Seminar Details
2009-04-21 (14.00) : Optimization methods for sparse principal component analysis
At Euler building
Organized by Mathematical Engineering
Speaker :
Journée, Michel
Abstract :
Given an m-by-n matrix that stores m samples of n random variables, sparse principal component analysis (sparse PCA)
is the problem of finding a linear combination of these variables that is sparse ( i.e., that involves as few as possible of the original variables)
but that still captures as much as possible of the variance in the data. Sparsity is typically enforced for the sake of interpretability: components
that are linear combinations of a small number of the original variables are easier to interpret. Sparse PCA has applications in virtually all areas
of science, where large data sets need to be analyzed.
This talk is organized in two parts. First, we formulate sparse PCA as the maximization of a convex function on a compact set and propose a simple
gradient-method for this class of problems. We then consider convex relaxations of sparse PCA, which are optimization problems for positive semidefinite
matrices of potentially large dimension. For the sake of numerical efficiency, we discuss an approach that rests on low-rank positive semidefinite matrices,
which entail a quotient manifold structure.