406 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
Sequential Convex Programming Methods for Solving Nonlinear Optimization Problems with DC constraints
This paper investigates the relation between sequential convex programming
(SCP) as, e.g., defined in [24] and DC (difference of two convex functions)
programming. We first present an SCP algorithm for solving nonlinear
optimization problems with DC constraints and prove its convergence. Then we
combine the proposed algorithm with a relaxation technique to handle
inconsistent linearizations. Numerical tests are performed to investigate the
behaviour of the class of algorithms.Comment: 18 pages, 1 figur
Conditional Gradient Algorithms for Rank-One Matrix Approximations with a Sparsity Constraint
The sparsity constrained rank-one matrix approximation problem is a difficult
mathematical optimization problem which arises in a wide array of useful
applications in engineering, machine learning and statistics, and the design of
algorithms for this problem has attracted intensive research activities. We
introduce an algorithmic framework, called ConGradU, that unifies a variety of
seemingly different algorithms that have been derived from disparate
approaches, and allows for deriving new schemes. Building on the old and
well-known conditional gradient algorithm, ConGradU is a simplified version
with unit step size and yields a generic algorithm which either is given by an
analytic formula or requires a very low computational complexity. Mathematical
properties are systematically developed and numerical experiments are given.Comment: Minor changes. Final version. To appear in SIAM Revie
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
Computing Large-Scale Matrix and Tensor Decomposition with Structured Factors: A Unified Nonconvex Optimization Perspective
The proposed article aims at offering a comprehensive tutorial for the
computational aspects of structured matrix and tensor factorization. Unlike
existing tutorials that mainly focus on {\it algorithmic procedures} for a
small set of problems, e.g., nonnegativity or sparsity-constrained
factorization, we take a {\it top-down} approach: we start with general
optimization theory (e.g., inexact and accelerated block coordinate descent,
stochastic optimization, and Gauss-Newton methods) that covers a wide range of
factorization problems with diverse constraints and regularization terms of
engineering interest. Then, we go `under the hood' to showcase specific
algorithm design under these introduced principles. We pay a particular
attention to recent algorithmic developments in structured tensor and matrix
factorization (e.g., random sketching and adaptive step size based stochastic
optimization and structure-exploiting second-order algorithms), which are the
state of the art---yet much less touched upon in the literature compared to
{\it block coordinate descent} (BCD)-based methods. We expect that the article
to have an educational values in the field of structured factorization and hope
to stimulate more research in this important and exciting direction.Comment: Final Version; to appear in IEEE Signal Processing Magazine; title
revised to comply with the journal's rul
Proximal Methods for Hierarchical Sparse Coding
Sparse coding consists in representing signals as sparse linear combinations
of atoms selected from a dictionary. We consider an extension of this framework
where the atoms are further assumed to be embedded in a tree. This is achieved
using a recently introduced tree-structured sparse regularization norm, which
has proven useful in several applications. This norm leads to regularized
problems that are difficult to optimize, and we propose in this paper efficient
algorithms for solving them. More precisely, we show that the proximal operator
associated with this norm is computable exactly via a dual approach that can be
viewed as the composition of elementary proximal operators. Our procedure has a
complexity linear, or close to linear, in the number of atoms, and allows the
use of accelerated gradient techniques to solve the tree-structured sparse
approximation problem at the same computational cost as traditional ones using
the L1-norm. Our method is efficient and scales gracefully to millions of
variables, which we illustrate in two types of applications: first, we consider
fixed hierarchical dictionaries of wavelets to denoise natural images. Then, we
apply our optimization tools in the context of dictionary learning, where
learned dictionary elements naturally organize in a prespecified arborescent
structure, leading to a better performance in reconstruction of natural image
patches. When applied to text documents, our method learns hierarchies of
topics, thus providing a competitive alternative to probabilistic topic models
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