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

TL;DR

This paper presents a linear dimensionality reduction method that preserves topological features of data using persistent homology, enabling shape comparison through filtration homomorphisms and new similarity measures.

Contribution

The paper introduces a novel algorithm for topologically faithful linear projection using persistent homology and defines measures for shape equivalence via filtration homomorphisms.

Findings

The method effectively preserves topological features in reduced dimensions.

Filtration homomorphisms enable direct shape comparison between original and projected data.

Validation with simple examples demonstrates the approach's potential.

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 $\mathbb{X}$ via simulated annealing. The projection $L$ induces a set of canonical simplicial maps from the Rips (or \v{C}ech) filtration of $\mathbb{X}$ to that of $L\mathbb{X}$. 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 $\mu_{\operatorname{quasi-iso}}$ or strong homotopy equivalence $\mu_{\operatorname{equiv}}$. These $\mu_{\operatorname{quasi-iso}}$ and $\mu_{\operatorname{equiv}}$ 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.