Optimización de la factorización de matrices no negativas en Bioinformática

En los últimos años se ha incrementado el interés de la comunidad científica en la Factorización de matrices no negativas (Non-negative Matrix Factorization, NMF). Este método permite transformar un conjunto de datos de grandes dimensiones en una pequeña colección de elementos que poseen semántica propia en el contexto del análisis. En el caso de Bioinformática, NMF suele emplearse como base de algunos métodos de agrupamiento de datos, que emplean un modelo estadístico para determinar el número de clases más favorable. Este modelo requiere de una gran cantidad de ejecuciones de NMF con distintos parámetros de entrada, lo que representa una enorme carga de trabajo a nivel computacional. La mayoría de las implementaciones de NMF han ido quedando obsoletas ante el constante crecimiento de los datos que la comunidad científica busca analizar, bien sea porque los tiempos de cómputo llegan a alargarse hasta convertirse en inviables, o porque el tamaño de esos datos desborda los recursos del sistema. Por ello, esta tesis doctoral se centra en la optimización y paralelización de la factorización NMF, pero no solo a nivel teórico, sino con el objetivo de proporcionarle a la comunidad científica una nueva herramienta para el análisis de datos de origen biológico. NMF expone un alto grado de paralelismo a nivel de datos, de granularidad variable; mientras que los métodos de agrupamiento mencionados anteriormente presentan un paralelismo a nivel de cómputo, ya que las diversas instancias de NMF que se ejecutan son independientes. Por tanto, desde un punto de vista global, se plantea un modelo de optimización por capas donde se emplean diferentes tecnologías de alto rendimiento...

Deformations of canonical triple covers

In this paper, we show that if X is a smooth variety of general type of dimension m≥3 for which the canonical map induces a triple cover onto Y, where Y is a projective bundle over P1 or onto a projective space or onto a quadric hypersurface, embedded by a complete linear series (except Q3 embedded in P4), then the general deformation of the canonical morphism of X is again canonical and induces a triple cover. The extremal case when Y is embedded as a variety of minimal degree is of interest, due to its appearance in numerous situations. For instance, by looking at threefolds Y of minimal degree we find components of the moduli of threefolds X of general type with KX3=3pg−9,KX3≠6, whose general members correspond to canonical triple covers. Our results are especially interesting as well because they have no lower dimensional analogues.

Deformations of canonical double covers

In this paper we show that if X is a smooth variety of general type of dimension m≥2 for which its canonical map induces a double cover onto Y, where Y is the projective space, a smooth quadric hypersurface or a smooth projective bundle over P1, embedded by a complete linear series, then the general deformation of the canonical morphism of X is again canonical and induces a double cover. The second part of the article proves the non-existence of canonical double structures on the rational varieties above mentioned. Our results have consequences for the moduli of varieties of general type of arbitrary dimension, since they show that infinitely many moduli spaces of higher dimensional varieties of general type have an entire “hyperelliptic” component. This is in sharp contrast with the case of curves or surfaces of lower Kodaira dimension.

Euclidean Distance Between Haar Orthogonal and Gaussian Matrices

In this work, we study a version of the general question of how well a Haar-distributed orthogonal matrix can be approximated by a random Gaussian matrix. Here, we consider a Gaussian random matrix (Formula presented.) of order n and apply to it the Gram–Schmidt orthonormalization procedure by columns to obtain a Haar-distributed orthogonal matrix (Formula presented.). If (Formula presented.) denotes the vector formed by the first m-coordinates of the ith row of (Formula presented.) and (Formula presented.), our main result shows that the Euclidean norm of (Formula presented.) converges exponentially fast to (Formula presented.), up to negligible terms. To show the extent of this result, we use it to study the convergence of the supremum norm (Formula presented.) and we find a coupling that improves by a factor (Formula presented.) the recently proved best known upper bound on (Formula presented.). Our main result also has applications in Quantum Information Theory.