Connecting Discrete Morse Theory and Persistence: Wrap Complexes and Lexicographic Optimal Cycles
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We study the connection between discrete Morse theory and persistent homology in the context of shape reconstruction methods. Specifically, we compare the Wrap complex, introduced by Edelsbrunner as a subcomplex of the Delaunay complex, and the lexicographic optimal homologous chains, considered by Cohen-Steiner, Lieutier, and Vuillamy. We show that given any cycle in a Delaunay complex at some radius threshold, the lexicographically optimal homologous cycle is supported on the Wrap complex at the same threshold, thereby establishing a close connection between the two methods. This result is obtained as a consequence of a general connection between reduction of cycles in the computation of persistent homology and gradient flows in the algebraic generalization of discrete Morse theory, which is of independent interest.
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