Computing Persistent Homology via Discrete Morse Theory

This report provides theoretical justification for the use of discrete Morse theory for the computation of homology and persistent homology, an overview of the state of the art for the computation of discrete Morse matchings and motivation for an interest in these computations, particularly from the point of view of topological data analysis. Additionally, a new simulated annealing based method for computing discrete Morse matchings is presented. For several problem instances this outperforms the best known heuristics for the task. The computation of homology and persistent homology has become an important task in computational topology, with applications in fields such as topological data analysis, computer vision and materials science. Unfortunately computing homology is currently infeasible for large input complexes. Discrete Morse theory enables the preprocessing of homology computation by reducing the size of the input complexes. This is advantageous from a memory and performance point of view. The key to making efficient use of discrete Morse theory is the quick computation of optimal, or good, discrete Morse matchings.