Stable Similarity Comparison of Persistent Homology Groups

TL;DR

This paper introduces a new pseudometric for comparing persistent homology groups that is invariant under similarity transformations, computationally efficient, and stable under conformal linear transformations, with demonstrated effectiveness on synthetic and real data.

Contribution

It defines a novel similarity-invariant pseudometric for persistent homology, linking Operator Theory and Topological Data Analysis, and shows its advantages over existing distances.

Findings

The pseudometric $d_{S}^{(2)}$ is stable under conformal linear transformations.

It is computationally faster than bottleneck distance.

The pseudometric is independent of frequency and amplitude in wave data.

Abstract

Classification in the sense of similarity is an important issue. In this paper, we study similarity classification in Topological Data Analysis. We define a pseudometric $d_{S}^{(p)}$ to measure the distance between barcodes generated by persistent homology groups of topological spaces, and we provide that our pseudometric $d_{S}^{(2)}$ is a similarity invariant. Thereby, we establish a connection between Operator Theory and Topological Data Analysis. We give the calculation formula of the pseudometric $d_{S}^{(2)}$ $(d_{S}^{(1)})$ by arranging all eigenvalues of matrices determined by barcodes in descending order to get the infimum over all matchings. Since conformal linear transformation is one representative type of similarity transformations, we construct comparative experiments on both synthetic datasets and waves from an online platform to demonstrate that our pseudometric $d_{S}^{(2)}$ $(d_{S}^{(1)})$ is stable under conformal linear transformations, whereas the bottleneck and Wasserstein distances are not. In particular, our pseudometric on waves is only related to the waveform but is independent on the frequency and amplitude. Furthermore, the computation time for $d_{S}^{(2)}$ $(d_{S}^{(1)})$ is significantly less than the computation time for bottleneck distance and is comparable to the computation time for accelerated Wasserstein distance between barcodes.