Monotonicity in Quadratically Regularized Linear Programs

TL;DR

This paper investigates the monotonicity of solutions in quadratically regularized linear programs, revealing geometric conditions for invariance and showing that monotonicity generally fails in higher dimensions, with implications for optimal transport.

Contribution

It provides a geometric characterization of monotonicity in regularized linear programs and demonstrates its failure in higher dimensions, advancing understanding of solution behavior in optimal transport.

Findings

Monotonicity corresponds to a geometric property of the polytope.

The property holds in low dimensions but fails in five or more points.

Support monotonicity does not hold in general for higher-dimensional cases.

Abstract

In optimal transport, quadratic regularization is a sparse alternative to entropic regularization: the solution measure tends to have small support. Computational experience suggests that the support decreases monotonically to the unregularized counterpart as the regularization parameter is relaxed. We find it useful to investigate this monotonicity more abstractly for linear programs over polytopes, regularized with the squared norm. Here, monotonicity can be stated as an invariance property of the curve mapping the regularization parameter to the solution: once the curve enters a face of the polytope, does it remain in that face forever? We show that this invariance is equivalent to a geometric property of the polytope, namely that each face contains the minimum norm point of its affine hull. Returning to the optimal transport problem and its associated Birkhoff polytope, we verify this property for low dimensions, but show that it fails for marginals with five or more point masses. As a consequence, the conjectured monotonicity of the support fails in general, even if experiments suggest that monotonicity holds for many cost matrices. Separately, we apply our geometric point of view to a problem of Erd\H{o}s, namely to characterize the doubly stochastic matrices whose maximal trace equals their squared norm.