Shape-Preserving Dimensionality Reduction : An Algorithm and Measures of Topological Equivalence
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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 𝕏 via simulated annealing. The projection L induces a set of canonical simplicial maps from the Rips (or Čech) filtration of 𝕏 to that of L𝕏. 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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