# On Geodesically Convex Formulations for the Brascamp-Lieb Constant

arXiv: https://arxiv.org/abs/1804.04051 · HTML: https://mlsanity.com/paper/1804.04051/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Apr 2018.

## Built on by (most cited first, 5 of 5)

- [Minimizing Dynamic Regret on Geodesic Metric Spaces](https://mlsanity.com/paper/2302.08652/) (2023)
- [Optimization and Sampling Under Continuous Symmetry: Examples and Lie Theory](https://mlsanity.com/paper/2109.01080/) (2021)
- [On a class of geodesically convex optimization problems solved via Euclidean MM methods](https://mlsanity.com/paper/2206.11426/) (2022)
- [Extragradient Type Methods for Riemannian Variational Inequality Problems](https://mlsanity.com/paper/2309.14155/) (2023)
- [Disciplined Geodesically Convex Programming](https://mlsanity.com/paper/2407.05261/) (2024)

## Abstract (arXiv)

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.