A Note on the Entropy/Influence Conjecture

The entropy/influence conjecture, raised by Friedgut and Kalai in 1996, seeks to relate two different measures of concentration of the Fourier coefficients of a Boolean function. Roughly saying, it claims that if the Fourier spectrum is "smeared out", then the Fourier coefficients are concentrated on "high" levels. In this note we generalize the conjecture to biased product measures on the discrete cube, and prove a variant of the conjecture for functions with an extremely low Fourier weight on the "high" levels.Comment: 12 page

Geometric Influences II: Correlation Inequalities and Noise Sensitivity

In a recent paper, we presented a new definition of influences in product spaces of continuous distributions, and showed that analogues of the most fundamental results on discrete influences, such as the KKL theorem, hold for the new definition in Gaussian space. In this paper we prove Gaussian analogues of two of the central applications of influences: Talagrand's lower bound on the correlation of increasing subsets of the discrete cube, and the Benjamini-Kalai-Schramm (BKS) noise sensitivity theorem. We then use the Gaussian results to obtain analogues of Talagrand's bound for all discrete probability spaces and to reestablish analogues of the BKS theorem for biased two-point product spaces.Comment: 20 page

A directed isoperimetric inequality with application to Bregman near neighbor lower bounds

Bregman divergences $D_\phi$ are a class of divergences parametrized by a convex function $\phi$ and include well known distance functions like $\ell_2^2$ and the Kullback-Leibler divergence. There has been extensive research on algorithms for problems like clustering and near neighbor search with respect to Bregman divergences, in all cases, the algorithms depend not just on the data size $n$ and dimensionality $d$, but also on a structure constant $\mu \ge 1$ that depends solely on $\phi$ and can grow without bound independently. In this paper, we provide the first evidence that this dependence on $\mu$ might be intrinsic. We focus on the problem of approximate near neighbor search for Bregman divergences. We show that under the cell probe model, any non-adaptive data structure (like locality-sensitive hashing) for $c$-approximate near-neighbor search that admits $r$ probes must use space $\Omega(n^{1 + \frac{\mu}{c r}})$. In contrast, for LSH under $\ell_1$ the best bound is $\Omega(n^{1+\frac{1}{cr}})$. Our new tool is a directed variant of the standard boolean noise operator. We show that a generalization of the Bonami-Beckner hypercontractivity inequality exists "in expectation" or upon restriction to certain subsets of the Hamming cube, and that this is sufficient to prove the desired isoperimetric inequality that we use in our data structure lower bound. We also present a structural result reducing the Hamming cube to a Bregman cube. This structure allows us to obtain lower bounds for problems under Bregman divergences from their $\ell_1$ analog. In particular, we get a (weaker) lower bound for approximate near neighbor search of the form $\Omega(n^{1 + \frac{1}{cr}})$ for an $r$-query non-adaptive data structure, and new cell probe lower bounds for a number of other near neighbor questions in Bregman space.Comment: 27 page

Noise stability of functions with low influences: invariance and optimality

In this paper we study functions with low influences on product probability spaces. The analysis of boolean functions with low influences has become a central problem in discrete Fourier analysis. It is motivated by fundamental questions arising from the construction of probabilistically checkable proofs in theoretical computer science and from problems in the theory of social choice in economics. We prove an invariance principle for multilinear polynomials with low influences and bounded degree; it shows that under mild conditions the distribution of such polynomials is essentially invariant for all product spaces. Ours is one of the very few known non-linear invariance principles. It has the advantage that its proof is simple and that the error bounds are explicit. We also show that the assumption of bounded degree can be eliminated if the polynomials are slightly ``smoothed''; this extension is essential for our applications to ``noise stability''-type problems. In particular, as applications of the invariance principle we prove two conjectures: the ``Majority Is Stablest'' conjecture from theoretical computer science, which was the original motivation for this work, and the ``It Ain't Over Till It's Over'' conjecture from social choice theory