# Two-sample goodness-of-fit tests on the flat torus based on Wasserstein distance and their relevance to structural biology

arXiv: https://arxiv.org/abs/2108.00165 · HTML: https://mlsanity.com/paper/2108.00165/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Jul 2021.

## Built on by (most cited first, 1 of 1)

- [An improved central limit theorem and fast convergence rates for entropic transportation costs](https://mlsanity.com/paper/2204.09105/) (2022)

## Abstract (arXiv)

This work is motivated by the study of local protein structure, which is defined by two variable dihedral angles that take values from probability distributions on the flat torus. Our goal is to provide the space P(ℝ2/mathbbZ2) with a metric that quantifies local structural modifications due to changes in the protein sequence, and to define associated two-sample goodness-of-fit testing approaches. Due to its adaptability to the space geometry, we focus on the Wasserstein distance as a metric between distributions. We extend existing results of the theory of Optimal Transport to the d-dimensional flat torus mathbbTd=ℝd/mathbbZd, in particular a Central Limit Theorem. Moreover, we propose different approaches for two-sample goodness-of-fit testing for the one and two-dimensional case, based on the Wasserstein distance. We prove their validity and consistency. We provide an implementation of these tests in textsfR. Their performance is assessed by numerical experiments on synthetic data and illustrated by an application to protein structure data.