Testing Surface Area

We consider the problem of estimating the surface area of an unknown n-dimensional set F given membership oracle access. In contrast to previous work, we do not assume that F is convex, and in fact make no assumptions at all about F . By necessity this means that we work in the property testing model; we seek an algorithm which, given parameters A and , satisfies: • if surf(F ) ≤ A then the algorithm accepts (whp); • if F is not -close to some set G with surf(G) ≤ κA, then the algorithm rejects (whp). We call κ ≥ 1 the “approximation factor” of the testing algorithm. The n = 1 case (in which “surf(F ) = 2m” means F is a disjoint union of m intervals) was introduced by Kearns and Ron [KR98], who solved the problem with κ = 1/ and O(1/ ) oracle queries. Later, Balcan et al. [BBBY12] solved it with with κ = 1 and O(1/ ) queries. We give the first result for higher dimensions n. Perhaps surprisingly, our algorithm completely evades the “curse of dimensionality”: for any n and any κ > 4 π ≈ 1.27 we give a test that uses O(1/ ) queries. For small n we have improved bounds. For n = 1 we can achieve κ = 1 withO(1/ ) queries (slightly improving [BBBY12]), or any κ > 1 with O(1/ ) queries (improving [KR98]). For n = 2, 3 we obtain κ ≈ 1.08, 1.125 respectively, with O(1/ ) queries. Getting an arbitrary κ > 1 for n > 1 remains an open problem. Finally, motivated by the learning results from [KOS08], we extend our techniques to obtain a similar tester for Gaussian surface area for any n, with query complexity O(1/ ) and any approximation factor κ > 4 π ≈ 1.27. ∗Supported by NSF grants CCF-1065106. †Supported by NSF grants CCF-0747250 and CCF-1116594 and a grant from the MSR–CMU Center for Computational Thinking. ‡This work was supported in part by the National Basic Research Program of China Grant 2011CBA00300, 2011CBA00301, the National Natural Science Foundation of China Grant 61033001, 61061130540. Research done while visiting Carnegie Mellon University.