Auto-encoders on chain complexes using algebraic discrete Morse theory

In this work, 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, with the aim of preserving parts of the global topological structure of both the complex and the signals.

We show that any deformation retract of a real degree-wise finite-dimensional based chain complexes is equivalent to a Morse matching. We 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 which locally minimizes reconstruction error.

This is joint work with Stefania Ebli and Celia Hacker.