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Abstract:We prove concentration inequalities for polynomials of independent, sparse $\alpha$-sub-exponential random variables. Specifically, we consider $X_i=\delta_i\xi_i$, where the Bernoulli selectors $\delta_i$ are independent with parameters $p_i$, and the variables $\xi_i$ are independent \(\alpha\)-sub-exponential random variables (not necessarily centered). For any polynomial $f:\mathbb R^n\to\mathbb R $ of degree at most $D$ and any $0<\alpha \le 1 $, we establish an $L_r$-moment bound for \(f(X)-\mathbb E f(X)\) in terms of partition norms of sparsity-weighted expected derivative tensors. The weights count distinct coordinates rather than multiplicities and therefore distinguish diagonal, partially diagonal, and off-diagonal contributions. This captures the sparse scaling in both collective fluctuation regimes and extreme-coordinate regimes.
When all sparsity parameters are equal to one, our result recovers the polynomial concentration inequality of Götze, Sambale, and Sinulis. In degree two, it recovers sparse Hanson-Wright bounds. As applications, we derive deviation inequalities for the distance between a sparse simple random tensor and a fixed subspace, and obtain lower bounds for the smallest singular value of matrices whose columns are independent sparse simple random tensors.

Submission history

From: Guozheng Dai [view email]
[v1] Tue, 23 Jun 2026 03:10:49 UTC (41 KB)