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

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 $\mathbb{R}$.

The title has been changed and the experimental section has been extended; 27 pages, 25 figures

References in corpus (3)

• Confidence sets for persistence diagrams
• Topological analysis of scalar fields with outliers
• Robust Persistence Diagrams using Reproducing Kernels