On canonical polyadic decomposition of overcomplete tensors of arbitrary even order

Abstract

Decomposition of tensors into summation of rank one components, known as Canonical Polyadic (CP) decomposition, has long been studied in the literature. Although the CP-rank of tensors can far exceed their dimensions (in which case they are called overcomplete tensors), there are only a handful of algorithms which consider CP-decomposition of such overcomplete tensors, and most of the CP-decomposition algorithms proposed in literature deal with simpler cases where the rank is of the same order as the dimensions of the tensor. In this paper, we consider symmetric tensors of arbitrary even order whose eigenvalues are assumed to be positive. We show that for a 2dth order tensor with dimension N, under some mild conditions, the problem of CP-decomposition is equivalent to solving a system of quadratic equations, even when the rank is as large as O(Nd). We will develop two different algorithms (one convex, and one nonconvex) to solve this system of quadratic equations. Our simulations show that successful recovery of eigenvectors is possible even if the rank is much larger than the dimension of the tensor.1