Shape-Preserving Dimensionality Reduction : An Algorithm and Measures of Topological Equivalence

2021/06/03 by Yu, Byeongsu, You, Kisung
#FOS: Computer and information sciences #Machine Learning (cs.LG) #Machine Learning (stat.ML)

paper · doi:10.48550/arxiv.2106.02096

Abstract

We introduce a linear dimensionality reduction technique preserving topological features via persistent homology. The method is designed to find linear projection L which preserves the persistent diagram of a point cloud \mathbbX via simulated annealing. The projection L induces a set of canonical simplicial maps from the Rips (or Čech) filtration of \mathbbX to that of L\mathbbX. In addition to the distance between persistent diagrams, the projection induces a map between filtrations, called filtration homomorphism. Using the filtration homomorphism, one can measure the difference between shapes of two filtrations directly comparing simplicial complexes with respect to quasi-isomorphism μ_quasi-iso or strong homotopy equivalence μ_equiv. These μ_quasi-iso and μ_equiv measures how much portion of corresponding simplicial complexes is quasi-isomorphic or homotopy equivalence respectively. We validate the effectiveness of our framework with simple examples.

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