Keywords: Topological Data Analysis, Algebraic Topology, Persistent Homology, Kernel Methods
TL;DR: In this article, we provide a general framework for representing multiparameter persistent homology with stability guarantees.
Abstract: Topological data analysis (TDA) is a new area of geometric data analysis that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for point clouds. One of the most important shape descriptors is persistent homology, which studies the topological variations as a filtration parameter changes; a typical parameter is the feature scale. For many data sets, it is useful to consider varying multiple filtration parameters at once, for example scale and density. While the theoretical properties of one-parameter persistent homology are well understood, less is known about the multiparameter case. Of particular interest is the problem of representing multiparameter persistent homology by elements of a vector space for integration with traditional machine learning. Existing approaches to this problem either ignore most of the multiparameter information to reduce to the one-parameter case or are heuristic and potentially unstable in the face of noise. In this article, we introduce a general representation framework for multiparameter persistent homology that encompasses previous approaches. We establish theoretical stability guarantees under this framework as well as efficient algorithms for practical computation, making this framework an applicable and versatile tool for TDA practitioners. We validate our stability results and algorithms with numerical experiments that demonstrate statistical convergence, prediction accuracy, and fast running times on several real data sets.
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