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

Tighter Relations Between Sensitivity and Other Complexity Measures

Sensitivity conjecture is a longstanding and fundamental open problem in the area of complexity measures of Boolean functions and decision tree complexity. The conjecture postulates that the maximum sensitivity of a Boolean function is polynomially related to other major complexity measures. Despite much attention to the problem and major advances in analysis of Boolean functions in the past decade, the problem remains wide open with no positive result toward the conjecture since the work of Kenyon and Kutin from 2004. In this work, we present new upper bounds for various complexity measures in terms of sensitivity improving the bounds provided by Kenyon and Kutin. Specifically, we show that deg(f)^{1-o(1)}=O(2^{s(f)}) and C(f) < 2^{s(f)-1} s(f); these in turn imply various corollaries regarding the relation between sensitivity and other complexity measures, such as block sensitivity, via known results. The gap between sensitivity and other complexity measures remains exponential but these results are the first improvement for this difficult problem that has been achieved in a decade.Comment: This is the merged form of arXiv submission 1306.4466 with another work. Appeared in ICALP 2014, 14 page