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Title: Goodness-of-fit tests on the flat torus based on Wasserstein distance and their relevance to structural biology Authors:  Alberto Gonzalez Sanz - Toulouse III (France)
Pierre Neuvial - CNRS and Toulouse Mathematics Institute (France)
Javier Gonzalez-Delgado - Toulouse Mathematics Institute and LAAS-CNRS (France) [presenting]
Juan Cortes - LAAS-CNRS (France)
Abstract: The motivation comes from the study of local protein structure, which is defined by two variable dihedral angles that take values from probability distributions on the flat torus. The goal is to provide the space $\mathcal{P}(\mathbb{R}^2/\mathbb{Z}^2)$ 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 $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$, in particular, a Central Limit Theorem. Moreover, we assess different techniques for two-sample goodness-of-fit testing for the two-dimensional case, based on the Wasserstein distance. We provide an implementation of these approaches in \textsf{R}. Their performance is illustrated by numerical experiments on synthetic data and protein structure data.