The condition number of join decompositions

TL;DR

This paper studies the numerical sensitivity of join decompositions, a generalization of tensor and polynomial decompositions, by defining and efficiently computing a condition number that measures their stability under perturbations.

Contribution

It introduces a condition number for join decompositions, characterizes its computation via singular values, and analyzes its behavior for special cases like tensor rank and Waring decompositions.

Findings

Condition number can be computed as the smallest singular value of an auxiliary matrix.

Sequences approaching boundary points of join sets are characterized.

Numerical experiments confirm the theoretical properties of the condition number.

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.