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Computing Brascamp-Lieb Constants through the lens of Thompson Geometry

Melanie Weber, Suvrit Sra

TL;DR

The paper tackles the problem of computing Brascamp-Lieb constants for feasible BL-data by recasting it as a nonlinear fixed-point problem on positive definite matrices. It introduces the map $G(X)=\bigl(\sum_j w_j A_j (A_j^* X A_j)^{-1} A_j^*\bigr)^{-1}$ and analyzes its convergence through Thompson geometry, establishing a fixed-point framework with non-expansive behavior and asymptotic regularity. A regularized variant $G_\mu$ is shown to be contractive, enabling a non-asymptotic convergence rate of $O\big(\log(1/\epsilon,\Delta/\delta^{3/2},\sqrt{d},1/R)\big)$ to obtain $\epsilon$-close BL constants, with data-dependent parameters $\delta,\Delta,R$. The work provides a transparent alternative to Riemannian optimization and connects nonlinear Perron-Frobenius theory with BL constant computation, offering a new fixed-point perspective and highlighting future directions for explicit initialization and broader applications such as operator scaling.

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.

Computing Brascamp-Lieb Constants through the lens of Thompson Geometry

TL;DR

The paper tackles the problem of computing Brascamp-Lieb constants for feasible BL-data by recasting it as a nonlinear fixed-point problem on positive definite matrices. It introduces the map and analyzes its convergence through Thompson geometry, establishing a fixed-point framework with non-expansive behavior and asymptotic regularity. A regularized variant is shown to be contractive, enabling a non-asymptotic convergence rate of to obtain -close BL constants, with data-dependent parameters . The work provides a transparent alternative to Riemannian optimization and connects nonlinear Perron-Frobenius theory with BL constant computation, offering a new fixed-point perspective and highlighting future directions for explicit initialization and broader applications such as operator scaling.

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.
Paper Structure (20 sections, 24 theorems, 90 equations, 1 figure)

This paper contains 20 sections, 24 theorems, 90 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our approach
  3. Summary of contributions
  4. Outline
  5. Background and Notation
  6. Thompson geometry on positive definite matrices
  7. Brascamp--Lieb inequalities
  8. Basic notation
  9. Examples
  10. Overview of our results
  11. Related work
  12. Proofs of the results
  13. Convergence of the fixed-point iteration using the map $G_t$
  14. Non-asymptotic convergence analysis of $G_\mu$
  15. Approximation error of $F_\mu$.
  16. ...and 5 more sections

Key Result

Proposition 3

Let $(\mathcal{A},\bm{w})$ denote a simple BL-datum and let $\bm{w}=(w_1, \dots, w_n)$ with $w_j \in (0,1)$ and $\sum_{j \in [m]} w_j =1$. Then Problem prob:BL is equivalent to solving the following optimization problem:

Figures (1)

  • Figure 1: Empirical performance of the map $G$ (Eq. \ref{['eq:map']}) in comparison with first-order Riemannian Optimization methods (using the Manopt implementation boumal_manopt_2014): Riemannian Steepest Descent (Riem-SD), Riemannian Trustregions (Riem-TR) and Riemannian Conjugate Gradient (Riem-CG). The Brascamp--Lieb datum consists of a set of $n$ linear transformations $A_j: \mathbb{R}^d \rightarrow \mathbb{R}^k$ and an $n$-dimensional vector that characterizes the exponents. The reported value of the loss function is the value of the objective \ref{['eq:F']}, i.e., the value attained by the BL constant.

Theorems & Definitions (42)

  • Definition 1: Brascamp--Lieb Constant
  • Proposition 3
  • Theorem 4
  • Definition 5: Thompson metric
  • Lemma 6: Properties of the Thompson metric
  • Definition 7
  • Theorem 8: lieb, Theorem 3.2
  • Theorem 9: tao-paper,Theorem 1.13
  • Definition 10: Simple Brascamp--Lieb data, tao-paper
  • Theorem 11
  • ...and 32 more