6 research outputs found
Regression on fixed-rank positive semidefinite matrices: a Riemannian approach
The paper addresses the problem of learning a regression model parameterized
by a fixed-rank positive semidefinite matrix. 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 gradient descent algorithms
adapted to the Riemannian geometry that underlies the set of fixed-rank
positive semidefinite matrices. In contrast with previous contributions in the
literature, no restrictions are imposed on the range space of the learned
matrix. The resulting algorithms maintain a linear complexity in the problem
size and enjoy important invariance properties. We apply the proposed
algorithms to the problem of learning a distance function parameterized by a
positive semidefinite matrix. Good performance is observed on classical
benchmarks
PETRELS: Parallel Subspace Estimation and Tracking by Recursive Least Squares from Partial Observations
Many real world data sets exhibit an embedding of low-dimensional structure
in a high-dimensional manifold. Examples include images, videos and internet
traffic data. It is of great significance to reduce the storage requirements
and computational complexity when the data dimension is high. Therefore we
consider the problem of reconstructing a data stream from a small subset of its
entries, where the data is assumed to lie in a low-dimensional linear subspace,
possibly corrupted by noise. We further consider tracking the change of the
underlying subspace, which can be applied to applications such as video
denoising, network monitoring and anomaly detection. Our problem can be viewed
as a sequential low-rank matrix completion problem in which the subspace is
learned in an on-line fashion. The proposed algorithm, dubbed Parallel
Estimation and Tracking by REcursive Least Squares (PETRELS), first identifies
the underlying low-dimensional subspace via a recursive procedure for each row
of the subspace matrix in parallel with discounting for previous observations,
and then reconstructs the missing entries via least-squares estimation if
required. Numerical examples are provided for direction-of-arrival estimation
and matrix completion, comparing PETRELS with state of the art batch
algorithms.Comment: submitted to IEEE Trans. Signal Processing. Part of the result was
reported at ICASSP 2012 and won the best student paper awar
Robust low-dimensional structure learning for big data and its applications
Ph.DDOCTOR OF PHILOSOPH