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Matrice aléatoire

En théorie des probabilités et en physique mathématique, une matrice aléatoire est une matrice dont les éléments sont des variables aléatoires. La théorie des matrices aléatoires a pour objectif de comprendre certaines propriétés de ces matrices, comme leur norme d'opérateur, leurs valeurs propres ou leurs valeurs singulières. Face à la complexité croissante des spectres nucléaires observés expérimentalement dans les années 1950, Wigner a suggéré de remplacer l'opérateur hamiltonien du noyau par une matrice aléatoire. Cette hypothèse féconde a conduit au développement rapide d'un nouveau champ de recherche très actif en physique théorique, qui s'est propagé à la théorie des nombres en mathématiques, avec notamment une connexion intéressante avec la fonction zêta de Riemann (voir par exemple l'article prépublié en par Michael Griffin, Ken Ono, Larry Rolen et Don Zagier et les commentaires (en anglais) de Robert C. Smith sur son blog). En plus de ces exemples on compte parmi les app

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Loi de Poisson

En théorie des probabilités et en statistiques, la loi de Poisson est une loi de probabilité discrète qui décrit le comportement du nombre d'événements se produisant dans un intervalle de temps fixé

Fonction de répartition

En théorie des probabilités, la fonction de répartition, ou fonction de distribution cumulative, d'une variable aléatoire réelle X est la fonction F{{sub|{{mvar|X}}}} qui, à

Variable aléatoire

vignette|La valeur d’un dé après un lancer est une variable aléatoire comprise entre 1 et 6. En théorie des probabilités, une variable aléatoire est une variable dont la valeur est déterminée après la

Statistical Applications of Random Matrix Theory: Comparison of Two Populations

Rémy Mariétan

During the last twenty years, Random matrix theory (RMT) has produced numerous results that allow a better understanding of large random matrices. These advances have enabled interesting applications in the domain of communication. Although this theory can contribute to many other domains such as brain imaging or genetic research, its has been rarely applied. The main barrier to the adoption of RMT may be the lack of concrete statistical results from probabilistic Random matrix theory. Indeed, direct generalisation of classical multivariate theory to high dimensional assumptions is often difficult and the proposed procedures often assume strong hypotheses on the data matrix such as normality or overly restrictive independence conditions on the data. This thesis proposes a statistical procedure for testing the equality of two independent estimated covariance matrices when the number of potentially dependent data vectors is large and proportional to the size of the vectors corresponding to the number of observed variables. Although the existing theory builds a very good intuition of the behaviour of these matrices, it does not provide enough results to build a satisfactory test for both the power and the robustness. Hence, inspired by spike models, we define the residual spikes and prove many theorems describing the behaviour of many statistics using eigenvectors and eigenvalues in very general cases. For example in the two central theorems of this thesis, the Invariant Angle Theorem and the Invariant Dot Product Theorem. Using numerous generalisations of the theory, this thesis finally proposes a description of the behaviour of a statistic under a null hypothesis. This statistic allows the user to test the equality of two populations, but also other null hypotheses such as the independence of two sets of variables. Finally, the robustness of the procedure is demonstrated for different classes of models and criteria for evaluating robustness are proposed to the reader. Therefore, the major contribution of this thesis is to propose a methodology both easy to apply and having good properties. Secondly, a large number of theoretical results are demonstrated and could be easily used to build other applications.

EPFL2019

Chargement

Modern Portfolio Theory Revisited

David Sébastien Morton de Lachapelle

In the first place the behavior of (online) traders on markets is analyzed and modeled, and it is shown that the average investor behaves as a mean-variance optimizer in finance. Within this description, transaction costs play a key role in explaining observed investment patterns and in particular an important uncovered relation between average investment and portfolio value. As online investors take into account transaction costs in their investment strategy, they are also sensitive to high portfolio rebalancing costs. Solutions to avoid high portfolio turnovers are investigated: first in the one-dimensional case, where it is shown that estimators with minimal-dispersion improve both the accuracy and the variance of volatility and Value-at-Risk forecasts; second, in a multi-asset environment where the ideal covariance matrix must have good conditioning properties to maintain reasonable portfolio turnover. Theoretical results are derived using Random Matrix Theory, as for instance an equation for the Stieltjes transform of the eigenvalue density of i.i.d. correlation matrices with general time-decreasing weight profiles. Results found in the one-dimensional case are generalized leading to long-memory covariance matrices. Finally, a "curse of dimensionality" in portfolio allocation is tackled by generalizing the Spectral Coarse Graining, a method of reduction for complex networks, that is extended to the simplification of the mean-variance optimization problem.

EPFL2012

Chargement

Subquadratic Overparameterization for Shallow Neural Networks

Volkan Cevher, Armin Eftekhari, Thomas Michaelsen Pethick, Chaehwan Song

Overparameterization refers to the important phenomenon where the width of a neural network is chosen such that learning algorithms can provably attain zero loss in nonconvex training. The existing theory establishes such global convergence using various initialization strategies, training modifications, and width scalings. In particular, the state-of-the-art results require the width to scale quadratically with the number of training data under standard initialization strategies used in practice for best generalization performance. In contrast, the most recent results obtain linear scaling either with requiring initializations that lead to the “lazy-training”, or training only a single layer. In this work, we provide an analytical framework that allows us to adopt standard initialization strategies, possibly avoid lazy training, and train all layers simultaneously in basic shallow neural networks while attaining a desirable subquadratic scaling on the network width. We achieve the desiderata via Polyak-Łojasiewicz condition, smoothness, and standard assumptions on data, and use tools from random matrix theory.

2021

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MATH-467: Probabilistic methods in combinatorics

We develop a sophisticated framework for solving problems in discrete mathematics through the use of randomness (i.e., coin flipping). This includes constructing mathematical structures with unexpected (and sometimes paradoxical) properties for which no other methods of construction are known.

FIN-525: Financial big data

The course's first part introduces modern methods to acquire, clean, and analyze large quantities of financial data efficiently. The second part expands on how to apply these techniques to financial analysis, in particular to intraday data and investment strategies.

PHYS-645: Physics of random and disordered systems

Introduction to the physics of random processes and disordered systems, providing an overview over phenomena, concepts and theoretical approaches Topics include: Random walks; Roughening/pinning; Localization; Random matrix theory; Spin glasses; Disorder and critical phenomena