Testing Sparse Functions over the Reals

TL;DR

This paper develops efficient algorithms and lower bounds for testing sparse function properties over the real numbers, focusing on k-linearity, k-sparse polynomials, and k-juntas, using Gaussian samples.

Contribution

It introduces the first efficient testers and lower bounds for property testing of sparse functions over the reals in the Gaussian setting.

Findings

Designed efficient property testers for k-linearity, k-sparse polynomials, and k-juntas.

Established (k) lower bounds for testing these properties.

Analyzed testing complexity under (1) Gaussian sample access.

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 $\Omega(k)$ lower bounds for testing each of these three properties.