Abstract. The tensor eigenproblem has many important applications, and both mathematical and applicationspecific communities have taken recent interest in the properties of tensor eigenpairs as well as methods for computing them. In particular, Kolda and Mayo [3] present a generalization of the matrix power method for symmetric tensors. We focus in this work on efficient implementation of their algorithm, known as the shifted symmetric higher-order power method, and on how a GPU can be used to accelerate the computation up to 70 × over a sequential implementation for an application involving many small tensor eigenproblems.

rank-k decompositions

Abstract. Well known connections exist between the singular value decomposition of a matrix A and the Schur decomposition of its symmetric embedding sym(A) = ([0A; A T 0]). In particular, if σ is a singular value of A then +σ and −σ are eigenvalues of the symmetric embedding. The top and bottom halves of sym(A)’s eigenvectors are singular vectors for A. Power methods applied to A can be related to power methods applied to sym(A). The rank of sym(A) is twice the rank of A. In this paper we develop similar connections for tensors by building on L-H. Lim’s variational approach to tensor singular values and vectors. We show how to embed a general order-d tensor A into an order-d symmetric tensor sym(A). Through the embedding we relate power methods for A’s singular values to power methods for sym(A)’s eigenvalues. Finally, we connect the multilinear and outer product rank of A to the multilinear and outer product rank of sym(A). Key words. tensor, block tensor, symmetric tensor, tensor rank AMS subject classifications. 15A18, 15A69, 65F15