Unbalanced Optimal Transport through Non-negative Penalized Linear Regression

Laetitia Chapel, Rémi Flamary, Haoran Wu, Cédric Févotte, Gilles Gasso

Keywords: Optimal Transport, Unbalanced Optimal Transport, Penalized Linear Regression, Lasso, MM algorithms

TL;DR: We propose a reformulation of the Exact Unbalanced Optimal Transport as a Non-negative Linear Regression problem which allows us to devise new algorithms such as the first regularization path for OT and multiplicative algorithms.

Abstract: This paper addresses the problem of Unbalanced Optimal Transport (UOT) in which the marginal conditions are relaxed (using weighted penalties in lieu of equality) and no additional regularization is enforced on the OT plan. In this context, we show that the corresponding optimization problem can be reformulated as a non-negative penalized linear regression problem. This reformulation allows us to propose novel algorithms inspired from inverse problems and nonnegative matrix factorization. In particular, we consider majorization-minimization which leads in our setting to efficient multiplicative updates for a variety of penalties. Furthermore, we derive for the first time an efficient algorithm to compute the regularization path of UOT with quadratic penalties. The proposed algorithm provides a continuity of piece-wise linear OT plans converging to the solution of balanced OT (corresponding to infinite penalty weights). We perform several numerical experiments on simulated and real data illustrating the new algorithms, and provide a detailed discussion about more sophisticated optimization tools that can further be used to solve OT problems thanks to our reformulation.

Code Of Conduct: I certify that all co-authors of this work have read and commit to adhering to the NeurIPS Statement on Ethics, Fairness, Inclusivity, and Code of Conduct.

Code: https://github.com/lchapel/UOT-though-penalized-linear-regression