Sparsity-adaptive concentration inequalities for random polynomials

Abstract

We prove concentration inequalities for polynomials of independent, sparse $α$-sub-exponential random variables. Specifically, we consider $X_i=δ_iξ_i$, where the Bernoulli selectors $δ_i$ are independent with parameters $p_i$, and the variables $ξ_i$ are independent \(α\)-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<α\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.