198 research outputs found
Schatten- Quasi-Norm Regularized Matrix Optimization via Iterative Reweighted Singular Value Minimization
In this paper we study general Schatten- quasi-norm (SPQN) regularized
matrix minimization problems. In particular, we first introduce a class of
first-order stationary points for them, and show that the first-order
stationary points introduced in [11] for an SPQN regularized
minimization problem are equivalent to those of an SPQN regularized
minimization reformulation. We also show that any local minimizer of the SPQN
regularized matrix minimization problems must be a first-order stationary
point. Moreover, we derive lower bounds for nonzero singular values of the
first-order stationary points and hence also of the local minimizers of the
SPQN regularized matrix minimization problems. The iterative reweighted
singular value minimization (IRSVM) methods are then proposed to solve these
problems, whose subproblems are shown to have a closed-form solution. In
contrast to the analogous methods for the SPQN regularized
minimization problems, the convergence analysis of these methods is
significantly more challenging. We develop a novel approach to establishing the
convergence of these methods, which makes use of the expression of a specific
solution of their subproblems and avoids the intricate issue of finding the
explicit expression for the Clarke subdifferential of the objective of their
subproblems. In particular, we show that any accumulation point of the sequence
generated by the IRSVM methods is a first-order stationary point of the
problems. Our computational results demonstrate that the IRSVM methods
generally outperform some recently developed state-of-the-art methods in terms
of solution quality and/or speed.Comment: This paper has been withdrawn by the author due to major revision and
correction
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
On the monotone and primal-dual active set schemes for -type problems,
Nonsmooth nonconvex optimization problems involving the quasi-norm,
, of a linear map are considered. A monotonically convergent
scheme for a regularized version of the original problem is developed and
necessary optimality conditions for the original problem in the form of a
complementary system amenable for computation are given. Then an algorithm for
solving the above mentioned necessary optimality conditions is proposed. It is
based on a combination of the monotone scheme and a primal-dual active set
strategy. The performance of the two algorithms is studied by means of a series
of numerical tests in different cases, including optimal control problems,
fracture mechanics and microscopy image reconstruction
Jump-sparse and sparse recovery using Potts functionals
We recover jump-sparse and sparse signals from blurred incomplete data
corrupted by (possibly non-Gaussian) noise using inverse Potts energy
functionals. We obtain analytical results (existence of minimizers, complexity)
on inverse Potts functionals and provide relations to sparsity problems. We
then propose a new optimization method for these functionals which is based on
dynamic programming and the alternating direction method of multipliers (ADMM).
A series of experiments shows that the proposed method yields very satisfactory
jump-sparse and sparse reconstructions, respectively. We highlight the
capability of the method by comparing it with classical and recent approaches
such as TV minimization (jump-sparse signals), orthogonal matching pursuit,
iterative hard thresholding, and iteratively reweighted minimization
(sparse signals)
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