In the process, we develop a novel analytic technique we dub multivariate FT-mollification. This provides a generic tool to approximate bounded (multivariate) functions by low-degree polynomials (with respect to several different notions of approximation). A univariate version of the method was introduced by Kane et al. (SODA 2010) in the context of streaming algorithms. In this work, we refine it and generalize it to the multivariate setting. We believe that our technique is of independent mathematical interest. To illustrate its generality, we note that it implies a multidimensional generalization of Jackson's classical result in approximation theory due to (Ganzburg 1979).
To obtain our main result, we combine the FT-mollification technique with several linear algebraic and probabilistic tools. These include the invariance principle of of Mossell, O'Donnell and Oleszkiewicz, anti-concentration bounds for low-degree polynomials, an appropriate decomposition of degree-2 polynomials, and a generalized hyper-contractive inequality for quadratic forms which takes the operator norm of the associated matrix into account. Our analysis is quite modular; it readily adapts to show that intersections of halfspaces and degree-2 threshold functions are fooled by bounded independence. From this it follows that Omega(1/eps^2)-wise independence derandomizes the Goemans-Williamson hyperplane rounding scheme.
Our techniques unify, simplify, and in some cases improve several recent results in the literature concerning threshold functions. For the case of ``regular'' halfspaces we give a simple proof of an optimal independence bound of Th