Implications of convergence rates in Sinkhorn balancing
type: Item
instance of: scholarly article
title: Implications of convergence rates in Sinkhorn balancing
zbMATH Open document ID: 0782.15012
DOI: 10.1016/0024-3795(93)90131-7
author: Eva Achilles
published in: Linear Algebra and its Applications
publication date: 1993-08-29T00:00:00Z
review text: A sequence of matrices \(\{A^{(t)}\}\), \(t=0,1,2,\ldots,\) is said to converge geometrically to a limit \(G\) if there exist positive constants \(M\) and \(\gamma\), where \(0<\gamma<1\), such that \(\| A^{(t)}- G\|\leq M\gamma^ t\) for some norm \(\|\cdot\|\), \(t=0,1,2,\ldots\). A nonnegative \(N\times N\) matrix \(A\) is said to have total support if there exists an \(N\times N\) doubly stochastic matrix \(B\) such that \(b_{ij}>0\) if and only if \(a_{ij}>0\) for all \(i,j=1,2,\ldots,N\). It is known that if \(A\) has total support then the sequence of matrices obtained beginning with \(A\) and alternately normalizing the column sums and then the row sums converges geometrically to a doubly stochastic limit. The author shows that if this iteration on \(A\) converges geometrically, then \(A\) has total support.
reviewed by: Richard Sinkhorn
Mathematics Subject Classification ID: 15B51
zbMATH DE Number: 399294
zbMATH Keywords: geometrical convergence
zbMATH Keywords: sequence of matrices
zbMATH Keywords: total support
zbMATH Keywords: doubly sotchastic matrix
zbMATH Keywords: doubly stochastic limit
MaRDI profile type: Publication
full work available at URL: 0024-3795(93)90131-7
OpenAlex ID: W2004972744
cites work: Q5330745
cites work: Concerning nonnegative matrices and doubly stochastic matrices
cites work: Diagonal Equivalence to Matrices with Prescribed Row and Column Sums
cites work: Continuous Dependence of A in the D 1 AD 2 Theorems
cites work: The rate of convergence of Sinkhorn balancing
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description: scientific article; zbMATH DE number 399294