A unified view of the area of sparse signal processing is presented in tutorial form by bringing together various fields in which the property of sparsity has been successfully exploited. For each of these fields, various algorithms and techniques, which have been developed to leverage sparsity, are described succinctly. The common potential benefits of significant reduction in sampling rate and processing manipulations through sparse signal processing are revealed. The key application domains of sparse signal processing are sampling, coding, spectral estimation, array processing, component analysis, and multipath channel estimation. In terms of the sampling process and reconstruction algorithms, linkages are made with random sampling, compressed sensing, and rate of innovation. The redundancy introduced by channel coding in finite and real Galois fields is then related to over-sampling with similar reconstruction algorithms. The error locator polynomial (ELP) and iterative methods are shown to work quite effectively for both sampling and coding applications. The methods of Prony, Pisarenko, and MUltiple SIgnal Classification (MUSIC) are next shown to be targeted at analyzing signals with sparse frequency domain representations. Specifically, the relations of the approach of Prony to an annihilating filter in rate of innovation and ELP in coding are emphasized; the Pisarenko and MUSIC methods are further improvements of the Prony method under noisy environments. The iterative methods developed for sampling and coding applications are shown to be powerful tools in spectral estimation. Such narrowband spectral estimation is then related to multi-source location and direction of arrival estimation in array processing. Sparsity in unobservable source signals is also shown to facilitate source separation in sparse component analysis; the algorithms developed in this area such as linear programming and matching pursuit are also widely used in compressed sensing. Finally, the multipath channel estimation problem is shown to have a sparse formulation; algorithms similar to sampling and coding are used to estimate typical multicarrier communication channels.
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Compressed sensing (CS) is an emerging field that has attracted considerable research interest over the past few years. Previous review articles in CS limit their scope to standard discrete-to-discrete measurement architectures using matrices of randomized nature and signal models based on standard sparsity. In recent years, CS has worked its way into several new application areas. This, in turn, necessitates a fresh look on many of the basics of CS. The random matrix measurement operator must be replaced by more structured sensing architectures that correspond to the characteristics of feasible acquisition hardware. The standard sparsity prior has to be extended to include a much richer class of signals and to encode broader data models, including continuous-time signals. In our overview, the theme is exploiting signal and measurement structure in compressive sensing. The prime focus is bridging theory and practice; that is, to pinpoint the potential of structured CS strategies to emerge from the math to the hardware. Our summary highlights new directions as well as relations to more traditional CS, with the hope of serving both as a review to practitioners wanting to join this emerging field, and as a reference for researchers that attempts to put some of the existing ideas in perspective of practical applications.
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Iterative constant modulus algorithms such as Godard and CMA have been used to blindly separate a superposition of co-channel constant modulus (CM) signals impinging on an antenna array. These algorithms have certain deficiencies in the context of convergence to local minima and the retrieval of all individual CM signals that are present in the channel. In this paper, we show that the underlying constant modulus fac-torization problem is, in fact, a generalized eigenvalue problem, and may be solved via a simultaneous diagonalization of a set of matrices. With this new, analytical approach, it is possible to detect the number of CM signals present in the channel, and to retrieve all of them exactly, rejecting other, non-CM signals. Only a modest amount of samples are required. The algorithm is robust in the presence of noise, and is tested on measured data, collected from an experimental setup .
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2003-07-10

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Finding sparse approximate solutions to large under-determined linear systems of equations is a common problem in signal/image processing and statistics. Basis pursuit, the least absolute shrinkage and selection operator (LASSO), wavelet-based deconvolution and reconstruction, and compressed sensing (CS) are a few well-known areas in which problems of this type appear. One standard approach is to minimize an objective function that includes a quadratic (2) error term added to a sparsity-inducing (usually 1) regularizater. We present an algorithmic framework for the more general problem of minimizing the sum of a smooth convex function and a nonsmooth, possibly nonconvex regularizer. We propose iterative methods in which each step is obtained by solving an optimization subproblem involving a quadratic term with diagonal Hessian (i.e., separable in the unknowns) plus the original sparsity-inducing regularizer; our approach is suitable for cases in which this subproblem can be solved much more rapidly than the original problem. Under mild conditions (namely con-vexity of the regularizer), we prove convergence of the proposed iterative algorithm to a minimum of the objective function. In addition to solving the standard 2 1 case, our framework yields efficient solution techniques for other regularizers, such as an norm and group-separable regularizers. It also generalizes immediately to the case in which the data is complex rather than real. Experiments with CS problems show that our approach is competitive with the fastest known methods for the standard 2 1 problem, as well as being efficient on problems with other separable regularization terms.
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This paper is a survey of the theory and methods of photogrammetric bundle adjustment, aimed at potential implementors in the computer vision community. Bundle adjustment is the problem of refining a visual reconstruction to produce jointly optimal structure and viewing parameter estimates. Topics covered include: the choice of cost function and robustness; numerical optimization including sparse Newton methods, linearly convergent approximations, updating and recursive methods; gauge (datum) invariance; and quality control. The theory is developed for general robust cost functions rather than restricting attention to traditional nonlinear least squares.
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Sparse Bayesian learning (SBL) and specifically relevance vector machines have received much attention in the machine learning literature as a means of achieving parsimonious representations in the context of regression and classification. The methodology relies on a parameterized prior that encourages models with few nonzero weights. In this paper, we adapt SBL to the signal processing problem of basis selection from overcomplete dictionaries, proving several results about the SBL cost function that elucidate its general behavior and provide solid theoretical justification for this application. Specifically, we have shown that SBL retains a desirable property of the 0-norm diversity measure (i.e., the global minimum is achieved at the maximally sparse solution) while often possessing a more limited constellation of local minima. We have also demonstrated that the local minima that do exist are achieved at sparse solutions. Later, we provide a novel interpretation of SBL that gives us valuable insight into why it is successful in producing sparse representations. Finally, we include simulation studies comparing sparse Bayesian learning with Basis Pursuit and the more recent FOCal Underdetermined System Solver (FOCUSS) class of basis selection algorithms. These results indicate that our theoretical insights translate directly into improved performance. Index Terms-Basis selection, diversity measures, linear inverse problems, sparse Bayesian learning, sparse representations.
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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 ban-dlimit, 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.
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2018-09-25

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Imaging spectrometers measure electromagnetic energy scattered in their instantaneous field view in hundreds or thousands of spectral channels with higher spectral resolution than multispectral cameras. Imaging spectrometers are therefore often referred to as hyperspectral cameras (HSCs). Higher spectral resolution enables material identification via spectroscopic analysis, which facilitates countless applications that require identifying materials in scenarios unsuitable for classical spectroscopic analysis. Due to low spatial resolution of HSCs, microscopic material mixing, and multiple scattering, spectra measured by HSCs are mixtures of spectra of materials in a scene. Thus, accurate estimation requires unmixing. Pixels are assumed to be mixtures of a few materials, called endmembers. Unmixing involves estimating all or some of: the number of endmembers, their spectral signatures, and their abundances at each pixel. Unmixing is a challenging, ill-posed inverse problem because of model inaccuracies, observation noise, environmental conditions, endmember variability, and data set size. Researchers have devised and investigated many models searching for robust, stable, tractable, and accurate unmixing algorithms. This paper presents an overview of unmixing methods from the time of Keshava and Mustard's unmixing tutorial [1] to the present. Mixing models are first discussed. Signal-subspace, geometrical, statistical, sparsity-based, and spatial-contextual unmixing algorithms are described. Mathematical problems and potential solutions are described. Algorithm characteristics are illustrated experimentally.
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This paper provides a review and commentary on the past, present, and future of numerical optimization algorithms in the context of machine learning applications. Through case studies on text classification and the training of deep neural networks, we discuss how optimization problems arise in machine learning and what makes them challenging. A major theme of our study is that large-scale machine learning represents a distinctive setting in which the stochastic gradient (SG) method has traditionally played a central role while conventional gradient-based nonlinear optimization techniques typically falter. Based on this viewpoint, we present a comprehensive theory of a straightforward, yet versatile SG algorithm, discuss its practical behavior, and highlight opportunities for designing algorithms with improved performance. This leads to a discussion about the next generation of optimization methods for large-scale machine learning, including an investigation of two main streams of research on techniques that diminish noise in the stochastic directions and methods that make use of second-order derivative approximations.
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Low-rank matrices play a fundamental role in modeling and computational methods for signal processing and machine learning. In many applications where low-rank matrices arise, these matrices cannot be fully sampled or directly observed, and one encounters the problem of recovering the matrix given only incomplete and indirect observations. This paper provides an overview of modern techniques for exploiting low-rank structure to perform matrix recovery in these settings, providing a survey of recent advances in this rapidly-developing field. Specific attention is paid to the algorithms most commonly used in practice, the existing theoretical guarantees for these algorithms, and representative practical applications of these techniques.
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2016-07-06
Tensors or {\em multi-way arrays} are functions of three or more indices$(i,j,k,\cdots)$ -- similar to matrices (two-way arrays), which are functionsof two indices $(r,c)$ for (row,column). Tensors have a rich history,stretching over almost a century, and touching upon numerous disciplines; butthey have only recently become ubiquitous in signal and data analytics at theconfluence of signal processing, statistics, data mining and machine learning.This overview article aims to provide a good starting point for researchers andpractitioners interested in learning about and working with tensors. As such,it focuses on fundamentals and motivation (using various application examples),aiming to strike an appropriate balance of breadth {\em and depth} that willenable someone having taken first graduate courses in matrix algebra andprobability to get started doing research and/or developing tensor algorithmsand software. Some background in applied optimization is useful but notstrictly required. The material covered includes tensor rank and rankdecomposition; basic tensor factorization models and their relationships andproperties (including fairly good coverage of identifiability); broad coverageof algorithms ranging from alternating optimization to stochastic gradient;statistical performance analysis; and applications ranging from sourceseparation to collaborative filtering, mixture and topic modeling,classification, and multilinear subspace learning.
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In many engineering areas, such as signal processing, practical results can be obtained by identifying approaches that yield the greatest quality improvement, or by selecting the most suitable computation methods. ABSTRACT | 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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analysis of discrete sigruls The signal is modeled as a linear combina-Absrruct-Thii paper gives an exposition of linear prediction in the tion of its past values d present and past values of a hypothetical input to a system whose output is the given signal. In the frequency domain, this is equivalent to modeliug the signal s p e c-by a pole-zero spectrum. The major part of the paper is devoted to all-pde models. The model parameters are obtained by a least squares nnnlysis in the time domain Two methods result, depending on whether the signal is wsumed to be stationay or nonstationary. The same results are then derived in the frequency domain. The resulting spectral matching formulation allows for the modeling of sehted poltiom of a s p e c t n u n , for arbitrary spectral shaping in the frequency domain, and for the modeling of continuous as well as discrete spectra. Thii also lepds to a dslssion o f the advrntnges and disodv~ntrges of the least quues mor criterion A spectral interpretation is given to the normalized minimum prediction error. Applications of the normalized error are given, including the determination of an "optimal" number of poles, The use of linear prediction in data compression is reviewed. For purposes of tmmnission, particular attention is given to the quantization and encoding of the reflection (or partial correlation)
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This paper studies a difficult and fundamental problem that arises throughout electrical engineering, applied mathematics, and statistics. Suppose that one forms a short linear combination of elementary signals drawn from a large, fixed collection. Given an observation of the linear combination that has been contaminated with additive noise, the goal is to identify which elementary signals participated and to approximate their coefficients. Although many algorithms have been proposed, there is little theory which guarantees that these algorithms can accurately and efficiently solve the problem. This paper studies a method called convex relaxation, which attempts to recover the ideal sparse signal by solving a convex program. This approach is powerful because the optimization can be completed in polynomial time with standard scientific software. The paper provides general conditions which ensure that convex relaxation succeeds. As evidence of the broad impact of these results , the paper describes how convex relaxation can be used for several concrete signal recovery problems. It also describes applications to channel coding, linear regression, and numerical analysis .
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2018-03-02

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We address the sparse signal recovery problem in the context of multiple measurement vectors (MMV) when elements in each nonzero row of the solution matrix are temporally correlated. Existing algorithms do not consider such temporal correlation and thus their performance degrades significantly with the correlation. In this work, we propose a block sparse Bayesian learning framework which models the temporal correlation. We derive two sparse Bayesian learning (SBL) algorithms, which have superior recovery performance compared to existing algorithms, especially in the presence of high temporal correlation. Furthermore, our algorithms are better at handling highly underdetermined problems and require less row-sparsity on the solution matrix. We also provide analysis of the global and local minima of their cost function, and show that the SBL cost function has the very desirable property that the global minimum is at the sparsest solution to the MMV problem. Extensive experiments also provide some interesting results that motivate future theoretical research on the MMV model.
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This paper presents an account of the current state of sampling, 50 years after Shannon's formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbert-space formulation, we reinterpret Shannon's sampling procedure as an orthogonal projection onto the subspace of band-limited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of "shift-in-variant" functions spaces, including splines and wavelets. Practically , this allows for simpler-and possibly more realistic-interpolation models, which can be used in conjunction with a much wider class of (anti-aliasing) prefilters that are not necessarily ideal low-pass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonban-dlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned.
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