Regularization by free additive convolution, square and rectangular cases

The free convolution is the binary operation on the set of probability measures on the real line which allows to deduce, from the individual spectral distributions, the spectral distribution of a sum of independent unitarily invariant square random matrices or of a sum of free operators in a non commutative probability space. In the same way, the rectangular free convolution allows to deduce, from the individual singular distributions, the singular distribution of a sum of independent unitarily invariant rectangular random matrices. In this paper, we consider the regularization properties of these free convolutions on the whole real line. More specifically, we try to find continuous semigroups $(\mu_t)$ of probability measures such that $\mu_0$ is the Dirac mass at zero and such that for all positive $t$ and all probability measure $\nu$, the free convolution of $\mu_t$ with $\nu$ (or, in the rectangular context, the rectangular free convolution of $\mu_t$ with $\nu$) is absolutely continuous with respect to the Lebesgue measure, with a positive analytic density on the whole real line. In the square case, we prove that in semigroups satisfying this property, no measure can have a finite second moment, and we give a sufficient condition on semigroups to satisfy this property, with examples. In the rectangular case, we prove that in most cases, for $\mu$ in a continuous rectangular-convolution-semigroup, the rectangular convolution of $\mu$ with $\nu$ either has an atom at the origin or doesn't put any mass in a neighborhood of the origin, thus the expected property does not hold. However, we give sufficient conditions for analyticity of the density of the rectangular convolution of $\mu$ with $\nu$ except on a negligible set of points, as well as existence and continuity of a density everywhere.Comment: 43 pages, to appear in Complex Analysis and Operator Theor

Large deviations of the extreme eigenvalues of random deformations of matrices

Consider a real diagonal deterministic matrix $X_n$ of size $n$ with spectral measure converging to a compactly supported probability measure. We perturb this matrix by adding a random finite rank matrix, with delocalized eigenvectors. We show that the joint law of the extreme eigenvalues of the perturbed model satisfies a large deviation principle in the scale $n$, with a good rate function given by a variational formula. We tackle both cases when the extreme eigenvalues of $X_n$ converge to the edges of the support of the limiting measure and when we allow some eigenvalues of $X_n$, that we call outliers, to converge out of the bulk. We can also generalise our results to the case when $X_n$ is random, with law proportional to $e^{- n Trace V(X)} X,$ for $V$ growing fast enough at infinity and any perturbation of finite rank.Comment: 44 page

Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices

Consider a deterministic self-adjoint matrix X_n with spectral measure converging to a compactly supported probability measure, the largest and smallest eigenvalues converging to the edges of the limiting measure. We perturb this matrix by adding a random finite rank matrix with delocalized eigenvectors and study the extreme eigenvalues of the deformed model. We give necessary conditions on the deterministic matrix X_n so that the eigenvalues converging out of the bulk exhibit Gaussian fluctuations, whereas the eigenvalues sticking to the edges are very close to the eigenvalues of the non-perturbed model and fluctuate in the same scale. We generalize these results to the case when X_n is random and get similar behavior when we deform some classical models such as Wigner or Wishart matrices with rather general entries or the so-called matrix models.Comment: 42 pages, Electron. J. Prob., Vol. 16 (2011), Paper no. 60, pages 1621-166

Atypicity Detection in Data Streams: a Self-Adjusting Approach

International audienceOutlyingness is a subjective concept relying on the isolation level of a (set of) record(s). Clustering-based outlier detection is a field that aims to cluster data and to detect outliers depending on their characteristics (i.e. small, tight and/or dense clusters might be considered as outliers). Existing methods require a parameter standing for the "level of outlyingness", such as the maximum size or a percentage of small clusters, in order to build the set of outliers. Unfortunately, manually setting this parameter in a streaming environment should not be possible, given the fast time response usually needed. In this paper we propose WOD, a method that separates outliers from clusters thanks to a natural and effective principle. The main advantages of WOD are its ability to automatically adjust to any clustering result and to be parameterless

Spectral Phase Transitions in Non-Linear Wigner Spiked Models

We study the asymptotic behavior of the spectrum of a random matrix where a non-linearity is applied entry-wise to a Wigner matrix perturbed by a rank-one spike with independent and identically distributed entries. In this setting, we show that when the signal-to-noise ratio scale as $N^{\frac{1}{2} (1-1/k_\star)}$, where $k_\star$ is the first non-zero generalized information coefficient of the function, the non-linear spike model effectively behaves as an equivalent spiked Wigner matrix, where the former spike before the non-linearity is now raised to a power $k_\star$. This allows us to study the phase transition of the leading eigenvalues, generalizing part of the work of Baik, Ben Arous and Pech\'e to these non-linear models.Comment: 27 page

Web Usage Mining : extraction de périodes denses à partir des logs

National audienceLes techniques de Web Usage Mining existantes sont actuellement basées sur un découpage des données arbitraire (e.g. "un log par mois") ou guidé par des résultats supposés (e.g. "quels sont les comportements des clients pour la période des achats de Noël ? "). Ces approches souffrent des deux problèmes suivants. D'une part, elles dépendent de cette organisation arbitraire des données au cours du temps. D'autre part elles ne peuvent pas extraire automatiquement des "pics saisonniers" dans les données stockées. Nous proposons d'exploiter les données pour découvrir de manière automatique des périodes "denses" de comportements. Une période sera considérée comme "dense" si elle contient au moins un motif séquentiel fréquent pour l'ensemble des utilisateurs qui étaient connectés sur le site à cette période

El glosario como herramienta en interpretación consecutiva. Estudio de un caso práctico: la conciliación en Ruanda

En el presente artículo buscamos, basándonos en un encargo real de interpretación consecutiva y bilateral, para las combinaciones lingüísticas francés-español y español-francés, poner de manifiesto la importancia del proceso de documentación previo al ejercicio de la interpretación en el ámbito de las Relaciones Internacionales. El testimonio de Yolande Mukagasana, superviviente del genocidio de Ruanda de 1994, es el eje temático de este trabajo que, a partir de la recopilación y el estudio de textos paralelos, ofrece como resultado un glosario ad hoc (francés-español y español-francés), una herramienta de referencia para el intérprete ante cuestiones relacionadas con el conflicto de Ruanda (1990-1994).G.I. HUM 767 (ayudas a Grupos de Investigación de la Junta de Andalucía) / Editorial Comares (colección interlingua