CS 880 : Advanced Complexity Theory 2 / 18 / 2008 Lecture 11 : Concentration on Influential Variables
نویسندگان
چکیده
Last time, we proved that the noise operator Tα is hypercontractive. Recall that the noise operator Tαf is defined as (Tαf)(x) = Ey∼ x[f(y)] (1) where = 1−α 2 , and y ∼ x means that we choose yi to be xi with probability 1− and −xi with probability . We have also shown that it has Fourier expansion Tαf = ∑ S⊆[n] α f̂(S)χS . The hypercontractivity of the noise operator states that not only is the norm of Tαf is no more than the norm of f , but that this holds even if we use the q-norm for Tαf and the p-norm for f where p < q. The latter is a stronger statement since the q-norm of a function is no smaller than its p-norm if p < q. Formally,
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