2021

1]Department of Mathematics, University of Bologna, Italy 2]ALMA-AI,
University of Bologna, Italy 3]AM2, University of Bologna, Italy 4]ARCES,
University of Bologna, Italy

# A probabilistic result on impulsive noise reduction in Topological Data
Analysis through Group Equivariant Non-Expansive Operators

Patrizio Frosini<EMAIL_ADDRESS>Ivan Gridelli
<EMAIL_ADDRESS>Andrea Pascucci<EMAIL_ADDRESS>[  [  [  [

###### Abstract

In recent years, group equivariant non-expansive operators (GENEOs) have
attracted attention in the fields of Topological Data Analysis and Machine
Learning. In this paper we show how these operators can be of use also for the
removal of impulsive noise and to increase the stability of TDA in the
presence of noisy data. In particular, we prove that GENEOs can control the
expected value of the perturbation of persistence diagrams caused by uniformly
distributed impulsive noise, when data are represented by $L$-Lipschitz
functions from $\mathbb{R}$ to $\mathbb{R}$.

###### keywords:

GENEO, impulsive noise, persistence diagram, persistent homology, machine
learning

###### pacs:

[

MSC Classification]MSC Primary: 55N31, 62R40; Secondary: 60-08, 65D18, 68T09,
68U05.

##  1 Introduction

In the last thirty years Topological Data Analysis (TDA) developed as a useful
mathematical theory for analyzing data, benefiting from the reduction of
dimensionality guaranteed by topology EdHa08 ; BiDFFaal08 ; Ca2009 . One of
the main tools in TDA is the concept of persistence diagram, which is a
collection of points in the real plane, describing the homological changes of
the sublevel sets of suitable continuous functions. These changes give
important information about the data of interest, focusing on some of their
most relevant properties. Persistence diagrams can be used in the presence of
noise, since a well-known stability theorem states that these topological
descriptors change in a controlled way when we know that the functions
expressing the filtrations we are interested in change in a controlled way
with respect to the sup-norm CSEdHa07 . Furthermore, $L^{p}$-stability of
persistence diagrams with respect to the sup-norm have been proved in
CoEdHaMi10 . Unfortunately, in many applications the sup-norm of the noise is
not guaranteed to be small, and hence these results cannot be directly
applied. In particular, these results cannot be directly used when data are
affected by impulsive noise.

Analogously, in the discrete setting of TDA the presence of outliers in cloud
points can drastically affect the corresponding persistence diagrams. The
problem of managing outliers has been studied by several authors by different
techniques. In FaLeRiWaBaSi14  an approach based on confidence sets has been
introduced. A method inspired from the $k$-nearest neighbors regression and
the local median filtering has been used in BuChDeFaOuWa15 , while the concept
of bagplot has been applied in AdAg19 . In ViFuKuSrBh20  an approach based on
reproducing kernels has been proposed.

In our paper we start exploring a different probabilistic approach, in the
topological setting. The main idea is that studying the properties of data is
a process that should be primarily based on the analysis of the observers that
are asked to examine the data, since we cannot ignore that different observers
can differently judge the same data. This approach has been initially proposed
in BeFrGiQu19  and requires both the definition of the space of Group
Equivariant Non-Expansive Operators (GENEOs) and the development of
geometrical techniques to move around in this space CoFrQu22 . In other words,
in this model we should not wonder how we have to manage the data but rather
how we have to manage the observers (i.e., GENEOs) analyzing the data. As an
initial step in this direction, in this paper we show how GENEOs can be used
to get stability of persistence diagrams of 1D-signals in the presence of
impulsive noise.

GENEOs have been studied in FrJa16  as a new tool in TDA, since they allow for
an extension of the theory that is not invariant under the action of every
homeomorphism of the considered domain. This is important in applications
where the invariance group is not the group of all homeomorphisms, such as the
ones concerning shape comparison. Interestingly, GENEOs are also deeply
related to the foliation method used to define the matching distance in
2-dimensional persistent homology CeDFFeal13 ; CeEtFr19  and can be seen as a
theoretical bridge between TDA and Machine Learning BeFrGiQu19 . Furthermore,
these operators make available lower bounds for the natural pseudo-distance
$d_{G}(\varphi_{1},\varphi_{2}):=\inf\limits_{g\in
G}\|\varphi_{1}-\varphi_{2}\circ g\|_{\infty}$, associated with a group $G$ of
self-homeomorphisms of the domain of the signals $\varphi_{1},\varphi_{2}$
FrJa16 .

In our paper, we prove that GENEOs can control the expected value of the
perturbation of persistence diagrams caused by uniformly distributed impulsive
noise, when data are represented by $L$-Lipschitz functions from $\mathbb{R}$
to $\mathbb{R}$. In order to do that, we choose a “mother” bump function
$\psi$, i.e. a continuous function that is non-negative, upper bounded by $1$
and compactly supported, and we assume that our noise is made up of finite
bumps each obtained by translating, heightening and/or widening $\psi$. The
function $\hat{\varphi}=\varphi+R$ represents the corrupted data, where the
noise is given by the function
$R=\sum\limits_{i=1}^{k}a_{i}\psi(b_{i}(x-c_{i}))$ for some
$a_{i},b_{i},c_{i}\in\mathbb{R}$, with $b_{i}>0$, and some positive integer
$k$.

In this situation, trying to use a convolution to approximate our starting
data is not effective, because even if it does contract the bumps, it does not
cut them, and hence does not improve the sup-norm distance from the original
data. A classical approach for the removal of impulsive noise is the one of
using a median filterVa08 . Although this approach would be quite efficient in
the discrete case, let us remark that in our setting we are considering
continuous functions. The analogous of the median for the continuous case
would be defined as the interval of those values $m$ such that

$\int\limits_{-\infty}^{m}f(x)dx=\int\limits_{m}^{+\infty}f(x)dx$

where $f$ is a density of probability. However, this operator is not stable: a
small alteration of the starting function could lead to a significant change
of the median.

The operators we consider in this paper are
$F^{\delta}=\max(\varphi(x-\delta),\varphi(x+\delta))$ and
$F_{\varepsilon}=\min(\varphi(x-\varepsilon),\varphi(x+\varepsilon))$. The
main idea is that $F^{\delta}$ cuts the noise “directed downwards” and
$F_{\varepsilon}$ the noise “directed upwards”, and hence their composition
should be able to eliminate all the bumps. These operators are GENEOs with
respect to isometries of the real line. We prove that moving in the space of
GENEOs by taking suitable values for $\varepsilon$ and $\delta$, we can get
quite close to restoring the original function $\varphi$, depending on how the
bumps are positioned. The closer the bumps are to being Dirac delta functions
and the further they are from each other, the better our approximation can be.
On the ground of this result, we finally get an estimate of the expected value
$E(\|F^{\delta}\circ F_{\varepsilon}(\hat{\varphi})-\varphi\|_{\infty})$.

The paper is structured as follows. In Section 2 the mathematical background
is laid. The case we consider and our notation are explained in Section 3. In
Section 4 we prove the results that are needed in order to demonstrate our
main result. In Section 5 the main theorem giving us a probabilistic upper
bound is formulated. In Section 6 some examples and experiments are presented
in order to better illustrate the use of our results. A brief discussion
concludes the paper.

##  2 Mathematical setting

In this section we will recall some basic concepts we will use in this paper.

###  2.1 Representing data as real functions

Let us consider a set $\varPhi$ of bounded functions from a set $X$ to
$\mathbb{R}$, which will represent the data we wish to compare. We shall call
$\varPhi$ the set of _admissible measurements_ on $X$. We endow $\varPhi$ with
the topology induced by the sup-norm $\|\cdotp\|_{\infty}$ and the
corresponding distance $D_{\varPhi}$. A pseudo-metric $D_{X}$ can be defined
on $X$ by setting
$D_{X}(x_{1},x_{2})=\sup\limits_{\varphi\in\varPhi}\lvert\varphi(x_{1})-\varphi(x_{2})\rvert$
for every $x_{1},x_{2}\in X$. We recall that a pseudo-metric on a set $X$ is a
distance $d$ without the property $d(x,y)=0\implies x=y$. We will consider the
topological space $(X,\tau_{D_{X}})$ where $\tau_{D_{X}}$ is the topology
induced by $D_{X}$. A base for this topology is given by the open balls
$\\{B(x,r):=\\{x^{\prime}\in X:D_{X}(x,x^{\prime})<r\\}$ with $x\in X$,
$r\in\mathbb{R}\\}$. The choice of this topology makes every function in
$\varPhi$ a continuous functions. As shown in BeFrGiQu19 , this fact enables
us to use persistence diagrams in the study of $\varPhi$.

###  2.2 GENEOs as operators acting on data

We are interested in considering transformations of data. Let
$\mathrm{Homeo}_{\varPhi}(X)$ be the set of $\varPhi$-preserving
homeomorphisms from $X$ to $X$ with respect to the topology $\tau_{D_{X}}$,
meaning that every $g$ in $\mathrm{Homeo}_{\varPhi}(X)$ is a homeomorphism of
$X$ such as both $\varphi\circ g$ and $\varphi\circ g^{-1}$ belong to
$\varPhi$ for every $\varphi$ in $\varPhi.$ Let $G$ be a subgroup of
$\mathrm{Homeo}_{\varPhi}(X)$. $G$ represents the set of transformations on
data for which we will require equivariance to be respected.

Under the previously stated assumptions we call the ordered pair $(\varPhi,G)$
a _perception pair_. We can now introduce the concept of GENEO.

######  Definition 1.

Let $(\varPhi,G)$ and $(\Psi,H)$ be perception pairs and assume that a
homomorphism $T:G\to H$ is given. A function $F:\varPhi\to\Psi$ is called a
_Group Equivariant Non-Expansive Operator (GENEO) from $(\varPhi,G)$ to
$(\Psi,H)$ with respect to $T$_ if the following properties hold:

  1. 1.

(Group Equivariance) $F(\varphi\circ g)=F(\varphi)\circ T(g)$ for every
$\varphi\in\varPhi,g\in G$;

  2. 2.

(Non-Expansivity) $D_{\Psi}(F(\varphi_{1}),F(\varphi_{2}))\leq
D_{\varPhi}(\varphi_{1},\varphi_{2})$ for every
$\varphi_{1},\varphi_{2}\in\varPhi$.

Let us now consider the set $\mathcal{F}_{\mathrm{all}}$ of all GENEOs from
$(\varPhi,G)$ to $(\Psi,H)$ with respect to $T:G\to H$. The space
$\mathcal{F}_{\mathrm{all}}$ is endowed with the extended pseudo-metric
$D_{\mathcal{F}_{\mathrm{all}}}$, defined by setting
$D_{\mathcal{F}_{\mathrm{all}}}(F_{1},F_{2})=\sup_{\varphi\in\varPhi}D_{\Psi}(F_{1}(\varphi),F_{2}(\varphi))$
for every $F_{1},F_{2}\in\mathcal{F}_{\mathrm{all}}$. The word _extended_
refers to the possibility that $D_{\mathcal{F}_{\mathrm{all}}}$ takes an
infinite value.

The following result can be proven BeFrGiQu19 :

######  Theorem 1.

If $(\varPhi,D_{\varPhi})$,$(\Psi,D_{\Psi})$ are compact and convex then the
metric space $(\mathcal{F}_{\mathrm{all}},D_{\mathcal{F}_{\mathrm{all}}})$ is
compact and convex.

If a non-empty set $\mathcal{F}\subseteq\mathcal{F}_{\mathrm{all}}$ is fixed,
we can define the following pseudo-distance $D_{\mathcal{F},\varPhi}$ on
$\varPhi$:

######  Definition 2.

For any $\varphi_{1},\varphi_{2}$ in $\varPhi$ we set

$D_{\mathcal{F},\varPhi}(\varphi_{1},\varphi_{2})=\sup_{F\in\mathcal{F}}\|F(\varphi_{1})-F(\varphi_{2})\|_{\infty}.$

This pseudo-distance allows us to compare data by taking into account how
agents operate on data. Notice how, if $G$ becomes larger, the natural pseudo-
distance $d_{G}$ becomes harder to compute but this new pseudo-distance
$D_{\mathcal{F},\varPhi}$ becomes easier to evaluate. In order to find a lower
bound for $D_{\mathcal{F},\varPhi}$ it is useful to introduce the notion of
persistence diagram.

###  2.3 Persistence Diagrams

We will now recall some basic definitions and results in persistent homology.
The interested reader can find more details in EdHa08 .

Let us consider an ordered pair $(X,\varphi)$ where $X$ is a topological space
and $\varphi:X\to\mathbb{R}$ is a continuous function. For any
$t\in\mathbb{R}$ we can set $X_{t}:=\varphi^{-1}(\left]-\infty,t\right])$. If
$u<v$ the inclusion $i_{u,v}:X_{u}\to X_{v}$ induces a homomorphism
$i^{k}_{u,v}:H_{k}(X_{u})\to H_{k}(X_{v})$ between the $k$th homology groups
of $X_{u}$ and $X_{v}$. We can define the _$k$ th persistent homology group_,
with respect to $\varphi$ and computed at the point $(u,v)$, as
$PH_{k}(u,v):=i^{k}_{u,v}(H_{k}(X_{u}))$. Moreover, we can define the _$k$ th
persistent Betti numbers function_ $r_{k}(u,v)$ as the rank of $PH_{k}(u,v)$.

The $k$th persistent Betti numbers function can be represented by the $k$th
persistence diagram. This diagram is defined as the multi-set of all the
ordered pairs $(u_{j},v_{j}),$ where $u_{j}$ and $v_{j}$ are the times of
birth and death of the $j$th $k$-dimensional hole in $X$, respectively. We
call _time of birth_ of a hole the first time at which the homology class
appears, and _time of death_ the first time at which the homology class merges
with an older one. When a hole never dies, we set its time of death equal to
$\infty$. We also add to this set all points of the form $(w,w)$ for
$w\in\mathbb{R}$.

In Figure 1 the filtration of the set $X:=\left[0,\frac{3}{4}\pi\right]$ given
by the function $\varphi(x)=2\sin x$ is illustrated. In this example, the
topology on $X$ is the one defined by the space $\varPhi$ of admissible
functions given by all functions $a\sin(x+b)$ with $a,b\in\mathbb{R}$. The
reader can easily check that this topology coincides with the Euclidean
topology. The persistence diagram in degree $k=0$ of the function $\varphi$ is
displayed in Figure 2.

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Figure 1: How the sublevel sets change with respect to the filtration induced
by $\varphi$. Figure 2: Persistence diagram of the function $2\sin x$ on
$\left[0,\frac{3}{4}\pi\right]$. The point $(0,\infty)$ describes the
existence of a connected component that is born at zero and never dies. The
point $(\sqrt{2},2)$ claims that there is a connected component born at
$\sqrt{2}$ that dies (merges with the other one) at 2. The trivial points on
the diagonal $u=v$ are not displayed.

###  2.4 Comparing Persistence Diagrams

Persistence diagrams can be efficiently compared by means of a suitable metric
$d_{\mathrm{match}}$. In order to define it, we first define the pseudo-
distance

$\delta((x,y),(x^{\prime},y^{\prime}))=\min\\{\max\\{\lvert
x-x^{\prime}\rvert,\lvert y-y^{\prime}\rvert\\},\max\\{\lvert
x-y\rvert/2,\lvert x^{\prime}-y^{\prime}\rvert/2\\}\\}$

for all $(x,y),(x^{\prime},y^{\prime})\in\\{(x,y)\in\mathbb{R}$ with $x\leq
y\\}\cup\\{(x,\infty)$ with $x\in\mathbb{R}\\}$ by agreeing that
$\infty-y=\infty,y-\infty=-\infty$ for
$y\not=\infty,\infty-\infty=0,\infty/2=\infty,\lvert\pm\infty\rvert=\infty,\min\\{\infty,c\\}=c,\max\\{\infty,c\\}=\infty$.

If two persistence diagrams $D,D^{\prime}$ are given, we can set

$d_{\mathrm{match}}(D,D^{\prime})=\inf\limits_{\sigma\in\Sigma}\sup\limits_{P\in
D}\delta(P,\sigma(P))$

where $\Sigma$ represents the set of all bijections between the multisets
$D,D^{\prime}$.

For every degree $k$ we can now define a new pseudo-metric:

$D_{\mathcal{F},\varPhi}^{\mathrm{match}}(\varphi_{1},\varphi_{2})=\sup_{F\in\mathcal{F}}d_{\mathrm{match}}(D_{F(\varphi_{1})},D_{F(\varphi_{2})})$

where $D_{F(\varphi_{1})},D_{F(\varphi_{2})}$ are the persistence diagrams at
degree $k$ of the functions $F(\varphi_{1}),F(\varphi_{2})$, respectively.

In this paper we will limit ourselves to considering data represented as
functions from $\mathbb{R}$ to $\mathbb{R}$, and we recall that for this kind
of data persistence diagrams are non-trivial only in degree $k=0$ (i.e., when
persistent homology is used to count connected components). For this reason,
in the following we will always assume $k=0$.

##  3 Our model

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Figure 3: How drastically impulsive noise can influence persistence diagrams.

In this paper we will be mainly interested in the set $\mathrm{Lip}_{L}$ of
all $L$-Lipschitz functions from $\mathbb{R}$ to $\mathbb{R}$, for some fixed
constant $L\in\mathbb{R}$, and in the set $C^{0}(\mathbb{R})$ of all functions
from $\mathbb{R}$ to $\mathbb{R}$ that are continuous with respect to the
Euclidean topology. We will set $X=\mathbb{R}$ and consider the perception
pairs $(\mathrm{Lip}_{L},G)$, $(C^{0}(\mathbb{R}),G)$, where $G$ is the group
of the Euclidean isometries of $\mathbb{R}$.

We will assume our noise to be made up of a finite number of copies of a
“mother” nonnegative continuous bump function $\psi:\mathbb{R}\to\mathbb{R}$,
such that $\mathrm{supp}(\psi)\subseteq\left]-\sigma,\sigma\right[$ for some
$\sigma>0$ and $\|\psi\|_{\infty}\leq 1$. We recall that the support of a
function is the closure of the set of points where $f$ is non-vanishing. After
fixing two positive real numbers $\eta,\beta$, the noise we will be adding is
a function $R$ belonging to the space $\mathcal{R}_{\eta,\beta}$ that contains
the null function and all functions of the form
$\sum\limits_{i=1}^{k}a_{i}\psi(b_{i}(x-c_{i}))$, where $k$ is a positive
integer and $a_{i},b_{i},c_{i}$ are real numbers such that $\lvert
c_{i}-c_{j}\rvert\geq\eta$ for $i\not=j$ and $b_{i}\geq\beta$ for every index
$i$. For any $R\in\mathcal{R}_{\eta,\beta}$ we define
$S(R):=\bigcup\limits_{i=1}^{k}\left]c_{i}-\frac{\sigma}{\beta},c_{i}+\frac{\sigma}{\beta}\right[$
and remark that if $x\not\in S(R)$ then $R(x)=0$.

Our purpose will be the one of recovering $\varphi\in\mathrm{Lip}_{L}$ as well
as possible from the function $\hat{\varphi}=\varphi+R$. An example of such
situation is depicted in Figure 4.

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---|---|---
Figure 4: In the figure on the left we have our original function $\varphi$,
in the middle a noise function R and in the right figure the corrupted
function $\hat{\varphi}:=\varphi+R$.

The following result will be of use.

######  Proposition 2.

Let $F_{1},F_{2}$ be GENEOs from $(C^{0}(\mathbb{R}),G)$ to
$(C^{0}(\mathbb{R}),G)$ with respect to the trivial homomorphism
$T=\mathrm{id}:G\to G$. Then $F_{1}\circ F_{2}$ is a GENEO from
$(C^{0}(\mathbb{R}),G)$ to $(C^{0}(\mathbb{R}),G)$ with respect to $T$.

Proof: For every $\varphi\in C^{0}(\mathbb{R}),g\in G$ we have that

$\displaystyle F_{1}\circ F_{2}(\varphi\circ g)$
$\displaystyle=F_{1}(F_{2}(\varphi\circ g))$
$\displaystyle=F_{1}(F_{2}(\varphi)\circ g)$
$\displaystyle=F_{1}(F_{2}(\varphi))\circ g$ $\displaystyle=(F_{1}\circ
F_{2})(\varphi)\circ g.$

Therefore, $F_{1}\circ F_{2}$ is $G$-equivariant. Moreover, for any
$\varphi_{1},\varphi_{2}\in C^{0}(\mathbb{R})$

$\displaystyle D_{C^{0}(\mathbb{R})}(F_{1}\circ F_{2}(\varphi_{1})),F_{1}\circ
F_{2}(\varphi_{2}))$
$\displaystyle=D_{C^{0}(\mathbb{R})}(F_{1}(F_{2}(\varphi_{1})),F_{1}(F_{2}(\varphi_{2})))$
$\displaystyle\leq
D_{C^{0}(\mathbb{R})}(F_{2}(\varphi_{1})),F_{2}(\varphi_{2}))$
$\displaystyle\leq D_{C^{0}(\mathbb{R})}(\varphi_{1},\varphi_{2}).$

It follows that $F_{1}\circ F_{2}$ is non-expansive.

##  4 Cutting off the noise by GENEOs

We start by introducing two families of GENEOs from $(C^{0}(\mathbb{R}),G)$ to
$(C^{0}(\mathbb{R}),G)$ with respect to the identical homomorphism.

######  Definition 3.

Let $\varphi\in\mathrm{Lip}_{L}$ and $\varepsilon>0$. For all $x\in\mathbb{R}$
we define:

  1. 1.

$F^{\varepsilon}(\varphi)(x)=\max(\varphi(x-\varepsilon),\varphi(x+\varepsilon))$;

  2. 2.

$F_{\varepsilon}(\varphi)(x)=\min(\varphi(x-\varepsilon),\varphi(x+\varepsilon))$.

######  Proposition 3.

The maps $F^{\varepsilon}$ and $F_{\varepsilon}$ are GENEOs from
$(C^{0}(\mathbb{R}),G)$ to $(C^{0}(\mathbb{R}),G)$ with respect to the
identical homomorphism.

Proof:

We start by proving that $F^{\varepsilon}$ is $G$-equivariant.

Let $g$ be the translation $x\mapsto x+k$ with $k\in\mathbb{R}$, then

$\displaystyle F^{\varepsilon}(\varphi\circ g)$
$\displaystyle=\max\\{\varphi((x+k)-\varepsilon),\varphi((x+k)+\varepsilon)\\}$
$\displaystyle=\max\\{\varphi((x-\varepsilon)+k),\varphi((x+\varepsilon)+k)\\}$
$\displaystyle=F^{\varepsilon}(\varphi)\circ g.$

Let $g$ be the symmetry $x\mapsto-x$, then

$\displaystyle F^{\varepsilon}(\varphi\circ g)$
$\displaystyle=\max\\{\varphi((-x)-\varepsilon),\varphi((-x)+\varepsilon)\\}$
$\displaystyle=\max\\{\varphi(-(x+\varepsilon)),\varphi(-(x-\varepsilon))\\}$
$\displaystyle=\max\\{\varphi(x+\varepsilon),\varphi(x-\varepsilon)\\}\circ g$
$\displaystyle=F^{\varepsilon}(\varphi)\circ g.$

Since every isometry in $G$ can be written as the composition of a symmetry
and a translation, our statement follows.

Furthermore, for any $x\in\mathbb{R}$

$\displaystyle\lvert
F^{\varepsilon}(\varphi_{1})(x)-F^{\varepsilon}(\varphi_{2})(x)\rvert=$
$\displaystyle=\lvert\max\\{\varphi_{1}(x-\varepsilon),\varphi_{1}(x+\varepsilon)\\}-\max\\{\varphi_{2}(x-\varepsilon),\varphi_{2}(x+\varepsilon)\\}\rvert$
$\displaystyle\leq\max\\{\lvert\varphi_{1}(x-\varepsilon)-\varphi_{2}(x-\varepsilon)\rvert,\lvert\varphi_{1}(x+\varepsilon)-\varphi_{2}(x+\varepsilon)\rvert\\}$
$\displaystyle\leq\max\\{\|\varphi_{1}-\varphi_{2}\|_{\infty},\|\varphi_{1}-\varphi_{2}\|_{\infty}\\}$
$\displaystyle=\|\varphi_{1}-\varphi_{2}\|_{\infty}.$

It follows that
$\|F^{\varepsilon}(\varphi_{1})-F^{\varepsilon}(\varphi_{2})\|_{\infty}\leq\|\varphi_{1}-\varphi_{2}\|_{\infty}$,
and hence $F^{\varepsilon}$ is non-expansive.

We observe that $\min\\{a,b\\}=-\max\\{-a,-b\\}$, and hence
$F_{\varepsilon}(\varphi)=-F^{\varepsilon}(-\varphi)$ for every $\varphi\in
C^{0}(\mathbb{R})$. Moreover, the map taking $\varphi$ to $-\varphi$ is a
GENEO from $(C^{0}(\mathbb{R}),G)$ to $(C^{0}(\mathbb{R}),G)$ with respect to
the identical homomorphism. Therefore, Proposition 2 guarantees that
$G$-equivariance and non-expansivity also hold for $F_{\varepsilon}$.

Since Proposition 2 shows that the composition of GENEOs is still a GENEO, the
operator $F^{\delta}\circ F_{\varepsilon}$ is a GENEO from
$(C^{0}(\mathbb{R}),G)$ to $(C^{0}(\mathbb{R}),G)$ with respect to the
identical homomorphism.

We want now to prove that if a function $\hat{\varphi}$ is obtained by adding
impulsive noise to a function $\varphi$ then the value of

$\|F^{\delta}\circ F_{\varepsilon}(\hat{\varphi})-\varphi\|_{\infty}$

is bounded, and possibly small, provided that $\delta$ and $\varepsilon$ are
suitably chosen. The main idea is that the operator $F_{\varepsilon}$ cuts the
noise “directed upwards” and $F^{\delta}$ cuts the noise “directed downwards”.

In order to proceed, we need two lemmas.

######  Lemma 1.

Let $R\in C^{0}(\mathbb{R},\mathbb{R})$, $\varphi\in\mathrm{Lip}_{L}$ for some
$L\in\mathbb{R}$, and set $\hat{\varphi}:=\varphi+R$. Then for any
$\varepsilon>0$ and $\delta>0$

$\mathrm{i)}$

$-L\varepsilon+F_{\varepsilon}(R)\leq
F_{\varepsilon}(\hat{\varphi})-\varphi\leq L\varepsilon+F_{\varepsilon}(R)$;

$\mathrm{ii)}$

$-L\delta+F^{\delta}(R)\leq F^{\delta}(\hat{\varphi})-\varphi\leq
L\delta+F^{\delta}(R)$;

$\mathrm{iii)}$

$-L(\delta+\varepsilon)+F^{\delta}\circ F_{\varepsilon}(R)\leq F^{\delta}\circ
F_{\varepsilon}(\hat{\varphi})-\varphi\leq
L(\delta+\varepsilon)+F^{\delta}\circ F_{\varepsilon}(R)$.

Proof: Since $\varphi$ is Lipschitz of constant $L$ we have that for any value
$x\in\mathbb{R}$ $\lvert\varphi(x-\varepsilon)-\varphi(x)\rvert\leq
L\varepsilon$ and $\lvert\varphi(x+\varepsilon)-\varphi(x)\rvert\leq
L\varepsilon$. Therefore,

$\displaystyle F_{\varepsilon}(\hat{\varphi})(x)$
$\displaystyle=F_{\varepsilon}(\varphi+R)(x)$
$\displaystyle=\min\\{\varphi(x-\varepsilon)+R(x-\varepsilon),\varphi(x+\varepsilon)+R(x+\varepsilon)\\}$
$\displaystyle\leq\min\\{\varphi(x)+L\varepsilon+R(x-\varepsilon),\varphi(x)+L\varepsilon+R(x+\varepsilon)\\}$
$\displaystyle=\varphi(x)+L\varepsilon+\min\\{R(x-\varepsilon),R(x+\varepsilon)\\}$
$\displaystyle=\varphi(x)+L\varepsilon+F_{\varepsilon}(R)(x).$

Analogously,
$F_{\varepsilon}(\hat{\varphi})(x)\geq\varphi(x)-L\varepsilon+F_{\varepsilon}(R)(x)$.
The same steps applied to $F^{\delta}$ yield the second statement of the
lemma. As for the last claim we can see that:

$\displaystyle F^{\delta}\circ F_{\varepsilon}(\hat{\varphi})(x)$
$\displaystyle=\max\\{F_{\varepsilon}(\hat{\varphi})(x-\delta),F_{\varepsilon}(\hat{\varphi})(x+\delta)\\}$
$\displaystyle\leq\max\\{\varphi(x-\delta)+L\varepsilon+F_{\varepsilon}(R)(x-\delta),\varphi(x+\delta)+L\varepsilon+F_{\varepsilon}(R)(x+\delta)\\}$
$\displaystyle\leq\max\\{\varphi(x)+L\delta+L\varepsilon+F_{\varepsilon}(R)(x-\delta),\varphi(x)+L\delta+L\varepsilon+F_{\varepsilon}(R)(x+\delta)\\}$
$\displaystyle=\varphi(x)+L\delta+L\varepsilon+\max\\{F_{\varepsilon}(R)(x-\delta),F_{\varepsilon}(R)(x+\delta)\\}$
$\displaystyle=\varphi(x)+L\delta+L\varepsilon+F^{\delta}\circ
F_{\varepsilon}(R)(x).$

Analogously, we can prove the lower bound. Henceforth we will assume that any
summation on an empty set of indexes is the null function.

######  Lemma 2.

Let $R\in\mathcal{R}_{\eta,\beta}$ and $\lambda\geq\frac{\sigma}{\beta}$. If
$\lambda\leq\rho\leq\frac{\eta}{2}-\lambda$ then

$\mathrm{a)}$

$F_{\rho}(R)(x)=\sum\limits_{a_{i}<0}\left[a_{i}\psi(b_{i}(x-\rho-
c_{i}))+a_{i}\psi(b_{i}(x+\rho-c_{i}))\right]\leq 0$

$\mathrm{b)}$

$F^{\rho}(R)(x)=\sum\limits_{a_{i}>0}\left[a_{i}\psi(b_{i}(x-\rho-
c_{i}))+a_{i}\psi(b_{i}(x+\rho-c_{i}))\right]\geq 0$

for all $x\in\mathbb{R}$. Moreover
$F_{\rho}(R),F^{\rho}(R)(x)\in\mathcal{R}_{2\lambda,\beta}$.

Proof: We will suppose without loss of generality that $c_{i}<c_{i+1}$ for all
$i=1,\dots,k-1$ and $a_{i}\not=0$ for all indexes $i$.

We want to show that at least one of $x-\rho$ and $x+\rho$ must always be
outside of $\bigcup\limits_{i=1}^{k}\left]c_{i}-\lambda,c_{i}+\lambda\right[$.
If by contradiction both $x-\rho$ and $x+\rho$ were in the same
$\left]c_{i}-\lambda,c_{i}+\lambda\right[$ then we would have $\rho<\lambda$,
against our hypotheses. Moreover $x-\rho$ and $x+\rho$ cannot belong to
different intervals $\left]c_{i}-\lambda,c_{i}+\lambda\right[$ and
$\left]c_{j}-\lambda,c_{j}+\lambda\right[$ for $i<j$: if by contradiction
$x-\rho\in\left]c_{i}-\lambda,c_{i}+\lambda\right[$ and
$x+\rho\in\left]c_{j}-\lambda,c_{j}+\lambda\right[$ for some $i<j$ then
$2\rho=x+\rho-(x-\rho)>(c_{j}-\lambda)-(c_{i}+\lambda)\geq
c_{j}-c_{i}-2\lambda\geq\eta-2\lambda$, and hence we would have
$\rho>\frac{\eta}{2}-\lambda$, against our hypotheses.

Since
$\left]c_{i}-\frac{\sigma}{\beta},c_{i}+\frac{\sigma}{\beta}\right[\subseteq\left]c_{i}-\lambda,c_{i}+\lambda\right[$,
it follows that at least one among the values $R(x-\rho)$ and $R(x+\rho)$ must
always be zero. Let us now set
$I^{-}_{i}:=\left](c_{i}-\rho)-\lambda,(c_{i}-\rho)+\lambda\right[$ and
$I^{+}_{i}:=\left](c_{i}+\rho)-\lambda,(c_{i}+\rho)+\lambda\right[$. These two
intervals must be disjoint, since
$(c_{i}+\rho-\lambda)-(c_{i}-\rho+\lambda)=2\rho-2\lambda\geq 0$.

Let us now consider $\\{I^{-}_{1},I^{+}_{1},\ldots,I^{-}_{k},I^{+}_{k}\\}$. We
will now prove that any two distinct elements from this set must be disjoint.
Since we have just proven that $I^{-}_{i}\cap I^{+}_{i}=\emptyset$ for
$i=1,\dots,k$, the following holds in the case $i<j$:

  1. 1.

$I^{+}_{i}\cap I^{+}_{j}=\emptyset$, since
$c_{j}+\rho-\lambda-(c_{i}+\rho+\lambda)\geq
c_{j}-c_{i}-2\lambda\geq\eta-2\lambda\geq 0$;

  2. 2.

$I^{-}_{i}\cap I^{-}_{j}=\emptyset$, since
$c_{j}-\rho-\lambda-(c_{i}-\rho+\lambda)\geq
c_{j}-c_{i}-2\lambda\geq\eta-2\lambda\geq 0$;

  3. 3.

$I^{+}_{i}\cap I^{-}_{j}=\emptyset$, since
$c_{j}-\rho-\lambda-(c_{i}+\rho+\lambda)\geq
c_{j}-c_{i}-2\rho-2\lambda\geq\eta-2\left(\frac{\eta}{2}-\lambda\right)-2\lambda=0$;

  4. 4.

$I^{+}_{j}\cap I^{-}_{i}$, since $c_{j}+\rho-\lambda-(c_{i}-\rho+\lambda)\geq
c_{j}-c_{i}+2\rho-2\lambda\geq\eta+2\lambda-2\lambda\geq 0$.

Let us now fix $k\in\\{1,\dots,k\\}$.

Since
$\mathrm{supp}(a_{k}\psi(b_{k}(x-c_{k})))\subseteq\left]c_{k}-\lambda,c_{k}+\lambda\right[$,
then $\mathrm{supp}(a_{k}\psi(b_{k}(x-\rho-c_{k})))\subseteq I^{-}_{k}$ and
$\mathrm{supp}(a_{k}\psi(b_{k}(x+\rho-c_{k})))\subseteq I^{+}_{k}$.

Hence, for any $x\in\mathbb{R}$ we have that

$\displaystyle F_{\rho}(R)(x)$ $\displaystyle=\min\\{R(x-\rho),R(x+\rho)\\}$
$\displaystyle=\begin{cases}0&\mbox{if
}x\notin\bigcup\limits_{i=1}^{k}(I_{i}^{-}\cup I_{i}^{+})\\\
a_{j}\psi(b_{j}(x+\rho-c_{j}))&\mbox{if }x\in I^{-}_{j}\text{ and }a_{j}<0\\\
a_{j}\psi(b_{j}(x-\rho-c_{j}))&\mbox{if }x\in I^{+}_{j}\text{ and
}a_{j}<0.\end{cases}$

This means that

$F_{\rho}(R)(x)=\sum\limits_{a_{i}<0}\left[a_{i}\psi(b_{i}(x-\rho-
c_{i}))+a_{i}\psi(b_{i}(x+\rho-c_{i}))\right]\leq 0.$

Now, given $c_{i}+\rho$ and $c_{j}-\rho$ centers of bumps of $F_{\rho}(R)$, we
have that

$\lvert(c_{i}+\rho)-(c_{j}-\rho)\rvert\geq\min\\{\eta-2\rho,2\rho\\}\geq
2\lambda.$

It follows that $F_{\rho}(R)\in\mathcal{R}_{2\lambda,\beta}$.

By noticing that $F^{\rho}(R)=-F_{\rho}(-R)$ we get the second part of the
thesis.

Let us remark that, in particular, if $\lambda=2\frac{\sigma}{\beta}$ then
$F_{\rho}(R),F^{\rho}(R)\in\mathcal{R}_{4\frac{\sigma}{\beta},\beta}$. We
observe that in order to exist a $\rho$ that satisfies the hypotheses of Lemma
2, the inequality $\eta\geq 4\lambda$ must hold.

We are now actually ready to prove a key result in our paper.

######  Theorem 4.

Given $\theta,\beta,L\in\mathbb{R}$, let
${\bar{R}}\in\mathcal{R}_{\theta,\beta}$, $\varphi\in\mathrm{Lip}_{L}$ and set
$\hat{\varphi}:=\varphi+{\bar{R}}$. If
$2\frac{\sigma}{\beta}\leq\varepsilon\leq\frac{\theta}{2}-2\frac{\sigma}{\beta}$
then for any
$\frac{\sigma}{\beta}\leq\delta\leq\frac{1}{2}\min\\{\theta-2\varepsilon,2\varepsilon\\}-\frac{\sigma}{\beta}$
the following inequality holds:

$\|F^{\delta}\circ F_{\varepsilon}(\hat{\varphi})-\varphi\|_{\infty}\leq
L(\varepsilon+\delta).$

Proof: Let $x\in\mathbb{R}$. Firstly, let us notice that the condition
$2\frac{\sigma}{\beta}\leq\varepsilon\leq\frac{\theta}{2}-2\frac{\sigma}{\beta}$
implies that
$\frac{\sigma}{\beta}\leq\frac{1}{2}\min\\{\theta-2\varepsilon,2\varepsilon\\}-\frac{\sigma}{\beta}$.
From Lemma 1 we know that

$-L(\delta+\varepsilon)+F^{\delta}\circ F_{\varepsilon}({\bar{R}})(x)\leq
F^{\delta}\circ F_{\varepsilon}(\hat{\varphi})(x)-\varphi(x)\leq
L(\delta+\varepsilon)+F^{\delta}\circ F_{\varepsilon}({\bar{R}})(x).$

Let us now prove that $F^{\delta}\circ F_{\varepsilon}({\bar{R}})(x)=0$.

By applying Lemma 2 with $\eta:=\theta$, $\lambda:=2\frac{\sigma}{\beta}$,
$\rho:=\varepsilon$ and $R:={\bar{R}}$ we get

$F_{\varepsilon}({\bar{R}})(x)=\sum\limits_{a_{i}<0}\left[a_{i}\psi(b_{i}(x-\varepsilon-
c_{i}))+a_{i}\psi(b_{i}(x+\varepsilon-c_{i}))\right]\leq 0.$

Let us remark that
$F_{\varepsilon}({\bar{R}})\in\mathcal{R}_{4\frac{\sigma}{\beta},\beta}$.
Moreover,

  1. 1.

$F^{\delta}\circ
F_{\varepsilon}({\bar{R}})(x)=\max\\{F_{\varepsilon}({\bar{R}})(x-\delta),F_{\varepsilon}({\bar{R}})(x+\delta)\\}\leq
0$ since both terms are negative;

  2. 2.

$F^{\delta}\circ
F_{\varepsilon}({\bar{R}})(x)=F^{\delta}(F_{\varepsilon}({\bar{R}}))(x)\geq
0$.

These inequalities follow from Lemma 2, by setting
$\eta:=4\frac{\sigma}{\beta}$, $\lambda:=\frac{\sigma}{\beta}$, $\rho:=\delta$
and $R:=F_{\varepsilon}({\bar{R}})$.

Therefore, we have proved that $\lvert F^{\delta}\circ
F_{\varepsilon}(\hat{\varphi})(x)-\varphi(x)\rvert\leq L(\delta+\varepsilon$)
for any $x\in\mathbb{R}$. It follows that $\|F^{\delta}\circ
F_{\varepsilon}(\hat{\varphi})-\varphi\|_{\infty}\leq L(\varepsilon+\delta)$.

Let us remark that Theorem 4 works under the (only) implicit assumption that
$\theta\geq 8\frac{\sigma}{\beta}$. In our setting this should not be
restrictive since it means that the noise added is made up of scattered, thin
bumps, without any reference to the height of the bumps: this is what we
expect when considering additive impulsive noise.

######  Corollary 5.

Given $\theta,\beta,L\in\mathbb{R}$, let $R\in\mathcal{R}_{\theta,\beta}$,
$\varphi\in\mathrm{Lip}_{L}$, and set $\hat{\varphi}:=\varphi+R$. If
$\theta\geq 8\frac{\sigma}{\beta}$, then

$\left\|F^{\frac{\sigma}{\beta}}\circ
F_{2\frac{\sigma}{\beta}}\left(\hat{\varphi}\right)-\varphi\right\|_{\infty}\leq
3L\frac{\sigma}{\beta}.$

Proof: The claim follows from Theorem 4 by taking
$\varepsilon:=2\frac{\sigma}{\beta}$ and $\delta:=\frac{\sigma}{\beta}.$

Corollary 5 and the well known stability of persistence diagrams with respect
to the max-norm CSEdHa07  immediately imply the following result, which is of
interest in TDA (the symbol $d_{\mathrm{match}}$ denotes the usual bottleneck
distance between persistence diagrams).

######  Corollary 6.

Given $\theta,\beta,L\in\mathbb{R}$, let $R\in\mathcal{R}_{\theta,\beta}$,
$\varphi\in\mathrm{Lip}_{L}$, and set $\hat{\varphi}:=\varphi+R$. If
$\theta\geq 8\frac{\sigma}{\beta}$, and $D$ and $D^{\prime}$ are the
persistence diagrams in degree $0$ of the filtering functions $\varphi$ and
$F^{\frac{\sigma}{\beta}}\circ F_{2\frac{\sigma}{\beta}}(\hat{\varphi})$,
respectively, then

$d_{\mathrm{match}}(D,D^{\prime})\leq 3L\frac{\sigma}{\beta}.$

##  5 Our main result

We are now ready to prove our main result. We start by stating a lemma
concerning the probability $p$ that any two distinct points in a randomly
chosen set of cardinality $k$ in an interval of length $\ell$ have a distance
greater than $\eta$. A proof of the following lemma is provided in 2001026 :
for the reader’s convenience, we report it here.

######  Lemma 3.

Let $X_{1},\dots,X_{k}$, with $k\geq 2$, be independent random variables,
uniformly distributed on the interval $[0,\ell]$, for some $\ell>0$. Let

$M:=\min_{1\leq i,j\leq k\atop i\neq j}\lvert X_{i}-X_{j}\rvert$

be the minimal distance between two distinct random variables. Then we have

$P(M>\eta)=\begin{cases}1&\text{if }\eta\leq 0,\\\
\left(1-\frac{(k-1)\eta}{\ell}\right)^{k}&\text{if
}0<\eta<\frac{\ell}{k-1},\\\ 0&\text{if }\eta\geq\frac{\ell}{k-1}.\end{cases}$

Proof: It suffices to consider the case $0<\eta<\frac{\ell}{k-1}$. By
symmetry, we have

$\displaystyle P(M>\eta)$
$\displaystyle=k!\,P((M>\eta)\cap(X_{1}<X_{2}<\cdots<X_{k}))$ (since
$X_{1},\dots,X_{k}$ are uniformly distributed)
$\displaystyle=k!\,\frac{\text{Leb}(S)}{\ell^{k}}$ (1)

where $S=\\{x\in[0,\ell]^{k}\mid
x_{1}<x_{2}-\eta<x_{3}-2\eta<\cdots<x_{k}-(k-1)\eta\\}$ and Leb denotes the
Lebesgue measure. Setting $y_{i}=x_{i}-(i-1)\eta$ for $i=1,\dots,k$, we have
that $\text{Leb}(S)=\text{Leb}(S^{\prime})$ where

$S^{\prime}=\\{y\in[0,\ell-(k-1)\eta]^{k}\mid y_{1}<y_{2}<\cdots<y_{k}\\}.$

On the other hand, again by symmetry, we have

$\text{Leb}(S^{\prime})=\frac{\text{Leb}([0,\ell-(k-1)\eta]^{k})}{k!}=\frac{(\ell-(k-1)\eta)^{k}}{k!}$

and plugging this last identity into (1) we get the thesis.

We can now prove the following result, concerning the expected value of the
error $\left\|F^{\frac{\sigma}{\beta}}\circ
F_{2\frac{\sigma}{\beta}}(\hat{\varphi})-\varphi\right\|_{\infty}$.

######  Theorem 7.

Let us choose a function $\varphi\in\mathrm{Lip}_{L}$, a non-negative
continuous function $\psi:\mathbb{R}\to\mathbb{R}$ with $\|\psi\|_{\infty}\leq
1$ and $\mathrm{supp}(\psi)\subseteq\left]-\sigma,\sigma\right[$ for some
$\sigma>0$, two positive numbers $\beta$ and $\ell$, and an integer $k\geq 2$.
For $i=1,\ldots,k$, let us fix $a_{i}\in\mathbb{R}$ and $b_{i}\geq\beta$, and
set $\bar{\alpha}:=\max\lvert a_{i}\rvert$. Moreover, let $c_{1},\dots,c_{k}$
be independent random variables, uniformly distributed on the interval
$[0,\ell]$. Let us consider the random variable $\hat{\varphi}:=\varphi+R$,
where $R(x):=\sum\limits_{i=1}^{k}a_{i}\psi(b_{i}(x-c_{i}))$ for any
$x\in\mathbb{R}$. If $\frac{\sigma}{\beta}<\frac{\ell}{8(k-1)}$, then

$E\left(\left\|F^{\frac{\sigma}{\beta}}\circ
F_{2\frac{\sigma}{\beta}}(\hat{\varphi})-\varphi\right\|_{\infty}\right)\leq
3L\frac{\sigma}{\beta}+k\bar{\alpha}\left(1-\left(1-8\frac{(k-1)}{\ell}\frac{\sigma}{\beta}\right)^{k}\right).$

Proof: By setting $\delta=\frac{\sigma}{\beta}$ and
$\varepsilon=2\frac{\sigma}{\beta}$ in statement $\mathrm{iii)}$ of Lemma 1,
we have that

$\left\|F^{\frac{\sigma}{\beta}}\circ
F_{2\frac{\sigma}{\beta}}(\hat{\varphi})-\varphi\right\|_{\infty}\leq
3L\frac{\sigma}{\beta}+\left\|F^{\frac{\sigma}{\beta}}\circ
F_{2\frac{\sigma}{\beta}}(R)\right\|_{\infty}.$

Since the operator $F^{\frac{\sigma}{\beta}}\circ F_{2\frac{\sigma}{\beta}}$
is non-expansive and $F^{\frac{\sigma}{\beta}}\circ
F_{2\frac{\sigma}{\beta}}(\mathbf{0})=\mathbf{0}$, it follows that

$\left\|F^{\frac{\sigma}{\beta}}\circ
F_{2\frac{\sigma}{\beta}}(R)\right\|_{\infty}\leq\left\|R\right\|_{\infty}\leq
k\bar{\alpha}\left\|\psi\right\|_{\infty}\leq k\bar{\alpha}.$

Therefore, $\left\|F^{\frac{\sigma}{\beta}}\circ
F_{2\frac{\sigma}{\beta}}(\hat{\varphi})-\varphi\right\|_{\infty}\leq
3L\frac{\sigma}{\beta}+k\bar{\alpha}$. If we apply Lemma 3 with
$\eta=8\frac{\sigma}{\beta}$, we obtain that
${R}\in\mathcal{R}_{8\frac{\sigma}{\beta},\beta}$ with probability
$p:=\left(1-8\frac{(k-1)}{\ell}\frac{\sigma}{\beta}\right)^{k}$. If
${R}\in\mathcal{R}_{8\frac{\sigma}{\beta},\beta}$, we can apply Theorem 4 by
setting $\delta=\frac{\sigma}{\beta}$, $\varepsilon=2\frac{\sigma}{\beta}$ and
$\theta=8\frac{\sigma}{\beta}$, and hence we obtain that
$\left\|F^{\frac{\sigma}{\beta}}\circ
F_{2\frac{\sigma}{\beta}}(\hat{\varphi})-\varphi\right\|_{\infty}\leq
3L\frac{\sigma}{\beta}$ with probability at least $p$. Since
$\left\|F^{\frac{\sigma}{\beta}}\circ
F_{2\frac{\sigma}{\beta}}(\hat{\varphi})-\varphi\right\|_{\infty}\leq
3L\frac{\sigma}{\beta}+k\bar{\alpha}$ in any case, it follows that

$\displaystyle E\left(\left\|F^{\frac{\sigma}{\beta}}\circ
F_{2\frac{\sigma}{\beta}}(\hat{\varphi})-\varphi\right\|_{\infty}\right)$
$\displaystyle\leq
3L\frac{\sigma}{\beta}p+\left(3L\frac{\sigma}{\beta}+k\bar{\alpha}\right)\left(1-p\right)$
$\displaystyle=3L\frac{\sigma}{\beta}+k\bar{\alpha}\left(1-p\right).$

Theorem 7 and the well known stability of persistence diagrams with respect to
the max-norm CSEdHa07  immediately imply the following result, which is of
interest in TDA (the symbol $d_{\mathrm{match}}$ denotes the usual bottleneck
distance between persistence diagrams).

######  Corollary 8.

Let us make the same assumptions of Theorem 7. Let $D$ and $D^{\prime}$ be the
persistence diagrams in degree $0$ of the filtering functions $\varphi$ and
$F^{\frac{\sigma}{\beta}}\circ F_{2\frac{\sigma}{\beta}}(\hat{\varphi})$,
respectively. If $\frac{\sigma}{\beta}<\frac{\ell}{8(k-1)}$, then

$E\left(d_{\mathrm{match}}(D,D^{\prime})\right)\leq
3L\frac{\sigma}{\beta}+k\bar{\alpha}\left(1-\left(1-8\frac{(k-1)}{\ell}\frac{\sigma}{\beta}\right)^{k}\right).$

This result shows that the use of suitable GENEOs can make TDA (relatively)
stable also in the presence of impulsive noise, under the assumptions we have
considered in this paper.

##  6 Examples and experiments

We will now validate our approach based on GENEOs by giving two examples and
illustrating some experimental results.

###  6.1 Examples

In order to verify how our approach works, we will set
$\tau_{n}:=\left(1-\frac{1}{n}\right)2\frac{\sigma}{\beta}+\frac{1}{n}\left(\frac{\theta}{2}-2\frac{\sigma}{\beta}\right)$
and consider the upper bound $\left\|F^{\frac{\tau_{n}}{2}}\circ
F_{\tau_{n}}\left(\hat{\varphi}\right)-\varphi\right\|_{\infty}\leq\frac{3}{2}L\tau_{n}$,
obtained by applying Corollary 5. We observe that $\tau_{n}\geq
2\frac{\sigma}{\beta}$ for every index $n$, and
$\lim\limits_{n\to+\infty}\tau_{n}=2\frac{\sigma}{\beta}$. We will examine two
examples that use the GENEOs $F^{\frac{\tau_{n}}{2}}\circ F_{\tau_{n}}$ and
show how our method based on such operators and the method based on
convolutions differ, as for their capability in removing additive impulsive
noise. Moreover, we will compare the actual error
$\left\|F^{\frac{\tau_{n}}{2}}\circ
F_{\tau_{n}}\left(\hat{\varphi}\right)-\varphi\right\|_{\infty}$ to its upper
bound $\frac{3}{2}L\tau_{n}$, by running several simulations. The convolutions
that will be applied in our examples use the functions
$T_{h}:\mathbb{R}\to\mathbb{R}$ defined by setting

$T_{h}(x):=\begin{cases}\frac{h}{2}&\mbox{if }-\frac{1}{h}\leq
x\leq\frac{1}{h}\\\ 0&\mbox{otherwise}\end{cases}$

for $h>0$. We will see that, although the convolution with such functions is
also a GENEO, it will not be able to efficiently remove the noise.

Our noise function $R$ will be constructed starting from the mother function
$\psi$ defined by setting $\psi(x):=e^{1-\frac{1}{1-x^{2}}}$ for
$x\in\left]-1,1\right[$ and $\psi(x):=0$ for $x\notin\left]-1,1\right[$. Using
the notation introduced in the previous sections, we will set $\sigma=1.1$,
thus satisfying the condition
$\mathrm{supp}(\psi)\subseteq\left]-\sigma,\sigma\right[$. The impulsive noise
will be added in an interval $\left[-{{\ell}},{{\ell}}\ \right]$. The
following parameters are considered, with these respective uniform
distributions:

  * •

$k\sim\mathrm{Unif}_{\\{1,\ldots,10\\}}$

  * •

$a_{i}\sim\mathrm{Unif}_{\left]-100,100\right[}$ for $i=1,\ldots,k$

  * •

$b_{i}\sim\mathrm{Unif}_{\left]0,100\right[}$ for $i=1,\ldots,k$

  * •

$c_{i}\sim\mathrm{Unif}_{\left]-{{\ell}}+\frac{\sigma}{\beta},{{\ell}}-\frac{\sigma}{\beta}\right[}$
for $i=1,\ldots,k$.

We set $\beta:=\min\limits_{i=1,\ldots,N}b_{i}$,
$\bar{\alpha}:=\max\limits_{i=1,\ldots,N}\lvert a_{i}\rvert$, and
$\eta:=\min\limits_{i\not=j}\lvert c_{i}-c_{j}\rvert$. After producing random
values for the parameters $N,a_{i},b_{i},c_{i}$, our algorithm checks whether
$\eta>8\frac{\sigma}{\beta}$, otherwise it generates another set of
parameters.

####  6.1.1 First example

Let us consider the function

$\varphi(x):=\begin{cases}\sin x&\mbox{if }-4\pi\leq x\leq 4\pi\\\
0&\mbox{otherwise}\end{cases}$

for $x\in\mathbb{R}$. We observe that $\varphi\in\mathrm{Lip}_{L}$, for $L=1$.
We will add noise in the interval $[-4\pi,4\pi]$ (meaning ${{\ell}}=4\pi$) and
visualize the results in such an interval. Figure 5 illustrates how the
function $\hat{\varphi}$ looks like compared to $\varphi$.

Figure 5: Example 1: Comparison between $\varphi$ and $\hat{\varphi}$.

We will start by considering how well the convolution $\hat{\varphi}*T_{n}$
can approximate the original function $\varphi$, when $n$ goes from $3$ to
$100$. From Figure 6, it is immediately apparent that the max-norm distance
between $\hat{\varphi}*T_{n}$ and $\varphi$ remains quite large.

|
---|---
|
Figure 6: Example 1: Denoising via convolution with $T_{h}$ ($h=n$).

If we apply a convolution with $T_{\frac{1}{n}}$, for $3\leq n\leq 100$, we
get the results displayed in Figure 7, showing that all information
represented by the function $\varphi$ is progressively destroyed.

|
---|---
|
Figure 7: Example 1: Denoising via convolution with $T_{h}$ ($h=\frac{1}{n}$).

In contrast, if we apply the operator $F^{\frac{\tau_{n}}{2}}\circ
F_{\tau_{n}}\left(\hat{\varphi}\right)$, for $n=3,5,20,100$, we get the
results displayed in Figure 8. As we can see, this operator is much more
efficient in removing the bumps and restoring the function $\varphi$.

|
---|---
|
Figure 8: Example 1: Denoising via the proposed GENEO
$F^{\frac{\tau_{n}}{2}}\circ F_{\tau_{n}}$.

As a matter of fact, when we apply a convolution with the function $T_{h}$ and
check the corresponding errors via the sup-norm, we get the results displayed
in Figure 9. Since
$\lim\limits_{h\to+\infty}\hat{\varphi}*T_{h}=\hat{\varphi}$ and
$\lim\limits_{h\to+\infty}\hat{\varphi}*T_{\frac{1}{h}}=0$, we get that the
errors tend to
$\|\hat{\varphi}-\varphi\|_{\infty}=\max\limits_{i=1,\ldots,N}\lvert
a_{i}\rvert=\bar{\alpha}$ and $\|\varphi\|_{\infty}$, respectively.

|
---|---
Figure 9: Example 1: Error made by applying a convolution with $T_{n}$ (left)
or $T_{\frac{1}{n}}$ (right).

In contrast, if we apply the operator $F^{\frac{\tau_{n}}{2}}\circ
F_{\tau_{n}}$, we get the results displayed in Figure 10, showing that the
upper bound for the error stated in Corollary 5 is quite tight. As we can
expect, we get the best denoising by replacing $\tau_{n}$ with
$2\frac{\sigma}{\beta}$ (see Figure 11).

Figure 10: Example 1: Error made by using the GENEO
$F^{\frac{\tau_{n}}{2}}\circ F_{\tau_{n}}$. Figure 11: Example 1: Denoising
via $F^{\frac{\sigma}{\beta}}\circ F_{2\frac{\sigma}{\beta}}$.

Finally, we executed 1000 simulations. In each of them we have produced a
function $\hat{\varphi}$ by adding random impulsive noise to the function
$\varphi$, then we have applied $F^{\frac{\sigma}{\beta}}\circ
F_{2\frac{\sigma}{\beta}}$ to $\hat{\varphi}$, in order to see how tight the
upper bound in Corollary 5 is to the actual error
$\|F^{\frac{\sigma}{\beta}}\circ
F_{2\frac{\sigma}{\beta}}(\hat{\varphi})-\varphi\|_{\infty}$. The same
parameters as in the beginning of this example have been used. As we can see
in Figure 12 the overestimation committed by our upper bound is often quite
close to zero, relatively to the Lipschitz constant $L$ of the function
$\varphi$.

Figure 12: Example 1: Histogram counting the number of cases in each of ten
bins, concerning the overestimation value
$3L\frac{\sigma}{\beta}-\|F^{\frac{\sigma}{\beta}}\circ
F_{2\frac{\sigma}{\beta}}(\hat{\varphi})-\varphi\|_{\infty}$ obtained in our
simulations. Most of the cases belong to the first bin, containing the
functions $\varphi$ for which $0\leq
3L\frac{\sigma}{\beta}-\|F^{\frac{\sigma}{\beta}}\circ
F_{2\frac{\sigma}{\beta}}(\hat{\varphi})-\varphi\|_{\infty}\leq 0.02$.

This suggests that such an upper bound is quite accurate.

####  6.1.2 Second example

Let us consider the function

$\varphi(x):=\begin{cases}\frac{1}{1000}(x-5)(x-3)(x+1)(x+4)(x+5)&\mbox{if
}-5\leq x\leq 5\\\ 0&\mbox{otherwise}\end{cases}$

for $x\in\mathbb{R}$. The coefficient $\frac{1}{1000}$ was chosen in order to
get that the Lipschitz constant $L$ is comparable to the one of the previous
example. In this example $L=\frac{27}{25}$. Figure 13 illustrates how the
function $\hat{\varphi}$ looks like compared to $\varphi$.

Figure 13: Example 2: Comparison between $\varphi$ and $\hat{\varphi}$.

We will start by considering how well the convolution $\hat{\varphi}*T_{n}$
can approximate the original function $\varphi$, when $n$ goes from $3$ to
$100$. From Figure 14, it is immediately apparent that the max-norm distance
between $\hat{\varphi}*T_{n}$ and $\varphi$ remains quite large.

|
---|---
|
Figure 14: Example 2: Denoising via convolution with $T_{h}$ ($h=n$).

If we apply a convolution with $T_{\frac{1}{n}}$, for $3\leq n\leq 100$, we
get the results displayed in Figure 15, showing that all information
represented by the function $\varphi$ is progressively destroyed.

|
---|---
|
Figure 15: Example 2: Denoising via convolution with $T_{h}$
($h=\frac{1}{n}$).

In contrast, if we apply the operator $F^{\frac{\tau_{n}}{2}}\circ
F_{\tau_{n}}\left(\hat{\varphi}\right)$, for $n=3,5,20,100$, we get the
results displayed in Figure 16. As we can see, this operator is much more
efficient in removing the bumps and restoring the function $\varphi$.

When we apply a convolution with the function $T_{h}$ and check the
corresponding errors via the sup-norm, we get the results displayed in Figure
17. As already seen, since
$\lim\limits_{h\to+\infty}\hat{\varphi}*T_{h}=\hat{\varphi}$ and
$\lim\limits_{h\to+\infty}\hat{\varphi}*T_{\frac{1}{h}}=0$ we get that the
errors tend to
$\|\hat{\varphi}-\varphi\|_{\infty}=\max\limits_{i=1,\dots,k}\lvert
a_{i}\rvert=\bar{\alpha}$ and $\|\varphi\|_{\infty}$, respectively.

|
---|---
|
Figure 16: Example 2: Denoising via the proposed GENEO $F^{\frac{\tau_{n}}{2}}\circ F_{\tau_{n}}$. |
---|---
Figure 17: Example 2: Error made by applying a convolution with $T_{n}$ (left)
or $T_{\frac{1}{n}}$ (right).

In contrast, if we apply the operator $F^{\frac{\tau_{n}}{2}}\circ
F_{\tau_{n}}$, we get the results displayed in Figure 18, showing that the
upper bound for the error stated in Corollary 5 is quite tight. As we can
expect, we get the best denoising by replacing $\tau_{n}$ with
$2\frac{\sigma}{\beta}$ (see Figure 19).

Figure 18: Example 2: Error made by using the GENEO
$F^{\frac{\tau_{n}}{2}}\circ F_{\tau_{n}}$. Figure 19: Example 2: Denoising
via $F^{\frac{\sigma}{\beta}}\circ F_{2\frac{\sigma}{\beta}}$.

Finally, we again executed 1000 simulations, using the same methodology as in
the previous case, this time considering the polynomial presented at the
beginning of this example. As we can see in Figure 20 the overestimation
committed by our upper bound is often quite close to zero, relatively to the
Lipschitz constant $L$ of the polynomial $\varphi$.

Figure 20: Example 2: Histogram counting the number of cases in each of ten
bins, concerning the overestimation value
$3L\frac{\sigma}{\beta}-\|F^{\frac{\sigma}{\beta}}\circ
F_{2\frac{\sigma}{\beta}}(\hat{\varphi})-\varphi\|_{\infty}$ obtained in our
simulations. Most of the cases belong to the first two bins, containing the
functions $\varphi$ for which $0\leq
3L\frac{\sigma}{\beta}-\|F^{\frac{\sigma}{\beta}}\circ
F_{2\frac{\sigma}{\beta}}(\hat{\varphi})-\varphi\|_{\infty}\leq 0.24$.

###  6.2 Experiments

In order to check how good the upper bound stated in Theorem 7 is, we have
made the following experiment. In a first step, we have fixed $\ell=20$ and
$\sigma=1.1$, and assumed as given the parameters $L>0$, $N\in\mathbb{N}$,
$\alpha>0$, $\beta>0$, $k\in\mathbb{N}$, $a_{1},\ldots,a_{k}\in]0,\alpha[$,
$b_{1},\ldots,b_{k}>\beta$, $c_{1},\ldots,c_{k}\in]0,\ell[$.

Firstly, we have used the parameters $L>0$, $N\in\mathbb{N}$ to generate a
random $L$-Lipschitz function $\varphi$ in the following way. We have randomly
chosen and sorted in ascending order $N$ points $x_{1},\ldots,x_{N}$ in the
open interval $]0,\ell[$, with uniform distribution. Hence we have obtained
the following decomposition:
$]0,\ell[=]0,x_{1}]\cup]x_{1},x_{2}]\cup\ldots\cup]x_{N},\ell[$. We have
defined our Lipschitz function $\varphi$ to be $0$ outside $]0,\ell[$. After
setting $(x_{0},y_{0})=(0,0)$ and $(x_{N+1},y_{N+1})=(\ell,0)$, for
$i\in\\{1,\ldots,N+1\\}$ the value $y_{i}$ of the function at $x_{i}$ has been
randomly chosen, with uniform distribution in an interval that allows for an
$L$-Lipschitz extension to $[0,\ell]$ of the function, i.e.,
$\left[\max\\{y_{i-1}-L(x_{i}-x_{i-1}),-L(\ell-
x_{i})\\},\min\\{y_{i-1}+L(x_{i}-x_{i-1}),L(\ell-x_{i})\\}\right]$. Finally,
the graph of the Lipschitz function $\varphi$ on $[0,\ell]$ has been obtained
by connecting each point $(x_{i-1},y_{i-1})$ to $(x_{i},y_{i})$ with a
segment, for $i\in\\{1,\ldots,N+1\\}$. We observe that $\varphi$ constructed
this way is an $L$-Lipschitz function.

Secondly, we have used the parameters $\alpha>0$, $\beta>0$, $k\in\mathbb{N}$,
$a_{1},\ldots,a_{k}\in]0,\alpha[$, $b_{1},\ldots,b_{k}>\beta$,
$c_{1},\ldots,c_{k}\in]0,\ell[$ to generate a noise function as follows. We
have considered the mother function $\psi$ defined by setting
$\psi(x):=e^{1-\frac{1}{1-x^{2}}}$ for $x\in\left]-1,1\right[$ and
$\psi(x):=0$ for $x\notin\left]-1,1\right[$. For each $L$-Lipschitz function
$\varphi$ produced in the previously described way, we have considered the
function
$\hat{\varphi}=\varphi+\sum\limits_{i=1}^{k}a_{i}\psi(b_{i}(x-c_{i}))$.

In Figures 21 and 22 some examples of the functions we have produced are
displayed, for $N=3$ and $N=7$, respectively. In each figure, the functions
are displayed without noise (left) and with added noise (right).

|
---|---
Figure 21: Five examples of the functions we have produced for $N=3$. In each figure, the functions are displayed without noise (left) and with added noise (right). |
---|---
Figure 22: Five examples of the functions we have produced for $N=7$. In each
figure, the functions are displayed without noise (left) and with added noise
(right).

In a second step, we have fixed $\ell=20$ and $\sigma=1.1$ once again, and
considered a probabilistic model assuming that $\alpha$, $\beta$, $L$ are
given and the values $N$, $k$, $a_{i}$, $b_{i}$, $c_{i}$ are random variables.
In this setting, we have compared the noise
$\|\hat{\varphi}-\varphi\|_{\infty}$ with the probabilistic upper bound stated
in Theorem 7 and the value $\left\|F^{\frac{\sigma}{\beta}}\circ
F_{2\frac{\sigma}{\beta}}(\hat{\varphi})-\varphi\right\|_{\infty}$,
representing the reduced noise that we can obtain by applying our method. In
order to average our results, for each triplet $(\alpha,\beta,L)$ with
$\alpha\in\\{50,55,60,\ldots,100\\}$, $\beta\in\\{3,4,5,\ldots,13\\}$, and
$L\in\\{1,2,\ldots,10\\}$, we have randomly generated $100$ examples of an
$L$-Lipschitz function $\varphi$ and its noisy version $\hat{\varphi}$, by
randomly choosing the parameters $N$, $k$, $a_{i}$, $b_{i}$, $c_{i}$ according
to the following distributions:

  * •

$N\sim\mathrm{Unif}_{\\{1,\ldots,10\\}}$

  * •

$k\sim\mathrm{Unif}_{\\{1,\ldots,10\\}}$

  * •

$a_{i}\sim\mathrm{Unif}_{\left]0,\alpha\right[}$ for $i=1,\ldots,k$

  * •

$b_{i}\sim\mathrm{Unif}_{\left]\beta,20\right[}$ for $i=1,\ldots,k$

  * •

$c_{i}\sim\mathrm{Unif}_{\left]3\frac{\sigma}{\beta},\ell-3\frac{\sigma}{\beta}\right[}$
for $i=1,\ldots,k$.

Then for the chosen values of each one of the three variables
$\alpha,\beta,L$, we have computed the mean, with respect to the other two
variables, of the average of the noise $\|\hat{\varphi}-\varphi\|_{\infty}$
and the average of the reduced noise $\left\|F^{\frac{\sigma}{\beta}}\circ
F_{2\frac{\sigma}{\beta}}(\hat{\varphi})-\varphi\right\|_{\infty}$ obtained by
our method, both evaluated for $(\varphi,\hat{\varphi})$ varying in the set of
cardinality $100$ that we have produced. We have also computed the mean of the
probabilistic upper bound stated in Theorem 7 with respect to the same two
variables. The results are displayed in Figures 23, 24, and 25. We remind the
reader that $\alpha$ and $\beta$ respectively express the maximum height and
the thinness of the noise bumps, while $L$ is a Lipschitz constant for each
function $\varphi$ we are interested in. These results illustrate the
effectiveness of the use of GENEOs in the reduction of impulsive noise.

Figure 23: Plots of the means of the averaged values of
$\|\hat{\varphi}-\varphi\|_{\infty}$ (yellow), the means of
$3L\frac{\sigma}{\beta}+k\alpha\left(1-\left(1-8\frac{(k-1)}{\ell}\frac{\sigma}{\beta}\right)^{k}\right)$
(brown), and the means of the averaged values of
$\left\|F^{\frac{\sigma}{\beta}}\circ
F_{2\frac{\sigma}{\beta}}(\hat{\varphi})-\varphi\right\|_{\infty}$ (blue) for
$\beta\in\\{3,4,5,\ldots,13\\}$ and $L\in\\{1,2,\ldots,10\\}$, when $\alpha$
varies in the set $\\{50,55,60,\ldots,100\\}$. Figure 24: Plots of the means
of the averaged values of $\|\hat{\varphi}-\varphi\|_{\infty}$ (yellow), the
means of
$3L\frac{\sigma}{\beta}+k\alpha\left(1-\left(1-8\frac{(k-1)}{\ell}\frac{\sigma}{\beta}\right)^{k}\right)$
(brown), and the means of the averaged values of
$\left\|F^{\frac{\sigma}{\beta}}\circ
F_{2\frac{\sigma}{\beta}}(\hat{\varphi})-\varphi\right\|_{\infty}$ (blue) for
$\alpha\in\\{50,55,60,\ldots,100\\}$ and $L\in\\{1,2,\ldots,10\\}$, when
$\beta$ varies in the set $\\{3,4,5,\ldots,13\\}$. Figure 25: Plots of the
means of the averaged values of $\|\hat{\varphi}-\varphi\|_{\infty}$ (yellow),
the means of
$3L\frac{\sigma}{\beta}+k\alpha\left(1-\left(1-8\frac{(k-1)}{\ell}\frac{\sigma}{\beta}\right)^{k}\right)$
(brown), and the means of the averaged values of
$\left\|F^{\frac{\sigma}{\beta}}\circ
F_{2\frac{\sigma}{\beta}}(\hat{\varphi})-\varphi\right\|_{\infty}$ (blue) for
$\alpha\in\\{50,55,60,\ldots,100\\}$ and $\beta\in\\{3,4,5,\ldots,13\\}$, when
$L$ varies in the set $\\{1,2,\ldots,10\\}$.

##  7 Conclusion

In our paper we have proved a stability property for persistence diagrams of
functions from $\mathbb{R}$ to $\mathbb{R}$, in the presence of impulsive
noise. This property shows that TDA can also be of use when noise drastically
changes the topology of the sublevel sets of the filtering functions we are
considering, and stresses some new possible interaction between TDA and the
theory of GENEOs. The experimental section shows that our approach is indeed
able to remove the impulsive noise, in most of the cases. It would be
interesting to check the possibility of extending our method to real-valued
functions defined on $n$-dimensional domains, by selecting suitable GENEOs. We
plan to devote our research to this topic in the future.

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## Statements and Declarations

This research has been partially supported by INdAM-GNSAGA. The authors
declare that they have no conflict of interest. P.F. devised the project. All
authors contributed to the manuscript. All authors read and approved the final
manuscript. The authors of this paper have been listed in alphabetical order.
Data available on request from the authors.


# Signatures of Parafermion Zero Modes in Fractional Quantum Hall-
Superconductor Heterostructures

Junyi Cao  Department of Physics, University of Illinois at Urbana-Champaign,
1110 West Green Street, Urbana, Illinois 61801, USA  Anthony J. Leggett
Institute for Condensed Matter Theory, University of Illinois at Urbana-
Champaign, 1110 West Green Street, Urbana, Illinois 61801, USA  Angela Kou
Department of Physics, University of Illinois at Urbana-Champaign, 1110 West
Green Street, Urbana, Illinois 61801, USA  Materials Research Laboratory,
University of Illinois at Urbana-Champaign, 104 S. Goodwin Ave, Urbana, IL
61801, USA  Eduardo Fradkin  Department of Physics, University of Illinois at
Urbana-Champaign, 1110 West Green Street, Urbana, Illinois 61801, USA  Anthony
J. Leggett Institute for Condensed Matter Theory, University of Illinois at
Urbana-Champaign, 1110 West Green Street, Urbana, Illinois 61801, USA

###### Abstract

Parafermion zero modes can arise in hybrid structures composed of $\nu=1/m$
fractional quantum Hall edges proximitized with an s-wave superconductor. Here
we consider parafermion and Cooper pair tunneling, and backscattering in a
junction formed in such hybrid structures. We find that the $4\pi m$
periodicity due to parafermion-only tunneling reduces, in the presence of
backscattering, to $4\pi$-periodic at zero temperature and $2\pi$-periodic at
finite temperature unless the fermion parity is fixed. Nevertheless, a clear
signature of parafermion tunneling remains in the shape of the current-phase
relation.

Non-Abelian topologically ordered phases are among the most promising
platforms for fault-tolerant quantum computation [1]. The excitations in these
phases are non-Abelian anyons that have non-trivial fusion rules and braiding
statistics [2]. These fusion rules provide a source of topological ground
state degeneracy, which allows for non-local storage of information and anyon
braiding that is topologically protected against decoherence [3, 4, 5, 6, 7].
While there has been significant interest in using Majorana zero modes (MZMs)
for topological quantum computation [1], the $\mathbb{Z}_{N}$ generalization
of MZMs, parafermion zero modes (PZMs) [8], is necessary to perform universal
topological quantum computation. It was shown in Ref. [9] that an array of
PZMs provides a realization of Fibonacci anyons that is capable of universal
topological quantum computation [1, 10, 11, 12].

It has been theoretically proposed that PZMs can arise in fractional
topological superconductors (FTSC) [13]; a key example is a hybrid structure
composed of edge states of a $\nu=1/m$ fractional quantum Hall (FQH) system
proximitized with an s-wave superconductor. A recent experiment has focused on
implementing such a structure in graphene and observed crossed Andreev
reflection (CAR), which was suggested to indicate the presence of PZMs [14]. A
renormalization group (RG) analysis in Ref. [15] showed that CAR is a
necessary but not a sufficient condition for the existence of PZMs. A need has
therefore arisen for a closer investigation into FTSCs and the identification
of additional observable signatures of PZMs in FTSCs. In this letter, we
theoretically consider an FTSC and the experimental signatures that arise when
two FTSCs are incorporated into a Josephson junction geometry. We identify the
PZMs at each end of the junction and determine the energy spectra and current-
phase relation of the junction in the presence of parafermion tunneling,
Cooper pair tunneling, and backscattering. We find, in the low-energy
effective Hamiltonian, that backscattering explicitly breaks the
$\mathbb{Z}_{m}$ symmetry present in the junction, which results in the
periodicity of the Josephson phase being the same for PZMs and MZMs. While the
periodicity is an insufficient distinguishing metric, additional features
arise in the thermally-averaged current-phase relation that discriminate
between PZMs and MZMs. We additionally identify, in the parity-projected
thermally-averaged current-phase relation, a temperature-dependent
$4\pi$-periodic fractional Josephson effect that occurs only under the
presence of parafermion tunneling.

Figure 1: The Josephson junction includes two copies of FTSCs, parafermion
tunneling $\Gamma_{P}$, Cooper pair tunneling $\Gamma_{\Delta}$ and
backscattering $\Gamma_{B}$. Each FTSC consists of two edges from $\nu=1/m$
FQH system (pink) proximity coupled by an s-wave SC (cyan). The spacing
between the two FTSCs is exaggerated to emphasize the presence of a junction.

Model for FTSC and PZM.—We first discuss our model for an FTSC, which consists
of two edges from a (fully spin-polarized) $\nu=1/m$ FQH system proximity
coupled by an s-wave superconductor (SC) finger as in Fig. 1. Since at each
finger, the edges are in proximity with an s-wave superconductor finger, we
include a density-density interaction term and a superconductor term
$\Delta\psi_{L}\psi_{R}+\text{h.c.}$ Here we denote by $L$ and $R$ the top and
bottom FQH edges states in contact with the SC finger. The corresponding
bosonized [16] Hamiltonian is,

$\displaystyle\begin{split}H=&\int_{-L}^{0}{\rm
d}x~{}\bigg{\\{}\frac{mv_{F}}{4\pi}\left[\left(\partial_{x}\phi_{R}\right)^{2}+\left(\partial_{x}\phi_{L}\right)^{2}\right]\\\
&+\frac{mU}{2\pi}\partial_{x}\phi_{L}\partial_{x}\phi_{R}-\frac{{\Delta}}{\ell_{0}^{2}}\cos\left[m\left(\phi_{R}-\phi_{L}\right)\right]\bigg{\\}},\end{split}$
(1)

where $v_{F}$ is the Fermi velocity of the edges, $\phi_{R/L}$ are the bosonic
fields on the top/bottom edge, $U$ is the interaction strength, ${\Delta}>0$
is the (dimensionless) proximity gap, and $\ell_{0}$ is the magnetic length
which plays the role of a UV cutoff.

It is useful to rewrite the bosonic fields as
$\phi_{R/L}=\varphi\pm\vartheta$, which obey the commutation relation,

$\displaystyle\commutator{\varphi(x)}{\vartheta(x^{\prime})}=i\frac{\pi}{m}\Theta(x-x^{\prime}),$
(2)

where $\rho\equiv\partial_{x}\varphi/\pi=\rho_{R}+\rho_{L}$ is the total
charge density operator. The Hamiltonian in the new basis is,

$\displaystyle\begin{split}H=\int_{-L}^{0}{\rm
d}x\bigg{\\{}&\frac{mu}{2\pi}\left[K\left(\partial_{x}\varphi\right)^{2}+\frac{1}{K}\left(\partial_{x}\vartheta\right)^{2}\right]\\\
&-\frac{{\Delta}}{\ell_{0}^{2}}\cos(2m\vartheta)\bigg{\\}},\end{split}$ (3)

with effective velocity $mu/\pi$ with $u=\sqrt{v_{F}^{2}-U^{2}}$ 111For this
letter, we will work in the units with $u=1$. and Luttinger parameter
$K=(v_{F}+U)/\sqrt{v_{F}^{2}-U^{2}}$, making this theory a sine-Gordon theory
on a finite interval. Hence in the vicinity of the critical point, the
proximity gap $\Delta$ and the distance from the Luttinger parameter to the
critical point $x\equiv 2-mK$ follow the leading terms of Kosterlitz RG [16,
18, 19],

$\displaystyle\derivative{{\Delta}}{l}=x{\Delta},~{}\quad\derivative{x}{l}=128m^{2}\pi^{5}{\Delta}^{2}.$
(4)

For the FTSC to be in the superconducting phase, either Luttinger parameter
must satisfy $K<K_{c}=2/m$ 222The Luttinger parameter we use here is related
to the Luttinger parameter in [15] by $K\rightarrow 1/K$., which requires a
sufficiently large attractive interaction, or strong pairing
${\Delta}>{\Delta}_{c}$ is required. Deep in the superconducting phase, the
$\vartheta$ field is pinned to one of the $2m$ minima of the cosine term in
the bulk, $\vartheta(x)=\frac{\tilde{n}\pi}{m}$, where
$\tilde{n}\in\mathbb{Z}_{2m}$ is an integer-valued operator which is related
to the clock operator in the $N$-state clock model on a single site with
$N=2m$ [2, 16].

This Hamiltonian has a $\mathbb{Z}_{2m}$ symmetry,
$\vartheta\rightarrow\vartheta+\frac{\pi}{m}$, representing charge
conservation modulo $e/m$. The corresponding symmetry generator is a
$\mathbb{Z}_{2m}$ generalization of the fermion parity operator $(-1)^{F}$,
which we call quasiparticle parity,

$\displaystyle\hat{P}\equiv\exp(i\pi\hat{Q})=\exp\left\\{i\left[\varphi(0)-\varphi(-L)\right]\right\\},$
(5)

where $Q\equiv\int_{-L}^{0}{\rm d}x~{}\rho(x)$ is the number operator. One can
check that this is indeed the symmetry generator by,

$\displaystyle\hat{P}^{-1}\vartheta\hat{P}=\vartheta+\frac{\pi}{m}.$ (6)

The parafermion operator localized at one end of the FTSC is,

$\displaystyle\alpha_{0}=\frac{1}{b}\int_{-b}^{0}{\rm
d}x~{}e^{i\vartheta(x)}\propto e^{i\vartheta_{0}},$ (7)

where $\vartheta_{0}\equiv\vartheta(x=0)$, $b$ denotes the length of the
region where a PZM is localized and is comparable to the coherence length
$\xi$. From solutions of the sine-Gordon equation, $\vartheta$ has
exponentially small fluctuation within $x\in[-b,0]$; hence, we can treat
$e^{i\vartheta(x)}$ as a constant in this region as well. From the form of the
PZM, we can interpret it as “half” of a quasiparticle pair, which reflects the
fractionalized nature of the system. The commutator between the Hamiltonian
and the PZM is,

$\displaystyle\commutator{H}{\alpha_{0}}$
$\displaystyle=-K\partial_{x}\varphi_{0}e^{i\vartheta_{0}}=0.$ (8)

The commutator equals zero because the total current density
$j=\partial_{x}\varphi/\pi$ is 0 at $x=0$, representing no total current
flowing from the edge to the FQH background. The commutation relation between
the parafermion zero mode operator $\alpha_{0}$ and the quasiparticle parity
$\hat{P}$ is,

$\displaystyle\hat{P}\alpha_{0}$
$\displaystyle=e^{i\frac{\pi}{m}}\alpha_{0}\hat{P}.$ (9)

Physically this means that the parafermion operator changes the number
operator by one.

Josephson junction and tunneling.—One way to experimentally identify
topologically nontrivial zero modes is the fractional Josephson effect [21,
13], i.e. $4m\pi$ periodic signal in the current-phase relation. The Josephson
junction of interest consists of two copies of FTSCs as in Fig. 1,

$\displaystyle\begin{split}H=\int_{-L}^{0}&{\rm
d}x~{}\bigg{\\{}\sum_{i=1,2}\frac{m}{2\pi}\left[K\left(\partial_{x}\varphi^{(i)}\right)^{2}+\frac{1}{K}\left(\partial_{x}\vartheta^{(i)}\right)^{2}\right]\\\
&-\frac{\Delta}{\ell_{0}^{2}}\cos(2m\vartheta^{(1)})-\frac{\Delta}{\ell_{0}^{2}}\cos(2m\vartheta^{(2)}-\delta\phi_{sc})\bigg{\\}},\end{split}$
(10)

where $\delta\phi_{sc}$ is the Josephson phase between the two SCs, and
$i=1,2$ describes the edges near the left/right SC, respectively. This system
has symmetries including an overall $\mathbb{Z}_{2m}$ quasiparticle parity and
charge conjugation
$\vartheta^{(i)}\rightarrow-\vartheta^{(i)},~{}\varphi^{(i)}\rightarrow\varphi^{(i)}$.
The allowed tunneling between two FTSCs are parafermion tunneling (charge
$e/m$), Cooper pair tunneling (charge $2e$), and backscattering (charge 0)
333Terms involving tunneling of a group of parafermions are also allowed, e.g.
Majorana fermion tunneling can be understood as tunneling of a group of $m$
parafermions. We will not study these here the tunneling amplitudes are in
general smaller than $\Gamma_{P}$ and the results can be derived from the
parafermion tunneling results..

The parafermion tunneling Hamiltonian can be written as,

$\displaystyle\begin{split}H_{P}&=\Gamma_{P}\alpha^{(1)\dagger}_{0}\alpha^{(2)}_{0}+\text{h.c.}\\\
&=\Gamma_{P}e^{i\left(\vartheta^{(1)}_{0}-\vartheta^{(2)}_{0}-\frac{\delta\phi_{sc}}{2m}\right)}+\text{h.c.},\end{split}$
(11)

where $\Gamma_{P}$ is the parafermion tunneling amplitude. Similarly, the
Cooper pair tunneling can be written as,

$\displaystyle H_{\Delta}$
$\displaystyle=\Gamma_{\Delta}e^{2mi\left(\vartheta^{(1)}_{0}-\vartheta^{(2)}_{0}-\frac{\delta\phi_{sc}}{2m}\right)}+\text{h.c.}$
(12)

We can see the Josephson phase $\delta\phi_{\text{sc}}$ is $4m\pi$ periodic.
Another process consistent with symmetry is the tunneling of “half” of a
quasiparticle-quasihole pair, which we call backscattering,

$\displaystyle H_{B}$
$\displaystyle=\Gamma_{B}e^{i\left(\varphi^{(1)}_{0}-\varphi^{(2)}_{0}\right)}+\text{h.c.}$
(13)

This tunneling term tunnels 0 charge; hence, it does not couple to the
electromagnetic field and does not contribute to tunneling current.

Effective Hamiltonian for tunneling through the Josephson junction.—In this
section, we consider energies below the superconducting potential,
$\absolutevalue{E}\ll\Delta$, where the system can be described by an
effective Hamiltonian including all of the tunneling processes. The effective
Hamiltonian can then be written in the basis where $e^{i\varphi^{(i)}_{0}}$
are diagonal, with eigenvalues $e^{in^{(i)}\pi/m}$, where $n^{(i)}$ can be
thought as the eigenvalue of the number operator $n^{(i)}\in\mathbb{Z}_{2m}$.
In this basis, the parafermion operator $e^{i\vartheta^{(i)}_{0}}$ shifts
$n^{(i)}$ by one and the Cooper pair tunneling term shifts $n^{(i)}$ by $2m$,
which is equivalent to not changing $n^{(i)}$. We find the effective
Hamiltonian,

$\displaystyle\begin{split}H_{\text{eff}}=\sum_{n^{(1)},n^{(2)}=0}^{2m-1}&\left\\{2\absolutevalue{\Gamma_{B}}\cos\left[\frac{(n^{(1)}-n^{(2)})\pi}{m}\right]+2\absolutevalue{\Gamma_{\Delta}}\cos(\delta\phi_{sc})\right\\}\outerproduct{n^{(1)},n^{(2)}}{n^{(1)},n^{(2)}}\\\
&+\absolutevalue{\Gamma_{P}}\left(e^{-i\frac{\delta\phi_{sc}}{2m}}\outerproduct{n^{(1)}+1,n^{(2)}-1}{n^{(1)},n^{(2)}}+e^{i\frac{\delta\phi_{sc}}{2m}}\outerproduct{n^{(1)},n^{(2)}}{n^{(1)}+1,n^{(2)}-1}\right).\end{split}$
(14)

Since $n^{(i)}\in\mathbb{Z}_{2m}$ and all tunneling terms conserve the total
quasiparticle parity $n^{(1)}+n^{(2)}\mod 2m$, this Hamiltonian can be block
diagonalized. In the following, we only consider the effective Hamiltonian in
the sector where the total quasiparticle parity is zero. The wavefunction
$\Psi_{r}(\delta\phi_{sc})$ satisfies a Harper-like equation [23],

$\displaystyle\begin{split}&\absolutevalue{\Gamma_{P}}\left[e^{-i\frac{\delta\phi_{sc}}{2m}}\Psi_{r+1}(\delta\phi_{sc})+e^{i\frac{\delta\phi_{sc}}{2m}}\Psi_{r-1}\left(\delta\phi_{sc}\right)\right]\\\
&+\left[2\absolutevalue{\Gamma_{B}}\cos(2\pi\frac{r}{m})+2\absolutevalue{\Gamma_{D}}\cos(\delta\phi_{sc})\right]\Psi_{r}(\delta\phi_{sc})\\\
&=E_{r}(\delta\phi_{sc})\Psi_{r}\left(\delta\phi_{sc}\right),\end{split}$ (15)

with $1\leq r\leq 2m$. The eigenstates satisfy the periodic boundary condition
$\Psi_{r}\left(\frac{\delta\phi_{sc}}{2m}+2\pi
k\right)=\Psi_{r}\left(\frac{\delta\phi_{sc}}{2m}\right)$ with
$k\in\mathbb{Z}$.

(a)

(b)

(c)

(d)

Figure 2: Spectra of tunneling effective Hamiltonian as functions of the
Josephson phase $\delta\phi_{sc}$ with (a) only parafermion tunneling, (b)
parafermion and Cooper pair tunneling, (c) parafermion tunneling and
backscattering and (d) backscattering, parafermion, and Cooper pair tunneling.
The dashed lines can be mapped to the solid line by symmetry transformations,
whereas the translucent lines cannot be mapped to the opaque ones.

We can see the effect of each tunneling term from Eq. (14) and Fig. 2. The
Cooper pair tunneling term is proportional to identity; hence it only provides
a $\delta\phi_{sc}$-dependent shift to all states. With no backscattering, the
eigenstates satisfy the boundary condition
$\Psi_{r}(\delta\phi_{sc}+2\pi)=\Psi_{r+1}(\delta\phi_{sc})$ and the
Hamiltonian is invariant under $\mathbb{Z}_{2m}$ transformation
$n^{(1)}\rightarrow n^{(1)}+1\mod 2m$. The backscattering term explicitly
breaks the $\mathbb{Z}_{2m}$ down to $\mathbb{Z}_{2}$, corresponding to
$n^{(1)}\rightarrow n^{(1)}+m\mod 2m$. The eigenstates now satisfy a different
boundary condition,
$\Psi_{r}(\delta\phi_{sc}+2\pi)=\Psi_{r+m}(\delta\phi_{sc})$.

These effects demonstrate that the $\mathbb{Z}_{m}$ part of the symmetry is
inherently different from the $\mathbb{Z}_{2}$ fermion parity. The unbroken
$\mathbb{Z}_{2}$ represents the topologically protected fermion parity and can
only be broken by nonlocal terms like creating a pair of parafermions at the
two ends of the FTSC, $\alpha_{-L}^{\dagger}\alpha_{0}$, whereas the
$\mathbb{Z}_{m}$ symmetry can be broken by local tunneling terms like
tunneling of a quasiparticle and a quasihole
$\psi_{R,qp}^{(1)\dagger}\psi_{L,qp}^{(1)}\psi_{R,qp}^{(2)}\psi_{L,qp}^{(2)\dagger}\sim\cos[2(\varphi_{0}^{(1)}-\varphi_{0}^{(2)})]$.
This suggests that in systems with backscattering, one cannot distinguish PZM
from MZM tunneling from the periodicity of energy-phase relation since both of
them have $\mathbb{Z}_{2}$ symmetry.

Tunneling current.—The difference between the $\mathbb{Z}_{2}$ and
$\mathbb{Z}_{m}$ part of the quasiparticle parity can also be seen in the
current-phase relation where the Josephson phase is $4m\pi$-periodic without
the backscattering term and $4\pi$-periodic with it as shown in Fig. 3. From
our effective Hamiltonian, the tunneling current operator is given by the
commutator between tunneling Hamiltonian and the total number operator
$\hat{N}^{(2)}=\left(\varphi^{(2)}_{-L}-\varphi^{(2)}_{0}\right)/\pi$ 444As an
operator, this definition is equivalent to
$\hat{I}(\delta\phi_{sc})=2e\derivative{H_{\text{eff}}}{\delta\phi_{sc}}$.,

$\displaystyle\hat{I}(\delta\phi_{sc})=$ $\displaystyle
e\derivative{\hat{N}^{(2)}}{t}=ie\commutator{H_{\text{eff}}}{\hat{N}^{(2)}}$
$\displaystyle\begin{split}=&\frac{2e}{m}\absolutevalue{\Gamma_{P}}\sin(\vartheta^{(1)}_{0}-\vartheta^{(2)}_{0}-\frac{\delta\phi_{sc}}{2m})\\\
&+4e\absolutevalue{\Gamma_{\Delta}}\sin\left[2m\left(\vartheta^{(1)}_{0}-\vartheta^{(2)}_{0}\right)-\delta\phi_{sc}\right].\end{split}$
(16)

The tunneling current for each eigenstate $\ket{\Psi_{r}}$ of Eq. (14) is then
given by,

$\displaystyle
I_{r}(\delta\phi_{sc})=\frac{\expectationvalue{\hat{I}}{\Psi_{r}}}{\innerproduct{\Psi_{r}}{\Psi_{r}}}=2e\derivative{E_{r}}{\delta\phi_{sc}}.$
(17)

We can see from Fig. 3(c,d) that the backscattering term explicitly breaks the
$\mathbb{Z}_{m}$ symmetry and results in a $4\pi$ periodicity in the Josephson
current.

(a)

(b)

(c)

(d)

Figure 3: Single-channel current-phase relations $I_{r}(\delta\phi_{sc})$ with
$\nu=1/3$ for (a) only parafermion tunneling, (b) parafermion and Cooper pair
tunneling, (c) parafermion tunneling and backscattering, (d) backscattering,
parafermion, and Cooper pair tunneling. Currents associated with additional
parafermion states that are not shown are related to the currents in this
figure by symmetry transformations of the eigenstates, represented by $2\pi$
shifts in the Josephson phase. The currents have the same color as the
corresponding energy states in Fig. 2.

If there are no terms violating the total quasiparticle parity at the
tunneling junction, e.g. terms proportional to
$\alpha^{(1)\dagger}_{L}\alpha^{(1)}_{0}$, there will be no transitions
between different channels of the tunneling current. However, at finite
temperature, the current thermalizes and becomes,

$\displaystyle\expectationvalue{\hat{I}(\delta\phi_{sc})}_{\beta}=\frac{\tr\left(e^{-\beta
H_{\text{eff}}}\hat{I}\right)}{\tr e^{-\beta H_{\text{eff}}}},$ (18)

at inverse temperature $\beta=1/T$. For $\nu=1$ with
$\absolutevalue{\Gamma_{\Delta}}=\absolutevalue{\Gamma_{B}}=0$, Eq. (18)
reduces to the known result [25, 26]. In the thermally-averaged current-phase
relation Eq. (18), each of the tunneling terms has different contributions.
The thermally-averaged current-phase relation for parafermion and Majorana
fermion tunneling are shown in Fig. 4(a,b). Both the parafermion and Majorana
fermion terms exhibit a zig-zag with a slope proportional to $\beta$ whereas
the Cooper pair tunneling term has a sine-wave contribution 555There is a
difference in the relative amplitude between Majorana fermion/parafermion
tunneling and Cooper pair tunneling contribution in Eq. (16).. Backscattering
does not contribute to the shape of the thermally-averaged current. All
thermally-averaged currents exhibit $2\pi$-periodicity due to the
contributions from states with different quasiparticle parity.

The fractional Josephson effect can arise at finite temperature if one is
capable of projecting into individual states. We can define the projected
current using a projection operator $\hat{P}_{r}$ in the definition of trace
in Eq. (18),

$\expectationvalue{I_{r}(\delta\phi_{sc})}_{\beta}=\frac{\tr(e^{-\beta
H_{\text{eff}}}\hat{P}_{r}\hat{I})}{\tr(e^{-\beta
H_{\text{eff}}}\hat{P}_{r})}.$ (19)

These projection operators $\hat{P}_{r}$ are generalizations of the fermion
parity projection operators $\hat{P}_{\pm}=[\mathbb{1}\pm(-1)^{F}]/2$, and $r$
depends on the parity symmetry of the system, i.e. it is $\pm$ if there is
finite backscattering and ranges from 1 to $2m$ otherwise. The projection
operators can be obtained by a linear combination of powers of the clock
matrix $\sigma$ [16]. In experiments, these projections can be realized by
fixing the charge difference between the two FTSCs, represented by
$n^{(1)}-n^{(2)}\mod 2m$ in Eq. (14). If there is no backscattering, the
projected currents are equivalent to the single-channel currents in Eq. (17),
similar to the $\nu=1$ case in Fig. 4(c). When there is backscattering, as in
Fig. 4(d), the parafermion tunneling adds a $4\pi$-periodic contribution to
the current with the amplitude that scales as a power-law of inverse
temperature $\beta$. This behavior is unique to parafermion tunneling, and
therefore, is a fingerprint of PZMs.

(a)

(b)

(c)

(d)

Figure 4: Thermal average of the current-phase relation with all tunneling
terms for different values of $\beta$ for (a) $\nu=1$ and (b) $\nu=1/3$.
Projected thermal average of the current-phase relation with all tunneling
terms for different values of $\beta$ for (c) $\nu=1$ and (d) $\nu=1/3$.
$\beta=0.01$ (blue), 0.1 (orange), 1 (green) in units of
$\absolutevalue{\Gamma_{M}}^{-1}$ in (a,c) and
$\absolutevalue{\Gamma_{P}}^{-1}$ in (b,d).

Discussion.—We have presented a model for FTSC consisting of two edge states
from a $\nu=1/m$ FQH system proximity coupled by an s-wave superconductor and
identified a PZM at one end of the FTSC. We have constructed an effective
Hamiltonian and shown the roles played by parafermion tunneling, Cooper pair
tunneling, and backscattering by identifying the symmetry of the ground states
and Josephson periodicity in different tunneling currents. We showed that for
a $\mathbb{Z}_{2m}$ FTSC, only the $\mathbb{Z}_{2}$ part of the symmetry is
topologically protected while the $\mathbb{Z}_{m}$ part can be explicitly
broken by local tunneling terms like tunneling of a quasiparticle and a
quasihole. We proposed using a projected thermal average of the current to
detect parafermion tunneling, which has a $4\pi$-periodic fractional Josephson
effect.

Although the results we have presented are true for general values of the
tunneling amplitudes, the relative strength of the tunneling terms is
important for experimental detection of the fractional Josephson effect, i.e.,
the parafermion tunneling term should be larger than the Cooper pair tunneling
and the backscattering term. For a Josephson junction with gap distance $l$,
the backscattering amplitude is expected to scale as $\Gamma_{B}\sim
e^{-l/\ell_{0}}$. The parafermion and Cooper pair tunneling amplitude are
expected to scale as $\Gamma_{P},~{}\Gamma_{\Delta}\sim e^{-l/\xi}$. As a
realistic example, we can consider niobium nitride, which under a magnetic
field of 6 T where $\nu=1/3$ should arise, has a larger coherence length
($\sim$50 nm) than the magnetic length ($\sim$10 nm). Hence, we expect the
backscattering to be much weaker than both Cooper pair and parafermion
tunneling. We naively expect the Cooper pairing amplitude to be much smaller
than the parafermion tunneling because the tunneling of multiple charges is
suppressed through an FQH background. The effects described in this letter
should therefore be observable in experiments. Additional screening layers may
also be beneficial for entering the regime where parafermion tunneling
dominates.

Acknowledgments.—We thank Erez Berg, Paul Fendley, Netanel Lindner, Yi-Zhuang
You, and Lucas Wagner for insightful discussions. This work was supported in
part by the US National Science Foundation (NSF) through the grant NSF
DMR-2225920 at the University of Illinois (JC and EF) and through the NSF
Quantum Leap Challenge Institute for Hybrid Quantum Architectures and
Networks, NSF OMA-2016136 (AK).

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