we use. This implies that the values of the correlations are multiplied by a
+factor that is 4 times larger at the later time. The same holds for the plots in Figure 5 where the
+correlations S0,0 and S2,0 are shown for fixed β = 1 and our three different choices for pressure.
+We now summarize our main findings:
+1. The deviations occur mostly near the sound peaks and amount to 1.5%-3.5% of the peaks’
+maximal values at time t = 600.
+2. There appear to be small but systematic deviations concerning the shape of the sound peak
+in all cases. One would need to conduct experiments with a higher resolution, i.e. more sites
+and consequently larger times and more trials, to determine whether there is indeed such
+a systematic deviation. With the resolution present in our experiments the question of a
+systematic deviation with respect to the shape of the peak cannot be decided.
+3. In some of the experiments the maximal deviations would be significantly smaller if a constant
+only depending on the values of β, P, m, n is added to all values of Sm,n(j, t), see e.g.
+correlations S0,0 and S2,0 for β = 1, P = 0.4 in Figure 5. This seems to be related to the
+approximation errors for the means ⟨r⟩, ⟨p⟩, and ⟨e⟩, that appear to be less pronounced in
+the case of momentum p. We have observed that these deviations decrease as the number
+of trials is increased and we do not expect a systematic deviation between GHD and MD in
+this respect.
+4. For (β; P) ∈ {(0.5; 0.95), (0.5; 1.21)} we observe that the size of the deviations is essentially
+the same for times t = 150 and t = 600 whereas for (β; P) ∈ {(0.5; 0.32), (1; 0.4), (2; 0.52),
+(2; 2.55), (2; 3.53)} these deviations are significantly larger at the smaller time. The remain-
+ing two cases (β; P) ∈ {(1; 1.5), (1; 2)} are somewhat in between, also depending on the
+correlation function that is considered, see Figure 5. This is an indication that the speed
+of convergence of tSm,n(yt, t) to the GHD prediction SLL
+m,n(y, 1) as t → ∞ depends on the
+values of β and P. As a rule we have observed that both increasing temperature or increasing
+pressure leads to a faster speed of convergence.
+5
+Conclusions and outlook
+As can be seen from Table 1, we picked the intermediate pressure such that ν ≃ 0. In the particle
+picture ν = 0 corresponds to the boundary condition q1 = qN. In thermal equilibrium the positions
+then perform an unbiased random walk with typical excursions of order
+√
+N. Thus the free volume
+is of order 1/
+√
+N. The particles are extremely dense and the picture of successive pair collisions
+breaks down completely. So one might wonder whether GHD is still valid under such extreme
+conditions. ν = 0 poses no particular difficulties for MD simulations. In GHD the factor 1/ν
+17
+
+appears in the expression for veff, see Eq. (31). This makes the numerical scheme slow and only
+values close to ν = 0 are accessible. However the correlator S changes smoothly through ν = 0.
+GHD also covers this seemingly singular point.
+Simultaneously A. Kundu [21] posted a somewhat puzzling note. He considers the parameter
+values β = 1, P = 1. When cutting the matrices Cm,n and Am,n at low orders, the resulting
+Sm,n consists of a few δ-peaks which move at constant velocity. After ballistic scaling, with high
+precission they turn out to lie on the curve obtained from GHD. A theoretical explanations seems
+to be missing.
+In [22] the molecular dynamics of Toda lattice correlations are simulated for the potential
+Vkd(x) = g
+γ e−γx
+with arbitrary γ, g > 0. To distinguish their parameters from ours, the variables in [22] are here
+denoted by ¯t, ¯r, ¯P, ¯β.
+¯P is the physical pressure and, comparing the Gibbs weights, one obtains
+the relations
+β = g
+γ
+¯β,
+P = 1
+γ
+¯P ¯β.
+From the equations of motions one deduces
+¯t =
+1
+√γgt,
+r(t) = γ¯r(¯t),
+p(t) = g
+γ ¯p(¯t).
+Thus, translating to our units, the MD simulations reported in [22] are (i) P = 0.01, β = 0.01,
+N = 1024, t = 400, (ii) P = 1, β = 1, N = 1024, t = 200, 300, and (iii) P = 400, β = 400,
+N = 256, t = 80. In fact, in all three cases the time scales are identical, t = ¯t. Since GHD was not
+available yet, no comparison could have been attempted.
+Case (i) is a very dilute chain. In this limit νρp is a unit Gaussian. The dressed functions
+become polynomials as ςdr
+0 (w) = a0, ςdr
+1 (w) = a1w, and ςdr
+2 (w) = a2w2 + a3 with coefficients
+a0, ..., a3 depending on (P, β). Note that for a noninteracting fluid a3 would vanish. As a result
+S0,0 is Gaussian, S1,1 has two peaks, and S2,2 has either two or three peaks.
+This is in good
+agreement with [22] and explains our motivation not to venture into the low density regime. Case
+(ii) interpolates between our β = 1, P = 0.40 and β = 1, P = 1.5. Note that now S0,0 has a local
+minimum at w = 0, which is very different from the structure in the dilute regime. On the other
+hand, S2,2 has a local maximum at w = 0, as is the case for low density/high temperature.
+The most interesting parameter value is (iii), which deserves more detailed studies. The issue
+is the behavior of the Toda chain at very low temperatures. Simply letting β → ∞ will freeze
+any motion. But the simultaneous limit β → ∞ with P = ¯Pβ at fixed physical pressure ¯P is
+meaningful, at least statistically. In this limit ν > 0 always. Also the density of states converges
+to the arcsine distribution,
+lim
+β→∞ νρp(w) =
+1
+π
+√
+4 ¯P − w2,
+|w| ≤ 2
+�
+¯P.
+To understand the dynamical behavior, the effective potential is expanded as
+e−r + ¯Pr ≃ 1
+2 ¯P(r − r0)2 + c0
+at its minimum r0. Since β is large, the initial fluctuations are of order 1/√β. Therefore the dy-
+namics can be approximated by a harmonic chain with ω2 = ¯P. The equilibrium time correlations
+18
+
+of the harmonic chain have intricate oscillatory behavior [14], which in the large β limit should
+match with the Toda lattice, as partially evidenced through case (iii). Clearly, GHD cannot repro-
+duce such fine details. Still, when averaged on suitable scales, the gross behavior of the harmonic
+chain oscillations might be visible.
+Acknowledgements
+This material is based upon work supported by the National Science Foundation under Grant
+No. 1440140, while five of the authors were in residence at the Mathematical Sciences Research
+Institute in Berkeley, California, during the fall semester of 2021.
+The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cam-
+bridge, for support and hospitality during the programme “Dispersive hydrodynamics: mathemat-
+ics, simulation and experiments, with applications in nonlinear waves” where some work on this
+paper was undertaken. This work was supported by EPSRC grant no EP/R014604/1. TG ac-
+knowledges the support of the European Union’s H2020 Marie Sk�lodowska–Curie grant No. 778010
+IPaDEGAN, of INdAM/GNFM and of the research project Mathematical Methods in NonLinear
+Physics (MMNLP), Gruppo 4-Fisica Teorica of INFN. GM is financed by the KAM grant number
+2018.0344. KTRM was supported by a Visiting Wolfson research fellowship from the Royal Society.
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+

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+Understanding the Effectiveness of Very Large
+Language Models on Dialog Evaluation
+Jessica Huynh, Cathy Jiao, Prakhar Gupta, Shikib Mehri, Payal Bajaj, Vishrav
+Chaudhary, Maxine Eskenazi
+Abstract Language models have steadily increased in size over the past few years.
+They achieve a high level of performance on various natural language processing
+(NLP) tasks such as question answering and summarization. Large language models
+(LLMs) have been used for generation and can now output human-like text. Due to
+this, there are other downstream tasks in the realm of dialog that can now harness the
+LLMs’ language understanding capabilities. Dialog evaluation is one task that this
+paper will explore. It concentrates on prompting with LLMs: BLOOM, OPT, GPT-
+3, Flan-T5, InstructDial and TNLGv2. The paper shows that the choice of datasets
+used for training a model contributes to how well it performs on a task as well as on
+how the prompt should be structured. Specifically, the more diverse and relevant the
+group of datasets that a model is trained on, the better dialog evaluation performs.
+This paper also investigates how the number of examples in the prompt and the type
+of example selection used affect the model’s performance.
+Jessica Huynh
+Carnegie Mellon University, e-mail: jhuynh@cs.cmu.edu
+Cathy Jiao
+Carnegie Mellon University, e-mail: cljiao@cs.cmu.edu
+Prakhar Gupta
+Carnegie Mellon University, e-mail: prakharg@cs.cmu.edu
+Shikib Mehri
+Amazon, e-mail: asmehri@amazon.com (work done while at Carnegie Mellon University)
+Payal Bajaj
+Microsoft Turing, e-mail: pabajaj@microsoft.com
+Vishrav Chaudhary
+Microsoft Turing, e-mail: vchaudhary@microsoft.com
+Maxine Eskenazi
+Carnegie Mellon University, e-mail: max@cs.cmu.edu
+1
+arXiv:2301.12004v1  [cs.CL]  27 Jan 2023
+
+2
+J. Huynh et al.
+1 Introduction
+In recent years, language models such as GPT-3 [5] have grown larger, and their per-
+formance on downstream natural language processing (NLP) tasks has significantly
+improved in low-resource settings where only a few instances per task are available
+(few-shot). The larger these models are, the higher their performances trend on tasks
+such as language generation and evaluation [39]. They can generate coherent, fluent
+and interesting responses. However, they can also produce responses that are repet-
+itive and un-engaging [29], in addition to being hard to control. Dialog evaluation is
+the task of assessing the quality of responses generated by dialog models in terms
+of properties like those mentioned above. However, one significant impediment for
+open-domain dialog generation research is the lack of meaningful automatic metrics
+for open-domain dialog evaluation. Standard language generation metrics have been
+shown to be ineffective for dialog evaluation [11], a large part of which is because
+conversations can be followed by multiple valid responses. Standard automatic met-
+rics (e.g. BLEU [24]), which use references for evaluation, cannot deal with this
+quality, known as the one-to-many response problem. Many recently introduced au-
+tomatic metrics for dialog evaluation [21, 12] have attained increasingly stronger
+correlations with human judgment. Since human dialog evaluation typically mea-
+sures multiple fine-grained properties (e.g. appropriate, interesting, consistent), au-
+tomatic evaluation metrics should be expected to do the same. This paper explores
+several fine-grained metrics that are measured both at turn-level (i.e. relevance and
+fluency), and dialog-level (i.e. consistency and coherence).
+Automatic dialog evaluation continues to be an evolving topic, but with fine-
+grained metrics and definitions varying across different human-annotated datasets
+[22, 46], it is important to be able to create reasonable automatic metrics with lim-
+ited data. Large language models (LLMs) that have been pre-trained on large-scale
+datasets are able to perform zero and few-shot inference [26, 32], and they have ex-
+hibited good reasoning skills [5, 39] in addition to having implicitly learned some
+notion of dialog quality [21]. This makes them suitable for open-domain dialog
+evaluation in zero-shot and extreme few-shot settings. While there have been a few
+attempts to use LLMs for dialog evaluation [36], there has not, to our knowledge,
+been a systematic study of LLMs for this task. This paper explores several aspects
+of LLM use in dialog evaluation: the effect of model type and size and the choice
+of training data as well as the use of in-context examples for dialog evaluation (the
+number and quality of the examples used). The experiments herein employ bench-
+marks to test both how well LLMs can be used for fine-grained evaluation, and how
+generalizable the models’ performance is across multiple domains and datasets.
+
+Very Large Language Models for Dialog Evaluation
+3
+2 Related Work
+2.1 LLMs
+Several LLMs have been released recently: T5 [27], GPT-3 [5], BLOOM [4], OPT
+[42], and TNLGv2 [34]. The following models, the sizes of which are shown in
+Figure 1, are explored here:
+•
+T5, trained on the 750B Colossal Clean Crawled Corpus (C4) contains heuristi-
+cally cleaned natural language English text from the web. Specific models con-
+sidered are:
+– Flan-T5 [8], T5 fine-tuned on 1836 tasks, including dialog tasks and data.
+– InstructDial [13], T5 fine-tuned specifically on 48 dialog tasks.
+•
+GPT-3 includes a 570B filtered CommonCrawl corpus [27] in addition to Web-
+Text [26], Books1, Books2, and Wikipedia [16].
+– InstructGPT (text-davinci-002) [23], GPT-3 fine-tuned with a prompting
+dataset and 175B parameters.
+•
+BLOOM was trained on 46 languages and 13 programming languages with a
+multilingual focus.
+•
+OPT contains data from the RoBERTa corpus [18], the Pile [9], and PushShift.io
+Reddit [2, 29].
+•
+TNLGv2 is trained on a subset of the Pile (notably excluding corpora classified
+as having natural dialog), two CommonCrawl snapshots [27], RealNews [40],
+and CC-Stories [37].
+Fig. 1 Large Language Models, comparison of select approximate sizes
+
+530B
+I size
+Model
+175B
+30B
+7B
+！
+0.5B
+BLOOM
+OPT
+InstructGPT Flan-T5 InstructDial TNLGv2
+LLM
+Fine-tuned LLM
+LLM4
+J. Huynh et al.
+As the number of parameters in these models increases, performance also in-
+creases: TNLGv2 530B, with around three times the number of parameters, out-
+performs the original GPT-3 on a variety of NLP tasks [34]. LLMs are also gener-
+alizable; they perform well on many NLP tasks in few-shot settings and zero-shot
+settings [38, 32]. However, several drawbacks and areas for exploration remain for
+LLMs that should be noted. Recent work has shown that performance on certain
+zero-shot tasks plateaus as model parameter size grows exponentially [5]. LLMs
+also struggle with parsing social situations [33] and correctly using context [1],
+which are important in dialog settings. This raises questions on the performance of
+LLMs for dialog evaluation, and how an LLM’s performance changes as it increases
+in size.
+The data that a model is trained on also influences the performance of down-
+stream tasks. T5 is fine-tuned on various subtasks, but pre-trained with C4. When
+pre-trained with domain-specific data, T5 performs better on tasks in that domain
+[3, 27]. Furthermore, adding several domains of data during pre-training makes the
+model likely to perform better [18, 42, 7]. Notably, BLOOM, OPT, Flan-T5, In-
+structGPT, and InstructDial are partially trained or fine-tuned on dialog datasets.
+Details on the content of these datasets can be found in Appendix A. This is impor-
+tant because natural dialog data is difficult to obtain, so either scripted conversations
+or Reddit threads are used since they are the most readily available. This dearth of
+data is the reason that few-shot prompting is of interest. While work such as [39]
+acknowledges emergent abilities in larger language models in few-shot prompting
+settings, this paper explores discrepancies in performance specifically for dialog
+evaluation.
+2.2 Dialog Evaluation
+Dialog evaluation presents a unique combination of challenges; it must consider
+multiple speakers [44], context that informs the current dialog turn, and the one-to-
+many aspect mentioned above [45].
+Metrics such as USR [22] and FED [21] were created to address some of
+these challenges; they are reference-free, capture complex aspects of dialog, and
+have good correlation with human evaluation. These metrics use models such as
+RoBERTa (125 million parameters) [18] and DialoGPT (345 or 762 million param-
+eters) [43] respectively. However, the best performing versions of these models are
+smaller than most models examined in this paper, and are fine-tuned on dialog data
+or on a specific dialog task. Other automatic evaluation metrics include GRADE
+[14] and DEB [31]. With current LLMs’ large increase in hyperparameters, their
+plethora of training data, and their promising generalizable performance on NLP
+tasks, these model-based metrics should improve as well.
+
+Very Large Language Models for Dialog Evaluation
+5
+2.3 Example selection for few-shot learning
+The example selection process for prompting LLMs is of great interest. Prompting
+an LLM with a task and a few examples enables the model to adapt to a new task
+without completely fine-tuning it. In particular, in-context examples can provide im-
+portant cues to help LLMs make predictions on tasks. Recent work has used a vari-
+ety of methods to examine example selection. Common methods measure semantic
+similarity between example embeddings [17, 35]. Alternatively, retrieval methods
+(e.g. BM25 [28]) have been used directly, or as a precursor to training a selection
+retriever [30].
+These example selection methods have shown promise in few-shot NLP tasks.
+In [35], the two-step framework for annotating and selecting in-context examples
+from large unlabeled data showed competitive performance across 10 tasks such as
+classification, commonsense reasoning, dialog state tracking, and code generation.
+[17] showed that selecting examples with similar sentence embeddings yields higher
+GPT-3 performance than random selection. However, the authors acknowledge that
+further investigation is required to find more efficient in-context example retrieval
+methods.
+Moreover, the wording and order of examples presented in prompts can also
+affect model performance [10, 17, 15]. Lu et al [19] observed order sensitivity across
+0.1B to 175B parameter GPT-2 and GPT-3 models when the models were probed
+with different text classification tasks and several in-context examples. Also, the
+wording of the in-context examples depends on the data used for model training;
+for unfamiliar prompt formats, model performance may decrease [15]. Increasing
+the size of the model and the amount of data does not resolve the issue since the
+same instability is still prevalent [47]. Thus this paper studies the effect of example
+selection on dialog evaluation.
+3 Evaluation Settings
+Two settings for dialog evaluation are explored: fine-grained evaluation and multi-
+domain evaluation. In-context examples are explored in both.
+3.1 Fine-Grained Evaluation
+Fine-grained metrics can be measured at both the turn level (e.g. informativeness
+and relevance), and the dialog level (e.g. coherence and diversity). The FED dataset
+[21] is used. It consists of 124 open-domain dialogs of humans with humans or
+with machines, for which each dialog has 3 responses that are chosen for annotation
+(8 turn-level and 10 dialog-level qualities along with overall turn- and dialog-level
+quality). This dataset was chosen due to the large number of previously studied fine-
+
+6
+J. Huynh et al.
+grained qualities as listed in Section 4.1, with the exception of correctness and error
+recovery, which are only specifically present in FED.
+In the experiments, the LM is prompted to output a rating (an integer value - see
+Appendix B) to evaluate each fine-grained quality in a response. The final rating
+for each fine-grained quality is a weighted sum of the K-top ratings outputted from
+the LM. Formally, given the K-top predicted ratings r1,r2,...,rK along with their
+corresponding log probabilities, p1,..., pK, the weight, wi, of each rating ri is derived
+as:
+wi =
+pi
+∑K
+j=1 pj
+The final rating, r, is calculated as:
+r =
+K
+∑
+i=1
+ri ∗wi
+In order to provide a more accurate view of the LM’s performance, K = 3 in the
+following experiments. Additionally, this scoring mechanism converts the LM pre-
+dictions onto a continuous scale, which more closely mirrors the average of human
+ratings. Results are reported with the Spearman correlations to the average human
+ratings for each fine-grained quality.
+3.2 Multi-domain Evaluation
+This task tests automatic dialog evaluation metrics for robustness across multiple
+dialog domains. The analysis uses only the overall quality metric since many of
+the domain datasets do not have fine-grained annotations. The Spearman correla-
+tion is used between human ratings and model predictions on the evaluation sets
+released by DSTC 10 Track 5 [6] “Automatic Evaluation and Moderation of Open-
+domain Dialogue Systems”. These sets contain human judgement ratings for dialog
+responses. In this setting, a model is shown a dialog context and a response, and it
+outputs “yes” if the response is a good response to that context, otherwise it outputs
+“no”. An example can be seen in Appendix C. The probability of the “goodness” of
+the response (i.e., the rating), g, is calculated as:
+g =
+pmodel(yes)
+pmodel(yes)+ pmodel(no)
+where pmodel(yes) and pmodel(no) are the log probilities of the model outputs
+for “yes” and “no”. Evaluation is carried out on 8 representative evaluation sets out
+of the 14 DSTC10 evaluation sets [6]. This subset was chosen because it covers
+multiple domains and datasets, such as persona, topic and chitchat-based responses.
+A robust dialog metric should perform well across all the domains and evaluation
+sets considered.
+
+Very Large Language Models for Dialog Evaluation
+7
+The evaluation sets used for fine-grained evaluation, FED-Turn (FT) and FED-
+Dial (FD) [21], are included as two of the eight datasets. The other datasets include:
+TopicalChat-USR (TU, knowledge-grounded open-domain conversations rated for
+six different dialog qualities) [22]; PersonaChat-USR (PU, persona-conditioned
+conversations annotated with the USR schema) [22]; DailyDialog-Zhao (DZ, more
+formal language conversations rated for appropriateness) [46]; DailyDialog-Gupta
+(DGU, rated for appropriateness) [11]; DailyDialog-GRADE (DGR; annotated for
+coherence) [14]; and Empathetic-GRADE (EG, emotionally grounded conversa-
+tions annotated for coherence) [14]. Although some of these datasets are not directly
+annotated for whether a response is good, the metric they use remains a component
+for overall quality, and thus it is treated as the indicator of the overall quality of the
+response in the experiments.
+3.3 In-Context Examples
+This paper uses two methods for example selection: random selection, and algorith-
+mic selection using BM25 [20] which calculates document similarity. The examples
+remain consistent for each evaluation test point. The random selection experiment is
+run three times, and the mean and standard deviation of the runs are reported. There
+are three configurations for BM25 between the test point and each possible example
+point - comparing the context only (BM25C), the response only (BM25R), and the
+concatenated context and response together (BM25C+R).
+With the FED dataset, an additional method, manual selection, is added for ex-
+ample selection. For each fine-grained dialog quality, a set of three dialogs which
+span a wide range of ratings is chosen that remains constant over every test point. In
+theory, the model should be able to show increased performance if it sees examples
+of very good, good and bad responses for fine-grained metrics. For the DSTC10
+datasets, an additional experiment tested how the number of examples used affects
+model performance.
+4 Experiments and Results
+The in-context example experiments are carried out on the largest available model,
+530B TNLGv2, to explore the ceiling of model performance on the dialog evalua-
+tion task. 6.7B TNLGv2 is used for a direct comparison of how much performance
+gain is provided by using more parameters.
+BLOOM and OPT are examined up to 7B and 30B respectively for the fine-
+grained metric evaluation task. 1 Smaller LLMs do not perform as well with in-
+1 Due to limitations in compute power, larger BLOOM and OPT models were not explored. How-
+ever, as the largest available GPT-3 model is explored, the comparisons appear sufficient to show
+the performance of a variety of LLMs.
+
+8
+J. Huynh et al.
+context examples unless they have been specifically tuned for the task, so only
+the 7B and 6.7B models for BLOOM and OPT respectively are explored for the
+DSTC10 datasets. Flan-T5 and InstructDial are analyzed in the 3B setting for con-
+sistency. Lastly, InstructGPT (text-davinci-002) is used, which has 175B parame-
+ters.
+4.1 Fine-grained Metric Evaluation
+FED is separated into turn-level and dialog-level metrics. The dataset has anno-
+tations for 8 different turn-level metrics, consisting of interestingness, engaging-
+ness, specificity, relevance, correctness, semantic appropriateness, understandabil-
+ity, and fluency, with the addition of overall quality. FED annotates three different
+responses for each dialog context; one FED dialog is treated as one example. The
+corresponding rating is inserted after the response statement in the prompt, an exam-
+ple of which can be seen in Appendix B. FED also looks at 10 different dialog-level
+metrics for a system’s responses: coherence, error recovery, consistency, diversity,
+topic depth, likeability, understandingness, flexibility, informativeness, and inquisi-
+tiveness, with overall quality included. The model is prompted with the full dialog
+context with the rating.
+The FED metric was previously evaluated with both fine-tuned (ft) and from-
+scratch 345M and 762M DialoGPT [43] models. In the following experiments on
+FED, 3 in-context examples were used for prompting in Tables 1, 2, 3 and 4 and
+Appendix D and E.
+4.1.1 In-Context Example Selection
+This setting evaluates 2 versions of the TNLGv2 model: 6.7B and 530B. These
+models are compared to the 762M ft DialoGPT model and the results are shown in
+Tables 1 and 2 and Appendix D.
+First, the performances of these models are compared over the three example
+selection methods: manual, random, and algorithmic. With manually chosen in-
+context examples, the 530B TNLGv2 model outperforms the DialoGPT model on
+almost all turn-level metrics except for understandability and fluency. There are
+significant gains in all of the dialog-level metrics as well. Since DialoGPT is fine-
+tuned on Reddit threads, more casual language is expected, compared to models
+like TNLGv2 where many of the training datasets consist of more formal language.
+Since the wording of conversational responses tends to be more casual, it is not sur-
+prising that the fine-tuned DialoGPT model outperforms even the largest TNLGv2
+model for fluency and understandability. However, the TNLGv2 models show large
+improvement on predicting turn- and dialog-level quality. This suggests that the
+TNLGv2 models have a strong grasp on overall quality, which may be due to train-
+ing on more formal language.
+
+Very Large Language Models for Dialog Evaluation
+9
+BM25C+R generally outperforms BM25C and BM25R. However, when choos-
+ing examples with BM25C+R, the correlation of understandability with human
+annotations increases significantly when using the 6.7B TNLGv2 model. 6.7B
+TNLGv2 consistently outperforms 530B TNLGv2 in this aspect with any BM25
+method. It appears that the smaller model is more influenced by the similarity of
+language in the examples than the larger one.
+Even when given random examples, the TNLGv2 models outperform the 762M
+ft DialoGPT model on a majority of the fine-grained metrics. This shows that larger
+models can better detect what constitutes a good response based on these metrics
+even if they are not given hand-picked examples. However, they generally do not
+outperform the manually or algorithmically chosen examples as expected.
+An additional observation is that there are certain factors that cause models to
+perform better or worse on specific metrics: number of parameters the model has,
+the type of training data, and the difficulty of the task. LLMs are able to provide
+increases in performance of over 50% for 15 out of 20 turn- and dialog-level met-
+rics compared to DialoGPT with 530B TNLGv2 and manually-chosen examples.
+However, if the 530B TNLGv2 model is compared to the 6.7B TNLGv2 model,
+this increase is only observed for 2 out of the 20 metrics: correctness and under-
+standability. LLMs can achieve high correlations with human judgement, but there
+is a limit to how much more performance gains can increase with extremely large
+models.
+Specificity, relevance, and correctness all relate to the context of the conversation
+while the other metrics are more turn-specific. It follows that relevance and correct-
+ness with BM25C+R on the 6.7B TNLGv2 model outperform the 530B TNLGv2
+model with manual examples. However, specificity performs worse. Choosing both
+diverse ratings and similar example points are important. This finding further sup-
+ports the idea that the nature of the data used to train these LLMs is important. Had
+the training data been more similar to conversational language, an increase could
+have been observed in the correlations for these metrics without choosing algorith-
+mically similar examples.
+TNLGv2 struggles with understandability; it performs the worst at the highest
+correlation of 0.193. It also has unstable performance; performing at significance
+with random examples and with algorithmically chosen examples on 6.7B, but not
+with manually chosen ones. This shows that choosing examples with diverse ratings
+helps a model less for metrics that it already performs poorly on; it would better
+benefit from examples that are similar.
+In general, even with the difference in training data, it is easier to obtain an overall
+sense of the conversation than a metric for a single turn for the larger models due
+to the large amount of parameters and variety of data that they have seen. When
+choosing examples based on context, the larger models generally perform worse; it
+appears that having different examples is more important for dialog-level metrics
+than for turn-level metrics.
+
+10
+J. Huynh et al.
+manual
+random
+BM25C+R
+Quality
+762M ft
+6.7B
+530B
+6.7B
+530B
+6.7B
+530B
+Interesting
+0.408
+0.455
+0.474
+0.293 ± 0.03
+0.398± 0.02
+0.358
+0.383
+Engaging
+0.318
+0.459
+0.484
+0.235± 0.04
+0.352± 0.02
+0.378
+0.383
+Specific
+0.267
+0.305
+0.450
+0.188± 0.02
+0.289± 0.01
+0.268
+0.322
+Relevant
+0.152
+0.214
+0.300
+0.179± 0.04
+0.299± 0.03
+0.392
+0.357
+Correct
+0.133
+0.195
+0.393
+0.171± 0.04
+0.338± 0.04
+0.399
+0.377
+Sem. Approp.
+0.155
+0.292
+0.395
+0.163± 0.03
+0.270± 0.01
+0.291
+0.294
+Understandable
+0.111
+0.021*
+0.036*
+0.146± 0.02
+0.129± 0.02
+0.193
+0.062*
+Fluent
+0.224
+0.164
+0.195
+0.052*± 0.03
+0.112*± 0.01
+0.096*
+0.178
+Overall
+0.209
+0.371
+0.475
+0.256± 0.02
+0.380± 0.01
+0.474
+0.514
+Table 1 Turn-level fine-grained metrics on the FED dataset for manually, randomly, and BM25
+chosen examples over the TNLGv2 6.7B and 530B models. BM25C+R stands for examples chosen
+by BM25 considering both the context and the response of the test point. Bold values indicate the
+best value for the metric and * values indicate correlations that are not statistically significant.
+manual
+random
+BM25C
+Quality
+762M ft
+6.7B
+530B
+6.7B
+530B
+6.7B
+530B
+Coherent
+0.251
+0.599
+0.727
+0.443± 0.03
+0.533± 0.02
+0.618
+0.512
+Error Recovery
+0.165*
+0.474
+0.578
+0.348± 0.04
+0.463± 0.06
+0.492
+0.419
+Consistent
+0.116*
+0.276
+0.382
+0.270± 0.02
+0.205* ± 0.04
+0.238
+0.046*
+Diverse
+0.420
+0.625
+0.620
+0.434± 0.06
+0.490± 0.02
+0.496
+0.548
+Topic Depth
+0.476
+0.640
+0.659
+0.361± 0.03
+0.531± 0.04
+0.559
+0.472
+Likeable
+0.262
+0.619
+0.686
+0.511± 0.03
+0.580± 0.01
+0.568
+0.515
+Understanding
+0.306
+0.517
+0.638
+0.479± 0.06
+0.496± 0.02
+0.567
+0.428
+Flexible
+0.293
+0.617
+0.656
+0.491± 0.05
+0.553± 0.03
+0.614
+0.451
+Informative
+0.288
+0.569
+0.547
+0.391± 0.04
+0.452± 0.04
+0.523
+0.419
+Inquisitive
+0.163
+0.537
+0.527
+0.436± 0.05
+0.444± 0.02
+0.334
+0.252
+Overall
+0.443
+0.630
+0.688
+0.479± 0.05
+0.570± 0.02
+0.607
+0.531
+Table 2 Dialog-level fine-grained metrics on the FED dataset for manually, randomly, and BM25
+chosen examples over the TNLGv2 6.7B and 530B models. BM25C stands for examples chosen
+by BM25 considering only the context of the test point.
+4.1.2 Comparisons Across LLMs
+These model comparisons are performed using manually chosen in-context exam-
+ples, since that is what generally performed the best in both turn-level and dialog-
+level metrics in Tables 3 and 4. Comparisons across smaller versions of BLOOM
+and OPT can be found in Appendix E.
+Even though the large versions of BLOOM and OPT could not be run, it is ap-
+parent that both of these models outperform TNLGv2 on understandability, and
+that OPT 6.7B can outperform TNLGv2 530B on fluency. Data dissimilarities were
+noted above in Section 4.1.1 between the TNLGv2 model and the FED data. Al-
+though BLOOM was only trained on some English data, it has still seen some ca-
+sual language, while OPT was partially trained on Reddit data. Thus the language
+appearing in the BLOOM and OPT training sets more closely matches that of the
+conversations used here. This explains the increase in performance.
+BLOOM 7B outperforms 6.7B TNLGv2 on correctness, while OPT 6.7B out-
+performs 6.7B TNLGv2 on relevance, correctness, semantic appropriateness and
+
+Very Large Language Models for Dialog Evaluation
+11
+TNLG
+BLOOM
+OPT
+Flan-T5
+InstructGPT
+Quality
+6.7B
+530B
+7B
+6.7B
+30B
+3B
+175B
+Interesting
+0.455
+0.474
+0.291
+0.429
+0.399
+0.519
+0.551
+Engaging
+0.459
+0.484
+0.435
+0.446
+0.349
+0.425
+0.489
+Specific
+0.305
+0.450
+0.296
+0.275
+0.207
+0.433
+0.421
+Relevant
+0.214
+0.300
+0.109
+0.272
+0.289
+0.435
+0.471
+Correct
+0.195
+0.393
+0.235
+0.342
+0.354
+0.378
+0.376
+Sem. Approp.
+0.292
+0.395
+0.258
+0.371
+0.382
+0.277
+0.374
+Understandable
+0.021*
+0.036*
+0.159
+0.131
+0.073*
+0.297
+0.382
+Fluent
+0.164
+0.195
+0.111
+0.201
+0.188
+0.200
+0.204
+Overall
+0.371
+0.475
+0.274
+0.368
+0.433
+0.445
+0.536
+Table 3 Turn-level fine-grained metrics on the FED dataset for manually chosen examples over
+the TNLGv2, BLOOM, OPT, Flan-T5, and InstructGPT models.
+TNLG
+BLOOM
+OPT
+FLAN-T5
+InstructGPT
+Quality
+6.7B
+530B
+7B
+6.7B
+30B
+3B
+175B
+Coherent
+0.599
+0.727
+0.613
+0.558
+0.584
+0.730
+0.707
+Error Recovery
+0.474
+0.578
+0.474
+0.377
+0.479
+0.398
+0.560
+Consistent
+0.276
+0.382
+0.323
+0.237
+0.309
+0.410
+0.517
+Diverse
+0.625
+0.620
+0.498
+0.454
+0.607
+0.544
+0.628
+Topic Depth
+0.640
+0.659
+0.637
+0.544
+0.609
+0.650
+0.680
+Likeable
+0.619
+0.686
+0.566
+0.544
+0.571
+0.659
+0.672
+Understanding
+0.517
+0.638
+0.484
+0.505
+0.483
+0.637
+0.694
+Flexible
+0.617
+0.656
+0.499
+0.528
+0.592
+0.595
+0.688
+Informative
+0.569
+0.547
+0.462
+0.497
+0.522
+0.662
+0.647
+Inquisitive
+0.537
+0.527
+0.539
+0.461
+0.537
+0.487
+0.578
+Overall
+0.630
+0.688
+0.531
+0.374
+0.530
+0.585
+0.690
+Table 4 Dialog-level fine-grained metrics on the FED dataset for manually chosen examples over
+the TNLGv2, BLOOM, OPT, Flan-T5, and InstructGPT models.
+fluency in addition. As previously noted, relevance and correctness are turn-level
+metrics that take more of the context into account, so with training data that is more
+similar to casual language, these models perform better. It should be noted that
+the overall turn- and dialog-level quality results were not surpassed by any smaller
+model, thus the very large models will have an advantage for overall metrics.
+Flan-T5 outperforms the largest model, TNLGv2 530B, on interestingness, rele-
+vance, and understandability at turn level and coherence, consistency, and informa-
+tiveness at dialog level. There is a larger performance drop for the semantic appro-
+priateness, error recovery, and overall dialog-level quality metrics. Error recovery
+is a relatively new metric [21]. Even though Flan-T5 was fine-tuned on many di-
+alog tasks, it may not have seen data that addresses this specific metric. Flan-T5
+only has 3B parameters, and the fact that it outperforms 530B TNLGv2 shows the
+importance of use of dialog data during pre-training or fine-tuning.
+InstructGPT, being fine-tuned with prompting at 175B parameters, is more suit-
+able for the present experiments. It performs very well on both turn- and dialog-level
+metrics, outperforming 530B TNLGv2 on almost all metrics. Since InstructGPT has
+already seen prompting, the model can better understand a task through only instruc-
+tions or combinations of instructions and in-context examples.
+
+12
+J. Huynh et al.
+Model
+TU
+DZ
+PU
+DGU
+DGR
+FT
+EG
+FD
+Experiments with Random Examples
+4ex
+0.112 ± 0.03 0.428 ± 0.01 0.403 ± 0.02 0.542 ± 0.00 0.338 ± 0.01 0.318 ± 0.02 0.248 ± 0.04 0.290 ± 0.05
+8ex
+0.169 ± 0.03 0.430 ± 0.03 0.331 ± 0.03 0.570 ± 0.01 0.429 ± 0.05 0.337 ± 0.01 0.200 ± 0.04 0.339 ± 0.18
+12ex
+0.148 ± 0.03 0.453 ± 0.02 0.384 ± 0.02 0.565 ± 0.01 0.410 ± 0.06 0.412 ± 0.03 0.160 ± 0.02 0.351 ± 0.08
+Experiments with Algorithmically Retrieved Examples
+4ex BM25R
+0.247
+0.424
+0.252
+0.482
+0.342
+0.364
+0.144
+0.264
+4ex BM25C
+0.129
+0.424
+0.339
+0.510
+0.370
+0.172
+0.192
+0.549
+4ex BM25C+R
+0.213
+0.441
+0.432
+0.479
+0.371
+0.137
+0.211
+0.479
+8ex BM25R
+0.309
+0.487
+0.275
+0.536
+0.304
+0.426
+0.121
+0.419
+8ex BM25C
+0.227
+0.564
+0.460
+0.627
+0.387
+0.323
+0.123
+0.518
+8ex BM25C+R
+0.185
+0.458
+0.439
+0.526
+0.308
+0.377
+0.171
+0.530
+12ex BM25R
+0.300
+0.474
+0.358
+0.570
+0.337
+0.393
+0.095*
+0.414
+12ex BM25C
+0.278
+0.688
+0.449
+0.674
+0.397
+0.377
+0.106*
+0.492
+12ex BM25C+R
+0.202
+0.491
+0.452
+0.465
+0.349
+0.358
+0.148
+0.493
+Best of DSTC10 baselines
+0.319
+0.532
+0.493
+0.596
+0.363
+0.247
+0.395
+0.555
+Table 5 Spearman correlation of model predictions for overall quality with human ratings for
+TNLGv2 530B model with algorithmically chosen examples. TU, PU, PZ, DZ, CG, DGU, DGR,
+EG, FT and FD are abbreviations for TopicalChat-USR, PersonaChat-USR [22], PersonaChat-
+Zhao [46], DailyDialog-Zhao [46], ConvAI2-GRADE [14], DailyDialog-Gupta [11], DailyDialog-
+GRADE [14], Empathetic-GRADE [14], FED-Turn and FED-Dial [21].
+Model
+TU
+DZ
+PU
+DGU
+DGR
+FT
+EG
+FD
+Few-shot in-context Experiments
+BLOOM-7B-4ex
+0.027*
+0.075
+0.123
+0.127
+0.131
+0.117
+0.012
+0.289
+OPT-6.7B-4ex
+0.115
+0.258
+0.444
+0.228
+0.091*
+0.486
+0.044*
+0.657
+TNLG-6.7B-4ex
+0.124
+0.198
+0.237
+0.209
+0.214
+0.296
+0.057*
+0.314
+TNLG-530B-4ex
+0.129
+0.424
+0.339
+0.510
+0.370
+0.172
+0.192
+0.549
+Flan-T5-3B-4ex
+0.447
+0.657
+0.578
+0.714
+0.379
+0.442
+0.396
+0.492
+InstructGPT-175B-4ex
+0.616
+0.716
+0.687
+0.746
+0.472
+0.506
+0.305
+0.412
+Zero-shot Experiments
+Flan-T5-3B-0ex
+0.357
+0.599
+0.533
+0.677
+0.351
+0.380
+0.418
+0.444
+InstructDial-3B-0ex
+0.446
+0.601
+0.376
+0.634
+0.286
+0.263
+0.475
+0.228
+Best of DSTC10 baselines
+0.319
+0.532
+0.493
+0.596
+0.363
+0.247
+0.395
+0.555
+Best TNLGv2 value
+0.309
+0.688
+0.460
+0.678
+0.429
+0.426
+0.248
+0.549
+Table 6 Spearman correlation of model predictions for overall quality with human ratings with 4
+examples chosen with BM25 using context. Macro average scores are also shown.
+4.2 DSTC10 Datasets
+The same set of experiments were carried out on the 8 datasets in the DSTC10 chal-
+lenge in Tables 5 and 6, and Appendix F. The previous best performing metrics on
+DSTC10 are compiled from [13], which include both reference-free and fine-tuned
+metrics (see Appendix G). Quality is evaluated in terms of how good a response is
+to the context.
+4.2.1 In-Context Example Selection
+Experiments are performed with randomly chosen examples and examples that were
+chosen by BM25 over 4, 8, and 12 examples in Table 5 and Appendix F. Higher
+correlation results are obtained on 4 datasets (DZ, DGU, DGR, and FT) with com-
+parable results on 3 datasets (TU, PU, and FD), as compared to the best DSTC10
+baselines. Most of the best results are on the 530B TNLGv2 model, which will be
+
+Very Large Language Models for Dialog Evaluation
+13
+discussed in this section, as compared to the 6.7B TNLGv2 model. Several factors
+are relevant here: the language of the dataset, the way the dataset was created, and
+how the dataset was annotated.
+DailyDialog contains more formal language, thus TNLGv2 should perform well
+since its training dataset includes data sources with formal language. DZ, DGU, and
+DGR almost always perform the best when examples are chosen from looking at
+the context; adding the response generally leads to poorer performance. Since these
+datasets are annotated for appropriateness and coherence, context is more important
+than a more turn-specific metric.
+TopicalChat was created through knowledge-grounding. The conversations could
+thus have more substance than a purely open-domain un-prompted conversation. It
+thus follows that response selection will work the best when choosing examples.
+PersonaChat has conversations that are persona-conditioned, so the quality of the
+conversation should take into account the entire conversation for each persona. It
+performs better with examples chosen for context and response or with just context.
+FED is split into turn- and dialog-level annotations, thus, for turn-level annota-
+tions choosing examples based on responses should work best, and for dialog-level
+annotations choosing examples based on either the context or the context and re-
+sponse should perform the best. Choosing examples with context and response per-
+forms the best for EG, but randomly choosing examples outperforms that result. It
+may be that with emotionally grounded conversations, the model needs more, or
+more diverse examples due to the different ways emotion can be expressed.
+In general, choosing examples algorithmically improves performance over ran-
+domly choosing examples. This is consistent with previous experiments above.
+However, randomly-chosen examples perform better on the DGR and EG datasets
+on the 530B TNLGv2 model. This may be because these two datasets were rated
+for coherence. Algorithmically, choosing examples based on context and response
+performs the best on EG, as was seen for coherence in FED in Section 4.1.1.
+4.2.2 Comparisons Across LLMs
+Table 6 compares the evaluation results across various LLMs. Due to model input
+length restrictions, the following experiments were carried out using 4 in-context ex-
+amples or in a zero-shot setting. BM25 is only used with the context as the example
+selection strategy, since it performed well with the TNLGv2 models.
+In the few-shot setting, models that were not fine-tuned or trained with prompting
+(BLOOM, OPT) did not have consistent results across the datasets. However, those
+that were fine-tuned or prompted (Flan-T5, InstructGPT, InstructDial) had results
+that were close to or surpassed the previous best DSTC10 baselines. InstructGPT
+performed the best. Even in the zero-shot setting, Flan-T5 outperforms the baseline
+in 6 of the datasets, and InstructDial in 5.
+These results clearly show that for dialog evaluation, it is insufficient to simply
+train on large amounts of general internet data. Specialized approaches such as in-
+struction tuning on multiple tasks improve the generalization capabilities of models
+
+14
+J. Huynh et al.
+in zero- and few-shot settings. It is not surprising that InstructGPT performs the best
+since it fine-tunes a very large language model with instructions.
+5 Conclusion
+LLMs have the potential to significantly contribute to dialog evaluation. Current
+LLMs perform well for this task in a few-shot setting. However, this performance
+varies greatly depending on the content of and number of examples in the prompt.
+Models prefer more similar examples for metrics that they struggle to evaluate,
+while preferring examples with more diverse ratings for metrics that they can evalu-
+ate well. Very large language models also still afford performance gains, especially
+for overall quality evaluation at the turn and dialog level. Even though large lan-
+guage models perform better at dialog-level fine-grained metrics, there are still pre-
+viously shown issues with how these models understand social situations and use
+context that may hinder further improvement if not addressed.
+Performance is also affected by the model’s training data. Smaller language mod-
+els that are fine-tuned on instructions, trained on dialog data, and/or trained on mul-
+tiple dialog tasks outperform larger language models. These smaller models also
+perform more consistently over different domains. This indicates that LLMs should
+have more diverse pre-training data in order to be able to handle a larger variety of
+tasks in few or zero-shot settings.
+More work needs to be done on understanding how a large language model mod-
+els different types of tasks. In-context example selection and example wording still
+remains unstable across large language models in many tasks, and the performance
+variation over different dialog domains in this paper demonstrates that as well.
+Presently, the LLMs explored in this paper have their own strengths. Smaller
+models such as BLOOM and OPT could share more training data similarity with
+dialog tasks based on their objective. TNLGv2 530B provides a very large lan-
+guage model that has shown improvement in dialog evaluation along with other
+NLP tasks. Flan-T5 and InstructDial show the efficacy of fine-tuning a LLM on
+dialog tasks, and InstructGPT shows the importance of training a model to better
+recognize prompts. The evaluations of these models provide suggestions for the
+characteristics of the best LLMs to use for dialog evaluation. Future work in using
+LLMs for other NLP tasks can benefit from such comprehensive analyses. Once a
+better understanding of LLMs is realized, the capabilities of large language models
+for zero- and few-shot tasks will increase greatly.
+6 Acknowledgements
+We would like to thank Microsoft for allowing us to use TNLGv2. J.H. was sup-
+ported by the NSF Graduate Research Fellowship under Grant Nos. DGE1745016
+
+Very Large Language Models for Dialog Evaluation
+15
+and DGE2140739. The opinions expressed in this paper do not necessarily reflect
+those of that funding agency.
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+
+18
+J. Huynh et al.
+A LLMs and Their Training/Fine-tuning Data
+Seen Dialog Fine-tuned
+Flan-T5
+✓
+✓
+InstructDial
+✓
+✓
+InstructGPT
+✓
+✓
+BLOOM
+✓
+×
+OPT
+✓
+×
+TNLGv2
+×
+×
+Table 7 LLMs with the datasets they were trained on. During training or fine-tuning: “Seen Dia-
+log” indicates that the model has explicitly seen dialog datasets and therefore elements of casual
+language, and “fine-tuned” indicates that the model was fine-tuned on dialog data. TNLGv2 has
+not seen datasets explicitly categorized as having dialog, but elements of casual language may be
+included in the Common Crawl snapshots and other internet-based corpora. Symbols: ✓means that
+the category is included and × means that the category is not included.
+B Prompt format examples FED
+Task: Given a dialog history and a response, rate how interesting the response is with regards
+to the dialog history.
+== Example 1 ==
+A: Hi!
+B: Hi. This is a pleasant surprise.
+A: Haha...thanks! how did you like the gift?
+Response: Currently unpacking it I guess. How’s your morning?
+Rating: 1/2
+A: Hope you like it! Morning is good. Busy finishing up stuff before the holidays.
+B: I think I traveled too much the last couple of months so no holiday for me. But I’m okay
+with that. Going anywhere exciting?
+A: Yes
+Response: Where to?
+Rating: 1/2
+A: Hawaii... looking forward to warm beaches.
+Response: WOW. Which island? I like Hawaii.
+Rating: 2/2
+Table 8 An example of a prompt with one example from FED [21]. Interestingness was rated in
+FED over three values corresponding to 0/2, 1/2, and 2/2. The resulting output is truncated to the
+integer value of 0, 1, or 2 to be used in evaluation.
+
+Very Large Language Models for Dialog Evaluation
+19
+C Prompt format examples DSTC10
+Instruction: Given a conversation and a response, choose if the response is a good response
+to the context
+Example
+Background info: none
+Conversation:
+Person A: did your meal meet with your approval ?
+Response: yes , i did . it was a good meal .
+Question: Is the above response a good response to the conversation?
+Answer: Yes
+Background info: none
+Conversation:
+Person B: i really do hate public transportation.
+Person A: i agree , it ’s just never on time.
+Response : you ’re right.
+Question: Is the above response a good response to the conversation?
+Answer:
+Table 9 An example of a prompt with examples from DSTC 10.
+D Additional algorithmically chosen FED examples
+BM25C
+BM25R
+Quality
+7B
+530B
+7B
+530B
+Interesting
+0.336 0.389
+0.355
+0.385
+Engaging
+0.308 0.332
+0.328
+0.389
+Specific
+0.217 0.224
+0.297
+0.329
+Relevant
+0.338 0.314
+0.311
+0.356
+Correct
+0.333 0.341
+0.300
+0.383
+Sem. Approp.
+0.261 0.270
+0.287
+0.337
+Understandable 0.141 0.028* 0.169 0.029*
+Fluent
+0.106 0.147 0.096* 0.121
+Overall
+0.435 0.438
+0.360
+0.407
+Table 10 Turn-level fine-grained metrics on the FED dataset for algorithmically chosen examples
+over the TNLGv2 6.7B and 530B models. BM25C stands for examples chosen by BM25 consid-
+ering the context and BM25R stands for examples chosen by BM25 considering the response.
+
+20
+J. Huynh et al.
+E Additional LLM sizes on FED
+BLOOM
+OPT
+Quality
+560M
+1.1B
+1.7B
+3B
+125M 350M 1.3B 2.7B
+Interesting
+0.282
+0.331
+0.336
+0.328
+0.187
+0.186 0.388 0.245
+Engaging
+0.217
+0.320
+0.278
+0.418
+0.121
+0.252 0.398 0.292
+Specific
+0.030* 0.065* 0.204
+0.353
+0.197 0.004* 0.217 0.222
+Relevant
+0.076* 0.056* 0.072* 0.091* 0.146
+0.105 0.231 0.177
+Correct
+0.106
+0.146
+0.124
+0.173
+0.119
+0.152 0.327 0.270
+Sem. Approp.
+0.048* 0.228
+0.205
+0.265
+0.148
+0.278 0.274 0.296
+Understandable -0.017* 0.043* -0.005* 0.087* 0.058* 0.021* 0.189 0.205
+Fluent
+0.158
+0.223 0.097* 0.091* 0.109 0.087* 0.158 0.163
+Overall
+0.086* 0.179 0.076* 0.285
+0.134
+0.219 0.338 0.197
+Table 11 Turn-level fine-grained metrics on the FED dataset for manually chosen examples over
+the smaller sizes of BLOOM and OPT.
+BLOOM
+OPT
+Quality
+560M
+1.1B
+1.7B
+3B
+125M 350M 1.3B 2.7B
+Coherent
+0.499
+0.533
+0.531 0.531 0.490 0.514 0.528 0.435
+Error Recovery 0.293
+0.298
+0.322 0.448 0.168 0.380 0.342 0.348
+Consistent
+0.217
+0.238 0.129* 0.264 0.193 0.191 0.250 0.268
+Diverse
+0.345
+0.430
+0.461 0.518 0.451 0.304 0.491 0.531
+Topic Depth
+0.418
+0.414
+0.519 0.462 0.228 0.302 0.462 0.454
+Likeable
+0.310
+0.374
+0.421 0.476 0.467 0.395 0.462 0.535
+Understanding
+0.276
+0.312
+0.257 0.371 0.389 0.283 0.414 0.494
+Flexible
+0.269
+0.432
+0.400 0.441 0.458 0.377 0.460 0.432
+Informative
+0.149* 0.384
+0.372 0.537 0.378 0.402 0.381 0.544
+Inquisitive
+0.198
+0.350
+0.318 0.339 0.489 0.300 0.439 0.413
+Overall
+0.262 0.146* 0.207 0.261 -0.000* 0.319 0.452 0.437
+Table 12 Dialog-level fine-grained metrics on the FED dataset for manually chosen examples over
+the smaller sizes of BLOOM and OPT.
+
+Very Large Language Models for Dialog Evaluation
+21
+F DSTC10 Results For TNLGv2 6.7B
+Model
+TU
+DZ
+PU
+DGU
+DGR
+FT
+EG
+FD
+Experiments with Random Examples
+4ex
+0.034* ± 0.05 0.117 ± 0.02 0.206 ± 0.02 0.080* ± 0.05 0.121± 0.05 0.191 ± 0.06 0.005* ± 0.04 0.228 ± 0.03
+8ex
+0.054* ± 0.05 0.160 ± 0.02 0.206 ± 0.03 0.109* ± 0.03 0.139 ± 0.08 0.178 ± 0.02 0.060* ± 0.06 0.238 ± 0.11
+12ex
+0.063* ± 0.03 0.149 ± 0.00 0.225 ± 0.01 0.114 ± 0.05 0.143 ± 0.06 0.210 ± 0.03 0.052* ± 0.02 0.127 ± 0.04
+Experiments with Algorithmically Retrieved Examples
+4ex BM25R
+0.148
+0.218
+0.223
+0.202
+0.094*
+0.273
+-0.012*
+0.335
+4ex BM25C
+0.124
+0.198
+0.237
+0.209
+0.214
+0.296
+0.057*
+0.314
+4ex BM25C+R
+0.05*
+0.142
+0.169
+0.167
+0.083*
+0.274
+0.038*
+0.339
+8ex BM25R
+0.077*
+0.270
+0.203
+0.222
+0.128
+0.199
+0.042*
+0.335
+8ex BM25C
+0.184
+0.328
+0.343
+0.526
+0.176
+0.363
+0.073*
+0.387
+8ex BM25C+R
+0.029*
+0.152
+0.020*
+0.092
+0.022*
+0.348
+0.024*
+0.440
+12ex BM25R
+0.069*
+0.338
+0.153
+0.213
+0.110*
+0.250
+0.026*
+0.401
+12ex BM25C
+0.285
+0.544
+0.325
+0.678
+0.208
+0.330
+0.042*
+0.365
+12ex BM25C+R
+0.035*
+0.168
+0.088*
+0.086*
+0.100*
+0.407
+0.092*
+0.343
+Table 13 Spearman correlation of model predictions with human ratings for TNLGv2 6.7B
+model with algorithmically chosen examples. TU, PU, PZ, DZ, CG, DGU, DGR, EG,
+FT and FD are abbreviations for TopicalChat-USR, PersonaChat-USR
+[22], PersonaChat-
+Zhao [46], DailyDialog-Zhao [46], ConvAI2-GRADE [14], DailyDialog-Gupta [11], DailyDialog-
+GRADE [14], Empathetic-GRADE [14], FED-Turn and FED-Dial [21].
+G DSTC10 Baseline Results
+Model
+Fine-Tuned on
+TU
+DZ
+PU DGU DGR
+FT
+EG
+FD
+DTSC 10 datasets
+USL-H [25]
+✓
+0.319
+0.385 0.493 0.481
+0.09 0.115 0.237 0.202
+GRADE [14]
+✓
+0.176
+0.532 0.329 0.596 0.254 0.048 0.300 0.106
+DynaEval [41]
+✓
+-0.013 0.169 0.148 0.038 0.122 0.247 0.159 0.555
+USR [22]
+×
+0.291
+0.363 0.140 0.353 0.066 0.055 0.268 0.084
+FED [21]
+×
+-0.090 -0.080 -0.004 0.025 -0.009 0.173 0.005 0.178
+DEB [31]
+×
+0.123
+0.486 0.351 0.579 0.363 0.044 0.395 0.141
+Best
+0.319
+0.532 0.493 0.596 0.363 0.247 0.395 0.555
+Table 14 Spearman correlation of model predictions with human ratings. The models fine-tuned
+on DSTC 10 datasets tend to perform better on the DSTC 10 datasets. TU, PU, PZ, DZ, CG,
+DGU, DGR, EG, FT and FD are abbreviations for TopicalChat-USR, PersonaChat-USR [22],
+PersonaChat-Zhao [46], DailyDialog-Zhao [46], ConvAI2-GRADE [14], DailyDialog-Gupta [11],
+DailyDialog-GRADE [14], Empathetic-GRADE [14], FED-Turn and FED-Dial [21].
+