An Alternating Direction Algorithm for Matrix Completion with Nonnegative Factors

This paper introduces a novel algorithm for the nonnegative matrix factorization and completion problem, which aims to nd nonnegative matrices X and Y from a subset of entries of a nonnegative matrix M so that XY approximates M. This problem is closely related to the two existing problems: nonnegative matrix factorization and low-rank matrix completion, … Read more

A Feasible method for Optimization with Orthogonality Constraints

Minimization with orthogonality constraints (e.g., $X^\top X = I$) and/or spherical constraints (e.g., $\|x\|_2 = 1$) has wide applications in polynomial optimization, combinatorial optimization, eigenvalue problems, sparse PCA, p-harmonic flows, 1-bit compressive sensing, matrix rank minimization, etc. These problems are difficult because the constraints are not only non-convex but numerically expensive to preserve during iterations. … Read more

Solving A Low-Rank Factorization Model for Matrix Completion by A Nonlinear Successive Over-Relaxation Algorithm

The matrix completion problem is to recover a low-rank matrix from a subset of its entries. The main solution strategy for this problem has been based on nuclear-norm minimization which requires computing singular value decompositions — a task that is increasingly costly as matrix sizes and ranks increase. To improve the capacity of solving large-scale … Read more

Sparse Signal Reconstruction via Iterative Support Detection

We present a novel sparse signal reconstruction method “ISD”, aiming to achieve fast reconstruction and a reduced requirement on the number of measurements compared to the classical l_1 minimization approach. ISD addresses failed reconstructions of l_1 minimization due to insufficient measurements. It estimates a support set I from a current reconstruction and obtains a new … Read more

Alternating Direction Augmented Lagrangian Methods for semidefinite programming

We present an alternating direction method based on an augmented Lagrangian framework for solving semidefinite programming (SDP) problems in standard form. At each iteration, the algorithm, also known as a two-splitting scheme, minimizes the dual augmented Lagrangian function sequentially with respect to the Lagrange multipliers corresponding to the linear constraints, then the dual slack variables … Read more

Analysis and Generalizations of the Linearized Bregman Method

This paper reviews the Bregman methods, analyzes the linearized Bregman method, and proposes fast generalization of the latter for solving the basis pursuit and related problems. The analysis shows that the linearized Bregman method has the exact penalty property, namely, it converges to an exact solution of the basis pursuit problem if and only if … Read more

A fast TVL1-L2 minimization algorithm for signal reconstruction from partial Fourier data

Recent compressive sensing results show that it is possible to accurately reconstruct certain compressible signals from relatively few linear measurements via solving nonsmooth convex ptimization problems. In this paper, we propose a simple and fast algorithm for signal reconstruction from partial Fourier data. The algorithm minimizes the sum of three terms corresponding to total variation, … Read more

A Fast Algorithm for Sparse Reconstruction based on Shrinkage, Subspace Optimization and Continuation

We propose a fast algorithm for solving the l1-regularized least squares problem for recovering sparse solutions to an undetermined system of linear equations Ax = b. The algorithm is divided into two stages that are performed repeatedly. In the first stage a first-order iterative method called “shrinkage” yields an estimate of the subset of components … Read more

An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise

We extend a recently proposed alternating minimization algorithm to the case of recovering blurry multichannel (color) images corrupted by impulsive rather than Gaussian noise. The algorithm minimizes the sum of a multichannel extension of total variation (TV), either isotropic or anisotropic, and a data fidelity term measured in the L1-norm. We derive the algorithm by … Read more

A Fast Algorithm For Image Deblurring with Total Variation Regularization

We propose and test a simple algorithmic framework for recovering images from blurry and noisy observations based on total variation (TV) regularization when a blurring point-spread function is given. Using a splitting technique, we construct an iterative procedure of alternately solving a pair of easy subproblems associated with an increasing sequence of penalty parameter values. … Read more