License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2018.25
URN: urn:nbn:de:0030-drops-94296
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9429/
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Sra, Suvrit ; Vishnoi, Nisheeth K. ; Yildiz, Ozan

On Geodesically Convex Formulations for the 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.

BibTeX - Entry

@InProceedings{sra_et_al:LIPIcs:2018:9429,
  author =	{Suvrit Sra and Nisheeth K. Vishnoi and Ozan Yildiz},
  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 =	{Eric Blais and Klaus Jansen and Jos{\'e} D. P. Rolim and David Steurer},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9429},
  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}
}

Keywords: Geodesic convexity, positive definite cone, geodesics, Brascamp-Lieb constant
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)
Issue Date: 2018
Date of publication: 13.08.2018

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