Convexity of the image of a quadratic map via the relative entropy distance
type: Item
instance of: scholarly article
title: Convexity of the image of a quadratic map via the relative entropy distance
zbMATH Open document ID: 1312.52005
published in: Beiträge zur Algebra und Geometrie
publication date: 2014-10-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1305.0243
review text: Let \(q_1, \dots, q_k :\mathbb{R}^n\to\mathbb{R}\) be positive definite quadratic forms and let \(\psi:\mathbb R^n\to\mathbb R^k\) be the corresponding map, \(\psi(x)=(q_1(x), \dots, q_k(x))\). Let \(a=(a_1, \dots, a_k)\) be a point in the convex hull of the image of \(\psi\) such that \(\sum^{k}_{i=1}a_i=1\). The author of the paper under review proves that there exists a point \(b= (b_1, \dots,b_k)\) in the image of \(\psi\) such that \(\sum^{k}_{i=1}b_i=1\) and such that \(\sum^{k}_{i=1}a_i \ln (a_i/ b_i)\leq \beta\), where \(\beta>0\) is an absolute constant. The author also proves that for any positive integer \(m\), there exists a point \(b= (b_1, \dots, b_k)\), such that \(\sum^{k}_{i=1}b_i=1\), the point \(b\) is a convex combination of at most \(m\) points of \(\psi(\mathbb{R}^n)\) and \(\sum^{k}_{i=1}a_i \ln (a_i/ b_i)<17/ \sqrt{m}\).
reviewed by: Monika Nowicka
Mathematics Subject Classification ID: 52A27
Mathematics Subject Classification ID: 52B55
Mathematics Subject Classification ID: 62B10
zbMATH DE Number: 6362516
zbMATH Keywords: Kullback-Leibler distance
zbMATH Keywords: relative entropy
zbMATH Keywords: quadratic convexity
zbMATH Keywords: positive semidefinite programming
zbMATH Keywords: Johnson-Lindenstrauss lemma
zbMATH Keywords: Gaussian measure
zbMATH Keywords: approximate Carathéodory theorem
zbMATH Keywords: Markov inequality
author: Alexander Barvinok
MaRDI profile type: Publication
OpenAlex ID: W2962694033
arXiv ID: 1305.0243
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DOI: 10.1007/S13366-013-0187-X
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description: scientific article; zbMATH DE number 6362516