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Proceedings of MTNS 2004, Leuven, Belgium Continuous-time flows on quotient spaces

Summary: Proceedings of MTNS 2004, Leuven, Belgium
Continuous-time flows on quotient spaces
for principal component analysis
P.-A. Absil
A novel matrix flow for principal or minor component analysis is
constructed. The development is based on decompositions of the
set of full-rank n × p matrices into orbits of Lie group actions.
Key words. Principal component analysis, principal subspace analysis, minor component
analysis, continuous-time matrix flows, quotient spaces, Lie group actions, semidefinite Lya-
punov functions, leftmost and rightmost eigenspaces.
1 Introduction
Given a symmetric matrix A, a flow on Rn×p
is said to have principal subspace analysis (PSA)
properties if the column space of the solution converges to the p-dimensional eigenspace of
A associated with the largest eigenvalues. If moreover the columns of the solution converge
to the p principal eigenvectors of A, i.e. those corresponding to the largest eigenvalues, then
the flow is said to achieve principal component analysis (PCA).
Several continuous-time dynamical systems on matrix spaces (also called matrix flows)
have been proposed in the literature that achieve PSA or even PCA. Interest for study-


Source: Absil, Pierre-Antoine - Département d'ingénierie Mathématique, Université Catholique de Louvain


Collections: Mathematics