384 research outputs found
R3MC: A Riemannian three-factor algorithm for low-rank matrix completion
We exploit the versatile framework of Riemannian optimization on quotient
manifolds to develop R3MC, a nonlinear conjugate-gradient method for low-rank
matrix completion. The underlying search space of fixed-rank matrices is
endowed with a novel Riemannian metric that is tailored to the least-squares
cost. Numerical comparisons suggest that R3MC robustly outperforms
state-of-the-art algorithms across different problem instances, especially
those that combine scarcely sampled and ill-conditioned data.Comment: Accepted for publication in the proceedings of the 53rd IEEE
Conference on Decision and Control, 201
Guarantees of Riemannian Optimization for Low Rank Matrix Completion
We study the Riemannian optimization methods on the embedded manifold of low
rank matrices for the problem of matrix completion, which is about recovering a
low rank matrix from its partial entries. Assume entries of an
rank matrix are sampled independently and uniformly with replacement. We
first prove that with high probability the Riemannian gradient descent and
conjugate gradient descent algorithms initialized by one step hard thresholding
are guaranteed to converge linearly to the measured matrix provided
\begin{align*} m\geq C_\kappa n^{1.5}r\log^{1.5}(n), \end{align*} where
is a numerical constant depending on the condition number of the
underlying matrix. The sampling complexity has been further improved to
\begin{align*} m\geq C_\kappa nr^2\log^{2}(n) \end{align*} via the resampled
Riemannian gradient descent initialization. The analysis of the new
initialization procedure relies on an asymmetric restricted isometry property
of the sampling operator and the curvature of the low rank matrix manifold.
Numerical simulation shows that the algorithms are able to recover a low rank
matrix from nearly the minimum number of measurements
Riemannian preconditioning
This paper exploits a basic connection between sequential quadratic programming and Riemannian gradient optimization to address the general question of selecting a metric in Riemannian optimization, in particular when the Riemannian structure is sought on a quotient manifold. The proposed method is shown to be particularly insightful and efficient in quadratic optimization with orthogonality and/or rank constraints, which covers most current applications of Riemannian optimization in matrix manifolds.Belgium Science Policy Office, FNRS (Belgium)This is the author accepted manuscript. The final version is available from The Society for Industrial and Applied Mathematics via http://dx.doi.org/10.1137/14097086
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