390 research outputs found
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
Generalized Nonconvex Nonsmooth Low-Rank Minimization
As surrogate functions of -norm, many nonconvex penalty functions have
been proposed to enhance the sparse vector recovery. It is easy to extend these
nonconvex penalty functions on singular values of a matrix to enhance low-rank
matrix recovery. However, different from convex optimization, solving the
nonconvex low-rank minimization problem is much more challenging than the
nonconvex sparse minimization problem. We observe that all the existing
nonconvex penalty functions are concave and monotonically increasing on
. Thus their gradients are decreasing functions. Based on this
property, we propose an Iteratively Reweighted Nuclear Norm (IRNN) algorithm to
solve the nonconvex nonsmooth low-rank minimization problem. IRNN iteratively
solves a Weighted Singular Value Thresholding (WSVT) problem. By setting the
weight vector as the gradient of the concave penalty function, the WSVT problem
has a closed form solution. In theory, we prove that IRNN decreases the
objective function value monotonically, and any limit point is a stationary
point. Extensive experiments on both synthetic data and real images demonstrate
that IRNN enhances the low-rank matrix recovery compared with state-of-the-art
convex algorithms.Comment: IEEE International Conference on Computer Vision and Pattern
Recognition, 201
A Splitting Augmented Lagrangian Method for Low Multilinear-Rank Tensor Recovery
This paper studies a recovery task of finding a low multilinear-rank tensor
that fulfills some linear constraints in the general settings, which has many
applications in computer vision and graphics. This problem is named as the low
multilinear-rank tensor recovery problem. The variable splitting technique and
convex relaxation technique are used to transform this problem into a tractable
constrained optimization problem. Considering the favorable structure of the
problem, we develop a splitting augmented Lagrangian method to solve the
resulting problem. The proposed algorithm is easily implemented and its
convergence can be proved under some conditions. Some preliminary numerical
results on randomly generated and real completion problems show that the
proposed algorithm is very effective and robust for tackling the low
multilinear-rank tensor completion problem
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