A Robust Persistent Homology : Trimming Approach

TL;DR

This paper introduces a robust persistent homology method using trimming to effectively capture geometric features despite outliers, applicable to both external and internal outliers, with theoretical guarantees and practical validation.

Contribution

It proposes a novel trimming-based robust persistent homology approach with proven convergence properties and demonstrates its effectiveness on simulated and real biological data.

Findings

The method achieves small Bottleneck distance to the population analogue with large samples.

It is effective in handling outliers both outside and inside the data cloud.

Validated on cellular biology datasets with promising results.

Abstract

This article studies the robust version of persistent homology based on trimming methodology to capture the geometric feature through support of the data in presence of outliers. Precisely speaking, the proposed methodology works when the outliers lie outside the main data cloud as well as inside the data cloud. In the course of theoretical study, it is established that the Bottleneck distance between the proposed robust version of persistent homology and its population analogue can be made arbitrary small with a certain rate for a sufficiently large sample size. The practicability of the methodology is shown for various simulated data and bench mark real data associated with cellular biology.