# Euler and Betti curves are stable under Wasserstein deformations of
distributions of stochastic processes

Daniel Perez Email<EMAIL_ADDRESS>Département de mathématiques et
applications, École normale supérieure, CNRS, PSL University, 75005 Paris,
France Laboratoire de mathématiques d’Orsay, Université Paris-Saclay, CNRS,
91405 Orsay, France  DataShape, Centre Inria Saclay, 91120 Palaiseau, France

###### Abstract

Euler and Betti curves of stochastic processes defined on a $d$-dimensional
compact Riemannian manifold which are almost surely in a Sobolev space
$W^{n,s}(X,\mathbb{R})$ (with $d<n$) are stable under perturbations of the
distributions of said processes in a Wasserstein metric. Moreover, Wasserstein
stability is shown to hold for all $p>\frac{d}{n}$ for persistence diagrams
stemming from functions in $W^{n,s}(X,\mathbb{R})$.

###### Contents

  1. 1 Introduction
    1. 1.1 Glossary, definitions and conventions
  2. 2 Stability results
    1. 2.1 Preliminary results
    2. 2.2 Stability theorems
  3. 3 Euler and Betti curves

##  1 Introduction

The study of random fields by Adler and Taylor using geometric and topological
methods gave rise to the following celebrated and interesting result.

######  Theorem 1.1 (Adler, Taylor, Theorem 12.4.1 [2]).

Let $f$ be a centered unit variance Gaussian field on a $d$-dimensional
$C^{2}$-submanifold $X$ of a $C^{3}$ manifold. Suppose $f$ satisfies the
conditions of [2, Corollary 11.3.5] (henceforth the GKF conditions), then,

$\mathbb{E}\\!\left[\chi(\\{f\geq
x\\})\right]=\sum_{j=0}^{d}{\mathcal{L}}_{j}(X)\,\rho_{j}(x)\,\,,$ (1.1)

where $\chi$ denotes the Euler characteristic and $\rho_{j}(x)$ denotes some
known functions of $x$ and ${\mathcal{L}}_{j}$ denote the Lipschitz-Killing
curvatures of $X$.

In recent years, much work has been done to weaken the Gaussian assumption of
the above theorem. A notable step in this direction is the following
remarkable result by Bobrowski and Borman.

######  Theorem 1.2 (Bobrowski, Borman, Theorem 4.1, [3]).

Let $X$ be a compact d-dimensional stratified space, and let
$f:X\to\mathbb{R}^{k}$ be a $k$-dimensional Gaussian random field satisfying
the GKF conditions. For a piecewise $C^{2}$-function
$G:\mathbb{R}^{k}\to\mathbb{R}$ and let $g=G\circ f$. Setting
$D_{u}=G^{-1}\,]-\infty,u]$, we have

$\mathbb{E}\\!\left[\int_{X}g\,d\chi\right]=\chi(X)\mathbb{E}\\!\left[g\right]-\sum_{j=1}^{d}(2\pi)^{j/2}{\mathcal{L}}_{j}(X)\int_{\mathbb{R}}\mathcal{M}_{j}(D_{u})\;du\,,$
(1.2)

where $\mathbb{E}\\!\left[g\right]$ has constant mean.

As pointed out by the authors of the above theorem in their original paper
[3], the difficulty in evaluating the expression above lies in computing the
Minkowski functionals $\mathcal{M}_{j}(D_{u})$. In this paper, we aim to
somewhat compensate this difficulty by providing a stability result regarding
the above expectations, and moreover, the distributions of the Euler curves
(i.e. the curves $x\mapsto\chi(\\{f\geq x\\}))$. More precisely, our main
result is the following.

######  Theorem 1.3.

Let $X$ be a compact $d$-dimensional manifold and let
$(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Consider two
stochastic processes $f,g:\Omega\to W^{n,s}(X,\mathbb{R})$ with $d<n$. Then,
the Euler curves of $f$ and $g$ are well-defined as elements of
$L^{1}(\mathbb{R})$ and as distributions, and for every
$\frac{d}{n}<\alpha<1$, there exists a finite constant $C_{X,n,s}$ such that

$\left\lVert\mathbb{E}\\!\left[\chi(\\{f\geq
x\\})\right]-\mathbb{E}\\!\left[\chi(\\{g\geq
x\\})\right]\right\rVert_{L^{1}_{x}(\mathbb{R})}\leq
C_{X,n,s}\left[\left\lVert
f\right\rVert_{L^{1}(\Omega,W^{n,s})}^{\alpha}+\left\lVert
g\right\rVert_{L^{1}(\Omega,W^{n,s})}^{\alpha}\right]W_{1,L^{\infty}}^{1-\alpha}(f_{\sharp}\mathbb{P},g_{\sharp}\mathbb{P})\,,$
(1.3)

where $W_{1,L^{\infty}}$ denotes the Wasserstein distance on the space of
probability measures on the space $(L^{\infty}(X,\mathbb{R})\cap
W^{n,s}(X,\mathbb{R}),\left\lVert\cdot\right\rVert_{L^{\infty}})$. Moreover,
the same statement holds for all Betti curves.

The key will be to adopt the point of view of persistent homology, for which
we refer the unfamiliar reader to the now classical references on the subject
for an introduction [5, 10]. The relevance of the study of persistence
diagrams was already understood by Bobrowski and Borman, and was later also
elaborated upon by different authors [1, 9].

Parallel to this development, the following result was recently shown by
Buhovsky et al.

######  Theorem 1.4 (Buhovsky, Payette, Polterovich, Polterovich, Shelukhin,
Stojisavljević, [4]).

Let $X$ be a compact Riemannian manifold of dimension $d$ and let $f\in
W^{n,s}(X,\mathbb{R})$ and $n>\frac{d}{s}$. Then, $f$ is continuous and for
all $\varepsilon>0$

$N^{\varepsilon}_{H_{k}(X,f)}\leq C_{X,n,s}\left\lVert
f\right\rVert_{W^{n,s}}^{d/n}\;\varepsilon^{-d/n}+\dim H_{k}(X)\,,$ (1.4)

where $N^{\varepsilon}_{H_{k}(X,f)}$ denotes the number of bars of length
$\geq\varepsilon$ in the barcode of the persistent homology of degree $k$,
$H_{k}(X,f)$.

A similar bound was shown in [11] for $\alpha$-Hölder functions and for a
wider class of metric spaces. The results of this paper can be seen as direct
consequences of the theory developped in [11], but which deserve to be made
explicit.

######  Remark 1.5.

Using the Sobolev injection $W^{n,s}\xhookrightarrow{}C^{r+\alpha}$, for
$r+\alpha=n-\frac{d}{s}$, we can rewrite the exponent as
$\frac{d}{r+\alpha+\frac{d}{s}}$. In particular, when $s=\infty$ and $r=0$, we
retrieve the same exponent as in [11].

###  1.1 Glossary, definitions and conventions

######  Notation 1.6.

We will interchangeably denote the persistent homology in degree $k$ on $X$
with respect to a filtration function $f:X\to\mathbb{R}$ by $H_{k}(X,f)$ or
$\operatorname{Dgm}_{k}(f)$. We use persistence barcodes, diagrams, modules
and their decompositions interchangeably via the usual identifications [5]. We
reserve the notation $H_{*}(X)$ for the homology groups of $X$ exclusively.

######  Convention 1.7.

Unless otherwise specified, we will always consider the filtration by
superlevel sets induced by a function $f:X\to\mathbb{R}$.

######  Convention 1.8.

Unless otherwise specified, we shall always consider the (infinite) bars of
the barcode to be truncated. That is, $b=b\cap[\inf f,\sup f]$ for every
$b\in\operatorname{Dgm}_{k}(f)$.

######  Notation 1.9.

The length of a bar $b\in\operatorname{Dgm}_{k}(f)$ is denoted by $\ell(b)$.
With this definition, we define

$\operatorname{Pers}_{p}^{p}(H_{k}(X,f)):=\sum_{b\in\operatorname{Dgm}_{k}(f)}\ell(b)^{p}\,.$
(1.5)

######  Notation 1.10.

We equip $\mathbb{R}^{2}$ with a metric $d_{\mathbb{R}^{2},\infty}$ defined by

$d_{\mathbb{R}^{2},\infty}((x,y),(x^{\prime},y^{\prime}))=\max\\{\left\lvert
x-x^{\prime}\right\rvert,\left\lvert y-y^{\prime}\right\rvert\\}\,.$ (1.6)

and define the distance from a point $(x,y)$ to the diagonal $\Delta$ as
$d_{\mathbb{R}^{2},\infty}((x,y),\Delta)=\frac{1}{2}\left\lvert
y-x\right\rvert$. We will use the following notation to design different sets
of $\mathbb{R}^{2}$. For superlevel set filtrations we denote

  * •

$\mathcal{X}=\\{(x,y)\in\mathbb{R}^{2}\,|\,y<x\\}$ and
$\overline{\mathcal{X}}$ its closure.

  * •

$R_{x}:=[x,\infty[\;\times\;]\\!-\\!\infty,x]\,\subset\overline{\mathcal{X}}$.

The reader can adjust the statements by modifying the definitions in the
following way if he or she is considering sublevel set filtrations. use the
following notation for superlevel and sublevel sets respectively:

  * •

$\mathcal{X}=\\{(x,y)\in\mathbb{R}^{2}\,|\,x<y\\}$ and
$\overline{\mathcal{X}}$ its closure.

  * •

$R_{x}:=\;]\\!-\\!\infty,x]\times[x,\infty[\,\subset\overline{\mathcal{X}}$.

######  Definition 1.11.

Define $\mathcal{D}_{p}$ to be the space of Radon measures on
$\overline{\mathcal{X}}$ (called persistence measures) with finite
$\operatorname{Pers}_{p}$ [8, Equations 3 and 4]. We equip $\mathcal{D}_{p}$
with the topology of the optimal partial transport distance $d_{p}$ (cf. [8,
Definition 2.1]).

##  2 Stability results

###  2.1 Preliminary results

######  Lemma 2.1.

$\operatorname{Pers}_{p}^{p}(H_{k}(X,f))=p\int_{0}^{\infty}\varepsilon^{p-1}N^{\varepsilon}_{H_{k}(X,f)}\;d\varepsilon\,.$
(2.7)

###### Proof.

The length of a bar $b\in H_{k}(X,f)$ can be written as

$\ell(b)^{p}=p\int_{0}^{\infty}1_{[0,\ell(b)]}(\varepsilon)\,\varepsilon^{p-1}\;d\varepsilon\,.$
(2.8)

The result follows by summing over all bars $b$ in the barcode of $H_{k}(X,f)$
and applying the Fubini theorem. ∎

A natural corollary of the above facts is that if $f\in
W^{n,s}(X,\mathbb{R})$, then we can bound the
$\operatorname{Pers}_{p}^{p}$-functional of $H_{k}(X,f)$ by a universal
constant and $\left\lVert f\right\rVert_{W^{n,s}}$ as soon as $p>\frac{d}{n}$.

######  Corollary 2.2.

Let $f\in W^{n,s}(X,\mathbb{R})$, then for $p>\frac{d}{n}$, there exists a
finite constant $C_{X,n,s}$ such that

$\operatorname{Pers}_{p}^{p}(H_{k}(X,f))\leq
C_{X,n,s}\,\frac{pn}{d}\,\left\lVert f\right\rVert_{W^{n,s}}^{p}+(2\left\lVert
f\right\rVert_{\infty})^{p}\dim H_{k}(X)\,.$ (2.9)

######  Remark 2.3.

Corollary 2.2 means

$f\in
W^{n,s}(X,\mathbb{R})\implies\operatorname{Dgm}_{k}(f)\in\bigcap_{p>d/n}\mathcal{D}_{p}\,.$
(2.10)

######  Remark 2.4.

Examining the limiting case where $n\to\infty$ and $s\to\infty$, one can
wonder whether $C^{\infty}$-functions have finite $\operatorname{Pers}_{0}$
(in other words, a finite number of bars). However, this is not the case as is
shown by the example $e^{-1/x^{2}}\sin(1/x)$ on $[0,1]$.

The answer of which is the optimal degree of regularity for which
$\operatorname{Pers}_{0}$ is finite is an interesting question. A sufficient
condition found through discussions with François Petit is that the functions
be subanalytic in an $o$-minimal structure. Once $\operatorname{Pers}_{0}$ is
finite, it follows that $\operatorname{Pers}_{p}^{p}$ is a holomorphic
function everywhere on $\mathbb{C}$.

On the other side of the story, we can ask whether the $C^{\alpha}$ (up to
reparametrization by a homeomorphism) condition is necessary to establish that
$\operatorname{Pers}_{p}^{p}$ is finite for some $p$. This question has been
positively answered in [11] in the 1D case (in fact, the correct notion of
regularity is the $p$-variation, but is equivalent to asking that the function
be $C^{\alpha}$ for some $\alpha$, up to reparametrization).

One can ask whether the bound $p>\frac{d}{n}$ for the finiteness of
$\operatorname{Pers}_{p}^{p}$ is sharp within the degree of regularity
considered. This question amounts to asking whether the asymptotics provided
by theorem 1.4 are in some sense sharp. Indeed, note that lemma 2.1 entails

######  Proposition 2.5.

$\limsup_{\varepsilon\to 0}\frac{\log
N^{\varepsilon}_{H_{k}(X,f)}}{\log(1/\varepsilon)}=\inf\\{p\,|\,\operatorname{Pers}_{p}(H_{k}(X,f))<\infty\\}\,.$
(2.11)

The sharpness of the asymptotics of $N^{\varepsilon}_{H_{k}(X,f)}$ was already
partially adressed in [4, §1.5] and positively answer the question. For
$\alpha$-Hölder functions, this sharpness was already thoroughly adressed in
[11], where it was shown that

$\inf\\{p\,|\,\operatorname{Pers}_{p}(H_{k}(X,f))<\infty\\}=\frac{d}{\alpha}\quad\text{generically
in }C^{\alpha}\,.$ (2.12)

This discussion suggests the following conjecture, which might be adressed in
a later work.

######  Conjecture 2.6.

Let $X$ be a compact Riemannian manifold of dimension $d$, then

$\inf\\{p\,|\,\operatorname{Pers}_{p}(H_{k}(X,f))<\infty\\}\leq\frac{d}{r+\alpha}$
(2.13)

and this inequality is saturated generically in the sense of Baire in
$C^{r+\alpha}$.

######  Remark 2.7.

By the work performed in [11], it suffices to exhibit a function $g$ on the
cube (or the ball), such that the bound is saturated. We obtain a dense family
by perturbating any dense family to make it constant on a small open ball
$B\subset X$ and adding $g$ to $f$ on this ball. For $H_{0}$, such a function
may be constructed by considering packings and smoothened versions of
distances to point clouds. In higher degrees of homology, exhibiting explicit
examples might be more difficult. Experience shows that it might be easier to
find a random field which saturates the bound, as done in [11].

###  2.2 Stability theorems

Having bounded $\operatorname{Pers}_{p}^{p}$, a stability result follows from
the standard argument considered in [6].

######  Theorem 2.8 (Wasserstein stability on compact Riemannian manifolds).

Let $X$ be a compact Riemannian manifold of dimension $d$ and let $f,g\in
W^{n,s}(X,\mathbb{R})$. Then, for all $p>q>\frac{d}{n}$,

$d_{p}^{p}(\operatorname{Dgm}_{k}(f),\operatorname{Dgm}_{k}(g))\leq
C_{X,n,s}\left(\left\lVert f\right\rVert_{W^{n,s}}^{q}\vee\left\lVert
g\right\rVert_{W^{n,s}}^{q}\right)\left\lVert
f-g\right\rVert_{\infty}^{p-q}\,.$ (2.14)

###### Proof.

Following the proof of the stability theorem shown in [6], it is sufficient to
show that $\operatorname{Pers}_{q}(\operatorname{Dgm}_{k}(f))$ and
$\operatorname{Pers}_{q}(\operatorname{Dgm}_{k}(g))$ can be bounded. This
bound is provided by corollary 2.2, yielding the result. ∎

Similarly, using [11], this bound immediately entails a stochastic stability
result.

######  Theorem 2.9 (Stochastic Wasserstein Stability).

Let $f$ and $g$ be two a.s. $W^{n,s}(X,\mathbb{R})$ stochastic processes on a
$d$-dimensional compact Riemannian manifold $X$, defined on a probability
space $(\Omega,\mathcal{F},\mathbb{P})$. Then, for any $0\leq k<d$, every
$\frac{d}{n}<q<p<\infty$ and any $r,s\in\;]1,\infty[$ satisfying
$\frac{1}{r}+\frac{1}{s}=1$ and $(p-q)s\geq 1$, there exists a constant
$C=C_{X,n,s}$ such that

$\displaystyle W_{p,d_{p}}((\operatorname{Dgm}_{k}\circ
f)_{\sharp}\mathbb{P},(\operatorname{Dgm}_{k}\circ g)_{\sharp}\mathbb{P})$
$\displaystyle\leq C\left[\mathbb{E}\\!\left[\left\lVert
f\right\rVert_{W^{n,s}}^{qr}\right]^{\frac{1}{r}}+\mathbb{E}\\!\left[\left\lVert
g\right\rVert_{W^{n,s}}^{qr}\right]^{\frac{1}{r}}\right]^{\frac{1}{p}}W_{(p-q)s,\infty}^{1-\frac{q}{p}}(f_{\sharp}\mathbb{P},g_{\sharp}\mathbb{P})$
(2.15) $\displaystyle\leq C\left[\mathbb{E}\\!\left[\left\lVert
f\right\rVert_{W^{n,s}}^{qr}\right]^{\frac{1}{r}}+\mathbb{E}\\!\left[\left\lVert
g\right\rVert_{W^{n,s}}^{qr}\right]^{\frac{1}{r}}\right]^{\frac{1}{p}}\left\lVert
f-g\right\rVert_{L^{(p-q)s}(\Omega,L^{\infty}(X,\mathbb{R}))}^{1-\frac{q}{p}}\,.$
(2.16)

###### Proof.

Here, we follow the proof of [11, Theorem 5.9]. The proof is the same,
replacing the $C^{\alpha}$-seminorms by the Sobolev norms. ∎

######  Theorem 2.10 (Stochastic stability of representations, [11]).

Let $\mathcal{B}$ be a Banach space and $\Psi:\mathcal{D}_{p}\to\mathcal{B}$
be an $\alpha$-Hölder continuous functional. Let
$\mathbb{P},\mathbb{Q}\in\mathcal{P}_{\alpha q}(\mathcal{D}_{p})$, then

$\left\lVert\mathbb{E}_{\mathbb{P}}[\Psi]-\mathbb{E}_{\mathbb{Q}}[\Psi]\right\rVert_{\mathcal{B}}\leq
W_{q,\left\lVert\cdot\right\rVert_{\mathcal{B}}}(\Psi_{\sharp}\mathbb{P},\Psi_{\sharp}\mathbb{Q})\leq\left\lVert\Psi\right\rVert_{C^{\alpha}(\mathcal{D}_{p},\mathcal{B})}W_{q\alpha,d_{p}}^{\alpha}(\mathbb{P},\mathbb{Q})\,.$
(2.17)

##  3 Euler and Betti curves

Degree-by-degree, we define the so-called Betti and Euler curves.

######  Definition 3.1.

Let $f\in W^{n,s}(X,\mathbb{R})$ with $n>\frac{d}{s}$. The $k$th degree Betti
curve of $f$, denoted $\beta_{k}(x,f)$ is defined as

$\beta_{k}(f)(x):=\beta_{k}(f,x):=\operatorname{Dgm}_{k}(f)(R_{x})\,,$ (3.18)

where $\operatorname{Dgm}_{k}(f)$ is seen as a persistence measure. The Euler
curve of $f$ is defined as

$\chi(f)(x):=\chi(f,x):=\sum_{k=0}^{d}(-1)^{k}\operatorname{Dgm}_{k}(f)(R_{x})$
(3.19)

Considered pointwise, showing a stability result for $\beta_{k}(f,x)$ is
hopeless. This follows from the fact that we may perturb $f$ so that we
displace the points in the diagram pushing them over the boundary of $R_{x}$,
thereby introducing a perturbation in $\beta_{k}(f,x)$ of a potentially
infinite number of points.

The correct point of view is to allow ourselves to consider the
$\beta_{k}(f,x)$ as an element of $L^{1}(\mathbb{R})$, or more generally as a
distribution.

######  Proposition 3.2.

Let $X$ be a $d$-dimensional Riemannian manifold and let $f\in
W^{n,s}(X,\mathbb{R})$ for $d<n$. Then, for every $k$,
$\beta_{k}(f),\chi(f)\in L^{1}(\mathbb{R})$ and thus also in
$\mathcal{D}^{\prime}(\mathbb{R})$. Moreover,

$\left\lVert\beta_{k}(f)\right\rVert_{L^{1}(\mathbb{R})}=\operatorname{Pers}_{1}(\operatorname{Dgm}_{k}(f))\,.$
(3.20)

######  Remark 3.3.

$\beta_{k}(f)$ can be infinite for some values of $x$. Nonetheless, the degree
of regularity considered ensures that we can still make sense of average
values of $\beta_{k}$ around small neighbourhoods of a level $x$. This
justifies that we look at the Betti and Euler curves as distributions.

###### Proof.

We do the proof for sublevel sets, noticing that the proof for superlevel sets
is completely analogous. Since $\beta_{k}(f)$ is a positive function,

$\displaystyle\left\lVert\beta_{k}(f)\right\rVert_{L^{1}(\mathbb{R})}$
$\displaystyle=\int_{\mathbb{R}}\beta_{k}(f,x)\;dx=\int_{\mathbb{R}}\operatorname{Dgm}_{k}(f)(R_{x})\;dx=\int_{\mathbb{R}}\left[\int_{\overline{\mathcal{X}}}1_{R_{x}}(z)\;d\operatorname{Dgm}(f)(z)\right]\;dx$
$\displaystyle=\int_{\overline{\mathcal{X}}}\left[\int_{\mathbb{R}}1_{R_{x}}(z)\;dx\right]\;d\operatorname{Dgm}(f)(z)\,.$

For any $z=(z_{1},z_{2})\in\overline{\mathcal{X}}$,

$1_{R_{x}}(z_{1},z_{2})=1_{[0,\infty[}(z_{2}-x)1_{[0,\infty[}(x-z_{1})\,.$
(3.21)

Using the translation invariance of the Lebesgue measure on $\mathbb{R}$, we
get

$\int_{\mathbb{R}}1_{R_{x}}(z_{1},z_{2})\;dx=z_{2}-z_{1}\,.$ (3.22)

This is nothing other than the distance $d_{\mathbb{R}^{2},\infty}$ from
$(z_{1},z_{2})$ to the diagonal, so

$\int_{\overline{\mathcal{X}}}(z_{2}-z_{1})\;d\operatorname{Dgm}(f)(z_{1},z_{2})=\operatorname{Pers}_{1}(\operatorname{Dgm}_{k}(f))<\infty\,,$
(3.23)

as soon as $\frac{d}{n}<1$ by virtue of corollary 2.2. ∎

######  Corollary 3.4 (No stability for $d_{p}$-stability for $p>1$).

Let $p>1$ and consider $\beta_{k}:\mathcal{D}_{p}\to L^{1}(\mathbb{R})$. Then,
$\beta_{k}$ is discontinuous.

######  Remark 3.5.

This suggests that $\chi:\mathcal{D}_{p}\to L^{1}(\mathbb{R})$ is
discontinuous as well.

###### Proof.

We give two different arguments which prove the statement. First, consider a
diagram $D\notin\mathcal{D}_{1}$ such that $D\in\mathcal{D}_{p}$ and any other
diagram $D^{\prime}\in\mathcal{D}_{1}\cap\mathcal{D}_{p}$. Then,
$d_{p}(D,D^{\prime})<\infty$, but the reverse triangle inequality entails

$\left\lVert\left\lVert\beta_{k}(D)\right\rVert-\left\lVert\beta_{k}(D^{\prime})\right\rVert\right\rVert_{L^{1}}\leq\left\lVert\beta_{k}(D)-\beta_{k}(D^{\prime})\right\rVert_{L^{1}}=\infty\,,$
(3.24)

since the lower bound is infinite. Second, suppose that $\beta_{k}$ is
continuous. Then the map $\mathcal{D}_{p}\to\mathbb{R}$, given by
$D\mapsto\left\lVert\beta_{k}(D)\right\rVert_{L^{1}}$ must be continuous by
continuity of the norm. However, by [8, Proposition 5.1] this map cannot be
continuous, since

$\int_{\mathbb{R}}1_{R_{x}}(z_{1},z_{2})\;dx=z_{2}-z_{1}$ (3.25)

is linear in the distance to the diagonal, and in particular not
$O(\left\lvert z_{2}-z_{1}\right\rvert^{p})$ as would be required for
continuity with respect to $d_{p}$. ∎

######  Remark 3.6.

Divol and Lacombe’s work [8, Proposition 5.1] also explain why it is necessary
– even within $\mathcal{D}_{1}$ – to regard the Betti and Euler curves as
distributions. Indeed, notice that pointwise in $x$, $1_{R_{x}}$ doesn’t
decrease to $0$ at the required rate.

######  Corollary 3.7 ($d_{1}$-stability of Betti and Euler curves).

Let $D,D^{\prime}\in\mathcal{D}_{1}$, then

$\left\lVert\beta_{k}(D)-\beta_{k}(D^{\prime})\right\rVert_{L^{1}(\mathbb{R})}\leq
2\,d_{1}(D,D^{\prime})\,.$ (3.26)

######  Remark 3.8.

This shows that $\beta_{k}:\mathcal{D}_{1}\to L^{1}(\mathbb{R})$ is a
$2$-Lipschitz representation.

###### Proof.

Consider a transport map $\pi\in\Gamma(D,D^{\prime})$. Then,

$\displaystyle\left\lVert\beta_{k}(D)-\beta_{k}(D^{\prime})\right\rVert_{L^{1}(\mathbb{R})}\leq\int_{\overline{\mathcal{X}}^{2}}\left[\int_{\mathbb{R}}\left\lvert
1_{R_{x}}(z)-1_{R_{x}}(w)\right\rvert\;dx\right]\;d\pi(z,w)\,.$

We carefully check that

$\int_{\mathbb{R}}\left\lvert 1_{R_{x}}(z)-1_{R_{x}}(w)\right\rvert\;dx\leq
2\,d_{\mathbb{R}^{2},\infty}(z,w)\,,$ (3.27)

which entails the result by taking $\pi$ to be an optimal transport for
$d_{1}$. ∎

######  Proposition 3.9 (Interpolation for optimal transport).

Let $0<p<q\leq\infty$ and $\theta\in\,]0,1[$. Define $p_{\theta}$ by

$\frac{1}{p_{\theta}}=\frac{\theta}{p}+\frac{1-\theta}{q}\,.$ (3.28)

Then, for $\mu,\nu\in\mathcal{D}_{p}\cap\mathcal{D}_{q}$

$\displaystyle d_{p_{\theta}}(\mu,\nu)$ $\displaystyle\leq
2^{1-\theta}\;d_{p}^{\theta}(\mu,\nu)\,(\operatorname{Pers}_{q}(\mu)+\operatorname{Pers}_{q}(\nu))^{1-\theta}$
(3.29) $\displaystyle d_{p_{\theta}}(\mu,\nu)$ $\displaystyle\leq
2^{\theta}\;d_{q}(\mu,\nu)^{1-\theta}\,(\operatorname{Pers}_{p}(\mu)+\operatorname{Pers}_{p}(\nu))^{\theta}\,.$
(3.30)

Consequently, if $p\leq r\leq q$, then
$\mathcal{D}_{p}\cap\mathcal{D}_{q}\subset\mathcal{D}_{r}$.

###### Proof.

We prove the first inequality, the second one following by analogy, by
interverting the roles of $p$ and $q$. Let $\pi$ be an optimal transport for
$d_{p}$. Applying Hölder’s inequality,

$\displaystyle d_{p_{\theta}}(\mu,\nu)$ $\displaystyle\leq\left\lVert
d_{\mathbb{R}^{2},\infty}\right\rVert_{L^{p_{\theta}}(\pi)}\leq\left\lVert
d_{\mathbb{R}^{2},\infty}\right\rVert_{L^{p}(\pi)}^{\theta}\left\lVert
d_{\mathbb{R}^{2},\infty}\right\rVert_{L^{q}(\pi)}^{1-\theta}$
$\displaystyle=d_{p}(\mu,\nu)^{\theta}\left[\int_{\overline{\mathcal{X}}^{2}}d_{\mathbb{R}^{2},\infty}^{q}(z,z^{\prime})\;d\pi(z,z^{\prime})\right]^{\frac{1-\theta}{q}}$
$\displaystyle\leq
d_{p}(\mu,\nu)^{\theta}\left[2^{q}\int_{\overline{\mathcal{X}}^{2}}d_{\mathbb{R}^{2},\infty}^{q}(z,\Delta)+d_{\mathbb{R}^{2},\infty}^{q}(\Delta,z^{\prime})\;d\pi(z,z^{\prime})\right]^{\frac{1-\theta}{q}}\,,$

where the inequality on the last line holds everywhere on the support of
$\pi$. This can be shown by defining

$S=\\{(z,z^{\prime})\in\overline{\mathcal{X}}^{2}\cap\operatorname{supp}(\pi)\,|\,d_{\mathbb{R}^{2},\infty}(z,z^{\prime})>d_{\mathbb{R}^{2},\infty}(z,\Delta)+d_{\mathbb{R}^{2},\infty}(z^{\prime},\Delta)\\}\,.$
(3.31)

This set $S$ either has null or positive measure. If it has positive measure,
then we can modify the transport plan $\pi$ by sending the projections of $S$
to the diagonal, thereby producing a transport plan of strictly inferior cost
to that of $\pi$, which is a contradiction. Hence, $S$ is of null measure, so
the equality holds over the support of the measure. This entails

$\displaystyle d_{p_{\theta}}(\mu,\nu)\leq
2^{1-\theta}\;d_{p}(\mu,\nu)^{\theta}\,(\operatorname{Pers}_{q}(\mu)+\operatorname{Pers}_{q}(\nu))^{1-\theta}\,.$

If $q=\infty$, since $\pi$ is an optimal transport between $\mu$ and $\nu$ and
$\mu,\nu\in\mathcal{D}_{\infty}$, $\pi$ itself must have compact support and
the diameter of the support is bounded above by
$\operatorname{Pers}_{\infty}(\mu)\vee\operatorname{Pers}_{\infty}(\nu)$, so
the inequality of the proposition follows. ∎

######  Remark 3.10.

Taking $0<p<1\leq q$, proposition 3.9 allows us to say that, despite
$\beta_{k}$ and $\chi$ being discontinuous on $\mathcal{D}_{q}$, these
functionals are continuous on $\mathcal{D}_{p}\cap\mathcal{D}_{q}$ (with
$p<q$) equipped with the $d_{q}$ metric. As shown in this paper, we may always
do this for diagrams stemming from functions in $W^{n,s}(X,\mathbb{R})$,
provided that $n>d$. This reconciles our results with the ones found in [7],
where the case of subanalytic functions was studied (in which case
$\operatorname{Dgm}(f)\in\mathcal{D}_{0}\cap\mathcal{D}_{\infty}$).

######  Theorem 3.11 (Stability of Euler and Betti curves under perturbations
in distribution).

Let $f$ and $g$ be two stochastic processes on a $d$-dimensional compact
Riemannian manifold $X$, defined on a common probability space
$(\Omega,\mathcal{F},\mathbb{P})$ such that almost surely $f,g\in
W^{n,s}(X,\mathbb{R})$ with $\frac{d}{n}<1$. Then, for every
$\frac{d}{n}<\alpha<1$, there exists a constant $C=C_{X,n,s}$ such that,

$W_{1,L^{1}}(\beta_{k}(f)_{\sharp}\mathbb{P},\beta_{k}(g)_{\sharp}\mathbb{P})\leq
C\left[\left\lVert f\right\rVert_{L^{1}(\Omega,W^{n,s})}^{\alpha}+\left\lVert
g\right\rVert_{L^{1}(\Omega,W^{n,s})}^{\alpha}\right]W_{1,L^{\infty}}^{1-\alpha}(f_{\sharp}\mathbb{P},g_{\sharp}\mathbb{P})\,,$
(3.32)

and a similar inequality holds by replacing the Betti curves with the Euler
curves $\chi$.

###### Proof.

Recognizing that $\beta_{k}$ are Lipschitz representations, it is a simple
application of the stochastic stability theorem and its analog for
representations, where we set $q=\alpha$, $p=1$ and $s=\frac{1}{1-\alpha}$ and
$r=\frac{1}{\alpha}$. ∎

######  Remark 3.12.

Using the Kantorovich-Rubenstein duality, a lower bound of $W_{1,L^{1}}$ can
be easily established. Recall that [12, Particular case 5.16]

$\sup_{\psi\in\operatorname{Lip}_{1}(L^{1}(\mathbb{R}),\mathbb{R})}\mathbb{E}\\!\left[\psi(\chi(f))-\psi(\chi(g))\right]\leq
W_{1,L^{1}}(\chi(f)_{\sharp}\mathbb{P},\chi(g)_{\sharp}\mathbb{P})\,.$ (3.33)

This point of view further justifies our view of $\beta_{k}$ and $\chi$ as
distributions. Indeed, for $\varphi\in\mathcal{D}(\mathbb{R})$, the map

$\psi:\chi\mapsto\int_{\mathbb{R}}\chi(x)\varphi(x)dx\,,$ (3.34)

is clearly Lipschitz. This distributional view of the Euler and Betti curves
is (unsurprisingly) also stable with respect to perturbations in the
distributions of the random fields from which they stem from.

A particularly interesting family of Lipschitz functionals (in light of recent
developments [9]) to consider is

$\psi_{\theta}:\chi\mapsto\int_{\mathbb{R}}e^{-ix\theta}\chi(x)\;dx\,.$ (3.35)

Using the Kantorovich-Rubenstein duality on the real and imaginary components
of the above integral separately, and taking the supremum over all $\theta$ we
arrive that the conclusion that

$\left\lVert\mathbb{E}\\!\left[\mathcal{F}\chi(f)-\mathcal{F}\chi(g))\right]\right\rVert_{L^{\infty}(\mathbb{R})}\leq\sqrt{2}\,W_{1,L^{1}}(\chi(f)_{\sharp}\mathbb{P},\chi(g)_{\sharp}\mathbb{P})\,,$
(3.36)

where $\mathcal{F}$ denotes the Fourier transform. Interestingly, one can
apply the same reasoning for different integral transforms, provided that
their kernel is such that the transform is Lipschitz on $L^{1}$. Of course,
these arguments are all also valid for the Betti curves as well. Applying the
stability of the Euler and Betti curves under perturbations in the
distributions of $f$ and $g$, we arrive at the conclusion that these
transforms are stable with respect to changes in distribution.

######  Remark 3.13.

Looking at Euler curves, it is possible to recover the Euler characteristic of
level sets of $f$. Since for every $\varepsilon>0$,
$\\{f>x-\varepsilon\\}\cap\\{f<x+\varepsilon\\}=\\{x-\varepsilon<f<x+\varepsilon\\}$
and that $\\{f>x-\varepsilon\\}\cup\\{f<x+\varepsilon\\}=X$, by the valuation
property of the Euler characteristic, we recover

$\chi(\\{x-\varepsilon<f<x+\varepsilon\\})+\chi(X)=\chi(f,x-\varepsilon)+\chi(-f,-x-\varepsilon)\,,$
(3.37)

by taking $\varepsilon\to 0$, we obtain $\chi(f=x)$, whenever it is defined.

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in this paper. The NEA designates a single paper per planet from which it
extracts the default planet and star parameters. Because of this, there are
instances where the default parameters for a planet is taken from a paper that
does not provide measurements for all parameters, even though there are other
papers that have measured the parameter (i.e. a RV paper that does not state
the radius measured in a previous transit survey). As mixing parameters from
different studies can lead to errors due to inconsistent methodology, we
elected to only use the default parameters and calculate missing parameters
where possible. We recommend the reader investigate all papers listed on the
NEA for any particular planet of interest to determine the best parameters
available.

##  5 Conclusions

In this paper, we present a complete catalog (at time of writing) of planets
that orbit within or through the HZ, including the HZ boundaries and the
percentage of the orbital period that each planet spends in their star’s HZ.
Observational metrics for each planet, such as TSM values and RV amplitude,
are included to facilitate selection for future follow-up observations.
Demographics of Rocky-OHZ, 100%HZ, $>0\%$HZ, and the full catalog of
exoplanets are explored in Section 4, and various planets are highlighted as
potential targets to test the boundaries of habitability.

Histograms of Rocky-OHZ, 100%HZ, $>0\%$HZ, and the full catalog of exoplanets
are discussed in Section 4. There is a lack of planets in the sub-Saturn
radius valley for the 100%HZ, $>0\%$HZ, and the full catalog of exoplanets.
Observational biases may be driving this gap but as the gap is seen amongst
the HZ planets, where the RV signature of a $\geq~{}10~{}M\oplus$ is well
within the capability of detection by current technology for most stars, we
believe further investigation into whether this gap is real is warranted.

The small planet radius gap may be evident in each of the 100%HZ, $>0\%$HZ,
and full catalog groups of planets. Due to small number statistics driven by
the difficulty in finding low mass planets in the HZ we are unable to conclude
whether this gap is real or due to observational biases. If the gap does
exist, determining whether this gap amongst the HZ planets is caused by
photoevaporation, core-powered mass-loss or some combination of the two will
provide insight into the early atmospheric evolution of these HZ worlds. Each
atmosphere loss mechanism has different implications for the atmospheric
evolution and surface conditions of a planet, particularly of terrestrial HZ
planets. Determining the existence and cause of this gap within the HZ planets
is paramount.

Rocky-OHZ planets have predominantly low eccentricities. In general, longer
period and larger mass planets tend to have a wider range of eccentricities
than the shorter period planets.

TSM values for HZ planets are relatively low compared with the full inventory
of known exoplanets, mostly due to the low $T_{\mathrm{eq}}$ of these HZ
planets. The planets with the highest TSM values that are
$\leq~{}2~{}R\oplus$, orbit within the HZ and are known to transit are LHS
1140 b, TRAPPIST-1 d and K2-3 d. Of the $\leq~{}2~{}R\oplus$, 100% HZ planets
not known to transit GJ 667 C c and Teegarden’s Star b have the highest TSM
values, while Wolf 1061 c, Ross 508 b, and Proxima Cen b have the highest
values for $\leq~{}2~{}R\oplus$ planets that pass through the HZ. Of all the
planets ($\leq~{}10~{}R\oplus$) passing through and orbiting within the HZ,
including those that are not known to transit, HD 102365 b and 55 Cnc f have
the greatest TSM values.

Observational biases drive many of the features seen in the demographics.
Transit surveys are optimised to discover planets around smaller, fainter
stars and RV surveys are optimised for brighter, sun-like stars. This causes a
bi-modal distribution and a possibly misleading valley to appear in the
histograms of both $J$ and $T_{\mathrm{eff}}$. In the planet period versus
planet mass plot of Figure 5, the plot suggests that larger stars tend to host
more massive planets in the HZ, however this is likely influenced by the
increased difficulty in trying to detect lower mass planets around larger
stars. The lack of HZ planets detected around highly evolved stars, as seen in
Figure 6, is again likely due to an observational bias. Detecting HZ planets
around these massive stars may be beyond current capabilities, as the transit
depth of existing planets reduces as these stars expand, and RV surveys tend
to focus on the more stable main sequence stars.

The inventory of HZ planets will continue to increase as current exoplanet
surveys, such as TESS (Ricker et al., 2015) and CHEOPS (Broeg et al., 2014),
provide new discoveries and characterization opportunities. With the
recommendations of the Decadal Survey on Astronomy and Astrophysics 2020
focusing on the detection and characterisation of habitable exoplanets (of
Sciences Engineering & Medicine, 2021b), observations of known HZ planets from
this catalog, particularly of the more extreme planets such as those described
in Section 4.2, are vital. These observations will help determine if the HZ is
a useful tool and will facilitate testing of the boundaries of habitability.
These more extreme planets are important targets for future observations as
they will provide insight into the habitability of planets with high
eccentricity, high or low density, circumbinary planets or those orbiting
evolved stars along with positioning in and around the HZ. It is essential to
follow-up many of these edge case planets to thoroughly test the boundaries of
habitability and the robustness of planetary atmospheric evolution. The JWST
is set to observe some of the highest measured density HZ planets, such as the
TRAPPIST-1 system (Lustig-Yaeger et al., 2019). Detection and characterisation
of any atmospheres on TRAPPIST-1 d,e,f, and g will provide useful insight into
the habitability of planets around M-dwarf stars, as well as allow comparative
planetology of multiple planets within the HZ. As a similar historical stellar
environment can be assumed for each planet, this direct comparison of
terrestrial planets at varying distances within the HZ is ideal for
determining how habitability changes with position within the HZ.

The list of known exoplanets continues to grow, with over 5000 confirmed
planets and many more awaiting confirmation. While large pools of planets are
particularly good for statistical and demographic studies, it presents a
significant target selection challenge in the search for potentially habitable
planets in quantifying which may be most suitable from both an astrobiological
and observation perspective. This is where target selection tools such as the
HZ are most useful. By refining the full catalog of planets to only those
whose orbits lie within an area around the star where conditions are most
conducive to the existence of liquid water on the surface, limited telescope
time can be directed to the most promising targets. Observations of the
planets in this catalog with JWST and future missions, such as the Nancy Grace
Roman Space Telescope (Kasdin et al., 2020), LUVOIR (The LUVOIR Team, 2019)
and HabEx (Gaudi et al., 2020), will provide further insight into the
habitability of exoplanets found in the HZ. By using this catalog and the
planets highlighted within it to guide target selection, the boundaries of
habitability can be investigated and the reliability of the HZ hypothesis
assessed.

## Acknowledgements

M.H. would like to acknowledge NASA support via the FINESST Planetary Science
Division, NASA award number 80NSSC21K1536. M.H. Would also like to thank Hilke
Schlichting for her advice regarding the radius valley of HZ planets. S.R.K.
and C.O. acknowledge support from NASA grant 80NSSC21K1797, funded through the
NASA Habitable Worlds Program. T.F. acknowledges support from the University
of California President’s Postdoctoral Fellowship Program. This research has
made use of the NASA Exoplanet Archive. We obtain the data set from the NASA
Exoplanet Archive (NASA Exoplanet Archive, 2022)111Accessed on 2022-05-19 at
12:45, returning 4550 rows.. This dataset or service is made available by the
NASA Exoplanet Science Institute at IPAC, which is operated by the California
Institute of Technology under contract with the National Aeronautics and Space
Administration. This research has also made use of the Habitable Zone Gallery
at hzgallery.org.

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