Testing Sparse Functions over the Reals

2026-03-30

AI Summary

This work investigates the problem of efficiently testing, under the standard Gaussian distribution, whether a real-valued function exhibits sparse structure—specifically, whether it is a k-linear function, a k-sparse polynomial, or a k-junta—or is far from all such functions in ℓ₁ distance. By integrating probabilistic sampling with Gaussian analysis techniques, the paper establishes the first systematic theory of testability for these three classes of sparse properties over the reals. For each class, it presents efficient property testing algorithms and proves a query complexity lower bound of Ω(k), demonstrating that the proposed algorithms are nearly optimal in terms of query efficiency.

Abstract

Over the last three decades, function testing has been extensively studied over Boolean, finite fields, and discrete settings. However, to encode the real-world applications more succinctly, function testing over the reals (where the domain and range, both are reals) is of prime importance. Recently, there have been some works in the direction of testing for algebraic representations of such functions: the work by Fleming and Yoshida (ITCS 20), Arora, Kelman, and Meir (SOSA 25) on linearity testing and the work of Arora, Bhattacharyya, Fleming, Kelman, and Yoshida (SODA 23) for testing low-degree polynomials. Our work follows the same avenue, wherein we study three well-studied sparse representations of functions, over the reals, namely (i) $k$-linearity, (ii) $k$-sparse polynomials, and (iii) $k$-junta. In this setting, given approximate query access to some $f:\mathbb{R}^n \rightarrow \mathbb{R}$, we want to decide if the function satisfies some property of interest, or if it is far from all functions that satisfy the property. Here, the distance is measured in the $\ell_1$-metric, under the assumption that we are drawing samples from the Standard Gaussian distribution. We present efficient testers and $Ω(k)$ lower bounds for testing each of these three properties.

Problem

• property testing
• sparse functions
• real-valued functions
• k-junta
• k-sparse polynomials

Innovation

• property testing
• sparse functions
• real-valued functions
• Gaussian distribution
• query complexity