We consider the class of Iterative Shrinkage-Thresholding Algorithms (ISTA) for solving linear inverse problems arising in signal/image processing. This class of methods is attractive due to its simplicity, however, they are also known to converge quite slowly. In this paper we present a Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) which preserves the computational simplicity of ISTA, but with a global rate of convergence which is proven to be significantly better, both theoretically and practically. Initial promising numerical results for wavelet-based image deblurring demonstrate the capabilities of FISTA.

Accurate signal recovery or image reconstruction from indirect and possibly undersampled data is a topic of considerable interest; for example, the literature in the recent field of compressed sensing is already quite immense. Inspired by recent breakthroughs in the development of novel first-order methods in convex optimization, most notably Nesterov’s smoothing technique, this paper introduces a fast and accurate algorithm for solving common recovery problems in signal processing. In the spirit of Nesterov’s work, one of the key ideas of this algorithm is a subtle averaging of sequences of iterates, which has been shown to improve the convergence properties of standard gradient-descent algorithms. This paper demonstrates that this approach is ideally suited for solving large-scale compressed sensing reconstruction problems as 1) it is computationally efficient, 2) it is accurate and returns solutions with several correct digits, 3) it is flexible and amenable to many kinds of reconstruction problems, and 4) it is robust in the sense that its excellent performance across a wide range of problems does not depend on the fine tuning of several parameters. Comprehensive numerical experiments on realistic signals exhibiting a large dynamic range show that this algorithm compares favorably with recently proposed state-of-the-art methods. We also apply the algorithm to solve other problems for which there are fewer alternatives, such as total-variation minimization, and

We consider regularized stochastic learning and online optimization problems, where the objective function is the sum of two convex terms: one is the loss function of the learning task, and the other is a simple regularization term such as ℓ1-norm for promoting sparsity. We develop extensions of Nesterov’s dual averaging method, that can exploit the regularization structure in an online setting. At each iteration of these methods, the learning variables are adjusted by solving a simple minimization problem that involves the running average of all past subgradients of the loss function and the whole regularization term, not just its subgradient. In the case of ℓ1-regularization, our method is particularly effective in obtaining sparse solutions. We show that these methods achieve the optimal convergence rates or regret bounds that are standard in the literature on stochastic and online convex optimization. For stochastic learning problems in which the loss functions have Lipschitz continuous gradients, we also present an accelerated version of the dual averaging method.

The goal of sparse approximation problems is to represent a target signal approximately as 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 wealth of applications.

We describe, analyze, and experiment with a framework for empirical loss minimization with regularization. Our algorithmic framework alternates between two phases. On each iteration we first perform an unconstrained gradient descent step. We then cast and solve an instantaneous optimization problem that trades off minimization of a regularization term while keeping close proximity to the result of the first phase. This view yields a simple yet effective algorithm that can be used for batch penalized risk minimization and online learning. Furthermore, the two phase approach enables sparse solutions when used in conjunction with regularization functions that promote sparsity, such as ℓ1. We derive concrete and very simple algorithms for minimization of loss functions with ℓ1, ℓ2, ℓ 2 2, and ℓ ∞ regularization. We also show how to construct efficient algorithms for mixed-norm ℓ1/ℓq regularization. We further extend the algorithms and give efficient implementations for very high-dimensional data with sparsity. We demonstrate the potential of the proposed framework in a series of experiments with synthetic and natural datasets.

Abstract. In this paper, we propose and study the use of alternating direction algorithms for several ℓ1-norm minimization problems arising from sparse solution recovery in compressive sensing, including the basis pursuit problem, the basis-pursuit denoising problems of both unconstrained and constrained forms, as well as others. We present and investigate two classes of algorithms derived from either the primal or the dual forms of the ℓ1-problems. The construction of the algorithms consists of two main steps: (1) to reformulate an ℓ1-problem into one having partially separable objective functions by adding new variables and constraints; and (2) to apply an exact or inexact alternating direction method to the resulting problem. The derived alternating direction algorithms can be regarded as first-order primal-dual algorithms because both primal and dual variables are updated at each and every iteration. Convergence properties of these algorithms are established or restated when they already exist. Extensive numerical results in comparison with several state-of-the-art algorithms are given to demonstrate that the proposed algorithms are efficient, stable and robust. Moreover, we present numerical results to emphasize two practically important but perhaps overlooked points. One point is that algorithm speed should always be evaluated relative to appropriate solution accuracy; another is that whenever erroneous measurements possibly exist, the ℓ1-norm fidelity should be the fidelity of choice in compressive sensing. Key words. Sparse solution recovery, compressive sensing, ℓ1-minimization, primal, dual, alternating direction method

The variational approach to signal restoration calls for the minimization of a cost function that is the sum of a data fidelity term and a regularization term, the latter term constituting a ‘prior’. A synthesis prior represents the sought signal as a weighted sum of ‘atoms’. On the other hand, an analysis prior models the coefficients obtained by applying the forward transform to the signal. For orthonormal transforms, the synthesis prior and analysis prior are equivalent; however, for overcomplete transforms the two formulations are different. We compare analysis and synthesis ℓ1-norm regularization with overcomplete transforms for denoising and deconvolution.

The heavy-tailed distribution of gradients in natural scenes have proven effective priors for a range of problems such as denoising, deblurring and super-resolution. These distributions are well modeled by a hyper-Laplacian ( p(x) ∝ e −k|x|α) , typically with 0.5 ≤ α ≤ 0.8. However, the use of sparse distributions makes the problem non-convex and impractically slow to solve for multi-megapixel images. In this paper we describe a deconvolution approach that is several orders of magnitude faster than existing techniques that use hyper-Laplacian priors. We adopt an alternating minimization scheme where one of the two phases is a non-convex problem that is separable over pixels. This per-pixel sub-problem may be solved with a lookup table (LUT). Alternatively, for two specific values of α, 1/2 and 2/3 an analytic solution can be found, by finding the roots of a cubic and quartic polynomial, respectively. Our approach (using either LUTs or analytic formulae) is able to deconvolve a 1 megapixel image in less than ∼3 seconds, achieving comparable quality to existing methods such as iteratively reweighted least squares (IRLS) that take ∼20 minutes. Furthermore, our method is quite general and can easily be extended to related image processing problems, beyond the deconvolution application demonstrated. 1

We describe, analyze, and experiment with a new framework for empirical loss minimization with regularization. Our algorithmic framework alternates between two phases. On each iteration we first perform an unconstrained gradient descent step. We then cast and solve an instantaneous optimization problem that trades off minimization of a regularization term while keeping close proximity to the result of the first phase. This yields a simple yet effective algorithm for both batch penalized risk minimization and online learning. Furthermore, the two phase approach enables sparse solutions when used in conjunction with regularization functions that promote sparsity, such as ℓ1. We derive concrete and very simple algorithms for minimization of loss functions with ℓ1, ℓ2, ℓ 2 2, and ℓ ∞ regularization. We also show how to construct efficient algorithms for mixed-norm ℓ1/ℓq regularization. We further extend the algorithms and give efficient implementations for very high-dimensional data with sparsity. We demonstrate the potential of the proposed framework in experiments with synthetic and natural datasets. 1

We present a new method for regularized convex optimization and analyze it under both online and stochastic optimization settings. In addition to unifying previously known firstorder algorithms, such as the projected gradient method, mirror descent, and forwardbackward splitting, our method yields new analysis and algorithms. We also derive specific instantiations of our method for commonly used regularization functions, such as ℓ1, mixed norm, and trace-norm. 1