Optimal fiber comparisons for random tensors
Proves optimal expectation bounds for tensor fiber norms using matrix norm comparisons.
Abstract
We extend Seginer’s matrix norm comparison to tensors with optimal dependence on the tensor order. For a real tensor $T$ of order $k\ge2$ with centered, integrable, independent and identically distributed entries, we prove $\mathbb{E} f(T)\le\mathbb{E}\lVert T\rVert_{\mathrm{inj}} \le C\sqrt{k\log k}\,\mathbb{E} f(T)$, where $f(T)$ is the largest Euclidean norm of a one-dimensional fiber and $C$ is universal. The comparison holds for every finite rectangular format, and the entry law may depend on both the dimensions and the tensor order. We show that the factor $\sqrt{k\log k}$ is optimal by considering Rademacher tensors on $[2]^k$.