Noise robustness of persistent homology on greyscale images, across filtrations and signatures

TL;DR

This paper evaluates how noise affects persistent homology in greyscale images, revealing that the robustness of topological features varies with different filtrations and signatures, impacting classification accuracy.

Contribution

It provides an empirical analysis of noise robustness in persistent homology applied to MNIST images, highlighting the influence of filtration choices on stability.

Findings

PH features are often sensitive to noise in classification tasks

The choice of filtrations and signatures affects noise robustness

Noise can significantly reduce classification performance

Abstract

Topological data analysis is a recent and fast growing field that approaches the analysis of datasets using techniques from (algebraic) topology. Its main tool, persistent homology (PH), has seen a notable increase in applications in the last decade. Often cited as the most favourable property of PH and the main reason for practical success are the stability theorems that give theoretical results about noise robustness, since real data is typically contaminated with noise or measurement errors. However, little attention has been paid to what these stability theorems mean in practice. To gain some insight into this question, we evaluate the noise robustness of PH on the MNIST dataset of greyscale images. More precisely, we investigate to what extent PH changes under typical forms of image noise, and quantify the loss of performance in classifying the MNIST handwritten digits when noise is added to the data. The results show that the sensitivity to noise of PH is influenced by the choice of filtrations and persistence signatures (respectively the input and output of PH), and in particular, that PH features are often not robust to noise in a classification task.