Title: Wasserstein convergence of Cech persistence diagrams for samplings of submanifolds
Venue: NeurIPS 2024
Accepted: yes
Keywords: topological data analysis; persistence diagrams; simplicial complexes; differential geometry; Wasserstein distance
Abstract: Cech Persistence diagrams (PDs) are topological descriptors routinely used to capture the geometry of complex datasets. They are commonly compared using the Wasserstein distances $\mathrm{OT}_p$; however, the extent to which PDs are stable with respect to these metrics remains poorly understood. 
We partially close this gap by focusing on the case where datasets are sampled on an $m$-dimensional submanifold of $\mathbb{R}^d$. Under this manifold hypothesis, we show that convergence with respect to the $\mathrm{OT}_p$ metric happens exactly when $p>m$. We also provide improvements upon the bottleneck stability theorem in this case and prove new laws of large numbers for the total $\alpha$-persistence of PDs. Finally, we show how these theoretical findings shed new light on the behavior of the feature maps on the space of PDs that are used in ML-oriented applications of Topological Data Analysis.

Wasserstein convergence of ˇ Cech persistence diagrams for samplings of submanifolds

Abstract

ˇ Cech Persistence diagrams (PDs) are topological descriptors routinely used to capture the geometry of complex datasets. They are commonly compared using the Wasserstein distances OT p ; however, the extent to which PDs are stable with respect to these metrics remains poorly understood. We partially close this gap by focusing on the case where datasets are sampled on an m -dimensional submanifold of R d . Under this manifold hypothesis, we show that convergence with respect to the OT p metric happens exactly when p > m . We also provide improvements upon the bottleneck stability theorem in this case and prove new laws of large numbers for the total α -persistence of PDs. Finally, we show how these theoretical findings shed new light on the behavior of the feature maps on the space of PDs that are used in ML-oriented applications of Topological Data Analysis.

1 Introduction

Topological Data Analysis (TDA) is a set of tools that aims at extracting relevant topological information from complex datasets, e.g. regarding connected components, loops, cavities, or higher dimensional features. These different notions are made formal through the use of homology theory , and in particular the i -th homology group H i ( A ) of a set A , which captures the i -dimensional topological features of A for i ≥ 0 , see e.g. [46] or Appendix B. TDA has been successfully applied in a variety of domains, including material science [51, 63, 30, 17], biology [21, 68, 8], real algebraic geometry [29, 35, 28, 44] and neuroscience [67, 61, 18], to name a few. When used in conjunction with more traditional approaches such as neural networks, TDA-based methods have outperformed state of the arts methods for tasks such as graph classifications [19, 50].

The most prominent techniques in TDA rely on multiscale approaches, in particular through the use of persistent homology [22]. Given a compact set A in R d , persistent homology tracks the evolution of the homology groups H i ( A t ) of the t -offset A t = ⋃ x ∈ A B ( x, t ) of A as t goes from 0 to + ∞ (where B ( x, t ) is the closed ball of radius t centered at x ). The process is summarized by the ˇ Cech persistence diagram (PD) of degree i of the set A : the PD dgm i ( A ) is a multiset 2 of points in the half-plane Ω := { ( u 1 , u 2 ) ∈ R 2 : u 1 < u 2 } , where each point ( u 1 , u 2 ) in the PD corresponds to a i -dimensional topological feature that appeared in A t at scale t = u 1 (its birth time) and disappeared at scale t = u 2 (its death time), see Figure 1. 3 Points close to the

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diagonal ∂ Ω = { ( u 1 , u 2 ) ∈ R 2 : u 1 = u 2 } correspond to topological features of small persistence pers( u ) = u 2 -u 1 2 , which have a short lifetime in the filtration ( A t ) t ≥ 0 .

A key property of PDs is their robustness to small perturbation in the data, making them suitable for analyzing real-world datasets. This stability property is phrased in terms of the bottleneck distance between PDs [36]. Let a and b be two PDs. A partial matching between a and b is a bijection of multisets γ : a ∪ ∂ Ω → b ∪ ∂ Ω , where each point ( u, u ) of ∂ Ω has infinite multiplicity. In other words, the points of a are either paired with a single point of b , or mapped to the diagonal ∂ Ω (and similarly for the points of b ). Let Γ( a, b ) be the set of all partial matchings between a and b . The bottleneck distance is defined as

The Bottleneck Stability Theorem [26, 22] states that if A 1 and A 2 are two compact sets, then for any integer i ≥ 0

where ε is the Hausdorff distance between the sets A 1 and A 2 , defined by d H ( A 1 , A 2 ) = sup x ∈ R d | d A 1 ( x ) -d A 2 ( x ) | and where d A is the distance function to a set A . An important property of the bottleneck distance is that it is blind to small-persistence topological features: if OT ∞ ( a, b ) = ε , then one can arbitrarily modify the PDs a and b on a slab of width ε above the diagonal (for the ℓ ∞ -metric) without changing the bottleneck distance between the two PDs.

Due to this phenomenon, the bottleneck distance turns out to be too weak in many situations of interest, where some topological features of small persistence can be as important as large-scale topological features in the PD (say, with a classification or a regression task in mind). For this reason, finer transport-like distances are often preferred to compare PDs. These distances, which we denote as OT p , are defined for 1 ≤ p < ∞ by

with OT p ≤ OT p ′ for 1 ≤ p ′ ≤ p < ∞ . They can be seen as modified versions of the Wasserstein distances used in optimal transport, with the diagonal ∂ Ω playing the role of a landfill of infinite mass, see e.g. [33].

The increased sensitivity to small perturbations of the OT p distances is of crucial importance in the standard TDA pipeline, which we briefly recall. Starting from a sample of sets A 1 , . . . , A n , one computes a sample of PDs a j = dgm i ( A j ) , j = 1 , . . . , n . Statistical methods to analyze this sample of PDs are typically awkward to define, due to the nonlinear geometry of the space of PDs [16, 75]. To overcome this issue, the space of PDs is mapped to a vector space through some map Φ called a feature map. Statistical method are then applied in the feature space on the transformed sample Φ( a 1 ) , . . . , Φ( a n ) . Various feature maps have been designed [23, 60, 25, 54, 20, 70, 53, 48], important examples including persistent images [4], PersLay [19], and PLLay [50]; in the latter two, the feature map is parametrized by a neural network. A good feature map should preserve as much as possible the geometry of the space of PDs [57]; in particular, Lipschitz (or Hölder) continuity of the feature map is a basic requirement. However, due to their (often desirable) sensitivity to small scale

features, most common feature maps are not regular with respect to the OT ∞ distance on the space of diagrams. Instead, they enjoy Lipschitz regularity with respect to either the finer OT 1 distance (see e.g. [33, Proposition 5.2]), or to the OT p distances (for p > 1 ) when restricted to diagrams a whose α -total persistence

is bounded for some α > 0 large enough, see [34] and [53]. Boundedness assumptions on the α -total persistence also yield a version of the Bottleneck Stability Theorem with respect to the finer OT p distance: if A 1 and A 2 are such that Pers α (dgm i ( A k )) ≤ M ( k = 1 , 2 ), then, for p ≥ α ,

where ε = d H ( A 1 , A 2 ) , see [27]. Hence, in addition to having intrinsic theoretical interest, controlling the total persistence and convergence with respect to the OT p distance of PDs is crucial to ensure the soundness of most methods commonly used in TDA. This is the subject of this article.

Contributions. We provide a deeper understanding of the structure of ˇ Cech PDs in the specific case where the underlying set A is a compact subset of am m -dimensional manifold M in R d , focusing in particular on the total persistence of the PDs and their convergence to the PDs of M with respect to the OT p distances. The importance of this case is supported by the manifold hypothesis , which often serves as a fundamental principle guiding the development of algorithms and models for data analysis [41, 76, 15]. Specifically, our main contributions are the following:

Theorem 2.2: When A ⊂ M is a compact set satisfying d H ( A , M ) ≤ ε for ε small enough, we provide a quadratic improvement upon the standard Bottleneck Stability Theorem (2). Namely, we show that there exists an optimal bottleneck matching γ such that the distance between a coordinate of a point u ∈ dgm i ( A ) and the coordinate of the matched point γ ( u ) ∈ dgm i ( M ) ∪ ∂ Ω is of order O ( ε 2 ) whenever the coordinate of u is larger than 2 ε .

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Theorem 3.3: In the case where the manifold M is generic and the set A is a δ -sparse point cloud (i.e. min x = y ∈ A ∥ x -y ∥ ≥ δ ), we provide a finer analysis by showing that the p -total persistence Pers p (dgm i ( A )) remains bounded and the distance OT p (dgm i ( A ) , dgm i ( M )) converges to 0 for all p > m whenever the ratio ε/δ is upper bounded.

Corollary 4.3: We then focus on a random context, by assuming that A = A n is obtained by sampling n i.i.d. random variables with positive bounded density f on a generic manifold M . We prove that OT p (dgm i ( A n ) , dgm i ( M )) converges in expectation to 0 for p > m . Furthermore, we obtain a law of large numbers for the α -total persistence of dgm i ( A n ) :

for all α > 0 , where C i is a constant that depends explicitly on M and f . In particular, for 0 ≤ i < m , Pers α (dgm i ( A n )) stays bounded if and only if α ≥ m .

Our contributions are to be compared to one of the only preexisting results regarding the α -total persistence of a PD: in [27], the authors proved that for all α strictly greater than the ambient dimension d , the α -total persistence of the ˇ Cech PD of a compact set A ⊂ B (0 , R ) ⊂ R d satisfies

for some constant C α,d depending on α and d . In the two scenarios we considered (either δ -sparse or random samples), the ambient dimension d in the constraint α > d for the control of the α -total persistence in (7) has been replaced by the smaller intrinsic dimension m of the problem: the manifold hypothesis has been successfully exploited.

To summarize, our work sheds light on the behaviour of PDs, provides new guarantees for commonly used ML methods (see e.g. Corollary 4.4), and suggests new heuristics (see Section 5). We also perform various experiments to illustrate the validity of our results and their relevance to the classic TDA pipeline. All proofs are deferred to the Appendix.

Related work. This work is part of a long ongoing effort to understand simplicial complexes and PDs in a random context [10, 12, 13, 11, 49, 32, 59, 14, 45]. Closest to our work, Hiraoka, Shirai

and Trinh gave limit laws for ˇ Cech PDs for random points in the cube [0 , 1] d [47], while Goel, Trinh and Tsunoda gave similar asymptotics in the case of samples on manifolds [42]. Limit laws for the total persistence have been obtained by Divol and Polonik in the case of random samples in the cube [34]. Among other contributions, this work generalizes this result to submanifolds: unlike the cube, a manifold has a nontrivial topology; a fact which considerably complicates the situation, for we have to take into account the presence of large topological features in the ˇ Cech PDs in order to control the total persistence. Note also that tools other than persistent homology exist for studying the geometry of point clouds. For instance, the authors in [77, 52] consider complete isometric invariants for point clouds that are computable in polynomial time.

2 ˇ Cech persistence diagrams for subsets of submanifolds

Recall that a fundamental result in TDA, the Bottleneck Stability Theorem (2), states that ˇ Cech PDs are stable with respect to Hausdorff perturbations. Consider the particular setting where one has access to a set A , obtained as an approximation of an unknown shape of interest S through some sampling procedure, with A ⊂ S and sup x ∈ S d A ( x ) ≤ ε . The Bottleneck Stability Theorem ensures that OT ∞ (dgm i ( A ) , dgm i ( S )) ≤ ε for any i ≥ 0 , a bound which cannot be improved in general. However, it turns out that a finer understanding of the proximity between dgm i ( A ) and dgm i ( S ) can be obtained if more regularity is assumed on the shape of interest S , namely in the situation where S = M is a compact submanifold with positive reach.

Let us first set some notation. Let M be an m -dimensional compact topological submanifold of R d ; we always assume that the boundary of M is empty. The orthogonal projection π M on M is defined for x close enough to M , and we define the reach τ ( M ) as the largest r > 0 such that the orthogonal projection π M is well (i.e. uniquely) defined for all x ∈ R d at distance strictly less than r from M . The reach is a key notion to quantify the regularity of a manifold, see e.g. [39] and [24] for more information.

Let A ⊂ M be such that d H ( A , M ) ≤ ε and let z ∈ R d . By definition of the Hausdorff distance, it holds that | d A ( z ) -d M ( z ) | ≤ ε . However, this naive bound can be quadratically improved as long as z stays far away from M .

Lemma 2.1. Let M ⊂ R d be a compact submanifold with positive reach and let A ⊂ M be a compact set with d H ( A , M ) ≤ ε for some ε > 0 . Let z ∈ R d \ M . Then, | d M ( z ) -d A ( z ) | ≤ ε 2 2 d M ( z ) ( 1 + d M ( z ) τ ( M ) ) .

We can build upon this basic remark to obtain a very precise control of the behavior of the ˇ Cech PD of the set A . Namely, we identify three regions in the upper halfplane Ω (displayed in Figure 2) which contain all points in the PD dgm i ( A ) (for some integer i ≥ 0 ). In the first region, corresponding to microscopic topological features disappearing at scales smaller than ε + ε 2 /τ ( M ) , the Bottleneck Stability Theorem cannot be improved. However, there exists an optimal matching (i.e. a matching γ : dgm i ( A ) ∪ ∂ Ω → dgm i ( M ) ∪ ∂ Ω that realizes the bottleneck distance (1)) such that at least one of the coordinates of any point in the other two regions is larger than τ ( M ) -ε 2 /τ ( M ) , and the proximity between a large coordinate of a point u ∈ dgm i ( A ) and the coordinate of the matched point γ ( u ) ∈ dgm i ( M ) ∪ ∂ Ω is of order O ( ε 2 ) . This yields a quadratic improvement upon the Bottleneck Stability Theorem.

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Theorem 2.2 (Improved Bottleneck Stability Theorem) . Let M ⊂ R d be a compact submanifold with positive reach and let A ⊂ M be a compact set such that d H ( A , M ) ≤ ε < τ ( M ) / 4 . Let i ≥ 0 be an integer. Then dgm i ( A ) is the union of three regions dgm (1) i ( A ) := dgm i ( A ) ∩{ u 1 , u 2 ≤ ε + ε 2 τ ( M ) } , dgm (2) i ( A ) := dgm i ( A ) ∩ { u 1 ≤ ε, u 2 ≥ τ ( M ) -ε 2 τ ( M ) } and dgm (3) i ( A ) := dgm i ( A ) ∩ { u 1 , u 2 ≥ τ ( M ) -ε 2 τ ( M ) } .

Furthermore, let C = 2 τ ( M ) ( 1 + R ( M ) τ ( M ) ) , where R ( M ) is the radius of the smallest ball that contains M . There exists an optimal matching γ : dgm i ( A ) ∪ ∂ Ω → dgm i ( M ) ∪ ∂ Ω for the bottleneck distance between dgm i ( A ) and dgm i ( M ) such that

Region (1): If u ∈ dgm (1) i ( A ) , then γ ( u ) ∈ ∂ Ω and ∥ u -γ ( u ) ∥ ∞ ≤ ε .

Region (2): If u ∈ dgm (2) i ( A ) , then γ ( u ) is of the form (0 , v 2 ) and | u 2 -v 2 | ≤ Cε 2 . The number of such points is finite and depends only on M .

Region (3): If u ∈ dgm (3) i ( A ) , then ∥ u -γ ( u ) ∥ ∞ ≤ Cε 2 .

Note that for any i ≥ d , the i -th PDs of M and A are actually trivial.

3 ˇ Cech persistence diagrams for subsets of generic submanifolds

The improved Bottleneck Stability Theorem (Theorem 2.2) yields information relative to the location of points in the ˇ Cech PD of A , but not about their numbers. However, both the α -total persistence of dgm i ( A ) and the distance OT p (dgm i ( A ) , dgm i ( M )) for p < ∞ crucially depend on the number of points in dgm i ( A ) having small persistence.

Unfortunately, no control on, say, the total persistence, can exist without additional assumptions. Indeed, in general, even the α -total persistence of the ˇ Cech PD of the submanifold M can be infinite.

Example 3.1. Let f : x ∈ R ↦→ 1 + x 4 sin(1 /x ) 2 . Consider the C 2 curve M in R 2 defined as the union of the graphs of the functions f and -f on [ -2 , 2] . Being C 2 , the curve has a positive reach [39]. For i = 1 , the ˇ Cech PD of M contains a sequence of points (1 , ℓ n ) for n ≥ 1 , where ℓ n = 1 + Θ( n -4 ) . In particular, as ∑ n ≥ 1 ( ℓ n -1) α = + ∞ for α < 1 / 4 , the α -total persistence of dgm 1 ( M ) is infinite for such a value of α . By considering the product M m ⊂ R 2 m , one can also build an m -dimensional C 2 compact submanifold without boundary such that dgm 1 ( M ) has an infinite α -total persistence for α < m/ 4 .

The existence of such counterexamples is explained by the fact that the distance function d M to a set M is not well-behaved in general, even when the set M is smooth. In contrast to this bleak general case, Song, Yim & Monod (in the case of surfaces in R 3 ) and Arnal, Cohen-Steiner & Divol (in the general case) studied the distance function d M to M when M is a generic submanifold [6, 71]. Their findings indicate that, although counterexamples such as the one presented in Example 3.1 exist, they are extremely uncommon in a sense which can be made precise.

Namely, given an abstract manifold M , Arnal, Cohen-Steiner and Divol show that the set of C 2 embeddings M of M into R d such that d M satisfies some desirable regularity conditions (described in Appendix C)

forms an open and dense set in the set of all C 2 embeddings equipped with the C 2 -Whitney topology [6, Theorem 1.1]. In what follows, we will simply refer to a C 2 compact submanifold M such that d M satisfies the regularity condition described in Appendix C as generic . See Figure 3 for an example.

Proposition 3.2 ( ˇ Cech PDs of generic submanifolds) . Let M be a generic submanifold of R d . Then, for any integer i ≥ 0 , the PD dgm i ( M ) contains finitely many points. In particular, Pers α (dgm i ( M )) < + ∞ for all α > 0 .

When M is a generic submanifold, it becomes a reasonable task to control the number of points in dgm i ( A ) where A ⊂ M is an approximation of M with d H ( A , M ) ≤ ε . The Bottleneck Stability Theorem implies that when ε is small enough (compared to the smallest persistence of a point in dgm i ( M ) ), every point of dgm i ( M ) is mapped to a point in dgm i ( A ) by the optimal bottleneck matching, leaving the points of dgm i ( A ) at ℓ ∞ -distance to ∂ Ω less than ε unmatched; those will be mapped to the diagonal ∂ Ω . The Improved Bottleneck Stability Theorem 2.2 (see Figure 2) shows that these points are of two kinds: those in Region (1), corresponding to small topological features in the set A (of size of order O ( ε ) ), and those in Region (3), corresponding to large topological features. There are many points in Region (1) (in fact, our proofs show that when A = A n is a random

sample of n points, the number of points in Region (1) is of order O ( n ) ). In contrast, the genericity hypothesis allows us to show that the number of points in Region (3) is small under reasonable sampling assumptions.

̸

We say that a point cloud A ⊂ M is ( δ, ε ) -dense in M if d H ( A , M ) ≤ ε and min x = y ∈ A ∥ x -y ∥ ≥ δ . Such point clouds naturally occur, e.g. as products of the farthest point sampling algorithm [2].

Theorem 3.3. Let M ⊂ R d be a generic compact submanifold and A ⊂ M be a ( δ, ε ) -dense set in M for some ε, δ > 0 . Let a ≥ ε/δ and let i ≥ 0 be an integer. There exist ε 0 > 0 depending only on M and C 0 , C 1 , C 2 , C 3 depending only on M and a such that if ε ≤ ε 0 , then dgm (3) i ( A ) has at most C 0 points and for all p ≥ 1 , α ≥ 0 ,

In particular, as long as the ratio ε/δ is larger than some constant a > 0 , the OT p distance between dgm i ( A ) and dgm i ( M ) converges to 0 for all p > m as ε → 0 , while the p -total persistence Pers p (dgm i ( A )) stays bounded.

Example 3.4. Consider two parallel line segments M in R 2 , and a finite set A consisting of two parallel grids of step 2 ε : the set A is (2 ε, ε ) -dense in M . Then, there are O ( ε -1 ) points in dgm 1 ( A ) with birth coordinates u 1 equal to 1 / 2 and persistence of order O ( ε 2 ) ; they all belong to dgm (3) 1 ( A ) , whose cardinality is thus not bounded by some C 0 = C 0 ( M ) . This example can be easily modified to make M a compact C 2 1 -dimensional manifold. This shows that the first conclusion of Theorem 3.3 cannot hold without a genericity assumption on M .

4 Random samplings of submanifolds

We now turn to the case of random samplings of (non-generic and generic) submanifolds. They tend to adopt configurations that are more regular than what can be expected from e.g. a general ε -dense sampling, yet their randomness gives rise to new technical difficulties. Let P be a probability measure having a density f (with respect to the volume measure) on a compact submanifold M of dimension m ≥ 1 . Let A = A n = { X 1 , . . . , X n } , where X 1 , . . . , X n is an i.i.d. sample from distribution P . Let i ≥ 0 be an integer; we consider the three regions described in Figure 2 and in the statement of Theorem 2.2, and write again dgm (1) i ( A n ) , dgm (2) i ( A n ) and dgm (3) i ( A n ) for the three corresponding PDs. This section is devoted to the study of the probabilistic asymptotic behaviour of these three random PDs, which can be decomposed into two almost independent questions: dgm (1) i ( A n ) only depends on small-scale phenomena and can essentially be reduced to the case of a cube, even if M is non-generic, while dgm (2) i ( A n ) and dgm (3) i ( A n ) are tightly connected to the macroscopic geometry of the submanifold and can be further controlled using genericity assumptions on M .

Describing the limit behavior of the random PD dgm (1) i ( A n ) requires extending the metric OT p between PDs to more general Radon measures. Indeed, a PD can equivalently be seen as an integervalued discrete Radon measure on Ω , by identifying a multiset a with the Radon measure ∑ u ∈ a δ u . Let M denote the space of Radon measures on Ω , that is the space of Borel measures on Ω which give finite mass to every compact set K ⊂ Ω . 4 The space of Radon measures is endowed with the vague topology , where a sequence ( µ n ) n of measures in M is said to converge vaguely to µ ∈ M if ∫ Ω ϕ d µ n → ∫ Ω ϕ d µ as n →∞ for all continuous functions ϕ : Ω → R with compact support.

The α -total persistence is defined for µ ∈ M by Pers α ( µ ) = ∫ Ω pers( u ) α d µ ( u ) . For p ≥ 1 , we let M p = { µ ∈ M : Pers p ( µ ) < + ∞} . The distance OT p , defined between PDs in (3), can be extended to the set M p , see [33]. The distance OT p between Radon measures is a variation of the Wasserstein distance, with the important difference that the standard Wasserstein distance is only defined for measures having the same mass, while the distance OT p is defined for measures having different (and even infinite) masses. We refer to Appendix F for the precise definition and the main properties of the OT p distance on the space M p .

For q > 0 , given a function f : N → R and a sequence of (nonnecessarily measurable) real maps ( Y n ) n defined on some probabilisitic space, the notation Y n = O L q ( f ( n )) means that E ∗ [ | Y n | q ] = O ( f ( n ) q ) , where E ∗ denotes the outer expectation [73, p.6] (and similarly for the little o notation).

4.1 Region (1)

Consider the rescaled Radon measure µ n,i = 1 n ∑ u ∈ dgm (1) i ( A n ) δ n 1 /m u , and note that µ n,i is a random measure, owing to the randomness of the set A n . Goel, Trinh and Tsunoda studied the vague convergence of the sequence ( µ n,i ) n [42, Remark 4.2]. Namely, assuming that the density f satisfies ∫ M f j < ∞ for all j ≥ 0 , they show that with probability 1 the sequence ( µ n,i ) n converges vaguely to some (non-random) Radon measure µ f,i . The limit measure µ f,i has support { 0 } × R + if i = 0 and Ω if 0 < i < m ; it is the zero measure if i ≥ m . We can further describe it as follows: let µ ∞ ,i,m be the limit of the sequence ( µ n,i ) in the case where the sample A n is uniform on the unit cube [0 , 1] m (see [34]). Then, for any continuous function ϕ : Ω → R with compact support,

Note that the vague convergence of Radon measures is only defined with respect to compactly supported functions; as such, it is blind to phenomena located increasingly close to the diagonal ∂ Ω as n goes to infinity. In particular, and except in the case of the uniform distribution on the unit cube [0 , 1] m (see [34]), it was not known whether µ n,i converges to µ f,i for the OT p distance as well, nor whether the sequences of total persistence (Pers α ( µ n,i )) converge. We close this gap with the following result.

Theorem 4.1 (Law of large numbers) . Assume that P has a density f on M bounded away from 0 and ∞ . Let i ≥ 0 be an integer and let 1 ≤ p < ∞ . Then µ f,i ∈ M p and E [OT p p ( µ n,i , µ f,i )] - - - - → n →∞ 0 .

Furthermore, for all α > 0 , Pers α (dgm (1) i ( A n )) n α m -1 = Pers α ( µ n,i ) = Pers α ( µ f,i ) + o L 1 (1) .

4.2 Regions (2)-(3)

It is a well-known fact that the Hausdorff distance ε = d H ( A n , M ) between a random sample and M is of order (log n/n ) 1 /m whenever the underlying density f is bounded away from zero and ∞ on M , see e.g. [38]. Hence the PDs dgm (2) i ( A n ) and dgm (3) i ( A n ) can be described using Theorem 2.2. However, in the case where M is a generic submanifold, one can actually obtain tighter results. We let # E denote the cardinality of a multiset E .

Proposition 4.2. Let M be a generic m -dimensional submanifold. Assume that P has a density f on M bounded away from 0 and ∞ . Let i ≥ 0 be an integer. There exists an optimal matching γ n : dgm i ( A n ) ∪ ∂ Ω → dgm i ( M ) ∪ ∂ Ω for the bottleneck distance between dgm i ( A n ) and dgm i ( M ) such that for any q ≥ 1 :

Region (2): It holds that max u ∈ dgm (2) i ( A n ) | u 2 -γ n ( u ) 2 | = O L q ( n -2 /m ) .

Region (3): It holds that max u ∈ dgm (3) i ( A n ) ∥ u -γ n ( u ) ∥ ∞ = O L q ( n -2 /m ) and #(dgm (3) i ( A n )) = O L q (1) .

We remark that the same bounds can be obtained almost surely (e.g. 'a.s. there exists C > 0 such that max u ∈ dgm (3) i ( A n ) ∥ u -γ n ( u ) ∥ ∞ ≤ Cn -2 /m '), rather than in expectation, using similar arguments.

Proposition 4.2 yields two distinct improvements upon direct applications of Theorem 2.2 and Theorem 3.3 to the random case. First, we obtain bounds of order n -2 /m instead of bounds of order (log n/n ) 2 /m . Second, the random sample A n is in general only δ -sparse for δ of order n -2 /m . Hence, A n is ( δ, ε ) -dense in M , but with a diverging ratio ε/δ . Therefore, Theorem 3.3 cannot be applied to control the total number of points in dgm i ( A n ) in Region (3).

4.3 Consequences for the Wasserstein convergence of persistence diagrams

As a simple consequence of Theorem 4.1 and Proposition 4.2, we obtain that for i < m , the p -Wasserstein convergence of (dgm i ( A n )) to dgm i ( M ) holds if and only if p > m , as well as precise asymptotics for the total persistence of (dgm i ( A n )) .

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Corollary 4.3. Let p ≥ 1 and let 0 ≤ i < d be an integer. Under the same assumptions as in Proposition 4.2, the following holds:

If p > m , then E [OT p p (dgm i ( A n ) , dgm i ( M ))] → 0 as n →∞ .

If p = m , E [OT p p (dgm i ( A n ) , dgm i ( M ))] → Pers p ( µ ∞ ,i,m )Vol( M ) as n → ∞ , where Vol( M ) is the volume of M .

If p < m and i < m , then E [OT p p (dgm i ( A n ) , dgm i ( M ))] → + ∞ as n →∞ .

Furthermore, for all α > 0 , Pers α (dgm i ( A n )) is equal to

As noted earlier, both dgm i ( A n ) and dgm i ( M ) are trivial if i ≥ d .

This corollary gives a precise answer to the questions raised in the introduction. First, when A n is a random subset of a m -dimensional generic manifold in R d , the α -total persistence of dgm i ( A n ) is not only bounded for α > d (as was shown by Cohen-Steiner & al. [27]), but for all α ≥ m . Moreover, the sequence dgm i ( A n ) converges for the OT p distance if p > m . Acurious phenomenon can be observed in the case p = m : the sequence does not converge to dgm i ( M ) as one would expect, but its distance to the power p to dgm i ( M ) converges to some constant-in that case, the cost to the power p of matching all the points in Region (1) to the diagonal ∂ Ω neither converges to 0 nor diverges, but is asymptotically equal to this constant.

Using these bounds on the total persistence, we obtain regularity guarantees for a large family of feature maps, called linear feature maps , which includes feature maps introduced in [23, 4, 60, 25, 54, 70, 19]. Let ( V, ∥· ∥ ) be a normed vector space, and let ϕ : Ω → V be a continuous bounded map. For α ≥ 0 , the linear feature map Φ α associated to ϕ and defined on the space D f of PDs having a finite number of points is defined for all a ∈ D f by Φ α ( a ) = ∑ u ∈ a pers( u ) α ϕ ( u ) ∈ V .

Corollary 4.4. Let α ≥ 1 and let 0 ≤ i < d be an integer. Under the same assumptions as in Proposition 4.2, it holds that Φ α (dgm i ( A n )) converges in probability to Φ α (dgm i ( M )) whenever α > m .

Remark that other weighting schemes are possible. For instance, [53] argued for using linear feature maps of the form Φ α ( a ) = ∑ u ∈ a arctan(pers( u ) α ) ϕ ( u ) . Similar results would hold for such feature maps, as the map u ↦→ arctan(pers( u ) α ) ϕ ( u ) / pers( u ) α is continuous and bounded whenever ϕ is.

5 Numerical experiments

We illustrate our results with synthetic experiments, full details are given in Appendix H. We create a generic submanifold of dimension m by applying a random diffeomorphism Ψ to a given m -dimensional submanifold M 0 (e.g. a torus). We then draw a sample of n i.i.d. observations sampled according to the pushforward P of the uniform distribution on M by Ψ .

102

101

102

101

101

10-1

p=1

100

p=2

100

10-3

p=3

10-1

p=4

10-1

10

-2

102

103

104

102

103

104

102

103

104

Continuity of feature maps. As a first experiment, we test the continuity of a feature map, the persistence image [4]. In Figure 4, we plot the persistence image of dgm 1 ( A n ) where A n is a sample of n = 10 4 points on a generic torus. We observe that the map is discontinuous for p < 2 : the two points with large persistence corresponding to the PD of the underlying torus are nonapparent in the image. For p > 2 , the two points are apparent, and the contribution of points with small persistence (close to the lower edge) has vanished. In the limit situation p = 2 , we see the contribution of both points with large and small persistence. This phenomenon suggests the following heuristics: when in presence of multiple datasets on m -dimensional objects whose global geometries need to be distinguished, feature maps with weights pers p with p > m should be used; when the relevant information is the underlying density of the datasets, the choice p < m should be preferred .

Convergence of total persistences. We verify the rate of convergence of the total persistence predicted by Theorem 4.1. For values of n ranging from 10 2 to 10 4 , we compute Pers p (dgm (1) i ( A n )) in three scenarios: points sampled on a circle for i = 0 , and points sampled on a torus for i = 0 and i = 1 . The correct rates of convergence are observed on a log-log plot, see Figure 5. For i = 1 , we remark that the asymptotic regime starts at larger values of n , above n = 10 3 .

Convergence of µ n,i . We sample n points on a torus by uniformly sampling the two angles ( θ, ϕ ) parametrizing the torus. We obtain a (nonnuniform) probability measure, having density f . We then compute, for various values of n , the measure µ n, 1 . The measure is approximated by kernel density estimation (see Figure 6). We approximate in a similar manner the measure µ ∞ , 1 , 2 by sampling n = 10 5 points on a square. We then apply the change of variable formula (9) to compute the theoretical limit µ f, 1 . The distance OT 2 ( µ n, 1 , µ f, 1 ) is then computed by approximating the measures on a grid: the distance converges to 0 as predicted by Theorem 4.1. See also Figure 6.

16

16

0.10

OT2(μn,i,μf,i)/OT2(O,μf,i)

14

14

0.08

12

12

Death

10

Death

10

0.06

8

8

0.04

6

6

4

4

0.02

2

2

0.00

0

0

0

5

10

15

0

5

10

15

0

20000400006000080000100000

n

Birth

Birth

6 Conclusion

Under the manifold hypothesis, we have greatly refined earlier work regarding the persistent homology of subsamples of compact sets, with especially strong results when the sampling is either random or well-behaved. In particular, we have precisely described the PDs of such samplings, and provided new convergence guarantees w.r.t. the p -Wassertein distances, as well as detailed asymptotics for their total α -persistence. This results in a deeper understanding of these objects, which play an important role in ML techniques applied to TDA. The main limitations of our work were the assumptions that the data is sampled from a submanifold, and without any noise. Relaxing those assumptions, as well as establishing similar guarantees for Vietoris-Rips complexes, could be the subject of future research. We also plan on exploring the consequences of our findings regarding the persistent homology dimension [3, 66, 9, 65] of submanifolds.

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Appendices

A Outline of the appendix

We recall some basic definitions regarding persistent homology in Appendix B. In Appendix C, we define topological Morse functions and discuss some of the properties of distance functions to compact sets. We also detail the genericity conditions from [6], and prove Proposition 3.2. We prove Lemma 2.1 and Theorem 2.2 in Appendix D, and Theorem 3.3 in Appendix E. We rigorously define the partial optimal transport distance OT p between Radon measures on Ω in Appendix F, before proving Theorem 4.1, Proposition 4.2, Corollary 4.3 and Corollary 4.4 in Appendix G. Finally, we provide some details on our experimental setup in Appendix H. Throughout the appendices, we write C = C ( a, b, . . . ) to implicitly state that a newly introduced constant C depends only on some objects a, b, . . . (e.g. C = C ( M ) ).

B Persistent homology

Although this work is only concerned with PDs with respect to the ˇ Cech filtration, it is more natural to define PDs for the sublevel sets of proper continuous functions that are bounded below. We refer to [22] for a thorough introduction to persistent homology and PDs in an even more general context.

Let f : R d → R be a proper continuous function that is bounded below-e.g., the distance function to a compact set. For t ∈ R , let X t = f -1 ( -∞ , t ] be the sublevel set of f at level t . The collection ( X t ) t ∈ R is called a filtration. Let i ≥ 0 be an integer. We let H i ( X t ) be the homology group of degree i with coefficients in any fixed field F (e.g. F = Z / 2 Z is a popular choice) of X t . For r < s , the inclusion between the sublevel sets at levels r and s induces a map ι i,r,s at the homology level. We define the persistent Betti number

As shown in [22, Corollary 3.34], this number is finite. Informally, it represents the number of i th dimensional topological features present in the sublevel set at level r that are still present at level s .

Define the extended half-plane Ω ∞ = { u = ( u 1 , u 2 ) ∈ ( R ∪ {-∞ , + ∞} ) 2 : -∞ ≤ u 1 < u 2 ≤ + ∞} and Ω = { u = ( u 1 , u 2 ) ∈ R 2 : u 1 < u 2 } . The collection of persistent Betti numbers ( β i,r,s ( f )) r<s defines a multiset of points in Ω ∞ , called the persistence diagram dgm i ( f ) of degree i of f . See e.g. [22] for its precise definition.

Though persistence diagrams can have points with infinite coordinates, these will be of little interest in the cases considered in this article 5 . To simplify our notation and definitions, we let from now on

dgm i ( f ) denote the finite part (i.e. the points whose coordinates are finite) of the diagram of degree i of f , and we assume that every diagram considered henceforth has no point with infinite coordinates. In particular, they are all multisets of the half-plane Ω := { u = ( u 1 , u 2 ) ∈ R 2 : u 1 < u 2 } , which leads to the following definition: the space D of persistence diagrams is the set of all multisets a in Ω that contain a finite number of points 6 in every quadrant Q u , u ∈ Ω .

Note that while persistence diagrams capture key topological features in an interpretable fashion, they are not complete metric invariants: different sets can have equal persistence diagrams, see e.g. [69]. As such, they can be seen as a form of topology-centered dimensionality reduction technique. Other topological invariants have been proposed, some of which are complete, such as those from [77, 52].

C Distance functions, topological Morse functions and generic submanifolds

In this section, we consider a special class of PDs -those yielded by the distance function to a compact set A ⊂ R d . We give a detailed description of such a PD when A = M is a generic submanifold.

The theory of persistent homology was historically developed for Morse functions f : R d → R . A Morse function f is a C 2 function whose critical points x (points for which d x f = 0 ) are nondegenerate (meaning that the Hessian of f at x is non-degenerate). The index of the critical point x is equal to the number of negative eigenvalues of the corresponding Hessian. The changes of topology of the sublevel sets of such a function are perfectly understood. First, the isotopy lemma states that two sublevel sets f -1 ( -∞ , u 1 ] and f -1 ( -∞ , u 2 ] are isotopic if no critical values are found in the interval [ u 1 , u 2 ] and if f -1 [ u 1 , u 2 ] is compact. Second, if f -1 [ u 1 , u 2 ] is compact and contains the critical points x 1 , . . . , x K , then f -1 ( -∞ , u 2 ] has the homotopy type of f -1 ( -∞ , u 1 ] with cells e k of dimension equal to the index of x k attached along their boundaries (see e.g. [56] for a much more in-depth treatment).

In such a situation, the PD dgm i ( f ) has a clear interpretation: the coordinates ( u 1 , u 2 ) of a point u ∈ dgm i ( f ) correspond to the critical value of a critical point of index i and i +1 respectively. Informally, the corresponding i -dimensional topological feature appears with the attachment of a i -dimensional cell at value u 1 , and is 'killed' by the attachment of a ( i +1) -dimensional cell at value u 2 .

The notion of Morse function extends to continuous functions with the following definition.

Definition C.1 (Topological Morse functions [58]) . Let U ⊂ R d be an open set and let f : U → R be a continuous function.

A point z ∈ U is said to be a topological regular point of f if there is a homeomorphism ϕ : V 1 → V 2 between open neighborhoods V 1 of 0 in R d and V 2 of U in R d with ϕ (0) = z and such that for all x = ( x 1 , . . . , x d ) ∈ V 1 ,

A point z ∈ U is said to be a topological critical point of f if it is not a topological regular point of f .

A point z ∈ U is said to be a non-degenerate topological critical point of f of index i if there exist an integer 0 ≤ i ≤ d and a homeomorphism ϕ : V 1 → U 2 between open neighborhoods V 1 of 0 in R d and V 2 of U in R d with ϕ (0) = z such that for all x = ( x 1 , . . . , x d ) ∈ V 1 ,

The function f is said to be a topological Morse function if all its topological critical points are non-degenerate.

For topological Morse functions, both the isotopy lemma and the handle attachment lemma stay valid:

Lemma C.2 (Isotopy Lemma) . Let f : R d → R be a proper topological Morse function. Let a < b be such that f -1 [ a, b ] contains no topological critical point. Then f -1 ( -∞ , a ] is a deformation retract of f -1 ( -∞ , b ] .

Lemma C.3 (Handle Attachment Lemma) . Let f : R d → R be a proper topological Morse function. Let c ∈ R and ε > 0 be such that f -1 [ c -ε, c + ε ] contains no topological critical point except for z 1 , . . . , z k ∈ f -1 ( c ) , with z j of index i j . Then f -1 ( -∞ , c + ε ] is homotopically equivalent to f -1 ( -∞ , c -ε ] with cells B i 1 , . . . , B i k of dimension i 1 , . . . , i k attached, i.e.

Their proofs are roughly the same as for smooth Morse functions -see [71, Theorems 4 and 5] for details. As a consequence, the description of PDs for Morse functions also stays valid for topological Morse functions.

In this paper, we are interested in the PDs of the distance function d A to various compact sets A , called the ˇ Cech persistence diagram 7 of A and denoted by dgm i ( A ) . In general, such a function is not a topological Morse function. Instead, changes in the topology of its sublevels can be partially (though less completely than for a Morse function) described in terms of zeros of its generalized gradient, which is defined at y ∈ R d \ A as

where σ A ( y ) = { x ∈ A : ∥ x -y ∥ = d A ( y ) } is the set of projections of y on A and c ( τ ) represents the center of the smallest enclosing ball of a set τ . When y ∈ R d \ A satisfies ∇ d A ( y ) = 0 , y is called a differential critical point of d A . We let Crit( A ) denote the set of differential critical points of d A . An adapted version of the Isotopy Lemma remains true, as shown in [43]:

Lemma C.4 (Isotopy Lemma for Distance Functions) . If 0 < a < b are such that d -1 A [ a, b ] contains no differential critical point of d A , then d -1 A ( -∞ , a ] is a deformation retract of d -1 A ( -∞ , b ] . Consequently, any ( u 1 , u 2 ) ∈ dgm i ( A ) is such that u 1 , u 2 ̸∈ [ a, b ] .

Without further assumptions, little else can be said regarding the topology of the sublevels of d A ; in particular, there is no equivalent to the Handle Attachment Lemma to control the changes occurring at critical values.

However, the distance function d M to a compact C 2 submanifold M ⊂ R d turns out to be a topological Morse function in a 'generic' sense, as was proven by Arnal, Cohen-Steiner and Divol.

Theorem C.5 (Genericity Theorem [6]) . Let M be a compact C 2 (abstract) manifold. Then the set of C 2 embeddings i : M → R d such that

the distance function d i ( M ) : R d \ i ( M ) → R is a topological Morse function,

for every z ∈ Crit( M ) , the projections σ M ( z ) are the vertices of a non-degenerate simplex of R d and z belongs to its relative interior,

the set Crit( M ) is finite,

for every z ∈ Crit( M ) and every x ∈ σ M ( z ) , the sphere S ( z, d M ( z )) is non-osculating M at x , in the sense that there exist δ > 0 and α > 0 such that for all y ∈ M ∩ B ( x, δ ) ,

there exist constants C > 0 and µ 0 ∈ (0 , 1) such that for every µ ∈ [0 , µ 0 ) , any point x such that ∥∇ d M ( x ) ∥ ≤ µ is at distance at most Cµ from Crit( M ) ,

is open and dense in the set of C 2 embeddings M → R d for the Whitney C 2 -topology.

When M is generic, the topological critical points of d M coincide with its differential critical points (see [6, Theorem 1.8]), and the ˇ Cech PD dgm i ( M ) can be related to the critical points of d M in the same way as for smooth Morse function. We are now in position to prove Proposition 3.2, which we restate for the reader's convenience:

Proposition 3.2 ( ˇ Cech PDs of generic submanifolds) . Let M be a generic submanifold of R d . Then, for any integer i ≥ 0 , the PD dgm i ( M ) contains finitely many points. In particular, Pers α (dgm i ( M )) < + ∞ for all α > 0 .

Proof. The proof is similar to that used in the case of smooth Morse functions, with the Isotopy Lemma and the Handle Attachment Lemma playing the same role; we briefly summarize it for completeness nonetheless. We do not distinguish between differential and topological critical values and points, as they coincide.

The set Crit( M ) is finite: this is simply Condition 3. from the Genericity Theorem. The Isotopy Lemma shows that there is no change in homotopy type between M a and M b if 0 < a < b and [ a, b ] contains no critical value. Similarly, M has the same homotopy type as M a if a > 0 is small enough. Hence changes in homology in the offsets can only occur at 0 , when the entire submanifold appears in the filtration, and at critical values of d M , and there can be no birth or death of interval between them.

Let us now consider a critical value c > 0 and 0 < ε < c such that d -1 M [ c -ε, c + ε ] contains no critical point except for z 1 , . . . , z k ∈ d -1 M ( c ) , where z j is of index i j . Then the Handle Attachment Lemma states that M c + ε is homotopically equivalent to M c -ε with cells B i 1 , . . . , B i k of dimension i 1 , . . . , i k attached, i.e.

Let i ≥ 1 , and let D i,b be the dimension of the cokernel of H i ( M c -ε ) → H i ( M c + ε ) (where the map is induced by the inclusion): it is precisely the number of births of intervals between c -ε and c + ε (hence precisely at c ) in the persistence module of degree i of the filtration. Similarly, the dimension D i -1 ,d of the kernel of H i -1 ( M c -ε ) → H i -1 ( M c + ε ) is the number of deaths of intervals at c in the persistence module of degree i -1 of the filtration. A straightforward application of the Mayer-Vietoris exact sequence yields that D i,b + D i -1 ,d is exactly equal to the number of i -dimensional cells among B i 1 , . . . , B i k , meaning that each i -cell corresponds exactly either to the birth of an interval for the homology of degree i , or to the death of an interval for the homology of degree i -1 (in particular, an i -1 -cell and an i -cell cannot 'cancel each other out'). This proves that for any i ≥ 1 , the multiset of critical values d M ( z ) of critical points z of M of index i is equal to the multiset

̸

This fact proves in turn that dgm i ( M ) is finite for any i ≥ 0 , which immediately implies that Pers α (dgm i ( M )) < + ∞ for all α > 0 .

D Proofs of Section 2

For M a m -dimensional differential submanifold of R d and x ∈ M , we let T x M be the tangent space of M at x , which is identified with a linear subspace of R d . We denote by π x : R d → T x M the orthogonal projection on this subspace and let π ⊥ x = id -π x be the orthogonal projection on the normal space at x . A key property, that we will repeatedly used, is that the reach of M controls the deviation of the manifold M from its tangent space. Namely, [39, Theorem 4.18] states that for all y ∈ M ,

We also define the weak feature size of M , denoted by wfs( M ) , as the minimal distance between a critical point of M and M . As by definition, the projection is not unique at a critical point, we must have wfs( M ) ≥ τ ( M ) .

We first prove Lemma 2.1:

Lemma 2.1. Let M ⊂ R d be a compact submanifold with positive reach and let A ⊂ M be a compact set with d H ( A , M ) ≤ ε for some ε > 0 . Let z ∈ R d \ M . Then, | d M ( z ) -d A ( z ) | ≤ ε 2 2 d M ( z ) ( 1 + d M ( z ) τ ( M ) ) .

Proof. As A ⊂ M , we have d M ( z ) ≤ d A ( z ) . Let x be a projection of z onto M , and let y ∈ A be a point at distance less than ε from x . Then, using (16) and the fact that z -x is orthogonal to T x M , we obtain that

Hence,

We can now prove Theorem 2.2:

Theorem 2.2 (Improved Bottleneck Stability Theorem) . Let M ⊂ R d be a compact submanifold with positive reach and let A ⊂ M be a compact set such that d H ( A , M ) ≤ ε < τ ( M ) / 4 . Let i ≥ 0 be an integer. Then dgm i ( A ) is the union of three regions dgm (1) i ( A ) := dgm i ( A ) ∩{ u 1 , u 2 ≤ ε + ε 2 τ ( M ) } , dgm (2) i ( A ) := dgm i ( A ) ∩ { u 1 ≤ ε, u 2 ≥ τ ( M ) -ε 2 τ ( M ) } and dgm (3) i ( A ) := dgm i ( A ) ∩ { u 1 , u 2 ≥ τ ( M ) -ε 2 τ ( M ) } .

Furthermore, let C = 2 τ ( M ) ( 1 + R ( M ) τ ( M ) ) , where R ( M ) is the radius of the smallest ball that contains M . There exists an optimal matching γ : dgm i ( A ) ∪ ∂ Ω → dgm i ( M ) ∪ ∂ Ω for the bottleneck distance between dgm i ( A ) and dgm i ( M ) such that

Region (1): If u ∈ dgm (1) i ( A ) , then γ ( u ) ∈ ∂ Ω and ∥ u -γ ( u ) ∥ ∞ ≤ ε .

Region (2): If u ∈ dgm (2) i ( A ) , then γ ( u ) is of the form (0 , v 2 ) and | u 2 -v 2 | ≤ Cε 2 . The number of such points is finite and depends only on M .

Region (3): If u ∈ dgm (3) i ( A ) , then ∥ u -γ ( u ) ∥ ∞ ≤ Cε 2 .

Proof. Proposition 5 from [7] states that if ε = d H ( A , M ) < ( √ 2 -1) τ ( M ) , then the offset A r deformation-retracts onto M for any

and ∆ = τ ( M ) 2 -2 ετ ( M ) -ε 2 . Under the stronger assumption that d H ( A , M ) < τ ( M ) / 4 , and using elementary calculus, we find that the offset A r deformation-retracts onto M for any

This means in particular that the homology type, hence the homology, of A r does not change in that interval; as a result, there can be no birth or death of intervals in the ˇ Cech persistence diagrams of A between ε + ε 2 τ ( M ) and τ ( M ) -ε 2 τ ( M ) , and all ( u 1 , u 2 ) ∈ dgm i ( A ) must either be such that u 1 , u 2 ≤ ε + ε 2 τ ( M ) , or u 1 ≤ ε + ε 2 τ ( M ) , u 2 ≥ τ ( M ) -ε 2 τ ( M ) , or u 1 , u 2 ≥ τ ( M ) -ε 2 τ ( M ) . This almost proves that the partition of dgm i ( A ) into dgm (1) i ( A ) , dgm (2) i ( A ) and dgm (3) i ( A ) as defined in the statement is correct, except that the definition of dgm (2) i ( A ) requires that u 1 ≤ ε , whereas we only have obtained that u 1 ≤ ε + ε 2 τ ( M ) .

Let γ : dgm i ( A ) ∪ ∂ Ω → dgm i ( M ) ∪ ∂ Ω be any optimal matching for the bottleneck distance: then the Bottleneck Stability Theorem states that any point u = ( u 1 , u 2 ) ∈ dgm i ( A ) is such that ∥ u -γ ( u ) ∥ ∞ ≤ d H ( A , M ) ≤ ε . Suppose that u is such that u 1 ≤ ε + ε 2 τ ( M ) and u 2 ≥ τ ( M ) -ε 2 τ ( M ) .

Its distance in the infinity norm to the diagonal ∂ Ω is equal to ( u 2 -u 1 ) / 2 ≥ ( τ ( M ) -ε -2 ε 2 /τ ( M )) / 2 ≥ 5 16 τ ( M ) > ε , where we use the fact that ε < τ ( M ) / 4 . Hence γ ( u ) must belong to dgm i ( M ) . Let ( v 1 , v 2 ) denote γ ( u ) . As in the proof of Proposition 3.2, the Isotopy Lemma shows that dgm i ( M ) only contains two types of points: points of the shape (0 , w 2 ) , which correspond to the homology of M itself, and points of the shape ( w 1 , w 2 ) , where in both cases w 1 , w 2 are critical values of d M . In particular, both w 1 and w 2 must be greater than wfs( M ) . Let dgm (2) i ( M ) denote the multiset of all points of the first type, and dgm (3) i ( M ) denote the multiset of all points of the second type. If v 1 was non-zero, it would have to be greater than wfs( M ) ≥ τ ( M ) , and we would have ∥ u -γ ( u ) ∥ ∞ ≥ | u 1 -v 1 | ≥ τ ( M ) -ε -ε 2 τ ( M ) ≥ τ ( M ) / 2 > ε , which would be a contradiction (we once again use that ε < τ ( M ) / 4 ). Hence v 1 must be 0 (i.e. γ ( u ) ∈ dgm (2) i ( M ) ), and u 1 = | u 1 -v 1 | ≤ ∥ u -γ ( u ) ∥ ∞ ≤ ε . This proves the correctness of the partition into regions from the statement.

Consider now u = ( u 1 , u 2 ) ∈ dgm (1) i ( A ) . All v = ( v 1 , v 2 ) ∈ dgm i ( M ) are such that

hence γ ( u ) must belong to ∂ Ω . This completes the proof of the first bullet point of the statement.

We have already shown that if γ is an optimal matching and u ∈ dgm (2) i ( A ) , then γ ( u ) ∈ dgm i ( M ) is of the form (0 , v 2 ) . As γ maps at most a single point of dgm (2) i ( A ) to each point of dgm i ( M ) , the number of such points is upper bounded by the number of points of the form (0 , v 2 ) with v 2 ≥ wfs( M ) in dgm i ( M ) . Though dgm i ( M ) need not be finite ( M is not assumed to be generic), applying Corollary 3.34 from [22] to d M shows that dgm i ( M ) is q-tame, and in particular that it contains only a finite number N of points ( v 1 , v 2 ) with v 1 ≤ wfs( M ) / 4 and v 2 ≥ wfs( M ) / 2 . Hence the cardinality of dgm (2) i ( A ) is bounded by N .

It only remains to show that γ can be chosen such that if u = ( u 1 , u 2 ) ∈ dgm (2) i ( A ) , respectively u ′ ∈ dgm (3) i ( A ) , then | u 2 -γ ( u ) 2 | ≤ Cε 2 , respectively ∥ u ′ -γ ( u ′ ) ∥ ∞ ≤ Cε 2 . To that end, remember first that as shown above, our starting optimal matching γ must be such that points in dgm (2) i ( A ) must be matched to points dgm (2) i ( M ) . Conversely and for the same reasons, points in dgm (2) i ( M ) must be matched to points in dgm (2) i ( A ) . Similarly, points u ∈ dgm (3) i ( A ) can only be matched to points in dgm (3) i ( M ) or to the diagonal ∂ Ω ; otherwise, ∥ u -γ ( u ) ∥ ∞ ≥ | u 1 -γ ( u ) 1 | = u 1 would be too large. Likewise, points in dgm (3) i ( M ) can only be matched to points in dgm (3) i ( A ) or to the diagonal. Hence γ defines disjoint submatchings

and

Now let R ( M ) be the radius of the smallest ball that contains M , and consider the functions

and

Let us compare the persistence diagrams dgm i ( a ) and dgm i ( m ) of the sublevel sets filtration of a and m and the ˇ Cech persistence diagrams dgm i ( A ) and dgm i ( M ) respectively (which are by definition the persistence diagrams of the sublevel sets filtration of d A and d M ).

Note first that d A and d M can have no critical values strictly greater than R ( M ) , as a critical point must belong to the convex hull of its projections. Note also that for any t ∈ [ τ ( M ) / 2 , R ( M )] , the sublevel set a -1 ( -∞ , t ] is exactly equal to d -1 A ( -∞ , t ] = A t . Consequently, dgm i ( a ) contains exactly two disjoint types of points. The first type are points of the form ( τ ( M ) / 2 , u 2 ) , which are in bijection with the points ( u 1 , u 2 ) ∈ dgm i ( A ) with u 1 ≤ τ ( M ) / 2 (the bijection maps ( u 1 , u 2 ) ↦→ ( τ ( M ) / 2 , u 2 ) ); those are exactly the points in dgm (2) i ( A ) . The second type are points of the form ( u 1 , u 2 ) with u 1 ≥ τ ( M ) / 2 , which are in trivial bijection (the bijection is the identity) with the points in dgm i ( A )

that satisfy the same condition; those are exactly the points of dgm (3) i ( A ) . The points of dgm (1) i ( A ) cannot be 'seen' in dgm i ( a ) . We will call dgm (2) i ( a ) the subdiagram comprised of the points of the first type, and dgm (3) i ( a ) the subdiagram of dgm i ( a ) comprised of the points of the second type.

Similarly, dgm i ( m ) contains two types of points: the first type are points of the form ( τ ( M ) / 2 , v 2 ) , which are in bijection with the points (0 , v 2 ) ∈ dgm i ( M ) (the bijection maps (0 , v 2 ) ↦→ ( τ ( M ) / 2 , v 2 ) ); those are exactly the points in dgm (2) i ( M ) . The second type are points of the form ( v 1 , v 2 ) with v 1 ≥ τ ( M ) / 2 , which are in trivial bijection (the bijection is the identity) with the points in dgm i ( M ) that satisfy the same condition; those are exactly the points of dgm (3) i ( M ) . We will call dgm (2) i ( m ) the subdiagram of dgm i ( m ) comprised of the points of the first type, and dgm (3) i ( m ) the subdiagram comprised of the points of the second type.

Recall that Lemma 2.1 states that | d M ( z ) -d A ( z ) | ≤ ε 2 2 d M ( z ) ( 1 + d M ( z ) τ ( M ) ) for any z ∈ R d \ M . If z ∈ R d is such that d A ( z ) ≤ τ ( M ) / 2 , then a ( z ) = m ( z ) = τ ( M ) / 2 ; if it is such that d M ( z ) ≥ R ( M ) , then a ( z ) = m ( z ) = R ( M ) . Otherwise, d M ( z ) ≥ d A ( z ) -d H ( A , M ) ≥ τ ( M ) / 4 and d M ( z ) ≤ R ( M ) , hence

where C is as defined in the proposition. This means that

Due to the Bottleneck Stability Theorem, the diagrams dgm i ( a ) and dgm i ( m ) must be at bottleneck distance less than Cε 2 .

Furthermore, let δ denote max u ∈ dgm i ( A ) ∪ ∂ Ω ∥ u -γ ( u ) ∥ ∞ . The matching γ (and in particular the submatchings γ (2) and γ (3) ) also induces (through the correspondence detailed above between the points of dgm i ( a ) and a subset of the points of dgm i ( A ) ) a matching γ ′ between dgm i ( a ) and dgm i ( m ) such that max u ∈ dgm i ( a ) ∪ ∂ Ω ∥ u -γ ′ ( u ) ∥ ∞ ≤ δ . Hence the bottleneck distance between dgm i ( a ) and dgm i ( m ) is at most min( δ, Cε 2 ) .

Let β : dgm i ( a ) ∪ ∂ Ω → dgm i ( m ) ∪ ∂ Ω be an optimal matching for the bottleneck distance. For similar reasons as for γ , the matching β can also be decomposed into two disjoint submatchings

and

We can use the two matchings β (2) and β (3) to define a new optimal matching ˜ γ : dgm i ( A ) ∪ ∂ Ω → dgm i ( M ) ∪ ∂ Ω as follows:

The points in dgm (1) i ( A ) are matched by ˜ γ to ∂ Ω as with γ .

Given u = ( u 1 , u 2 ) ∈ dgm (2) i ( A ) , let u ′ = ( τ ( M ) / 2 , u 2 ) be the point of dgm (2) i ( a ) with which u is in bijection. We let ˜ γ match u with the point v = (0 , v 2 ) ∈ dgm (2) i ( M ) which is in bijection with β (2) ( u ′ ) = ( τ ( M ) / 2 , v 2 ) ∈ dgm (2) i ( m ) . Then | u 2 -v 2 | ≤ min ( δ, Cε 2 ) due to the optimality of β , and this defines a bijective matching ˜ γ (2) : dgm (2) i ( A ) → dgm (2) i ( M ) . Note also that max u ∈ dgm (2) i ( A ) | u 1 -˜ γ (2) ( u ) 1 | = max u ∈ dgm (2) i ( A ) u 1 = max u ∈ dgm (2) i ( A ) | u 1 -γ (2) ( u ) 1 | , hence max u ∈ dgm (2) i ( A ) ∥ u -˜ γ (2) ( u ) ∥ ∞ ≤ δ .

We have seen that dgm (3) i ( a ) = dgm (3) i ( A ) and dgm (3) i ( m ) = dgm (3) i ( M ) . We simply define the restriction and corestriction ˜ γ (3) : dgm (3) i ( A ) ∪ ∂ Ω → dgm (3) i ( M ) ∪ ∂ Ω of ˜ γ as being equal to β (3) : dgm (3) i ( a ) ∪ ∂ Ω → dgm (3) i ( m ) ∪ ∂ Ω . The optimality of β implies that max u ∈ dgm (3) i ( A ) ∥ u -˜ γ ( u ) ∥ ∞ ≤ min ( δ, Cε 2 ) .

Thus the global matching ˜ γ : dgm i ( A ) ∪ ∂ Ω → dgm i ( M ) ∪ ∂ Ω is well-defined, is optimal for the bottleneck distance, and satisfies the conditions stated in the proposition. This completes the proof.

E Proof of Theorem 3.3

We prove Theorem 3.3, which we restate for the reader's convenience:

Theorem 3.3. Let M ⊂ R d be a generic compact submanifold and A ⊂ M be a ( δ, ε ) -dense set in M for some ε, δ > 0 . Let a ≥ ε/δ and let i ≥ 0 be an integer. There exist ε 0 > 0 depending only on M and C 0 , C 1 , C 2 , C 3 depending only on M and a such that if ε ≤ ε 0 , then dgm (3) i ( A ) has at most C 0 points and for all p ≥ 1 , α ≥ 0 ,

Proof. Let us first prove the bound on the cardinality of dgm (3) i ( A ) . As M is generic, Theorem 1.6 from [6] states that if ε is smaller than some ε 0 = ε 0 ( M ) , then each point in Crit( A ) at distance more than τ ( M ) / 2 from A must be at distance at most K 1 ε from one of the finitely many points of Crit( M ) for some K 1 = K 1 ( M ) . Corollary 1.7 from the same article then states that the number of points in Crit( A ) at distance less than K 1 ε from a given point of Crit( M ) is upper bounded by some constant that depends on M and the ratio ε/δ , and is decreasing in this ratio; hence it is upper bounded by some constant that depends on M and a . The proof of this corollary also shows that the maximum number of projections on A of each of these points of Crit( A ) is also upper bounded by some constant that depends on M and a . Hence there exist constants K 2 = K 2 ( M , a ) and K 3 = K 3 ( M , a ) such that if ε ≤ ε 0 , then there are at most K 2 points in Crit( A ) at distance more than τ ( M ) / 2 from A , and each has at most K 3 projections on A .

Lemma E.1 below, applied to the interval [ τ ( M ) / 2 , ∞ ) and the set A , then states that the number of points in dgm i ( A ) such that at least one of their coordinates is greater than τ ( M ) / 2 is bounded by K 2 ( ( K 3 i +1 ) + ( K 3 i +2 ) ) . Hence there are at most C 0 := K 2 2 K 3 ≥ K 2 ( ( K 3 i +1 ) + ( K 3 i +2 ) ) points in dgm (3) i ( A ) when ε ≤ ε 0 .

Now let us prove the bounds on OT p p (dgm i ( A ) , dgm i ( M )) and Pers α (dgm i ( A )) . As stated in Theorem 2.2, which applies as ε 0 < τ ( M ) / 4 , each point u = ( u 1 , u 2 ) ∈ dgm (1) i ( A ) is such that its coordinates satisfy 0 ≤ u 1 , u 2 ≤ ε + ε 2 /τ ( M ) ≤ 2 ε . In particular, they must correspond to the birth or the death of an interval of the ˇ Cech persistence module of A that occurs before filtration time 2 ε . The homology of the offsets A t can be computed using the ˇ Cech simplicial complex of A (see e.g. [37]). In particular, each change in the homology of the offsets, hence each birth or death in the ˇ Cech persistence module of A , is induced by the apparition of some simplex σ at the corresponding filtration value in the ˇ Cech complex, and each such apparition causes at most a single death or birth. If a simplex σ appears before filtration time 2 ε , it is by definition contained in a ball of radius 2 ε , hence it is of diameter at most 4 ε . Let us assume from now on that ε 0 ≤ τ ( M ) / 16 . Consider x ∈ A ; then [2, Proposition 8.7] states that the intersection B ( x, 4 ε ) ∩ A contains at most K 4 ( ε/δ ) m ≤ K 4 a m points for some constant K 4 = K 4 ( M ) . Hence x belongs to at most 2 K 4 a m simplices that appear before ε , and there are at most # A · 2 K 4 a m such simplices. As the cardinality # A can be bounded by K 5 /δ m for some K 5 = K 5 ( M ) , we find that #(dgm (1) i ( A )) ≤ K 6 /δ m for some K 6 = K 6 ( M , a ) .

Furthermore, when M is generic, Proposition 3.2 states that its PD dgm i ( M ) has a finite number of points. Let γ be an optimal matching between dgm i ( A ) and dgm i ( M ) for the bottleneck distance that satisfies the conclusions of Theorem 2.2. We find that any point u ∈ dgm (1) i ( A ) is matched to a point of ∂ Ω at distance at most ε from u . Moreover, the number of points in dgm (2) i ( A ) ∪ dgm (3) i ( A ) is bounded by some constant K 7 = K 7 ( M , a ) , and they are all matched to a point of dgm i ( M ) or ∂ Ω at distance at most ε . In particular, these finitely many points are at distance at most K 8 = K 8 ( M ) from ∂ Ω . Furthermore, this matching is surjective, in the sense that γ matches all points of dgm i ( M ) to a point of dgm i ( A ) . As a result, for any p ≥ 1 , we find that

for some C 1 = C 1 ( M , a ) . Likewise, for any α ≥ 0 , we have that

Lemma E.1. Let A ⊂ R d be a finite set, and let a < b ∈ R ∪ {-∞ , + ∞} and i ≥ 0 . Let

be a covering of the set of critical points of d A whose critical value belongs to [ a, b ] , 8 and let

be the cardinality of the union of the projections of the critical points of C j . Then the number of points in dgm i ( A ) such that at least one of their coordinates belongs to [ a, b ] is upper-bounded by ∑ L j =1 ( N j i +1 ) + ( N j i +2 ) , hence by ∑ L j =1 N i +2 j .

Proof. It is shown in [5] that the distance function to any finite point cloud is a topological Morse function. Hence its topological critical points are in bijection (via their critical values) with the non-zero coordinates of the points in (the union over all degrees of) the ˇ Cech persistence diagrams of A . The number of points in dgm i ( A ) such that at least one of their coordinates belongs to [ a, b ] must then be upper bounded by the number of topological critical points z of topological Morse index i or i +1 such that d A ( z ) ∈ [ a, b ] . Moreover, it is also shown that the topological critical points of d A are a subset of A ∪ Crit( A ) , though not all differential critical points need be topologically critical. Hence any such z belongs to C j for some j ∈ { 1 , . . . , L } , and σ A ( z ) ⊂ ⋃ z ′ ∈ C j σ A ( z ′ ) .

As shown in [5], if z is of critical index i , then the linear span Span( σ A ( z ) -z ) is of dimension i . By Carathéodory's theorem, there exists i +1 affinely independent points in σ A ( z ) ⊂ ⋃ z ′ ∈ C j σ A ( z ′ ) such that z belongs to their convex hull. Those i +1 points uniquely identify z among the critical points of A , as it is the only point equidistant to them that belongs to their convex hull. Hence there is an injection from the topological critical points of A of index i that belong to C j into the set of subsets of cardinality i +1 of ⋃ z ′ ∈ C j σ A ( z ′ ) . Applying the same reasoning to the points of index i +1 , we find that there are at most ∑ n j =1 ( N j i +1 ) + ( N j i +2 ) points in dgm i ( A ) with at least one of their coordinates in [ a, b ] , as desired.

F The OT p distance between Radon measures

Let M denote the space of Radon measures on Ω , and let us define ¯ Ω := { u = ( u 1 , u 2 ) ∈ R 2 : u 1 ≤ u 2 } . We call π an admissible transport plan between ν 1 , ν 2 ∈ M if it is a Radon measure on ¯ Ω × ¯ Ω such that for all Borel sets A,B ⊂ Ω ,

For p ∈ [1 , + ∞ ) , we define

where ∥ u ∥ ∞ = max( | u 1 | , | u 2 | ) and Adm ( ν 1 , ν 2 ) is the set of all admissible transport plans between ν 1 and ν 2 . We also define

For all p ∈ [1 , ∞ ] , the infimum is in fact a minimum (see [33]). We call OT p ( ν 1 , ν 2 ) the p -Wasserstein distance between ν 1 and ν 2 , though it differs from the usual Wasserstein distance, which is defined between measures with equal finite mass, while OT p is defined between measures that can have different (and even infinite) masses. Intuitively, OT p allows for some of the mass of ν 1 and ν 2 to be transported to the diagonal ∂ Ω := { ( u, u ) ∈ R 2 } , which acts as an infinitely deep landfill. The p -Wasserstein distance is a distance on the space M p = { ν ∈ M : OT p ( ν, 0) < ∞} , where 0 denotes the null measure.

As explained in Section 4, a PD a can be identified with the Radon measure ∑ u ∈ a δ u . Let D p be the set of PDs being in M p . Then, Divol and Lacombe show in [33] that the OT p distance defined in (18) coincides with the OT p distance defined between PDs in Equation (3), and likewise for the case p = ∞ .

We require the following lemma from [33]:

Lemma F.1. A sequence of measures ( ν n ) n ≥ 1 converges with respect to OT p to some measure ν if and only if the sequence ( ν n ) n ≥ 1 converges vaguely towards ν and Pers p ( ν n ) → Pers p ( ν ) as n →∞ .

G Proofs of Section 4

We start with the proof Theorem 4.1, which is split into a series of lemmas, before proving Proposition 4.2, Corollary 4.3 and Corollary 4.4.

Let us restate Theorem 4.1 for the reader's convenience:

Theorem 4.1 (Law of large numbers) . Assume that P has a density f on M bounded away from 0 and ∞ . Let i ≥ 0 be an integer and let 1 ≤ p < ∞ . Then µ f,i ∈ M p and E [OT p p ( µ n,i , µ f,i )] - - - - → n →∞ 0 .

Furthermore, for all α > 0 , Pers α (dgm (1) i ( A n )) n α m -1 = Pers α ( µ n,i ) = Pers α ( µ f,i ) + o L 1 (1) .

Before proving Theorem 4.1, we state a simple lemma which allows us to control the mass of balls on M .

Lemma G.1. Let M be a compact submanifold with positive reach. Let P be a probability measure having a density f on M satisfying f min ≤ f ≤ f max for two strictly positive constants f min , f max . There exist constants c m , C m depending only on m such that for all 0 ≤ r ≤ τ ( M ) / 4

Let A n be a sample of n i.i.d. observations of law P . Then, there exists C = C ( M ) depending on M such that for all x ∈ M and all r > 0 ,

Proof. For the first statement, see [2, Proposition 31]. Let us prove the second one. Remark that the probability is zero for r > diam( M ) . Hence, we can assume that r ≤ diam( M ) . When r ≤ τ ( M ) / 4 , it holds that

When τ ( M ) / 4 ≤ r ≤ diam( M ) , we write

Hence, the result holds with C = c m min ( 1 , ( τ ( M ) / 4) m diam( M ) m ) .

Proof of Theorem 4.1. Recall that Goel, Trinh and Tsunoda [42] have shown that almost surely, the sequence of Radon measures ( µ n,i ) n vaguely converges to the Radon measure µ f,i . We start by showing that, using this vague convergence and Lemma F.1, it is enough to prove the convergence of the p -total persistence.

Lemma G.2. Let ( ν n ) n ≥ 1 be a sequence of random measures in M p that converges vaguely almost surely to a Radon measure ν ∈ M p , and such that E [ | Pers p ( ν n ) -Pers p ( ν ) | ] → n →∞ 0 . Then, E [OT p p ( ν n , ν )] → n →∞ 0 .

Proof. Let us first show that ( D n ) n ≥ 1 = (OT p p ( ν n , ν )) n ≥ 1 converges in probability to 0 . We use the following standard result: if for every subsequence ( Z n k ) k ≥ 1 of ( Z n ) n ≥ 1 , one can extract a subsequence ( Z n k l ) l ≥ 1 that converges almost surely to 0 , then the sequence ( Z n ) n ≥ 1 converges in probability to 0 . Let ( D n k ) k ≥ 1 be a subsequence of ( D n ) n ≥ 1 = (OT p p ( ν n , ν )) n ≥ 1 . Then, as (Pers p ( ν n )) n ≥ 1 converges in L 1 to Pers p ( ν ) , it also converges in probability. In particular, there exists a subsequence ( n k l ) l ≥ 1 such that (Pers p ( ν n k l )) n ≥ 1 converges almost surely to Pers p ( ν ) . When restricting ourselves to this subsequence, we have both vague convergence of the measures and convergence of the p -total persistence. Hence, according to Lemma F.1, we have D n k l = OT p p ( ν n k l , ν ) → l →∞ 0 almost surely, proving that we actually have that ( D n ) n ≥ 1 converges in probability to 0 . To prove that E [ D n ] → n →∞ 0 , it remains to show that the sequence ( D n ) n ≥ 1 is uniformly integrable. By considering the trivial transport plan that sends all probability mass to ∂ Ω , we have for all n ≥ 1

But the sequence (Pers p ( ν n )) n ≥ 1 is uniformly integrable, as it converges in L 1 . Hence, so is the sequence ( D n ) n ≥ 1 , concluding the proof.

Using Lemma G.2, Theorem 4.1 would follow from the facts that µ f,i ∈ M p and that E [ | Pers p ( µ n,i ) -Pers p ( µ f,i ) | ] converges to 0 .

Recall that C c (Ω) is the set of continuous functions f : Ω → R with compact support (i.e. the support is bounded and at positive distance from ∂ Ω ). For s ≥ 0 , let T s = { ( u 1 , u 2 ) ∈ Ω : u 2 ≥ s } .

Lemma G.3. Let α > 0 . Let ( ν n ) n ≥ 1 be a sequence of random measures in M α that converges vaguely almost surely to a Radon measure ν ∈ M . Assume that the sequence of random variables ( ν n (Ω)) n ≥ 1 is uniformly integrable and that

Then, ν ∈ M α and E [ | Pers α ( ν n ) -Pers α ( ν ) | ] → n →∞ 0 .

Proof. We divide the proof into several steps.

Let ϕ ∈ C c (Ω) . We first show that ( ∫ ϕ d ν n )) n ≥ 1 converges in L 1 to ∫ ϕ d ν . By assumption, the convergence holds almost surely. Furthermore, as ϕ is bounded and as the sequence ( ν n (Ω)) n ≥ 1 is uniformly integrable, so is the sequence ( ∫ ϕ d ν n ) n ≥ 1 . Hence, E [ | ∫ ϕ d( ν n -ν ) | ] → n →∞ 0 .

Let ( ϕ k ) k ≥ 1 be an increasing sequence of functions in C c (Ω) that converge pointwise to the function pers α . Then, almost surely,

By Fatou's lemma, E [lim inf n →∞ ∫ pers α d ν n ] ≤ lim inf n →∞ E [Pers α ( ν n )] = C < + ∞ by assumption. Hence, by letting k →∞ and applying the monotone convergence theorem, we obtain Pers α ( ν ) ≤ C , proving that ν ∈ M α .

The same argument can be applied to the constant function equal to 1 , showing that ν (Ω) < + ∞ .

Let s ≥ 1 . The function pers α can be decomposed into a sum of three positive continuous functions pers α = ϕ (1) s + ϕ (2) s + ϕ (3) s , where ϕ (1) s has compact support, the support of ϕ (2) s

is included in the band { u ∈ Ω : pers( u ) ≤ 1 /s } and the support of ϕ (3) s is included in T s . Hence,

The first term in the above sum is equal to zero because of the first item, the second one is smaller than s -α (sup n E [ ν n (Ω)] + ν (Ω)) , and the third one is smaller than

Using the hypotheses of the lemma, the second and the third term converges to 0 as s goes to ∞ . We obtain that lim sup n → + ∞ E [ | Pers α ( ν n ) -Pers α ( ν ) | ] = 0 .

Our goal is to show that the conditions of Lemma G.3 holds for the sequence ( µ n,i ) n ≥ 1 to conclude. Remark that for any Radon measure ν ∈ M α and s ≥ 0

where we use Fubini's theorem for the first equality and the fact that { u : pers( u ) ≥ t } ⊂ T 2 t for the first inequality. We also have ν (Ω) = ν ( T 0 ) . Hence, the different conditions of Lemma G.3 can all be obtained by controlling the random variable µ n,i ( T s ) for s ≥ 0 .

Proposition G.4. Let M be a compact submanifold with positive reach. Assume that P has a density f on M satisfying f min ≤ f ≤ f max for two positive constants f min , f max . Then there exist c, C > 0 that depend on M , i , f min and f max such that for all integer n ≥ 1 and all s ≥ 0 ,

Before proving Proposition G.4, let us show how to use it to conclude the proof of Theorem 4.1. First, it implies that the random variables µ n,i (Ω) = µ n,i ( T 0 ) for n ≥ 1 have a uniformly bounded second moment, and are therefore uniformly integrable. Second, we have E [ µ n,i ( T s )] ≤ E [ µ n,i ( T s ) 2 ] 1 / 2 using Hölder's inequality. Hence, (22) implies that for any α > 0 , we have

In particular, sup n E [Pers α ( µ n,i )] < + ∞ . Likewise,

so that lim s → + ∞ sup n E [ ∫ T s pers α ( u )d µ n,i ( u )] = 0 . We are therefore in position to apply Lemma G.3, proving the convergence of the α -total persistence. Together with Lemma G.2 with α = p ≥ 1 , we also obtain the OT p -convergence of µ n,i . It remains to prove Proposition G.4.

Proof of Proposition G.4. Write ε n = d H ( A n , M ) . Let s ≥ 0 and let ε 0 < τ ( M ) / 2 be a small parameter, to be fixed later. Recall that we write # S for the cardinality of a multiset S . Notice that µ n,i ( T s ) = 0 if sn -1 /m > ε n + ε 2 n /τ ( M ) . In particular, we may assume without loss of generality that s ≤ n 1 /m diam( M )(1 + diam( M ) /τ ( M )) = n 1 /m ε max , for otherwise there is nothing to prove. Consider the event E = { ε n + ε 2 n /τ ( M ) < ε 0 } . By definition of Region (1), if E is satisfied, then all the coordinates of points of the PD dgm (1) i ( A n ) are smaller than ε 0 . Notice that the cardinality of dgm i ( A n ) is smaller than the number of i -dimensional simplices in the ˇ Cech complex of A n , which is itself smaller than n i +1 (as each simplex corresponds uniquely to a choice of i +1 vertices of A n ). Hence

Werequire the following lemma, which bounds the upper tail of the random variable ε n = d H ( A n , M ) .

Lemma G.5. If r ≤ τ ( M ) / 2 , then

for two positive constants c m , C m depending only on m . In particular, for any q ≥ 1 , d H ( A n , M ) = O L q ((ln n/n ) 1 /m ) .

Proof. The bound P ( d H ( A n , M ) > r ) is given in [1, Lemma III.23]. Furthermore, [1, Lemma III.23] also states that for any q > 0 , there exists C q depending on f min and m such that, with probability at least 1 -n -q/m , d H ( A n , M ) ≤ C q ( ln n n ) 1 /m . In particular, we obtain that

proving the second claim of the lemma.

Note that ε n ≤ diam( M ) , so P ( ε n + ε 2 n /τ ( M ) ≥ ε 0 ) ≤ P ( ε n ≥ c 0 ε 0 ) , with c 0 = (1 + diam( M ) /τ ( M )) -1 . Apply Lemma G.5 with r = c 0 ε 0 to obtain that

for some positive constants c 1 , C 0 . Furthermore, recall that s ≤ n 1 /m ε max . Hence,

for some positive constant c 2 .

It remains to bound E [ µ n,i ( T s ) 2 1 { E } ] . For each j = 1 , . . . , n , consider the set Ξ j n,s of critical points z ∈ Crit( A n ) such that σ A n ( z ) contains the point X i and sn -1 /m ≤ d A n ( z ) ≤ ε n + ε 2 n /τ ( M ) .

Lemma G.6. It holds that µ n,i ( T s ) is smaller than

where L j is the cardinality of the set ⋃ z ∈ Ξ j n,s σ A n ( z ) .

Proof. Lemma E.1 applied to a realization of A n and the interval [ sn -1 /m , ε n + ε 2 n /τ ( M )] yields that the number of points in dgm i ( A n ) with at least one of their coordinates in [ sn -1 /m , ε n + ε 2 n /τ ( M )] is upper bounded by ∑ n j =1 L i +2 j . By definition, this means that µ n,i ( T s ) ≤ 1 n ∑ n j =1 L i +2 j , as desired.

One can show that the set Ξ j n,s is localized, in the sense that it is included in a ball centered at X j , with a radius depending on the sample A n , that is small with high probability. Hence, the number L j is controlled by the number of points in A n found in a small (random) neighborhood of X j . Let us make this idea rigorous.

For r ≥ 0 , define the shape

We build a partition of C ( r ) in the following way. Consider a finite partition W of the unit sphere in R m ×{ 0 } d -m into sets of diameters smaller than θ = π/ 4 (for the geodesic distance on the sphere), which we fix for a given dimension m . Let W ( r ) be the partition of C ( r ) consisting of the sets

where W is an element of the partition W .

For x ∈ M , consider an isometry ι x : R d → R d sending R m × { 0 } d -m to T x M . Let C ( x, r ) = x + ι x ( C ( r )) . Likewise, we define a partition W ( x, r ) by applying the affine transformation W ↦→ x + ι x ( W ) to each W ∈ W ( r ) .

For j = 1 , . . . , n , let R jn be the smallest radius r ≤ ε 0 such that every W ∈ W ( X j , r ) contains a point of A n other than X j . By convention, we let R jn = ε 0 if such a radius does not exist. R jn can be made measurable with a good choice of x ↦→ ι x ; we assume it to be the case henceforth.

Lemma G.7. Let ε 0 ≤ τ ( M ) / √ 2 . For all j = 1 , . . . , n , if z ∈ Crit( A n ) is such that X j ∈ σ A n ( z ) and d A n ( z ) ≤ ε 0 , then ∥ X j -z ∥ ≤ c 0 R jn for some positive absolute constant c 0 .

̸

Proof. Recall that for x ∈ M , π x is the orthogonal projection on T x M while π ⊥ x is the orthogonal projection on the normal space at x . Let j = 1 , . . . , n and let z ∈ Crit( A n ) be such that X j ∈ σ A n ( z ) . The direction e = π X j ( z -X j ) / ∥ π X j ( z -X j ) ∥ belongs to the unit sphere in T X j M ; note that π X j ( z -X j ) = 0 due to Lemma G.8 below, which applies as d A n ( z ) ≤ ε 0 ≤ τ ( M ) / √ 2 . Hence, ι -1 x ( e ) belongs to an element W 0 of the partition W . Consider the corresponding element W of the partition W ( X j , R jn ) .

̸

If R jn = ε 0 , then the conclusion of the lemma holds (for c 0 = 1 ): indeed we have ∥ X j -z ∥ = d A n ( z ) ≤ ε 0 ≤ R jn . Otherwise, by assumption, there exists a point X k ∈ W for some k = j . As X j ∈ σ A n ( z ) , it holds that ∥ X j -z ∥ ≤ ∥ X k -z ∥ . Hence,

and

We write

By construction, as the diameter of W 0 is less than θ , we have ⟨ π X j ( X k -X j ) , π X j ( z -X j ) ⟩ ≥ cos( π/ 4) ∥ π X j ( X k -X j ) ∥∥ π X j ( z -X j ) ∥ . On the other hand, according to [39, Theorem 4.18],

which also implies that ∥ π X j ( X k -X j ) ∥ ≥ ∥ X j -X k ∥ √ 1 -ε 2 0 4 τ ( M ) 2 . Similarly, Lemma G.8 below states that ∥ π X j ( z -X j ) ∥ ≥ ∥ z -X j ∥ / √ 2 and ∥ π ⊥ X j ( z -X j ) ∥ ≤ ∥ z -X j ∥ 2 /τ ( M ) (using our assumption that d A n ( z ) ≤ ε 0 ≤ τ ( M ) / √ 2 ).

Hence we obtain that

Dividing by ∥ X j -X k ∥ and using that ∥ X j -X k ∥ ≤ R jn and that ε 0 ≤ τ ( M ) / √ 2 , we see that ∥ z -X j ∥ is smaller than R jn up to an absolute multiplicative constant.

We now prove the lemma used above:

Lemma G.8. Let A ⊂ M , z ∈ Crit( A ) and x ∈ σ A ( z ) . Then ∥ π ⊥ x ( z -x ) ∥ ≤ ∥ z -x ∥ 2 /τ ( M ) . Furthermore, if d A ( z ) ≤ τ ( M ) / √ 2 , then ∥ π x ( z -x ) ∥ ≥ ∥ z -x ∥ / √ 2 .

Proof. The point z can be written as a convex combination z = ∑ k λ k y k where the points y k are in A ⊂ M . Then, using [39, Theorem 4.18],

as ∥ y k -z ∥ 2 = ∥ x -z ∥ 2 and ∑ k λ k ( y k -z ) = 0 . The second inequality ∥ π x ( z -x ) ∥ ≥ ∥ z -x ∥ / √ 2 from the statement follows from the first one through a direct computation.

Let us now show that the random variable R jn has controlled tails.

Lemma G.9. For all x ∈ M , r ≥ 0 , B ( x, r ) ∩ M ⊂ C ( x, r ) .

Proof. Let y ∈ M be such that ∥ π x ( y -x ) ∥ ≤ r . Then, according to [39, Theorem 4.18], ∥ π ⊥ x ( y -x ) ∥ ≤ ∥ y -x ∥ 2 2 τ ( M ) . In particular, B ( x, r ) ∩ M ⊂ C ( x, r ) .

Lemma G.10. For 0 < t ≤ τ ( M ) / 4 and j = 1 , . . . , n , we have P ( R jn > t ) ≤ C m e -c m f min ( n -1) t m for some positive constants c m , C m .

Proof. If R jn is larger than t , then there exists at least one set W ∈ W ( X j , t ) such that its intersection with A n contains only X j . Hence,

Let π 0 be the orthogonal projection from R d to R m ×{ 0 } d -m . The image of a set W ∈ W ( X j , t ) by the projection y ↦→ π X j ( y -X j ) is equal to ι X j ( π 0 ( W 0 )) ⊂ T X j M for some W 0 ∈ W ( t ) . For t ≤ τ ( M ) / 4 , the orthogonal projection y ∈ B ( X j , t ) ∩ M ↦→ π X j ( y -X j ) ∈ T X j M is a diffeomorphism on its image, with Jacobian lower bounded by a constant c m that depends only on m , see e.g. [31, Lemma 2.2]. According to Lemma G.9, the preimage of ι X j ( π 0 ( W 0 )) ⊂ T X j M by this diffeomorphism is equal to W ∩ M . Hence, by a change of variable,

for some c ′ m > 0 . Hence,

We conclude by taking the expectation.

Let us now control the number of points found in a ball B ( X j , κR jn ) for some κ ≥ 0 . Let ρ 0 > 0 be small enough such that ∫ B ( x,ρ 0 ) ∩ M f < 1 / 2 for any x ∈ M .

Lemma G.11. Let l ≥ 0 , κ > 0 be such that κε 0 ≤ min( ρ 0 , τ ( M ) / 4) . For j = 1 , . . . , n , let K jn ( κ ) be the number of elements of A n found in B ( X j , κR jn ) . Then, E [ K jn ( κ ) l ] ≤ C m,l (1+ ( f max f min ) l κ lm ) for some constant C m,l which depends on m and l .

Proof. Let us write { W 1 , . . . , W K } = W ( X j , κR jn ) . Without loss of generality, we can assume that n ≥ K +1 , as otherwise the bound is trivial. As in [34, Lemma 5], we remark that there is at least one sample point in every W i , and that there is (almost surely) one single element W i ∗ of the partition with exactly one sample point on its boundary. Let N i be the cardinality of ( W i ∩ A n ) \{ X j } , and N -1 be the cardinality of A n \ B ( X j , κR jn ) . Define

and α i := ˜ α i 1 -˜ α i ∗ for all i = i ∗ , as well as α -1 = 1 -∑ K i =1 ˜ α i 1 -˜ α i ∗ . Note that as κR jn ≤ ρ 0 , we have ˜ α i < 1 / 2 for all i = 1 , . . . , K . As κR jn ≤ κε 0 ≤ τ ( M ) / 4 , we may use once again that the orthogonal projection y ∈ B ( X j , κR jn ) ∩ M ↦→ π X j ( y -X j ) ∈ T X j M is a diffeomorphism on its image, with Jacobian upper and lower bounded by constants depending only on m (see [31, Lemma 2.2]) to also obtain that

for all i = 1 , . . . , K and some constants c = c ( m ) , C = C ( m ) . Hence there exists C ′ = C ′ ( m ) > 0 such that

for all i = 1 , . . . , K . Consider a multinomial random variable L = ( L 1 , . . . , ̂ L i ∗ , . . . , L K , L -1 ) of parameters n -2 and ( α 1 , . . . , ̂ α i ∗ , . . . , α K , α -1 ) , and let E denote the event

Then conditionally on X j , R jn and i ∗ , the variable N = ( N 1 , . . . , ̂ N i ∗ , . . . , N K , N -1 ) follows the same distribution as L | E . Thus, conditionally on X j , R jn and W i ∗ , the variable K jn ( κ ) = 1 + ∑ K i =1 N i = 2 + ∑ K i =1 ,i = i ∗ N i (where the initial 1 comes from X j ) has the same distribution as

̸

Note that as κR jn ≤ ρ 0 , we have α -1 ≥ 1 / 2 .

As a result, Lemma G.12 below yields that

̸

where C K,l , C ′ K,l and C m,l are constants whose dependencies are indicated by their indices (remember that K = # W depends only on m ). We can now conclude by considering

̸

̸

̸

̸

̸

The quantity n l E [ R ml jn ] is bounded by a constant, which is proved by integrating the tail bound found in Lemma G.10. Indeed, Lemma G.10 implies that the random variable nR m jn is subexponential with a subexponential norm m independent of n , of order O (1 /f min ) ; the moment of order l of such a random variable is bounded by C l m l , see [74, Section 2.7].

Let us now prove the technical lemma used above.

Lemma G.12. Let K ≥ 1 and L = ( L 1 , . . . , L K , L K +1 ) be a random multinomial variable of parameters n and α 1 , . . . , α K , α K +1 . Then there exists C = C ( K,l ) such that

Proof. Let X 1 , . . . , X n be i.i.d. categorical variables of parameters α 1 , . . . , α K , α K +1 . Define L k ( p ) = ∑ r ≤ p 1 X r = k ; then ( L 1 ( n ) , . . . , L K ( n ) , L K +1 ( n )) has the same distribution as ( L 1 , . . . , L K , L K +1 ) , and we identify the two in our notations. For a fixed n , and for any injective function ι : { 1 , . . . , k } → { 1 , . . . , n } , consider the event

i.e. ι ( i ) is the first appearance of i among the variables X 1 , . . . , X n . Note that E := { L 1 , . . . , L K ≥ 1 } = ⊔ ι E ι where the sum is taken over all such injective functions. Then

̸

Fix a function ι , and assume without loss of generality that ι (1) < ι (2) < . . . < ι ( K ) . Conditioned by A ι , the variable Y i := ∑ ι ( i +1) -1 r = ι ( i )+1 1 X r = K +1 is a binomial variable of parameters ι ( i +1) -ι ( i ) -2 ≤ n and ∑ i i =1 α i α 1 + ... + α i + α K +1 ≤ ∑ K i =1 α i . Hence E [ Y l i | E ι ] ≤ C 1 ( l )( n ∑ K i =1 α i ) l using classical bounds on the l -th moment of a binomial variable, and we see that

̸

where we write ι (0) = 0 and ι ( K +1) = n to simplify notations.

Let us wrap things up. Recall Lemma G.6: it holds that

where L j is the cardinality of the set ⋃ z ∈ Ξ j n,s σ A n ( z ) . But according to Lemma G.7, if z ∈ Ξ j n,s , then ∥ X j -z ∥ ≤ c 0 R jn . In particular, d A n ( z ) ≤ c 0 R jn , and any point y ∈ σ A n ( z ) is at distance less than 2 c 0 R jn from X j . Hence, L j is smaller than K jn ( κ ) for κ = 2 c 0 . Choose ε 0 so that κε 0 ≤ κε 0 ≤ min( ρ 0 , τ ( M ) / 4) . We are in position to apply Lemma G.11. We further remark that Lemma G.7 implies that Ξ j n,s is empty if c 0 R jn < sn -1 /m .

̸

Hence, by Jensen's inequality,

for some constants c m , C m,i > 0 , where we apply Lemma G.10 and Lemma G.11 at the last line.

This completes the proof of Theorem 4.1.

We now prove Proposition 4.2:

Proposition 4.2. Let M be a generic m -dimensional submanifold. Assume that P has a density f on M bounded away from 0 and ∞ . Let i ≥ 0 be an integer. There exists an optimal matching γ n : dgm i ( A n ) ∪ ∂ Ω → dgm i ( M ) ∪ ∂ Ω for the bottleneck distance between dgm i ( A n ) and dgm i ( M ) such that for any q ≥ 1 :

Region (2): It holds that max u ∈ dgm (2) i ( A n ) | u 2 -γ n ( u ) 2 | = O L q ( n -2 /m ) .

Region (3): It holds that max u ∈ dgm (3) i ( A n ) ∥ u -γ n ( u ) ∥ ∞ = O L q ( n -2 /m ) and #(dgm (3) i ( A n )) = O L q (1) .

Proof. Let ε n = d H ( A n , M ) . Let Π M := { x ∈ σ M ( z ) : z ∈ Crit( M ) } be the (finite) set of projections of critical points z ∈ Crit( M ) . The proof of Theorem 3.3 relied on the use of Theorem 1.6 in [6]. This theorem states roughly that both critical points z ∈ Crit( A n ) far from M and their projections x ∈ σ A n ( z ) are stable with respect to the Hausdorff distance, meaning that every such point z is at distance O ( ε n ) from a critical point z ′ ∈ Crit( M ) , with x being at distance O ( ε n ) from a point x ′ ∈ Π M . Thus, the number of critical points of A n located close to a given z ′ ∈ Crit( M ) is crudely upper bounded by the number of subsets which can be formed by selecting elements in neighborhoods of size O ( ε n ) around x ′ ∈ σ M ( z ′ ) . We used the same idea to bound the cardinality of dgm (3) i ( A ) in Theorem 3.3.

In a random setting, the distance ε n is of order (ln n/n ) 1 /m , as suggested by Lemma G.5, while the number of points found in a ball of radius r is typically of order nr m , yielding a logarithmic number of elements in a neighborhood of size O ( ε n ) around a given point x ∈ M . Hence, our earlier strategy is not tight enough to bound in expectation the cardinality of dgm (3) i ( A ) by a constant.

We improve upon this strategy with the following intuition: the maximal distance between a point x of M and the point cloud A n does not really matter in [6, Theorem 1.6], but only the density of the point cloud A n around the (finitely many) projections x ∈ Π M . Although ε n is of order (ln n/n ) 1 /m , the distance between a fixed point x ∈ M and A n is known to be of order n -1 /m (see (21)). This remark explains how we can intuitively replace neighborhoods of radii (ln n/n ) 1 /m by radii of size n -1 /m in the previous arguments.

We now make these ideas rigorous. Let Crit > ( A n ) = { z ∈ Crit( A n ) : d A n ( z ) ≥ τ ( M ) / 2 } denote the set of critical points of A n 'far' from M and let Π A n := { x ∈ σ A n ( z ) : z ∈ Crit > ( A n ) } be the corresponding set of projections. We let ε 0 be a small constant to be fixed later. Theorem 1.6 from [6] states that, thanks to the genericity of M , there exist K 1 , K 2 > 0 such that for ε 0 small enough, if ε n < ε 0 :

there exists a map ϕ : Crit > ( A n ) → Crit( M ) such that d ( z, ϕ ( z )) ≤ K 1 ε n for all z ∈ Crit > ( A n ) , and

there exists a map ψ : Π A n → Π M such that d ( x, ψ ( x )) ≤ K 2 ε n for all x ∈ Π A n .

Furthermore, the map ψ is such that for each z ∈ Crit > ( A n ) and each x ′ ∈ σ M ( ϕ ( z )) , there exists x ∈ σ A n ( z ) such that ψ ( x ) = x ′ (if ε 0 is chosen small enough). Indeed, due to the genericity of M , each critical point z ′ ∈ Crit( M ) belongs to the relative interior of σ M ( z ′ ) (as stated in Appendix C). In particular, it cannot belong to the convex hull of any strict subset of σ M ( z ′ ) . By continuity of the convex hull for the Hausdorff distance, as soon as z ∈ Crit > ( A n ) is close enough to ϕ ( z ) and its projections x ∈ σ A n ( z ) are close enough to σ M ( ϕ ( z )) (i.e. as soon as ε n is small enough), the point z cannot belong to the convex hull of any subset S ⊂ σ A n ( z ) that does not contain for each x ′ ∈ σ M ( ϕ ( z )) at least one point x such that ψ ( x ) = x ′ . As z must belong to the convex hull of σ A n ( z ) , we conclude that each x ′ ∈ σ M ( ϕ ( z )) has at least one preimage by ψ in σ A n ( z ) . As Crit( M ) is finite, taking the minimum distance such that this property holds for z ′ over all z ′ ∈ Crit( M ) proves the claim.

We introduce the random function defined as

where Π r M is as usual the r -offset of Π M . This random variable measures the density of the point cloud A n in neighborhoods of size r of points x ′ ∈ Π M .

Let proving (34).

One can also check that ε n can be replaced by E ( η n ) in the end of the proof of Theorem 1.6 in [6], so that (35) holds.

Let us now prove (33). Let z ∈ Crit > ( A n ) . Consider x A n ∈ σ A n ( z ) ; we know that ∥ ψ ( x A n ) -x A n ∥ ≤ K 5 E ( η n ) using (33). As ϕ ( z ) -ψ ( x A n ) is orthogonal to T ψ ( x A n ) M , [39] states that ∥ π ⊥ ψ ( x ) ( x A n -ψ ( x A n )) ∥ ≤ ∥ x A n -ψ ( x A n ) ∥ 2 2 τ ( M ) ≤ K 2 5 E ( η n ) 2 2 τ ( M ) , hence

where L 4 = K 2 5 ( R ( M ) τ ( M ) +1 ) , R ( M ) is the radius of the smallest closed ball that contains M , and d M ( ϕ ( z )) ≤ R ( M ) because a critical point must belong to the convex hull of its projections. As

give (when ε n < ε 0 for ε 0 small enough) the largest distance between a projection x ∈ Π A n and the corresponding projection ψ ( x ) ∈ Π M . We also define

We require the following controls on the random variables ρ n and η n .

Lemma G.13. There exist positive constants ε 0 = ε 0 ( M ) , K 3 = K 3 ( M ) , K 4 = K 4 ( M ) and K 5 = K 5 ( M ) such that if ε n < ε 0 , it holds that for any z ∈ Crit > ( A n )

Proof. The lemma follows from a careful read of the proof of Theorem 1.6 in [6]. First, remark that in the proof of Lemma 5.1 in [6] (applied with r = τ ( M ) / 2 , R = diam( M ) ), the Hausdorff distance ε = d H ( A n , M ) can be replaced by E ( η n ) . Hence, if z ∈ Crit > ( A n ) , there exists a µ -critical point z ′ of M at distance less than E ( η n ) , with µ ≤ L 1 ( M ) E ( η n ) . By genericity, for a choice of ε 0 small enough, this point z ′ is at distance L 2 ( M ) µ from a critical point z 0 ∈ Crit( M ) . For ε 0 small enough, this point z 0 is necessarily equal to ϕ ( z ) , with ∥ z -ϕ ( z ) ∥ ≤ (1 + L 2 L 1 ) E ( η n ) . Due to the Lipschitz property of the projection onto M around z 0 (see the arguments found at the bottom of p. 19 in [6]), any point x ∈ π M ( z ) is such that ∥ x -x ′ ∥ ≤ L 3 ( M ) ∥ z -ϕ ( z ) ∥ for some x ′ ∈ σ M ( ϕ ( z )) . By taking the supremum over all such points x , we obtain that

the same bound applies to each of the projections x ′ A n ∈ σ A n ( z ) , we find that the closed ball B ( ϕ ( z ) , √ d M ( ϕ ( z )) 2 + E ( η n ) 2 L 4 ) contains σ A n ( z ) . Since z is the center of the smallest ball that contains σ A n ( z ) , whose radius is d A n ( z ) , we find that

Similarly, consider now x M ∈ σ M ( ϕ ( z )) , and let x A n ∈ ψ -1 ( x M ) ∩ σ A n ( z ) (we have seen below the definition of ϕ and ψ that its existence is guaranteed). We have shown earlier that there exists x ∈ σ M ( z ) such that ∥ x -x M ∥ , ∥ x -x A n ∥ ≤ O ( E ( η n )) , and the same reasoning as above yields that ∥ z -x A n ∥ 2 = ∥ z -x ∥ 2 + O ( E ( η n ) 2 ) and ∥ z -x M ∥ 2 = ∥ z -x ∥ 2 + O ( E ( η n ) 2 ) (with all big O constants depending only on M ), hence that ∥ z -x M ∥ 2 ≤ ∥ z -x A n ∥ 2 + E ( η n ) 2 L 5 ( M ) = d A n ( z ) 2 + E ( η n ) 2 L 5 .

As before, this shows that the closed ball B ( z, √ d A n ( z ) 2 + E ( η n ) 2 L 5 ) contains σ M ( ϕ ( z )) , hence that

This, together with (36) and the fact that d A n ( z ) ≥ τ ( M ) / 2 , shows that | d A n ( z ) -d M ( ϕ ( z )) | ≤ K 3 ( M ) E ( η n ) 2 .

Consider a (random) optimal matching γ n : dgm i ( A n ) ∪ ∂ Ω → dgm i ( M ) ∪ ∂ Ω such that max u ∈ dgm (2) i ( A n ) | u 2 -γ n ( u ) 2 | ≤ Cε 2 n and max u ∈ dgm (3) i ( A n ) ∥ u -γ n ( u ) ∥ ∞ ≤ Cε 2 n whose existence is guaranteed by Theorem 2.2 (for some C = C ( M ) > 0 ). For r ≥ 0 , we let N ( r ) be the number of points of A n found in Π r M (i.e. at distance less than r from a point in Π M ).

Lemma G.14. There exists ε 0 = ε 0 ( M ) such that if ε n ≤ ε 0 , then

Furthermore, #(dgm (3) i ( A n )) is smaller than N ( ρ n ) i +2 .

Proof. Let us assume from now on that ε 0 is small enough that 2( K 3 + C ) ε 2 0 is smaller than the smallest difference between two distinct critical values of d M .

Consider a point ( u 1 , u 2 ) ∈ dgm (3) i ( A n ) that is mapped by γ n to the diagonal. The coordinates u 1 and u 2 differ by at most Cε 2 n , hence they are critical values of d A n that correspond to critical points that are mapped by ϕ to two critical points of M that have the same critical value (due to 2( K 3 + C ) ε 2 being smaller than the smallest difference between two distinct critical values of d M ). But as stated in Equation (33) of Lemma G.13, these critical values must then be K 3 E ( η n ) 2 -close to that of the two critical points of M , hence ∥ u -γ n ( u ) ∥ ∞ = d ( u, ∂ Ω) = ( u 2 -u 1 ) / 2 ≤ K 3 E ( η n ) 2 , as desired.

Likewise, consider a point ( u 1 , u 2 ) of dgm (3) i ( A n ) that is mapped by γ n to a point ( v 1 , v 2 ) of dgm (3) i ( M ) . The birth coordinates u 1 and v 1 differ by at most Cε 2 , hence the associated critical values must correspond to a critical point z of A n and a critical point z ′ of M such that ϕ ( z ) has the same filtration value as z ′ (again due to 2( K 3 + C ) ε 2 being smaller than the smallest difference between two distinct critical values of d M ). But as above, we have | u 1 -v 1 | = | d A n ( z ) -d M ( z ′ ) | = | d A n ( z ) -d M ( ϕ ( z )) | ≤ K 3 E ( η n ) 2 . As the same applies to u 2 and v 2 , we find that ∥ u -γ n ( u ) ∥ ∞ ≤ K 3 EE ( η n ) 2 and max u ∈ dgm (3) i ( A n ) ∥ u -γ n ( u ) ∥ ∞ ≤ K 3 E ( η n ) 2 . The same reasoning shows that max u ∈ dgm (2) i ( A n ) | u 2 -γ n ( u ) 2 | ≤ K 3 E ( η n ) 2 .

Finally, and by definition, Π A n = { x ∈ σ A n ( z ) : z ∈ Crit( A n ) , d A n ( z ) ≥ τ ( M ) / 2 } is included in A n ∩ Π ρ n M , hence #(Π A n ) ≤ N ( ρ n ) . Lemma E.1 applied to A n and the interval [ τ ( M ) / 2 , + ∞ ) then yields that the number of points in dgm (3) i ( A n ) is smaller than N ( ρ n ) i +2 .

Lemma G.15. Let ε 0 be the parameter defined in Lemma G.14. It holds that for all q ≥ 1 , E ( η n ) 1 { ε n ≤ ε 0 } = O L q ( n -1 /m ) and N ( ρ n ) 1 { ε n ≤ ε 0 } = O L q (1) .

Assume for a moment that Lemma G.15 holds. Then, we write (for a choice of ε 0 small enough)

where we recall that R ( M ) is the radius of the smallest enclosing ball of M . Because of Lemma G.5, the random variable 1 { ε n > ε 0 } is a O L q ( n -2 /m ) for all q ≥ 1 . But then, because of Lemma G.15, we obtain that the right-hand side in (38) is a O L q ( n -2 /m ) for all q ≥ 1 . Likewise, we obtain that max u ∈ dgm (3) i ( A n ) ∥ u -γ n ( u ) ∥ ∞ = O L q ( n -2 /m ) for all q ≥ 1 . At last, we have

The first term is a O L q (1) for all q ≥ 1 because of Lemma G.15, while the second one is a O L q (1) for all q ≥ 1 because of Lemma G.5. This concludes the proof of Proposition 4.2. Let us now prove Lemma G.15.

Proof of Lemma G.15. Let t > 0 . The key observation to obtain Lemma G.15 is that (34) implies that if η n > t , then there exists r > t with r ≤ K 4 E ( r ) .

Lemma G.16. For all λ > 0 , there exist positive constants C λ , c λ (depending on λ , M and f min ) such that for all r > 0 , P ( r ≤ λE ( r )) ≤ C λ exp( -c λ nr m ) .

Proof. Remark that if r ≤ λE ( r ) , then r ≤ λF ( r ) . The set M ∩ Π r M is the union of a finite number of balls of radius r . Hence, it can be covered by C λ = C λ ( M ) open balls of radius r/ (2 λ ) , with centers x 1 , . . . , x C λ ∈ M . Note that if all these balls intersect A n , then for all y ∈ M ∩ Π r M , d A n ( y ) < r/λ . Hence, if λF ( r ) ≥ r , then the intersection of one of these balls with A n is empty. Hence, according to Lemma G.1,

Remark that the fonction E is nondecreasing and 1 -Lipschitz continuous: we have for r < s , E ( r ) ≤ E ( s ) ≤ E ( r ) + ( r -s ) . Fix t > 0 and consider the sequence t k = a k t for some a > 1 to fix. Assume that η n > t and that ε n ≤ ε 0 . Then η n is between two values t k < t k +1 . But then, according to Lemma G.14,

Choose a > 1 such that 1 K 4 -( a -1) > 0 , and let

We have proven that if ε n ≤ ε 0 and η n > t , then there exists k ≥ 0 with E ( t k ) ≥ λt k . Hence,

A standard comparison between this sum and an integral shows that this sum is at most of order

for two positive constants K 6 , K 7 depending on M and f min . But when nt m ≤ 1 , we can simply use the bound P ( η n > t, ε n ≤ ε 0 ) ≤ 1 . Hence,

for two other constants K 8 , K 9 .

To summarize, we have shown that the random variable nη m n 1 { ε n ≤ ε 0 } is subexponential, with subexponential norm depending only on M and f min , see e.g. [74, Section 2.7]. We will now simply say that a random variable is subexponential to indicate that it is subexponential with a norm depending only on M and f min .

Lemma G.1 shows that the random variable nd A n ( x ) m for a fixed x ∈ M is also subexponential. Thus, so is nE (0) m = n max x ∈ Π M d A n ( x ) m , as a maximum of a finite number of subexponential random variables. As nE ( η n ) m ≤ n ( η n + E (0)) m ≤ 2 m -1 ( nη m n + nE (0) m ) , the random variable nE ( η n ) m 1 { ε n ≤ ε 0 } is also subexponential. In particular, we have E ( η n ) 1 { ε n ≤ ε 0 } = O L q ( n -1 /m ) for all q ≥ 1 .

It remains to bound N ( ρ n ) . First, because of (35), the random variable nρ m n 1 { ε n ≤ ε 0 } is subexponential. Also, for a fixed t , N ( t ) follows a binomial distribution of parameter n and P (Π t M ) . As long as t ≤ τ ( M ) / 4 , a ball of radius t is of mass smaller than C m f max t m (see Lemma G.1). Let 0 ≤ k ≤ n be an integer. For any t ≤ τ ( M ) / 4 , we bound

where K 10 is proportional to the subexponential norm of nρ m n 1 { ε n ≤ ε 0 } , see [74, Section 2.7]. Let t = ( k/n ) 1 /m min( τ ( M ) / 4 , 1 / (#Π M · 2 C m f max ) 1 /m ) . This choice of t ensures that t ≤ τ ( M ) / 4 and that E [ N ( t )] ≤ n (#Π M ) C m f max t m ≤ k/ 2 . Then, by Bernstein's inequality [74, Theorem 2.8.4],

for some constant K 11 depending on m , M and f max . We have proven that for all k ≥ 0 ,

for some constant K 12 depending on m , M , f min and f max . Hence, the random variable N ( ρ n ) 1 { ε n ≤ ε 0 } is subexponential, with a subexponential norm depending on M , f min and f max . This implies in particular that N ( ρ n ) 1 { ε n ≤ ε 0 } = O L q (1) for all q ≥ 1 .

Corollary 4.3 is a simple consequence of Proposition 4.2:

Corollary 4.3. Let p ≥ 1 and let 0 ≤ i < d be an integer. Under the same assumptions as in Proposition 4.2, the following holds:

If p > m , then E [OT p p (dgm i ( A n ) , dgm i ( M ))] → 0 as n →∞ .

If p = m , E [OT p p (dgm i ( A n ) , dgm i ( M ))] → Pers p ( µ ∞ ,i,m )Vol( M ) as n → ∞ , where Vol( M ) is the volume of M .

If p < m and i < m , then E [OT p p (dgm i ( A n ) , dgm i ( M ))] → + ∞ as n →∞ .

Furthermore, for all α > 0 , Pers α (dgm i ( A n )) is equal to

Proof. Let ε n = d H ( A n , M ) . Consider the event E = { ε n < τ ( M ) / 2 } . According to Theorem 2.2, on E , the number of points of dgm (2) i ( A n ) is bounded by some constant N 0 depending only on M .

Consider the optimal matching γ n given by Proposition 4.2. On the event E , this optimal matching sends all the points of dgm (1) i ( A n ) to the diagonal ∂ Ω . Thus, we have (when E is

satisfied)

Furthermore, note that

We take the expectation and apply the Cauchy-Schwarz inequality to obtain (together with Theorem 4.1, Proposition 4.2 and Lemma G.5):

When E is not realized, we crudely bound OT p p (dgm i ( A n ) , dgm i ( M )) by considering the matching sending every point to the diagonal. The cost of this matching is bounded by C M (1 + n i +1 ) , where n i +1 is an upper bound on the number of points in dgm i ( A n ) and C M is some constant depending on M . Hence,

according to Lemma G.5. This proves the first bullet point.

The second bullet point is proved likewise from (41). Indeed, in that case, it holds that

Also, using the same crude bound, we obtain that

So far, we have proved that OT p p (dgm i ( A n ) , dgm i ( M )) = Pers p (dgm (1) i ( A n )) + o L 1 (1) . According to Theorem 4.1, it holds that Pers p (dgm (1) i ( A n )) = Pers p ( µ f,i ) + o L 1 (1) for p = m , proving the second bullet point.

For the third bullet point, we use that on the event E ,

The latter is equal to n 1 -p/m Pers p ( µ f,i ) + o L 1 ( n 1 -p/m ) , with Pers p ( µ f,i ) > 0 : indeed, the support of the measure µ f,i is nontrivial for i < m , see [47, 42]. Hence,

The second term goes to zero (use the crude bound Pers p (dgm (1) i ( A n )) ≤ C M (1 + n i +1 ) ), while the first one diverges. This proves the third bullet point.

At last, we prove the formula for the asymptotic expansion of

The first term is equal to n 1 -α/m Pers α ( µ ∞ ,i,m ) ∫ M f ( x ) 1 -α m d x + o L 1 ( n 1 -α/m ) according to Theorem 4.1. Remark that for u, v ∈ Ω , | pers α ( u ) -pers α ( v ) | ≤ 2 α (pers α ( u ) + pers α ( v )) ∥ u -v ∥ ∞ . Hence, on the event E = { ε n < τ ( M ) / 2 } ,

We crudely bound the persistence of a point in dgm i ( A n ) by R ( M ) , the radius of the smallest enclosing ball of M , to obtain that

We have already established that #(dgm (2) i ( A n )) = O L 2 (1) (actually, it is larger than N 0 with only exponentially small probability). Furthermore, Proposition 4.2 states that #(dgm (3) i ( A n )) = O L 2 (1) . Lemma G.5 also states that ε n = O L 2 (((log n ) /n ) 1 /m ) . Hence,

This concludes the proof.

Finally, remember that we defined linear feature maps as follows (at the end of Section 4): we let ( V, ∥ · ∥ ) be a normed vector space and ϕ : Ω → V be Lipschitz continuous. For α ≥ 0 , the linear feature map Φ α associated to ϕ and defined on the space D f of finite PDs is

Let us prove the associated Corollary 4.4:

Corollary 4.4. Let α ≥ 1 and let 0 ≤ i < d be an integer. Under the same assumptions as in Proposition 4.2, it holds that Φ α (dgm i ( A n )) converges in probability to Φ α (dgm i ( M )) whenever α > m .

Proof. According to [33, Proposition 5.1], the feature map Φ α is continuous with respect to the OT α distance. But we have shown in Corollary 4.3 that E [OT α α (dgm i ( A n ) , dgm i ( M ))] → 0 as n →∞ . In particular, OT α (dgm i ( A n ) , dgm i ( M )) converges in probability to 0 . By continuity, so does ∥ Φ α (dgm i ( A n )) -Φ α (dgm i ( M )) ∥ .

H Details on numerical experiments

Weprovide in this section additional details on the numerical experiments conducted in Section 5. PDs are computed using GUDHI [62]. PDs are plotted using the giotto-tda library [72] and persistence images using scikit-tda [64]. All experiments can be easily run on a standard office laptop over a few hours.

Simulation of generic tori. Let R > r be two positive numbers. The (standard) torus M 0 of major radius R and minor radius r is given as the image of [0 , 2 π ] 2 by the map

Consider a list y 1 , . . . , y k of pairs of angles in [0 , 2 π ] 2 , together with positive numbers a 1 , . . . , a K . We let r ( y ) = r + ∑ K k =1 a k ψ ( d ( y, y k ) /σ ) , where ψ ( t ) = 1 { t < 1 } exp(1 / ( t 2 -1)) , 0 < σ < 2 π and d is the Euclidean distance on the flat torus. We then define

Lemma H.1. Assume that max ( θ,ϕ ) ∈ [0 , 2 π ] 2 r ( θ, ϕ ) < R -r . Then, the image of ˜ F is a compact smooth submanifold, homeomorphic to a torus.

Proof. The condition on the function r ensures that the function ˜ F is injective and (2 π ) -periodic in each of its variables. Let ( θ, ϕ ) ∈ [0 , 2 π ] 2 . Wewrite r as a shorthand for r ( θ, ϕ ) . Let S = R + r cos( θ ) and U = -r sin( θ ) + ∂ θ r cos( θ ) . The differential of ˜ F is equal to

The determinant of the two first rows is equal to SU , and is therefore only equal to 0 if U = 0 . But in that case,

is a rank 2 matrix. Indeed, note that U is equal to the dot product of e θ = ( -sin( θ ) , cos( θ )) with the (nonzero) vector ( r , ∂ θ r ) . Hence, if U = 0 , then the dot product of ( r , ∂ θ r ) with the vector (cos( θ ) , sin( θ )) (that is perpendicular to e θ ) is nonzero. Furthermore, one of the two top entries in the second column is also nonzero. Indeed, this is clear if ∂ ϕ r = 0 or if sin( ϕ ) = 0 (we have S > 0 by assumption). Otherwise, having the two entries equal to zero would imply that S 2 = -( ∂ ϕ r ) cos 2 ( θ ) , a contradiction.

In all cases, d ˜ F ( θ, ϕ ) is of maximal rank. This implies the conclusion.

Let R = 2 . 4 , r = 0 . 8 , σ = 2 and K = 30 . We create a random torus by sampling pairs of angles Y = { y 1 , . . . , y K } uniformly at random, conditioned on the fact that max y ∈ [0 , 2 π ] 2 d Y ( y ) < σ . We then draw the numbers a 1 , . . . , a K as exponential random variables of scale 0 . 5 , conditioned on the fact that max ϕ ∈ [0 , 2 π ] r ( ϕ ) < R -r . We let M be the image of [0 , 2 π ] 2 by ˜ F . Although we do not rigorously prove that the manifold M is almost surely generic, we conjecture that it is the case.

A probability measure on P on M is obtained in the following way. Sample ( θ, ϕ ) ∈ [0 , 2 π ] 2 so that F ( θ, ϕ ) is uniform on the torus. Then, let x = ˜ F ( θ, ϕ ) . To put it another way, P is the pushforward of the uniform measure on the torus by ˜ F ◦ F -1 .

Continuity of feature maps We sample n = 10 4 points according to P , and compute the ˇ Cech PDs of the corresponding set A n . For i = 1 , we plot the persistence images with weights pers p , p = 0 , 1 , 2 , 4 , with both birth and persistence values ranging between 0 and 1 , and grid step equal to 0 . 002 . See Figure 4.

Convergence of the total persistences. We run three experiments. First, we compute the ˇ Cech PD for i = 0 of n uniform points on a (nongeneric) circle. Second, we compute the ˇ Cech PD for i = 0 of n uniform points on a (nongeneric) torus. Third, we compute the ˇ Cech PD for i = 1 of n uniform points on a (nongeneric) torus. Each experiment is ran for values of n ranging from n = 10 2 to n = 10 4 . We observe in Figure 5 the convergence rate of the total persistence of dgm (1) i ( A n ) predicted by Theorem 4.1. We then repeat each experiment 10 times, with similar rates of convergence observed for each run, although with a large relative standard deviation (around 30% ) for very small values of Pers p (dgm (1) i ( A n )) (the relative standard deviation being defined by the ratio between the standard deviation and the mean).

Convergence of µ n,i . Let i = 1 . Let Q be the probability distribution obtained as the pushforward of the uniform distribution on [0 , 2 π ] 2 by the map F defined above. Using Equation (49), one can show that the density f of Q is given by

where ( θ, ϕ ) = F -1 ( x ) . Assume that the Radon measure µ ∞ ,i, 2 has a density g 0 on Ω (this fact is not proven, but conjectured and strongly supported by experiments). Then, using (9), one can make a change of variables and observe that µ f,i has a density g f , given by

We sample a set A n of n = 10 5 points on a square and consider the convolution of the measure 1 n ∑ u ∈ dgm i ( A n ) δ n 1 /m u with a Gaussian kernel, so that we obtain an estimation ˆ g 0 of the density g 0 . We then define an estimator ˆ g f of g f by approximating the integral over a regular grid of 100 points on [0 , 2 π ] .

We then place a grid of size 100 × 100 over the square [0 , 17] 2 . The measure µ f,i is approximated by the measure ˆ µ f,i with piecewise constant density on each square of the grid, with density in the square centered at x given by ˆ g f ( x ) . The corresponding measure is displayed in the center of Figure 6.

For a given value of n , we sample a set A n of n points according to Q , and compute the corresponding measure µ n,i . The same transformations are applied to µ n,i , so that we obtain a piecewise constant measure ˆ µ n,i on the same 100 × 100 grid. The heatmap of this measure is shown on the left of Figure 6.

At last, we approximate the distance OT 2 ( µ n,i , µ f,i ) by computing the distance OT 2 (ˆ µ n,i , ˆ µ f,i ) using the POT library, see [40, 55]. The distance is then normalized by OT 2 (0 , ˆ µ f,i ) = √ Pers 2 (ˆ µ f,i ) . The evolution of the normalized distance with respect to n is plotted in Figure 6. As predicted by Theorem 4.1, the OT 2 distance converges to 0 as n gets larger. We then repeat each experiment 10 times, with a convergence to 0 observed each time, the relative standard deviation being always smaller than 7% .

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper provides a more fine-grained analysis of stability results for persistent homology for data sampled from manifolds, proving new theoretical guarantees for methods in topological data analysis.\n\nSTRENGTHS:\nThe results of the paper are in this reviewers opinion strong and interesting. Topological methods, in particular persistent homology, has become a widely used and powerful tool for the analysis of data, in particular data zhat is sampled from an underlying manifold. In many cases, the use of persistent homology can add an additional layer of explainability. For this, mathematical guarantees are fundamental. The new guarantees proven in this paper are significantly stronger than previous ones. In this reviewers opinion the proofs are correct and the precise arguments as well as the underlying ideas are nicely presented.\n\nWEAKNESSES:\nAs the authors note, the proofs require some assumptions, which are not always guaranteed: they assume the manifold hypothesis and that the points are sampled without noise. Also, their analysis is based on Cech persistence and not Vietoris-Rips persistence, which is used more often in practice. In this reviewers opinion it is however still reasonable to make these assumptions, as they provide the necessary structure to thoroughly mathematically analyze the used tool (in this case persistent homology) in an „ideal world setting“, as is often done for mathematical guarantees.\n\nQUESTIONS:\n-Did you perform any experiments on data with noise or with Vietoris-Rips persistence? If so, how did the results compare to the ones you present in the paper?\n\nLIMITATIONS:\nThe limitations (that are also mentioned above) are addressed and explained at multiple points in the paper:\n-in line 169 the authors argue why some assumptions are always needed.\n-in line 216 they explain why the genericity assumption is needed for their Theorem 3.3.\n- in line 353 they again mention the assumptions as the main limitation of their work.", "TITLE": "Official Review", "RECOMMENDATION": 9, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-06-24"}

comments: SUMMARY:
The paper describes three new theorems on the stability of persistence diagrams with respect to Bottleneck and Wasserstein distance under certain additional assumptions.

STRENGTHS:
All three results are interesting fundamental contributions to the field of topological data analysis

WEAKNESSES:
I am not sure about how much the results are in scope for a machine learning conference. I do agree that a solid foundational layer is important but my feeling is that the connection could be described in more length (Sec 4.4 seems to serve this purpose partially).

The authors also admit themselves in the conclusion that the results rely on the manifold hypothesis and the absence of noise. I guess one could argue in length about the manifold hypothesis and how realistic it is in practice. But my hunch is that not allowing any noise is a more serious limitation.

QUESTIONS:
Why do you think that the paper is an adequate fit for Neurips, as opposed to, say, a journal specialized on TDA?

LIMITATIONS:
As mentioned, they address the limitations of their results adequately
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 4
DATE: 2024-07-05

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe $p$-optimal transport convergence of the Cech persistence diagram of a sample of a closed embedded manifold is studied, both in a deterministic and a probabilistic setting. The paper has three main results:\n1. An improvement of the classical Cech bottleneck stability result for sufficiently good samples of closed embedded manifolds.\n2. A bound for the total $p$-persistence, and a $p$-optimal transport-convergence result, for the Cech persistence diagram of sufficiently good (deterministic) samples of generically embedded closed manifolds; in the case $p$ is strictly larger than the dimension of the manifold.\n3. A probabilistic convergence result for the Cech persistence diagram, and a law of large numbers for its total persistence. The probabilistic convergence result says in particular that $p$-optimal transport convergence of the Cech persistence diagram occurs if and only if $p$ is strictly larger than the manifold dimension.\n\nSTRENGTHS:\n- The paper is very well written: it is clear and easy to follow, it contains many useful and relevant references, and the results are clear and well abstracted.\n- The results are (to the best of my understanding) novel, as well as relevant for both theoretical and applied purposes. The results are connected to a lot of previous literature, so they should be of great interest to researchers in stochastic topology as well as the Topological Data Analysis public in general.\n- The probabilistic result identifies precisely the choices of $p$ that lead to convergence.\n\nWEAKNESSES:\n- The intersection of Topological Data Analysis and Machine Learning is non-trivial but not huge, restricting the audience of the paper to some extent.\n- The results have several restrictive hypothesis: strict manifold hypothesis (data lies exactly on a manifold), genericity of the manifold, no noise of any kind.\n\nQUESTIONS:\n1. Is there any hope that (at least part of) the main probabilistic result (Corollary 4.3) still holds for non-generic manifolds? Could the genericity assumption be removed if we instead assume that the data is sampled from a manifold plus noise?\n2. Theorem 3.3 fails for non-generic manifolds. Is it possible to quantify (say, with a number or function) how generic a manifold is, in such a way that the constants of Theorem 3.3 only depend on the level of genericity of the manifold?\n3. In your summary of Corollary 4.3, in the introduction, why don't you say that optimal transport convergence occurs if and only if $p > m$?\n\nLIMITATIONS:\nThe manifold being generic also seems like a limitation to me. It seems genericity could fail in highly structured scenarios (data sampled from a configuration space or a simulation of a dynamical system). Please correct me if I am wrong.", "TITLE": "Official Review", "RECOMMENDATION": 8, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-07-11"}

comments: SUMMARY:
The authors study the behaviour of persistent homology under subsampling of compact sets. They have provided new convergence guarantees with respect to the p-Wasserstein distance and asymptotic results for their $\alpha$-persistence.

STRENGTHS:
(S1)	The paper addresses a relevant problem in TDA, and proves a number of significant results

(S2)	The paper is comprehensive and well-written

WEAKNESSES:
(W1) While the main results of this paper are theoretically solid and of strong interest to the TDA community, I do not see direct applications of this work to AI/ML. The authors have referenced a few works where their work might be applied (persistent homology dimension), but they have not elaborated enough on how exactly their method can be used, what kind of significant questions it can answer, and how successful that will be. 

(W2) The experiments are very limited and do not demonstrate the implications of the proved results in ML

QUESTIONS:
(Q1) Are there any results about real-world applications of the theory?

LIMITATIONS:
Yes, all limitations are addressed adequately by the authors.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-12