Wideband analog signals push contemporary analog-to-digital conversion systems to their performance limits. In many applications, however, sampling at the Nyquist rate is inefficient because the signals of interest contain only a small number of significant frequencies relative to the bandlimit, although the locations of the frequencies may not be known a priori. For this type of sparse signal, other sampling strategies are possible. This paper describes a new type of data acquisition system, called a random demodulator, that is constructed from robust, readily available components. Let K denote the total number of frequencies in the signal, and let W denote its bandlimit in Hz. Simulations suggest that the random demodulator requires just O(K log(W/K)) samples per second to stably reconstruct the signal. This sampling rate is exponentially lower than the Nyquist rate of W Hz. In contrast with Nyquist sampling, one must use nonlinear methods, such as convex programming, to recover the signal from the samples taken by the random demodulator. This paper provides a detailed theoretical analysis of the system’s performance that supports the empirical observations.

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.

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

We consider the problem of estimating a sparse signal from a set of quantized, Gaussian noise corrupted measurements, where each measurement corresponds to an interval of values. We give two methods for (approximately) solving this problem, each based on minimizing a differentiable convex function plus an regularization term. Using a first order method developed by Hale et al, we demonstrate the performance of the methods through numerical simulation. We find that, using these methods, compressed sensing can be carried out even when the quantization is very coarse, e.g., 1 or 2 bits per measurement.

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.

Many combinatorial problems arising in machine learning can be reduced to the problem of minimizing a submodular function. Submodular functions are a natural discrete analog of convex functions, and can be minimized in strongly polynomial time. Unfortunately, state-of-the-art algorithms for general submodular minimization are intractable for larger problems. In this paper, we introduce a novel subclass of submodular minimization problems that we call decomposable. Decomposable submodular functions are those that can be represented as sums of concave functions applied to modular functions. We develop an algorithm, SLG, that can efficiently minimize decomposable submodular functions with tens of thousands of variables. Our algorithm exploits recent results in smoothed convex minimization. We apply SLG to synthetic benchmarks and a joint classification-and-segmentation task, and show that it outperforms the state-of-the-art general purpose submodular minimization algorithms by several orders of magnitude. 1

We provide a comprehensive review of five representative ℓ1minimization methods, i.e., gradient projection, homotopy, iterative shrinkage-thresholding, proximal gradient, and augmented Lagrange multiplier. The repository is intended to fill in a gap in the existing literature to systematically benchmark the performance of these algorithms using a consistent experimental setting. The experiment will be focused on the application of face recognition, where a sparse representation framework has recently been developed to recover human identities from facial images that may be affected by illumination change, occlusion, and facial disguise. The paper also provides useful guidelines to practitioners working in similar fields. 1.

We tackle the problem of detecting occluded regions in a video stream. Under assumptions of Lambertian reflection and static illumination, the task can be posed as a variational optimization problem, and its solution approximated using convex minimization. We describe efficient numerical schemes that reach the global optimum of the relaxed cost functional, for any number of independently moving objects, and any number of occlusion layers. We test the proposed algorithm on benchmark datasets, expanded to enable evaluation of occlusion detection performance, in addition to optical flow.

statistical recovery

Seismic imaging typically begins with the removal of multiple energy in the data, out of fear that it may introduce erroneous structure. However, seismic multiples have effectively seen more of the earth’s structure, and if treated correctly can potential supply more information to a seismic image compared to primaries. Past approaches to accomplish this leave ample room for improvement; they either require extensive modification to standard migration techniques, rely too much on prior information, require extensive pre-processing, or resort to full-waveform inversion. We take some valuable lessons from these efforts and present a new approach balanced in terms of ease of implementation, robustness, efficiency and well-posedness, involving a sparsity-promoting inversion procedure using standard Born migration and a data-driven multiple modeling approach based on the focal transform.