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Sharp spectral norm concentration of sparse random tensors

arXiv · AI, language, vision and robotics · article · Sep 17, 2026 · UTC

We prove a sharp concentration inequality for the spectral norm of sparse random tensors with independent Bernoulli entries. Let $T$ be an order-$k$ tensor of dimension $n\times\cdots\times n$ with independent Bernoulli$(p)$ entries, where $k$ is fixed. For any $c,r>0$, we show that $\|T-\mathbb E T\|\le C_{k,r,c}\sqrt{np}$ with probability at least $1-n^{-r}$ whenever $np\ge c\log n$. We extend this bound to inhomogeneous Bernoulli sampling with deterministic entrywise weights. This removes the logarithmic factor in the work of Zhou and Zhu (2021). The proof follows the Kahn--Szemerédi light-

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