Time: 4:30-5:30 p.m.
Date: Wednesday, October 8, 2025
Speaker: Alberto González Sanz
Title: Quadratically regularized optimal transport
Abstract: Optimal transport (OT) has become a central tool in data science and statistics. A common approach, either to design more efficient numerical methods or to mitigate the curse of dimensionality, is to regularize OT with an entropy. The most widely used choice is the logarithmic entropy, which leads to the celebrated entropic optimal transport (EOT). From a practical standpoint, EOT is successful for two main reasons: (i) the dual solutions are very smooth, and (ii) the dual problem is strongly concave. Property (i) helps overcome the curse of dimensionality in empirical estimation, while property (ii) yields algorithms with linear convergence, such as the Sinkhorn algorithm. In this talk, we will discuss alternative entropy-based regularizations of optimal transport. In particular, we will focus on the quadratically regularized optimal transport problem (QOT), which uses the $L^2$ entropy and has recently emerged as a sparse alternative to EOT. Unlike EOT, whose solutions always have full support—even for small regularization parameters—QOT solutions (or QOT plans) tend to concentrate on the support of the unregularized transport problem. However, the dual of QOT is not strongly concave, and its dual solutions are not necessarily smoother than those of classical OT. This raises several natural questions: Do the supports decrease monotonically? At what rate does sparsification occur? How quickly does the QOT cost converge to the classical OT cost? Are there algorithms with linear convergence? And does QOT still suffer from the curse of dimensionality? We will review recent theoretical results that provide answers to these questions.
Bio: Alberto González Sanz is an Assistant Professor in the Department of Statistics at Columbia University. He earned his Ph.D. in Mathematics in 2023 through a joint program between the University of Valladolid and the University of Toulouse III (Paul Sabatier), under the supervision of Professors Eustasio del Barrio and Jean-Michel Loubes. His research interests include optimal transport, empirical processes, multivariate quantiles, optimization, kernel methods, robust statistics, and fairness in machine learning.