Seminar Details
2011-09-20 (14.00) : Linear regression on fixed-rank matrices: a Riemannian approach
At Euler building
Organized by Mathematical Engineering
Speaker :
Meyer, Gilles
Abstract :
In this work, we address the problem of learning a linear regression
model parameterized by a fixed-rank matrix. We deal with the case of
fixed-rank symmetric positive semidefinite matrices and the case of
fixed-rank non-symmetric matrices. The focus is on the nonlinear
nature of the search space and on scalability to high-dimensional
problems. The mathematical developments rely on the theory of batch
and online gradient descent algorithms adapted to the Riemannian
geometry that underlies the set of fixed-rank matrices. The resulting
algorithms scale to high-dimensional problems, enjoy local convergence
properties, favorably compete with the state-of-the-art and confer a
geometric basis to recent contributions on learning fixed-rank
matrices. The considered linear regression framework encompasses many
modern machine learning applications including the learning of a
low-rank distance, low-rank matrix completion, learning on data pairs,
ranking or multi-task learning.