Title: On the estimation of  persistence intensity functions and linear representations of persistence diagrams
Venue: NeurIPS 2023
Accepted: no
Keywords: topological data analysis

persistence diagram

betti numbers

non-parametric density estimation

persistence surface
Abstract: Persistence diagrams are one of the most popular types of data summaries used in Topological Data Analysis. The prevailing statistical approach to analyzing persistence diagrams is concerned with filtering out topological noise. In this paper, we adopt a different viewpoint and aim at estimating the actual distribution of a random persistence diagram, which captures both topological signal and noise. To that effect, [CD19] has shown that, under general conditions, the expected value of a random persistence diagram is a measure admitting a Lebesgue density, called the persistence intensity function. In this paper, we are concerned with estimating the persistence intensity function and a novel, normalized version of it -- called the persistence density function. We present a class of kernel-based estimators based on an i.i.d. sample of persistence diagrams and derive estimation rates in the supremum norm. As a direct corollary, we obtain uniform consistency rates for estimating linear representations of persistence diagrams, including Betti numbers and persistence images. Interestingly, the persistence density function delivers stronger statistical guarantees. 

Abstract

Persistence diagrams are one of the most popular types of data summaries used in Topological Data Analysis. The prevailing statistical approach to analyzing persistence diagrams is concerned with filtering out topological noise. In this paper, we adopt a different viewpoint and aim at estimating the actual distribution of a random persistence diagram, which captures both topological signal and noise. To that effect, [CD19] has shown that, under general conditions, the expected value of a random persistence diagram is a measure admitting a Lebesgue density, called the persistence intensity function. In this paper, we are concerned with estimating the persistence intensity function and a novel, normalized version of it - called the persistence density function. We present a class of kernel-based estimators based on an i.i.d. sample of persistence diagrams and derive estimation rates in the supremum norm. As a direct corollary, we obtain uniform consistency rates for estimating linear representations of persistence diagrams, including Betti numbers and persistence images. Interestingly, the persistence density function delivers stronger statistical guarantees.

1 Introduction

Topological Data Analysis (TDA) is a field at the interface of computational geometry, algebraic topology and data science whose primary objective is to extract topological and geometric features from possibly high-dimensional, noisy and/or incomplete data. The literature on the statistical analysis of TDA summaries has mainly focused on distinguishing topological signatures from the unavoidable topological noise resulting from the data sampling process. Toward that goal, the primary objective in designing statistical inference methods for TDA is to isolate points on the sample persistence diagrams that are sufficiently far from the diagonal to be deemed statistically significant in the sense of expressing underlying topological features instead of randomness. This paradigm is entirely natural when the target of inference is the unobservable persistence diagram arising from a filtration of interest, and the sample persistent diagrams are noisy and imprecise approximations to it. On the other hand, empirical evidence has also demonstrated that topological noise is not unstructured and, in fact, may also carry expressive and discriminative power that can be leveraged for various machine-learning tasks. In some applications, the distribution of the topological noise itself is of interest; in cosmology, see e.g., [WNv + 21]. As a result, statistical summaries able to express the properties of both topological signal and topological noise in a unified manner have also been proposed and investigated: e.g., persistence images and

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linear functional of the persistence diagrams. In a recent contribution, [CD19] has derived 35 sufficient conditions to ensure that the expected persistent measure - the expected value 36 of the random counting measure corresponding to a noisy persistent diagram - admits a 37 Lebesgue density, hereafter called the persistence intensity function; see also [CWRW15]. 38 The significance of this result is multifaceted. First, the persistent intensity function provides 39 an explicit and highly-interpretable representation of the entire distribution of the persistence 40 homology of random filtrations. Secondly, it allows for a straightforward calculation of the 41 expected linear representation of a persistent diagram as a Lebesgue integral. Finally, the 42 representation provided by the persistence intensity function is of functional, as opposed 43 to algebraic, nature and thus analytically simpler. It is amenable to statistical analysis via 44 well-established theories and methods from the non-parametric statistics literature. 45

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In this paper we derive consistency rates of estimation of the persistence intensity function and of a novel variant called persistence density function in the /lscript ∞ norm based on a sample of i.i.d. persistent diagrams. As we argue below in Theorem 3.1, controlling the estimation error for the persistence intensity function in the /lscript ∞ norm is stronger than controlling the optimal transport measure OT q for any q > 0 and, under mild assumptions, immediately implies uniform control and concentration of any bounded linear representation of the persistence diagram including (persistent) Betti numbers and persistence surfaces.

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2 Background and definitions

In this section we introduce fundamental concepts from TDA that we will use throughout the paper. We refer the reader to [CM21, CD19] for detailed background and extensive references.

Persistence diagrams. We define a persistence diagram to be a locally finite multiset of points D = { r i = ( b i , d i ) | 1 ≤ i ≤ N ( D ) } belonging to the set

consisting of all the points on the plane in the positive orthant above the identity line and of 59 coordinate values no larger than a fixed constant L > 0. The coordinates of each point of 60 D correspond to the birth and death times of a persistent homology feature, where time is 61 measured with respect to the totally ordered set indexing a filtration. The restriction that 62 the persistence diagrams be contained in a box of side length L is a mild assumption that is 63 widely used in the TDA literature; see [DL21] and the discussion therein. To simplify our 64 notation, we will omit the dependence on L , but we will keep track of this parameter in our 65 error bounds. Some related quantities used throughout are 66

That is, ∂ Ω is a segment on the diagonal in R 2 and Ω /lscript consists of all the points in Ω at a 67 Euclidean distance of /lscript or smaller from it. 68

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The expected persistent measure and its normalization. A persistence diagram D = { r i = ( b i , d i ) ∈ Ω | 1 ≤ i ≤ N ( D ) } can be equivalently represented as a counting measure µ on Ω given by

where B = B (Ω) is the class of all Borel subsets of Ω and δ r denotes the Dirac point mass at 72 r ∈ Ω. We will refer to µ as the persistence measure corresponding to D and, with a slight 73 abuse of notation, will treat persistence diagrams as counting measures. If D is a random 74 persistence diagram, then the associated persistence measure is also random. In addition 75 to the persistence measure µ associated to a persistence diagram D , we will also study its 76 normalized measure ˜ µ , which is the persistence measure divided by the total number of points 77

N ( D ) in the persistence diagram. In detail, ˜ µ is the (possibly random) probability measure 78 on Ω given by 79

The normalized persistence measure may be desirable when the number of points N ( D ) 80 in the persistence diagram is not of direct interest but their spatial distribution is. This 81 is typically the case when the persistence diagrams at hand contain many points or are 82 obtained from large random filtrations (e.g. the Vietoris-Rips complex built on point clouds), 83 so that the value of N ( D ) will mostly accounts for noisy topological fluctuations due to 84 sampling. 85

We will consider the setting in which the observed persistence diagram D is a random draw 86 from an unknown distribution. Then, the (non-random) measures 87

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are well defined. We will refer to E [ µ ] and E [˜ µ ] as the expected persistence measure and the expected persistence probability, respectively. Notice that typically, neither is a discrete measure, and that the expected persistence probability is a probability measure by construction.

The interpretations of the measure E [ µ ] and the probability measure E [˜ µ ] is straightforward: for any Borel set A ⊂ Ω, E [ µ ]( A ) is the expected number of points from the random persistence diagram falling in A , while E [˜ µ ]( A ) is the probability that a random persistence diagram will intersect A . As a result, they are able to directly express the randomness of the distribution of persistence diagram including structural properties of the topological noise. Despite their interpretability, the expected persistence measure and probability are not yet standard concepts in the practice and theory of TDA. As a result, they have not been thoroughly investigated.

The persistence intensity and density functions and linear representations. In a recent, important contribution, [CD19] derived conditions - applicable to a wide range to problems - that ensure that the expected persistence measure E [ µ ] and its normalization E [˜ µ ] both admit densities with respect to the Lebesgue measure on Ω. Specifically, under fairly mild and general conditions detailed in [CD19] there exist measurable functions p : Ω → R ≥ 0 and ˜ p : Ω → R ≥ 0 , such that for any Borel set A ⊂ Ω,

In fact, [CD19] provided explicit expressions for p and ˜ p (see Section D.5). Notice that, by construction, ˜ p integrates to 1 over Ω. We will refer to the functions p and ˜ p as the persistence intensity and the persistence density functions, respectively. We remark that the notion of a persistence intensity function was originally put forward by [CWRW15].

The persistence intensity and density functions 'operationalize' the notions of expected persistence measure and expected persistence probability introduced above, allowing to evaluate, for any Borel set A , E [ µ ]( A ) and E [˜ µ ]( A ) in a straightforward way as Lebesgue integrals.

The main objective of the paper is to construct estimators ̂ p and ˇ p of the persistence intensity p and persistence density ˜ p , respectively, and to provide high probability error bounds with respect to the L ∞ norm. As we show below in Theorem 3.1, L ∞ -consistency for the persistence intensity function is a stronger guarantee than consistency in the OT p metric, for any p < ∞ . Interestingly, we find that estimation of the persistence probability density function is statistically easier, in the sense that uniform estimation error bounds can be obtained for all points in Ω. In contrast, estimating the persistence intensity function becomes progressively more difficult for points near ∂ Ω. See Theorem 3.6 below.

Linear representations of persistence diagrams. As noted in [CD19], the persistence intensity and density functions are naturally suited to compute the expected value of linear

representations of random persistence diagrams. A linear representation Ψ of the persistence 124 diagram D = { r i = ( b i , d i ) ∈ Ω | 1 ≤ i ≤ N ( D ) } with corresponding persistence measure µ 125 is a summary statistic of D of the form 126

for a given measurable function f on Ω. (An analogous definition can be given for the normalized persistence measure ˜ µ instead). Then,

where the second identity follows from (3). Linear representations include persistent Betti 129 numbers, persistence surfaces, persistence silhouettes and persistence weighted Gaussian 130 kernels. 131

The persistence surface is an especially popular linear representation introduced by [AEK + 17]. 132 In detail, for a kernel function K ( · ) : R 2 → R ≥ 0 and any x ∈ R 2 , let K h ( x ) = 1 h 2 K ( x h ), 133 where h > 0 is the bandwidth parameter 1 . The persistence surface of a persistence measure 134 µ is defined as 135

where f ( ω ): R 2 → R is the user-defined weighting function, chosen to ensure stability of the representation. Our analysis allows to immediately obtain consistency rates for the expected persistence surface in L ∞ norm, which, for brevity, we present in the supplementary material (see Theorem B.5). Instead we focus on the estimation error the expected Betti numbers.

Betti and the persistent Betti numbers. The Betti number at scale x ∈ [0 , L ] is the number of persistent homologies that are in existence at 'time" x . Furthermore, the persistent Betti number at a certain point x = ( x 1 , x 2 ) ∈ Ω measures the number of persistent homologies that are born before x 1 and die after x 2 . In our notation, given a persistence diagram D and its associated persistence measure µ , for x ∈ [0 , L ] and x = ( x 1 , x 2 ) ∈ Ω, the corresponding Betti number and persistent Betti number are given by

respectively, where B x = [0 , x ) × ( x, L ] and B x = [0 , x 1 ) × ( x 2 , L ]. Though Betti numbers are among the most prominent and widely used TDA summaries, relatively little is known about the statistical hardness of estimating their expected values when the sample size is fixed and the number of persistence diagrams increases. Our results will yield error bounds of this type. We will also consider normalized versions of the Betti numbers defined using the persistence probability ˜ µ of the persistence diagram:

Notice that, by definition, ˜ β x ( D ) ≤ 1. While their interpretation is not as direct as the Betti numbers computed using persistence diagrams, the expected normalized (persistence) Betti numbers are informative topological summaries while showing favorable statistical properties (see Theorem 3.12 below).

3 Main results

3.1 The OT distance between measures and L ∞ distance between intensity functions

A popular and, arguably, natural metric for persistence diagrams - and, more generally, locally finite Radon measures such as normalized persistence measures and probabilities - is

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the optimal transport distance; see, e.g., [DL21]. In detail, for two Radon measures µ and ν supported on Ω, an admissible transport from µ to ν is defined as a function π : Ω × Ω → R , such that for any Borel sets A,B ⊂ Ω,

Let adm ( µ, ν ) denote all the admissible transports from µ to ν . For any q ∈ R + ∪ {∞} , the q -th order Optimal Transport (OT) distance between µ and ν is defined as

When µ and ν are persistent diagrams the choice of q = ∞ corresponds to the widely-used bottleneck distance. The OT distance is widely used for good reasons: by transporting from and to the diagonal ∂ Ω, it captures the distance between two measures that have potentially different total masses, taking advantage of the fact that points on the diagonal have arbitrary multiplicity in persistent diagrams. It also proves to be stable with respect to perturbations of the input to TDA algorithms. However, for expected persistent measures with intensity functions with respect to the Lebesgue measure, we will show next that the L ∞ distance between intensity functions provides a tighter control on the difference between two persistent measures. Below, for a real-valued function on Ω, we let ‖ f ‖ ∞ = sup x ∈ Ω | f ( x ) | be its L ∞ norm.

Theorem 3.1 Let µ , ν be two expected persistent measures on Ω with intensity functions p µ and p ν respectively. Then

Furthermore, there exists two sequences of expected persistence measures { µ n } n ∈ N and { ν n } n ∈ N with intensity functions { p µ n } n ∈ N and { p ν n } n ∈ N respectively such that, as n →∞ ,

The bottleneck distance For the case q = ∞ , which yields the bottleneck distance when applied to persistence diagrams, there can be no meaningful upper bound in the form of (7): we show in Section D.1 of the supplementary material that there exist two sequences of measures such that their bottleneck distance converges to a finite number while the L ∞ distance between their intensity functions vanishes. Existing contributions [Pey18, NGK21] also upper bound the optimal transport distance by a Sobolev-type distance between density functions. It is noteworthy that these bounds require, among other things, the measures to have common support and the same total mass, two conditions that are not assumed in Theorem 3.1.

3.2 Non-parametric estimation of the persistent intensity and density functions

In this section, we analyze the performance of kernel-based estimators of the persistent intensity function p ( · ) and the persistent density function ˜ p ( · ). We adopt the setting where we observe n i.i.d. persistent measures µ 1 , µ 2 , . . . , µ n . The procedures we proposed are directly inspired by kernel density estimators for probability densities traditionally used in the non-parametric statistics literature; see, e.g., [GN21]. Specifically, we consider the following estimator for p ( · ) and ˜ p ( · ), respectively:

where K ( · ) is the kernel function, which we assume to satisfy a number of standard regularity 196 conditions used in non-parametric literature, discussed in detail in Section B.2 of the 197 supplementary material. 198

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Assumptions. We will impose a number of regularity conditions on the expected persistent measures, the persistence intensity and density functions and the kernel function. Of course, we will assume throughout that both p and ˜ p (see (3)) are well-defined as densities with respect to the Lebesgue measure, though we point out that this is not strictly necessary for Theorems 3.6 and 3.9.

Our first assumption of smoothness of both p and ˜ p is needed to control the point-wise bias of our estimators and is a standard assumption in non-parametric density estimation.

Assumption 3.2 (Smoothness) The persistence intensity function p and persistence probability density function ˜ p are Hölder smooth of the order of s > 0 with parameters L p and L ˜ p respectively 2 .

In our next assumption, we impose boundedness conditions on p and ˜ p , which are needed in order to apply a key concentration inequality for empirical processes.

Assumption 3.3 (Boundedness) For some q > 0 , let ¯ p ( ω ) := ‖ ω -∂ Ω ‖ q 2 p ( ω ) . Then,

Notice that instead of assuming a bound on the L ∞ norm of the intensity function p , we are only requiring the weaker condition that the weighted intensity function ¯ p ( ω ) = ‖ ω -∂ Ω ‖ q 2 p ( ω ) has finite L ∞ norm, due to the fact that the total mass of the persistence measure may not be uniformly bounded in a number of common data-generating mechanisms. Indeed, it is not a priori clear that Assumption 3.3 itself is realistic; in the supplementary material we prove that this assumption holds for the Vietoris-Rips filtration built on i.i.d. samples. On the other hand, assuming that the persistence density is uniformly bounded poses no problems. See Theorems B.1 and B.2 in the supplementary material for formal arguments. This fact is the primary reason why the persistence probability density function - unlike the persistence intensity function - can be estimated uniformly well over the entire set Ω - see (3.6) below. We refer readers to Section B.1 of the supplementary materials for details and a discussion on this subtle but consequential point.

In our last assumption, we require a uniform bound on the q -th order total persistence, though not on the total number of points in the persistence diagram. As elucidated in [CSEHM10] and discussed in [DP19] and [DL21], this is a relatively mild assumption, which should be expected to hold under a broad variety of data-generating mechanisms.

Assumption 3.4 (Bounded total persistence) There exists a constant M > 0 , such that, for the value of q as in Assumption 3.3, it holds that, almost surely,

We will denote with Z q L,M the set of persistent measures on Ω L satisfying Assumption 3.4.

We are now ready to present our first result concerning the bias of the kernel estimators, whose proof is relatively standard.

Theorem 3.5 Under Assumption 3.2, for any ω ∈ Ω , 233

The next result provides high-probability uniform bounds on the fluctuations of the kernel 234 estimators around their expected values. 235

Theorem 3.6 Suppose that Assumptions 3.3 and 3.4 hold. Then, 236

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(a) there exists a positive constant C depending on M, ‖ K ‖ ∞ , ‖ K ‖ 2 , ‖ ¯ p ‖ ∞ and q such that for any δ ∈ (0 , 1) , it can be guaranteed with probability at least 1 -δ that

(b) there exists a positive constant C depending on M, ‖ K ‖ ∞ , ‖ K ‖ 2 , ‖ ˜ p ‖ ∞ and q such 240 that for any δ ∈ (0 , 1) , it can be guaranteed with probability at least 1 -δ that 241

Remark. The dependence of the constants on problem related parameters is made explicit in the proofs; see the supplementary material.

There is an important difference between the two bounds in Theorem 3.6: while the variation of ˇ p h ( ω ) is uniformly bounded everywhere on Ω, the variation of ˆ p h ( ω ) is uniformly bounded only when ω is at least 2 h away from the diagonal ∂ Ω, and may increase as ω approaches the diagonal. The difficulty in controlling the variation of ˆ p h near the diagonal comes from the fact that we only assume the total persistence of the persistent measures to be bounded; in other words, the number of points near the diagonal in the sample persistent diagrams can be prohibitively large, since their contribution to the total persistence is negligible. This is to be expected in noisy settings in which the sampling process will result in topological noise consisting of many points in the persistence diagram near the diagonal. The above result suggests that it is advantageous to rely on density-based, instead of intensity-based representations of the persistent measures.

Bias-variance trade-off and minimax lower bound. If follows from Theorems 3.5 and 3.6 that the choice h /equivasymptotic n -1 2( s +1) for the bandwidth will optimize the bias-variance trade-off, yielding high-probability estimation errors

The following theorem shows that the above rate is minimax optimal for the persistence density function. For brevity, we here omit a similar result for the persistence intensity function (see Theorem B.4 in the supplementary material).

Theorem 3.7 Let F denote the set of functions on Ω with Besov norm bounded by B > 0 :

where the infimum is taken over estimator ˇ p n mapping µ 1 , . . . , µ n to an intensity function in F , the supremum is over the set of all probability distributions on Z q L,M and ˜ p is the intensity function of E P [˜ µ ] .

3.3 Kernel-based estimators for linear functionals of the persistent measure

The kernel estimators (8) can serve as a basis for estimating bounded linear representations of the expected persistence measure E [ µ ] and its normalized counterpart E [˜ µ ]. Specifically, for R > 0, let F 2 h,R and ˜ F R denote the set of linear representations of the form

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Then,

Then, any linear representations Ψ ∈ F 2 h,R and ˜ Ψ ∈ ˜ F R can be estimated by 270

respectively. The following theorems provide uniform bounds on the bias and variation of 271 these kernel-based estimators. 272

Theorem 3.8 Under Assumption 3.2, it holds that 273

Theorem 3.9 Assume that Assumptions 3.2 and 3.3 hold. Then,

(a) there exists a constant C depending on M, ‖ K ‖ ∞ , ‖ K ‖ 2 , ‖ ¯ p ‖ ∞ and q such that for any δ ∈ (0 , 1) , it can be guaranteed with probability at least 1 -δ that

(b) there exists a constant C depending on M, ‖ K ‖ ∞ , ‖ K ‖ 2 , ‖ ˜ p ‖ ∞ and q such that for any δ ∈ (0 , 1) , it can be guaranteed with probability at least 1 -δ that

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It is important to highlight the fact that the above bounds hold uniformly over the choice of 279 linear representations under only mild integrability assumptions. 280

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Theorems 3.8 and 3.9 are direct corollaries of Theorems 3.5 and 3.6. We again stress the difference between the two upper bounds of Theorem 3.9: part (a) shows that for a linear functional of the original persistent measure to have controlled variation, we need the field of integral to be at least 2 h away from the diagonal ∂ Ω, a requirement that is not necessary for linear functionals of the normalized persistent measure, as is shown in part (b).

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Next, we apply Theorems 3.9 and 3.9(a) to the persistent Betti number, which, for any x ∈ Ω, can be estimated by

Corollary 3.10 Under Assumption 3.2, it holds that

Corollary 3.11 Under Assumptions 3.2 and 3.3(a), there exists a constant C depending on M, ‖ K ‖ ∞ , ‖ K ‖ 2 , ‖ ¯ p ‖ ∞ and q > 2 such that for any δ ∈ (0 , 1) ,

holds with probability at least 1 -δ . 292

Notice that in order for the variation of ˆ β x ,h to be bounded, we need x to be at least 2 h 293 away from the diagonal ∂ Ω, and that the upper bound for the variation increases as x 294

approaches the diagonal. Therefore, based on our analysis, the kernel-based estimator ˆ p h 295 will not be guaranteed to yield a stable estimation of the Betti number β x . As remarked 296 above, this issue arises as the intensity function may not be uniformly bounded near the 297 diagonal. Indeed, in the supplementary material, we describe an alternative proof technique 298 based on an extension of the standard VC inequality and arrive at a very similar rate. 299

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If instead we target the normalized Betti numbers ˜ β x , this issue disappears when we deploy the analogous estimator ˇ β x,h = ∫ B x ˇ p h ( ω )d ω , constructed using ˇ p h . Indeed, Theorem 3.9(b) leads to the following uniform bounds.

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Corollary 3.12 Assume that Assumptions 3.2 and 3.3 hold true. Then there exist a constant C > 0 depending on M, ‖ K ‖ ∞ , ‖ K ‖ 2 , ‖ ˜ p ‖ ∞ and q such that for any δ ∈ (0 , 1) , it can be guaranteed with probability at least 1 -δ that

As a direct consequence of the previous result, we obtain a uniform error bound for the 306 expected normalized Betti curve , i.e. 307

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To the best of our knowledge this is the first result of this kind, as typically one can only establish pointwise and not uniform consistency of Betti numbers.

4 Numerical Illustration and discussion

To illustrate our methodology and highlight the differences between the persistence intensity and density functions, we consider the MNIST handwritten digits dataset and the ORBIT5K dataset. The ORBIT5K dataset contains independent simulations for the linked twist map, dynamical systems for fluid flow as described in [AEK + 17]; see also Appendix G.2 of [KKZ + 20]. In Section E of the supplementary material, we show the estimated persistence intensity and density functions computed from persistence diagrams obtained over a varying number of random samples from the ORBIT5K datasets, for different model parameters. The figures confirm our theoretical finding that the values of the persistence density function near the diagonal are not as high (on a relative scale) as those of the persistence intensity function. An analogous conclusion can be reached when inspecting the persistence intensity and density functions for different draws of the MNIST datasets for the digits 4 and 8. We further include plots of the average Betti and normalized Betti curves from the ORBIT5K dataset, along with the curves of the empirical point-wise 5% and 95% quantiles. These plots reveal the different scales of the Betti curves and normalized Betti curves, and of their uncertainty.

In this paper, we have taken the first step towards developing a new set of methods and theories for statistical inference for TDA based on samples of persistence diagrams. Our main focus is on the estimation of the persistence intensity function [CD19, CWRW15], a TDA summary of a functional type that encodes the entire distribution of a random persistence diagram and is naturally suited to handle linear representations. We have analyzed a simple kernel estimator and derived uniform consistency rates that hold under very mild assumptions. We also propose the persistence density function, a novel functional TDA summary that enjoys stronger statistical guarantees. Though our results guarantee that the proposed estimators are consistent, in order to carry out statistical inference, it is necessary to develop more sophisticated procedures that quantify the uncertainty of our estimators. Towards that goal, it would be interesting to develop bootstrap-based methods for constructing confidence bands for both the persistence intensity and density functions.

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[WNv + 21] Georg Wilding, Keimpe Nevenzeel, Rien van de Weygaert, Gert Vegter, Pratyush Pranav, Bernard J T Jones, Konstantinos Efstathiou, and Job Feldbrugge. Persistent homology of the cosmic web - I. Hierarchical topology in ΛCDM cosmologies. Monthly Notices of the Royal Astronomical Society , 507(2):2968-2990, 2021.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper considers the problem of estimating _persistence intensity (resp. density) functions_, which are topological summaries arising when considering multiples realizations $\\mu_1,\\dots,\\mu_n$ of persistence diagrams---which are counting measures supported on an open-half-plane $\\Omega$. Namely, (Chazal and Divol, 2019) proved that $\\frac{1}{n} \\sum_{i=1}^n \\mu_i$ converges toward a limit object $E[\\mu]$ called an _expected persistence diagram_ which, under mild assumptions, admits a density $p$ wrt the Lebesgue measure on $\\Omega$, called a persistence _intensity_ function. Note that $p$ may not integrate to $1$ (the $\\mu_i$s are *not* probability measures, only Radon measures). \n\nIt is natural to wonder how fast one can estimate $E[\\mu]$ given an i.i.d. sample $\\mu_1,\\dots,\\mu_n$. Given that $E[\\mu]$ has a smooth density $p$, while the $(\\mu_i)$ are discrete, it is tempting to adopt a kernel based approach (i.e. considering the convolution of the $\\mu_i$ by a kernel $K_h$, with $h>0$ being the bandwidth), yielding an estimator $\\hat{p} = \\hat{p}_{h,n}$ of $p$. This is the purpose of the current paper which shows that, in brief, assuming $p$ is $s$-Hölder,\n\n$$\\| E[\\hat{p}] - p \\|_\\infty \\leq O(h^s), \\qquad \\text{(bias)},$$\n\nand (with high probability)\n\n$$ \\| \\hat{p} - E[\\hat{p}] \\|_{\\infty , h} \\leq O(n^{-\\frac{1}{2}}h^{-1}), \\qquad \\text{(variance)},$$\n\nwhere $\\| \\cdot \\|_{\\infty,h}$ denote the sup-norm but only accounting for points in $\\Omega$ that are at distance $> 2h$ from the diagonal $\\partial \\Omega = \\{ (t,t), t \\in \\mathbb{R} \\}$. \n\nAs an alternative, they also study the persistence _density_ function, which is substantially similar to the above, but considering the quantity $\\frac{1}{n} \\sum_{i=1}^n \\frac{\\mu_i}{\\mu_i(\\Omega)}$ as a starting point, hence the limit object is a _probability_ distribution. This first normalisation step allows the authors to obtain similar results as those mentioned above, but stronger in the sense that they do not need to ignore points close to $\\partial \\Omega$.\n\nSTRENGTHS:\n## Clarity\n\nThe paper is fairly well written, though it may be hard to understand for the reader that is not familiar with statistical topological data analysis problems. Theorems are precisely stated, and, with few exceptions, mathematical quantities are well-defined.  \n\n## Originality and Significance\n\nThe authors provides quantitative results for the convergence of kernel-based density estimator for expected PD, which is new to the best of my knowledge. The introduction of the persistence _density_ may also be worth of interest, if further motivated (see below). \n\n## Quality\n\nThis is a competent paper in terms of the results provided (which seem correct as far as I can tell). However, the motivation behind the type of results themselves is arguably questionable.\n\nWEAKNESSES:\nThere are several points which fail to convince me and prevent me from supporting this paper yet. \n\n## 1. The use of the sup-norm. \n\nThe paper provides results in the sup-norm on $\\Omega$. It is important to stress that this norm does not account for the peculiar role played by the diagonal in the geometry of persistence diagrams, in contrast with the standard $\\mathrm{OT}_p$ metric. The authors justify this by an inequality of the form $\\mathrm{OT}_p^p(\\cdot, \\cdot) \\leq C | \\cdot - \\cdot|$, \nmeaning that being small in sup-norm is (strictly, as proved by the authors) more demanding that being close in $\\mathrm{OT}_p$ metric. [Edit: fix the rendering by removing the infty symbol; the norm in the rhs is the sup-norm.]\n\nSure, it means that the rate obtained for the sup-norm implies the same rate for the $\\mathrm{OT}_p$ metric (up to the role played by the exponent), but it also means that the task is _harder_, and that this norm does not induce the same topology as the natural metrics on persistence diagrams. Recall (as suggested in $\\ell$167-171) that accounting for the diagonal is not simply a trick to compare measures with different total masses, but also has an algebraic meaning (from which we get the mentioned stability results). The sup-norm induces a topology that fails to capture the fact that one can compare diagrams \"downweighting points close to the diagonal\". \n\nTo me, this is what prevents, for the persistence _intensity_ function, to obtained \"global\" sup bound, and get only bounds valid $2h$-away from the diagonal. The sup-norm cannot handle the noise properly. \n\nIn addition, because performing estimation in sup-norm is _harder_ than estimation in $\\mathrm{OT}_p$-metric, one only get a convergence rate of $\\frac{s}{2(s+1)}$, which is natural for the sup-norm, but quite _slow_ for the $\\mathrm{OT}_p$ metric. Indeed, (DIvol and Lacombe, 2021) prove that the empirical expected persistence diagram (i.e. without any convolution by a kernel involved) converges toward the persistence _intensity_ function at (the faster) parametric rate $O(n^{-\\frac{1}{2}})$. Of course, the latter result considers the weaker (but more natural) metric $\\mathrm{OT}_p$, but as long as there is no very strong motivation to compare persistence intensity functions using a sup-norm, it is reasonable to wonder why should we struggle to obtain slower convergence rates. \n\n## 2. Motivations behind the persistence _density_ function\n\nAs (interestingly) observed by the authors, statistical estimation improves when considering the normalized persistence _density_ function. However, here as well, I fail to be fully convinced by the proposed motivation, namely \"the normalized persistence measure may be desirable when the number of points (...) is not of direct interest but their spatial distribution is\" ($\\ell$80-81). \n\nI do not agree with this claim because this normalization typically discard points away from the diagonal, asymptotically. For, consider a $N$-sample on a sphere + some tubular noise, and the Vietoris-Rips filtration on it. Then (with high probability), the corresponding (random) persistence diagram in $H_2$ (degree-$2$ homology) will have one point away from the diagonal, and a bunch of points close to the diagonal accounting for the noise---so does the corresponding persistence _intensity_ function. As $N$ increases, the points accounting for the noise get closer and closer to the diagonal, but also more abundant. As such, if one normalize the persistence measure by its mass/number of points, the bump/point accounting for the \"robust\" topological information will asymptotically be erased. In particular, this normalized persistence measure is not continuous (for, say, the vague topology) with respect to the Hausdorff distance, a central property satisfied by the Vietoris-Rips filtration. (Note : this is what we can observe in Figure 3 vs Figure 2). \n\nTherefore, (i) it is not surprising that one obtains stronger (this time) results with this weaker representation and (ii) it is not clear to me why would one actually consider this representation at is losses its topological interpretation, as far as I can tell. \n\n## 3. About the experiments\n\nThe numerical illustrations have all been deferred to the supplementary material. \n\nTo me, the (main body of the) paper should be self-contained, in sense that one should not _have to_ look at the supplementary material to understand it at high level. Proofs, complementary results, _complementary_ experimental report can be deferred to the supplementary material, but having a **Numerical illustration** section without any numerical illustration, mostly saying \"look at the supplementary material\", is not serving the paper. Right now, the paper can be considered as experiment-less, and while numerical illustrations are not mandatory, this clearly does not support the paper. \n\nNote that I looked at the experiments nonetheless. While they are fairly interesting, they do not bring more motivation to support the paper (with, e.g., a ML experiment where using the persistence _density_ function is much better than using the more standard persistence intensity function).  \n\n## 4. Complementary minor comments\n- I think that there is a typo in the definition of $\\Omega_\\ell$, which is (I think) inconsistent with the description made below ($\\ell$67) and its use in section 3. \n- More references could have been cited through the paper, e.g., when listing different linear representations ($\\ell$129-131), it may be nice to credit their respective authors.  Similarly, a more precise comparison with related work (mostly Divol's line of work with Polonik, Chazal and then Lacombe), would be helpful to understand what is the paper contribution and how it differs from these works.\n\nQUESTIONS:\nMy main interrogation is about the motivation of the proposed objects (sup-norm, normalisation wrt the mass), as detailed in points 1. and 2. above.\n\nThe motivation can come from theoretical considerations, but also from numerical ones; e.g. an experiment showing that considering the sup-norm is more informative that the $\\mathrm{OT}_p$ metric (I mean, it theoretically is---as discussed in section D1, but I do not see a _practical_ situation where it is important to observe that the sup-norm diverges while the $\\mathrm{OT}_p$ converges).\n\nLIMITATIONS:\nI do not see any negative societal impact _specific_ to this work.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2023-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proves several theoretical inequalities involving the optimal transport distance, intensity, and density functions on a plane triangle with a non-standard boundary motivated by persistent homology.\n\nSTRENGTHS:\nThe paper rigorously proves in appendix C six theorems and three corollaries from section 3. All definitions, statements, and proofs are written in great detail. Also, the paper is well-written overall.\n\nWEAKNESSES:\nStarting already from section 2 about the background, it seems that the term \"persistence\" is not really needed because there is no connection with real data. \n\nBorel sets, measures, densities, and other concepts of classical probability theory can be considered on a plane triangle without the \"persistent\" adjective. \n\nHence it is strange to read lines 97-99 saying that \"measure and probability are not yet standard concepts in the practice and theory of TDA. As a result, they have not been thoroughly investigated\" because measure and probability are standard concepts in probability theory for nearly 100 years. \n\nThe conclusions reveal the main theoretical weaknesses in lines 328 and 332: \"Our main focus is on the estimation of the persistence intensity function [CD19, CWRW15].\" More explicitly, line 106-109 accepts that \"[CD19] provided explicit expressions for p and p˜  ... We will refer to the functions p and p˜ as the persistence intensity and the persistence density functions, respectively. We remark that the notion of a persistence intensity function was originally put forward by [CWRW15].\"\n\nQUESTIONS:\nTaking into account lines 106-109, what is the theoretical advance in the paper over the past work [CD19, CWRW15]?\n\nAre Assumptions 3.2, 3.3, and 3.4 essential for the proven results? Do you have counter-examples to the theorems and corollaries when one of these assumptions fails?\n\nEven if we accept Assumptions 3.2, 3.3, and 3.4, can the word \"persistence\" be removed from sections 2-3 so that all results are proved for measures on any plane triangle? The words \"Betti numbers\" can be easily defined for any triangle bounded by the diagonal x=y. Then will the paper become much more suitable for a more theoretical venue in statistics?\n\nDo Assumptions 3.2, 3.3, and 3.4 hold for persistence diagrams obtained from the experiments in section 4?\n\nThe main practical weakness is the lack of a problem statement for the data mentioned in section 4. Is this data real or simulated? Some pictures would be helpful.\n\nLIMITATIONS:\nThough the paper doesn't include the required keyword \"limitation\", the limitations appear in Assumptions 3.2, 3.3, 3.4. For instance, Assumption 3.4 essentially requires that there is not too much \"little noise\". \n\nIn a simple case of the sublevel persistence of a scalar function, this function can be perturbed only by introducing a \"bounded amount\" of pairs of adjacent local maxima and minima. More exactly, the persistence diagram allows only a bounded sum of \"noisy artefacts\" near the diagonal.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work develops a set of methods and theories for statistical inference for TDA based on samples of persistence diagrams:  \na. The work focuses on the estimation of the persistence intensity function. The work also proposes the novel persistence density function, which is the normalized version of the persistence intensity function.\nb. The work present a class of kernel-based estimators based on an iid sample of persistence diagrams and derive estimation rates in the supremum norm, which is stronger than the optimal transport distance norm.\nc. The work obtains uniform consistency rates of estimating linear representations of persistence diagrams, including Betti numbers, persistent surfaces, persistent silhouttes and weighted Gaussian kernels.\nd. The work presents several theorems, theorem 3.1 compares the L^\\infty norm and the optimal transport distance in terms of controlling the estimation error; theorem 3.5, 3.6 show the kernel estimation error bound for the persistent density function and the persistent intensity function; theorem 3.8, 3.9 show the estimation error bounds for the linear representations. \n\nThese theoretic results are fundamental and important, they lead to novel direction for statistical inference for TDA based on random persistent diagrams.\n\nSTRENGTHS:\nThis work is very solid, and gives rigorous mathematical proofs. The formulations of key concepts, main theorems are clear and rigorous, the mathematical deductions for lemmas, theorems, corollaries are thorough and in detail. The theoretic results are convincing and impressive.\n\nWEAKNESSES:\nThe work is highly theoretical, the heavy mathematical deductions are abstract. It will be more helpful for readers to digest if the authors further explain the motivations, the main proof approaches, the interpretations of the theorem, the potential direct applications of the results. More specifically,\n\n1. It will be helpful for readers to better understand the article to give a table of symbols, list the major symbols and their meanings;\n2. It will be helpful to give some figure to illustrate the concepts, such as persistent diagram;\n3. Some math symbols and operators can be further explained, such as:\na. The two symbols in line 145 are hard to differentiate, especially on a laptop screen, maybe the authors can emphasize the shuttle differences, or use different symbols;\nb. The formula in line 165, \\|x-y\\|_2^q needs more explanation\nc. The formula in line 247 in the supplementary, the operator Proj_{\\partial\\Omega} needs more explanation\n\nQUESTIONS:\n1. The interpretation of the concept of random persistent diagram. For example, if we consider the MNIST data sets. Do we treat each image as a point cloud and build a persistent diagram ? Do we only consider the images of one digit or different digits together? Where does the randomness come from? Different writing styles of different people ? random noises in the imaging process ? \n\n2. Theorem 3.1, the result is very general and can be applied in much broader fields. Does the inequality hold on general compact domains ? How tight is the bound ?\n\n3. In the proof of theorem 3.1, please explain the current admissible transport plan . Is there other admissible transport scheme, which can lead to tighter estimation? \n\n4. Please give some direct applications of the estimated persistent intensity/density functions, can we use it for generating persistent diagrams？ recognize， authenticate， classify persistent diagrams in TDA？\n\nLIMITATIONS:\nThis work is theoretical, it mainly focuses on theoretical deductions. The limitations are not adequately addressed. It will be helpful if the limitations in practical applications are further discussed. \n1. In reality, how difficult to satisfy all the assumptions listed in the paper ?\n2. If the point cloud include several homology generators with similar birth and death times, may the current approach mix them and cause confusion ?\n3. In theorem 3.6, from samples close to the diagonal, can we get more precise estimation ?", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-07"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper tackles the problem of estimating the persistence intensity function, describing the distribution of rando persistence diagrams, and proposes a variant called persistence probability function that integrates to one.  The paper starts with a theoretical analysis of the estimation error bound of the intensity function using the OT measure and the L-infinite norm, showing that the latter allows the definition of stricter bounds. The paper also proposes a method to estimate the persistence intensity function and the persistence probability function under the assumption of i.i.d. samples using a kernel density estimation approach.\n\nSTRENGTHS:\nPersistent diagrams are an important tool for characterizing topological structures (e.g. surfaces and graphs). Being able to estimate with high accuracy the distribution of such structures could indeed be beneficial to their analysis, with applications also to the learning domain.\n\nThe paper, at least from a not-so-expert reader like I am, seems very rigorous in the theoretical analysis and provides in the sup. mat. all the proofs of the introduced theorems.\n\nWEAKNESSES:\nThe main weakness (if we want to call it so) of the paper is that it is not easy to read by nonexperts of the specific topic. It is quite dense and mostly mathematical and does not provide many intuitive explanations of why some properties could be important from a practical perspective.\nIn general, I would have appreciated a more gentle introduction to the problem and a wider introduction/literature review on the practical application/advantages of persistent intensity functions.\n\nQUESTIONS:\nI think that it would help to add some details in the introduction about the importance and applicability of the proposed tool, which is still not clear to me. \nIn the conclusions, you state that statistical inference is not yet possible with the proposed method. Does this mean that it cannot be used in practice?\n\nRow 101: of problems\nRow 139: of the expected\n\nLIMITATIONS:\nI don’t foresee any particular negative societal impact. A discussion about the practical applicability of the method would be interesting.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-09"}