Friday, March 23, 2007

How much are increasing sets positively correlated?

There's a paper of Talagrand, one of the masters of Fourier analysis of boolean functions, called "How much are increasing sets positively correlated?" which I've always kind of known about but never really fully read. A few days ago I studied it more closely, and it turns out to be really nice, with proofs that are not too hard (although the notation took me a while to decipher). Maybe I'll show some of it in a future class.

It proves some nice facts, including:
  • If $f : \{-1,1\}^n \to \{-1,1\}$ has $Var[f] \leq p$, then $W_1(f) = \sum_{|S| = 1} \hat{f}(S)^2 \leq O(p \log (1/p))$. (This is a fact that I seem to use all the time in my work...)
  • If $\mathbb{I}\mathbb{I}(f)$ denotes $\sum_i Inf_i(f)^2$, then $W_2(f) = \sum_{|S| = 2} \hat{f}(S)^2 \leq O(\mathbb{I}\mathbb{I}(f) \log(1/\mathbb{I}\mathbb{I}(f)))$. (This is the main technical lemma/theorem in the paper.)
  • There is a monotone function $f : \{-1,1\}^n \to \{-1,1\}$ for which a constant fraction of all $x$ have the property that $f$ has $\Omega(\sqrt{n})$ sensitive coordinates on $x$. This immediately implies that the function has $\mathbb{I}(f) \geq \Omega(\sqrt{n})$, but in a much different way than $\mathbb{I}(Maj_n) \geq \Omega(\sqrt{n})$. With Majority, most $x$'s have $0$ sensitive coordinates, but a $\Omega(1/\sqrt{n})$ fraction of them have $(n+1)/2$ sensitive coordinates. With this function, the distribution on the number of sensitive coordinates is much simpler -- a constant fraction have $\Omega(\sqrt{n})$. The function is constructed randomly -- a random DNF of width $\sqrt{n}$ and size $(\ln 2) 2^{\sqrt{n}}$.
Actually, this last function, one can check, gives an example of a monotone function that has $NS_{O(1/\sqrt{n})}(f) \geq \Omega(1)$; and as we saw when studying hardness amplification, $\Theta(1/\sqrt{n})$ is the least noise rate to which a monotone function can be sensitive.

An important improvement to the second bullet point above came from a notable paper of Benjamini, Kalai, and Schramm: They showed that in fact $W_k(f) \leq O(\mathbb{I}\mathbb{I} (f) \log^{k-1}(1/\mathbb{I}\mathbb{I}(f))$. I'm trying to use this fact right now for a problem I'm working on...

No comments: