{"ID":23507325,"CreatedAt":"2026-09-18T02:21:44.056544415Z","UpdatedAt":"2026-09-20T18:11:56.143995915Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.20520","arxiv_id":"2609.20520","title":"Sharp spectral norm concentration of sparse random tensors","abstract":"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 $c,r\u003e0$, 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--heavy decomposition with a refined estimate on the heavy tuple part. We also obtain a log-free second eigenvalue bound for the random hypergraph model of Friedman and Wigderson (1995).","short_abstract":"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 $c,r\u003e0$, we show that $\\|T-\\mathbb E T\\|\\le C_{k,r,c}\\sqrt...","url_abs":"https://arxiv.org/abs/2609.20520","url_pdf":"https://arxiv.org/pdf/2609.20520v1","authors":"[\"Zhixin Zhou\",\"Yizhe Zhu\"]","published":"2026-09-17T14:58:31Z","proceeding":"math.PR","tasks":"[\"math.PR\",\"math.CO\",\"math.ST\",\"stat.ML\"]","methods":"[]","has_code":false}