Discrete signature tensors for persistence landscapes

TL;DR

This paper introduces the discrete landscape feature map (DLFM), a new method for analyzing persistence landscapes via discrete signature tensors, improving injectivity and stability for data analysis tasks.

Contribution

It proposes the DLFM, a discrete alternative to continuous signature maps for persistence landscapes, with proven injectivity, stability, and applicability to biological data.

Findings

DLFM captures sequence similarity and knot depth in protein data

The method is stable and computationally feasible

It enhances analysis of persistence landscapes in machine learning

Abstract

Signature tensors of paths are a versatile tool for data analysis and machine learning. Recently, they have been applied to persistent homology, by embedding barcodes into spaces of paths. Among the different path embeddings, the persistence landscape embedding is injective and stable, however it loses injectivity when composed with the signature map. Here we address this by proposing a discrete alternative. The critical points of a persistence landscape form a time-series, of which we compute the discrete signature. We call this association the discrete landscape feature map (DLFM). We give results on the injectivity, stability and computability of the DLFM. We apply it to a knotted protein dataset, capturing sequence similarity and knot depth with statistical significance.