Generalized projections for non-negative functions
type: Item
instance of: scholarly article
title: Generalized projections for non-negative functions
zbMATH Open document ID: 0837.62006
DOI: 10.1007/BF01874442
author: Imre Csiszár
published in: Acta Mathematica Hungarica
publication date: 1996-05-20T00:00:00Z
review text: Given a strictly convex function \(f\) and a \(\sigma\)-finite measure space \((X, {\mathcal X}, \mu)\), the \(f\)-divergence \(D_f\) and the Bergman distance are used as distances on the set \(S\) of \({\mathcal X}\)-measurable nonnegative functions. An important problem in many applications of such distances in probability theory, statistics and information theory is the question of the existence of projections of elements \(s \in S\) onto convex subsets \(E \subseteq S\). The aim of the paper is to generalize results for \(I\)-projections \((f = x \ln x)\) earlier obtained by the author to the distances \(D_f\) and \(B_f\) for general \(f\). The existence of projections and an inequality being a generalization of Pythagoras' theorem are established in Theorem 1. The convergence in \(\mu\)-measure and in \(L_1 (\mu)\) of minimizing sequences is investigated in Theorem 2. Convex subsets \(E \subseteq S\) defined by a system of linear inequalities are considered in Theorem 3.
Mathematics Subject Classification ID: 62B10
Mathematics Subject Classification ID: 62B99
zbMATH DE Number: 794425
zbMATH Keywords: generalized projections
zbMATH Keywords: \(f\)-divergence
zbMATH Keywords: measurable nonnegative functions
zbMATH Keywords: \(I\)-projections
zbMATH Keywords: strictly convex function
zbMATH Keywords: Bergman distance
zbMATH Keywords: existence of projections
zbMATH Keywords: convex subsets
zbMATH Keywords: generalization of Pythagoras' theorem
zbMATH Keywords: linear inequalities
MaRDI profile type: Publication
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cites work: Q5737389
cites work: Q5539510
cites work: I-divergence geometry of probability distributions and minimization problems
cites work: Sanov property, generalized I-projection and a conditional limit theorem
cites work: Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems
cites work: Approximation–Theoretic Derivation of Logarithmic Entropy Principles for Inverse Problems and Unique Extension of the Maximum Entropy Method to Incorporate Prior Knowledge
cites work: General entropy criteria for inverse problems, with applications to data compression, pattern classification, and cluster analysis
cites work: Q3803964
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cites work: Q4184753
description: scientific article; zbMATH DE number 794425