487 research outputs found
Fast Low-Rank Matrix Learning with Nonconvex Regularization
Low-rank modeling has a lot of important applications in machine learning,
computer vision and social network analysis. While the matrix rank is often
approximated by the convex nuclear norm, the use of nonconvex low-rank
regularizers has demonstrated better recovery performance. However, the
resultant optimization problem is much more challenging. A very recent
state-of-the-art is based on the proximal gradient algorithm. However, it
requires an expensive full SVD in each proximal step. In this paper, we show
that for many commonly-used nonconvex low-rank regularizers, a cutoff can be
derived to automatically threshold the singular values obtained from the
proximal operator. This allows the use of power method to approximate the SVD
efficiently. Besides, the proximal operator can be reduced to that of a much
smaller matrix projected onto this leading subspace. Convergence, with a rate
of O(1/T) where T is the number of iterations, can be guaranteed. Extensive
experiments are performed on matrix completion and robust principal component
analysis. The proposed method achieves significant speedup over the
state-of-the-art. Moreover, the matrix solution obtained is more accurate and
has a lower rank than that of the traditional nuclear norm regularizer.Comment: Long version of conference paper appeared ICDM 201
Nonconvex Nonsmooth Low-Rank Minimization via Iteratively Reweighted Nuclear Norm
The nuclear norm is widely used as a convex surrogate of the rank function in
compressive sensing for low rank matrix recovery with its applications in image
recovery and signal processing. However, solving the nuclear norm based relaxed
convex problem usually leads to a suboptimal solution of the original rank
minimization problem. In this paper, we propose to perform a family of
nonconvex surrogates of -norm on the singular values of a matrix to
approximate the rank function. This leads to a nonconvex nonsmooth minimization
problem. Then we propose to solve the problem by Iteratively Reweighted Nuclear
Norm (IRNN) algorithm. IRNN iteratively solves a Weighted Singular Value
Thresholding (WSVT) problem, which has a closed form solution due to the
special properties of the nonconvex surrogate functions. We also extend IRNN to
solve the nonconvex problem with two or more blocks of variables. In theory, we
prove that IRNN decreases the objective function value monotonically, and any
limit point is a stationary point. Extensive experiments on both synthesized
data and real images demonstrate that IRNN enhances the low-rank matrix
recovery compared with state-of-the-art convex algorithms
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