11affiliationtext: Université Paris-Saclay, CNRS, Inria, Laboratoire de
Mathématiques d’Orsay22affiliationtext: Université Côte d’Azur, Inria Sophia
Antipolis-Mediterranée33affiliationtext: CEREMADE, Université Paris-Dauphine,
Université PSL**footnotetext: The authors contributed equally to this work.

# Wasserstein convergence of Čech persistence diagrams for samplings of
submanifolds

Charles Arnal∗ David Cohen-Steiner∗ Vincent Divol∗

###### Preamble

Čech 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.

##  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 $H_{i}({\mathsf{A}})$ with coefficients
in some field $\mathcal{F}$ of a set ${\mathsf{A}}$, which captures the
$i$-dimensional topological features of ${\mathsf{A}}$ for $i\geq 0$, see e.g.
[Hat00]. TDA has been successfully applied in a variety of domains, including
material science [KKP+16, STR+17, DKR+18, BHO18], biology [CR20, SL22,
ACC+22], and neuroscience [SPCGB19, RYB+20, CPH21], 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 [CCI+20, KKZ+20].

The most prominent techniques in TDA rely on multiscale approaches, in
particular through the use of persistent homology [CDSGO16]. Given a compact
set ${\mathsf{A}}$ in ${\mathbb{R}}^{d}$, persistent homology tracks the
evolution of the homology groups $H_{i}({\mathsf{A}}^{t})$ of the $t$-offset
${\mathsf{A}}^{t}=\bigcup_{x\in{\mathsf{A}}}\overline{B}(x,t)$ of
${\mathsf{A}}$ as $t$ goes from $0$ to $+\infty$ (where $\overline{B}(x,t)$ is
the closed ball of radius $t$ centered at $x$). The process is summarized by
the Čech persistence diagram (PD) of degree $i$ of the set ${\mathsf{A}}$: the
PD $\mathrm{dgm}_{i}({\mathsf{A}})$ is a multiset111A multiset is a set where
each element appears with some non-zero multiplicity. of points in the half-
plane $\Omega:=\\{(u_{1},u_{2})\in{\mathbb{R}}^{2}:\ u_{1}<u_{2}\\}$, where
each point $(u_{1},u_{2})$ in the PD corresponds to a $i$-dimensional
topological feature that appeared in ${\mathsf{A}}^{t}$ at scale $t=u_{1}$
(its birth time) and disappeared at scale $t=u_{2}$ (its death time), see
Figure 1.222In general, PDs can have points with infinite coordinates. For
Čech PDs, this will only be the case for a single point of the diagram for
$i=0$, of coordinate $(0,+\infty)$. We discard this point in the following.
Points close to the diagonal
$\partial\Omega=\\{(u_{1},u_{2})\in{\mathbb{R}}^{2}:\ u_{1}=u_{2}\\}$
correspond to topological features of small persistence
$\mathrm{pers}(u)=\frac{u_{2}-u_{1}}{2}$, which have a short lifetime in the
filtration $({\mathsf{A}}^{t})_{t\geq 0}$.

Figure 1: The Čech PD of a point cloud ${\mathsf{A}}$ in ${\mathbb{R}}^{2}$
for $i=1$ and its $t$-offsets. The two points far from the diagonal
$\partial\Omega$ in $\mathrm{dgm}_{i}({\mathsf{A}})$ correspond to the two
large cycles in the set ${\mathsf{A}}$.

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 [ELZ02].
Let $a$ and $b$ be two PDs. A partial matching between $a$ and $b$ is a
bijection of multisets $\gamma:a\cup\partial\Omega\rightarrow
b\cup\partial\Omega$, where each point $(u,u)$ of $\partial\Omega$ has
infinite multiplicity. In other words, the points of $a$ are either paired
with a single point of $b$, or mapped to the diagonal $\partial\Omega$ (and
similarly for the points of $b$). Let $\Gamma(a,b)$ be the set of all partial
matchings between $a$ and $b$. The bottleneck distance is defined as

$\mathrm{OT}_{\infty}(a,b)=\inf_{\gamma\in\Gamma(a,b)}\max_{u\in
a\cup\partial\Omega}\|u-\gamma(u)\|_{\infty}.$ (1)

The Bottleneck Stability Theorem [CSEH05, CDSGO16] states that if
${\mathsf{A}}_{1}$ and ${\mathsf{A}}_{2}$ are two compact sets, then for any
integer $i\geq 0$

$\mathrm{OT}_{\infty}(\mathrm{dgm}_{i}({\mathsf{A}}_{1}),\mathrm{dgm}_{i}({\mathsf{A}}_{2}))\leq\varepsilon,$
(2)

where $\varepsilon$ is the Hausdorff distance between the sets
${\mathsf{A}}_{1}$ and ${\mathsf{A}}_{2}$, defined by

$d_{H}({\mathsf{A}}_{1},{\mathsf{A}}_{2})=\sup_{x\in{\mathbb{R}}^{d}}|d_{{\mathsf{A}}_{1}}(x)-d_{{\mathsf{A}}_{2}}(x)|$

and where $d_{\mathsf{A}}$ is the distance function to a set ${\mathsf{A}}$.
An important property of the bottleneck distance is that it is blind to small-
persistence topological features: if $\mathrm{OT}_{\infty}(a,b)=\varepsilon$,
then one can arbitrarily modify the PDs $a$ and $b$ on a slab of width
$\varepsilon$ above the diagonal (for the $\ell_{\infty}$-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 $\mathrm{OT}_{p}$, are defined for $1\leq
p<\infty$ by

$\mathrm{OT}_{p}(a,b)=\inf_{\gamma\in\Gamma(a,b)}\left(\sum_{u\in
a\cup\partial\Omega}\|u-\gamma(u)\|_{\infty}^{p}\right)^{1/p},$ (3)

with $\mathrm{OT}_{p}\leq\mathrm{OT}_{p^{\prime}}$ for $1\leq p^{\prime}\leq
p<\infty$. They can be seen as modified versions of the Wasserstein distances
used in optimal transport, with the diagonal $\partial\Omega$ playing the role
of a landfill of infinite mass, see e.g. [DL21].

The increased sensitivity to small perturbations of the $\mathrm{OT}_{p}$
distances is of crucial importance in the standard TDA pipeline, which we
briefly recall. Starting from a sample of sets
${\mathsf{A}}_{1},\dots,{\mathsf{A}}_{n}$, one computes a sample of PDs
$a_{j}=\mathrm{dgm}_{i}({\mathsf{A}}_{j})$, $j=1,\dots,n$. Statistical methods
to analyze this sample of PDs are typically awkward to define, due to the
nonlinear geometry of the space of PDs [BW20, Wag21]. To overcome this issue,
the space of PDs is mapped to a vector space through some map $\Phi$ called a
feature map. Statistical method are then applied in the feature space on the
transformed sample $\Phi(a_{1}),\dots,\Phi(a_{n})$. Various feature maps have
been designed [CFL+14, RHBK15, CWRW15, KHF16, CCO17, STR+18, KFH18, HKN19],
important examples including persistent images [AEK+17], PersLay [CCI+20], and
PLLay [KKZ+20]; 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 [MV21]; 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 $\mathrm{OT}_{\infty}$ distance on the space
of diagrams. Instead, they enjoy Lipschitz regularity with respect to either
the finer $\mathrm{OT}_{1}$ distance (see e.g. [DL21, Proposition 5.2]), or to
the $\mathrm{OT}_{p}$ distances (for $p>1$) when restricted to diagrams $a$
whose $\alpha$-total persistence

$\mathrm{Pers}_{\alpha}(a)=\sum_{u\in a}\mathrm{pers}(u)^{\alpha}$ (4)

is bounded for some $\alpha>0$ large enough, see [DP19] and [KFH18].
Boundedness assumptions on the $\alpha$-total persistence also yield a version
of the Bottleneck Stability Theorem with respect to the finer
$\mathrm{OT}_{p}$ distance: if ${\mathsf{A}}_{1}$ and ${\mathsf{A}}_{2}$ are
such that $\mathrm{Pers}_{\alpha}(\mathrm{dgm}_{i}({\mathsf{A}}_{k}))\leq M$
($k=1,2$), then, for $p\geq\alpha$,

$\mathrm{OT}_{p}^{p}(\mathrm{dgm}_{i}({\mathsf{A}}_{1}),\mathrm{dgm}_{i}({\mathsf{A}}_{2}))\leq
M\varepsilon^{p-\alpha},$ (5)

where $\varepsilon=d_{H}({\mathsf{A}}_{1},{\mathsf{A}}_{2})$, see [CSEHM10].
Hence, in addition to having intrinsic theoretical interest, controlling the
total persistence and convergence with respect to the $\mathrm{OT}_{p}$
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 Čech PDs
in the specific case where the underlying set ${\mathsf{A}}$ is a compact
subset of an $m$-dimensional manifold ${\mathsf{M}}$ in ${\mathbb{R}}^{d}$,
focusing in particular on the total persistence of the PDs and their
convergence to the PDs of ${\mathsf{M}}$ with respect to the $\mathrm{OT}_{p}$
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 [GPPVW12, WB19,
BCR+22]. Specifically, our main contributions are the following:

  * •

2.3: When ${\mathsf{A}}\subset{\mathsf{M}}$ is a compact set satisfying
$d_{H}({\mathsf{A}},{\mathsf{M}})\leq\varepsilon$ for $\varepsilon$ small
enough, we provide a quadratic improvement upon the standard Bottleneck
Stability Theorem (2). Namely, we show that there exists an optimal bottleneck
matching $\gamma$ such that the distance between a coordinate of a point
$u\in\mathrm{dgm}_{i}({\mathsf{A}})$ and the coordinate of the matched point
$\gamma(u)\in\mathrm{dgm}_{i}({\mathsf{M}})\cup\partial\Omega$ is of order
$O(\varepsilon^{2})$ whenever the coordinate of $u$ is larger than
$2\varepsilon$.

  * •

3.7: In the case where the manifold ${\mathsf{M}}$ is generic and the set
${\mathsf{A}}$ is a $\delta$-sparse point cloud (i.e. $\min_{x\neq
y\in{\mathsf{A}}}\|x-y\|\geq\delta$), we provide a finer analysis by showing
that the $p$-total persistence
$\mathrm{Pers}_{p}(\mathrm{dgm}_{i}({\mathsf{A}}))$ remains bounded and the
distance
$\mathrm{OT}_{p}(\mathrm{dgm}_{i}({\mathsf{A}}),\mathrm{dgm}_{i}({\mathsf{M}}))$
converges to $0$ for all $p>m$ whenever the ratio $\varepsilon/\delta$ is
upper bounded.

  * •

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

$\mathrm{Pers}_{\alpha}(\mathrm{dgm}_{i}({\mathsf{A}}_{n}))=\mathrm{Pers}_{\alpha}(\mathrm{dgm}_{i}({\mathsf{M}}))+C_{i}n^{1-\alpha/m}+o_{L^{1}}(n^{1-\alpha/m})+O_{L^{1}}\Big{(}\left(\frac{\log
n}{n}\right)^{\frac{1}{m}}\Big{)}$ (6)

for all $\alpha>0$, where $C_{i}$ is a constant that depends explicitly on
${\mathsf{M}}$ and $f$. In particular, for $0\leq i<m$,
$\mathrm{Pers}_{\alpha}(\mathrm{dgm}_{i}({\mathsf{A}}_{n}))$ stays bounded if
and only if $\alpha\geq m$.

Our contributions are to be compared to one of the only preexisting results
regarding the $\alpha$-total persistence of a PD: in [CSEHM10], the authors
proved that for all $\alpha$ strictly greater than the ambient dimension $d$,
the $\alpha$-total persistence of the Čech PD of a compact set
${\mathsf{A}}\subset B(0,R)\subset{\mathbb{R}}^{d}$ satisfies

$\mathrm{Pers}_{\alpha}(\mathrm{dgm}_{i}({\mathsf{A}}))\leq
C_{\alpha,d}R^{\alpha}$ (7)

for some constant $C_{\alpha,d}$ depending on $\alpha$ and $d$. In the two
scenarios we considered (either $\delta$-sparse or random samples), the
ambient dimension $d$ in the constraint $\alpha>d$ for the control of the
$\alpha$-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 4.18), 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.

Related work. This work is part of a long ongoing effort to understand
simplicial complexes and PDs in a random context [BA14, BKS17, BW17, BK18,
KM13, DC19, Owa22, BH22, HL23]. Closest to our work, Hiraoka, Shirai and Trinh
gave limit laws for Čech PDs for random points in the cube $[0,1]^{d}$
[HST18], while Goel, Trinh and Tsunoda gave similar asymptotics in the case of
samples on manifolds [GTT19]. Limit laws for the total persistence have been
obtained by Divol and Polonik in the case of random samples in the cube
[DP19]. 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 Čech PDs in order to control
the total persistence.

Notation. To simplify notation, we write $K=K({\mathsf{M}})$ (or
$K=K({\mathsf{M}},a,\ldots)$) to implicitly state that some constant $K$
depends only on ${\mathsf{M}}$ (or only on ${\mathsf{M}}$, $a$, etc.).

##  2 Čech persistence diagrams for subsets of submanifolds

Although this work is only concerned with PDs with respect to the Čech
filtration, it is more natural to define PDs for the sublevel sets of proper
continuous functions that are bounded below. We refer to [CDSGO16] for a
thorough introduction to persistent homology and PDs in an even more general
context.

Let $f:{\mathbb{R}}^{d}\to{\mathbb{R}}$ be a proper continuous function that
is bounded below–e.g., the distance function to a compact set. For
$t\in{\mathbb{R}}$, let $X_{t}=f^{-1}(-\infty,t]$ be the sublevel set of $f$
at level $t$. The collection $(X_{t})_{t\in{\mathbb{R}}}$ is called a
filtration. Let $i\geq 0$ be an integer. We let $H_{i}(X_{t})$ be the homology
group of degree $i$ with coefficients in any fixed field ${\mathcal{F}}$ (e.g.
${\mathcal{F}}={\mathbb{Z}}/2{\mathbb{Z}}$ is a popular choice) of $X_{t}$.

The persistence diagram $\mathrm{dgm}_{i}(f)$ of degree $i$ of $f$ tracks the
evolution of the homology groups of degree
$(H_{i}(X_{t}))_{t\in{\mathbb{R}}}$. Define the extended half-plane
$\Omega_{\infty}=\\{u=(u_{1},u_{2})\in({\mathbb{R}}\cup\\{-\infty,+\infty\\})^{2}:\
-\infty\leq u_{1}<u_{2}\leq+\infty\\}$ and
$\Omega=\\{u=(u_{1},u_{2})\in{\mathbb{R}}^{2}:\ u_{1}<u_{2}\\}$. Then
$\mathrm{dgm}_{i}(f)$ presents itself as a multiset (a set whose elements can
have multiplicity greater than $1$) in $\Omega_{\infty}$. We refer to
[CDSGO16] for its precise definition, but it can be intuitively understood as
follows: for $u=(u_{1},u_{2})\in\Omega_{\infty}$, the number of points in the
persistence diagram $\mathrm{dgm}_{i}(f)$ found in the upper-left quadrant
$Q_{u}=\\{v\in({\mathbb{R}}\cup\\{-\infty,+\infty\\})^{2}:\ v_{1}\leq
u_{1}<u_{2}\leq v_{2}\\}$ is informally equal to the number of $i$th
dimensional features present in the sublevel set $f^{-1}(-\infty,u_{1}]$
(captured through its $i$th homology group $H_{i}(f^{-1}(-\infty,u_{1}])$)
that are still present in the sublevel set $f^{-1}(-\infty,u_{2}]$. Note that
this number of points is always finite [CDSGO16, Corollary 3.34]. Though
persistence diagrams can have points with infinite coordinates, these will be
of little interest in the cases considered in this article.333The persistence
diagram of degree $i$ of the distance function to a set has no point with
infinite coordinates if $i>0$, and a single such point if $i=0$ whose
coordinates are $(0,+\infty)$. To simplify our notation and definitions, we
let from now on $\mathrm{dgm}_{i}(f)$ 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
$\Omega:=\\{u=(u_{1},u_{2})\in{\mathbb{R}}^{2}:\ u_{1}<u_{2}\\}$, which leads
to the following definition: the space $\mathcal{D}$ of persistence diagrams
is the set of all multisets $a$ in $\Omega$ that contain a finite number of
points444This corresponds to the set of diagrams of q-tame persistence modules
as defined in [CDSGO16]. in every quadrant $Q_{u}$, $u\in\Omega$.

In this paper, we are interested in the PDs of the distance function
$d_{\mathsf{A}}$ to various compact sets ${\mathsf{A}}$, called the Čech
persistence diagram555The distance function $d_{\mathsf{A}}$ is proper due to
the compacity of ${\mathsf{A}}$, hence its persistence module is $q$-tame and
the associated persistence diagram is well-defined –see [CDSGO16]. of
${\mathsf{A}}$ and denoted by $\mathrm{dgm}_{i}({\mathsf{A}})$.

Changes in the topology of the sublevels of $d_{\mathsf{A}}$ can be partially
described in terms of zeros of its generalized gradient, which is defined at
$y\in{\mathbb{R}}^{d}\backslash{\mathsf{A}}$ as

$\nabla
d_{\mathsf{A}}(y)=\frac{y-c(\sigma_{\mathsf{A}}(y))}{d_{\mathsf{A}}(y)},$ (8)

where $\sigma_{\mathsf{A}}(y)=\\{x\in{\mathsf{A}}:\
\|x-y\|=d_{\mathsf{A}}(y)\\}$ is the set of projections of $y$ on
${\mathsf{A}}$ and $c(\tau)$ represents the center of the smallest enclosing
ball of a set $\tau$. When $y\in{\mathbb{R}}^{d}\backslash{\mathsf{A}}$
satisfies $\nabla d_{\mathsf{A}}(y)=0$, $y$ is called a differential critical
point of $d_{\mathsf{A}}$. We let $\mathrm{Crit}({\mathsf{A}})$ denote the set
of differential critical points of $d_{\mathsf{A}}$. An adapted version of the
classical Isotopy Lemma for Morse functions is true for distance functions to
compact sets as well, as shown in [Gro93]:

######  Lemma 2.1 (Isotopy Lemma for Distance Functions).

If $0<a<b$ are such that $d_{\mathsf{A}}^{-1}[a,b]$ contains no differential
critical point of $d_{\mathsf{A}}$, then $d_{\mathsf{A}}^{-1}(-\infty,a]$ is a
deformation retract of $d_{\mathsf{A}}^{-1}(-\infty,b]$. Consequently, any
$(u_{1},u_{2})\in\mathrm{dgm}_{i}({\mathsf{A}})$ is such that
$u_{1},u_{2}\not\in[a,b]$.

A fundamental result in TDA, the Bottleneck Stability Theorem (2), states that
Čech PDs are stable with respect to Hausdorff perturbations. Consider the
particular setting where one has access to a set ${\mathsf{A}}$, obtained as
an approximation of an unknown shape of interest ${\mathsf{S}}$ through some
sampling procedure, with ${\mathsf{A}}\subset{\mathsf{S}}$ and
$\sup_{x\in\mathsf{S}}d_{\mathsf{A}}(x)\leq\varepsilon$. The Bottleneck
Stability Theorem ensures that
$\mathrm{OT}_{\infty}(\mathrm{dgm}_{i}({\mathsf{A}}),\mathrm{dgm}_{i}({\mathsf{S}}))\leq\varepsilon$
for any $i\geq 0$, a bound which cannot be improved in general. However, it
turns out that a finer understanding of the proximity between
$\mathrm{dgm}_{i}({\mathsf{A}})$ and $\mathrm{dgm}_{i}({\mathsf{S}})$ can be
obtained if more regularity is assumed on the shape of interest
${\mathsf{S}}$, namely in the situation where ${\mathsf{S}}={\mathsf{M}}$ is a
compact submanifold with positive reach.

Let us first set some notation. Let ${\mathsf{M}}$ be an $m$-dimensional
compact topological submanifold of ${\mathbb{R}}^{d}$; we always assume that
the boundary of ${\mathsf{M}}$ is empty and that $m\geq 1$. The orthogonal
projection $\pi_{\mathsf{M}}$ on ${\mathsf{M}}$ is defined for $x$ close
enough to ${\mathsf{M}}$, and we define the reach $\tau({\mathsf{M}})$ as the
largest $r>0$ such that the orthogonal projection $\pi_{\mathsf{M}}$ is well
(i.e. uniquely) defined for all $x\in{\mathbb{R}}^{d}$ at distance strictly
less than $r$ from ${\mathsf{M}}$. The reach is a key notion to quantify the
regularity of a manifold, see e.g. [Fed59] and [CCSL06] for more information.
In particular, any compact $C^{2}$ submanifold has positive reach. For
$x\in{\mathsf{M}}$, we let $T_{x}{\mathsf{M}}$ be the tangent space of
${\mathsf{M}}$ at $x$, which is identified with a linear subspace of
${\mathbb{R}}^{d}$. We denote by $\pi_{x}:{\mathbb{R}}^{d}\to
T_{x}{\mathsf{M}}$ the orthogonal projection on this subspace and let
$\pi_{x}^{\bot}=id-\pi_{x}$ be the orthogonal projection on the normal space
at $x$. A key property, that we will repeatedly used, is that the reach of
${\mathsf{M}}$ controls the deviation of the manifold ${\mathsf{M}}$ from its
tangent space. Namely, [Fed59, Theorem 4.18] states that for all
$y\in{\mathsf{M}}$,

$\|\pi_{x}^{\bot}(x-y)\|\leq\frac{\|x-y\|^{2}}{2\tau({\mathsf{M}})}.$ (9)

We also define the weak feature size of ${\mathsf{M}}$, denoted by
$\mathrm{wfs}({\mathsf{M}})$, as the minimal distance between a critical point
of ${\mathsf{M}}$ and ${\mathsf{M}}$. As by definition, the projection is not
unique at a critical point, we must have
$\mathrm{wfs}({\mathsf{M}})\geq\tau({\mathsf{M}})$.

Let ${\mathsf{A}}\subset{\mathsf{M}}$ be such that
$d_{H}({\mathsf{A}},{\mathsf{M}})\leq\varepsilon$ and let
$z\in{\mathbb{R}}^{d}$. By definition of the Hausdorff distance, it holds that
$|d_{{\mathsf{A}}}(z)-d_{\mathsf{M}}(z)|\leq\varepsilon$. However, this naive
bound can be quadratically improved as long as $z$ stays far away from
${\mathsf{M}}$.

######  Lemma 2.2.

Let ${\mathsf{M}}\subset{\mathbb{R}}^{d}$ be a compact submanifold with
positive reach and let ${\mathsf{A}}\subset{\mathsf{M}}$ be a compact set with
$d_{H}({\mathsf{A}},{\mathsf{M}})\leq\varepsilon$ for some $\varepsilon>0$.
Let $z\in{\mathbb{R}}^{d}\backslash{\mathsf{M}}$. Then,
$|d_{\mathsf{M}}(z)-d_{\mathsf{A}}(z)|\leq\frac{\varepsilon^{2}}{2d_{\mathsf{M}}(z)}\left(1+\frac{d_{\mathsf{M}}(z)}{\tau({\mathsf{M}})}\right)$.

###### Proof.

As ${\mathsf{A}}\subset{\mathsf{M}}$, we have $d_{\mathsf{M}}(z)\leq
d_{\mathsf{A}}(z)$. Let $x$ be a projection of $z$ onto ${\mathsf{M}}$, and
let $y\in{\mathsf{A}}$ be a point at distance less than $\varepsilon$ from
$x$. Then, using (9) and the fact that $z-x$ is orthogonal to
$T_{x}{\mathsf{M}}$, we obtain that

$\displaystyle d_{\mathsf{A}}(z)^{2}$
$\displaystyle\leq\|z-y\|^{2}=\|z-x\|^{2}+\|x-y\|^{2}+2\langle z-x,x-y\rangle$
$\displaystyle\leq d_{\mathsf{M}}(z)^{2}+\varepsilon^{2}+2\langle
z-x,\pi_{x}^{\bot}(x-y)\rangle$ $\displaystyle\leq
d_{\mathsf{M}}(z)^{2}+\varepsilon^{2}+\frac{d_{\mathsf{M}}(z)\varepsilon^{2}}{\tau({\mathsf{M}})}.$

Hence,

$d_{\mathsf{A}}(z)-d_{\mathsf{M}}(z)=\frac{d_{\mathsf{A}}(z)^{2}-d_{\mathsf{M}}(z)^{2}}{d_{\mathsf{A}}(z)+d_{\mathsf{M}}(z)}\leq\frac{\varepsilon^{2}}{2d_{\mathsf{M}}(z)}\left(1+\frac{d_{\mathsf{M}}(z)}{\tau({\mathsf{M}})}\right).\qed$

Figure 2:  PDs of ${\mathsf{M}}$ (red) and of ${\mathsf{A}}$ (black).

We can build upon this basic remark to obtain a very precise control of the
behavior of the Čech PD of the set ${\mathsf{A}}$. Namely, we identify three
regions in the upper halfplane $\Omega$ (displayed in Figure 2) which contain
all points in the PD $\mathrm{dgm}_{i}({\mathsf{A}})$ (for some integer $i\geq
0$). In the first region, corresponding to microscopic topological features
disappearing at scales smaller than
$\varepsilon+\varepsilon^{2}/\tau({\mathsf{M}})$, the Bottleneck Stability
Theorem cannot be improved. However, there exists an optimal matching (i.e. a
matching
$\gamma:\mathrm{dgm}_{i}({\mathsf{A}})\cup\partial\Omega\to\mathrm{dgm}_{i}({\mathsf{M}})\cup\partial\Omega$
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
$\tau({\mathsf{M}})-\varepsilon^{2}/\tau({\mathsf{M}})$, and the proximity
between a large coordinate of a point $u\in\mathrm{dgm}_{i}({\mathsf{A}})$ and
the coordinate of the matched point
$\gamma(u)\in\mathrm{dgm}_{i}({\mathsf{M}})\cup\partial\Omega$ is of order
$O(\varepsilon^{2})$. This yields a quadratic improvement upon the Bottleneck
Stability Theorem.

######  Theorem 2.3 (Improved Bottleneck Stability Theorem).

Let ${\mathsf{M}}\subset{\mathbb{R}}^{d}$ be a compact submanifold with
positive reach and let ${\mathsf{A}}\subset{\mathsf{M}}$ be a compact set such
that $d_{H}({\mathsf{A}},{\mathsf{M}})\leq\varepsilon<\tau({\mathsf{M}})/4$.
Let $i\geq 0$ be an integer. Then $\mathrm{dgm}_{i}({\mathsf{A}})$ is the
union of three regions
$\mathrm{dgm}_{i}^{(1)}({\mathsf{A}}):=\mathrm{dgm}_{i}({\mathsf{A}})\cap\\{u_{1},u_{2}\leq\varepsilon+\frac{\varepsilon^{2}}{\tau({\mathsf{M}})}\\}$,
$\mathrm{dgm}_{i}^{(2)}({\mathsf{A}}):=\mathrm{dgm}_{i}({\mathsf{A}})\cap\\{u_{1}\leq\varepsilon,u_{2}\geq\tau({\mathsf{M}})-\frac{\varepsilon^{2}}{\tau({\mathsf{M}})}\\}$
and
$\mathrm{dgm}_{i}^{(3)}({\mathsf{A}}):=\mathrm{dgm}_{i}({\mathsf{A}})\cap\\{u_{1},u_{2}\geq\tau({\mathsf{M}})-\frac{\varepsilon^{2}}{\tau({\mathsf{M}})}\\}$.

Furthermore, let
$C=\frac{2}{\tau({\mathsf{M}})}\left(1+\frac{R({\mathsf{M}})}{\tau({\mathsf{M}})}\right)$,
where $R({\mathsf{M}})$ is the radius of the smallest ball that contains
${\mathsf{M}}$. There exists an optimal matching
$\gamma:\mathrm{dgm}_{i}({\mathsf{A}})\cup\partial\Omega\to\mathrm{dgm}_{i}({\mathsf{M}})\cup\partial\Omega$
for the bottleneck distance between $\mathrm{dgm}_{i}({\mathsf{A}})$ and
$\mathrm{dgm}_{i}({\mathsf{M}})$ such that

  * •

Region (1): If $u\in\mathrm{dgm}_{i}^{(1)}({\mathsf{A}})$, then
$\gamma(u)\in\partial\Omega$ and $\|u-\gamma(u)\|_{\infty}\leq\varepsilon$.

  * •

Region (2): If $u\in\mathrm{dgm}_{i}^{(2)}({\mathsf{A}})$, then $\gamma(u)$ is
of the form $(0,v_{2})$ and $|u_{2}-v_{2}|\leq C\varepsilon^{2}$. The number
of such points is finite and depends only on ${\mathsf{M}}$.

  * •

Region (3): If $u\in\mathrm{dgm}_{i}^{(3)}({\mathsf{A}})$, then
$\|u-\gamma(u)\|_{\infty}\leq C\varepsilon^{2}$.

Note that for any $i\geq d$, the $i$-th PDs of ${\mathsf{M}}$ and
${\mathsf{A}}$ are actually trivial. Furthermore, under the hypotheses of 2.3,
the sub-diagrams $\mathrm{dgm}_{i}^{(2)}({\mathsf{A}})$ and
$\mathrm{dgm}_{i}^{(2)}({\mathsf{M}})$ are empty if $i$ is strictly greater
than the dimension of ${\mathsf{M}}$.

Note also that Regions (2) and (3) can be made smaller as follows: there
exists $\delta:{\mathbb{R}}_{+}\rightarrow{\mathbb{R}}_{+}$ such that
$\lim_{\varepsilon\xrightarrow{>}0}\delta(\varepsilon)=0$ and such that
$\mathrm{dgm}_{i}^{(2)}({\mathsf{A}})\subset\mathrm{dgm}_{i}({\mathsf{A}})\cap\\{u_{1}\leq\varepsilon,u_{2}\geq\mathrm{wfs}({\mathsf{M}})-\delta(\varepsilon)\\}$
and
$\mathrm{dgm}_{i}^{(3)}({\mathsf{A}})\subset\mathrm{dgm}_{i}({\mathsf{A}})\cap\\{u_{1},u_{2}\geq\mathrm{wfs}({\mathsf{M}})-\delta(\varepsilon)\\}$
if $d_{H}({\mathsf{A}},{\mathsf{M}})\leq\varepsilon$. These inequalities are
proved using the lower semicontinuity of the generalized gradient, as well as
the Isotopy Lemma for distance functions. As
$\mathrm{wfs}({\mathsf{M}})\geq\tau({\mathsf{M}})$, this can yield tighter
constraints than in the statement of Proposition 2.3, but there is sadly no
simple description of the function $\delta$ (e.g. in terms of
$\tau({\mathsf{M}})$ or $\mathrm{wfs}({\mathsf{M}})$).

###### Proof.

Proposition 5 from [AKF+22] states that if
$\varepsilon=d_{H}({\mathsf{A}},{\mathsf{M}})<(\sqrt{2}-1)\tau({\mathsf{M}})$,
then the offset ${\mathsf{A}}^{r}$ deformation-retracts onto ${\mathsf{M}}$
for any

$r\in\left[\frac{1}{2}(\tau({\mathsf{M}})+\varepsilon-\sqrt{\Delta}),\frac{1}{2}(\tau({\mathsf{M}})+\varepsilon+\sqrt{\Delta})\right]$

and
$\Delta=\tau({\mathsf{M}})^{2}-2\varepsilon\tau({\mathsf{M}})-\varepsilon^{2}$.
Under the stronger assumption that
$d_{H}({\mathsf{A}},{\mathsf{M}})<\tau({\mathsf{M}})/4$, and using elementary
calculus, we find that the offset ${\mathsf{A}}^{r}$ deformation-retracts onto
${\mathsf{M}}$ for any

$r\in\left[\varepsilon+\frac{\varepsilon^{2}}{\tau({\mathsf{M}})},\tau({\mathsf{M}})-\frac{\varepsilon^{2}}{\tau({\mathsf{M}})}\right].$

This means in particular that the homology type, hence the homology, of
${\mathsf{A}}^{r}$ does not change in that interval; as a result, there can be
no birth or death of intervals in the Čech persistence diagrams of
${\mathsf{A}}$ between
$\varepsilon+\frac{\varepsilon^{2}}{\tau({\mathsf{M}})}$ and
$\tau({\mathsf{M}})-\frac{\varepsilon^{2}}{\tau({\mathsf{M}})}$, and all
$(u_{1},u_{2})\in\mathrm{dgm}_{i}({\mathsf{A}})$ must either be such that
$u_{1},u_{2}\leq\varepsilon+\frac{\varepsilon^{2}}{\tau({\mathsf{M}})}$, or
$u_{1}\leq\varepsilon+\frac{\varepsilon^{2}}{\tau({\mathsf{M}})},u_{2}\geq\tau({\mathsf{M}})-\frac{\varepsilon^{2}}{\tau({\mathsf{M}})}$,
or
$u_{1},u_{2}\geq\tau({\mathsf{M}})-\frac{\varepsilon^{2}}{\tau({\mathsf{M}})}$.
This almost proves that the partition of $\mathrm{dgm}_{i}({\mathsf{A}})$ into
$\mathrm{dgm}_{i}^{(1)}({\mathsf{A}})$, $\mathrm{dgm}_{i}^{(2)}({\mathsf{A}})$
and $\mathrm{dgm}_{i}^{(3)}({\mathsf{A}})$ as defined in the statement is
correct, except that the definition of $\mathrm{dgm}_{i}^{(2)}({\mathsf{A}})$
requires that $u_{1}\leq\varepsilon$, whereas we only have obtained that
$u_{1}\leq\varepsilon+\frac{\varepsilon^{2}}{\tau({\mathsf{M}})}$.

Let
$\gamma:\mathrm{dgm}_{i}({\mathsf{A}})\cup\partial\Omega\to\mathrm{dgm}_{i}({\mathsf{M}})\cup\partial\Omega$
be any optimal matching for the bottleneck distance: then the Bottleneck
Stability Theorem states that any point
$u=(u_{1},u_{2})\in\mathrm{dgm}_{i}({\mathsf{A}})$ is such that
$\|u-\gamma(u)\|_{\infty}\leq
d_{H}({\mathsf{A}},{\mathsf{M}})\leq\varepsilon$. Suppose that $u$ is such
that $u_{1}\leq\varepsilon+\frac{\varepsilon^{2}}{\tau({\mathsf{M}})}$ and
$u_{2}\geq\tau({\mathsf{M}})-\frac{\varepsilon^{2}}{\tau({\mathsf{M}})}$. Its
distance in the infinity norm to the diagonal $\partial\Omega$ is equal to
$(u_{2}-u_{1})/2\geq(\tau({\mathsf{M}})-\varepsilon-2\varepsilon^{2}/\tau({\mathsf{M}}))/2\geq\frac{5}{16}\tau({\mathsf{M}})>\varepsilon$,
where we use the fact that $\varepsilon<\tau({\mathsf{M}})/4$. Hence
$\gamma(u)$ must belong to $\mathrm{dgm}_{i}({\mathsf{M}})$. Let
$(v_{1},v_{2})$ denote $\gamma(u)$. The Isotopy Lemma for distance functions
shows that $\mathrm{dgm}_{i}({\mathsf{M}})$ only contains two types of points:
points of the shape $(0,w_{2})$, which correspond to the homology of
${\mathsf{M}}$ itself, and points of the shape $(w_{1},w_{2})$, where in both
cases $w_{1},w_{2}$ are critical values of $d_{\mathsf{M}}$. In particular,
both $w_{1}$ and $w_{2}$ must be greater than $\mathrm{wfs}({\mathsf{M}})$.
Let $\mathrm{dgm}_{i}^{(2)}({\mathsf{M}})$ denote the multiset of all points
of the first type, and $\mathrm{dgm}_{i}^{(3)}({\mathsf{M}})$ denote the
multiset of all points of the second type. If $v_{1}$ was non-zero, it would
have to be greater than $\mathrm{wfs}({\mathsf{M}})\geq\tau({\mathsf{M}})$,
and we would have
$\|u-\gamma(u)\|_{\infty}\geq|u_{1}-v_{1}|\geq\tau({\mathsf{M}})-\varepsilon-\frac{\varepsilon^{2}}{\tau({\mathsf{M}})}\geq\tau({\mathsf{M}})/2>\varepsilon$,
which would be a contradiction (we once again use that
$\varepsilon<\tau({\mathsf{M}})/4$). Hence $v_{1}$ must be $0$ (i.e.
$\gamma(u)\in\mathrm{dgm}_{i}^{(2)}({\mathsf{M}})$), and
$u_{1}=|u_{1}-v_{1}|\leq\|u-\gamma(u)\|_{\infty}\leq\varepsilon$. This proves
the correctness of the partition into regions from the statement.

Consider now $u=(u_{1},u_{2})\in\mathrm{dgm}_{i}^{(1)}({\mathsf{A}})$. All
$v=(v_{1},v_{2})\in\mathrm{dgm}_{i}({\mathsf{M}})$ are such that

$\|u-v\|_{\infty}\geq
v_{2}-u_{2}\geq\mathrm{wfs}({\mathsf{M}})-\varepsilon-\frac{\varepsilon^{2}}{\tau({\mathsf{M}})}\geq\tau({\mathsf{M}})\left(1-\frac{1}{4}-\frac{1}{16}\right)>\varepsilon\geq\|u-\gamma(u)\|_{\infty},$

hence $\gamma(u)$ must belong to $\partial\Omega$. This completes the proof of
the first bullet point of the statement.

We have already shown that if $\gamma$ is an optimal matching and
$u\in\mathrm{dgm}_{i}^{(2)}({\mathsf{A}})$, then
$\gamma(u)\in\mathrm{dgm}_{i}({\mathsf{M}})$ is of the form $(0,v_{2})$. As
$\gamma$ maps at most a single point of $\mathrm{dgm}_{i}^{(2)}({\mathsf{A}})$
to each point of $\mathrm{dgm}_{i}({\mathsf{M}})$, the number of such points
is upper bounded by the number of points of the form $(0,v_{2})$ with
$v_{2}\geq\mathrm{wfs}({\mathsf{M}})$ in $\mathrm{dgm}_{i}({\mathsf{M}})$.
Though $\mathrm{dgm}_{i}({\mathsf{M}})$ need not be finite, applying Corollary
3.34 from [CDSGO16] to $d_{\mathsf{M}}$ shows that
$\mathrm{dgm}_{i}({\mathsf{M}})$ is q-tame, and in particular that it contains
only a finite number $N$ of points $(v_{1},v_{2})$ with
$v_{1}\leq\mathrm{wfs}({\mathsf{M}})/4$ and
$v_{2}\geq\mathrm{wfs}({\mathsf{M}})/2$. Hence the cardinality of
$\mathrm{dgm}_{i}^{(2)}({\mathsf{A}})$ is bounded by $N$.

It only remains to show that $\gamma$ can be chosen such that if
$u=(u_{1},u_{2})\in\mathrm{dgm}_{i}^{(2)}({\mathsf{A}})$, respectively
$u^{\prime}\in\mathrm{dgm}_{i}^{(3)}({\mathsf{A}})$, then
$|u_{2}-\gamma(u)_{2}|\leq C\varepsilon^{2}$, respectively
$\|u^{\prime}-\gamma(u^{\prime})\|_{\infty}\leq C\varepsilon^{2}$. To that
end, remember first that as shown above, our starting optimal matching
$\gamma$ must be such that points in $\mathrm{dgm}_{i}^{(2)}({\mathsf{A}})$
must be matched to points $\mathrm{dgm}_{i}^{(2)}({\mathsf{M}})$. Conversely
and for the same reasons, points in $\mathrm{dgm}_{i}^{(2)}({\mathsf{M}})$
must be matched to points in $\mathrm{dgm}_{i}^{(2)}({\mathsf{A}})$.
Similarly, points $u\in\mathrm{dgm}_{i}^{(3)}({\mathsf{A}})$ can only be
matched to points in $\mathrm{dgm}_{i}^{(3)}({\mathsf{M}})$ or to the diagonal
$\partial\Omega$; otherwise,
$\|u-\gamma(u)\|_{\infty}\geq|u_{1}-\gamma(u)_{1}|=u_{1}$ would be too large.
Likewise, points in $\mathrm{dgm}_{i}^{(3)}({\mathsf{M}})$ can only be matched
to points in $\mathrm{dgm}_{i}^{(3)}({\mathsf{A}})$ or to the diagonal. Hence
$\gamma$ defines disjoint submatchings

$\gamma^{(2)}:\mathrm{dgm}_{i}^{(2)}({\mathsf{A}})\rightarrow\mathrm{dgm}_{i}^{(2)}({\mathsf{M}})$

and

$\gamma^{(3)}:\mathrm{dgm}_{i}^{(3)}({\mathsf{A}})\cup\partial\Omega\to\mathrm{dgm}_{i}^{(3)}({\mathsf{M}})\cup\partial\Omega.$

Now let $R({\mathsf{M}})$ be the radius of the smallest ball that contains
${\mathsf{M}}$, and consider the functions

$a:{\mathbb{R}}^{d}\rightarrow{\mathbb{R}},x\mapsto\min(\max(d_{\mathsf{A}}(x),\tau({\mathsf{M}})/2),R({\mathsf{M}}))$

and

$m:{\mathbb{R}}^{d}\rightarrow{\mathbb{R}},x\mapsto\min(\max(d_{\mathsf{M}}(x),\tau({\mathsf{M}})/2),R({\mathsf{M}})).$

Let us compare the persistence diagrams $\mathrm{dgm}_{i}(a)$ and
$\mathrm{dgm}_{i}(m)$ of the sublevel sets filtration of $a$ and $m$ and the
Čech persistence diagrams $\mathrm{dgm}_{i}({\mathsf{A}})$ and
$\mathrm{dgm}_{i}({\mathsf{M}})$ respectively (which are by definition the
persistence diagrams of the sublevel sets filtration of $d_{\mathsf{A}}$ and
$d_{\mathsf{M}}$).

Note first that $d_{\mathsf{A}}$ and $d_{\mathsf{M}}$ can have no critical
values strictly greater than $R({\mathsf{M}})$, as a critical point must
belong to the convex hull of its projections. Note also that for any
$t\in[\tau({\mathsf{M}})/2,R({\mathsf{M}})]$, the sublevel set
$a^{-1}(-\infty,t]$ is exactly equal to
$d_{\mathsf{A}}^{-1}(-\infty,t]={\mathsf{A}}^{t}$. Consequently,
$\mathrm{dgm}_{i}(a)$ contains exactly two disjoint types of points. The first
type are points of the form $(\tau({\mathsf{M}})/2,u_{2})$, which are in
bijection with the points $(u_{1},u_{2})\in\mathrm{dgm}_{i}({\mathsf{A}})$
with $u_{1}\leq\tau({\mathsf{M}})/2$ (the bijection maps
$(u_{1},u_{2})\mapsto(\tau({\mathsf{M}})/2,u_{2})$); those are exactly the
points in $\mathrm{dgm}_{i}^{(2)}({\mathsf{A}})$. The second type are points
of the form $(u_{1},u_{2})$ with $u_{1}\geq\tau({\mathsf{M}})/2$, which are in
trivial bijection (the bijection is the identity) with the points in
$\mathrm{dgm}_{i}({\mathsf{A}})$ that satisfy the same condition; those are
exactly the points of $\mathrm{dgm}_{i}^{(3)}({\mathsf{A}})$. The points of
$\mathrm{dgm}_{i}^{(1)}({\mathsf{A}})$ cannot be “seen” in
$\mathrm{dgm}_{i}(a)$. We will call $\mathrm{dgm}_{i}^{(2)}(a)$ the subdiagram
comprised of the points of the first type, and $\mathrm{dgm}_{i}^{(3)}(a)$ the
subdiagram of $\mathrm{dgm}_{i}(a)$ comprised of the points of the second
type.

Similarly, $\mathrm{dgm}_{i}(m)$ contains two types of points: the first type
are points of the form $(\tau({\mathsf{M}})/2,v_{2})$, which are in bijection
with the points $(0,v_{2})\in\mathrm{dgm}_{i}({\mathsf{M}})$ (the bijection
maps $(0,v_{2})\mapsto(\tau({\mathsf{M}})/2,v_{2})$); those are exactly the
points in $\mathrm{dgm}_{i}^{(2)}({\mathsf{M}})$. The second type are points
of the form $(v_{1},v_{2})$ with $v_{1}\geq\tau({\mathsf{M}})/2$, which are in
trivial bijection (the bijection is the identity) with the points in
$\mathrm{dgm}_{i}({\mathsf{M}})$ that satisfy the same condition; those are
exactly the points of $\mathrm{dgm}_{i}^{(3)}({\mathsf{M}})$. We will call
$\mathrm{dgm}_{i}^{(2)}(m)$ the subdiagram of $\mathrm{dgm}_{i}(m)$ comprised
of the points of the first type, and $\mathrm{dgm}_{i}^{(3)}(m)$ the
subdiagram comprised of the points of the second type.

Recall that Lemma 2.2 states that
$|d_{\mathsf{M}}(z)-d_{\mathsf{A}}(z)|\leq\frac{\varepsilon^{2}}{2d_{\mathsf{M}}(z)}\left(1+\frac{d_{\mathsf{M}}(z)}{\tau({\mathsf{M}})}\right)$
for any $z\in{\mathbb{R}}^{d}\backslash{\mathsf{M}}$. If
$z\in{\mathbb{R}}^{d}$ is such that
$d_{\mathsf{A}}(z)\leq\tau({\mathsf{M}})/2$, then
$a(z)=m(z)=\tau({\mathsf{M}})/2$; if it is such that $d_{\mathsf{M}}(z)\geq
R({\mathsf{M}})$, then $a(z)=m(z)=R({\mathsf{M}})$. Otherwise,
$d_{\mathsf{M}}(z)\geq
d_{\mathsf{A}}(z)-d_{H}({\mathsf{A}},{\mathsf{M}})\geq\tau({\mathsf{M}})/4$
and $d_{\mathsf{M}}(z)\leq R({\mathsf{M}})$, hence

$|d_{\mathsf{M}}(z)-d_{\mathsf{A}}(z)|\leq\frac{\varepsilon^{2}}{2d_{\mathsf{M}}(z)}\left(1+\frac{d_{\mathsf{M}}(z)}{\tau({\mathsf{M}})}\right)\leq\frac{2\varepsilon^{2}}{\tau({\mathsf{M}})}\left(1+\frac{R({\mathsf{M}})}{\tau({\mathsf{M}})}\right)=C\varepsilon^{2},$

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

$\|m-a\|_{\infty}\leq C\varepsilon^{2}.$

Due to the Bottleneck Stability Theorem, the diagrams $\mathrm{dgm}_{i}(a)$
and $\mathrm{dgm}_{i}(m)$ must be at bottleneck distance less than
$C\varepsilon^{2}$.

Furthermore, let $\delta$ denote
$\max_{u\in\mathrm{dgm}_{i}({\mathsf{A}})\cup\partial\Omega}\|u-\gamma(u)\|_{\infty}$.
The matching $\gamma$ (and in particular the submatchings $\gamma^{(2)}$ and
$\gamma^{(3)}$) also induces (through the correspondence detailed above
between the points of $\mathrm{dgm}_{i}(a)$ and a subset of the points of
$\mathrm{dgm}_{i}({\mathsf{A}})$) a matching $\gamma^{\prime}$ between
$\mathrm{dgm}_{i}(a)$ and $\mathrm{dgm}_{i}(m)$ such that
$\max_{u\in\mathrm{dgm}_{i}(a)\cup\partial\Omega}\|u-\gamma^{\prime}(u)\|_{\infty}\leq\delta$.
Hence the bottleneck distance between $\mathrm{dgm}_{i}(a)$ and
$\mathrm{dgm}_{i}(m)$ is at most $\min(\delta,C\varepsilon^{2})$.

Let
$\beta:\mathrm{dgm}_{i}(a)\cup\partial\Omega\to\mathrm{dgm}_{i}(m)\cup\partial\Omega$
be an optimal matching for the bottleneck distance. For similar reasons as for
$\gamma$, the matching $\beta$ can also be decomposed into two disjoint
submatchings

$\beta^{(2)}:\mathrm{dgm}_{i}^{(2)}(a)\rightarrow\mathrm{dgm}_{i}^{(2)}(m)$

and

$\beta^{(3)}:\mathrm{dgm}_{i}^{(3)}(a)\cup\partial\Omega\to\mathrm{dgm}_{i}^{(3)}(m)\cup\partial\Omega.$

We can use the two matchings $\beta^{(2)}$ and $\beta^{(3)}$ to define a new
optimal matching
$\tilde{\gamma}:\mathrm{dgm}_{i}({\mathsf{A}})\cup\partial\Omega\to\mathrm{dgm}_{i}({\mathsf{M}})\cup\partial\Omega$
as follows:

  * •

The points in $\mathrm{dgm}_{i}^{(1)}({\mathsf{A}})$ are matched by
$\tilde{\gamma}$ to $\partial\Omega$ as with $\gamma$.

  * •

Given $u=(u_{1},u_{2})\in\mathrm{dgm}_{i}^{(2)}({\mathsf{A}})$, let
$u^{\prime}=(\tau({\mathsf{M}})/2,u_{2})$ be the point of
$\mathrm{dgm}_{i}^{(2)}(a)$ with which $u$ is in bijection. We let
$\tilde{\gamma}$ match $u$ with the point
$v=(0,v_{2})\in\mathrm{dgm}_{i}^{(2)}({\mathsf{M}})$ which is in bijection
with
$\beta^{(2)}(u^{\prime})=(\tau({\mathsf{M}})/2,v_{2})\in\mathrm{dgm}_{i}^{(2)}(m)$.
Then $|u_{2}-v_{2}|\leq\min\left(\delta,C\varepsilon^{2}\right)$ due to the
optimality of $\beta$, and this defines a bijective matching
$\tilde{\gamma}^{(2)}:\mathrm{dgm}_{i}^{(2)}({\mathsf{A}})\rightarrow\mathrm{dgm}_{i}^{(2)}({\mathsf{M}})$.
Note also that
$\max_{u\in\mathrm{dgm}_{i}^{(2)}({\mathsf{A}})}|u_{1}-\tilde{\gamma}^{(2)}(u)_{1}|=\max_{u\in\mathrm{dgm}_{i}^{(2)}({\mathsf{A}})}u_{1}=\max_{u\in\mathrm{dgm}_{i}^{(2)}({\mathsf{A}})}|u_{1}-\gamma^{(2)}(u)_{1}|$,
hence
$\max_{u\in\mathrm{dgm}_{i}^{(2)}({\mathsf{A}})}\|u-\tilde{\gamma}^{(2)}(u)\|_{\infty}\leq\delta$.

  * •

We have seen that
$\mathrm{dgm}_{i}^{(3)}(a)=\mathrm{dgm}_{i}^{(3)}({\mathsf{A}})$ and
$\mathrm{dgm}_{i}^{(3)}(m)=\mathrm{dgm}_{i}^{(3)}({\mathsf{M}})$. We simply
define the restriction and corestriction
$\tilde{\gamma}^{(3)}:\mathrm{dgm}_{i}^{(3)}({\mathsf{A}})\cup\partial\Omega\rightarrow\mathrm{dgm}_{i}^{(3)}({\mathsf{M}})\cup\partial\Omega$
of $\tilde{\gamma}$ as being equal to
$\beta^{(3)}:\mathrm{dgm}_{i}^{(3)}(a)\cup\partial\Omega\to\mathrm{dgm}_{i}^{(3)}(m)\cup\partial\Omega$.
The optimality of $\beta$ implies that
$\max_{u\in\mathrm{dgm}_{i}^{(3)}({\mathsf{A}})}\|u-\tilde{\gamma}(u)\|_{\infty}\leq\min\left(\delta,C\varepsilon^{2}\right)$.

Thus the global matching
$\tilde{\gamma}:\mathrm{dgm}_{i}({\mathsf{A}})\cup\partial\Omega\to\mathrm{dgm}_{i}({\mathsf{M}})\cup\partial\Omega$
is well-defined, is optimal for the bottleneck distance, and satisfies the
conditions stated in the proposition. This completes the proof. ∎

##  3  Čech persistence diagrams for subsets of generic submanifolds

The improved Bottleneck Stability Theorem (2.3) yields information relative to
the location of points in the Čech PD of ${\mathsf{A}}$, but not about their
numbers. However, both the $\alpha$-total persistence of
$\mathrm{dgm}_{i}({\mathsf{A}})$ and the distance
$\mathrm{OT}_{p}(\mathrm{dgm}_{i}({\mathsf{A}}),\mathrm{dgm}_{i}({\mathsf{M}}))$
for $p<\infty$ crucially depend on the number of points in
$\mathrm{dgm}_{i}({\mathsf{A}})$ having small persistence. Unfortunately, no
control on, say, the total persistence, can exist without additional
assumptions. Indeed, in general, even the $\alpha$-total persistence of the
Čech PD of the submanifold ${\mathsf{M}}$ can be infinite.

######  Example 3.1.

Let $f:x\in{\mathbb{R}}\mapsto 1+x^{4}\sin(1/x)^{2}$. Consider the $C^{2}$
curve ${\mathsf{M}}$ in ${\mathbb{R}}^{2}$ 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 [Fed59]. For $i=1$, the Čech PD of ${\mathsf{M}}$ contains a
sequence of points $(1,\ell_{n})$ for $n\geq 1$, where
$\ell_{n}=1+\Theta(n^{-4})$. In particular, as $\sum_{n\geq
1}(\ell_{n}-1)^{\alpha}=+\infty$ for $\alpha<1/4$, the $\alpha$-total
persistence of $\mathrm{dgm}_{1}({\mathsf{M}})$ is infinite for such a value
of $\alpha$. By considering the product
${\mathsf{M}}^{m}\subset{\mathbb{R}}^{2m}$, one can also build an
$m$-dimensional $C^{2}$ compact submanifold without boundary such that
$\mathrm{dgm}_{1}({\mathsf{M}})$ has an infinite $\alpha$-total persistence
for $\alpha<m/4$.

The existence of such counterexamples is explained by the fact that the
distance function $d_{\mathsf{M}}$ to a set ${\mathsf{M}}$ is not well-behaved
in general, even when the set ${\mathsf{M}}$ is smooth. In contrast to this
bleak general case, Song, Yim & Monod (in the case of surfaces in
${\mathbb{R}}^{3}$) and the authors (in the general case) studied the distance
function $d_{\mathsf{M}}$ to ${\mathsf{M}}$ when ${\mathsf{M}}$ is a generic
submanifold [ACSD23, SYM23]. 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. To do so, we first
need to introduce topological Morse functions.

The theory of persistent homology was historically developed for Morse
functions $f:{\mathbb{R}}^{d}\to{\mathbb{R}}$. A Morse function $f$ is a
$C^{2}$ function whose critical points $x$ (points for which $d_{x}f=0$) are
non-degenerate (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 $f^{-1}(-\infty,u_{1}]$ and $f^{-1}(-\infty,u_{2}]$ 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 $x_{1},\dots,x_{K}$, then
$f^{-1}(-\infty,u_{2}]$ has the homotopy type of $f^{-1}(-\infty,u_{1}]$ with
cells $e_{k}$ of dimension equal to the index of $x_{k}$ attached along their
boundaries (see e.g. [Mil63] for a much more in-depth treatment).

In such a situation, the PD $\mathrm{dgm}_{i}(f)$ has a clear interpretation:
the coordinates $(u_{1},u_{2})$ of a point $u\in\mathrm{dgm}_{i}(f)$
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 3.2 (Topological Morse functions [Mor59]).

Let $U\subset{\mathbb{R}}^{d}$ be an open set and let $f:U\to{\mathbb{R}}$ be
a continuous function.

  * •

A point $z\in U$ is said to be a topological regular point of $f$ if there is
a homeomorphism $\phi:V_{1}\to V_{2}$ between open neighborhoods $V_{1}$ of
$0$ in ${\mathbb{R}}^{d}$ and $V_{2}$ of $U$ in ${\mathbb{R}}^{d}$ with
$\phi(0)=z$ and such that for all $x=(x_{1},\dots,x_{d})\in V_{1}$,

$f\circ\phi(x)=f(z)+x_{d}.$ (10)

  * •

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

  * •

A point $z\in U$ is said to be a non-degenerate topological critical point of
$f$ of index $i$ if there exist an integer $0\leq i\leq d$ and a homeomorphism
$\phi:V_{1}\to U_{2}$ between open neighborhoods $V_{1}$ of $0$ in
${\mathbb{R}}^{d}$ and $V_{2}$ of $U$ in ${\mathbb{R}}^{d}$ with $\phi(0)=z$
such that for all $x=(x_{1},\dots,x_{d})\in V_{1}$,

$f\circ\phi(x)=f(z)-\sum_{j=1}^{i}x_{j}^{2}+\sum_{j=i+1}^{d}x_{j}^{2}.$ (11)

  * •

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 3.3 (Isotopy Lemma).

Let $f:{\mathbb{R}}^{d}\rightarrow{\mathbb{R}}$ be a proper topological Morse
function. Let $a<b$ be such that $f^{-1}[a,b]$ contains no topological
critical point. Then $f^{-1}(-\infty,a]$ is a deformation retract of
$f^{-1}(-\infty,b]$.

######  Lemma 3.4 (Handle Attachment Lemma).

Let $f:{\mathbb{R}}^{d}\rightarrow{\mathbb{R}}$ be a proper topological Morse
function. Let $c\in{\mathbb{R}}$ and $\varepsilon>0$ be such that
$f^{-1}[c-\varepsilon,c+\varepsilon]$ contains no topological critical point
except for $z_{1},\ldots,z_{k}\in f^{-1}(c)$, with $z_{j}$ of index $i_{j}$.
Then $f^{-1}(-\infty,c+\varepsilon]$ is homotopically equivalent to
$f^{-1}(-\infty,c-\varepsilon]$ with cells $B^{i_{1}},\ldots,B^{i_{k}}$ of
dimension $i_{1},\ldots,i_{k}$ attached, i.e.

$f^{-1}(-\infty,c+\varepsilon]\simeq f^{-1}(-\infty,c-\varepsilon]\cup
B^{i_{1}}\cup\ldots\cup B^{i_{k}}.$

Their proofs are roughly the same as for smooth Morse functions –see [SYM23,
Theorems 4 and 5] for details. As a consequence, the description of PDs for
Morse functions also stays valid for topological Morse functions.

Figure 3: A generic torus.

It is not true in general that the distance function $d_{\mathsf{A}}$ to a
compact set ${\mathsf{A}}$ is a topological Morse function, as shown by
Example 3.1. Though the Isotopy Lemma for distance functions (Lemma 2.1) is an
analogue of the Isotopy Lemma for Morse functions, there is no equivalent to
the Handle Attachment Lemma to control the changes occurring at critical
values, and as a result little can be said regarding the topology of the
sublevel sets of $d_{\mathsf{A}}$ without further assumptions. However, the
distance function $d_{\mathsf{M}}$ to a compact $C^{2}$ submanifold
${\mathsf{M}}\subset{\mathbb{R}}^{d}$ turns out to be a topological Morse
function in a “generic” sense, as was proven by the authors.

######  Theorem 3.5 (Genericity Theorem [ACSD23]).

Let $M$ be a compact $C^{2}$ (abstract) manifold. Then the set of $C^{2}$
embeddings $i:M\to{\mathbb{R}}^{d}$ such that

  1. 1.

the distance function $d_{i(M)}:{\mathbb{R}}^{d}\backslash
i(M)\to{\mathbb{R}}$ is a topological Morse function,

  2. 2.

for every $z\in\mathrm{Crit}({\mathsf{M}})$, the projections
$\sigma_{\mathsf{M}}(z)$ are the vertices of a non-degenerate simplex of
${\mathbb{R}}^{d}$ and $z$ belongs to its relative interior,

  3. 3.

the set $\mathrm{Crit}({\mathsf{M}})$ is finite,

  4. 4.

for every $z\in\mathrm{Crit}({\mathsf{M}})$ and every
$x\in\sigma_{\mathsf{M}}(z)$, the sphere $S(z,d_{\mathsf{M}}(z))$ is non-
osculating ${\mathsf{M}}$ at $x$, in the sense that there exist $\delta>0$ and
$\alpha>0$ such that for all $y\in{\mathsf{M}}\cap B(x,\delta)$,

$\|y-z\|^{2}\geq\|x-z\|^{2}+\alpha\|y-x\|^{2},$ (12)

  5. 5.

there exist constants $C>0$ and $\mu_{0}\in(0,1)$ such that for every
$\mu\in[0,\mu_{0})$, any point $x$ such that $\|\nabla
d_{\mathsf{M}}(x)\|\leq\mu$ is at distance at most $C\mu$ from
$\mathrm{Crit}({\mathsf{M}})$,

is open and dense in the set of $C^{2}$ embeddings $M\to{\mathbb{R}}^{d}$ for
the Whitney $C^{2}$-topology.

Put another way, given an abstract manifold $M$, “almost all” $C^{2}$
embeddings ${\mathsf{M}}$ of $M$ into ${\mathbb{R}}^{d}$ are such that
$d_{\mathsf{M}}$ is a topological Morse functions, and satisfies other
regularity conditions whose usefulness will become apparent in some of the
coming proofs. In what follows, we will simply describe a $C^{2}$ submanifold
${\mathsf{M}}$ that satisfies Conditions 1-5 from Theorem 3.5 as generic. When
${\mathsf{M}}$ is generic, the topological critical points of $d_{\mathsf{M}}$
coincide with its differential critical points (see [ACSD23, Tmh 1.8]), and
the Čech PD $\mathrm{dgm}_{i}({\mathsf{M}})$ can be related to the critical
points of $d_{\mathsf{M}}$ in the same way as for smooth Morse function. We
summarize this fact in the next proposition.

######  Proposition 3.6.

Let ${\mathsf{M}}$ be a generic compact submanifold of ${\mathbb{R}}^{d}$.

  * •

The set $\mathrm{Crit}({\mathsf{M}})$ is finite, and so is
$\mathrm{dgm}_{i}({\mathsf{M}})$ for any $i\geq 0$. In particular, for all
$\alpha>0$,

$\mathrm{Pers}_{\alpha}(\mathrm{dgm}_{i}({\mathsf{M}}))<+\infty.$

  * •

If $z\in\mathrm{Crit}({\mathsf{M}})$ is of index $i$, the set of its
projections $\sigma_{\mathsf{M}}(z)$ forms a non-degenerate simplex of
dimension at most $i$.

  * •

For any $i\geq 1$, the multiset of critical values $d_{\mathsf{M}}(z)$ of
critical points $z$ of ${\mathsf{M}}$ of index $i$ is equal to the multiset

$\\{u_{1}:\ (u_{1},u_{2})\in\mathrm{dgm}_{i}({\mathsf{M}}),\ u_{1}\neq
0\\}\cup\\{u_{2}:\ (u_{1},u_{2})\in\mathrm{dgm}_{i-1}({\mathsf{M}})\\}.$ (13)

Note that as for the Čech persistence diagram of any compact set,
$\mathrm{dgm}_{0}({\mathsf{M}})$ contains only points of the form $(0,u_{2})$
for some $u_{2}>0$. Note also that the proof below shows that the Čech
persistence modules of ${\mathsf{M}}$ are point-wise finite dimensional, and
can be decomposed into intervals.

###### 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 $\mathrm{Crit}({\mathsf{M}})$ is finite: this is simply Condition 3.
from the Genericity Theorem. The Isotopy Lemma shows that there is no change
in homotopy type between ${\mathsf{M}}^{a}$ and ${\mathsf{M}}^{b}$ if $0<a<b$
and $[a,b]$ contains no critical value. Similarly, ${\mathsf{M}}$ has the same
homotopy type as ${\mathsf{M}}^{a}$ 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 $d_{\mathsf{M}}$, and
there can be no birth or death of interval between them.

Let us now consider a critical value $c>0$ and $0<\varepsilon<c$ such that
$d_{\mathsf{M}}^{-1}[c-\varepsilon,c+\varepsilon]$ contains no critical point
except for $z_{1},\ldots,z_{k}\in d_{\mathsf{M}}^{-1}(c)$, where $z_{j}$ is of
index $i_{j}$. Then the Handle Attachment Lemma states that
${\mathsf{M}}^{c+\varepsilon}$ is homotopically equivalent to
${\mathsf{M}}^{c-\varepsilon}$ with cells $B^{i_{1}},\ldots,B^{i_{k}}$ of
dimension $i_{1},\ldots,i_{k}$ attached, i.e.

${\mathsf{M}}^{c+\varepsilon}\simeq{\mathsf{M}}^{c-\varepsilon}\cup
B^{i_{1}}\cup\ldots\cup B^{i_{k}}.$

Let $i\geq 1$, and let $D_{i,b}$ be the dimension of the cokernel of
$H_{i}({\mathsf{M}}^{c-\varepsilon})\rightarrow
H_{i}({\mathsf{M}}^{c+\varepsilon})$ (where the map is induced by the
inclusion): it is precisely the number of births of intervals between
$c-\varepsilon$ and $c+\varepsilon$ (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 $H_{i-1}({\mathsf{M}}^{c-\varepsilon})\rightarrow
H_{i-1}({\mathsf{M}}^{c+\varepsilon})$ 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 $B^{i_{1}},\ldots,B^{i_{k}}$, 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 $i\geq 1$, the multiset of critical values $d_{\mathsf{M}}(z)$ of
critical points $z$ of ${\mathsf{M}}$ of index $i$ is equal to the multiset

$\\{u_{1}:\ (u_{1},u_{2})\in\mathrm{dgm}_{i}({\mathsf{M}}),\ u_{1}\neq
0\\}\cup\\{u_{2}:\ (u_{1},u_{2})\in\mathrm{dgm}_{i-1}({\mathsf{M}})\\}.$ (14)

This fact proves in turn that $\mathrm{dgm}_{i}({\mathsf{M}})$ is finite for
any $i\geq 0$, which immediately implies that
$\mathrm{Pers}_{\alpha}(\mathrm{dgm}_{i}({\mathsf{M}}))<+\infty$ for all
$\alpha>0$.

Finally, the bound on the dimension of a critical simplex associated to a
critical point of index $i$ is a consequence of Remark 7.3 in [ACSD23] –we
give some additional details further below. ∎

Considering a well-chosen ellipse and a hyperbole gives a simple illustration
of the fact that the index of a critical point ($2$ for the center of the
ellipse, $1$ for the “center” of the hyperbole) cannot be directly deduced
from the dimension of the associated simplex ($1$ in both cases). Instead, and
as explained in Section 7 of [ACSD23], the index of a critical point
$z\in\mathrm{Crit}({\mathsf{M}})$ can be further decomposed as follows: the
restriction of the distance function $d_{\mathsf{M}}$ to a small neighborhood
of $z$ in $\\{x\in{\mathbb{R}}^{d}\backslash{\mathsf{M}}:\
\dim(\sigma_{\mathsf{M}}(x))=\dim(\sigma_{\mathsf{M}}(z))\\}$ is a $C^{2}$
Morse function, and $z$ if a critical point of it. Let $j$ be its index with
respect to this restriction; then the index $i$ of $z$ with respect to
$d_{\mathsf{M}}$ satisfies

$i=j+\dim(\sigma_{\mathsf{M}}(z)).$

When ${\mathsf{M}}$ is a generic submanifold, it becomes a reasonable task to
control the number of points in $\mathrm{dgm}_{i}({\mathsf{A}})$ where
${\mathsf{A}}\subset{\mathsf{M}}$ is an approximation of ${\mathsf{M}}$ with
$d_{H}({\mathsf{A}},{\mathsf{M}})\leq\varepsilon$. The Bottleneck Stability
Theorem implies that when $\varepsilon$ is small enough (compared to the
smallest persistence of a point in $\mathrm{dgm}_{i}({\mathsf{M}})$), every
point of $\mathrm{dgm}_{i}({\mathsf{M}})$ is mapped to a point in
$\mathrm{dgm}_{i}({\mathsf{A}})$ by the optimal bottleneck matching, leaving
the points of $\mathrm{dgm}_{i}({\mathsf{A}})$ at $\ell_{\infty}$-distance to
$\partial\Omega$ less than $\varepsilon$ unmatched; those will be mapped to
the diagonal $\partial\Omega$. The Improved Bottleneck Stability Theorem 2.3
(see Figure 2) shows that these points are of two kinds: those in Region (1),
corresponding to small topological features in the set ${\mathsf{A}}$ (of size
of order $O(\varepsilon)$), and those in Region (3), corresponding to large
topological features. There are many points in Region (1) (in fact, our proofs
show that when ${\mathsf{A}}={\mathsf{A}}_{n}$ 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 ${\mathsf{A}}\subset{\mathsf{M}}$ is
$(\delta,\varepsilon)$-dense in ${\mathsf{M}}$ if
$d_{H}({\mathsf{A}},{\mathsf{M}})\leq\varepsilon$ and $\min_{x\neq
y\in{\mathsf{A}}}\|x-y\|\geq\delta$. Such point clouds naturally occur, e.g.
as products of the farthest point sampling algorithm [AL18].

######  Theorem 3.7.

Let ${\mathsf{M}}\subset{\mathbb{R}}^{d}$ be a generic compact submanifold and
${\mathsf{A}}\subset{\mathsf{M}}$ be such that
$d_{H}({\mathsf{A}},{\mathsf{M}})\leq\varepsilon$. Let $i\geq 0$ be an
integer. There exist $\varepsilon_{0}=\varepsilon_{0}({\mathsf{M}})>0$ and
$C_{0}=C_{0}({\mathsf{M}})$ such that if $\varepsilon\leq\varepsilon_{0}$,
then:

  1. 1.

For each $u=(u_{1},u_{2})\in\mathrm{dgm}_{i}^{(3)}({\mathsf{A}})$ with
$\gamma(u)\in\partial\Omega$, one of the finitely many critical values $c$ of
$d_{\mathsf{M}}$ is such that
$u_{1},u_{2}\in[c-C_{0}\varepsilon^{2},c+C_{0}\varepsilon^{2}]$.

Furthermore, assume that ${\mathsf{A}}$ is a $(\delta,\varepsilon)$-dense set
in ${\mathsf{M}}$ for some $\delta>0$. Let $a\geq\varepsilon/\delta$. Then,
there exist constants $C_{1},C_{2},C_{3},C_{4}$ depending on ${\mathsf{M}}$
and $a$ such that:

  1. 2.

The persistence diagram $\mathrm{dgm}_{i}^{(3)}({\mathsf{A}})$ has at most
$C_{1}$ points.

  2. 3.

For all $p\geq 1$, $\alpha\geq 0$,

$\begin{split}&\mathrm{OT}_{p}^{p}(\mathrm{dgm}_{i}({\mathsf{A}}),\mathrm{dgm}_{i}({\mathsf{M}}))\leq
C_{2}\varepsilon^{p-m}\\\
&\mathrm{Pers}_{\alpha}(\mathrm{dgm}_{i}({\mathsf{A}}))\leq
C_{3}(C_{4}^{\alpha}+\varepsilon^{\alpha-m}).\end{split}$ (15)

In particular, as long as the ratio $\varepsilon/\delta$ is larger than some
constant $a>0$, the $\mathrm{OT}_{p}$ distance between
$\mathrm{dgm}_{i}({\mathsf{A}})$ and $\mathrm{dgm}_{i}({\mathsf{M}})$
converges to $0$ for all $p>m$ as $\varepsilon\to 0$, while the $p$-total
persistence $\mathrm{Pers}_{p}(\mathrm{dgm}_{i}({\mathsf{A}}))$ stays bounded.

######  Example 3.8.

Consider two parallel line segments ${\mathsf{M}}$ in ${\mathbb{R}}^{2}$, and
a finite set ${\mathsf{A}}$ consisting of two parallel grids of step
$2\varepsilon$: the set ${\mathsf{A}}$ is $(2\varepsilon,\varepsilon)$-dense
in ${\mathsf{M}}$. Then, there are $O(\varepsilon^{-1})$ points in
$\mathrm{dgm}_{1}({\mathsf{A}})$ with birth coordinates $u_{1}$ equal to $1/2$
and persistence of order $O(\varepsilon^{2})$; they all belong to
$\mathrm{dgm}_{1}^{(3)}({\mathsf{A}})$, whose cardinality is thus not bounded
by some $C_{0}=C_{0}({\mathsf{M}})$. This example can be easily modified to
make ${\mathsf{M}}$ a compact $C^{2}$ $1$-dimensional manifold. This shows
that the first conclusion of 3.7 cannot hold without a genericity assumption
on ${\mathsf{M}}$.

###### Proof.

  1. wide

Though the distance function to ${\mathsf{A}}$ need not be topologically
Morse, it remains true (due to the Isotopy Lemma for distance functions) that
the changes in topology of the sublevel sets of $d_{\mathsf{A}}$ occur at its
critical values. Hence
$u=(u_{1},u_{2})\in\mathrm{dgm}_{i}^{(3)}({\mathsf{A}})$ is such that
$u_{1},u_{2}$ are critical values of $d_{\mathsf{A}}$, with
$u_{1},u_{2}\geq\tau({\mathsf{M}})/2$. Theorem 1.6 from [ACSD23] states that
there exists $K_{1}=K_{1}({\mathsf{M}})>0$ such that if $\varepsilon$ is
smaller than some $\varepsilon_{1}$, then each point in
$\mathrm{Crit}({\mathsf{A}})$ at distance more than $\tau({\mathsf{M}})/2$
from ${\mathsf{A}}$ must be at distance at most $K_{1}\varepsilon$ from one of
the finitely many points of $\mathrm{Crit}({\mathsf{M}})$. Let us consider
$z_{\mathsf{A}}\in\mathrm{Crit}({\mathsf{A}})$ and
$z_{\mathsf{M}}\in\mathrm{Crit}({\mathsf{M}})$ such that
$\|z_{\mathsf{A}}-z_{\mathsf{M}}\|\leq K_{1}\varepsilon$ and
$d_{\mathsf{A}}(z_{\mathsf{A}})\geq\tau({\mathsf{M}})/2$; we are going to show
that there exists $C_{2}=C_{2}({\mathsf{M}})>0$ such that
$|d_{\mathsf{A}}(z_{\mathsf{A}})-d_{\mathsf{M}}(z_{\mathsf{M}})|\leq
C_{2}\varepsilon^{2}$. Let us first consider
$x_{\mathsf{M}}\in\sigma_{\mathsf{M}}(z_{\mathsf{M}})$; the proof of Theorem
1.6 from [ACSD23] also shows that there exists
$x_{\mathsf{A}}\in\sigma_{\mathsf{A}}(z_{\mathsf{A}})$ such that
$\|x_{\mathsf{A}}-x_{\mathsf{M}}\|\leq K_{2}\varepsilon$ for some constant
$K_{2}=K_{2}({\mathsf{M}})$. As $z_{\mathsf{M}}-x_{\mathsf{M}}$ is orthogonal
to $T_{x_{\mathsf{M}}}{\mathsf{M}}$, Inequality (9) states that
$\|\pi_{x_{\mathsf{M}}}^{\bot}(x_{\mathsf{A}}-x_{\mathsf{M}})\|\leq\frac{\|x_{\mathsf{A}}-x_{\mathsf{M}}\|^{2}}{2\tau({\mathsf{M}})}\leq\frac{K_{2}^{2}\varepsilon^{2}}{2\tau({\mathsf{M}})}$,
hence

$\displaystyle\|z_{\mathsf{M}}-x_{\mathsf{A}}\|^{2}$
$\displaystyle=\|z_{\mathsf{M}}-x_{\mathsf{M}}\|^{2}+2\langle
z_{\mathsf{M}}-x_{\mathsf{M}},x_{\mathsf{M}}-x_{\mathsf{A}}\rangle+\|x_{\mathsf{M}}-x_{\mathsf{A}}\|^{2}$
$\displaystyle\leq
d_{\mathsf{M}}(z_{\mathsf{M}})^{2}+\frac{K_{2}^{2}\varepsilon^{2}}{\tau({\mathsf{M}})}d_{\mathsf{M}}(z_{\mathsf{M}})+K_{2}^{2}\varepsilon^{2}\leq
d_{\mathsf{M}}(z_{\mathsf{M}})^{2}+\varepsilon^{2}K_{3},$

where
$K_{3}=K_{2}^{2}\left(\frac{R({\mathsf{M}})}{\tau({\mathsf{M}})}+1\right)$,
$R({\mathsf{M}})$ is the radius of the smallest ball that contains
${\mathsf{M}}$, and $d_{\mathsf{M}}(z_{\mathsf{M}})\leq R({\mathsf{M}})$
because a critical point must belong to the convex hull of its projections. As
the same bound applies to each of the projections
$x_{\mathsf{A}}^{\prime}\in\sigma_{\mathsf{A}}(z_{\mathsf{A}})$, we find that
the closed ball
$\overline{B}\left(z_{\mathsf{M}},\sqrt{d_{\mathsf{M}}(z_{\mathsf{M}})^{2}+\varepsilon^{2}K_{3}}\right)$
contains $\sigma_{\mathsf{A}}(z_{\mathsf{A}})$. Since $z_{\mathsf{A}}$ is the
center of the smallest ball that contains
$\sigma_{\mathsf{A}}(z_{\mathsf{A}})$, whose radius is
$d_{\mathsf{A}}(z_{\mathsf{A}})$, we find that

$d_{\mathsf{A}}(z_{\mathsf{A}})^{2}\leq
d_{\mathsf{M}}(z_{\mathsf{M}})^{2}+\varepsilon^{2}K_{3}.$ (16)

Similarly, the proof of Theorem 1.6 from [ACSD23] shows that there exists
$y\in\sigma_{\mathsf{M}}(z_{\mathsf{A}})$ such that
$\|y-x_{\mathsf{M}}\|,\|y-x_{\mathsf{A}}\|\leq O(\varepsilon)$, and the same
reasoning as above yields that
$\|z_{\mathsf{A}}-x_{\mathsf{A}}\|^{2}=\|z_{\mathsf{A}}-y\|^{2}+O(\varepsilon^{2})$
and
$\|z_{\mathsf{A}}-x_{\mathsf{M}}\|^{2}=\|z_{\mathsf{A}}-y\|^{2}+O(\varepsilon^{2})$
(with all big $O$ constants depending only on ${\mathsf{M}}$), hence that
$\|z_{\mathsf{A}}-x_{\mathsf{M}}\|^{2}\leq\|z_{\mathsf{A}}-x_{\mathsf{A}}\|^{2}+\varepsilon^{2}K_{4}=d_{\mathsf{A}}(z_{\mathsf{A}})^{2}+\varepsilon^{2}K_{4}$
for some $K_{4}=K_{4}({\mathsf{M}})$. As before, this shows that the closed
ball
$\overline{B}\left(z_{\mathsf{A}},\sqrt{d_{\mathsf{A}}(z_{\mathsf{A}})^{2}+\varepsilon^{2}K_{4}}\right)$
contains $\sigma_{\mathsf{M}}(z_{\mathsf{M}})$, hence that

$d_{\mathsf{M}}(z_{\mathsf{M}})^{2}\leq
d_{\mathsf{A}}(z_{\mathsf{A}})^{2}+\varepsilon^{2}K_{4}.$ (17)

This, together with (16) and the fact that
$d_{\mathsf{A}}(z_{\mathsf{A}})\geq\tau({\mathsf{M}})/2$, shows that
$|d_{\mathsf{A}}(z_{\mathsf{A}})-d_{\mathsf{M}}(z_{\mathsf{M}})|\leq
C_{2}\varepsilon^{2}$ for some $C_{2}=C_{2}({\mathsf{M}})$. We have shown that
any critical value of $d_{\mathsf{A}}$ greater than $\tau({\mathsf{M}})/2$ is
at distance less than $C_{2}\varepsilon^{2}$ from a critical value of
$d_{\mathsf{M}}$, which concludes the proof of the first point (by taking
$\varepsilon_{0}$ small enough for $2C_{2}\varepsilon_{0}^{2}$ to be smaller
than the minimum distance between two distinct critical values of
$d_{\mathsf{M}}$).

  2. wiide

Let us now prove the bound on the cardinality of
$\mathrm{dgm}_{i}^{(3)}({\mathsf{A}})$. As ${\mathsf{M}}$ is generic, Theorem
1.6 from [ACSD23] states that if $\varepsilon$ is smaller than some
$\varepsilon_{0}=\varepsilon_{0}({\mathsf{M}})$, then each point in
$\mathrm{Crit}({\mathsf{A}})$ at distance more than $\tau({\mathsf{M}})/2$
from ${\mathsf{A}}$ must be at distance at most $K_{5}\varepsilon$ from one of
the finitely many points of $\mathrm{Crit}({\mathsf{M}})$ for some
$K_{5}=K_{5}({\mathsf{M}})$. Corollary 1.7 from the same article then states
that the number of points in $\mathrm{Crit}({\mathsf{A}})$ at distance less
than $K_{1}\varepsilon$ from a given point of $\mathrm{Crit}({\mathsf{M}})$ is
upper bounded by some constant that depends on ${\mathsf{M}}$ and the ratio
$\varepsilon/\delta$, and is decreasing in this ratio; hence it is upper
bounded by some constant that depends on ${\mathsf{M}}$ and $a$. The proof of
this corollary also shows that the maximum number of projections on
${\mathsf{A}}$ of each of these points of $\mathrm{Crit}({\mathsf{A}})$ is
also upper bounded by some constant that depends on ${\mathsf{M}}$ and $a$.
Hence there exist constants $K_{6}=K_{6}({\mathsf{M}},a)$ and
$K_{7}=K_{7}({\mathsf{M}},a)$ such that if $\varepsilon\leq\varepsilon_{0}$,
then there are at most $K_{6}$ points in $\mathrm{Crit}({\mathsf{A}})$ at
distance more than $\tau({\mathsf{M}})/2$ from ${\mathsf{A}}$, and each has at
most $K_{7}$ projections on ${\mathsf{A}}$.

Lemma A.1, applied to the interval $[\tau({\mathsf{M}})/2,\infty)$ and the set
${\mathsf{A}}$, then states that the number of points in
$\mathrm{dgm}_{i}({\mathsf{A}})$ such that at least one of their coordinates
is greater than $\tau({\mathsf{M}})/2$ is bounded by
$K_{6}\binom{K_{7}}{i+2}$. Hence there are at most $C_{0}:=K_{6}2^{K_{7}}\geq
K_{6}\binom{K_{7}}{i+2}$ points in $\mathrm{dgm}_{i}^{(3)}({\mathsf{A}})$ when
$\varepsilon\leq\varepsilon_{0}$.

  3. wiiide

Let us bound
$\mathrm{OT}_{p}^{p}(\mathrm{dgm}_{i}({\mathsf{A}}),\mathrm{dgm}_{i}({\mathsf{M}}))$
and $\mathrm{Pers}_{\alpha}(\mathrm{dgm}_{i}({\mathsf{A}}))$. As stated in
2.3, which applies as $\varepsilon_{0}<\tau({\mathsf{M}})/4$, each point
$u=(u_{1},u_{2})\in\mathrm{dgm}_{i}^{(1)}({\mathsf{A}})$ is such that its
coordinates satisfy $0\leq
u_{1},u_{2}\leq\varepsilon+\varepsilon^{2}/\tau({\mathsf{M}})\leq
2\varepsilon$. In particular, they must correspond to the birth or the death
of an interval of the Čech persistence module of ${\mathsf{A}}$ that occurs
before filtration time $2\varepsilon$. The homology of the offsets
${\mathsf{A}}^{t}$ can be computed using the Čech simplicial complex of
${\mathsf{A}}$ (see e.g. [EH10]). In particular, each change in the homology
of the offsets, hence each birth or death in the Čech persistence module of
${\mathsf{A}}$, is induced by the apparition of some simplex $\sigma$ at the
corresponding filtration value in the Čech complex, and each such apparition
causes at most a single death or birth. If a simplex $\sigma$ appears before
filtration time $2\varepsilon$, it is by definition contained in a ball of
radius $2\varepsilon$, hence it is of diameter at most $4\varepsilon$. Let us
assume from now on that $\varepsilon_{0}\leq\tau({\mathsf{M}})/16$. Consider
$x\in{\mathsf{A}}$; then [AL18, Proposition 8.7] states that the intersection
$\overline{B}(x,4\varepsilon)\cap{\mathsf{A}}$ contains at most
$K_{8}(\varepsilon/\delta)^{m}\leq K_{8}a^{m}$ points for some constant
$K_{8}=K_{8}({\mathsf{M}})$. Hence $x$ belongs to at most $2^{K_{8}a^{m}}$
simplices that appear before $\varepsilon$, and there are at most
$\\#{\mathsf{A}}\cdot 2^{K_{8}a^{m}}$ such simplices. As the cardinality
$\\#{\mathsf{A}}$ can be bounded by $K_{9}/\delta^{m}$ for some
$K_{9}=K_{9}({\mathsf{M}})$, we find that
$\\#(\mathrm{dgm}_{i}^{(1)}({\mathsf{A}}))\leq K_{10}/\delta^{m}$ for some
$K_{10}=K_{10}({\mathsf{M}},a)$.

Furthermore, when ${\mathsf{M}}$ is generic, Proposition 3.6 states that its
PD $\mathrm{dgm}_{i}({\mathsf{M}})$ has a finite number of points. Let
$\gamma$ be an optimal matching between $\mathrm{dgm}_{i}({\mathsf{A}})$ and
$\mathrm{dgm}_{i}({\mathsf{M}})$ for the bottleneck distance that satisfies
the conclusions of 2.3. We find that any point
$u\in\mathrm{dgm}_{i}^{(1)}({\mathsf{A}})$ is matched to a point of
$\partial\Omega$ at distance at most $\varepsilon$ from $u$. Moreover, the
number of points in
$\mathrm{dgm}_{i}^{(2)}({\mathsf{A}})\cup\mathrm{dgm}_{i}^{(3)}({\mathsf{A}})$
is bounded by some constant $K_{11}=K_{11}({\mathsf{M}},a)$, and they are all
matched to a point of $\mathrm{dgm}_{i}({\mathsf{M}})$ or $\partial\Omega$ at
distance at most $\varepsilon$. In particular, these finitely many points are
at distance at most $K_{12}=K_{12}({\mathsf{M}})$ from $\partial\Omega$.
Furthermore, this matching is surjective, in the sense that $\gamma$ matches
all points of $\mathrm{dgm}_{i}({\mathsf{M}})$ to a point of
$\mathrm{dgm}_{i}({\mathsf{A}})$. As a result, for any $p\geq 1$, we find that

$\displaystyle\mathrm{OT}^{p}_{p}(\mathrm{dgm}_{i}({\mathsf{A}}),\mathrm{dgm}_{i}({\mathsf{M}}))$
$\displaystyle\leq\sum_{u\in\mathrm{dgm}_{i}^{(1)}({\mathsf{A}})}\|u-\gamma(u)\|_{\infty}^{p}+\sum_{u\in\mathrm{dgm}_{i}^{(2)}({\mathsf{A}})\cup\mathrm{dgm}_{i}^{(3)}({\mathsf{A}})}\|u-\gamma(u)\|_{\infty}^{p}$
$\displaystyle\leq K_{10}\delta^{-m}\varepsilon^{p}+K_{11}\varepsilon^{p}\leq
K_{10}a^{m}\varepsilon^{p-m}+K_{11}\varepsilon^{p}\leq C_{1}\varepsilon^{p-m}$

for some $C_{1}=C_{1}({\mathsf{M}},a)$. Likewise, for any $\alpha\geq 0$, we
have that

$\displaystyle\mathrm{Pers}_{\alpha}(\mathrm{dgm}_{i}({\mathsf{A}}))$
$\displaystyle=\sum_{u\in\mathrm{dgm}_{i}^{(1)}({\mathsf{A}})}\mathrm{pers}(u)^{\alpha}+\sum_{u\in\mathrm{dgm}_{i}^{(2)}({\mathsf{A}})\cup\mathrm{dgm}_{i}^{(3)}({\mathsf{A}})}\mathrm{pers}(u)^{\alpha}$
$\displaystyle\leq
K_{10}\delta^{-m}\varepsilon^{\alpha}+K_{11}K_{12}^{\alpha}\leq
K_{10}a^{m}\varepsilon^{\alpha-m}+K_{11}K_{12}^{\alpha}\leq
C_{2}(C_{3}^{\alpha}+\varepsilon^{\alpha-m})$

for some $C_{2}=C_{2}({\mathsf{M}},a),C_{3}=C_{3}({\mathsf{M}})$. This
completes the proof. ∎

##  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 $\varepsilon$-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 ${\mathsf{M}}$ of dimension $m\geq 1$. Let
${\mathsf{A}}={\mathsf{A}}_{n}=\\{X_{1},\dots,X_{n}\\}$, where
$X_{1},\dots,X_{n}$ is an i.i.d. sample from distribution $P$. Let $i\geq 0$
be an integer; we consider the three regions described in Figure 2 and in the
statement of 2.3, and write again $\mathrm{dgm}_{i}^{(1)}({\mathsf{A}}_{n})$,
$\mathrm{dgm}_{i}^{(2)}({\mathsf{A}}_{n})$ and
$\mathrm{dgm}_{i}^{(3)}({\mathsf{A}}_{n})$ 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: $\mathrm{dgm}_{i}^{(1)}({\mathsf{A}}_{n})$ only depends on small-
scale phenomena and can essentially be reduced to the case of a cube, even if
${\mathsf{M}}$ is non-generic, while
$\mathrm{dgm}_{i}^{(2)}({\mathsf{A}}_{n})$ and
$\mathrm{dgm}_{i}^{(3)}({\mathsf{A}}_{n})$ are tightly connected to the
macroscopic geometry of the submanifold and can be further controlled using
genericity assumptions on ${\mathsf{M}}$.

Describing the limit behavior of the random PD
$\mathrm{dgm}_{i}^{(1)}({\mathsf{A}}_{n})$ requires extending the
$\mathrm{OT}_{p}$ metrics between PDs to more general Radon measures. Indeed,
a PD can equivalently be seen as an integer-valued discrete Radon measure on
$\Omega$, by identifying a multiset $a$ with the Radon measure $\sum_{u\in
a}\delta_{u}$. Let $\mathcal{M}$ denote the space of Radon measures on
$\Omega$, that is the space of Borel measures on $\Omega$ which give finite
mass to every compact set $K\subset\Omega$.666A compact set $K\subset\Omega$
is at positive distance from the diagonal. Hence, a measure
$\mu\in\mathcal{M}$ can have an accumulation of mass close to
$\partial\Omega$. The space of Radon measures is endowed with the vague
topology, where a sequence $(\mu_{n})_{n}$ of measures in $\mathcal{M}$ is
said to converge vaguely to $\mu\in\mathcal{M}$ if
$\int_{\Omega}\phi\mathrm{d}\mu_{n}\to\int_{\Omega}\phi\mathrm{d}\mu$ as
$n\to\infty$ for all continuous functions $\phi:\Omega\to{\mathbb{R}}$ with
compact support.

The $\alpha$-total persistence is defined for $\mu\in\mathcal{M}$ by
$\mathrm{Pers}_{\alpha}(\mu)=\int_{\Omega}\mathrm{pers}(u)^{\alpha}\mathrm{d}\mu(u)$.
For $p\geq 1$, we let $\mathcal{M}^{p}=\\{\mu\in\mathcal{M}:\
\mathrm{Pers}_{p}(\mu)<+\infty\\}$. The distance $\mathrm{OT}_{p}$, defined
between PDs in (3), can be extended to the set $\mathcal{M}^{p}$ as follows
(see [DL21] for details): let us define
$\bar{\Omega}:=\\{u=(u_{1},u_{2})\in{\mathbb{R}}^{2}:\ u_{1}\leq u_{2}\\}$. We
call $\pi$ an admissible transport plan between
$\nu_{1},\nu_{2}\in\mathcal{M}$ if it is a Radon measure on
$\bar{\Omega}\times\bar{\Omega}$ such that for all Borel sets
$A,B\subset\Omega$,

$\pi(A\times\bar{\Omega})=\nu_{1}(A)\quad\text{ and }\pi(\bar{\Omega}\times
B)=\nu_{2}(B).$ (18)

For $p\in[1,+\infty)$, we define

$\mathrm{OT}_{p}^{p}(\nu_{1},\nu_{2})=\inf_{\pi\in\text{Adm}(\nu_{1},\nu_{2})}\iint\|u-v\|_{\infty}^{p}\mathrm{d}\pi(u,v)\in{\mathbb{R}}\cup\\{+\infty\\},$
(19)

where $\text{Adm}(\nu_{1},\nu_{2})$ is the set of all admissible transport
plans between $\nu_{1}$ and $\nu_{2}$. We also define

$\mathrm{OT}_{\infty}(\nu_{1},\nu_{2})=\inf_{\pi\in\text{Adm}(\nu_{1},\nu_{2})}\sup\\{\|u-v\|_{\infty}\,:\,(u,v)\in\text{Support}(\pi)\\}$
(20)

For all $p\in[1,\infty]$, the infimum is in fact a minimum. We call
$\mathrm{OT}_{p}(\nu_{1},\nu_{2})$ the $p$-Wasserstein distance between
$\nu_{1}$ and $\nu_{2}$, though it differs from the usual Wasserstein
distance, which is defined between measures with equal finite mass, while
$\mathrm{OT}_{p}$ is defined between measures that can have different (and
even infinite) masses. Intuitively, $\mathrm{OT}_{p}$ allows for some of the
mass of $\nu_{1}$ and $\nu_{2}$ to be transported to the diagonal
$\partial\Omega:=\\{(u,u)\in{\mathbb{R}}^{2}\\}$, which acts as an infinitely
deep landfill. The $p$-Wasserstein distance is a distance on the space
$\mathcal{M}_{p}=\\{\nu\in\mathcal{M}:\ \mathrm{OT}_{p}(\nu,0)<\infty\\}$,
where $0$ denotes the null measure, and its restriction to the space of PDs
coincides with the $\mathrm{OT}_{p}$ distance defined earlier.

We require the following lemma from [DL21]:

######  Lemma 4.1.

A sequence of measures $(\nu_{n})_{n\geq 1}$ converges with respect to
$\mathrm{OT}_{p}$ to some measure $\nu$ if and only if the sequence
$(\nu_{n})_{n\geq 1}$ converges vaguely towards $\nu$ and
$\mathrm{Pers}_{p}(\nu_{n})\to\mathrm{Pers}_{p}(\nu)$ as $n\to\infty$.

For $q>0$, given a function $f:{\mathbb{N}}\rightarrow{\mathbb{R}}$ and a
sequence of (non-necessarily measurable) real maps $(Y_{n})_{n}$ defined on
some probabilistic space, the notation $Y_{n}=O_{L^{q}}(f(n))$ means that
${\mathbb{E}}^{*}[|Y_{n}|^{q}]=O(f(n)^{q})$, where ${\mathbb{E}}^{*}$ denotes
the outer expectation [VDVW97, p.6] (and similarly for the little $o$
notation).

###  4.1 Region (1)

Consider the rescaled Radon measure
$\mu_{n,i}=\frac{1}{n}\sum_{u\in\mathrm{dgm}_{i}^{(1)}({\mathsf{A}}_{n})}\delta_{n^{1/m}u}$
–this rescaling is natural, as the number of points in $a_{n,i}^{(1)}$ is
typically of order $n$, while their coordinates are of order $n^{-1/m}$ – and
note that $\mu_{n,i}$ is a random measure, owing to the randomness of the set
${\mathsf{A}}_{n}$. Goel, Trinh and Tsunoda studied the vague convergence of
the sequence $(\mu_{n,i})_{n}$ [GTT19, Remark 4.2]. Namely, assuming that the
density $f$ satisfies $\int_{\mathsf{M}}f^{j}<\infty$ for all $j\geq 0$, they
show that with probability $1$ the sequence $(\mu_{n,i})_{n}$ converges
vaguely to some (non-random) Radon measure $\mu_{f,i}$. The limit measure
$\mu_{f,i}$ has support $\\{0\\}\times{\mathbb{R}}_{+}$ if $i=0$ and $\Omega$
if $0<i<m$; it is the zero measure if $i\geq m$. We can further describe it as
follows: let $\mu_{\infty,i,m}$ be the limit of the sequence $(\mu_{n,i})$ in
the case where the sample ${\mathsf{A}}_{n}$ is uniform on the unit cube
$[0,1]^{m}$ (see [DP19]). Then, for any continuous function
$\phi:\Omega\to{\mathbb{R}}$ with compact support,

$\int_{\Omega}\phi(u)\mathrm{d}\mu_{f,i}(u)=\int_{\Omega}\int_{\mathsf{M}}f(x)\phi(f(x)^{-1/m}u)\mathrm{d}x\mathrm{d}\mu_{\infty,i,m}(u).$
(21)

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 $\partial\Omega$ as $n$ goes to infinity.
In particular, and except in the case of the uniform distribution on the unit
cube $[0,1]^{m}$ (see [DP19]), it was not known whether $\mu_{n,i}$ converges
to $\mu_{f,i}$ for the $\mathrm{OT}_{p}$ distance as well, nor whether the
sequences of total persistence $(\mathrm{Pers}_{\alpha}(\mu_{n,i}))$ converge.
We close this gap with the following result.

######  Theorem 4.2 (Law of large numbers).

Assume that $P$ has a density $f$ on ${\mathsf{M}}$ bounded away from $0$ and
$\infty$. Let $i\geq 0$ be an integer and let $1\leq p<\infty$. Then
$\mu_{f,i}\in\mathcal{M}_{p}$ and

${\mathbb{E}}[\mathrm{OT}_{p}^{p}(\mu_{n,i},\mu_{f,i})]\xrightarrow[n\to\infty]{}0.$

Furthermore, for all $\alpha>0$,
$\mathrm{Pers}_{\alpha}(\mathrm{dgm}_{i}^{(1)}({\mathsf{A}}_{n}))n^{\frac{\alpha}{m}-1}=\mathrm{Pers}_{\alpha}(\mu_{n,i})=\mathrm{Pers}_{\alpha}(\mu_{f,i})+o_{L^{1}}(1)$.

Note that the statement becomes clearly wrong when $p=\infty$, as the support
of $\mu_{n,i}$ is bounded for all $n$, while that of $\mu_{f,i}$ is not. Note
also that without the assumption from Theorem 4.2 that $f$ be bounded away
from $0$, it is easy to find examples where
$\int_{\mathsf{M}}f(x)^{1-\frac{p}{m}}\mathrm{d}x=\infty$ for $p$ large
enough, hence $\mathrm{Pers}_{p}(\mu_{f,i})=\infty$ and
$\mu_{f,i}\not\in\mathcal{M}_{p}$; consider e.g. the segment $[-1/2,1/2]$,
$f(x)=x$ and $p\geq 2$ (boundaryless examples can be similarly constructed).

###### Proof of 4.2.

Recall that Goel, Trinh and Tsunoda [GTT19] have shown that almost surely, the
sequence of Radon measures $(\mu_{n,i})_{n}$ vaguely converges to the Radon
measure $\mu_{f,i}$. We start by showing that, using this vague convergence
and Lemma 4.1, it is enough to prove the convergence of the $p$-total
persistence.

######  Lemma 4.3.

Let $(\nu_{n})_{n\geq 1}$ be a sequence of random measures in
$\mathcal{M}_{p}$ that converges vaguely almost surely to a Radon measure
$\nu\in\mathcal{M}_{p}$, and such that
${\mathbb{E}}[|\mathrm{Pers}_{p}(\nu_{n})-\mathrm{Pers}_{p}(\nu)|]\to_{n\to\infty}0$.
Then, ${\mathbb{E}}[\mathrm{OT}_{p}^{p}(\nu_{n},\nu)]\to_{n\to\infty}0$.

###### Proof.

Let us first show that $(D_{n})_{n\geq
1}=(\mathrm{OT}_{p}^{p}(\nu_{n},\nu))_{n\geq 1}$ converges in probability to
$0$. We use the following standard result: if for every subsequence
$(Z_{n_{k}})_{k\geq 1}$ of $(Z_{n})_{n\geq 1}$, one can extract a subsequence
$(Z_{n_{k_{l}}})_{l\geq 1}$ that converges almost surely to $0$, then the
sequence $(Z_{n})_{n\geq 1}$ converges in probability to $0$. Let
$(D_{n_{k}})_{k\geq 1}$ be a subsequence of $(D_{n})_{n\geq
1}=(\mathrm{OT}_{p}^{p}(\nu_{n},\nu))_{n\geq 1}$. Then, as
$(\mathrm{Pers}_{p}(\nu_{n}))_{n\geq 1}$ converges in $L^{1}$ to
$\mathrm{Pers}_{p}(\nu)$, it also converges in probability. In particular,
there exists a subsequence $(n_{k_{l}})_{l\geq 1}$ such that
$(\mathrm{Pers}_{p}(\nu_{n_{k_{l}}}))_{n\geq 1}$ converges almost surely to
$\mathrm{Pers}_{p}(\nu)$. 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 4.1, we have
$D_{n_{k_{l}}}=\mathrm{OT}_{p}^{p}(\nu_{n_{k_{l}}},\nu)\to_{l\to\infty}0$
almost surely, proving that we actually have that $(D_{n})_{n\geq 1}$
converges in probability to $0$. To prove that
${\mathbb{E}}[D_{n}]\to_{n\to\infty}0$, it remains to show that the sequence
$(D_{n})_{n\geq 1}$ is uniformly integrable. By considering the trivial
transport plan that sends all probability mass to $\partial\Omega$, we have
for all $n\geq 1$

$D_{n}\leq\mathrm{Pers}_{p}(\nu_{n})+\mathrm{Pers}_{p}(\nu).$

But the sequence $(\mathrm{Pers}_{p}(\nu_{n}))_{n\geq 1}$ is uniformly
integrable, as it converges in $L^{1}$. Hence, so is the sequence
$(D_{n})_{n\geq 1}$, concluding the proof. ∎

Using Lemma 4.3, 4.2 would follow from the facts that
$\mu_{f,i}\in\mathcal{M}_{p}$ and that
${\mathbb{E}}[|\mathrm{Pers}_{p}(\mu_{n,i})-\mathrm{Pers}_{p}(\mu_{f,i})|]$
converges to $0$.

Recall that $C_{c}(\Omega)$ is the set of continuous functions
$f:\Omega\to{\mathbb{R}}$ with compact support (i.e. the support is bounded
and at positive distance from $\partial\Omega$). For $s\geq 0$, let
$T_{s}=\\{(u_{1},u_{2})\in\Omega:\ u_{2}\geq s\\}$.

######  Lemma 4.4.

Let $\alpha>0$. Let $(\nu_{n})_{n\geq 1}$ be a sequence of random measures in
$\mathcal{M}_{\alpha}$ that converges vaguely almost surely to a Radon measure
$\nu\in\mathcal{M}$. Assume that the sequence of random variables
$(\nu_{n}(\Omega))_{n\geq 1}$ is uniformly integrable and that

$\sup_{n}{\mathbb{E}}[\mathrm{Pers}_{\alpha}(\nu_{n})]<+\infty\text{ and
}\lim_{s\to+\infty}\limsup_{n}{\mathbb{E}}[\int_{T_{s}}\mathrm{pers}^{\alpha}(u)\mathrm{d}\nu_{n}(u)]=0.$

Then, $\nu\in\mathcal{M}_{\alpha}$ and
${\mathbb{E}}[|\mathrm{Pers}_{\alpha}(\nu_{n})-\mathrm{Pers}_{\alpha}(\nu)|]\to_{n\to\infty}0$.

###### Proof.

We divide the proof into several steps.

  1. 1.

Let $\phi\in C_{c}(\Omega)$. We first show that
$(\int\phi\mathrm{d}\nu_{n}))_{n\geq 1}$ converges in $L^{1}$ to
$\int\phi\mathrm{d}\nu$. By assumption, the convergence holds almost surely.
Furthermore, as $\phi$ is bounded and as the sequence
$(\nu_{n}(\Omega))_{n\geq 1}$ is uniformly integrable, so is the sequence
$(\int\phi\mathrm{d}\nu_{n})_{n\geq 1}$. Hence,
${\mathbb{E}}[|\int\phi\mathrm{d}(\nu_{n}-\nu)|]\to_{n\to\infty}0$.

  2. 2.

Let $(\phi_{k})_{k\geq 1}$ be an increasing sequence of functions in
$C_{c}(\Omega)$ that converge pointwise to the function
$\mathrm{pers}_{\alpha}$. Then, almost surely,

$\int\phi_{k}\mathrm{d}\nu\leq\liminf_{n\to\infty}\int\phi_{k}\mathrm{d}\nu_{n}\leq\liminf_{n\to\infty}\int\mathrm{pers}_{\alpha}\mathrm{d}\nu_{n}.$

By Fatou’s lemma,
${\mathbb{E}}[\liminf_{n\to\infty}\int\mathrm{pers}_{\alpha}\mathrm{d}\nu_{n}]\leq\liminf_{n\to\infty}{\mathbb{E}}[\mathrm{Pers}_{\alpha}(\nu_{n})]=C<+\infty$
by assumption. Hence, by letting $k\to\infty$ and applying the monotone
convergence theorem, we obtain $\mathrm{Pers}_{\alpha}(\nu)\leq C$, proving
that $\nu\in\mathcal{M}_{\alpha}$.

  3. 3.

The same argument can be applied to the constant function equal to $1$,
showing that $\nu(\Omega)<+\infty$.

  4. 4.

Let $s\geq 1$. The function $\mathrm{pers}_{\alpha}$ can be decomposed into a
sum of three positive continuous functions
$\mathrm{pers}_{\alpha}=\phi_{s}^{(1)}+\phi_{s}^{(2)}+\phi_{s}^{(3)}$, where
$\phi_{s}^{(1)}$ has compact support, the support of $\phi_{s}^{(2)}$ is
included in the band $\\{u\in\Omega:\ \mathrm{pers}(u)\leq 1/s\\}$ and the
support of $\phi_{s}^{(3)}$ is included in $T_{s}$. Hence,

$\displaystyle\limsup_{n\to+\infty}{\mathbb{E}}[|\mathrm{Pers}_{\alpha}(\nu_{n})$
$\displaystyle-\mathrm{Pers}_{\alpha}(\nu)|]\leq\limsup_{n\to+\infty}{\mathbb{E}}[|\int\phi_{s}^{(1)}\mathrm{d}(\nu_{n}-\nu)|]$
$\displaystyle+\limsup_{n\to+\infty}{\mathbb{E}}[|\int\phi_{s}^{(2)}\mathrm{d}(\nu_{n}-\nu)|]+\limsup_{n\to+\infty}{\mathbb{E}}[|\int\phi_{s}^{(3)}\mathrm{d}(\nu_{n}-\nu)|].$

The first term in the above sum is equal to zero because of the first item,
the second one is smaller than
$s^{-\alpha}(\sup_{n}{\mathbb{E}}[\nu_{n}(\Omega)]+\nu(\Omega))$, and the
third one is smaller than

$\limsup_{n}{\mathbb{E}}[\int_{T_{s}}\mathrm{pers}_{\alpha}(u)\mathrm{d}\nu_{n}(u)]+\int_{T_{s}}\mathrm{pers}_{\alpha}(u)\mathrm{d}\nu(u).$

Using the hypotheses of the lemma, the second and the third term converges to
$0$ as $s$ goes to $\infty$. We obtain that
$\limsup_{n\to+\infty}{\mathbb{E}}[|\mathrm{Pers}_{\alpha}(\nu_{n})-\mathrm{Pers}_{\alpha}(\nu)|]=0$.
∎

Our goal is to show that the conditions of Lemma 4.4 holds for the sequence
$(\mu_{n,i})_{n\geq 1}$ to conclude. Remark that for any Radon measure
$\nu\in\mathcal{M}_{\alpha}$ and $s\geq 0$

$\begin{split}\int_{T_{s}}\mathrm{pers}^{\alpha}(u)\mathrm{d}\nu(u)&=\alpha\int_{0}^{\infty}t^{\alpha-1}\nu(T_{s}\cap\\{u:\
\mathrm{pers}(u)\geq t\\})\mathrm{d}t\\\
&\leq\alpha\int_{s/2}^{\infty}t^{\alpha-1}\nu(T_{2t})\mathrm{d}t+\alpha\int_{0}^{s/2}t^{\alpha-1}\nu(T_{s})\mathrm{d}t\\\
&\leq\alpha\int_{s/2}^{\infty}t^{\alpha-1}\nu(T_{2t})\mathrm{d}t+(s/2)^{\alpha}\nu(T_{s}),\end{split}$
(22)

where we use Fubini’s theorem for the first equality and the fact that $\\{u:\
\mathrm{pers}(u)\geq t\\}\subset T_{2t}$ for the first inequality. We also
have $\nu(\Omega)=\nu(T_{0})$. Hence, the different conditions of Lemma 4.4
can all be obtained by controlling the random variable $\mu_{n,i}(T_{s})$ for
$s\geq 0$.

######  Proposition 4.5.

Let ${\mathsf{M}}$ be a compact submanifold with positive reach. Assume that
$P$ has a density $f$ on ${\mathsf{M}}$ satisfying $f_{\min}\leq f\leq
f_{\max}$ for two positive constants $f_{\min}$, $f_{\max}$. Then there exist
$c,C>0$ that depend on ${\mathsf{M}}$, $i$, $f_{\min}$ and $f_{\max}$ such
that for all integer $n\geq 1$ and all $s\geq 0$,

${\mathbb{E}}[\mu_{n,i}(T_{s})^{2}]\leq C\exp(-cs^{m}).$ (23)

Before proving Proposition 4.5, let us show how to use it to conclude the
proof of 4.2. First, it implies that the random variables
$\mu_{n,i}(\Omega)=\mu_{n,i}(T_{0})$ for $n\geq 1$ have a uniformly bounded
second moment, and are therefore uniformly integrable. Second, we have
${\mathbb{E}}[\mu_{n,i}(T_{s})]\leq{\mathbb{E}}[\mu_{n,i}(T_{s})^{2}]^{1/2}$
using Hölder’s inequality. Hence, (22) implies that for any $\alpha>0$, we
have

$\displaystyle{\mathbb{E}}[\mathrm{Pers}_{\alpha}(\mu_{n,i})]$
$\displaystyle\leq\alpha\int_{0}^{\infty}t^{\alpha-1}{\mathbb{E}}[\mu_{n,i}(T_{2t})]\mathrm{d}t$
$\displaystyle\leq\alpha\sqrt{C}\int_{0}^{\infty}t^{\alpha-1}e^{-c2^{m-1}t^{m}}\mathrm{d}t.$

In particular,
$\sup_{n}{\mathbb{E}}[\mathrm{Pers}_{\alpha}(\mu_{n,i})]<+\infty$. Likewise,

$\displaystyle\sup_{n}{\mathbb{E}}\left[\int_{T_{s}}\mathrm{pers}_{\alpha}(u)\mathrm{d}\mu_{n,i}(u)\right]$
$\displaystyle\leq\alpha\int_{s/2}^{\infty}t^{\alpha-1}{\mathbb{E}}[\mu_{n,i}(T_{2t})]\mathrm{d}t+(s/2)^{\alpha}{\mathbb{E}}\left[\mu_{n,i}(T_{s})\right]$
$\displaystyle\leq\alpha\sqrt{C}\int_{0}^{\infty}t^{\alpha-1}e^{-c2^{m-1}t^{m}}\mathrm{d}t+(s/2)^{\alpha}\sqrt{C}\exp(-c/2s^{m})$

so that
$\lim_{s\to+\infty}\sup_{n}{\mathbb{E}}[\int_{T_{s}}\mathrm{pers}_{\alpha}(u)\mathrm{d}\mu_{n,i}(u)]=0$.

We are therefore in position to apply Lemma 4.4, proving the convergence of
the $\alpha$-total persistence. Together with Lemma 4.3 with $\alpha=p\geq 1$,
we also obtain the $\mathrm{OT}_{p}$-convergence of $\mu_{n,i}$. It remains to
prove Proposition 4.5.

###### Proof of Proposition 4.5.

Write $\varepsilon_{n}=d_{H}({\mathsf{A}}_{n},{\mathsf{M}})$. Let $s\geq 0$
and let $\varepsilon_{0}<\tau({\mathsf{M}})/2$ be a small parameter, to be
fixed later. Recall that we write $\\#S$ for the cardinality of a multiset
$S$. Notice that $\mu_{n,i}(T_{s})=0$ if
$sn^{-1/m}>\varepsilon_{n}+\varepsilon^{2}_{n}/\tau({\mathsf{M}})$. In
particular, we may assume without loss of generality that $s\leq
n^{1/m}\mathrm{diam}({\mathsf{M}})(1+\mathrm{diam}({\mathsf{M}})/\tau({\mathsf{M}}))=n^{1/m}\varepsilon_{\max}$,
for otherwise there is nothing to prove. Consider the event
$E=\\{\varepsilon_{n}+\varepsilon_{n}^{2}/\tau({\mathsf{M}})<\varepsilon_{0}\\}$.
By definition of Region (1), if $E$ is satisfied, then all the coordinates of
points of the PD $\mathrm{dgm}_{i}^{(1)}({\mathsf{A}}_{n})$ are smaller than
$\varepsilon_{0}$. Notice that the cardinality of
$\mathrm{dgm}_{i}({\mathsf{A}}_{n})$ is smaller than the number of
$i$-dimensional simplices in the Čech complex of ${\mathsf{A}}_{n}$, which is
itself smaller than $n^{i+1}$ (as each simplex corresponds uniquely to a
choice of $i+1$ vertices of ${\mathsf{A}}_{n}$). Hence

$\displaystyle{\mathbb{E}}[\mu_{n,i}(T_{s})^{2}\mathbf{1}\\{E^{c}\\}]\leq
n^{-2}n^{2i+2}{\mathbb{P}}(\varepsilon_{n}+\varepsilon_{n}^{2}/\tau({\mathsf{M}})\geq\varepsilon_{0}).$

We require the following lemma, which bounds the upper tail of the random
variable $\varepsilon_{n}=d_{H}({\mathsf{A}}_{n},{\mathsf{M}})$ and which we
prove in the Appendix.

######  Lemma 4.6.

If $r\leq\tau({\mathsf{M}})/2$, then

${\mathbb{P}}(d_{H}({\mathsf{A}}_{n},{\mathsf{M}})>r)\leq\frac{C}{f_{\min}r^{m}}\exp(-ncf_{\min}r^{m})$
(24)

for two positive constants $c=c(m),C=C(m)$. In particular, for any $q\geq 1$,
$d_{H}({\mathsf{A}}_{n},{\mathsf{M}})=O_{L^{q}}((\ln n/n)^{1/m})$.

Note that $\varepsilon_{n}\leq\mathrm{diam}({\mathsf{M}})$, so
${\mathbb{P}}(\varepsilon_{n}+\varepsilon_{n}^{2}/\tau({\mathsf{M}})\geq\varepsilon_{0})\leq{\mathbb{P}}(\varepsilon_{n}\geq
c_{0}\varepsilon_{0})$, with
$c_{0}=(1+\mathrm{diam}({\mathsf{M}})/\tau({\mathsf{M}}))^{-1}$. Apply Lemma
4.6 with $r=c_{0}\varepsilon_{0}$ to obtain that

$\displaystyle{\mathbb{E}}[\mu_{n,i}(T_{s})^{2}\mathbf{1}\\{E^{c}\\}]$
$\displaystyle\leq n^{2i}\frac{C}{f_{\min}r^{m}}\exp(-ncf_{\min}r^{m})\leq
C_{0}\exp(-c_{1}n)$

for some positive constants $c_{1}$, $C_{0}$. Furthermore, recall that $s\leq
n^{1/m}\varepsilon_{\max}$. Hence,

$\displaystyle{\mathbb{E}}[\mu_{n,i}(T_{s})^{2}\mathbf{1}\\{E^{c}\\}]$
$\displaystyle\leq
C_{0}\exp(-c_{1}\varepsilon_{\max}^{-m}s^{m})=C_{0}\exp(-c_{2}s^{m})$

for some positive constant $c_{2}$.

It remains to bound ${\mathbb{E}}[\mu_{n,i}(T_{s})^{2}\mathbf{1}\\{E\\}]$. For
each $j=1,\dots,n$, consider the set $\Xi_{n,s}^{j}$ of critical points
$z\in\mathrm{Crit}({\mathsf{A}}_{n})$ such that $\sigma_{{\mathsf{A}}_{n}}(z)$
contains the point $X_{i}$ and $sn^{-1/m}\leq
d_{{\mathsf{A}}_{n}}(z)\leq\varepsilon_{n}+\varepsilon_{n}^{2}/\tau({\mathsf{M}})$.

######  Lemma 4.7.

It holds that $\mu_{n,i}(T_{s})$ is smaller than

$\frac{1}{n}\sum_{j=1}^{n}L_{j}^{i+2},$ (25)

where $L_{j}$ is the cardinality of the set
$\bigcup_{z\in\Xi_{n,s}^{j}}\sigma_{{\mathsf{A}}_{n}}(z)$.

###### Proof.

Lemma A.1 applied to a realization of ${\mathsf{A}}_{n}$ and the interval
$[sn^{-1/m},\varepsilon_{n}+\varepsilon_{n}^{2}/\tau({\mathsf{M}})]$ yields
that the number of points in $\mathrm{dgm}_{i}({\mathsf{A}}_{n})$ with at
least one of their coordinates in
$[sn^{-1/m},\varepsilon_{n}+\varepsilon_{n}^{2}/\tau({\mathsf{M}})]$ is upper
bounded by $\sum_{j=1}^{n}L_{j}^{i+2}$. By definition, this means that
$\mu_{n,i}(T_{s})\leq\frac{1}{n}\sum_{j=1}^{n}L_{j}^{i+2}$, as desired. ∎

One can show that the set $\Xi_{n,s}^{j}$ is localized, in the sense that it
is included in a ball centered at $X_{j}$, with a radius depending on the
sample ${\mathsf{A}}_{n}$, that is small with high probability. Hence, the
number $L_{j}$ is controlled by the number of points in ${\mathsf{A}}_{n}$
found in a small (random) neighborhood of $X_{j}$. Let us make this idea
rigorous.

For $r\geq 0$, define the shape

$\mathcal{C}(r)=\\{y=(y_{m},y_{d-m})\in{\mathbb{R}}^{m}\times{\mathbb{R}}^{d-m}:\
\|y\|\leq r,\ \|y_{d-m}\|\leq\frac{\|y\|^{2}}{2\tau({\mathsf{M}})}\\}.$ (26)

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

$\\{y=(y_{m},y_{d-m})\in\mathcal{C}(r):\ y_{m}/\|y_{m}\|\in W\\},$

where $W$ is an element of the partition $\mathcal{W}$.

For $x\in{\mathsf{M}}$, consider an isometry
$\iota_{x}:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ sending
${\mathbb{R}}^{m}\times\\{0\\}^{d-m}$ to $T_{x}{\mathsf{M}}$. Let
$\mathcal{C}(x,r)=x+\iota_{x}(\mathcal{C}(r))$. Likewise, we define a
partition $\mathcal{W}(x,r)$ by applying the affine transformation $W\mapsto
x+\iota_{x}(W)$ to each $W\in\mathcal{W}(r)$.

For $j=1,\dots,n$, let $R_{jn}$ be the smallest radius $r\leq\varepsilon_{0}$
such that every $W\in\mathcal{W}(X_{j},r)$ contains a point of
${\mathsf{A}}_{n}$ other than $X_{j}$. By convention, we let
$R_{jn}=\varepsilon_{0}$ if such a radius does not exist. $R_{jn}$ can be made
measurable with a good choice of $x\mapsto\iota_{x}$; we assume it to be the
case henceforth.

######  Lemma 4.8.

Let $\varepsilon_{0}\leq\tau({\mathsf{M}})/\sqrt{2}$. For all $j=1,\dots,n$,
if $z\in\mathrm{Crit}({\mathsf{A}}_{n})$ is such that
$X_{j}\in\sigma_{{\mathsf{A}}_{n}}(z)$ and
$d_{{\mathsf{A}}_{n}}(z)\leq\varepsilon_{0}$, then $\|X_{j}-z\|\leq
c_{0}R_{jn}$ for some positive absolute constant $c_{0}$.

###### Proof.

Recall that for $x\in{\mathsf{M}}$, $\pi_{x}$ is the orthogonal projection on
$T_{x}{\mathsf{M}}$ while $\pi_{x}^{\bot}$ is the orthogonal projection on the
normal space at $x$. Let $j=1,\dots,n$ and let
$z\in\mathrm{Crit}({\mathsf{A}}_{n})$ be such that
$X_{j}\in\sigma_{{\mathsf{A}}_{n}}(z)$. The direction
$e=\pi_{X_{j}}(z-X_{j})/\|\pi_{X_{j}}(z-X_{j})\|$ belongs to the unit sphere
in $T_{X_{j}}{\mathsf{M}}$; note that $\pi_{X_{j}}(z-X_{j})\neq 0$ due to
Lemma B.2 below, which applies as
$d_{{\mathsf{A}}_{n}}(z)\leq\varepsilon_{0}\leq\tau({\mathsf{M}})/\sqrt{2}$.
Hence, $\iota_{x}^{-1}(e)$ belongs to an element $W_{0}$ of the partition
$\mathcal{W}$. Consider the corresponding element $W$ of the partition
$\mathcal{W}(X_{j},R_{jn})$.

If $R_{jn}=\varepsilon_{0}$, then the conclusion of the lemma holds (for
$c_{0}=1$): indeed we have
$\|X_{j}-z\|=d_{{\mathsf{A}}_{n}}(z)\leq\varepsilon_{0}\leq R_{jn}$.
Otherwise, by assumption, there exists a point $X_{k}\in W$ for some $k\neq
j$. As $X_{j}\in\sigma_{{\mathsf{A}}_{n}}(z)$, it holds that
$\|X_{j}-z\|\leq\|X_{k}-z\|$. Hence,

$\displaystyle\|X_{j}-z\|^{2}\leq\|X_{k}-z\|^{2}=\|X_{j}-z\|^{2}+\|X_{j}-X_{k}\|^{2}+2\langle
X_{k}-X_{j},X_{j}-z\rangle$

and

$\langle X_{k}-X_{j},z-X_{j}\rangle\leq\frac{\|X_{j}-X_{k}\|^{2}}{2}.$ (27)

We write

$\langle
X_{k}-X_{j},z-X_{j}\rangle=\langle\pi_{X_{j}}(X_{k}-X_{j}),\pi_{X_{j}}(z-X_{j})\rangle+\langle\pi_{X_{j}}^{\bot}(X_{k}-X_{j}),\pi_{X_{j}}^{\bot}(z-X_{j})\rangle$

By construction, as the diameter of $W_{0}$ is less than $\theta$, we have
$\langle\pi_{X_{j}}(X_{k}-X_{j}),\pi_{X_{j}}(z-X_{j})\rangle\geq\cos(\pi/4)\|\pi_{X_{j}}(X_{k}-X_{j})\|\|\pi_{X_{j}}(z-X_{j})\|$.
On the other hand, according to [Fed59, Theorem 4.18],

$\|\pi_{X_{j}}^{\bot}(X_{k}-X_{j})\|\leq\frac{\|X_{k}-X_{j}\|^{2}}{2\tau({\mathsf{M}})},$
(28)

which also implies that
$\|\pi_{X_{j}}(X_{k}-X_{j})\|\geq\|X_{j}-X_{k}\|\sqrt{1-\frac{\varepsilon_{0}^{2}}{4\tau({\mathsf{M}})^{2}}}$.
Similarly, Lemma B.2 from the Appendix states that
$\|\pi_{X_{j}}(z-X_{j})\|\geq\|z-X_{j}\|/\sqrt{2}$ and
$\|\pi_{X_{j}}^{\bot}(z-X_{j})\|\leq\|z-X_{j}\|^{2}/\tau({\mathsf{M}})$ (using
our assumption that
$d_{{\mathsf{A}}_{n}}(z)\leq\varepsilon_{0}\leq\tau({\mathsf{M}})/\sqrt{2}$).

Hence we obtain that

$\displaystyle\cos(\pi/4)\|X_{j}-X_{k}\|\|z-X_{j}\|\frac{\sqrt{1-\frac{\varepsilon_{0}^{2}}{4\tau({\mathsf{M}})^{2}}}}{\sqrt{2}}$
$\displaystyle\leq\cos(\pi/4)\|\pi_{X_{j}}(X_{k}-X_{j})\|\|\pi_{X_{j}}(z-X_{j})\|$
$\displaystyle\leq\frac{\|X_{k}-X_{j}\|^{2}}{2}+\frac{\|X_{k}-X_{j}\|^{2}\|z-X_{j}\|^{2}}{2\tau({\mathsf{M}})^{2}}$
$\displaystyle\leq\|X_{k}-X_{j}\|^{2}\left(\frac{1}{2}+\frac{\varepsilon_{0}^{2}}{2\tau({\mathsf{M}})^{2}}\right).$

Dividing by $\|X_{j}-X_{k}\|$ and using that $\|X_{j}-X_{k}\|\leq R_{jn}$ and
that $\varepsilon_{0}\leq\tau({\mathsf{M}})/\sqrt{2}$, we see that
$\|z-X_{j}\|$ is smaller than $R_{jn}$ up to an absolute multiplicative
constant. ∎

Let us now show that the random variable $R_{jn}$ has controlled tails.

######  Lemma 4.9.

For all $x\in{\mathsf{M}}$, $r\geq 0$,
$B(x,r)\cap{\mathsf{M}}\subset\mathcal{C}(x,r)$.

###### Proof.

Let $y\in{\mathsf{M}}$ be such that $\|\pi_{x}(y-x)\|\leq r$. Then, according
to [Fed59, Theorem 4.18],
$\|\pi_{x}^{\bot}(y-x)\|\leq\frac{\|y-x\|^{2}}{2\tau({\mathsf{M}})}$. In
particular, $B(x,r)\cap{\mathsf{M}}\subset\mathcal{C}(x,r)$. ∎

######  Lemma 4.10.

For $0<t\leq\tau({\mathsf{M}})/4$ and $j=1,\dots,n$, we have
${\mathbb{P}}(R_{jn}>t)\leq Ce^{-cf_{\min}(n-1)t^{m}}$ for some positive
constants $c=c(m)$, $C=C(m)$.

###### Proof.

If $R_{jn}$ is larger than $t$, then there exists at least one set
$W\in\mathcal{W}(X_{j},t)$ such that its intersection with ${\mathsf{A}}_{n}$
contains only $X_{j}$. Hence,

$\displaystyle{\mathbb{P}}(R_{jn}>t|X_{j})\leq\sum_{W\in\mathcal{W}(X_{j},t)}{\mathbb{P}}({\mathsf{A}}_{n}\cap
W=X_{j}|X_{j})=\sum_{W\in\mathcal{W}(X_{j},t)}(1-P(W))^{n-1}.$

Let $\pi_{0}$ be the orthogonal projection from ${\mathbb{R}}^{d}$ to
${\mathbb{R}}^{m}\times\\{0\\}^{d-m}$. The image of a set
$W\in\mathcal{W}(X_{j},t)$ by the projection $y\mapsto\pi_{X_{j}}(y-X_{j})$ is
equal to $\iota_{X_{j}}(\pi_{0}(W_{0}))\subset T_{X_{j}}{\mathsf{M}}$ for some
$W_{0}\in\mathcal{W}(t)$. For $t\leq\tau({\mathsf{M}})/4$, the orthogonal
projection $y\in B(X_{j},t)\cap{\mathsf{M}}\mapsto\pi_{X_{j}}(y-X_{j})\in
T_{X_{j}}{\mathsf{M}}$ is a diffeomorphism on its image, with Jacobian lower
bounded by a constant $c(m)$ that depends only on $m$, see e.g. [Div21, Lemma
2.2]. According to Lemma 4.9, the preimage of
$\iota_{X_{j}}(\pi_{0}(W_{0}))\subset T_{X_{j}}{\mathsf{M}}$ by this
diffeomorphism is equal to $W\cap M$. Hence, by a change of variable,

$P(W)=P(W\cap M)\geq f_{\min}c(m)\mathrm{Vol}_{m}(W_{0})\geq
f_{\min}c^{\prime}(m)t^{m}$

for some $c^{\prime}(m)>0$. Hence,

$\displaystyle{\mathbb{P}}(R_{jn}>t|X_{j})\leq\\#\mathcal{W}e^{-f_{\min}c^{\prime}(m)(n-1)t^{m}}.$

We conclude by taking the expectation. ∎

Let us now control the number of points found in a ball $B(X_{j},\kappa
R_{jn})$ for some $\kappa\geq 0$. Let $\rho_{0}>0$ be small enough such that
$\int_{B(x,\rho_{0})\cap{\mathsf{M}}}f<1/2$ for any $x\in{\mathsf{M}}$.

######  Lemma 4.11.

Let $l\geq 0,\kappa>0$ be such that
$\kappa\varepsilon_{0}\leq\min(\rho_{0},\tau({\mathsf{M}})/4)$. For
$j=1,\dots,n$, let $K_{jn}(\kappa)$ be the number of elements of
${\mathsf{A}}_{n}$ found in $B(X_{j},\kappa R_{jn})$. Then,
${\mathbb{E}}[K_{jn}(\kappa)^{l}]\leq
C(1+\left(\frac{f_{\max}}{f_{\min}}\right)^{l}\kappa^{lm})$ for some constant
$C=C(m,l)$.

###### Proof.

Let us write $\\{W_{1},\ldots,W_{K}\\}=\mathcal{W}(X_{j},\kappa R_{jn})$.
Without loss of generality, we can assume that $n\geq K+1$, as otherwise the
bound is trivial. As in [DP19, 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
$(W_{i}\cap{\mathsf{A}}_{n})\backslash\\{X_{j}\\}$, and $N_{-1}$ be the
cardinality of ${\mathsf{A}}_{n}\backslash B(X_{j},\kappa R_{jn})$. Define

$\tilde{\alpha}_{i}:=\int_{W_{i}}f$

and $\alpha_{i}:=\frac{\tilde{\alpha_{i}}}{1-\tilde{\alpha_{i^{*}}}}$ for all
$i\neq i^{*}$, as well as
$\alpha_{-1}=\frac{1-\sum_{i=1}^{K}\tilde{\alpha}_{i}}{1-\tilde{\alpha}_{i^{*}}}$.
Note that as $\kappa R_{jn}\leq\rho_{0}$, we have $\tilde{\alpha_{i}}<1/2$ for
all $i=1,\ldots,K$. As $\kappa
R_{jn}\leq\kappa\varepsilon_{0}\leq\tau({\mathsf{M}})/4$, we may use once
again that the orthogonal projection $y\in B(X_{j},\kappa
R_{jn})\cap{\mathsf{M}}\mapsto\pi_{X_{j}}(y-X_{j})\in T_{X_{j}}{\mathsf{M}}$
is a diffeomorphism on its image, with Jacobian upper and lower bounded by
constants depending only on $m$ (see [Div21, Lemma 2.2]) to also obtain that

$cf_{\min}(\kappa R_{jn})^{m}\leq\tilde{\alpha_{i}}\leq Cf_{\max}(\kappa
R_{jn})^{m}$

for all $i=1,\ldots,K$ and some constants $c=c(m)$, $C=C(m)$. Hence there
exists $C^{\prime}=C^{\prime}(m)>0$ such that

$cf_{\min}(\kappa R_{jn})^{m}\leq\alpha_{i}\leq C^{\prime}f_{\max}(\kappa
R_{jn})^{m}$

for all $i=1,\ldots,K$. Consider a multinomial random variable
$L=(L_{1},\ldots,\widehat{L_{i^{*}}},\ldots,L_{K},L_{-1})$ of parameters $n-2$
and
$(\alpha_{1},\ldots,\widehat{\alpha_{i^{*}}},\ldots,\alpha_{K},\alpha_{-1})$,
and let $E$ denote the event

$\\{L_{i}\geq 1\ \forall i\in\\{1,\ldots,K\\}\backslash\\{i^{*}\\}\\}.$

Then conditionally on $X_{j}$, $R_{jn}$ and $i^{*}$, the variable
$N=(N_{1},\ldots,\widehat{N_{i^{*}}},\ldots,N_{K},N_{-1})$ follows the same
distribution as $L\ |\ E$. Thus, conditionally on $X_{j}$, $R_{jn}$ and
$W_{i^{*}}$, the variable
$K_{jn}(\kappa)=1+\sum_{i=1}^{K}N_{i}=2+\sum_{i=1,i\neq i^{*}}^{K}N_{i}$
(where the initial $1$ comes from $X_{j}$) has the same distribution as

$2+\sum_{i=1,i\neq i^{*}}^{K}L_{i}|\ E.$

Note that as $\kappa R_{jn}\leq\rho_{0}$, we have $\alpha_{-1}\geq 1/2$.

As a result, Lemma B.3 in the Appendix yields that

$\displaystyle{\mathbb{E}}[K_{jn}(\kappa)^{l}|X_{j},R_{jn},W_{i^{*}}]$
$\displaystyle={\mathbb{E}}[(2+\sum_{i=1,i\neq i^{*}}^{K}L_{i})^{l}\ |\ E]\leq
2^{l}(2^{l}+{\mathbb{E}}[(\sum_{i=1,i\neq i^{*}}^{K}L_{i})^{l}\ |\ E])$
$\displaystyle\leq 2^{l}(2^{l}+C(K,l)(1+(n-2)\sum_{i=1,i\neq
i^{*}}^{K}\alpha_{i})^{l})$ $\displaystyle\leq
C^{\prime}(K,l)(1+(n\sum_{i=1,i\neq i^{*}}^{K}\alpha_{i})^{l})\leq
C^{\prime}(K,l)(1+(nKC^{\prime}f_{\max}(\kappa R_{jn})^{m})^{l})$
$\displaystyle\leq C(m,l)(1+\kappa^{ml}f_{\max}^{l}n^{l}R_{jn}^{ml}).$

We can now conclude by considering

$\displaystyle{\mathbb{E}}[K_{jn}(\kappa)^{l}]$
$\displaystyle={\mathbb{E}}_{X_{j},R_{jn},W_{i^{*}}}[{\mathbb{E}}[K_{jn}(\kappa)^{l}|X_{j},R_{jn},W_{i^{*}}]]\leq{\mathbb{E}}_{R_{jn}}[C(m,l)(1+\kappa^{ml}f_{\max}^{l}n^{l}R_{jn}^{ml})]$

The quantity $n^{l}{\mathbb{E}}[R_{jn}^{ml}]$ is bounded by a constant, which
is proved by integrating the tail bound found in Lemma 4.10. Indeed, Lemma
4.10 implies that the random variable $nR_{jn}^{m}$ 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
[Ver18, Section 2.7]. ∎

Let us wrap things up. Recall Lemma 4.7: it holds that

$\mu_{n,i}(T_{s})\leq\frac{1}{n}\sum_{j=1}^{n}L_{j}^{i+2},$ (29)

where $L_{j}$ is the cardinality of the set
$\bigcup_{z\in\Xi^{j}_{n,s}}\sigma_{{\mathsf{A}}_{n}}(z)$. But according to
Lemma 4.8, if $z\in\Xi^{j}_{n,s}$, then $\|X_{j}-z\|\leq c_{0}R_{jn}$. In
particular, $d_{{\mathsf{A}}_{n}}(z)\leq c_{0}R_{jn}$, and any point
$y\in\sigma_{{\mathsf{A}}_{n}}(z)$ is at distance less than $2c_{0}R_{jn}$
from $X_{j}$. Hence, $L_{j}$ is smaller than $K_{jn}(\kappa)$ for
$\kappa=2c_{0}$. Choose $\varepsilon_{0}$ so that
$\kappa\varepsilon_{0}\leq\kappa\varepsilon_{0}\leq\min(\rho_{0},\tau({\mathsf{M}})/4)$.
We are in position to apply Lemma 4.11. We further remark that Lemma 4.8
implies that $\Xi^{j}_{n,s}$ is empty if $c_{0}R_{jn}<sn^{-1/m}$.

Hence, by Jensen’s inequality,

$\displaystyle{\mathbb{E}}[\mu_{n,i}(T_{s})^{2}\mathbf{1}\\{E\\}]$
$\displaystyle\leq{\mathbb{E}}\left[\left(\frac{1}{n}\sum_{j=1}^{n}\mathbf{1}\\{2c_{0}R_{jn}\geq
sn^{-1/m}\\}K_{jn}(2c_{0})^{i+2}\right)^{2}\right]$
$\displaystyle\leq{\mathbb{E}}\left[\mathbf{1}\\{2c_{0}R_{1n}\geq
sn^{-1/m}\\}K_{1n}(2c_{0})^{2i+4}\right]$
$\displaystyle\leq\sqrt{{\mathbb{P}}(2c_{0}R_{jn}\geq
sn^{-1/m}){\mathbb{E}}[K_{1n}(2c_{0})^{4i+8}]}$ $\displaystyle\leq
C(m,i)\exp(-c(m)f_{\min}s^{m}),$

where we apply Lemma 4.10 and Lemma 4.11 at the last line. ∎

This completes the proof of 4.2. ∎

###  4.2 Regions (2)-(3)

It is a well-known fact that the Hausdorff distance
$\varepsilon=d_{H}({\mathsf{A}}_{n},{\mathsf{M}})$ between a random sample and
${\mathsf{M}}$ is of order $(\log n/n)^{1/m}$ whenever the underlying density
$f$ is bounded away from zero and $\infty$ on ${\mathsf{M}}$, see e.g.
[FLR+14]. Hence the PDs $\mathrm{dgm}_{i}^{(2)}({\mathsf{A}}_{n})$ and
$\mathrm{dgm}_{i}^{(3)}({\mathsf{A}}_{n})$ can be described using 2.3.
However, in the case where ${\mathsf{M}}$ is a generic submanifold, one can
actually obtain tighter results. We let $\\#E$ denote the cardinality of a
multiset $E$.

######  Proposition 4.12.

Let ${\mathsf{M}}$ be a generic $m$-dimensional submanifold. Assume that $P$
has a density $f$ on ${\mathsf{M}}$ bounded away from $0$ and $\infty$. Let
$i\geq 0$ be an integer. There exists an optimal matching
$\gamma_{n}:\mathrm{dgm}_{i}({\mathsf{A}}_{n})\cup\partial\Omega\to\mathrm{dgm}_{i}({\mathsf{M}})\cup\partial\Omega$
for the bottleneck distance between $\mathrm{dgm}_{i}({\mathsf{A}}_{n})$ and
$\mathrm{dgm}_{i}({\mathsf{M}})$ such that for any $q\geq 1$:

  * •

Region (2): It holds that
$\max_{u\in\mathrm{dgm}_{i}^{(2)}({\mathsf{A}}_{n})}|u_{2}-\gamma_{n}(u)_{2}|=O_{L^{q}}(n^{-2/m})$.

  * •

Region (3): It holds that
$\max_{u\in\mathrm{dgm}_{i}^{(3)}({\mathsf{A}}_{n})}\|u-\gamma_{n}(u)\|_{\infty}=O_{L^{q}}(n^{-2/m})$
and $\\#(\mathrm{dgm}_{i}^{(3)}({\mathsf{A}}_{n}))=O_{L^{q}}(1)$.

We remark that the same bounds can be obtained almost surely (e.g. “a.s. there
exists $C>0$ such that
$\max_{u\in\mathrm{dgm}_{i}^{(3)}({\mathsf{A}}_{n})}\|u-\gamma_{n}(u)\|_{\infty}\leq
Cn^{-2/m}$”), rather than in expectation, using similar arguments.

Proposition 4.12 yields two distinct improvements upon direct applications of
2.3 and 3.7 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
${\mathsf{A}}_{n}$ is in general only $\delta$-sparse for $\delta$ of order
$n^{-2/m}$. Hence, ${\mathsf{A}}_{n}$ is $(\delta,\varepsilon)$-dense in
${\mathsf{M}}$, but with a diverging ratio $\varepsilon/\delta$. Therefore,
3.7 cannot be applied to control the total number of points in
$\mathrm{dgm}_{i}({\mathsf{A}}_{n})$ in Region (3).

###### Proof.

Let $\varepsilon_{n}=d_{H}({\mathsf{A}}_{n},{\mathsf{M}})$. Let
$\Pi_{\mathsf{M}}:=\\{x\in\sigma_{\mathsf{M}}(z):\
z\in\mathrm{Crit}({\mathsf{M}})\\}$ be the (finite) set of projections of
critical points $z\in\mathrm{Crit}({\mathsf{M}})$. The proof of 3.7 relied on
the use of Theorem 1.6 in [ACSD23]. This theorem states roughly that both
critical points $z\in\mathrm{Crit}({\mathsf{A}}_{n})$ far from ${\mathsf{M}}$
and their projections $x\in\sigma_{{\mathsf{A}}_{n}}(z)$ are stable with
respect to the Hausdorff distance, meaning that every such point $z$ is at
distance $O(\varepsilon_{n})$ from a critical point
$z^{\prime}\in\mathrm{Crit}({\mathsf{M}})$, with $x$ being at distance
$O(\varepsilon_{n})$ from a point $x^{\prime}\in\Pi_{\mathsf{M}}$. Thus, the
number of critical points of ${\mathsf{A}}_{n}$ located close to a given
$z^{\prime}\in\mathrm{Crit}({\mathsf{M}})$ is crudely upper bounded by the
number of subsets which can be formed by selecting elements in neighborhoods
of size $O(\varepsilon_{n})$ around
$x^{\prime}\in\sigma_{{\mathsf{M}}}(z^{\prime})$. We used the same idea to
bound the cardinality of $\mathrm{dgm}_{i}^{(3)}({\mathsf{A}})$ in 3.7.

In a random setting, the distance $\varepsilon_{n}$ is of order $(\ln
n/n)^{1/m}$, as suggested by Lemma 4.6, 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 $O(\varepsilon_{n})$ around a
given point $x\in{\mathsf{M}}$. Hence, our earlier strategy is not tight
enough to bound in expectation the cardinality of
$\mathrm{dgm}_{i}^{(3)}({\mathsf{A}})$ by a constant.

We improve upon this strategy with the following intuition: the maximal
distance between a point $x$ of ${\mathsf{M}}$ and the point cloud
${\mathsf{A}}_{n}$ does not really matter in [ACSD23, Theorem 1.6], but only
the density of the point cloud ${\mathsf{A}}_{n}$ around the (finitely many)
projections $x\in\Pi_{\mathsf{M}}$. Although $\varepsilon_{n}$ is of order
$(\ln n/n)^{1/m}$, the distance between a fixed point $x\in{\mathsf{M}}$ and
${\mathsf{A}}_{n}$ is known to be of order $n^{-1/m}$ (see (40)). 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
$\mathrm{Crit}_{>}({\mathsf{A}}_{n})=\\{z\in\mathrm{Crit}({\mathsf{A}}_{n}):d_{{\mathsf{A}}_{n}}(z)\geq\tau({\mathsf{M}})/2\\}$
denote the set of critical points of ${\mathsf{A}}_{n}$ “far” from
${\mathsf{M}}$ and let
$\Pi_{{\mathsf{A}}_{n}}:=\\{x\in\sigma_{{\mathsf{A}}_{n}}(z):\
z\in\mathrm{Crit}_{>}({\mathsf{A}}_{n})\\}$ be the corresponding set of
projections. We let $\varepsilon_{0}$ be a small constant to be fixed later.
Theorem 1.6 from [ACSD23] states that, thanks to the genericity of
${\mathsf{M}}$, there exist $K_{1},K_{2}>0$ such that for $\varepsilon_{0}$
small enough, if $\varepsilon_{n}<\varepsilon_{0}$:

  * •

there exists a map
$\phi:\mathrm{Crit}_{>}({\mathsf{A}}_{n})\rightarrow\mathrm{Crit}({\mathsf{M}})$
such that $d(z,\phi(z))\leq K_{1}\varepsilon_{n}$ for all
$z\in\mathrm{Crit}_{>}({\mathsf{A}}_{n})$, and

  * •

there exists a map $\psi:\Pi_{{\mathsf{A}}_{n}}\rightarrow\Pi_{\mathsf{M}}$
such that $d(x,\psi(x))\leq K_{2}\varepsilon_{n}$ for all
$x\in\Pi_{{\mathsf{A}}_{n}}$.

Furthermore, the map $\psi$ is such that for each
$z\in\mathrm{Crit}_{>}({\mathsf{A}}_{n})$ and each
$x^{\prime}\in\sigma_{\mathsf{M}}(\phi(z))$, there exists
$x\in\sigma_{{\mathsf{A}}_{n}}(z)$ such that $\psi(x)=x^{\prime}$ (if
$\varepsilon_{0}$ is chosen small enough). Indeed, due to the genericity of
${\mathsf{M}}$, each critical point $z^{\prime}\in\mathrm{Crit}({\mathsf{M}})$
belongs to the relative interior of $\sigma_{\mathsf{M}}(z^{\prime})$ (as
stated in 3.5). In particular, it cannot belong to the convex hull of any
strict subset of $\sigma_{\mathsf{M}}(z^{\prime})$. By continuity of the
convex hull for the Hausdorff distance, as soon as
$z\in\mathrm{Crit}_{>}({\mathsf{A}}_{n})$ is close enough to $\phi(z)$ and its
projections $x\in\sigma_{{\mathsf{A}}_{n}}(z)$ are close enough to
$\sigma_{\mathsf{M}}(\phi(z))$ (i.e. as soon as $\varepsilon_{n}$ is small
enough), the point $z$ cannot belong to the convex hull of any subset
$S\subset\sigma_{{\mathsf{A}}_{n}}(z)$ that does not contain for each
$x^{\prime}\in\sigma_{\mathsf{M}}(\phi(z))$ at least one point $x$ such that
$\psi(x)=x^{\prime}$. As $z$ must belong to the convex hull of
$\sigma_{{\mathsf{A}}_{n}}(z)$, we conclude that each
$x^{\prime}\in\sigma_{\mathsf{M}}(\phi(z))$ has at least one preimage by
$\psi$ in $\sigma_{{\mathsf{A}}_{n}}(z)$. As $\mathrm{Crit}({\mathsf{M}})$ is
finite, taking the minimum distance such that this property holds for
$z^{\prime}$ over all $z^{\prime}\in\mathrm{Crit}({\mathsf{M}})$ proves the
claim.

We introduce the random function defined as

$\forall r\geq 0,\ E(r):=\min(\sup\\{d_{{\mathsf{A}}_{n}}(y):\
y\in{\mathsf{M}}\cap\Pi_{\mathsf{M}}^{r}\\},\varepsilon_{0})=:\min(F(r),\varepsilon_{0}),$
(30)

where $\Pi_{\mathsf{M}}^{r}$ is as usual the $r$-offset of $\Pi_{\mathsf{M}}$.
This random variable measures the density of the point cloud
${\mathsf{A}}_{n}$ in neighborhoods of size $r$ of points
$x^{\prime}\in\Pi_{\mathsf{M}}$.

Let

$\rho_{n}:=\sup\\{d_{\Pi_{\mathsf{M}}}(x):x\in\Pi_{{\mathsf{A}}_{n}}\\}$ (31)

give (when $\varepsilon_{n}<\varepsilon_{0}$ for $\varepsilon_{0}$ small
enough) the largest distance between a projection $x\in\Pi_{{\mathsf{A}}_{n}}$
and the corresponding projection $\psi(x)\in\Pi_{\mathsf{M}}$. We also define

$\eta_{n}:=\sup\\{d_{\Pi_{\mathsf{M}}}(x):\ x=\pi_{\mathsf{M}}(z),\
z\in\mathrm{Crit}_{>}({\mathsf{A}}_{n})\\}.$ (32)

We require the following controls on the random variables $\rho_{n}$ and
$\eta_{n}$.

######  Lemma 4.13.

There exist positive constants
$\varepsilon_{0}=\varepsilon_{0}({\mathsf{M}})$, $K_{3}=K_{3}({\mathsf{M}})$,
$K_{4}=K_{4}({\mathsf{M}})$ and $K_{5}=K_{5}({\mathsf{M}})$ such that if
$\varepsilon_{n}<\varepsilon_{0}$, it holds that for any
$z\in\mathrm{Crit}_{>}({\mathsf{A}}_{n})$

$\displaystyle|d_{{\mathsf{A}}_{n}}(z)-d_{\mathsf{M}}(\phi(z))|\leq
K_{3}E(\eta_{n})^{2},$ (33) $\displaystyle\eta_{n}\leq K_{4}E(\eta_{n}),$ (34)
$\displaystyle\rho_{n}\leq K_{5}E(\eta_{n}).$ (35)

###### Proof.

The proof mostly follows from a careful read of the proof of Theorem 1.6 in
[ACSD23]. First, remark that in the proof of Lemma 5.1 in [ACSD23] (applied
with $r=\tau({\mathsf{M}})/2$, $R=\mathrm{diam}({\mathsf{M}})$), the Hausdorff
distance $\varepsilon=d_{H}({\mathsf{A}}_{n},{\mathsf{M}})$ can be replaced by
$E(\eta_{n})$. Hence, if $z\in\mathrm{Crit}_{>}({\mathsf{A}}_{n})$, there
exists a $\mu$-critical point $z^{\prime}$ of ${\mathsf{M}}$ at distance less
than $E(\eta_{n})$, with $\mu\leq L_{1}({\mathsf{M}})E(\eta_{n})$. By
genericity, for a choice of $\varepsilon_{0}$ small enough, this point
$z^{\prime}$ is at distance $L_{2}({\mathsf{M}})\mu$ from a critical point
$z_{0}\in\mathrm{Crit}({\mathsf{M}})$. For $\varepsilon_{0}$ small enough,
this point $z_{0}$ is necessarily equal to $\phi(z)$, with
$\|z-\phi(z)\|\leq(1+L_{2}L_{1})E(\eta_{n})$. Due to the Lipschitz property of
the projection onto ${\mathsf{M}}$ around $z_{0}$ (see the arguments found at
the bottom of p. 19 in [ACSD23]), any point $x\in\pi_{\mathsf{M}}(z)$ is such
that $\|x-x^{\prime}\|\leq L_{3}({\mathsf{M}})\|z-\phi(z)\|$ for some
$x^{\prime}\in\sigma_{\mathsf{M}}(\phi(z))$. By taking the supremum over all
such points $x$, we obtain that

$\eta_{n}\leq L_{3}(1+L_{2}L_{1})E(\eta_{n}),$

proving (34).

One can also check that $\varepsilon_{n}$ can be replaced by $E(\eta_{n})$ in
the end of the proof of Theorem 1.6 in [ACSD23], so that (35) holds.

The proof of (33) is essentially the same as that of the first point of 3.7–we
omit it. ∎

Consider a (random) optimal matching
$\gamma_{n}:\mathrm{dgm}_{i}({\mathsf{A}}_{n})\cup\partial\Omega\rightarrow\mathrm{dgm}_{i}({\mathsf{M}})\cup\partial\Omega$
such that
$\max_{u\in\mathrm{dgm}_{i}^{(2)}({\mathsf{A}}_{n})}|u_{2}-\gamma_{n}(u)_{2}|\leq
C\varepsilon^{2}_{n}$ and
$\max_{u\in\mathrm{dgm}_{i}^{(3)}({\mathsf{A}}_{n})}\|u-\gamma_{n}(u)\|_{\infty}\leq
C\varepsilon^{2}_{n}$ whose existence is guaranteed by 2.3 (for some
$C=C({\mathsf{M}})>0$). For $r\geq 0$, we let $N(r)$ be the number of points
of ${\mathsf{A}}_{n}$ found in $\Pi_{\mathsf{M}}^{r}$ (i.e. at distance less
than $r$ from a point in $\Pi_{\mathsf{M}}$).

######  Lemma 4.14.

There exists $\varepsilon_{0}=\varepsilon_{0}({\mathsf{M}})$ such that if
$\varepsilon_{n}\leq\varepsilon_{0}$, then

$\displaystyle\max_{u\in\mathrm{dgm}_{i}^{(2)}({\mathsf{A}}_{n})}|u_{2}-\gamma_{n}(u)_{2}|\leq
K_{3}E(\eta_{n})^{2},$
$\displaystyle\max_{u\in\mathrm{dgm}_{i}^{(3)}({\mathsf{A}}_{n})}\|u-\gamma_{n}(u)\|_{\infty}\leq
K_{3}E(\eta_{n})^{2}.$

Furthermore, $\\#(\mathrm{dgm}_{i}^{(3)}({\mathsf{A}}_{n}))$ is smaller than
$N(\rho_{n})^{i+2}$.

###### Proof.

Let us assume from now on that $\varepsilon_{0}$ is small enough that
$2(K_{3}+C)\varepsilon_{0}^{2}$ is smaller than the smallest difference
between two distinct critical values of $d_{\mathsf{M}}$.

Consider a point $(u_{1},u_{2})\in\mathrm{dgm}_{i}^{(3)}({\mathsf{A}}_{n})$
that is mapped by $\gamma_{n}$ to the diagonal. The coordinates $u_{1}$ and
$u_{2}$ differ by at most $C\varepsilon_{n}^{2}$, hence they are critical
values of $d_{{\mathsf{A}}_{n}}$ that correspond to critical points that are
mapped by $\phi$ to two critical points of ${\mathsf{M}}$ that have the same
critical value (due to $2(K_{3}+C)\varepsilon^{2}$ being smaller than the
smallest difference between two distinct critical values of $d_{\mathsf{M}}$).
But as stated in Equation (33) of Lemma 4.13, these critical values must then
be $K_{3}E(\eta_{n})^{2}$-close to that of the two critical points of
${\mathsf{M}}$, hence
$\|u-\gamma_{n}(u)\|_{\infty}=d(u,\partial\Omega)=(u_{2}-u_{1})/2\leq
K_{3}E(\eta_{n})^{2}$, as desired.

Likewise, consider a point $(u_{1},u_{2})$ of
$\mathrm{dgm}_{i}^{(3)}({\mathsf{A}}_{n})$ that is mapped by $\gamma_{n}$ to a
point $(v_{1},v_{2})$ of $\mathrm{dgm}_{i}^{(3)}({\mathsf{M}})$. The birth
coordinates $u_{1}$ and $v_{1}$ differ by at most $C\varepsilon^{2}$, hence
the associated critical values must correspond to a critical point $z$ of
${\mathsf{A}}_{n}$ and a critical point $z^{\prime}$ of ${\mathsf{M}}$ such
that $\phi(z)$ has the same filtration value as $z^{\prime}$ (again due to
$2(K_{3}+C)\varepsilon^{2}$ being smaller than the smallest difference between
two distinct critical values of $d_{\mathsf{M}}$). But as above, we have
$|u_{1}-v_{1}|=|d_{{\mathsf{A}}_{n}}(z)-d_{\mathsf{M}}(z^{\prime})|=|d_{{\mathsf{A}}_{n}}(z)-d_{\mathsf{M}}(\phi(z))|\leq
K_{3}E(\eta_{n})^{2}$. As the same applies to $u_{2}$ and $v_{2}$, we find
that $\|u-\gamma_{n}(u)\|_{\infty}\leq K_{3}EE(\eta_{n})^{2}$ and
$\max_{u\in\mathrm{dgm}_{i}^{(3)}({\mathsf{A}}_{n})}\|u-\gamma_{n}(u)\|_{\infty}\leq
K_{3}E(\eta_{n})^{2}$. The same reasoning shows that
$\max_{u\in\mathrm{dgm}_{i}^{(2)}({\mathsf{A}}_{n})}|u_{2}-\gamma_{n}(u)_{2}|\leq
K_{3}E(\eta_{n})^{2}$.

Finally, and by definition,
$\Pi_{{\mathsf{A}}_{n}}=\\{x\in\sigma_{{\mathsf{A}}_{n}}(z):\
z\in\mathrm{Crit}({\mathsf{A}}_{n}),\
d_{{\mathsf{A}}_{n}}(z)\geq\tau({\mathsf{M}})/2\\}$ is included in
${\mathsf{A}}_{n}\cap\Pi_{\mathsf{M}}^{\rho_{n}}$, hence
$\\#(\Pi_{{\mathsf{A}}_{n}})\leq N(\rho_{n})$. Lemma A.1 applied to
${\mathsf{A}}_{n}$ and the interval $[\tau({\mathsf{M}})/2,+\infty)$ then
yields that the number of points in $\mathrm{dgm}_{i}^{(3)}({\mathsf{A}}_{n})$
is smaller than $N(\rho_{n})^{i+2}$. ∎

######  Lemma 4.15.

Let $\varepsilon_{0}$ be the parameter defined in Lemma 4.14. It holds that
for all $q\geq 1$,
$E(\eta_{n})\mathbf{1}\\{\varepsilon_{n}\leq\varepsilon_{0}\\}=O_{L^{q}}(n^{-1/m})$
and
$N(\rho_{n})\mathbf{1}\\{\varepsilon_{n}\leq\varepsilon_{0}\\}=O_{L^{q}}(1)$.

Assume for a moment that Lemma 4.15 holds. Then, we write (for a choice of
$\varepsilon_{0}$ small enough)

$\displaystyle\max_{u\in\mathrm{dgm}_{i}^{(2)}({\mathsf{A}}_{n})}|u_{2}-\gamma_{n}(u)_{2}|$
$\displaystyle\leq
K_{3}E(\eta_{n})^{2}\mathbf{1}\\{\varepsilon_{n}\leq\varepsilon_{0}\\}+R({\mathsf{M}})\mathbf{1}\\{\varepsilon_{n}>\varepsilon_{0}\\}$
$\displaystyle\leq
K_{3}E(\eta_{n})^{2}\mathbf{1}\\{\varepsilon_{n}\leq\varepsilon_{0}\\}+R({\mathsf{M}})\mathbf{1}\\{\varepsilon_{n}>\varepsilon_{0}\\}.$
(36)

where we recall that $R({\mathsf{M}})$ is the radius of the smallest enclosing
ball of ${\mathsf{M}}$. Because of Lemma 4.6, the random variable
$\mathbf{1}\\{\varepsilon_{n}>\varepsilon_{0}\\}$ is a $O_{L^{q}}(n^{-2/m})$
for all $q\geq 1$. But then, because of Lemma 4.15, we obtain that the right-
hand side in (36) is a $O_{L^{q}}(n^{-2/m})$ for all $q\geq 1$. Likewise, we
obtain that
$\max_{u\in\mathrm{dgm}_{i}^{(3)}({\mathsf{A}}_{n})}\|u-\gamma_{n}(u)\|_{\infty}=O_{L^{q}}(n^{-2/m})$
for all $q\geq 1$. At last, we have

$\displaystyle\\#(\mathrm{dgm}_{i}^{(3)}({\mathsf{A}}_{n}))\leq
N(\rho_{n})^{i+2}\mathbf{1}\\{\varepsilon_{n}\leq\varepsilon_{0}\\}+n^{i+2}\mathbf{1}\\{\varepsilon_{n}>\varepsilon_{0}\\}.$

The first term is a $O_{L^{q}}(1)$ for all $q\geq 1$ because of Lemma 4.15,
while the second one is a $O_{L^{q}}(1)$ for all $q\geq 1$ because of Lemma
4.6. This concludes the proof of Proposition 4.12. Let us now prove Lemma
4.15. ∎

###### Proof of Lemma 4.15.

Let $t>0$. The key observation to obtain Lemma 4.15 is that (34) implies that
if $\eta_{n}>t$, then there exists $r>t$ with $r\leq K_{4}E(r)$.

######  Lemma 4.16.

For all $\lambda>0$, there exist positive constants $C_{\lambda}$,
$c_{\lambda}$ (depending on $\lambda$, ${\mathsf{M}}$ and $f_{\min}$) such
that for all $r>0$, ${\mathbb{P}}(r\leq\lambda E(r))\leq
C_{\lambda}\exp(-c_{\lambda}nr^{m})$.

###### Proof.

Remark that if $r\leq\lambda E(r)$, then $r\leq\lambda F(r)$. The set
${\mathsf{M}}\cap\Pi_{\mathsf{M}}^{r}$ is the union of a finite number of
balls of radius $r$. Hence, it can be covered by
$C_{\lambda}=C_{\lambda}({\mathsf{M}})$ open balls of radius $r/(2\lambda)$,
with centers $x_{1},\dots,x_{C_{\lambda}}\in{\mathsf{M}}$. Note that if all
these balls intersect ${\mathsf{A}}_{n}$, then for all
$y\in{\mathsf{M}}\cap\Pi_{\mathsf{M}}^{r}$,
$d_{{\mathsf{A}}_{n}}(y)<r/\lambda$. Hence, if $\lambda F(r)\geq r$, then the
intersection of one of these balls with ${\mathsf{A}}_{n}$ is empty. Hence,
according to Lemma B.1,

${\mathbb{P}}(r\leq\lambda
F(r))\leq\sum_{k=1}^{C_{\lambda}}{\mathbb{P}}(d_{{\mathsf{A}}_{n}}(x_{k})\geq
r/(2\lambda))\leq
C_{\lambda}\exp(-C({\mathsf{M}})f_{\min}n(r/\lambda)^{m}).\qed$

Remark that the fonction $E$ is nondecreasing and $1$-Lipschitz continuous: we
have for $r<s$, $E(r)\leq E(s)\leq E(r)+(r-s)$. Fix $t>0$ and consider the
sequence $t_{k}=a^{k}t$ for some $a>1$ to fix. Assume that $\eta_{n}>t$ and
that $\varepsilon_{n}\leq\varepsilon_{0}$. Then $\eta_{n}$ is between two
values $t_{k}<t_{k+1}$. But then, according to Lemma 4.14,

$\displaystyle\frac{E(t_{k})}{t_{k}}\geq\frac{E(\eta_{n})-(\eta_{n}-t_{k})}{t_{k}}\geq\frac{1}{a}\frac{E(\eta_{n})}{\eta_{n}}\frac{t_{k+1}}{t_{k}}-(a-1)\geq\frac{1}{K_{4}}-(a-1)$
$\displaystyle\frac{E(t_{k+1})}{t_{k+1}}\geq\frac{1}{a}\frac{E(\eta_{n})}{t_{k}}\geq\frac{1}{a}\frac{E(\eta_{n})}{\eta_{n}}\geq\frac{1}{K_{4}a}.$

Choose $a>1$ such that $\frac{1}{K_{4}}-(a-1)>0$, and let

$\lambda=\min\Big{(}\frac{1}{K_{4}}-(a-1),\frac{1}{K_{4}a}\Big{)}>0$

We have proven that if $\varepsilon_{n}\leq\varepsilon_{0}$ and $\eta_{n}>t$,
then there exists $k\geq 0$ with $E(t_{k})\geq\lambda t_{k}$.

Hence,

$\displaystyle{\mathbb{P}}(\eta_{n}>t,\varepsilon_{n}\leq\varepsilon_{0})\leq
C_{\lambda}\sum_{k\geq 0}\exp(-c_{\lambda}a^{km}nt^{m}).$

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

$K_{6}(nt^{m})^{-1}\exp(-K_{7}nt^{m}).$ (37)

for two positive constants $K_{6}$, $K_{7}$ depending on ${\mathsf{M}}$ and
$f_{\min}$. But when $nt^{m}\leq 1$, we can simply use the bound
${\mathbb{P}}(\eta_{n}>t,\varepsilon_{n}\leq\varepsilon_{0})\leq 1$. Hence,

${\mathbb{P}}(\eta_{n}>t,\varepsilon_{n}\leq\varepsilon_{0})\leq\min(1,K_{6}(nt^{m})^{-1}\exp(-K_{7}nt^{m}))\leq
K_{8}\exp(-K_{9}nt^{m}))$ (38)

for two other constants $K_{8}$, $K_{9}$.

To summarize, we have shown that the random variable
$n\eta_{n}^{m}\mathbf{1}\\{\varepsilon_{n}\leq\varepsilon_{0}\\}$ is
subexponential, with subexponential norm depending only on ${\mathsf{M}}$ and
$f_{\min}$, see e.g. [Ver18, 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 ${\mathsf{M}}$ and $f_{\min}$.

Lemma B.1 shows that the random variable $nd_{{\mathsf{A}}_{n}}(x)^{m}$ for a
fixed $x\in{\mathsf{M}}$ is also subexponential. Thus, so is
$nE(0)^{m}=n\max_{x\in\Pi_{\mathsf{M}}}d_{{\mathsf{A}}_{n}}(x)^{m}$, as a
maximum of a finite number of subexponential random variables. As
$nE(\eta_{n})^{m}\leq n(\eta_{n}+E(0))^{m}\leq
2^{m-1}(n\eta_{n}^{m}+nE(0)^{m})$, the random variable
$nE(\eta_{n})^{m}\mathbf{1}\\{\varepsilon_{n}\leq\varepsilon_{0}\\}$ is also
subexponential. In particular, we have
$E(\eta_{n})\mathbf{1}\\{\varepsilon_{n}\leq\varepsilon_{0}\\}=O_{L^{q}}(n^{-1/m})$
for all $q\geq 1$.

It remains to bound $N(\rho_{n})$. First, because of (35), the random variable
$n\rho_{n}^{m}\mathbf{1}\\{\varepsilon_{n}\leq\varepsilon_{0}\\}$ is
subexponential. Also, for a fixed $t$, $N(t)$ follows a binomial distribution
of parameter $n$ and $P(\Pi_{\mathsf{M}}^{t})$. As long as
$t\leq\tau({\mathsf{M}})/4$, a ball of radius $t$ is of mass smaller than
$C_{m}f_{\max}t^{m}$ (see Lemma B.1). Let $0\leq k\leq n$ be an integer. For
any $t\leq\tau({\mathsf{M}})/4$, we bound

$\displaystyle{\mathbb{P}}(N(\rho_{n})\geq
k,\varepsilon_{n}\leq\varepsilon_{0})$ $\displaystyle\leq{\mathbb{P}}(N(t)\geq
k,\rho_{n}\leq t)+{\mathbb{P}}(\rho_{n}>t,\varepsilon_{n}\leq\varepsilon_{0})$
$\displaystyle\leq{\mathbb{P}}(N(t)\geq k)+2\exp(-K_{10}nt^{m}),$

where $K_{10}$ is proportional to the subexponential norm of
$n\rho_{n}^{m}\mathbf{1}\\{\varepsilon_{n}\leq\varepsilon_{0}\\}$, see [Ver18,
Section 2.7]. Let
$t=(k/n)^{1/m}\min(\tau({\mathsf{M}})/4,1/(\\#\Pi_{\mathsf{M}}\cdot
2C_{m}f_{\max})^{1/m})$. This choice of $t$ ensures that
$t\leq\tau({\mathsf{M}})/4$ and that ${\mathbb{E}}[N(t)]\leq
n(\\#\Pi_{\mathsf{M}})C_{m}f_{\max}t^{m}\leq k/2$. Then, by Bernstein’s
inequality [Ver18, Theorem 2.8.4],

$\displaystyle{\mathbb{P}}(N(t)\geq k)$
$\displaystyle\leq{\mathbb{P}}(N(t)-{\mathbb{E}}[N(t)]\geq
k/2)\leq\exp\left(-\frac{k^{2}/8}{n(\\#\Pi_{\mathsf{M}})C_{m}f_{\max}t^{m}+k/6}\right)$
$\displaystyle\leq\exp\left(-K_{11}k\right)$

for some constant $K_{11}$ depending on $m$, ${\mathsf{M}}$ and $f_{\max}$. We
have proven that for all $k\geq 0$,

$\displaystyle{\mathbb{P}}(N(\rho_{n})\geq
k,\varepsilon_{n}\leq\varepsilon_{0})$ $\displaystyle\leq
2\exp\left(-K_{11}k\right)+2\exp(-K_{10}nt^{m})$ $\displaystyle\leq
2\exp\left(-K_{11}k\right)+2\exp(-K_{12}k)$

for some constant $K_{12}$ depending on $m$, ${\mathsf{M}}$, $f_{\min}$ and
$f_{\max}$. Hence, the random variable
$N(\rho_{n})\mathbf{1}\\{\varepsilon_{n}\leq\varepsilon_{0}\\}$ is
subexponential, with a subexponential norm depending on ${\mathsf{M}}$,
$f_{\min}$ and $f_{\max}$. This implies in particular that
$N(\rho_{n})\mathbf{1}\\{\varepsilon_{n}\leq\varepsilon_{0}\\}=O_{L^{q}}(1)$