37 research outputs found
A Nonconvex Projection Method for Robust PCA
Robust principal component analysis (RPCA) is a well-studied problem with the
goal of decomposing a matrix into the sum of low-rank and sparse components. In
this paper, we propose a nonconvex feasibility reformulation of RPCA problem
and apply an alternating projection method to solve it. To the best of our
knowledge, we are the first to propose a method that solves RPCA problem
without considering any objective function, convex relaxation, or surrogate
convex constraints. We demonstrate through extensive numerical experiments on a
variety of applications, including shadow removal, background estimation, face
detection, and galaxy evolution, that our approach matches and often
significantly outperforms current state-of-the-art in various ways.Comment: In the proceedings of Thirty-Third AAAI Conference on Artificial
Intelligence (AAAI-19
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