We consider the problem of testing whether an unknown low-degree polynomial $p$ over $\mathbb{R}^n$ is sparse versus far from sparse, given access to noisy evaluations of the polynomial $p$ at \emph{randomly chosen points}. This is a property-testing analogue of classical problems on learning sparse low-degree polynomials with noise, extending the work of Chen, De, and Servedio (2020) from noisy \emph{linear} functions to general low-degree polynomials. Our main result gives a \emph{precise characterization} of when sparsity testing for low-degree polynomials admits constant sample complexity independent of dimension, together with a matching constant-sample algorithm in that regime. For any mean-zero, variance-one finitely supported distribution $\boldsymbol{X}$ over the reals, degree $d$, and any sparsity parameters $s \leq T$, we define a computable function $\mathrm{MSG}_{\boldsymbol{X},d}(\cdot)$, and: - For $T \ge \mathrm{MSG}_{\boldsymbol{X},d}(s)$, we give an $O_{s,\boldsymbol{X},d}(1)$-sample algorithm that distinguishes whether a multilinear degree-$d$ polynomial over $\mathbb{R}^n$ is $s$-sparse versus $\varepsilon$-far from $T$-sparse, given examples $(\boldsymbol{x},\, p(\boldsymbol{x}) + \mathrm{noise})_{\boldsymbol{x} \sim \boldsymbol{X}^{\otimes n}}$. Crucially, the sample complexity is \emph{completely independent} of the ambient dimension $n$. - For $T \leq \mathrm{MSG}_{\boldsymbol{X},d}(s) - 1$, we show that even without noise, any algorithm given samples $(\boldsymbol{x},p(\boldsymbol{x}))_{\boldsymbol{x} \sim \boldsymbol{X}^{\otimes n}}$ must use $Ω_{\boldsymbol{X},d,s}(\log n)$ examples. Our techniques employ a generalization of the results of Dinur et al. (2007) on the Fourier tails of bounded functions over $\{0,1\}^n$ to a broad range of finitely supported distributions, which may be of independent interest.

翻译：本文研究在给定含噪声多项式求值样本的条件下，检验未知低次多项式 $p$ 在 $\\mathbb{R}^n$ 上是否具有稀疏性（或远离稀疏性）的问题。该问题是经典含噪声稀疏低次多项式学习问题的性质检验版本，将 Chen、De 和 Servedio（2020）针对含噪声线性函数的研究推广至一般低次多项式。我们的主要结果精确刻画了低次多项式稀疏性检验在何种条件下可实现与维度无关的常数样本复杂度，并给出了该情形下匹配的常数样本算法。对于任意均值零、方差一的有限支撑实数分布 $\\boldsymbol{X}$、次数 $d$ 及稀疏性参数 $s \\leq T$，我们定义了一个可计算函数 $\\mathrm{MSG}_{\\boldsymbol{X},d}(\\cdot)$，并证明：- 当 $T \\ge \\mathrm{MSG}_{\\boldsymbol{X},d}(s)$ 时，存在一个样本复杂度为 $O_{s,\\boldsymbol{X},d}(1)$ 的算法，能够基于样本 $(\\boldsymbol{x},\\, p(\\boldsymbol{x}) + \\mathrm{noise})_{\\boldsymbol{x} \\sim \\boldsymbol{X}^{\\otimes n}}$ 区分 $\\mathbb{R}^n$ 上的 $d$ 次多重线性多项式是 $s$-稀疏还是 $\\varepsilon$-远离 $T$-稀疏。关键之处在于样本复杂度完全独立于环境维度 $n$。- 当 $T \\leq \\mathrm{MSG}_{\\boldsymbol{X},d}(s) - 1$ 时，我们证明即使在没有噪声的情况下，任何基于样本 $(\\boldsymbol{x},p(\\boldsymbol{x}))_{\\boldsymbol{x} \\sim \\boldsymbol{X}^{\\otimes n}}$ 的算法都需要至少 $\\Omega_{\\boldsymbol{X},d,s}(\\log n)$ 个样本。我们的技术方法推广了 Dinur 等人（2007）关于 $\\{0,1\\}^n$ 上有界函数傅里叶尾部的结果，将其扩展至广泛的有限支撑分布，这一推广可能具有独立的研究价值。