Concentration inequalities for random tensors

Abstract

We show how to extend several basic concentration inequalities for simple random tensors X = x1⊗· · ·⊗xd where all xk are independent random vectors in Rn with independent coefficients. The new results have optimal dependence on the dimension n and the degree d. As an application, we show that random tensors are well conditioned: (1 - o(1))nd independent copies of the simple random tensor X ∈ Rnd are far from being linearly dependent with high probability. We prove this fact for any degree d = o(√n/log n) and conjecture that it is true for any d = O(n).