Spectral norm of random tensors

07/07/2014

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by   Ryota Tomioka, et al.

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We show that the spectral norm of a random n_1× n_2×...× n_K tensor (or higher-order array) scales as O(√((∑_k=1^Kn_k)(K))) under some sub-Gaussian assumption on the entries. The proof is based on a covering number argument. Since the spectral norm is dual to the tensor nuclear norm (the tightest convex relaxation of the set of rank one tensors), the bound implies that the convex relaxation yields sample complexity that is linear in (the sum of) the number of dimensions, which is much smaller than other recently proposed convex relaxations of tensor rank that use unfolding.