Stabilizing the unstable output of persistent homology computations

TL;DR

This paper introduces a technique to extract more stable and informative features from persistent homology computations, improving robustness for practical applications like brain imaging analysis.

Contribution

The authors develop a method to stabilize unstable information in persistent homology outputs by convolving discontinuous functions, enabling more reliable data analysis.

Findings

Enhanced stability of persistent homology features

Successful localization of homology generators in brain data

Applicable to various practical data analysis scenarios

Abstract

We propose a general technique for extracting a larger set of stable information from persistent homology computations than is currently done. The persistent homology algorithm is usually viewed as a procedure which starts with a filtered complex and ends with a persistence diagram. This procedure is stable (at least to certain types of perturbations of the input). This justifies the use of the diagram as a signature of the input, and the use of features derived from it in statistics and machine learning. However, these computations also produce other information of great interest to practitioners that is unfortunately unstable. For example, each point in the diagram corresponds to a simplex whose addition in the filtration results in the birth of the corresponding persistent homology class, but this correspondence is unstable. In addition, the persistence diagram is not stable with respect to other procedures that are employed in practice, such as thresholding a point cloud by density. We recast these problems as real-valued functions which are discontinuous but measurable, and then observe that convolving such a function with a suitable function produces a Lipschitz function. The resulting stable function can be estimated by perturbing the input and averaging the output. We illustrate this approach with a number of examples, including a stable localization of a persistent homology generator from brain imaging data.