Pull-back Geometry of Persistent Homology Encodings
Authors: Shuang Liang
Renata Turkes
Jiayi Li
Nina Otter
Guido Montufar
Venue: Accepted by TMLR ()

Abstract: Persistent homology (PH) is a method for generating topology-inspired representations of data. Empirical studies that investigate the properties of PH, such as its sensitivity to perturbations or ability to detect a feature of interest, commonly rely on training and testing an additional model on the basis of the PH representation. To gain more intrinsic insights about PH, independently of the choice of such a model, we propose a novel methodology based on the pull-back geometry that a PH encoding induces on the data manifold. The spectrum and eigenvectors of the induced metric help to identify the most and least significant information captured by PH. Furthermore, the pull-back norm of tangent vectors provides insights about the sensitivity of PH to a given perturbation, or its potential to detect a given feature of interest, and in turn its ability to solve a given classification or regression problem. Experimentally, the insights gained through our methodology align well with the existing knowledge about PH. Moreover, we show that the pull-back norm correlates with the performance on downstream tasks, and can therefore guide the choice of a suitable PH encoding.

Published in Transactions on Machine Learning Research (02/2024)
Pull-back Geometry of Persistent Homology Encodings
Shuang Liang
kangshuunngg ucla edu
Deportment of Statistics & Data Science
UCLA
Renata Turkes
rendta.turkcsBuantteeer
Department of Mathematics & Computer Science University of Antwery
Jiayí Li
jiayi liGg ucla edu
Deportment of Statistics Data Science
UCLA
Nina Otter
mona-lisannnotteeriiiee
DalaShape, Inria-Saclay;
Laboratoire de Mathématiques d' d'Orsay, Université Paris-Sacie Saris-Saclay
Guido Montúfar
montufor@matth ucla. edu
Departments of Mathematics and Statistics & Data Science, UCLA; Max Planck Institute for Mathematics in the Sciences
Reviewed on OpenReview: https: // openreview. net/ pet/forum? id- 7ysuRA8zzu
Abstract
Persistent homology (PH) is a method for generating topology-inspired representations of data. Empirical studies that investigate the propertics of PH, such as sonsitivity to perturbations ability to detect feature of interest, commonly rely on training and testing additional model the basis of the PH representation. To gain more intrinsic insights about PH, independently of the choice of such model, propose novel methodology based on the pull-back geometry that PH encoding induces on the data manifold. The spectrum and eigenvectors of the induced metric help to identify the most and least significant information captured by PH. Furthermore, the pull-back norm of tangent vectors provides insights about the sensitivity of PH to given perturbation, or its potential to detect a given feature of interest, and in turn its ability solve a given classification or regression problem. Experimentally, the insights gained through methodology align well with the existing knowledge about PH. Morcover, show that the pull-back norm correlates with the performance on downstream lasks, and can therefore guide the choice of suitable PH encoding.
Keywords:
Persistent homology, data representation, Jacobian spectrum, pull-back geometry, sensitivity analysis
1 Introduction
Persistent homology (PH) is well-ostablished technique in applied and computational topology (Carlsson 2009 Oudol 2015). At its core, PH seeks to create representations of data that highlight topological aspects. Recently, they have been found to also capture purely geometric aspects the data, such as curvature (Collins et al. 2004 Bubenik al., (2200) and convexity (Turkeš et al.) (022). PH representations have been used particularly in applications where multiscale homological features can be expected capture relevant. information, such in the prediction of biomolecular properties (Cang & Weil 2017 Cang al.

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2018
Wang et al. 2020), quantification of similarity in materials (Lee et al.) 2017]. medical imaging (iigh et al. 2023) or the analysis of tree-like brain artery structures (Bendich et al. 2016). PH also increasingly interfaces with machine learning (see, e.g., Carriere et al. 2020 Hensel et al. 2021). Beside serving as a data representation tochnique, PH has been used, for instance, to investigate the dccision boundaries of neural networks (Ramamurthy et al. 2019) or the transformations of data sets across the layers of a deep neural network (Naitzat et al. 2020). Other work has explored training neural networks to approximate PH features (Hofer et al., 2019 Montúfar et al. 2020; de Surrel et al. 2022), which can facilitate faster computations or serve basis for fine-tuning PH representations. Along this thread, a topology-encodde neural network based on PH was proposed by Haft-Javaherian et al. (2020).
In applications of PH to data classification or regression tasks, it is common to employ a model, such as a support vector machine (SVM) or a neural network, on the PH features. Such performance-basee based testing comes with two main drawbacks. Firstly, additional time and effort are needed to choose the classifier or regression model and tune their hyperparameters: this typically involves a grid scarch over all PH parameters, bul also over models (e.g., SVM and neural networks), and over the model's parameters (e.g., regularization parameter of SVM, and much larger list of hyperparameters for neural networks), which also requires training and testing for each combination of the three groups of parameter values. Socondly, the conclusions drawn regarding the effectiveness of PH are contingent upon the choice of the model and its specific parameters.
In this work, we propose a novel methodology aimed at gaining a more intrinsic understanding of PH encodings, irrespective of a particular classification or regression model. Here, a PH encoding denotes the mapping from data to a vectorized PH representation. We use the pull-back geometry induced by the PH encoding map to investigate its sensitivity to any particular data variation. A data variation is represented by a vector field in the data space, and can therefore be understood perturbation vector fields reflecting data perturbations (e.g., translation or dilation of a point cloud) and gradient vector fields resulting from data features (e.g., a label indicating the presence of an anomaly or discaso).
umbrella term that includes both
The Jacobian of an encoding mapping characterizes the behavior of the encoding in response to data variations. Specifically, the rank and eigenvectors of the Jacobian characterize the number of independent data variations and the most significant data variations captured by the encoding, respectively. The average pull-back norm of a vector field quantifies the sensitivily of PH to the corresponding data variation, and thus it helps assess to what extent PH is sensitive to a given perturbation or how effective it is at detecting a given feature (which in turns translates to its ability to solve a given problem). Furthermore, optimizing the average pull-back norm can guide the choice of a suitable PH encoding (choice of filtration, PH representation, and their paramcters): one only needs to evaluate the pull-back norm over the different choices of PH parameters. This approach eliminates the need to train and select a classifier on top of PH features, at the same time providing insights that are more intrinsic to the underlying problem. Indeed, if the performance of particular model on PH features is poor, one can hardly make any claims about the PH representation itself (since the problem could be that the model is poor). On the other hand, if the pull-back norm of the vector field is close to zero, we are more confident that the representation cannot recognize the given perturbation or feature. We provide a schematic diagram in Figure| [ that illustrates the pipeline of our proposed method and compares it with performance-basse methods.
We center our attention on a widely used PH representation known as the persistence image (PI) (Adams et al., 2017), which is an image-like representation of the input data. in terms of multiscale homological features. Our methodology, however, extends to other PH representations and, more broadly, to any differentiable encoding whose representation space can be endowed with a Riemannian manifold structure. In our experiments, we illustrate this generality by applying our approach to the Point.et encoding, a benchmark deep learning model for point clouds. We note that the insights about the (PH) encodings obtained through our approach depend on the specifics of the data set. Nonetheless, by evaluating different datasets one may be able to draw certain conclusions that hold with some generality: for instance, conclude that particular encoding captures a particular feature in datasets of particular type. This is an interesting prospect that can be facilitated by our proposed approach.
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INPUT
PUL-BACK GROMETRY METHOD
Encoding mup J:M
Rank of Jacobian din[
Amount information captured by
Data varistion (tangent field M) V:M UNeMTy.M
Eigenvector Jacubian:
Jacobian mapping I TyM + Tixy
The mosd, /loust signifirant information captured by f
Average pull bock morn Vil Bx (VX(V()
Perturbation T:
Data Feature p: Gradient field Up: X xp(X)
Edfectiveness of in capturing a data variation
Perturbation
V+ - =(X)-X
Intrinsic insight
APPLICATION
Interpreting the most significant information captured by f
Quantifying the seusitivity of to perturberions
PERFORMANCE METHOD
Train & test Model; on J(M): Performances Train Modela on f(M): Performanceg
(Section
(Section
Quantilying the ability of to dotret data features
Selecting peprpparmmetess For the eneoding f
Thin & test Model,r on f(M): Performanmeer
(Section
(Section 5
Depeuds ou the choice of Models
Figure 1: Schematic pipeline of our proposed method (comparing with performance-based testing).
Main contributions
We present an approach that can be used to investigate persistence images and their induced pull-back geometry on the manifold of input data sets in terms of the rank, spectrum, and eigenvectors of the Jacobian, as well as the pull-back norm of langent vectors on the data manifold (Section
We show how the above approach can be used to identify which data perturbations are captured by the encodings and which are ignored on given data sets. We also show how this facilitates an intrinsic comparison of PH encodings built with different filtrations. We experimentally demonstrate the insights gained via our approach align well with the existing knowledge about PH (Section
We show how the above approach be used to quantify what extent PH encoding can recognize data feature of interest given data set (sez feature in data set of brain artery trees). We also show how this quantitative evaluation can guide the selection of hyperparameters for the encodings. Finally, we show that the pull-back norm is predictive for the performance on a downstream task (Section
Related work Our discussion falls within the general subject of interpreting complex nonlinear map by investigating the effect that local input perturbations have on the output. This is conceptually related to topics such sensitivity analysis (Saltelli 2002), interpretable machine learning (Samek al. 2021).
sensitivity of outputs to input perturbations Moolnar et. al. 2020), activation maximization (Simonyan
ct al. 2013 relevance propagation (Montavon et al. 2018). adversarial robustness (Engstrom et al. 2019),
interpretable controls in implicit generative models (Härkönen et al. 2020), or function parametrizations in
artificial neural networks (Du et al.] 2019 Kornblith et al.] 2019).
Hauser Ray 2017) investigated the Riemannian geometry on the data manifold that induced by representations learned using artificial neural networks and show, in particular, that the metric tensor can be
Data and code developed in this
research
available
hetp:://gestbb..om///hanngg.i..
pul lback geteleeggooerryyeeo parsssstt-priitttthhhhehme
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found by backpropagating the coordinate representations learned by the network. This shares similarities with our approach, as the induced geometry is essentially the pull-back geometry from representation space by the Jacobian map. Meller & Berkouk (2023) proposed graph representation for neural networks using a singular value decomposition of the weight matricos. While they tackle the nonlinearity by studying linear maps contained in the nonlinear map consecutively, our emphasis lies in the local linear approximation of the nonlinear map.
There exist few studies that investigate the sensitivity of PH representations to perturbations and their ability to recognize specific features, number of PH representations to different types of transformations (such as rotation, translation, change of image brightness or contrast, as well as Gaussian and salt and pepper noise). However, the main method to assess sensitivity is the performance of an SVM trained on the representations of the original data and tested on the representations of data under transformations, and thus requires training and testing, and depends on the choice of a particular classifier. Bubenik ct al. (2020) showed in theory and experiments that persistence landscapes can be used to detect curvature of an underlying set based on sampled point cloud. Turkes el. al. (2022) conducted investigations towards identifying fundamental types of tasks for which PH representations might be most uscful. They showed that beside curvature and number of holes, PH representations can be used for detecting convexity. However, they focus on three specific tasks (detecting number of holes, convexity, and curvature), whereas we study the alignment of the representations with any given feature defined on the space of imput data sets. Moreover, in that work, the performance PH is experimentally evaluated via the SVM accuracy, whereas our approach does not require training and testing of any model.
do in this work. For example, Turkeš et al. 2021 study the sensitivity of
More generally, in the context of inverse problems in persistence theory, there are several lines of work that study conditions under which persistence diagram maps are surjective or injective; see the survey of Oudot & Solomon (220) and references therein. Within this context, our work can be seen as providing framework related to the study of the injectivity of specific PH encodings. The work of Xenopoulos et al. (2022) deals with local explainability using topological represontations. In contrast, we deal with the representations. McGuire et al. (2023) measured the dissimilarity between representations learned by neural net norksrks trained on PH encodings and networks trained on raw data. They experimentally demonstrate that networks learn considerably different reprosentations when processing PH encodings instead of raw data. RRiec (2023) compared the expressivity of PH against the Peiifeiler-Lehman hierarchy of graph isomorphism tests, and explored the potential of PH to capture certain graph structures and characteristic properties. Finally, an important line of work in the study of PH encodings is concerned with developing computable notions of optimal representative cycles for persistent homology classes; see, c.g., the survey of Li et al. (021).
2 Preliminaries on Persistent Homology
The key idea behind PH is to construct a filtration, i.e., a sequence of topological spaces by gradually adding simplices (vertices, edges, triangles, etc.) to the data, and to study the evolution of topological features (components, holes, voids, and higher-dimensional voids) across the filtration. In this section, we give a brief overview of the different filtrations that we consider in this work (Scction 2.1 and the persistence image that we will use to represent PH information (Section 2.2
2.1 Filtration
Let X (a, e i= .N} denote a point cloud consisting of N points in RRP. To build a filtration on X, we commonly construct a sequence of simplicial complexes on X.
A simplicial complex can be thought of as a space obtained by taking union of vertices, edges, triangles, tetrahedra and higher-dimensional simplices. Formally, a collection K c 2 of subsets of X is called a simplicial compler if a e K and T c a imply T e K. An element a in a simplicial complex is called a (|a] - simple,,,, where |or| is the cardinality of a. Specifically, one can think of O-simplices as vertices (i.e., elements of X); 1-simplices as edges (i.c., pairs of elements of X); 2-simplices as triangles, etc. We note that according this definition of a simplicial complex, not every element of X is necessarily a O-simplex; this is important for the types of filtrations that we consider in our work, such the DTM filtration.
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In the following we focus simplicial complexes obtained the clique complex of an R-neighborhood graph.
The clique compler of the R-neighborhood graph on point cloud X consists of all subsets a of X such that the distance between any pair of points in a is at most R:
CI(X, R) to 2 Bx( R) n Br(Uj, R) 2, V2i, CO).
where (2 R) (y e R dr(s,y) s R) denotes the Euclidean R-ball centered at x and dg denotes the
Euclidean distance.
A filtration with respect to the clique complex is an indexed collection {Ky of subsets K, CI(X, R) satisfying the condition that K,, c Kr, if rı 5 r-- The construction of a filtration is equivalent to assigning a filtration value ④(o) to each simplex a in CI(X, R) in the following sense. Given {Kr)rese,, one can define the filtration value for any simplex a as Φ(o) = inf
K,). Conversely, given a filtration value for every simplex, one can define the collection of subsets as K, = {or CI(X, : (()) sr..
We will focus on some common filtrations built upon CI(X, R)):
1. The Vietoris- Rips filtration (Vietoris 1927 p. This defines the filtration value for each simplex as its diameter: (()) = Diam(o)= = maxx.yee (r)r))).
2. The distance-to vertex as ((x;))) R EzjeKNN(H.)) dx((,, where KNN(,;) denotes the set of k nearest neighbors of Ij in X, so that the outliers have a large filtration function value and appear late in the filtration. Then it defines the filtration value for edges as o({x,,,,))= = )() + ((x,]) + dx(x,, x,2/2, and for simplices with degree (lo) 1) greater than one as Φ(r) = max,, { x(),,,)))
(DTM) filtration (Anai et al 2020
This defines the filtration value for each
3. The height. filtration with respect to hyperplane. Let ee R² be the unit normal vector of hyperplane. The corresponding height filtration defines the filtration value of vertices as d({x))) (æ,, e), and the filtration value of any other simplices as Φ(a)= max {p({x))} We note that this means that, in order to capture features of interest, one needs to set the maximum length for an edge to be present in the height filtration (for details, see Appendix F.3
2.2 Persistence image
The k-dimensional homology group, k-dimensional homology, of simplicial complex characterizes the k- dimensional holes in the complex. Each non-zero k-dimensional homology class in the k-dimensional homology group uniquely characterizes a k-dimensional hole. As we introduced earlier, the set K, includes more and more simplicos as the parameter r increases. In persistent homology, we are interested in how the homology groups of K, change as we vary the parameter r. By the matrix reduction algorithm (Edelsbrunner et al., 2002), one can identify a birth-death pair (b, d) E [0, R7]2 for every non-trivial homology class that appears the filtration. Roughly, the homology class first "appears" in K₁ and "persists" until Ka, degenerating to the trivial class afterwards. In addition to the bounded intervals, the matrix reduction algorithms gives infinite intervals (b, oo), which one can interpret as homology classes that appear at K₂ and do not degenerate to the trivial class for any filtration value considered. The persistence diagram (PD) is a summary of such information.
Formally, the k-dimensional PD the multiset of birth-death pairs for all k-dimensional homology classes that appear in the filtration. Although the space of PDs can be endowed with a metric structure, PDs do not lend themselves to processing with techniques that require a Hilbert space structure, including support vector machines and principal component analysis (PCA) (Reininghaus et al. 2015). Hence, one often considers vector representations of PDs. The most commonly used ones include persistence images (Adams et al. 2017 and persistence landscapes [Bbbenik] 2015
In our work, we will focus on persistence images (PIs). Let PD be a k-dimensional persistence diagram in birth-death coordinates. One converts this to a multiset r/(D) birth-lifespan coordinates by applying the linear map 7(b, d)= = (b,1) with I= d- b to each birth-death pair (b, d). Given an infinite interval (b, oo), in practice, one sets the death value to the maximum filtration value R. Given kernel function I(..)) (2. y) on
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R² and weighting function a(b,l), the persistent surface is the function bR:R. R defined by
(xx,))
Σ ((b.)))..n(...)..
(b.))e((P)
A persistence image (PI) is finite-dimensional representation of obtained follows. One splits subdomain of vé by Px P grid of regions. Then the PI of resolution P is the matrix whose (i,j)-th entry pixel is the integration value of vo over the (i,j)-th region. Note that, since the death time d cannot be smaller than the birth time b, the birth-death pairs (b, d) always lie above the diagonal line, ie., PD _ ((x, y) E IR? y x 0). The transformed birth-lieepaal pairs (b, I) lie in the first quadrant, i.., ()D) _ {(2, v) IR x 0, 0.
(sec an illustration in Figure
Point cloud
Wetorts Rips fitration
didmerdional PD
(NPD)
P1
0.4
0.3
Deeth
Ko
Kgg
C9
108 ete 550 ais Cle ods 835 6:4
(8)
(D)
(d)
Figure 2: The pipeline for constructing a persistence image described in Section 2.2 From left to right: (a) input point cloud; (b) Vietoris-Rips filtration built on the point cloud; (c) 1-dimensional persistence diagram; (d) buth-lifespan pairs (transformed 1-dimensional persistence diagram); and (e) persistence image.
For constructing Pls, one needs to choose 1) the resolution P, 2) the kernel function 90.. (z, y) and its associated parameters, and 3) the weighting function a(b, l). One of the main difficulties in working with Pls is that there is no canonical way to choose these hyperparameters {Adams et al. 2017). Adams et al (2017) studied the effects of PI parameters on the performance of certain classifiers (K-medoids classifiers) that take PIs as inputs. However, we note this approach heavily depend on the choice of downstreem model. This motivates us to investigate what kind of information of the input data intrinsically captured by the PI under different choices of the hyperparameters.
A motivation for considering PIs is that they provide differentiable PH representations and that, with an appropriate choice of metric, the space of PIs has Fuclidean structure, which simplifies computations (sce Section 3.1]. We will later consider derivatives of the mapping from input data to Pls. These can be obtained using existing automatic differentiation packages and librarics, such as topologylayer (Gabrielsson et al. 2020 and Gudhi The GUDHI Project 2020 We provide further details about this in Appendix
3 Methods: Sensitivity of PH Encoding to Data Variations
We consider an encoding map f M + N, where M is a space of point clouds and M is the space of persistence images (Section 3.1 We conceptualize input data variations (Section 3.2) and the resulting changes of the encoding output (Section 3.3
3.1 Input space and output space
We let M be the space of point clouds in D that contain exactly N points,
M M= [ [X|=N]
A point cloud X is a finite subset RD with cardinality |X] = N. A point cloud may be regarded as an unordered list of points, determined only up to permutation. It can also be regarded as a probability
Prrecisely, the map from point clouds Pls generically differentiahle (see detailed discussion Appendix

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distribution on R Hence we equip M with the 2- Wasserstein distance (sce, e.g., Peyré et al. 2019
[ (EB(aww)))) WEN(X.Y) XEE
dw (x,
min
Here Q(X,Y) denotes the set of bijections w: X + Y between the sets X and Y, and dg denotes the Euclidean distance on I D. The 2- 2-Wassenstein distance induces a metric topology on M. Further, M can be endowed with a Riemannian manifold structure with dim(M) é m = D N. For simplicity of presentation, in the following we treat M as an Euclidean space. Nonetheless, our discussion is consistent with the Riemannian manifold structure and applies in that level of generality (see Appendix B). For a detailed discussion regarding the Riemannian structure and differential calculus on Wasserstein space refer the reader to Ambrosio et al. (Chapter 8, 2005) and Villani et al. (0009)
We let / be the space of persistence images of fixed resolution P. Thus we can interpret W as submanifold embedded in PPP and endowed with the canonical Euclidean distance, with dim(N) = n p2 3 Here again, other choices of metric on the space of PIs are possible
3.2 Data variations
To characterize local variations of the input data, we consider tangent vectors on the data manifold. We conceptualize the intuitive concepts of perturbations and feature variations in terms of corresponding vector fields on the data manifold.
The tangent space at X .,, denoted M,M, is the vector space of all vectors emanating from X and langential to the data manifold M. The dimension of T is equal to the dimension of the data manifold, dim(TxM) = dim(M). Each tangent vector e T characterizes a local variation of a single point cloud X. A vector field specifies a variation for each point cloud in M. More specifically, a vector field V on M is a smooth map V .M- MM, assigning to each X in M a tangent vector V(X) e M.
A perturbation is a modification of point cloud in the data manifold, e-g-; by rotation or shearing. This can be described by a map π: MM M taking data X M to perturbed data =(X) E M. The perturbation vector field V* associates to each X e M tangent vector (x()) capturing the difference between π (X) and X (see Figure 3 l
Definition (Perturbation vector field). Let M be manifold, TxM the langent space at X e M, and T :: ::M- + M a perturbation map. The perturbation vector field induced by ı is defined as
:M uxTxx; X V*() = (X) -
A feature p is a real-valued smooth function defined on the data manifold, p:M. M value to each X. Discrete-valued (categorical) features can be converted to continuous ones by considering probability distributions or logits of the fcature values. For instance, the "cat-or-dog" feature can be converted to continuous feature p(X)= Prob(X cat) [0.
- R, assigning a fcature
The gradient of feature introduces vector field the data manifold. The gradient vectors point in the direction of steepost increase of the feature, with magnitude indicating the rate (see Figure 3 right).
Definition 2 (Gradient vector field). Let M be a manifold, TxM the tangent space at X &M, and p: P:M + R a real-valued feature. The gradient vector field of p is the vector field on M defined as
M
X x+(P(X)
such that (x(X) [T wrgmaa ( (X)) and [x((X] max 20 Here de
0 is the
directional derivative along c.
We note that if one considers the 1-Wasserstein distance on the space of PDs, and any of the L1, L2 or Los norms the space of Pls, then Pls are known to be stable (Adams et al. 2017 Theorem 5). On the other hand, Pls, together with the La norm, unstable if one instead considers p- -assersteii distances on the space of PDs (Adams et al. 2017 Remark 6).
4This be of interest to try establish more general stability results for Pls. An example would Wasserstcin distance
between persistence images assigning appropriate cost b aud directions.
5 Figure g schematic illustration. It is not intended to imply that the kind of depicted point clouds indeed form torus.
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Figure 3: The space of point clouds forms manifold, which in this figure is depicted as lorus; each point on this manifold is point cloud, Left: vector fields on the data manifold correspond to variations of the point clouds; in this illustration, the red arrows correspond to "rotation" and the blue arrows to "shcaring". Right: a continuous feature on the data manifold induces a gradient vector field; the figure illustrates a binary feature, where the dashed line the class boundary, and the continuous feature value represents the probability of the data point belonging the "red" class.
Definition 1 and Definition are given for the case that M is a Euclidean space. We provide definitions of perturbation vector fields and gradient vector fields for the case general Riemannian manifolds in Appendix B.3 Further, we provide details on how to estimate such vector fields using finite data sets in Appendices and F.4
3.3 Encoding variations
Having characterized data variations in terms of vector fields, the next step to describe the behavior of the encoding map f in response to these variations. Specifically, we are going to introduce the average pull-back norm of vector field to quantify the sensitivity of the encoding map to the corresponding data variation. At the outset of this subsection, we emphasize that whether not it is desirable to have an encoding that is sensitive to a particular data variation depends on the specific practice scenario and whether this variation is perceived as valuable information or as noise that one would like to filter out in the encoding.
Jacobian The Jacobian of an encoding map f, denoted by J& is a linear transformation between tangent spaces that characterizes the local behavior of f. While tangent vector ve M describes one type of data variation at X, the image tangent vector Jx(1) describes the resulting variation of the encoding f(X),
JK TxM + Txx;;; U ((()
We may wrile this linear transformation in terms of a Jacobian matrix Jk e Rxx with respect to a basis. If there is no risk of confusion, we will omit the super-/subscrips f and X. We provide visualization for the Jacobian map in the left panel of Figure
The rank of the Jacobian is the dimension of the image of TxM under the Jacobian map, rank(J{) = dim (T). It corresponds the number of degrecs of froodom of the data that are captured by the encoding. For instance, (ank('/) = dim(Tx indicates that f is sensitive to all local data variations, whereas rank (Jx) 0 means that f is approximately invariant under all local variations and thus approximately constant near X.
Pull-back norm To measure the encoding's effectiveness in capturing a data variation, we introduce the average pull-back norm of a vector field. The pull-back norm of a tangent vector V(X) at is defined as
[V(X),, -1a Vv(xy o V(X).
SStrictly speaking this is semi-norm, may vanish non-sero tangent vectors.

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M
N
M
AV
X' ViN
T
(x(
Figure 4: A visualization of the Jacobian map and the pull-back norm. Here f denotes an encoding map from the imput space M to the output space W. Left: the Jacobian of the encoding sends tangent vectors in the tangent space M M to tangent vectors in the tangent space TxxyN N. Right: the pull-back norm of a tangent vector on M measures by what amount the output of the encoding would change in response to the variation of the imput by that tangent vector. In this schematic illustration, the pull-back back norm of the red vector ("noising") is larger than the pull-back norm of the blue vector ("shearing").
Here |. Ill denotes the vector norm in output space N and = is the Gram matrix of the encoding f at X. While in the above definition, we consider the Euclidean metric for the output space of Pls, our approach can be applied for other choices of metric as well. We present the definition of pull-back norm for any diffcrential encoding mapping between Riemannian manifolds in Appendix B.4. We also provide a visualization for the pull-back norm the right panel of Figure
The pull-back norm of V(X) measures the sensitivity of to the variation V(X) X. To measure the scnsitivity across different inputs, we take the average with respect to distribution on M. In practice, we use the empirical distribution of a given data set D (X:)=1....Ip)
Definition 3 (Average pull-bact norm). The average pull-back norm of vector field V with respect to an encoding map and data set D (Xi)i........ of cardinality [D] is
IV Σ IV(X), D
XED
Please note that in Definition 3 V(X) denotes tangent vector at X in the space of point clouds. Specifically, vector V(X) e T corresponds to "vector field" on X which assigns a vector to each point x e X in the point cloud X. We say that an encoding can detect a data variation characterized by vector field V if the encoding scnsitive to V, which alludes to the average pull-back norm of V.
Singular value decomposition To gain more fino-grained insight into the properties of an encoding, we consider the singular value decompasition (SVD) of the Jacobian matrix,
J QAQA
Here Q "x"" Q I mom are orthogonal matrices, and E Raxm is a diagonal matrix containing in its diagonal the singular values in decreasing order A 2 A2 ... Amin(m,x)) This sequence of ordered singular value is the spectrum of the Jacobian J. Accordingly, the Gram matrix has eigendecomposition G J QA We denote the right singular vectors, ie., the columns Q, by .m.... We will refer to these qi 's as the eigenvectors of the encoding. Any tangent vector u e TxM can be written as v = Σ(e, q,)4, and its pull-back norm as || = VΣ (f(v, With this, the pull-back norm is decomposed as two parts: the spectrum of Jacobian and the alignment between v and eigenvectors, which is described by the inner product. particular, the pull-back norm is large implies that the eigenvectors with top largest cigenvalues can be regarded the data variations that the encoding considers most "important". We offer visualization in Appendix D
aligned with q, that have large singular values. This also

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Comparison between encodings Later will compare different encodings by examining their sensitivity to specific data variations. To place different encodings on the same scaling level, we consider the normalized average pull-back norm [Σ (X)s/A///// We provide details about this normalization technique in Appendix F.2 Another way of comparing different encodings is via the Burcs-Wasserse Wasserstein distance (Bhatia et al. 2019 between their Gram matrices. The Bures- Wasserstein distance quantifies the aligmment between the eigandccompastiions of the Gram matrices. For two positive definite matrices and B, the Bures- Wasserstein distance is computed as:
dew (A, = [TrA+ TBB BA
For matricos that are not strictly positive definite we use the same definition after adding a small multiple of the identity matrix.
4 Identifying What Is Recognized
In this section, we seek to identify, for fixed PH encodings, which data variations are recognized and which are ignored. Specifically, we investigate the total amount of data variations that are captured by PH encodings (Section 4.1). and among those captured variations we interpret the most significant ones (Section 4.2) Then, we quantify the "importance" for any data variation (Section 4.3) and measure the dissimilarity of the captured information across different PH encodings (Section 4.4h. It. is important to notice that, even though our approach eliminates potential inference biases induced by downstream models, our results still significantly depend on the data set under consideration. Therefore, we emphasize that this section is dedicated to investigating "what is recognized by PH encodings within specific data sets.
Synthetic data Throughout this section we consider synthetic data set of point clouds in R² sampled from curves in the Radial Frequency Pattern (RFP) lamily. A point cloud of this type is shown in Figure 2 The curve (FP((, is parametrized by p(0)= =1 + cos(w0), 0 E (0, 2x]. Loosely speaking it represents the shape of a flower with w petals of size characterized by a. We take in (3,4.... 10) and 10 values of a evenly distributed on the interval [0.5, 0.9]. For each curve RFP (a.a)) we evenly sample
150 points
obtain point cloud, which is then scaled to the unit square [0, 1] (sce examples in Appendix F.3 Notably, each curve in the RFP family has the same topology. This allows to validate the ability of PH to capture information beyond topology. The RFP data set has also been used in studying the importance of specific shape features in shape recognition and object perception (Schmidtmann et al. 2015
PH encodings We investigate PH encodings constructed on different filtrations: Vietoris-Rip Ritoris-Riis (Rips) filtration, DDM filtration, and Height. filtration (with respect to the hyperplane with normal vector [1,0]"). For each filtration we extract the I-dimensional PDs, and convert them to Pls with the same PI parameters. In the following discussion, we sometimes refer to these encodings by the name of the filtration on which they are constructed, denoting for instance the PH encoding constructed on Rips filtration simply as Rips. For reference we also include a PointNet encoding. PointNet (Qi et al. 2017) is a deep neural network architecture designed for processing point clouds directly. We train the network to predict the number of petals w, achieving a test accuracy of 100%. We take the output of the second-to-last layer of the trained PointNet as the output of the PointNet encoding. More details concerning the filtration and PT parameters, and the PointNet encoding are provided in Appendix F
4.1 Spectrum of the Jacobian
As explained in Section 3.3] the rank of the Jacobian indicates the maximum amount of information, in a dimension sense, that can be captured by the encoding. The spectrum, i.c., the sequence of ordered singular values X₂ 2 A2 2 Ams provides a fuller picture, indicating to what extent different eigenvectors are highlighted.
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Spectrum of Jacobian
1st eigenvectors
2nd eigenvectors
Rips
DTM
Normaiized Magnitude
Height
PointNet
11
0.6
Eigenvalue Index
Figure Left: The normalized spectrum of the Jacobian for different encodings. Shown is the mean and standard error of the ordered normalized singular values over different input point clouds. Right: The top two eigenvectors of the Jacobian for the PH encoding constructed on the Rips filtration at particular inpul point cloud.
The left plot in Figure reports the normalized spectrum of the Jacobian for different encodings, which is the sequence of ordered normalized singular values
Aa 2
Arank(J) > 0.
A1
A1
The first observation is that the rank of the Jacobian much smaller than the dimension the point cloud space and the PI space. Indeed, note that the dimension of M is Dx N 2x 150 300, and the dimension of N is x P 20 20 400. On the other hand, the normalized singular values decay to 10-5 before index 40. We conchude that the four encodings under consideration capture only small set of variations in the input data and discard many others. Secondly, while Rips, DTM, and Height all have a similar number of singular values larger than 10-5, Rips and DTM exhibit sharper initial decay than Height. This implies that Height has a larger effective rank while Rips and DTM concentrate their attention more specifically on a few variations. This difference in decay rates may stem from the fact that the size of holes, which is captured by Rips and DTM, is influenced by fewer variations compared to the position of holes, which is information retained by Height.
The middle and right plots in Figure show the top two eigenvectors for Rips at example point cloud X. An eigenvector corresponds to list of vectors attached to all individual points in the point cloud. These eigenvectors provide insight into the nature of the "important" data variations. More technically, these variations correspond to the most effective way to change the birth/death parameters of cortain homology classes For instance, the middle plot. of Figure 5 the vectors on the petals depict variations that narrow, broaden the petals, which in turn change the death parameter the corresponding homology classes. The eigenvectors can also be used to obtain point saliency maps, which we discuss Appendix[i However, we observe that the eigenvectors do not necessarily have an obvious intuitive description. Hence, interpretations are needed to bridge the gap between abstract eigenvectors and hunan-understanddbbe concepts.
4.2 Alignment between eigenvectors and perturbation tangent vectors
To interpret the eigenvectors of the encoding, consider their alignment with different perturbation vector fields.
Perturbations on the data manifold We consider eight types of partorbations applied to the input data, illustrated in Figure D The first two, rotation and translation, are Euclidean motions, i.c., transformations that preserve the Euclidean distances between the points a point cloud. They are used to test the fact that
Thee effective rank is the number of singular values that have similar order of magnitude the top singular value. 8This hears some resemblance adversarial perturbations considered in neural networks.
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pointwise distance-based encodings, namely Rips and DTM, should remain invariant under such variations. The dilation, streeth and shearing variations serve to test the sensitivity of the encoding to invertible linear transformations of the point clouds. The next two variations are used to test the robustness of PH encoding against noise: the noising variation adds coordinate-wise Gaussian noise at cach point in the point cloud; the wiggly variation adds a sine-lype noise al every point in the point cloud along the normal direction. Lastly, the convez variation transforms the point cloud towards the boundary of its convex hull through a linear interpolation. We present visualization of the effects of shearing and conver the PH associated with Rips filtration and Height filtration in Appendix
Rotation
Translation
Dilation
Strech
Shearing
Noising
Wigaly
Convex
Figure 6: Eight types of perturbations on a RFP pointcloud.
We examine the angles between the top eigenvectors of the different encodings and different perturbation tangent vectors. In Figure 7 we record the average inner product between the perturbation tangent vectors and the top four eigenvectors each encoding method:
(X) (X)
Σ
1,,,,44,4
DRFP
XeDRFF
The eigenvectors of depend on X. Since each encoding map induces a different orthonormal basis (of.gg..... ........h ) on the langent. space of the data manifold, a fixed langent vector will have different coordinates with respect to the different encodings. Figure serves as a sort dictionary, showing how each perturbation (rotation, translation, dilation, etc.) expressod the language of cach encoding.
Height
Rotation
Translation
0.0
Dilation
Strech
Shearing
Noising
0.44
Wiggly
Convex
0.00
Figure 7: Absolute inner product between perturbation vectors and top four eigenvectors of different encodings. A higher value implies greater alignment, i.e., greater sensitivity to a perturbation.
For the PH encoding constructed on Rips and DTM filtration, we find that the top cigenvectors exhibit relatively strong alignment with conver. The corresponding average inner product is around 0.1. This indicates that the most "important" data variation for Rips and DTM closely related to convexity. This is consistent with the fact that these encodings capture geometric properties, such as birth values of holes in the filtration (that increase under the concer perturbation, see Figure 6). At the same time, the top eigenvectors of Rips and DTM are orthogonal to rolation and Iranslation. This indicates that Fuclidean motion is not. as relevant in the Rips and DTM, as to be expected from the definitions of these encodings.
For the PH encoding constructed on Height filtration, the top cigenvectors have a significant alignment with Euclidean motions and stretch This makes sense, since Height is designed to collect information the
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position of holes. On the other hand, we do not observe strong alignment between top eigenvectors of Height and conves. This might seem to be in contradiction with Tuurke et al. (2022) who demonstrated that PH on height filtration can be used for detecting convexity. Note, however, that they use O-dimensional PH on cubical complexes (or analogously, on Rips complexes on geodesic distances), which recognizes concave shapes by their multiple connected components (O-dimensional cycles) for at least some height. filtration directions. 0-dimensional PH with respoct to such a filtration would see two connected components (two petals) for a while, that would then merge into one at some point. This would happen earlier under conver perturbation (i.e,, one of the connected components would die sooner), so that the alignment can be expected to be more significant in that case.
For Nei encoding, the top eigenvector has a relatively strong aligmment with translation. This is consistent with a previous observation by Tuukee et al. (2022) where the PointNet did not perform well in classification tasks when the test data was corrupted by translations. Note akin to the approach adopted by Turkes et al. (2022), we do not use data augmentation techniques during the training for PointNet. This might. lead to the sensitivity of the trained Point.et to Iranslation. On the other side, Point.et is robust. under conver perturbations. This can be attributed to the nature of RPF data set. Recall the RPF data set comprises point clouds defined by two independent parameters, and w, which charactcrize the size and the number of the petals, respectively. While the Point.et is trained to identify the number of petals, it can easily learn from the data set to ignore the size of petals. Notice increasing the petal size bears strong resemblance to conver perturbation (see examples of RPF point clouds in Appendix F.3). Therefore, one can loosely infer that the RPF data set is "inherently" augmented by conver perturbation, and consequently the trained NeintNet might learn from the data to ignore conver information.
4.3 Pull-back norm of perturbation vector fields
In some scenarios, one interested in the sensitivity of an encoding to certain types of perturbations (Ren ltt al.] 2020). In Figure g (left) we evaluate the average pull-back norm of different perturbation vector fields with respect different encodings. The pull-back norm takes into account not only the alignment with the encoding eigenvectors but also the magnitude of the corresponding singular values. We provide numerical function in htttps///iithb: wwt. son/shuang] wawpgg//ppllbackkccm automatic computation of the average pull-back norm with respect to Vietoris-Rips filtration for any given perturbation on a given data set. Note that it is not necessary to have an explicit function description of the perturbation ı : .M- M, since we only require the set of perturbed point clouds, i.c., x xxX) for all X in the data set.
that allows
For DTM and Rips, find that noising, wiggly, and conver, have a significantly larger pull-back norm than the other data variations. This is consistent with the Jacobian spectrum in Figure 5 (which indicates DTM and Rips have faster- dectei-d spectrum and therefore capture only few data variations), and the alignment information in Figure 7 (which indicates alignment of the top eigenvectors with these particular variations). The PH encoding constructed on Height filtration has a relatively large pull-back norm for many of the considered data variations, including dilation, stretch, and shearing. Also the pull-back norm associated with conver perturbations with respect to Height exhibits a moderate average value and a large variance. This implics Height is sensitive to conver perturbations in certain point clouds, while being less sensitive in others. This aligus with the alignment information in Figure 7 which indicates the most "important" data variations seen by Height, on average, are not closely related with conver. Similar to Height, PointNet also leads to relatively large pull-back norms, but with a different profile and with exception of convez, which has a small pull-back norm under PointNet. Rips and DTM have a faster-decaying Jacobian spectrum than Height and PointNet, indicating that they are sensitive to fewer data variations.
4.4 Distance between Gram matrices
Along this thread, we can also investigate the relationship between encodings. The right panel of Figure S shows the average Bures- Wasserstein distance between the Gram matrices of different encodings.
The average distance matrix shown in the right part of Figure indicates that all encodings are different, whereby some are more similar and some are more dissimilar (see also alignment pattern in Figure 7 Rips
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Rips
DTM
Height
PointNet
Rips
0.00
1.45
2.11
1.95
with Height fibration Point Nei
#3.00
=0.08
Pulbbck norms
Pointtee Heightt DTM
1.97
1:0.23
20.00
1:011
3.10
2,11
2.13
0,00
2.29
+0.45
20.45
=0.00
=:0.51
1.95
0.00
Rotation
Translation
Diletion
Strech_x
Sheering
Noising
Wiggly
Figure 8: Left: Average pull-back norm different perturbation vector fields with respect different encodings Right: Bures- wasessssserttnnn distance between Gram matrices J J of different encodings.
and DTM are closest each other, while PointNet and Height both differ significantly from Rips and DTM. This indicates that Rips and DTM capture similar information which is different from the information that is captured by Height and PointNet. This makes sense, since the DTM filtration function is the average distance to neighbors, which approximates the distance function that underlies the Rips filtration; moreover, the data we consider does not contain outliers. We also find that although Height and PointNet give relatively similar pull-back norms for rotation, translation, and dilation, overall these two encodings are very different.
5 Selecting Hyperparameters
In this section, we shift our focus to the problem: how do we select the hyperparameters of the encoding in order to detect a data feature of interest. Recall that there are three major hyperparameters to choose when constructig Pls: 1) the resolution P, 2) the kernel function 9(a.)) (z. y) and its associated parameters and 3) the weighting function abb,). We first focus on the first two PI parameters, namely the resolution and the variance for the Gaussian kernels. We examine their impact on the rank and spectrum of Jacobian (Section 5.1.1) and the pull-back norm of gradient vector fields of data features of interest. (Section 5.1.2) We demonstrate there is strong correlation between the pull-back norms of gradient vector fields and the downstream task performance, where the task objective is to predict that data feature (Section 5.1.3 We then investigate the impact of weighting functions on the pull-back geometry. We introduce the beta weighting function, which allows highlighting persistence intervals with different length (persistence time). Then we examine the effects of the mean parameter beta weighting function on the pull- back geometry (Section 5.2.1 Finally, again we show a significant correlation between the pull-bace back norm of gradient vector fields and the downstream task performance (Section 5.2.2
Real-world data In this section we utilize the brain artery tree data (Bendich et al. 2016). This data set comprises 96 artery trees in R (see Figure g left). These artery trees are obtained by applying a tube-tracking algorithm to Magnetic Resonance Angiography (MRA) images from 96 human subjects. We randomly subsample three point clouds from the vertices of each artery tree, with each point cloud containing N 500 points. Then we normalize the sampled point clouds to the unit cube 1]3.
Feature Each point cloud is labeled with binary feature, based the corresponding human subjects' medical information.
5.1 Resolution and variance Gaussian kernel
We focus on the 1-dimensional PIs on the Vietoris-Rips filtration. We investigate two hyperparameters involved the construction of PIs: 1) the resolution P. and 2) the variance 72 of the Gaussian kernel (see Figure g right). We set. the baseline PI parameters P= 20, 72 10 and consider
PH encoding
linear weighting function a(b.l) max(I] AAdms et al. 2017
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Rosiazton 30
ResoMttonn 13
Figure 9: Left: point cloud sampled from a brain artery tree, where the color represents the z-coordinate. Right: the corresponding PI representation with different paramcter settings.
5.1.1 Spectrum of the Jacobian
We begin by analyzing the effects of the resolution P and variance 72 of the Gaussian kernel on the spectrum of the Jacobian. In each plot of Figure [O] we varied one parameter while keeping the other fixed at the baseline setting, and present the normalized spectrum. We again observe a low-rank phenomenon, since the average rank is always below 160, while in the baseline setting the dimension of the point cloud and PI spaces are respectively m D N = 3 500 1500 and n P P = 20 20 = 400.
Resolution
Variance Gaussian
1.00c
Normelized Magnitude
Megnitude
1011
1.00e-05
1.00e-04
1 1.00--33
2017
101
1077
140
160
10a1
140
140
Eigenvalue Index
Elgervalue Index
Figure 10: Spectrum of the Jacobian matrix depending on the PI parameters. Left: The effect of changing the resolution P of PI. Right: The effect of changing the variance 72 of Gaussian kernel in PT.
In the left part of Figure 10 we see that the spectrum decays slower the resolution increases, indicating increase in rank. This implies that, as one would expect, higher resolution allows the PI to capture more information.
Interestingly, we observe that, as the variance 72 increases, the rank of the Jacobian initially increases and then decreases. For fixed resolution, very small variance results in sparse PIs (see the first column in the right panel of Figure 0 where multiple PD points (b,() may fall into one pixel and can only highlight that pixel; very large variance leads to blurred PIs (sce the third column in the right panel of Figure 9). where it also becomes difficult to distinguish between PD points.
5.1.2 Pull-back norm of feature gradient vector fields
We now explore the effects of the resolution and variance on the pull-back back norm of gradient vector fields of the following data feature.
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Rank Jacobian mappinn
Pull-back nom gradient (x10
128-57
158.30
338.51
10:50
11:23
$33.28
$14.26
$33.45
114.53
22.64
22.77
22.37
22.36
22.31
$2.20
Resolution
+.00.3
04:11 17
9.0095
86
+11.14
109.8
114.00
111.32
811.42
136.98
$1..09
119.00
$.1.82
$11.67
121.04
Recooluee
$2.93
$2.72
+2.36
11:34
$2.44
11,26
$2.38
11.07
+2.36
$2.31
10.27
=2.19
10
m0.00
70.97
P4:18
00.03
01:13
93.00
25.53
42.20
9:10
10:45
10.07
16.40
30.48
10.08
10.27
10.61
22 $4.30
$4.81
24.77
24.77
$5.04
25.12
R-
+2.79
9.20
$2.50
82
22.42
$2.40
22.37
22.34
1.0
2.29
3.57
4.86
6.14
7.3
8.71
10.0
1.0
2.29
3.57
4.66
6.4
7.43
8.1
13.0
Variance in Gaussian Kernel (x10 (x10-1)
Variance of Gaussian Kernel (x10 5)
Validation accuracy (%)
Robust validetion accuracy (%)
24.30
69.04
17:54
40:75
70.02 18:43
70.47
-7.57
70:47
37.69
70.13
17.01
28.32
73.47
15.00
71.17
8-
15.42
69.45
27:11
25.01
71.17
71.37
17.57
28.25
70.13
14.20
20:47
10
18.52
23.30
70.02
Resolution 7
09:22
69.45
71.15
11.85
90.18
69-78
75.13
70.47
Aesolution
-4-
68.01
09:45
71.53
71.31
10.47
N5.13
70 1B
71.17
a4.20
14.07
66:34
07.00
8.743
18.50
68.30
70
10.00
14:27
60.34
87.53
88.57
10.52
88.52
88.30
2-
56.55 55
25.83
24.04
65.30
67.71
28.82
89:15
$5.95
19-19
89
27:44
27.34
$2.44
25.02
62:29
A-
55.24
27.85
25.07
64.25
57.70
60.
60:30 70
23.83
62:64
07:01
23.03
59.00
23.63
70.15
10
3.57
4:86
7.43
100
2.29
4:66
6.14
7.43
8.71
10.0
Variance of Gaussian Kernel (x10 5)
Varlance of Gaussian Kernel (x10
9)
Figure 11: For the brain artery data set, shown is the effect of the resolution (vertical axis) and variance of the Gaussian kernel (horizontal axis) of the PI on the average rank of the Jacobian (upper left), average pull-back norm of the gradient vector field the see feature (upper right), as well as the test accuracy (lower left) and robust test accuracy (lower right) of the logistic regression model predicting sex based on the PI.
Our goal is to locate the optimal Pl parameters of the PH encoding to effectively detect the seæ feature in the brain artery point clouds. We consider the domain (P. E [15, 30] x 110-$.10-4 Here the ranges for P and 72 are selected based on the values where the spectra in Figure 10 exhibit the slowest decay.
In the upper right plot of Figure I1 we present the average pull-back norm of the gradient field of the ser feature under different PI parameter choices. The gradient fields are estimated via numerical methods detailed in Appendix F.4 We observe that the pull-back norm generally increases as the resolution P increases, whereas the pull-back norm is not monotonic with 7². Moreover, the optimal value for y² varies depending on the choice of resolution. Also, comparison with the upper left part of Figure reveals that the maximum pull- norm is not necessarily attained for parameters where the rank of the Jacobian is maximal, ie., when PIs capture the most information about the point cloud.
5.1.3 Correlation with downstream task performance
We investigate the hypothesis that a high pull-back norm correlates with the performance of a predictor trained on the encoding. To this end we feed PIs generated with different choices of the parameters into logistic regression models and train these to predict the sez feature. Here we use logistic regression as the downstream model bocause its simplicity, with the intention to minimize the impact of model complexity and training techniques on the task performance. We provide results for convolutional neural networks (CNN) in Appendix F.4
We evaluate the performance in terms of validation accuracy and robust validation accuracy using cross- validation, which are presented in the lower left and lower right plots in Figure II The validation accuracy and robust validation accuracy exhibit similar pattern to the pucl-back norm. Notably, all three quantities reach their maximum at around P = 30 and n² 3.57 x 10-5, and their minimum at the lower-left corner.
Robust validation accuracy evaluates the accuracy on data set subject additive Gaussian noise on the inputs.
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Voa ddlation accuracy
30-
Rors
Robuse
PCD:6.52
PCC: 0.36
POC: 0.82
POC: 0.85
162
-22 22 UA
-82
02.
UA
jocobion 1b)
Pull bollbbakroom
[c]
Figure 12: Gaussian kernel density estimation of the joint distribution of four pairs of variables: (a) Jacobian rank vs. validation accuracy; (b) Jacobian rank vs. robust validation accuracy; (c) pull-back norms vs. validation accuracy; and (d) pull-back norms vs. robust validation accuracy, where the downstream models are chosen as logistic regression models.
For more quantitative comparison, Figure 12 shows a kernel density estimate of the joint distribution of four pairs of variables: Jacobian rank vs. validation accuracy, Jacobian rank vs. robust validation accuracy, pull-back norm validation accuracy, and pull-back norm vs. robust validation accuracy. The plots clearly indicate strong correlation between the pull-back norm and the performance Pearson's 's correlation coefficient (PCC) between the four pairs of variables, along with the p-value for a two-sided Lest,, are presented in the lower right corner of each plot in Figure 12
the downstream task. The
We conclude that for the considered task, there is a significant correlation between the pull-back norm and the downstream task performance. is also interesting to interpret these results together with the rank of the Jacobian (upper left in Figure 11). The results demonstrate that the improvement in downstream performance is only somewhat correlated with including more information, but it is strongly correlated with including the most relevant information, which is precisely quantified by the pull-back norm. Therefore, we suggest that the proposed framework can be used to select appropriate PH encodings in practice. Note that the procedure independent of the downstream model architectures and training techniques.
5.2 Weighting function
PH encoding We maintain our focus on the 1-dimensional Pls with respect to the Vietoris-Rips filtration. We set the bascline PI parameters as P= 20 and 72 3 10-5. For the weighting function, we consider the beta weighting function induced by the probability density function of a beta distribution:
T(a+ 3) (x=)a=1(11 = I(a:I(())
ab,
where I(-) is the Gamma function and н is a scaling factor. We consider the mean-variance parameterization for the beta weighting function: = 1: (((=)) -1) and 3 (1 - k.) (k(2-)) -1). Here k is the mean parameter, which controls the concentration of the weighting function, and s² is the variance parameter, which controls the "degree" of concentration. We set 8² as 0.065 and as 1.
We consider beta weighting function for several reasons: 1) beta weighting function is compactly supported, which is more suitable for PI; 2) it assigns zero weight to the horizontal axis, which aligns with the stability criteria proposed by Adams et al.) (2017): 3) by tuning the mean parameter, one can highlight persistence intervals with different length (see Figure 13 for an illustration). This allows to investigate questions such "are short persistence intervals more crucial to this application than long persistence intervals?"
5.2.1 Pull-back norm of feature gradient vector fields
We investigate the effects of the mean parameter on the rank of Jacobian and pull-back norm of the gradient vector field of the see feature (see the left and middle panel in Figure 14). In the left panel of Figure 14 observe that as the weighting function assigns more importance to longer persistence intervals, the rank
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km0.00 0.07
km0
km 0.
7-
6
5-
5-
6-1
3-
2-
0.0
0.2
0.4
0.5 0.0
0.2
0.4
0.5 0.0 0.1
0.2
0.4
0.5
Length of the interval
Length of the interval
Length of the interval
Figure 13: As a weighting function for PIs for the brain artery data, we use the beta weighting function with different values of mean parameter Larger assigns more importance to longer persistence intervals. The top right corner of each panel shows the 1-dimensional PI derived from the Rips filtration on one point cloud, illustrating the impact of the weighting function depicted in the main plot.
of the Jacobian monotonically decreases. This aligns with the fact that the mumber of longer persistence intervals is generally smaller than the number of shorter persistence intervals. However, the pull-back norm peaks when & is set to 0.1 and then decreases as k increases further.
5.2.2 Correlation with downstream task performance
We again examine the correlation between the pull-back norm and the performance of the logistic regression models trained on Pls. We present the validation accuracy in the right panel of Figure [4] Notably, we observe that the validation accuracy demonstrates a similar pattern to the pull-back norm. Quantitatively, the Pearson's 's correlation coefficient (PCC) between pull-back norm and validation accuracy is 0.839, with a two-sided test p-value of 0.009, In contrast, the PCC between rank and validation accuracy is 0.338, with p-value 0.413, which indicates once again that including more information in the data representation does not necessarily improve the downstream performance. These findings reinforce conclusion that the pull-back norm is highly predictive for the downstream task performance in this task.
Pul-back
Validiatoon accuracy
0.16
0.675
100
0.650
05
0.12
85
50
0.08
0.06
0.07
0.08
0.09
0.30
6.31
0.22
0.13
0.14
0.07
0.68
0.99
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6.11
0.12
0.13
0.14
6.57
0.68
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0.10
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0.13
0.14
Maan parametar (k)
Slean parameter (O)
Muan parornstat (k
Figure 14: The impact of the mean parameter for the beta weighting function the rank of Jacobian (left), pull-back norm of gradient field of feature ser (middle), and the 7-folded validation classification accuracy (right). The pull-back norms and validation accuracy are strongly correlated, and they both indicate persistence intervals with medium length are vital to classify the see feature.
Interestingly, we observe that both the pucl-back norm and the validation accuracy reach their maximal at an intermediate value for the mean parameter. This implies that persistence intervals of medium length are most crucial for classifying the ser feature, which is consistent with the observation in the original paper Bendich al. 2016). We note that this an example of application for which medium length intervals the barcode contain the most information for the problem at hand. In Appendix G we complement this discussion by considering real-world data set of point clouds sampled from human body meshes, on which
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compare long and short persistence intervals from different perspective: which part of the point clouds is the focus of long intervals and which is the focus of short ones.
6 Conclusions
The methods and observations presented in this work contribute to addressing some of the main bottlenecks in the practical application of PH, namely how to identify which data variations are captured by PH encodings, how to quantify the effectiveness of these encodings in detecting particular data features, and how to select the parameters of PH encodings in order to obtain data representations that are suitable for solving particular task.
We presented ways to analyze the most relevant features on the data manifold that are captured by persistence images with different choices of the filtration and compared the results with neural-network- based encodings. For example, in the RFP data set we found that while a pretrained PointNet had a relatively high alignment with translation and dilation, the 1-dimensional persistence image encoding with Height filtration had a high alignment with stretch, and the 1-dimensional persistence image encoding with Rips filtration had a high alignment with data variation that makes the point clouds more convex. At the same time we observe that the response of the encodings to these perturbations is less than 10% as strong as for other more abstract data variations captured by the singular vectors of the Jacobian; for instance, the maximal value taken by inner products between unit langent vectors representing perturbations and unit singular vectors is less than 0.1.
We demonstrated on the real-world brain artery tree data set that feature aligmment as measured by the Jacobian permits PH parameter tuning without the need to train classifier top of the data representation in order to select the parameters based on the test accuracy. Rather, one can select the parameters based on the pull-back norm of the features of interest, and perform training using the data representation with the highest pull-back norm. Mcanwhile, we found that the persistence intervals of modium length are crucial for classifying the ser feature on this data set. This goes against the popular belief that long intervals are the most important, at the same time confirming the findings from the original paper that employs PH on this data set.
Limitations and future work Our analysis is based on the structure of the Jacobian of the data encoding, which by nature focuses only on local variations of the input data, In future it will be interesting to further advance these methods in rogard to non-local data variations, where synthetic notions of derivatives such as our empirical evaluation of the vector fields, and ideas such as the application of the iterated closest point method could serve as point of departure. Further, the analysis of non-linear transformations via Gram matrices has seen a number of recent advances in the context of artificial neural networks, and it will be interesting to explore possible synergies between those investigations and PH data encodings.
A limitation of the proposed methodology is the assumption about the differentiability of the encoding, and the need for a Riemannian manifold structure for the representation space. For this reason, our methodology cannot be directly applied to analyze certain common PH representations, such as persistence diagrams, since these cannot be endowed with a smooth structure (Leygonie et al.] 2022). We note, however, that a Riemannian framework for approzimated PDs has been introduced by Anirudh et al. (2016).
The pull-back geometry approach also faces some computational challenges, since calculating the average pull-back norm for either perturbation vector fields or gradient vector fields requires the computation of the Jacobian matrix Jx. which is of size (P x P. D. x N), for cach data point X, where P, D, N are respectively the resolution for PI, the dimension of points in the point cloud and the number of points in the point cloud. On the positive side, this enables insights that are more intrinsic to the problem rather than being dependent on the choice of a classifier. Porformanco-based methods can also involve computational challenges, due the need to choose the downstream models and tune their hyperparameters. We regard the proposed methods not as substitute but complementary to berformance-based methods.
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Acknowledgements
We thank anonymous TMLR referees for helpful feedback. This project has been supported in part by NSF CAREER 21 2145630, NSF 2212520, DFG SPP 2298 grant 464109215, ERC Starting Grant 757983, and BMBF in DAAD project 57616814.
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Appendix
The appendix is organized into the following sections.
Appendix
A
Notation
Appendix
B
Riemannian manifold structure of the space of point clouds
Appendix
Diffcrentiability of the mapping from point clouds to Pls
Appendix
D
Visualizing the Jacobian of the encoding over the data manifold
Appendix
Point saliency maps for PH encodings
Appendix
Details on the experiments
Appendix
G
Investigating which part of the data is highlighted by PH encodings
A
Notation
Table provides a summary of the notation.
B Riemannian manifold structure of the space of point clouds
Let M denote the collection of all point clouds in RD' that contain exactly points,
M= (x =
Recall that the 2- Wasserstein distance dw between two point clouds of the same size in R.D. is defined as
Σ Edb((s((a))) 2e
dw (X,Y)= wER(X,))
min
where is bijection between X and Y, ?(X,Y) contains all bijections, and de denotes the Euclidean metric on D. The 2- Wasserstein distance defined above induces metric topology on M.
Compared to other distances in the space of point clouds, for instance the Gromov-Hauudorff distance which is commonly used in the study of stability theory of persistent homology (see, e.g., Blumberg & Lesnick 2022), the 2- Wassersstnn distance endows the space of point clouds with favorable manifold structure. This manifold structure ensures that every small neighborhood isomctric to an Euclidean open set.
We discuss the topological manifold structure (Appendix B.1 and Riemannian manifold structure (Ap- pendix B.2) the space of point clouds. Then introduce the Riemannian style definition for perturbation vector fields and gradient vector fields (Appendix B.3
B.1 Manifold structure
We proceed to establish a manifold structure on M.
Proposition 4. Let M be the set containing all point clouds in TRP with N distinct points, and dw be the 2- Wasserstein distance on M. For any point cloud X E M, there crists Wasserstein ball Bw (X,sx) and an injective mapping εx BwwBB(XKr (x,((X)))+ + DxN such that
dw d=((x(())xxxx2)) VY, Ze Bw(X,ex)
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Table
Notations and definitions
Notation
Definition
M
The manifold of point clouds
m
Dimension of manifold M
Dimension the space where point clouds are located Number of points point clouds
N
X
Point cloud
Point in point cloud
dw
Wasserstein distance between point clouds The set (1,
[N]
Y(
The set of bijections between point clouds X and Y Bijection wet point clouds X and Y Euclidean distance
wi wiX-1
dø
K,
Simplicial complex
Simplex
Φ: - IR
Filtration function
Birth parameter of homology classes
Death parameter of homology classes
Lifespan paramcter of homology classes
TxM
Tangent space at X on M Tangent vector Vector field
V
V:M+up TxM-
V(X)
Tangent vector at X assigned by V Perturbation mapping on the space of point clouds Vector field induced by perturbation One-dimensional feature function on the space of point clouds Gradient vector field of function
T:M+M
V
p: M
Vp
N
The manifold of persistence images Dimension of manifold NV
Transformation PD points from birth-death to birth-lifespan coordinate Gaussian kernel located at (b,1) Resolution of persistence images
9b.t
P
??
Variance of Gaussian kernel in persistence images
abb,
Weighting function Persistence surface
bb
k
Mean parameter for the beta weighting function
I:M+N
Encoding map from M to N
Jacobian mapping (Jacobian matrix) of map f at X
Gy,
Gram matrix of map f at X
The i-th largest singular value the Jacobian mapping
4.44
The i-th eigenvector of the PH encoding mapping
1.Is
Pull-back norm induced by mapping f Finite data set of point clouds
D
daw
Bures- Wasscrstein distance between positivo-defiite matrices
Proof. Consider point cloud X = N in M. For arbitrary é 0, we construct an injective mapping Ex from EX) to D*N then choose a radius fx such that the above equation holds. Denote [N] = (1,2...., The map (X) can be characterized by total order X, r [N]] X, where
(())))
[NN] DNN
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reorders X by assigning [N] (1, to {a:) For any other point cloud Y e Bw (X,,,) there exists an optimal transport plan between X and Y, denoted by wxy : Y, satisfying
Σ de(z,(()))
WXY= = arg min w x
This maps assigns each element in X to distinct element in Y. We define an embedding &x from Y to R DxN as follows:
Ex [wxy o T()), ((2),...., o { DxN
We proceed to show that ςx is an injective embodding. For any Y, Bw (X,e) with Ex(Y) Ex(2Z) consider optimal transport plans wxy between X and Y, and wxz between X and Z. Since (Y) Ex(Z), we have wxy(x) (x) Vx X. Hence, for any Y
y wxY o (wxY) (v) = wxz (wxx)- E Z.
Notice w (w bijection between Y and Z. Therefore, Y Z and εx is injective.
Next calculate the radius &x that preserves the distance between any two point clouds. The goal is to find radius e such that for any Y, Bw (X,e), the Wasserstein distance between the point clouds Y and Z,
Σ [(,uuur(()) Yye
dw (2, 2)
is equal to the Euclidean distance between the embedding Ex(Y) and Ex(2),
N Σ d o (()))yor()))xx 1=1
de(Ex(Y),Ex(22)
Σ (B(wxx()) xeX
Σ (z, (wxr)
Notice it suffices to find radius E such that for any Y, Ze Bw (X,e), wyz = wxz (wxx)- Equivalently, the optimal bijection between Y and Z is given by the composition wxz o (WXY) : - - z. The key idea is that if and are both sufficiently close to X in the sense of the Wasserstein distance, then the distance between y e and (wxy)~'(s) (y) and the distance between 2 and (wx2) (2) will be small. Hence, each point y Y will be close wxz (wxx)-=(() and thus will have wyz wxz (=xx)- The situation is illustrated in Figure 15
Denote wxz (WXY)- as Wyz. For any point cloud Y Bw(X, we have
dw(X) Σ (c(l(wxx(=))) <E. xeX
Hence,
max de ( (x CE.
2EX
This suggests for any two point clouds Y,Z Bw (X,E),
maxds (x) 2z.
mex
Equivalently,
maxde(y.uyz(0 < <s.
yeY
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t wyz (v)
6 2
(x)(x)
wxz(xx)
WYZ (1)
Figure 15: Shown are point clouds Y (orange triangles) and Z (green squares) in a Wasserstein neighborhood of point cloud X (blue circles). Left: both (xy E and wxz(x) are close to a e X, and hence they are close each other. Right: the distance between y and # 4yz (v) Z lower boundod.
Let denote the minimal pairwise distance of points X:
8 min x2( 21,22eX
Note is strictly greater than zero since points are mutually different. For a fixed point y Y, any point z in Z other than (y) has a lower-bounded distance from y::
min (DVZ(1))))
d 2s, Vy YyeY.
Now consider ex= = We have
dE(1, 25x
- - 22x
min dE(U,)); VyeY.
4
4
#z (wyz(a) 2*
Equivalently,
wyZ aig (2, VyeY.
26%
This means wyz Wyz, which completes the proof.
Corollary 5. Let M be the set containing all point clouds in R with N distinct points. M, together with the metric topology induced by Wasserstein distance, forms manifold of dimension D N.
Proof. By Proposition for every X M one can find neighborhood Bw (Xx,xx) and an injective mapping Ex: (xx,eex) R D satisfying
VY, Bw (wxxx dw (Y, 2) dx()
Notice εx is a bijective isometry between (Bw (X, and (Ex(Bw (X,,)))) dg). Hence, &x is open and continuous. Consequently, εx homeomorphism, and {Ex: (w(xxxx) - RDNN xeM, serving as an atlas, endows M with the manifold structure.
B.2 Riemannian metric structure
Next we introduce the Riemannian metric structure for the manifold of point clouds, and show the distance induced by the Riemannian metric coincides with the Wasserstein distance. To this end, consider an alternative definition for the space of point clouds:
M' = [X:[N]] + -R' |X is injective ~Sx
The equivalence relation is defined by:
X1 Sy X₂
Ev SN X1 X₂
Im(Xi) Im(X2).
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Figure 16: Shown is a curve in the point cloud space [n]] E C, which maps the set (1, x [0,1] (left) to the plane I2 (right). In the right panel, the collection of black dots represent the image of y at a specific time [0, 1], which forms a point cloud in R2; the collection of orange arrows represents the velocity tangent. vector at " along the curve 7.
Here Sy denotes the N-symmetric group and Im(X) denotes the image of mapping X. Two mappings are deemed equivalent. when their images identical. Note the image of each mapping X [N] R is a point
cloud in IR we defined earlier, i.e. Im(X) ε M. In fact, the mapping M' - [M;[[]]] lm(X) gives the identification between the original and new definitions for the space of point clouds. For simplicity, we use one representative X to denote the equivalence class [X] and also refer to X's as point clouds.
This definition allows defining smooth curves in the point clouds space. Specifically, a smooth curves in M' is an element of the following set:
C= c [N] 1 RD (:;t) is injective, V (k,(k; C² [NJM/ ~Sx
Here I denote the closed unit interval [0, 1] and the equivalence relation is defined by
Eue e SN : (v(-).-) 72(1-)) <> Im(7)())) Im(r2((;)),, VL.
A curve in point cloud space [>] C essentially collection of N curves in Rº such that each time the N points on those curves form point cloud (see Figure I6 For simplicity we use y to denote the equivalence class [?].
The langent space at point cloud X is defined as
TxM'= = (V Im(x) Im(X) - D V a vector field on RD).
A Riemannian metric manifold structure that smoothly assigns positive-defiite inner product gx (:.)) the tangent space T at each X e M. We introduce Riemannian metric for M follows: TxM
gx (), Σ (V()) zelm(X)
Endowed with this Riemannian metric, M' is Riemannian manifold, denoted by (M',g). We now discuss the distance induced by this Riemannian metric. For a curve in point cloud space y C, the velocity tangent vector at Yt along curve y is defined as (see an example in the right panel in Figure 16):
te 1m(m)) RR:; x (ko,).
Here Kg e [N] is the preimage of point x under Te, iee, (e(k)) The length of curve is given by the length functional L:
L:C R; THe V In (in, dl.
The distance induced by the Riemannian metric is the minimal length of curves between two points on the manifold.
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Definition 6 (Riemannian distance). Let (M.9) be Riemannian manifold. The Riemannian distance between two points X,Y V. e .M is defined as
d,(X,)) (L7) y smooth curve in M connecting X and Y).
We point out the Riemannian distance coincides with the Wasserstein distance. To see this, we need the following result by Do Carmo & Flaherty Francis (Lemma 2.3, Chap. 9, 1992).
Lemma 7 (Do Carmo& & Flaherty Francis 1992. Let Xo, X1 be two points in Riemannian manifold M, and curve joining Xo to X1. Then y minimizes the length functional if and only if y minimizes the energy functional defined as follows:
E:C. + I;; 7 S, gh (trs dt)
Moreover, when y is the minimizer, L(n) VE().
Now we procoed to introduce the main statement of this subscction. Proposition 8. Let gM' be the Riemannian manifold of point clouds. Then the Riemannian distance is equivalent to Wasserstein distance, dg dw.
Proof. Consider two point clouds Xo, X1 E MM. Let L be the length functional, and E' be the energy functional of all curves connecting X, and X,. For any curve y joining Xo to X1, we have
E(7) f, gn (frrft) di
Si Σ (f(()),<(<))))
zelm(1))
Σ (k,n(k,, ddt
xelm(re)
N
= Σ (0r(ktt)
Σ J, (k,
N
=
= Σ El>(k,))]
The above equations along with Lemma 7 indicate the minimizer of the length functional L coincides with the minimizer of E E(r(k,-),. Recall for each k, ((,-)) )) is curve in IRD joining Xo(k) to X1 (k).. In Euclidean spaces, it is known that straight lines minimize the length functional, which implics, by Lemma 7 they also minimize (k)(k,))) Specifically for fixed end points Xa(k) and Xi()), E(>(k,)) has minimal value de (Xo(k), (k))). Therefore, finding the minimizer of ΣN E()(k,)) equivalent finding the bijection between Im (Xp) and Im(X,) that produces the minimal value of the sum of squared distances between points paired by the bijection. This exactly coincides with the optimal transport problem. In conclusion, we have
dg(Xo, X1) min I((r)
-min v E(r)
1
-min Σ [l((k,)]
min
Σ &(z,w()))
(e)) (x))
dw (mmmmm(X)) Im Im))))..
This is what was claimed.
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B.3 Vector fields
In this section, we provide definitions for perturbation vector fields and gradient vector fields for the manifold of point clouds.
Assume T: M + M is a perturbation mapping. As introduced in Section 3.2 the perturbation vector fields V in the Euclidean space defined as:
VL (x) = x E.EM.
The main idea behind this definition is that each tangent vector (X) specifies the direction of the straight line connecting and (XX). For general Riemannian manifolds, the notion of straight lines is generalized by minimizing geodesics. Formally, the curve between two points on manifold that minimizes the length functional is minèmizing geodesic. Then the question arises whether there exists minimizing geodesic between any two points on the manifold. To address this, we introduce the concepts of geodesic completeness and the Hopf-Rinow theorem.
As introduced in Appendix B.2 the Riemannian metric induces a distance dg on the manifold M. A sequence Cauuchy sequence if for any positive number e there exists a positive
(Xi) of points on (M, dg) integer N such that (X, X,) < <E, Vi,j N.
Definition 9 (Geodesically complete manifold). The Riemannian manifold (M,g) is geodesically complete if any Cauchy sequence {Xi)iez+ converges in M: BY M such that lim,- dg(X,, Y) =
Since the Euclidean space R a complete metric space, automatically the Riemannian manifold point clouds is geodesically complete. Also notice this manifold is connected, since there exists a path connecting any two point clouds. The Hopf-Rinow theorem ensures the existence of minimizing geodesics between any two point clouds.
Theorem 10 (Hopf-Rinow theorem). Let (M,g) be a connected Riemannian manifold. If (M,dg) is geodesically complete, there ezists minimizing geodesic between any two points on M.
Now we formally define the perturbation vector fields.
Definition 11 (Perturbation vector field). Let (M,g) be a geodesically complete manifold, and म M M be a perturbation mapping. The perturbation vector field Vx is defined as
V - ux X V+(X) (x),,
where (X,((X) is a minimizing geodesic 7x.r(X) - M with (x)+(x) X and (x,=(x) x(X).
We proceed define the Riemannian gradient vector field. Assume p is a real-valued smooth function on M. In the cases of Euclidean spaces, the gradient vector can be characterized by the following property:
a (V)(X), " VXERm. " " R
For general Riemannian manifolds, the notion of directional derivatives is generalized by derivations. Definition 12 (Derivative). Let M be manifold, and C (M) be the space of smooth functions on M. A derivative at Xe M a linear map Ø: - (M) + R satisfying the Leibniz identity:
a(fg)= -000g9(()+-) + a())
For a fixed point X E M, it turns out that each langent vector o M can be uniquely associated with a derivative, denoted by d,, in the sense that (p(p) measures the rate change of the function value p(X), moving through X with the velocity specified by v. Detailed discussion regarding the equivalence between
tangent vectors and derivations can be found in the work of Tu (Chapter 8, 2011). Now we provide the definition of the Riemannian gradient.
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Definition 13 (Gradient vector field). Let (M,g) be a Riemannian manifold, and p M R smooth function. The gradient vector field of denoted by vp, is defined as the vector field
Vp:M-uxTxMM X vx))
satisfying the property:
9x ox(V VX MXEM,
Note that Definition 11 and Definition
13 applicable to other types of data, provided that the data space can be equipped with a geodesically complete Riemannian manifold structure.
B.4 Pull-back metric
In this section, we provide definitions for pull-back metric for general encoding mappings between Riemannian manifolds.
Let gM, be the Riemannian manifold for input data, (N gx') be the Riemannian manifold for output data, and f + W be a differential encoding mapping between the input space and output space. Recall that for any X in .M the Jacobian of f at X, denoted by J is a linear mapping between TxM and Tςxx).V The pull-back metric induced by f. denoted by g is the structure that assigns the following inner-product on the langent space TxM for each X e <MM:
g (V (V.W)= gN (J),
Then the pull-back norm for any tangent vector V X is defined
| = (V.VV
Please note that when considers N as vector space and IN as the Euclidean metric, the above definition for pull-back metric reduces to the one that we introduced in Section 3.3 Meanwhile, we point out that, while the Riemannian metric on M does not affect the pull-back metric, it is still necessary in our approach since it's essential in defining the perturbation vector field and gradient vector field (see Definition I1 and Definition 13
C Differentiability of the mapping from point clouds to Pls
We provide details for the differentiability of the mapping from point cloud data to PIs and the computation of the Jacobian. The encoding mapping from the point clouds to PIs can be viewed as the composition of the following two maps:
M Bar + +
X PD, (Filt(X)) (I(PD((FI((X))).
Here Bar denotes the set of PDs, Fil((X) denotes filtration on a point cloud X, and PD2 denotes the k-dimensional PD on filtration Filt(X).
Recall that by definition, a persistence diagram is a multiset consisting of bounded and infinite persistence intervals, with its elements in x R where İ. Ru (+oo). As a space multisets, Bar does not naturally come equipped with differential structure. However, one can study the differentiability for maps from or Bar by equipping Bar with diffeeoogy structure (Leygonie et. al. 2022 Section 3.5). Specifically, for any
positive integers and j. Bar is covered by the map R x Seg Bar, where 2 R can be thought of as the space of ordered PDs consisting of fixed number (resp. j) of boundod (resp. infinite) persistence intervals and Sty the quotient map modulo the order. Note the action of St,, would turn vectors into multisets. Then map A M Bar is said to be differentiable at X e M if there exists an open neighborhood U of X,
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positive integers i.j and differential map Ā:U + R²i R such that A Sa,5 A:
R 28
x
R
S
A
M
Bar
Similarly, map B Bar W is said to be differentiable at Z Bar if for all positive integers i,j and all vectors Z E R x IR such that (2 Z, the map Bo Stg I x R + N is differential open neighborhood of Z:
2i x R
S
B
Bar
N
We refer reader to Leygonie el al. [2022] for a detailed introduction to the analysis above.
Differentiability for the map from point clouds to PDs Let X M be a point cloud, U be an open neighborhood of X, and CI(X, R) be the clique complex on the R-neighborhood graph of X. A filtration Filt would induce a total order on the simplices in CL(X, Rt) in the sense that o I T if and only d(o) s ((T) where e is the filtration function determining the filtration. First let us assume that 1) Filt induces a strict total order at point cloud X, i.e., simplices in CI(X, R) have distinct filtration values, and 2) the distance between any two points in X does not equal to R. The filtration Filt then defines map sending point cloud to vector of filtration values as follows:
d X
IR (I(I))))))
where ((01)
(0(2) <... (()))))).)). Note that, according our second assumption, i.e., pairwise distances of points in X do not equal to R, one can assume that all point clouds in U would have the same number of simplices in the corresponding clique complexes. Specifically, consider any point cloud Y e U and the optimal transport plan wxy from X to Y (see Appendix B.1 There exists a bijection between CI(X, It) and R):
AXY : xx + (wxx (Z): R).
Hence, the map Å₁ is well-defined on U. Note also that A is differentiable at X if the filtration value d(a) is differentiable at X for any simplex a CI(X, Rt). Then given homology degree k the filtration would define a barcode template, which is multiset Pa of pairs of simplices in CI(X, R) and multiset (X) of simplices in CI(X, R), such that (Leygonie et al. 2022):
(F)l((()))) =
((o(o')++oo))
This defines a map Ag sending the vector of ordered filtration values to a vector of ordered PD:
dz: - 2 ((((()))))))))))
((T:))
e R2 x R
In the case where the order induced by the filtration a strict total order, the barcode template is uniquely defined, and hence the map Ag is well defined (see Babrie-aa Meanwhile, one can assume that the barcode template would be stable on U in the sense that if (α,T) Px(X) (resp. o' e (X)), then (Axy(o), (x)(r)) P&(Y) (resp. Axy(o') U+)Y)). Hence, the map Åg can be viewed as fixed permutation on (U) and is hence differentiable. In practice, the barcode template can be obtained via the matrix reduction alg; we refer readers to Leygonie et al.] (2022) for more details regarding the existence and construction of the barcode template. Please note that the map Á = Åg Å would be the desired differentiable map such that PD (Fil((X)) St, Á(X) and with this map we have shown the map PD₃ (Fil(()) is differentiable at X if the filtration defines strict total order on CI(X, and the pairwise distances of points in X do not equal to R.
al. 2019 for a detailed discussion).
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At certain point clouds, the filtration could induce non-strict total order since multiple simplices can have the same filtration value. In such cases, the map A2 may no longer be well-defined since the barcode template is not unique. Consequently, the derivative of A2 should be considered as a subderivative -abrie-Ga ct al. 2019) and the map (D2(Filt()) is not differentiable at such point clouds. In practice, this can be resolved by refining the total order into a strict order deterministically or randomly (see, e-g., "t,,Skraba el, al. 2017), which corresponds to selecting an element in the subderivative. On the other side, at point cloud X in which there are two points with a distance between them equal to R, a small perturbation on these critical points can result in removing simplices in CI(X, R). Hence, the the map Å1 may not be continuous at X even if one considers its codomain as a space of sequences, and the map PD, (Filt(()))) would be non-differentiable at such point cloud. However, please note that the set of point clouds with such property has a dense and open complement in M.
Differentiability for the map from PDs to PIs The map from PDs to PIs is guaranteed to be differentiable everywhere in Bar (Leygonie et al., 2022 Proposition 7.3). We note that here one needs to consider the subset of Bar consisting of only bounded intervals. In practice, for computations, one sets the death value of infinite intervals to the maximum filtration value R.
Generic differentiabilly, for PH encoding As shown above, the PH encoding is differentiable at point clouds satisfying 1) the filtration defines strict total order and 2) the pairwise distances of points in the point cloud do not equal to R. We now point out that for Victoris-Rips filtration and Height filtration, the PH encoding is generically differentiable, Le., the set where PH encoding is differentiable is dense in M. As we indicated above, the set of point clouds that satisfies the second requirement. open and dense in M. Hence, it is suffices to show the set of point clouds where the filtration defines a strict total order is also dense and open, as the interscction of open, dense sets would still be dense.
For Height filtration, by Leygonie et al. (Proposition 5.2. 2022 we claim that the filtration defines a strict. total order for point clouds outside the following set
M= E M: 301 02 E ()(X,R),((o) (
M:
where Clo(X, It) denotes the set of all vertices in CI(X, R). Hence, the PH encoding would be generically differentiable since the complement of M is dense in M. We point out that Leygonic et al. (022) consider a slightly different setting where the input space M is considered as a parameter space for the map do + (C)((X For example, consider M M- and for each f M define (o(1; (), Vu e Cl,(X, However, one can easily adapt our setting to be consistent with the their setting as follows. Assume the height filtration function du is determined by the vector w, ie., Φo(c) (c. w)). Consider any point cloud X E . and a neighborhood U of X. The map do can be viewed as parametrized by U in the sense that for any Y e U, define (u;;; Y)= x (v), w), VuE Clo(X, R), where Wxy is the optimal transport plan from X to Y (see Appendix Bi Intuitively, as moving a vertex U E Clo(X X, R) would result in a change in (o(v), the coordinates of all the vertices in Clo(X, R) can be viewed as a parametrization for the map Φo-
For Victoris- Dictoris-Rips filtration, we will need the concept of general position for point clouds. Specifically, point cloud X (x1,-..,,NN) is said to be in general position if v (i.j) * (k,1), where k, (1,.....2), |z, - 2z * | 21122- Following Leygonie et al. (Corollary 5.9, 2022 we claim that the Rips filtration would define a strict order for point clouds in general position. As the subspace of such point clouds is dense in M, the PH encoding is generically differentiable. We point out that Leygonie et al. (Corollary 5.9, 2022) consider the domain of the barcode valued map as the vector space RDAN_ However, their results can be extended to the above claim by noticing that the manifold of point clouds in our setting is locally homeomorphic to D (see Appendix B.1.
Computation for the partial derivatives We now derive the partial derivatives for the PH encoding f:M + N c PxP, First, assume f is differentiable at X e M. As discussed in Appendix B.1 there exists a neighborhood U of X and a coordinate map U D Specifically, assume 7: [N] + X is the total order that characterizes {(X), i.c., {(X) = [[T(1),(() (2).... (N)] (see Appendix BJ. Recall that when constructing PIs from transformed PDs, one needs to choose a smooth kernel I(.)), e.g., Gaussian kernel, and smooth
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weighting function a(b,l), e.g., linear function of I. The persistence surface a would then be defined as:
(xx1))= Σ ((.11))(1))(
(b.))e((PD)
Then, one can obtain a persistence image in the form of a P P matrix whose (i,j)-th entry is:
Phy S Spixel (x,
Here Uij pixel ij forms grid subdivision of subdomain of tb. In our experiments, consider rectangle subdomain [mmin Fmax] x [thnin Vmax] and evenly-spaced rectangle pixels. Specifically, consider grid points Zmin + kÂx for h- = 0, I,..,,P, where Ar Emis-Emin and yn Vmin + lyy for | = 0, o....... Ay = Mmax--Wii Set pixcl; = [[; [Uj. 35+1 for i 0,1..... .......--. P-
......,, P, where
and j = 0, 0.1,..., P-1 1. We then
estimate the integral value by
Ply - (xi,
where is the area of pixely: i.e., = Ardy For any (i,j) [P] [P], & [NV, the partial derivative @PI; can be formulated as
(r(k)
OPL (T(k)
Σ OP.; do(g)
(F)I)) do(a) 0T(k)
We point out that (Filt((()) is a multisct consisting of bounded and infinite persistence intervals and the notation ((r) E (FI(((((X)) is used to denote the sunmation is taken over all the filtration values that appear in these intervals.
For point clouds where the filtration only defines a non strict total order, 0r(R) should be viewed as a subderivative as discussed above and one needs to select element in the subderivative. For example, consider a point cloud Xu consisting of points evenly-spaced on S' (sce Figure 17) and the PH encoding that outputs the 1-dimensional PI on the Rips filtration. Notice the distance bet ween any two adjacent points takes the same value. Therefore, the barcode template is not unique. In the first row of Figure 17 we show the top eigenvector and the Jacobian matrix at Xo and other two point clouds near Xo. As shown in the figure, both eigenvectors and Jacobian matrices are indeed unstable. We point out, however, we perturb X, even slightly, then clouds for which the encoding is diffcrenniable is dense in M. As shown in the second row of Figure 17 the eigenvector and the Jacobian matrix become stable near point cloud X1, which is obtained by adding slight noise on Xo. A similar result can be observed for point cloud whose points are uniformly sampled from S (sce the third row of Figure 17). We also note that in practice, due to numerical approximation errors, it might still be the case that, e.g,, even for points in generic position, one might have that two edges in the Vietoris-Rips complex might. have the same filtration value. This is an issue that needs to be considered applications, and that could inform specific choices in computations.
Bd(a)
expect the information contained in the Jacobian to be reliable, because the set of points
Meanwhile, point out that while the information of Jacobian specific point clouds can be unstable, the computational results in this work are still reliable since these quantities, e.g., average inner-product between perturbation vector fields and the top eigenvectors (Section 4.2) and average pull-back norm (Section 4.3), are computed over diverse point clouds within a dataset and those undesirable point clouds are very sparse in the input space as we have previously shown. To demonstrate this clearer, we experimentally investigate the relation between average pull-back norm and the number of point clouds considerod the computation (see Figure 18]. We obeserve that, on both RPF and brain artery tree data set, the pull-back norm quickly converges to stable value once a certain amount of point clouds are included in the computation. This demonstrates the stability and reliability of the insights obtained via our methodology.
To compute the Jacobian in our experiments, we use the Gudhi library The GUDHI Project 2020) version 3.8.0 and Tensorflow version 2.12.0. Specifically, we use Gudhi library to collect PDs, especially Gudhi. . tensorflow. Satorffoo..ipssayee.e class to collect PDs with respect to the Rips filtration. We then mamually compute PIs as described above using Tensorflow. Finally, we collect the Jacobian for the whole pipeline with tensorflow. GradientTape. jacobian function.
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Point Cloud X, [Evenly sampled from digerivector
Point Cloud Noise;
Point Cloud Xç Noie;
Jocablan
1st
jocabian
digenveztor
jocosian
68
00
4.5
45
Point Cloud
Point
eigenvector
Jecobian matrix
1st eigenvector
Jacobian
eigenvector
Jacobian matrix
04
44
02
-02
-34
4A4
44
Point Cloud Ky (Linifammly sampled from x+1 Jacablan
Point Cloud X; Noise;
Point Cloud X; + Noisey
1st
1st eigenvector
jocablan matrix
1st eigenvector
jacoolan matrix
2.5
65
62
85
-22
4,2
33
4,8
08
Stability of the Jacobian at three point clouds: Xo, comprising points evenly spaced on S (first row); X1, obtained by adding slight noise to Xn (second row); and X2, consisting of points uniformly sampled from Si (third row). We consider the PH encoding that outputs the 1-dimensional PI on the Vietoris-Rips filtration. In the first two columns each row, we show the top eigenvector of the Jacobian and the Jacobian matrix for the original point cloud. In the next two the point cloud obtained by adding Noise; resp. Noisez to the original point cloud. Noise1 and Noise2 are consistent across rows. Note that Xo is not sampled randomly from the circle, but in deterministic manner which ensures the points are equidistant, so that the pair of simplices that yields the birth and death of the loop is not unique. This causes the non differentiiibbil of the encoding map and the instability of the Jacobian information. However, the Jacobian is stable at X1 and X2, which illustrates that the Jacobian is generically stable the input space.
Figure 17:
of columns we show the same information for
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0.13
0.09
0.12
Average pul-bbcck
0.08
Auerage mul--aack
0.10
0.06
0.09
0.05
0.08
Size of the sub data set
Size of the sub: data set
Figure 18: Evaluation of the average pull-back norm of the noising perturbation vector field with respect to the I-dimensional PH encoding on Rips filtration over subsets of the RPF data set (left) and the average pull-back norm of the gradient vector field of the see feature with respect to the same PH encoding over subsets the brain artery tree data set (right). both cases, we begin from one random point cloud and iteratively include one more random point cloud until the entire dataset is exhausted. We record the average pull-back norm the size of the sub- data set increases. This process repeated 10 times. We observe that. more point clouds are included, the average pull-back norm converges to stable value.
Visualizing the Jacobian of the encoding over the data manifold
In this section we visualize the singular value decomposition (SVD) of Jacobian on a toy data set, aiming at providing further intuition for the cigenvectors of Jacobian mappings. We consider point clouds that are uniformly sampled from axis-aligned ellipses of width w and height h. These are illustrated in the left panel of Figure 19 To visualize the Jacobian an imput point cloud X, we plot the pull-back unit ball around in the data manifold,
B' (x {u Mx ||0ly= 13.
This corresponds to the preimage of a unit ball in A Notably, the equation 1 = |ulls Av(u 9:)² indicates that the pull-back unit ball forms an m-dimensional ellipsoid with semi-axes gı, 92...... Im, and the lengths of these semi-axes are given by Vai Vxg" "....
neget
Figure 19: Visualization of the pull-back metric on toy data set. Left: The space of point clouds sampled from ellipses of width and height h. Right: The pull-back unit ball different locations on the space of ellipses. Shorter semi-axes of the pull-back unit ball correspond to larger eigenvalues of the Jacobian.
In the right panel of Figure 19 we plot the pull-back unit balls for the encoding given by the 1-dimensional PI with respect the Vietoris-Rips filtration. The plot reveals that the eigenvectors of the encoding associated with the larger eigenvalue consistently align with the direction of increasing min(w, h). This alignment is
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in accordance with what would expect the Victoris-Rii Rips filtration to capture this specific data set, since the death value depends on the radius of the inner circumcircle of a hole. Consequently, variations in the length of the major axis have minimal impact on Pl. Conversely, altering the length of the minor axis directly affects the death paramcter and changes the PI.
E
Point saliency maps for PH encodings
In the context of interpretability, point saliency maps are commonly used tools explain the decisions made by trained models (Montavon al. 2018 Zheng al. 2019 Those maps assign importance scores to cach point in inpul point cloud (or to each pixel in an imput image), indicating their significance in relation to the model's prediction. In our study, we employ a similar strategy to visualize the importance of each point in an input point cloud with respect to the PH encoding.
Given a point cloud and an encoding map f, we define the encoding saliency score for each point 24 E X
as Ox.
s5(2;)
#5 where || || denotes the Frobenius norm, which is the generalized Fuclidean norm for matrices. Here, de represents the Jacobian matrix of the encoding, which is matrix of format p2 x D obtained by flattening the PI into vector in R and laking the partial derivative of each output pixel with respect to each coordinate of each point in the imput point cloud. Similarly, dz: respect to the coordinates of the i-th point in the point cloud. This score quantifies the sensitivity of the reprosentation to variations on the coordinates of z.
Øf is the vector of partial derivatives of the encoding with
In Figure 20 we plot the point saliency score with respect to the Rips used Section for point clouds sampled from eight synthetic curves in IR2. For each point cloud we highlight individual points according their saliency scores. We observe that endpoints and points related the inner circumcycle are often highlighted. These points correspond to the simplices that create/ destroy certain homology classes in the filtration. For instance, in the curved rhombus in the first-row, sccond-column pancl, the endpoints (ycllow) the upper right side are highlighted. Note that the edge between these two endpoints will connect the gap and create a homology class. Therefore, variations on these two points can significantly change the birth parameter of the corrosponding homology class. Mcanwhile, the two points (green) on the upper left and lower right sides are highlighted. They are related to the inner circumcycle of the rhombus, and are vertices of the triangle that destroys the homology class. Therefore variations on these two points can change the death parameter of the homology class significantly.
Before concluding this section, it's important to note that the saliency scores shown in Figure 20 are heavily influenced by the sampling process. For example, many of the highlighted endpoints shown in Figure 20
400
Figure 20: Point saliency score with respect to 1-dimensional PIs on Vietoris-Rips filtration for synthetic 2-dimensional point clouds.
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are result of sparse sampling. Consequently, the resulting saliency scores might be more indicative of the sampling rather than the underlying shape itself. To better comprehend the shape, one potential approach is to compute the average saliency score across multiple different samplings.
F Details on the experiments
F.1
Reproducibility
The data and code developed for this research are made available at salb
F.2 Jacobian normalization
Throughout our work, we use pull-back norm to quantify the sensitivity of the encoding method to data variations. One needs to be careful when comparing the pull-back norms induced by different encodings, since encodings may have different scaling levels. For example, consider Euclidean data space M = EP, and two encoding mappings f1 ===+ 0.1y, f2 = x+ 10y. The pull-back norms of tangent vector = [1.0]* 0]" with respect to fi and f2 are
[1
(1
1 0
|ulls |1 0l
[1. [1,0]*
-,,
0.1)
(t 10 [1,0]
Jullsa -| =
=1.
In terms of "absolute sensitivity", fi and f2 has the same level of sensitivity to variation v. More specifically, when variation is applied to a data point X, both fi fi(X) and f2(X) would change with distance approximately the representation space. However, in terms "relative scnsitivity", fi more sensitive v. The reason is that for f1, the vector is the cigenvector of the Jacobian with the largest eigenvalue; while for f2, the eigenvector with the smallest eigenvalue. Equivalently, for f1, v has the largest. pull-back norm among all tangent vectors with the same norm v, whereas for f2, has the smallest pull-back norm.
In Section 4.3 our goal is to study and compare the focus of different encodings. In Section 5.1.2 we search for the encoding whose primary focus is on the data variations of interest. Hence, in both sections we remove the scaling factor by considering the normalized Jacobian. Specifically, we divide the Jacobian matrix by its largest singular value, Jr =
=// J For vector fields V we consider the normalized average pull-back norm:
1 D
Σ (J.V(X)- XED
Returning to the previous example, the normalized pull-back norm for vector with respect to fı and fa are To 0.1, respectively.
|d = = and | XP"
X
F.3 Identifying what is recognized
We provide details for the experiments in Section
Radial Frequency Pattern data set Figure 21 shows some examples in the Radial Frequency Patterns (RFP) data set DRFF.
PH parameters For the Vietoris-Rips filtration, we set marimal edge length as 1. For the DTM filtration, we set maximal_ edge__leggeeeeeeeeeeee as 0.5 and parameter m as 0.02. For the Height filtration, we set Leaiimaa__deeeeevev length as 0.1. We note that we need to set the maximum edge length to a small value to be able to capture topological features of interest (for instance, one wants avoid connecting different outer regions of petals); alternatively, one could use geodesic distances cubical complexes, also Turkes et al.
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W
W=4
W
Figure 21: Visualization of the Radial Frequency Pattern (RFP) data.
2022 Fig.20) for a detailed discussion. Notice there's no need to set small maximum edge length for Rips and DTM filtrations since the filtration value of edge in Rips and DTM filtrations takes the distance between its two vertices into account. In contrast, in Height filtration the filtration value of an edge is defined by the maximal filtration value of its vertices, which implies any two points will be immediately connected by an edge between them, if exists, after they appear in the filtration. For the construction of PI, we set the resolution P as 20, variance 7² of the Gaussian kernel as 10-4, and the range of the image as [0, 1] x [o. 1].
The weighting function is set as a(b, ()= = L. The implementation utilized Tensorflow version 2.12.0 and Gudhi (The GUDHI Project 2020) version 3.8.0.
Visualization of the effects of perturbation on PH We visualizes the effects of shearing perturbation and conver perturbation on the persistent homology with respect to Rips filtration (see the upper panel in Figure 22) and Height filtration (see the lower panel in Figure 22). We omit plots associated with DTM filtration as they closcly resemble those associated with Rips filtration. Please note that while these visualizations provide insights into the effects of perturbations on PH, they are highly dependent on the specific point clouds under consideration. In fact, the effects of perturbations on PH can vary significantly across different point clouds.
As shown in the upper penal in Figure 22 the shearing changes the sparsity of points in the point clouds and hence changes the birth values (I: coordinates) of certain points in PD with respect to Rips filtration. Meanwhile, shearing changes the size of petals and hence changes the death values (y coordinates) of certain PD points with respect to Rips filtration. On the other hand, convez has the effect of "opening up" the central region of the point clouds. Consequently, some loops appear later in the filtration, notably the five loops that already exist in the second column of the first row but do not show up in the second column of the third row. Consequently, conver also induces changes in the birth values of PD points associated with these loops. We note that, in comparison to shearing, the conver perturbation has a more significant impact on PL. This is consistent with the findings illustrated in Figure g where Rips is more sensitive to conver compared to shearing.
In the lower panel of Figure 22 we can observe distinct effects of these two perturbations on the PH associated with the Height filtration. The shearing perturbation directly changes the I coordinates of the points in the original points clouds, which correspond to their filtration values, and consequently changes the PI noticeably. On the other hand, the conver perturbation changes the PH in a more significant way. Specifically, certain small loops that initially appeared in the filtration (as seen in the first row of the lower panel in Figure 22 longer appear in the entire filtration after perturbation (as seen in the third row of the lower panel in Figure 22). This observation arises from the small value assigned to the mazimal eege__eeegeeeeee parameter for Height filtration. In fact, for some point clouds, the central points originally have distance smaller than mazimal edge length. Then conver perturbation can pull these central points apart, causing their distance Lereiaa___ee__eeeeee
to exceed the from the filtration, resulting in the disappearance of certain small loops. In such cases, as shown in Figure 22 the conver can significantly change the PI associated with Height filtration. However, for other point clouds where the ren points initially have distance greater than maximal edge length, the corresponding PT
threshold. Consequently, edges connecting these central points vanish
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Figure 22: Figure (a) visualizes the effects of different perturbations persistent homology with respect to Rips filtration. The first four columns illustrate the simplicial complex K, in the Rips filtration with parameter r ranging from 0.0 to 0.25. The fifth and sixth columns show the corresponding PDs and PIs, respectively. The first row is associated with the original point cloud; while the second and third rows are associated with the point cloud perturbed by shearing and convez, respectively. Figure (b) visualizes the effects of perturbations on persistent homology with respect to Height filtration, following the same row-column arrangement in Figure (a).
may remain relatively unchanged after the conver perturbation. This is again consistent with the results shown in Figure where the pull-back norms of the conver perturbation associated with Height filtration exhibit significant range of variation.
PointNet architecture and training details In our experiments, we labeled each point cloud data X DRFP with the number of petals, i.e. the parameter of the curve RFP a. from which X is sampled.
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16
with Rics fitration
witth fitration
nomm
Rotation
Transiation
Dilation
Stroch_x x
Smearing
Noising
Wigely
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Figure 23: Unnormalized average pull-back norm of different perturbation vector fields with respect different encodings.
This produccs 8 classes total and we train the PointNet model to classify cach point cloud in the data set. The PointNet model consists of I-dimensional convolutional layer with 64 filters and kernel size followed by batch normalization and rectified linear unit (ReLU) activation. Then global max pooling applied to obtain permutation-invarian representation. This is followed by two fully connected layers with 128 and 64 hidden units, respectively, with ReLU activation. The final output layer uses softmax activation to produce theprobability distribution over the output classes. We trained the model with a batch size of 32 for 100 epochs, using a learning rate of 0.001. The optimization algorithm used was Adam, and the model was trained using the cross-entropy loss function. We note data augmentation techniques are used for training Point.Net. in literatures (see, e.g.; Qi al., 2017). However, we do not augment the data during training, in order ensure fair comparison with other encodings. The implementation utilized Tensorflow version 2.12.0.
Perturbation vector field estimation Let D= (X, be finite data set of point clouds, and π M - M a perturbation mapping defined on the data manifold. In the case where the data lies in Fuclidean space, i.e. M=R" Rm one can compute the perturbation vectors as following:
V=(X)= = (XX X, VX
However, in the case of general Riemannian data manifold, the subtraction between any two points on the manifold may not be well-defined. To address this, we control the perturbation mapping such that for every E D the perturbed point cloud =(X) lies in a small neighborhood X and calculate the perturbation vector via the local coordinate system. Specifically, as shown in Appendix B.1 for each point cloud D, one find neighborhood X e Ux c M and injective isometry çx Ux R m We control the perturbation mapping sends every point X to Ux, ie.,
(X)eUs Ux, VX
Then the perturbation vectors can computed with the subtraction on the Euclidean domain Ex(Ux) follows:
V (x) ( (X),,, VX
Unnormalized pull-back norms As discussed in Appendix we consider the normalized average
pull-back norms for perturbation tangent vector fields in Section 4.3 We present the results of unnormalized average pull-back norms in Figure 23 Note that one could conclude from Figure 23 that, on average, Height is more sensitive convez than Rips terms of "absolute sensitivity". However, we reach the opposite conclusion from Figure in terms of "relative sensitivity".
F.4 Selecting hyperparameters
We provide further details Section
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03
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Figure 24: Left: shown are two point clouds in the plane. Right: the orange point cloud is transformed by iterated closest point algorithm and the grey vectors represent the difference vector between blue point clouds and the transformed orange point cloud.
PH parameters In this section, we focus on PH encoding constructed on Vietoris-Rips filtration. We set the parameter marimal edge legnth as 0.25, and the range of Pl as [0.0, 0.25] [0.0, 0.25]. The implementation utilized Gudhi (The GUDHI Project 2020 version 3.8.0 and Tensorflow version 2.10.0.
Gradient vector field estimation Let D [e be a finite data set of point clouds, and p(X;) be the corresponding feature values of point clouds D. In the case when the data lies in Euclidean space, we can estimate the gradient vectors with the finite difference method (FDM) follows:
(x))
Vp(X= XX, X' argmax YED dx(X,X') VXED.
For binary categorical feature, i.e. p(X) (0, the formula can be modified as
vp(X)= X'=
argmin Ve(ZeD--(()+o(X))
dE((,Y), VX
However, in the case of general Riemannian data manifold, the subtraction between any two points on M may not be well-defined. To address this, when estimating gradient vector located at X, we send other point clouds in the data set to a neighborhood X via transformation that preserves the feature value. In our experiment, we use Euclidean transformation since the ser feature irrelevant to the position or oricntation of the brain artery trees. Then FDM can be applied the transformed data set, via the coordinate system that neighborhood.
Specifically, utilize the iterated closest point (ICP) method (Chen & Medioni 1992 Let E(RP) be the collection of all Euclidean transformations in R
Besl & McKay
1992
E(RD) RR + 1R (()
=x Va,
Note cach t can be naturally extended to transformation on point clouds: (X) (((=)==(())) Let Ç be a error function in the sense that <(X, Y) measures the "difference" between X and Y. Given two point clouds X and Y, the ICP algorithm searches the Euclidean transformation that gives the minimal error value:
LX,Y axg (=B(R))
Here 4x,y is Euclidean transformation found by ICP algorithm for point clouds X and Y. Define the ICP discrepancy between point clouds X and Y (not necessarily distance) as
dicr(X,Y) dw (x(XxxxxxX))
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Figure 25: Right: test accuracy of the convolutional neural network (CNN) trained with PIs produced by different paramcter settings. Left: robust test accuracy of the convolutional neural network (CNN) trained with PIs produced by different parameter settings.
Noto DE voltt accura0
Potta ualidattinn accuracy
walldaatoon accuracc
86
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PCC 0:00 5.6X00
PCC 0.81
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Ronk Jhootian
Rank jacobian
Pul Bul-backk norm
Pul wol-backkoot norm
Figure 26: Gaussian kernel density estimation of the joint distribution of four pairs of variables: (a) Jacobian rank vs. validation accuracy; (b) Jacobian rank vs. robust validation accuracy; (c) pull-back norms vs. validation accuracy; and (d) pull-back norms vs. robust validation accuracy, where the downstream models are chosen as convolutional neural notworks.
where dw is Wasserstein distance between point clouds. Let Ux be a neighborhood of X and &x a coordinate map ex : Ux Th. Estimate the gradient vector field for binary categorical feature p follows (see illustration in Figure 24
vx(X) Ex - (X), X' =
argmin dx), VX Ye(ZzD:o(Z)<x(()
The implementation utilized python library Open3D Zhou et 2018 version 0.17.0 and POT (Flamary et al.
2021 version 0.9.0.
Downstream tasks and performance In Section 5 we fed PIs produced by different parameter settings to logistic regression models to predict the ser feature. Specifically, we normalize the PIs such that the pixel values range within [0, 1]. We implemented logistic regression models using Scikit-learn (Pedregosa et al. 2011h version 1.2.2 with default hyperparameters. When evaluating the model, we use a 7-fold cross validation. And for robust evaluation in Section 5.1.3) we add identically and independent. distributed Gaussian noise with variance 10-2 2 each coordinate of cach point input point clouds.
Here, we also investigate the effects of resolution and variance of Gaussian kernel the downstream performance of convolutional neural network (CNN). The CNN model takes Pls inputs, then begins with a convolutional layer with 32 filters and 3 x 3 kernel, followed by a ReLU activation function. Then max pooling with a pool size of 2 x 2 is applied. Subsequently, another convolutional layer with 64 filters and a 3x3 kernel is added, also followed by a ReLU activation and max pooling. The resulting outputs are then flattened and passed through fully connected layer with 64 neurons and ReLU activation. Finally, a single neuron with a sigmoid activation function is used for binary classification. To train the model, we employ Adam optimizer with the cross-entropy loss function. The implementation utilized Tensorflow version 2.12.0.
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We represent the downstream performance of the CNN model in Figure 25 Shown in Figure 26 is the kernel density estimation of the joint distribution of four pairs of variables: Jacobian rank vs. validation accuracy, Jacobian rank vs. robust validation accuracy, pull-back norm vs. validation accuracy, and pull-back norm vs. robust validation accuracy. Additionally, the Pearson correlation coefficient and p-value a two-sided test are presented at the lower right corner of each point in Figure 26 We observe that the correlation bet ween pull-back norms and downstream performances remains significant.
We conjecture that complex models, such as CNNss are able to obtain good downstream performance even if the average pull-back norm is low, long as it is not zero. Intuitively, when the feature information is indeed contained in the encoded representation but is not significantly pronounced, a simple model may not be able to extract this information, but a complex model, along with appropriate training techniques, can still learn to extract and utilize this information.
G Investigating which part of the data is highlighted by PH encodings
In this section, we demonstrate how our method facilitates investigating which part of the data is the focus of PH encodings. To this end, we will introduce noising perturbation on different parts of the point clouds and examine the average pull-back norm of these perturbation vector fields. Moreover, we also consider the beta weighting function for PIs and investigate the effects of the beta mean parameter on the pull-back geometry, which allows comparing the focus of long and short persistence intervals.
Human body data We utilize the benchmark mesh segmentation data Chen et al. 2009). This data set consists of moshes representing 19 different types of 3-dimensional shapes, each annotated with mannally added segmentation labels. For our analysis, we focus on the subset of meshes representing the hunan body, which encompasses various gestures such as standing, walking, and sitting. We randomly subsample three point clouds from the vertices of each human body mesh, with each point cloud containing N 500 points.
PH encoding We focus the PH encoding that sends each point cloud to its 2-dimensional PI with respect to the Vietoris-Rips filtration. We choose 2-dimensional PH because we note that the underlying geometric objects of human body meshes are 2-dimensional surfaces whose 1-dimensional homology is typically zero, and whose reduced O-dimensional homology is zero as well. For the construction of PIs, we set the P1 hyperparameters as P 20, x 10-4, mazimal Laazima_edge______ hagge_legnth== 0.3, and the range of PI as [0.0, [0.0, 0.3]. For the weighting function, we again employ the beta weighting function that we introduced in Section 5.2 We set the variance parameter for the beta weighting function s² as 0.04 and consider the mean parameter & ranging from 0.1 to 0.5. We present a visualization of the impact of the mean parameter on the PIs in Figure 27
PC with vegmeniation
F1 0.11
1t 0:4)
Pilx -ixx-.o 0.71
C13
4.3
03 275
Figure 27: Illustatioon of the human body data set: (a) Point cloud with body parts segmentation; (b) 2-dimensional persistence diagram with respect to the Vietoris-Rips filtration; (c) to (e): persistence images obtained with beta functions with the beta mean parameter set as 0.1, 0.4, and 0.7 respectively.
Perturbation We merge the segmentation labels for the sampled human body point clouds into categories: head, torso, arms, hands, legs and feet (sce panel (a) Figure 27 for an illustration). Accordingly, we consider 6 types of perturbations, each adding independent Gaussian noise to points in one body part. We provide visualizations of some of the perturbations in Figure
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Farso
Pt perturted by free
AM- -x---.1)
Figure 28: Illustration of the effects of noising perturbation on Head, Torso, and Arms on the point cloud, PD and PIs with different weighting functions..
Pull-back norm We evaluate the average pull-back norms of the noising perturbation on 6 body parts and present the results in Figure 29
It is noticeable in Figure 29 that persistence intervals of varying lengths capture distinct aspects of the data. Specifically, when the mean parameter is set to 0.1, the encoding exhibits significant sensitivity to perturbations legs, when k = 0.2, the focus of the encoding switches to head. For larger mean parameters, the encoding becomes most sensitive to perturbations on lorso. This could be explained by observing that shorter persistence intervals in 2-dimensional PDs on Rips filtration are more related to smaller hollow shapes in the data, such as arms and head, while longer intervals relate more to larger hollow shapes, such as the torso.
These findings can also guide the selection of hyperparameters for PH encodings. For instance, in face recognition lask, know from above that shorter persistence intervals are sensitive to data variations on head. Hence, should choose value around 0.2 for the beta mean parameter in order to obtain data presentation that is most suitable for this task. At the same time, we note once again that long persistence intervals do not always contain the most important information and that the optimal choice of the weighting function (and other hyperparameters) depend on the specific task at hand.
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Figure 29: Averaged pull-back norms of noising perturbations 6 body parts.
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Reviews:
[{"review_id": "chsE3XEdbE", "reviewer": "Reviewer_8VK4", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "# Global summary\n\nThis is a competent paper supported by nice experiments. However, my main feeling is that it is somewhat oversimplifying the problem it considers and only scratches the surface. I am not saying that the paper is inherently bad, but that it could be much better by bringing much more insights (especially from a theoretical viewpoint), formalizing and hopefully addressing important questions. \n\nTo me, the paper looks like ''interesting experimental findings'', while it could be ''bringing deep insights and important questions on the variational sturcture of the persistent homology map'', and the latter would be significantly more impactful and useful for the TDA community in my opinion. \n\n(edit on 01/13 : removing backquotes that yielded improper formatting)\n\n# Strengths\n\n- Very nice illustrations!\n- Overall well written (though some choices are disputable, see below).\n- Extensive set of experiments.\n\n# Major Weaknesses\n\nWhile I like its main purpose, I feel that the paper is frustratingly superficial from a theoretical standpoint and does not bring proper insight on the topic it covers. Many non-trivial aspects or questions are put under the rug. For instance:\n- The paper relies on an (arbitrary) embedding of PDs in an Euclidean space to avoid the much harder though much more insightful question _how to quantify the variation of $X \\mapsto \\mathrm{PH}(f(X,\\cdot))$ for a given filtration $f$?_ This requires to equip the space of PDs with a differentiable structure (and understanding the impact of the non-smoothness of the map $\\mathrm{PH}$). The (significant) benefit would be to get an analysis that would be independent of the choice of the vectorization.\n- As said above, the map $X \\mapsto \\mathrm{PH}(f(X,\\cdot))$ is not smooth (only differentiable a.e.), which means in practice that for a very small perturbation $V$, the generators of $\\mathrm{PH}(X)$ and $\\mathrm{PH}(X + V(X))$ can be quite different, i.e. eigenvectors of the Jacobian/Gram matrix may not be stable. As such I am not sure how the proposed interpretation is reliable. \n- I would like having more insight on the relation between perturbations and filtrations. That would help the practitioner to address the question _how should I design a filtration $f$ that would be sensitive/unsensitive to a given type of perturbation $V$?_ or, conversely, _given this $f$, which are the perturbations $V$ that are detected/undetected by $f$?_ To me, the important point is thus to formalize, in a general abstract setting, a way to quantify the variation of $V \\mapsto J_f(V(X))$. Typically, I would like to have an insight on _why_ the Rips filtration is good at reflecting ``convexification'' (this may be related to the notion of _convexity defect_), or to know _in advance_ that a given filtration is insensitive to some transformations (e.g. the Rips being insensitive to translation-rotation; though it is obvious in that particular case). That requires to consider the variations of the map-valued map $X \\mapsto f(X,\\cdot)$. For instance, the Rips PI is insensitive to translations and rotations because the map $X \\mapsto \\mathrm{dist}_X(\\cdot)$ is equivariant under this action and the operator $\\mathrm{PH}$ has its own invariances. To me, that's an important question raised by the paper, but not addressed nor even formalized. \n\n# Minor comments\n\n- In section 3, the choice of mixing the Manifold viewpoint and the Euclidean viewpoint on the space of point clouds is a bit confusing: its too sophisticated for the Euclidean case and too simplistic for the Manifold setting. E.g. Definition 1 and 2 \"are for the case that $\\mathcal{M}$ is a Euclidean space\" : in that case, the notion of tangent space is not required, etc. On the other hand, as acknowledged by the authors, this definition is too simplistic for the actual manifold setting. \n- Peyré et al., --> Should be Peyré and Cuturi. It is published in _Foundations and Trends in machine Learning_. Other citations may be checked as well. \n- Not sure that considering the Wasserstein distance between persistence images (end of $\\S$3.1) is meaningful/useful. If one use the standard Wasserstein distance (as defined in Optimal Transport), then one has to normalize the images, which is dubious in my opinion. If one use the Wasserstein distance as defined in TDA (using the diagonal as a trash bin), then why not simply doing that in the space of persistence diagrams directly? (And this boils down to the harder problem of computing variations in a metric space that I mention above)", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "Persistent Homology (PH) provides a way to extract topological information from structured object, such as point clouds represented as elements of $\\mathbb{R}^{N \\times D}$ ($N$ points in dimension $D$) by the mean of a _filtration_ which, in the case of point clouds, may be encoded as a map $f : \\mathbb{R}^{N \\times D} \\times \\mathbb{R}^D \\to \\mathbb{R}$. \nOne such example is the _Rips_ filtration defined as $f(X, x) \\coloneqq \\min_{i=1,\\dots,N} |x - x_i|$, where $X = (x_1,\\dots,x_N)$.  \n\nEach choice of filtration $f(X,\\cdot)$ for a given object $X$ yields a topological descriptors called a _persistence diagram_ (PD). The space of PDs lacking linear structure, it is usual to embed these PDs in a Euclidean space $\\mathbb{R}^n$ by the mean of a vectorization. The _persistence image_ (PI) embedding is one standard option to do so. \n\nEventually, one obtains a map $\\Phi_f : \\mathbb{R}^{N \\times D} \\ni X \\mapsto \\mathrm{PI}(\\mathrm{PH}(f(X,\\cdot)) \\in \\mathbb{R}^n$. A natural question, considered by this work, is then _given a perturbation of $X$, how is $\\Phi_f(X)$ affected?_ Or, somewhat equivalently, what kind of information $\\Phi_f(X)$ is ``encoding'' about $X$ given the choice of filtration $f$? \n\nThis map is known to be locally Lipschitz hence differentiable almost everywhere. Therefore, the natural idea considered in this work is to consider the Jacobian $J_f$ of $\\Phi_f$ and quantify how a perturbation of $X$ encoded as a (``tangent'', if points belong to a manifold) vector fields $V$ are changing $\\Phi_f$, which can directly be deduced from looking at the spectral composition of $J_f(V(X))$ and quantified using appropriated norms. \n\nThe current paper investigates this idea and illustrates it through various numerical experiments.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I would like to see an updated version of the work that would be much stronger on the formalism / theoretical side, bringing much more general insights and as such being more impactful and useful for the targeted audience. \n\nI can admit that it is too much work for the current paper. In that case, I let the other reviewers and the editors assessing whether it is worth for publication in TMLR in its current state (up to minor modifications).", "broader_impact_concerns": "No specific concerns in terms of ethical implications for the current work.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "DRHcnXl5ky", "reviewer": "Reviewer_4uUj", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "+++++**Strengths**\n\n1.  The paper introduces a novel method to analyze persistent homology (PH) encodings by investigating the Jacobian spectrum and pull-back geometry. This approach is a significant contribution to the field of topological data analysis, offering a new lens to understand PH.\n\n2. By focusing on how different PH encodings capture or ignore data variations, the study has practical implications. The ability to predict performance on downstream tasks using the pull-back norm is particularly useful for selecting suitable PH encodings.\n\n3. The paper includes experimental demonstrations that bolster the theoretical findings. This empirical approach not only validates the proposed methods but also demonstrates their applicability in real-world scenarios, such as in the analysis of brain artery trees.\n\n4. The research is presented with clarity and thoroughness, making complex concepts accessible while providing depth in analysis. This balance enhances the paper's appeal to both experts and those newer to the field.\n\n-----**Weaknesses**\n\n1. While the paper is well-explained, the methods' complexity could be a barrier for some readers, particularly those who are not well-versed in advanced topological concepts or mathematical formulations.\n\n2. The focus on PH and its specific applications might limit the paper's appeal to a broader audience. Its relevance is primarily confined to specialists in topological data analysis and related fields.\n\n3.While the paper situates itself well within its specific research domain, it could benefit from a broader contextualization in terms of its implications for other areas of data analysis or machine learning.\n\n4. While the experiments conducted are robust, additional validation across more diverse datasets and scenarios could strengthen the findings. This would ensure the generalizability of the research.\n\nIn summary, the paper makes a significant contribution to the understanding of persistent homology encodings, with practical applications for selecting PH encodings based on the pull-back norm. Its innovative approach and thorough experimental validation are notable strengths. However, the complexity of the methods and the niche focus of the application might limit its accessibility and appeal. Further validation in diverse scenarios could enhance the robustness of the findings. Overall, the paper's strengths make it a valuable addition to the field, and its acceptance is recommended.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper delves into the complex realm of data representation using persistent homology (PH), a technique that accentuates topological features of data. It uniquely investigates how PH encodings affect data's manifold geometry by analyzing the Jacobian spectrum and pull-back geometry. This approach allows for assessing which data features are emphasized or overlooked by PH without needing specific task training. A significant finding is the correlation between the pull-back norm and performance in downstream tasks, suggesting a new method for selecting appropriate PH encodings. The study stands out in the field of topological data analysis, as it not only enhances the understanding of PH properties but also provides practical guidance for optimizing PH representations for varied applications.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I think this is a good paper. The following general changes are recommended to enhance its contribution to the field:\n\n1. Given the paper's advanced mathematical content, particularly around Jacobian spectrum and pull-back geometry, a more detailed explanation or simplification of these concepts would aid comprehension. This could include illustrative examples or diagrams that break down these complex ideas.\n\n2. While the study's experimental validation, such as the analysis of brain artery trees, is robust, extending these experiments to a wider range of datasets would demonstrate the versatility of the proposed methods.\n\n3. A section comparing the paper's approach with other existing methods in topological data analysis, especially in terms of recognizing or ignoring data variations, could contextualize its efficacy and novelty.\n\n4. The paper would benefit from an expanded discussion on the implications of its findings in broader data science contexts, particularly in how PH encodings could influence machine learning models or data representation strategies.\n\n5. Incorporating detailed case studies where the proposed method is applied, especially in contexts different from the brain artery trees dataset, would provide practical insights into the method's application.\n\n6.. A more in-depth analysis of potential limitations, such as computational demands or scenarios where the method might not perform optimally, would provide a balanced view of the approach.\n\n7. Given the paper's focus on geometry and topology, additional visual aids or enhanced graphical representations would significantly aid in understanding the complex methodologies and results.", "broader_impact_concerns": "The paper's focus on specialized aspects of persistent homology could limit its broader impact, as its highly technical nature might restrict its accessibility and applicability beyond the niche field of topological data analysis. Expanding its scope to demonstrate relevance and potential applications in wider areas of data science and machine learning could significantly enhance its overall impact.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "UNJPugJjny", "reviewer": "Reviewer_LaLp", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. The presentation is sensibly organized, with much background given on persistence homology/diagrams, and various filters.\n2. The basic perspective is sound, and it's clear that some insights might be garnered by looking at the Jacobian and its spectrum.\n\nWeaknesses:\n1. The contribution is limited by the fact that they only look at particular datasets, and the insights garnered are apparent or might be more easily gained with performance-based testing. As a result, I feel like the work is overclaiming slightly with the generality that it alludes to in its abstract and intro.\n    1. I think it's a stretch to claim that the interpretations of what persistence homology and its representations capture is unclear. It is even stated several times that it captures \"holes\" of various dimensions in the data, as they are formed with respect to the filtration at hand. Each of the filtrations is also relatively easily understood. This leads to conclusions in many of the experiments that are relatively obvious: that higher-resolution persistence images lead to higher-rank Jacobians and that Rips and DTM are invariant to rotation and translation, for example.\n    2. It's unclear how many empirical datasets or point clouds are to be considered before one has a fair idea of what perturbations are valued. Doing a performance-based evaluation would simultaneously achieve this (empirical) aim and train the model at hand. This can be seen in the evaluation of the PointNet model in the RFP experiments.\n2. While the presentation is well-organized, the writing of the definitions and the perspective of the work leaves something to be desired.\n    1. It would seem relevant to at least note briefly that persistence diagrams lie above the diagonal $y=x$ by definition, and to show some examples, to give readers intuition and so that they know the information in a $P \\times P$ image is half of what they'd expect it to be. Actually, it's a little unclear to me how the kernel smoothing behaves with respect to this limited domain, and this should be clarified.\n    2. In Def. 3, I think it'd be useful to remind readers that $X$ denotes a point cloud and that $\\mathcal{D}$ denotes a collection of point clouds that form your data. Given that a tangent vector in the space of point clouds corresponds to a \"vector field\" in the sense of a perturbation vector per point, it's easy for someone to think one might be calculating norms of these individual perturbation vectors.\n    3. When considering the input space of point clouds, I think it's important to note the fact that these point clouds are determined up to permutation, and they must be quotiented by the action of $S_n$. On the other hand, it's also unclear to me why the Wasserstein distance is referenced, as one intends to simply consider evaluation of the pullback metric. Lastly, as footnote 2 indicates, one should really probably consider the Wasserstein metric on the persistence images, and it's unfortunate that the method sticks to the less natural and informative Euclidean distance on persistence images. \n    4. A perturbation vector field that simply points directly from a point cloud to its perturbed point cloud is only feasible when the target manifold is Euclidean, as it is here. It also may only be reflective of the perturbation if it is small (a 180 degree rotation would not generate a rotational vector field). \n    5. Figure 1 is highly confusing without prior introduction of RFPs and what subset of them (or datasets sampled from them) is represented by a torus. Please clarify.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors consider some case studies in evaluating the Jacobian of persistence images generated by particular sets of point cloud data. In particular, they consider the RFP and brain artery tree datasets, and look at the spectrum and eigenvectors of the Jacobian. Roughly speaking, high rank is desired for the Jacobian and high magnitude eigenvectors provide some intuition of the point cloud perturbations that are detected by persistence images.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I do not currently feel like recommending the submission for acceptance due to the above questions and weaknesses. The major point to address is the need for more impactful applications and insights in those particular instances. \n\nOverall, I do find the positioning of the work to be a little perplexing. The method of analysis is not at all restricted to embeddings generated by persistence homology, so it might be useful to consider its use in other settings (like the neural network ones considered). Moreover, embeddings via persistence homology are some of the most interpretable, I feel, at least relative to those from deep learning.", "broader_impact_concerns": "Not applicable.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "SSHS8InWbj", "reviewer": "Reviewer_DuMc", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper addresses important TDA questions and brings value to both TDA and machine learning community. While it is well organized and written, there are a few improvements that can be made to enhance the paper's clarity.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper focuses on examining persistence homology (PH) encodings, which are mappings from data to vectorized PH representations. Persistence homology is a powerful tool in topological data analysis (TDA) that captures structural features in data (e.g., holes, voids) at multiple scales. The output of PH is often mapped into a vector space for further analysis or machine learning tasks. Persistence images (PIs) represent a popular example of such mapping. This paper considers PH encodings derived from PIs based on three distinct filtrations: Vietoris-Rips, distance-to-measure, and height filtration. Each of these filtrations offers a different perspective on the data, capturing different aspects of its topological structure at different scales.\n \nThe authors explore how the choice of filtration impacts the resulting PH encoding and, consequently, the information captured about the data. They demonstrate that through the measurement of various perturbations on the data manifold in relation to pull-back geometry, it becomes possible to identify which of them are captured and which are overlooked by the PH encodings.\n\nFurthermore, this paper addresses the issue of parameter selection for the PH encodings. Effectively selecting parameters is crucial in obtaining meaningful and task-specific representations of the data. The authors demonstrate, on real-world data, the effectiveness of selecting PH encoding parameters by taking into account the pull-back norm of features of interest.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "On page 1, PH encoding is defined as the mapping from data to vectorized representations of the associated PH. In a few sentences later persistence images (PIs) are called examples of PH encodings, but PIs are mappings from persistence diagrams and not from the original data. Also, on page 8, the three filtrations were also called PH encodings.  While I understand what the authors meant by that, at first it caused some confusion, thus all these instances have to be carefully rephrased.\n\nOn page 2, I suggest changing \"vectorized representations of the associated PH\" to \"vectorized PH representations\".\n\nIn the paper, the authors say “it is unclear which aspects of the data are highlighted or which ones are suppressed in the PH representations’’ and there are similar statements throughout the paper. This is not completely true since the representative cycles can help interpret the PH results by identifying and visualizing essential topological features and patterns in data. I suggest rephrasing these sentences and maybe the authors can consider adding a short discussion of representative cycles.\n\nOn page 4, the authors mention “k-dimensional PD” and “k-dimensional homology class”. However, the paper lacks the definition of k-dimensional homology. Including the definition of k-dimensional homology would be beneficial, particularly for readers who may not be familiar with TDA.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper's main claim and contribution are based on the perturbation analysis and the tools required to obtain novel insights. All the reviewers, including this AE, agreed that this is a valuable contribution to the community. However, there were some concerns, which I mention below.

1. One reviewer suggested that it would be interesting to evaluate how the embedding Jacobian satisfies the general goals of persistence homology: capturing holes of different dimensions in a way that is robust to rotations and minor noise.
2. Another reviewer noted the difficulty in deriving good insights, while also acknowledging the advantages of this work.

The paper has done a fair amount of revision to accommodate the concerns. 

Overall, we have decided to **accept** the work.
Decision: Accept (poster)