Alpha shapes and optimal transport on the sphere

TL;DR

This paper extends topological data analysis techniques using optimal transport theory on the sphere, demonstrating applications to neural data with complex structures, and simplifying preprocessing steps.

Contribution

It introduces a variant of the Legendre transform for cosine similarity kernels using the c-transform, enhancing TDA methods on spherical data.

Findings

Applied the method to brain activity data showing toroidal structures

Replaced complex preprocessing with transport map and density transformation

Validated the approach on real neural datasets

Abstract

In [3], the authors used the Legendre transform to give a tractable method for studying Topological Data Analysis (TDA) in terms of sums of Gaussian kernels. In this paper, we prove a variant for sums of cosine similarity-based kernel functions, which requires considering the more general "$c$-transform" from optimal transport theory [16]. We then apply these methods to a point cloud arising from a recent breakthrough study, which exhibits a toroidal structure in the brain activity of rats [11]. A key part of this application is that the transport map and transformed density function arising from the theorem replace certain delicate preprocessing steps related to density-based denoising and subsampling.