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

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 vCech) filtration of mathbbX to that of LmathbbX. 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_operatornamequasi-iso or strong homotopy equivalence μ_operatornameequiv. These μ_operatornamequasi-iso and μ_operatornameequiv 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.