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

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 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.

Topology-Preserving Dimensionality Reduction via Interleaving Optimization

“…Several non-differentiable methods incorporating persistent homology have been developed [12,36,13]. With the development of optimization techniques for persistent homology, several differentiable methods have been proposed based on optimization of the 2-Wasserstein metric on persistence diagrams [20,34] and an approach based on simulated annealing [37]. While not employed on Vietoris-Rips filtrations, the work of [29] uses the bottleneck distance for optimization of functional maps on shapes.…”