# Spectral Persistent Homology: Persistence Signals

Michael E. Van Huffel
Department of Mathematics
ETH Zurich
Zurich, Switzerland
<EMAIL_ADDRESS>
&Matteo Palo
Department of Mathematics
ETH Zurich
Zurich, Switzerland
<EMAIL_ADDRESS>

###### Abstract

In this paper, we present a novel family of descriptors for persistence
diagrams, reconceptualizing them as signals in $\mathbb{R}^{2}_{+}$. This
marks a significant advancement in Topological Data Analysis. Our methodology
transforms persistence diagrams into a finite-dimensional vector space through
functionals of the discrete measures induced by these diagrams. While our
focus is primarily on frequency-based transformations, we do not restrict our
approach exclusively to this types of techniques. We term this family of
transformations as Persistence Signals and prove stability for some members of
this family against the 1-Kantorovitch-Rubinstein metric, ensuring its
responsiveness to subtle data variations. Extensive comparative analysis
reveals that our descriptor performs competitively with the current state-of-
art from the topological data analysis literature, and often surpasses, the
existing methods. This research not only introduces a groundbreaking
perspective for data scientists but also establishes a foundation for future
innovations in applying persistence diagrams in data analysis and machine
learning.

_Keywords_ Topological Data Analysis $\cdot$ Persistent Homology $\cdot$
Spectral Persistence $\cdot$ Frequency Transformations

##  1 Introduction

Topological Data Analysis (TDA) has emerged as a transformative approach in
data science, providing unique insights into the underlying structure of
complex datasets through the capture of their topological features. The
effectiveness of machine learning algorithms, particularly in pattern
recognition and feature extraction, underscores the importance of
understanding data geometry. TDA offers a more sophisticated exploration of
this geometric landscape, leading to numerous successful applications across
various fields. Notable examples include neuroscience (Anderson et al. (2018),
Dabaghian et al. (2012)), where TDA has also been instrumental in
understanding complex neural networks and brain connectivity patterns. In the
field of environmental science, TDA has been applied for climate data
analysis, as seen in Nunez et al. (2019). Additionally, TDA has made
significant contributions to materials science, particularly in the study of
the structure of materials, as exemplified by Donato et al. (2016). These
examples underscore the broad applicability and potential of TDA in enhancing
our understanding of complex data across diverse scientific domains.

Persistent homology, a core methodology within TDA, systematically keeps track
of the appearing and disappearing of topological characteristics across a
sequence of nested topological spaces, a process documented in seminal works
by Edelsbrunner and Harer (2008), and Zomorodian and Carlsson (2005). These
topological features are typically represented through persistence diagrams
(PDs). However, the space of these diagrams is notably unstructured: they vary
in the number of points they contain, and fundamental operations like addition
and scalar multiplication are not clearly defined. This lack of structure,
including the absence of a Hilbert space, as proved in Bubenik and Wagner
(2020) and Mileyko et al. (2011), poses significant challenges in integrating
persistence diagrams into machine learning workflows, where such a space is
often crucial for diverse tools and techniques including classification,
neural networks, and feature selection.

Our contribution in this study is multi-faceted and significant in the field
of TDA. We introduce a novel family of descriptors, termed Persistence Signals
(PS), which represent a paradigm shift in how persistence diagrams are
interpreted and utilized. By reconceptualizing persistence diagrams as signals
within the $\mathbb{R}^{2}$ space, we enable their transformation into a
finite-dimensional vector space. This is achieved through the application of
functionals to the discrete measures induced by these diagrams. Our
methodology is not limited to frequency-based techniques, allowing for a
broader scope of analysis and application. A key aspect of our contribution is
the proof of stability of some member of the PS family against the
1-Kantorovitch-Rubinstein metric. This crucial property underscores the
robustness of our approach, ensuring that the PS descriptors are sensitive and
responsive to subtle variations in data.

Our approach not only provides a novel method for vectorizing persistence
diagrams but also bridges two crucial and burgeoning realms: TDA and signal
processing theory. This fusion opens the doors to an entirely new perspective
in data analysis, offering inspiring ways to unravel complex data patterns and
extract meaningful insights. Our methodology stands as a testament to the
synergy between these two fields, enabling a more intuitive and accessible
approach to understanding and manipulating TDA structures. As such, we see
this work as a pioneering step into a new domain of study that we call
Spectral Persistence Homology. To the best of our knowledge, this is the first
instance where tools from classical signal processing theory have been
employed to achieve a nearly lossless information vectorization method. Our
method requires minimal tuning and is designed to be user-friendly, making it
accessible even to those with limited knowledge of TDA. This groundbreaking
approach not only simplifies the application of TDA but also expands its
potential to a wider audience, promising significant advancements in various
fields that rely on complex data analysis.

The remainder of this paper is organized as follows: in Section 2, we review
related work that connects TDA and ML, focusing particularly on classical
methods of embedding persistence diagrams using vectorization or kernel
methods. Section 3 delves into the background of persistent homology and
persistent measures, offering foundational knowledge crucial for grasping the
ideas from this work. In Section 4, we present our methodology along with the
stability theorems that underpin it, providing theoretical insights into our
approach. Section 5 is dedicated to showcasing our experimental results. Here,
we not only present our findings but also compare them with standard methods
from the TDA literature, highlighting the advancements our method brings.
Finally, Section 6 is devoted to summarizing our key findings and outlining
ideas for further work, suggesting potential avenues for future research and
development based upon the groundwork established in this paper. The
implementation of Persistence Signals has been made available to the public as
an open-source resource111https://github.com/majkevh/spectral-master.

##  2 Related Work

In the realm of TDA, efforts to reconcile statistics and machine learning have
taken shape through two principal strategies. Initially, direct engagement
with persistence diagrams involved exploring concepts such as the Fréchet mean
Mileyko et al. (2011), confidence sets Perea et al. (2019), and metric
learning methods employing distances like Wasserstein or bottleneck distances
Fasy et al. (2014). However, challenges arose due to the complex geometry
inherent in persistence diagrams Wagner (2019).

To overcome these challenges, an alternative strategy seeks to map the space
of persistence diagrams to more manageable alternatives where established
machine learning algorithms can be applied. Two notable families have emerged:

Vectorization Methods. Among these methods, Persistence Images Adams et al.
(2017), which partially inspired this work, are particularly notable. This
approach first applies a Gaussian kernel function to the persistence diagram
and then discretizes it over a grid. Another method for embedding persistence
diagrams is the use of Persistence Landscapes Bubenik (2015), which represent
the diagrams as landscapes of functions across various scales. Worth
mentioning are their multi-parameter extensions, Carrier̀e and Blumberg
(2020), Vipond (2020) , that enhance robustness using multi-parameter
persistent homology.

More modern approaches include ATOL Royer et al. (2021), who utilizes a two-
step process, computing a discrete measure supported on $b$ points to
approximate the average measure of the input data distribution and generating
contrast functions to convert observed measures into $b$-dimensional vectors,
and PersLay Carriere et al. (2020), that introduces a specialized neural
network layer for vectorizing persistence diagrams.

Designing Specific Kernels. A second family of approaches involves crafting
specific kernels on the space of persistence diagrams, such as the multi-scale
kernel of Reininghaus et al. (2014), the weighted Gaussian kernel Kusano et
al. (2016) and sliced Wasserstein kernel Carrière et al. (2017), all these
methods captures topological features at different scales and often their
performances are comparable to state-of- arts, albeit facing challenges in
explicit representation and scalability.

##  3 Background

In this section we briefly provide an overview of persistent homology, along
with its various output forms, specifically focusing on persistence diagrams
and persistence measures.

###  3.1 Persistent Homology

Persistent homology, a cornerstone of TDA, offers a robust framework for
analyzing the topological structure of datasets, such as point clouds or time-
series. This methodology is adept at monitoring the evolution of homological
features across various scales. For a comprehensive overview, readers are
referred to Edelsbrunner and Morozov (2017). The development of persistent
homology has been significantly advanced by seminal contributions in the
field, including the works of Zomorodian and Carlsson (2005), and Edelsbrunner
et al. (2000). The essence of persistent homology lies in its ability to
effectively capture and track the changes in topological features across
various scales. This is achieved through a filtration process, which involves
a series of progressively inclusive topological spaces $X_{0}\subseteq
X_{1}\subseteq\cdots\subseteq X_{n}=X$, allowing for a detailed and granular
analysis of the dataset’s topological properties.

A standard approach for illustrating the insights obtained from persistent
homology is the use of a persistence diagram. In our study, we adopt the
notation for persistence diagrams as delineated in Monod (2022). The detailed
definition of persistence diagrams is provided in Definition 3.1.

###### Definition 3.1

A persistence diagram $D$ is a locally finite multiset of points in the half-
plane $\Omega=\\{(x,y)\in$ $\left.\mathbb{R}^{2}\mid x<y\right\\}$ together
with points on the diagonal
$\partial\Omega=\left\\{(x,x)\in\mathbb{R}^{2}\right\\}$ counted with infinite
multiplicity. Points in $\Omega$ are called off-diagonal points. We denote as
persistence of a feature, the value $y-x$.

###### Remark 3.2

In this study, we operate under the assumption that all points on the diagram
exhibit finite persistence.

The space of persistence diagrams can be endowed with a parametrized metric,
denoted as $d_{s}$, where $1\leq s\leq\infty$. This metric is known as the
Kantorovich-Rubinstein distance, as detailed in Definition 3.3. For the sake
of simplicity, we will henceforth refer to this metric as the Wasserstein
distance.

###### Definition 3.3

Let $\|\cdot\|_{q}$ denote the $q$-norm on $\mathbb{R}^{2}$ for $1\leq
q\leq\infty$. Let $D_{1},D_{2}$ be any two persistence diagrams. For $1\leq
p<\infty$, the $p$-Wasserstein distance is

$\mathrm{W}_{p,q}\left(D_{1},D_{2}\right)=\inf_{\gamma}\left(\sum_{x\in
D_{1}}\|x-\gamma(x)\|_{q}^{p}\right)^{\frac{1}{p}}\,,$

where $\gamma$ ranges over all bijections between $D_{1}$ and $D_{2}$.

###### Remark 3.4

If $p=q$ we denote $W_{p,p}$ as $W_{p}$ for simplicity of notation.

###### Remark 3.5

For $p=\infty$, $W_{\infty,q}$ becomes the so called bottleneck distance.

The collection of all persistence diagrams that possess finite $p$-total
persistence is represented by $\mathcal{D}_{p}$. As mentioned in Monod (2022),
the metric space $(\mathcal{D}_{p},\mathrm{W}_{p,q})$ occupies a pivotal role
in TDA. Particularly noteworthy is the case when $q=\infty$. In this scenario,
the metric space $(\mathcal{D}_{p},\mathrm{W}_{p,\infty})$ constitutes a
complete and separable metric space—essentially, a Polish space—for any $1\leq
p<\infty$. This characteristic endows the space with significant analytical
power, as it ensures that statistical and probabilistic concepts such as
probability measures, expectations, and variances are well-defined within this
context, as elucidated by Mileyko et al. (2011).

###  3.2 Persistence Measures

In Chazal et al. (2013), an alternative perspective is presented, suggesting
the definition of a persistence diagram as a measure expressed by
$\mu=\sum_{\mathbf{x}\in D}m(\mathbf{x})\delta_{\mathbf{x}}$, where
$D\subset\Omega$ is locally finite, and $m(\mathbf{x})\in\mathbb{N}$ is the
multiplicity of each $\mathbf{x}$, for all $\mathbf{x}\in D$. This results in
$\mu$ being a locally finite measure supported on $\Omega$ with an integer
mass on each point of its support. This perspective encourages the
consideration of more general Radon measures supported on the upper half-plane
$\Omega$, Chazal et al. (2013). By incorporating such measures, a statistical
viewpoint can be introduced into the realm of persistent homology.

Expanding upon the aforementioned measure-centric perspective, in Divol and
Lacombe (2019), the authors propose metrics applicable to the space
$\mathcal{M}$ of Radon measures with support on $\Omega$. Specifically, for
finite values of $p\geq 1$, the authors articulate the definition of
persistence for a measure $\mu\in\mathcal{M}$ as
$\operatorname{Pers}_{p}(\mu):=\int_{\Omega}d(x,\partial\Omega)^{p}\mathrm{~{}d}\mu(x)$.
Here, the term $d(x,\partial\Omega):=\inf_{y\in\partial\Omega}d(x,y)$
signifies the distance from a point $x\in\Omega$ to its orthogonal projection
onto the diagonal $\partial\Omega$. Furthermore, the authors establish the set
$\mathcal{M}^{p}$ as the collection encompassing all persistence measures with
finite $p$.

In the realm of persistence measures, the concept of the optimal partial
transport distance plays a pivotal role. This distance has been formally
introduced by Figalli and Gigli (2010) and is detailed in Definition 3.6. In
our study, we use the same notation as in Monod (2022).

###### Definition 3.6

Given two persistence measures $\mu$ and $\nu$, their optimal partial
transport distance is defined as

$\operatorname{OT}_{p,q}(\mu,\nu)=\left(\inf_{\pi}\int_{\bar{\Omega}\times\bar{\Omega}}\|x-y\|_{q}^{p}\,\mathrm{d}\Pi(x,y)\right)^{1/p}\,,$

where $\bar{\Omega}=\Omega\cup\partial\Omega$, and $\Pi$ is a Radon measure on
$\bar{\Omega}\times\bar{\Omega}$ that satisfies
$\Pi(P\times\bar{\Omega})=\mu(P)$ and $\Pi(\bar{\Omega}\times Q)=\nu(Q)$ for
all Borel sets $P,Q\subseteq\Omega$.

###### Remark 3.7

For the case where $p=q$, we simplify the notation to $\mathrm{OT}_{p}$.

Building on this foundational concept, we can then consider the metric space
defined as $\left(\mathcal{M}^{p},\mathrm{OT}_{p}\right)$. Here,
$\mathcal{M}^{p}$ is equipped with the optimal partial transport distance
$\mathrm{OT}_{p}$, serving as its metric. As established in Divol and Lacombe
(2019), the subset $\mathcal{D}^{p}$ within $\mathcal{M}^{p}$ maintains its
closed subspace property and inherits the key characteristics of
$\mathcal{M}^{p}$. This includes being a Polish space, applicable for all
$1\leq p<\infty$ and $1\leq q\leq\infty$. Consequently, the space
$(\mathcal{M}_{p},\mathrm{OT}_{p,q})$ emerges as a robust framework for
statistical and probabilistic applications. Moreover, within this framework,
$\mathrm{OT}_{p,q}$ coincide with $\mathrm{W}_{p,q}$ when restricted to
$\mathcal{D}_{p}$. As also mentioned in Monod (2022), this alignment
underscores the role of persistence measures and the optimal partial transport
distance as comprehensive extensions to traditional persistence diagrams and
the $p$-Wasserstein distance.

##  4 Methodology

In this section, we introduce the Spectral Persistent Homology framework for
embedding persistence measures in vector spaces along with the stability
results.

As an overview, the process of vectorizing a persistence diagram using
Persistence Signals, depicted in Algorithm 1, involves two distinct steps:
first, we compute the discretized persistence diagram (refer to Section 4.2),
transforming the persistence diagram into a format suitable for further
processing. Following this, a functional is applied to this discretized
representation to achieve an embedding within a vector space. As detailed in
Section 4.1, this work primarily focuses on functionals that define valid
frequency and time-frequency distribution functions. Notably, these include
the Fourier Transform for frequency distribution, and the Wavelet and Gabor
Transforms for time-frequency distribution. These transforms will be
introduced in the subsequent subsection. It’s important to note, however, that
our framework is versatile and not exclusively limited to these transforms.

Algorithm 1 Persistence Signals vectorization

1:$f$: Transformation function with hyperparameters, Grid size: $\\{N,M\\}$,
PDs $\triangleright$ Section 4.1

2:Discretize PDs on grid $\triangleright$ Section 4.2

3:Compute Functional on grid $\triangleright$ Section 4.1.1

4:Embedding of PDs

###  4.1 Functionals on Persistence Measures

Given $\mu\in\mathcal{M}^{p}$, we consider a functional $\Psi$ defined by

$\Psi(\mu)(f):=\Psi_{\mu}(f)=\int_{\Omega}f(\mathbf{x},\cdot)d\mu(\mathbf{x})\,.$

This functional represents a mapping from the measure space $\mathcal{M}^{p}$
to a space of functions $\mathcal{F}(\mathcal{M}^{p})$. In our case, since the
measure $\mu$ is discrete and is a measure induced by a persistence diagram
$D$, have that

$\Psi_{\mu}(f)=\sum_{\mathbf{x}\in D}m(\mathbf{x})f(\mathbf{x},\cdot)\,.$

In this paper, we concentrate on functionals that establish valid frequency
and time-frequency distribution functions (hereinafter referred to, for
simplicity, as frequency transforms). Both families of these transforms are
mapped onto the complex hyperplane (frequency domain from now on).
Consequently, $\mathcal{F}(\mathcal{M}^{p})$ can be characterized as a vector
space. The choice of selecting functionals that map into the frequency domain
is motivated by the fact that persistent diagrams summarize topological
features across various length scales of a filtration parameter. This
parameter can be seen as a timestamp, indicating when topological features
emerge or vanish, therefore it seems natural to interpret them as signals.

In this work, we focus on three possible choices of $f$: two time-frequency
distributions, namely the Gabor Transform and the Wavelet Transform, and one
frequency transform, specifically the Fourier Transform (which can be regarded
as a special case of the Gabor Transform).

Fourier Transform. Let $f(\mathbf{x},\boldsymbol{\omega})=e^{-2\pi
j\boldsymbol{\omega}^{\top}\mathbf{x}}$. We have that
$\Psi_{\mu,{F}}(\boldsymbol{\omega}):\mathcal{M}^{p}\mapsto\mathbb{C}$
represents the Fourier Transform of the measure $\mu$. This choice is
motivated by the Fourier Transform’s ability to decompose signals into their
constituent frequencies, providing a global frequency perspective that is
essential in understanding the overall structure of persistence diagrams.

Wavelets Transform. Let
$f(\mathbf{x},a,\mathbf{b})=\frac{1}{a}\phi_{a,\mathbf{b}}\left(\mathbf{x}\right)$,
where $\phi_{a,\mathbf{b}}$ is a orthonormal wavelet, $a$ is a scalar
representing the scale factor, and $\mathbf{b}$ is a 2D vector representing
the translation. Then it holds that
$\Psi_{\mu,\mathcal{W}}(a,\mathbf{b}):\mathcal{M}^{p}\rightarrow\mathbb{C}$
represents the Wavelet Transform of the measure $\mu$. Chosen for its spatial-
frequency localization, the Wavelet Transform has the ability to adjust focus
between fine and broad details aligning well with the varying scales inherent
in topological data.

Gabor Transform. Let
$f(\mathbf{t},\mathbf{x},\lambda,\theta,\psi,\sigma,\gamma)=g_{\lambda,\theta,\psi,\sigma,\gamma}(t_{1}-x_{1},t_{2}-x_{2})$,
where $g_{\lambda,\theta,\psi,\sigma,\gamma}(x,y)$ is the Gabor kernel defined
as:

$g_{\lambda,\theta,\psi,\sigma,\gamma}(x,y)=\exp\left(-\frac{x^{\prime
2}+\gamma^{2}y^{\prime
2}}{2\sigma^{2}}\right)\cdot\exp\left(j\left(2\pi\frac{x^{\prime}}{\lambda}+\psi\right)\right)\,,$

with $x^{\prime}=x\cos\theta+y\sin\theta$ and
$y^{\prime}=-x\sin\theta+y\cos\theta$. Here, $\lambda\geq 0$ is the wavelength
of the sinusoidal factor, $\theta\in[0,2\pi]$ is the orientation of the normal
to the parallel stripes of a Gabor function, $\psi\in[0,2\pi]$ is the phase
offset, $\sigma\geq 0$ is the standard deviation of the Gaussian envelope, and
$\gamma\in[0,1]$ is the spatial aspect ratio. We have that
$\Psi_{\mu,\mathcal{G}}(\mathbf{t},\lambda,\theta,\psi,\sigma,\gamma):\mathcal{M}^{p}\rightarrow\mathbb{C}$
represents the Gabor Transform of the measure $\mu$. The Gabor Transform,
combining both time and frequency information, provides a localized frequency
analysis of the signals arising from the persistence diagrams. This transform
is utilized for its proficiency in localized time-frequency analysis,
essential for understanding the dynamic evolution of topological features in
persistence diagrams. In our setup, we take $\theta=0,\psi\in\\{0,\pi/2\\}$
and omit it from the parameters from now on.

####  4.1.1 Grid Based Approach in Computational Persistence Theory

Considering the practical computational requirements of all transforms under
consideration, our theory is formulated based on the computation of
discretized persistence diagrams (for details, refer to Section 4.2). In this
framework, measures are confined to allocating mass exclusively at points on a
grid (for a definition of the grid measure subspace $\Gamma_{p}$, refer to
Lemma 4.1). Consequently, to compute the embedding, we employ a discretized
version of the transforms discussed in the previous section. For a
comprehensive literature review on the Discrete Fourier Transform (DFT),
Discrete Wavelet Transform (DWT), and Discrete Gabor Transform (DGT), we
direct the reader to Oppenheim and Schafer (2009) and Søndergaard (2007).
Given a persistence measure $\mu\in\Gamma_{p}$, the three functionals on the
transforms depicted in Section 4.1 are now formulated as follows:

Discrete Fourier Functional. $\Psi_{\mu,F}(x_{i},x_{j})=\sum_{x_{m},x_{n}\in
G_{N,M}}m(x_{m},x_{n})e^{-2\pi j\left(\frac{x_{m}}{x_{M}}\cdot
x_{i}+\frac{x_{n}}{x_{N}}\cdot x_{j}\right)\,.}$. This functional is computed
using the widely known Fast Fourier Transform (FFT) algorithm, Cooley and
Tukey (1965). The magnitude and phase of the FFT output are extracted,
flattened, and concatenated to form the FFT vectorization.

Discrete Waivelets Functional.
$\Psi_{\mu,\mathcal{W}}\left(a,\mathbf{b},x_{i},x_{j}\right)=\sum_{x_{m},x_{n}\in
G_{N,M}}m(x_{m},x_{n})\cdot\phi_{a,\mathbf{b}}(x_{i}-x_{m},x_{j}-x_{n})\,.$
This functional is computed for standard orthonormal wavelets (Haar, Coiflet,
Daubechies ,$\ldots$) using Fast Wavelet Transform (FWT) algorithm, Shensa
(1992). Coefficients of FWT are then extracted, concatenated and used as
vectorization.

Discrete Gabor Functional.
$\Psi_{\mu,\mathcal{G}}\left(\lambda,\psi,\sigma,\gamma,\mathcal{N}_{G}(\cdot),x_{i},x_{j}\right)=\sum_{x_{m},x_{n}\in\mathcal{N}_{G}(x_{i},x_{j})}m(x_{m},x_{n})\cdot
g_{\lambda,\psi,\sigma,\gamma}(x_{i}-x_{m},x_{j}-x_{n})\,.$ In the above
formula, $\mathcal{N}_{G}\left(x_{i},x_{j}\right)$ represents the neighborhood
grid around the point $\left(x_{i},x_{j}\right)$. This neighborhood is used to
limit the computation to a local region around each point, enhancing
efficiency and focusing on local features. The DGT employs convolution
algorithms for computation. To achieve vectorization of the DGT, both the
magnitude and phase of the DGT output are extracted. These components are then
flattened and concatenated, forming the comprehensive DGT embedding.

###  4.2 Discretized Persistence Diagrams

The concept of discretization of persistence diagrams naturally arises from
the grid-based algorithms used in real-life computations of the transforms
outlined in Section 4.1, and it forms a critical aspect of our methodology.

This discretization process in our methodology involves two primary steps.

####  4.2.1 Shifting Step

Taking inspiration from the work by Adams et al. (2017), we first apply a
shifting process to the measure induces by a persistence diagram using shift
$\tau$. Specifically, let $\mu\in\mathcal{M}^{p}$ represent a persistence
measure in birth-death coordinates. We define
$\tau:\mathcal{M}^{p}\rightarrow\mathcal{M}^{p}$ as a linear shifting
operation $\tau(\mu)=(x_{1},x_{2}-x_{1})$. Upon applying this shift, we obtain
$\tau(\mu)$, the transformed measure now represented in birth-persistence
coordinates in the space $\tilde{\Omega}$. This shifting is applied with the
intention of better aligning the shifted diagram with the subsequent step of
discretization on a grid. The motivation for this shifting lies in the
distribution of points in the persistence diagram. Initially, in birth-death
coordinates, points are located along the upper diagonal $\partial\Omega$,
leaving the bottom diagonal empty. This distribution can be suboptimal for
certain analyses, especially when considering a square grid for discretization
aligned with the axis. The shift to birth-persistence coordinates changes
this, as the points no longer lie solely on the upper diagonal. Instead, this
shifting process redistributes the points more uniformly across the grid,
making better use of the available space.

####  4.2.2 Mapping Step

###### Lemma 4.1

Let ${G}_{N,M}$ be a regular grid on $\mathbb{R}^{2}_{+}$ consisting of points
$\\{(x_{i},y_{j})\mid
x_{i}=i\Delta,y_{j}=j\Delta,i=0,\ldots,N-1,j=0,\ldots,M-1\\}$ where $\Delta$
is the grid finesse. Define $\mathcal{S}$ as the $\sigma$-algebra containing
all subsets of ${G}_{N,M}$, and $\mu:\mathcal{S}\rightarrow[0,\infty]$ as a
measure such that for any $A\in\mathcal{S}$, $\mu(A)=\sum_{(x_{i},y_{j})\in
A}m_{ij}\in\mathcal{M}^{p}$ where $m_{ij}$ is a integer mass assigned to the
point $(x_{i},y_{j})$. The triple $\Gamma_{p}=({G}_{N,M},\mathcal{S},\mu)$
constitutes a discrete measure space.

The proof of Lemma 4.1 is straightforward and follows directly from the axioms
defining a measure space.

###### Proposition 4.2

Let $\Gamma_{p}\subset\mathcal{M}^{p}$ be a discrete measure space as defined
in Lemma 4.1. For a persistence diagram $D$ and a measure
$\mu=\sum_{\mathbf{x}\in\tau(D)}m(\mathbf{x})\delta_{\mathbf{x}}\in\mathcal{M}^{p}$,
consider the $1$-Optimal partial transport distance ($1$-OT) between $\mu$ and
any measure $\nu\in\Gamma_{p}$. Consider the optimization problem

$\nu^{\star}=\arg\min_{\nu^{\prime}\in\Gamma_{p}}OT_{1}(\mu,\nu^{\prime}),\quad
s.t.\quad\nu^{\prime}(G_{N,M})=\mu(D)\,.$

If we choose
$\nu=\sum_{\mathbf{x}\in\tau(D)}m(\Theta(\mathbf{x}))\delta_{\Theta(\mathbf{x})}$,
where $\Theta:\mathbb{R}^{2}_{+}\rightarrow G_{N,M}$ is a mapping that assigns
each point $\mathbf{x}\in\tau(D)$ to the closest point in $G_{N,M}$ minimizing
$\|\mathbf{x}-\mathbf{x}^{\prime}\|_{1}$, then it holds that $\nu=\nu^{\star}$
is the solution to the optimization problem.

The proof of Proposition 4.2 is provided in Appendix A.1.

By Proposition 4.2, we therefore obtain a persistence measure $\nu^{\star}$ on
$\Gamma_{p}$. We refer to $\nu^{\star}$ as the measure induced by the so-
called discretized persistence diagram.

###  4.3 Stability

Given the inevitable presence of noise or measurement errors, it is essential
for data analysis tools to be stable against minor input perturbations.

In line with this, we have established stability results with respect to the
1-Wasserstein metric for two of the transforms we considered, namely the
Fourier and Gabor Transforms. For the Wavelet Transform, there does not always
exist a closed-form expression, which makes finding stability results very
challenging. Extending these results to the Wavelet Transform remains a
challenge for future work.

###### Theorem 4.3

Let $\mu=\sum_{\mathbf{x}\in\Theta_{\tau}(D)}\delta_{\mathbf{x}}$ and
$\nu=\sum_{\mathbf{x}\in\Theta_{\tau}(D^{\prime})}\delta_{\mathbf{x}}$ be two
measures on the discretized persistence diagrams of $D,D^{\prime}$, let $f$ be
a functional, either Gabor or Fourier (refer to Section 4.1). Then, there
exists a finite constant $c$ such that:

$\left\|\Psi_{\mu}(f)-\Psi_{\nu}(f)\right\|_{\infty}\leq c\cdot
W_{1}(D,D^{\prime}).$

This statement proposes a bound on the difference between the functionals of
the two measures, scaled by a constant and the 1-Wasserstein distance between
the original persistence diagrams.

###### Remark 4.4

The following expressions are equivalent:

$\mu=\sum_{\mathbf{x}\in\Theta(\tau(D))}m(\mathbf{x})\delta_{\mathbf{x}}=\sum_{\mathbf{x}\in\Theta_{\tau}(D)}\delta_{\mathbf{x}}.$

For convenience, we used the latter and cleaner notation in Theorem 4.3, where
the multiplicity of each point is implicitly accounted for within the
summation. For example, if a point appears with a multiplicity of 2, it is
effectively included twice in the summation.

The proof of Theorem 4.3 is provided in Appendix A.2.

##  5 Results

In this section, we demonstrate how PS effectively preserve topological
information, proving to be as competitive as state-of-the-art methods from the
TDA litarature. Our experiments are conducted on two types of classification
tasks: graph-based classification and classification of point clouds derived
from dynamical systems. In our experiments, we utilized a Vanilla Random
Forest classifier from scikit-learn. The choice of a random forest classifier
was driven by two primary considerations: Firstly, it facilitates a direct
comparison with the study conducted by Royer et al. (2021). Secondly, in line
with our objective to showcase the overall effectiveness of our methods
relative to existing literature, employing this type of classifier helps to
circumvent the necessity for elaborate infrastructure or extensive fine-
tuning. Such an approach ensures that the results are not excessively
influenced by particular design choices.

###  5.1 Graph Classification

Given the growing interest in graph data within learning problems, the
standard graph classification framework, denoted as $\Omega:=(G,y)$,
represents a finite collection of graphs and their corresponding labels in
$\mathcal{G}\times\mathcal{Y}$.

###### Definition 5.1

For a graph $\mathcal{G}=(V,E)$, the HKS function with diffusion parameter $t$
is defined for each vertex $v\in V$ as

$HKS_{t}(v)=\sum_{k=1}^{|V|}\exp\left(-t\lambda_{k}\right)\psi_{k}(v)^{2},$

where $\lambda_{k}$ is the $k$-th eigenvalue of the normalized graph
Laplacian, and $\psi_{k}$ is the corresponding eigenfunction.

A significant advancement in TDA for graph classification is attributed to the
work in Carriere et al. (2020), which introduced a method for extracting
topological information from graphs using heat kernel signatures (HKS), as
detailed in Definition 5.1 and further explored in Hu et al. (2014). This
approach involves extended persistence, encoding the topological structure of
graphs and resulting in four types of finite persistence topological features.

These extended persistence diagrams not only offer richer information than
conventional ones but also ensure data points have finite coordinates,
enhancing their utility and reliability.

In our study, we benchmarked our methods against a variety of graph datasets
that are widely recognized as standard benchmarks in graph classification
research. The datasets include IMDB-B and IMDB-M, which consist of social
graphs and are sourced from Tran et al. (2018), as well as COX2, DHFR, MUTAG,
and PROTEINS. These latter datasets, also obtained from Tran et al. (2018),
are derived from chemoinformatics and bioinformatics fields.

We adopt the methodology from Royer et al. (2021) for generating extended
persistence diagrams, using two specific HKS diffusion times ($t_{1}=.1$ and
$t_{2}=10$), focusing solely on the topological structure without considering
additional graph attributes. For the evaluation, we used three distinct square
grid sizes—$20\times 20$, $32\times 32$, and $50\times 50$—across all our
proposed transforms when dealing with both Biomedical and Social Networking
graph problems.

Regarding the Wavelet Transform, we experimented with six different wavelet
types: Daubechies ($dbN$) and Coiflet ($coifN$) wavelets for $N=1,2,3$. For
the Gabor Transform, we implemented it with a kernel size that is half of the
grid size and set the parameters to
$\sigma=5,\theta=0,\lambda=10,\gamma=0.5,\psi=0$. This setup effectively
utilizes only the real part of the output (since $\psi=0$). It is important to
note that these parameters were not individually fine-tuned for each task in
this pre-print, but rather were used as default settings. We refer to this
specific configuration as the Real Vanilla Gabor Transform.

The highest accuracies achieved with these transforms are presented in Table
1. Detailed observations for each transform across the different grids can be
found in the Appendix B.Performance is assessed through 10-fold evaluations on
each dataset, reporting both average and best 10-fold results.

All computations were conducted on a single laptop, an Apple M3 Pro equipped
with an 11-core CPU.

Dataset | SV† | RetGK⋆ | FGSD† | GCNN† | GIN† | P† | MP† | Perslay⋆ | ATOL⋆ | BBA† | PS
---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  |  |  |  | Mean⋆ | Max†
MUTAG | $88.3$ | $90.3$ | $92.1$ | $86.7$ | $89.0$ | $79.2$ | $86.1$ | $89.8$ | $88.3$ | $90.4$ | $89.8$ | $91.7$
COX2 | $78.4$ | $81.4$ | - | - | - | $76.0$ | $79.9$ | $80.9$ | $79.4$ | $81.2$ | $80.6$ | $82.4$
PROTEINS | $72.6$ | $75.8$ | $73.4$ | $76.3$ | $75.8$ | $65.4$ | $67.5$ | $74.8$ | $71.4$ | $74.7$ | $72.8$ | $73.6$
DHFR | $78.4$ | $81.5$ | - | - | - | $70.9$ | $81.7$ | $80.3$ | $82.7$ | $80.5$ | $81.8$ | $83.1$
IMDB-B | $72.9$ | $71.9$ | $73.6$ | $73.1$ | $74.3$ | $54.0$ | $68.7$ | $71.2$ | $74.8$ | $69.4$ | $68.4$ | $69.8$
IMDB-M | $50.3$ | $47.7$ | $52.4$ | $50.3$ | $52.1$ | $36.3$ | $46.9$ | $48.8$ | $47.8$ | $46.7$ | $43.7$ | $44.4$

Table 1: Comparative Analysis of Classification Accuracy on Benchmark Graph
Datasets. Note: The symbol † indicates that the performance should be compared
with the Max metric, while the symbol $\star$ denotes that the comparison
should be made with the Mean metric.

In line with Carriere et al. (2020), Royer et al. (2021), and Hacquard (2023),
we benchmark our performance against the leading methods for these problems,
as well as against traditional topological methods documented in the
literature. For the latter, P signifies the superior method between the
Persistence Images and Persistence Landscapes, while MP indicates the best
between Multiparameter Persistence Images and Landscape. All these
persistence-based vectorizations are coupled with a Vanillla XGBoost
classifier for the learning task. In contrast, the top performing methods for
these graph problems predominantly utilize graph kernel methods specifically
designed for graph problems. These include: a scale-variant topological method
(SV from Tran et al. (2018)), a graph kernel method based on random walks
(RetGK from Zhang et al. (2018)), a graph embedding method using spectral
distances (FGSD from Verma and Zhang (2017)), a Capsule Graph Neural Network
(GCNN from Zhang and Chen (2019)), and a Graph Isomorphism Network (GIN from
Xu et al. (2019)). Additionally, we also compare our results with those of
PersLay (from Carriere et al. (2020)), a neural network-driven topological
vectorization method encapsulating most topological frameworks in the
literature, and ATOL (from Royer et al. (2021)), which introduces a fast,
learned, unsupervised vectorization method for measures in Euclidean spaces.
Lastly, we compare with BBA from Hacquard (2023), which employs rectangle-
based classification with boosting on persistence measures. Notably, both
PersLay and ATOL use the same eight diagrams with HKS parameters as ours.
However, in a few cases for PersLay, the authors employed two HKS diffusion
times and then discarded one of them, without any loss in performance.

The accuracy of competitors, as reported in their respective publications
(except for P and MP, which are cited from Hacquard (2023)), should be
interpreted as follows: For RetGK, ATOL, and PersLay, the evaluation procedure
was carried out over ten 10-fold validations, similar to our approach, making
the results directly comparable to our Mean in Table 1. For FGSD, SV, GIN,
GCNN, BBA, P, and MP, the average accuracy was calculated using a single
10-fold validation; therefore, their performance should be compared with our
Max (maintaining consistency with Carriere et al. (2020)).

In Table 1, our results are at the state-of-the-art for the Biomedical
benchmark datasets. Our methods consistently surpass all traditional TDA
methods, such as P and MP and moreover prove parallel or sometimes better
performance in some task with more advanced technique such as ATOL, PersLay,
SV and BBA.

Significantly, on the MUTAG dataset, our method outperforms all comparable
techniques, achieving the same mean accuracy as Perslay but with a superior
maximum accuracy222The maximum accuracies of Perslay are not included in Table
1; for these details, please refer to Carriere et al. (2020)..

Moreover, our performance on the COX2 and DHFR datasets is particularly
noteworthy, with our methods achieving the second-highest average accuracy in
both instances. Specifically for topological methods, our approach attains the
highest maximum accuracy on the DHFR dataset.

It is important to note, however, that a direct comparison of our results with
Perslay’s findings for the IMDB and PROTEINS datasets is not feasible, owing
to Perslay’s unique preprocessing approach that focuses exclusively on the 500
points furthest from the diagonal.

Although our methods demonstrate exceptional performance overall, it should be
acknowledged that for the Social Networking datasets, they are slightly below
the current best methods.

###  5.2 Dynamical systems (Orbit5K Dataset)

The Orbit5K dataset, a commonly used benchmark in TDA for classification
methods and for modeling DNA microarray flows, is characterized by chaotic
trajectories within the unit cube $[0,1]^{2}$, exhibiting unique topological
features dependent on a parameter $\rho>0$. Notably, this dataset is the same
as referenced in various studies Adams et al. (2017); Le and Yamada (2018);
Carrière et al. (2017); Hacquard (2023); Carriere et al. (2020); Royer et al.
(2021).

Figure 1: Representative Point Cloud Samples from the Orbit5K Dataset.

For each $\rho$ class, we form point clouds by iterating the following
recursive equations for a sequence of 1000 points, beginning from a random
initial point $(x_{0},y_{0})$ in $[0,1]^{2}$:

$\displaystyle x_{n+1}$ $\displaystyle=x_{n}+\rho y_{n}(1-y_{n})\mod 1,$
$\displaystyle y_{n+1}$ $\displaystyle=y_{n}+\rho x_{n+1}(1-x_{n+1})\mod 1.$

Figure 1 displays one sample per class, illustrating the diversity and
challenge involved in this classification task.

We generated 700 training and 300 testing datasets for each $\rho$ class,
conducting a one-versus-one classification and using persistence diagrams for
both $H_{1}$ and $H_{0}$ homologies, following the approach described in Le
and Yamada (2018). We employ the same transforms with hyperparameter settings
as described in the previous subsection. We adopt a regular square grid of
$64\times 64$ and $128\times 128$ for all transforms in this learning task.
The highest accuracies achieved with our transforms are presented in Table 2.
Detailed observations for each transform across the different grids are
available in Appendix B. For this dynamical particles classification task, we
report the accuracy averaged over 10 runs.

In line with Carriere et al. (2020) and Hacquard (2023), we compare our
methods with four kernel methods on persistence diagrams: Persistence Scale
Space Kernel (PSS-K from Reininghaus et al. (2015)), Persistence Weighted
Gaussian Kernel (PWG-K from Kusano et al. (2016)), Sliced Wasserstein Kernel
(SW-K from Carrière et al. (2017)), and Persistence Fisher Kernel (PF-K from
Le and Yamada (2018)). Moreover, we compare with two neural network
architectures, Perslay and Persformer (respectively from Carriere et al.
(2020) and Reinauer et al. (2021)), Persistence Images (PI from Adams et al.
(2017)), and a rectangle-based classification algorithm on persistence
measures (BBA from Hacquard (2023)). The accuracies in Table 2 for our
competitor methods are sourced from Le and Yamada (2018) for the kernel-based
approaches, and from the respective original publications for the non-kernel
methods. Our methods outperform most of the kernel methods, Persistence
Images, and BBA classification, but are slightly less effective than the
neural networks.

PSS-K† | PWG-K† | SW-K† | PF-K† | PI⋆ | Perslay† | Persformer∗ | BBA⋆ | PS⋆
---|---|---|---|---|---|---|---|---
$72.38(\pm 2.4)$ | $76.63(\pm.7)$ | $83.6(\pm.9)$ | $85.9(\pm.8)$ | $82.5$ | $87.7(\pm 1.0)$ | $91.2(\pm.8)$ | $83.3(\pm.5)$ | $84.6(\pm.8)$

Table 2: Comparative Classification Accuracy on the Orbit5K Dataset. Notes: †
denotes methods with accuracy averaged over 100 runs; ⋆ indicates averaging
over 10 runs; ∗ signifies an average over 5 runs. For Persistence Images, no
standard deviation is reported in Adams et al. (2017).

##  6 Conclusion and further work

In this study, we introduced a new vectorization framework for persistence
diagrams by treating them as signals on $\mathbb{R}^{2}_{+}$. We employed a
functional-based transforms, focusing particularly on frequency methods. This
approach has proven to be stable and effective in embedding information into a
vector space, preserving key topological features. Its performance, either
comparable or superior to leading methods, is demonstrated across various
graph classification tasks and synthetic dynamical particle classifications.
The simplicity of our method, along with its potential for enhancement, makes
it a strong contender in the field of TDA vectorization techniques.

However, our approach is not without limitations. A significant challenge lies
in the choice of grid. Presently, the grid’s fine granularity entails high
computational expense for subsequent use of our vectorization in classical ML
pipelines due to high dimensionality of the embedding. Analysis of various
persistence diagrams reveals a concentration of points near the origin,
contrasting with extended areas of low point density. Consequently, many grid
points remain unassociated with any persistence diagram points, resulting in
numerous zeros in integral transforms. These zeros contribute little to no
information gain, serving primarily to inflate the dimensionality of the
embedding.

To address this, a non-uniform, adaptive grid that aligns with the
distribution of points in persistence diagrams appears to be a more efficient
alternative.

Furthermore, our methodology is not confined to the frequency domain. Future
research could explore other domains, aiming to identify additional
functionals that effectively retain the information within persistence
diagrams.

Another prospective direction for future work involves incorporating our
vectorization concept into frameworks akin to those described in Carriere et
al. (2020) or Reinauer et al. (2021). This integration could leverage neural
networks or transformer architectures to refine the dimensionality of our
embeddings, potentially enhancing both performance and applicability.

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##  Appendix A Proofs

This section encompasses all the mathematical proofs presented in this paper.

###  A.1 Proof of Proposition 4.2

We start with observing that since both measures $\nu$ and $\mu$ have the same
mass and the cost of transport is $\|\mathbf{x}-\mathbf{x}^{\prime}\|_{1}$,
then $OT_{1}(\mu,\nu)=W_{1}(\mu,\nu)$. Let $\mathbf{x}$ be a point on the
shifted diagram and let $\mathbf{g}$ be a point on the grid. Given the
equivalence of the $1$-optimal partial transport cost to the $1$-Wasserstein
metric, the cost transporting mass from $\mathbf{x}$ to $\mathbf{g}$ is
$\|\mathbf{x}-\mathbf{g}\|_{1}$. The transport plan $\gamma=\Theta$ should
minimize the total cost, which is the sum of the product of the transported
mass and the distance it’s transported. Formally, therefore we want to
minimize
$\sum_{i=1}^{|D|}\sum_{j=1}^{NM}\Theta_{ij}\|\mathbf{x}_{i}-\mathbf{g}_{j}\|_{1}$.

For each point $\mathbf{x}_{i}$ in $\tau(D)$, find the closest grid point
$\mathbf{g}_{j^{*}}$ such that
$\|\mathbf{x}_{i}-\mathbf{g}_{j^{*}}\|_{1}\leq\|\mathbf{x}_{i}-\mathbf{g}_{j^{*}}\|_{1}$
for all $j\in\\{NM\\}$. We state that this assignment ensures that the
transport cost for each point is as small as possible. Based on this
assignment we define
$\nu^{\star}=\sum_{\mathbf{x}\in\tau(D)}m(\Theta(\mathbf{x}))\delta_{\Theta(\mathbf{x})}$.

To prove that the assignment is indeed optimal suppose by contradiction that
there exist an other transport plan $\Theta^{\prime}$ that reaches a lower
total transport cost. There must be at least one pair
$\left(\mathbf{x}_{i},\mathbf{g}_{j}\right)$ in $\Theta^{\prime}$ where
$\mathbf{x}_{i}$ is not transported to its closest grid point. The cost of
$\Theta^{\prime}$ is therefore greater than the cost of the transport plan
where each $\mathbf{x}_{i}$ is matched with its nearest grid point,
contradicting the optimality of $\Theta^{\prime}$. $\square$

###  A.2 Proof of Theorem 4.3

We begin by stating three auxiliary lemmata, which will be proven in the
upcoming sections and will be instrumental in proving Theorem 4.3.

###### Lemma A.1

Let $D$ and $D^{\prime}$ be persistence diagrams with finitely many points,
and consider the mapping $\Theta$ defined in Proposition 4.2. For any bounded
function $f$, the following inequality holds:

$\left\|\sum\limits_{\mathbf{x}\in
D}f\left(\Theta(\mathbf{x},\boldsymbol{\omega})\right)-\sum\limits_{\mathbf{x^{\prime}}\in
D^{\prime}}f\left(\Theta(\mathbf{x^{\prime}},\boldsymbol{\omega})\right)\right\|_{\infty}\leq
C\left\|\sum\limits_{\mathbf{x}\in
D}f\left(\mathbf{x},\boldsymbol{\omega}\right)-\sum\limits_{\mathbf{x^{\prime}}\in
D^{\prime}}f\left(\mathbf{x^{\prime}},\boldsymbol{\omega}\right)\right\|_{\infty},$

where $C$ is a constant.

###### Lemma A.2

Consider the Fourier transform function $f(\mathbf{x},\boldsymbol{\omega})$
from Section 4.1. Assume that $\left\|\boldsymbol{\omega}\right\|_{\infty}\leq
M$.333This assumption is without loss of generality since the Fourier
transform here is computed on a two-dimensional finite grid, and therefore the
frequencies assume discrete finite values. For all
$\mathbf{x},\mathbf{x}^{\prime}\in\mathbb{R}^{2}_{+}$, we have:

$\left\|f(\mathbf{x},\boldsymbol{\omega})-f(\mathbf{x}^{\prime},\boldsymbol{\omega})\right\|_{\infty}\leq
K\left\|\mathbf{x}-\mathbf{x}^{\prime}\right\|_{2},$

where $K$ is a finite constant value.

###### Lemma A.3

Consider the Gabor transform function
$f(\mathbf{t},\mathbf{x},\lambda,\psi,\sigma,\gamma)$ from Section 4.1. For
ease of notation we will refer to it just as
$f_{\lambda,\psi,\sigma,\gamma}(\mathbf{t},\mathbf{x})$ since the other
variables are just hyperparameters to tune . For all
$\mathbf{x},\mathbf{x}^{\prime}\in\mathbb{R}^{2}_{+}$, we have:

$\left\|f_{\lambda,\psi,\sigma,\gamma}(\mathbf{t},\mathbf{x})-f_{\lambda,\psi,\sigma,\gamma}(\mathbf{t},\mathbf{x^{\prime}})\right\|_{\infty}\leq
K^{\prime}\left\|\mathbf{x}-\mathbf{x}^{\prime}\right\|_{2},$

where $K^{\prime}$ is a finite constant value.

Using an argument similar to the one in Adams et al. [2017], which is aimed at
proving the stability of Persistence Images, we note that, given $D$ and
$D^{\prime}$ are comprised of a finite number of points (Remark 3.2), there
exists a matching $\gamma$ that attains the minimum value in the Wasserstein
metric.

$\displaystyle\left\|\Psi_{\mu}(f)-\Psi_{\nu}(f)\right\|_{\infty}$
$\displaystyle=\left\|\sum\limits_{\mathbf{x}\in\tau(D)}f\left(\Theta(\mathbf{x}),\boldsymbol{\omega}\right)-\sum\limits_{\mathbf{x}^{\prime}\in\tau(D^{\prime})}f\left(\Theta(\mathbf{x}^{\prime}),\boldsymbol{\omega}\right)\right\|_{\infty}$
$\displaystyle\leq
C\left\|\sum\limits_{\mathbf{x}\in\tau(D)}f\left(\mathbf{x},\boldsymbol{\omega}\right)-\sum\limits_{\mathbf{x}^{\prime}\in\tau(D^{\prime})}f\left(\mathbf{x}^{\prime},\boldsymbol{\omega}\right)\right\|_{\infty}\quad\text{by
Lemma \ref{lemma:lipchitz:mapping} on $\tau(D),\tau(D^{\prime})$}$
$\displaystyle=C\left\|\sum\limits_{\mathbf{x}\in\tau(D)}f\left(\mathbf{x},\boldsymbol{\omega}\right)-f\left(\gamma(\mathbf{x}),\boldsymbol{\omega}\right)\right\|_{\infty}\quad\text{since
$\exists$ finite matching $\gamma$}$ $\displaystyle\leq
C\sum_{\mathbf{x}\in\tau(D)}\left\|f\left(\mathbf{x},\boldsymbol{\omega}\right)-f\left(\gamma(\mathbf{x}),\boldsymbol{\omega}\right)\right\|_{\infty}\quad\text{by
triangle inequality}$ $\displaystyle\leq
C^{\prime}\sum_{\mathbf{x}\in\tau(D)}\left\|\mathbf{x}-\gamma(\mathbf{x})\right\|_{2}\quad\text{by
Lemma \ref{lemma:lipchitz:fourier} and \ref{lemma:lipchitz:gabor} for Fourier
and Gabor transforms resp.}$ $\displaystyle\leq
C^{\prime\prime}\sum_{\mathbf{x}\in
D}\left\|\mathbf{x}-\gamma(\mathbf{x})\right\|_{\infty}\quad\text{ since
}\|\tau(\cdot)\|_{2}\leq\sqrt{5}\|\cdot\|_{\infty}\text{ (see Thm 1
\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{adams2017persistence}{\@@citephrase{[}}{\@@citephrase{]}}})}$
$\displaystyle=C^{\prime\prime}W_{1}(D,D^{\prime})\,.$

$\square$

####  A.2.1 Proof of Auxiliary Lemma A.1

We start by considering the shifted diagrams $\tau(D)$ and $\tau(D^{\prime})$.
Applying the triangle inequality to the $L^{\infty}$ norm, we can decompose
the left side of the given inequality into three parts:

$\left\|\sum\limits_{\mathbf{x}\in\tau(D)}f\left(\Theta(\mathbf{x},\omega)\right)-\sum\limits_{\mathbf{x^{\prime}}\in\tau(D^{\prime})}f\left(\Theta(\mathbf{x^{\prime}},\omega)\right)\right\|_{\infty}$

$\begin{aligned}
&=\left\|\sum\limits_{\mathbf{x}\in\tau(D)}\left(f\left(\Theta(\mathbf{x},\boldsymbol{\omega})\right)-f\left(\mathbf{x},\boldsymbol{\omega}\right)\right)+\sum\limits_{\mathbf{x}\in\tau(D)}f\left(\mathbf{x},\boldsymbol{\omega}\right)-\sum\limits_{\mathbf{x^{\prime}}\in\tau(D^{\prime})}\left(f\left(\Theta(\mathbf{x^{\prime}},\boldsymbol{\omega})\right)-f\left(\mathbf{x^{\prime}},\boldsymbol{\omega}\right)\right)-\sum\limits_{\mathbf{x^{\prime}}\in\tau(D^{\prime})}f\left(\mathbf{x^{\prime}},\boldsymbol{\omega}\right)\right\|_{\infty}\\\
&\leq\left\|\sum\limits_{\mathbf{x}\in\tau(D)}\left(f\left(\Theta(\mathbf{x},\boldsymbol{\omega})\right)-f\left(\mathbf{x},\boldsymbol{\omega}\right)\right)\right\|_{\infty}+\left\|\sum\limits_{\mathbf{x^{\prime}}\in\tau(D^{\prime})}\left(f\left(\Theta(\mathbf{x^{\prime}},\boldsymbol{\omega})\right)-f\left(\mathbf{x^{\prime}},\boldsymbol{\omega}\right)\right)\right\|_{\infty}+\left\|\sum\limits_{\mathbf{x}\in\tau(D)}f\left(\mathbf{x},\boldsymbol{\omega}\right)-\sum\limits_{\mathbf{x^{\prime}}\in\tau(D^{\prime})}f\left(\mathbf{x^{\prime}},\boldsymbol{\omega}\right)\right\|_{\infty}\end{aligned}$

Now, let $K_{max}$ :=
$\sup\limits_{\tilde{D}\in\mathcal{D}}\left\|\sum\limits_{\mathbf{x}\in\tau(\tilde{D})}\left(f\left(\Theta(\mathbf{x},\boldsymbol{\omega})\right)-f\left(\mathbf{x},\boldsymbol{\omega}\right)\right)\right\|_{\infty}\,.$
We have that

$\left\|\sum\limits_{\mathbf{x}\in\tau(D)}f\left(\Theta(\mathbf{x},\boldsymbol{\omega})\right)-\sum\limits_{\mathbf{x^{\prime}}\in\tau(D^{\prime})}f\left(\Theta(\mathbf{x^{\prime}},\boldsymbol{\omega})\right)\right\|_{\infty}\leq
2K_{max}+\left\|\sum\limits_{\mathbf{x}\in\tau(D)}f\left(\mathbf{x},\boldsymbol{\omega}\right)-\sum\limits_{\mathbf{x^{\prime}}\in\tau(D^{\prime})}f\left(\mathbf{x^{\prime}},\boldsymbol{\omega}\right)\right\|_{\infty}$
$\leq\left(\frac{2K_{max}}{K_{min}}+1\right)\left\|\sum\limits_{\mathbf{x}\in\tau(D)}f\left(\mathbf{x},\boldsymbol{\omega}\right)-\sum\limits_{\mathbf{x^{\prime}}\in\tau(D^{\prime})}f\left(\mathbf{x^{\prime}},\boldsymbol{\omega}\right)\right\|_{\infty}\,,$

where we defined
$K_{min}:=\inf\limits_{\tilde{D},\tilde{D^{\prime}}\in\mathcal{D}}\left\|\sum\limits_{\mathbf{x}\in\tau(\tilde{D})}f\left(\Theta(\mathbf{x},\boldsymbol{\omega})\right)-\sum\limits_{\mathbf{x^{\prime}}\in\tau(\tilde{D^{\prime}})}f\left(\mathbf{x^{\prime}},\boldsymbol{\omega}\right)\right\|_{\infty}$.
Without loss of generality we assume that $K_{min}\neq 0$. By setting
$C=\left(\frac{2K_{max}}{K_{min}}+1\right)$, we conclude the proof.

####  A.2.2 Proof of Lemma A.2

Using the Fourier function transform from Section 4.1 we compute:

$\displaystyle\left\|e^{-2\pi i\mathbf{x}^{\top}\mathbf{\omega}}-e^{-2\pi
i{\mathbf{x}^{\prime}}^{\top}\mathbf{\omega}}\right\|_{\infty}$
$\displaystyle=\left\|\cos\left(2\pi\mathbf{x}^{\top}\mathbf{\omega}\right)-\cos\left(2\pi{\mathbf{x}^{\prime}}^{\top}\mathbf{\omega}\right)-i\left(\sin\left(2\pi\mathbf{x}^{\top}\mathbf{\omega}\right)-\sin\left(2\pi{\mathbf{x}^{\prime}}^{\top}\mathbf{\omega}\right)\right)\right\|_{\infty}\quad\text{by
Euler's formula}$
$\displaystyle=\left\|-2\sin\left(\pi\left(\mathbf{x}-{\mathbf{x}}^{\prime}\right)^{\top}\mathbf{\omega}\right)\sin\left(\pi\left(\mathbf{x}-{\mathbf{x}}^{\prime}\right)^{\top}\mathbf{\omega}\right)-2\sin\left(\pi\left(\mathbf{x}-{\mathbf{x}}^{\prime}\right)^{\top}\mathbf{\omega}\right)i\cos\left(\pi\left(\mathbf{x}+{\mathbf{x}}^{\prime}\right)^{\top}\mathbf{\omega}\right)\right\|_{\infty}\quad\text{by
basic trig. identities}$ $\displaystyle\leq
2\left\|\sin\left(\pi\left(\mathbf{x}-{\mathbf{x}}^{\prime}\right)^{\top}\mathbf{\omega}\right)\right\|_{\infty}\cdot\left\|\sin\left(\pi\left(\mathbf{x}-{\mathbf{x}}^{\prime}\right)^{\top}\mathbf{\omega}\right)+i\cos\left(\pi\left(\mathbf{x}+{\mathbf{x}}^{\prime}\right)^{\top}\mathbf{\omega}\right)\right\|_{\infty}$
$\displaystyle\leq
4\left\|\sin\left(\pi\left(\mathbf{x}-{\mathbf{x}}^{\prime}\right)^{\top}\mathbf{\omega}\right)\right\|_{\infty}$
$\displaystyle\leq
4\pi\left\|\left(\mathbf{x}-\mathbf{x}^{\prime}\right)^{\top}\mathbf{\omega}\right\|_{\infty}\quad\text{by
$|\sin(x)|\leq|x|$}$ $\displaystyle\leq\sqrt{2}\pi
4M\left\|\mathbf{x}-\mathbf{x}^{\prime}\right\|_{2}\quad\text{by Cauchy
Schwarz and by $\left\|\mathbf{\omega}\right\|_{\infty}\leq\sqrt{2}M$}$
$\displaystyle\leq 8\pi
M\left\|\mathbf{x}-\mathbf{x}^{\prime}\right\|_{\infty}\,.$

We can conclude the proof by taking $K=8\pi M$.

####  A.2.3 Proof of Lemma A.3

We commence the proof by introducing a more concise notation. As delineated in
Section 4.1, the function $f_{\lambda,\psi,\sigma,\gamma}$ is defined by the
equation:

$f_{\lambda,\psi,\sigma,\gamma}(\mathbf{t},\mathbf{x})=\underbrace{\exp\left(-\frac{(t_{1}-x_{1})^{2}+\gamma^{2}(t_{2}-x_{2})^{2}}{2\sigma^{2}}\right)}_{:=\alpha_{\mathbf{x}}(\mathbf{t})}\cdot\underbrace{\exp\left(j\left(2\pi\frac{t_{1}-x_{1}}{\lambda}+\psi\right)\right)}_{:=\beta(\mathbf{x},\mathbf{t})}.$

###### Remark A.4

It is noteworthy that $\|\beta\|_{\infty}=1$, and $\alpha$ is a non-negative,
bounded by 1, and differentiable function. Consequently, it can be deduced
that for all $\mathbf{x},\mathbf{x^{\prime}}\in\tau(D)$:

$\left\|\alpha_{\mathbf{x}}-\alpha_{\mathbf{x^{\prime}}}\right\|_{\infty}\leq|\nabla\alpha|\|\mathbf{x}-\mathbf{x^{\prime}}\|_{2},$

where $\nabla\alpha$ denotes the maximal directional derivative of
$\alpha_{\mathbf{x}}(\mathbf{t})$. This property is derived from the fact that
$\left|\alpha_{\mathbf{x}}(\mathbf{t})-\alpha_{\mathbf{x^{\prime}}}(\mathbf{t})\right|=\left|\alpha_{\mathbf{x}}(\mathbf{t})-\alpha_{\mathbf{x}}(\mathbf{t}+\mathbf{x}-\mathbf{x^{\prime}})\right|\leq|\nabla\alpha|\|\mathbf{x}-\mathbf{x^{\prime}}\|_{2}$.

###### Remark A.5

Recall that we operate under the assumption that all points in $D$ are finite.
This assumption consequently implies the existence of a finite constant $L$
such that $\|\nabla\alpha\|_{\infty}\leq L$.

We now compute:

$\displaystyle\left\|f_{\lambda,\psi,\sigma,\gamma}(\mathbf{t},\mathbf{x})-f_{\lambda,\psi,\sigma,\gamma}(\mathbf{t},\mathbf{x^{\prime}})\right\|_{\infty}$
$\displaystyle=\left\|\alpha_{\mathbf{x}}(\mathbf{t})\cdot\beta(\mathbf{x},\mathbf{t})-\alpha_{\mathbf{x}^{\prime}}(\mathbf{t})\cdot\beta(\mathbf{x}^{\prime},\mathbf{t})\right\|_{\infty}$
$\displaystyle=\left\|\alpha_{\mathbf{x}}(\mathbf{t})\cdot\beta(\mathbf{x},\mathbf{t})-\alpha_{\mathbf{x}}(\mathbf{t})\cdot\beta(\mathbf{x}^{\prime},\mathbf{t})+\alpha_{\mathbf{x}}(\mathbf{t})\cdot\beta(\mathbf{x}^{\prime},\mathbf{t})-\alpha_{\mathbf{x}^{\prime}}(\mathbf{t})\cdot\beta(\mathbf{x}^{\prime},\mathbf{t})\right\|_{\infty}$
$\displaystyle\leq\|\beta\|_{\infty}\|\alpha_{\mathbf{x}}(\mathbf{t})-\alpha_{\mathbf{x}^{\prime}}(\mathbf{t})\|_{\infty}+\|\alpha\|_{\infty}\|\beta(\mathbf{x},\mathbf{t})-\beta(\mathbf{x}^{\prime},\mathbf{t})\|_{\infty}$
$\displaystyle\leq|\nabla\alpha|\|\mathbf{x}-\mathbf{x}^{\prime}\|_{2}+2\quad\text{from
Remark \ref{appendix:remark:gabor}}$ $\displaystyle\leq
L\|\mathbf{x}-\mathbf{x}^{\prime}\|_{2}+2\quad\text{from Remark
\ref{appendix:remark:gabor2}}$ $\displaystyle\leq
K\|\mathbf{x}-\mathbf{x}^{\prime}\|_{2}\quad\text{ for some constant $K$}\,.$

##  Appendix B Complementary material

This section includes extensive tables that provide a comparative analysis of
graph data and dynamical particles classification scores for various methods
studied in this paper.

Dataset/Grid FOURIER Real V. GABOR WT-$db1$ WT-$db2$ WT-$db3$ WT-$coif1$
WT-$coif2$ WT-$coif3$ Mean Max Mean Max Mean Max Mean Max Mean Max Mean Max
Mean Max Mean Max MUTAG 20 $87.8(\pm.8)$ $88.9$ $88.2(\pm.9)$ $89.4$
$87.8(\pm.8)$ $88.9$ $87.8(\pm.8)$ $88.9$ $88.7(\pm 1.0)$ $90.6$ $88.6(\pm.8)$
$90.0$ $87.7(\pm 1.2)$ $90.0$ $87.5(\pm.9)$ $89.4$ 32 $89.1(\pm 1.2)$ $91.1$
$89.7(\pm 1.0)$ $91.1$ $88.4(\pm 1.3)$ $90.0$ $88.6(\pm 1.0)$ $90.0$ $88.9(\pm
1.0)$ $90.6$ $88.9(\pm.7)$ $90.0$ $89.7(\pm.5)$ $90.6$ $\mathbf{89.8(\pm.9)}$
$\mathbf{91.7}$ 50 $88.1(\pm 1.1)$ $89.4$ $87.8(\pm 1.2)$ $89.4$ $86.9(\pm
1.1)$ $88.9$ $87.6(\pm.9)$ $88.9$ $88.3(\pm 1.4)$ $91.1$ $88.2(\pm.9)$ $89.4$
$88.3(\pm 1.1)$ $90.0$ $88.1(\pm.6)$ $88.9$ DHFR 20 $77.8(\pm.7)$ $78.9$
$76.8(\pm.8)$ $78.3$ $77.8(\pm.6)$ $78.9$ $78.3(\pm.7)$ $78.9$ $79.3(\pm.6)$
$80.0$ $78.5(\pm.6)$ $79.6$ $79.2(\pm 1.0)$ $81.2$ $79.0(\pm.8)$ $80.4$ 32
$79.1(\pm 1.0)$ $80.5$ $78.3(\pm.7)$ $79.5$ $81.5(\pm.7)$ $\mathbf{83.1}$
$\mathbf{81.8(\pm.7)}$ $82.9$ $81.1(\pm 1.0)$ $82.4$ $81.4(\pm.5)$ $82.1$
$80.7(\pm.5)$ $82.4$ $80.6(\pm.7)$ $81.9$ 50 $79.6(\pm.7)$ $81.2$
$78.5(\pm.8)$ $80.0$ $80.6(\pm.4)$ $81.2$ $80.2(\pm.4)$ $80.8$ $80.7(\pm.4)$
$81.5$ $81.0(\pm.5)$ $81.9$ $81.5(\pm.5)$ $82.3$ $80.9(\pm.9)$ $82.7$ COX2 20
$78.3(\pm.6)$ $78.9$ $\mathbf{80.6(\pm 1.0)}$ $\mathbf{82.4}$ $79.2(\pm.7)$
$80.4$ $79.7(\pm.7)$ $81.1$ $79.7(\pm.7)$ $80.9$ $80.0(\pm.4)$ $80.4$
$79.7(\pm.5)$ $80.4$ $79.5(\pm.8)$ $81.5$ 32 $76.7(\pm.7)$ $77.8$
$78.4(\pm.8)$ $80.0$ $77.3(\pm 1.0)$ $79.3$ $78.0(\pm 1.0)$ $79.3$ $78.0(\pm
1.1)$ $79.3$ $78.0(\pm.6)$ $78.9$ $78.2(\pm.6)$ $78.9$ $77.4(\pm.8)$ $78.5$ 50
$77.1(\pm.9)$ $78.3$ $79.5(\pm.8)$ $81.3$ $77.5(\pm.7)$ $78.5$ $78.2(\pm.5)$
$78.9$ $77.7(\pm.6)$ $78.9$ $78.7(\pm.7)$ $80.0$ $78.2(\pm 1.0)$ $79.3$
$78.8(\pm 1.2)$ $80.2$ PROTEINS 20 $72.5(\pm.6)$ $73.2$ $72.7(\pm.3)$ $73.1$
$72.4(\pm.4)$ $72.8$ $72.5(\pm.7)$ $73.4$ $72.6(\pm.6)$ $73.4$
$\mathbf{72.8(\pm.6)}$ $\mathbf{73.6}$ $72.3(\pm.5)$ $73.0$ $72.1(\pm.4)$
$72.9$ 32 $71.9(\pm.5)$ $72.8$ $72.3(\pm.5)$ $73.1$ $72.1(\pm.5)$ $72.9$
$72.3(\pm.5)$ $73.1$ $72.3(\pm.5)$ $73.3$ $72.3(\pm.6)$ $73.4$ $71.9(\pm.5)$
$73.0$ $72.1(\pm.8)$ $73.0$ 50 $71.9(\pm.6$ $72.7$ $72.3(\pm.6)$ $73.4$
$72.1(\pm.5)$ $73.1$ $72.4(\pm.6)$ $73.2$ $72.4(\pm.6)$ $73.2$ $72.7(\pm.5)$
$73.5$ $72.6(\pm.5)$ $73.5$ $72.6(\pm.5)$ $73.3$ IMDB-B 20 $67.6(\pm.6)$
$68.6$ $67.7(\pm.5)$ $68.6$ $67.5(\pm.8)$ $68.7$ $67.1(\pm.6)$ $68.3$
$67.0(\pm.6)$ $67.7$ $67.1(\pm.3)$ $67.5$ $66.8(\pm.7)$ $67.9$ $66.4(\pm.9)$
$68.3$ 32 $66.7(\pm.5)$ $67.7$ $67.4(\pm.7)$ $68.6$ $66.4(\pm.9)$ $68.3$
$66.9(\pm.8)$ $67.9$ $67.2(\pm.6)$ $67.9$ $66.7(\pm.8)$ $67.7$ $66.8(\pm.6)$
$67.7$ $66.8(\pm.6)$ $67.8$ 50 $67.5(\pm.8)$ $69.1$ $\mathbf{68.4(\pm.7)}$
$\mathbf{69.8}$ $68.2(\pm.6)$ $69.3$ $68.1(\pm.7)$ $69.2$ $67.4(\pm.8)$ $68.5$
$68.2(\pm.7)$ $69.1$ $67.7(\pm.4)$ $68.3$ $67.7(\pm.5)$ $68.5$ IMDB-M 20
$42.2(\pm.5)$ $43.1$ $42.0(\pm.4)$ $42.7$ $42.4(\pm.4)$ $42.9$ $42.2(\pm.5)$
$42.9$ $42.5(\pm.3)$ $43.3$ $42.3(\pm.7)$ $43.5$ $42.3(\pm.3)$ $42.8$
$42.5(\pm.4)$ $43.2$ 32 $42.8(\pm.5)$ $43.7$ $42.5(\pm.4)$ $43.2$
$42.8(\pm.5)$ $43.9$ $42.5(\pm.4)$ $43.5$ $43.1(\pm.5)$ $43.9$ $42.9(\pm.5)$
$43.6$ $42.7(\pm.4)$ $43.5$ $42.6(\pm.5)$ $43.8$ 50 $42.9(\pm.5)$ $43.7$
$42.9(\pm.2)$ $43.3$ $43.0(\pm.5)$ $43.6$ $43.3(\pm.6)$ $\mathbf{44.4}$
$43.1(\pm.5)$ $43.9$ $42.9(\pm.3)$ $43.5$ $\mathbf{43.7(\pm.5)}$ $43.9$
$43.3(\pm.4)$ $44.1$

Table 3: Comparative analysis of graph data classification scores of the
methods presented in this study.

Dataset/Grid FOURIER Real V. GABOR WT-$db1$ WT-$db2$ WT-$db3$ WT-$coif1$
WT-$coif2$ WT-$coif3$ Orbit5K 64 $81.0(\pm 1.1)$ $82.2(\pm.8)$ $82.8(\pm.9)$
$80.8(\pm.7)$ $81.1(\pm.9)$ $81.0(\pm.9)$ $81.6(\pm 1.0)$ $81.3(\pm.6)$ 128
$83.4(\pm 1.0)$ $82.7(\pm.7)$ $82.2(\pm.7)$ $83.4(\pm.6)$ $83.8(\pm.7)$
$83.6(\pm 1.1)$ $83.9(\pm 1.0)$ $84.6(\pm.8)$

Table 4: Comparative analysis of Orbit5K classification scores of the methods
presented in this study.