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Revision #1 to TR09-117 | 18th February 2010 17:59

Bounded Independence Fools Degree-2 Threshold Functions

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Revision #1
Authors: Ilias Diakonikolas, Daniel Kane, Jelani Nelson
Accepted on: 18th February 2010 17:59
Downloads: 692
Keywords: 


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 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.



Changes to previous version:

-- Fixed various typos + other minor changes
-- (Using the lemma numbering from the first version.) Removed Lemma G.5 from the Appendix (it was wrong). The net effect is that Theorem G.6 in the Appendix reduces the m^6 dependence of Theorem 8.1 to m^4, not m^2.


Paper:

TR09-117 | 18th November 2009 02:47

Bounded Independence Fools Degree-2 Threshold Functions





TR09-117
Authors: Ilias Diakonikolas, Daniel Kane, Jelani Nelson
Publication: 18th November 2009 04:13
Downloads: 1030
Keywords: 


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 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.



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