Bounded Independence Fools degree-2 Threshold Functions
Ilias Diakonikolas, Daniel Kane, Jelani Nelson

 Manuscript, 2009

Abstract: Let x be a random vector coming from any k-wise independent distribution over {-1, 1}n . For an n-variate degree-2 polynomial p, we prove that E[sgn(p(x))] is determined up to an additive ε for k = poly(1/ε). This answers an open question of Diakonikolas et al. (FOCS 2009). Using standard constructions of k-wise independent distributions, we obtain a broad class of explicit generators that ε -fool the class of degree-2 threshold functions with seed length log n ∙poly(1/ε).

 

Our approach is quite robust: it easily extends to yield that the intersection of any constant number of degree-2 threshold functions is ε-fooled by poly(1/ ε)-wise independence. Our results also hold if the entries of x are k-wise independent standard normals, implying for example that bounded independence derandomizes the Goemans-Williamson hyperplane rounding scheme.

 

To achieve our results, we introduce a technique we dub multivariate FT-mollification, a generalization of the univariate form introduced by Kane et al. (SODA 2010) in the context of streaming algorithms. Along the way we prove a generalized hypercontractive inequality for quadratic forms which takes the operator norm of the associated matrix into account. These techniques may be of independent interest.

 

 

Preliminary version: [PDF]

Arxiv report: [arXiv]

 

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