Testing Sparse Functions over the Reals
This work addresses function testing over real numbers, which is important for real-world applications, but it is incremental as it extends prior work on algebraic representations to sparse functions.
The paper tackles the problem of testing whether a real-valued function is close to being k-linear, a k-sparse polynomial, or a k-junta, given query access and using an ℓ1-distance metric under a Gaussian distribution. It presents efficient testers and proves Ω(k) lower bounds for each property.
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.