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Projecting onto rectangular matrices with prescribed row and column sums Bauschke, Heinz H.; Singh, Shambhavi; Wang, Xianfu

Abstract

In 1990, Romero presented a beautiful formula for the projection onto the set of rectangular matrices with prescribed row and column sums. Variants of Romero’s formula were rediscovered by Khoury and by Glunt, Hayden, and Reams for bistochastic (square) matrices in 1998. These results have found various generalizations and applications. In this paper, we provide a formula for the more general problem of finding the projection onto the set of rectangular matrices with prescribed scaled row and column sums. Our approach is based on computing the Moore–Penrose inverse of a certain linear operator associated with the problem. In fact, our analysis holds even for Hilbert–Schmidt operators, and we do not have to assume consistency. We also perform numerical experiments featuring the new projection operator.

Item Metadata

Title
Projecting onto rectangular matrices with prescribed row and column sums
Creator
Bauschke, Heinz H.; Singh, Shambhavi; Wang, Xianfu
Publisher
Springer International Publishing
Date Issued
2021-12-06
Description
In 1990, Romero presented a beautiful formula for the projection onto the set of rectangular matrices with prescribed row and column sums. Variants of Romero’s formula were rediscovered by Khoury and by Glunt, Hayden, and Reams for bistochastic (square) matrices in 1998. These results have found various generalizations and applications. In this paper, we provide a formula for the more general problem of finding the projection onto the set of rectangular matrices with prescribed scaled row and column sums. Our approach is based on computing the Moore–Penrose inverse of a certain linear operator associated with the problem. In fact, our analysis holds even for Hilbert–Schmidt operators, and we do not have to assume consistency. We also perform numerical experiments featuring the new projection operator.
Subject
Generalized bistochastic matrices; Hilbert–Schmidt operators; Moore–Penrose inverse; Projection; Rectangular matrices; Transportation polytope
Genre
Article
Type
Text
Language
eng
Date Available
2022-01-19
Provider
Vancouver : University of British Columbia Library
Rights
Attribution 4.0 International (CC BY 4.0)
DOI
10.14288/1.0406304
URI
http://hdl.handle.net/2429/80688
Affiliation
Science, Faculty of; Mathematics, Department of
Citation
Fixed Point Theory and Algorithms for Sciences and Engineering. 2021 Dec 06;2021(1):23
Publisher DOI
10.1186/s13663-021-00708-1
Peer Review Status
Reviewed
Scholarly Level
Faculty
Copyright Holder
The Author(s)
Rights URI
http://creativecommons.org/licenses/by/4.0/
Aggregated Source Repository
DSpace

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Attribution 4.0 International (CC BY 4.0)