On Geodesically Convex Formulations for the Brascamp-Lieb Constant

TL;DR

This paper introduces two geodesically log-concave formulations for computing the Brascamp-Lieb constant, with polynomial dimension complexity, advancing the theoretical understanding and potential computational approaches for this problem.

Contribution

It presents two new geodesically convex formulations for the Brascamp-Lieb constant with polynomial dimension complexity, improving upon previous exponential-dimensional reductions.

Findings

Formulations are geodesically log-concave on the positive definite manifold.

Dimensions of the formulations are polynomial in input bit complexity.

Advances the theoretical framework for efficient computation of the Brascamp-Lieb constant.

Abstract

We consider two non-convex formulations for computing the optimal constant in the Brascamp-Lieb inequality corresponding to a given datum, and show that they are geodesically log-concave on the manifold of positive definite matrices endowed with the Riemannian metric corresponding to the Hessian of the log-determinant function. The first formulation is present in the work of Lieb and the second is inspired by the work of Bennett et al. Recent works of Garg et al.and Allen-Zhu et al. also imply a geodesically log-concave formulation of the Brascamp-Lieb constant through a reduction to the operator scaling problem. However, the dimension of the arising optimization problem in their reduction depends exponentially on the number of bits needed to describe the Brascamp-Lieb datum. The formulations presented here have dimensions that are polynomial in the bit complexity of the input datum.