Original: S ANITY C HECK FOR P ERSISTENCE D IAGRAMS
Revised: S ANITY C HECK FOR P ERSISTENCE D IAGRAMS

Original: Anonymous authors

Original: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Recently many efforts have been made to incorporate persistence diagrams, one of the major tools in topological data analysis (TDA), into machine learning pipelines.
Revised: Recently many efforts have been made to incorporate persistence diagrams, one of the major tools in topological data analysis (TDA), into machine learning pipelines.

Original: To better understand the power and limitation of persistence diagrams, we carry out a range of experiments on both graph and shape data, aiming to decouple and inspect the effects of different factors involved.
Revised: To better understand the power and limitation of persistence diagrams, we carry out a range of experiments on both graph and shape data, aiming to decouple and inspect the effects of different factors involved.

Original: To this end, we propose a sanity check for persistence diagrams to delineate critical values and pairings of critical values (structure of persistence module).
Revised: To this end, we propose a sanity check for persistence diagrams to delineate critical values and pairings of critical values (structure of persistence module).

Original: For graph classiﬁcation tasks, we note that while persistence pairing yields consistent improvement over various benchmark datasets, it appears that for various ﬁltration functions tested, most discriminative power comes from critical values.
Revised: For graph classiﬁcation tasks, we note that while persistence pairing yields consistent improvement over various benchmark datasets, it appears that for various ﬁltration functions tested, most discriminative power comes from critical values.

Original: For shape segmentation and classiﬁcation, however, we note that persistence pairing shows signiﬁcant power on most of the benchmark datasets, and improves over summaries based on merely critical values.
Revised: For shape segmentation and classiﬁcation, however, we note that persistence pairing shows signiﬁcant power on most of the benchmark datasets, and improves over summaries based on merely critical values.

Original: Our results help provide insights on when persistence diagram based summaries could be more suitable.
Revised: Our results help provide insights on when persistence diagram based summaries could be more suitable.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Topological data analysis (TDA) is an emerging ﬁeld that aims to characterize the shape of low and high dimensional data via methods stemming from algebraic topology.
Revised: Topological data analysis (TDA) is an emerging ﬁeld that aims to characterize the shape of low and high dimensional data via methods stemming from algebraic topology.

Original: One of the major tools of
Revised: One of the major tools of

Original: TDA is persistence diagrams (PDs).
Revised: TDA is persistence diagrams (PDs).

Original: In recent years, many efforts have been made to utilize PDs as features for downstream machine learning tasks, such as material science (Buchet et al., 2018), signal analysis (Perea & Harer, 2015) , cellular data (C ´ amara, 2017) and shape recognition (Li et al., 2014).
Revised: In recent years, many efforts have been made to utilize PDs as features for downstream machine learning tasks, such as material science (Buchet et al., 2018), signal analysis (Perea & Harer, 2015) , cellular data (C ´ amara, 2017) and shape recognition (Li et al., 2014).

Original: However, the geometry of the PD does not lend itself easily to well-adopted classiﬁers due to the lack of Hilbert space structure.
Revised: However, the geometry of the PD does not lend itself easily to well-adopted classiﬁers due to the lack of Hilbert space structure.

Original: To handle this issue, a natural way is to apply vectorization (Bubenik, 2015; Carri`ere et al., 2015; Chazal et al., 2014; Kaliˇsnik, 2019) or kernelization (Reininghaus et al., 2015; Carriere et al., 2017) to PDs, i.e., embedding PDs either to a Euclidean space R d or a reproducing kernel Hilbert space (RKHS) associated with certain kernels.
Revised: To handle this issue, a natural way is to apply vectorization (Bubenik, 2015; Carri ` ere et al., 2015; Chazal et al., 2014; Kali ˇ snik, 2019) or kernelization (Reininghaus et al., 2015; Carriere et al., 2017) to PDs, i.e., embedding PDs either to a Euclidean space R d or a reproducing kernel Hilbert space (RKHS) associated with certain kernels.

Original: However, these approaches still have limitations.
Revised: However, these approaches still have limitations.

Original: First, it has been shown that ﬁnite-dimension embedding can miss information about PDs (Carriere & Bauer, 2018).
Revised: First, it has been shown that ﬁnite-dimension embedding can miss information about PDs (Carriere & Bauer, 2018).

Original: Second, the time of computing a kernel is quadratic in the number of PDs, which is prohibitive for large-scale applications.
Revised: Second, the time of computing a kernel is quadratic in the number of PDs, which is prohibitive for large-scale applications.

Original: Third, choosing the right vectorization/kernelization and its associated hyper-parameters is not straightforward and usually requires multiple rounds of trial and error.
Revised: Third, choosing the right vectorization/kernelization and its associated hyper-parameters is not straightforward and usually requires multiple rounds of trial and error.

Original: Due to these extra complexities, one may wonder whether the beneﬁts of using PDs outweigh the extra cost.
Revised: Due to these extra complexities, one may wonder whether the beneﬁts of using PDs outweigh the extra cost.

Original: Speciﬁcally, PDs have two major components: 1) ﬁltration function and 2) persistence pairing (decomposition of 1D persistence module).
Revised: Speciﬁcally, PDs have two major components: 1) ﬁltration function and 2) persistence pairing (decomposition of 1D persistence module).

Original: In this paper, we ask a simple yet fundamental question: How much extra power can persistence pairing bring in?
Revised: In this paper, we ask a simple yet fundamental question: How much extra power can persistence pairing bring in?

Original: Our contributions.
Revised: Our contributions.

Original: In this work, we propose a simple sanity check that can shield light on the power of PDs.
Revised: In this work, we propose a simple sanity check that can shield light on the power of PDs.

Original: Speciﬁcally, we permute the persistence points in such a way that only coordinates of
Revised: Speciﬁcally, we permute the persistence points in such a way that only coordinates of

Original: PDs remain the same but the original pairing is completely destroyed.
Revised: PDs remain the same but the original pairing is completely destroyed.

Original: These fake PDs have the same form as the original PDs and therefore the same kernels for true diagrams can also be applied to fake ones.
Revised: These fake PDs have the same form as the original PDs and therefore the same kernels for true diagrams can also be applied to fake ones.

Original: We use these fake diagrams as the input for various tasks and check their effectiveness.
Revised: We use these fake diagrams as the input for various tasks and check their effectiveness.

Original: As we will see, this simple trick brings various insights on the use of PDs for different problems.
Revised: As we will see, this simple trick brings various insights on the use of PDs for different problems.

Original: To the best of our knowledge, our paper is the ﬁrst work systematically quantifying (empirically) the power of persistence pairing for various applications.
Revised: To the best of our knowledge, our paper is the ﬁrst work systematically quantifying (empirically) the power of persistence pairing for various applications.

Original: Speciﬁcally, our contributions are the following.
Revised: Speciﬁcally, our contributions are the following.

Original: • We propose the sanity check for PDs that decouples the statistics of the critical values of ﬁltration function and the persistence pairing.
Revised: • We propose the sanity check for PDs that decouples the statistics of the critical values of ﬁltration function and the persistence pairing.

Original: Using the proposed method, we ﬁnd that in graph classiﬁ- cation, even fake diagrams perform quite well compared to original PDs.
Revised: Using the proposed method, we ﬁnd that in graph classiﬁcation, even fake diagrams perform quite well compared to original PDs.

Original: As a byproduct of our extensive experiments, we also provide some rules of thumb for using PDs in graph classiﬁcation.
Revised: As a byproduct of our extensive experiments, we also provide some rules of thumb for using PDs in graph classiﬁcation.

Original: • For shape segmentation and classiﬁcation, we ﬁnd the power of persistence pairing depends on the particular featurization chosen.
Revised: • For shape segmentation and classiﬁcation, we ﬁnd the power of persistence pairing depends on the particular featurization chosen.

Original: With the right choice of featurization, utilizing persistence pairing brings in signiﬁcant improvement.
Revised: With the right choice of featurization, utilizing persistence pairing brings in signiﬁcant improvement.

Original: Intuitively, we think that the shape models have more prominent geometric features in them, which are effectively captured by PDs.
Revised: Intuitively, we think that the shape models have more prominent geometric features in them, which are effectively captured by PDs.

Original: In contrast, PDs seem to be less effective at capturing features for graphs, partly due to the choice of descriptor functions as well as the nature of noise in graph (random insertions) which makes PDs less stable.
Revised: In contrast, PDs seem to be less effective at capturing features for graphs, partly due to the choice of descriptor functions as well as the nature of noise in graph (random insertions) which makes PDs less stable.

Original:  E XPERIMENT
Revised:  E XPERIMENT

Original: Setup.
Revised: Setup.

Original: On a high level, we use PDs obtained from different ﬁltration functions as features.
Revised: On a high level, we use PDs obtained from different ﬁltration functions as features.

Original: Depending on the task, we choose a proper featurization method followed by SVM for ﬁnal classiﬁcation.
Revised: Depending on the task, we choose a proper featurization method followed by SVM for ﬁnal classiﬁcation.

Original: We use kernel methods (see D.2) when possible since they tend to perform better, but for large-scale applications, computing kernels is not feasible so we use vector methods (see D.1).
Revised: We use kernel methods (see D.2) when possible since they tend to perform better, but for large-scale applications, computing kernels is not feasible so we use vector methods (see D.1).

Original: We maintain the same experiment settings for original PDs and permuted ones (see below).
Revised: We maintain the same experiment settings for original PDs and permuted ones (see below).

Original: For a comprehensive evaluation, we perform experiments on various tasks (graph classiﬁcation, shape segmentation, and object classiﬁcation) and diverse data types including social networks, molecules/proteins, and meshes from 3D shapes.The code is available on github.
Revised: For a comprehensive evaluation, we perform experiments on various tasks (graph classiﬁcation, shape segmentation, and object classiﬁcation) and diverse data types including social networks, molecules/proteins, and meshes from 3D shapes.

Original: Filtration function.
Revised: Filtration function.

Original: We test degree (deg), Ricci Curvature (Lin et al., 2011) (Ricci), closeness centrality (cc), and square of Fiedler vector (the eigenvector corresponding to the second smallest eigenvalue of graph Laplacian, denoted as Fiedler-s) for graph classiﬁcation.
Revised: We test degree (deg), Ricci Curvature (Lin et al., 2011) (Ricci), closeness centrality (cc), and square of Fiedler vector (the eigenvector corresponding to the second smallest eigenvalue of graph Laplacian, denoted as Fiedler-s) for graph classiﬁcation.

Original: On shape datasets, we use geodesic distance as ﬁltration function for shape segmentation and closeness centrality for shape classiﬁcation.
Revised: On shape datasets, we use geodesic distance as ﬁltration function for shape segmentation and closeness centrality for shape classiﬁcation.

Original: See details about kernels and datasets at appendix A.1.
Revised: See details about kernels and datasets at appendix A.1.

Original: Sanity check for PDs.
Revised: Sanity check for PDs.

Original: We introduce a sanity check that aims to preserve statistics of ﬁltration function but destroy the persistence pairing of critical values of the ﬁltration function.
Revised: We introduce a sanity check that aims to preserve statistics of ﬁltration function but destroy the persistence pairing of critical values of the ﬁltration function.

Original: For a diagram
Revised: For a diagram

Original: P = { p 1 , p 2 , ..., p n } of size n , denote the coordinates of point p i by ( x i , y i ) where x i (cid:54)
Revised: P = { p 1 , p 2 , ..., p n } of size n , denote the coordinates of point p i by ( x i , y i ) where x i (cid:54)

Original: =
Revised: =

Original: y i for all i .
Revised: y i for all i .

Original: Take the multiset P multiset = { x 1 , y 1 , x 2 , y 2 , ..., x n , y n } as input.
Revised: Take the multiset P multiset = { x 1 , y 1 , x 2 , y 2 , ..., x n , y n } as input.

Original: We then randomly sample two values, denoted by a and b , without replacement from P multiset as the coordinates of the persistence point in the fake diagram.
Revised: We then randomly sample two values, denoted by a and b , without replacement from P multiset as the coordinates of the persistence point in the fake diagram.

Original: For diagrams from sublevel ﬁltrations, we ensure all persistence points in permuted diagrams lie above the diagonal by taking min ( a, b ) as x-coordiniate and max ( a, b ) as y-coordinate of the point in the permuted diagram.
Revised: For diagrams from sublevel ﬁltrations, we ensure all persistence points in permuted diagrams lie above the diagonal by taking min ( a, b ) as x-coordiniate and max ( a, b ) as y-coordinate of the point in the permuted diagram.

Original: Note that it is possible that two birth values can be paired in fake diagrams.
Revised: Note that it is possible that two birth values can be paired in fake diagrams.

Original: Similar strategy is applied for superlevel ﬁltration as well.
Revised: Similar strategy is applied for superlevel ﬁltration as well.

Original: Repeat the same procedure n times and get a fake diagram called P fake of size n .
Revised: Repeat the same procedure n times and get a fake diagram called P fake of size n .

Original: Baseline.
Revised: Baseline.

Original: As permuting diagrams adds stochasticity in persistence diagrams, we also introduce two deterministic baselines to better quantify the power of persistence pairing.
Revised: As permuting diagrams adds stochasticity in persistence diagrams, we also introduce two deterministic baselines to better quantify the power of persistence pairing.

Original: Pervec (vector obtained from coordinates of points in the per muted diagram) is a histogram vector of the coordinates of
Revised: Pervec (vector obtained from coordinates of points in the per muted diagram) is a histogram vector of the coordinates of

Original: PD, i.e., the histogram of P .
Revised: PD, i.e., the histogram of P .

Original: Filvec is a histogram vector of all the ﬁl tration function values on the graph.
Revised: Filvec is a histogram vector of all the ﬁl tration function values on the graph.

Original: The length of Pervec and Filvec is a hyper-parameter chosen from { 100 , 200 , 300 } by cross-validation.
Revised: The length of Pervec and Filvec is a hyper-parameter chosen from { 100 , 200 , 300 } by cross-validation.

Original:  SANITY CHECK FOR G RAPHS
Revised:  SANITY CHECK FOR G RAPHS

Original: [Figure]
Revised: [Figure]

Original: In this section, we perform graph classiﬁcation on common benchmark datasets with different ﬁltration functions and featurization.
Revised: In this section, we perform graph classiﬁcation on common benchmark datasets with different ﬁltration functions and featurization.

Original: Through extensive experiments (see full table on synthetic and real graphs at appendix A.2), we draw the following conclusions.
Revised: Through extensive experiments (see full table on synthetic and real graphs at appendix A.2), we draw the following conclusions.

Original: Choice of ﬁltration function.
Revised: Choice of ﬁltration function.

Original: The choice of ﬁltration function clearly matters.
Revised: The choice of ﬁltration function clearly matters.

Original: For datasets such as BZR , COX2 , DD , DHFR , FRANKENSTEIN , NCI1 ,
Revised: For datasets such as BZR , COX2 , DD , DHFR , FRANKENSTEIN , NCI1 ,

Original: REDDIT 5K using Ricci curvature as the ﬁltration function yields the best accuracy as opposed to other ﬁltration functions.
Revised: REDDIT 5K using Ricci curvature as the ﬁltration function yields the best accuracy as opposed to other ﬁltration functions.

Original: If we ﬁx  method to be Sliced Wasserstein kernel (sw), for graphs like IMDB-B (69.5 for degree vs. 69.2 for Ricci) and IMDB-M (43.1 vs. 43.7), PROTEINS (73.6 vs. 73.8), D&D (76.1 vs. 76.9), even degree performs as well as Ricci.
Revised: If we ﬁx the method to be Sliced Wasserstein kernel (sw), for graphs like IMDB-B (69.5 for degree vs. 69.2 for Ricci) and IMDB-M (43.1 vs. 43.7), PROTEINS (73.6 vs. 73.8), D&D (76.1 vs. 76.9), even degree performs as well as Ricci.

Original: This is consistent with the ﬁndings in the paper (Cai & Wang, 2018) where it is shown simple statistics based on node degree can perform on par with the state of the art.
Revised: This is consistent with the ﬁndings in the paper (Cai & Wang, 2018) where it is shown simple statistics based on node degree can perform on par with the state of the art.

Original: This raises the concern that the current benchmark datasets might be limited in evaluating different methods.
Revised: This raises the concern that the current benchmark datasets might be limited in evaluating different methods.

Revised: [Table]

Original: [Table]
Revised: [Table]

Original: Sliced Wasserstein kernel + Ricci is powerful.
Revised: Sliced Wasserstein kernel + Ricci is powerful.

Original: Choosing the best accuracy for sw among different ﬁltrations yields decent performance.
Revised: Choosing the best accuracy for sw among different ﬁltrations yields decent performance.

Original: COX2 : 80.5 (best accuracy when using sw) vs. 82.0 (best accuracy among all ﬁltration functions and featurizations for a single dataset).
Revised: COX2 : 80.5 (best accuracy when using sw) vs. 82.0 (best accuracy among all ﬁltration functions and featurizations for a single dataset).

Original: BZR : 88.4 vs. 88.4.
Revised: BZR : 88.4 vs. 88.4.

Original: DD : 76.9 vs. 77.4.
Revised: DD : 76.9 vs. 77.4.

Original: DHFR : 82.8 vs. 82.8.
Revised: DHFR : 82.8 vs. 82.8.

Original: FRANKENSTEIN : 72.0 vs. 72.3.
Revised: FRANKENSTEIN : 72.0 vs. 72.3.

Original: IMDB-B : 69.2 vs. 69.5.
Revised: IMDB-B : 69.2 vs. 69.5.

Original: IMDB-M : 45.2 vs. 46.5.
Revised: IMDB-M : 45.2 vs. 46.5.

Original: NCI1 : 77.8 vs. 77.8.
Revised: NCI1 : 77.8 vs. 77.8.

Original: PROTEINS : 73.8 vs. 74.0.
Revised: PROTEINS : 73.8 vs. 74.0.

Original: REDDIT 5K : 54.1 vs.
Revised: REDDIT 5K : 54.1 vs.

Original: As a corollary, to achieve good performance for graph classiﬁcation, a rule of thumb is to use
Revised: As a corollary, to achieve good performance for graph classiﬁcation, a rule of thumb is to use

Original: Ricci curvature as ﬁltration plus Sliced Wasserstein kernel.
Revised: Ricci curvature as ﬁltration plus Sliced Wasserstein kernel.

Original: Sanity check.
Revised: Sanity check.

Original: Looking at the mean accuracies over four different ﬁltration functions, sw performs better than sw-p/pervec/ﬁlvec, although the amount of improvement depends on the dataset.
Revised: Looking at the mean accuracies over four different ﬁltration functions, sw performs better than sw-p/pervec/ﬁlvec, although the amount of improvement depends on the dataset.

Original: For
Revised: For

Original: BZR , DHFR , FRANKENSTEIN , IMDB-B , NCI1 , PROTEIN , REDDIT 5K sw is better than ﬁlvec, pervec, and sw-p.
Revised: BZR , DHFR , FRANKENSTEIN , IMDB-B , NCI1 , PROTEIN , REDDIT 5K sw is better than ﬁlvec, pervec, and sw-p.

Original: For COX2 , DD , sw is no better than the best of ﬁlvec/pervec/sw-p, but the gap (0. for COX2 and 1.05 for DD ) is small.
Revised: For COX2 , DD , sw is no better than the best of ﬁlvec/pervec/sw-p, but the gap (0. for COX2 and 1.05 for DD ) is small.

Original: Pervec and ﬁlvec are used as baselines.
Revised: Pervec and ﬁlvec are used as baselines.

Original: The performance of pervec, ﬁlvec and sw-p is expected to be close to each other since none of them is using persistence pairing.
Revised: The performance of pervec, ﬁlvec and sw-p is expected to be close to each other since none of them is using persistence pairing.

Original: This is indeed the case.
Revised: This is indeed the case.

Original: The best of pervec and ﬁlvec is close to sw-p for all the graphs.
Revised: The best of pervec and ﬁlvec is close to sw-p for all the graphs.

Original: (The difference is less than 2.5 percent.)
Revised: (The difference is less than 2.5 percent.)

Original: Note hyper-parameter choice for pervec/ﬁlvec and sw-p will also result in different performance.
Revised: Note hyper-parameter choice for pervec/ﬁlvec and sw-p will also result in different performance.

Original: After all, the input for pervec/ﬁlvec are vectors while for sw-p
Revised: After all, the input for pervec/ﬁlvec are vectors while for sw-p

Original: the input is fake PDs.
Revised: the input is fake PDs.

Original: We believe that the difference between pervec, ﬁlvec and sw-p is reasonable, and the conclusion that most discriminative power comes from function values is robust.
Revised: We believe that the difference between pervec, ﬁlvec and sw-p is reasonable, and the conclusion that most discriminative power comes from function values is robust.

Original: Learning for PDs.
Revised: Learning for PDs.

Original: We compare the accuracy obtained from sw + best ﬁltration function with
Revised: We compare the accuracy obtained from sw + best ﬁltration function with

Original: Perslay (Carriere et al., 2019) /WKPI
Revised: Perslay (Carriere et al., 2019) /WKPI

Original: (Zhao & Wang, 2019) where learning is involved.
Revised: (Zhao & Wang, 2019) where learning is involved.

Original: See B. for details of Perslay and WKPI.
Revised: See B. for details of Perslay and WKPI.

Original: As shown in Table 2, our method is comparable with Perslay.
Revised: As shown in Table 2, our method is comparable with Perslay.

Original: We conjecture replacing original ﬁltration (based on heat kernel signature) used in Perslay with Ricci curvature may improve its performance for some datasets.
Revised: We conjecture replacing original ﬁltration (based on heat kernel signature) used in Perslay with Ricci curvature may improve its performance for some datasets.

Original: WKPI (WKPI-kM and WKPI-kC differ in how they initialize weights.)
Revised: WKPI (WKPI-kM and WKPI-kC differ in how they initialize weights.)

Original: learns the weights of different points in diagrams [Table] which results in much better performance.
Revised: learns the weights of different points in diagrams, which results in much better performance.

Original: This conﬁrms the belief that to fully utilize the power of PDs, a proper weighting scheme is crucial
Revised: This conﬁrms the belief that to fully utilize the power of PDs, a proper weighting scheme is crucial.

Original: [Table]

Original: [Table]
Revised: [Table]

Original: The effect of adding loops.
Revised: The effect of adding loops.

Original: We analyze the effect of adding loops, denoted as Ext − 1 (see B. for details on extended PDs), for graph classiﬁcation.
Revised: We analyze the effect of adding loops, denoted as Ext − 1 (see B. for details on extended PDs), for graph classiﬁcation.

Original: Since extended PDs capture loops in graphs with respect to the ﬁltration function, the hope is that adding Ext − 1 in the PD will make it more discriminative.
Revised: Since extended PDs capture loops in graphs with respect to the ﬁltration function, the hope is that adding Ext − 1 in the PD will make it more discriminative.

Original: But one can perhaps argue the improvement may come from the coordinate values of extended PDs, so we ﬁx featurization as pervec (as opposed to sw) to see the difference.
Revised: But one can perhaps argue the improvement may come from the coordinate values of extended PDs, so we ﬁx featurization as pervec (as opposed to sw) to see the difference.

Original: Tableshows that adding coordinates of extended PDs increase the accuracy, no matter whether we utilize pairing (sw) or not (pervec).
Revised: Table shows that adding coordinates of extended PDs increase the accuracy, no matter whether we utilize pairing (sw) or not (pervec).

Original:  SANITY CHECK FOR S HAPES
Revised:  SANITY CHECK FOR S HAPES

Original: We now perform sanity check on the problem of supervised 3D shape segmentation and object classiﬁcation (see appendix A.3 due to space limitation).
Revised: We now perform sanity check on the problem of supervised 3D shape segmentation and object classiﬁcation (see appendix A.3 due to space limitation).

Original: As we will see, the choice of permuting diagrams makes a much bigger difference, shown in Table 7 and Table 8.
Revised: As we will see, the choice of permuting diagrams makes a much bigger difference, shown in Table 7 and Table 8.

Original: For each vertex x , we use the geodesic distance (distance to x ) as the ﬁltration function and compute the super-level 0-PDs as features.
Revised: For each vertex x , we use the geodesic distance (distance to x ) as the ﬁltration function and compute the super-level 0-PDs as features.

Original: We convert PDs into feature vectors via either persistence landscape (Bubenik, 2015) or persistence image (Adams et al., 2017) and use SVM (Ma & Belkin, 2017) as our classiﬁer.
Revised: We convert PDs into feature vectors via either persistence landscape (Bubenik, 2015) or persistence image (Adams et al., 2017) and use SVM (Ma & Belkin, 2017) as our classiﬁer.

Original: We use Princeton shape benchmark as dataset.
Revised: We use Princeton shape benchmark as dataset.

Original: This benchmark contains several different ground truth segmentations for each shape.
Revised: This benchmark contains several different ground truth segmentations for each shape.

Original: For each shape in the training set, we use the same ground truth segmentation as Kalogerakis et al.
Revised: For each shape in the training set, we use the same ground truth segmentation as Kalogerakis et al.

Original: For each category, we use 50% data for training and the rest for testing.
Revised: For each category, we use 50% data for training and the rest for testing.

Original: The partial results are shown in Table 3 (see Table 8 for full results), from which we have following observations.[Table] [Table] [Table]
Revised: The partial results are shown in Table 3 (see Table 8 for full results), from which we have following observations.

Original: [Table]

Original: [Table]
Revised: [Table]

Original: Results.
Revised: Results.

Original: The effects of sanity check clearly depend on the vectorization selected.
Revised: The effects of sanity check clearly depend on the vectorization selected.

Original: For Persistence Landscape (PL), there are some categories (such as cup, vase...) where permuting
Revised: For Persistence Landscape (PL), there are some categories (such as cup, vase...) where permuting

Original: PDs yields better results than the original diagrams.
Revised: PDs yields better results than the original diagrams.

Original: Why?
Revised: Why?

Original: We ﬁnd corresponding diagrams for those categories all have very few persistence points (less than 2) far away from the diagonal on average.
Revised: We ﬁnd corresponding diagrams for those categories all have very few persistence points (less than 2) far away from the diagonal on average.

Original: See diagram statistics in Table 11.
Revised: See diagram statistics in Table 11.

Original: In the formulation of persistence landscape, points close to diagonal are treated less important and thus play a less important role in ﬁnal shape segmentation.
Revised: In the formulation of persistence landscape, points close to diagonal are treated less important and thus play a less important role in ﬁnal shape segmentation.

Original: In contrast, permuting PDs will “pull” those points away from diagonal with high probability and make them more important under PL’s framework, therefore explaining the fact that permuting diagrams coupled with PL yields better results for those categories.
Revised: In contrast, permuting PDs will “pull” those points away from diagonal with high probability and make them more important under PL’s framework, therefore explaining the fact that permuting diagrams coupled with PL yields better results for those categories.

Original: In contrast, Persistence Image (PI) takes a very different way to vectorize PDs.
Revised: In contrast, Persistence Image (PI) takes a very different way to vectorize PDs.

Original: In particular, PI assigns a weight for each point and points near the diagonal are not necessarily assigned small weight.
Revised: In particular, PI assigns a weight for each point and points near the diagonal are not necessarily assigned small weight.

Original: We use the same weight (proportional to death time) as in the original paper.
Revised: We use the same weight (proportional to death time) as in the original paper.

Original: It turns out that 1) permuting PDs for PI always gives worse results except for categories bearing and bust where a very small gap exists and 2) PI yields better results on all shapes compared to
Revised: It turns out that 1) permuting PDs for PI always gives worse results except for categories bearing and bust where a very small gap exists and 2) PI yields better results on all shapes compared to

Original: PL.
Revised: PL.

Original: Interestingly, given that PL gives smaller weights to points closer to the diagonal, while PI does not necessarily do (as in this experiments), we think this suggests that while PD can identify “features” of input shapes in a canonical way, the importance of these features (especially in terms of the tasks they are used for) may not be consistent with their persistence.
Revised: Interestingly, given that PL gives smaller weights to points closer to the diagonal, while PI does not necessarily do (as in this experiments), we think this suggests that while PD can identify “features” of input shapes in a canonical way, the importance of these features (especially in terms of the tasks they are used for) may not be consistent with their persistence.

Original: This point has been made earlier in Kusano et al.
Revised: This point has been made earlier in Kusano et al.

Original: ; Adams et al.
Revised: ; Adams et al.

Original: ; Zhao & Wang (2019), which allow assigning different weights to persistence points.
Revised: ; Zhao & Wang (2019); Cai & Wang (2020), which allow assigning different weights to persistence points.

Original: In our case, sanity check is proved to be a handy trick that can be applied to quickly test if the downstream featurization is effective.
Revised: In our case, sanity check is proved to be a handy trick that can be applied to quickly test if the downstream featurization is effective.

Original:  C ONCLUSION AND F UTURE W ORK
Revised:  C ONCLUSION AND F UTURE W ORK

Original: By introducing sanity check for PDs , we ﬁnd persistence pairing is crucial for shape datasets.
Revised: By introducing sanity check for PDs , we ﬁnd persistence pairing is crucial for shape datasets.

Original: For graph datasets, although the small (yet consistent) improvements pairing brings may be unexpected and unsatisfying, we interpret this due to the challenging nature of graph classiﬁcation and dataset problem.
Revised: For graph datasets, although the small (yet consistent) improvements pairing brings may be unexpected and unsatisfying, we interpret this due to the challenging nature of graph classiﬁcation and dataset problem.

Original: In the future, we are interested in understanding the discriminative power of PDs in a principle way.
Revised: In the future, we are interested in understanding the discriminative power of PDs in a principle way.

Original: We believe developing a connection between the structure of the persistence module and generalization error is important for the application of PDs in machine learning.
Revised: We believe developing a connection between the structure of the persistence module and generalization error is important for the application of PDs in machine learning.

Original: A M ISSING DETAILS A. M ISSING D ETAILS AT S ECTION 2: D ATASETS AND C HOICE OF K ERNELS
Revised: A M ISSING DETAILS A. M ISSING D ETAILS AT S ECTION 2: D ATASETS AND C HOICE OF K ERNELS

Original: We apply sanity check on diverse data types, whose quantitative summaries are in the appendix.
Revised: We apply sanity check on diverse data types, whose quantitative summaries are in the appendix.

Original: Graph Datasets : We perform experiments on common benchmark datasets for graph classiﬁcation.
Revised: Graph Datasets : We perform experiments on common benchmark datasets for graph classiﬁcation.

Original: IMDB-B , IMDB-M , REDDIT 5K are composed of social networks.
Revised: IMDB-B , IMDB-M , REDDIT 5K are composed of social networks.

Original: BZR , COX2 , DD , DHFR ,
Revised: BZR , COX2 , DD , DHFR ,

Original: FRANKENSTEIN , NCI1 , PROTEINS , PTC are graphs from medical or biological applications.
Revised: FRANKENSTEIN , NCI1 , PROTEINS , PTC are graphs from medical or biological applications.

Original: Shape Datasets : We use Princeton Shape Benchmark (Chen et al., 2009) for shape segmentation.
Revised: Shape Datasets : We use Princeton Shape Benchmark (Chen et al., 2009) for shape segmentation.

Original: This benchmark contains 19 categories of different objects (human, cup, glasses...) with 20 shapes for each category.
Revised: This benchmark contains 19 categories of different objects (human, cup, glasses...) with 20 shapes for each category.

Original: For shape classiﬁcation, we use ModelNet10/ModelNet40 (Wu et al., 2015) where there are 4899/12,311 CAD models from 10/40 man-made object categories (bathtub, bed, chair...) respectively.
Revised: For shape classiﬁcation, we use ModelNet10/ModelNet40 (Wu et al., 2015) where there are 4899/12,311 CAD models from 10/40 man-made object categories (bathtub, bed, chair...) respectively.

Original: Choice of Kernels : We test four kernels for PDs i.e., sliced Wasserstein kernel (sw), persistence scale space kernel (pss), persistence weighted Gaussian kernel (pwg) and persistence Fisher kernel (pf) for different ﬁltration functions on PROTEINS .
Revised: Choice of Kernels : We test four kernels for PDs i.e., sliced Wasserstein kernel (sw), persistence scale space kernel (pss), persistence weighted Gaussian kernel (pwg) and persistence Fisher kernel (pf) for different ﬁltration functions on PROTEINS .

Original: As shown in Table 4 in the appendix, the performance of different kernels are close in terms of accuracy.
Revised: As shown in Table 4 in the appendix, the performance of different kernels are close in terms of accuracy.

Original: We prefer Sliced Wasserstein kernel mainly because it is fast ( O ( mlogm ) as opposed to O ( m 2 ) for pss), easy to tune (search over 5 values for bandwidth as opposed to 45 hyper-parameter combinations for pwg), and still yields decent performance.
Revised: We prefer Sliced Wasserstein kernel mainly because it is fast ( O ( mlogm ) as opposed to O ( m 2 ) for pss), easy to tune (search over 5 values for bandwidth as opposed to 45 hyper-parameter combinations for pwg), and still yields decent performance.

Original: Note that in this paper we focus on understanding the extra power persistence pairing brings in, not achieving the state of the art.
Revised: Note that in this paper we focus on understanding the extra power persistence pairing brings in, not achieving the state of the art.

Original: [Table]
Revised: [Table]

Original: 73.58 sw 74.
Revised: 73.58 sw 74.

Original: 73.72 pwg 74.
Revised: 73.72 pwg 74.

Original: A. M ISSING D ETAILS AT S ECTION 3: S YNTHETIC G RAPH D ATA
Revised: A. M ISSING D ETAILS AT S ECTION 3: S YNTHETIC G RAPH D ATA

Original: In this section, we perform sanity check on synthetic graph dataset where graphs sampled from 2 stochastic block models are classiﬁed.
Revised: In this section, we perform sanity check on synthetic graph dataset where graphs sampled from 2 stochastic block models are classiﬁed.

Original: In particular, denote sbm ( n 1 , n 2 , p, q ) as stochastic block model of two blocks of size n 1 and n 2 , and within each block, the edge probability is p , while between blocks, the edge probability is q .
Revised: In particular, denote sbm ( n 1 , n 2 , p, q ) as stochastic block model of two blocks of size n 1 and n 2 , and within each block, the edge probability is p , while between blocks, the edge probability is q .

Original: We sample 1000 graphs for classiﬁcation from sbm (100 , 50 , 0 . 5 , 0 . 1) and sbm (75 , 75 , 0 . 4 , 0 . 2) .
Revised: We sample 1000 graphs for classiﬁcation from sbm (100 , 50 , 0 . 5 , 0 . 1) and sbm (75 , 75 , 0 . 4 , 0 . 2) .

Original: This is a simple classiﬁcation problem and there are many ways to achieve perfect results.
Revised: This is a simple classiﬁcation problem and there are many ways to achieve perfect results.

Original: To make the classiﬁcation harder, we randomly ﬂip some labels.
Revised: To make the classiﬁcation harder, we randomly ﬂip some labels.

Original: For example, when label noise is 0.1, we randomly ﬂip 10% of labels.
Revised: For example, when label noise is 0.1, we randomly ﬂip 10% of labels.

Original: We are interested in the behaviors of whether to perform sanity check or not under different label noise levels.
Revised: We are interested in the behaviors of whether to perform sanity check or not under different label noise levels.

Original: As shown in Figure 3, independent from ﬁltration functions used, applying sanity check has rather small effects on the ﬁnal performance for different noise levels.
Revised: As shown in Figure 3, independent from ﬁltration functions used, applying sanity check has rather small effects on the ﬁnal performance for different noise levels.

Original: [Figure]
Revised: [Figure]

Original: [Table]
Revised: [Table]

Original: A. M ISSING D ETAILS AT S ECTION 4: 3D O BJECT CLASSIFICATION
Revised: A. M ISSING D ETAILS AT S ECTION 4: 3D O BJECT CLASSIFICATION

Original: Next, we evaluate our method for shape classiﬁcation on ModelNet10/ModelNet40.
Revised: Next, we evaluate our method for shape classiﬁcation on ModelNet10/ModelNet40.

Original: For each shape, 1024 points are uniformly sampled on mesh faces according to face area and normalized into a unit sphere.
Revised: For each shape, 1024 points are uniformly sampled on mesh faces according to face area and normalized into a unit sphere.

Original: We then construct a 8-neighborhood graph from sampled points and use closeness centrality as our ﬁltration function.
Revised: We then construct a 8-neighborhood graph from sampled points and use closeness centrality as our ﬁltration function.

Original: For each shape, we use the resulting PDs as the shape representations and use sw/pf/PI/PL for object classiﬁcation.
Revised: For each shape, we use the resulting PDs as the shape representations and use sw/pf/PI/PL for object classiﬁcation.

Original: [Table]
Revised: [Table]

Original: Table 7 shows that if we do not utilize persistence pairing, we get very similar results for three methods (P, Pervec, Filvec), which are much worse than result using PDs (NP), regardless of featurizations.
Revised: Table 7 shows that if we do not utilize persistence pairing, we get very similar results for three methods (P, Pervec, Filvec), which are much worse than result using PDs (NP), regardless of featurizations.

Original: A detailed performance breakdown on ModelNet 10 is also shown in Table 6 where permuting diagrams yields much worse results on all shape categories except dresser.
Revised: A detailed performance breakdown on ModelNet 10 is also shown in Table 6 where permuting diagrams yields much worse results on all shape categories except dresser.

Original: NP
Revised: NP

Original: P
Revised: P

Original: Pervec
Revised: Pervec

Original: Filvec
Revised: Filvec

Original: ModelNet55 . 2 / 53 .
Revised: ModelNet55 . 2 / 53 .

Original: [Table]
Revised: [Table]

Original: Results both in Table 8 and 7 suggest that persistence pairings are effective and meaningful for 3D models, which partly could be due to that 3D models tend to have clear geometric features that are also stable under the typical types of (Hausdorff) noise added to these models.
Revised: Results both in Table 8 and 7 suggest that persistence pairings are effective and meaningful for 3D models, which partly could be due to that 3D models tend to have clear geometric features that are also stable under the typical types of (Hausdorff) noise added to these models.

Original: In contrast, persistent homology seems to be less effective at capturing features for graphs: this could partly be due to the choice of descriptor functions used for graphs are not effective at capturing features.
Revised: In contrast, persistent homology seems to be less effective at capturing features for graphs: this could partly be due to the choice of descriptor functions used for graphs are not effective at capturing features.

Original: Another potential reason could be that PDs are more sensitive to the common types of noise in graphs (random insertions), and hence resulting persistence-based features are less stable and meaningful.
Revised: Another potential reason could be that PDs are more sensitive to the common types of noise in graphs (random insertions), and hence resulting persistence-based features are less stable and meaningful.

Original: A. A NALYSIS OF T RUE AND F AKE D IAGRAMS A.4. S EPARATION FROM T RUE AND F AKE D IAGRAMS
Revised: A. A NALYSIS OF T RUE AND F AKE D IAGRAMS A.4. S EPARATION FROM T RUE AND F AKE D IAGRAMS

Original: Due to the decent result without using persistence pairing on graph datasets, one may wonder whether there is any useful structure contained in persistence pairing for graph classiﬁcation.
Revised: Due to the decent result without using persistence pairing on graph datasets, one may wonder whether there is any useful structure contained in persistence pairing for graph classiﬁcation.

Original: To better understand the implications of the sanity check , we try to separate true PDs from fake PDs.
Revised: To better understand the implications of the sanity check , we try to separate true PDs from fake PDs.

Original: For each dataset and ﬁltration function, we compute PDs and generate fake diagrams by applying sanity check .
Revised: For each dataset and ﬁltration function, we compute PDs and generate fake diagrams by applying sanity check .

Original: We compute the Sliced Wasserstein kernel and train SVM to discriminate the true diagrams from the fake ones.
Revised: We compute the Sliced Wasserstein kernel and train SVM to discriminate the true diagrams from the fake ones.

Original: We observe in Table 9 consistently that regardless of the datasets and ﬁltration functions, we can easily get 85%-100% accuracy.
Revised: We observe in Table 9 consistently that regardless of the datasets and ﬁltration functions, we can easily get 85%-100% accuracy.

Original: This suggests that true diagrams have some structure that is very different from fake ones.
Revised: This suggests that true diagrams have some structure that is very different from fake ones.

Original: A.4. C ONFUSION M ATRIX A NALYSIS
Revised: A.4. C ONFUSION M ATRIX A NALYSIS

Original: We also examine the confusion matrix in cases where we need to additionally differentiate true/fake diagrams.
Revised: We also examine the confusion matrix in cases where we need to additionally differentiate true/fake diagrams.

Original: In particular, for each graph G i with label y
Revised: In particular, for each graph G i with label y

Original: i
Revised: i

Original: (assume all labels are represented as positive numbers), we compute the true PD of each graph and generate a fake diagram.
Revised: (assume all labels are represented as positive numbers), we compute the true PD of each graph and generate a fake diagram.

Original: In scenario 1) we assign both diagrams as same label y i and in scenario 2), we assign two diagram with different label y i and − y i .
Revised: In scenario 1) we assign both diagrams as same label y i and in scenario 2), we assign two diagram with different label y i and − y i .

Original: In other words, in the second scenario, a classiﬁer has to differentiate both true/fake
Revised: In other words, in the second scenario, a classiﬁer has to differentiate both true/fake

Original: [Table]
Revised: [Table]

Original: [Table]
Revised: [Table]

Original: diagrams and diagrams of different types of graphs.
Revised: diagrams and diagrams of different types of graphs.

Original: We want to know whether scenario 2 will make problem harder.
Revised: We want to know whether scenario 2 will make problem harder.

Original: In particular, we use node degree as ﬁltration function and Sliced Wasserstein kernel.
Revised: In particular, we use node degree as ﬁltration function and Sliced Wasserstein kernel.

Original: It can be seen in Table 10 that fake diagrams never get confused with true diagrams.
Revised: It can be seen in Table 10 that fake diagrams never get confused with true diagrams.

Original: We can recover the confusion matrix on the left from the matrix on the right by merging true and fake diagrams.
Revised: We can recover the confusion matrix on the left from the matrix on the right by merging true and fake diagrams.

Original: B B ACKGROUND B. P ERSISTENT H OMOLOGY
Revised: B B ACKGROUND B. P ERSISTENT H OMOLOGY

Original: The deﬁnition of our proposed method relies on the so-called persistence diagram induced by a scalar function.
Revised: The deﬁnition of our proposed method relies on the so-called persistence diagram induced by a scalar function.

Original: We refer readers to resources such as (Edelsbrunner & Harer, 2010; Oudot, 2015) for formal discussions on persistent homology and related developments.
Revised: We refer readers to resources such as (Edelsbrunner & Harer, 2010; Oudot, 2015) for formal discussions on persistent homology and related developments.

Original: Below we only provide
Revised: Below we only provide

Original: [Table]
Revised: [Table]

Original: an intuitive and informal description of the persistent homology induced by a function under a simple setting.
Revised: an intuitive and informal description of the persistent homology induced by a function under a simple setting.

Original: Let f : X → R be a continuous real-valued function deﬁned on a topological space X .
Revised: Let f : X → R be a continuous real-valued function deﬁned on a topological space X .

Original: We want to understand the structure of X from the perspective of f .
Revised: We want to understand the structure of X from the perspective of f .

Original: Speciﬁcally, let
Revised: Speciﬁcally, let

Original: X α :=
Revised: X α :=

Original: { x ∈ X | f ( x ) < α } denote the sublevel set of X w.r.t.
Revised: { x ∈ X | f ( x ) < α } denote the sublevel set of X w.r.t.

Original: α ∈ R .
Revised: α ∈ R .

Original: Now as we sweep X bottom-up (top down) by increasing the value, the sequence of sublevel (superlevel) sets connected by natural inclusion maps gives rise to a ﬁltration of X induced by f :
Revised: Now as we sweep X bottom-up (top down) by increasing the value, the sequence of sublevel (superlevel) sets connected by natural inclusion maps gives rise to a ﬁltration of X induced by f :

Original: [Equation]
Revised: [Equation]

Original: We track how the topological features of sublevel sets change in terms of homology classes.
Revised: We track how the topological features of sublevel sets change in terms of homology classes.

Original: In particular, as α increases, sometimes new topological features are born at time α , that is, new families of homology classes are created in H k
Revised: In particular, as α increases, sometimes new topological features are born at time α , that is, new families of homology classes are created in H k

Original: ( X α ) , the k -th homology group of X .
Revised: ( X α ) , the k -th homology group of X .

Original: Sometimes, existing topological features disappear, i.e, some homology classes become trivial in H k ( X β ) for some β > α .
Revised: Sometimes, existing topological features disappear, i.e, some homology classes become trivial in H k ( X β ) for some β > α .

Original: The persistent homology captures such birth and death events, and summarizes them in the so-called persistence diagram Dg k ( f ) (We will call it diagram for short when no ambiguity is raised).
Revised: The persistent homology captures such birth and death events, and summarizes them in the so-called persistence diagram Dg k ( f ) (We will call it diagram for short when no ambiguity is raised).

Original: Speciﬁcally, Dg k ( f ) consists of a set of persistence points ( α, β ) ∈ R 2 , where each ( α, β ) indicates a k -th homological feature created at α and killed at β .
Revised: Speciﬁcally, Dg k ( f ) consists of a set of persistence points ( α, β ) ∈ R 2 , where each ( α, β ) indicates a k -th homological feature created at α and killed at β .

Original: we also call ( α, β ) persistence pairing since it pairs two critical values α and β of the ﬁltration function.
Revised: we also call ( α, β ) persistence pairing since it pairs two critical values α and β of the ﬁltration function.

Original: In particular, 0 homology is just connected components and can be computed efﬁciently in
Revised: In particular, 0 homology is just connected components and can be computed efﬁciently in

Original: O ( α ( n ) n ) using union-ﬁnd data structure where α ( n ) is an extremely slow-growing inverse Ackermann function.
Revised: O ( α ( n ) n ) using union-ﬁnd data structure where α ( n ) is an extremely slow-growing inverse Ackermann function.

Original: B. E XTENDED P ERSISTENCE H OMOLOGY
Revised: B. E XTENDED P ERSISTENCE H OMOLOGY

Original: Ordinary persistence sometimes may be insufﬁcient to encode the topology of an object X .
Revised: Ordinary persistence sometimes may be insufﬁcient to encode the topology of an object X .

Original: For example, when X is a graph, the loops persist forever since they are not ﬁlled during the sublevel ﬁltration.
Revised: For example, when X is a graph, the loops persist forever since they are not ﬁlled during the sublevel ﬁltration.

Original: Similarly, upfork branching points (w.r.t.
Revised: Similarly, upfork branching points (w.r.t.

Original: the ﬁltration function f ) are not captured (while those pointing downwards are detected), since they do not create connected components in the sublevel ﬁltration.
Revised: the ﬁltration function f ) are not captured (while those pointing downwards are detected), since they do not create connected components in the sublevel ﬁltration.

Original: To address the issues above, extended persistence reﬁnes the analysis by also including the superlevel set
Revised: To address the issues above, extended persistence reﬁnes the analysis by also including the superlevel set

Original: X α = { x ∈ X : f ( x ) ≥ α } into the ﬁltration in Eqn (1).
Revised: X α = { x ∈ X : f ( x ) ≥ α } into the ﬁltration in Eqn (1).

Original: Similarly, letting α decrease from ∞ to −∞ gives a sequence of increasing subsets, for which structural changes can be recorded.
Revised: Similarly, letting α decrease from ∞ to −∞ gives a sequence of increasing subsets, for which structural changes can be recorded.

Original: In particular, assuming we have a sequence of reals α 1 < α 2 < · · · < α m such that X α 1 = ∅
Revised: In particular, assuming we have a sequence of reals α 1 < α 2 < · · · < α m such that X α 1 = ∅

Original: ( X α 1 = X ) and X α m =
Revised: ( X α 1 = X ) and X α m =

Original: X
Revised: X

Original: ( X α m = ∅ ), we consider the following extended sequence:
Revised: ( X α m = ∅ ), we consider the following extended sequence:

Original: [Equation]
Revised: [Equation]

Original: where ( A ; B ) denotes a pair of space, and at the homology level, note that the natural map from
Revised: where ( A ; B ) denotes a pair of space, and at the homology level, note that the natural map from

Original: X α m → ( X ; X α m ) induces an isomorphism.
Revised: X α m → ( X ; X α m ) induces an isomorphism.

Original: One can then consider the resulting persistent homology induced by the above extended sequence, and the resulting PDs are called extended PDs.
Revised: One can then consider the resulting persistent homology induced by the above extended sequence, and the resulting PDs are called extended PDs.

Original: Persistence pairings in such diagrams have four types, depending on whether the birth and death happen during the upward ﬁltration (ﬁrst line in Eqn (2)) or downward ﬁltration (second line in
Revised: Persistence pairings in such diagrams have four types, depending on whether the birth and death happen during the upward ﬁltration (ﬁrst line in Eqn (2)) or downward ﬁltration (second line in

Original: Eqn (2)).
Revised: Eqn (2)).

Original: In the context of graphs, these types are denoted as Ord 0 , Rel 1 , Ext +0 and Ext − 1 for downwards branches, upwards branches, connected components and loops respectively.
Revised: In the context of graphs, these types are denoted as Ord 0 , Rel 1 , Ext +0 and Ext − 1 for downwards branches, upwards branches, connected components and loops respectively.

Original: Overall, we denote
Revised: Overall, we denote

Original: [Equation]
Revised: [Equation]

Original: [Figure]
Revised: [Figure]

Original:  OTTLENECK D 
Revised: B OTTLENECK D ISTANCE

Original: [Figure]

Original: Given two PDs D 1 and D 2 , let Γ : D 1 ⊇
Revised: Given two PDs D 1 and D 2 , let Γ : D 1 ⊇

Original: A → B ⊆ D 2 be a partial bijection between D 1 and
Revised: A → B ⊆ D 2 be a partial bijection between D 1 and

Original: D 2 .
Revised: D 2 .

Original: Then for any point x ∈ A , the p-cost of x is deﬁned as c p ( x ) = (cid:107) x − Γ( x ) (cid:107) p ∞ and for y ∈
Revised: Then for any point x ∈ A , the p-cost of x is deﬁned as c p ( x ) = (cid:107) x − Γ( x ) (cid:107) p ∞ and for y ∈

Original: ( D 1 ∪ D 2 ) \
Revised: ( D 1 ∪ D 2 ) \

Original: ( A ∪ B ) , the p -cost of y is deﬁned as c (cid:48) p
Revised: ( A ∪ B ) , the p -cost of y is deﬁned as c (cid:48) p

Original: ( y )
Revised: ( y )

Original: = (cid:107) y − π ∆ ( y ) (cid:107) p ∞ , where π ∆ the projection onto the diagonal ∆ = { ( x, x ) : x ∈ R } .
Revised: = (cid:107) y − π ∆ ( y ) (cid:107) p ∞ , where π ∆ the projection onto the diagonal ∆ = { ( x, x ) : x ∈ R } .

Original: The cost of this partial bijection Γ is deﬁned as c p (Γ) =
Revised: The cost of this partial bijection Γ is deﬁned as c p (Γ) =

Original: (cid:16)(cid:80) x
Revised: (cid:16)(cid:80) x

Original: c p ( x ) + (cid:80) y c (cid:48) p ( y ) (cid:17) 1 /p .
Revised: c p ( x ) + (cid:80) y c (cid:48) p ( y ) (cid:17) 1 /p .

Original: Finally, deﬁne the p -th diagram distance d p as the cost of the best partial bijection:
Revised: Finally, deﬁne the p -th diagram distance d p as the cost of the best partial bijection:

Original: [Equation]
Revised: [Equation]

Original: (3) where Γ ranges over all partial bijections between D 1 and D 2 .
Revised: (3) where Γ ranges over all partial bijections between D 1 and D 2 .

Original: In the particular case when p = ∞ , the cost of Γ is therefore c (Γ) =
Revised: In the particular case when p = ∞ , the cost of Γ is therefore c (Γ) =

Original: max { max x c 1 ( x ) + max u c (cid:48) 

Revised: max { max x c 1 ( x ) + max u c (cid:48)

Original: ( y ) } ) .
Revised: ( y ) } ) .

Original: The corresponding distance d ∞ is often called the bottleneck distance between diagrams D 1 and D 2 .
Revised: The corresponding distance d ∞ is often called the bottleneck distance between diagrams D 1 and D 2 .

Original: B. PD S FOR G RAPHS
Revised: B. PD S FOR G RAPHS

Original: For graph classiﬁcation, (Hofer et al., 2017) is the ﬁrst work introducing a neural network framework to convert PDs into feature vectors for graph classiﬁcation in an end-to-end data-dependent way.
Revised: For graph classiﬁcation, (Hofer et al., 2017) is the ﬁrst work introducing a neural network framework to convert PDs into feature vectors for graph classiﬁcation in an end-to-end data-dependent way.

Original: Perslay (Carriere et al., 2019) uniﬁes many existing featurization such as persistence landscape, persistence silhouette (Chazal et al., 2014) and persistence surface as different instances of a single permutation invariant neural network based on DeepSets (Zaheer et al., 2017).
Revised: Perslay (Carriere et al., 2019) uniﬁes many existing featurization such as persistence landscape, persistence silhouette (Chazal et al., 2014) and persistence surface as different instances of a single permutation invariant neural network based on DeepSets (Zaheer et al., 2017).

Original: Weighted
Revised: Weighted

Original: Persistence Image Kernel (Zhao & Wang, 2019) (WKPI) is a recently proposed weighted kernel based on persistence image.
Revised: Persistence Image Kernel (Zhao & Wang, 2019) (WKPI) is a recently proposed weighted kernel based on persistence image.

Original: WKPI assigns weights for different locations in PDs where weights are learned from data via gradient descent.
Revised: WKPI assigns weights for different locations in PDs where weights are learned from data via gradient descent.

Original: Empirically, improved performance over other kernels is shown for the task of graph classiﬁcation.
Revised: Empirically, improved performance over other kernels is shown for the task of graph classiﬁcation.

Original: C D IAGRAM S TATISTICS
Revised: C D IAGRAM S TATISTICS

Original: We list diagram statistics for the shape segmentation task in Table 11.
Revised: We list diagram statistics for the shape segmentation task in Table 11.

Original: Ave # of PD points (ﬁrst row) stands for the average number of persistence points in the diagram for a shape category.
Revised: Ave # of PD points (ﬁrst row) stands for the average number of persistence points in the diagram for a shape category.

Original: A persistence point is considered as near diagonal if its lifetime is less than one-tenth of the lifetime of the furthest point in the same diagram.
Revised: A persistence point is considered as near diagonal if its lifetime is less than one-tenth of the lifetime of the furthest point in the same diagram.

Original: As we can see in Table
Revised: As we can see in Table

Original: [Table]
Revised: [Table]

Original: most persistence points for mech, bust, cup, bearing, glasses, ﬁsh, vase, teddy are concentrated near diagonal.
Revised: most persistence points for mech, bust, cup, bearing, glasses, ﬁsh, vase, teddy are concentrated near diagonal.

Original: Those are exactly the same categories on which permuting diagram yields much better results than not permuting diagrams when PI is used
Revised: Those are exactly the same categories on which permuting diagram yields much better results than not permuting diagrams when PI is used

Original: [Table]
Revised: [Table]

Original: [Table]
Revised: [Table]

Original: D PD S FOR M ACHINE L EARNING D. V ECTOR M ETHODS FOR PD S
Revised: D PD S FOR M ACHINE L EARNING D. V ECTOR M ETHODS FOR PD S

Original: Persistence Landscape (PL) (Bubenik, 2015) is the ﬁrst proposed vectorization method for PDs to overcome some undesirable property of the space of PDs , such as lacking a unique Frechet mean.
Revised: Persistence Landscape (PL) (Bubenik, 2015) is the ﬁrst proposed vectorization method for PDs to overcome some undesirable property of the space of PDs , such as lacking a unique Frechet mean.

Original: This construction is mainly intended for statistical computations, enabled by the vector space structure of L p .
Revised: This construction is mainly intended for statistical computations, enabled by the vector space structure of L p .

Original: Given a PD D = { ( b i , d i ) } mi =1 , persistence landscape can be thought of as a sequence of functions λ k : R → ¯ R where λ k ( t ) =
Revised: Given a PD D = { ( b i , d i ) } mi =1 , persistence landscape can be thought of as a sequence of functions λ k : R → ¯ R where λ k ( t ) =

Original: k th largest value of min ( t − b i , d i − t )
Revised: k th largest value of min ( t − b i , d i − t )

Original: + .
Revised: + .

Original: Persistent Image (PI) (Adams et al., 2017) produces a persistence surface ρ B from a PD by taking a weighted sum of Gaussians centered at each point.
Revised: Persistent Image (PI) (Adams et al., 2017) produces a persistence surface ρ B from a PD by taking a weighted sum of Gaussians centered at each point.

Original: The vector representation, named by persistence image, is created by integrating persistence surface over a grid.
Revised: The vector representation, named by persistence image, is created by integrating persistence surface over a grid.

Original: In particular, they ﬁx a grid in the plane with n pixels and assign to each the integral of ρ B over that region.
Revised: In particular, they ﬁx a grid in the plane with n pixels and assign to each the integral of ρ B over that region.

Original: There are at least three parameters involved in the construction of persistence image: 1) a nonnegative weighting function 2) the bandwidth of Gaussian kernel (many other functions can be chosen but in the original paper only Gaussian is considered) and 3) the resolution of the grid put over the persistence surface.
Revised: There are at least three parameters involved in the construction of persistence image: 1) a nonnegative weighting function 2) the bandwidth of Gaussian kernel (many other functions can be chosen but in the original paper only Gaussian is considered) and 3) the resolution of the grid put over the persistence surface.

Original: The authors report that in classiﬁcation experiments they conducted, the accuracy is insensitive to the choice of resolution and bandwidth.
Revised: The authors report that in classiﬁcation experiments they conducted, the accuracy is insensitive to the choice of resolution and bandwidth.

Original: D. K ERNEL M ETHODS FOR PD S
Revised: D. K ERNEL M ETHODS FOR PD S

Original: Persistence Weighted Gaussian Kernel (pwg) (Kusano et al., 2016) essentially utilizes the idea of kernel mean embedding of distribution, where persistence diagram, treated as a special case of distribution, can be embedded into RKHS.
Revised: Persistence Weighted Gaussian Kernel (pwg) (Kusano et al., 2016) essentially utilizes the idea of kernel mean embedding of distribution, where persistence diagram, treated as a special case of distribution, can be embedded into RKHS.

Original: In particular, Let K, ρ > 0 and D 1 and D 2 be two PDs.
Revised: In particular, Let K, ρ > 0 and D 1 and D 2 be two PDs.

Original: Let K ρ be the Gaussian kernel with parameter ρ > 0 .
Revised: Let K ρ be the Gaussian kernel with parameter ρ > 0 .

Original: Let H ρ be the RKHS associated to k ρ .
Revised: Let H ρ be the RKHS associated to k ρ .

Original: Let µ 1 = Σ x ∈ D 1 arctan ( K pers ( x ) p
Revised: Let µ 1 = Σ x ∈ D 1 arctan ( K pers ( x ) p

Original: k
Revised: k

Original: ρ ( ∗ , x ))
Revised: ρ ( ∗ , x ))

Original: ∈ H p be the kernel mean embedding of D 1 weighted by the diagonal distances.
Revised: ∈ H p be the kernel mean embedding of D 1 weighted by the diagonal distances.

Original: Let µ 2 be deﬁned similarly.
Revised: Let µ 2 be deﬁned similarly.

Original: Let τ > 0 , the persistence weighted gaussian kernel K pwg is deﬁned as the gaussian kernel with bandwidth τ on H p :
Revised: Let τ > 0 , the persistence weighted gaussian kernel K pwg is deﬁned as the gaussian kernel with bandwidth τ on H p :

Original: [Equation]
Revised: [Equation]

Original: Persistence Scale Space Kernel (pss) (Reininghaus et al., 2015) represents persistence diagram as sum of Dirac’s delta measure.
Revised: Persistence Scale Space Kernel (pss) (Reininghaus et al., 2015) represents persistence diagram as sum of Dirac’s delta measure.

Original: The persistence scale space kernel is deﬁned as the scalar product of the solution of the heat diffusion equation with the persistence diagram as an initial value.
Revised: The persistence scale space kernel is deﬁned as the scalar product of the solution of the heat diffusion equation with the persistence diagram as an initial value.

Original: The closed form
Revised: The closed form

Original: [Equation]
Revised: [Equation]

Original: can be computed exactly in O ( | D 1 | ∗ | D 2 | ) time where ¯ q = ( y, x ) is the symmetric of q = ( x, y ) along the diagonal.
Revised: can be computed exactly in O ( | D 1 | ∗ | D 2 | ) time where ¯ q = ( y, x ) is the symmetric of q = ( x, y ) along the diagonal.

Original: | D 1 | and | D 2 | denote the cardinality of the multisets D 1 and D
Revised: | D 1 | and | D 2 | denote the cardinality of the multisets D 1 and D

Original: Sliced Wasserstein Kernel (sw) (Carriere et al., 2017) uses Sliced Wasserstein approximation of the Wasserstein distance to deﬁne a new kernel for PDs.
Revised: Sliced Wasserstein Kernel (sw) (Carriere et al., 2017) uses Sliced Wasserstein approximation of the Wasserstein distance to deﬁne a new kernel for PDs.

Original: Different from previous multiple kernels, it is provable not only stable but also discriminative (with a bound depending on the number of points in the PDs) w.r.t.
Revised: Different from previous multiple kernels, it is provable not only stable but also discriminative (with a bound depending on the number of points in the PDs) w.r.t.

Original: the ﬁrst diagram distance w ∞ 1 between PDs.
Revised: the ﬁrst diagram distance w ∞ 1 between PDs.

Original: In particular, the kernel has the following closed-form:
Revised: In particular, the kernel has the following closed-form:

Original: [Equation]
Revised: [Equation]

Original: ) where SW ( D 1 , D 2 ) , is deﬁned as the sliced Wasserstein distance between PDs.
Revised: ) where SW ( D 1 , D 2 ) , is deﬁned as the sliced Wasserstein distance between PDs.

Original: Persistence Fisher Kernel (pf) (Le & Yamada, 2018) differs from slice Wasserstein kernel in the sense that the Wasserstein geometry is replaced by Fisher information geometry (metric), which induces a negative deﬁnite distance.
Revised: Persistence Fisher Kernel (pf) (Le & Yamada, 2018) differs from slice Wasserstein kernel in the sense that the Wasserstein geometry is replaced by Fisher information geometry (metric), which induces a negative deﬁnite distance.

Original: The form of Persistence ﬁsher kernel is
Revised: The form of Persistence ﬁsher kernel is

Original: [Equation]
Revised: [Equation]

Original: where t is a positive scalar and d FIM is the Fisher information metric.
Revised: where t is a positive scalar and d FIM is the Fisher information metric.

Original: Weighted Persistence Image Kernel (WKPI) (Zhao & Wang, 2019) is recently proposed weighted kernel that is based on persistence image.
Revised: Weighted Persistence Image Kernel (WKPI) (Zhao & Wang, 2019) is recently proposed weighted kernel that is based on persistence image.

Original: In particular, WKPI assigns weight for different location in the diagram where the weight is learned via gradient descent.
Revised: In particular, WKPI assigns weight for different location in the diagram where the weight is learned via gradient descent.

Original: The form of WKPI is
Revised: The form of WKPI is

Original: [Equation]
Revised: [Equation]

Original: (8) where PI, PI’ are persistence image, s is the location for each pixel in PI, w is the weight function that will be learned from data.
Revised: (8) where PI, PI’ are persistence image, s is the location for each pixel in PI, w is the weight function that will be learned from data.

Original: Note that majority of time to compute WKPI is spent on learning w ( p s ) via stochastic gradient descent.
Revised: Note that majority of time to compute WKPI is spent on learning w ( p s ) via stochastic gradient descent.

Original: pss pwg sw 
Revised: pss pwg sw pf

Original: [Equation]

Original: WKPI # of Param
Revised: WKPI # of Param

Original: Exact Comp.
Revised: Exact Comp.

Original: Time
Revised: Time

Original: O
Revised: O

Original: [Equation]
Revised: [Equation]

Original: O
Revised: O

Original: [Equation]
Revised: [Equation]

Original: O
Revised: O

Original: [Equation]
Revised: [Equation]

Original: O
Revised: O

Original: [Equation]
Revised: [Equation]

Original: Approx.
Revised: Approx.

Original: TimeO
Revised: TimeO

Original: [Equation]
Revised: [Equation]

Original: n
Revised: n

Original: [Equation]
Revised: [Equation]

Original: O
Revised: O

Original: [Equation]
Revised: [Equation]

Original: n
Revised: n

Original: [Equation]
Revised: [Equation]

Original: O
Revised: O

Original: [Equation]
Revised: [Equation]

Original: mn
Revised: mn

Original: [Equation]
Revised: [Equation]

Original: E H YPER - PARAMETERS C HOICE
Revised: E H YPER - PARAMETERS C HOICE

Original: We chose our hyper-parameters by 10-fold cross-validation on training set.
Revised: We chose our hyper-parameters by 10-fold cross-validation on training set.

Original: We list the speciﬁc search range for all methods below.
Revised: We list the speciﬁc search range for all methods below.

Original: All experiments are performed on a single
Revised: All experiments are performed on a single

Original: Persistence Landscape: the number of λ k ( t ) are chosen from { 3, 4, 5, 6, 7, 8 } and each function λ k ( t ) is discretized into { 50, 100, 200, 300 } bins.
Revised: Persistence Landscape: the number of λ k ( t ) are chosen from { 3, 4, 5, 6, 7, 8 } and each function λ k ( t ) is discretized into { 50, 100, 200, 300 } bins.

Original: Persistence Image: we use the same weight function (Gaussian) in the original paper, the resolution chosen from { 20*20, 30*30 } , and bandwidth of Gaussian is selected from { 0.01, 0.1, 1, 10, 100 } .
Revised: Persistence Image: we use the same weight function (Gaussian) in the original paper, the resolution chosen from { 20*20, 30*30 } , and bandwidth of Gaussian is selected from { 0.01, 0.1, 1, 10, 100 } .

Original: Persistence Scale Space Kernel: the parameter t is chosen from 13 values: { 0.001, 0.005, 0.01, 0.05, 0.1, 0.5, 1, 5, 10, 100, 500, 1000 } .
Revised: Persistence Scale Space Kernel: the parameter t is chosen from 13 values: { 0.001, 0.005, 0.01, 0.05, 0.1, 0.5, 1, 5, 10, 100, 500, 1000 } .

Original: Sliced Wasserstein Kernel: following the original paper (Carriere et al., 2017), we grid search bandwidth from the 5 values: { 0.01, 0.1, 1, 10, 100 } and the number of slices, i.e., M 2 is set to be 10.
Revised: Sliced Wasserstein Kernel: following the original paper (Carriere et al., 2017), we grid search bandwidth from the 5 values: { 0.01, 0.1, 1, 10, 100 } and the number of slices, i.e., M 2 is set to be 10.

Original: Persistence Weighted Gaussian Kernel: we try all the combinations of 5 values from { 0.01, 0.1, 1, 10, 100 } for bandwidth, and 3 values from { 0.1, 1, 10 } for both K and ρ , leading to 45 different sets of parameters.
Revised: Persistence Weighted Gaussian Kernel: we try all the combinations of 5 values from { 0.01, 0.1, 1, 10, 100 } for bandwidth, and 3 values from { 0.1, 1, 10 } for both K and ρ , leading to 45 different sets of parameters.

Original: Persistence Fisher Kernel: there are two hyper-parameters t and τ (for smoothing persistence diagram), both of which are selected from { 0.01, 0.05, 0.1, 0.5, 1, 5, 10, 50, 100 } .
Revised: Persistence Fisher Kernel: there are two hyper-parameters t and τ (for smoothing persistence diagram), both of which are selected from { 0.01, 0.05, 0.1, 0.5, 1, 5, 10, 50, 100 } .

Original: Pervec/Filvec : The length of the vector is
Revised: Pervec/Filvec : The length of the vector is

Original: Choice of SVM hyper-parameter:
Revised: Choice of SVM hyper-parameter:

Original: For kernel SVM, the only hyper-parameter C is selected from { 0.01, 1, 10, 100, 1000 } .
Revised: For kernel SVM, the only hyper-parameter C is selected from { 0.01, 1, 10, 100, 1000 } .

Original: For Pervec and Filvec, Gaussian kernel is utilized where the bandwidth is selected from { 0.01, 0.1, 1, 10, 100 } .
Revised: For Pervec and Filvec, Gaussian kernel is utilized where the bandwidth is selected from { 0.01, 0.1, 1, 10, 100 } .

Original: Graph Classiﬁcation: We follow the standard protocol in graph classiﬁcation literature, i.e., 10-fold cross-validations, using 9 folds for training and the rest for testing, and repeat the experimentstimes.
Revised: Graph Classiﬁcation: We follow the standard protocol in graph classiﬁcation literature, i.e., 10-fold cross-validations, using 9 folds for training and the rest for testing, and repeat the experimentstimes.

Original: We report the average classiﬁcation accuracies.
Revised: We report the average classiﬁcation accuracies.

Original: F D ATASETS D ESCRIPTION
Revised: F D ATASETS D ESCRIPTION

Original: The statistics of the benchmark graph datasets used in the paper are reported in Table 6.
Revised: The statistics of the benchmark graph datasets used in the paper are reported in Table 6.

Original: Due to space limitation, we only list the dataset statistics here.
Revised: Due to space limitation, we only list the dataset statistics here.

Original: We provide detailed dataset description in the data appendix.
Revised: We provide detailed dataset description in the data appendix.

Original: [Table]
Revised: [Table]

Original: F. N ON - ATTRIBUTED G RAPH D ATASETS
Revised: F. N ON - ATTRIBUTED G RAPH D ATASETS

Original: IMDB-BINARY (Yanardag & Vishwanathan, 2015) is a movie collaboration dataset that consists of the ego-networks of 1,000 actors/actresses who played roles in movies in IMDB.
Revised: IMDB-BINARY (Yanardag & Vishwanathan, 2015) is a movie collaboration dataset that consists of the ego-networks of 1,000 actors/actresses who played roles in movies in IMDB.

Original: In each graph, nodes represent actors/actresses, and there is an edge between them if they appear in the same movie.
Revised: In each graph, nodes represent actors/actresses, and there is an edge between them if they appear in the same movie.

Original: These graphs are derived from the Action and Romance genres.
Revised: These graphs are derived from the Action and Romance genres.

Original: IMDB-MULTI is generated in a similar way to IMDB-BINARY.
Revised: IMDB-MULTI is generated in a similar way to IMDB-BINARY.

Original: The difference is that it is derived from three genres: Comedy, Romance, and Sci-Fi.
Revised: The difference is that it is derived from three genres: Comedy, Romance, and Sci-Fi.

Original: REDDIT-BINARY consists of graphs corresponding to online discussions on Reddit.
Revised: REDDIT-BINARY consists of graphs corresponding to online discussions on Reddit.

Original: In each graph, nodes represent users, and there is an edge between them if at least one of them responds to the other’s comment.
Revised: In each graph, nodes represent users, and there is an edge between them if at least one of them responds to the other’s comment.

Original: There are four popular subreddits, namely, IAmA, AskReddit, TrollXChromosomes, and atheism.
Revised: There are four popular subreddits, namely, IAmA, AskReddit, TrollXChromosomes, and atheism.

Original: IAmA and AskReddit are two question/answer based subreddits, and TrollXChromosomes and atheism are two discussion-based subreddits.
Revised: IAmA and AskReddit are two question/answer based subreddits, and TrollXChromosomes and atheism are two discussion-based subreddits.

Original: A graph is labeled according to whether it belongs to a question/answer-based community or a discussion-based community.
Revised: A graph is labeled according to whether it belongs to a question/answer-based community or a discussion-based community.

Original: REDDIT-MULTI (5K) is generated in a similar way to REDDIT-BINARY.
Revised: REDDIT-MULTI (5K) is generated in a similar way to REDDIT-BINARY.

Original: The difference is that there are ﬁve subreddits involved, namely, worldnews, videos, AdviceAnimals, aww, and mildlyinteresting.
Revised: The difference is that there are ﬁve subreddits involved, namely, worldnews, videos, AdviceAnimals, aww, and mildlyinteresting.

Original: Graphs are labeled with their corresponding subreddits.
Revised: Graphs are labeled with their corresponding subreddits.

Original: F. A TTRIBUTED G RAPHS
Revised: F. A TTRIBUTED G RAPHS

Original: PTC (Helma et al., 2001) consists of graph representations of chemical molecules.
Revised: PTC (Helma et al., 2001) consists of graph representations of chemical molecules.

Original: In each graph, nodes represent atoms, and edges represent chemical bonds.
Revised: In each graph, nodes represent atoms, and edges represent chemical bonds.

Original: Graphs are labeled according to carcinogenicity on rodents, divided into male mice (MM), male rats (MR), female mice (FM), and female rats (FR).
Revised: Graphs are labeled according to carcinogenicity on rodents, divided into male mice (MM), male rats (MR), female mice (FM), and female rats (FR).

Original: PROTEINS (Borgwardt et al., 2005) consist of graph representations of proteins.
Revised: PROTEINS (Borgwardt et al., 2005) consist of graph representations of proteins.

Original: Nodes represent secondary structure elements (SSE), and there is an edge if they are neighbors along the amino acid sequence or one of three nearest neighbors in space.
Revised: Nodes represent secondary structure elements (SSE), and there is an edge if they are neighbors along the amino acid sequence or one of three nearest neighbors in space.

Original: The discrete attributes are SSE types.
Revised: The discrete attributes are SSE types.

Original: The continuous attributes are the 3D length of the SSE.
Revised: The continuous attributes are the 3D length of the SSE.

Original: Graphs are labeled according to which EC top-level class they belong to.
Revised: Graphs are labeled according to which EC top-level class they belong to.

Original: DD (Dobson & Doig, 2003) consists of graph representations of 1,178 proteins.
Revised: DD (Dobson & Doig, 2003) consists of graph representations of 1,178 proteins.

Original: In each graph, nodes represent amino acids, and there is an edge if they are less than six Angstroms apart.
Revised: In each graph, nodes represent amino acids, and there is an edge if they are less than six Angstroms apart.

Original: Graphs are labeled according to whether they are enzymes or not.
Revised: Graphs are labeled according to whether they are enzymes or not.

Original: NCI1 (Shervashidze et al., 2011; Kriege & Mutzel, 2012) consists of graph representations of 4, chemical compounds screened for activity against non-small cell lung cancer and ovarian cancer cell lines, respectively.
Revised: NCI1 (Shervashidze et al., 2011; Kriege & Mutzel, 2012) consists of graph representations of 4, chemical compounds screened for activity against non-small cell lung cancer and ovarian cancer cell lines, respectively.

Original: FRANK (Kazius et al., 2005) is a chemical molecule dataset that consists of 2,401 mutagens and 1,936 nonmutagens.
Revised: FRANK (Kazius et al., 2005) is a chemical molecule dataset that consists of 2,401 mutagens and 1,936 nonmutagens.

Original: Originally, nodes are associated with chemical atom symbols.
Revised: Originally, nodes are associated with chemical atom symbols.

Original: BZR , COX2 , and DHFR (Sutherland et al., 2003) all are chemical compound datasets.
Revised: BZR , COX2 , and DHFR (Sutherland et al., 2003) all are chemical compound datasets.

Original: Still, in each graph, nodes represent atoms, and edges represent chemical bonds.
Revised: Still, in each graph, nodes represent atoms, and edges represent chemical bonds.

Original: The discrete attributes correspond to atom types.
Revised: The discrete attributes correspond to atom types.

Original: The continuous attributes are 3D coordinates.
Revised: The continuous attributes are 3D coordinates.