On the Information Processing of One-Dimensional Wasserstein Distances with Finite Samples

TL;DR

This paper analyzes how the one-dimensional Wasserstein distance captures pointwise density differences and support variations with finite samples, using theoretical analysis and neural data applications.

Contribution

It provides a theoretical characterization of the information processing capabilities of 1D Wasserstein distances with finite samples, especially regarding density and support differences.

Findings

Wasserstein distances can detect pointwise density differences.

The analysis uses Poisson processes and neural data.

Results confirm the ability to identify meaningful differences.

Abstract

Leveraging the Wasserstein distance -- a summation of sample-wise transport distances in data space -- is advantageous in many applications for measuring support differences between two underlying density functions. However, when supports significantly overlap while densities exhibit substantial pointwise differences, it remains unclear whether and how this transport information can accurately identify these differences, particularly their analytic characterization in finite-sample settings. We address this issue by conducting an analysis of the information processing capabilities of the one-dimensional Wasserstein distance with finite samples. By utilizing the Poisson process and isolating the rate factor, we demonstrate the capability of capturing the pointwise density difference with Wasserstein distances and how this information harmonizes with support differences. The analyzed properties are confirmed using neural spike train decoding and amino acid contact frequency data. The results reveal that the one-dimensional Wasserstein distance highlights meaningful density differences related to both rate and support.