Filtration learning in exact multi-parameter persistent homology and classification of time-series data

TL;DR

This paper introduces a framework for optimizing filtrations in exact multi-parameter persistent homology to improve time-series data classification, offering a direct gradient computation method for efficient learning.

Contribution

It proposes a novel filtration learning approach for EMPH with an explicit gradient formula, enabling direct filtration updates without automatic differentiation.

Findings

Effective filtration optimization for EMPH in classification tasks

Explicit gradient formula accelerates the learning process

Improved topological analysis of time-series data

Abstract

To analyze the topological properties of the given discrete data, one needs to consider a continuous transform called filtration. Persistent homology serves as a tool to track changes of homology in the filtration. The outcome of the topological analysis of data varies depending on the choice of filtration, making the selection of filtration crucial. Filtration learning is an attempt to find an optimal filtration that minimizes the loss function. Exact Multi-parameter Persistent Homology (EMPH) has been recently proposed, particularly for topological time-series analysis, that utilizes the exact formula of rank invariant instead of calculating it. In this paper, we propose a framework for filtration learning of EMPH. We formulate an optimization problem and propose an algorithm for solving the problem. We then apply the proposed algorithm to several classification problems. Particularly, we derive the exact formula of the gradient of the loss function with respect to the filtration parameters, which makes it possible to directly update the filtration without using automatic differentiation, significantly enhancing the learning process.