On the Largest Multilinear Singular Values of Higher-Order Tensors.

Let an denote the largest mode-n multilinear singular value of an I-1 x ... x I-N tensor tau. We prove that sigma(2)(1) + ... + sigma(2)(n-1) +sigma(2)(n+1) + ... + sigma(2)(N) <= (N- 2) parallel to tau parallel to(2) + sigma(2)(n), n = 1, ... , N, where parallel to.parallel to denotes the Frobenius norm. We also show that at least for third-order cubic tensors the inverse problem always has a solution. Namely, for each sigma(1), sigma(2), and sigma(3) that satisfy sigma(2)(1) + sigma(2)(2) <= parallel to tau parallel to(2) + sigma(2)(3), sigma(2)(1) + sigma(2)(3) <= parallel to tau parallel to(2) + sigma(2)(2), sigma(2)(2) + sigma(2)(3) <= parallel to tau parallel to(2) + sigma(2)(1), and the trivial inequalities sigma(1) >= 1/root n parallel to tau parallel to, sigma(2) >= 1/root n parallel to tau parallel to, sigma(3) >= 1/root n parallel to tau parallel to, there always exists an n x n x n tensor whose largest multilinear singular values are equal to sigma(1), sigma(2), and sigma(3). We also show that if the equality sigma(2)(1) + sigma(2)(2) = parallel to tau parallel to(2) + sigma(2)(3) holds, then tau is necessarily equal to a sum of multilinear rank-(L-1, 1, L-1) and multilinear rank-(1, L-2, L-2) tensors and we give a complete description of all its multilinear singular values. We establish a connection with honeycombs and eigenvalues of the sum of two Hermitian matrices. This seems to give at least a partial explanation of why results on the joint distribution of multilinear singular values are scarce.