Small Ball Probabilities for Simple Random Tensors

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