The condition number of join decompositions

Join decompositions generalize some ubiquitous decompositions in multilinear algebra, namely tensor rank, Waring, and block term decompositions. A join decomposition is a fixed-length sum of points from several smooth submanifolds embedded in a mutual vector space. The set of points admitting a join decomposition is called a join set. Given a point in the ambient vector space, the join approximation problem consists of finding the closest point in the join set. We examine the geometric condition number of 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. We show how the join approximation problem may be formulated as a Riemannian optimization problem, despite the fact that join sets are not habitually Riemannian manifolds. We show that the geometric condition number governs the convergence of the Riemannian Gauss-Newton method for solving join approximation problems. For some special join sets, we characterize the behavior of sequences in the join set converging to the latter’s boundary points. Finally, the discussion is specialized to the tensor rank decomposition and we provide several numerical experiments confirming the main theoretical results.status: publishe