Consider a network of agents connected by communication links, where each agent holds a real value. The gossip problem consists in estimating the average of the values diffused in the network in a distributed manner. Current techniques for gossiping are designed to deal with worst-case scenarios, which is irrelevant in applications to distributed statistical learning and denoising in sensor networks. We design second-order gossip methods tailor-made for the case where the real values are i.i.d. samples from the same distribution. In some regular network structures, we are able to prove optimality of our methods, and simulations suggest that they are efficient in a wide range of random networks. Our approach of gossip stems from a new acceleration framework using the family of orthogonal polynomials with respect to the spectral measure of the network graph (joint work with Raphaël Berthier, and Pierre Gaillard).

# Colloquia

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

The moments of characteristic polynomials play a central role in Random Matrix Theory. They appear in many applications, ranging from quantum mechanics to number theory. The mixed moments of the characteristic polynomials of random unitary matrices, i.e. the joint moments of the polynomials and their derivatives, can be expressed recursively in terms of combinatorial sums involving partitions. However, these combinatorial sums are not easy to compute, and so this does not give an effective method for calculating the mixed moments in general. I shall describe an alternative evaluation of the mixed moments, in terms of solutions of the Painlevé V differential equation, that facilitates their computation and asymptotic analysis.

A matrix M of real numbers is called totally positive if every minor of M is nonnegative. This somewhat bizarre concept from linear algebra has surprising connections with analysis - notably polynomials and entire functions with real zeros, and the classical moment problem and continued fractions - as well as combinatorics. I will explain briefly some of these connections, and then introduce a generalization: a matrix M of polynomials (in some set of indeterminates) will be called coefficientwise totally positive if every minor of M is a polynomial with nonnegative coefficients. Also, a sequence (an)n≥0 of real numbers (or polynomials) will be called (coefficientwise) Hankel-totally positive if the Hankel matrix H = (ai+j)i,j ≥= 0 associated to (an) is (coefficientwise) totally positive. It turns out that many sequences of polynomials arising in enumerative combinatorics are (empirically) coefficientwise Hankel-totally positive; in some cases this can be proven using continued fractions, while in other cases it remains a conjecture.