Michel Journée

Low-Rank Optimization on the Cone of Positive Semidefinite Matrices

We propose an algorithm for solving nonlinear convex programs defined in terms of a symmetric positive semidefinite matrix variable X. This algorithm rests on the factorization X = Y Y T , where the number of columns of Y fixes the rank of X. It is thus very effective for solving programs that have a low rank solution. The factorization X = Y Y T evokes a reformulation of the original problem as an optimization on a particular quotient manifold. The present paper discusses the geometry of that manifold and derives a second order optimization method. It furthermore provides some conditions on the rank of the factorization to ensure equivalence with the original problem. The efficiency of the proposed algorithm is illustrated on two applications: the maximal cut of a graph and the sparse principal component analysis problem.We propose an algorithm for solving optimization problems defined on a subset of the cone of symmetric positive semidefinite matrices. This algorithm relies on the factorization

Geometric Optimization Methods for the Analysis of Gene Expression Data

X=YY^T

From subspace learning to distance learning: A geometrical optimization approach

, where the number of columns of

Geometric Optimization Methods for Independent Component Analysis Applied on Gene Expression Data

Y

Refining Sparse Principal Components

fixes an upper bound on the rank of the positive semidefinite matrix

Optimization on the orthogonal group for independent component analysis

X

Generalized Power Method for Sparse Principal Component Analysis

. It is thus very effective for solving problems that have a low-rank solution. The factorization

Elucidating the Altered Transcriptional Programs in Breast Cancer using Independent Component Analysis

X=YY^T

Gradient-optimization on the orthogonal group for Independent Component Analysis

leads to a reformulation of the original problem as an optimization on a particular quotient manifold. The present paper discusses the geometry of that manifold and derives a second-order optimization method with guaranteed quadratic convergence. It furthermore provides some conditions on the rank of the factorization to ensure equivalence with the original problem. In contrast to existing methods, the proposed algorithm converges monotonically to the sought solution. Its numerical efficiency is evaluated on two applications: the maximal cut of a graph and the problem of sparse principal component analysis.

Comparative assessment of old and new suboptimal control schemes on three example processes

DNA microarrays provide such a huge amount of data that unsupervised methods are required to reduce the dimension of the data set and to extract meaningful biological information. This work shows that Independent Component Analysis (ICA) is a promising approach for the analysis of genome-wide transcriptomic data. The paper first presents an overview of the most popular algorithms to perform ICA. These algorithms are then applied on a microarray breast-cancer data set. Some issues about the application of ICA and the evaluation of biological relevance of the results are discussed. This study indicates that ICA significantly outperforms Principal Component Analysis (PCA).