A Probabilistic Result on Impulsive Noise Reduction in Topological Data Analysis through Group Equivariant Non-Expansive Operators

Patrizio Frosini; Ivan Gridelli; Andrea Pascucci

doi:10.3390/e25081150

A Probabilistic Result on Impulsive Noise Reduction in Topological Data Analysis through Group Equivariant Non-Expansive Operators

Entropy (Basel). 2023 Jul 31;25(8):1150. doi: 10.3390/e25081150.

Authors

Patrizio Frosini¹, Ivan Gridelli¹, Andrea Pascucci¹

Affiliation

¹ Department of Mathematics, University of Bologna, 40126 Bologna, Italy.

Abstract

In recent years, group equivariant non-expansive operators (GENEOs) have started to find applications 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 R to R.

Keywords: GENEO; impulsive noise; machine learning; persistence diagram; persistent homology.

Grants and funding

This research is supported by the European Union in the context of the H2020 EU Marie Curie Initial Training Network project named WAKEUPCALL, and by INdAM-GNSAGA.