Testing noisy low-degree polynomials for sparsity

2025-11-11

AI Summary

This paper studies property testing of low-degree polynomial sparsity under noise: given noisy evaluations of a degree-$d$ polynomial at random points, determine whether it is $s$-sparse or $varepsilon$-far from all $T$-sparse low-degree polynomials. Methodologically, the work integrates multilinear polynomial analysis with Fourier tail estimation, extending Dinur et al.’s techniques—originally developed for the Boolean hypercube—to arbitrary finite-support distributions. The main contribution is the first exact characterization of constant-sample complexity for this problem: when $T geq mathrm{MSG}$, the query complexity is $O_{s,X,d}(1)$, independent of dimension $n$; whereas when $T < mathrm{MSG}-1$, a lower bound of $Omega(log n)$ holds, revealing the critical role of the monomial support gap ($mathrm{MSG}$) as a sharp threshold. These results break the sample-size bottleneck in high-dimensional sparsity testing and generalize noise-tolerant linearity testing to general low-degree polynomials.

Technology Category

• Machine Learning: Learning Theory
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

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 $oldsymbol{X}$ over the reals, degree $d$, and any sparsity parameters $s leq T$, we define a computable function $mathrm{MSG}_{oldsymbol{X},d}(cdot)$, and: - For $T ge mathrm{MSG}_{oldsymbol{X},d}(s)$, we give an $O_{s,oldsymbol{X},d}(1)$-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 $(oldsymbol{x},, p(oldsymbol{x}) + mathrm{noise})_{oldsymbol{x} sim oldsymbol{X}^{otimes n}}$. Crucially, the sample complexity is emph{completely independent} of the ambient dimension $n$. - For $T leq mathrm{MSG}_{oldsymbol{X},d}(s) - 1$, we show that even without noise, any algorithm given samples $(oldsymbol{x},p(oldsymbol{x}))_{oldsymbol{x} sim oldsymbol{X}^{otimes n}}$ must use $Omega_{oldsymbol{X},d,s}(log n)$ 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.

Problem

• Testing sparsity of noisy low-degree polynomials
• Characterizing constant sample complexity conditions
• Distinguishing sparse versus far-from-sparse polynomials

Innovation

• Testing sparsity of noisy polynomials via random samples
• Using computable MSG function to determine sample complexity
• Extending Fourier analysis techniques to finitely supported distributions