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On Geodesically Convex Formulations for the Brascamp-Lieb Constant

Authors Suvrit Sra, Nisheeth K. Vishnoi, Ozan Yildiz

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Subject Classification

ACM Subject Classification
  • Theory of computation → Nonconvex optimization
  • Theory of computation → Convex optimization
Keywords
  • Geodesic convexity
  • positive definite cone
  • geodesics
  • Brascamp-Lieb constant

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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 [Lieb, 1990] and the second is new and inspired by the work of Bennett et al. [Bennett et al., 2008]. Recent work of Garg et al. [Ankit Garg et al., 2017] also implies 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.

Cite As Get BibTex

Suvrit Sra, Nisheeth K. Vishnoi, and Ozan Yildiz. On Geodesically Convex Formulations for the Brascamp-Lieb Constant. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 25:1-25:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.25

Author Details

Suvrit Sra
  • Massachusetts Institute of Technology (MIT), Cambridge, MA, USA
Nisheeth K. Vishnoi
  • École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
Ozan Yildiz
  • École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland

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