Homology groups of filtrations

TL;DR

This paper introduces a new homology group for filtrations that captures all possible homology classes across a filtration, aiding in noise robustness in topological data analysis.

Contribution

It defines the homology group of a filtration as a product of kernels of homology maps, providing a comprehensive feature set for analyzing noisy data.

Findings

Defines homology group of filtration as a product of kernels

Enables selection of features within noise tolerance

Enhances robustness in topological data analysis

Abstract

Such modern applications of topology as data analysis and digital image analysis have to deal with noise and other uncertainty. In this environment, topological spaces often appear equipped with a real valued function. Persistence is a measure of robustness of the homology classes of the filtration of the lower level sets of this function. In this paper we introduce the homology group of filtration as the product of the kernels of the homology maps of the inclusions. This group contains all possible homology classes in all elements of the filtration so that we can later pick the features that lie within the user's choice of the acceptable level of noise.