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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 thecombinatorial Laplacian, and the resultant Hodge decomposition. Meanwhile,discrete Morse theory has been widely used to speed up computations by reducingthe size of complexes while preserving their global topological properties. Inthis paper, we provide an approach to signal compression and reconstruction onchain complexes that leverages the tools of algebraic discrete Morse theory.The main goal is to reduce and reconstruct a based chain complex together witha set of signals on its cells via deformation retracts, preserving as much aspossible the global topological structure of both the complex and the signals.We first prove that any deformation retract of real degree-wisefinite-dimensional based chain complexes is equivalent to a Morse matching. Wewill then study how the signal changes under particular types of Morsematching, showing its reconstruction error is trivial on specific components ofthe Hodge decomposition. Furthermore, we provide an algorithm to compute Morsematchings with minimal reconstruction error.

Morse Theoretic Signal Compression and Reconstruction on Chain Complexes