Small Ball Probabilities for Simple Random Tensors

Abstract

We study the small ball probability of an order-$\ell$ simple random tensor $X=X^{(1)}\otimes \cdots \otimes X^{(\ell )}$ where $X^{(i)},1\le i\le \ell$ are independent random vectors in ${\mathbb{R}}^n$ that are log-concave or have independent coordinates with bounded densities. We show that the probability that the projection of $X$ onto an $m$-dimensional subspace $F$ falls within an Euclidean ball of length $\epsilon$ is upper bounded by $\frac{\epsilon }{(\ell -1)!}{\left(C\log \left(\frac{e}{\epsilon }\right)\right)}^{\ell }$ and also this upper bound is sharp when $m$ is small. We also established that a much better estimate holds true for a random subspace.