Randomness Efficient Noise Stability and Generalized Small Bias Sets
We present a randomness efficient version of the linear noise operator T_ρ from boolean function analysis by constructing a sparse linear operator on the space of boolean functions {0,1}ⁿ → {0,1} with similar eigenvalue profile to T_ρ. The linear operator we construct is a direct consequence of a generalization of ε-biased sets to the product distribution 𝒟_p on {0,1}ⁿ where the marginal of each coordinate is p = 1/2-1/2ρ. Such a generalization is a small support distribution that fools linear tests when the input of the test comes from 𝒟_p instead of the uniform distribution. We give an explicit construction of such a distribution that requires log n + O_{p}(log log n + log1/(ε)) bits of uniform randomness to sample from, where the p subscript hides O(log² 1/p) factors. When p and ε are constant, this yields a support size nearly linear in n, whereas previous best known constructions only guarantee a size of poly(n). Furthermore, our construction implies an explicitly constructible "sparse" noisy hypercube graph that is a small set expander.
pseudorandomness
derandomization
epsilon biased sets
noise stability
Theory of computation~Pseudorandomness and derandomization
31:1-31:16
Regular Paper
Dana
Moshkovitz
Dana Moshkovitz
University of Texas at Austin, Department of Computer Science, TX, USA
Supported in part by NSF Grant CCF-1705028 and CCF-1648712.
Justin
Oh
Justin Oh
University of Texas at Austin, Department of Computer Science, TX, USA
Supported by NSF Grant CCF-1705028.
David
Zuckerman
David Zuckerman
University of Texas at Austin, Department of Computer Science, TX, USA
Supported in part by NSF Grant CCF-1705028 and a Simons Investigator Award (#409864).
10.4230/LIPIcs.FSTTCS.2020.31
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Dana Moshkovitz, Justin Oh, and David Zuckerman
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