Distribution-Free Testing of Linear Functions on ℝⁿ

We study the problem of testing if a function depends on a small number of linear directions of its input data. We call a function $f$ a linear $k$-junta if it is completely determined by some $k$-dimensional subspace of the input space. In this paper, we study the problem of testing whether a given $n$ variable function $f : \mathbb{R}^n \to \{0,1\}$, is a linear $k$-junta or $ε$-far from all linear $k$-juntas, where the closeness is measured with respect to the Gaussian measure on $\mathbb{R}^n$. Linear $k$-juntas are a common generalization of two fundamental classes from Boolean function analysis (both of which have been studied in property testing) $\textbf{1.}$ $k$- juntas which are functions on the Boolean cube which depend on at most k of the variables and $\textbf{2.}$ intersection of $k$ halfspaces, a fundamental geometric concept class. We show that the class of linear $k$-juntas is not testable, but adding a surface area constraint makes it testable: we give a $\mathsf{poly}(k \cdot s/ε)$-query non-adaptive tester for linear $k$-juntas with surface area at most $s$. We show that the polynomial dependence on $s$ is necessary. Moreover, we show that if the function is a linear $k$-junta with surface area at most $s$, we give a $(s \cdot k)^{O(k)}$-query non-adaptive algorithm to learn the function up to a rotation of the basis. In particular, this implies that we can test the class of intersections of $k$ halfspaces in $\mathbb{R}^n$ with query complexity independent of $n$.