Persistent Homology-Based Projection Pursuit

Abstract

Dimensionality reduction problem is stated as finding a mapping $f:X \in {{\mathbb{R}}^m} \to Z \in {{\mathbb{R}}^n}$, where ⪡ m while preserving some relevant properties of the data. We formulate topology-preserving dimensionality reduction as finding the optimal orthogonal projection to the lower-dimensional subspace which minimizes discrepancy between persistent diagrams of the original data and the projection. This generalizes the classic projection pursuit algorithm which was originally designed to preserve the number of clusters, i.e. the 0-order topological invariant of the data. Our approach further allows to preserve k-th order invariants within the principled framework. We further pose the resulting optimization problem as the Riemannian optimization problem which allows for a natural and efficient solution.