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Ehemalige Mitarbeiter

Dr. Paul Breiding


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The condition number of join decompositions.
Citation key BV-The-Condition-Number-Of-Join-Decompositions
Author Paul Breiding and Nick Vannieuwenhoven
Pages 287–309
Year 2018
DOI 10.1137/17M1142880
Journal SIAM J. Matrix Anal. Appl.
Volume 39
Number 1
Month 02
Abstract The join set of a finite collection of smooth embedded submanifolds of a mutual vector space is defined as their Minkowski sum. Join decompositions generalize some ubiquitous decompositions in multilinear algebra, namely, tensor rank, Waring, partially symmetric rank, and block term decompositions. This paper examines the numerical sensitivity of join decompositions to perturbations; specifically, we consider the condition number for general join decompositions. It is characterized as a distance to a set of ill-posed points in a supplementary product of Grassmannians. We prove that this condition number can be computed efficiently as the smallest singular value of an auxiliary matrix. For some special join sets, we characterized the behavior of sequences in the join set converging to the latter's boundary points. Finally, we specialize our discussion to the tensor rank and Waring decompositions and provide several numerical experiments confirming the key results.
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