The condition number of join decompositions
For researchers in multilinear algebra and tensor decompositions, this work provides a theoretical and computational framework for assessing the numerical stability of join decompositions, which is a novel contribution to the field.
This paper studies the numerical sensitivity of join decompositions, characterizing the condition number as a distance to ill-posed points in a product of Grassmannians and showing it can be computed efficiently via the smallest singular value of an auxiliary matrix. Numerical experiments on tensor rank and Waring decompositions confirm the results.
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.