Les thèmes abordés incluent
Vendredi 4 avril à l'Institut Henri Poincaré.
Abstract: This talk will review results from the theory of pseudo-random graphs, that are (non-random) graphs sharing properties with random graphs. For dense graphs, we will describe equivalent characterizations of pseudo-randomness, including one based on spectral separation. For d-regular (sparse) graphs, we will show how “spectral expansion” implies pseudo-randomness, and a converse in the case of transitive graphs, a family which contains Cayley graphs among others. If time permits we may discuss additional topics such as quasi-random groups (that are groups whose non-trivial representations have large dimension), whose Cayley graphs turn out to be expanders, and consequences for the maximal size of “product-free” subsets of quasi-random groups.
Abstract: Recent works in time-frequency analysis proposed to switch the focus from the maxima of the spectrogram toward its zeros, which form a random point pattern with a very stable structure. Several signal processing tasks, such as component disentanglement and signal detection procedures, have already been renewed by using modern spatial statistics onthe pattern of zeros. Tough, they require cautious choice of both the discretization strategy and the observation window in the time-frequency plane. To overcome these limitations, we propose a generalized time-frequency representation: the Kravchuk transform, especially designed for discrete signals analysis, whose phase space is the unit sphere, particularly amenable to spatial statistics. We show that it has all desired properties for signal processing, among which covariance, invertibility and symmetry, and that the point process of the zeros of the Kravchuk transform of complex white Gaussian noise coincides with the zeros of the spherical Gaussian Analytic Function. Elaborating on this theorem, we finally develop a Monte Carlo envelope test procedure for signal detection based on the spatial statistics of the zeros of the Kravchuk spectrogram.
Outline: After, reviewing the unorthodox path focusing on the zeros of the standard spectrogram and the associated theoretical results on the distribution of zeros in the case of white noise, I will introduce the Kravchuk transform and study the random point process of its zeros from a spatial statistics perspective. Then I will present the designed Monte Carlo envelop test, and illustrate its numerical performance in adversarial settings, with both low signal-to-noise ratio and small number of samples, and compare it to state-of-the-art zeros-based detection procedures.
Abstract: Understanding the structure and stability of large ecosystems is a central problem in theoretical ecology, where interaction matrices are typically non-symmetric and heterogeneous. This motivates the use of Approximate Message Passing (AMP), a family of algorithms introduced by Donoho, Maleki & Montanari (2009) and Bayati & Montanari (2011), central to high-dimensional inference problems like compressed sensing. AMP comes with a tractable analytical framework - state evolution - that rigorously describes the asymptotic behavior of iterates in the high-dimensional limit, often recovering predictions from non-rigorous statistical physics methods.
Originally developed for symmetric random matrices, such as Wigner type ensembles, AMP has since been extended to more general classes of matrices. In this talk, I will briefly review the AMP framework and its connection to random matrix models. I will then present an AMP algorithm adapted to elliptic ensembles, and discuss recent developments aimed at handling non-symmetric matrices with sparse variance profiles and structured correlations, with a focus on applications to theoretical ecology.
Le séminaire MEGA a été créé en 2014 par Djalil Chafaï et Camille Male avec l'aide de Florent Benaych-Georges.
Image est tirée de https://www.mat.tuhh.de/forschung/aa/forschung.html.