Morse Theoretic Signal Compression and Reconstruction on Chain Complexes

TL;DR

This paper introduces a novel method combining algebraic discrete Morse theory with topological data analysis to efficiently compress and reconstruct signals on chain complexes while preserving their topological features.

Contribution

It develops a framework for signal compression on chain complexes using Morse matchings, with algorithms to minimize reconstruction error and preserve topological structure.

Findings

Reconstruction error is trivial on specific Hodge components.

Any deformation retract of chain complexes corresponds to a Morse matching.

Algorithm for minimal-error Morse matchings is provided.

Abstract

At the intersection of Topological Data Analysis (TDA) and machine learning, the field of cellular signal processing has advanced rapidly in recent years. In this context, each signal on the cells of a complex is processed using the combinatorial Laplacian, and the resultant Hodge decomposition. Meanwhile, discrete Morse theory has been widely used to speed up computations by reducing the size of complexes while preserving their global topological properties. In this paper, we provide an approach to signal compression and reconstruction on chain complexes that leverages the tools of algebraic discrete Morse theory. The main goal is to reduce and reconstruct a based chain complex together with a set of signals on its cells via deformation retracts, preserving as much as possible the global topological structure of both the complex and the signals. We first prove that any deformation retract of real degree-wise finite-dimensional based chain complexes is equivalent to a Morse matching. We will then study how the signal changes under particular types of Morse matching, showing its reconstruction error is trivial on specific components of the Hodge decomposition. Furthermore, we provide an algorithm to compute Morse matchings with minimal reconstruction error.