## Bounded Independence Fools Degree-2 Threshold Functions

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Diakonikolas, Ilias

Kane, Daniel M.

Nelson, Jelani

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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 epsilon for k = poly(1/epsilon). 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 epsilon-fool the class of degree-2 threshold functions with
seed length log(n)*poly(1/epsilon).
Our approach is quite robust: it easily extends to yield that the
intersection of any constant number of degree-2 threshold functions is
epsilon-fooled by poly(1/epsilon)-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.

Comment: Using v1 numbering: removed Lemma G.5 from the Appendix (it was wrong). Net effect is that Theorem G.6 reduces the m^6 dependence of Theorem 8.1 to m^4, not m^2

Comment: Using v1 numbering: removed Lemma G.5 from the Appendix (it was wrong). Net effect is that Theorem G.6 reduces the m^6 dependence of Theorem 8.1 to m^4, not m^2

##### Keywords

Computer Science - Computational Complexity