3,300 research outputs found
Super-Linear Convergence of Dual Augmented-Lagrangian Algorithm for Sparsity Regularized Estimation
We analyze the convergence behaviour of a recently proposed algorithm for
regularized estimation called Dual Augmented Lagrangian (DAL). Our analysis is
based on a new interpretation of DAL as a proximal minimization algorithm. We
theoretically show under some conditions that DAL converges super-linearly in a
non-asymptotic and global sense. Due to a special modelling of sparse
estimation problems in the context of machine learning, the assumptions we make
are milder and more natural than those made in conventional analysis of
augmented Lagrangian algorithms. In addition, the new interpretation enables us
to generalize DAL to wide varieties of sparse estimation problems. We
experimentally confirm our analysis in a large scale -regularized
logistic regression problem and extensively compare the efficiency of DAL
algorithm to previously proposed algorithms on both synthetic and benchmark
datasets.Comment: 51 pages, 9 figure
Fast Image Recovery Using Variable Splitting and Constrained Optimization
We propose a new fast algorithm for solving one of the standard formulations
of image restoration and reconstruction which consists of an unconstrained
optimization problem where the objective includes an data-fidelity
term and a non-smooth regularizer. This formulation allows both wavelet-based
(with orthogonal or frame-based representations) regularization or
total-variation regularization. Our approach is based on a variable splitting
to obtain an equivalent constrained optimization formulation, which is then
addressed with an augmented Lagrangian method. The proposed algorithm is an
instance of the so-called "alternating direction method of multipliers", for
which convergence has been proved. Experiments on a set of image restoration
and reconstruction benchmark problems show that the proposed algorithm is
faster than the current state of the art methods.Comment: Submitted; 11 pages, 7 figures, 6 table
Computational Methods for Sparse Solution of Linear Inverse Problems
The goal of the sparse approximation problem is to approximate a target signal using a linear combination of a few elementary signals drawn from a fixed collection. This paper surveys the major practical algorithms for sparse approximation. Specific attention is paid to computational issues, to the circumstances in which individual methods tend to perform well, and to the theoretical guarantees available. Many fundamental questions in electrical engineering, statistics, and applied mathematics can be posed as sparse approximation problems, making these algorithms versatile and relevant to a plethora of applications
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