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

TL;DR

This paper demonstrates how group equivariant non-expansive operators (GENEOs) can effectively reduce impulsive noise in topological data analysis, enhancing stability and robustness of persistence diagrams under noisy conditions.

Contribution

It introduces a novel application of GENEOs for impulsive noise reduction in TDA and provides a theoretical proof of their effectiveness in controlling perturbations.

Findings

GENEOs can control expected perturbation of persistence diagrams

The approach improves stability of TDA with noisy data

The method is applicable to Lipschitz functions from real to real

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 $\mathbb{R}$.