6 papers
Empirical optimal transport potentials: fast rates and a functional central limit theorem
Alberto González-Sanz, Gilles Mordant, Shunan Sheng
Optimal transport potentials are fundamental objects in statistics, economics, and machine learning: their gradients generate optimal transport maps, while the potentials themselve…
The entropic optimal (self-)transport problem: Limit distributions for decreasing regularization with application to score function estimation
Gilles Mordant
We study the statistical properties of the entropic optimal (self) transport problem for smooth probability measures. We provide an accurate description of the limit distribution f…
The Catastrophic Failure of The k-Means Algorithm in High Dimensions, and How Hartigan's Algorithm Avoids It
Roy R. Lederman, David Silva-Sánchez, Ziling Chen +3
Lloyd's k-means algorithm is one of the most widely used clustering methods. We prove that in high-dimensional, high-noise settings, the algorithm exhibits catastrophic failure: wi…
Estimation of Algebraic Sets: Extending PCA Beyond Linearity
Alberto González-Sanz, Gilles Mordant, Ãlvaro Samperio +1
An algebraic set is defined as the zero locus of a system of real polynomial equations. In this paper we address the problem of recovering an unknown algebraic set fr…
Manifold Learning with Sparse Regularised Optimal Transport
Stephen Zhang, Gilles Mordant, Tetsuya Matsumoto +1
Manifold learning is a central task in modern statistics and data science. Many datasets (cells, documents, images, molecules) can be represented as point clouds embedded in a high…
Infinitesimal behavior of Quadratically Regularized Optimal Transport and its relation with the Porous Medium Equation
Alejandro Garriz-Molina, Alberto González-Sanz, Gilles Mordant
The quadratically regularized optimal transport problem has recently been considered in various applications where the coupling needs to be \emph{sparse}, i.e., the density of the…