Computing Brascamp-Lieb Constants through the lens of Thompson Geometry

TL;DR

This paper introduces a new algorithmic approach for computing Brascamp-Lieb constants using Thompson geometry, providing polynomial time guarantees and an efficient alternative to existing methods based on Riemannian optimization.

Contribution

It develops a fixed-point iteration method analyzed via Thompson metric, offering a novel, transparent, and efficient approach with polynomial time guarantees.

Findings

Achieves weakly polynomial time complexity

Provides an efficient alternative to Riemannian optimization methods

Offers a transparent analysis framework using Thompson geometry

Abstract

This paper studies algorithms for efficiently computing Brascamp-Lieb constants, a task that has recently received much interest. In particular, we reduce the computation to a nonlinear matrix-valued iteration, whose convergence we analyze through the lens of fixed-point methods under the well-known Thompson metric. This approach permits us to obtain (weakly) polynomial time guarantees, and it offers an efficient and transparent alternative to previous state-of-the-art approaches based on Riemannian optimization and geodesic convexity.