Testing noisy low-degree polynomials for sparsity

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.