Discrete Morse Theory for Computing Zigzag Persistence

We introduce a theoretical and computational framework to use discrete Morse theory as an efficient preprocessing in order to compute zigzag persistent homology. From a zigzag filtration of complexes $(K_i)$, we introduce a {\em zigzag Morse filtration} whose complexes $(A_i)$ are Morse reductions of the original complexes $(K_i)$, and we prove that they both have same persistent homology. This zigzag Morse filtration generalizes the {\em filtered Morse complex} of Mischaikow and Nanda~\cite{MischaikowN13}, defined for standard persistence. The maps in the zigzag Morse filtration are forward and backward inclusions, as is standard in zigzag persistence, as well as a new type of map inducing non trivial changes in the boundary operator of the Morse complex. We study in details this last map, and design algorithms to compute the update both at the complex level and at the homology matrix level when computing zigzag persistence. We deduce an algorithm to compute the zigzag persistence of a filtration that depends mostly on the number of critical cells of the complexes, and show experimentally that it performs better in practice.