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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2018.25

On Geodesically Convex Formulations for the Brascamp-Lieb Constant

Authors: Suvrit Sra, Nisheeth K. Vishnoi, and Ozan Yildiz

Published in: LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

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.

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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)

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@InProceedings{sra_et_al:LIPIcs.APPROX-RANDOM.2018.25,
  author =	{Sra, Suvrit and Vishnoi, Nisheeth K. and Yildiz, Ozan},
  title =	{{On Geodesically Convex Formulations for the Brascamp-Lieb Constant}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
  pages =	{25:1--25:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-085-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{116},
  editor =	{Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.25},
  URN =		{urn:nbn:de:0030-drops-94296},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2018.25},
  annote =	{Keywords: Geodesic convexity, positive definite cone, geodesics, Brascamp-Lieb constant}
}

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On Geodesically Convex Formulations for the Brascamp-Lieb Constant

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