Géométrie des matrices positives semi-définies de rang fixé : un peu de théorie et beaucoup d’applications

in Proceedings of Sixième Conférence Internationale Francophone d'Automatique (CIFA 2010) (2010, June)

Cet article est une introduction au calcul et `a l’optimisation sur les matrices sym´etriques positives semid ´efinies de rang (faible) fix´e. L’approche propos´ee est bas´ee sur deux g´eom´etries ... [more ▼]

Cet article est une introduction au calcul et `a l’optimisation sur les matrices sym´etriques positives semid ´efinies de rang (faible) fix´e. L’approche propos´ee est bas´ee sur deux g´eom´etries riemanniennes quotient, qui permettent de calculer efficacement tout en pr´eservant le rang et le caract` ere positif des matrices consid´er´ees. Le champ d’applications est vaste, et l’article survole quelques d´eveloppements r´ecents qui illustrent l’int´erˆet de l’approche consid´er´ee dans les probl`emes de tr`es grande taille rencontr´es en contrˆole, statistiques, et apprentissage. [less ▲]

Gene expression data analysis using spatiotemporal blind source separation

in Verleysen, Michel (Ed.) ESANN'2009 proceedings, European Symposium on Artificial Neural Networks - Advances in Computational Intelligence and Learning. (2009, April)

We propose a “time-biased” and a “space-biased” method for spatiotemporal independent component analysis (ICA). The methods rely on computing an orthogonal approximate joint diagonalizer of a collection ... [more ▼]

We propose a “time-biased” and a “space-biased” method for spatiotemporal independent component analysis (ICA). The methods rely on computing an orthogonal approximate joint diagonalizer of a collection of covariance-like matrices. In the time-biased version, the time signatures of the ICA modes are imposed to be white, whereas the space-biased version imposes the same condition on the space signatures. We apply the two methods to the analysis of gene expression data, where the genes play the role of the space and the cell samples stand for the time. This study is a step towards addressing a question first raised by Liebermeister, on whether ICA methods for gene expression analysis should impose independence across genes or across cell samples. Our preliminary experiment indicates that both approaches have value, and that exploring the continuum between these two extremes can provide useful information about the interactions between genes and their impact on the phenotype. [less ▲]