  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.
