Small Ball Probabilities for Simple Random Tensors

We study the small ball probability of an order-$\mathrm{\ell }$ simple random tensor $X=X^{(1)}\otimes \cdots \otimes X^{(\mathrm{\ell })}$ where $X^{(i)},1\le i\le \mathrm{\ell }$ 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 $\frac{\epsilon }{(\mathrm{\ell }-1)!}{(C\log \left(\frac{e}{\epsilon }\right))}^{\mathrm{\ell }}$ 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.