A characterization of constant‐sample testable properties

Abstract

We characterize the set of properties of Boolean‐valued functions f:X→{0,1} on a finite domain scriptX that are testable with a constant number of samples (x,f(x)) with x drawn uniformly at random from scriptX. Specifically, we show that a property scriptP is testable with a constant number of samples if and only if it is (essentially) a k‐part symmetric property for some constant k, where a property is k‐part symmetric if there is a partition X1,…,Xk of scriptX such that whether f:X→{0,1} satisfies the property is determined solely by the densities of f on X1,…,Xk. We use this characterization to show that symmetric properties are essentially the only graph properties and affine‐invariant properties that are testable with a constant number of samples and that for every constant d≥1, monotonicity of functions on the d‐dimensional hypergrid is testable with a constant number of samples.

Sample-based high-dimensional convexity testing

“…Such an algorithm has access to independent draws (x, S(x)) ∈ R n × {0, 1}, where x is drawn from N (0, 1) n and S ⊆ R n is the unknown set being tested for convexity (so in particular the algorithm cannot select points to be queried) with S(x) = 1 if x ∈ S. We say such an algorithm is an ε-tester for convexity if it accepts S with probability at least 2/3 when S is convex and rejects with probability at least 2/3 when it is ε-far from convex, i.e., dist(S, C) ≥ ε for all convex sets C ⊆ R n . The model of samplebased testing was originally introduced by Goldreich, Goldwasser, and Ron almost two decades ago [GGR98], where it was referred to as "passive testing;" it has received significant attention over the years [KR00, GGL + 00, BBBY12, GR16], with an uptick in research activity in this model over just the past year or so [AHW16,BY16,BMR16c,BMR16b,BMR16a].…”