Concentration inequalities for random tensors

Abstract

We show how to extend several basic concentration inequalities for simple random tensors $X=x_{1}\otimes\cdots\otimes x_{d}$ where all $x_{k}$ are independent random vectors in ${\mathbb{R}}^{n}$ 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: $(1-o(1))n^{d}$ independent copies of the simple random tensor $X\in{\mathbb{R}}^{n^{d}}$ are far from being linearly dependent with high probability. We prove this fact for any degree $d=o(\sqrt{n/\log n})$ and conjecture that it is true for any $d=O(n)$.