IMPROVED OPTIMAL TESTING RESULTS FROM GLOBAL HYPERCONTRACTIVITY

The problem of testing low-degree polynomials has received significant attention over the years due to its importance in theoretical computer science. The problem is specified by three parameters, the field size q, the degree d, and the proximity parameter δ, and the goal is to design a test that makes as few queries as possible to a given function and distinguishes between the case the function has degree at most d and the case it is δ-far from any degree d function. We say that a test is optimal if it makes Q queries and rejects any function which is δ-far from degree d with probability \Omega(min(1, Qδ)). The most natural tester to consider is the t-flat test, wherein one picks an affine subspace A of dimension t (chosen appropriately) uniformly at random, and checks that de\sansg(f|_A) ≤ d. The t-flat test was shown to be optimal by Bhattacharyya et al. [Proceedings of FOCS, 2010, pp. 488-497] for q = 2, and later by Haramaty, Shpilka, and Sudan [SIAM J. Comput., 42 (2013), pp. 536-562] for all prime powers q. Their analyses, however, has a tower type dependency on the field size q (i.e., in the hidden constant in the big \Omega notation). We improve the result of Haramaty, Shpilka, and Sudan, showing that the dependency on the field size is polynomial in q. Our technique also applies in the more general setting of lifted affine invariant codes and gives the same polynomial dependency on the field size. This answers a problem raised by Haramaty, Ron-Zewi, and Sudan [Theory Comput., 11 (2015), pp. 299-338]. Our approach significantly deviates from the strategy taken in earlier works and is based on studying the structure of the collection of erroneous subspaces, i.e., subspaces A such that f|_A has degree greater than d. Toward this end, we observe that these sets are poorly expanding in the affine Grassmann graph and use that to establish structural results on them via global hypercontractivity. We then use this structure to perform local correction on f.