Séminaire Matrices et graphes aléatoires (MEGA)

Les thèmes abordés incluent

Prochaine séance

Vendredi 7 février à l'Université Paul Sabatier (Toulouse), bâtiment 1R3 (code 4823A), 1er étage, salle K. Johnson.

Abstract: Quantum channels are the objects that describe the dynamics of quantum systems (just as transition matrices for classical ones). In this lecture we will mainly investigate the following problem: what does the spectrum of a quantum channel typically look like? This is interesting as it allows to grasp important properties of the corresponding system, such as mixing or decay of correlations. From a mathematical point of view, we will see that this question boils down to understanding the spectral distribution of a random matrix model with a specific tensor product structure. We will show that, under very general assumptions, the latter generically has a large spectral gap, between its first and second largest eigenvalues, and a bulk spectrum that converges towards a well-defined limit, which can be described using tools from free probability. If time allows, we will explain how these results can be extended to other tensor product random matrix models, beyond those describing random quantum channels, thanks to a central limit theorem for tensor products of free variables. The presentation will be mostly based on the works: arXiv:2302.07772 (with P. Youssef) and arXiv:2311.12368, arXiv:2404.19662 (with P. Oliveira Santos and P. Youssef).

Abstract: We consider the Brown measure of $a + \mathfrak c$, where $a$ lies in a commutative tracial von Neumann algebra $\mathcal B$ and $\mathfrak c$ is a $\mathcal B$-valued circular element. Under certain conditions on $a$ and the covariance of $\mathfrak c$, we show that the Brown measure of $a + \mathfrak c$ has a density with respect to the Lebesgue measure on the complex plane. This density is real analytic apart from jump discontinuities at the boundary of its support. Furthermore, we fully classify the types of local growth of this density, which may occur wherever it vanishes in the interior of its support. The boundary of the support is one-dimensional and we fully classify the types of singularities, which appear wherever the boundary is not smooth. We also prove that all these singularity types can emerge even if $\mathfrak c$ is a standard circular element. Moreover, we establish that these Brown measures arise as limits of the empirical spectral measures of large, diagonally deformed non-Hermitian random matrices with independent entries, whose variance profiles are allowed to have zero blocks. This is based on joint works with Torben Krüger.

Abstract: The behavior of the random feature model in a high-dimensional framework has recently become a popular topic of interest in the machine learning literature. This model is generally considered for feature vectors composed of independent and identically distributed (iid) entries. We move beyond this specific assumption, which may be restrictive in various applications. To this end, we propose studying the performance of the random feature model with non-iid data by introducing a variance profile to the feature matrix. The performance of this model is linked to the spectrum of the random feature matrix, which turns out to be a nonlinear mixture of random variance profiled matrices. We have computed the limiting traffic distribution of such matrices using an extension of the method of moments. Knowledge of this distribution allowed us to introduce a new random matrix, which we call the « linear plus chaos » matrix, and which shares the same limiting spectrum as the random feature matrix. This linear plus chaos model proves to be simpler to study and has enabled us to derive deterministic equivalents that describe the asymptotic behavior of the performance of the random feature model.

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Histoire

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.