# Topological Optimal Transport for Geometric Cycle Matching

Stephen Y Zhang  School of Mathematics and Statistics, the University of
Melbourne, Parkville, Melbourne, 3010, VIC, Australia  Melbourne Integrative
Genomics, the University of Melbourne, Parkville, Melbourne, 3010, VIC,
Australia  Michael P H Stumpf  School of Mathematics and Statistics, the
University of Melbourne, Parkville, Melbourne, 3010, VIC, Australia  Melbourne
Integrative Genomics, the University of Melbourne, Parkville, Melbourne, 3010,
VIC, Australia  School of Biosciences, the University of Melbourne, Parkville,
Melbourne, 3010, VIC, Australia  Tom Needham  Mathematics Department, Florida
State University, Academic Way, Tallahassee, 32306, FL, USA  Agnese Barbensi
School of Maths and Physics, University of Queensland, St Lucia, Brisbane,
4067, QLD, Australia

###### Abstract

Topological data analysis is a powerful tool for describing topological
signatures in real world data. An important challenge in topological data
analysis is matching significant topological signals across distinct systems.
In geometry and probability theory, optimal transport formalises notions of
distance and matchings between distributions and structured objects. We
propose to combine these approaches, constructing a mathematical framework for
optimal transport-based matchings of topological features. Building upon
recent advances in the domains of persistent homology and optimal transport
for hypergraphs, we develop a transport-based methodology for topological data
processing. We define measure topological networks, which integrate both
geometric and topological information about a system, introduce a distance on
the space of these objects, and study its metric properties, showing that it
induces a geodesic metric space of non-negative curvature. The resulting
Topological Optimal Transport (TpOT) framework provides a transport model on
point clouds that minimises topological distortion while simultaneously
yielding a geometrically informed matching between persistent homology cycles.

##  1 Introduction

Topological data analysis (TDA) is a quickly growing field in computational
and applied topology. In recent years, TDA has established itself as an
effective framework to analyse, cluster, and detect patterns in complex data
[52]. One of the key algorithms in TDA is persistent homology (PH), a
computational tool that describes the structure of data based on topological
features persisting across different scales [35, 18, 22]. In a nutshell, the
PH algorithm proceeds by building a nested sequence of discrete spaces
describing the input data at increasingly coarse scales, known as a filtration
of simplicial complexes. The topological features of this filtration are then
quantified by computing the homology groups of each simplicial complex in the
sequence. A structural theorem [56] guarantees that the birth, death, and
evolution of homology classes in the filtration can be summarised as a multi-
set, called a persistence diagram (PD). PDs have been shown to contain rich
information about the initial data, and PH-based data analysis approaches
appear in an ever-increasing number of applications across multiple fields in
modern science [40, 42, 51, 44].

Optimal transport (OT) is a far-reaching mathematical theory that, in its most
basic and classical formulation, formalises the problem of finding a matching
of a given probability distribution to another as efficiently as possible, in
terms of a cost function [50, 36]. When the cost function is induced by the
distance function on a metric space $(X,d)$, optimal transport provides a
natural lifting of the ground metric to a metric on the space of probability
distributions supported on $X$, commonly known as the Wasserstein distance. An
extension of the optimal transportation problem which has recently received
considerable attention is the _Gromov-Wasserstein_ (GW) problem [31, 32, 8,
43], in which the probability distributions to be matched are defined over
different metric spaces, rather than supported on a common space. The
objective is then to find matchings that minimise _distortion_ of pairwise
distances. While the GW distance was originally introduced to compare metric
measure spaces and can be viewed as a relaxation of the well known Gromov-
Hausdorff distance [31], some of its most successful applications address the
problem of matching more general structured objects such as (labelled) graphs
[54, 53, 10, 48]. Among its numerous advantages, the GW distance has brought
substantial improvements to graph processing tasks over classical methods, and
can be efficiently approximated by a range of numerical algorithms in practice
[53, 37, 20, 9]. On the theoretical front, the space of _gauged measure
spaces_ (i.e., a measure space $(X,\mu)$ together with a symmetric gauge
function $k:X\times X\to\mathbb{R}$) endowed with the GW distance has the
geometry of an Alexandrov space, and a Riemannian orbifold structure can be
developed in this setting [43, 10]. More recently, inspired by the goal of
encoding relations in complex systems as higher-order networks, a variant of
the GW problem, referred to as _co-optimal transport_ [46], was applied to
model hypergraphs both from a theoretical and applied point of view [11].

Figure 1: The Topological Optimal Transport (TpOT) problem. From left to
right: two input point clouds, their persistence diagrams, and corresponding
PH-hypergraphs. The objective of the TpOT problem combines a Gromov-
Wasserstein distortion on the point clouds (geometric information), (partial)
Wasserstein matching of points in the persistence diagrams (topological
information) and HyperCOT on the PH-hypergraphs (coupling of geometric and
topological information).

In this paper, we combine the TDA and OT approaches described above to develop
a transport-based theory for topological data processing. Our approach is
inspired by recent extensions built upon persistent homology computations: the
hyperTDA framework [2] encodes each point in a persistence diagram, together
with a geometric realisation of its homology class as a generating cycle, as a
higher-order network called a _PH-hypergraph_. In a PH-hypergraph, vertices
corresponds to data points used as input to the PH computation, and hyperedges
are given by the generators of features in the persistence diagram. We define
the _topological optimal transport (TpOT) problem_ , by considering a trade-
off between preservation of geometric relationships, preservation of
topological features, and coupling the two via the PH-hypergraph structure
(see Figure 1). The output is (1) a matching between points that is
topologically driven and geometrically informed, coupled with (2) a matching
between persistent homology classes that is geometrically driven and
topologically informed.

Finding meaningful ways of matching topological features across distinct
systems is an important challenge in TDA. The past few years have seen
considerable effort in addressing this question, with a number of different
solutions proposed [13, 3, 25, 38, 55, 21], many of which bridge between
topological data analysis and optimal transport [23, 27, 30]. A key aspect of
the TpOT method is leveraging existing transportation theory for persistence
diagrams to drive a matching between topological features, while also taking
into account geometric information and the spatial interconnectivity of
homology generators. By endowing PH-hypergraphs with probability measures, we
introduce the concept of measure topological networks, and we formalise a
framework for studying the TpOT induced distance $d_{\mathrm{TpOT},p}$. Our
main contributions are as follows:

  * •

We develop a flexible measure-theoretic formalism for simultaneously encoding
the geometry and topology of a finite point cloud, in the form of the
aforementioned measure topological networks, as well as a family of distances
$d_{\mathrm{TpOT},p}$, $p\geq 1$, between these objects; see Definitions 1 and
2, respectively.

  * •

The distance $d_{\mathrm{TpOT},p}$ is shown to be a pseudometric on the space
of measure topological networks $\mathcal{P}$, and the zero distance
equivalence relation is completely characterised; see Theorem 1.

  * •

We show that the metric induced by $d_{\mathrm{TpOT},p}$ is geodesic,
characterise the exact form of the geodesics and use this characterisation to
show that this metric space is non-negatively curved, in the sense of
Alexandrov; see Theorems 2, 3 and 4, respectively.

  * •

Our framework is centred on an optimisation problem for which we provide
efficient numerical algorithms, which we demonstrate on a variety of examples;
see Section 5.

The paper is organised as follows. Section 2 describes the necessary
mathematical background on persistent homology and optimal transport. In
Section 3, after motivating and constructing the TpOT problem, we define
_measure topological networks_ and the family of (pseudo-)metrics
$d_{\mathrm{TpOT},p}$. We then discuss theoretical properties (Subsection 3.2)
of the distance $d_{\mathrm{TpOT},p}$, and provide a characterisation of
geodesics in Section 4. Finally, we provide details of the computational
implementation (Subsection 3.4), as well as examples (Section 5).

##  2 Mathematical background

Our formalisation of the TpOT problem relies on constructions in topological
data analysis and measure theory. The aim of this section is to outline the
mathematical background needed to define measure topological networks and the
family of distances $d_{\mathrm{TpOT},p}$ described in the introduction.

###  2.1 Persistent homology and persistent homology-hypergraphs

Let $(X,d)$ be a finite metric space – for example, a point cloud
$X=\\{x_{1},\cdots,x_{N}\\}\subset\mathbb{R}^{n}$ as illustrated in Figure 2A.
For any $\varepsilon>0$, the Vietoris-Rips complex $K(X)_{\varepsilon}$ (see
e.g. [18, Ch.III.2]) is the simplicial complex obtained from $X$ by adding a
$k$-simplex $[x_{i_{0}},x_{i_{1}},\cdots,x_{i_{k}}]$ whenever the distance
between all pairs of points in $\\{x_{i_{0}},x_{i_{1}},\cdots,x_{i_{k}}\\}$ is
less than $\varepsilon$. Note that $K(X)_{\varepsilon}$ is a sub-complex of
$K(X)_{\varepsilon^{\prime}}$ whenever $\varepsilon\leq\varepsilon^{\prime}$.
Thus, as $\varepsilon$ grows, this yields a nested sequence of simplicial
complexes:

$\mathcal{K}_{\rm VR}(X):=K(X)_{\varepsilon_{0}}\hookrightarrow
K(X)_{\varepsilon_{1}}\hookrightarrow\cdots\hookrightarrow
K(X)_{\varepsilon_{M}}.$

Here, the numbers $\varepsilon_{i}$, $i=1,\ldots,M$, are the parameters
corresponding to the creation of new simplices; see Figure 2B.

Figure 2: Computing PH and the PH-hypergraph (A) A point cloud and (B) a
filtration of simplicial complexes built on it. (C) The $1$-dimensional
persistence diagram of the filtration in (B). (D) The PH-hypergraph
constructed after computing a representative cycle (red cycles in (B)) for
each class in (C).

More generally, for a finite metric space $X$, a _filtration_ over $X$ is a
finite sequence $\mathcal{K}=(K_{\varepsilon_{1}},\ldots,K_{\varepsilon_{M}})$
of simplicial complexes, indexed by an increasing sequence of real numbers
$\varepsilon_{i}\leq\varepsilon_{i+1}$, together with simplicial maps
$K_{\varepsilon_{i}}\to K_{\varepsilon_{i+1}}$ for all $i$, such that the
vertex set of each $K_{\varepsilon_{i}}$ is a subset of $X$. Common
alternatives to the Vietoris-Rips complex in the literature with a similar
interpretation include the Čech complex [18, Ch.III.2] and the Alpha complex
[18, Ch.III.4]. Clearly, different constructions of the filtration
$\mathcal{K}$ will influence the results of the persistent homology
computation and their interpretation.

A filtration $\mathcal{K}$ over $X$ can be analysed by computing the
simplicial homology groups of each simplicial complex in the sequence and the
corresponding induced maps. Collectively, this information is called the
_persistent homology_ (PH) of $\mathcal{K}$. The Structural Theorem [57],
guarantees that, when using coefficients in $\mathbb{Z}/2\mathbb{Z}$, the
birth, death, and evolution of homology classes in a filtration of simplicial
complexes can be summarised in a multi-set, called a _persistence diagram_. We
use $\Lambda$ to denote the upper-diagonal cone in the upper-half plane,
$\Lambda=\\{(x,y)\in\mathbb{R}^{2},y>x>0\\}$. The persistence diagram $D$
associated to the filtration $\mathcal{K}$ is then a collection of points in
$\Lambda$: $D=\\{(b_{i},d_{i})\\}_{i=1}^{|D|}\subset\Lambda$.

Each point $(b_{i},d_{i})\in D$ represents a homology class in the filtration
$\mathcal{K}$, with coordinates corresponding to the birth $b_{i}$ and death
$d_{i}$ parameters of the class (see Figure 2C). The persistence
$d_{i}-b_{i}>0$ of a homology class is a measure of significance of the
underlying topological feature. The persistence diagram $D$ can be interpreted
as a topological descriptor of $X$. A local, geometric realisation of each
homology class can be obtained by computing a representative cycle generating
the class: see for instance the red cycles in Figure 2B. Computing cycles
depends on a number of choices [5, 16, 34, 49, 29], and therefore different
representatives may lead to differing downstream results.

The recently introduced PH-hypergraph [2] encodes the algebraic information of
a persistence diagram, together with the local geometric interpretation of
generating cycles, in a single higher-order network. Recall that a
_hypergraph_ is defined by a pair $(V,E)$ consisting of a set of vertices $V$
and a set of hyperedges $E$. Given a point cloud $X$, we may construct an
associated filtration $\mathcal{K}$ together with a choice
$g=\\{c_{1},\ldots,c_{|D|}\\}$ of generating cycles for classes in the
corresponding persistence diagram $D$. These data can be summarised in a PH-
hypergraph $H=H(X,\mathcal{K},g)$ in which we take $X$ to be the vertex set,
and one hyperedge added for each cycle in $g$ (see Figure 2D). Moreover, for
every cycle $c\in g$, a point $x_{i}\in X$ belongs to the corresponding
hyperedge $e(c)$ if it is a vertex of any simplex in $c$. While the hypergraph
structure $H$ clearly depends on the choice of generators $g$ and the choice
of filtration $\mathcal{K}$, for a fixed filtration $\mathcal{K}$ the
qualitative (hyper)network structure of the PH-hypergraph has been empirically
shown to be stable to noise and different choices of generating cycles [2].

###  2.2 Optimal transport on persistence diagrams

Consider now two Borel probability measures $\mu$ and $\mu^{\prime}$ supported
on a Polish metric space $(X,d)$. For $p\in[1,\infty)$, the _$p$ -Wasserstein
distance_ [50, Definition 6.1] between $\mu$ and $\mu^{\prime}$ is then
defined as

$d_{\mathrm{W},p}(\mu,\mu^{\prime}):=\left(\inf_{\pi\in\Pi(\mu,\mu^{\prime})}\int_{X\times
X}d(x,y)^{p}\>\mathrm{d}\pi(x,y)\right)^{1/p}.$ (1)

This metric is the central object of study in the optimal transport (OT)
literature. It is a standard result that the infimum in (1) is realised (i.e.
it is in fact a _minimum_) and a minimiser $\pi$ of (1) is referred to as an
_optimal coupling_. Intuitively, this is a probabilistic matching between
points in the supports of $\mu$ and $\mu^{\prime}$.

The optimal transport approach of finding matchings has inspired different
notions of distance on the space of persistence diagrams: the most popular
construction is known as the bottleneck distance [12], which is related to the
$\infty$-Wasserstein distance. In this article, we are more interested in
similar adaptations of the $p$-Wasserstein distances for $1\leq p<\infty$. The
usual modification of the Wasserstein matching problem to compare persistence
diagrams is to allow points to be matched to a virtual point
$\partial_{\Lambda}$, representing the diagonal line $\\{(a,a)\mid
a\in\mathbb{R}\\}\subset\mathbb{R}^{2}$ (i.e. homology classes with vanishing
persistence). Formally, for $p\in[1,\infty)$, the _$p$ -Wasserstein distance_
between two persistence diagrams $D$ and $D^{\prime}$ (see [28, Equation 6])
is defined as

$d_{\mathrm{W},p}^{\mathrm{PD}}(D,D^{\prime})^{p}=\min_{\pi\in\Pi(D,D^{\prime})}\left(\sum_{(a,b)\in\pi}\|a-b\|^{p}_{p}+\sum_{s\in
U_{\pi}}\|s-\operatorname{Proj}_{\partial_{\Lambda}}(s)\|^{p}_{p}\right),$ (2)

where the various notations are described as follows:

  * •

Slightly abusing notation, to illustrate the analogy to the OT Wasserstein
distance, we denote by $\Pi(D,D^{\prime})$ the set of all _partial matchings_
between points representing homology classes in $D$ and $D^{\prime}$. For any
$\pi\in\Pi(D,D^{\prime})$, $\pi\subset D\times D^{\prime}$ has the property
that each $a\in D$ and $b\in D^{\prime}$ appear in at most one ordered pair in
$\pi$.

  * •

The set $U_{\pi}$ is the set of _unmatched points_ for $\pi$: $s\in U_{\pi}$
if and only if $s\in D$ and $(s,b)\not\in\pi$ for any $b\in D^{\prime}$ or
$s\in D^{\prime}$ and $(a,s)\not\in\pi$ for any $a\in D$.

  * •

The map $\operatorname{Proj}_{\partial_{\Lambda}}$ is the metric projection
(w.r.t. any $\|\cdot\|_{p}$) of $\Lambda$ onto the diagonal, as represented by
the point $\partial_{\Lambda}$. Explicitly, for $s=(x,y)$,

$\operatorname{Proj}_{\partial_{\Lambda}}(s)=((x+y)/2,(x+y)/2).$

The output from solving (2) is a partial matching $\pi$ between homology
classes in $D$ and $D^{\prime}$ which is optimal with respect to the cost
induced by the $\|\cdot\|_{p}$-norm in $\mathbb{R}^{2}$, restricted to
$\Lambda$. By construction, this matching only depends on the coordinates of
points in the persistence diagram that correspond to homology classes.
Notably, the matching is agnostic to geometric and topological aspects of the
data that are not captured in the persistence diagram representations. While
the generality of the persistence diagram representation may contribute to its
wide applicability, in many contexts this may be a limitation.

Observe that the notation and concepts used to define
$d_{\mathrm{W},p}^{\mathrm{PD}}$ are similar to those that appear in the
optimal transport framework described above; indeed, the family of persistence
diagram distances (2) can be generalised via the language of _partial_ OT
[17]. We define a _measure persistence diagram_ to be a Radon measure $\mu$
supported on $\Lambda$. Then, any finite persistence diagram
$D=\\{(b_{i},d_{i})\\}_{i=1}^{|D|}$ determines a measure persistence diagram:

$\nu_{D}=\sum_{i=1}^{|D|}\delta_{x_{i}},$ (3)

where each $\delta_{x_{i}}$ denotes a Dirac supported at
$x_{i}=(b_{i},d_{i})$.

Let $\overline{\Lambda}:=\Lambda\cup\\{\partial_{\Lambda}\\}$ be obtained by
adjoining to $\Lambda$ the virtual point $\partial_{\Lambda}$, with the
resulting space endowed with the disjoint union topology. Given two measure
persistence diagrams $\mu$ and $\mu^{\prime}$, we say a Radon measure $\pi$ on
$\overline{\Lambda}\times\overline{\Lambda}$ is _admissible_ if it has
marginals $\mu$ and $\mu^{\prime}$ (we consider $\mu$ and $\mu^{\prime}$ to be
measures on $\overline{\Lambda}$ which are supported on $\Lambda$) and it
satisfies the additional constraint
$\pi((\partial_{\Lambda},\partial_{\Lambda}))=0$, and write
$\pi\in\Pi_{\mathrm{adm}}(\mu,\mu^{\prime})$. Then for $p\in[1,\infty)$, we
define the _$p$ -Wasserstein distance between measure persistence diagrams_
[17] as:

$d_{\mathrm{W},p}^{\mathrm{MPD}}(\mu,\mu^{\prime})=\inf_{\pi\in\Pi_{\rm
adm}(\mu,\mu^{\prime})}\left(\int_{\overline{\Lambda}\times\overline{\Lambda}}\|x-x^{\prime}\|^{p}_{p}\>\mathrm{d}\pi(x,x^{\prime})\right)^{1/p}.$
(4)

If $\nu_{D}$ and $\nu_{D^{\prime}}$ arise from finite persistence diagrams $D$
and $D^{\prime}$, respectively, as in (3), then it is not hard to show that
$d_{\mathrm{W},p}^{PD}(D,D^{\prime})=d_{\mathrm{W},p}^{\mathrm{MPD}}(\nu_{D},\nu_{D^{\prime}})$
[17, Prop. 3.2]. In general, $d_{\mathrm{W},p}^{\mathrm{MPD}}$ is finite,
provided that we work in the space of measure persistence diagrams $\mu$ with
_finite $p$-persistence_, in the sense that

$\int_{\Lambda}\|x-\mathrm{Proj}_{\partial_{\Lambda}}(x)\|^{p}_{p}d\mu(x)<\infty.$

In what follows, for any given $p$, we will restrict to the space of measure
persistence diagrams $\mu$ with finite $p$-persistence. With an abuse of
notation, we will still indicate their space as MPD.

###  2.3 Gromov-Wasserstein and co-optimal transport distances

A _metric measure space (mm-space)_ [43, 32] is a triple $M=(X,d,\mu)$, where
$X$ is a space endowed with a complete separable metric $d$ and with a fully
supported Borel measure $\mu$. Given two mm-spaces $M=(X,d,\mu)$ and
$M^{\prime}=(X^{\prime},d^{\prime},\mu^{\prime})$ and a coupling
$\pi\in\Pi(\mu,\mu^{\prime})$ and $p\in[1,\infty)$, we introduce the
$p$-distortion $\mathrm{dis}_{p}^{\mathrm{GW}}(\pi)$:

$\mathrm{dis}_{p}^{\mathrm{GW}}(\pi)=\left(\int_{(X\times
X^{\prime})^{2}}|d(x,y)-d^{\prime}(x^{\prime},y^{\prime})|^{p}\>\mathrm{d}\pi(x,x^{\prime})\>\mathrm{d}\pi(y,y^{\prime})\right)^{1/p}=\|d-d^{\prime}\|_{L^{p}(\pi\otimes\pi)}.$

By taking the infimum over all feasible couplings, we obtain the _Gromov-
Wasserstein (GW) $p$-distance_ between $M$ and $M^{\prime}$:

$d_{\mathrm{GW},p}(M,M^{\prime}):=\inf_{\pi\in\Pi(\mu,\mu^{\prime})}\mathrm{dis}_{p}^{\mathrm{GW}}(\pi).$

This quantity is finite if we work in the space of mm-spaces whose distance
functions have finite $p$-th moment; let us make the simplifying convention
that we always work with bounded mm-spaces, in order to avoid this technical
issue. The GW-distance induces a pseudo-metric on the space of mm-spaces [31,
32], such that $d_{\mathrm{GW},p}(M,M^{\prime})=0$ if and only if there is a
measure-preserving isometry from $M$ to $M^{\prime}$.

For our purposes, we relax the requirement on the function $d:X\times
X\to\mathbb{R}$ to be a metric and instead allow any symmetric, measurable and
bounded function $k\in L^{p}_{\mathrm{sym}}(X\times X)$. Then, the structure
$M=(X,k,\mu)$ is instead referred to as a gauged measure space [43] or a
measure network [10] (in fact, the definition of a measure network in [10]
even drops the symmetry condition). This allows us to consider pairwise
relations on $X$ to be modelled by a wider class of functions, such as
affinity kernels. In this case, the space $(\mathcal{M},d_{\mathrm{GW},p})$ of
equivalence classes of measure networks (under the equivalence relation $M\sim
M^{\prime}\Leftrightarrow d_{\mathrm{GW},p}(M,M^{\prime})=0$) is a complete,
geodesic, metric space [43, 10].

Recently, a GW-like distance has been developed for comparing hypergraph
structures [11], based on ideas from the _co-optimal transport problem_
introduced in [45]. Recall that a hypergraph is defined by a pair $H=(V,E)$,
where $V$ is a finite set of vertices and $E$ is a set of hyperedges; each
$e\in E$ is a subset of $V$. A general structure which encompasses the notion
of a hypergraph is a _measure hypernetwork_ , which is a quintuple
$H=(X,\mu,Y,\nu,\omega)$, where $(X,\mu)$ and $(Y,\nu)$ are respectively
Polish spaces with fully supported Borel probability measures, and $\omega$ is
a non-negative, measurable and bounded function $\omega:X\times
Y\to\mathbb{R}$. Indeed, from a hypergraph $(V,E)$, one obtains a measure
network $(V,\mu,E,\nu,\omega)$, where $\mu$ and $\nu$ are uniform probability
measures and $\omega:V\times E\to\mathbb{R}$ is a binary incidence function
($\omega(v,e)=1\Leftrightarrow v\in e$).

Given two measure hypernetworks $H=(X,\mu,Y,\nu,\omega)$ and
$H^{\prime}=(X^{\prime},\mu^{\prime},Y^{\prime},\nu^{\prime},\omega^{\prime})$,
the _$p$ -co-optimal distortion_ for a pair of couplings
$(\pi^{v},\pi^{e})\in\Pi(\mu,\mu^{\prime})\times\Pi(\nu,\nu^{\prime})$ (the
notation here is intended to evoke the idea that $\pi^{v}$ is a coupling of
vertices and $\pi^{e}$ is a coupling of hyperedges) is defined to be

$\mathrm{dis}^{\mathrm{COOT}}_{p}(\pi^{v},\pi^{e}):=\left(\int_{X\times
X^{\prime}\times Y\times
Y^{\prime}}|\omega(x,y)-\omega^{\prime}(x^{\prime},y^{\prime})|^{p}\>\mathrm{d}\pi^{v}(x,x^{\prime})\>\mathrm{d}\pi^{e}(y,y^{\prime})\right)^{1/p}=\|\omega-\omega^{\prime}\|_{L^{p}(\pi^{v}\otimes\pi^{e})}.$

Similarly to the GW distance, the _hypernetwork $p$-co-optimal transport
distance_ is given by

$\displaystyle d_{\mathrm{COOT},p}(H,H^{\prime}):=$
$\displaystyle\inf_{\pi^{v}\in\Pi(\mu,\mu^{\prime}),\>\pi^{e}\in\Pi(\nu,\nu^{\prime})}\mathrm{dis}^{\mathrm{COOT}}_{p}(\pi^{v},\pi^{e}).$
(5)

The expression $d_{\mathrm{COOT},p}(H,H^{\prime})$ defines a pseudo-metric on
the space of measure hypernetworks, and induces a metric on the space of
equivalence classes of measure hypernetworks up to isomorphism, which is also
shown to be geodesic and complete in [11, Theorem 1].

##  3 Topological Optimal Transport

Having discussed various matching-based comparison approaches for persistence
diagrams, point clouds, and hypergraphs, we turn to the aim of this article,
which is to develop an optimal transportation framework for matching point
clouds that couples geometric and topological information. Our starting point
is to represent a point cloud $X$ together with its topological features as a
PH-hypergraph $H=H(X,\mathcal{K},g)$. Given another point cloud and PH-
hypergraph, our objective is to find a coupling between them that minimises
topological distortion and optimally preserves topological features. A natural
attempt to a find a topology-informed matching between point clouds $X$ and
$X^{\prime}$ is to simply apply the HyperCOT framework [11] on their PH-
hypergraphs $H=H(X,\mathcal{K},g)$ and
$H^{\prime}=H^{\prime}(X^{\prime},\mathcal{K}^{\prime},g^{\prime})$. However,
this simplistic approach does not work due to two problems we list below.

  * •

Problem 1. PH-hypergraphs do not contain any information on the significance
(measured by persistence) of homology classes. One could naturally weight
hyperedges by the persistence value of their corresponding homology classes
[2]. However, in an optimal transport setting, the HyperCOT would promote
“splitting” of mass from hyperedges with higher persistence to hyperedges with
lower persistence, rather than matching significant hyperedges.

  * •

Problem 2. PH-hypergraphs can be disconnected, and there might be points which
do not belong to any hyperedge [2]. In this context, the desired property is
to have geometric (spatial) information co-driving the transport plan.

The goal of this section is to develop a framework for geometric cycle
matching which addresses these problems.

###  3.1 Measure topological networks

To solve the problems described above, we first introduce an appropriate
general model for the structures that we wish to compare. Here we introduce
the following notation. For a locally compact Polish metric space $Y$, we
denote by $\overline{Y}$ the augmented space obtained as the disjoint union
$\overline{Y}=Y\cup\\{\partial_{Y}\\}$ (with the disjoint union topology),
where $\partial_{Y}$ is an abstract point.

######  Definition 1 (Measure Topological Network).

A _measure topological network_ , or simply _topological network_ , is a
triple $P=\left((X,k,\mu),(Y,\iota,\nu),\omega\right)$, where

  * •

$(X,k,\mu)$ is a gauged measure space (see Section 2.3),

  * •

$Y$ is a locally compact Polish space,

  * •

$\nu$ is a Radon measure supported on $Y$,

  * •

$\iota:Y\longrightarrow\Lambda$ is a continuous function,

  * •

$\iota_{\\#}\nu$ is a measure persistence diagram (see Section 2.2), and

  * •

$\omega$ is a measurable and bounded function $\omega:X\times Y\to\mathbb{R}$,
so that the quintuple $\left(X,\mu,Y,\nu,\omega\right)$ is a measure
hypernetwork (see Section 2.3).

Let us denote by $\mathcal{P}$ the class of all measure topological networks,
with the simplifying conventions that gauged measure spaces have bounded
kernels and that measure persistence diagrams have finite
$\infty$-persistence.

######  Example 1 (Topological Network from a Metric Measure Space).

A measure topological network arises from the data of a PH-hypergraph defined
over a metric measure space, and this is inspiration for the definition and
the source of all computational examples. Indeed, given a finite metric
measure space $(X,d,\mu)$, an associated persistence diagram $D$ (obtained
via, say, Vietoris-Rips persistent homology) and a choice of a set of
generating cycles $g$ for $D$, we construct an associated topological network
as follows. First, we take $Y=D$ as a multiset of points in $\Lambda$ (one can
consider this as a proper set by indexing its points, so that $Y$ can be
considered a finite topological space). Taking $\nu$ to be the uniform (i.e.,
counting) measure and $\iota:Y\to\Lambda$ to be inclusion, we have that
$\iota_{\\#}\nu=\nu_{D}$ as in (3). Finally, $\omega$ is a kernel representing
the hypergraph structure $(X,g)$. In our computational experiments, we use the
binary incidence function: for $x\in X$ and a point $(a,b)\in D$, represented
by a cycle $c\in g$, $\omega(x,c)=1$ if and only if $x$ is a vertex of any
simplex in $c$. The flexibility in the definition allows for other hypergraph
kernels, such as those studied in [11]. We can also replace the metric $d$
with a symmetric gauge function $k$ that may better encode local geometry
without issue—see Section 5 for examples.

######  Remark 1.

In full generality, topological measure networks are not restricted to
describing topological features of the underlying space. The construction
described in Example 1 motivates the use of “topology” in the name.

Although computational examples of topological networks in this paper will
always be constructed as in Example 1, we provide the following additional
construction to motivate the level of generality of our definition.

######  Example 2 (Topological Network from a Curvature Set).

Let $(X,d,\mu)$ be a finite metric space. For fixed $k<|X|$, consider the map
$d_{k}:X^{k}\to\mathbb{R}^{k\times k}$ defined by
$d_{k}(x_{1},\ldots,x_{k})=(d(x_{i},x_{j}))_{i,j=1}^{k}$; that is, this map
takes a $k$-tuple of points to its distance matrix. The image of this map is
an invariant of $X$ introduced by Gromov in [26], called the $k$th _curvature
set_ of $X$. Given a $k\times k$ distance matrix, one can apply degree-$\ell$
Vietoris-Rips persistent homology, so that the composition yields a map
$D_{k,\ell}$ from $X^{k}$ into the space of persistence diagrams—this
invariant and other related invariants were thoroughly studied in the recent
paper [24]. This structure gives rise to a topological network as follows. Let
$\widetilde{Y}$ be the multiset consisting of all persistence points arising
in diagrams in the image of $D_{k,\ell}$, let $Y$ be its underlying set, let
$\nu$ be the pushforward of uniform measure on $\widetilde{Y}$ to $Y$ (so that
$\nu$ counts persistence points with multiplicity), and let
$\iota:Y\to\Lambda$ be the inclusion map. We then define $\omega:X\times
Y\to\mathbb{R}$ as

$\omega(x,y)=\mathbb{P}_{\mu^{\otimes(k-1)}}\big{(}y\in
D_{k,\ell}(x,x_{1},\ldots,x_{k-1})\big{)}.$

We now define a notion of distance between measure topological networks. This
distance will be our main object of study throughout the rest of the paper.

######  Definition 2 (Topological Optimal Transport (TpOT)).

Given two topological networks $P=\left((X,k,\mu),(Y,\iota,\nu),\omega\right)$
and
$P^{\prime}=\left((X^{\prime},k^{\prime},\mu^{\prime}),(Y^{\prime},\iota^{\prime},\nu^{\prime}),\omega^{\prime}\right)$,
we formulate the _Topological Optimal Transport (TpOT) problem_ as:

$\displaystyle d_{\mathrm{TpOT},p}(P,P^{\prime})$
$\displaystyle:=\inf_{\begin{subarray}{c}\pi^{v}\in\Pi(\mu,\mu^{\prime})\\\
\pi^{e}\in\Pi_{\rm
adm}(\nu,\nu^{\prime})\end{subarray}}\left(\int_{\overline{Y}\times\overline{Y^{\prime}}}\|\iota(y)-\iota^{\prime}(y^{\prime})\|_{p}^{p}\>\mathrm{d}\pi^{e}(y,y^{\prime})\right.$
(6) $\displaystyle\qquad\qquad\qquad\qquad+\int_{(X\times
X^{\prime})^{2}}|k(x,y)-k^{\prime}(x^{\prime},y^{\prime})|^{p}\>\mathrm{d}\pi^{v}(x,x^{\prime})\>\mathrm{d}\pi^{v}(y,y^{\prime})$
(7) $\displaystyle\qquad\qquad\qquad\qquad\left.+\int_{X\times
X^{\prime}\times\overline{Y}\times\overline{Y^{\prime}}}|\omega(x,y)-\omega^{\prime}(x^{\prime},y^{\prime})|^{p}\>\mathrm{d}\pi^{v}(x,x^{\prime})\>\mathrm{d}\pi^{e}(y,y^{\prime})\right)^{\frac{1}{p}}.$
(8)

Letting $d_{p}$ denote the $\ell^{p}$-distance on $\mathbb{R}^{2}$, this can
be expressed more concisely as

$d_{\mathrm{TpOT},p}(P,P^{\prime})=\inf_{\begin{subarray}{c}\pi^{v}\in\Pi(\mu,\mu^{\prime})\\\
\pi^{e}\in\Pi_{\rm
adm}(\nu,\nu^{\prime})\end{subarray}}\left(\|d_{p}\circ(\iota,\iota^{\prime})\
\|^{p}_{L^{p}(\pi^{e})}+\|k-k^{\prime}\|^{p}_{L^{p}(\pi^{v}\otimes\pi^{v})}+\|\omega-\omega^{\prime}\|^{p}_{L^{p}(\pi^{v}\otimes\pi^{e})}\right)^{\frac{1}{p}}.$
(9)

Similar to Section 2.2, we say a Radon measure $\pi$ on
$\overline{Y}\times\overline{Y^{\prime}}$ is _admissible_ , and we write
$\pi\in\Pi_{\mathrm{adm}}(\nu,\nu^{\prime})$, if it has marginals $\nu$ and
$\nu^{\prime}$, and it satisfies
$\pi((\partial_{Y},\partial_{Y^{\prime}}))=0$. Here we are implicitly
extending $\omega$ and $\iota$ to functions
$\omega:X\times\overline{Y}\to\mathbb{R}$,
$\iota:\overline{Y}\to\overline{\Lambda}$ by setting
$\omega_{|X\times\partial_{Y}}=0$ and
$\iota(\partial_{Y})=\partial_{\Lambda}$. The restriction
$\omega_{|X\times\partial_{Y}}$ can be interpreted as (trivial) a membership
function (via the map $\iota$) for the diagonal $\partial_{\Lambda}$.

Let us interpret the various components of this definition, when the measure
topological spaces arise from PH-hypergraphs, as described in Example 1.

  * •

The coupling $\pi^{v}$ defines a probabilistic matching between the points of
the gauged measure spaces (see Figure 1, left column) and the admissible
coupling $\pi^{e}$ defines a probabilistic matching between generating cycles
of the persistence diagrams (see Figure 1, middle column).

  * •

The integral in (6) is a Wasserstein term which measures the quality of the
matching of generating cycles. This term is intended to address Problem 1.

  * •

The integral in (7) is a Gromov-Wasserstein term which matches the quality of
the matching of points in the gauged measure spaces. This term is intended to
address Problem 2.

  * •

The integral in (8) is a co-optimal transport term which addresses the basic
problem of matching PH-hypergraphs.

######  Remark 2.

In applications, it is frequently useful to include tunable weights on the
terms of (9), which can be used as hyperparameters to accentuate topology or
geometry as is appropriate for a given task. To keep the exposition clean, we
suppress these weights from theoretical considerations. They are explicitly
included in the exposition of the computational pipeline—see Section 3.3.

###  3.2 Metric properties of TpOT

Next we will show that $d_{\rm TpOT,p}$ defines a metric on the space of
(certain equivalence classes of) topological networks. As a first step, we
have the following proposition.

######  Proposition 1.

The infimum of Equation (9) is realised.

###### Proof of Proposition 1.

It is a classic result that the coupling space $\Pi(\mu,\mu^{\prime})$ is
(sequentially) compact in $P(X\times X^{\prime})$, where $P(X\times
X^{\prime})$ is the set of Borel probability measures on $X\times X^{\prime}$,
see, e.g., [50, Lemma 4.4] and [11, proof of Lemma 24], where we topologise
spaces of measures with the weak topology. Similarly, $\Pi_{\rm
adm}(\nu,\nu^{\prime})$ is sequentially compact in the space
$M(\overline{Y}\times\overline{Y^{\prime}})$ of Radon measures on
$\overline{Y}\times\overline{Y^{\prime}}$ (see [17, proof of Prop 3.1], which
extends to Polish spaces as in Definition 1), with the vague topology. These
two together imply that $\Pi(\mu,\mu^{\prime})\times\Pi_{\rm
adm}(\nu,\nu^{\prime})$ is sequentially compact in $P(X\times
X^{\prime})\times M(\overline{Y}\times\overline{Y^{\prime}})$ with the product
topology.

To complete the proof, we just need to show lower-semicontinuity of the
function $C_{p}:\Pi(\mu,\mu^{\prime})\times\Pi_{\rm
adm}(\nu,\nu^{\prime})\rightarrow\mathbb{R}$ defined by

$C_{p}(\pi,\xi)=\ \int_{X\times
X^{\prime}\times\overline{Y}\times\overline{Y^{\prime}}}|\omega-\omega^{\prime}|^{p}\>\mathrm{d}(\pi\otimes\xi)+\int_{(X\times
X^{\prime})^{2}}|k-k^{\prime}|^{p}\>\mathrm{d}(\pi\otimes\pi)+\int_{\overline{Y}\times\overline{Y^{\prime}}}|d_{p}\circ(\iota,\iota^{\prime})|^{p}\>\mathrm{d}\xi.$
(10)

Continuity of each of the second two terms follows exactly as in the proofs of
[11, Lemma 24] and [17, Proposition 3.1], respectively. For the first term, we
consider a weakly converging
$\pi_{n}^{v}\overset{\tiny\text{w}}{\longrightarrow}\pi^{v}$, and a vaguely
converging $\pi_{n}^{e}\overset{\tiny\text{v}}{\longrightarrow}\pi^{e}$. Then,
by the same argument as in [17, Proposition 3.1], the sequence
$(\pi^{v}\otimes\pi^{e})_{n}^{\prime}:U\mapsto\int_{U}|\omega-\omega^{\prime}|^{p}\>\mathrm{d}(\pi^{v}_{n}\otimes\pi^{e}_{n})$
converges to
$(\pi^{v}\otimes\pi^{e})^{\prime}:U\mapsto\int_{U}|\omega-\omega^{\prime}|^{p}\>\mathrm{d}(\pi^{v}\otimes\pi^{e})$.
By using the Pormanteau Theorem ([17, Proposition A.4], which also holds for
weak convergence) as in [17, Proposition 3.1], continuity follows. ∎

######  Proposition 2.

The function $d_{\mathrm{TpOT},p}$ defines a pseudo-metric on the space of
topological networks $\mathcal{P}$.

To prove Prop. 2 we only need to show that the triangle inequality holds, as
symmetry is obvious from the definition of $d_{\mathrm{TpOT},p}$. We follow
the usual approach, which relies on the following standard result (see e.g.
[43, Lemma 1.4]).

######  Lemma 1 (Gluing Lemma).

Let $(X_{i},\mu_{i}),i=1,\cdots,n$, be Polish probability spaces. Consider
couplings $\xi_{i}\in\Pi(\mu_{i-1}\mu_{i})$, for $i=2,\cdots,n$. There exists
a unique probability measure $\xi$ on $X_{1}\times\cdots\times X_{n}$ such
that $(p_{i-1}\times p_{i})_{\\#}\xi=\xi_{i}$ for every $i=2,\cdots,n$, where
$p_{i}:X_{1}\times\cdots\times X_{n}\longrightarrow X_{i}$ is the projection
on the $i$th factor.

The unique measure guaranteed by the lemma will be referred to as the _gluing_
of $\xi_{2},\ldots,\xi_{n}$ and will sometimes be denoted as
$\xi_{2}\boxtimes\xi_{3}\boxtimes\cdots\boxtimes\xi_{n}$. Lemma 1 can be
adapted to the case of Radon measures and admissible couplings [17, Prop.
3.2].

###### Proof of Proposition 2.

The proof follows adapts techniques used in the proof of [17, Prop. 3.2] and
[11, Thm. 1]. To prove that the triangle inequality holds for
$d_{\mathrm{TpOT},p}$, consider three topological networks $P_{1}$, $P_{2}$
and $P_{3}$, and let $(\pi_{12}^{v},\pi^{e}_{12})$ and
$(\pi_{23}^{v},\pi^{e}_{23})$ be the couplings realising
$d_{\mathrm{TpOT},p}(P_{1},P_{2})$ and $d_{\mathrm{TpOT},p}(P_{2},P_{3})$,
which exist by Prop. 1. Then, using the gluing lemma, we construct a
probability measure $\pi^{v}$ on $X_{1}\times X_{2}\times X_{3}$ with
marginals $\pi^{v}_{12}$ and $\pi^{v}_{23}$ on $X_{1}\times X_{2}$ and
$X_{2}\times X_{3}$, respectively. We denote by $\pi^{v}_{13}$ the marginal on
$X_{1}\times X_{3}$. Similarly, the gluing lemma adapted to Radon measures and
admissible couplings yields a measure $\pi^{e}$ on
$\overline{Y_{1}}\times\overline{Y_{2}}\times\overline{Y_{3}}$ with marginals
that agree with $\pi^{e}_{ij}$ when restricted to
$\overline{Y_{i}}\times\overline{Y_{j}}\setminus\\{(\partial_{Y_{i}},\partial_{Y_{j}})\\}$
and that induces a zero cost on $(\partial_{Y_{i}},\partial_{Y_{j}})$. Then,
we have

$\displaystyle d_{\mathrm{TpOT},p}(P_{1},P_{3})$
$\displaystyle\leq\left(\|d_{p}\circ(\iota_{1},\iota_{3})\|^{p}_{L^{p}(\pi^{e}_{13})}+\|k_{1}-k_{3}\|^{p}_{L^{p}(\pi^{v}_{13}\otimes\pi^{v}_{13})}+\|\omega_{1}-\omega_{3}\|^{p}_{L^{p}(\pi^{v}_{13}\otimes\pi^{e}_{13})}\right)^{1/p}$
(11)
$\displaystyle=\left(\|d_{p}\circ(\iota_{1},\iota_{3})\|^{p}_{L^{p}(\pi^{e})}+\|k_{1}-k_{3}\|^{p}_{L^{p}(\pi^{v}\otimes\pi^{v})}+\|\omega_{1}-\omega_{3}\|^{p}_{L^{p}(\pi^{v}\otimes\pi^{e})}\right)^{1/p}$
(12)
$\displaystyle=\left\|\begin{pmatrix}\|d_{p}\circ(\iota_{1},\iota_{3})\|_{L^{p}(\pi^{e})}\\\
\|k_{1}-k_{3}\|_{L^{p}(\pi^{v}\otimes\pi^{v})}\\\
\|\omega_{1}-\omega_{3}\|_{L^{p}(\pi^{v}\otimes\pi^{e})}\end{pmatrix}\right\|_{p}$
$\displaystyle\leq\left\|\begin{pmatrix}\|d_{p}\circ(\iota_{1},\iota_{2})\|_{L^{p}(\pi^{e})}+\|d_{p}\circ(\iota_{2},\iota_{3})\|_{L^{p}(\pi^{e})}\\\
\|k_{1}-k_{2}\|_{L^{p}(\pi^{v}\otimes\pi^{v})}+\|k_{2}-k_{3}\|_{L^{p}(\pi^{v}\otimes\pi^{v})}\\\
\|\omega_{1}-\omega_{2}\|_{L^{p}(\pi^{v}\otimes\pi^{e})}+\|\omega_{2}-\omega_{3}\|_{L^{p}(\pi^{v}\otimes\pi^{e})}\end{pmatrix}\right\|_{p}$
(13)
$\displaystyle\leq\left\|\begin{pmatrix}\|d_{p}\circ(\iota_{1},\iota_{2})\|_{L^{p}(\pi^{e})}\\\
\|k_{1}-k_{2}\|_{L^{p}(\pi^{v}\otimes\pi^{v})}\\\
\|\omega_{1}-\omega_{2}\|_{L^{p}(\pi^{v}\otimes\pi^{e})}\end{pmatrix}\right\|_{p}+\left\|\begin{pmatrix}\|d_{p}\circ(\iota_{2},\iota_{3})\|_{L^{p}(\pi^{e})}\\\
\|k_{2}-k_{3}\|_{L^{p}(\pi^{v}\otimes\pi^{v})}\\\
\|\omega_{2}-\omega_{3}\|_{L^{p}(\pi^{v}\otimes\pi^{e})}\end{pmatrix}\right\|_{p}$
(14)
$\displaystyle=\left\|\begin{pmatrix}\|d_{p}\circ(\iota_{1},\iota_{2})\|_{L^{p}(\pi^{e}_{12})}\\\
\|k_{1}-k_{2}\|_{L^{p}(\pi^{v}_{12}\otimes\pi^{v}_{12})}\\\
\|\omega_{1}-\omega_{2}\|_{L^{p}(\pi^{v}_{12}\otimes\pi^{e}_{12})}\end{pmatrix}\right\|_{p}+\left\|\begin{pmatrix}\|d_{p}\circ(\iota_{2},\iota_{3})\|_{L^{p}(\pi^{e}_{23})}\\\
\|k_{2}-k_{3}\|_{L^{p}(\pi^{v}_{23}\otimes\pi^{v}_{23})}\\\
\|\omega_{2}-\omega_{3}\|_{L^{p}(\pi^{v}_{23}\otimes\pi^{e}_{23})}\end{pmatrix}\right\|_{p}$
(15)
$\displaystyle=d_{\mathrm{TpOT},p}(P_{1},P_{2})+d_{\mathrm{TpOT},p}(P_{2},P_{3}).$

Here, we use $\|\cdot\|_{p}$ to indicate the standard $\ell^{p}$ norm on
$\mathbb{R}^{3}$. The inequality (11) follows by suboptimality, while (12) and
(15) by marginal conditions. The triangle inequalities in $L^{p}$ and $d_{p}$
give (13), and the triangle inequality in $\ell^{p}$ gives (14). ∎

In analogy with measure (hyper)networks [8, 11], we define an equivalence
class on $\mathcal{P}$ as follows.

######  Definition 3 (Weak Isomorphism).

A _weak isomorphism_ of topological networks $P,P^{\prime}$ is a pair of
measure-preserving maps $\phi:X\longrightarrow X^{\prime}$ and
$\psi:Y\longrightarrow Y^{\prime}$ such that
$\iota(y)=\iota^{\prime}(\psi(y))$ for $\nu$-almost every $y\in Y$,
$\omega(x,y)=\omega^{\prime}(\phi(x),\psi(y))$ for $\mu\otimes\nu$-almost
every $(x,y)\in X\times Y$, and
$k(x_{1},x_{2})=k^{\prime}(\phi(x_{1}),\phi(x_{2}))$ for
$\mu\otimes\mu$-almost every $(x_{1},x_{2})\in X\times X$. We say that
$P,P^{\prime}$ are _weakly isomorphic_ if there exists a topological network
$\overline{P}$ and weak isomorphisms from $\overline{P}$ to $P$ and to
$P^{\prime}$. We use $P\sim_{w}P^{\prime}$ to denote that $P$ and $P^{\prime}$
are weakly isomorphic; this is an equivalence relation on $\mathcal{P}$.

######  Proposition 3.

Two topological measure networks
$P=\left((X,k,\mu),(Y,\iota,\nu),\omega\right)$ and
$P^{\prime}=\left((X^{\prime},k^{\prime},\mu^{\prime}),(Y^{\prime},\iota^{\prime},\nu^{\prime}),\omega^{\prime}\right)$
have distance $d_{\mathrm{TpOT},p}(P,P^{\prime})=0$ if and only if
$P\sim_{w}P^{\prime}$.

###### Proof.

The “if” direction is clear, by the triangle inequality. Conversely, assume
$d_{\mathrm{TpOT},p}(P,P^{\prime})=0$ and let $\pi^{v}$ and $\pi^{e}$ be
couplings realising this distance (Prop. 1).

We construct $\tilde{P}$ by setting $\tilde{X}=X\times X^{\prime}$,
$\tilde{\mu}=\pi^{v}$. Similarly, we take $\tilde{Y}=\left((Y\times
Y^{\prime})\cup(Y\times\partial_{Y^{\prime}})\cup(\partial_{Y}\times
Y^{\prime})\right)$ (where we write $Y\times\partial_{Y^{\prime}}$ rather than
$Y\times\\{\partial_{Y^{\prime}}\\}$ for the sake of cleaner notation, and
take similar conventions elsewhere), augmented to
$\overline{Y}\times\overline{Y^{\prime}}=\tilde{Y}\cup\partial_{Y}\times\partial_{Y^{\prime}}$,
and with measure $\tilde{\nu}=\pi^{e}$. Next, we define $\phi,\phi^{\prime}$
and $\psi,\psi^{\prime}$ as coordinate projection maps.

Given
$\tilde{x}_{1}=(x_{1},x_{1}^{\prime}),\tilde{x}_{2}=(x_{2},x_{2}^{\prime})\in
X\times X^{\prime}$, we set
$\tilde{k}(\tilde{x}_{1},\tilde{x}_{2})=k(\phi(\tilde{x}_{1}),\phi(\tilde{x}_{2}))=k(x_{1},x_{2})$.
By optimality of $\pi^{v}$ and since $d_{\mathrm{TpOT},p}(P,P^{\prime})=0$, we
have that
$k(\phi(\tilde{x}_{1}),\phi(\tilde{x}_{2}))=k^{\prime}(\phi^{\prime}(\tilde{x}_{1}),\phi^{\prime}(\tilde{x}_{2}))$
for almost every $\tilde{x}_{1},\tilde{x}_{1}$.

Similarly, for
$\tilde{y}_{1}=(y_{1},y_{1}^{\prime}),\tilde{y}_{2}=(y_{2},y_{2}^{\prime})\in
Y\times Y^{\prime}$ if we define
$\tilde{\omega}(\tilde{x},\tilde{y})=\omega(\phi(x),\psi(y))$, by optimality
of $\pi^{v}$ we have
$\tilde{\omega}(\tilde{x},\tilde{y})=\omega(\phi(x),\psi(y))=\omega^{\prime}(\phi^{\prime}(x),\psi^{\prime}(y))$
for $\pi^{v}\otimes\pi^{e}$-almost every
$(\tilde{x},\tilde{y})\in\tilde{X}\times\tilde{Y}$. Finally, we set
$\tilde{\iota}:\tilde{Y}\longrightarrow\Delta$ as
$\tilde{\iota}(\tilde{y})=\iota(\psi(y))=\iota^{\prime}(\psi^{\prime}(y^{\prime}))$
for almost every $\tilde{y}$, again by optimality of $\pi^{e}$. This implies
that $P$ and $P^{\prime}$ are weakly isomorphic. ∎

The work above immediately implies the following.

######  Theorem 1.

The pseudometric $d_{\mathrm{TpOT},p}$ induces a metric on the quotient space
$\faktor{\mathcal{P}}{\sim_{w}}$.

We abuse notation and also denote by $d_{\mathrm{TpOT},p}$ the induced metric
on the quotient space.

###  3.3 A topology-driven metric on point clouds

We now summarise the metric described above in the discrete setting of point
clouds with finite number of points. Consider a point cloud
$X=\\{x_{1},\cdots,x_{N}\\}$ in $\mathbb{R}^{n}$. Any choice of filtration
$\mathcal{K}$ over $X$ yields a corresponding persistence diagram
$D=D(X,\mathcal{K})$. Choosing a representative cycle for each homology class
in $D$ yields the set of generators $g$, and the corresponding PH-hypergraph
$H=H(X,\mathcal{K},g)$. We can now endow $X$ with a topological measure
hypergraph structure by considering

$P_{X}=\left((X,k,\mu_{X}),(Y,\iota,\nu),\omega_{H}\right),$

where $k$ is a gauge function of choice defined on $X$ (for instance, a kernel
function, or pairwise distances), $\mu_{X}$ is a chosen measure on $X$ (for
instance, uniform), $Y$ is a set with $|D|$ elements (counted with
multiplicity), $\nu$ and $\iota$ are such that $\iota_{\\#}\nu=\nu_{D}$, where
$\nu_{D}$ is as in (3), and $\omega_{H}$ is the binary incidence function for
$H$ (see Example 1).

Generating cycles are far from being unique, and there are currently several
different algorithms and software available to compute them [2, 5, 14, 15, 19,
16, 29, 34, 49]. As explained in Section 5, here persistent homology
computations are performed using the Julia software Ripserer.jl [14], which
implements the involutive algorithm [14, 15] to compute homology and
representatives.

Given two point clouds $X=\\{x_{1},\cdots,x_{N}\\}$ and
$X^{\prime}=\\{x^{\prime}_{1},\cdots,x^{\prime}_{N^{\prime}}\\}$ in
$\mathbb{R}^{n}$ endowed with filtrations, let $P_{X}$ and $P_{X^{\prime}}$ be
their associated topological networks. In this concrete setting, we focus on
the $p=2$ version of the metric. As was described in Remark 2, it is useful in
applications to include tunable weights on the terms of TpOT. With this
notation, the distance $d_{\mathrm{TpOT},2}=d_{\mathrm{TpOT},\alpha,\beta}$
can be reformulated, using notation similar to that of [10], as

$d_{\mathrm{TpOT},\alpha,\beta}(P_{X},P_{X^{\prime}})^{2}=\min_{\pi^{v}\in\Pi(\mu,\mu^{\prime}),\pi^{e}\in\Pi(\widetilde{\nu},\widetilde{\nu}^{\prime})}\alpha\left\langle
L(C,C^{\prime}),\pi^{v}\otimes\pi^{v}\right\rangle+(1-\alpha)\left\langle\widetilde{C}(D,D^{\prime}),\pi^{e}\right\rangle+\beta\left\langle
L(\omega_{H},\omega_{H^{\prime}}),\pi^{v}\otimes\pi^{e}\right\rangle.$ (16)

The notations used here are as follows:

  * •

All inner products in this expression are Frobenius inner products of
matrices.

  * •

The terms $C,C^{\prime}$ are the pairwise (Euclidean) distance matrices, and
$D,D^{\prime}$ are the persistence diagrams associated to the given
filtrations.

  * •

The term $\widetilde{C}(D,D^{\prime})$ denotes the augmented $L^{2}$ cost
matrix of dimension $(|D|+1)\times(|D^{\prime}|+1)$ for persistence diagrams,
whose last column and row correspond to transport to and from
$\partial_{\Lambda}$ respectively, following the diagonal projection
$\pi_{\partial_{\Lambda}}$ (see [28, Equation 8]).

  * •

The expression $L(C,C^{\prime})$ denotes the fourth-order distortion tensor,
whose term with index $(i,j,k,\ell)$ is given by
$L(C,C^{\prime})_{ijk\ell}=\frac{1}{2}|C_{ik}-C^{\prime}_{j\ell}|^{2}$. The
term $\langle L(C,C^{\prime}),\pi^{v}\otimes\pi^{v}\rangle$ can be understood
by considering the measure $\pi^{v}$ as a matrix, so that the product
$\pi^{v}\otimes\pi^{v}$ is also naturally identified with a fourth order
tensor (with the same dimensions as $L(C,C^{\prime})$). Then the inner product
is computed by isomorphically identifying these fourth order tensors with
matrices.

  * •

We denote by $\omega_{H}$ and $\omega_{H^{\prime}}$ the hypernetwork functions
of the PH-hypergraphs $H$ and $H^{\prime}$, respectively. Specifically,
$\omega_{H}$ and $\omega_{H^{\prime}}$ are represented as matrices of size
$N\times|D|+1$ , $N^{\prime}\times|D^{\prime}|+1$ respectively, that are
obtained as the hypergraph incidence matrices with an additional $0$-row each
representing membership for the diagonal edge.

  * •

The expression $L(\omega_{H},\omega_{H^{\prime}})$ is the fourth-order given
by:

$\displaystyle\big{(}L(\omega_{H},\omega_{H^{\prime}})\big{)}_{i,k,j,l}$
$\displaystyle=\frac{1}{2}|\omega_{ik}-\omega_{jl}|^{2}$ $\displaystyle\text{
for }\ 1\leq i\leq|D|,1\leq j\leq|D^{\prime}|$
$\displaystyle\big{(}L(\omega_{H},\omega_{H^{\prime}})\big{)}_{|D|+1,k,j,l}$
$\displaystyle=\frac{1}{2}|\omega_{jl}|^{2}$ $\displaystyle\text{ for }\ 1\leq
j\leq|D^{\prime}|$ (17)
$\displaystyle\big{(}L(\omega_{H},\omega_{H^{\prime}})\big{)}_{i,k,|D^{\prime}|+1,l}$
$\displaystyle=\frac{1}{2}|\omega_{ik}|^{2}$ $\displaystyle\text{ for }\ 1\leq
i\leq|D|,j=|D^{\prime}|+1$ (18)
$\displaystyle\big{(}L(\omega_{H},\omega_{H^{\prime}})\big{)}_{|D|+1,k,|D^{\prime}|+1,l}$
$\displaystyle=0$ (19)

In other words, the cost of matching any real edge to the diagonal is given by
the $L^{2}$ norm of the edge (17 and 17), while matching the diagonal with the
diagonal has zero cost (19).

  * •

The parameter $\alpha\in[0,1]$ controls the tradeoff between the cost of
matchings in the persistence diagram space (coefficient $1-\alpha$) and
matchings in terms of the Gromov-Wasserstein distortion (coefficient
$\alpha$).

  * •

The parameter $\beta\in[0,\infty)$ controls the degree to which the geometric
and topological matchings are coupled by the hypergraph structure.

The solution of the TpOT problem in 16 involves determining an optimal pair of
couplings $\pi^{v},\pi^{e}$. The coupling $\pi^{v}$ induces a transport plan
between points in the point clouds $X,X^{\prime}$. When
$\beta>0,\alpha\in[0,1]$, this transport balances between preserving
topological features and pairwise distances. The coupling $\pi^{e}$ induces a
matching between homology classes in the persistence diagrams $D,D^{\prime}$.
For $0<\alpha<1$, this matching is driven by the Wasserstein matching, and it
is informed by pairwise Euclidean proximity of points forming representative
cycles of classes in $D,D^{\prime}$. Implementation details are described in
Section 5, where we also provide numerical examples.

###  3.4 Numerical algorithms

We now aim at numerical algorithms for approximating the solution to the TpOT
problem in practice. Starting from (16), we consider a regularised variant of
the TpOT problem by adding an entropic regularisation to the transport plans
$(\pi^{e},\pi^{v})$. Writing

$\mathsf{L}(\pi^{v},\pi^{e})=\alpha\langle
L(C,C^{\prime}),\pi^{v}\otimes\pi^{v}\rangle+(1-\alpha)\langle\widetilde{C}(D,D^{\prime}),\pi^{e}\rangle+\beta\langle
L(\omega_{H},\omega_{H^{\prime}}),\pi^{v}\otimes\pi^{e}\rangle,$

we have

$\displaystyle\min_{\pi^{v}\in\Pi(\mu,\mu^{\prime}),\>\pi^{e}\in\Pi(\widetilde{\nu},\widetilde{\nu}^{\prime})}$
$\displaystyle\mathsf{L}(\pi^{v},\pi^{e})+\varepsilon_{v}\operatorname{KL}(\pi^{v}|\mu\otimes\mu^{\prime})+\varepsilon_{e}\operatorname{KL}(\pi^{e}|\widetilde{\nu}\otimes\widetilde{\nu}^{\prime}).$

For $\varepsilon_{v},\varepsilon_{e}>0$, we can take advantage of the entropic
regularisation to utilise fast, smooth optimisation techniques. In particular,
we can find a local minimum by projected gradient descent [37, Section 2.3],
where both the gradient and the projection are calculated with respect to the
KL-divergence. This leads to the following iterative scheme:

$\displaystyle\begin{split}\pi^{v}_{t+1}&\leftarrow\operatorname{Proj}_{\Pi(\mu,\mu^{\prime})}^{\operatorname{KL}}\left[-\varepsilon_{v}^{-1}\nabla_{v}\mathsf{L}(\pi^{v}_{t},\pi^{e}_{t})(\mu\otimes\mu^{\prime})\right]\\\
\pi^{e}_{t+1}&\leftarrow\operatorname{Proj}_{\Pi(\widetilde{\nu},\widetilde{\nu}^{\prime})}^{\operatorname{KL}}\left[-\varepsilon_{e}^{-1}\nabla_{e}\mathsf{L}(\pi^{v}_{t},\pi^{e}_{t})(\widetilde{\nu}\otimes\widetilde{\nu}^{\prime})\right].\end{split}$
(20)

Each of these projections can be calculated by matrix scaling using the
Sinkhorn algorithm [36, Chapter 4], and the gradients of $\mathsf{L}$ have
closed-form expressions:

$\displaystyle\begin{split}\nabla_{v}\mathsf{L}(\pi^{v},\pi^{e})&=2\alpha
L(C,C^{\prime})\otimes\pi^{v}+\beta
L(\omega_{H},\omega_{H}^{\prime})\otimes\pi^{e}\\\
\nabla_{e}\mathsf{L}(\pi^{v},\pi^{e})&=(1-\alpha)\widetilde{C}[D,D^{\prime}]+\beta
L(\omega_{H}^{\top},{\omega_{H}^{\prime}}^{\top})\otimes\pi^{v}.\end{split}$
(21)

In the unregularised case when $\varepsilon_{v}=\varepsilon_{e}=0$, an
alternating minimisation in $(\pi^{v},\pi^{e})$ can be used to find a local
minimum. Fixing $\pi^{e}$, the minimisation problem in
$\pi^{v}\in\Pi(\mu,\mu^{\prime})$ is

$\displaystyle\min_{\pi^{v}\in\Pi(\mu,\mu^{\prime})}\alpha\langle
L(C,C^{\prime}),\pi^{v}\otimes\pi^{v}\rangle+\beta\langle
L(\omega_{H},\omega_{H^{\prime}})\otimes\pi^{e},\pi^{v}\rangle.$ (22)

This falls within the fused Gromov-Wasserstein framework which was introduced
and studied in detail by [45, 48]. A local minimum can be found using a
conditional gradient method [45, Algorithm 1].

Fixing $\pi^{v}$ and minimising in
$\pi^{e}\in\Pi(\widetilde{\nu},\widetilde{\nu}^{\prime})$, we have

$\displaystyle\min_{\pi^{e}\in\Pi(\widetilde{\nu},\widetilde{\nu}^{\prime})}\>\langle
M,\pi^{e}\rangle,\quad M=\pi^{v}\otimes
L(\omega_{H},\omega_{H^{\prime}})+(1-\alpha)\widetilde{C}(D,D^{\prime})$ (23)

and we have the identity

$\langle\pi\otimes L(X,X^{\prime}),\xi\rangle=\langle
L(X^{\top},(X^{\prime})^{\top})\otimes\pi,\xi\rangle\rangle=\langle-X^{\top}\pi(X^{\prime}),\xi\rangle.$

This amounts to a standard optimal transport problem.

##  4 Characterisation of geodesics

Geodesic properties have been extensively studied for the space of measure
networks under Gromov-Wasserstein distance [43, 10, 33] and the space of
persistence diagrams under Wasserstein distance [47, 6]. In this section, we
derive similar results for the $\mathrm{TpOT}$ metric when $p=2$. These
results are of theoretical interest, but we plan to explore practical
implications in future work; for example, the geodesic structure and curvature
properties can be used to study Fréchet means of ensembles of topological
networks, extending ideas in [47, 10]. We begin by recalling some definitions
from metric geometry—see [4] as a general reference.

###  4.1 Metric Geometry Concepts

Consider a metric space $(X,d)$. A _geodesic_ between points $x,y\in X$ is
defined as a path $\gamma:[0,1]\longrightarrow X$ with $\gamma(0)=x$,
$\gamma(1)=y$ and such that, for every $0\leq s\leq t\leq 1$, we have
$d(\gamma(s),\gamma(t))=(t-s)d(x,y)$. In fact, it suffices to show that
$d(\gamma(s),\gamma(t))\leq(t-s)d(x,y)$ always holds, as the reverse
inequality then follows for free—see, e.g., [7, Lemma 1.3].

We say that $(X,d)$ is a geodesic metric space if, for any pair of points
$x,y\in X$, there exists a geodesic $\gamma$ with $\gamma(0)=x$ and
$\gamma(1)=y$. Further, we say that $(X,d)$ is uniquely geodesic if for any
pair of points $x,y\in X$ the geodesic connecting them is unique.

For $(X,d)$ a geodesic space, we say that it has curvature bounded below by
zero [43, Section 4.2] if, for every geodesic $\gamma:[0,1]\longrightarrow X$
and every point $x\in X$, the following holds:

$d(\gamma(t),x)^{2}\geq(1-t)d(\gamma(0),x)^{2}+td(\gamma(1),x)^{2}-t(1-t)d(\gamma(0),\gamma(1))^{2},\quad
0\leq t\leq 1.$

Intuitively, this says that geodesic triangles in $X$ are always “thicker”
than the corresponding triangles in (flat) Euclidean space.

###  4.2 Geodesics between measure persistence diagrams

The space of persistence diagrams with the $p$-Wasserstein distance is well
studied, and it is known that it admits geodesics which are essentially linear
interpolations of diagrams [47, 6]. What follows is a discussion of existence
and uniqueness of geodesics in the case of measure persistence diagrams; these
results are a straightforward generalisation of known results for persistence
diagrams.

Consider two measure persistence diagrams $\nu_{0}$ and $\nu_{1}$, and
$\xi\in\Pi_{\text{adm}}(\nu_{0},\nu_{1})$ a coupling between them. For
$t\in[0,1]$ consider the map
$\phi_{t}:\overline{\Lambda}\times\overline{\Lambda}\to\overline{\Lambda},\>\phi_{t}(\lambda,\lambda^{\prime})=(1-t)\lambda+t\lambda^{\prime}$.
Finally, we define

$\nu_{t}^{\xi}:=(\phi_{t})_{\\#}\xi.$ (24)

######  Proposition 4.

Consider measure persistence diagrams $\nu_{0},\nu_{1}$, and $\xi$ an optimal
coupling that realises $d^{\mathrm{MPD}}_{\mathrm{W},p}(\nu,\nu^{\prime})$.
The path $\gamma(t)=\nu_{t}^{\xi}$ defines a geodesic between
$\nu_{0},\nu_{1}$.

###### Proof.

Set $\xi_{st}$ as the coupling in $\Pi_{\rm adm}(\nu_{s}^{\xi},\nu_{t}^{\xi})$
given by $\xi_{st}=(\phi_{s}\times\phi_{t})_{\\#}\pi^{e}$, where we use the
map

$\displaystyle\phi_{s}\times\phi_{t}:(\overline{\Lambda}\times\overline{\Lambda})$
$\displaystyle\rightarrow(\overline{\Lambda}\times\overline{\Lambda})$
$\displaystyle(\lambda,\lambda^{\prime})$
$\displaystyle\mapsto(\phi_{s}(\lambda,\lambda^{\prime}),\phi_{t}(\lambda,\lambda^{\prime})).$

By sub-optimality, we have that

$d^{\mathrm{MPD}}_{\mathrm{W},p}(\nu_{s}^{\xi},\nu_{t}^{\xi})^{p}\leq\int_{\overline{\Lambda}^{2}}\|\phi_{s}(\overline{\lambda})-\phi_{t}(\overline{\lambda})\|^{p}_{p}\xi_{st}(d\phi_{s}(\overline{\lambda})\times
d\phi_{t}(\overline{\lambda}))=\int_{\overline{\Lambda}^{2}}\|\phi_{s}-\phi_{t}\|^{p}_{p}\;\;\xi_{st},$

where $\overline{\lambda}=(\lambda,\lambda^{\prime})$. From the equality

$\|\phi_{s}(\overline{\lambda})-\phi_{t}(\overline{\lambda})\|^{p}_{p}=\|(t-s)\lambda-(t-s)\lambda^{\prime}\|^{p}_{p}$

we have

$\int_{\overline{\Lambda}^{2}}\|\phi_{s}-\phi_{t}\|^{p}_{p}\;\;\xi_{st}=(t-s)^{p}\int_{\overline{\Lambda}^{2}}\|d_{p}\|^{p}_{p}\;\;\xi=(t-s)^{p}d^{\mathrm{MPD}}_{\mathrm{W},p}(\nu_{0},\nu_{1})^{p}.$

Putting everything together, we have
$d^{\mathrm{MPD}}_{\mathrm{W},p}(\nu_{s}^{\xi},\nu_{t}^{\xi})\leq(t-s)d^{\mathrm{MPD}}_{\mathrm{W},p}(\nu_{0},\nu_{1})$,
and this completes the proof. ∎

######  Remark 3.

Consider two measure persistence diagrams $\nu_{0},\nu_{1}$. Take any path
between them of the form $\nu_{t}=(\phi_{t})_{\\#}\xi$, where $\xi$ is an
optimal coupling between $\nu_{0},\nu_{1}$, and for every
$(\lambda,\lambda^{\prime})\in\overline{\Lambda}\times\overline{\Lambda}$, the
function $t\mapsto\phi_{t}(\lambda,\lambda^{\prime})$ traces out a geodesic
between $\lambda$ and $\lambda^{\prime}$ in
$(\overline{\Lambda},\|\cdot\|_{p})$. Then, the same argument as in
Proposition 4 shows that $\nu_{t}$ is a geodesic between $\nu_{0},\nu_{1}$. In
particular, this shows that $d^{\mathrm{MPD}}_{\mathrm{W},1}$ admits geodesics
which are not of the form $\phi_{t}$.

######  Remark 4.

This formula for geodesics in the space of measure persistence diagrams takes
the same form as the well-known formula for geodesics in classical Wasserstein
space—see, e.g., [1, Theorem 7.2.2].

###  4.3 Geodesics in the space of measure topological networks

We now focus on geodesics in $\faktor{\mathcal{P}}{\sim_{w}}$, using
techniques which follow the those employed by Sturm in the Gromov-Wasserstein
setting [43]. Let $P=((X,k,\mu),(Y,\iota,\nu),\omega)$ and
$P^{\prime}=((X^{\prime},k^{\prime},\mu^{\prime}),(Y^{\prime},\iota^{\prime},\nu^{\prime}),\omega^{\prime})$
be topological networks and let $(\pi^{v},\pi^{e})$ be a pair of optimal
couplings that realise the infimum in Definition 2. We construct the
topological network

$P_{t}=\left((\tilde{X},k_{t},\pi^{v}),(\tilde{Y},\iota_{t},\pi^{e}),\omega_{t}\right),$
(25)

where $\tilde{X}=X\times X^{\prime},\tilde{Y}=\left((Y\times
Y^{\prime})\cup(Y\times\partial_{Y^{\prime}})\cup(\partial_{Y}\times
Y^{\prime})\right)$ are as in Proposition 3 (we once again simplify notation
by using expressions such as $Y\times\partial_{Y^{\prime}}$ rather than
$Y\times\\{\partial_{Y^{\prime}}\\}$),

$\displaystyle\omega_{t}((x,x^{\prime}),(y,y^{\prime}))$
$\displaystyle=(1-t)\omega(x,y)+t\omega^{\prime}(x^{\prime},y^{\prime}),$
$\displaystyle k_{t}((x_{1},x_{1}^{\prime}),(x_{2},x_{2}^{\prime}))$
$\displaystyle=(1-t)k(x_{1},x_{2})+tk^{\prime}(x^{\prime}_{1},x^{\prime}_{2}),$
$\displaystyle\iota_{t}(y,y^{\prime})$
$\displaystyle=(1-t)\iota(y)+t\iota^{\prime}(y^{\prime}).$

Note that
$(\iota_{t})_{\\#}\pi^{e}=\nu_{t}^{(\iota\times\iota^{\prime})_{\\#}\pi^{e}}$
as in (24). In the following, we use $[P]$ to denote the equivalence class of
$P\in\mathcal{P}$ with respect to $\sim_{w}$.

######  Theorem 2.

For $P,P^{\prime}\in\mathcal{P}$, the path
$\gamma:[0,1]\to\faktor{\mathcal{P}}{\sim_{w}}$ defined by
$\gamma_{t}=[P_{t}]$ defines a geodesic between $[P]$ and $[P^{\prime}]$, with
respect to $d_{\mathrm{TpOT},p}$.

###### Proof.

Throughout the proof, let $\pi^{v}\in\Pi(\mu,\mu^{\prime})$ and
$\pi^{e}\in\Pi_{\rm adm}(\nu,\nu^{\prime})$ be optimal couplings for
$P,P^{\prime}$. We fix $0\leq s\leq t\leq 1$, and we use the notation
$\tilde{y}=(y,y^{\prime})\in\tilde{Y}$,
$\tilde{x}=(x,x^{\prime})\in\tilde{X}$, and
$\partial=(\partial_{Y},\partial_{Y^{\prime}})$. Consider now any pair of
couplings $\pi,\xi$ for $P_{s},P_{t}$. By sub-optimality, we have

$\displaystyle\begin{split}d_{\mathrm{TpOT},p}(P_{s},P_{t})^{p}&\leq\int_{\tilde{X}^{2}\times\tilde{Y}^{2}}|\omega_{s}(\tilde{x}_{1},\tilde{y}_{1})-\omega_{t}(\tilde{x}_{2},\tilde{y}_{2})|^{p}\>\mathrm{d}\pi(\tilde{x}_{1},\tilde{x}_{2})\>\mathrm{d}\xi(\tilde{y}_{1},\tilde{y}_{2}))\\\
&\qquad\qquad+\int_{\tilde{X}^{4}}|k_{s}(\tilde{x}_{1},\tilde{x}_{2})-k_{t}(\tilde{x}_{3},\tilde{x_{4}})|^{p}\>\mathrm{d}\pi(\tilde{x}_{1},\tilde{x}_{3})\>\mathrm{d}\pi(\tilde{x}_{2},\tilde{x}_{4})\\\
&\qquad\qquad+\int_{(\tilde{Y}\cup\partial)^{2}}\|\iota_{s}(\tilde{y}_{1})-\iota_{t}(\tilde{y}_{2})\|^{p}_{p}\>\mathrm{d}\xi(\tilde{y}_{1},\tilde{y}_{2}).\end{split}$
(26)

Let $\mathbf{1}_{\pi^{v}}$ denote the identity coupling of $\pi^{v}$ to
itself, that is,
$\mathbf{1}_{\pi^{v}}=(\mathrm{diag}_{\tilde{X}})_{\\#}(\pi^{v})$, where
$\mathrm{diag}_{\tilde{X}}$ is the map

$\displaystyle\mathrm{diag}_{\tilde{X}}:\tilde{X}$
$\displaystyle\rightarrow\tilde{X}\times\tilde{X}$ $\displaystyle\tilde{x}$
$\displaystyle\mapsto(\tilde{x},\tilde{x}).$

Similarly, set
$\mathbf{1}_{\pi^{e}}=(\mathrm{diag}_{\tilde{Y}})_{\\#}(\pi^{e})$, where
$\mathrm{diag}_{\tilde{Y}}$ is defined analogously. Note that we have that
$\xi_{st}\coloneqq(\iota_{s}\times\iota_{t})_{\\#}\mathbf{1}_{\pi^{e}}=(\phi_{s}\times\phi_{t})_{\\#}\pi^{e}$
as in the proof of Prop. 4.

By the same argument as in the proof of Prop. 4, we have that

$\displaystyle\int_{(\tilde{Y}\cup\partial)^{2}}\|\iota_{s}(\tilde{y})-\iota_{t}(\tilde{y})\|^{p}_{p}\>\mathrm{d}\mathbf{1}_{\pi^{e}}(\tilde{y},\tilde{y})$
$\displaystyle=(t-s)^{p}\int_{\overline{Y}\times\overline{Y^{\prime}}}\|\iota(y)-\iota^{\prime}(y^{\prime})\|^{p}_{p}\>\mathrm{d}\pi^{e}(y,y^{\prime}).$

Similarly, from the definition of $k_{s}$ and $k_{t}$, we have

$|k_{s}(\tilde{x}_{1},\tilde{x}_{2})-k_{t}(\tilde{x}_{1},\tilde{x}_{2})|^{p}=|(t-s)k(x_{1},x_{2})-(t-s)k^{\prime}(x^{\prime}_{1},x^{\prime}_{2})|^{p},$

and it follows that

$\displaystyle\int_{\tilde{X}^{4}}|k_{s}(\tilde{x}_{1},\tilde{x}_{2})-k_{t}(\tilde{x}_{1},\tilde{x}_{2})|^{p}\>\mathrm{d}\mathbf{1}_{\pi^{v}}(\tilde{x}_{1},\tilde{x}_{1})\>\mathrm{d}\mathbf{1}_{\pi^{v}}(\tilde{x}_{2},\tilde{x}_{2})$
$\displaystyle\hskip
108.405pt=(t-s)^{p}\int_{\tilde{X}^{4}}|k(x_{1},x_{2})-k^{\prime}(x^{\prime}_{1},x^{\prime}_{2})|^{p}\>\mathrm{d}\mathbf{1}_{\pi^{v}}(\tilde{x}_{1},\tilde{x}_{1})\>\mathrm{d}\mathbf{1}_{\pi^{v}}(\tilde{x}_{2},\tilde{x}_{2})$
$\displaystyle\hskip
108.405pt=(t-s)^{p}\int_{\tilde{X}^{2}}|k(x_{1},x_{2})-k^{\prime}(x_{1}^{\prime},x_{2}^{\prime})|^{p}\>\mathrm{d}\pi^{v}(x_{1},x_{1}^{\prime})\>\mathrm{d}\pi^{v}(x_{2},x_{2}^{\prime}),$

where we have applied the definition of $\mathbf{1}_{\pi^{v}}$ in the last
line. Applying the same arguments to the remaining term in (26), we have

$|\omega_{s}(\tilde{x},\tilde{y})-\omega_{t}(\tilde{x},\tilde{y})|^{p}=|(t-s)\omega(x,y)-(t-s)\omega^{\prime}(x^{\prime},y^{\prime}))|^{p}$

and

$\displaystyle\int_{\tilde{X}^{2}\times\tilde{Y}^{2}}|\omega_{s}(\tilde{x},\tilde{y})-\omega_{t}(\tilde{x},\tilde{y})|^{p}\>\mathrm{d}\mathbf{1}_{\pi^{v}}(\tilde{x},\tilde{x})\>\mathrm{d}\mathbf{1}_{\pi^{e}}(\tilde{y},\tilde{y})$
$\displaystyle\hskip
108.405pt=(t-s)^{p}\int_{\tilde{X}^{2}\times\tilde{Y}^{2}}|\omega(x,y)-\omega^{\prime}(x^{\prime},y^{\prime})|^{p}\>\mathrm{d}\mathbf{1}_{\pi^{v}}(\tilde{x},\overline{x})\>\mathrm{d}\mathbf{1}_{\pi^{e}}(\tilde{y},\tilde{y})$
$\displaystyle\hskip
108.405pt=(t-s)^{p}\int_{\tilde{X}\times(\overline{Y}\times\overline{Y^{\prime}})}|\omega(x,y)-\omega^{\prime}(x^{\prime},y^{\prime})|^{p}\>\mathrm{d}\pi^{v}(x,x^{\prime})\>\mathrm{d}\pi^{e}(y,y^{\prime}).$

Putting these together, we have that

$\displaystyle d_{\mathrm{TpOT},p}(P_{s},P_{t})^{p}$
$\displaystyle\leq(t-s)^{p}\left(\int_{\tilde{X}\times(\overline{Y}\times\overline{Y^{\prime}})}|\omega(x,y)-\omega^{\prime}(x^{\prime},y^{\prime})|^{p}\>\mathrm{d}\pi^{v}(x,x^{\prime})\>\mathrm{d}\pi^{e}(y,y^{\prime})\right.$
$\displaystyle\hskip
36.135pt+\int_{\tilde{X}^{2}}|k(x_{1},x_{2})-k^{\prime}(x_{1}^{\prime},x_{2}^{\prime})|^{p}\>\mathrm{d}\pi^{v}(x_{1},x_{1}^{\prime})\>\mathrm{d}\pi^{v}(x_{2},x_{2}^{\prime})$
$\displaystyle\left.\hskip
36.135pt+\int_{\overline{Y}\times\overline{Y^{\prime}}}\|\iota(y)-\iota^{\prime}(y^{\prime})\|^{p}_{p}\>\mathrm{d}\pi^{e}(y,y^{\prime})\right)$
$\displaystyle=\left((t-s)d_{\mathrm{TpOT},p}(P_{s},P_{t})\right)^{p},$

and the result follows.

∎

A geodesic of the form described in Equation 25 will be called a convex
geodesic. A natural question is whether all geodesics in
$\faktor{\mathcal{P}}{\sim_{w}}$ are convex. We will show below (Theorem 3)
that this is the case for $p=2$. As a first observation, the proof of Theorem
2 easily generalises to give the following corollary. This corollary, in
particular, shows that if $p=1$ then there are geodesics which are not convex
(c.f. Remark 3).

######  Corollary 1.

Consider the path $Q^{t}$ between topological networks $Q^{0}$ and $Q^{1}$
given by

$Q^{t}=\left((\tilde{X},k_{t},\pi^{v}),(\tilde{Y},\iota_{t},\pi^{e}),\omega_{t}\right),$

where $k_{t},\omega_{t},\pi^{v},\pi^{e},\tilde{X},\tilde{Y}$ are as in Prop.
2, and $\iota_{t}$ is such that the map

$t\mapsto\iota_{t}((y^{0},y^{1}))$

defines a geodesic between $\iota^{0}(y)$ and $\iota^{1}(y^{1})$ in
$(\overline{\Lambda},\|\cdot\|_{p})$ as $t$ varies in $[0,1]$ (as in Remark 3)
for $\pi^{e}$-almost every $\tilde{y}=(y^{0},y^{1})\in\tilde{Y}$. Then $Q^{t}$
defines a geodesic $[Q^{t}]$ in $\faktor{\mathcal{P}}{\sim_{w}}$.

While some of the results below extend to $p>1$, from now on, we focus on the
case where $p=2$, and write $d_{\mathrm{TpOT}}$ in place of
$d_{\mathrm{TpOT},p}$. This is for the sake of simplifying notation, and
because we will use $p=2$ in computational implementations and examples below.

######  Theorem 3.

All geodesics in
$\left(\faktor{\mathcal{P}}{\sim_{w}},d_{\mathrm{TpOT}}\right)$ are convex.

###### Proof.

The proof follows by adapting techniques from [43, Thm. 3.1] and [6, Thm. 10].
Let $[Q^{t}]$ be an arbitrary geodesic, with

$Q^{t}=\left((X^{t},k^{t},\mu^{t}),(Y^{t},\iota^{t},\nu^{t}),\omega^{t}\right)$

Pick a dyadic decomposition of the unit interval
$t_{0}=0,t_{1}=\frac{1}{2^{n}},\cdots,t_{i}=\frac{i}{2^{n}},\cdots,t_{2^{n}}=1$.
For each $i\in\\{0,\cdots,2^{n}\\}$, let $\pi^{v}_{i},\pi^{e}_{i}$ be optimal
couplings for $Q^{t_{i}},Q^{t_{i-1}}$. Consider the gluings

$\overline{\pi^{v}}=\pi^{v}_{0}\boxtimes\pi^{v}_{\frac{1}{2^{n}}}\boxtimes\cdots\boxtimes\pi^{v}_{1}\quad\mbox{and}\quad\overline{\pi^{e}}=\pi^{e}_{0}\boxtimes\pi^{e}_{\frac{1}{2^{n}}}\boxtimes\cdots\boxtimes\pi^{e}_{1}$

(see Lemma 1 and the ensuing discussion) and the couplings

$\pi^{v}=(p_{0}\times
p_{1})_{\\#}\overline{\pi^{v}}\quad\mbox{and}\quad\pi^{e}=(p_{0}\times
p_{1})_{\\#}\overline{\pi^{e}},$

with $p_{i}:X^{0}\times X^{t_{1}}\times\cdots\times X^{1}\rightarrow X^{i}$
projection on the $i$th factor. Then, by suboptimality, we have

$\displaystyle d_{\rm TpOT}(Q^{0},Q^{1})^{2}$
$\displaystyle\leq\|\omega^{0}-\omega^{1}\|^{2}_{L^{2}(\pi^{v}\otimes\pi^{e})}+\|k^{0}-k^{1}\|^{2}_{L^{2}(\pi^{v}\otimes\pi^{v})}+\|d_{2}\circ(\iota^{0},\iota^{1})\|^{2}_{L^{2}(\pi^{e})}$
$\displaystyle=:A+B+C$ (27)

For any choice of $t\in\\{0,1/2^{n},2/2^{n},\cdots,1\\}$, let

$\xi^{v}_{t}=(p_{0}\times p_{1}\times
p_{t})_{\\#}\overline{\pi^{v}}\quad\mbox{and}\quad\xi^{e}_{t}=(p_{0}\times
p_{1}\times p_{t})_{\\#}\overline{\pi^{e}}.$

We now estimate (27) term-by-term. First observe that

$\displaystyle A$
$\displaystyle=\|\omega^{0}-\omega^{1}\|^{2}_{L^{2}(\pi^{v}\otimes\pi^{e})}=\left\|t\left(\frac{1}{t}(\omega^{0}-\omega^{t})\right)+(1-t)\left(\frac{1}{1-t}(\omega^{t}-\omega^{1})\right)\right\|^{2}_{L^{2}(\xi^{v}_{t}\otimes\xi^{e}_{t})}$
$\displaystyle=\frac{1}{t}\|\omega^{0}-\omega^{t}\|^{2}_{L^{2}(\xi^{v}_{t}\otimes\xi^{e}_{t})}+\frac{1}{1-t}\|\omega^{t}-\omega^{1}\|^{2}_{L^{2}(\xi^{v}_{t}\otimes\xi^{e}_{t})}-\frac{1}{t(1-t)}\|(1-t)(\omega^{0}-\omega^{t})-t(\omega^{t}-\omega^{1})\|^{2}_{L^{2}(\xi^{v}_{t}\otimes\xi^{e}_{t})},$
(28)

with the last line following by the general identity

$|ta+(1-t)b|^{2}=t|a|^{2}+(1-t)|b|^{2}-t(1-t)|a-b|^{2},$

applied to $a=\frac{1}{t}(\omega^{0}-\omega^{t})$ and
$b=\frac{1}{1-t}(\omega^{t}-\omega^{1})$, pointwise. Similarly,

$\displaystyle B$
$\displaystyle=\frac{1}{t}\|k^{0}-k^{t}\|^{2}_{L^{2}(\xi^{v}_{t}\otimes\xi^{v}_{t})}+\frac{1}{1-t}\|k^{t}-k^{1}\|^{2}_{L^{2}(\xi^{v}_{t}\otimes\xi^{v}_{t})}-\frac{1}{t(1-t)}\|(1-t)(k^{0}-k^{t})-t(k^{t}-k^{1})\|^{2}_{L^{2}(\xi^{v}_{t}\otimes\xi^{v}_{t})}$
(29)

and

$\displaystyle C$
$\displaystyle=\frac{1}{t}\|\iota^{0}-\iota^{t}\|^{2}_{L^{2}(\xi^{e}_{t})}+\frac{1}{1-t}\|\iota^{t}-\iota^{1}\|^{2}_{L^{2}(\xi^{e}_{t})}-\frac{1}{t(1-t)}\|(1-t)(\iota^{0}-\iota^{t})-t(\iota^{t}-\iota^{1})\|^{2}_{L^{2}(\xi^{e}_{t})}.$
(30)

Recalling that $t=k2^{-n}$ for some $k$, the first term in (28) satisfies

$\displaystyle\frac{1}{t}\left\|\omega^{0}-\omega^{t}\right\|_{L^{2}(\xi_{t}^{v}\otimes\xi_{t}^{e})}^{2}=2^{n}\cdot\frac{1}{k}\left\|\omega^{0}-\omega^{k2^{-n}}\right\|_{L^{2}(\xi_{t}^{v}\otimes\xi_{t}^{e})}^{2}$
$\displaystyle=2^{n}\cdot\frac{1}{k}\left\|\sum_{\ell=1}^{k}(\omega^{(\ell-1)2^{-n}}-\omega^{\ell
2^{-n}})\right\|_{L^{2}(\xi_{t}^{v}\otimes\xi_{t}^{e})}^{2}$
$\displaystyle\leq
2^{n}\cdot\frac{1}{k}\left(\sum_{\ell=1}^{k}\left\|\omega^{(\ell-1)2^{-n}}-\omega^{\ell
2^{-n}}\right\|_{L^{2}(\xi_{t}^{v}\otimes\xi_{t}^{e})}\right)^{2}$ (31)
$\displaystyle\leq
2^{n}\sum_{\ell=1}^{k}\left\|\omega^{(\ell-1)2^{-n}}-\omega^{\ell
2^{-n}}\right\|_{L^{2}(\xi_{t}^{v}\otimes\xi_{t}^{e})}^{2},$ (32)

where (31) follows by the triangle inequality for the $L^{2}$-norm and (32) is
Jensen’s inequality. Applying the same argument to the second term of (28), as
well as to (29) and (30), it follows that

$\displaystyle A$ $\displaystyle\leq
2^{n}\sum_{j=1}^{2^{n}}\|\omega^{(j-1)2^{-n}}-\omega^{j2^{-n}}\|_{L^{2}(\pi^{v}_{j}\otimes\pi^{e}_{j})}-\frac{1}{t(1-t)}\|(1-t)(\omega^{0}-\omega^{t})-t(\omega^{t}-\omega^{1})\|^{2}_{L^{2}(\xi^{v}_{t}\otimes\xi^{e}_{t})},$
(33) $\displaystyle B$ $\displaystyle\leq
2^{n}\sum_{j=1}^{2^{n}}\|k^{(j-1)2^{-n}}-k^{j2^{-n}}\|_{L^{2}(\pi^{v}_{j}\otimes
pi^{v}_{j})}-\frac{1}{t(1-t)}\|(1-t)(k^{0}-k^{t})-t(k^{t}-k^{1})\|^{2}_{L^{2}(\xi^{v}_{t}\otimes\xi^{v}_{t})},$
(34) $\displaystyle C$ $\displaystyle\leq
2^{n}\sum_{j=1}^{2^{n}}\|\iota^{(j-1)2^{-n}}-\iota^{j2^{-n}}\|_{L^{2}(\pi^{e}_{j})}-\frac{1}{t(1-t)}\|(1-t)(\iota^{0}-\iota^{t})-t(\iota^{t}-\iota^{1})\|^{2}_{L^{2}(\xi^{e}_{t})}.$
(35)

From (33),(34), (35), we deduce that

$\displaystyle d_{\rm TpOT}(Q^{0},Q^{1})^{2}$ $\displaystyle\leq A+B+C$
$\displaystyle\leq\left(\sum_{j=1}^{2^{n}}\left(\|\omega^{(j-1)2^{-n}}-\omega^{j2^{-n}}\|_{L^{2}(\pi^{v}_{j}\otimes
pi^{e}_{j})}+\|k^{(j-1)2^{-n}}-k^{j2^{-n}}\|_{L^{2}(\pi^{v}_{j}\otimes\pi^{v}_{j})}+\|\iota^{(j-1)2^{-n}}-\iota^{j2^{-n}}\|_{L^{2}(\pi^{e}_{j})}\right)\right)$
$\displaystyle\qquad-\frac{1}{t(1-t)}\left(\|(1-t)(\omega^{0}-\omega^{t})-t(\omega^{t}-\omega^{1})\|^{2}_{L^{2}(\xi^{v}_{t}\otimes\xi^{e}_{t})}+\|(1-t)(\omega^{0}-\omega^{t})-t(\omega^{t}-\omega^{1})\|^{2}_{L^{2}(\xi^{v}_{t}\otimes\xi^{e}_{t})}\right.$
$\displaystyle\left.\hskip
216.81pt+\|(1-t)(\iota^{0}-\iota^{t})-t(\iota^{t}-\iota^{1})\|^{2}_{L^{2}(\xi^{e}_{t})}\right)$
(36) $\displaystyle=d_{\rm TpOT}(Q^{0},Q^{1})^{2}$
$\displaystyle\qquad-\frac{1}{t(1-t)}\left(\|(1-t)(\omega^{0}-\omega^{t})-t(\omega^{t}-\omega^{1})\|^{2}_{L^{2}(\xi^{v}_{t}\otimes\xi^{e}_{t})}+\|(1-t)(\omega^{0}-\omega^{t})-t(\omega^{t}-\omega^{1})\|^{2}_{L^{2}(\xi^{v}_{t}\otimes\xi^{e}_{t})}\right.$
$\displaystyle\left.\hskip
216.81pt+\|(1-t)(\iota^{0}-\iota^{t})-t(\iota^{t}-\iota^{1})\|^{2}_{L^{2}(\xi^{e}_{t})}\right),$
(37)

where (37) follows by the assumption that $[Q^{t}]$ is a geodesic and by the
optimality of $\pi^{e}_{t},\pi^{v}_{t}$. This inequality implies that
$\pi_{v},\pi_{e}$ are optimal for $Q^{0},Q^{1}$, and that

$\displaystyle\|(1-t)(\omega^{0}-\omega^{t})-t(\omega^{t}-\omega^{1})\|^{2}_{L^{2}(\xi^{v}_{t}\otimes\xi^{e}_{t})}$
$\displaystyle+\|(1-t)(\omega^{0}-\omega^{t})-t(\omega^{t}-\omega^{1})\|^{2}_{L^{2}(\xi^{v}_{t}\otimes\xi^{e}_{t})}$
$\displaystyle\hskip
72.26999pt+\|(1-t)(\iota^{0}-\iota^{t})-t(\iota^{t}-\iota^{1})\|^{2}_{L^{2}(\xi^{e}_{t})}=0$

for every dyadic number $t$. This implies that $d_{\rm TpOT}(Q^{t},P_{t})=0$
for every dyadic number, with $P_{t}$ as in Theorem 2. Continuity of the
function $t\mapsto Q^{t}$ and density of dyadic numbers in $[0,1]$ complete
the proof. ∎

From the characterisation of geodesics provided above, we easily deduce
curvature bounds for $\faktor{\mathcal{P}}{\sim_{w}}$.

######  Theorem 4.

The metric space
$\left(\faktor{\mathcal{P}}{\sim_{w}},d_{\mathrm{TpOT}}\right)$ has curvature
bounded below by zero.

###### Proof.

Let $[P^{t}]$ be a geodesic in $\faktor{\mathcal{P}}{\sim_{w}}$ between
$P^{0}$ and $P^{1}$. By Theorem 3 we can assume that

$P_{t}=\left((\tilde{X},k_{t},\pi^{v}),(\tilde{Y},\iota_{t},\pi^{e}),\omega_{t}\right),$

as in (25); in particular, $\pi^{v},\pi^{e}$ are optimal for $P^{0},P^{1}$.
Let
$P^{\prime}=\left((X^{\prime},k^{\prime},\pi^{\prime}),(Y^{\prime},\iota^{\prime},\nu^{\prime}),\omega^{\prime}\right)$
be an arbitrary topological network, and let $\xi^{v},\xi^{e}$ be optimal for
$P^{t}$ and $P^{\prime}$. Then we have

$\displaystyle d_{\rm TpOT,2}(P^{t},P^{\prime})^{2}+t(1-t)d_{\rm
TpOT,2}(P^{0},P^{1})^{2}$
$\displaystyle\qquad\qquad=\|\omega_{t}-\omega^{\prime}\|^{2}_{L^{2}(\xi^{v}\otimes\xi^{e})}+\|k_{t}-k^{\prime}\|^{2}_{L^{2}(\xi^{v}\otimes\xi^{v})}+\|d_{2}\circ(\iota_{t},\iota^{\prime})\|^{2}_{L^{2}(\xi^{e})}$
$\displaystyle\hskip
144.54pt+t(1-t)\left(\|\omega_{0}-\omega_{1}\|^{2}_{L^{2}(\pi^{v}\otimes\pi^{e})}+\|k_{0}-k_{1}\|^{2}_{L^{2}(\pi^{v}\otimes\pi^{v})}+\|d_{2}\circ(\iota_{0},\iota_{1})\|^{2}_{L^{2}(\pi^{e})}\right)$
$\displaystyle\qquad\qquad=\|\omega_{t}-\omega^{\prime}\|^{2}_{L^{2}(\xi^{v}\otimes\xi^{e})}+\|k_{t}-k^{\prime}\|^{2}_{L^{2}(\xi^{v}\otimes\xi^{v})}+\|d_{2}\circ(\iota_{t},\iota^{\prime})\|^{2}_{L^{2}(\xi^{e})}$
$\displaystyle\hskip
144.54pt+t(1-t)\left(\|\omega_{0}-\omega_{1}\|^{2}_{L^{2}(\xi^{v}\otimes\xi^{e})}+\|k_{0}-k_{1}\|^{2}_{L^{2}(\xi^{v}\otimes\xi^{v})}+\|d_{2}\circ(\iota_{0},\iota_{1})\|^{2}_{L^{2}(\xi^{e})}\right)$
(38)
$\displaystyle\qquad\qquad=(1-t)\left(\|\omega_{0}-\omega^{\prime}\|^{2}_{L^{2}(\xi^{v}\otimes\xi^{e})}+\|k_{0}-k^{\prime}\|^{2}_{L^{2}(\xi^{v}\otimes\xi^{v})}+\|d_{2}\circ(\iota_{0},\iota^{\prime})\|^{2}_{L^{2}(\xi^{e})}\right)$
$\displaystyle\hskip
144.54pt+t\left(\|\omega_{1}-\omega^{\prime}\|^{2}_{L^{2}(\xi^{v}\otimes\xi^{e})}+\|k_{1}-k^{\prime}\|^{2}_{L^{2}(\xi^{v}\otimes\xi^{v})}+\|d_{2}\circ(\iota_{1},\iota^{\prime})\|^{2}_{L^{2}(\xi^{e})}\right)$
(39) $\displaystyle\qquad\qquad\geq(1-t)d_{\rm
TpOT,2}(P^{0},P^{\prime})^{2}+td_{\rm TpOT,2}(P^{1},P^{\prime})^{2},$ (40)

where 38 follows by marginal properties of $\xi^{v}$ and $\xi^{e}$, 39 by
definition of $w_{t},k_{t}$ and $\nu_{t}$, and 40 by suboptimality. ∎

##  5 Examples

In this section, we demonstrate our computational framework and the
capabilities of TpOT on a range of numerical examples. In what follows,
persistent homology computations are performed using the Julia software
Ripserer.jl [14]. Specifically, we compute the Vietoris-Rips filtration and we
use the involutive algorithm [14, 15] to compute homology and representatives.
We emphasise that the choice of filtration and representatives influences the
resulting transport plan and analysis.

Figure 3: (A) Source point cloud: four circles in $\mathbb{R}^{2}$. (B) Target
point cloud, with points coloured according to the Gromov-Wasserstein
coupling. (C) Target point cloud, coloured according to the TpOT coupling with
$\alpha=0.5$ and $\beta=1$. (D) As $\alpha$ varies, TpOT interpolates between
a coupling driven by geometry (Gromov-Wasserstein, $\alpha\uparrow 1$) and
topological features (persistence diagram matchings, $\alpha\downarrow 0$).
(E) Geodesic between the point clouds for $\beta=1$.

###  5.1 Matching point clouds

In Figure 3, we consider a source point cloud $X$ consisting of four circles
of uniform size (Figure 3(A)) and a target point cloud $X^{\prime}$ resembling
a “flower” with four ellipses as “petals” (Figure 3(B)). The point clouds $X$
and $X^{\prime}$, together with the uniform distribution on points, are then
measure spaces. We choose to endow these measure spaces with Gaussian kernels
$k,k^{\prime}$ so that $k_{ij}=K(x_{i},x_{j})$ and
$k^{\prime}_{ij}=K(x_{i}^{\prime},x_{j}^{\prime})$ where

$K(x,y)=\exp\left(-\frac{\|x-y\|_{2}^{2}}{h}\right),$

where the bandwidth $h$ is chosen for each set of points
$\\{x_{1},\ldots,x_{N}\\}$ such that
$h^{-1}\left(N^{-2}\sum_{ij}\|x_{i}-x_{j}\|_{2}^{2}\right)=1$.

In Figure 3(B), points of the target point cloud are coloured by their
corresponding source points under the Gromov-Wasserstein matching. Clearly,
topological features are not preserved by this matching. Instead, each of the
circles in the source point cloud is split up among multiple ellipses in the
target point cloud.

The $1$-dimensional persistence diagram of the Vietoris-Rips filtration
$\mathcal{K}_{\rm VR}(X)$ contains 4 significant, almost identical, homology
classes (one per circle), and the same is true for $\mathcal{K}_{\rm
VR}(X^{\prime})$ (one class per ellipse). A choice of generating cycle for
each class in the diagrams yields the PH-hypergraphs
$H_{X}=H(X,\mathcal{K}_{\rm VR}(X),g_{X})$ and
$H_{X^{\prime}}=H(X^{\prime},\mathcal{K}_{\rm VR}(X^{\prime}),g_{X^{\prime}})$
and the corresponding topological networks (see Section 3.3)

$\displaystyle P_{X}$
$\displaystyle=\left((X,d_{\mathbb{R}^{n}},\mu_{X}),(Y,\iota,\nu),\omega_{H}\right)$
$\displaystyle P_{X^{\prime}}$
$\displaystyle=\left((X^{\prime},d_{\mathbb{R}^{n}},\mu_{X^{\prime}}),(Y^{\prime},\iota^{\prime},\nu^{\prime}),\omega_{H^{\prime}}\right).$

Using the algorithms we developed in Section 3.3, we are able to
simultaneously find matchings between points and topological features in $X$
and $X^{\prime}$ as the parameters $(\alpha,\beta)$ vary.

Figure 4: (A) Source point cloud and target point cloud have almost identical
persistence diagrams. (B) Target point cloud, matching induced by GW distance.
The colour code represents the matching. (C) Target point cloud, matching
induced by TpOT with $\alpha=0.2,\beta=5$, and (D) matching induced by TpOT
with $\alpha=0.8,\beta=5$.

Intuitively, the parameter $\alpha$ controls the relative contributions of the
geometric distortion (measured in terms of a Gromov-Wasserstein loss, with
coefficient $\alpha$) and topological distortion (measured in terms of
transport cost on persistence features, with coefficient $1-\alpha$). When
$\beta=0$, point clouds and persistence diagrams are matched independently.
For $\beta>0$, the hypergraph distortion $(\pi^{v},\pi^{e})\mapsto\left\langle
L(\omega_{H},\omega_{H^{\prime}}),\pi^{v}\otimes\pi^{e}\right\rangle$ couples
the matching of the point clouds and persistence diagrams, allowing geometric
information to inform the topological matching and vice versa. In the limit as
$\beta\to\infty$, this term dominates the overall objective in (16) and in the
limit we recover a partial matching variant of HyperCOT [11] in which
hyperedges can be transported to a null edge (corresponding to the diagonal in
the persistence diagram setting) for a cost equal to the squared $L^{2}$-norm
of the corresponding incidence function.

Theorem 2 allows us to explicitly construct a geodesic $[P_{t}],t\in[0,1]$
between $P_{X}$ and $P_{X^{\prime}}$.

At each time $t\in[0,1]$, the corresponding gauged measure space $P_{t}$ can
be represented as a point cloud with a gauge function $\omega_{t}$ given by
interpolation of $\omega$ and $\omega^{\prime}$. In order to visualise the
family of interpolating points, we consider the function
$C_{t}((x,x^{\prime}),(y,y^{\prime}))\mapsto(1-t)C(x,y)+tC^{\prime}(x^{\prime},y^{\prime})$
where $C(x,y)$ and $C^{\prime}(x^{\prime},y^{\prime})$ are the squared
Euclidean distances on the source and target point clouds respectively; this
amounts to linear interpolation of $C$ and $C^{\prime}$ in
$L^{2}_{\pi^{v}\otimes\pi^{v}}((X\times X^{\prime})^{2})$. At each value of
$t$, positions of points in the interpolating point cloud are obtained by
applying multidimensional scaling (MDS) algorithm to $C_{t}$ followed by an
alignment step to remove issues due to the invariance of MDS under rigid
transformation. Figure 3(E) shows a snapshot of this geodesic computed
numerically for $\alpha=0.5,\beta=1$. We note that the geodesic almost
perfectly recovers a homeomorphisms connecting each loop with their matched
petal.

Figure 5: (A) Source point cloud: a noisy mug in $\mathbb{R}^{3}$. (B) Target
point cloud, a noisy solid torus in $\mathbb{R}^{3}$. Matching induced by GW
distance. The colour code represents the matching. (C) Target point cloud,
matching induced by TpOT with $\alpha=0.1$ and $\alpha=1$. The colour code
represents the matching. (D) Persistence diagrams for source (blue) and target
(yellow) point clouds. Both diagrams contain a unique significant homology
class.

By construction of $X$, representative cycles for each of the four persistent
homology classes span the circles almost entirely, and similarly for
$X^{\prime}$. This makes the example in Figure 3 a relatively simple one, as
most of the points in $X$ and $X^{\prime}$ are part of “topologically
significant” cycles. We next investigate what happens for noisy point clouds
which contain topological features with short lifetimes, or “topological
noise”. Figure 4 shows an example of transport between point clouds containing
noisy disks, and in this case the significant features in each of the
persistent diagrams all have different lifespans. For small values of $\alpha$
(e.g. $\alpha=0.2$, Figure 4(C)) the influence of the topological distortion
in the PD-space is greater. As a result, TpOT matches points in cycles with
similar lifespan. On the other hand, for higher values (e.g. $\alpha=0.8$,
Figure 4(D)) the geometric distortion has greater influence, and we observe
that the matching tends to preserve local proximity between loops, rather than
their relative sizes.

Figure 6: (A) A point cloud $X$ consisting of 3 noisy loops of various sizes.
(B) A point cloud $X$ consisting of the same 3 noisy loops as $X$, placed in
different ways. Points forming the three persistent cycles in
$\mathcal{K}_{\rm VR}(X)$ and $\mathcal{K}_{\rm VR}(X)$ are coloured in blue,
green, and yellow. (C) Persistence diagrams of $\mathcal{K}_{\rm VR}(X)$ and
$\mathcal{K}_{\rm VR}(X)$ plotted super-imposed to each other and showing
Wasserstein (left) and Geometric (right) matchings. The three highly
persistent classes are coloured as the corresponding cycles in (A) and (B).
Geometric and Wasserstein matchings are shown with arrows in (A) and (B) as
well. (D) A point cloud $X_{1}$ consisting of a chain of 9 noisy circles. (E)
Persistence diagrams of $X_{1}$ super-imposed to that of $X_{2}$ (point cloud
in (F-G)). Colour indicates points forming the 9 second-most persistent cycles
in each point cloud. (F) The result of cycle matching induced by Wasserstein
distance. (G) The result of geometric cycle matching. Colour indicates matched
cycles.

A more challenging example is shown in Figure 5. Here the objective is to
transport a noisy point cloud $M$ resembling a mug (see Figure 5(A)), into a
noisy solid torus $T$ (see Figure 5(B)) whilst preserving the topological
features. In this context, matching of persistent homology features amounts to
mapping the “handle” in $M$ into an essential (i.e. not bounding a disk)
closed curve in $T$. As shown in Figure 5(D), the persistence diagrams of $T$
and $M$ have only one significant class each. That implies that outside of the
points involved in the two chosen representative cycles, points in $M$ and $T$
have almost trivial topological signal. For this example, we set $\beta=5$.
For low values of $\alpha$, the $\pi^{v}$ matches points successfully maps the
handle in an essential closed curve spanning the doughnut hole (Figure 5(C),
left), while for $\alpha=1$ (Figure 5(C), right) this effect is lost. This can
be explained by looking at the matching between cycles induced by $\pi^{e}$,
Figure 5(D): for low values of $\alpha$, the Wasserstein component promotes
matching the highly persistent classes together, while for $\alpha=1$ these
are split into less significant classes.

Figure 7: (A) Snapshots of a trefoil undergoing thermal relaxation. (B) Top
rows shows the trefoil at step 4 of the simulation, while mid and bottom rows
show the trefoil at step 5. Each column correspond to one of the most
persistent homology classes of the trefoil at step 4. Mid and bottom rows
shows the matched class at step 5, according to Wasserstein distance (mid) and
TpOT (bottom). (C) Box plot of correlation values between cycles and their
images.

###  5.2 Geometric cycle matching

The past few years have seen increasing efforts in addressing the problem of
matching topological features across different systems, and many solutions
have been proposed, including some inspired by techniques in optimal transport
[30, 27, 23]. Perhaps, the simplest approach is to straightforwardly use the
matching induced by Wasserstein-like distances on persistence diagrams, such
as the bottleneck distance[12] and the $d_{\mathrm{W},p}^{\mathrm{PD}}$
Wasserstein distances [6]. More nuanced solutions rely on statistical
considerations [13, 38, 21], on the existence of maps between the initial data
[3, 25], or, when such a map is unknown, on leveraging the algebraic topology
of the PH construction to define a notion of dissimilarity between the
underlying complexes [55].

In our case, the topological feature coupling $\pi^{e}$ yields a matching
between homology classes which is informed by proximity of points creating the
corresponding geometric cycles. Figure 6 shows comparisons between the
Wasserstein matching on persistence diagrams (PD Wasserstein) and joint
geometric-topological matching (TpOT). Similarly to Figure 4, the first
examples concerns mapping classes between point clouds $X$ (Figure 6(A)) and
$X^{\prime}$ (Figure 6(B)) having almost identical persistence diagrams
(Figure 6(C)). In this case, joint matching of geometric and topological
features by TpOT allows us to preserve spatial proximity between
representative cycles, see Figure 6(C). This can be particularly important
when homology classes in a persistence diagram are indistinguishable, as shown
in the example in Figure 6(D-G). Here the TpOT matching (Figure 6(G))
correctly maintains spatial proximity between representative cycles, in
contrast with PD Wasserstein matching (Figure 6(F)) which is essentially
random.

Finally, we consider the problem of tracking topological features as a system
evolves over time. Figure 7(A) shows a piecewise linear spatial curve forming
a trefoil knot as it undergoes thermal relaxation. The simulation consists of
$20$ steps and it was generated using KnotPlot [41]. At each time $t_{i}$, the
knot takes a spatial configuration $C_{i}$, that can be represented as the
point cloud $X_{i}$ of its vertices. At each step, we interpret $X_{i}$ as the
topological network obtained by computing 1-dimensional PH and PH-hypergraph.
We can then match homology classes between consecutive persistence diagrams
$D_{i},D_{i+1}$ using PD Wasserstein distance and $d_{\rm TpOT}$,
respectively, see Figure 7(B). The TpOT matching significantly improves the
accuracy of generator matchings compared to PD Wasserstein, as shown in Figure
7(B). Since the ground truth correspondence of points in the curve is known,
in Figure 7(C) we show the correlation between generators and their images
under the PD Wasserstein as well as TpOT matchings. We find that the TpOT
matching provides a clear improvement over the PD Wasserstein matchings, which
appears to be close to random.

######  Remark 5.

Consider two point clouds, and the convex $GW$ geodesic connecting them (see
[43]). Consider now the Vietoris-Rips filtration along the geodesic path. A
natural question is whether the path induced by the persistent diagrams
follows a convex geodesic. A counterexample to this question is given by
Figure 3, as the GW-mapping between points suggests that the GW-geodesic
“breaks” the loops, and thus, the four persistent homology classes. This can
be verified by computing the path of Vietoris-Rips PDs obtained by inputting
the various interpolation metrics. The same question applies for
$TpOT$-geodesics. In this context, the answer depends on whether the $GW$-term
contributes to cycle matching or not. When it doesn’t (e.g. Figure 3(E)),
persistent diagrams computed from Vietoris-Rips filtration along the TpOT-
geodesic do indeed follow the convex-geodesic in the persistent diagram space.
When GW contributes (e.g. Figure 4 and Figure 6), then the persistent diagram
path is non geodesic.

##  6 Conclusion

Inspired by recent advances in computational topology [2] and optimal
transport for structured objects [48, 39, 11], we propose to encode geometric
and topological information about a point cloud jointly in a hypergraph
structure. Our approach lays the foundation for a topology-driven analysis of
the geometry of gauged measure spaces. As in the co-optimal setting [39, 11],
the TpOT problem outputs a pair of coupled transport plans. The first one
provides a matching of the underlying metric spaces (point clouds) that
preservers topological features. The second one matches the topological
features (cycles representing persistent homology classes) in a way that
maintains local metric structure. This geometric cycle matching represents a
key novelty of our method, and, to the best of our knowledge, it is the first
one taking into account the geometric nature of data, and the spatial
interconnectivity of generating cycles.

Constructing PH-hypergraphs, and thus the TpOT pipeline, depends on computing
explicit cycles representing homology classes. It is well known that, even in
the case of simplicial complexes, generating cycles are far from being unique,
and that different choices can potentially bring important biases in the
subsequent analysis [29]. Given their potential and demonstrate usefulness,
the search for new methods of producing “good” representatives, and the
research into homology generators in general [5, 16, 19, 29, 34, 49, 2], is a
very active sub-field of topological data analysis. New, important
advancements in this area currently provide a wide variety of algorithms to
compute representatives, that can all be implemented in the TpOT pipeline,
depending on the specific goal or application. In addition to that, the
experimentally shown robustness of the hyperTDA method [2] suggests that, when
dealing with systems sufficiently complex from a topological point of view,
the qualitative behaviour of a TpOT induced matching should be robust by
changes to the representative cycles.

### Data availability

Data and software used to produce results in this paper are available at the
GitHub repository https://github.com/zsteve/TPOT

### Acknowledgments

SYZ gratefully acknowledge funding from the Australian Government Research
Training Program and an Elizabeth and Vernon Puzey Scholarship. MPHS is funded
through the University of Melbourne DRM initiative, through an ARC Laureate
Fellowship (FL220100005) and acknowledges financial support from the
Volkswagen Foundation through a “Life?” program grant. TN was partially
supported by NSF grants DMS 2107808 and DMS 2324962.

### Declarations

The authors declare no conflict of interest or competing interests.

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