Sample-Based High-Dimensional Convexity Testing

In the problem of high-dimensional convexity testing, there is an unknown set S ⊆ R which is promised to be either convex or ε-far from every convex body with respect to the standard multivariate normal distribution N (0, 1). The job of a testing algorithm is then to distinguish between these two cases while making as few inspections of the set S as possible. In this work we consider sample-based testing algorithms, in which the testing algorithm only has access to labeled samples (x, S(x)) where each x is independently drawn from N (0, 1). We give nearly matching sample complexity upper and lower bounds for both one-sided and twosided convexity testing algorithms in this framework. For constant ε, our results show that the sample complexity of one-sided convexity testing is 2Θ̃(n) samples, while for two-sided convexity testing it is 2Θ̃( √ n). 1998 ACM Subject Classification F.2 Analysis of Algorithms and Problem Complexity