On the Information Processing of One-Dimensional Wasserstein Distances with Finite Samples
This work addresses a theoretical gap in understanding Wasserstein distances for practitioners in fields like neuroscience and bioinformatics, though it appears incremental as it builds on existing analysis methods.
The paper tackled the problem of whether the one-dimensional Wasserstein distance can accurately identify pointwise density differences when supports overlap, by analyzing its information processing capabilities with finite samples. The results demonstrated that it captures meaningful density differences related to both rate and support, as confirmed using neural spike train decoding and amino acid contact frequency data.
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.