Maharam's problem

We construct an exhaustive submeasure that is not equivalent to a measure. This solves problems of J. von Neumann (1937) and D. Maharam (1947)

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

Parisi measures

AbstractIn the Parisi theory of spin glasses, the limiting free energy of the system is computed by optimizing over a “functional order parameter”. In mathematical terms this amounts to construct certain functions F(μ) of a probability measure μ on [0,1] and to compute the infimum over μ. The study of the maps μ↦F(μ) is a challenging problem of functional analysis. Progress on this problem seems required for further advances in the theory of spin glasses. The main objective of this paper is to explain the functional analysis part of the problems to the reader with no background (or interest) in spin glasses. As a first step in the study of these functions F(μ), we prove certain differentiability properties, that allow in certain cases to interpret (as conjectured by physicists) the Parisi measure (i.e. the probability μ at which F(μ) is minimum) in terms of spin glasses

Concentration of Measure and Isoperimetric Inequalities in Product Spaces

The concentration of measure prenomenon roughly states that, if a set $A$ in a product $\Omega^N$ of probability spaces has measure at least one half, ``most'' of the points of $\Omega^N$ are ``close'' to $A$. We proceed to a systematic exploration of this phenomenon. The meaning of the word ``most'' is made rigorous by isoperimetric-type inequalities that bound the measure of the exceptional sets. The meaning of the work ``close'' is defined in three main ways, each of them giving rise to related, but different inequalities. The inequalities are all proved through a common scheme of proof. Remarkably, this simple approach not only yields qualitatively optimal results, but, in many cases, captures near optimal numerical constants. A large number of applications are given, in particular in Percolation, Geometric Probability, Probability in Banach Spaces, to demonstrate in concrete situations the extremely wide range of application of the abstract tools