On the Effectiveness of Persistent Homology
Authors: Renata Turkes
Guido Montufar
Nina Otter
Venue: NeurIPS 2022 Accept (2022)
Keywords: persistent homology
Betti numbers
curvature
convexity
topology of data
geometry of data
point clouds

Abstract: Persistent homology (PH) is one of the most popular methods in Topological Data Analysis. Even though PH has been used in many different types of applications, the reasons behind its success remain elusive; in particular, it is not known for which classes of problems it is most effective, or to what extent it can detect geometric or topological features. The goal of this work is to identify some types of problems where PH performs well or even better than other methods in data analysis. We consider three fundamental shape analysis tasks: the detection of the number of holes, curvature and convexity from 2D and 3D point clouds sampled from shapes. Experiments demonstrate that PH is successful in these tasks, outperforming several baselines, including PointNet, an architecture inspired precisely by the properties of point clouds. In addition, we observe that PH remains effective for limited computational resources and limited training data, as well as out-of-distribution test data, including various data transformations and noise. For convexity detection, we provide a theoretical guarantee that PH is effective for this task in $\mathbb{R}^d$, and demonstrate the detection of a convexity measure on the FLAVIA dataset of plant leaf images. Due to the crucial role of shape classification in understanding mathematical and physical structures and objects, and in many applications, the findings of this work will provide some knowledge about the types of problems that are appropriate for PH, so that it can --- to borrow the words from Wigner 1960 --- ``remain valid in future research, and extend, to our pleasure", but to our lesser bafflement, to a variety of applications. 

On the Effectiveness of Persistent Homology
Renata Turkeš
Guido Montúfar
University of Antwerp
University of California, Los Angeles montufar contufar@mathee www.uclleeee..o edu
renata. tur.esshaantwwerrnn.nn be
Nina Otter
Queen Mary University of London otter&qmul ac .k
Abstract
Persistent homology (PH) is one of the most popular methods in Topological Data Analysis. Even though PH has been used many different types applications, the reasons behind its success remain elusive; in particular, it is not known for which classes of problems is most effective, or to what extent it can detect geometric topological features. The goal of this work is to identify types of problems where PH performs well or even better than other methods data analysis. We consider three fundamental shape analysis tasks: the detection of the number of holes, curvature and convexity from 2D and 3D point clouds sampled from shapes. Experiments demonstrate that PH is successful in these tasks, outperforming several baselines, including PointNet, an architecture inspired precisely by the properties of point clouds. In addition, we observe that PH remains effective for limited computational resources and limited training data, as well out-of-diitributtin test data, including various data transformations and noise. For convexity detection, we provide theoretical guarantee that PH is effective for this task RR,, and demonstrate the detection of convexity measure on the FLAVIA data set of plant leaf images. Due to the crucial role of shape classification in understanding mathematical and physical structures and objects, and many applications, the findings of this work will provide some knowledge about the types of problems that are appropriate for PH, so that can - to borrow the words from Wigner 1960- - "remain valid in future research, and extend, to our pleasure", but to our lesser bafflement, to a variety applications.
Introduction
Persistent homology (PH) is an extension of homology, which gives way to capture topological information about connectivity and holes in a geometric object. PH can regarded as a framework inputs to learning algorithms. There have been numerous successful applications of PH in the last decade, from
to compute representations of raw data that can be used for further processing
prediction of biomolecular properties [18 TTO), face, gait and activity recognition (58 66| 70 122
or digital forensics [6]. to discriminating breast-cancer subtypes 197). quantifying the porosity of
nanoporous materials [69]. classifying fingerprints (50), or studying the morphology of leaves (7]
At the same time, the reasons behind these successes are not yet well understood. Indeed, the data used in real-world applications is complex, so that there numerous effects at play and one is often left unsure why PH worked, i.e., what type of topological or geometric information it captured that facilitated the good performance.
A database of applications of persistent homology is being maintained at
51
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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The title of our manuseript is inspired by a famous paper from 1960, "The unreasonable effec- tiveness of mathematics in the natural sciences" (TI2), in which Wigner discusses, with wonder, how mathematical concepts have applicability far beyond the context in which they were originally developed. The same, we believe, true for persistent homology. While this method has been applied successfully to wide range of application problems, we believe that for PH to remain relevant, there is a need to better understand why it is so successful. Thus, we distinguish between the usefulness of PH for applications, which has been attested in hundreds of applications and publications, and its effectiveness, namely that PH is capable of producing an intended or desired result. Thus, here we initiate investigation into the effectiveness of PH, or in other words, we investigate what is seen by persistent homology: Given a data set, i.c., a point cloud, which underlying topological and geometric features can we detect with PH? This question is related to manifold learning and, specifically, topological and geometric inference: Given a finite point cloud X of (noisy) samples from unknown manifold M, how can one infer properties of M (11 12 25 29 Obtaining
representation of a shape that can be used in statistical models is an important task in data analysis and numerous approaches to modeling surfaces and shapes (104).
To pursue our investigation, we set out to identify some fundamental data-analysis tasks that can be solved with PH. Since PH is inspired by homology, which provides measure for the number of components, holes, voids, and nigher-dimensional cycles of space - to which we collectively refer as "topological features" -, we start with the obvious question of whether PH applied to point cloud sampled from a geometric object can detect the number of (1-dimensional) holes of the underlying object. Unlike homology, however, PH registers also the persistence of topological features across scales, and can thereby capture geometric information, such as size or position of holes. We therefore also investigate how well PH can detect fundamental geometric notions of curvature and convexity. For each of the three problems, we first discuss theoretical results that provide a guarantee that PH can solve these tasks. Detection of convexity with PH has not been investigated in the literature to date, and we prove a new result. To investigate how well the PH pipeline works in practice, we compare its performance against several baselines on synthetic point-cloud data sets. As first machine learning (ML) baseline we take an SVM trained on the distance matrices of point clouds. We further consider fully-connected neural networks (NN) with a single multiple hidden layers, also trained on distance matrices. As a stronger baseline we consider a PointNet trained on the point clouds directly. PointNet [8] is designed specifically for point cloud data. Similar architectures with convolutional (and fully-connected and pooling) layers have heen applied for Betti-number and curvature estimation [T). For convexity detection, we also evaluate the performance of PH on real-world data. The theoretical guarantees above imply that the results for PH would generalize to new data.
Finally, we note that our goal is not to claim the superiority of PH compared to other approaches in the literature, in particular, with the ofte-of-thh--rt methods for each of the problems. We do not necessarily expect that on well-speeified mathematical problems PH will beat -f-te-of-the-art algorithms that have been specifically designed for those tasks. Instead, what we think is interesting and remarkable is that PH can in fact solve tasks it is not specifically or uniquely designed for. Moreover, an advantage of PH is that it can reveal, e.g., both topology and curvature at the same time, avoiding the need to employ and combine ofate-of-the-an models for each of the tasks.
Related work In spite of the growing interest in PH, so far there is only limited work in the direction that we pursue here. There is indeed theoretical evidence that the number of holes of the underlying space can be detected from PH (under some conditions about the target space, the sample
density and closeness the space) 32 63 77 and there is significant interest in investigating how well this works in practice [30]. However, so far there are only few available results. Some works demonstrate that PH can be used detect the number of holes, but only on individual toy examples (c.g. (90). [25] Figure 19]. (65] Figures 9-20], (29) Figures 2, 3, 6, 11, 12]) without looking into the statistical significance between different classes of data, or the accuracy of some classification algorithms on a comprehensive data set. There are also some works where PH is used to estimate the Betti numbers on possibly larger data set, but only with the goal of using this information to e.g., study the behavior of deep neural networks (76) or ensure topologically correct dimensionality
reduction [82) or image segmentation (56), so that the soundness of this estimation is not investigated, which is the focus of our work. Some insights about PH and curvature have been obtained in the
literature, starting with an illustrative example in (39, Figure 12] which shows that PH on the filtered
tangent complex can distinguish between letters (C and 1) that have the same topology, since their
2

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curvature is different. Recently [7] show both theoretically and experimentally that PH can predict curvature (with computational experiments replicated in [T07]. which inspired us to investigate this problem in more detail. Regarding the important geometric problem of classification between convex and concave shapes, we were not able identify any previous works investigating the applicability of PH to this task.
Some further recent work investigating the topological and geometric features seen by PH are the following. Bubenik and Diotko [16] show that using PH of points sampled from spheres one can determine the dimension of the underlying spheres. A connection has also been established between PH and the magnitude of a metric space (an isometric invariant)
[79). There have been several efforts in using PH to estimate fractal dimensions, such as [95] in which Schweinhart proves that the fractal dimension of some metric spaces can be recovered from the PH of random samples.
Main contributions Our contributions can be summarized as follows.
We prove that PH can detect convexity in Rd (Theorem 1b.
We define a new tubular filtration function (medium through which PH is extracted from data), that
is crucial for the detection of convexity (Definition D
We demonstrate experimentally that PH can detect the number of holes (Section curvature (Section 4). and convexity (Section 5) from symthetic point clouds in R2 or R2,, outperforming SVMs and tully-connected networks trained on distance matrices, and PointNet trained on point clouds. For convexity detection, we also show that PH obtains good performance on a real- world data set of plant leaf images.
We demonstrate experimentally that PH features allow to solve the above tasks even in the case of limited training data (Section B) noisy (Section 3) and out-of-distribution (Section (5) test data, and limited computational resources (Section 3 Section 4 Section 5
We provide insights about the topological and geometric features that are captured with long and short persistence intervals (Section 6), and formulate guidelines for applications that suitable for PH (Section]) 7
We provide data sets that can be directly used as a benehmark for our tasks or other related point- cloud-analysis or classification problems. We provide computer code to construct more data and replicate our experiments.
2 Background on persistent homology
Homology is a topological concept that attempts to distinguish between topological spaces by constructing algebraic invariants that reflect their connectivity properties [90), i.e., k-dimensional cycles (components, holes, voids, ...). The number of independent k-dimensional cycles is called k-th Betti number and denoted by B,. For example, the circle has Betti numbers 3n 1, 31 = 1.
B2 = 0, and for a torus, we have f0g- = 1, 31 = 2, B3 =1
Persistent homology is an extension of this idea [13] that has found success in applications to data.
To calculate PH from some data X, we must first build filtration, i.e., family of nested topological spaces {K, FreR which, in suitable sense, approximate X at different scales r E R. Typically, X = {I1, .2,..., } is a point cloud in Rd, and K, is a simplicial complex, a set of simplices think of vertices, edges, triangles, ...) such that if a K, and S then T E K, [39]. A common choice is K, = V R (1 r), where VR X r) is the Vietoris- simplicial complex, in which PH can then be summarized with a persistence diagram (PD), a scatter plot with the x and y/ axes respectively depicting the scale r E R at which each cycle is born and dies or is identified (i.e.. merges) with another cycle within a filtration. The length / = d b of a persistence interval (b, d) measures the lifespan the so-called "persistence" of the corresponding cycle in the filtration.
(which
{=1,--- ......... Fm } V VR(XX" r) when dist my) < for all < i, & m.
Instead of working directly with persistence diagrams, these are often represented by other signatures that are better suited for machine learning frameworks. A common choice is persistence image (PI) [2) (discretized sum of Gaussian kernels centered at the PD points), or a persistence landscape (PL) (functions obtained by "stacking isosceles triangles" above persistence intervals, with height reflecting their lifespan) (15). The steps for extracting PH features are visualized in Appendix B.2 For good choices of filtration and signature [103), there are theoretical results that guarantee that PH

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is stable under small perturbations [26 98 After the PH signature is calculated, statistical hypothesis testing [50] or machine learning techniques such as SVM or k-NN 48 can be used on these features to study the differences within the data set of interest. It is important to note that PH is very flexible, as different choices can be made in every step of the pipeline, regarding the input and output of PH, as detailed in the remainder of this section.
2.1 Approximation of space at scale E IR.
Instead of the Vietoris-Rips complex, other types of complexes can be used to approximate the data X at the given scale E R. For instance, since Vietoris-Rips simplicial complex is large [80], one might rather choose the alpha complex (46) (for a visualization, see Appendix C.2). which is closely related to the Vietoris-Rip complex [63] but consists of significantly less simplices and is faster to construct when the dimension of the ambient space is 2 or 3 (for details, see Appendix (A.4). If data X is image rather than a point cloud, if the point cloud can be seen an image without losing important information, cubical complexes (59) might be more suitable choice, where vertices, edges and triangles are replaced by vertices, edges and squares (for visualization, see Appendix E.
2.2 Filtration
A filtration {Kr of point cloud in Rd can be constructed from any function f Rd + R by considering K, to be the sublevel set of thresholded by E lR: {y E IRd [(y) S r). The underlying filtration function in the common PH pipeline introduced above the distance function ōx _2 + I. where (1) min (dist(y) x) x E X} is the distance to point cloud X. Indeed, the Vietoris -ips simplicial complex V VR(Xrr) approximates the sublevel set
Kr 6x' ((-oc, r|) ty Idd (8x(2) S r) Uxexx(x,)))
where B(x, r) is ball with radius centered around X 27
However, PH on such a filtration is very sensitive outliers, since even a single outlier changes ōx significantly. In the presence of outliers, it is better to replace the distance function with the Distance-to-Measun (DTM) 8x,m Rdd R, where (x))(ar) the average distance from a number of neighbors on the point cloud (5 [27] (for visualization, see Appendix C.2) However, depending on the task, there are many other filtration functions one could choose, such as rank (84). height, radial, erosion, dilation (48), and the resulting PH captures completely different information about the cycles [103]. For example, whereas PH with respect to the Vietoris- Rips filtration encodes the size of the hole, PH on the height filtration informs about the position of the hole. For more detailed discussion about the influence of filtration, see Appendix
2.3 Persistence signature
Next to PIs and PLs, plethora of persistence signatures has been introduced in the literature, e.g..
Betti numbers 57 106 or Euler characteristic (72) (across scales), or even scalar summaries such as
amplitude (48). entropy 35 93 or algebraic functions of the birth and death values 4 62 Some of these signatures summarize the same information, but lie in different metric spaces [O5]. Others, however, such the scalar summaries listed above, discard information compared to PDs, might sometimes be useful to, c.g., only capture the (total or maximum) persistence of intervals, but not all the detailed information about all birth and death values (see Appendix F).
3 Number of holes
In this section, we focus on the task of (ordinal) classification of point clouds by the number of 1-dimensional holes. Research in psychology shows that global properties often dominate perception, and, in particular, that topological invariants such as number of holes, inside versus outside, and connectivity can be effective primitives for recognizing shapes (85) Extracting such topological information can therefore prove useful for many computer vision tasks. There are theoretical results in the literature that ensure that PH with respect to the alpha simplicial complex can be successful for this problem (Appendix A.2) and the computational experiments that follow demonstrate this success in practice.

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Data We consider 20 different shapes in R² and R2, with four different shapes having the same number of holes (0, 2, 4 or 9). For each shape, we construct 50 point clouds each consisting of 1 0000 points sampled from uniform distribution over the shape, resulting in a balanced data set of 1 000 20 50 point clouds. few examples of these point clouds are shown in Figure [ The label of point cloud is the number of holes in the underlying shape.
DO
shapes
number of holes
2
4
9
Figure I: Number of holes data set.
PH pipeline For each point cloud X and scale r IR, we consider its alpha complex. We will look into scenarios in which data contains noise, and therefore, instead of the standard distance function, Serancceto-MMasuee (DTM) the filtration function. We extract 1-dimensional PDs. that are then transformed to Pls, PLs, or simple signature consisting only of lifespans d - b of the 10 most persisting cycles (as there are at maximum 9 holes of interest in the given data set and classified with an SVM. We consider a "PH simple" pipeline, which relies on the 10 lifespans, and a "PH" pipeline wherein grid search is employed to choose the best out of the three aforementioned persistent signatures and the values of their parameters. For more details on the pipeline, see Appendix B.2 and Appendix
we consider
Results We investigate the clean and robust test accuracy under four types of transformations (translation, rotation, stretch, shear) and two types of noise (Gaussian noise, outliers). For more details on these transformations. see Appendix C. We train the classifier on 80% of the original point clouds, and test on the remaining 20% of the data either in its original form or subject to transformations and noise. The results reported in Figure [ (with detailed results across multiple runs in Appendix C.3) show that PH obtains very good test accuracy on this classification task, even in the presence of affine transformations or noise, outperforming baseline machine- and deep-learning techniques We reach a similar conclusion in case of limited training data and computational resources ( (Appendix C.4 Appendix C.6). Firstly, the evolution of the test accuracy across different amounts of training data demonstrates that PH achieves good performance for a small number of training point clouds, which is not the case for other pipelines. Secondly, although the hyperparameter tuning of the PH pipeline does take time (as we consider a wide range parameters for the different persistence signatures), it is still less than for PointNet. Moreover, Figure 2 shows that even the simple PH pipeline, where the SVM is used directly on the lifespans of the 10 most persisting cycles (without any tuning of PH-related parameters) performs well.
4 Curvature
This section considers a regression task to predict the curvature of an underlying shape based point cloud sample. Estimating curvature-related quantities is of prime importance in computer vision, computer graphics, computer-aided design computational geometry, e.g., for surface segmentation, surface smoothing or denoising, surface reconstruction, and shape design (33 For continuous
Althoouhh sphere has no 1-dimensional holes, its PD might consist of many short intervals which correspond the small holes on the surface. In addition, in the presence of noise, additional small holes might appear for any point cloud. Hence, it is not a good idea consider the cardinality |PD] of the PD the signature.
"Interestingly, although PointNet was designed with the idea to be invariant to affine transformations, it
performs poorly when the test data is translated or rotated (and this is consistent with some previous results when it contains outliers. Traditional neural networks perform very poorly, which might not come as big surprise, since it was recently demonstrated that they transform topologically complicated
66 III T15 T27
data into topologically simple one as it passes through the layers, vastly reducing the Betti numbers (nearly
always even reducing to their lowest possible values: Bx = 0 for k > 0, and So 1) (6).. Of course, the choice of activation function and hyperparameters might have important influence on performance (76).
5

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PH simple
PH
ML
shallow
NN deep
Poi ntNet
original
translation
rotation
stretch
shear
gaussian
outliers
Figure 2: Persistent homology can detect the number of holes.
surfaces, normals and curvature are fundamental geometric notions which uniquely characterize local geometry up rigid transformations [52]. Recently, it has been shown that, using PH, curvature can be both recovered in theory (Appendix A.3) and effectively estimated in practice (17). We run similar experiment, evaluating the PH pipeline against our baselines, and also taking closer look into the importance of short intervals.
Data A balanced data set is generated in the same way in [77] We consider unit disks D on surfaces of constant curvature 6: (i) &. 0, Euclidean plane, (ii) k >> 0, sphere with radius 1/vR, and (iii) к < O, Poincaré disk model of the hyperbolic plane. Curvature lies in the interval sphere with constant curvature к (as it spherical cap). For each E {-2, 1-2,61.96. 1.96. ..... -0.04, 0.04, 1.96), we construct 10 point clouds by sampling 500 points from the unit disk D2 with the probability measure proportional the surface area measure [13] Section 2.7, Section 4. 4.11. A few examples with -1,
[- so that a disk with radius one can be embedded on the upper hemisphere
E 11, -0. 0.1, O, 0, 1, 2} are illustrated in Figure E These 101 x 10 = 1 010 point clouds are considered as the training data, whereas the test data set built similar way for 100 values of к chosen uniformly at random from [-2, 2). The label of a point cloud is the curvature и of the underlying disk D Note that all these disks are homoemorphic: they are contractible, that their
homology is trivial and homology is thus unable to distinguish between them
shapes
curvature
-2.00
-1.00
-0.10
0.00
0.10
1.00
2.00
Figure 3: Curvature data set.
PH pipeline For each point cloud X, first caleulate the suitable matrix of pairwise distances between the point-cloud points: hyperbolic, Euclidean or spherical, respectively for negative, zero and positive curvature 15 Section 2.7]. The input for PH is the filtered Vietoris-Rip simplicial complex We extract () and 1-dimensional PDs, which are then transformed into Pls, PLs or lifespans, to be fed SVM. More details the pipeline provided in Appendix B.2 2 and Appendix D..I
Results Figure shows the mean squared errors for the PH and other pipelines, together with their regression lines, with detailed results across multiple runs listed in Appendix D.2 The results show that PH indeed detects curvature, outperforming other methods 0 Next to the PH pipeline discussed
"The unit disks with negative curvature here visualized hyperbolic paraholoids. These saddle surfaces have non-constant curvature, they locally resemble the hyperbolic plane.
'Alpha complex is faster to compute, but involves Delaunay triangulation, whose unique existence is guaranteed only Euclidean spaces. To calculate PH. rely the Ripser software [102] which is the time the most efficient library to compute PH with Vietoris-Ripe complex [80].
"Simple machine and deep learning techniques are able to differentiate between positive and negative curvature, but perform poorly in predicting the actual value of the curvature of the underlying surface.

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above, wherein grid search is used to tune the parameters (Appendix B.2). we also consider SVM on the lists of lifespans of all persistence intervals (PH simple), and SVM only on the 10 longest lifespans (PH simple 10), in order to investigate if all persistence intervals contribute to prediction. We see that the performance drops if we only focus on the longest 10 intervals, so that the many short intervals together capture the geometry of interest for this problem. Similarly as in Section [3] the grid search across the different parameters for persistence signatures does take time (Appendix D.3), but Figure 4 shows that SVM on simple signature of all (O-dimensional) lifespans performs well. We highlight that the data used here, as was the data in Bubenik's 's work [17], is sampled from surfaces with constant curvature. In future work it would be interesting to conduct similar experiments shapes with non-constant curvature.
O-dim PH simple
O-dim PH simple 10
O-dim PH
ML
PointNet
Prodicted labol
Predicted label b y t
predicted label
x
8
1: 1::3
label
label
True
True
MSE = 0.06
MSE = 0.21
MSE = 0.08
MSE = 0.34
MSE 578.28
1-dim PH simple
1-dim PH simple 10
1-dim PH
NN shallow
NN deep
Predicted label
8 Bxis b
Predicted label
b y s.
Prodicted labal
Prodicted label
Prodicted label
True label
True label
True
True label
True
MSE = 0.34
MSE 0.29
MSE = 0.18
MSE = 0.66
MSE = 0.43
Figure Persistent homology can detect curvature.
5
Convexity
In this section, we consider the binary classification task that consists of detecting whether a point cloud is sampled from a convex set. Convexity is a fundamental concept in geometry (40), which plays an important role in learning, optimization (9]. numerical analysis, statistics, information theory, and economics [9T]. Furthermore, points convexities and concavities have been demonstrated as crucial for human perception of shapes across many experiments [94]
To the best of our knowledge, prior to work PH has not been employed analyze convexity, and it a task for which PH's effectiveness might seem surprising. In the first decade after the introduction of PH, it was seen primarily the descriptor of global topology. Recently, there have been many discussions and greater understanding that PH also captures local geometry (3). However, it is still suggested that the long persistence intervals capture topology (as was the case with the detection of holes in Section 3). and many-even too many for the human eye to count-short persistence intervals capture geometrical properties (as was the case with curvature prediction in Section 4 However, as we show in Theorem (proof in Appendix I) and as our experiments suggest, it is a single, and the loonee-l persistence interval that enables us to detect concavity. crucial ingredient in our result is the introduction of tubular filtrations (Definition [ which, to the best of our knowledge, are novel contribution to the TDA literature (details in Appendix A.
Definition 1. Given a line a C Rd, we define the tubular function with respect to as follows:
Ta : IR +
dist(x, (x)
where dist(æ; ar) is the distance of the point x from the line a. Given X C E2 and line a, we are interested in studying the sublevel sets of Ta, i.e., the subsets of X consisting of points at a specific distance from the line. We define
X X fo X [x) 5r} =f E X dist(æ, sr).
7

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We call {X FrERgo the tubular filtration with respect to
Theorem 1. Let X C. Rd be triangulizable. We have that X convex if and only if for every line a in Rd the persistence diagram in degree 0 with respect to the tubular filtration (Xea. contains exactly one interval.
Data We construct balanced data by sampling 5 000 points from convex and concave (non- convex) shapes in R.2. First, we consider the "regular" convex shapes of triangle, square, pentagon and circle, and their concave variants, sampling 60 point clouds of each of the eight shapes, 480 point clouds in total. Next, we build 480 "random" convex and concave shapes, in order to be able investigate if an algorithm is actually detecting convexity, or only the different basic shapes. A few examples are shown in Figure To construct a random convex shape, we generate 10 points at random, and then build their convex hull using the quickhull algorithm (T08). We construct random concave shapes in similar way, but instead of the convex hull, we build the alpha shape [45 [45|47] 47]
with the optimized alpha parameter, which gives finer approximation of shape from given of points. If the alpha shape is convex (i.e., if the alpha shape and its convex hull are the same), we reconstruct the concave shape from seratch. point cloud has label it is sampled from a convex shape, and 0 otherwise.
shapes
convexity
Figure 5: Convexity data set.
PH pipeline To build a filtration, we consider cubical complexes filtered by tubular functions that measure the distance of points from a certain line, see Definition] [or a precise definition. For good choice of line, multiple components would be seen in the filtration concave shape, at least for some values r E IR (see also the illustrations in Appendix 1. For this reason we consider the cubical complex, rather than the standard Vietoris- simplicial complex wherein these separate components could be connected with an edge (for details, see Appendix EE.1) To build an image from the point cloud, we construct contains any point-cloud points, and white otherwise.
point cloud sampled from
x grid and define pixel as black if
Since sources of concavity can lie anywhere the point cloud, consider nine different lines for the tubular filtration function (for a visualization of the pipeline, see Appendix(E.1 For each of the nine lines, we extract O-dimensional PD, it captures information about the components. If the point cloud is thus sampled from a convex shape, its PD will only see single component for any line, whereas there will be multiple components at least for some lines for points clouds sampled from concave shapes. For this reason, for each of the nine lines, we focus our attention only on the lifespan of the second most persisting eycle. We consider this d-dimensional vector signature, but in our experiment choose an even simpler summary: the maximum of these lifespans, since we only care if there multiple components for at least line. This scalar could even be used as some measure of a level of concavity of shape.
PH
Results As already indicated, gain some insights into how well the different approaches discrim- inate convexity from concavity rather than differentiating between the different basic shapes, we look at the classification accuracies under different conditions (Figure 6 with detailed results across multiple runs in Appendix E.2) We start with the easiest case, where both the train and test data consist simple regular convex and concave shapes (Figure first row), and then proceed the scenario where both train and test data are random shapes (Figure second row). Next we proceed to out-of-distr test data, where we train on the regular and test on random shapes, or vice versa. In every case, train 400 and test on 80 point clouds. The results show that PH is able to detect

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The PH pipeline above makes a wrong prediction when concavity is barely pronounced, or if missed by the selected tubular filtration lines (for details see Appendix [..4. However, the accuracy of PH can easily be improved simply by considering a finer resolution for the cubical complexes and/or convexity, surpassing other methods significantly in all scenarios except for PointNet in the scenario on the data set of regular shapes which performs on par. Results reported in Appendix E.3 show PH is also computationally efficient.
NN shallow
NN cleap
test-roguaar
test-rardom
test=random
test=negular
Figure 6: Persistent homology can detect convexity.
additional tubular filtration lines. The particular PH pipeline summarized in this section would also make a wrong prediction if the data set would include shapes that have small or non-central holes, e.g., square with hole in the top left corner. In this case, the accuracy could also be improved by considering a finer cubical complex resolution and by considering additional non-central tubular filtration lines within shapes, by adding (the maximum lifespan of the) 1-dimensional PH which captures holes. The pipeline is not limited to polygons, or connected shapes, and it can be generalized surfaces in higher dimensions (Theorem TT. In Appendix G also consider the real-world data set FLAVIA for which we demonstrate that a PH pipeline is effective in detecting a continuous measure of convexity.
6 Implications for PH interpretation: topology vs. geometry
Here we discuss how our results contribute to the important and ongoing discussion about the interpretation of long versus short persistence intervals. When PH was first introduced in the literature, the long intervals were commonly considered as important or "signal", and short intervals as irrelevant "noise" (88 Subsequently the discussion has refined when was shown that short and leniuh persistence intervals have the most distinguishing power for specific types of applications 199. The current understanding is roughly that long intervals reflect the topological signal, and (many) short intervals can help in detecting geometric features 3 We believe that our
work brings new insight into this discussion. We give summary the implications of our work in this section, and we provide a more detailed discussion in Appendix
Topology and long persistence Stability results guarantee that a number of longest persistence intervals reflect the topological signal, i.c., the number of cycles (32). These theorems give information about the threshold that differentiates between long and short persistence intervals. In Section 3 where we focus on the topology of underlying shapes, the experiments demonstrate that this threshold can be learned with simple machine learning techniques. However, is important to highlight that the distinction between long and short persistence is vague in practice. Indeed, seemingly short persistence intervals capture the topology Section 3 but the second-longest interval is topological noise in Section [5] since every shape in the data set has only single component (although this second-longest interval captures important geometric information, what enabled us to discriminate between convex and concave shapes). These two problems also clearly indicate how the long intervals that encode topology might or might not be irrelevant, depending on the signal the particular application domain.
Geometry and short persistence The current understanding that (many) short persistence intervals detect geometry. Section confirms that this indeed can be the case. However, we highlight that all cycles can encode geometric information, such as information about their size (with respect to the Pettoris-Rips and related filtrations, as in Sections 3 and 4 their position (with

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respect to the height or tubular filtration, as in Section 5). This further implies that, depending on the application, any number of short or intervals of any persistence can be important, which was clearly demonstrated in Section 5 where we show that single interval detects convexity.
7 Conclusions
Main contribution The goal of this work is to gain a better understanding of the topoloical and geometric features that can be captured with persistent homology. We focus on the detection of number of holes (Section curvature (Section and convexity (Section(5). Theoretical evidence for the first two classes of problems has been established in the literature, and we prove new result that guarantees that PH can detect convexity (Theorem [T. We also experimentally demonstrate that PH can solve all three problems for synthetic point clouds in R2 and R outperforming a few baselines. This true even when there is limited training data and computational resources, and for noisy out of distribution test data. For convexity detection, we also show the effectiveness of PH in a real world plant morphology application.
Relevance Firstly, the findings point the way to further advances in utilizing the potential of PH in applications: we can expect PH to be successful for classification or regression problems where the data classes differ with respect to number of holes, curvature and/or convexity. Detailed guidelines are discussed in Appendix F Due the crucial role of shape classification in understanding and recognizing physical structures and objects, image processing and computer vision (73), our results demonstrate that PH can-to borrow the words from Wigner 1122]"rrmani valid in future research, and extend, to our pleasure". and lesser bafflement, to variety of applications. Secondly, the results advance the discussion about the importance of long and short persistence intervals, and their relationship topology and geometry (Section(6) Topology is captured by the long intervals, geometry is encoded in all persistence intervals, and any interval can encode the signal in the particular application domain.
The results focus on three selected problems and data sets, and would therefore be interesting to consider other tasks. In addition, we do not have an extensive comparison of the state-of-the-art for the given problems. Our work seeks to understand if PH successful for a selected set of tasks by benchmarking it against some well- well-performing methods.
Limitations
Future research An in-depth analysis of the hypothetical applications discussed in this paper (Appendix F) and selected success stories of PH from the literature could further improve our understanding of the topological and geometric information encoded PH, and the interpretation of persistence intervals of different lengths. Alternative approaches for detection of convexity with PH (relying on higher homological dimensions, or multiparameter persistence) are particularly interesting avenues for further work. Furthermore, even though our results imply that PH features are recommended over bascline models for the three selected classes of problems, they also provide inspiration on how to improve existing learning architectures. Further work could investigate deep learning models on PH (and standard) features or kernels (14 89 [16). an additional network layer for topological signatures, or PH-based priors, regularization or loss functions 113 121
34 36 37 56
Potential negative societal impact While we recognize that the applications of shape analysis can take many different directions, we do not foresee a direct path of this research to negative societal impacts.
Funding transparency statement This research was conducted while RT was a Fulbright Visiting Researcher at UCLA. GM acknowledges support from ERC grant 757983, DFG grant 464109215, 15, and NSE-CAREER grant DMS-2145630. NO acknowledges support from the Royal Society, under grant RGS1R22212169.
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Checklist
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(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes] both the introductory and concluding section, we clearly point to the relevant sections for each of the contributions.
(b) Did you describe the limitations your work? [Yes] The limitations are highlighted in the concluding Section 7 and in the end of curvature and convexity Sections 4 and we provide more details about the interesting types of point clouds that could be considered.
(c) Did you discuss any potential negative societal impacts of your work? [Yes] There is paragraph in Section addressing these impacts.
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ments multiple times)? [Yes] See Appendix C Appendix Appendix B
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Appendix B
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) your work uses existing assets, did you cite the creators? [Yes] We use libraries
described and acknowledged in Appendix B
(b) Did you mention the license of the assets? [Yes] See Appendix B
(c) Did you include any new assets either in the supplemental material as URL? Yes] The data and code are provided in the Supplemental and referenced in Section [ and Appendix B
(d) Did you discuss whether and how consent was obtained from people whose data you're using/curating? [N/A] We do not use any personal data.
(c) Did you discuss whether the data you using/curating contains personally identifiable information or offensive content? [N/A] We construct data sets of point clouds that do contain any personal information.
5. If you used crowdsourcing conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]
(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]
(c) Did you include the estimated hourly wage paid participants and the total amount spent on participant compensation? [N/A]
17


Reviews:
[{"review_id": "x7rx-d-01En", "reviewer": "Reviewer_pzbB", "summary": "In this article, the authors propose to investigate the data science tasks where persistent homology performs actually better than standard methods. They focus on three different problems: holes, curvature and convexity detection, and they show that, for each of them, persistent homology leads to better accuracies than neural nets.", "questions": "---What are the variances of the accuracies in the figures? Since some experiments involve adding random noise, and since training and test sets are drawn randomly, accuracies should always be computed on several runs, and their means and variances across the runs should be reported.\n\n---Why is PH suited for curvature detection? One can easily get the intuition why PH is good for hole and convexity, but it is less clear concerning curvature. In particular, why is 0-dimensional PH efficient, but not 1-dimensional PH? Why do the short bars matter more than the long ones? It would be nice if the authors provided some more intuition for this task.\n\n---The paper should be more explicit about the parameters: in the hole detection experiment, what are the DTM parameters? In the convexity detection experiment, what is the resolution of the image computed from the point cloud? \n\n---Instead of computing an image from the point cloud in the convexity experiment, could one use scalar field analysis instead (https://link.springer.com/article/10.1007/s00454-011-9360-x)?\n\n---What is the influence of the point cloud dimension? Is there any curse of dimensionality? In other words, is there any reason to believe that the proposed topological methods would still be efficient?", "limitations": "I think the authors properly identified the limitations of their experiments.", "rating": 3, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 1, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "I think the question of identifying the tasks where computational topology improves on existing methods is definitely important, making the paper well motivated. Moreover,  the writing of the paper is very good and clear.\n\nHowever, the significance of the paper is limited by two major weaknesses.\n\nFirst, there is no theoretical statements or results about the topological methods used to solve the tasks. This makes the claims of the paper harder to believe, since there is no way of estimating the generalization capability of topological methods on data sets that are different from those used in the paper.\n\nSecond, the experiments of the paper are only done on synthetic data sets in two dimensions. When there is no theory, it is necessary to benchmark many different data sets, from both synthetic and real data, in order for the claims to be convincing, but the proposed experiments are much too synthetic and simple.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "vNPWqdF78Za", "reviewer": "Reviewer_6onG", "summary": "The paper mainly discusses what can be detected by persistent homology (PH) features by introducing three shape analysis tasks. Experiments on these tasks show that PH can effectively detect these shapes even with limited computational resources and limited training data.", "questions": "See \"Weakness\".", "limitations": "The limitations are adequately addressed.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "In general, the paper does add some interesting and new discussions, but it seems that some discussions have been covered in previous works.\n\nStrengths:\n\nThe paper is generally well-written, and not hard to follow, except for some minor mistakes. E.g., “0- and 1-dim PH” in Figure 1 should be replaced by “0- and 1-dim PI”; In Line 282, it seems that the longest interval should be topological noise, while the second-longest one is useful.\n\nThe discussion in the paper is interesting. Recently, there are many works integrating PH into various downstream deep learning tasks and achieving state-of-the-art results. However, many works lack insight on why PH can boom performance on these tasks. Therefore, researching what can be detected by PH quite interests me. The discussion on long/short persistence is also interesting.\n\nThe discussion on using PH to detect convexity is new.\n\nWeakness:\n\nFirst, detecting the number of holes with PH is not new. Actually, we can directly compute the betti number (the number of holes) of a given point cloud data. (E.g., in [56], researchers compute the betti number with several preprocessing steps.) Then what is the superiority of the proposed method: (to infer/learn the number of holes with PH features)? \n\nSecond, detecting curvature is not new either. Actually, a similar experiment that uses PH to detect curvature is done in [17]. In fact, they also provide theoretical guarantees and experiments on unsupervised learning.\n\nAlso, there are some comparably minor problems. (1) building simplicial complexes and computing PH is quite computational costly, therefore better add comparison on times in the experiment part; (2) adding persistent homology on downstream tasks that require topological information such as curvature/convexity/betti number can better support the experiment part of the paper. But considering the limited rebuttal time, this can be future work.\n\n--**Post rebuttal**--\n\nGenerally speaking, the topic of the paper is worth researching, and some of the discussions are interesting. Although most of the experiments are based on synthetic datasets, it should be admitted that very few real-world graphs totally rely on the number of holes/convexity/curvature to predict certain attributes, e.g., their label.  Therefore I would like to raise my score, but certain improvements should be considered:\n\n(1) Add PH to deep learning models and evaluate whether it can boost performance on real-world tasks. For example, over-squashing is a curvature-related attribute [1], therefore it can be evaluated whether adding PH in GNN can solve the problem.\n\n(2) Some works combining PH with deep learning models are missing, e.g., [2], [3]. \n\n[1] Topping J, Di Giovanni F, Chamberlain B P, et al. Understanding over-squashing and bottlenecks on graphs via curvature. ICLR 2021.\n\n[2] Hu X, Li F, Samaras D, et al. Topology-preserving deep image segmentation. Neurips 2019.\n\n[3] Yan Z, Ma T, Gao L, et al. Link prediction with persistent homology: An interactive view. ICML 2021.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "aeJyyPlEhT", "reviewer": "Reviewer_786m", "summary": "In this paper, the author(s) identify special types of problems where persistent homology (PH)  performs well or even better than other state-of-the-art methods in data analysis. They found that PH can work better in three fundamental shape analysis tasks: the detection of the number of holes, curvature and convexity. They have carried out  different experiments to demonstrate the advantage of PH than existing models. ", "questions": "To demonstrate the \"effectiveness\" of PH, realistic problems should be considered and a through comparison with the state-of-the-art models should be carried out. In fact, there are many excellent papers, in which these comparison is carried out together with a detailed descriptions of the advantage and limitations of PH models.  For instance, \n\nZ. X. Cang and Guo-Wei Wei, TopologyNet: Topology based deep convolutional and multi-task neural networks for biomolecular property predictions, PLOS Computational Biology, 13(7), e1005690 (2017).\n\nZ. X. Cang, Lin Mu and Guo-Wei Wei, Representability of algebraic topology for biomolecules in machine learning based scoring and virtual screening, PLOS Computational Biology,  14(1), e100592 (2018).\n\nDuc Duy Nguyen, Zixuan Cang, Kedi Wu, Menglun Wang, Yin Cao  and Guo-Wei Wei, Mathematical deep learning for pose and binding affinity prediction and ranking in D3R Grand Challenges, Journal of Computer Aided Molecular Design, 33, 71-82   (2019).\n\nDuc Nguyen, Zixuan Cang, and Guo-Wei Wei, A review of mathematical representations of biomolecular data, Physical Chemistry Chemical Physics, 22, 4343-4367 (2020).\n\nMenglun Wang, Z.  X. Cang, and Guo-Wei Wei,   A topology-based network tree for the prediction of protein-protein binding free energy changes following mutation, Nature Machine Intelligence, 2, 116-123 (2020).\n\nZhenyu Meng and Kelin Xia, \"Persistent spectral–based machine learning (PerSpect ML) for protein-ligand binding affinity prediction.\" Science Advances, 7 (19), eabc5329 (2021)\n\nZhao, Qi, Ze Ye, Chao Chen, and Yusu Wang. \"Persistence enhanced graph neural network.\" In International Conference on Artificial Intelligence and Statistics, pp. 2896-2906. PMLR, 2020.\n\nWong, Chi-Chong, and Chi-Man Vong. \"Persistent homology based graph convolution network for fine-grained 3D shape segmentation.\" In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 7098-7107. 2021.", "limitations": "As stated above, to demonstrate the \"effectiveness\" of PH, realistic problems should be considered and a through comparison with the state-of-the-art models should be carried out. In general, it should be easy to show that PH models can characterize certain curvature and convexity related information. However, there are gigantic amount of models that can have much better accuracy in the characterization of curvature and convexity. Only when the PH models are fully compared with state-of-the-art models, the authors can argue about the \"Effectiveness of Persistent Homology\". Otherwise, it is just the potential application!", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper has discussed about the advantage of PH in several different areas. It is interesting to see a summerization of these advantages of PH.\n\nPH-based machine learning models have already been widely used in different areas. In fact, the advantage, strength and limitations of PH models have been extensively discussed in various excellent application papers and reviewer papers. The paper lacks a detailed discussion of these models and the proof-of-concept type of experimental tests are not convincing at all! ", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ["I don’t know"]}, "extra_text": {}}, {"review_id": "Fzn-HWkuoCt", "reviewer": "Reviewer_sPjT", "summary": "This paper studies the effectiveness of PH for detection of number of holes, curvature and convexity using point clouds samples from shapes. The analyses are performed on various simple datasets in comparison with baselines providing promising initial results.", "questions": "Training curves show that NNs overfit data easily. Could you please repeat these experiments using networks endowed with normalization and regularization methods such as ResNets?", "limitations": "Limitations were briefly addressed.", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "PH is an important method used for topological data analysis. The paper introduces a nice analysis of the employment of PH for several tasks on point clouds. The paper is well written, and analyses are clearly presented. In several benchmark, PH outperforms baseline with promising results for the future work.\n\nThe main weakness of the paper is the datasets. The paper can have more impact if the analyses are extended to point clouds of more complicated shapes, especially shapes of real-world objects. Although the current results are interesting, their generalization to more complicated datasets is not clear.  As mentioned in the limitation, a more detailed comparison with state-of-the-art would improve the paper.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept