Persistent and Zigzag Homology: A Matrix Factorization Viewpoint
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Over the past two decades, topological data analysis has emerged as a young field of applied mathematics with new applications and algorithmic developments appearing rapidly. Two fundamental computations in this field are persistent homology and zigzag homology. In this paper, we show how these computations in the most general case reduce to finding a canonical form of a matrix associated with a type-A quiver representation, which in turn can be computed using factorizations of associated matrices. We show how to use arbitrary induced maps on homology for computation, providing a framework that goes beyond the capabilities of existing software for topological data analysis. Furthermore, this framework offers multiple opportunities for parallelization which have not been previously explored.
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